Characterization of Highly Cross-linked Polymers 9780841208247, 9780841210714, 0-8412-0824-7

Table of contents :
Title Page......Page 1
Half Title Page......Page 3
Copyright......Page 4
ACS Symposium Series......Page 5
FOREWORD......Page 6
PdftkEmptyString......Page 0
PREFACE......Page 7
1 Formation and Properties of Polymer Networks Experimental and Theoretical Studies......Page 8
Pre-Gel Intramolecular Reaction......Page 9
Intramolecular Reaction and Gelation......Page 11
Network Properties......Page 14
Theoretical Correlations between Gel Point and Shear Modulus......Page 20
Literature Cited......Page 26
2 Computer Simulation of End-linked Elastomers Sol-Gel Distributions at High Extents of Reaction......Page 28
ALGORITHM......Page 29
EFFECT OF FINITENESS......Page 30
SOL FRACTIONS......Page 31
CONSTITUENTS OF THE SOL......Page 33
NETWORK IMPERFECTIONS......Page 35
SUMMARY......Page 37
Literature Cited......Page 39
3 Rheological Changes During the Copolymerization of Vinyl and Divinyl Monomers......Page 40
Experimental......Page 41
Theory......Page 42
Results and Discussion......Page 48
Literature Cited......Page 52
4 Elastomeric Poly(dimethylsiloxane) Networks with Numerous Short-Chain Segments......Page 54
Short-Chain Unimodal PDMS Networks......Page 55
Bimodal PDMS Networks......Page 56
Literature Cited......Page 60
5 Light Scattering of Randomly Cross-linked Polystyrene......Page 62
Theoretical Background......Page 63
Experimental......Page 66
Results and Discussion......Page 67
Literature Cited......Page 75
6 Correlation Networks in Polymeric Materials Determined by Small-Angle Neutron Scattering......Page 77
Theory......Page 78
Experimental......Page 80
Correlation Networks......Page 81
The Delta Deuterated Fraction Method......Page 83
Results......Page 84
Discussion......Page 91
Acknowledgments......Page 93
Literature Cited......Page 94
7 Carboxyl-Terminated Butadiene-Acrylonitrile-Modified Epoxy Resin and Its Graphite Fiber-Reinforced Composite Morphology and Dynamic Mechanical Properties......Page 96
Experimental......Page 99
Results and Discussion......Page 101
Literature Cited......Page 112
8 Mechanical Behavior of Some Epoxies with Epoxidized Natural Oils as Reactive Diluents......Page 114
Materials......Page 115
Polymerization......Page 117
Viscoelastic Response......Page 119
Ultimate Mechanical Behavior......Page 125
Literature Cited......Page 128
9 Volume Recovery in Aerospace Epoxy Resins Effects on Time-Dependent Properties of Carbon Fiber-Reinforced Epoxy Composites......Page 130
Experimental......Page 131
Results and Discussion......Page 135
Stress-Strain Analysis......Page 136
Dynamic Mechanical Analysis......Page 139
Differential Scanning Calorimetry......Page 143
Thermal Mechanical Analysis......Page 148
Moisture Sorption Kinetics......Page 155
Conclusions......Page 162
Literature Cited......Page 167
10 Structure and Fracture of Highly Cross-linked Networks......Page 170
Cross-link Density......Page 171
Fracture......Page 172
Experimental......Page 173
Results and Discussion......Page 181
Acknowledgments......Page 186
Literature Cited......Page 187
11 Fractographic Effect of Glassy Organic Networks......Page 189
Fractography of Brittle Materials......Page 190
Phenol-Formaldehydes......Page 191
Some Other Glassy Networks......Page 194
Methacrylate Networks......Page 195
Mechanism of Formation of Stries......Page 198
Localized Plastic Deformation and Strength......Page 202
Concluding Remarks......Page 203
Literature Cited......Page 206
12 Peroxide Cross-linked Natural Rubber and cis-Polybutadiene Characterization by High-Resolution Solid-State Carbon-13 NMR......Page 208
Carbon-13 NMR Pulse Sequence......Page 210
Fourier Transform Infrared Analysis of the Curing Process......Page 212
EXPERIMENTAL......Page 213
Instrument Analysis......Page 214
RESULTS AND DISCUSSION......Page 215
LITERATURE CITED......Page 234
13 Carbon-13 Magic Angle NMR Spectroscopic Studies of an Epoxy Resin Network......Page 236
RESULTS AND DISCUSSION......Page 237
Literature Cited......Page 241
14 NMR Kinetic Analysis of Polyethylene-Peroxide Cross-linking Reactions......Page 243
Experimental......Page 246
Results......Page 247
Literature Cited......Page 257
15 Degradation Chemistry of Primary Cross-links in High-Solids Enamel Finishes Solar-Assisted Hydrolysis......Page 259
Experimental Section......Page 260
Results......Page 261
Discussion......Page 269
Literature Cited......Page 271
16 IR Spectroscopic Studies of Degradation in Cross-linked Networks Photoenhanced Hydrolysis of Acrylic-Melamine Coatings......Page 272
Experimental......Page 274
Results and Discussion......Page 276
Conclusion......Page 282
Literature Cited......Page 285
17 Electron Spin Resonance (ESR) of Photodegradation in Polymer Networks Photoinitiation Rate Measurement by Nitroxide Termination......Page 286
ESR and Free Radical Quantification......Page 287
Experimental......Page 288
Kinetics of Nitroxide Decay......Page 291
Results......Page 294
Discussion......Page 296
Conclusion......Page 299
Literature Cited......Page 300
18 Highly Cross-linked CR-39 Polycarbonate and Its Degradation by High-Energy Radiation......Page 302
Results and Discussion......Page 304
Conclusions......Page 307
Literature Cited......Page 311
Author Index......Page 312
A......Page 313
C......Page 314
Ε......Page 315
G......Page 316
M......Page 317
Ρ......Page 318
R......Page 319
Τ......Page 320
Y......Page 321

Citation preview

Characterization of Highly Cross-linked Polymers

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

ACS

SYMPOSIUM

SERIES

Characterization of Highly Cross-linked Polymers S. S. Labana, EDITOR Ford Motor Company

R Ford Motor Company

Based on a symposium sponsored by the Division of Organic Coatings and Plastics Chemistry at the 185th Meeting of the American Chemical Society, Seattle, Washington, March 20-25, 1983

American Chemical Society, Washington, D.C. 1984

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

243

Library of Congress Cataloging in Publication Data Characterization of highly cross-linked polymers. (ACS symposium series, ISSN 0097-6156; 243) Includes papers presented at the Symposium on Highly Cross-linked Polymers sponsored by the Division of Organic Coatings and Plastic t th 185th meeting of the American Chemica Wash., March 20-25, 1983." Bibliography: p. Includes indexes. 1. Polymers and polymerization—Congresses. I. Labana, Santokh S., 1936. II. Dickie, R. Α., 1940. III. Symposium on Highly Cross-linked Polymers (1983: Seattle, Wash.) IV. American Chemical Society. Division of Organic Coatings and Plastics Chemistry. V. Series. QD380.C45 1984 ISBN 0-8412-0824-7

547.7

83-25733

Copyright © 1984 American Chemical Society All Rights Reserved. The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner's consent that reprographic copies of the chapter may be made for personal or internal use or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to copying or transmission by any means—graphic or electronic—for any other purpose, such as for general distribution, for advertising or promotional purposes, for creating a new collective work, for resale, or for information storage and retrieval systems. The copying fee for each chapter is indicated in the code at the bottom of the first page of the chapter. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by A C S of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of any right or permission, to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. Registered names, trademarks, etc., used in this publication, even without specific indication thereof, are not to be considered unprotected by law. PRINTED IN THE UNITED STATES

OF

AMERICA

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

ACS Symposium Series M. Joan Comstock, Series Editor Advisory Board Robert Baker

y

U.S. Geological Survey

Carnegie-Mellon University

Martin L. Gorbaty

Theodore Provder

Exxon Research and Engineering Co.

