Magnetic Fluids 9783110807356, 9783110143904

Table of contents :
Preface to the English edition
Foreword
Contents
Introduction
1 Magnetic Properties of Colloidal Ferromagnetics
1.1 Magnetism of Small Particles
1.2 Magnetization of Single-Domain Particle Dispersions
1.3 Magnetization Kinetics of Colloidal Ferromagnetics
1.4 Particle Interactions and Magnetic Properties of Colloids
1.5 Dispersions of Ferromagnetic Particles in Liquid Crystals
2 Structure of Ferromagnetic Colloids
2.1 Stabilization of Colloids
2.2 Thermodynamics of Colloids and Magnetic Interactions
2.3 Periodic Structures of Ferromagnetic Colloids in Thin Layers
2.4 2-Dimensional Structures of Non-Magnetic Dispersions in Magnetic Fluids
2.5 Diffraction and Scattering of Light in Structured Colloids
2.6 Magneto-Optical Effects
2.7 Small-Angle Thermal Neutron Scattering
3 Models of Magnetizing Fluids
3.1 General Principles for Developing Models of Magnetizable Media
3.2 Selection and Approbation of Magnetic Fluid Models by Statistical Modelling
3.3 Models of Magnetizable Fluids with Regard to Gyromagnetic Effects
4 Quasi-Equilibrium Hydrodynamics of Magnetic Fluids
4.1 Basic Equations
4.2 Levitation in Magnetic Fluids
4.3 Hydrostatics of a Magnetizable Fluid Drop
4.4 Structure of Magnetic Fluid Surface in a Vertical Magnetic Field
4.5 Parametric Oscillations of Free Surface of a Magnetic Fluid
4.6 Hydrodynamics of Magnetic Fluids in Thin Layers
4.7 Numerical Modelling of Two-Dimensional Hydrodynamics of Magnetic Fluid in Flat Gaps
5 Internal Rotations in Hydrodynamics of Magnetic Colloids
5.1 Magneto-Viscous Effect
5.2 Hydrodynamics of Magnetic Fluids in Rotating Fields
5.3 Rheology of Magnetic Colloids in a Field
6 Heat- and Mass-Transfer in Magnetic Fluids
6.1 Peculiarities of Thermal Processes in Magnetic Fluids
6.2 Thermomagnetic Convection
6.3 Heat-Transfer Characteristics in Magnetic Colloids and Suspensions
6.4 Mass-Transfer under Non-Stationary Diffusion in Magnetic Field
6.5 Magnetodifflusion Convection
7 Preparation of Magnetic Fluids
7.1 Preparation of Magnetic Fluids by Size Reduction
7.2 Preparation of Magnetic Fluids by Decomposing Carbonyls and Other Compounds
7.3 Preparation of Magnetic Fluids by Electrodeposition Technique
7.4 Preparation of Magnetic Colloids by Evaporation of Metal in a Liquid
7.5 Chemical Coprecipitation Technique
7.6 Method of Carrier Liquid Exchange
Appendix
References
Subject index

Citation preview

Blums . Cebers . Maiorov Magnetic Fluids

E. Blums . A. Cebers . M. M. Maiorov

Magnetic Fluids

Walter de Gruyter . Berlin· New York· 1997

Authors Professor Dr. E. Blums Dr. A. Cebers Dr. M. M. Maiorov Institute of Physics Latvian Academy of Sciences Salaspils-I, LV-2169 Latvia This book contains 226 figures.

