Molecular Dynamics of Additives in Polymers 9789067642590

Table of contents :
Cover
Half Title
New Concepts in Polymer Science
Molecular Dynamics of Additives in Polymers
Copyright
Contents
Dedication
Preface
1. Introduction. Basic concepts and parameters
References
2. Theories and Equations
2.1 Empirical Equations
2.2 Hydrodynamic Model
2.3 Activation Theories
2.3.1 Kinetic Theory of Liquids
2.3.2 Transition State Theory
2.3.3 Other Theories
2.4 Free Volume Models
2.4.1 Theory of Cohen and Turnbull /31/
2.4.2 Bueche Theory
2.5 “Hybrid” Model
2.6 Some Important Thermodynamic Equations
2.7 Models for Motional Scale
References
3. Experimental Techniques
3.1 Spin Probe Technique
3.1.1 the Ways of Estimation of Translational Mobility and Local Spin Probe Concentrations
3.2 Electrophysical Techniques
3.2.1 Analysis of Dielectric Losses
3.2.2 Thermstimulated Depolarization Technique
3.2.3 Electrical Conductivity Investigations
3.3 Evaluation of Diffusion Constants by Sorption-Desorption Kinetic Data
3.4 Other Techniques
References
4. Molecular Structure and Motional Frequencies of Additives
4.1 Rotational Frequencies
4.2 Translational Frequencies
4.3 Ratio of Rotational and Translational Frequencies
4.4 Scale of Molecular Motions
4.4.1 Rotational Dynamics
4.4.2 Translational Diffusion
References
5. Temperature Effects and Energetics of Molecular Dynamics
5.1 Small-Scale Rotational Dynamics
5.2 Large-Scale Rotational Dynamics
5.3 Translational Dynamics
5.4 Apparent and True Values of Activation Energy
References
6. Pressure Effect and Activation Volumes
6.1 Pressure Dependences of Molecular Motion
6.2 Activation Volumes
6.2.1 High Frequency Rotational Dynamics
6.2.2 Low Frequency Rotational Dynamics
6.2.3 Activation Volumes for Translational Dynamics
6.3 The Ratio Between Activation Parameters
References
7. Additive Motion and Macromolecular Dynamics
7.1 Segmental Dynamics of Macromolecules
7.1.1 Peculiarities of Temperature Dependences of Frequencies of α- and β-processes
7.1.2 Volume and Pressure Effects in Segmental Dynamics
7.1.3 Differences in Mechanisms of Relaxation Processes
7.2 Correlations Between Segmental and Particle Dynamics
7.3 Cases of Correlation Break: Porous Systems
7.4 Comparison of Dynamic Behavior of Particles in Polymers and Liquids
References
8. Effect of Polymer Structure and Physical State
8.1 Effect of Chemical Structure
8.2 Transitions in Polymers
8.3 Crystalline Polymers
8.4 Mechanical Deformations
References
9. Dynamics of Water in Polymers
9.1 Phase Equilibria of Polymer-Water Systems
9.2 Concentration Dependence of the Coefficients of Diffusion
9.3 Effect of the Nature of Polymers on the Translational Mobility of the Molecules of Water
9.4 Kinetics of the Sorption of Vapours and the Swelling of Polymers in Water
9.5 Kinetics of Swelling, Complicated by Chemical Reaction
References
Subject index

Citation preview

New Concepts in Polymer Science Molecular Dynamics of Additives in Polymers

New Concepts in Polymer Science Previous titles in this book series: Structure and Properties o f Conducting Polym er Composites V.E. GuV Interaction o f Polym ers with Bioactive and Corrosive M edia A.L. Iordanskii, T.E. Rudakova and G.E. Zaikov Im m obilization on Polymers M . I. Shtilman Radiation Chem istry o f Polymers V.S. Ivanov Polym eric Com posites R. B. Seym our Reactive Oligom ers S . G. Entelis, V. V. Evreinov and A.I. K uzaev D iffusion o f Electrolytes in Polymers G.E. Zaikov, A.L. Iordanskii and V.S. M arkin Chem ical Physics o f Polym er Degradation and Stabilization N . M Em anuel and A.L. Buchachenko O f related interest: Journal o f A dhesion Science and Technology Editors: K.L. M ittal an d W.J. van Ooij N ew Polym eric M aterials Editor-in-Chief: F.E. Karasz Journal o f Biom aterials Science, Polymer Edition Editors: C.H. Bamford, S.L. Cooper and T. Tsuruta Composite Interfaces Editor-in-Chief: H. Ishida

New Concepts in Polymer Science

Molecular Dynamics of Additives in Polymers

A.L. Kovarski

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

First published 1997 by VSP BY Published 2021 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton. FL 33487-2742 © 1997 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN 13: 978-90-6764-259-0 (hbk)

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CONTENTS Preface Chapter 1. Introduction. Basic concepts and parameters Chapter 2. Theories and equations 2.1 Empirical equations 2.2 Hydrodynamic model 2.3 Activation theories 2.3.1 Kinetic theory of liquids 2.3.2 Transition state theory 2.3.3 Other theories 2.4 Free volume models 2.4.1 Theory of Cohen and Turnbull /31/ 2.4.2 Bueche theory 2.5 “Hybrid” model 2.6 Some important thermodynamic equations 2.7 Models for motional scale Chapter 3. Experimental techniques 3.1 Spin probe technique 3.1.1 The ways of estimation of translational mobility and local spin probe concentrations 3.2 Electrophysical techniques 3.2.1 Analysis of dielectric losses 3.2.2 Thermstimulated depolarization technique 3.2.3 Electrical conductivity investigations 3.3 Evaluation of diffusion constants by sorption-desorption kinetic data 3.4 Other techniques

xi 1 8 9 10 15 16 23 31 36 36 40 42 45 48 53 53 63 65 67 70 77 80 83

VI

contents

Chapter 4. Molecular structure and motional frequencies of additives 4.1 Rotational frequencies 4.2 Translational frequencies 4.3 Ratio of rotational and translational frequencies 4.4 Scale of molecular motions 4.4.1 Rotational dynamics 4.4.2 Translational diffusion

89 89 97 108 112 112 114

Chapter 5. Temperature effects and energetics of molecular dynamics 5.1 Small-scale rotational dynamics 5.2 Large-scale rotational dynamics 5.3 Translational dynamics 5.4 Apparent and true values of activation energy

121 122 126 127 135

Chapter 6. Pressure effect and activation volumes 6.1 Pressure dependences of molecular motion 6.2 Activation volumes 6.2.1 High frequency rotational dynamics 6.2.2 Low frequency rotational dynamics 6.2.3 Activation volumes for translational dynamics 6.3 The ratio between activation parameters

142 142 148 149 151 152 154

Chapter 7. Additive motion and macromolecular dynamics 7.1 Segmental dynamics of macromolecules 7.1.1 Peculiarities of temperature dependences of frequencies of a- and P-processes 7.1.2 Volume and pressure effects in segmental dynamics 7.1.3 Differences in mechanisms of relaxation processes 7.2 Correlations between segmental and particle dynamics 7.3 Cases of correlation break: porous systems 7.4 Comparison of dynamic behavior of particles in polymers and liquids Chapter 8. Effect of polymer structure and physical state 8.1 Effect of chemical structure 8.2 Transitions in polymers 8.3 Crystalline polymers 8.4 Mechanical deformations

160 160 161 164 166 170 177 179 186 186 194 201 208

contents

Chapter 9. Dynamics of water in polymers 9.1 Phase equilibria of polymer-water systems 9.2 Concentration dependence of the coefficients of diffusion 9.3 Effect of the nature of polymers on the translational mobility of the molecules of water 9.4 Kinetics of the sorption of vapours and the swelling of polymers in water 9.5 Kinetics of swelling, complicated by chemical reaction Subject index

225 226 236 250 252 265 271

Dedicated to my mother, Prof. B.M. Kovarskaya

PREFACE

The aim of this book is to consider dynamic behavior of low molecular particles in solid polymeric medium. In contrast to the majority of existing books on polymer permeability, mass-exchange, or diffusion, this monograph is concerned with behavior of an individual particle in a polymer matrix rather than with the aspects of sorption of these particles or their transport through the matrix. Small particles play a very important role in physical chemistry of polymers and affect significantly the utility properties of the polymer materials. Such particles are an indispensable component of any polymer. These are the ingredients of the synthesis reaction, products of destruction, the particles deliberately added to the polymer to improve its performance, and particles infiltrating polymer from the environment. The majority of modem polymer materials are composites that incorporate low molecular components. The processes which result in polymer enrichment with low molecular components are listed below: 1. Synthesis - monomers, catalysts, initiators, chain growth regulators, solvents, etc.; 2. Chemical modification - cross-linking agents (vulcanizators) and other reagents; 3. Physical modification - structure formers, intensifiers, fillers, plasticizers, antistatics, etc.; 4. Stabilization, antistabilization - stabilizers, antioxidants, initiators of degradative process; 5. Degradation - fragments of macromolecules, free radicals, charged particles, organic and inorganic products; 6. Refining of articles - dyes, pigments; 7. Sorption of particles by a polymer from the environment - charged particles, gases, vapours, water. It follows from the above that polymers can accomodate the molecules of virtually all the types of chemical substances, from trapped electrons and gas molecules up to complex organic molecules, dyes and inhibitors. In terms of polymer modification, all the low molecular additives fall into two groups which either modify or does not modify the properties of the polymer matrix. Plasticizers and fillers belong to the first group, the second group consists of stabilizers, dyes, etc. These groups of additives differ by their chemical structure

Chapter 1

Introduction. Basic Concepts and Parameters

Thermal motion of particles in all aggregation states of matter (gases, liquids, and solids) includes oscillation about an equilibrium, and translational and rotation motions. The latter two types of motion form the base of the term ‘molecular dynamics’. Molecular dynamics is that particular element that joins all transport processes: ‘mass transfer’ (diffusion), ‘viscosity’ and ‘thermal conductivity’. The first process in the list is connected to transfer of particles and levelling of their concentrations, in response to gradient of concentration or chemical potential. Viscosity is the transfer of momentum (impulse). Heat transfer under conditions of temperature gradient is named thermal conductivity. To describe these processes, a general approach is used, named ‘phenomenological’ (as distinguished from the ‘molecular-kinetic’ one, discussed in Chapter 2) /1 -13/. It consists of the following points. Let a change of a property 0, defined by motion of molecules, happen along X axis only (socalled one-dimensional transfer). If a specific square disposed perpendicular to the X axis is considered, it is evident that the molecules will pass through it in both directions along the X axis with equal probabilities. However, in the conditions of concentration gradient

, the transport proceeds only in one

direction. The transport of 0 value through specific square per specific time is 50 called ‘flux density’ or ‘flux’, J q -------. The general expression which connects dt an one-dimension flux to the gradient of value 0 looks as follows: J 0 = - k |^ . OX

(1.1)

Here K is a coefficient dependent on properties of the matrix. Minus in Eq. 1.1 means that the flow is directed opposite to the gradient.

1

All transfer processes are connected to chaotic thermal motion of molecules. In conditions of concentration gradient, this chaotic motion causes levelling of concentration

=> oj . This process is called diffusion. Colliding

in this translational motion, the molecules transfer a fraction of their energy to each other. If a temperature gradient exists in the system, heat flux occurs which is directed from the sites with higher temperature to award cooler ones. This process is called ‘heat transfer’. According to Eq. 1.1, the density of the heat flux, J q , is described by the following expression: 0T

Jq

M

ÔX

.

(1-2)

Here X is the constant called ‘thermal conductivity coefficient’. According to the International System of Units (SI), heat amount is measured in joules (J), heat flux density, J/m2-s or W/m2. Consequently, in accord to Eq. 1.2 the dimension of the thermal conductivity coefficient is: W/m-K or J/m-K-s. In the heat transfer process, the temperature change of a body is the quotient of the heat transferred divided by the heat capacity. The rate of temperature levelling off is determined by the ‘coefficient of temperature conductivity’: A = —c p (m2/s). P

(1.3)

Here p is the density; Cp is the heat capacity of body mass unit at constant pressure. ‘Viscosity’ is another important parameter that characterizes the transport processes. In a flow not only heat is transferred from more hot regions to cooler ones, but momentum I = mu is also transferred from faster to slower regions of the flow. This process is called ‘internal friction’ or viscosity, and provides levelling of particle motion rates in gas or liquid flows. The density of momentum flow Jj represents the total momentum transferred during specific time through specific square perpendicular to the flow direction - the X axis:

2

(1.4)

JI

Here u is the flow rate; t| is a viscosity coefficient. The dimension of momentum flow is [kg/m-s2]; the dimension of viscosity coefficient is Pa s [kg/m-s] (Puaz); N/m2-s. The viscosity coefficient defines the rate of momentum transfer from one region of the flow to another. The flow rate itself is levelled with the rate proportional to the ratio of momentum to mass of specific volume (density): v=-· P

(1.5)

Parameter v is called ‘kinetic viscosity’, contrary to the coefficient rj, which is called ‘dynamic viscosity’. The dimension of the kinetic viscosity is the same as for diffusion coefficient: m2/s (see below). Let us now consider in more detail the process of diffusion, which is directly connected to the subject of the book. Eq. 1.1 for a flow of particles J n looks as follows: J n = - D fOX :·

O·6)

Here c is concentration of diffusing particles; D is the coefficient of diffusion. Eq. 1.6 is called ‘the first Fick law’. For a stationary flow through a plate of thickness x and area s, the solution of the Fick equation gives the following expression: 0 = D st— . x

(1.7)

Here 0 is amount of the substance that passed through the plate during time t , r . A c at the gradient of concentration — . x The product of the diffusion coefficient and gas solubility in the substance investigated, a, is called the coefficient of permeability P: P = Da.

3

(1.8)

It follows from Eqs. 1.6 and 1.7 that diffusion coefficient is the number of particles, which pass specific area during specific time at specific gradient of concentration. The dimension of the diffusion coefficient in ISU units is [m2/s]. Thus, diffusion coefficient defines the rate of concentration levelling and depends on structure and physical properties of substance. As it has been mentioned already, Eqs. 1.6 and 1.7 are suitable just for stationary flow. Nonstationary process of diffusion is described by ‘the second Fick’s law’, which defines change of substance concentration in different points of space as a time-dependent function: ôc

(1.9)

ôT

If diffusion proceeds in three directions of an isotropic medium, the Eq. 1.9 can be recast as: — =D

at

(^dx2

dy2

d i} )

( 1 . 10 )

or — = D V ^. at In the case of anisotropic medium, the nonstationary transfer of a substance is described by the following equation: dc _ a 2c ^ a 2c ^ a 2c - D X 2 +Dy 2 +D Z 2 at dz ay dx

(1.11)

To calculate diffusion coefficients it is necessary to solve these equations with definite initial and boundary conditions (forms and sizes of samples, homogeneity, initial concentration of penetrant, etc.). The reader can find the mathematical methods of solving these equations and numerous examples in the monographs /5 - 7, 11 - 13/. Consider two important ratios, assocoated with the name of Einstein. The first among them allows to determine the time of particle transfer to distance x. It looks as follows:

4

t = X

( 1. 12)

The second ratio connects the diffusion coefficient of a particle to its ‘mobility’ 5: D = 5kT .

(1.13)

Here k is the Boltzmann constant. By mobility we call the proportionality constant between particle rate u and force F, suppressing it: u = ÔF.

(1.14)

As it follows from Eq. 1.14, mobility is inversely proportional to the friction coefficient 8 = 1/4·

(1.15)

The ratio between the mobility and diffusion coefficient is fulfilled for any particles, dissolved or suspended in gas or liquid and moving under the influence of any external field (electrical or gravitational force). Diffusion is often regarded as mere mass transfer only, although the latter is performed just in a flow of particles at gradient of concentrations or other external forces such as potential difference. For example, the mass transfer through an interface surfaces (at contact of two solids, on gas-liquidsolid interfaces in heterogeneous mixtures, etc.) or directed transfer of charged particles in electric fields (so-called electrodiffusion) can be mentioned. The diffusion term includes mass transfer, and in this case it is called ‘interdiffusion’, and thermal motion of particles in the absence of the concentration gradient, the so-called ‘self diffusion’. Self diffusion represents a chaotic translational motion of particles induced by thermal energy. This process is described by Eqs. 1.6 - 1.14, in where diffusion coefficient is substituted by ‘self diffusion coefficient’ D s . As mentioned above, thermal motion of particles includes translational and rotational components. The intensity of rotational motion is characterized by relaxation times r o r rotation frequency v. Moreover, rotation of particles can be described phenomenologically on the basis of usual diffusional Eqs. 1.6 - 1.14, if the flow is considered as the total number of particles in 1 cm3 (a volume unit) that passes through the angle (p during specific time in direction

5

to positive cp. Then the Eq. 1.6 can be recast in a form similar to the Fick equation for translational diffusion: Jcp “

D(p

dp() dq>

(1.16)

The variable of rotation p(cp) was selected in a manner that the number of particles in specific volume, which possess orientation in angles between (p and tp + dcp, equals p(cp)5cp . The value p( the expression for difussion coefficient of molecules in liquid is obtained as follows:

Dm =

6 ro

exp

w + U) kT J

b ( E_" -----exp kT/ 6 x0

(2.43)

According to Frenkel, activation energy E is composed, in general case, of two terms: energy of hole formation, w, and activation energy of hole translation, U. However, it is supposed that U is significantly lower than w. In this case, the main contribution to the activation energy of self diffusion is made by the energy of hole formation, i.e. E s w . Eq. 2.43 holds for P => 0. According to 2.30, at increased pressure the expression for self diffusion coefficient is: wo + PAv

D=

iff

.

(2.44)

Here Av is the minimum size of hole, the parameter also known as the ‘activation volume’. It will be recalled that by physical sense it is the minimum size of hole, required for nearest particle to jump into it. Let us now discuss the problem, to which Frenkel paid great attention: the dependence of activation energy on variation of intermolecular distance and intermolecular interaction. That variation is induced by change of volume V with pressure and temperature changes. If w changes in a narrow range of pressure, its dependence on volume can be expressed by a linear equation: w = w q - c ( v - v q ).

(2.45)

Here v = V/N is the volume per one particle, the index ’0’ belongs to the parameters at normal pressure; c is the coefficient. v —vn P Using the definition of the bulk modulus K : -------- = ------, we obtain: v0 K

21

w = W o_ i - v op . iv

(2.46)

Thermal change of volume can be taken into account by the term (x v q T. In this case, we obtain:

w = W0 - v q c |

(2.47)

In these equations v q and w q are the values of v and w, respectively, at P => 0 and T => 0. Substitution of this expression into Eq. 2.44 gives: D=

exp

f v

wq

+ aP^

kT

,

(2.48)

Here a = — c + Av. K The above analysis was made for self diffusion of particles, i.e. for their translational mobility. Let’s now discuss the specificity of rotational motion of particles basing on the Frenkel’s theory. Rotational motion in simple liquids at low temperatures is similar to rotational dynamics in lattice. The period between two acts of reorientation of a particle, xr , is significantly higher than the period of rotation oscillations, tq . These values are connected by the equation, similar to Eq. 2.34: Tr = T 0 e x p ^ j .

(2.50)

At high temperatures, when thermal energy kT becomes comparable with the activation energy, rotation may become free as in gases. Dislocation of a hole near a particle is required for both particle reorientation and its translation. It should be mentioned that, at least, for symmetric particles the rotational dynamics can be reduced to usual

22

translational diffusion of points on a sphere surface. Using the formula by Einstein for rotational motion: (2.51) in accord with Eq. 2.50, we obtain: (2.52) To sum up the results of the above analysis of the molecular dynamics from positions of the theory of Frenkel, it should be noted that its basic features are: quazicrystalline model of liquid, hole concept of molecular dynamics, and tight connection of activation parameters of particle dynamics to the energy of formation, amount and size of holes. 2.3.2 Transition State Theory

The theory of transition state, also called as the theory of rate processes, was primarily developed by Eyring, and also by Polanyi and Ewans in the middle 1930’s for the description of kinetics of chemical processes, and then was applied to viscosity and diffusion in liquids /21/. The theory is based on the idea that before transferring into final products, reacting particles are activated and form a so-called ‘transition state’ or ‘activated complex’. Activated complex can be considered as an usual molecule, characterized by specific thermodynamical properties. The fourth degree of freedom, connected to the movement along the reaction coordinate, exists in this complex alongside with three usual degrees of freedom. Initial reagents are in balance with activated complexes, and approaches of thermodynamics and statistical mechanics are used in the analysis of this balance. These approaches are suitable for a great number of particles, and one of its advantages is the possibility to make calculations of energetic barrier height and preexponential factor, and applicabilicity to a wide range of chemical and physical processes. In this section we discuss the fundamentals of this theory in a part, related to viscosity, translational and rotational diffusion of particles in

23

liquid. We also show results of later theoretical studies, which use similar approaches. The main equation of the theory appears for the rate constant k as follows: (2.53) Here K* is the equilibrium constant of formation of a transition state; h is the Plank constant; x is the transmission coefficient, i.e. parameter calculating probability for a particle to continue its way along the reaction coordinate after ‘climbing’ up to the top of the potential barrier. It is usually accepted that x = 1. Eq. 2.53 is general for all types of reactions. Molecular transfer (diffusion) and momentum transfer (viscosity) is considered as a monomolecular chemical reaction in the framework of the transition state theory. Taking into account that to form a transitional state in a monomolecular process, a particle must pass a e high potential barrier, the equilibrium constant is determined by the following expression: (2.54)

q

Here q and q* are the sums of states per specific volume for a particle in the initial or activated state. Taking Eq. 2.54 into account, the expression for the rate constant appears as follows: (2.55) The rate constant can also be expressed through thermodynamical characteristics of transition state, basing on the known equations for the free energy: 0* * AGU = -R T lnK ,

(2.56) (2.57)

0* 0* Here AH and AS are the standard heat and entropy of activation, respectively.

24

Finally, we obtain: kT — exp h F

(

AH

(

O

AS

l rtJ l r J exp

(2.58)

Let us point out once again that Eqs. 2.55 -2.58 are valid for the rate constant of monomolecular chemical reaction. Basing on Eq. 2.55, for the rate constant of diffusion we obtain: . kT q k = X— ·— exp h q

e dif RT

(2.59)

8^ Using the Einstein equation D = — (where 8 is a distance equal to the 6t

molecular diameter) and taking into account that k = 1/t , we obtain: ~ _2 kT q ( e^if D = X5 -r -·— exp— ^ h q V RT

(2.60)

or p. r 2 kT q D = X§ - ¡ - - — exP h q

AH RT

exp

AS

*\

R J

(2.61)

Equation for viscosity can be written as follows: ^1 =

h

q f 7 exP %S2r q

e vis RT

(2.62)

2 Assuming that 8 r equals the molecular volume, and in this case equals N/V (V is molar volume; N is Avogadro number), we obtain the equation for viscosity:

25

hN

T| =

q

*VIS ‘ V RT,

XV q *

(2.63)

Using equations of thermodynamics (2.56, 2.57) we obtain: hN q = — exp XV

—

AS

RJ

r exp

AH

*\

[ r t J

(2.64)

Let us now discuss the physical sense of the main parameters of Eqs. (2.60 - 2.64): activation energy and preexponential factor, its term q/q*, in more detail. As it has been already mentioned, the transition state theory is based on the hole model of liquid, which suggests that formation of a hole requires an energy equal to the energy of molecule vaporization?. This energy is: Aevap = A H v ap - R T .

(2.65)

AHvap represents the latent heat of vaporization; RT is the coefficient of correction for external work, extended in vaporizing mole of a liquid. The correlation between the values of free volume, Vf, and the volume of spheric particles, v, was determined by the authors of the theory with the following equation: v f = c 3 ( v 1/ 3 - d ) 3 .

(2 .66 )

Here d is the particle diameter; c is the coefficient dependent on particle 1/3 packing (c = 2 for the wide-spread cubic type of lattice). Value v is determined from the expression 1/3

V

2RTV 1/3 = --------------N Aevap

(2.67)

In accord with the concepts of statistic mechanics, the ratio of sums of states for translational motion is determined from the following equation:

26

q (2^mkT)1/2 1/3 T = ■— r 1— v f

( 2 . 68)

Combining Eqs. 2.60, 2.63, 2.68, the following expressions could be obtained: T) =

'N '

2/3

,V .

2R T XAeVap

(27tmkt)^2exp|

D=

(2.69)

(2.70)

Because the viscous flow of a liquid and diffusion are closely connected to the process of hole formation, the activation energy of these processes is also connected to the work of hole formation. According to the ideas of the authors of the theory Aevap edjf (or e vis) = ------ - · n

(2-71)

It was shown empirically that the value of coefficient n falls within the range 3 < n 0.5 the following equation is fulfilled: At — coo

Q t. Qoo

8

— •exp n

2

71 Dt ,2

A (3.54) J

The method of the substance distribution by layers (the partitioning technique) is mainly used for the analysis of Dt of liquids and solids in polymers. After a polymer sample had been exposed to diffusant solution, it is sliced into thin layers, and the diffusant concentration is measured in each layer separately. The data obtained are usually displayed as the concentration versus distance curve. To determine diffusion coefficients of dyes, solvents, acids, paramagnetic particles (free radicals), luminescent substances the method of determining the motion rate of the constant concentration front is widely used. The diffusion border is determined by the change of colour (dyes), intensity of luminecsence (luminophores), ESR. signal (free radicals), etc. This method is also applied for determination of diffusion coefficients of oxygen. In this case, the process of free radical oxidation is studied, which leads to transformation of the ESR spectrum or the change of colour, or the process of quenching of phosphorescence of special additives. To determine Dt of liquids in polymers the technique of the concentration change rate in the particular place of the sample by variation of any physical parameter is also widely used. For example, electrical conductivity, refraction index, radioactivity, etc. are measured. To sum up the discussion, let us note that virtually all techniques of physicochemical analysis - optical, spectroscopic, magnetic, electrophysical, barometric, etc. - can be used in the study of sorption-desorption kinetics of particles in polymers.

82

3.4 OTHER TECHNIQUES

In parallel with widely applied techniques of the spin probe and dielectric relaxation, we should mention the studies of rotational mobility of additives in polymers, made with the help of optical or electrooptical techniques, extensively developed in recent times. These techniques are based on the analysis of electrochromism effects /51 - 53/, nonlinear optical effects (NLO) /54 - 56/, and various luminescent (fluorescent) effects, dependent on rotational mobility of optically active additives (probes) /57 - 68/. These effects are, for example, polarized and double band luminescence, decay and depolarization of fluorescence, photobleaching, transient optical dichroism. Some of these techniques are discussed in brief below. Electrochromism is the combination of electrooptical effects, appearing under the application of an external electrical field to solution of polar organic molecules /51/. The following effects belong to the list: splitting of energy levels (Shtark effect), change of electron transition moment, change of distribution of orientations of dipole moments. Electrochromism is displayed in deformation of absorption bands in spectra of electron transitions. Analysis of the deformation of absorption bands is used for studying parameters of molecules in the excited state. It was shown in /52, 53/ that this technique can be successfully used also for studying rotational dynamics of polar particles in liquids and solids, including polymers. The experimental procedure consists in applying a direct voltage and light beam with an electrical vector at a definite angle to a solution or polymeric sample with added polar molecules as probes. Spectra of the electrochromism are obtained by slow scanning the light wave-length and measuring the increase of optical density in the electric field by contour of the absorption band. Molecules of azodyes are usually used as probes, which have high dipole moment in the ground and excited states. They also possess an intensive absorption band in visible and near UV-range. The range of dye concentrations in polymers is lO 3 - 1(H mol/kg. The electrochromism technique gives information about the low-frequency area of the spectrum of rotational motions ( 103 - 10 6 s). The NLO technique is close to the electrochromism technique by its physical principles. Nonlinear optical effects are analyzed in samples prepolarized with an electrical field /54 - 56/. The technique can be used for studying the rotational mobility of polar additives in amounts from 3 to 30%. Changes in polarization, fluorescence yield, position of fluorescence maximum, efficiency of intramolecular eximer formation were used for evaluation of rotational mobility of particles. It has been already mentioned

83

that various mechanisms may contribute to these phenomena. For example, some fluorescent techniques are based on the principle of absorption of incident polarized light beam by the particles, in which direction of the vector of chromophore’s transition dipoles is close to that of the light beam polarization. As a result, initially isotropic distribution of particles in orientations is transformed into anisotropic one. Levelling of the distribution happens through rotational diffusion of particles. This effect is used for the analysis of rotational mobility frequencies /6 , 59/. Photoselection of photobleached particles occurs as a result of a chemical reaction. The technique is used for determination of rotational correlation times in the range of 103 > t > 10·' s, and there are possibility to broaden this range. Complex aromatic molecules, such as rubrene and tetracene, are used as the chromophores-probes in the concentration of 10 100 ppm /62, 63/. Perene, naphthalenes, carbazoles, dianthryls, and some others are also used as probes. Higher concentrations of particles (» 2500 ppm) are used in the transient optical dichroism technique, concenptionally and methodically close to the photoblleaching technique /64/. All the techniques mentioned above require speacial treatment of samples, that includes preparation of films or disks. Electrooptical techniques require a creation of electrically conductive coatings, as well as the technique of thermostimulated depolarization (Sec. 3.2.2).