Glidden Coatings and Resins

Herbert D. Kaesz

James C. Randall

University of California—Los Angeles

Phillips Petroleum Company

Rudolph J. Marcus

Charles N. Satterfield

Office of Naval Research

Massachusetts Institute of Technology

Marvin Margoshes

Dennis Schuetzle

Technicon Instruments Corporation

Ford Motor Company Research Laboratory

Donald E. Moreland USDA, Agricultural Research Service

W. H. Norton J. T. Baker Chemical Company

Robert Ory USDA, Southern Regional Research Center

Davis L. Temple, Jr. Mead Johnson

Charles S. Tuesday General Motors Research Laboratory

C. Grant Willson IBM Research Department

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

FOREWORD The ACS SYMPOSIUM SERIES was founded in 1974 to provide

a medium for publishing symposia quickly in book form. The format of the Series parallels that of the continuing ADVANCES IN CHEMISTTIY SERIE

papers are not typese mitted by the authors in camera-ready form. Papers are reviewed under the supervision of the Editors with the assistance of the Series Advisory Board and are selected to maintain the integrity of the symposia; however, verbatim reproductions of previously published papers are not accepted. Both reviews and reports of research are acceptable since symposia may embrace both types of presentation.

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

PREFACE have long been an important class of materials and are used in a diverse assortment of applications including organic coatings,fiber-reinforcedplastics, elastomers, and adhesives. Characterization of these materials has always been difficult, especially for the more highly cross-linked materials general intractability. Considerabl progres year developing theoretical approaches to the description of the molecular structure of cross-linked polymers; several chapters describe the recent progress in this important area. Light scattering and rheological characterization techniques have been applied to cross-linking systems in the pre-gel state. Macroscopic mechanical characterization of cross-linked polymers has been the subject of many investigations; fracture behavior, relationships between molecular structure, morphology, and mechanical properties, and the dependence of properties on thermal history are discussed by several authors. Not all network formation occurs through formation of chemical bonds; in one chapter in this volume, neutron scattering results suggest the formation of correlation networks in certain polymer blends. Characterization of the chemical structure of highly cross-linked polymers, and of the chemical changes that accompany degradation processes, relies on spectroscopic methods. Solid-state nuclear magnetic resonance techniques have the potential to allow a more detailed characterization than before possible of the chemical environment and structure of chemical crosslinks in elastomers and thermoset epoxies. Degradation processes in crosslinked systems have been studied by using infrared spectroscopy, solid-state NMR, and electron spin resonance. It is a pleasure to acknowledge the support of the Ford Motor Company. We also wish to thank A. Oslanci and M. Dvonch for their secretarial assistance. Finally, sincere thanks to the authors who have made this volume possible through their hard work and cooperation. CROSS-LINKED POLYMERS

S. S. LABANA R. A. DICKIE

Ford Motor Company Dearborn, Michigan November 3, 1983 ix In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

1 Formation and Properties of Polymer Networks Experimental and Theoretical Studies J. L STANFORD, R. F. T. STEPTO, and R. H. STILL Department of Polymer Science and Technology, The University of Manchester Institute of Science and Technology, Manchester, M6O 1QD, England

Experimental results on reactions forming tri- and tetrafunctional polyurethan ester networks ar particula ation of intramolecular reaction and its effect on shear modulus of the networks formed at complete reaction. The amount of pre-gel intramolecular reaction i s shown to be significant for non-linear polymerisations, even for reactions in bulk. Gel-points are delayed by an amount which depends on the dilution of a reaction system and the functionalities and chain structures of the reactants. Shear moduli are generally markedly lower than those expected for the perfect networks corresponding to the various reaction systems, and are shown empirically to be closely related to amounts of pre-gel intramolecular reaction. Deviations from Gaussian stress-strain behaviour are reported which relate to the low molar-mass of chains between junction points. Finally, a rate theory of random polymerisation i s described which enables the moduli of networks to be predicted from the molar mass, functionality, chain structure and initial dilution of the reactants used for network formation. This paper presents a survey o f published and more recent work on c o r r e l a t i o n s between network p r o p e r t i e s and r e a c t a n t s t r u c t u r e s and r e a c t i o n c o n d i t i o n s , and extends the work presented i n recent p u b l i c a t i o n s (1 2 2)· ^ r e a c t i o n systems used have been p o l y oxypropylene (POP) t r i o l s o r t e t r o l s and mixtures o f d i o l s and t r i o l s o f v a r i o u s molar masses r e a c t i n g w i t h d i i s o c y a n a t e s ( t o g i v e polyurethanes) o r d i a c i d c h l o r i d e s ( t o g i v e p o l y e s t e r s ) . Systems have been chosen so t h a t l i k e groups had equal r e a c t i v i t i e s and r e a c t i o n s have been c a r r i e d out i n bulk and a t v a r i o u s d i l u t i o n s i n i n e r t s o l v e n t s using equimolar amounts o f the d i f f e r e n t r e a c t i v e groups. E x p e r i m e n t a l l y , emphasis has been placed on the extent t o which p r e - g e l i n t r a m o l e c u l a r r e a c t i o n n e

9

9

0097-6156/ 84/0243-0001 $06.00/0 © 1984 American Chemical Society In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

2

HIGHLY CROSS-LINKED POLYMERS

and the consequent delay i n the g e l p o i n t beyond the i d e a l , Flory-Stockmayer g e l p o i n t (4,5) d e f i n e s the p h y s i c a l p r o p e r t i e s of the networks formed at complete r e a c t i o n . Intramolecular r e a c t i o n can introduce e l a s t i c a l l y i n e f f e c t i v e loops i n t o a rubbery network. In g e n e r a l , loops produce the opposite e f f e c t s on p h y s i c a l p r o p e r t i e s to those expected from entanglements. T h e o r e t i c a l approaches are o u t l i n e d which attempt to account f o r i n t r a m o l e c u l a r r e a c t i o n i n terms o f reactant s t r u c t u r e ( f u n c t i o n ­ a l i t y , molar mass, and chain s t r u c t u r e ) and r e a c t i o n c o n d i t i o n s (concentrations o f r e a c t a n t s ) . The approaches a l l o w the p r e d i c t i o n o f g e l p o i n t s accounting f o r p r e - g e l i n t r a m o l e c u l a r r e a c t i o n . A d d i t i o n a l l y , account of p r e - g e l and p o s t - g e l i n t r a ­ molecular r e a c t i o n allows the p r e d i c t i o n o f shear modulus at complete r e a c t i o n . Pre-Gel Intramolecular

Reactio

Previous studies(6») have shown how the number f r a c t i o n of r i n g s t r u c t u r e s formed during i r r e v e r s i b l e l i n e a r random polymerisa­ t i o n s l e a d i n g to polyurethanes may be measured. The work has been extended(7,8) t o n o n - l i n e a r polyurethane formation using hexamethylene diisocyanate(HDI) and POP t r i o l s . For n o n - l i n e a r p o l y m e r i s a t i o n s , i t i s found t h a t the number of r i n g s t r u c t u r e s per molecule(Np) i s always s i g n i f i c a n t , even i n bulk r e a c t i o n s . For example, Figure 1 shows N versus extent o f r e a c t i o n ( p ) , f o r l i n e a r and n o n - l i n e a r polyurethane-forming bulk r e a c t i o n s w i t h approximately equimolar concentrations o f r e a c t i v e groups(2,,6^,1). The much l a r g e r values o f N i n the n o n - l i n e a r compared with the l i n e a r p o l y m e r i s a t i o n are due t o the l a r g e r number o f o p p o r t u n i t i e s per molecule f o r i n t r a m o l e c u l a r r e a c t i o n i n the former type o f p o l y m e r i s a t i o n . However, the other f a c t o r s i n f l u e n c i n g i n t r a ­ molecular r e a c t i o n i n the two systems, p a r t i c u l a r l y the number of bonds(v) i n the chain forming the s m a l l e s t r i n g s t r u c t u r e p r e d i c t more i n t r a m o l e c u l a r r e a c t i o n i n the l i n e a r system. A d e t a i l e d d i s c u s s i o n of these f a c t o r s has been given elsewhere(2.). It should be noted t h a t i t i s not p o s s i b l e t o reduce the number of r i n g s t r u c t u r e s formed i n such r e a c t i o n systems as the amounts o f i n t e r m o l e c u l a r r e a c t i o n r e l a t i v e to i n t r a m o l e c u l a r r e a c t i o n are at a maximum f o r r e a c t i o n s i n bulk. The g e l p o i n t o f the n o n - l i n e a r system shown i n Figure 1 was at ρ = 0.765 compared with the value of 0.707 expected i n the absence o f i n t r a m o l e c u l a r r e a c t i o n . Thus, although ρ at g e l i s only about Q% higher than expected, N = 0.3 at ρ = 0.765, showing t h a t at g e l about one molecule i n three contained a r i n g structure. Such r i n g s t r u c t u r e s or loops can have marked e f f e c t s on the p r o p e r t i e s of networks formed at complete r e a c t i o n ( 1 , 2 , 9 12). Developments i n the t h e o r e t i c a l aspects of the work, a l l o w i n g p r e d i c t i o n of N , the g e l p o i n t , and the shear moduli of networks formed at complete r e a c t i o n are presented i n the l a s t s e c t i o n of the present paper. r