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Cataloging-in-Puhlication Data

BlOms, Elmars. [Magnitnye zhidkosti. English] Magnetic fluids / E. Blums, A. Cebers, M. M. Maiorov. - English ed. p. em. Includes bibliographical references and index. ISBN 3-11-014390-9 (alk. paper) 2. Ferromagnetism. 3. HydroI. Magnetic fluids. dynamics. I. TSebers, A. O. (AndreI Osval'dovich), 1947. II. Ma'iorov, M. M. (Mikhail Mikha'ilovich) III. Title. QC766.M36B5813 1996 530A'2-dc2\ 96-47046 CIP

Die Deutsche Bihliothek - Cataloging-in-Puhlication Data Blums, Elmars: Magnetic fluids / E. Blums ; A. Cebers ; M. M. Maiorov. Berlin; New York: de Gruyter, 1997 ISBN 3-11-014390-9 NE: Cebers, Andrejs:; Maiorov, Mikael M.:

© Copyright 1996 by Walter de Gruyter & Co., 0-10785 Berlin All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Printing: Ratzlow Druck, Berlin. Binding: Mikolai GmbH, Berlin. Cover design: Hansbernd Lindemann, Berlin

Preface to the English edition Methods of preparation of novel materials - magnetic colloids, their structure, physical properties, transfer phenomena, hydrodynamics, thermomechanics and a variety of applications have been discussed extensively in recent scientific literature. Studies of magnetic fluids require a multi-disciplinary approach based on latest achievements in physical and colloidal chemistry, magnetism, physics of fluids and disperse systems, hydro- and thermomechanics, as well as in engineering science. However, there are still comparatively few books aimed at providing a generic view on extensive and diverse research activities. Although a couple of monographs on electrodynamics, hydrodynamics and thermomechanics of magnetic fluids have been published, these are basically concerned with macroscopical problems and fail to reflect fully the entire scope of properties of magnetic fluids as colloidal systems and their interactions with an external field. To some extent we have sought to fill this gap in a book "Magnetic fluids" by E.Blums, A.Cebers and M.Maiorov published by Zinatne Publishing House, Riga in Russian in 1989. The printed copies were sold out soon, moreover, we received many requests from our colleagues world-wide to publish a similar book in English. We would like to convey our gratitude to publishers from WaIter de Gruyter & Co who expressed their interest in publishing our manuscript. The book is based on the Russian edition of our book, therefore not all references to latest publications are included here. Despite a relatively large number of publications appearing every year, principal concepts of this book have retained interest among research community. It is evident that many ideas have been developed further and are corroborated experimentally. In order not to overextend the volume of a book we appended it only with some of our latest results, which we regarded of principal significance. The chapter describing preparation methods have been profoundly updated (Chapter 7). The book is primarily addressed to wide circles of readers having appropriate knowledge in physics, magnetism and hydrodynamics. We hope that the book can be valuable to lecturers in universities and post-graduate students in order to look for new information, as well as for research scientists in facilitating generation of new ideas. Certainly we are fully aware that the book is not free from limitations, therefore we would appreciate any constructive remarks and recommendations. We would like to acknowledge the contributions of our colleagues and friends, particularly Dr.Ronald E.Rosensweig (USA) and Prof.Klaus Stierstadt (Germany) for interest in our work, valuable remarks and support in publication of the book. The authors are grateful to their colleagues for assistance in preparation of the book: to Dr.Juris Plavins for translating and editing the manuscript, as well as for revising the bibliography, and also to Inna Yatchenko, Dina Grube, Gunta Svikle and Silvija Viluma for preparing the manuscript and drawings.