REFERENCES

1. Wasserman A.M., Kovarskii A.L., Spin Probes and Labels in Physical Chemistry o f Polymers, Nauka, Moscow, 1986. (Rus) 2. Spin Labeling. Theory and Application, Ed. Berliner L.J., Academic Press, New York, 1 (1976), 2 (1986). 3. Kuznetsov A.N., Spin Probe Technique, Nauka, Moscow, 1974. (Rus) 4. Buchachenko A.L., Kovarskii A.L., Wasserman A.M., The Study o f Polymers by Paramagnetic Probe Technique, In: Advances in Polymer Sciences, Ed. Rogovin Z.A., Wiley New York, 1974, 26-42. 5. Tormala P. and Lindberg J.J., Spin Labels and Probes in Dynamic and Structural Studies of Synthetic and Modified Polymers, In: Structural Studies o f Mocromolecules by Spectroscopic Methods, Ed. Ivin K.J., Wiley, London, 1976, 255-271.

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6. Kovarskii A.L., Spin Probes and Labels. A Quater of a Century of Application to Polymer Studies, In: Polymer Yearbook, Ed. Pethrick R.A., Harwood, Switzerland, 1996,13, 113-139. 7. Buchachenko A.L., Wasserman A.M., Aleksandrova T.A., Kovarskii A.L., Spin Probe Studies in Polymer Solids, In: Molecular Motions in Polymers by ESR, Ed. Boyer R.F., Keinath S.E., Harwood Academic Publishers, Chur, 1980, 33-42. 8. Atlas o f Spin Probes and Labels ESR Spectra, Ed. Antsiferova L.I., Wasserman A.M. et al., Nauka, Moscow, 1977. (Rus) 9. Kuznetsov A.N., Wasserman A.M., Volkov A.U., Korst N.N., Determination of Rotational Correlation Times of Nitric Oxide Radicals in Viscous Medium, Chem. Phys. Lett., 1971,12, 103-106. 10. Goldman G.F., Bruno G.V., Freed J.H., Estimating Slow-motional Rotational Correlation Times for Nitroxides by ESR, J. Phys. Chem., 1972,76,1858-1860. 11. Poluektov O.G., Dubinsky A.A., Grinberg O.Ya., Lebedev S.Ya., Application of 2 mm Range ESR for Investigation of Rotational Motion by Spin Probe Technique, Khim. Fizika, 1982,11, 1480-1484. (Rus) 12. Hide J.S., Dalton L.R., see Ref. 17,2, 1. 13. Hide J.S., Fronzisz W., Mottley C., Pulsed ELDOR Measurement of Nitrogen Ti in Spin Labels, Chem. Phys. Lett., 1984,110, 621-625. 14. Livshitz V.A., Kuznetsov V.A., Barashkova I.I., Wasserman A.M., “Superslow” Rotations of Spin Probes in Solid Polymers, Vysokomolekuliarnye Soedineniya, 1982, 24A, 1085-1093. (Rus) 15. Dzuba S.A., Tsvetkov Yu.D., Magnetization Transfer in Pulsed EPR of l5N Nitroxides: Reorientational Motion Model of Molecules in Glassy Liquids, Chem. Phys., 1988,120, 291-298. 16. Dzuba S.A., Tsvetkov Yu.D., Maryasov A.G.,Echo-induced EPR Spectra of Nitroxides in Organic Glasses: Model of Orientational Molecular Motions Near Equilibrium Position, Chem. Phys. Lett., 1992, 188,217-222. 17. Gorcester J., Millhouse G.L., and Freed J.H., Two-dimensional ESR, In: Modern Pulsed and Continuous Wave Electron Spin Resonance, Ed. Kevan L. and Bowman M.K., Wiley, New York, 1990, 119-194. 18. Maresch G.G., Weber M., Dubinskii A.A., Spiess H.W., 2D-ELDOR Detection of Magnetization Transfer in Disordered Solid Polymers, Chem. Phys. Lett., 1992,193, 134-140. 19. Grinberg O.Ya., Dubinskii A.A., Lebedev Ya.S., ESR of Free Radicals in 2 mm Wavelength Region, Uspekhi Khimii, 1983,52, 1490-1513. (Rus) 20. Andreozzi L., Giordano M., Leporini D., Pardi L., Phys. Lett., 1991, A160, 309.

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21. Andreozzi L., Giordano M., Leporini D., Pardi L., Angeloni A.S., Mol. Cryst. Liq. Cryst., 1992,212, 107. 22. Wasserman A.M., Kovarskii A.L., Yasina L.L., Buchachenko A.L.,Exchange and Dipole Interactions and Local Concentrations of Stable Radicals in Polymers and Liquids, Teor. i Experim. Khimiya, 1977, 13, 30-34. (Rus) 23. Kokorin A.I., Kirsh Yu.E., Zamaraev K.I., Kabanov V.A., Investigation of Molecular Mobility of Poly-4-vinylpyridine by Spin Label Technique Dokl. A N S S S R , 1973, 208, 1391-1394. (Rus) 24. Martini G., Ottaviani M.F., Pedocchi L., Ristori S., The State of Hydrated Vanadyl Ions Absorbed on a Perflorinated Ionomer as Studied by ESR and ENDOR, Macromolecules, 1989, 22, 1743-1748. 25. Barklie R.C., Girard O., Bradell O., EPR of Vo2+ in a Naftion Membrane, J. Phys. Chem., 1988,92, 1371-1377. 26. McCrum N.G., Read B.E., Williams G., Anelastic and Dielectric Effects in Polymeric Solids, Wiley London, 1967. 27. Hill N.E., Vaughan W.F., Price A.H., Dielectric Properties and Molecular Behaviour, Van Nostrand-Reinhold, London, 1969. 28. Electrical Properties o f Polymers, Ed. Sazhin B.I., Khimiya, Moscow, 1970. (Rus) 29. Havriliak S., Negami S., Complex Plan Representation of Dielectric and Mechanical Processes in Some Polymers, Polymer, 1967, 8, 161-210. 30. Gross B., Charge Storage in Solid Dielectrics, Elsevier, Amsterdam, 1964. 31. Van Tumhout J., Thermally Stimulated Discharge o f Polymer Electrets, Elsevier, Amsterdam, 1975. 32. Electrets, Ed. Sessler G.M., Springer-Verlag, Berlin, 1980. 33. Lushcheikin G.A., Polymer Electrets, Khimiya, Moscow, 1976; 2nd Ed., 1984. (Rus) 34. Kovarskii A.L., Thermally Stimulated Depolarization Technique for Studying Polymer Relaxation, In: Encyclopedia o f Engineering Materials, Marcel Dekker, New York, 1988,1, 643-676. 35. Williams G., Watts D.C., Nonsymmetrical Dielectric Relaxation Behavior Arizing a Simple Decay Function, Trans. Far. Soc., 1970, 66, 80-91. 36. Kobeko P.P., Amorphous Substances, AN SSSR, Moscow, 1952. (Rus) 37. Little W., Superconductivity of Organic Polymers, J. Polym. Sci., Part C, 1967,17, 3-12. 38. Methes K.H., Electrical Properties, in Engineering Design fo r Plastics, Ed. Baer E., Reinhold Pub. Corp., New York, 1961. 39. Kobeko P.P., Shishkin N.I., The Dependence of Electroconductivity of Supercooled Liquids on Pressure, Volume and Temperature, Zh. Tekh. Fiz., 1947,17, 27-32. (Rus)

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40. Tubandt C., Z. Electrochem., 1920, 26, 358; Handbuch der experimental Phizik, B12, Leipzig, 1932. 41. Frohlich P., Theory o f Dielectrics, Clarendon Press, New York, 1950. 42. Zlatkevich L., Radiothermoluminescence and Transitions in Polymers, Springer, New York, 1987. 43. Kuleshov I.V., Nikolskii V.G., Radiothermoluminescence o f Polymers, Khimiya, Moscow, 1991. (Rus) 44. Barrer R.M., Diffusion in and through Solids, Cambridge University Press, 1941. 45. Crank J., The Mathematics o f Diffusion, Oxford Univ. Press, 1956. 46. Jost W., Diffusion in Solids, Liquids and Gases, Acad. Press, N.Y., 1960. 47. Diffusion in Polymers, Ed. Crank J., Park S., N.Y., Academic Press, 1968, 483 p. 48. Reitlinger S.A., Permeability o f Polymer Materials, Khimiya, Moscow, 1972, 269 p. (Rus) 49. Malkin A.Ja., Chalykh A.E., Diffusion and Viscosity o f Polymers. Methods o f Measurement, Khimiya, Moscow, 1973, 303 p. (Rus) 50. Chalukh A.E., Diffusion in Polymer Systems, Khimiya, Moscow, 1987. (Rus) 51. Labhart H., Electrochromism, Adv. Chem. Phys., 1968,13, 179-204. 52. Blumenfeld L.A., Cherniakovskii F.P., Gribanov V.A., and Kanevskii I.M., On the Motion of Polymer Molecules Studies by the Electrochromism Method, J. Macromol. Sci., 1972, A6, 1201-1225. 53. Chernyakovskii F.P., Electrochromism as a Method for Investigation of Slow Motions in Macromolecules, Uspekhi Khimii, 1979, 48, 563-582. (Rus) 54. Hamph H.L., Yang J., Wong J.K., Torkelson J.M., Dopant Orientation Dynamics in Doped Second-order Nonlinear Optical Amorphous Polymers. I. Effects Of Temperature Above and Below Tg in Corona Poled Films, Macromolecules, 1990, 23, 3640-3647; II. Effects of Physical Aging on Poled Films, ibid, 3648-3654. 55. Dhinojwala A., Wong J.K., Torkelson J.M., Quantitative Analysis of Rotational Dynamics in Doped Polymers Above and Below Tg: A Novel Application of Second Order Nonlinear Optics, Macromolecules, 1992, 25, 7395-7395. 56. Stahelin M., Burland D.M., Ebert M., et al., Re-evolution of the Thermal Stability of Optically Nonlinear Polymeric Guest-host Systems, Appl. Phys. Lett., 1992,61, 1626-1628. 57. Anufrieva E.V., Gotlib Yu.Ya., Investigation of Polymers in Solution by Polarized Luminescence, Adv. Polym. Sci., 1981,40, 1-68.

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58. Chigginno K.P., Roberts A.J., Phillips D., Time-resolved Fluorescence Techniques in Polymer and Biopolymer Studies, Adv. Polym. Sci., 1981, 40,69-168. 59. Fleming G.R., Chemical Applications o f Ultrafast Spectroscopy, Oxford, New York, 1986. 60. Haugland R.P., Molecular Probes. Handbook o f Fluorescent Probes and Research Chemicals, Molecular Probes Inc., Eugene, OR, 1990. 61. Waldow D.A., Ediger M.D., Yamaguchi Y., Matsushita Y., and Noda I., Viscosity Dependence of Local Segmental Dynamics of Anthracenelabeled PS in Diluted Solution, Macromolecules, 1991, 24, 3147-3153. 62. Cicerone M.T., and Ediger M.D., Photobleaching Technique for Measuring Ultraslow Reorientation Near and Below the Glass Transition: Tetracene in O-terphenyl, J. Phys. Chem., 1993, 97, 1048910497. 63. Inoue T., Cicerone M.T., and Ediger M.D., Molecular Motions and Viscoelasticity of Amorphous Polymers Near Tg, Macromolecules, 1995, 28, 3425-3433. 64. Jones P., Jones W.J., and Williams G.J., J. Chem. Soc. Faraday Trans., 1990, 86,1013. 65. Paczkowski J. and Neckers D.C., Developing Fluorescence Probe Technology for Monitoring the Photochemical Polymerization of Polyacrylates, Chemtracts-Macromol. Chem., 1992, 3, 75-94. 66. Kochetkov I.N. and Neckers D.C., Fluorescent Probe Studies of UV Laser-induced Polymerization of Polyacrylates, J. Imaging Sci. Technol., 1993,37, 156-162. 67. Anufrieva E.V., The Structure and Intermolecular Mobility of Macromolecules in Solution as Studied by Time Polarized Kuminescence, Pure and Appl. Chem., 1982, 54, 533-548. 68. Krakovyak N.G., Ananyeva T.T., and Anufrieva E.V., Bridge Bond Formation in Polymer Systems, Rev. Macromol. Chem. and Phys., 1993, 33, 181-236.

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Chapter 4

Molecular Structure and Motional Frequencies of Additives

As noted above, the heat motion of low molecular particles in polymer combines rotational and translational components. Beside the external conditions, the intensity and amplitude of these motions depend on the structure of particles and properties of the polymer matrix. This chapter discusses specific features of connection between the mobility of particles with their size and structure.

4.1 ROTATIONAL FREQUENCIES

First of all, we’ll consider an important feature of rotational motion of particles in amorphous substances, because to our point of view, this problem has not been adequately studied yet. It shows that in glass-forming liquids and polymers at low temperatures, there are at least two types of rotational motion of molecules, which are distinguished by frequencies and activation energies. This effect was originally observed, probably, by Denis and Edwards for anthrone relaxation in polystyrene /1/. More thoroughly, this effect was investigated by Johari and Goldstain in a cycle of studies on dielectric relaxation of rigid aromatic molecules in glass-forming liquids /2 4/, and later it was found and discussed for various organic molecules in polyethylene /5/ and polystyrene /6 - 8/. According to the data on dielectric relaxation, Fig. 4.1 shows an Arrhenius plot of rotational frequencies of rigid aromatic molecules in pyridine. The temperature interval tested is 100 - 170 K and includes the vitrification temperature of pyridine. As it is seen, there are two branches of temperature dependences differing by temperature ranges, character and

89

slope. High-temperature branches of the rotational relaxation are not described by a linear dependence on the logf - 1/T coordinates and undoubtedly connected with the transition glass-liquid. Lowtemperature relaxation is successfully described by the Arrhenius law.

Fig. 4.1. Temperature dependence of rotational frequency of chloronaphthalene (1), toluene (2), and brombenzene (3) molecules in pyridine, based on the dielectric relaxation data /2 - 4/. The nature of temperature dependences of particle rotational frequencies is essentially the same as for a- and p-processes in polymers /9 12/. The thermostimulated depolarization technique was applied to observe this effect for dye molecules in polystyrene /6 - 8/. Fig. 4.2 summarizes the data on the frequencies of additives rotation in PS obtained by different techniques. It should be noted that van-der-Waals volumes are close for all molecules ranging between 105 and 150 cm3/mol. The data in Fig. 4.2 show that there are two types of rotational motion of particles - high-frequency (small-scale) and low-frequency (large-scale). Mechanisms responsible for these effects are discussed in Chapter 7. Let us mention just in brief that at T < Tg rigid structure of a cage in glasses (in both polymeric and low molecular ones) allows reorientation of particles, but only within a narrow range of angles. Molecular motion in this range may be characterized as the rotational trumbling (local reorientation). With increasing temperature, the large-scale cooperative restructurization is performed near Tg, which enables rotations by large angles. This process may be characterized as a transition and it does not obey the Arrhenius equation. The transition process degenerates at T »

90

Tg, where a single type of the molecular rotation is observed (combined aprelaxation).

Fig. 4.2. Rotational relaxation (correlation) times of organic molecules in PS obtained through thermostimulated depolarization (TSD), dielectric relaxation (DR), and spin probe ESR spectroscopy techniques, plotted as a function of l/T. o

91

Thus, the analysis of rotational frequency of additives in polymers and their dependence on sizes and structure requires consideration of the type of the motion, related to the experimental data, and the frequency (temperature) range, in which they were recorded. For instance, the values obtained by the traditional spin probe technique gives information only on the high-frequency region where differences betwen frequencies of both types of motion disappear. Let us note that application of the pulse ESR techniques, the spin echo technique, for example - allows registration of low frequencies of the probe rotations (Sec. 3.1). In this case, the probe gives the information about low-temperature range of the relaxation. Unfortunately, the works performed in accordance with the technique /13 - 15/ fails to bring the frequency parameters of the probe into correspondence with one or another relaxational process in glasses. The data reported in the literature on the connection between rotational frequencies of particles and their structure was mainly obtained by dielectric techniques (in alternating fields and TSD), and by the spin probe technique. In recent years, photobleaching techique has been successfully used for this purpose (Sec. 3.4). The advantage of the dielectric techniques is a very wide range of frequencies detected. The more so, these methods allow to get information on large-scale modes of rotation (a-process). TSD technique is the most promising one, differing from dielectric relaxation technique in alternating fields by high resolution (Sec. 3.2.2). The advantages of the spin probe technique are, first of all, the possibility of the analysis of particle rotation anisotropy. However, the limited range of characteristic frequencies of the method allows to study only the processes of the local reorientation of particles (P- and aP-processes). Now we are to discuss the analysis of the frequency dependences for both types of particle rotation on their structure. Such analysis for correlation times of spin probe rotation was made in /16 - 20/. The data were obtained with the nitroxideradicals containing heavy mercury atoms in their structure (Fig. 4.3). This allowed to change the molecular mass of particles without significant change of their volume. The linear dependence of log-c on the vander-Waals volume of particles (Vw) is observed for the probes of a rigid structure barring from any intramolecular rotations. Thus, the free volume model gives a true description of these dependences. The only exception are the particles with intramolecular motion. In this case, the nature of this dependence differs greatly, i.e. at Vw values above a critical point the frequencies of rotation does not depend on the molecule size. This effect may be attributed to the fact that the correlation tmes of these radicals are mainly defined by rotation of a fragment containing a nitroxyl group of about 150 o

A 3 in volume.

92

Fig. 4.3. Dependence of rotational correlation times for spin probes in natural rubber at 313 K on Vw /18/. Circle points denote probes with rigid structure, and triangles denote probes with internal rotations. Mercury-containing probes fit well the volume dependence of mobility, common for rigid particles, despite their molecular mass exceeding the M value of hydrocarbon nitroxides. This evidences that the molecular volume is the main factor determining its rotational mobility (at least for O particles of the volume no less than 150 A 3). Table 4.1 Parameters of Eq. 4.1 for rotational diffusion of organic molecules in polymers /8, 18/. Polymer Diffusant Type of rotation Spin probes High-frequency NR rotation Spin probes High-frequency PE rotation Low-frequency Aromatic PS dyes rotation

93

T, K 313

logA -10.9

BxlO2, mol/cm3 3.9

313

-11.6

3.93

273

-7.15

11.4

10

.5 1—----- — — ------------------ --------- —----- — ----too 150 200 250 300

350

Vw, cnrVtaof

Fig. 4.5. Dependence of rotational correlation times for dyes on Vwin PS at 273 K according to the TSD data /81. I

II

o'5

c h

= ch2

c h

= ch2

o

c h

3

o*

c h

3

94

Similar results were obtained for a low-frequency type of rotation of dye molecules (shown above) in PS by the thermostimulated depolarization (TSD) techique (Fig. 4.4) /8/. Volume dependences of rotation period both for probes and dye molecules are well described by the equation similar to that of the free volume theory (Eq. 2.111 ): x = A -exp(BV w ),

(4.1)

where A and B are constants, dependent on the rotation type (Table 4.1). Thus, the free volume model can be considered as the one satisfactorily describing the dependence of rotation frequencies of small particles on their volume. As follows from Eqs. 2.109 - 2.112, the coefficient B in Eq. 4.1 may be determined by the ratio: (4.2)

The

ratio can be experimentally obtained under the bulk

compression of the system providing to calculate the free volume and, hence, to verify the satisfiability of Eq. 2.112 with the accuracy up to the coefficient y. This calculation is given in Chap. 6. As it was mentioned above, the study of dielectric relaxation of aromatic additives in PE had been carried out in /51. This study confirmed the existence of two types of the rotational motion of particles. However, the data obtained did not allow the correlation between the particle motion intensity and their sizes to be disclosed. It is caused, apparently, by a low accuracy of the relaxation times determination by complex peaks of the loss factor. The attention should be paid to the circumstance that analysis of the data on rotational frequencies of spin probes, shown in Fig. 4.3, was performed by the equations of the isotropic rotational model and did not take into account asymmetry of a tensor of the particle rotational diffusion. Hence, frequencies close to the highest frequency of rotation around a large radical axis were determined, in fact. The analysis of anisotropy of paramagnetic particle rotation in liquids and polymers is performed in works /20 - 23/. The investigations were fulfilled by the ESR technique in the 3 cm and 2 mm ranges of wavelengths. The values of T| and xj_ for two orthogonal axes of the radical were determined in the approximation of axialsymmetric tensor of rotational diffusion. The data obtained allowed to

95

for rod-like particles VIII

conclude that the anisotropy parameter N

- XII virtually does not depend on the matrix and is generally defined by the asymmetry degree. The more so, anisotropy parameter does not depend on temperature. This means that the activation energies of particle rotation around different axes are equal to each other. The length of the half-axes ni and rj_of rotation ellipsoid, the ratio of van-der-Waals volumes of particles Vrei, and values of the anisotropy parameter N are shown in Table 4.2. As expected, rotational anisotropy increases with the increase of the radical length. The parameter N can also be calculated theoretically on the basis of a hydrodynamic model. As the data from Table 4.2 show, in this case the results of calculations by the free volume model are much closer to the empirical ones. Table 4.2 Parameters for spin probe rotation anisotropy /16/. Radical

r±(A)

r,|(I) 4.0 4.3 4.3 4.3 4.3

Vrel

N,

n

2

1 1 1 3.8 TEMPO 1.5 1.1 VIII 4.6 1.5 1.4 IX 6.4 2.5 1.3 XIII 7.4 1.5 4.0 1.6 1.7 XII 8.5 6.0 1.8 Note: Ni - experimental values; N 2 - the values calculated by hydrodynamic equation. The question about distribution in rotation frequencies of low molecular particles in solid matrices is also of interest. Such information was obtained by the spin probe technique and TSD /16, 24/. Analysis of the form and line width of ESR spectrum of the spin probes shows that at T >> Tg in polymers a distribution in correlation times of rotation (t ) exists. However, the spectrum width of t does not exceed one decade in most homogeneous polymers. The method of fractional TSD was used for analysing the distribution in rotational relaxation times of paranitropara-dimethylamino azobenzene in polystyrene in the temperature range of 253 - 340 K. It was found that distribution in t is of a bell-like shape with the width depending on temperature (Fig. 4.5). The half width of the spectrum (on the half height) decreases from ~ 2.5 orders of magnitude at 273 K to 1 order of magnitude at 313 K.

96

Fig. 4.5. Half-width of the t spectrum for paranitroparadimethylamino azobenzene in PS plotted as a function of temperature /24/.

4.2 TRANSLATIONAL FREQUENCIES

As it follows from the previous Section, the methods of analysis of the molecular rotation allow to determine the frequencies of rotation around different molecular axises, and thus to obtain information on the components of the rotational motion tensor. Unfortunately, no similar methods are available for studying translational diffusion. In most cases, the information accessible concerns only the influence of particle structure on the rate of their passing through a polymeric membrane, and nothing can be said about the D value in relation to different molecular axes of the particle. Numerous experiments with polymeric matrices showed that the diffusion rate depends on the size, shape, and chemical origin of the particles. It seems very difficult to take all these factors into account. As shown in Chap. 2, classical diffusion theories (activation and free volume ones) consider the motion of spherically symmetric particles. Asymmetry is taking into account through the introduction of empirical equations or corrections. Most often, these equations are exponential. Analysis of data on diffusion of low molecular particles was performed by many authors /25 - 32/. Such analyses cause a conclusion that dependences of the diffusion coefficient for different particles on their sizes cannot be described by a general equation. The type of these dependences and their quantitative characteristics are defined by the shape (degree of asymmetry) and size of the particles, chemical

97

structure of the particle and matrix, i.e. by parameters of intermolecular interaction. To analyse these dependences and calculations of self diffusion coefficients of particles, theoretical and empirical equations are used. The most wide-spread among the model equations is the one of the free volume theory, which assumes an exponential dependence of D on particle volume (or linear dependence of lgD vs. V).

vw icns /noli

Fig. 4.6. Dependence of translational diffusion on Vw for: a - spin probes I, VIII, X, XV in PE at 313 K /19/; b - n-paraffins in NR at 298 K; c - nparaffins in PB at 298 K and acrylic monomers in PVC at 333 K (according to the data from Table 4.3). In many cases, and especially in homological series of particles, the exponential dependence of diffusion coefficient on the particle volume is successfully fulfilled /27, 26, 30, 33 - 34/. Let us consider the examples of such dependences basing on the data obtained by Chalykh /35/ and shown in Table 4.3 and Fig. 4.6. It is seen that dependences of logD on Vw are almost linear. The equation that describes these dependences is similar to the equation for rotational diffusion:

98

lgD = lg A - B % 3.