r

r

r

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

1.

STANFORD ET

3

Formation and Properties of Networks

AL.

ΟΛ

0.3

-

non-linear HDI + L G 5 6

0.2

-

0.1

linear

—

0

HDI+PEG200

C\

Ο

0.2

0.4

0.6

0.8

1.0

Ρ Figure 1. Number o f r i n g s t r u c t u r e per molecule (N ) as a f u n c t i o n o f extent o f r e a c t i o n ( p ) f o r l i n e a r and n o n - l i n e a r polyurethane forming r e a c t i o n s i n bulk w i t h approximately equimolar concentrations o f r e a c t i v e groups, r = [ N C O l / [ p H l S 1) (6,7). r

0

0

0 - l i n e a r p o l y m e r i s a t i o n , HDI + p o l y ( e t h y l e n e g l y c o l ) (PEG200) at 70OC,[NCOj = 5.111 mol k g - , [ O H ] = 5.188 mol kg- ; number-average o f bonds i n chain forming s m a l l e s t r i n g s t r u c t u r e (v) = 25.2. • - n o n - l i n e a r p o l y m e r i s a t i o n , HDI + POP t r i o l (LG56) at 7QoC,[NCqIo = 0.9073 mol k g ' S J 0 H j = 0.9173 mol k g - ; v= 115. Reproduced w i t h permission, from Ref. 2. Copyright 1982, American Chemical S o c i e t y . 1

0

0

1

o

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

4

HIGHLY CROSS-LINKED POLYMERS

Intramolecular Reaction and G e l a t i o n An expression has been d e r i v e d ( 4 ) f o r the extent o f r e a c t i o n a t g e l a t i o n i n R A 2 + RBf random(13) or condensation p o l y m e r i s a t i o n which accounts more completely than e a r l i e r expressions(14-16)for i n t r a m o l e c u l a r r e a c t i o n . I t may be rearranged t o g i v e o ( f - l ) = (1 - X ' c

a b

)

(1)

2

Here, a = P Pu> where p and p^ are the extents o f r e a c t i o n o f A and Β groups at g e l , r e s p e c t i v e l y , and X'ab * ring-forming parameter. When λ ^ = 0, the c l a s s i c a l Flory-Stockmayer c o n d i ­ t i o n f o r g e l a t i o n i s obtained. X'g^ i s p r e d i c t e d (4) to be p r o p o r t i o n a l to the d i l u t i o n o f a r e a c t i o n system, to i n c r e a s e w i t h f u n c t i o n a l i t y , an w i t chai s t i f f n e s molar mass o f r e a c t a n t s c

a

a

s

a

1

X

c

'ab

c

2

= int/ ext

s

where c i n t ( e r n a l ) * the c o n c e n t r a t i o n o f groups which can r e a c t i n t r a m o l e c u l a r l y w i t h a given group on a molecule and c t(ërnal) i s the c o n c e n t r a t i o n o f groups which can r e a c t i n t e r m o l e c u l a r l y w i t h the same group. ex

c

with

int

=

(f-2)Pab.(l,3/2) 2

Pab

(3)

3/2

= (3/2iTvb ) /N

(4)

where ν i s the number o f bonds i n the chain t h a t can form the s m a l l e s t r i n g , w i t h b i t s e f f e c t i v e bond l e n g t h , d e f i n e d such t h a t i t s mean-square end-to-end d i s t a n c e equals v b , and Ν i s the Avogadro constant. The p o s s i b i l i t y o f forming r i n g s o f a l l s i z e s i s accounted f o r by φ(1,3/2), w i t h 2

φ(1,3/2) =

Σ i=l

lV

3 / 2

=

2.612

(5)

Values o f c t have to be chosen a r b i t r a r i l y s i n c e λ' i s assumed to be constant f o r a given system. In p r a c t i v e , the two extreme experimental v a l u e s , c t = c + c and c t = c + c^ r e p r e s e n t i n g the i n i t i a l and g e l - p o i n t c o n c e n t r a t i o n s , are used i n the t h e o r e t i c a l treatment d e s c r i b e d ( 4 ) . The dependence o f X b on f u n c t i o n a l i t y i s allowed f o r by the f a c t o r (f-2) i n Equation 3. S i m i l a r l y , chain s t i f f n e s s and molar mass of r e a c t a n t s are allowed f o r by the f a c t o r ( v b ) '2 i Equation 4 and the dependence o f X on d i l u t i o n i s represented by i t s p r o p o r t i o n a l i t y to c t m Equation 2. F i g u r e 2 i l l u s t r a t e s r e s u l t s obtained from t r i - and t e t r a f u n c t i o n a l polyurethane-forming r e a c t i o n systems, w i t h X' b p l o t t e d against ( c + C ^ Q ) " , the i n i t i a l d i l u t i o n o f r e a c t i v e groups. I t i s apparent t h a t the p l o t s are curved r a t h e r than e x

b

e x

a 0

D 0

e x

C9

a c

f

a

2

n

f

a b

- 1

e x

a

1

a 0

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

1.

STANFORD ET AL.

Formation and Properties of Networks

0.4

0.3

0.2 1 ^

0.1

2

m

So* bo 9 -1 ._ _ .1 0.3 OA (

Ο

0.1

0.2

- -

r 1 / k

c

m

o

f

1

-

1, 0.5

Figure 2. Ring forming parameter (A'ab) versus i n i t i a l d i l u t i o n o f r e a c t i v e groups ( ( c + c b o ) - ). Experimental values o f a νι/ere used t o evaluate X b according t o Eq. 1· Systems: 1 and 2, HDI+POP t r i o l s ; 3 , 4,4 -diphenyl methane diisocyanate(MDI)+PQP t r i o l ; 4 and 5, HDI+POP t e t r o l s . Reactions c a r r i e d out a t 80°C i n bulk and i n nitrobenzene s o l u t i o n . System 1, HDI+LHT240, v=33; system 2, HDI+LHT112, v=61; system 3, MDI+LHT240, v=30; system 4, HDI+OPPE-NHI, v=29; system 5, HDI+0PPE-NH2, v=33.(LHT240 and LHT112oxypropylated 1,2,6 hexane t r i o l s ; OPPE-NHI and 0PPE-NH2 oxypropylated p e n t a e r y t h r i t o l s . ) Reproduced, w i t h permission,from Réf. 1. Copyright 1982, Plenum P u b l i s h i n g C o r p o r a t i o n . 1