July, 1996 Riga

E. Blums A. Cebers M. Maiorov

Foreword For many years classical magnetohydrodynamics has been concerned with phenomena occurring solely due to interaction between an electric current and a magnetic field. Despite the fact, that a number of remarkable characteristic phenomena pertaining to magnetisation of natural media (paramagnetic solutions of electrolytes, air, liquid oxygen etc.) have been reported, the effects related to the magnetisation due to interaction between an electromagnetic field and a medium in virtue of the negligible . size of these, generally are not taken into consideration. Interest towards the branch of magnetohydrodynamics specifically exploring phenomena related with interactions between an electromagnetic field and a media upon the magnetisation of the latter emerges in the middle of the 1960s when artificial, easily magnetisable liquid colloidal media - magnetic fluids are synthesised. The interest is twofold. First of all, the outstanding diversity of effects of a magnetic field upon a structure and properties of colloidal dispersions of ferromagnetic materials give rise to a novel class of fundamental problems of physical, physico-chemical and hydrodynamic nature. Secondly, by virtue of advances in technology of synthesis of stable magnetic fluids on different basis, a number of various proposals concerning practical applications of magnetic fluids are set forth. Magnetofluidic seals, oscillation dampers, fluid bearings and magnetic lubricants, new sensors and elements for automatisation systems, magnetogravimetric analysers and separators, thermomagnetic converters of energy, ideas pertaining to systems for magnetic delivery of drugs, new methods of medical diagnostics - this is only a sketchy picture of vast opportunities of potential applications of magnetic fluids. In order to materialise the proposals mentioned above it is necessary not only to synthesise stable colloidal systems exhibiting high magnetisation, but also to study their physico-chemical and thermophysical properties, as well as to describe the new class of physical and hydrodynamic phenomena, taking place in magnetic fluids in the presence of an external magnetising field. A list of references relevant to the considered scope of phenomena is quite an extensive one and usually is opened by a study undertaken by Nouringer and Rosensweig (1964) in which a term of ferrohydrodynamics is attached to hydrodynamics of liquid magnetic materials. A considerable influence upon the development of research in this field is exerted by a review of early activities aimed at studying magnetic colloids, written by M.I.Shliomis [561] where a term "magnetic fluid" is proposed to denote a class of colloid-base magnetising liquids. Up to now a number of extensive studies dealing with a wide scope of physicochemical, hydrodynamic and thermophysical characteristics have been carried out, however, in fact there are no monographs offering a generic interpretation of research activities. The book presents a systematised picture of data pertaining to the interaction between magnetic fluids and a field. Moreover, alongside with exercising this approach an extensive scope of physical, hydrodynamic and thermophysical problems are considered. For instance, magnetisation peculiarities of ensembles of ferromagnetic particles are dealt with, the existing concepts of the stability of magnetic colloids related to the effect of magnetic forces are presented. Description of the mechanisms bringing about the emergence of non-uniform concentration structures, as well as formation of complicated spatial configurations of magnetic fluids is attempted on this

vrn

Foreword

basis. Authors are focusing their attention on orientation processes of magnetic colloidal particles, as well as to related phenomena these processes give rise to, including effects of the pondennotive moment of forces. A description of thennophysical phenomena is offered, caused by interaction between magnetic fluids and a field; particular interest is taken in studying specific phenomena of mass transfer occurring due to development of stratification of concentration in the presence of a non-unifonn magnetic field. A definite coverage in the book is allotted to fonnulation of models aimed at describing the behaviour of real magnetic fluids. General principles defining interactions between an electromagnetic field and a matter are treated in quite a concise way. A more elaborate consideration of these principles in relation to magnetic fluids is conveyed in a monograph by RE.Rosensweig "Ferrohydrodynamics" [521], as well as in a review by Y.Y.Gogosov, Y.ANalyetova and G.AShaposhnikova "Hydrodynamics of magnetisable liquids" [261], where a wide scope of models for magnetisable disperse systems are considered. As an introduction into topics dealt with hereafter, authors may recommend the following monographs: "Heat and mass transfer in magnetic fields" by E.Blums, Yu.AMikhailov, ROzols [78], "Introduction to thennomechanics of magnetic fluids" by Y.G.Bashtovoy, B.M.Berkovsky, AN.Vislovitch [39], whereas concerning thennophysical aspects a book "Magnetic fluids - natural convection and heat transfer" by Y.E.Fertman [232] may be suggested. Results of the theoretical and research work carried out by the authors and their colleagues at Institute of Physics of Latvian Academy of Sciences lay the foundation of the book. However, in order to achieve entirety and substantiate the conclusions authors have arrived at, the results obtained and published by other scientists in various research periodicals are made wide use of. To alleviate the main body of a book from detailed mathematical computations in a number of cases the necessary derivation of theoretical relations is included as a problem presented at the end of each corresponding chapter. Issues related to physico-chemical aspects of stabilisation of colloids in solutions of surfactants are treated in a cursory manner. Partly, this is caused by the fact that at present many aspects of this problem relevant to dispersions in non-aqueous media are beyond the scope of this monograph. However, we reckon it to be reasonable to present a short review of key methods of preparation and stabilisation of magnetic fluids, thus enabling the reader showing its initial interest to choose the most appropriate technique for preparation of experimental samples. It is worth noting that problems linked with applications of magnetic fluids are not dealt with in the book due to their extensive diversity: during a research and design stage a number of specific and technically complicated issues have to be solved, which in their tum merit to be the topic of a separate monograph. We are most grateful to our teacher Yu.AMikhailov for continuous interest in our work and numerous suggestions. Special thanks are owed to our colleagues G.Kronkalns, ROzols, AYu.Chukrov and ARimsa the results of whose research work are made use of in the book. Finally, the authors appreciate the contributions of many colleagues from different countries, who actively participated in discussing several scientific issues and kindly shared their ideas and practices, as well as placed at our disposal selected results of their research.