(4.3)

Table 4.3 Diffusion constants for some organic molecules in polymers /35/. Polymer Natural rubber

Polyisobutylene

Polybutadiene

T, K

Diffusant Hexane Nonane Decane Dodecane Tetradecane Hexadecane Benzene Toluene Nitrobenzene Dioctilphthalate Amlin Cyclohexane Tetrachlorocarbon Heptane Hexane Octane Isooctane Undecane Dodecane Hexadecane Benzene Toluene Xylene Ethylbenzene Chlorobenzene Cyclohexane Tetralin Decalin T e^rachloro^arbon_ Hexane Oktane Decane Dodecane Hexadecane

99

Vw», cm3/mol 298 69 100 298 298 110 298 131 298 151 172 298 53 298 298 63 298 63 257 298 59 298 298 62 53 298 59 298 69 298 89 298 89 298 126 298 298 131 172 298 298 53 298 63 74 298 74 298 62 298 62 298 86 298 298 95 53 298 69 298 ‘ 89 298 110 298 298 131 172 298

M

logD, cm2/s

86 128 142 170 198 226 78 92 123 418 93 84 154 72 86 114 114 156 170 226 78 92 106 114 113 84 132 138 154 86 114 142 170 226

-6.15 -6.25 -6.45 -6.66 -6.82 -6.89 -6.05 -6.07 -6.8 -7.92 -7.0 -6.41 -6.33 -7.8 -7.6 -8.12 -8.4 -8.7 -8.9 -9.24 -8.3 -8.7 -9.25 -8.92 -8.9 -9.15 -9.63 -9.7 -8.88 -6.19 -6.4 -6.53 -6.62 -6.74

PS

Benzene Cyclohexane Decalin Didecylphthalate Diheptylphthalate Dibutylphthalate Diamilphthalate a-Methylnaphthalene Benzene Toluene Xylene Ethylbenzene Chlorobenzene Decalin Dibutylphthalate Diethylphthalate Diheptylphthalate

ecu

PVC

PVAc2>

CH 2CI2 CHCb CH2Br2 CHBr3 CH 3J CH 2J 2 CH3(CH2) 2C1 CH3 (CH2) 3C1 CH 2CI-CH 2CI CCh=CHCl Benzene Acrylic acid Methylacrylate Ethylacrylate Butylacrylate Pentylacrylate Methacrylic acid Methylmethacrylate Butylmethacrylate 2-Ethylhexylmethacrylate Aceton Ethylacetate Propylacetate Butylacetate

100

298 298 298 298 298 298 298 298 323 323 323 323 323 323 323 323 323 293 293 293 293 293 293 293 293 293 293 293 293 333 333 333 333 333 333 333 333 333 303 303 303 303

53 62 95 319 216 175 195 53 63 74 74 62 95 175 134 216 52 30 41 41 58 33 56 47 57 38 51 53 38 51 61 83 94 48 61 92 125 39 52 72 62

78 84 138 474 334 278 306 142 78 92 106 114 113 138 278 222 334 154 85 119 174 253 142 268 78 92 99 63 78 72 86 100 115 129 86 100 142 184 58 88 102 116

-6.19 -6.46 -6.7 -7.18 -6.86 -6.3 -6.2 -6.5 -9.4 -9.5 -9.75 -9.7 -9.8 -12.0 -10.15 -9.7 -11.5 -14.05 -10.71 -12.24 -11.6 -12.54 -11.84 -12.5 -13.0 -12.7 -12.2 -12.45 -13.1 -7.92 -8.2 -8.45 -8.89 -9.14 -8.1 -8.5 -9.3 -10.1 -6.6 -6.7

-6.85

-7.09

h 2o

313 9 18 -6.08 CH 3 0 H 313 22 32 -7.3 C 2H 5 0 H 313 32 46 -8.4 C 3H 7 0 H 313 44 60 -9.02 n-C4H9OH 313 56 74 -9.6 1S0 -C4H 9OH 313 53 74 -10.0 tert-C4H90H 313 53 74 -10.2 Note: 1) Calculated from group increments with zero order accuracy /36/; 2) from the data of /37/. The values of coefficients A and B are shown in Table 4.4. Table 4.4 Parameters of Eq. 4.3 for translational diffusion of organic molecules in polymers. Polymer NR

PIB

PB PVC PVAc PE PS

Diffusant n-paraffins (n > 4) (n < 4) iso-paraffins n-paraffins (n > 4) (n < 4) iso-paraffins cycles n-paraffins cycles derivatives of acrylic and metacrylic acids cetones alcohols nitroxide spin probes aromatics

T, K

logA

BxlO2, mol/cm3

Ref.

298 313 313

-5.56 -5.5 -5.76

1.84 4.94 4.71

35 27 27

298 308 308 298 298 328 298

-6.72 -7.58 -8.55 -6.64 -6.0 -5.3 -5.69

3.52 4.28 4.5 7.56 1.18 1.26 2.48

35 27 27 35 35 35 35

333 303 313 313 323

-6.95 -5.98 -5.52 -6.42 -8.65

5.6 3.42 1.88 1.8 2.5

35 37 35 19 35

The data on self diffusion of saturated hydrocarbons in rubbers, presented in /27/, show that the dependences become more sharp with the reduction in the number of carbon atoms, n, to 4 - 5, and with the increase in the values of A and B. Most probably, this effect is connected with deviations from sphericity, raising with the increase in the number of carbon atoms.

101

Thus, the wide-spread opinion that linear molecules move along the large axis is confirmed. The slope angle of lgD = f(Vw) dependences, and consequently the value of B for cyclic compounds which symmetry is closer to the spheric one, is significantly higher than for long molecules of saturated hydrocarbons. As the cross-section of such particles does not depend on n, the ratio between the values of activation volume V* and particle volume Vw differs from the ratio for spheric particles. Evidently, the coefficient B depends on this ratio. Indeed, comparison of Eq. 4.3 with the equation of the free volume model gives:

B = y

4) in PIB, the value of B is twice higher than for NR. Hence, the above data lead to the conclusion that equations of free volume model are fulfilled for many cases, at least qualitatively. However, there are lots of exclusions from this rule. The

102

examples are the forementioned data on diffusion of paraffins with deviation of lgD - V dependences from linearity at small n values. The data shown in Table 4.3 on the diffusion of benzene derivatives is another example. They do not obey a linear dependence on the lgD - V coordinates. The introduction of substitutors causes an abrupt change of the diffusion coefficients at a small change of volume. Despite a trend towards an increase in the D value with the increase in the molecule volume, Eq. 4.3 is unable to describe the translational diffusion of haloid-derivatives of saturated hydrocarbons (Table 4.4). It can be concluded that intermolecular potentials, not taken into account in the free volume model, are of significant role. Therefore, to calculate diffusion coefficients Eq. 4.3 should be used only in the cases, when its aplicability is proved experimentally. Otherwise, the empirical ratios shown below can be used. Because these equations use not only the particle volume but also their molecular mass, let us discuss the data on the diffusion coefficient dependence on M. As Chap. 2 shows, the main parameters of theoretical models, describing dynamics of particles in a viscous medium, are their volumes or linear dimensions. It is evident that these parameters can be substituted by the molecular mass in the case, if it is proportional to the size of the particles. This is illustrated quite well by the data on the translational diffusion of normal paraffins. In these compounds, volume and molecular mass increase with the number of carbon atoms in chain, and dependences lgD - V, lg D M and lg D - n are similar. Studies /38 - 39/ displayed the following empirical equation connecting the diffusion coefficient of alkans with the number of carbon atoms, n: (4.5) Here Di is the diffusion coefficient of the first homologous series member; a is a coefficient dependent on the matrix properties. However, in the cases, when the connection between sizes of particles and their molecular mass is not observed anymore, parameter M could not be used for estimating D. For example, at the introduction of heavy halogen atoms into hydrocarbons, their volume changes significantly smoother than their mass. Diffusion coefficients are also little affected. For example, in molecules CH 2CHCI, CH 2J 2, CHBr3 the values of vw are 50.7, 56 and 57.7 cm3/mol, respectively. The values of M are 63, 268 and 253, and diffusion coefficients in PS at 293 K are 2.8, 3.2 and 3 .4 x l0 12 cm2/s. It is seen that volume rather than molecular mass determines the value of D. This conclusion is consistent with the above mentioned study of rotational dynamics of particles with mercury atoms.

103

For the systems with linear dependence of D on M, the empirical equation of Auerbach is used /40/: D = KMb .

(4.6)

log D = log K + b log M .

(4.7)

or in logarithmic form:

Table 4.5 Parameters of Eq. 4.6 /30/. Matrix PB

NR

PIB

PVC

Diffusant aromatic hydrocarbons aliphatic cyclic hydrocarbons phthalates sebacinates n-paraffins aromatic hydrocarbons phthalates chloroderivatives of hydrocarbons n-paraffins aromatic hydrocarbons aliphatic cyclic hydrocarbons chloroderivatives of hydrocarbons phthalates sebacinates acrylates

T, K

KxlO4, cm2/s

b

298

1.8

1.3

298 298 298 323

4.0 2.0 2.0 5.0

1.5 1.4 1.35 1.5

323 323

3.7 3.7

1.5 1.5

323 323

3.0 16

1.75 2.2

323

6

2.2

323

18

2.8

323 343 343 343

16 0.005 0.005 0.32

2.6 3.7 3.5 3.5

The results of experimental data processing by this equation, borrowed from /30/, are shown in Table 4.5. Eq. 4.6 is usually fulfilled for translational diffusion of homologous series of particles with different n values and the same functional groups. The values of b usually range from 1.3

104

to 3.7 and depend on the matrix and the type of homologous sequence of molecules. It should be noted that the value of b is close to unity for the self diffusion of particles in liquid /41 - 44/. In many cases, the values of b for aromatic and chlorine-substituted compounds are higher than for linear paraffins. The connection between the coefficient b and molar cohesive energy of the matrix, 8, was derived in /30/: b = 0.21 ·- —— , S0

(4.8)

where 8o = 6.8 kcal/mol. Eq. 4.8 adequately describes the elastic state of polymers. Constant K in Eq. 4.6 also depends on the structure of diffusing particles and the matrix. It was suggested that this parameter is bound to the friction coefficient \ and the molecule volume V /30/. The value of K increases with temperature and decreases with pressure.

Fig. 4.7. Dependence of b on T-Tg for low molecular particles diffusion in polymers: PB, PIP, PMMA, PIB, PVC, and PVAc /30/. It was also shown /30/ that for treatments of the data on diffusion of spheric particles, and the particles with low molecular mass, the exponential function can be successfully used: D = K'V b , or in the logarithmic form:

105

(4.9)

ln D = l n K '- b l n V .

(4.10)

It is evident that in the simpliest case, when correlation between molecular mass and volume exists, Eqs. 4.9 and 4.6 are identical. It was shown that the value of b decreases with temperature, and the unitary dependence of b on the reduced temperature T - Tg is observed for different matrixes (Fig. 4.7). These data show that the parameter b is most probably bound to the free volume of polymer (b « Vj· ^). Thus, tangles of Eqs. 4.3 and 4.9 are of the same physical sense, i.e. they express the value of Vc. Consider another series of semiempirical equations, used for calculations of diffusion coefficients. The following expression was suggested in /45 - 46/ for linear molecules with n repeating groups:

D = £n=L aan

( 4 .1 1 )

or D=^ ^ - ,

M a

(4.12)

where D n=i is the value of D for the first member of the homologous series; ct is the relative increase of the cross-section of the molecule when passing to the next member of the series, a is a constant. It was shown that these equations are valid for great amount of systems, and that the value of a correlates with the energy of intermolecular interaction in the matrix. In its turn, this interaction defines the viscosity of the matrix. The following expression can be used for molecules with q double bonds and p branchings /45/: D n= i( 2 n - q ) 0 2( n - p ) a n (n -q )

(4.13)

For complex molecules with heavy side fragments the equation takes the form:

106

D=

Dl

(4.14)

Here M' is the mass of the molecule without side groups; Mi is the molecular mass of the first member of the series. Let us now discuss the point about the ratio of diffusion coefficients of smaller particles, such as atoms and molecules of gases with diameter below O 5 A . The decrease of D with size increase is also characteristic for these particles. However, their main parameters are diameter or cross-section of molecule /25, 29, 30, 32, 34/. The examples of such dependences, similar to those reviewed in / 25,47/, are shown in Fig. 4.8. However, it should be noted that these dependences are not apparently universal, because diffusion of gases (He, H 2, O 2, N 2, and CO 2) in rubbers is not described by linear dependences in the coordinates mentioned /27/. The results of the analysis of gas diffusion, made in /48/, give the opportunity to use the Brandt theory equation (Sec. 2.3.3) as the universal one. In the frames of this theory, a linear correlation must occur between the logarithm of reduced diffusion constant (lo g -^ -) and reduced molecular diameter (molecular diameter reduced d2 further by one-half the square root of the free volume per the length of -CH 2group). This correlation is confirmed by experimental data, obtained for gases with the molecular volume V < 40 cm3/mol /25, 48/.

7

I < I__ _ j__ 1__ 1— l x .. -9 1,772.13 2,66 3,21 1,77 2J3 2,68 3,2 A He Ne Ar Xe He Ne Ar Xe

Fig. 4.8. Dependences of D t on the diameter of gas particles diffusing through PE (a) and polyamide (b) at different temperatures /29/.

107

To sum up this part, it should be noted that the intensity of particle motion in the same polymer matrix depends on their sizes, shape (symmetry), and chemical structure, the last parameter defining the energy of interaction with the matrix /25 - 35, 45 - 51/. An important role, especially for the rotational diffusion of particles, is played by available degrees of intramolecular freedom. The equations of the free volume model (exponential dependence of D and x on volume) have some advantages over the equations connecting motion frequencies with the mass and linear sizes of the particles with the volume over 40 cm3/mol. In other words, basing on the structure of particles, their volume is the more preferable parameter for calculation of rotational and translational mobility. Translational dynamcs of smaller particles of gases (V < 40 cm3/mol) is characterized by the exponential dependence of the diffusion coefficient on linear size (diameter).

4.3 RATIO OF ROTATIONAL AND TRANSLATIONAL FREQUENCIES

The question about correlation between frequencies and activation parameters of rotational and translational motion of particles is very important for undestanding the mechanisms of molecular motions and kinetics of chemical reactions in condensed phase. For example, diffusional kinetics is controlled by the translational mobility of particles, and the reaction rate in a ‘cell’ generally depends on the frequencies of the rotational motion. The ratio of coefficients of translational (Dt) and rotational (D r = — ) 6x diffusion is determined from the following equations. In the frames of the hydrodynamic model (Eqs. 2.8 and 2.17): Dr Dt

3r

2 ’

(4.15)

where r is the radius of a spheric particle. The activation model gives: In

=A+

AE* + PAV*

108

RT

(4.16)

(O

where A = In

D?

n 0t JI v. D At P = 0 we obtain:

* * * * ; AE = E t - E r ; AV

In

=A +

AE

RT

(4.17)

The following correlation should be fulfilled for the free volume model: AV In Dr = A + vDt / Vf

(4.18)

As seen from Eqs. 4.15 - 4.18, the ratio of the diffusion coefficients for both types of motions depends on the square of particle radius, or activation energy difference, or on the difference of the activation volumes.

Fig. 4.9. Dependences of Dt (a) and n (b) on D r for spin probe TEMPO in NR (1), butadiene-nitrile rubber (2), butadiene-styrene rubbers with different content of styrene (3 - 5), PIB (6), butyl rubber (7), and atactic PP (8).

109

Let us discuss experimetal results. In accord with the NM R data, the ratio of correlation times for translational and rotational motions of pyridine molecules in pyridine is 83 at 293 K /51/. This value shows that during a single event of transition into new position the particle changes its orientation an average of 83 time. Rotational and translational mobilities of five nitroxides (spin probes) in liquid and in polyethylene were discussed in /16, 18, 19, 52/. The value, used as a parameter characterizing the ratio of intensities of these types of motions, was taken as:

6D t Let us remind that v r = 6D r , and v t = ——L (Sec. 2.2). The value of l2 n shows how many times the particle changes its orientation during the translation over the distance 1. The data shown below used the value of 1 = 5 o

A . This value is close to the diameters of the majority of additive molecules. The more so, the distances of elementary jumps of such particles are of the same degree (Sec. 4.4). Fig. 4.9 shows dependences of Dt on D r in some polymers at T > Tg. Linear dependences in logarithmic coordinates point out the correlation of experimental data to Eqs. (4.16) and (4.18). These dependences can be also described by the following general expression: D t = a ( D r )P .

(4.19)

Dependences of n on matrix, particle size and temperature are shown in Table 4.6 and Fig. 4.10. It is seen that the ratio of frequencies of rotational and translational mobility of particles depends on their size: n decreases with the growth of the particle volume. This complies with the hydrodynamic model (Eq. 4.15). However, another consequence of the model, the independence of n on the matrix properties and temperature, are not fulfilled. For example, n is 40-fold decreased for spin probe in PE at temperature increase by 57°. In liquid (m-xylol), the values of n of the same nitroxides vary within the range 1.3 < n < 0.4, i.e. significantly lower than those for PE. These data better correlate with the equations of the activation and freevolume models. In the frames of the activation model, n depends on AE*, and in the free-volume model, on AV* and free volume. Activation parameters of

110

molecular motion are discussed in detail in Chaps. 5 and 6. Here we just note that free volume in liquid is greater than in polymer, and therefore, the value of n should be lower. This is observed in experiments. Table 4.6 Ratio of rotational and translational frequencies for spin probes in liquid and in polymer. Matrix

Particle

m-xylene

probe TEMPO probe TEMPO

PE PE

probes: XV VIII TEMPO TEMPO TEMPO

Vw, cm3/mol

T, K

101

248

101 198 156 101 101 101

D ,,

n

Ref.

1.7x10'°

6x1 O'6

7.5

69

313 353

5x10» 3.2xl09

10-8 io-7

125 80

53

350 350 350 323 293

1.7x10« 7.9x10'° 1x 109 1.2x10'9 5.2x109 5 x l0 9 10-9 1.7xl09 10 -io 1.7x10«

D r,

S

cm2/s

5.3 22 29 416 1040

Vw, cnp/mol

Fig. 4.10. Dependence of n on Vw for spin probes in PE at 313 K /19/.

Ill

54

4.4 SCALE OF M OLECULAR M O T IO N S

The scale of molecular motion is usually considered as the angle of rotation or jump length in the elementary act of reorientation or translation of a particle. Models and experimental data on motional scale of particles are discussed below. 4.4.1 Rotational Dynamics

Analysis of scale (amplitude) of rotational motion is performed more often in the frames of two models: continuous (Brownian) rotational diffusion and jumps /55 - 59/. The first model suggests that the angular orientation of a particle changes continuously. The second model states that orientation change is jump-like. If rotations through different angles are equally probable, and this probability does not depend on the rotation angle in the previous act of reorientation, such model is called ‘the model of noncorrelated jumps’. ‘The model of fixed jum ps’ suggests that orientations of a radical are rigidly fixed in space because of matrix structural peculiarity. In this case, rotations can be performed just through angles between these orientations. The first of these models (continuous diffusion) can be considered as the jump-like diffusion, in which elementary angle of reorientation is infinitely small. The techniques of analysis of rotational models are described in Secs. 2.7 and 3.1. A significant part of experimental data was obtained with spin probes (nitroxide radicals) the structure of which is shown in Sec. 3.1. Investigations were performed both in liquids and polymers. The amplitude of rotational diffusion of nitroxide radicals in vitrified dekaline and n-butanol depends on the particle size. Radicals smaller by size (TEMPO, TEMOL) rotate in a jump-like manner, whereas rotation of larger radicals is better described by the continuous diffusion model /58, 60/. Thus, the character of rotation of particles varies from jump-like type to continuous diffusion with the increase in their size. Fremy salt

/r

1

\

rotation 2 7 in ice is not described by both models; the authors /55/ suggested that in this system an intermediate situation is manifested. A wide group of liquids was studied at room temperature by Rayleigh light-scattering technique and dielectric relaxation technique /61/. The data obtained show that rotation of molecules in quinoline, nitrobenzene, chlorobenzene, anilin, and pyridine is in

112

S03

NO*

agreement with the jump-like mechanism. On the other hand, rotation of nitrobenzene and camphor molecules in CCL and CS2 is described by the diffusion model. Thus, the character of rotation of the same particle (nitrobenzene) changes in different matrices. These data show that the character of rotational diffusion depends not only on the particle size but also on the ratio of particle size and surrounding molecules. It also depends on the solvent viscosity. Large asymmetric particles rotate in viscous solvents according to the continuous diffusion mechanism. Small particles in lowviscosity media rotate by the jump-like mechanism. These conclusions correlate with the hydrodynamic model. More rigorous analysis, that allows to estimate the value of the rotation angle 0 can be made for some paramagnetic particles with the help of ELDOR technique. It was reported in /62, 63/ that the value of 0 for sulfur radical in oleum glass is ~ 0.4 rad. Smaller values of 0 were found for dithiolate nickel complex in n-butanol and chloroform glasses (0.15 rad), and for peroxide radical in polystyrene (0.1 rad). It was shown that the probability of elementary rotations of particles through angles, different from those mentioned above, decreases abruptly (exponentially). The 0 value rises with temperature increase. The results obtained show that the model of fixed jumps for glassy matrixes is quite correct. At the same time, the character of particle motion comes closer to Brownian diffusion with temperature reduction, and with temperature growth - to equiprobable noncorrelated jumps. The electron spin echo technique also gives information about elementary angles of paramagnetic particle reorientation /13/. Studying rotations of nitroxide radicals in toluene at T < Tg, the authors concluded that the minimum angle of reorientation is 35°. In polymeric matrixes such investigations were performed by the methods of spin probes /64, 65/ and thermostimulated depolarization (TDS) /66, 67/. The results of the spin probe technique (radicals TEMPO and XIV) showed that in polymers, similar to liquids, rotation of large particles is more adequately described by the diffusion model. Moreover, it was found that the rotation scale of the same particle depends on polymer. Rotation of TEMPO radical in PVC and PMMA is closer to diffusion, and in PP, PE and PS, to jump-like model. How could these differences be explained? An extended analysis showed that correlation times of probe rotation in these polymers are different in vitrification point: second group of polymers characterized by lower values of t comparing with the first one. This means that rotations through greater angles are characteristic for matrixes with higher rotational mobility of particles. Moreover, the second group of polymer matrices possesses higher specific surface (Chap. 7, Table 7.5). As a consequence, it possesses higher microporosity. It may be suggested [concluded] that higher

113

porous packing of polymers from the second group manifests conditions for quite fast reorientation of particles through large angles. The above mentioned data are cosistent with the temperature range below Tg. High rate of cooperative matrix restructurization is observed in the vicinity of vitrification temperature, so-called a-process. Moreover, particles assume a possibility for reorientation through large angles (large-scale rotation). This is confirmed by the data on thermostimulated depolarization (TSD). The essence of this method is the heat-induced depolarization of a sample prepolarization by a direct electric field (Sec. 3.3.2). Depolarization of polar systems occurs as a result of unfreezing of rotational mobility of dipoles. The results of the study of polymeric matrixes show that sample heating in the temperature range of T < Tg (p-relaxation) cannot destroy polarization completely. Destruction of residual polarization occurs at T > Tg, i.e. in the range of a-relaxational process. Since total dipole moment (polarization) depends on orientational angles of polar fragments, it is evident that the effectiveness of depolarization process depends on the reorientational angle of dipoles. Complete destruction of residual polarization in the range of the a-process shows that reorientational angles of dipoles grow sharply in this range. It was shown that conclusion is correct both for segmental dynamics of macromolecules, and for rotational dynamics of additives /67/. The analysis of charges, released during depolarization, and activation volumes of depolarization showed that reorientational angles of dipoles at P-relaxation fall within the range of 5 - 30° /67/. This range of 0 values includes the data on the ELDOR technique, shown above. Minimal values of 0, obtained by the electron spin echo technique, are close to the upper boundary of the range. An additional information about rotation scale is given by the analysis * of activation volumes Vr . This parameter defines the minimum of the fluctuational hole size, enough for particle reorientation. It is analyzed in detail in Sec. 6.2. The studies under high pressure showed that small-scale reorientation of particles (P-process) requires fluctuational holes of small sizes. The volume of such holes is about 20 - 50% of particle volume. The large-scale reorientation of particles (a-relaxation) requires much greater activation volume, equal and even greater that the particle volume. 4.4.2 Translational Diffusion

As shown in the previous Section, there are various methods of scale estimation for rotational motion of particles. The opportunity of the experimental analysis of elementary length of particle translation is rather

114

limited. It seems most probable that the only experimental method, suitable for this purpose, is the study of baric coefficients of translational mobility the activation volumes. The values of V* are discussed in detail in Sec. 6.2.2 and shown in Table 6.5. For evaluating of the particle translation scale, the following results are most important: 1. Activation volumes sharply depend on the shape and structure of particles, and range from 20 to 200% of inherent volume of the molecules. 2. In same polymer, higher values of V* are observed for diffusion of cyclic molecules, the lower - for linear molecules. 3. Homologous series of n-paraffins (hexane, octane and decane) displays y* reduction of ----- (from 68 to 47%) at close values of the activation ^w volume. The last mentioned means that the length of elementary jump changes insignificantly with increase in the particle size. Thus, linear particles can move by distances much shorter than their lengths, i.e. by rather short jumps. Calculation of the elementary jump length for n-paraffins in PE from V* o

gives the average value 1 = 5.5 A under the condition of spheric activation volume /68/. For fluctuational hole of a cylindric form calculation gives 1 = O

8.2 A . For diffusion of a cyclic compounds, cyclohexane for example, in the o

same polymer we get in approximation of spheric hole 1 = 7.3 A , and for O

benzene -1 = 6.6 A . Such is an order of elementary translations of particles in a polymeric matrix. Higher values of 1 were obtained for the diffusion of gases by calculations from the activation energy using Eq. 2.89 /68/. In case of CO 2, O

diffusion I = 11 A in approximation of spheric hole; for cylindric hole 1 = 84 O

A . The last value seems to be overestimated. For H 2 and He particles the o

calculation gives 1 « 13 A (cylindric hole). Differences in jump distances between experimental data for multiatomic molecules and calculations for gas particles can be attributed to the differences in estimation techiques and in characters of translations. For small particles of gases polymer medium most probably represents a porous matrix, the motion character in which is closer to the gas-phase one.

115

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116

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

Molecular Motions Near Equilibrium Position, Chem. Phys. Lett., 1992, 188,217-222. Wasserman A.M., Kovarskii A.L., Spin Probes and Labels in Physical Chemistry o f Polymers, Nauka, Moscow, 1986. (Rus) Kovarskii A.L., Placek J., Szocs F., Study of Rotational Mobility of Stable Radicals in Solid Polymers, Polymer, 1978,19, 1137-1141. Barashkova 1.1., Wasserman A.M., Kovarskii A.L., Effects of Spin Probe Size on Rotational Mobility in Polymers, Vysokomol. Soed., 1982, 24A, 91-95. (Rus.) Barashkova 1.1., Wasserman A.M., Kovarskii A.L., The Effect of Spin Probe Size on the Ratio of Rotational and Translational Mobility in Polyethylene, Vysokomol. Soed., 1981,23B, 434. (Rus) Wasserman A.M., Kuznetsov A.N., and Kovarskii A.L., Anisotropic Rotational Diffusion of Nitroxide Radicals, Zh.Struct.Khim., 1971, 4, 609-616. (Rus) Smith P.M., Anisotropic Rotation of Spin-probes in Polymers, Eur.Polym.J., 1979,15, 147-151. Dadali A.A., Wasserman A.M., Kirillov S.T., Rotational Anisotropy of Spin Probes in Polymers, Vysokomol. Soed., 1980, 22A, 1321-1326. (Rus) Poluektov O.G., Dubinskii A.A., Greenberg O.Y., and Lebedev Y.S., Application of 2mm Range ESR for Investigation of Rotational Motion by Spin Probe Technique, Khim.Fizika, 1982,11, 1480-1484. (Rus) Kovarskii A.L., Saprygin V.N., Study of Rotational Mobility of a Polar Molecule in PS by TDS-Method, Polymer, 1982,23(Suppl.), 974-978. Stannett V., Simple Gases, In: Diffusion in Polymers, Eds. Crank J. and Park S„ 1968,41-73. Fujita H., Organic Vapors Above Glass Transition Temperature, ibid, 75-105. van Amerongen G.J., Diffusion in Elastomers, Rubb. Chem. Techno!., 1964,37, 1065-1152. Rodgers K., Solubility and Diffusion, In: Physics and Chemistry o f Organic Solid State, Eds. Fox D., Labes M.M., and Weissberger A., Wiley, New York, 1965,2,289-328. Reitlinger S.A., Permeability o f Polymer Matrials, Khimiya, Moscow, 1974. (Rus) Chalykh A.E., Diffusion in Polymer Systems, Khimiya, Moscow, 1987. (Rus) Moisan J.Y., Effects of Oxygen Permiation and Stabilizer Migration on Polymer Degradation, In: Polymer Permeability, Ed. Comyn J., Elsevier Applied Science, London, 1985, 119-175. Schlotter N.E. and Furlan P.Y., A Review of Small Molecule Diffusion in Polyolefins, Polymer, 1992,33, 3323-3342.