ao

f

c

a

f

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

5

6

HIGHLY CROSS-LINKED POLYMERS

l i n e a r as p r e d i c t e d by Equation!?. D e t a i l e d d i s c u s s i o n s o f the r e s u l t s shown i n F i g u r e 1 and o f s i m i l a r r e s u l t s f o r p o l y e s t e r forming systems have been given else\i/here( 1,2,4,5). In g e n e r a l , p o l y e s t e r - f o r m i n g systems are found t o g i v e more l i n e a r p l o t s than polyurethane-forming systems and, w i t h regard t o the choice °f ext> the use o f c + c b g i v e s more l i n e a r p l o t s than c t = c + cbc« Thus, the f u n c t i o n a l dependence o f X,' b on d i l u t i o n appears t o be b e t t e r described by theory i f i n i t i a l d i l u t i o n ((c + C^Q)- ) i s used. From F i g u r e 2 i t i s c l e a r t h a t i n t r a m o l e c u l a r r e a c t i o n i n c r e a s e s w i t h d i l u t i o n and, as i n d i c a t e d i n Figure 1, w i t h f u n c t i o n a l i t y . I n a d d i t i o n , the p o i n t s on the curves a t the low­ e s t d i l u t i o n s r e f e r t o bulk r e a c t i o n mixtures, i n d i c a t i n g again ( c f . Figure 1) t h a t i n t r a m o l e c u l a r r e a c t i o n always occurs. The e f f e c t s o f chain s t i f f n e s s can be seen by comparing systems 1 and 3, which have s i m i l a r value t h a t o f system 3 c o n t a i n The i n i t i a l slopes o f the curves i n F i g u r e 2 and o f the corresponding p l o t s w i t h f c + c b c ^ s a b s c i s s a can be analysed according t o Equations 3 and 4, and values o f b found. The values obtained are given i n Table I . The two values o f b f o r each system g e n e r a l l y encompass the value expected from s o l u t i o n c

a 0

0

e x

a c

a

1

a o

a c

Table I . Values o f E f f e c t i v e Bond Length (b) o f Chains Forming the Smallest Ring S t r u c t u r e s ( o f ν bonds), (i) c t = c + c ; (ii) c t = c + cbc- \>DI i s the f r a c t i o n o f bonds due t o the d i i s o c y a n a t e r e s i d u e i n the chain o f ν bonds, Reproduced, w i t h permission, from R e f . l . Copyright 1982, Plenum P u b l i s h i n g Corp. e x

System 1. 2. 3. 4. 5.

HDI/LHT240 HDI/LHT112 MDI/LHT240 HDI/0PPE-NH1 HDI/0PPE-NH2

3 o

b Q

e x

a c

f

V

v /v

b/nm(i)

b/nm(ii)

3 3 3 4 4

33 61 30 29 33

0.303 0.164 0.233 0.345 0.303

0.247 0.222 0.307 0.240 0.237

0.400 0.363 0.488 0.356 0.347

D I

p r o p e r t i e s ( l , 2 , 4 , 5 ) . Thus the e f f e c t i v e average value o f C g £ l i e s somewhere bêtween~(c + cbo) and (cbc + cbc)» and probably nearer t o ( c + cbc)* The g e n e r a l l y s m a l l e r values o f b f o r the a l i p h a t i c t e t r a f u n c t i o n a l systems (4 and 5) compared w i t h the a l i p h a t i c t r i f u n c t i o n a l systems (1 and 2) probably i n d i c a t e a r e l a t i v e undercounting o f o p p o r t u n i t i e s f o r i n t r a m o l e c u l a r r e a c t i o n f o r growing s p e c i e s from t e t r a f u n c t i o n a l compared w i t h t r i f u n c t i o n a l reactants. X

ao

a c

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

1.

STANFORD ET AL.

Formation and Properties of Networks

7

Comparison o f systems 1 and 2 and systems 4 and 5 show t h a t s m a l l e r values o f b are obtained f o r the l a r g e r values o f ν or the s m a l l e r values o f Vnj/v, i n d i c a t i n g t h a t the chains w i t h the l a r g e r p r o p o r t i o n s of oxypropylene u n i t s are the more f l e x i b l e . Hence although system 1 g i v e s higher values o f X' b than system 2 because i t has a s m a l l e r value o f v, the d i f f e r e n c e between the curves f o r the two systems i n F i g u r e 2 i s reduced because b f o r system 2 i s s m a l l e r . S i m i l a r c o n s i d e r a t i o n s h o l d t r u e f o r the r e l a t i v e values o f X'ab f o r systems 4 and 5. Other aspects o f g e l a t i o n s t u d i e s which have been reported are the determination o f e f f e c t i v e f u n c t i o n a l i t i e s ^ ) and the use o f d i o l - t r i o l m i x t u r e s ^ ) to i n v e s t i g a t e the e f f e c t s o f v a r i a t i o n o f average f u n c t i o n a l i t y . The former work used a t r i o l which had been independently c h a r a c t e r i s e d w i t h respect to f u n c t i o n a l i t y and showed the shortcoming f u s i n g e l a t i o dat alon deduce the chemical f u n c t i o n a l i t i e work used mixtures o f a g sebacoy c h l o r i d e at d i f f e r e n t i n i t i a l d i l u t i o n s i n diglyme as s o l v e n t . The hydroxyl groups had equal r e a c t i v i t i e s and the r e a c t i o n mixtures were equimolar i n hydroxyl and a c i d c h l o r i d e qroups. At zero d i l u t i o n , the equation o f Stockmayer (5>17)> a = (fyj"\l), where f i s the weight-average f u n c t i o n a l i t y " o f the p o l y o l mixture, i s obeyed. The r e s u l t s are i l l u s t r a t e d i n F i g u r e 3, where a~ i s p l o t t e d versus i n i t i a l d i l u t i o n . The i n t e r c e p t s i n a at zero d i l u t i o n are equal t o the values o f ( f " l ) c a l c u l a t e d from the amounts o f d i o l and t r i o l i n the r e a c t i o n mixtures, and the decreases i n o^- w i t h i n i t i a l d i l u t i o n are due t o i n t r a m o l e ­ cular reaction. a

c

w

l

c

- 1

c

w

1

Network P r o p e r t i e s C o r r e l a t i o n s between Gel P o i n t and Shear Modulus. The r e a c t i o n systems i n F i g u r e 2 were used to form networks a t complete reaction(1,2,10,11). S o l f r a c t i o n s were removed and shear moduli were determined i n the dry and e q u i l i b r i u m - s w o l l e n s t a t e s at given temperatures u s i n g u n i a x i a l compression or a t o r s i o n pend­ ulum at 1Hz. The procedures used have been d e s c r i b e d i n d e t a i l elsewhere(11,12). The shear moduli(G) obtained were i n t e r p r e t e d according t o Gaussian theory(18-20) t o g i v e values o f M , the e f f e c t i v e molar mass between j u n c t i o n p o i n t s , c o n s i s t e n t w i t h the a f f i n e behaviour expected at the s m a l l s t r a i n s used (20). Equation 6 was used w i t h ρ c

G = ARTpφ

1 / 3 2

(V /V ) u

2 / 3

F

/M

c

(6)

the d e n s i t y o f the dry networK Φ2 the volume f r a c t i o n o f s o l v e n t present i n a swollen network, V the volume o f the dry, u n s t r a i n e d network, and Vp the volume at formation. A has the value (1-2/f) f o r networks showing phantom behaviour and 1 f o r networks showing a f f i n e behaviour (19,20). u

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

8

HIGHLY CROSS-LINKED POLYMERS

1.01 Ο

I 0.2

1 1 0.4 0.6 ao* bo

{ c

c

, 1 / k g

l-J

1 0.6 m

o

f

1.0

1

χ

F i g u r e 3. α " versus i n i t i a l d i l u t i o n o f r e a c t i v e groups ( ( c + c b ) - l ) f o r mixtures o f diol(PPG1025) and t r i o l ( L H T 112) r e a c t i n q w i t h sebacoyl c h l o r i d e a t 6Q0C i n diglyme. 0

a 0

0

rrpeOo/IMo = 1. PPG1025 - POP d i o l ; LHT112 - POP t r i o l (see c a p t i o n Figure 2). Curves 1, f = 2.99; 2, f = 2.82; 3, f = 2.65; 4, f = 2.50; 5, f = 2.35. Reproduced, w i t h permission, from Ref. 3. Copyright 1982, S o c i e t y o f Polymer Science, Japan. w

w

w

w

w

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

I.