Foreword

IX

Introduction, Chapters 1 - 5 and Appendix are written by A. Cebers, Chapter 6 and Chapter 7 are written by E.Blums and M.M.Maiorov, respectively. Overall editing was carried out by E.Blums.

Contents Introduction 1 1.1 1.2 1.3 1.4 1.5

Magnetic Properties of Colloidal Ferromagnetics Magnetism of Small Particles Magnetization of Single-Domain Particle Dispersions Magnetization Kinetics of Colloidal Ferromagnetics Particle Interactions and Magnetic Properties of Colloids Dispersions of Ferromagnetic Particles in Liquid Crystals

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Structure of Ferromagnetic Colloids Stabilization of Colloids Thermodynamics of Colloids and Magnetic Interactions Periodic Structures of Ferromagnetic Colloids in Thin Layers 2-Dimensional Structures of Non-Magnetic Dispersions in Magnetic Fluids Diffraction and Scattering of Light in Structured Colloids Magneto-Optical Effects Small-Angle Thermal Neutron Scattering

3 Models of Magnetizing Fluids 3.1 General Principles for Developing Models of Magnetizable Media 3.2 Selection and Approbation of Magnetic Fluid Models by Statistical Modelling 3.3 Models of Magnetizable Fluids with Regard to Gyromagnetic Effects 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Quasi-Equilibrium Hydrodynamics of Magnetic Fluids Basic Equations Levitation in Magnetic Fluids Hydrostatics of a Magnetizable Fluid Drop Structure of Magnetic Fluid Surface in a Vertical Magnetic Field Parametric Oscillations of Free Surface of a Magnetic Fluid Hydrodynamics of Magnetic Fluids in Thin Layers Numerical Modelling of Two-Dimensional Hydrodynamics of Magnetic Fluid in Flat Gaps

1 11 11 14 32 48 61 67 67 72 84 94 100 104 116 129 129 141 156 163 163 169 177 193 200 206 230

5 Internal Rotations in Hydrodynamics of Magnetic Colloids 5.1 Magneto-Viscous Effect 5.2 Hydrodynamics of Magnetic Fluids in Rotating Fields 5.3 Rheology of Magnetic Colloids in a Field

241 241 253 277

6 6.1 6.2 6.3 6.4 6.5

289 289 295 312 322 332

Heat- and Mass-Transfer in Magnetic Fluids Peculiarities of Thermal Processes in Magnetic Fluids Thermomagnetic Convection Heat-Transfer Characteristics in Magnetic Colloids and Suspensions Mass-Transfer under Non-Stationary Diffusion in Magnetic Field Magnetodiffusion Convection

XII Contents

7 7.1 7.2 7.3 7.4 7.5 7.6

Preparation of Magnetic Fluids Preparation of Magnetic Fluids by Size Reduction Preparation of Magnetic Fluids by Decomposing Carbonyls and Other Compounds Preparation of Magnetic Fluids by Electrodeposition Technique Preparation of Magnetic Colloids by Evaporation of Metal in a Liquid Chemical Coprecipitation Technique Method of Carrier Liquid Exchange