117

33. Ryskin G.Ya., Activation Energy and Temperature Dependence of Diffucion in Polymers, Zh. Tekhn.Fiz., 1955,25,458-465. (Rus) 34. M.Saleem M., Asfour A.-F.A., De Kee D., and Harrison B.H., Diffusion of Organic Penetrants through LDPE Films: Effect of Size and Shape of the Penetrant Molecules, J.Appl.Polym.Sci., 1989,37,617-625. 35. The diffusion constants were obtained in the Institute of Physical Chemistry of Russian Academy of Sciences by Chalykh A.E. 36. Askadskii A.A. and Matveev Yu.I., Structure and Physical Properties o f Polymers, Khimiya, Moscow, 1983. (Rus) 37. Hutchinson A.T., Kokes R.J., Hoard J.L., Long F.A., Interdiffusion of PVA with a Series of Solvents, J. Chem. Phys., 1952,20, 1231-1236. 38. Yampolskii Yu.P., Gladkova N.K., Filippova V.G, Durgar’yan S.G., Permeability of Silane-siloxane Block Copolymers for Hydrocarbons, Vysokomol. Soed., 1985,27A, 1917-1925. (Rus) 39. Yampolskii Yu.P., Durgar’yan S.G., Diffusional Separation of Hydrocarbons by Means of Polymer Membranes, Neftekhimiya, 1983, 23, 435-453. (Rus) 40. Auerbach J., Miller W.R., and Kuryla W.G., A Diffusivity Approach for Studying Polymer Structure J. Polym. Sci., 1958,28, 129-150. 41. Frenkel S.Ya., Elyashevich G.K., Panov Yu.N., Uspekhi Khimii i Fiziki Polimerov, Khimiya, Moscow, 1970, 87-138. (Rus) 42. Hellwege K.H., Knappe W., and Lohe D., Koll. Ztschr., 1961,179,40. 43. Gary-Bobo C.P., Weber H.W., Diffusion of Alcohols and Amides in Water from 4 to 37°, J. Phys. Chem., 1969,73, 1155-1157. 44. Budtov V.P., Fokanov V.P., and Yanovskaya N.K., Calculation of Coefficients of Rotational and Translational Friction of Low Molar Chain Molecules, Zh. Fiz. Khim., 1974,48, 265-270. (Rus) 45. Vasenin R.M., Coefficient of Diffusion and the Nature of Diffusing Molecules, Vysokomol. Soed., 1960, 2, 857-863. (Rus) 46. Vasenin R.M., Diffusion Coefficient and the Nature of Diffusing Molecules, Vysokomol. Soed., 1961,3, 1220-1223. (Rus) 47. Permeability and Diffusion Data, Polymer Handbook, Eds. Braundrup J. and Junrgut E., Pergamon Press, New York, 1989. 48. Michaels A.S., and Bixler H.J., Flow of Gases through PE, J. Polym. Sci., 1961,50,413-421. 49. Moisan J.Y., Diffusion des Additives du Polyethylene, Eur.Polym.J., 1980,16, 979; 1980,16,989;1980,16,997;1981,17, 857-864. 50. Asfour A.-F.A., Saleem M., De Kee D., and Harrison B.H., Diffusion and Saturated Hydrocarbons in LDPE Films, J. Appl. Polym. Sci., 1989, 38, 1503-1514.

118

51. Assink R., De Zwaan J., Jonas J., Pressure Effects on Coupling between Rotational and Translational Motions in Liquids, J. Chem. Phys., 1972, 56,4975-4983. 52. Wasserman A.M., Barashkova 1.1., Yasina L.L., Pudov V.S., Rotational and Translational Diffusion of Nitroxide Radicals in Amorphous Polymers, Vysokomol. Soed., 1977,19A, 2083-2089. (Rus) 53. Stryukov V.B., About Diffusion Constants of Paramagnetic Molecule in Polymer Media, Vysokomol. Soed., 1968,179, 641-644. (Rus) 54. Barashkova 1.1., Dadaly A.A., Aliev 1.1., Zhorin V.A., Kovarskii A.L., Wasserman A.M., and Buchachenko A.L., Effects of Pressure on Molecular Dynamics of Spin Probes, Vysokomol. Soed., 1983, 25A, 840847. (Rus) 55. Goldman S.A., Bruno G.V., Polneszek C.F., and Freed J.H., An ESR Study of Anisotropic Rotational Reorientation and Slow Tumbling in Liquid Media, J. Chem. Phys., 1972,56, 716-735. 56. Valiev K.A., Ivanov E.N., Brownian Rotational Motion, Uspekhi Fiz. Nauk, 1973,109,31-37. (Rus) 57. Buchachenko A.L., Dynamics of Elementary Processes in Liquids, Uspekhi Khim. Nauk., 1973,48, 1714-1737. (Rus) 58. Kuznetsov A.N., Spin Probe Technique, Nauka, Moscow, 1976. (Rus) 59. Korst N.N., and Antciferova L.I., Study of Slow Molecular Motion of Stable Radicals by ESR, Uspekhi Fiz. Nauk, 1978,126,67-91. (Rus) 60. Livshitz V.A., Krinichny V., Kuznetsov A.N., A Study of Character of Rotation of Nitroxide Radicals in Liquids, Chem. Phys. Lett., 1977, 45, 541-544. 61. Gershon N., Zamir E., BenReuven A., Ber., 1971,75, 316. 62. Benderskii V.A., Piven N.P., Electron-electron Double Resonance Investigations of Slow Molecular Motion, Zh.Fiz.Khim., 1985, 59, 13291334.(Rus) 63. Piven N.P., Benderskii V.A., Electron-electron Double Resonance Investigations of Slow Molecular Motion, Khim.Fiz., 1984, 3, 386-393. (Rus) 64. Kovarskii A.L., Wasserman A.M., Buchachenko A.L., Spin Probe Studies in Polymer Solids, In: Studies o f Molecular Motion in Polymers by ESR, Eds. Boyer R.F. and Keinath S.E., Harwood Academic Press, Chur, Switzerland, 1979, 177-189. 65. Wasserman A.M., Alexandrova T.A., Buchachenko A.L., The Study of Rotational Mobility of Stable Nitroxide Radicals in PVAc, Europ. Polym. J., 1976,12,691-695. 66 . Kovarskii A.L., Mansimov S.A., Buchachenko A.L., “Thermostimulated Depolarization and Mechanism of Relaxation Processes in Polymers”, Dokl. A N SSSR, 1986,291(5), 1142-1146. (Rus)

119

67. Kovarskii A.L., Mansimov S.A., On the Amplitude of Molecular Motion and Mechanism of Relaxational Processes in Polymers, Polymer Bull., 1989,21,613-619. 68 . Kumins C.A., and Roteman J., Diffusion of Gases and Vapours through PVC-PVAc Films, J. Polym. Sci., 1961,55, 699-711. 69. Barashkova 1.1., and Wasserman A.M., Rotational and Translational Mobilities of a Spin Probe in Polymer-solvent System, Vysokomol. Soed., 1980,22A, 2540-2544. (Rus).

120

Chapter 5

Temperature Effects and Energetics of Molecular Dynamics

Temperature dependence of particle mobility in polymeric medium shares a number of common features of all types of motions tested. They are: — Exponential (or close to exponential) increase of frequencies of molecular motions with temperature; — Dependence of temperature coefficient (activation energy) of the molecular mobility on structural features of particles and the matrix; — Deviations from linearity of Arrhenius plot at T > Tg, i.e. existence of a temperature dependence of the activation energy in this'region; — Compensation effect - the linear dependence between the activation energy and preexponential factor; — Inflexion points in the temperature dependences in the range of relaxation and phase transitions of the polymer. The subject of this chapter is the analysis of temperature dependences of additive motions in relation to their structure and the type of motion; the role of structural and physical peculiarities of the matrix are discussed in Chap. 8 . The activation energy is one of the main parameters determined from temperature dependence:

In the present case, this expression, written down for the diffusion coefficient, is correct for other dynamic parameters: correlation times, rotational relaxation times, or rotation frequencies.

121

In the frames of activation theories, discussed in Sec. 2.3, the activation energy defines the minimal value of the enegry, required for a particle translation or reorientation. In fact, E* is equal to the potential barrier height, which separates two equilibrium positions of the particle. However, there are some cases, discussed in Sec. 5.4, according to which the experimantal values of E* are not always equal to model ones. In these cases, the notions of apparent or effective activation energy are used. Most often, experiments are performed under constant normal pressure. In this case, isobaric value Ep* is measured. Additional important information is given by the activation energy value, determined under isochoric conditions (Ev*). The connection between two activation parameters is determined from Eqs. 2.142 and 2.153. Moreover, the analysis of temperature dependences allows to obtain values of preexponential factors of activation theory formulas and thermodynamic parameters of the transition state theory equation.

5.1 SM ALL-SCALE ROTATIONAL DYN AM ICS

Table 5.1 Parameters for spin probe rotation in polymers /1,2/. Polymer

E*, Probe TEMPO NR PIB PE PP PMMA PS PVC Probe BZONO PMMA PP

T > Tg kJ/mol logt, s 29 45 42 46 21

84 42 21

46

E*, kJ/mol

T < Tg

logt, s

-15.73 -16.06 -17.20 -17.69 -12.69 -19.04 -15.84

6.3 3.8 4.2 8.0 8.8

-11.48 - 10.11 -10.50 -9.04 -9.30 - 10.20 -10.62

-11.30 -16.50

4.2 3.8

-9.60 -9.46

11 8

Temperature dependences of frequencies of small-scale rotation were studied for spin probes by ESR spectroscopy /1 - 8 / and for dye molecules by TSD technique (Sec. 3.3.2) /9 -12/. It should be noted that these particles

122

belong to polar and even highly polar type (p = 3 - 8 D). Temperature dependences of frequencies are of the exponential type (or close to exponential), and their characteristic feature is an inflexion point near Tg (Fig. 5.1). More particularly, the inflexion range, covering the 20 - 30° interval, should be discussed. By the way, these deviations from the linearity cause one of the main sources of errors in E* determination.

Fig. 5.1. Temperature dependences of correlation times for spin probe TEMPO rotation in PE (1), PP (2), PS (3), PMMA (4), and PVC (5)/2/. At T > Tg, the values Ep* make up 20 - 60 kJ/mol depending on the polymer type (Table 5.1). Activation energies are weakly dependent on the particle size (Table 5.2). For example, for the probes III, V, XI, XVII in PP and NR, van-der-Waals volumes of which are more than two-fold different, activation energies of rotation differ only by 4 - 5 kJ/mol. This is close to the experimental error. Apparently, the main contribution into E* of the rotational motion of particles is made by the temperature coefficient of the segmental motion frequencies of surrounding macromolecules. This question is discussed in detail in Sec. 7.2. Preexponential factors in the Arrhenius equation

123

*N

T = XQexp

RT

V

J

range within 10 15 - 10 18 s at T > Tg and exceed ‘normal’ values determining the period of orientational oscillations of molecules in a condensed phase ( 10' 2 - 10-'3S). Table 5.2 Activation energies for spin probes rotation in PE at T > Tg (kJ/mol) /12, 13/. Probe I II IV V

F * 38 40 35 35

Ev* experimental

calculated 12

—

13

13 16

12

14

12

Ev*/Ep* 0.32 0.33 0.34 0.4

The linear dependence between Into and Ep*, a so-called ‘compensation effect’, was revealed as well (Fig. 5.2). All these facts indicate that activation energies at a constant pressure are effective (apparent) and interrelated in a complicated manner with the potential barrier for molecular rotation. Among the causes of the observed peculiarities is the temperature dependence of the activation energy due to increased intermolecular distances (Sec. 5.4). At T < Tg, activation energies of the rotational motion of particles are significantly lower. According to ESR spectroscopy, the value of E* for rotation of spin probes are particularly low (4-11 kJ/mol). It was shown that the ESR data on the temperature coefficients of spin probe rotation frequencies below Tg are too low due to a specific effect produced by a wide spectrum of correlation times /1/. More correct data for these particles are given by the spin echo technique and for dye molecule rotation by TSD technique. Both these techniques give values of Ep* in the range of 20 - 40 kJ/m ol/12, 15/. Such decrease in activation parameters while crossing the glasstransition point is observed not only for rotational diffusion but also for all processes caused by molecular mobility (diffusion, ion conductivity, Prelaxation processes, etc.). It will be shown in Sec. 5.4 that these changes are usually proportional to that in the thermal expansion coefficient.

124

Fig. 5.2. Compensational effect for rotation of probes in polymers /1,5/. Activation energies at a constant volume are markedly lower than those at a constant pressure (Table 5.2). Preexponential factors of the Arrhenius equation at a constant volume are of a ‘normal’ value and much lower than at P = const /12, 14/. Thus, it can be assumed that the values of Ev* define the height of the barrier of the small-scale rotational diffusion of particles. The contribution of the barrier to the general enthalpy of activation ^ E v^ E p j is 0.3 - 0.4 above Tg and 0.7 - 0.8 at T < Tg / 8/. According to the hybride model (Eq. 2.143) this increase corresponds to transition from the mode in which motion is regulated by free volume fluctuations ( p e / p v > 0 to the energetically limited one ( p e / p v < 0 · Within increasing pressure Ev* undergoes weak changes while Ep* is markedly decreased. Thus, for probe I rotation in PVAc Ep* falls from 34 to 21 kJ/mol with increasing pressure up to 400 MPa. These effects contradict the Frenkel-Eyring activation theory stating that the energy of hole formation increases with pressure. A decrease in Ep* is obviously due to a connection existing between this parameter and a polymer thermal expansion coefficient. The analytical data show that the ratio Ep*/a is practically independent of pressure within the range up to 200 MPa, being 1.5xl05 kJ/mol deg for probe TEMPO rotation in PE /11/.

125

5.2 LARGE-SCALE ROTATIONAL DYNAM ICS

The information on large-scale rotational dynamics was received by dielectric relaxation techniques for rigid aromatic molecules in liquids /16, 17/ and polystyrene /10, 11, 18/. This type of motion is characterized by nonlinear temperature dependences in the Arrhenius coordinates. Taking into account the data shown in Fig. 4.2, it can be elasily suggested that in the range of high tempratures (T » Tg) values of E* for both types of rotational motion are brought together. However, in contrast to the small-scale rotational dynamics, the activation energy is increased rather sharply with the particle volume. The data on activation energies Ep* for T < Tg region are listed in Table 5.3. It is seen that the activation energy is much higher (5 - 10-fold) than that for the small-scale rotational motion and is comparable with the Ep* values for a-relaxational process of the segmental motion of macromolecules (Sec. 7.1). Table 5.3 Activation parameters for low-frequency rotation of molecules in polymers and liquids (in vicinity of Tg) /10, 11, 15, 18/. Molecule Matrix Vw, cmVmol Ep*, kJ/mol V*, cm3/mol V*/Vw PS 123 101 » Dye I 100.5 1.0 PS Dye II 156.3 142 155» 1.0 PS 367 Dye III 304.6 306') 1.0 Anthrone PS 2062) 108 248 1.9 ChloronaphDecalin 252 1572) 1.6 thalene 1992) o-Terphenyl o-Terphenyl 319 1.5 1572) 252 IsopropylIsopropyl1.9 benzene benzene ') Experimental values /10, 11/. 2>Calculated from the data o f/15, 18/ by Eq. 2.153. The experiments performed in isochoric conditions showed that the Ev* values do not differ significantly from Ep* ^ E v/ E p = 0.8-0.9^ . According to hybrid model (Eq. 2.143), that type of rotational motion is energetically limited ( p e / p v < 0 ·

126

5.3 TRANSLATIONAL DYN AM ICS

In the majority of the cases, the activation energy of diffusion grows with the particle size. The slope of this dependence are defined by the structure and shape of the particles. The situation is similar to that of dependence of the diffusion coefficients on the size of particles (Sec. 4.2). The illustrative example of this are the data on diffusion of linear molecules of nparaffins /19, 20/. The dependence of activation energy (Ep*) on the number of carbon atoms in the chain is shown in Fig. 5.3. For lowest alkanes (n < 4) a sharp growth of Ep* is observed, and at n > 10 the rate of Ep* increase strives to the activation energy value of a viscous flow. The following equation is valid for polyethylene: E

♦

-B ln

M vM V

(5.1)

where M ' is the molecular mass of polyethylene macromolecules, M is the molecular mass of diffusant. E*, kJ/moI

* Fig. 5.3. Dependence of E( for diffusion of n-paraffins through PE on carbon atoms number (a) and molecular mass (b) / 20/. As follows from Eq. 5.1, the dependence of the activation energy on InM is linear for chain molecules, which is supported by the experimental data / 20/.

127

Fig. 5.4. Dependence of E( on molar volume of organic vapours diffusing through PVAc (a) and PS (b) /21, 24/. There is no doubt that nonlinearity of E* - n dependences, as well as lgD - n dependences, is stipulated by one and the same reason - translational motion of long linear molecules is performed by small jumps, the length of which does not depend on the molecule length. This is confirmed by the data on the activation volumes shown in the next Chapter.

128

For bulk organic molecules close to linear, dependence of E* on V is most often observed /21 - 23/ (Fig. 5.4). It can be described by the empiric equation: E* = a + b V .

(5.2)

Eq. 5.2 is valid for the diffusion of organic molecules, different by chemical structure. I.e., it is apparently universal. It is also suitable for describing molecules of paraffins with the chain length, at least, up to 5 carbon atoms. Let us mention that the experimental data relate to the T > Tg region. Coefficients a and b in Eq. 5.2 for diffusion of particles in PVAc and PS are close (a 0, b = 1.98 kJ/cm3, taking into account that V is the molar volume). logD

Fig. 5.5. Temperature dependence of diffusion coefficient for diphenylene in polyisoprene ( 1), methylnaphthalene in polybutadiene (2 ), diheptylphthalate in polybutadiene (3), tributylphosphate in polybutadiene (4), hexadecane in polyisobutylene (5) /20/. Ryskin and Zhurkov suggested a different equation for description of such dependences /21/: E* = E o (l + a V ),

129

(5.3)

where Eo* and a are the coefficients dependent on the matrix properties. Eq. 5.3 was obtained on the basis of the experimental material shown in Fig. 5.4, where E* is rather different from zero at V => 0. However, it is probable that this result is bound to a rather high error in determination of E*, which can easily reach 20 - 30% and is connected not only with the errors in determination of diffusion coefficients but also with often met nonlinearity of the temperature dependences (in the Arrhenius coordinates) /20, 25/. Examples of such dependences are shown in Fig. 5.5. Reasons of these nonlinearities are most probably connected with the existence of a temperature variations of the potential barrier height of diffusion. The same reason forms the base of the so-called compensation effect, i.e. connection between the preexponential factor and the activation energy /24, 27, 28/. An example of such dependences is shown in Fig. 5.6. The analysis of these phenomena in more detail is discussed in Sec. 5.4.

Fig. 5.6. Compensation effect for gas diffusion in elastomers /27/. We should also note that some authors reported the diffusional activation energy to be virtually independent on the particle size were observed. This may be illustrated by the example of the above shown data on n-paraffins with more than 4 - 5 carbon atoms in molecule and also those on polyolefins /29/ and spin probes 141. However, it should also be mentioned that for clear searching of this dependence it is required to have a selection of particles, significantly different by sizes. In opposite case, changes of E* may fall within the error range. For example, addition of a single CH 2 group to nparafins results in the change of the diffusion activation energy in polyethylene according to the data from / 20/ fits the range from 0.1 kcal/mol (at n > 10) to 1.5 kcal/mol (at n < 4). According to /30/, the changes in the

130

value of E* range from 1 to 4 kcal/mol. The diffusion activation energy of methyl esters of n-paraffins with aliphatic chain length of 9 to 26 carbon atoms changed by 6 kcal/mol only /31/. In this case, the dependence of E* on the chain length passed its maximum at n = 6 - 7. As temperature increases, the dependences of E* on particle sizes become weaker.

T.'fc Fig. 5.7. Temperature dependence for diffusion in PS of: CH3 OH (1), C 2H 5OH (2), CH 2CI2 (3), C 2H 5Br (4) /21/. Table 5.4 Activation energy for diffusion of organic molecules in rubbery and glassy state of polymers /32/. Polymer PS

PMMA PVAc

Diffusant CH 3OH C 2H 5OH CH 2CI2 C 2H 5Br H 20 CH 3OH C 2H 5OH H 2O CH 3OH

Ep*, kJ/mol T > Tg T < T* 40.7 73.5 41.1 88.2 42.0 100.8 52.1 109.2 43.7 62.6 52.1 90.7 119.7 77.3 26.9 58.0 31.9 86.1

lnD°, cm2/s T < Tg T > Tg - 1.1 3.6 2.4 4.4 5.8 -2.6 - 1.8 6.6

In the vicinity of glass-transition temperature, inflexion points are observed in In D versus 1/T dependences, the activation energy below Tg being significantly lower in this case (Fig. 5.7). In most cases, the activation energy falls into the range of 10 - 30 kcal/mol (40 - 120 kJ/mol) at T > Tg and

131

5- 12 kcal/mol (20-50 kJ/mol) at T < Tg /32/ (Table 5.4).As it is seen, values of E* are close to the activation energy of the large-scale rotational motion. According to the data from Table 5.4, the ratio of activation energies below and above Tg is 0.55 ±0.15. Preexponential factors are decreased also below the glass transition region. The analysis of multiple data on diffusion of particles shows that practically all transitions in polymers of both first and second order can affect the temperature dependence of diffusion coefficients (Sec. 8.2). Table 5.5 Activation energies for molecules diffusion in PE at T > Tg (kJ/mol) /33/. Molecule n-hexane n-octane n-decane cyclohexane benzene ') Calculated by Eq. 2.153.

Ev* ■> 44 38 46 10 28

Ep* 65 64 66 61 65

Ev*/Ep* 0.68 0.6 0.7 0.2 0.38

Table 5.5 lists the activation energies of translational diffusion for organic substances in polymers at a constant volume and pressure /33/. They usually exceed the values of Ep* and Ev* for a small-scale rotational diffusion. The Ev*/Ep* ratio for translational diffusion changes in a wide range (0.43 0.6). Translation of cyclic compounds (benzene, cyclohexane, spin probes) is characterized by lower values of Ev*/Ep*as compared with linear hydrocarbons. Due to asymmetry of the shape of molecules their translation primarily occurs in the direction of the large axis by reptation mechanism and requires relatively small fluctuations of free volume. In this case, the contribution of a potential barrier to the activation enthalpy increases. Let us now discuss the features of translational dynamics of gas O particles (d < 4 A ). The activation energy of gas diffusion at T < Tg is 15 - 40 kJ/mol and at T > Tg, 20 - 60 kJ/mol /25, 28, 36/. As it is seen, the activation energy of small gas particles is not significantly lower than E* values for diffusion of organic vapours. It was shown in /24/ that the ratio of activation energies in a glassy state and in a rubbery state equals 0.75. Let us note also that in work /37/ inflexion points at Tg were observed for large gas molecules only, such as Ar and Kr. For smaller particles no inflexion points were observed. This result is, apparently, consistent with the presence of pores of a certain size. By contrast, several inflexion points are observed in many other

132

cases, which are associated with relaxationa! transitions in complex polymeric systems (Sec. 7.3). Table 5.6 Parameters of Eq. 5.4 for gas diffusion in polymers /36/. Polymer LDPE HDPE Polyisoprene PVAc PVC unplast. PEMA

T, K 298 298 298 303 298 298

a -3.52 -10.3 16.0 -106.0 -66.8 -38,3

b 12.2 13.4 5.89 49.7 34.9 21.6

Contrary to larger particles, the activation energy of diffusion of gases is a linear function of their diameter rather than volume(Fig. 5.8) /28, 35 - 37/. Table 5.6 shows parameters of the linear equation E* = a + bd.

(5.4)

It is seen that coefficients a and b depend on both polymer structure and temperature. Basing on general considerations, it would be expected E* = 0 at d = 0. As values of a are quite far from zero, it can be concluded that these dependences are of a smooth curve type. It should be noted that in some cases deviations of the E* - d dependence from linearity were observed /28, 35/. Basing on such data, van Krevelen concluded that in reality the dependence E* = f(dn) takes place with n changing from 1 to 2 /24/. However, it should be noted that with rare exception, the dependences E* - d can be successfully approximated by linear dependence (n * 1). Similar to the case of diffusion of bulky organic molecules, the compensation effect - linear connection between E* and lgDo, is observed for gas diffusion. This dependence is described by equations /24/: * E r -8 lg D 0 = 0.5 V1000 * lg D 0 *0.5 V

1000

133

10 )

.

Here Er* and Eg* are the activation energies of rubbery and glassy state, respectively. Et'.kJfeal

Fig. 5.8. Dependences of E( on d for gas diffusion in PVAc (a), PVC (b), LDPE (c), and PIP (d) /35/. The compensational dependence allows to calculate the diffusion coefficient basing on the single parameter, activation energy. Composing the above mentioned equations with the Arrhenius equation we get for elastomers:

and for glasses:

* logD = - E r 2.3R

134

435.

4;

*

lo g D « -

is .f i

2.3R vT

545.

-4 .

Note that van Crevelen derived a nomogram for calculating the activation energy by gas particle diameter and temperature of glass transition of polymers.

5.4 APPARENT AND TRUE VALUES O F ACTIVATION ENERGY

Let us now discuss the question, if experimental values of the activation energy are apparent (effective) or ‘true’, i.e. to what degree the activation energies obtained by temperature dependences of motional frequencies are correspondent to the model concept of the potential barrier height. In many cases, experimental values of E* are effective, i.e. they have no clear physical sense. First of all, it concerns the rubbery state of polymers. The following criteria point out the effectiveness of the activation parameters: 1. Overestimated values of the preexponential factor and the activation energy. For example, for rotational dynamics of particles, the values of to at T > Tg falls into the range of 1015 - 10'18 s (Table 5.1), that is quite different from normal values (1 0 12 - 1013 s) characterizing the orientational oscillation period of particles in the condensed phase. Values of E* are overestimated also and exceed the calculated values for potential barriers of the molecular motion, which range within 10 - 20 kJ/mol. Apparent values of E* reach 80 kJ/mol for small-scale rotational motion, 300 - 400 kJ/mol for large-scale one, and 100 - 150 kJ/mol for the diffusion. Similar estimates were obtained for translational dynamics. The following example should be mentioned. In accordance with the activation theory (Sec. 2.3), the preexponential factor is determined from the expression: D0 =

135

v d -1

6

’

(5.5)

where v = 1012 - 1013 s·' is the frequency of particles oscillations in the O condensed phase; d = 5 - 10 A is the length of elementary diffusional jump. It follows from this that Do = 102 - 104 cm2/s. The experimental values of Do may exceed significantly these values, especially at T > Tg (Table 5.4). 2. Deviations from linearity of the temperature dependence in the Arrhenius coordinates (Fig. 5.5). These deviations are observed in investigations of the molecular dynamics in a wide range of temperatures and testify a decrease of E* with temperature. 3. Existence of the compensation effect, i.e. the linear dependence between the activation energy and preexponent logarithm, which does not follow from activation theories. All three features mentioned are closely connected and are the consequence of the motion barrier change with temperature. Frenkel was the first who predicted such effects. The author of the kinetic theory of liquids (Sec. 2.3.1) pointed out that increase of intermolecular distances with temperature causes a decrease in the motion barrier w in accordance with the following equation: (5.6) where v = V/N is the volume per a particle; index ‘0’ relates to parameters at P -> 0 and T -» 0; C is a coefficient; R is the bulk modulus. Equations connecting the activation barrier or ‘true’ activation energy E*(T) with apparent one were obtained in /32, 38/. Lebedev et al. /38/ found that (5.7) For preexponential factors the authors derived the following expression: (5.8) where logt0 is the true value of x°.