9

Formation and Properties of Networks

STANFORD ET A L .

The r e s u l t s a r e shown i n Figure 4, where M / M ° i s p l o t t e d versus p p. The molar mass between j u n c t i o n p o i n t s o f the p e r f e c t nètwork(M °) i s c a l c u l a b l e from the molar mass and s t r u c t u r e o f the r e a c t a n t s (1,2.) and M was evaluated from the measured modulus using Equation 6 with A=l. p ^ ^ extent o f i n t r a m o l e c u l a r r e a c t i o n a t g e l a t i o n (1,2) given by the expression c

c

r

c

c

s

n e

r ? c

f

=

Pr,c 0 corresponds t o the ideal(Flory-Stockmayer) g e l - p o i n t , and M / M ° = 1 t o the p e r f e c t , a f f i n e network. In a l l cases i n Figure 4, M / M ° exceeds 1 and tends t o 1 as p -*· 0. Thus, only i n the l i m i t o f a p e r f e c t g e l l i n g system i s a p e r f e c t network achieved, f o r which a f f i n behaviou i predicted equal t o 3 and 2 on th phantom networks o f f u n c t i o n a l i t i e , respectively p r e - g e l i n t r a m o l e c u l a r r e a c t i o n , which causes α t o exceed l / ( f - l ) i n value, a l s o produces some e l a s t i c a l l y i n e f f e c t i v e loops which have marked e f f e c t s on the moduli o f the dry networks. In f a c t , M / M ° i s equal t o the p r o p o r t i o n a l r e d u c t i o n i n modulus compared w i t h t h a t expected f o r the p e r f e c t , dry network. Thus, M °/M = 10 correpsonds t o a 1 0 - f o l d r e d u c t i o n . Any e f f e c t s due t o entanglements are i n a l l cases overshadowed by the reductions i n moduli due t o l o o p s . c

c

c

c

ç c

c

c

c

c

The points at the lowest values of p . r

c

c

f o r the various

systems are those f o r bulk r e a c t i o n s and even f o r these s i g n i ­ f i c a n t reductions i n moduli a r e apparent. In a d d i t i o n , such r e d u c t i o n s can be produced by r e l a t i v e l y s m a l l values o f p . Thus, system 1 shows a 5 - f o l d r e d u c t i o n i n modulus f o r an excess extent o f r e a c t i o n a t g e l a t i o n o f only 0.05, and system 5 a 3-fold reduction f o r p = 0.10. The r e l a t i v e p o s i t i o n s o f the l i n e s f o r the v a r i o u s systems can be r e l a t e d t o M o(or v ) , f , and the chain s t r u c t u r e s o f the reactants(1,2,9-12). The slopes o f the l i n e s show t h a t the r e ­ duction i n modulus w i t h p r e - g e l i n t r a m o l e c u l a r r e a c t i o n i s l a r g e r f o r t r i f u n c t i o n a l compared w i t h t e t r a f u n c t i o n a l networks ( c f . systems 1 and 2 w i t h 4 and 5 ) , although higher values o f p« o b t a i n f o r t e t r a f u n c t i o n a l r e a c t i o n systems ( c f . Figure 2;! In a d d i t i o n , f o r a given f u n c t i o n a l i t y , the r e d u c t i o n i s l a r g e r f o r s m a l l e r values o f M ° ( c f . systems 1 with 2 and 4 w i t h 5 ) ; t h a t i s , f o r a given amount o f i n t r a m o l e c u l a r r e a c t i o n (or value of p ) systems with s m a l l e r loops have l a r g e r p r o p o r t i o n s o f those loops e l a s t i c a l l y i n e f f e c t i v e . The networks f o r system 3, based o f MDI, g i v e values o f M / M ° near u n i t y , corresponding t o r e l a t i v e l y high values o f t h e i r rubbery moduli. The reasons f o r t h i s phenomenon are not completely understood but a r e o b v i o u s l y r e l a t e d t o the s t i f f e r , aromatic chain s t r u c t u r e between j u n c t i o n p o i n t s i n these networks. r

c

r c

c

c

c

r > c

c

c

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

10

HIGHLY CROSS-LINKED POLYMERS

F i g u r e 4. Molar mass between e l a s t i c a l l y e f f e c t i v e j u n c t i o n p o i n t s (M ) r e l a t i v e t o t h a t f o r the p e r f e c t network(M °) versus extent o f i n t r a m o l e c u l a r r e a c t i o n a t gelation(p ) . Reaction systems as f o r F i g u r e 2. l i n e s through experimental p o i n t s f o r systems 1,2,4,5; and t h e o r e t i c a l curves f o r t r i - and t e t r a f u n c t i o n a l networks (see t e x t , l a s t s e c t i o n ) . System 1, HDI+LHT240, M °=0.635 kg mol"" , v=33; system 2, HDI+LHT112, M °=1.168 kg m o l " , v=61; system 3, MDI+LHT240, M °=0.705 kg m o l " , v=30; system 4, HDI+0PPE-NH1, M °=0.500 ! M . The presence o f s t a t e s 4 and 6 i n a network a t complete r e a c t i o n i s presented s c h e m a t i c a l l y i n Figure 7. I t can be seen t h a t over the complete r e a c t i o n system the number o f j u n c t i o n p o i n t s l o s t i s Ν .Ρβ> where N i s the number o f monomer u n i t s i n i t i a l l y and P i s the f r a c t i o n o f u n i t s i n s t a t e 6. Hence a t complete r e a c t i o n , the number o f e l a s t i c a l l y e f f e c t i v e f u n c t i o n p o i n t s i n N ( P i » - Ρε), where Ρι» + Ρβ = 1. Thus, r

c

c

c

c

8

c

a

6

a

M /M ° c

=

c

1/(1-2P ) 6

(10)

The a n a l y t i c a l expression f o r Ρβ i s Pe =

2

2

3

2

Ρ(-Ρ λ+ (2λ - 4λ /3)ρ - (8λ /9Ηη(1 - 3ρ/(2λ + 3))) + 3λρ /2 2

2

3

+ 3λ ρ/2 + (3λ /2 + 3λ /4Ηη(1 - 2ρ/(λ + 2))

(11)

Here, λ i s a ring-forming parameter given by the equation ( c f . Equations 2 t o 4) λ

=

P

a b

/c

(12)

a 0

with P defined by Equation 4 and c the i n t i a l c o n c e n t r a t i o n o f A groups. U n l i k e λ'φ o f Equation 2, λ i s a uniquely defined parameter; i n the d e f i n i t i o n o f X , the denominator, c ^ , had to be chosen a r b i t r a r i l y . A corresponding equation t o Equation 11 has been d e r i v e d t o d e f i n e t h e extent o f r e a c t i o n a t q e l a t i o n ( 2 4 ) . I t enables p (see Equation 7) t o be evaluated as a f u n c t i o n o f λ and a l l o w ! p r e d i c t i o n o f the c o r r e l a t i o n between g e l p o i n t and r e d u c t i o n i n shear modulus ( v i z . M / M ° ) . The c o r r e l a t i o n i s shown as curve 1 i n Figure 8, which may be compared w i t h the experimental curves i n Figure 4 f o r RA2 + RBf p o l y m e r i s a t i o n s . a D

a 0

f

a D

e x

r

c

c

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

c

HIGHLY CROSS-LINKED POLYMERS

16

Table I I . Smallest Subset o f States o f Monomer U n i t s i n an RA3 Polymerisation. A- denotes a c o n t i n u i n g chain o f undefined l e n g t h .

State

1.

)-A A

A

1,2 1,5

2.

-AA. )-A A

2,3

2,6 3

-

A

\ >-A

A

>-A A

+ A-* A

A

A

-AA

3,4

>^AA-

A

v

A >

A

A

*

A

A > ;

V A + AX

A

V-AA— W

v

>A A

+

-AA

-ΑΑ'

4.

Reaction

Reaction Route

-AA

>-A Al^Y

-AA )>-A + A- -> V A A -AA -AA v

-AAv )-AA-AA 7

5.

( V

6. f V

N-A

A'

A

5,6

S

A

(*> * *(*>

A A

^>-AA-

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

-

STANFORD ET AL.

Formation and Properties of Networks

Figure 7. Rate theory -occurrence of s t a t e s 4 and 6 i n network at complete r e a c t i o n from an RA3 p o l y m e r i s a t i o n . AA/A State 4: -AA-/ ; State 6: ( \-AAAAΚ V

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

18

HIGHLY CROSS-LINKED POLYMERS

M

C

/

M

C

Figure 8. P r e d i c t e d correlations(24)between r e d u c t i o n i n shear modulus a t complete reactionTM /M °) and extent o f i n t r a m o l e c u l a r r e a c t i o n a t g e l a t i o n ( p ) f o r an R A 3 polymerisation. Curve 1: From r a t e theory, accounting f o r p r e - g e l and postg e l i n t r a m o l e c u l a r r e a c t i o n (Equation 10). Curve 2: Accounting f o r p r e - g e l i n t r a m o l e c u l a r r e a c t i o n only (Equation 13). c

c

r > c

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

1.

STANFORD ET A L .

19

Formation and Properties of Networks

I f only p r e - g e l i n t r a m o l e c u l a r r e a c t i o n i s considered, then the number o f s m a l l e s t loops i n an RA3 p o l y m e r i s a t i o n i s 3Ν^ρ / 2 . The number o f j u n c t i o n p i n t s l o s t a t complete r e a c t i o n i s ' twice t h i s number (see Figure 7) and Γc

M

/ M

c c° =

^

^

Γ

,

Ο

'

·

(

1

3

)

This equation corresponds t o Equation 8 f o r an RA2 + RB3 polymer­ i s a t i o n . The r e s u l t i n g r e l a t i o n s h i p between M / M ° and p i s shown by curve 2 i n Figure 8, which may be compared w i t h t n e c a l c u l a t e d curves i n Figure 4. In a r e a l network o f f u n c t i o n a l i t y four o r l e s s , the s m a l l e s t loops apparently l e a d t o e l a s t i c a l l y i n e f f e c t i v e j u n c t i o n p o i n t s . In a d d i t i o n , l a r g e r loops can a l s o c o n t r i b u t e t o such d e f e c t s . The r e l a t i v e p o s i t i o n s the b a s i s o f the s m a l l e s cannot be neglected, with approximately t h e same number o f loops o c c u r r i n g p o s t - g e l as p r e - g e l . The importance o f both p o s t - g e l and p r e - g e l i n t r a m o l e c u l a r r e a c t i o n i s a l s o apparent from F i g u r e 4 f o r RA2 + RB3 systems, where,apart from t h e data f o r the aro­ matic system 3, t h e c a l c u l a t e d curves g e n e r a l l y l i e w e l l below the experimental curves and have d i f f e r e n t shapes therefrom. To a l l o w d i r e c t comparison w i t h such experimental data, developments o f the r a t e theory t o evaluate M / M ° f o r RA2 + RB3 systems a r e p r e s e n t l y i n progress. c

c

c

r c

c

Literature Cited 1. Stanford, J.L.; Stepto, R.F.T.; Still, R.H., in "Reaction Injection Moulding and Fast Polymerisation Reactions"; Kresta, J.E., Ed.; Plenum Publishing Corp: New York, 1982; p.31. 2. Stanford, J.L.; Stepto, R.F.T., in "Elastomers and Rubber Elasticity"; Mark, J.E.; Lal, J., Eds.; ACS SYMPOSIUM SERIES No. 193, American Chemical Society: Washington D.C., 1982; Chap. 20. 3. Ahmed, Z.; Stepto, R.F.T. Polymer J. 1982, 14, 767. 4. Ahmad, Z.; Stepto, R.F.T. Colloid and Polymer Sci. 1980, 258, 663. 5. Stepto, R.F.T., in "Developments in Polymerisation - 3"; Haward, R.N., Ed.; Applied Science Publishers Ltd.: London, 1982; Chap. 3. 6. Stepto, R.F.T.; Waywell, D.R. Makromol. Chem. 1972, 152, 247, 263. 7. Stanford, J.L.; Stepto, R.F.T. Brit. Polymer J . 1977, 9, 124. 8. Ahmad, Z. Ph.D. Thesis, University of Manchester, England, 1978. 9. Stepto, R.F.T. Polymer 1979, 20, 1324. 10. Hunt, N.G.K.; Stepto, R.F.T.; Still, R.H. Proc. 26th IUPAC Int. Symp. on Macromolecules, Mainz, 1979, p.697.

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

HIGHLY CROSS-LINKED POLYMERS

20

11. Cawse, J.L. Ph.D. Thesis, University of Manchester, England, 1979. 12. Fasina, A.B.; Stepto, R.F.T. Makromol.Chem., 1981, 182, 2479. 13. Stanford, J.L.; Stepto, R.F.T. J. Chem. Soc. Faraday Trans.I 1975, 71, 1292. 14. Frisch, H.L. 128th Meeting Amer. Chem. Soc., Polymer Di v., Minneapolis, 1955. 15. Kilb, R.W. J. Physic. Chem. 1958, 62, 969. 16. Stepto, R.F.T. Faraday Disc. Chem. Soc. 1974, 57, 69. 17. Stockmayer, W.H. J. Polymer Sci. 1952, 9, 69; 1953, 11, 424 18. Dusek, K.; Prins, W. Adv. Polymer Sci. 1969, 6, 1. 19. Flory, P.J. Polymer 1979, 20, 1317. 20. Mark, J.E. Pure and Applied Chem. 1981, 53, 1495. 21. Stanford, J.L.; Stepto R.F.T.; Waywell, D.R. J. Chem. Soc. Faraday Trans.I. 1975 71 1308 22. Askitopoulos, V. England, 1981. 23. Cawse, J.L.; Stanford, J.L.; Stepto, R.F.T. Proc. 26th IUPAC Int. Symp. on Macrmolecules, Mainz, 1979, p.393. 24. Lloyd, A.C. M.Sc. Dissertation, University of Manchester, England, 1981. 25. Gordon, M.; Temple, W.B. Makromol. Chem. 1972, 160, 263. RECEIVED

September 22, 1983

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

2 Computer Simulation of End-linked Elastomers Sol-Gel Distributions at High Extents of Reaction YU-KWAN LEUNG and Β. E. EICHINGER Department of Chemistry, University of Washington, Seattle, WA 98195

The end-linking tri- and tetrafunctional cross-linkers i n the bulk has been simulated on the computer. The algorithm places the molecules at random i n an image container and then joins their ends together at junctions. The spanning forest for the constructed graph i s then found, and the s o l and gel components are identified. The results that are reported here include, amongst other things, the distribution of c y c l i c species i n the s o l and the proportion of defects i n the gel. In the traditional gelation theory formulated by Flory (1) and Stockmayer (2), i t i s assumed that l i k e functional groups are equally reactive and a l l reactions occur intermolecularly before the gel point. Subsequently, beyond gelation, f i n i t e species formed in the sol portion are limited to acyclic trees. This i s not correct because intramolecular reaction leading to the formation of ring structures i n a random polycondensation must occur. The effect of cyclization has been treated by various approaches, e.g. Jacobson-Stockmayer ring-chain factors (3), cascade theory (4), and rate theory (5). Since the probability of ring structures increases with the extent of reaction, the sol must eventually contain significant amounts of c y c l i c s . Neglect of intramolecular reactions, especially in the post-gel region, can yield distorted distributions of s o l species and inaccurate gelation conditions. Since many of the present studies on cyclization are directed to the pre-gel stage of the reaction, l i t t l e i s known about the molecular constituents of the sol species and the gel structures i n the later stages of the reaction. The importance of intramolecular reactions i n random polymerization was extensively discussed by Stepto i n a recent review a r t i c l e ( 6 ) ,

0097-6156/ 84/0243-0021 $06.00/ 0 © 1984 American Chemical Society In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