136

The values of

a[E*(T)1

are negative. That is why, values of Ea* and ÔT logx0 significantly exceed the true ones (the latter by modulus). If the temperature-induced change of E*(T) is sufficiently smooth, the deviation from the linear dependence on the Arrhenius coordinates can be negligible. As an example, let us consider the data on rotational dynamics of spin probes. Values of

a[E *(T )]

for this type of molecular motion range within 5T - (5 - 20)xl0 2 kJ/mol grad. It follows from this that the second term of the right part of Eq. 5.7, which equals 15-60 kJ/mol (at T = 293 K) significantly d[E*(T)] defines the Ea* value (20 - 80 kJ/mol). While is connected with the ÔT temperature change of the matrix free volume, the reason of small changes in apparent activation energy of probes motion with their sizes becomes quite clear. Ryskin /32/ derived an equation which connects Ea* with the activation energy at Tg (Eg*): E a = E ^ ( l + aTg ) .

(5.9)

For preexponential factor:

0

0

*

a ‘E a

In Da = ln D u + ----- . R

(5.10)

Parameter a is determined as a coefficient of temperature change of the activation barrier: a=

1 dE E

dl

(5.11)

Eqs. 5.9 and 5.10 give a possibility to calculate values of lnD° Eg* and a, if ‘normal’ values of D°, calculated by the Eq. 5.5, are substituted into them. Calculations performed by these equations for multiple polymer solvent systems have shown that the true activation energy is much lower than Ea* and is close to solution heat of organic compounds (25 - 50 kJ/mol), and value of a depends on the polymer structure and is abruptly decreased

137

with the polymer transition from rubbery into glassy state /32/. This effect is undoubtedly connected with the decrease of thermal expansion coefficient of polymers below Tg. In accordance with the state equation by Frenkel (Eq. 2.31), the activation energy is proportional to the thermal expansion coefficient, * E a « — —. As it is known, the value of a in the vitreous state of polymers is kT2 much lower than in the rubbery state /40/. The more so, at T > Tg an increase of a with temperature is observed /39/. All the data mentioned allow to make the following conclusion: 1. Sharp change in the apparent activation energy above the glass transition temperature is connected with the change in a. 2. The higher the temperature range of investigation inside the polymer rubbery state is, the higher is the apparent activation energy deviation from its true value. 3. Below Tg, the barrier height weakly depends on temperature. As a consequence, apparent and true values of activation energies are close. 4. The reason of sharp changes of a at T > Tg is bound to an increase of molecular motions amplitude, as it is shown in Sec. 4.4. More correct data can be obtained by studying the temperature dependences of the molecular mobility under isochoric conditions. This conclusion correlates well with concept developed by Weymann (Sec. 2.3.3). Let us consider the data obtained with high-pressure technique (Sec. 6.2). The ratio Ev*/Ep* is 0.3 - 0.6 for rubbery state and 0.6 - 0.8 at T < Tg. Thus, isobaric and isochoric values of the activation energy at low temperatures are brought closer to each other. The equation binding both these values is the following (Sec. 2.6): E p = E v +aKTV ,

(5.12)

where a is the thermal expansion coefficient; K is the bulk modulus. Let us note that preexponential factors obtained under isochoric conditions are also close to ‘normal’ ones. To conclude the Section, let us mention that Eqs. 5.7, 5.9 and 5.12 can be written in the general form: Eg —E{ + c T ,

138

(5.13)

where Ea* and Et* are the apparent and true values of the activation energy, respectively; c is the coefficient significantly dependent on matrix properties. Existence of two terms of apparent activation energy is displayed in the fact that a change of particle characteristics (size, for example) does not necessarily lead to a noticeable change of activation parameters of its motion.

REFERENCES

1. 2. 3. 4. 5. 6. 7.

8. 9. 10.

Wasserman A.M., Kovarskii A.L., Spin Probes and Labels in Physical Chemistry o f Polymers, Moscow, Nauka, 1986. (Rus) Kovarskii A.L., Plachek J., Szocs F., Study of Rotational Mobility of Stable Radicals in Solid Polymers, Polymer, 1978,19, 1137-1141. Kovarskii A.L., Wasserman A.M., Buchachenko A.L., Vysokomol. Soed., 1971,13A, 1647-1651. (Rus) Barashkova 1.1., Waserman A.M., Kovarskii A.L., Effect of Spin Probe Size on Rotational and Translational Mobility in Polyethylene, Vysokomol Soed., 1981,23B, 434-438. (Rus) Wasserman A.M., Buchachenko A.L., Kovarskii A.L., and Neiman M.B., Paramagnetic Probe Technique for Studying Molecular Motions in Polymers, Vysokomol. Soed., 1968,10A, 1930-1936. (Rus) Buchachenko A.L., Wasserman A.M., Kovarskii A.L., Paramagnetic Probe Technique for Studying Molecular Motions in Polymers, Europ. Polym. J., Suppl., 1969,473. Tormala P., and Lindberg J.J., Spin Labels and Probes in Dynamic and Structural Studies of Synthetic and Modified Polymers, In: Structural Studies o f Macromolecules by Spectroscopic Methods, Ed. Ivin K.J., Wiley, London, 1976, 256-271. Kovarskii A.L., Aliev I.I., Peculiarities of Rotational Dynamics of Small Molecules in the Vicinity of Tg of Polymers, Vysokomol. Soed., 1986, 28B, 843. (Rus) Kovarskii A.L. and Saprygin V.N., Study of Rotational Mobility of a Polar Molecule in PS by TSD-Method, Polymer, 1982, 23(Suppl.), 974978. Kovarskii A.L., Aliev I.I., On the Amplitude of Molecular Motion and Mechanism of Relaxational Processes in Polymers, Polym. Bull., 1989, 21,613-619.

139

11. Kovarskii A.L., Molecular Dynamics and Radical Reactions in Polymers under High Pressures, Doctoral Thesis, Institute of Chemical Physics, RAS, Moscow, 1988. (Rus) 12. Kovarskii A.L., PVT-Effects in Molecular Dynamics of Polymers, In: High Pressure Chemistry and Physics o f Polymers, Ed. Kovarskii A.L., CRC-Press, Boca Raton, 1994, 117-151. 13. Kovarskii A.L., Molecular Dynamics in Polymers under High Pressures, Vysokomol. Soed., 1986,28A, 1347-1360. (Rus) 14. Barashkova 1.1., Dadali A.A., Aliev 1.1., Zhorin V.A., Kovarskii A.L., Wasserman A.M., Buchachenko A.L., Effects of Pressure on Molecular Dynamics of Spin Probes, Vysokomol. Soed., 1983, 25B, 840-847. (Rus) 15. Dzuba S.A., Tsvetkov Yu.D. and Maryasov A.G., Echo-induced EPR spectra of Nitroxides in organic Glasses: Model of Orientational molecular Motions near Equilibrium Position, Chem. Phys. Lett., 1992, 188,217-222. 16. Johari G.P., Goldstain M., Secondary Relaxation in Glasses of Rigid Molecules, / Chem. Phys., 1970,53,2372-2379. 17. Johari G.P., Intrinsic Mobility of Molecular Glasses, /. Chem. Phys., 1973,58,1766-1770. 18. Davis M., Edwards A., Dielectric Studies of Mobility of Polar Molecules in Polystyrene, Trans. Far. Soc., 1967,63, 2162-2167. 19. Barrer R.M., Permeability in Relation to Viscosity and Structure of Rubber, Trans. Far. Soc., 1982, 38, 322-331. 20. Chalykh A.E., Diffusion in Polymer Systems, Khimiya, Moscow, 1987. (Rus) 21. Zhurkov S.N., Ryskin G.Y., Zh. Tekhn. Fiz., 1954, 24, 797. (Rus) 22. Duda J.L., and Vrentas J.S., Diffusion in Atactic Polystyrene above Glass Transition Point, /. Polym. Sci., 1968,6(A-2), 675-685. 23. Saleem M., Asfour A., and DeKee D., Diffusion of Organic Penetrants through Low Density Polyethylene Films, /. Appl. Polym. Sci., 1989, 38, 1503-1514. 24. Van Krevelen D.W., Properties o f Polymers. Correlation with Chemical Structure, Elsevier, Amsterdam, 1972. 25. van Amerongen G.J., Diffusion in Elastomers, Rub. Chem. Techno!., 1964, 37,1065-1152. 26. Rodgers K., Solubility and Diffusion, In: Physics and Chemistry o f Organic Solid State, 2, Eds. Fox D., Labes M.M. and Weissberger A., Wiley, New York, 1965,289-328. 27. Barrer R.M., Scirrow G.J., Transport and Equilibrium Phenomena in Gas-Elastomer Systems, I, Kinetic Phenomena, Polym. Sci., 1948, 3, 549563.

140

28. Stannett V., Simple Gases In: Diffusion in Polymers, Eds. Crank J. and Park G.S., Academic Press, New York, 1968,41-73. 29. Jackson R.A., Oldland S.R.D. and Rajaczkowski A., J. Appl. Polym. Sei., 1968,12, 1297. 30. Asfour A., Saleem M., and DeKee D., Diffusion of Saturated Hydrocarbons in Low Density Polyethylene Films, J. Appl. Polym. Sei., 1989,38, 1503-1514. 31. Moisan J.Y., Diffusion des Additifs du Polyethylene I-IV, Europ. Polym. J., 1980,16,979,989; 1981,17, 857-864. 32. Ryskin G.Ya., Activation Energy and Temperature Dependence of Diffusion in Polymers, Zh. Tekhn. Fiz., 1955,25, 458-465. (Rus) 33. McCall D.W., Slichter W.P., Diffusion in Ethylene Polymers, III. Effects of Temperature and Pressure, J. Amer. Chem. Soc., 1958,80, 1861. 34. Meares P., The Diffusion of Gases therough Polyvinyl Acetate, J. Amer. Soc., 1954,76, 3415-3422. 35. Permeability and Diffusion Data, Polymer Handbook, Eds. Braundrup J. and Junrgut E., Pergamon Press, New York, 1989. 36. Tikhomirov B.P., Hopfenberg H.B., Stannett V.T., and Williams J.L., Permeation, Diffusion and Solution of Gases and Water in Unplasticized PVC, Macromol. Chem., 1968,118, 117-188. 37. Tikhomirova N.S., Malinski Yu.M., Karpov V.L., Studies of Diffusion Processes in Polymers, Vysokomol. Soed., 1960,2,221-229. (Rus) 38. Lebedev Ya.S., Tsvetkov Yu.D., and Voevodskii V.V., On the Nature of Compensation Effect in Radical Recombination Reactions in Polymers, Kin. Kat., 1960,1,496-500. (Rus) 39. Tsirule K.I., and Tyunina E.L., Compressibility of Polymers, In: HighPressure Chemistry and Physics o f Polymers, Ed. Kovarskii A.L., CRCPress, Boca Raton, 1994, 1-22.

141

Chapter 6

Pressure effect and activation volumes

6.1 PR ESSU R E DEPENENCES OF M OLECULAR M O TIO N

Bulk compression of polymers causes a decrease of the free volume and is accompanied by a reduction of the intensity of macromolecules segmental motion and the motion of introduced particles. Consequently, the activation volume is always positive which agrees with the prediction of the activation theory and free volume model. Another consequence of these theories, namely, linearity of baric dependences of logr is often unsatisfied, i.e. with increasing pressure they tend to be weaker /1 - 7/ (Figs. 6.1a and 6.1b). One should note the similarity of baric dependences of logx and the curves of polymer compressibility, i.e. V dependences on P /8/ (Fig. 6.2). It is known that a deviation of compressibility curves from linearity is due to an increase in the bulk modulus with pressure. Fig. 6.3 presents volume dependences of the parameters of particle rotational mobility under polymer uniform compression /3, 9 - 11/. As it is seen, these dependences are linear, and so the change in volume is evidently the main parameter determining particle mobility under isothermal compression of polymers. The experimental value of activation volume calculated from an angle of the slope of baric dependences decreases with pressure. It is determined that there is a linear correlation between activation volume and polymer compressibility wich has the following form /1/: V*=ap = ^ ,

(6.1)

where p is compressibility; K is the bulk modulus; a is a proportionality coefficient.

142

Fig. 6.4 gives an example of this dependence for rotational dynamics of spin probes in PE. Thus, compressibility (bulk modulus) is one of the parameters determining the experimental values of V*. It is well known that the baric dependence of the modulus K is linear /8/. According to Eq. 6.1 there should be a linear correlation between 1/V* and P which is experimentally supported (Fig. 6.5)/12/.

Fig. 6.1. Pressure dependences of: a) rotational correlation times for probe I in PIB at 326 K (1), probe I in PE at 293 K (2), probe I in PDMS at 297 K (3) /1/, pyridine molecules in pyridine at 297 K (4) /14/; b) diffusion constants for benzene in PE at 293 K (1) /4/, probe I in PE at 293 K (2) /3/, p-nitroaniline in PET at 293 K (3) /5/, hexane in natural rubber at 293 K (4) 161, p-aminoazobenzene in nylon 6 at 353 K (5) 111, and pyridine molecules in pyridine at 297 K (6) 121.

143

¿S

SO

P

7S

100

(MPa.)

Fig. 6.2. Compressibility curves for LDPE at 293 K (1), PIB at 326 K (2), and PDMS at 297 K (3).

Fig. 6.3. Volume dependences of rotational correlation times (1) and diffusion constants (2) for probe I in PE /3, 9, 10/, and relaxation times for reprocesses in PVC (3) /11/ on the relative changes of specific volume with pressure (dark points) and temperature (bright points).

144

60 v* o

H

Fig. 6.4. Dependence of rotational activation volume for probe I on a change in compressibility factor with temperature and pressure for PE (1 - 7), PIB (8-11), PDMS (12), and nitrile rubber (13) 191.

Fig. 6.5. Pressure dependences of the activation volume for the probe I rotation in PIB at 326 K /12/. So, the experimental values of the activation volume depend on pressure (compressibility). Basing on Eq. 6.1, one can write:

145

( 6 . 2)

Using Eqs. 6.2, the Frenkel-Eyring equation may be recast to describe a system with the pressure-dependent modulus:

(6.3) To use the Eq. 6.3 it is not necessary to have a PVT-diagram of the

of diffusion coefficients or frequencies of particle rotation determining Vo*

baric dependences of diffusion coefficients and correlation times of particle rotation coincide or are close to the values obtained from PVT-diagrams /1, 9 1.

Let us now discuss the possibilities of the free volume model for the analysis of baric dependences of dynamic parameters. It has been shown already in Sec. 2.3 that mobility of particles in the framework of this model is V defined by the probability of free volume formation p « ------ . From this it Vf follows that the free volume is the main characteristic parameter of matrix regardless on the type of external influence, temperature or pressure, which causes its change. However, experimental data show that this consequence of the model is not fulfilled in many cases. Fig. 6.4 presents the dependence of molecular mobility parameters on the relative change of polymer volume with temperature and pressure. It is seen that the change of volume with temperature leads to a more sharp change of the molecular mobility, than the same change of volume with pressure. This effect is a consequence of the existence of temperature dependence of movility at constant volume, i.e. Ev * O.This point is discussed in detail in the next Section. Here let us mention that the main condition of applicabilicity of the Cohen-Turnbull equation is Ev « Ep which is fulfilled only for the self diffusion of a small number of

146

particles, for example, cyclic hydrocarbons (Table 5.5). It was concluded in /13, 14/ that this model gives overestimated values of V* and Vf. Table 6.1 Calculated and experimental values of free volume fractions Vf/15/. Polymer PB

NR

Diffusant hexane hexane dodecane dodecane hexane hexane

Vf, exp. 0.070 0.045 0.060 0.052 0.064 0.050

P, Mpa 8 40 10 20 8 20

Vf, calc. 0.072 0.050 0.063 0.059 0.065 0.059

P/log DP 50

Dp=o

40 30 20 10

0 P, MPa

Fig. 6.6. Pressure dependence of diffusion coefficient for methylmethacrylate in polymethylmethacrylate (l), nitrogen in nitrile rubber (2) and dodecane in polyisobutylene (3) in coordinates of Eq. 6.4 / 151. To treat the experimental data on baric dependences of organic substances, the diffusion coefficient in polymers Eq. 2.102 was used in /15/. For the sake of convenience, it was transformed as follows:

147

In

DP

\

-1

( V f ) p = o , ( V f ) p = o p-1

D P=0y

YV

yV*p

and further on: DP

In V D p =o /

-1

(V f)p=o

yv*p

( V f ) p = 0 P. yV

(6.4)

Baric dependences in coordinates of Eq. 6.4 are shown in Fig. 6.6. It is seen that this equation for the isothermal compression of the system is fulfilled, at least, qualitatively. Calculation of free volumes by this equation also gives satisfactory results obtained, however, within only a narrow pressure range (Table 6.1).

6.2 ACTIVATION VOLUMES

As shown in the previous Section, the experimental values of the activation volumes depend on pressure (compressibility coefficient). To compare V* for different types of molecular motion in future we will use its value at P -> 0. Namely, these values are shown in tables and figures. A close relationship between the V* and K values is evident from their symbatic increase with temperature. As seen from Fig. 6.4, there is a common linear dependence of activation volume for spin probes rotation in various polymers on their compressibility changing with temperature and pressure. The evaluation of the possible temperature and pressure changes of the experimental values of V* requires the data on the change in polymer compressibility p (or bulk modulus K = p-1) with external parameters (PTVdiagrams) 191. Values of p fall within 104 - 10 3 MPa·' range and depend on the polymer physical state, temperature and pressure. In the glassy state the temperature coefficient of compressibility is negligible and decreases with increasing pressure. The strongest dependence of p on P and T is typical for the polymer elastic state. In most polymers a two-fold increase in p is observed from Tg up to Tg + 100°C /8/. Hence, maximum differences in activation volumes associated with the temperature change of p may reach

148

100%. In the same physical state various polymers insignificantly differ in their compressibility factors. For example, p values in most of rubbers differ by no more than 20% at the same temperature. To avoid errors when analyzing the data on different polymeric systems one should apply V* values obtained at temperatures equidistant from Tg or correlate the V*/p ratio. Now we consider the dependence of activation volumes on particle size and structure for various types of molecular motion. 6.2.1 High-frequency Rotational Dynamics The V* values for spin probes rotation in polymers are 20 - 70 cm3/mol (Table 6.2) and exceed the activation volumes for molecular rotation in liquids ( 7 - 1 5 cm3/mol). The activation volumes increase with the van-derWaals volume of particles (Fig. 6.7), the V*/Vw ratio for rotation of particles of different size in the same polymer remaining practically constant. For example, this value makes up 0.3 ± 0.08 for probe rotation in the amorphous regions of PE. V*/Vw ranges within 0.2 - 0.5 for different polymers in the elastic state (0.15 - 0.2 for liquids). Hence, the fluctuational volume sufficient for reorientation is 2-5-fold lower than the intrinsic volume of particles, i.e. their rotational dynamics is controlled by the small-scale motions of macromolecular kinetic elements. 3

Fig. 6.7. Dependences of activation volumes for rotation of probes I-V in PE at 343 K (1) /3, 16/, diffusion of azo dyes (2) and dyes of anthraquinone series (3) in PET at 403 K /13/ on molecular volume.

149

Table 6.2 Activation volumes for high-frequency rotation of small molecules in polymers and liquids at P -» 0 /1 - 3,16/. Matrix PDMS PIB NR

Nitrile rubber

PE

Pyridine Fluorobenzen e Nitrobenzene Toluene

Molecule probe I probe I probe I probe I probe V probe V probe I probe I probe IV probe IV probe I probe I probe II probe III probe IV probe V pyridine fluorobenzene

T, K 297 326 293 343 293 343 293 343 293 343 293 343 343 343 343 343 303 303

V*, cm3/mol 60 35 23 33 28 48 39 53 51 72 23 36 55 73 37 46 7.5 8.7

V*/Vw 0.57 0.33 0.22 0.32 0.17 0.28 0.38 0.52 0.35 0.49 0.22 0.35 0.35 0.37 0.25 0.38 0.14 0.14

nitrobenzene toluene

303 303

14.3 10.4

0.19 0.15

The higher values of V* and V*/Vw are observed in two cases: the system with a relatively high compressibility factor (polydimethylsiloxane, PDMS) and a polar polymer (nitrile rubber). The different causes of high values of V* for these two polymers become evident if one compares the ratio of activation volumes and compressibility. V*/p = lxlO5 cm3MPa/mol for probe rotation in a polar rubber, which is twice higher than that for PDMS. The differences are due to the stronger intermolecular interaction of polar probes (p = 3 D) with a polar polymer. For this reason the experimental data for a nitrile rubber do not fit a common dependence. With decreasing temperature V* sharply falls from 25 - 30 cm3/mol to 8 - 10 cm3/mol in the vicinity of Tg (Table 6.3) /17/. In the same temperature range the p value is about two-fold decreased. The V*/p ratio remains unchangeable at temperatures above and below Tg. In a translation region this ratio is changed by 1.7 times. These results show that the change in compressibility is not a single cause of sharp alternations of V*. Interpreting

150

activation volume as a fluctuating hole necessary for motion, one can easily suppose that it depends not only on a particle size but also on the amplitude of its turn (Sec. 4.4). Evidently, the amplitude (an angle of turn) of rotational motion decreases below Tg which is accompanied by reducing the V*/Vw value from 0.3 at T > Tg to 0.1 at T < Tg. Table 6.3 Temperature changes of compressibility and activation volume for probe TEMPO rotation in PVAc /17/. T, K 293 303 313 333 343 353 363 373

V*, cm3/mol 8.5 8.9 10.3 18.1 25.3 28.2 28.9 30.5

pxlO4, MPa-' 2.47 2.59 3.23 3.94 4.60 5.00 5.11 5.42

(V*/p)xl04, cm3/M Pam ol 3.4 3.4 3.2 4.6 5.5 5.6 5.6 5.6

6.2.2 Low Frequency Rotational Dynamics As mentioned above, the data on a large amplitude low-frequency component of particle rotational dynamics in polymers have been obtained by the thermostimulated depolarization technique using polar dye molecules. One can find the data on activation parameters in Table 5.3. In the same table one can also find the data on dielectric relaxation of rigid organic molecules in liquids and polymers /18 - 20/. It is clear that large activation volumes, comparable or even exceeding a molecular size, are typical for a low frequency component of rotational motion. These results allow us to prove the considerable differences in rotation amplitudes of these two types of motion. The above mentioned results show that the experimental values of activation volumes are dependent on a number of parameters: particle size, energy of intermolecular interaction, matrix compressibility and, apparently, the amplitude of an elementary act of reorientational motion of particles. The rotation amplitude is more thoroughly discussed in Secs. 4.4 and 7.1.

151

6.2.3 Activation Volumes for Translational Dynamics

Translational diffusion of particles as well as high-frequency rotational dynamics are characterized by the similar peculiarities, namely, nonlinearity of baric dependences (Fig. 6.1b), connection of activation volumes with the polymer compressibility, and a linear correlation between logD and volume / 1, 10/. Table 6.4 Activation volumes for translational diffusion of molecules in polymers and liquids/2 -7 , 14,21,22/. Molecule probe I probe II probe III n-hexane n-octane n-decane 3-methylpentane neohexane hexene-2 cyclohexane benzene NR n-hexane dichlorodifluorom PDMS ethane ____ PETP____ ___ J>irutro a ni li n e_____ fluorobenzene Fluorobenzene Pyridine pyridine cyclohexane Cyclohexane Matrix PE

T, K 293 343 343 293 293 293 293 293 293 293 293 293 298

V*, cm3/mol 56 110 110 47 54 47 64 70 79 116 84 146 21

V*/Vw 0.52 0.7 0.56 0.68 0.6 0.47 0.89 0.94 1.25 2.0 1.9 2.1 0.46

367 293 293 303

93 13 13 24

0.96 0.22 0.23 0.41

Activation volumes for translational diffusion are 1.5 - 2.5 times higher than those for small-amplitude rotational motion of molecules both in * Vt polymers and in liquids (Table 6.4). Thus, the — ratio is 2.4 for TEMPO

Vr

probe in PE and 1.7 for pyridine molecules in pyridine. The V* values for molecules diffusion in liquids are close to activation volumes of their viscous flow 121.

152

Let now consider the dependences of activation volumes for particles translational motion on their size and structure, which were obtained both for spin probes /3, 16/ and for alkane and dye molecules /4, 5, 13/. The analytical data show that diffusion of branched and cyclic hydrocarbons requires a larger activation volume than that of normal ones with the same amount of carbon atoms. The study of two groups of dyes - anthraquinone and azo derivatives - indicates that an activation volume increases with a larger size of *

V, particles (Fig. 6.7). The — — ratio is 0.7 for the first group and 0.5 - 0.6 for Vw the second one. These divergences may be due to a different shape of dye molecules of both groups. The asymmetry of shape, namely, the ratio of longitudinal and cross sections is higher in molecules of azoderivatives than in those of anthraquinone derivatives, the first group of molecules requiring smaller fluctuations of free volumes for their movement (provided that jumps are primarily directed along the ellipsoid large axis). The ratio of activation volumes of translational diffusion of molecules and their intrinsic volume ranges within 0.5 - 2.0. The A . ratio for linear Vw molecules of normal paraffins decreases with growth of the number of carbon atoms. It is explained, apparently, by the fact that translational motion of these particles happens along the large axis according to the reptational mechanism. In other words, the length of an elementary jump for such particles is much smaller than the molecule length and remains constant for the whole of homologous series. *

Vt The — — ratios are, on average, twice higher for translational Vw diffusion than for rotational one. It means that the small-amplitude rotation of particles requires lower fluctuations of the free volume. Activation volumes of translational diffusion and large-amplitude rotation of partuicles are of the same order of magnitude and do not differ sharply from a molecule volume. Probably, both types of motion are regulated by free volume fluctuations of the same range.

153

6.3 THE RATIO BETWEEN ACTIVATION PARAMETERS

Activation parameters, Ep*, Ev* and V*, are connected with each other, and each of them can be calculated through another parameter. Experimental values of E* and V* for various types of molecular motion are *

Ep given in Table 6.5. The data presented in Table 6.5 show that the — ratio V depends weakly on the type of motion and physical state of matrix, and is not * Ey far from unity. The value of ——, vice versa, depends on these factors and EP changes in a more wide range. Table 6.5 Ratio of activation parameters for molecular motion of additives in polymers. Type of motion High-frequency rotation (spin probes) T < Tg T > Tg Low-frequency rotation T Tg 50 150

- 100 200

Ev*/ E p (V*/p)xl04, cm3/MPa-mol

0.9- 1.1 0.6-0.8 3-3.5 0.7- 1.5 0.3-0.6 12-15 - 0.7- 1.4 0.8 - 0.9 —

- 50-150 1

0.4-0.7 —

Let us now consider equations which connect activation parameters. One of them is the empiric equation, obtained for the diffusion in metals /23/ and introduced into physical chemistry of polymers by Eby /24/: V* = kpEp*.

154

(6.5)

The coefficient k is taken equal 4. However, the data in Table 6.5 show that k ♦ Ep value has to be a function of P, because the —^ ratio weakly depends on the V motional type and physical state of polymer. More regorous calculation of the activation parameters can be performed on the basis of the equation deduced with the help of the thermodynamics (Sec. 2.6): EP* = Ev* +TcxKV*.