22

HIGHLY CROSS-LINKED POLYMERS

The complexity of the post-gel problem renders theoretical treatment rather d i f f i c u l t , but computer simulation offers an attractive means to obtain some information on the distributions of interest. A random stepwise polyreaction was simulated by Falk and Thomas (7) to examine the polymer size distributions with and without consideration of ring formation. Their system was composed of monômeric units RA , which were represented by an array of random numbers, ana during the reaction process details of connectivity were not recorded. Recently, a Monte Carlo simulation of network formation has been reported by Mikes and Dusek (8), who assumed that the s o l molecules in the post-gel period were acyclic. In our model, the spatial arrangement of reactive groups i s taken into account and molecules of different shapes are sorted and counted Simulations were done ring formation i n a l l largest particle obtained i s identified as the gel. The distribution of other f i n i t e species and network imperfections are analyzed and discussed. f

ALGORITHM A number Ν of primary molecules or prepolymers (A ) having η bonds were*ai8tributed randomly i n a cubical box whose length i s L, where 1 , 1 / 3

η M H ο Ρ [1] * a Μ i s the molecular weight of one bond unit, ρ i s the density or the polymer and Ν i s Avogadro's number. End-to-end distances of the primary molecules were generated as random three-dimensional vectors with a Gaussian distribution characterized by a one-dimensional variance, σ 2 2 N

a

« c n l /3 x

[2]

where 1 i s the length of one bond and C i s the characteristic ratio (11)· The Ν molecules of f-functional cross-linkers were also randomly Sistributed i n the reaction cube, with the number of cross-linking agents determined from N - 2rN /f [3] c

p

where r i s the stoichiometric ratio, defined as the ratio of the number of Β functional groups to the number of A functional groups. The growth of polymeric molecules was accomplished by joining the ends of the prepolymers with available Β functional groups i n the nearby cross-linkers. Junctions were formed by starting with the nearest neighbor and

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

2.

End-linked Elastomers

LEUNG AND EICHINGER

proceeded i n the order of increasing r , the distance between A and Β functional groups. Reactive groups were not allowed to form a bond i f their distances r were greater than a set distance parameter. Hence, the distance parameter controls the extent of reaction i n the polymerization. Ά vertex i s defined as a condensed point of a graph, which may be a free end, joint or cross-link; i t s degree i s the number of prepolymers attached to i t . As each -AB- bond was formed, the degree of the cross-linker involved increased by one (maximum allowable number i s f ) and the end was labelled 'reacted*. The indices of ends to which the connections were made through the shared cross-linker, called the connecting index, were recorded. The process of netformation was closely monitored by keeping track of the degrees and the connecting indices of the visited ends Finally, the connecte large random graphs b written by Nijenhuis and Wilf (12). Schematic diagrams i l l u s t r a t i n g representative structures are depicted in Figure 1. Beyond the gel point, only one large particle was observed, which i s consistent with results obtained from other methods (8,13). The remaining molecules are f i n i t e species whose molecular sizes seldom exceed 20 prepolymer units. The smallest ring that can be formed i s the one-chain loop. Dangling ends of the molecules can be identified as those vertices whose degree equals one. The structures of s o l molecules could be recognized by referring to the set of the degrees assigned to i t s vertices and the number of one-chain loops formed (10). EFFECT OF FIBUTENESS Computation were made for trifunctional and tetrafunctional end-linked poly(dimethylsiloxane) systems (14,15). Values of the parameters |or PDMS are: M -37 g/mol, Ο -6.3, 1-1.64 & and p«0.97 g/cm (11). Edge effects were investigated by performing calculations on systems of different sizes. Table 1 shows the results for a series of simulation with f-4, n«50 and Ν -5000, 7500, 10000, 12500, 15000. The number of configurations for each Ν were chosen such that the t o t a l sample consisted of ca. βδοοο functional groups. The values of the s o l fractions w and the cycle rank per chain ξ ', averaging over different Ν , were 0.0886 and 0.303 respectively, and the average extent of reaction of A functional groups, P , was 0.806. Individual w and £ ' showed no trend that could be attributed to an edge çffect except for the case of Ν -5000. Values for M and Ν , the mole fractions of x-mer and that of the bow-tie trimer (see figure 1) respectively, are also included i n Table 1. Given these results, i t was concluded that 4 different Q

χ

fl

A

g

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

HIGHLY CROSS-LINKED POLYMERS

24

configurations of 10000 A molecules each would constitute an adequate representation o i thermodynamically large system; a l l subsequent results were obtained with simulations of this size. For systems of this size, a comparison of the concentrations of free ends i n the core and i n the rim of the reaction box indicated that edge effects result i n an overestimation of the concentration of free ends by no more than ca. 4%.

Table I. The E f f e c t of Reaction Box Sizes f o r A

N

p 5000 7500 10000 12500 15000

L,A

*A

251 288 316 341 362

0.803 0.807 0.807 0.810 0.803

w

N

0.09 0.08 0.087 0.086 0.089

l

0.870 0.880 0.888

N

2

N

9

+ B, System'

3

0.03 0.033 0.043 0.032 0.292 0.034 0.045 0.034 0.308 0.038 0.034 0.026 0.294

* MW-1850 and r«1.0 mole fraction of bow-tie trimer cycle rank per chain of the interior gel

SOL FRACTIONS Accurate prediction of the dependence of s o l fraction on the extent of reaction requires information on the proportion of ring structures formed by intramolecular reactions. The weight fractions w of soluble material that were generated for a variety of runs for trifunctional networks are presented i n Table 2. For comparison, s o l fractions w^', calculated by means of a recursive method (16) allowing only treelike s o l molecules are entered i n column 5 of Table 2. It can be seen that for low molecular weight prepolymers, the inclusion of cyclics boosts the s o l fraction by 2 to 3 % for P over 0.80. Sol fractions are plotted against the extent of reaction for the case of an A + B copolymerization i n Figure 2. The broken curves represent results obtained from our simulation program whereas the result of the treelike model (16) i s indicated by the solid curve. It i s noted that deviations between the two models increase as chain length decreases and as the extent of reaction increases. g

ft

2

4

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

LEUNG AND E1CHINGER

End-linked Elastomers

Figure 1. Schematic diagram i l l u s t r a t i n g various structures i n the post-gel stage of the reaction. sol molecule drawn i s a bow-tie trimer.

The

Figure 2. Variation of s o l fraction with the extent of reaction for an A + B copolymerization: MW-1850 ( c i r c l e s ) and 450O0 (apples). j i curve corresponds to results of the treelike model . 4

s

i

d

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

26

HIGHLY CROSS-LINKED POLYMERS Table I I . Sol F r a c t i o n s f o r A

MIf

a

r

1850

1.0

4700

1.0

18500

1.0

9

+ B« System

w s 0.809 0.864 0.897 0.920 0.883 0.910 0.851 0.877 0.897

0.172 0.083 0.051 0.037 0.051 0.032 0.069 0.047 0.033

s 0.152 0.054 0.022 0.013 0.036 0.018 0.070 0.045 0.024

Ref. 16

COMSTITUENTS OP THE SOL Figure 3 shows the histogram of the weight fraction of x-mers versus χ for a trifunctional cross-linker at P^-0.897. The prepolymer chosen has 50 Si-0 bonds. The shaded bars represent weight fractions of molecular cyclics whereas the open bars indicate those of the treelike structures. For example, there are two types of dimer: the linear and the c y c l i c , where the latter has the shape of a tadpole. Our results show that the cyclics outnumber the linear by a ratio of 8si. This i s because the tadpole containing only one reactive group i s a stable structure which has less chance to be absorbed by the gigantic gel particle. On the other hand, the branched trimer i s predominant i n the class of trimer. This can be understood to imply that the probability of double edge formation, which requires that the two chain vectors involved be located i n the same volume element, i s very small. In Figure 4, the mole fractions of selected c y c l i c graphs, Ν , are plotted against for different molecular weights of the prepolymer. The results for MW-1850, 4700, 18500 and 32900 are given by the circles, triangles, squares and apples respectively. As expected, the population of cyclics increases with higher degrees of conversion and shorter chain lengths. Typical molecular weight distributions for the tetrafunctional system are depicted i n Figure 5. In contrast to the monotonie w function predicted by the acyclic model (1), our findings show that the weight fractions have a maximum at the trimer. The high proportion of c y c l i c trimer,

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

End-linked Elastomers

LEUNG A N D EICHINGER

0.004

Figure 3. Size distribution of the s o l for the trifunctional system with n«50 and P^-0.897. The shaded bars and the open bars represent weight fractions of c y c l i c and acyclic molecules respectively. I

ι

- ? N

g

0/

/*

0.08

*//

.

0

.

ι

ι

ι

1

1 ο

N

g

0.02

- τ 1

Δ

ο

-

Δ

α ι

0.8

ι 0.9

ΡΑ

Figure 4. Dependence of the mole fraction of eyelies the s o l upon the extent of reaction for different molecular weights: MW*1850 ( c i r c l e s ) , 4700 (triangles), 18500 (squares) and 32900 (apples). In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

HIGHLY CROSS-LINKED POLYMERS

28

a 'bow-tie' graph consisting of one loop and two chains attached to a single junction, results from the favorable conversion of c y c l i c dimer (tadpole) to c y c l i c trimer due to the a v a i l a b i l i t y of an unreacted Β functional group attached to the dimer. The population of tadpole trimer, resulting from the attachment of a third chain to the t a i l of the 'tadpole' dimer, i s small because i t s formation requires another cross-linker, which i s scarce at high conversion. I t i s evident that i n the post-gel stage of the reaction, loop formation has a great influence on the pattern of molecular size distributions. We have not considered the difference between the s o l that might be extracted from a real network and that which i s permanently incarcerated by virtue of i t s concatenation with the network. For more elaborate calculations chain trajectories would be molecules of these two e.g. n*50, the proportion of incarcerated sol molecules should be small since the loops that are formed are too small to be wrapped around by another chain. NETWORK IMPERFECTIONS For trifunctional networks, three types of dangling ends were identified by the program. They are the dangling loop (see Figure 6), 'I' end and 'Y' end (see Figure 7). Symbols used in Figure 6 and 7 are the same as that of Figure 4 . The population of dangling ends, 7), i s expressed i n terms of the number of end configurations per number of prepolymers incorporated in the gel. The populations of the Ί ' and Ύ ' free ends are independent of molecular weight, as illustrated i n Figure 7. On the other hand, i t can be seen i n Figure 6 that the occurrence of one-chain loops i n the network agrees with the trend shown by the s o l cyclics as described i n the previous section. The observed increase in one-chain loop probabilities with shorter chain lengths i s consistent with the Gaussian s t a t i s t i c s assumed by the molecules. Network imperfections do not vanish at complete conversion because of the loops. It i s estimated by extrapolation that at 100% conversion, ca. 3% of the primary chains react to form loops for n«50. The cycle rank of a network i s defined as the number of cuts required to reduce the network to a tree (17). I t i s a structural factor characteristic of the perfection of a network (18); for a perfect network, the cycle rank per chain (' i s given by €'«l-2/f. The {' values for various networks are l i s t e d i n Table 3. The gels produced at very high extents of reaction s t i l l exhibit various kinds of structural imperfections: for example, the cycle rank of the

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

End-linked Elastomers

LEUNG AND EICHINGER 0.030 ι

1

1

1

1

,

1

0.020 -

0.003 -

Figure 5. Histogram of the weight fractions of x-raer versus χ for the tetrafunctional system with n*50 and Ρ «0.886 (symbols same as i n Figure 3).

0.03

0 I 0.7

1

1

1

0.8

1

0.9

1

1

1.0

PA

Figure 6. Plots of dangling end population versus the extent of reaction (symbols same as i n Figure 4). Broken lines represent non-stoichiometric systems with r values indicated.

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

HIGHLY CROSS-LINKED POLYMERS

30

tetrafunctional network formed at complete conversion i s estimated to be 0.48, which i s about 4% below the value 0.50 for the perfect network.

Table

I I I . Cycle Rank per Chain g

0.740 3 4

0.160

f

f o r Various Networks

*

0.825

0.875

0.915

1.0

0.113

0.182

0.223

1/3

0.300

0.365

0.410

1/2

Theoretica

DEPEDEBCE OP LOOP PROBABILITY OH CHAIN LEWGTH The weight fraction T) of one-chain loops i n the polymerization system i s defined as the percentage of primary molecules whose ends are connected d i r e c t l y to each ^ y The log-log plots of T) vs. rms end-to-end distance of the primary chain at different functionalities and conversions yield pare1lei straight lines. The average least-squares slope of these lines i s found to be -0.76, which i s taken to be -3/4 for subsequent calculations. The standard deviation of the slope i s 0.04. The relationship between ν and rms end-to-end distance i s cast i n Figure 8 as a representation of the discovered relation 0

o t

e r

1

2

Q

ο

V

2

0

« k = p S

Z P P

(q

2 }

P

In these equations α denotes the extent o f c r o s s l i n k i n g , ! , e . the f r a c t i o n o f r e p e a t i n g u n i t s b e a r i n g a c r o s s l i n k , and y = DP i s the weight average degree o f p o l y m e r i z a t i o n o f the primary ^ chains, i . e . the chains before c r o s s l i n k i n g . DP i s the weightaverage degree o f p o l y m e r i z a t i o n o f the t o t a l , c r o s s l i n k e d p o l y ­ mer, and Ρ (q^) and Ρ (q ) are the p a r t i c l e s c a t t e r i n g f a c t o r s of the c r o s s l i n k e d ancPof the primary chains r e s p e c t i v e l y . The v a r i a b l e q gives the magnitude o f the s c a t t e r i n g v e c t o r q as q = (4Tr/X)sin(9/2)

(9)

and the v a r i a b l e φ i s d e f i n e d as 2

2

φ = exp(-q b /6)

(10)

2 with b denoting the mean square d i s t a n c e between two adjacent r e p e a t i n g u n i t s . Then the weight-average degree o f p o l y m e r i z a t i o n i s given by DP

w

= S**(0) = y ( l + a ) / ( l - a ( y - l ) )

and the p a r t i c l e s c a t t e r i n g

factor

In Characterization of Highly Cross-linked Polymers; Labana, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

(11)

58

HIGHLY CROSS-LINKED POLYMERS

2 * 2 * P ( q ) = S* > •z

1

>

64

HIGHLY CROSS-LINKED POLYMERS

l y i n c r e a s e s the value o f ρ t o 0.8. A l i s t o f the ρ parameters f o r d i f f e r e n t models i s given i n Table J I . The reduced f i r s t cumulant Γ/q e x h i b i t s f o r the higher a convexed^urve i n the good s o l v e n t when p l o t t e d a g a i n s t q whereas the Γ/q versus q curve i n the t h e t a s o l v e n t i s w e l l a p p r o x i ­ mated by a l i n e a r l i n e . A convexed curve was p r e d i c t e d f o r the s o f t sphere model (17) and was found f o r PVAç .microgels (16) . The molecular weight dependence o f R^ and \ S J £ o f the correspon­ ding c r o s s l i n k e d p o l y s t y r e n e sampïes i n the good s o l v e n t toluene are shown i n Figure 4. The behavior i s s i m i l a r to that i n cyclohe­ xane, b u t the p o i n t s o f measurement from the two s e r i e s no longer seem t o form a common curve. The ρ parameters are about 20 t o 45% l a r g e r than i n the t h e t a s o l v e n t , a behavior t h a t i s found a l s o f o r l i n e a r p o l y s t y r e n e and which was p r e d i c t e d by theory (8,18,19). I t should be mentioned found by experiment ar ry when the Kirkwood -Oseen approach f o r the hydrodynamic i n t e r a c t i o n (1,3)is taken. 2

2

2 (b) Angular Dependence o f Γ/q . Equation (14) reduces f o r s m a l l q to

D

a

/ P

P

D

-

1

+

«/3)