( 6 . 6)

Here K = — is the bulk modulus. P

Experimental data, as a rule, confirm correctness of this ratio. Eq. 6.6 can be presented in the simplified form as follows: Ep*(l - tp)= TaKV*, where (p =

(6.7)

Ey Ep

Table 6.6 Activation volumes for spin probe TEMPO rotation, obtained through free volume (Vi*) and activation (V2*) models. Vr, cm3/g a> Vi*, cm3/g b) V2*, cm3/g c> T, K Ta,K Polymer 1.4-2.8 23 0.058 293 233 PE 1.8-3.6 35 0.091 326 208 PIB 1.3-2.6 60 0.096 297 153 PDMS a>Calculated by Eq. 2.115. b>Calculated by Eq. 2.116 with y = 0.5 and 1. c) Evaluated taking into account the molecular mass of the probe (156). This ratio allows to estimate the activation volume through the Ep* value. The aK value depends weakly on pressure and is 1 ± 0.3 MPa/K. Thus, we get:

155

EP =

V T

( 6 . 8)

1-9

The values of

· 0 (cp —> 0). *

Ep However, calculations using Eq. 6.11 yield the values of — by several times V lower than the experimental ones. This result reveals the major shortcoming of activation theories ignoring the existence of a temperature dependence of the molecular motion frequencies in isochoric conditions (Ev*). For example, Frenkel’s theory states that a number of holes determining molecular mobility is increased with tempreature proportionally to the rise in volume (Sec. 2.3.1). This theory assumes a number of holes and, hence, motion frequencies are constant under heating in isochoric conditions. The same shortcoming is typical for the free volume model. As mentioned in Sec. 2.4, the system volume (free volume) is a key parameter determining mobility in terms of this model. Such assumption is true under the isothermal compression of the system. However, according to this model, an increase in temperature at a constant volume should not result in the mobility change. Therefore, Eqs. 2.111, 2.112 are satisfactorily fitted only if

156

Ev* = 0 or at least Ev* « Ep*. As follows from Tables 5.2, 5.5, 6.5, 6 .6 , this condition is observed neither for a- and p-relaxational processes of segmental diffusion, nor for rotational dynamics of small particles in polymers. This model yields a lesser error if used to analyze the translational diffusion of cyclic molecules (see Sec. 5.3). The treated experimental data presented in Table 6.7 show that the activation volumes exceed the V* values calculated according to the activation model. The similar conclusion was made by the authors of /13, 14/. Table 6.7 Parameters of Eqs. 2.142, 2.143 for dynamic processes in polymers (y = 0.5) /1 ,3 ,9 , 11, 12, 17/. Process v f, Ep*, v*. axlO4, v m, Ev*, and kJ/mol kJ/mol cm3/mol K-< cm3/mol cm3/mol polymer Viscosity PIB 25.2 52.5 6.8 Rotation of probe I in PVA at: 72.4 6.2 T > Tg 33.6 16.0 4.2 30.5 T < Te 7.6 72.4 3.8 10.5 8.5 1.8 P-relaxation in: 4.1 PVC 63 53.5 25 1.2 45.1 PM MA 88 75 21 1.96 85.6 6.1 60.7 PCIFjE 60 4.0 71 31 0.75 a-relaxation in: 4.3 PVC 422 300 342 1.2 45.1 85.6 4.5 PM MA 248 398 1.96 355 72.4 87.4 5.5 PVA 182 186 1.8 60.7 3.8 PCIF 3E 205 143 162 0.75

Vf/Vm ln(pE/pv) 0.13

0.5

0.083 0.055

-2.0

0.092 0.071 0.065

-18.2 -26.3 -19.8

0.095 0.052 0.076 0.062

-39.4 -54.0 -20.7 -35.4

0.2

The above mentioned data indicate justification to combine the activation model and the free volume model as it is done in the hybrid model (Sec. 2.5). The advantage of this approach is that such combination takes into account the activation energy at a constant volume /25/. Table 6.7 gives calculations of some parameters by Eqs. 2.142, 2.143 using the experimental values of activation parametrs. The Table shows that the fraction of a free

157

volume in polymers at room temperature and normal pressure ranges within 5 - 10%, that seems quite reasonable. The pE/pv values for a- and p-processes are less than 1, i.e. these processes are energetically limited. As for rotational and translational dynamics of small molecules in polymers, their mode changes during transition through the glass point, i.e. both types of motion, are restricted by the rate of the free volume formation above Tg and energetically limited below Tg.

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Kovarskii A.L., Aliev 1.1., The Dependence of Diffusion Constants of Small Molecules in Polymers on Pressure, Vysokomol. Soed., 1983, 25A, 2293-2299. (Rus) Assink R., Zwaan J., Jonas J., Pressure Effects on Coupling between Rotational and Translational Motions in Liquids, J. Chem. Phys., 1972, 56,4975-4981. Barashkova 1.1., Dadaly A.A., Aliev 1.1., Zhorin V.A., Kovarskii A.L., Effect of Pressure on the Molecular Dynamics in Polymers, Vysokomol. Soed., 1983, 25A, 840-847. (Rus) McCall D.W., Slichter W.P., Diffusion in Ethylene Polymers, III. Effects of Temperature and Pressure, / Amer. Chem. Soc., 1958,80, 1861-1868. Ito T., Seta J., Urukawa H., Fujita Sh., Diffusion of Dye Molecule in Polymer under High Pressure, Sen-i Gakkaishi, 1981,119, 37-44. Chalykh A.E., Diffusion in Polymers, Thesis, Institute of Physical Chemistry, Moscow, 1975. (Rus) Seta J., Ito T., Effect of Pressure on Dying Equilibrium in pAminoazobenzene-nylon System, Sen-i Gakkaishi, 1984,221, 40-48. Tsirule K.I., and Tyunina E.L., Compressibility of Polymers, In: HighPressure Chemistry and Physics of Polymers, Ed. Kovarskii A.L., CRCPress, Boca Raton, USA, 1994, 1-22. Kovarskii A.L., Molecular Dynamics in Polymers at High Pressures, Vysokomol. Soed., 1986,28A, 1347-1360. (Rus) Kovarskii A.L., Aliev 1.1., The Analysis of Pressure Dependences on the Mobility of Small Molecules in Polymers, Vysokomol. Soed., 1985, 27B, 110-113. (Rus) Saito S., Sasabe H., Nakajama T., Yada K., Dielectric Relaxation and Electrical Conduction of Polymers as a Function of Pressure and Tem perature,/. Polymer Sci., 1968,6(A-2), 1297-1306.

158

12. Kovarskii A.L., Molecular Dynamics and Radical Reactions in Polymers Under High Pressure, Thesis, Institute of Chemical Physics, AN SSSR, Moscow, 1988. (Rus) 13. Ito T., Seta J., Urukawa H., Fujita Sh., Effect of Pressure on the Diffusion of Organic Penetrants in Polymers, Proc. IUPAC, 28th Macromol. Symp., 1982,721. 14. Assink R., The Concentration and Pressure Dependence of the Diffusion in Polydimethylsiloxane, Polymer Sci., Polym. Phys. Ed., 1977, 15, 227234. 15. Chalykh A.E., Diffusion in Polymer Systems, Khimiya, Moscow, 1987, 66 p. (Rus) 16. Dadaly A.A., Wasserman A.M., Buchachenko A.L., Irzhak V.I., Effect of pressure on the Rotational Mobility of Spin Probes in Polymers, Europ. Polym. J., 1981,17, 525-530. 17. Kovarskii A.L., Aliev 1.1., Peculiarities of Rotational Dynamics of Small Molecules in the Vicinity of Tg of Polymers, Vysokomol. Soed., 1986, 28B, 843-847. (Rus) 18. Davis M., Edwards A., Dielectric Studies of Mobility of Polar Molecules in Polystyrene, Trans. Far. Soc., 1967,63, 2162-2169. 19. Johari G.P., Goldstain M., Secondary Relaxation in Glasses of Rigid Molecules,/. Chem. Phys., 1970,53,2372-2376. 20. Kovarskii A.L., and Mansimov S.A., On the Amplitude of Molecular Motion and Mechanism of Relaxational Processes in Polymers, Polym. Bull., 1989,21,613-619. 21. Seta J., Takeno M., Ito T., Effect of Solvent on Activation Volume for the Diffusion of Aminobenzene in Cellulose Diacetate, Sen-i Gakkashi, 1983,221,40-46. 22. Gaisin N.K., Idijatullin Z.Sh., Samigullin F.M., The Evaluation of Activation Parameters for Self diffusion in Liquids, Zh. Fiz. Khim., 1985, 59,2703-2709. (Rus) 23. Keyes R.W., Volumes of Activation for Diffusion in Solids, J. Chem. Phys., 1958,29,467-475. 24. Eby R.K., Empirical Relations of Thermodynamic ‘Constants’ to the Activation Parameters of Polymer Relaxations, J. Chem. Phys., 1962, 37, 2785-2788. 25. Macedo P.B. and Litovitz T.A., On the Relative Roles of Free Volume and Activation Energy in the Viscosity of Liquids, J. Chem. Phys., 1965, 42, 245-259.

159

Chapter 7

Additive Motion and Macromolecular Dynamics

The motion of low molecular particle introduced into polymer matrix correlates with that of kinetically independent elements of macromolecules (segments) surrounding the particle. First of all, the degree of this correlation is defined by the size ratio of the particle and the segment and by the character of intermolecular interaction. Setting of rigorous quantitative descriptions of this connection is a complicated problem, and approaches to its solution are in progress. Moreover, the study of mechanisms of the relaxation processes in polymers is still in its infancy. It should be noted that the data on low molecular particle dynamics help in clearing up the mechanisms of segmental motions of polymer chians. In the present Chapter using the experimental data we try to answer the question, the motion of what types of medium kinetic units define the frequencies and activation parameters of the particle motion; in what cases the connection between the dynamics of the particles and segments is broken. However, initally we will discuss in brief the modern ideas on the mechanism of relaxational processes, based on segmental dynamics of macromolecules.

7.1 SEGM ENTAL DYNAM ICS OF M ACRO M O LECULES

A specific feature of polymeric systems is the complexity and diversity of molecular motions /1 - 4/. Unlike simple polymers, liquids always contain some set of kinetic elements whose motion defines a relaxational spectrum. Moreover, one can hardly divide the intramolecular dynamics in solid polymers into rotational and translational components as in simple liquids, since both types of motion are closely interrelated. Rotational motions and torsional oscilations in the chain give rise to translational movements and

160

reorientations of large fragments of macromolecules. Fundamental tasks of research in this field are to determine the type and size of kinetically independent units, amplitude of motion, and regularities of distribution in dynamic parameters. The spectrum of relaxation times in polymers embraces a wide range of frequencies and is continuous-discrete, i.e. consists of overlapping peaks of separate relaxational processes. Maxima of these peaks are conventionally designated by letters a, p, y, 5, etc. in the order of decreasing temperature. In this case, the same relaxational processes (for example, glass transition) in various polymers can be designated in different ways due to discrepancies in spectra of amorphous and crystalline polymers and more complexive polymer system. To avoid confusion let us asume the relaxational process related with glass transition in amorphous polymers or amorphous regions of crystalline polymers to be an a-process and the relaxational process occuring at a lower temperature to be a P-relaxation. One should note that both a- and pprocesses are most well-studied ones, and these are the processes defining the main polymer properties. 7.1.1 Peculiarities of Temperature Dependences o f Frequencies of a - and pProcesses

Sharp differences in the nature of a- and p-processes are supported by their temperature dependences (Fig. 7.1). The logT =y(l/T) dependences for P-process are usually linear, i.e. formally governed by the Arrhenius law. Activation energies at T < Tg make up 40 - 100 kJ/mol (Table 7.1) and the values of preexponential factors (to) are within 1CH0 - 1015 s. According to some data, there is a discontinuity on the Arrhenius dependence in the region of Tg 15/. Activation energies of P-process at T > Tg are much higher than in the vitreous state. In PMMA, for example, values of Ep* are 180 and 77 kJ/mol, respectively. Temperature dependences of the a-relaxation is of quite different, nonArrhenius, nature. Activation energy (measured in a narrow temperature range) decreases with temperature (Fig. 7.2). These data show that the aprocess is of a complicated nature resembling a phase transition. At the temperature range of above Tg + 50° the frequencies of a- and P-processes become closer, a single temperature dependence being observed for both of them. This region of the so-called ‘combined ap-process’ is determined by the Ep* values close to the activation energy of the P-process.

161

It should be noted that the similar regularities are typical for glassforming liquids, namely, occurrence of a- and p-relaxational processes, analogous temperature dependences, close values of activation energies /14 16/. All these data indicate that the nature of molecular motions determining a- and p-relaxational processes is common for all amorphous substances, both polymer and nonpolymer nature. A specific feature of the polymer state, in terms of molecular dynamics, is primarily observed in the relaxational processes associated with the motion of large molecular aggregates, chain segments linked with crystalline regions, as well as in the parameters of distribution in molecular mobility. Table 7.1 Activation parameters for relaxation processes in polymers /4 - 12/. V*, cm3/mol Ep*, kJ/mol Technique1* T, K 400 53 DR 398 355 390 - 430 TSD 21 DR 310-385 88 3 DR >400 63 24 PEMA ap DR 300 - 400 22 24 P 44 PBMA DR 345 370 114 ap DR 310-325 105 314 a 12 114 DR 310-325 3 DR 376 321 525 PVC a TSD 350 - 380 342 422 a 25 DR 295 63 3 162 PCIF 3E TSD 320 - 360 205 a DR 355 31 71 3 PETPh DR 356 518 830 a 2) DR 35 295 63 3 PVAc DR 360 139 294 a TSD 186 300 - 340 182 a PCS 419 318 PS a Nylon 6 TSD 310-340 93 150 a NMR 273 - 323 30 NR 28 (ap) NMR 223 273 2 20 PB («31 PIB NMR 323 373 20 25 (aP) ') DR is dielectric relaxation; TSD is thermostimulated depolarization; PCS is photon correlation spectroscopy; NMR is nuclear magnetic resonance 2>in amorphous regions. Polymer PMMA

Process ap a

—

—

162

T -i Fig. 7.1. Diagram of Arrhenius dependences of a- and p-relaxation frequencies.

Fig. 7.2. Temperature dependences of the activation energy (I) and activation volume (2) for a-relaxation in polychloroethylene /13/.

163

Activation energies at a constant volume for a- and P-processes obey the same regularities as Ev* for two types of rotational motion of additives (Chapter 5). The Ev* values for relaxational processes are lower than Ep* (Table 7.2). According to the hybrid model equations, the molecular dynamics providing these relaxational processes is contributed both by the probability of activation volume foramtion and by the probability of fluctuation of the energy necessary for barrier overcoming. The second factor is limiting for a- and P-processes, while for aP-process both contributions are approximately the same. However, it should be noted that the a-relaxation is not of an Arrhenius type and one should not expect satisfactory results while applying the equations of the activation theory. Table 7.2 The Ev*/Ep* values for relaxation processes in polymers /4, 12, 13, 17 - 19/. Polymer PETPh PVC PCIFjE

Process P P P a NR (a p ) PB (aP ) PIB (a p ) ') See the footnote to Table 7.1.

Technique') DR DR DR DR NMR NMR NMR

Ev*/EP* 0.86 0.88 0.83-0.85 0.7-0.73 0.53 0.5 0.66

7.1.2 Volume and Pressure Effects ¡n Segmental Dynamics The bulk compression of polymers is accompanied by an increase in the glass transition temperature and relaxation times of molecular motions. For most polymeric systems, pressure derivative of the glass transition point (a-transition)

aia d?

is 0.2 ± -0.1 deg/MPa /20/. The temperature of the p-

transition is more weakly dependent on pressure

A

= 0.1 deg/ MPa . In ) this respect, baric dependences of a-process relaxation times are sharper than those of the P-process. Activation volumes for the a-relaxation make up 100500 cm3/mol and are on the average by an order higher than those for the p-

164

process (10 - 40 cm3/mol). The ratio of activation energies for these relaxational processes are of the same order. Temperature dependences of activation volumes are of the common nature for p-processes and rotational small-scale motion of particles, i.e. with higher temperature the V* value raises proportionally to an increase in compressibility /21/. The connection between an activation volume and compressibility is also supported by the fact that during polymer transition from the elastic into vitreous state V* is abruptly decreased /5/. As mentioned above, the similar effects are observed for rotational dynamics of spin probes. Fig. 7.3 presents the dependences of V* on polymer compressibility for Pprocesses. One can see that a single linear dependence is typical for different polymers. Section 6.3 describes the possible reasons responsible for the observed correlation.

Fig. 7.3. Dependence of the activation volume for p-relaxation on compressibility of polymers /22/: 1 - PMMA, 2 - PVC, 3 - PCTFE, 4 - PETPh, 5 - rotation of spin probe I in nitrile rubber. An inverse relationship is observed for the a-relaxation, i.e. activation volumes decrease with temperature (Fig. 7.2). In the high temperature region the values of the V* and E* activation parameters for a- and P-processes become closer, a combined aP-process being observed. Table 7.1 shows that the activation volumes of this process are closer to the values of V* for the Pprocess than for the a-process. Taking into acoount that V* increases with temperature for the P-process, the Vp* and Vap* values can be assumed to coincide. The following peculiarity is important to note: when using the high frequency techniques, such as pulse NMR, the values of V* (calculated by the

165

shift under pressure of the minimum on the temperature dependence of Ti) appear to be much lower than the activation volumes of the a-relaxation obtained by the low frequency methods (Table 7.1). The NM R data obviously refer to the high temperature ap-process characterized by V* lower than those of the a-process. 7.1.3 Differences in Mechanisms of Relaxational Processes

There are different explanations of the mechanisms of a- and pprocesses. The model attributing the differences in frequencies of both processes to those in the sizes of kinetically independent elements has prevailed for a long time. Thus, for example, the theory of dielectric relaxation of polymeric systems has widely used the concepts that P-processes are caused by the motion of side groups and small fragments of the main chain, while a-processes are connected with the dynamics of large segments whose motion is unfreezed in the glass transition region. The analysis of pressure effects of a- and p-processes in polymers has shown that the observed peculiarities are unlikely to be attributed to the differences in the segment sizes (Sec. 7.2). This conclusion is supported by the data on the dielectric relaxation in glass-forming liquids consisted of rigid molecules (Sec. 4.1). The analytical data on activation volumes also agree with those for the dependences of activation energies for a- and P-processes on the polymer molecular mass obtained by Bernstein et al /23/. These authors have found that a chain has its critical length (making up about 10 units for PS and PMS, about 5 units for PDMS and 2 - 4 units for PC) when activation energies of both relaxational processes cease to change. It means that in this region of chain lengths an independent kinetic unit is formed, which is the same for both processes. The TSD data on polymer bulk compression lead to certain conclusions on the nature of a- and P-processes. The specific feature of the TSD technique (in contrast to dielectric relaxation in alternating fields) is the single irreversible nature of depolarization processes. Thus, under heating of an electret, containing only one type of dipoles (or kinetic elements) unfreezing motions in the region of a low temperature p-peak should result in a complete degeneration of residual polarization irrespective of the existence of other types of motion unfreezed at higher temperatures. In this case, one should observe only a single TSD peak (providing the polymer structural and dynamic homogeneity). Note that for dielectric relaxation in alternating fields all types of dipole motion are active if differing in frequencies. When heating

166

is applied to an electrete, a complete depolarization can be induced only by a high-frequency type of motion. Others will be unable to generate current. C6

Fig. 7.4. Diagram of reorientation of dipole p in a- and P-processes. Two (or more) maxima of depolarizing current in the systems with dipoles of the same type can arise if there are the barriers preventing from a total dipole disorientation in a low temperature relaxational process. In other words, there should be a critical turning angle of dipole 0 (Fig. 7.4) with a low probability of dipole extension beyond it in a low temperature region. In this case, polymer heating causes only a partial destruction of polarization. The rest of it can be destroyed only at higher temperatures when extension of dipole beyond the critical angle 0 becomes more probable.Therefore, the existence of two peaks on TSD curves can be easily attributed to the difference in amplitudes of rotational motion of macromolecular segments. This conclusion is supported by the correlation between activation volumes of a- and P-processes and isolated charges Q (Fig. 7.5) /24/. The correlation tends to mean the following. In a vitreous polymer the free volume is small and permits only low amplitude rotational motions restricted by a small value of 0. The extension of the dipole moment vector beyond the critical angle is associated with an overcoming of the high potential barrier and hardly probable. Those motions require low activation energies and volumes and can destroy only a part of polarization. The temperature increase provokes the conditions enabling the turning of dipoles to larger angles (Fig. 7.4). These large amplitude rotations result in a complete destruction of polarization and require large activation volumes. Hence, both the volumes and charges depend on the same parameter, i.e. the amplitude of rotational motions. This is the reason of the correlation observed.

167

Fig. 7.5. Dependence of ratio of charges isolated in a- and P-processes in PCTFE (1), PVC, PETPh and PMMA (2) on the ratio of activation volumes. What is the value of 0 for solid systems? The comparison of experimental and theoretical values of Qa/Qp has shown that this angle ranges within 5 - 30° /24/. The ESR-spectroscopy of spin-labelled polymers also provides the information on the value of the angle 0. It is shown /25/ that if a spin label rotates around various molecular axises with different frequencies, an ESR spectrum depends on the angle at the top of the fast rotation cone. The analysis of ESR spectra of a spin label rigidly bound with a PE macromolecule has shown that the angle 0 is 10 - 25° at T < Tg and sharply tends to increase above Tg. Thus, the spin label method data provide both qualitative and quantitative support of the TSD results. One should note one more peculiarity of a-relaxation, namely, that the activation parameters of this process are much closer to the parameters of translational dynamics of particles than to those of rotational one, though they do not coincide completely. One can not exclude that the molecular dynamics of a-processes involves along with a rotational component a translational one. Let consider for the conclusion the arguments in favour of alternative explanations in terms of the data on the activation volumes. The model of differing segments has already been critisized above. The structural heterogeneity of polymer glasses is another evidence for a- and p-processes occurrence. This model assumes the presence of the regions sharply differing

168

in packing density (or in fluctuations of density of various sizes). It should be added that a-processes have to occur in more dense and ^-processes in loosened regions. This model explains the experimentally obtained differences in charges and fails to do it for the divergensies in activation volumes reaching an order of magnitude. According to the above mentioned data on the dynamics of small molecules, the differences in activation volumes of the same particles (segment) at various matrix sites can be associated only with those in compressibility factors. But even for amorphous and crystalline regions in the same polymer, the compressibility factors are no more than four times different. Besides, one can hardly explain the correlation between charges and activation volumes. This model does not agree with the TSD results under varying the mode of sample cooling. A sharp cooling (quenching) of samples after their polarization at an increased temperature results in lower a-peaks as compared with those in the slowly cooled samples /4/. At the same time, p-peaks are not practically affected by changes in the temperature mode. The ratio of areas below peaks for both relaxational processes (charges) also remains unchangeable. However, the logic of the model attributing the occurrence of a- and fl-processes to the structuraldynamic heterogeneity implies the existence of antibatic changes in a- and Ppeaks due to an alternation of the ratio of regions differing in a packing density. However, the mechanism of the relaxational processes should not be understood as ignoring the structural heterogeneity of organic glasses. The latter is undoubtedly of great importance and observed, when studying many physical and chemical processes, in broadening of peaks obtained by relaxational techniques, in particular. This chapter does not aim at the detailed description of the distribution functions. In this respect, it is noted only that according to the data obtained by various methods the width of the function of distribution in frequencies of relaxational processes does not exceed 3 - 4 orders (at a half-height) and narrows rather sharply with increasing tempereature 16, 26/. Structural heterogeneity of a polymeric matrix determines the width of distribution in dynamic parameters, but can hardly explain the occurrence of two relaxational processes even because aand p-processes are of a radically different nature and described by different functions. Since volume effcts are the main subject of the present study, let us give an example of distributions in activation volumes for a- and P-processes 161. Analysis of factors determining the V* value allows us to conclude that these distributions are determined by distribution in polymer compressibility. Recurring to the nature of relaxational processes one should note that P-processes in polymers are governed by the same regularities as the rotational dynamics of molecules in liquids. As for the a-process, it rather has

169

the nature of transition showing in the non-Arrhenius temperature dependence, enormous values of activation parameters, and relatively narrow temperature range of observance. In kinetical terms, the a-transition is similar to crystallization and melting processes since it requires an optimal mobility of macromolecular segments. In these conditions the structure typical for glasses can transform into a structure of the elastic state. The rearrangement is accompanied by large-scale reorientation and movement of segments to longer distances. The width of the temperature range of this transition depends on the structural heterogeneity determining a set of polymer glass transition temperatures.

7.2 CORRELATIONS BETWEEN SEGMENTAL AND PARTICLE DYNAMICS

Motion of a particle in a condensed medium is of a cooperative type, i.e. its reorientation or translation is possible only if a movement of surrounding kinetic units has occurred /27/. Such kinetic units in polymer matrix are segments of macromolecules. It is evident that both frequencies and activation parameters of additives motion should be connected with the corresponding parameters of the segmental motion. As it has been mentioned above, it is impossible now to stipulate a strong quantitative correlation taking into account all structural-dynamic features of particle and matrix. Using the experimental material of the previous Chapters, we will try to answer the following questions: what is the type of the relaxational processes that controls rotational and translational dynamics of particles, and what is the size of kinetic elements of macromolecules (segments) that control one or another type of particle motion. Activation parameters of particles dynamics and relaxational processes are compared in Table 7.3. It is clearly seen that activation parameters of the large-scale rotation of particles coincide with the parameters of the a-process relaxation. In their turn, parameters of the smallscale rotation are close to those of P- and aP-relaxation. As a- and Pprocesses are based on rotational motions of macromolecule segments, the data mentioned allow to conclude that the rotational dynamics of additives is completely controlled by rotation of surrounding segments. If the ideas of the free volume model are used, the above mentioned data mean that free volumes are formed by the rotational movement of segments, which are comparable with the activation volumes of additive molecule rotation.

170

Moreover, these data show that two types of particle rotation are caused by two relaxational processes in polymer. Let us remind that as mentioned above, the degeneration of the a-process occurs in the range of high temperatures, and general relaxational process is observed, usually called as the ‘combined ap-relaxation’. As follows from Table 7.3 the activation parameters of this process are very close to those of P-process. It can be safely stated that the mechanisms of both processes are the same, i.e. we are dealing with the same process but in different temperature regions. As seen from the Table, activation volumes of the combined process may be twice higher than the V* values for the P-process. This difference coincides with the difference in the compressibility coefficients as measuredabove and below Tg. It was already shown that the compressibility coefficient affects the V* value. It should be noted that the data from the spin probe technique belong to the high-frequency high-temperature range, i.e. to the range of the ap-process. Table 7.3 Activation parameters for relaxational processes and particle dynamics in polymers. Process Ep*, kJ/mol V*, cm3/mol V*/Vw Relaxation: 100 - 350 200 - 400 a-process (T « Tg) 20-40 20 - 100 P-process (T < Tg) 20 - 100 __ _20j 60___ _aj3_-process fT >_>_Tg)________ Additive dynamics: 120-370 100-300 1-2 large-scale rotation (T « Tg) 20-70 20-50 0.1-0.5 small-scale rotation (T » Tg) 50-150 0.5-2.0 50-150 translation (T > Tg)

P e/P v

« «

1 1

Tg /11/. No. 1. 2. 3. 4. 5. 6. 7. 8.

Polymer

\

Rotation

\ E*, ' kJ/mol Natural rubber ! 29.3 Nitryl rubber ! 37.6 Butadiene-styrene rubber I 1 with styrene content: I 65% ! 38.5 50% ! 36.4 30% ! 30.1 Polyisobutylene T 44.7 Butylrubber | 33.4 Atactic polypropylene ! 59.4

Translation ♦ , cm2/s D ?,s-' 1 E t . 1 1 kJ/mol 3.1xl013! 53.5 8.0x10' 3.0xl0'4! 76.9 1.6xl04 1 1 1 \

1

2.6xl014 I 3.6xl014! 6.7x10*3; 6.7xlO'3~T 7.2xl013! 1.7xl0181

90.3 74.0 60.2 74.4 51.4 106.6

4.5x 105 8.0xl03 4.0x102 l.lxlO 3 2.3xl03 l.OxlO9

According to the coefficients a and b, the polymers studied fall into three groups. Polvdienes: natural rubber, nitrile and butadiene-styrene rubber. Despite differences in glass transition temperatures and intensity of molecular motions, the ratio of rotational and translational frequencies of probe in them is described by the uniform dependence with parameters: loga = -25.5 ± 1.0 and b = 2.05 ± 0.16. Butvlrubber and polvisobutvlene. Parameters in Eq. 8.1 for these polymers are loga = -22.6 ± 0.6 and b = 1.6 ± 0.1. Atactic polypropylene: loga = -23.7 ± 0.6 and b = 1.8 ± 0.1. Fig. 4.9b shows the ratio of rotational and translational frequencies vr = — Tt in · dependence on Dr. This ratio shows how many times a n=— vt xr

188

particle changes its orientation during translation by the distance, equal to diameter. The results obtained allow to make the following conclusions. The ratio of rotation and translation frequencies of particles in polymers at T > Tg depends on macromolecule structure and is described by a uniform equation for polymers with similar structures only. Translational mobility of particles in the sequence of polymers: polydienes - atactic polypropylene - butylrubber and polyisobutylene, decreases much more abruptly, than rotational one (n increases). One may suggest that distributed mean values of fluctuational hole size for such polymers as butylrubber and polyisobutylene exceed the activation volume of probe rotation (35 cm3/mol for PIB), but are smaller than that of translation (50 - 100 cm3/mol). This leads to an increase in the waiting time for the particle for formation of a fluctuational hole of necessary size in the nearest surrounding. At the same time, rotation is easily provided by the necessary activation volume. The ratio of rotation and translation frequencies decreases with temperature (increase of Dr) and approaches to the n value, typical for liquids. In contrast, temperature decrease leads to a sharp increase of n.

Fig. 8.1. Temperature dependence for TEMPO probe rotation in PEG /12/. As an illustration of the effect of polymer side group structure on molecular mobility of an additive, consider the results of investigations,

189

performed by the spin probe technique on the polyethyleneglycoles with -OH and -OCOCH 3 end groups /12/.Both types of polymers were characterized by close values of molecular mass (M n = 1450 - 1650) and molecular-mass Mw

and by close temperatures of crystalline / regions melting (317 - 318 K). Fig. 8.1 presents the temperature dependences of the correlation times for the probe. Inflexion points of these dependences are associated with the glass transition and melting points. Substitution of OCOCH 3 end groups by -OH leads to a decrease of the rotation frequencies for a particle in the rubbery region (III) and increase of glass transition temperature (inflexion point temperature). At the same time, the frequencies of probe rotation in melt (region I) and glass (region II) are practically equal for both polymers. Thus, the formation of hydrogen bonds in polymer restricts mobility of molecules; this effect is most clearly displayed in the rubbery state of polymer. distribution

Mr

= 1.23-1.28

The effect of molecular mass. Investigation of the mobility of molecules in dependence on the molecular mass M allows to get an information on sizes of the kinetic element of a macromolecule, which determines intensity of additive motion, and on the role of end groups in this motion. Main characteristics of polymer changes with molecular mass increase up to Miim, after which they strive for a constant value. This relates to the glass transition temperature, relaxation times, viscosity, strength, etc. The value of M, at which Tg becomes constant, depends on the chain flexibility. The Miim values for flexible-chain polymers fit the 1000 - 5000 range, and for rigid-chain polymers this range is 10000 - 20000 /13, 14/. Differences in Miim values for these groups of polymers are determined by the length of the chain kinetic segment. The following equation was obtained on the basis of inveastigation of polyacrylonitrile /15/: rp

rp OO

Ag = j g

M,

( 8 . 2)

where a is the constant; T^° is the value of Tg at M n => 00. Molecular mobility of additives, as well as permeability of polymer decreases with molecular mass in accordance with the analogous equation:

190

InD = InDoo

a

(8.3)

Mn

Dependence of dynamic parameters for additives on M was not observed only in the investigations, in which the molecular mass range was limited. Table 8.2 shows the values of Mum for translational diffusion of organic molecules. Table 8.2 Values of Mum for organic molecules diffusion in polymers 111. Polymer PDMS Polybutadiene Polyisobutylene PVC PS PMMA

Diffusant Benzene Methylnaphthalene Benzene Hexadecane Octane Hydrogen Dioctylphthalate Decaline Methylacrylate

MlimXlO·3 1.3 4.1 3.7 10-13 8 -1 0 5 70 30-50 12

As expected, Mim, for rubbers is lower than for rigid-chain polymers. For one and the same polymer (PIB) the lower values of Mumare observed for hydrogen. Apparently, this is connected to a different type of diffusion process. Rotational diffusion of additives obeys the same regularities that translational ones. For example, the values of Miim, obtained in investigations of spin probe rotational mobility in polyethyleneglycole with different molecular masses fall within the 200 - 1000 range /16/. So, the interconnection of additive mobility and polymer structure is rather elaborate and defined by many parameters. This is the reason that attempts failed to get the quantitative parameters, which characterize this interconnection, density of cohesive energy, for example 161. The role of cross links. Formation of cross links in a polymer due to chemical and physicochemical processes (radiation, for example) leads to a decrease in the rotational and translational dynamics of additives. The value of these changes depends on the cross links concentration, structure of a bridge connecting the macromolecules, and additive molecule size. There are numerous data in the literature on studies of permeability of cross-linked polymers. Less works has been devoted to the study of diffusion coefficients,

191

and only a part of them displays investigations performed in a broad range of cross-links density, and the data on the average distance between cross links are shown. The examples, shown below, were taken from these particular works. The dependence of rotation correlation time for spin probe TEMPO in rubbers, vulcanized by y-radiation, rapid electrons, and with the help of peroxy compounds, was studied in /17, 18/. The results obtained showed that the frequencies of the probe rotation do not change until high concentrations of cross-links. By way of example, an increase of x for probe in peroxide vulcanizates of isoprene rubber was observed, when the average molecular mass of a chain section between cross-links, M s , becomes smaller than 104. A sharp jump of x was observed in the M s range from 2000 to 3000. In this range, the distance between links approaches to the size of polymer kinetic segment, consisted of several monomeric units. Similar results were obtained for translational diffusion. Investigations of butane, benzene and hexadecane diffusion in sulfuric vulcanizates of cysisoprene showed that a decrease of D for these particles is observed at M s < ( 4- 5)xl 0-3 111. Thus, for the polymer linked by short cross bonds the effects are observed, similar to the increase in the molecular mass of linear polymer molecular mobility changes most sharply in the range of M, comparable with the kinetic segment length. The resemblance of these processes is also displayed by similar equations, used for their description. Main equations for these processes, empirically or semiempirically obtained, are presented below. The equation connecting diffusion coefficient with M s is similar to Eq. 8.3: lnD = lnD oo+JU · Ms

(8.4)

The following equations were obtained on the basis of the free volume model (Eqs. 2.114). For the change of D with molecular mass we get: In

f D \-l __ v f , (v f ) ' M r vDoo; B Ba

(8.5)

Here vf is the fraction of free volume for a polymer with infinitely high molecular mass; D,*, is the diffusion coefficient in such polymer; B is a constant.

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At M n >> 103, Eq. 8.5 looks similar to Eq. 8.3: InD = In Don + = Mr To describe the changes of the diffusion coefficient with increase of the lattice density, analysis of the free volume model led to the expression:

InD = InDoo

B

a '(M s) 1

vf

v f - a ' ( M s)_1

( 8. 6)

Eq. 8.6 is most often used in a more simple form: InD = InD«,

a'B 1 Mo Vf00 lvls

(8.7)

Eqs. 8.4, 8.6 and 8.7 can be used for diffusion of particles in lattices with short cross links. The same rules as for dynamics of partcles in linear polymers are observed at the increase of cross bridge length, i.e. the main role is played by bridge flexibility. The conclusion is supported by many of experimental data. Let us discuss some examples. Coefficients of benzene diffusion in natural rubber vulcanizates with M s = 3300 increase in the following sequence of cross links: carbon-carbon (radiational vulcanization) monosulfide - polysulfide (sulfuric vulcanization) 161. The length of the latter link is much longer than the others (it contains 3 - 6 sulfur atoms). Results of investigation of oligoesteracrylate nets /19/ show that frequencies of spin probe rotation depend on the nature of oligomeric block disposed between main methacrylic chains. If this block is long and flexible, frequencies of probe rotation increase due to plasticizing effect. Shorter and more rigid cross links cause no change of the molecular mobility. The effect of various chemical groups, introduced into the cross chain, was studied for polyurethanes and polyepoxides in /20, 21/. Particle size is one of parameters, which affects the value of changes in additive mobility under polymer vulcanization. For example, it has been shown /22/ that diffusion coefficients for gases in polymer decrease with the growth of net density in proportion to gas molecule diameter.

193

8.2 TRANSITIONS IN POLYMERS

Let us consider the effect of relaxational and phase transitions on parameters of rotational and translational dynamics of particles. Recall that glass-rubber transition is the main relaxational transition. There are also socalled secondary relaxational transitions, observed at T < Tg. Not so wellstudied liquid-liquid transition at T > Tg also belong to this type of transitions /10/. Contrary to the phase transitions of the first order, relaxational transitions are not accompanied by changes of the system volume. Jumps are observed only for volume derivatives with respect to temperature and pressure, i.e. for coefficients of thermal expansion and compressibility. It should be intimated that the terms a, P, y-, ô-transitions are used in the literature. For amorphous polymers, a is a synonym of the glass-rubber transition, p Transition is related to peaks of mechanical and dielectric losses at T < Tg. Comprehensive investigations of additives dynamics in the regions of secondary transitions were not performed. Moreover, the use of the term ‘transition’ for secondary processes seems to be rather conditional, because most of them are described by Arrhenius dependences, in contrast with the a-processes possessing all features of transitions (Sec. 7.1). That is why, we restrict the discussion to the analysis of the data of rubber-glass transition, melting and crystallization. The experimental material accumulated thus far shows that frequential characteristics of particle motion - correlation times and diffusion coefficients - change smoothly in the regions of relaxational transitions, i.e. no jumps of x and D happen. However, the slope of the temperature dependences of these ^ In t parameters changes. This means that jumps in derivatives -------- and SlnD

happen, i.e. the values of apparent activation energy. These features

can be explained in the frames of conclusions, made in Chap. 5. Frequential characteristic of the molecular motion are bound to the system volume. It has already been mentioned that volume changes smoothly in these transitions. At the same time, apparent activation energies are bound to the temperature bulk coefficient, i.e. thermal expansion coefficient. The latter value sharply changes in the transition region. The more so, if thermal expansion coefficient in glasses can be considered constant, in the rubbery region it 2 - 3-fold increases with temperature. In its turn, this leads to a much higher increase of apparent activation energy.

194

Examples on changes in temperature coefficients of spin probe rotation frequencies and diffusion coefficients of particles in the Tm range of polymers are presented in Figs. 5.1 and 5.7. One can see that the change of the temperature dependences slope occurs within a specific temperature range. As a result, an error in determination of the transition point exists, which may reach 10 - 20°. Difference in activation energies above and below Tg (AE*) for translational diffusion is 20 - 60 kJ/mol (Table 5.4). The same difference for the small-scale rotational dynamics of particles falls into the 15 - 40 kJ/mol range (Table 5.1). From this it follows that the ranges of AE* for rotational and translational mobility of particles are virtually equal. The AE* value depends on polymer and also on particle. Bending temperature Tb does not depend or depends weakly on the structure of particles and is bound to the glass transition temperature of the matrix. Attention should be paid to the fact that the activation energy of p-relaxational processes also grows at T > Tg /23/. The model concepts which describe the changes observed for the activation energy were discussed in Sec. 5.4. According to Eqs. 5.7 - 5.13, * 5E activation energy increases at T > Tg due to increase of the absolute value, determined by the thermal expansion coefficient of the medium. In other words, experimental values of E* in rubbery state sharply exceed the potential barrier height of molecular motion. Changes in mechanism of molecular motion at passing the glass transition point consist in increase not only of intensity, but also of amplitude of segmental motion of macromolecules. This process is accompanied by more sharp than in glasses growth of fluctuational free volume with temperature, large-scale cooperative movements of segments, repacking of macromolecules. The process displays all features of transition (a-relaxation), is not described by Arrhenius dependences and terminates at temperatures 50 - 80° above Tg. Analysis of ESR spectra allows to conclude that the spectrum of correlation times for spin probes rotation is significantly broadened at temperatures below Tg /24/. Rough estimations have shown that the width of the T-set in natural rubber decreases from 1 order at T > Tg to 2.5 orders at T < Tg /24 - 26/. The conclusion about broad spectrum of correlation times is supported by the data on thermostimulated depolarization of n-para-ndimethylaminoazobenzene in PS /25/. Recall that this method gives an information about large-scale rotational dynamics of particles. The TSD data showed that the spectrum width increases from 1 order at T « Tg to 4 - 5 orders at T = Tg -100°.

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We will now look at the data on the ratio of rotation and translation frequencies of particles near polymer glass-rubber transition. Both types of molecular motion for the TEMPO spin probe in polystyrene with Tg * 363 K were studied in /27/. The results are shown in Table 8.3. Table 8.3 Rotational and translational diffusion coefficients for spin probe TEMPO in PS /27/. T, K

Dr, s·'

Dt, cm2/s

353 373 393

1.5xl07 5.5xl07 1.7xl08

1.2xl0‘2 3.0x10" 5.0x10'°

n=— Tr 5200 700 130

Parameter n shows how many times a particle changes its orientation during translation by the distance, equal to its diameter. The results shown in Table 8.3 display that n increases with temperature fall below Tg. Unfortunately, the data shown do not allow to make a clear conclusion whether this increase proceeds smoothly or by a jump. However, one can say that the main contribution to the increase of n is made by a decrease in translational mobility. Intensity of translational motion of a particle 416-fold decreases in the 353 - 393 K temperature range, whereas rotational motion decreases by 11-fold only. Thus, below Tg the particle spends much more time in a ‘cage’, which virtually does not restrict its rotational mobility. The result becomes understandable, if one takes into account that translational motion requires higher activation volume, probability of formation of which is low in rigid systems. Let us note that similar results were obtained in investigation of rotational and translational dynamics for a spin probe in low molecular liquid - dibutylphthalate - below Tg /28/. Attention should be paid to the fact that in the case, when no connection between mobility of additives and segmental mobility of macromolecules, and position of the inflexion point on temperature dependence of motion frequencies in the glass transition point are observed. This situation was discussed in Sec. 7.3. It relates to diffusion of small gases and to porous systems, as well. Let us discuss results of lowfrequential methods - thermostimulated polarization and optical methods, mentioned in Chap. 3. These methods give information about large-amplitude rotational dynamics of particles in polymers. Mobilities of rubrene and tetracene were studied in /29/ near glass transition temperature of three polymers by photobleaching technique in the

196

10-' - 103 s range of correlation times. The results obtained have shown that rotation of these particles at temperatures near Tg is characterized by very big values of apparent activation energy. They are ~ 700 kJ/mol for polystyrene, > 1000 kJ/mol for polysulfone, and ~ 200 kJ/mol for poly isobutylene. Temperature dependence of x for these particles is sutisfactorily described by the WLF type equation (Eq. 2.4). Temperature dependence of correlation times in the WLF equation coordinates is shown in Fig. 8.2. These dependences represent smooth curves, passing the glass transition point without discontinuity. The WLF equation parameters are shown in Table 8.4.

Fig. 8.3. Temperature dependence of rubrene and tetracene rotation in PS, PIB and PSF /29/. As follows from Table 8.4, coefficients Ci and C 2 weakly depend on matrix and particle. Taking into account probable error due to narrow temperature range of the investigation, one may think these coefficients to be equal, excluding the data on rubrene rotation in PS. The reason of deviations is not clear yet. It should be noted that parameters of WLF equation, shown in Table 8.3, are close to the correspondent parameters, obtained by mechanical relaxation technique. For example, Ci for PS equals 12, and C 2 equals 41.6-49.9 K /30/.

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Table 8.4 Parameters of WLF equation for rubrene and tetracene rotation in polymers and rotational correlation times at Tg (xg) /29/. Polymer PIB

Tg, K 205

PS

375

PSF

459

c,

Particle rubrene tetracene rubrene tetracene rubrene tetracene

C2,K 57.2 54.3 54.9 85.3 47.8

13.0 13.3 13.8 24.3 12.8

logxe 3.8 2.1 1.1 0.2 0.8 -1.2

Recall, it follows from Eq. 2.100 that Cl

V 2.3fg

and C 2 =

Aa

( 8. 8)

where V* is the fractional activation volume of molecular motion; fg is the fractional free volume at Tg; Aa is the difference of thermal expansion coefficients above and below Tg. Good correlation between the values of these coefficients for polymer viscosity and large-amplitude rotational dynamics of particles means that both processes require a comparable activation volume, as confirmed by the V* values presented in Sec. 6.2.1. Inoue et al. /29/ performed analysis of rotational mobility of particles under relaxation of PS volume. For this purpose samples were rapidly cooled from 388 K to T < Tg. Dependence of x on isothermal heating time was measured for rubrene at 368.5 K and 372.7 K. Correlation times increased up to a constant value within ~ 4000 s and 3000 s for two mentioned temperatures, respectively. The results obtained testify a close connection of additive molecular mobility with polymer volume. Now consider an important question about intensity of additives motion near glass transition point. The data of spin probe and photobleaching techiques for small-scale and large-scale rotational diffusion are shown in Tables 7.5 and 8.4. It follows from these Tables that glass transition temperature is not the isofrequency point for both types of motion. Frequencies of spin probe rotation (TEMPO) in polymers are 25-fold different. Much higher differences in xg are observed for the large-scale

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rotation: they reach three orders for one and the same particle. Moreover, linear dependences of xg on glass transition temperature are observed from three studied polymers. Similar effects are obtained in the same polymeric systems for viscoelastic relaxation times of the fastest Rouse modes. It is evident that the fraction of free volume in the glass transition point vr(g) is different for polymers. The vt(g) value increases with Tg. As we have ascertained on the above mentioned data, analysis of application of free volume model equations, W LF equation, in particular, to the large-scale rotational mobility of particles gives a suitable information about properties of polymeric matrix. The W LF equation was repeatedly used also for small-scale rotation and translational diffusion of particles 16, 7, 31 - 36/. It is widely used for dynamics of particles in polymer solutions 111. Coefficients Ci and C 2 for various types of particle rotation are shown below: - for small-scale rotation of spin probes Ci = 0.15 - 0.3, C 2 = 30 - 50 K /31/; - for large-scale rotation Ci = 13 - 14, C 2 = 47 - 57 K. Comparison of these data with the data in Table 8.3 shows that differences in Ci values for various types of molecular motion are much higher, than for C 2. As follows from Eq. 8.8, the ratio of Ci values for different types of motion of the same matrix is defined only by the ratio of free volumes. This ratio of Ci satisfactorily correlate with the V* ratio, obtained at bulk compression of polymer (Chap. 6). For example, at average values of V* of 30 cm3/mol and 200 cm3/mol for small-scale and large-scale rotations (Tables 6.2 and 6.4), respectively, their ratio equals ~ 7, which is close to the Ci ratio for the same types of motions. Connection between glass transition temperature, diffusion coefficients for particles, and activation energy of diffusion was studied in /4,6, 37, 38/. It has been shown that linear dependences are observed between coefficients of gas diffusion in polymers and T - Tg, and between activation energy of diffusion and T - Tg /4/. Let us now discuss the data on high temperature transitions. Rotation of spin probe BZONO in the region of liquid-liquid transition ( T l l ) in polystyrene was studied in /39/. An inflexion point on the Arrhenius plot of correlation times was observed at 325 K. The values of activation energy are 61 kJ/mol at T < T l l and 28 kJ/mol at T > T l l , respectively. Thus, contrary to glass-rubber transition, liquid-liquid transition is accompanied by increase decrease rather than increase in the apparent activation energy. Melting of crystalline regions causes a jump-like increase of molecular mobility of additives (Fig. 8.2). Correlation times of spin probe rotation decrease abruptly within a narrow range of temperatures /12, 40/. For polyethyleneglycole with M = 4000 this range is 4° /12/,and for PEG with M = 1500 - about 10° /40/. The differences may be connected with those not only in molecular mass, but also in molecular-mass distribution of polymers. The

199

results of both investigations showed that change of x started at temperatures 10 - 20° below Tm, measured by traditional techniques. Thus, low molecular particles, which are localized in amorphous regions and in defects of crystalline phase, are sensitive to the changes in mobility of local surrounding before ‘macroscopic’ melting set-in. Activation energies of probe rotation at temperatures below and above the inflexion point are 52 and 34 kJ/mol, respectively /12/. So, melting leads to decrease of apparent activation energy similar to liquid-liquid transition. The same results were obtained in investigations of translational diffusion of gases in gutta-percha /4/ and large amount of antioxidants in polyethylene /41/. In both cases activation energies in melt are lower than at T < Tm. Differences in E* values for diffusion of antioxidants in PE fall into 10 - 80 kJ/mol range. Crystallization, the process opposite to melting, is also followed by change in molecular mobility of additives. As bulky organic molecules do not enter crystalline phase due to their size, their local concentrations in amorphous regions increasing. Investigations of local concentrations change for nitroxide radicals at crystallization of liquids, performed in /42/, support this suggestion. Beside decrease of local concentrations, mobility of additives in amorphous phase decreases also. Crystallites represent physical knots, similar to cross links, which limit the conformational set of macromolecules and decrease segmental mobility. Consider the results of investigation of PETPh crystallization from glassy state by the spin probe technique /43/. Crystallization of amorphous PETPh film proceeds with increase of sample temperature over Tg (343 K). The growth of correlation times of the probe TEMPOL rotation was observed at crystallization, proportional to the polymer density. The x - p dependence allowed the authors to analyze crystallization kinetics using the Avrami equation:

Here xa, xt and xc are the correlation times at the beginning of crystallization, at time t and in the end of the process; k is the rate constant of crystallization; n is the coefficient, dependent on morphology. Values of k in the 383 - 413 K range, n = 1, and activation energy of crystallization, equal to 84 kJ/mol, were obtained. The data obtained correlate with the data got by independent techniques.

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8.3 CRYSTALLINE PO LYM ERS

As mentioned in the previous Section, crystallization of a polymer causes a decrease of particle rotation frequencies, which is proportional to the increase in the polymer density (spin probe in PETPh). Similar dependences were also observed in many investigations for translational diffusion coefficients of particles, for example, for organic additives / 8/ and gases / 5 , 44/ in PE, for water in polyacetales /45/, gases in polyethyleneterephthalate /46/, water permeability in natural rubber 161, etc. However, before interpretaion of the results of diffusional experiments, it is necessary to clear up the question about the character of the defects in crystalline polymer and regions of additives localization. Crystalline polymers represent a typical example of two-phase systems, in which crystallites are embedded in an amorphous matrix /47 - 49/. The crystallinity degree of widely used synthetic polymers, such as polyethylene, polypropylene, polyamides, is 40 -50%. That is why, they are often called semicrystalline polymers. Higher crystallinity degree is observed in polymeric single crystals and ‘mats’, grown up from a melt or solution, and extended chain crystals (ECC), prepared from melt under high pressure /48, 50/. Crystalline polymers are characterized by a high variety of morphological forms. The most wide-spread structural units of crystalline polymer are lamellae - flat crystalline plates, sizes and amount of which depend on structural features of macromolecules and crystallization conditions. Lamellae are linked to each other by macromolecular segments of various length. These segments may be in a stressed state, i.e. in the state with restricted mobility. From this point of view, crystallites play the role of physical knots, and crystallization is analogous to vulcanization. However, crystallization differs by the fact that growing crystal ejects the defects and additives to the surface. In concequence with this, the surface of the lamellae becomes a zone of increased defectiveness. It contains the end groups and loops of the macromolecules, and adsorbed admixture particles. Lamellae are often disposed symmetrically, forming large structures, so-called spherulites. Spherulites represent one of the most wide-spread types of supermolecular structures in crystalline polymers. Another wide-spread type of structures - fibrilles - are also composed by lamellae. Therefore, the main part of amorphous phase in crystalline polymer is composed of interlamella parts. A specific role in physical and chemical processes can be played by surface defects of lamellae and interstructural zones. At the same time, the mean density of interlamella zones, pu, is higher than that of melt, pm, extrapolated to the same temperature. For example,

201

according to the data of X-ray diffraction for PE pii = 0.89, and pm =0.84 g/cm3 /51/. Thus, amorphous zones of semicrystalline polymers differ from noncrystalline matrixes by higher structural and dynamic heterogeneity (densily packed interlamella zones, numerous defects on the lamellae surface, defective interspherulitic and interfibrillar zones). As shown in many investigations, performed by various techniques /9, 47, 52 -54/, organic additives, such as antioxidants, dyes, initiators, probes O 3 with molecular volume above 100 A , do not enter the crystalline phase of polymer and localize in amorphous interlayers. For example, it was shown that the surfaces of lamellae and spherulites are of most intensive colour after dye sorption /54/. Analysis of ESR spectra for spin probes with molecular O 3 volume of ~ 170 A has shown that their local concentrations in polymer increase in proportion to the crystallinity degree /24,42, 54, 55/. Two main reasons stipulate impermeability of polymer crystalline zones: extremely low diffusion coefficients of defects and small size of crystalline cell (Table 8.5). Table 8.5 Parameters of crystalline cell for some polymers /47/ and permeability P to oxygen at 25°C /52/. Polymer Polyethylene Polypropylene Polyoxymethylene PoIy-4-methylpentene -1

Type of cell Orthorombic Monoclinic Hexagonal Tetragonal

O O O a, A b, A c, A 7.4 4.9 2.5 6.6 20.9 6.5 4.4 17.3 — 18.7 18.7 13.8

PxlO10, cm/ cm-s-cm Hg 0.51 2.3 — 42.3

For comparison, diameters of some gases and organic compounds are O shown below (in A ): He - 2.55; H 2O - 3.7 (effective); H 2 - 2.82; N H 3 - 2.9; O 2 3.47; CH 3OH - 3.63; CH 4 - 3.76; N 2 - 3.8; C 2H 4 - 4.16; C 2H 6 - 4.44; C3H 5.12; C 6H 6 - 5.35; nitroxide spin probes - 8-10; antioxidants - >10. Permeability of crystalline zones for small molecules of gases depends on the particular structure of a crystalline cell. Broad experimental material on this point was obtained for oxygen diffusion in investigations of macroradical oxidation in crystalline phase /52/. It became known, for example, that crystalline zones in polyethylene are inaccessible for oxygen. In polypropylene crystallites are also hardly accessible for oxygen, its solubility

202

being practically independent on sizes of spherulites. On the contrary, oxidation rate of poly-4-methylpentene-l crystallites is higher than that of amorphous phase. Crystals of this polymer possess loose packing and are accessible for oxygen molecules. Crystals of gutta-percha and polyalkenomers have good accessibility for ozone. Three main factors can be named, which affect the molecular dynamics of particles in amorphous regions of crystalline polymer: 1. Decrease of rotational and translational diffusion coefficients as a result of restricted chain segment mobility via crystalline crosslinking; 2. Decrease of polymeric membrane permeability due to diffusion path length increase; 3. Increase of width of distribution in dynamic parameters due to structural heterogeneity of the amorphous phase. In addition to the features of crystalline system mentioned above, another very important feature should be mentioned. This feature of dynamics of particles was detected by various techniques. It consists in directing a flow of particles through the most defective regions of crystalline polymeric membrane. These highly defective regions include not only above mentioned interlamellar layers and interspherulitic zones, but also the microcracks, arized under crystallization and heat treatment of polymeric material. In consequence, two opposite factors affect the rate of particle flow: increasing of the average length of the diffusion paths due to existence of impermeable crystalline formations and acceleration of the flow by the breaks in the material continuity. Thus, the study of crystalline polymer permeability determines the effective value of the diffusion coefficient, Deff. To analyze the diffusion path elongation, the following equation is used /56 - 58/: D = D a i|/ = I^ L .

y

(8.9)

Here vp is the volume fraction of the amorphous phase; y is the coefficient of diffusion path elongation. The value of y depends not only on the geometrical size of crystallites, but also on their morphological features (spherulites, fibrilles, etc.), and on existence of microcavities, as well. That is why, the analysis of the y factor is a complex task. To determine this factor, it is suitable to obtain the data on the change of self diffusion coefficient or molecular mobility in the amorphous phase at an increase of the crystallinity. By way of example, consider a number of investigations on the effect of polypropylene supermolecular structure on coefficients of rotational and

203

translational diffusion of spin probes /53, 55, 59/. It has been found in /53/ that the spin probes, introduced in low amounts (up to 5 x l0 19 spin/cm3) into crystalline polymers, including polypropylene, are distributed randomly in the amorphous phase. Moreover, it has been shown that the change in the radius of spherulites in PP samples, ranged within 10 - 500 pm, causes no change in rotation frequencies for the spin probe /24, 55/. This result displays that even if spherulitic interlayers have a loose packing and microcavities, this does not affect the dynamics of the additives uniformly distributed over the amorphous phase, because the number of particles in interlamellar parts is much greater than that in spherulitic ones. It follows from these data that the analysis of rotational dynamics of particles allows to obtain an information on the change of molecular mobility in amorphous zones and, consequently, provides a basis for estimating the self diffusion coefficient Da and parameter y in Eq. 8.9. This approach was used in /59/ for investigations of D t and D r in PP samples, differed by tacticity and crystallinity degree (Table 8.6). As follows from Fig. 8.3, coefficients D t and D r decrease with increase of 200% they increase /31,77/. Such type of dependences is explained by crystallization of NR at high stretching degrees, which leads to restriction of molecular mobility. To observe changes of molecular mobility, it is necessary to fix tensions with the help of structural changes in the sample. Crystallization is one of such fixing processes. The results on cold stretching of polymeric glasses obtained by different authors do not allow an unambiguous conclusion about change of additive mobility. It has been shown /31, 43/ that D r values of spin probe in oriented amorphous PETPh are significantly lower than in nonoriented one.

208

The rate of crystallization under annealing of oriented PETPh is also significantly lower, which is connected with the restriction of segmental mobility of macromolecules. A small increase of diffusion coefficients at orientation in glasses was observed for methylenechloride in PS /79/ and oxygen in PC and PMMA /80/. In some investigations no stretching-induced changes of gas permeability of polymeric glasses were observed /6/. Thus, we clearly see the opposite effects of glass orientation on rotational and translational dynamics of particles. This is apparently connected to the fact that stretching of glasses is accompanied by a decrease of molecular mobility and formation of a significant amount of defects. The former is confirmed, for example, by deceleration of PETPh crystallization /43/, and the latter - by numerous data on decrease of density and increase of sorption ability of polymers 16, 81/. That is why, the permeability, closely connected with the structural features of the matrix, increases and rotational mobility dependent on segmental dynamics decreases. Stretching of crystalline polymers was always accompanied by a decrease of rotational and translational diffusion of particles /65, 82 - 99/. Most often, the differences in the results concern X ranges, in which changes of mobility are sharply defined. Dynamics of TEMPO probe in isotropic and oriented at room temperature films of isotactic PP were studied /89/. The spin probe was introduced by two methods - before and after stretching of polymer. Changes in D r and Dt with X are shown in Fig. 8.5. Activation energies are shown in Table 8.7.

Dr, S’1

Dt, cm2*8

2 4 6 8

A,

Fig. 8.5. Dependence of Dt and Dr for spin probe TEMPO on X of PP.

209

Table 8.7 Activation energies and preexponential factors for rotational translational diffusion of the TEMPO probe in oriented PP /89/. * E t , kJ/mol D^, cm2/s 1 94.5 5xl05 Probe introduced before stretching 94.5 4.5 1.3xl07 7.0 134.5 5.0x10'° 10.0 — — Probe introduced after stretching 4.5 94.5 3.2xl05 7.0 94.5 2.0x105 10.0 — — X

♦ E r , kJ/mol 52.1

4.0xl016

46.2 42.8 36.1

4 .5 x l0 '5 l.OxlO15 9.4xl013

52.1 52.1 36.1

4 x l0 16 4 x l0 16 9.9xl013

and

D ?,s-'

Fig. 8.5 shows that both rotational and translational mobilities of the particle introduced into polymer before stretching decreases with X increase. This effect is followed by decreasing of the activation energy of particle rotation. For translational diffusion E* is constant at X < 4.5 and sharply increases at X = 7. Rotational diffusion coefficients for the probe, introduced into the sample after stretching, decrease only at X > 7, whereas D t decreases in X = 1 - 7 range, but not so abruptly than for the probes introduced into isotropic samples. The reasons of the differences observed in kinetic parameters are that additive localization regions depend on the stage of its introduction into polymer. The probes, introduced phase an isotropic polymer, are found distributed more uniformly over the amorphous part of the sample, than those introduced after orientation. In the last case smaller part of amorphous phase is accessible for particles, mainly those parts of it, open from the the surface and in which orientation degree of macromolecules is minimal. This is testified by Dt and Dr values higher, than in the case of the probe introduction into isotropic samples. Analysis of local concentrations of spin probes by ESR line widths has shown that the part of amorphous phase, accessible for low molecular particles, decreases from 39.6% in isotropic PP to 21.4% in the sample with X = 11.5 /92/. In the latter case, in accord with the IR-spectroscopy data, total fraction of the amorphous phase changes insignificantly (from 39.3 to 38.0%). One can suggest that interfibrillar zones are most accessible for an additive in oriented polymer.

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The ratio of diffusion coefficients Dt for isotropic (X = 1) and Dr oriented samples are shown in Table 8.8. Table 8.8 Ratio of diffusion constants and y-factors for spin probe TEMPO in PP /89/. -, ^ - x l O 18, cm2/s2 À Dr 1 6.4 Probe introduced after stretching 7 3.0 Probe introduced before stretching 7 1.5

y 1 2.1 4.3

Translational diffusion coefficient for particles introduced into isotropic sample much more abruptly decreases at orientation, than Dt of particles, introduced after orientation. This is apparently connected with the fact that under stretching of polymer most part of particles dislocates in closed zones, surrounded by crystallites. Desorption of these particles from the sample is retarded, thus the experimental values of D t are underestimated. In accord with Eq. 8.9, apparent Dt value are lower by y than diffusion coefficients inside localization zones. Table 8.8 shows that y is 2.1-fold increased in the sample with X = 7, and 4.3-fold - if the probe was introduced before stretching. It is reasonable that desorption of probes, introduced after stretching and localized in easily accessible zones, occurs much faster and y values are lower. It should also be noted that decrease of rotational diffusion coefficients under polymer orientation with the probe introduced supports a sufficient modification of intrafibrillar amorphous inerlayers. Simultaneous analysis of rotational mobility of particles and orientation degree of macromolecules when stretching PE and isotactic PP was perfomed in /86/. Orientation degree was controlled by the methods of birefringence and IR-spectroscopy. It was found that orientation degree for chains is higher in samples with narrow molecular-mass distribution, than at wide MMD. The reason is that at stretching of a sample with wide MMD low molecular fraction promotes development of plastic deformation. Correlation times of probes rotation much more sharply increase at stretching of samples with narrow MMD. These results show that the orientation degree of macromolecules is the main parameter determining mobility of probes under crystalline polymer drawing. Therefore, nonlinear dependences of molecular

211

mobility of particles on stretching degree, observed in many studies, are most likely connected with the absence of chain orientation at small X. Annealing of oriented polymers leads to smoothing differences in mobilities of particles comparing with isotropic sample /65, 82, 83, 91/. Complete disappearance of the differences in mobilities may be reached at increase of annealing temperature. The fraction of crystalline polymer zones, accessible for particles, does not change at annealing /92/. It was found that the effect of orientational drawing on permeability decreased gradually with an increase of the initial crystallinity degree of the sample /5/. The effect of orientation for samples with various supramolecular structures on mobility of additives was analyzed in detail in /86/ on the basis of the spin probe technique data. Characteristics of samples and correlation times of spin probe rotation are shown in Table 8.9. Table 8.9 Rotational correlation times r for PE samples differing in structure and elongation degree X /87/. Sample cp, % X,% p, cm3/g t at 297 K, ns 0 0.92 65 l.L D P E 0.3 0 0.932 45 0.4 2. HDPE 56 0 0.945 0.54 3. HDPE 0.962 1500 1.15 4. HDPE 2400 0.968 1.58 5. HDPE 0.996 97 0.79 6. HDPE - ECC 80 0.91 7. HDPE - ‘mats’ Notes: HDPE with extended chains crystallites (sample 6) was obtained by annealing of the sample 4 at T = 538 K under 700 MPa pressure during 2 hours; single crystal mats (sample 7) were obtained from solution in tetrachloroethylene at 351 K. The data in Table 8.9 display that in highly oriented samples 4 and 5 rotational mobility of particles is lower than in sample 6, prepared from sample 4, but possessing much higher density and crystallinity. It is evident that during annealing under high pressure formation of large crystallites is accompanied by a relaxation of stressed chains and an increase of the molecular mobility in amorphous regions. Amorphous regions in ‘mats’ occupy an intermediate position by rotation frequencies of particles. Nonoriented samples 1 and 2 demonstrate significantly higher mobility. Molecular mobility of additives in fibers is much lower, than in an isotropic sample /100, 101/. For example, rotation frequency of a spin probe in PP fibers is lower by an order of maltitude /100/. A very wide distribution

212

over dynamic parameters and important role of humidity can be attributed to the main features of particle dynamics in fibers. Superposition of lines in ESR spectrum was observed in investigations of spin probe rotational mobility in fibers of terylene and nylon /100/. This superposition testifies a large difference in mobility of particles. The authors concluded that this result is bound to a difference in the segmental mobilities in inter- and intrafibrillar amorphous zones. The mobility of the particles increases with water content / 101 /. t x 10» s

Fig. 8 .6 . Dependence of rotational correlation times for spin probe TEMPO on deformational degree of preoriented PP films with narrow (a) and wide (b) molacular mass distribution and different orientational degree X: 13.5(1), 13(2), 12 (3), 8.5 (4), 6 (5), 5 (6 ), and 3.5 (7)/90/. One more characteristic feature of oriented polymers is anisotropy of translational diffusion of particles. This effects consists in lower values of diffusion coefficient along sample stretching, than in perpendicular direction /6 , 8 , 41, 102/. For example, diffusion rate of stabilizer Irganox 1076 along HDPE orientation axis decreases, and in perpendicular direction it increases with X /102/. Diffusion anisotropy ¡increases with the orientation degree. In such highly oriented systems as fibers, molecules - benzidine dyes, for example - can diffuse in perpendicular direction only, but not along the fibers /103/.

213

One may name two main reasons, which lead to anisotropy of translational diffusion of particles - anisotropy of stressed chain segmental motion and specificity of orientation impermeable crystalline structures, which leads to a significantly higher complication of the particle translation path along stretching axis. The data on orientation of additives along stretching axis can be an argument in favour of the second factor. Orientation of additive particles was observed by the ESR technique /76, 84/. Orientation degree of particles is determined by X and also by asymmetry of the particle shape. By way of example, the fraction of oriented probes A in PE with X = 5 - 6 is very high (90 - 95%). A bit lower part of oriented states (~85%) is observed for the probe B. Moreover, in these investigations an abrupt decrease of nitroxide spin probes solubility was observed after PE orientation /84/. Similar result was obtained in investigation on tolyene sorption in oriented PE /95/. Samples with X = 10 displayed a 4-fold decrease of particle concentration comparing with isotropic sample. As mentioned in Chap. 4, asymmetric rod-like particles translate along the large axis. Since such particles are oriented along the stretching axis, their motion should be faster in this direction. The fact that high diffusion coefficients are observed in perpendicular direction testifies existence of substantial structural hindrances for their motion along the stretching axis. In other words, it is more likely that structural rather than dynamic factor forms the base of translational motion anisotropy of particle diffusion. The data shown above represent a comparative analysis of molecular mobility of particles in polymer before and after orientational stretching. However, they give no answer to the question about changes in molecular mobility during loading. The search for this answer was performed in /74, 85, 86 , 90/. Rotational mobility of spin probe TEMPO during loading of preoriented films of isotactic PP and HDPE was studied. It is known that loading of polymers is accompanied by conformational transitions in the amorphous phase /49/. Loaded oriented polymers (PE, PETPh, PA) display a decrease of the part of gosh-conformations and an increase of the fraction of trans-conformations of macromolecules /104/. The initial set of conformations restores after removal of elastic load. Moreover, high stretching loads may cause additional irreversible orientation of polymer. The results obtained have shown that the type of dependence of correlation time x for probe rotation on the deformation degree e is dissimilar to samples with different molecular-mass distribution and different X (Fig. 8 .6 ). In samples with narrow MMD and high orientation degree (7. > 8 for PE and X> 13 for PP) rotation frequencies grow linearily with e. A minimum is observed on curves of x = f(s) dependence at loading samples with lower values of X. The authors explain the observed effects by changes in density of amorphous interlayers under loading. The loosening of the structure is

214

observed on the initial stage of loading of low-oriented samples. It is accompanied by an increase of segmental mobility. Further increase of deformations causes a sealing of amorphous zones and a rigidity increase of the chains under load. Highly oriented samples are characterized by a high amount of extended chains, so the load is distributed more uniformely over the macromolecules. This causes an increase of the polymer density even on the initial stage of loading. Loosening of low-oriented and nonoriented polymers on the initial stage is supported by the data of the X-ray-structural analysis and endothermal effect. It should be noted that variation of deformation procedure exerted no effect on x = f(e) dependence. These dependences were the same under creep deformation (a = const), stepwise loading and stress relaxation loading (e = const). The same characteristics, as for low-oriented polymers with narrow MMD, were observed under loading of samples with wide MMD for all 1 values, i.e. two parts of the x = f(e) curve. This is explained by the promotion of plastic deformation of the sample by high concentration of low molecular fraction. Influence of other types of deformation - uniaxial compression and shift - on mobility of particles is not studied well. Attention should be paid to the work on studying combined effect of high pressure and shift on rotational and translational dynamics of a spin probe in HDPE /105/. The films and powders prepared from the polymer were compressed with a Bridgeman anvils with 2 GPa pressure, and then anvils were turned by 1000°. It is known /106, 107/ that treatments of polymer under such conditions causes plastic flow, accompanied by structural changes. Intensive degradation of crystalline phase, accompanied by decrease of crystallinity degree and sizes of crystallites, belongs to these changes. Polymorphous transformations are also possible. In particular, PE lattice transforms from orthorombic into monoclinic type. The number of defects of various size is significantly increased. Amorphous phase is also seriously changed: large residual tensions are accumulated, and packing density increases. On the other hand, microcavities occur in amorphous zones, and some fraction of chain break. As a result, the amorphous phase becomes heterogeneous: zones with higher and lower density appear, comparing with the amorphous phase of the initial sample. Fig. 8.7 and Table 8.10 presents the data obtained by the spin probe technique.

215

Table 8.10 Activation energies and preexponential factors for probe TEMPO rotation and translation in HDPE treated under HP and shear deformation at room temperature /105/. Sample Untreated Treated:

E r ±2, kJ/mol 32

lg D ?,s·' 14.0

E t ±17, kJ/mol 67

lgD {\ cm2/s 2.1

26 12.6 88 4.6 24 138 12.6 12.0 Note: sample 1 - the probe introduced before treating; sample 2 - the probe introduced after treating. 1 2

D», cn^/s

Fig. 8.7. Temperature dependences of rotational (a) and translational (b) diffusion coefficients of spin probe TEMPO before (1) and after (2) shear deformation of PE samples under P = 2 GPa /105/. It follows from Fig. 8.7 that plastic flow leads to a decrease of rotational and translational diffusion coefficients for particles. The value of the effect depends on the stage of probe introduction - changes are more serious for the samples with the probe introduced before treatment. The results are quite similar to those obtained in investigations of particle mobility in oriented PP. They are most likely stipulated by the same reasons: particles introduced after treatment enter the easier accessible defect regions. This is confirmed by the estimation of the complication factor for diffusion path y. The calculation was made in the same manner as for oriented polymers: changes of molecular mobility in amorphous phase were controlled by change

216

Dr of D r, and changes of the —- values were attributed to changes of y. The Dt results obtained have shown that y is higher for the samples with probe introduced before treatment, than for those with probe introduced after the treatment (y = 8 and y = 3.6, respectively; T = 313 K). Activation energies for rotational mobility of particles are weakly decreased, and those of translational mobility grow due to polymer treatment. The reasons are apparently in defects accumulated in amorphous interlayers. The size of these defects is comparable or exceeds the activation volumes, required for rotation of particles, but is lower than V* for their * translation. The sharp increase of E t , observed for diffusion of particles introduced after treatment, is most likely connected with relaxation and repacking of loosen regions of the amorphous phase under sample heating. The results obtained display the way of molecular mobility change for polymer as a result of shear deformation under one-axis compression. However, the question about changes of particle mobility during plastic flow is also of interest. There were no direct measurements of this effect. Thus, kinetic studies have shown that the rate of diffusional chemical reactions (free radical recombination) sharply increases during the flowing /106, 107/. From all this, a conclusion can be made about acceleration of translational diffusion under these conditions.

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Regions of PP in Stressed Conditions, Vysokomol. Soed., 1980, 22A(4), 868-875. (Rus) 91. Kovarskii A.L., Saprygin V.N., Rapoport N.Ya., Diagnostics of Microcracks in Oriented PP by Spin Probe Technique, Mech. Komp. Mater., 1979, 2, 351-353. (Rus) 92. Krisyuk B.E., Rogov Yu.N., Griva A.P., and Denisov E.T., Effect of Orientation and Annealing on the Fraction of PP Films Accessible for Spin Probes, Vysokomol. Soed., 1982, 24B, 634-637. (Rus) 93. Mitchenko Yu.I., Gribanov S.A., Dyachkov A.N., and Aizenstain E.M., Molecular Mobility in PETPH under Orientation, Vysokomol. Soed., 1975,17B, 547-550. (Rus) 94. Kovarskii A.L., Wasserman A.M., Buchachenko A.L., Study of PETPh Crystallization Kinetics by Probe Technique, Vysokomol. Soed., 1970, 128,211-214. (Rus) 95. Ng H.C., Leung W.P., and Choy C.L., Sorption and Diffusionof Toluene in Isotropic and Oriented Linear PE, J. Polym. Sci., Polym. Phys. Ed., 1985,23,973-989. 96. Peterlin A., Williams J.L., and Stannett V., Sorption of Organic Vapours into Drawn and Undrawn PE, J. Polym. Sci., A-2, 1967, 5,957-963. 97. Peterlin A., Dependence of DiffusiveTransport on Morphology of Crystalline Polymers, J. Macromol. Sci., 1975,11B, 57-87. 98. Peterlin A., Drawing and Annealing of Fibrous Materials, J. Appl. Phys., 1977,48, 4099-4108. 99. De Candia F., Perullo A.S., Vittoria V., and Peterlin A., Mechanical and Transport Properties of Drawn LDPE, J. Appl. Polym. Sci., 1983, 28, 1815-1817. lOO.Stryukov V.B., Rozantsev E.G., and Kashlinskii A.I., Investigation of Oriented Polymers by Means of Spin Probe Method, Doklady A N SSSR, 1970,190, 895-897. (Rus) 101. Tormala P., Colloid and Polym. Sci., 1977,22, 209-211. 102. Moisan J.Y., Diffusion des Additives du Polyethylene, Eur. Polym. J., 1981,17,857-864. 103. Frey-Wissling A., Anisotropic Diffusion of Direct Dyes in Fibres, J. Polym. Sci., 1947,2,314-317. 104. Pakhomov P.M., Shermatov M., Korsukov V.E. et al., Connection of Conformational Transitions with Large Period of Deformation in PE, Vysokomol. Soed., 1976,18A, 132-139. (Rus) 105. Zhorin V.A., Saprygin V.N., Barashkova I.I. et al., Molecular Mobility in PE after Plastic Flow under Pressure, Vysokomol. Soed., 1989, 31A, 1311-1315. (Rus) 106. Zhorin V.A., Processes Acompanying Plastic Flow under High Pressure in Polymers, Vysokomol. Soed., 1994,36, 559-579. (Rus)

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107.Zharov A.A., Reactions of Solid Monomers and Polymers under Shear Deformation and High Pressure, In: High Pressure Chemistry and Physics o f Polymers, Ed. Kovarskii A.L., CRC-Press, Boca Raton, 1994, 265-300.

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Chapter 9

Dynamics of Water in Polymers

The problem “water in polymers” is one of the most actively and fruitfully developing problems of the physical chemistry of polymers. We do completely agree with Rowland’s opinion /1/ that “the unabating for more than 80 years interest to the nature of the interaction between water and polymers has been due to many causes, two of them being considered as the main ones”. The first is associated with its practical significance. The polymer materials have come to a very extensive use as protective coatings, adhesives, insulating shells, packing materials, artificial leather, membranes, medicinal materials for different purposes, etc. Now, water is the medium to be most frequently handled when using these materials and the articles manufactured of them under real conditions. By the way, in most cases of reprocessing, testing, and storing materials, not merely polymers, but their aqueous solutions ought to be spoken of1. Therefore, the data on the translational mobility of the molecules of water, coefficients of solubility and permeability, sorption isotherms, and diagrams of state of polymer-water systems are necessary both for predicting the behavior of these materials in humid media, for assessing their operating capacity, and for selecting the specific materials. Here, it should be noted that the number of polymers and polymerbased composites that are synthesized, applied, and developed at various polymer-material science centers, considerably exceeds the number of polymers, systems, and materials, for which the aforementioned parameters have already been defined, the mechanisms of diffusion and dissolution of water described, its states identified, and the specific behavior of the functional groups of macromolecules during their interaction with the molecules of water revealed. All this enables us to state that the task of accumulation and generalization of the experimental data on water-polymer systems will always be of current interest. 1 The solutions of water in polymers are thereby meant, resulting from the contact of polymers with water or its vapour.

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The second cause of the interest of investigators to these systems is associated with an “ unusual - specific or abnormal”, as is generally considered, character of variation of diffusion and sorption parameters of polymer-water systems dependent on the humidity of the surrounding medium 111. This primarily concerns the negative concentration dependence of interdiffusion coefficients, the closeness of miscibility heats to zero value, the formation of associates and clusters by the molecules of water in polymer media. All these anomalies are conventionally explained by the specific interactions facilitated by the carbon bonds between the molecules of water and polar groups of polymers. In our opinion, it would be unjustified to isolate water as a specific diffusant in polymers, since this should “compel” the investigators to abstract from the general regularities of diffusion in polymer-low molecular substance systems, emphasize the behavior of water, thus hindering one from revealing the specific features inherent to the water-polymer systems at the background of the general physicochemical pattern of the interaction between components in polymer systems. At present, it is possible to state that in the late 1980s a fairly integral concept on the description of the translational mobility of the molecules of water dissolved in polymer has been formed in the practice of the investigation of diffusion in water-polymer systems. Let us note that from the physicochemical viewpoint this concept is a fairly general one, and is applied to examine the inter- and self diffusion of gases, solvents, plasticizers, and oligomers in polymers 121. Underlying this concept are notions on the phase equilibrium in the polymer-water systems, the structure of the aqueous solutions in polymers, and the nonequilibrium state of the polymer diffusion medium. At the first glance, it may seem puzzling that, when speaking about the translational mobility of the molecules of water dissolved in polymers, we have given priority to the notion on the phase equilibrium of these systems. However, it should be borne in mind that the translational mobility, or, more exactly, diverse forms of its manifestation, are associated with and reflect just the structure of solutions and its transformation with temperature, composition, nature, and the thermal prehistory of a polymer.

9.1 PH ASE EQUILIBRIA OF POLYMER-W ATER SY STE M S

In accordance with a formal feature - the quantity of absorbed water (C max ) under normal conditions - all polymer materials are subdivided into

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three groups /31: hydrophilic (C max > 10 wt.%), hydrophobic (C max < 4 wt.%), and moderately absorbing (4 < C max < 10 wt.%). Such a classification is obviously conventional, since the sorption capacity of polymers with regard to low molecular substances in general and to water in particular substantially depends on temperature, pressure, the thermal prehistory of materials, the presence of admixtures in sorbate, of various functional additives, and may serve merely as its technical and operational characteristic. It would be more advisable to analyze the sorption-diffusion behavior of water in polymers from the standpoint of the phase equilibrium of these systems /2, 4/. The generalized diagrams of phase state are known to characterize the position in the temperature-concentration field of the following coexistence regions: • • • •

different phases - amorphous, crystalline, liquid-crystalline; labile and metastable solutions; phase inversion zones; glassy, high-elastic, and viscofluid states of the system.

Fig. 9.1. Diagrams of the phase state of the systems: (a) polyoxypropylenediol-water; molecular masses of polyoxypropylenediol: 1 - 480, 2 - 700, 3 - 1000, 4 - 1500; (b) polyvinylacetatemethanol; (c) polyvinyl alcohol-water. Fig. 9.1 exemplifies several types of the diagrams of phase states of binary polymer-water systems. It is obvious that they do not essentially differ

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from the diagrams of state of other polymer-solvent systems /5/, and in the case of amorphous lamination these diagrams may have both the lower critical miscibility temperature (LCMT) and the higher critical miscibility temperature (HCMT). For partly crystalline polymers, the diagrams of state are characterized by a combination of the crystalline and amorphous equilibria. For most of the investigated systems, the curves of vitrification temperatures for polymers and their solutions run, as a rule, below those of critical miscibility temperatures. Therefore, it is not surprising at all that water in some areas of the diagrams and, accordingly, at some humidity values2 causes amorphization of a polymer, whereas at other values it serves as a kinetic stimulator of the processes of devitrification, crystallization or polymorphous transformations. There may also be cases where it forms inclusion compounds with the fragments of macromolecules - molecular complexes. Similar effects are also observed in other polymer systems 161. From the standpoint of statistical thermodynamics of solutions, the diagrams of state in the area of compositions corresponding to one-phase state of the system should be supplemented by the lines confining the areas of existence of different associative structures 121: heterophase fluctuations, clusterization of molecules, and the filling of accessive active centers of macromolecules by the molecules of water. Note that essentially just these zones of the diagrams of phase state are experimentally investigated, and the information thus obtained is used to characterize the state and translational mobility of the molecules of water in polymers, and, hence, to establish the mechanism of interaction of components. Now, the binodals and liquidus lines are determined by direct physicochemical methods /5/; while the concentration boundaries of these areas are conventionally calculated from the isotherms of the sorption of water by polymers /1, 2/. To this end, for example, when plotting the uppermost boundary of the saturation area (C sa{) of the accessible polar groups of macromolecules by the molecules of water, the experimental isotherms (C), measured at different temperatures, are approximated within the framework of the “double sorption” model /7, 8/ by a combination of Langmuir isotherms Cl

= c sat

bp 1 + bp ’

(9.1)

2 Note that, since the composition of water-polymer systems is connected

through the sorption isotherm with the activity of water vapour, the diagrams of phase state can be plotted in the coordinates “temperature-humidity of the surrounding medium”.

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and by that of Flory-Huggins isotherms C 2 = - g - · Pwater .where

T g l), i.e., the polymer-water system is found in a highly elastic state, the value of D v -cpi should be *

calculated with Eq. 9.10 and the reference data of D j^ · Here f2 and f, are the fractions of the free volumes of polymer and water at ^exp- And finally, when ( (Pl,cr (Tg,l < Texp < Tg,2)i that is> when the transition is found in the middle region of the sorption isotherm, then, in determining D v -cpj in the region of (p /P s) < (p /p s )cr > w'^ possible to avail oneself of the assumptions as made above for the vitrified state of the system, whereas in the region of ( p /p s ) > (p /P s )c r . of the above

247

assumptions made for the highly elastic state of the system. In the latter case, > tpi —