Appendix Distribution Function Moments ofParticles in Orientation Space at Thermodynamic Equilibrium References Subject index

343 344 347 352 356 361 369

377 379 413

Introduction Natural liquid media exhibiting magnetic order are not known at present. The strongest magnetic properties are inherent in saturated molecular solutions of transition and rare-earth metals the magnetic susceptibility of which do no exceed the order of 104 • In virtue of this, specific effects detennined by the combination of magnetic properties with fluidity until recently have attracted little attention of research community. The situation changes markedly after artificial strongly magnetic liquids - colloidal solutions of ferromagnetic materials (about 10 nm in size) are synthesised in the middle of 60-ies. Small ferromagnetic particles consist of a single domain, consequently, the magnetic moment of an individual particle is about 105 times larger than magnetic moments of transition or rare-earth metal ions. Therefore, eventual value of the Langevin paramagnetic susceptibility of a colloidal solution will exceed that of ions by the same order of magnitude, whereas saturation of magnetisation is reached at field values as low as about 1 kOe. The field can serve as an effective means to exercise control over a magnetic fluid. For instance, volumetric pondermotive forces exceeding those of gravity by several orders of magnitude can be achieved. The forces are employed in magnetofluidic separators, acceleration and position probes and elsewhere. Feasibility of localisation of the magnetic fluid by using a field leads to development of a novel class of sealing and damping devices, puts forth new ideas in medicine and other fields of technology. Consequently the problem of magnetic fluids is brought into a scope of interest of research scientists. First research communications [459] are based on the assumption of the magnetic fluid as a one-component magnetizable medium, in which pondermotive forces can arise exclusively in the presence of a non-uniform field, however, quite soon it turned out that by virtue of particle rotation in moving colloids non-equilibrium processes of magnetic relaxation, as well as the moment of volumetric pondermotive forces shall be taken into consideration. Thus, a concept of internal rotation in magnetic fluids is formulated, description [109, 280, 372, 510] of increase of the viscosity of a colloid in the presence of a field reported in [423, 447], as well as mechanisms of transformation of the latent intrinsic rotation of colloidal particles into macroscopic motion are offered [133] (concept is explicated in Chapter 5). Necessity to include effects of internal rotation into description of magnetic fluids has led to the revision of classical models of magnetizable media [583] and has required devising of several new phenomenological ones [104, 128, 261, 303, 314, 349, 412, 560, 588]. It turned out that the offered models under definite conditions can yield contradicting results. The issue of selecting the most appropriate model pertaining to magnetic colloids has been solved by employing a numerical simulation technique just recently [388, 135] (Chapter 3). Concept of deceleration of the rotation of particles in a field allows to interpret experimental data demonstrating an increase in the viscosity of a colloid in a field. However, situations may occur when the anisotropy of rheological behaviour is observed in the presence of a field [395], which contradicts the initial concept. The next model in order of sophistication is the model of diluted suspension of ellipsoidal ferroparticles (Chapter 5). A technique of Brownian dynamics can be regarded as an efficient means of obtaining quantitative information concerning peculiarities of

2

Introduction

rheological behaviour in a field. On its basis pursuant to the experimental study reported in [395] conclusion is arrived at concerning a more pronounced effective viscosity dependence of a suspension upon the shear rate provided a field is oriented transversally with respect to the flow direction and longitudinally to the velocity gradient than in a case of the longitudinal orientation of a field. However, even this model seems to be a quite simplified one. For instance, an increase in the effective viscosity of a magnetic fluid in the field directed along the vorticity vector is observed in [404] and can not be adequately explained within a framework of a model of suspended particles with a fixed ratio of semi-axes [129]. Evidently, a more accurate model shall take into account the growth of anisotropic aggregates under the influence of a field and their break-down due to viscous stresses. The models pertaining to coarse-dispersed suspensions, in which effects of the Brownian rotation do not playa significant role, are considered in [570, 571]. By virtue of a likelihood of the phase stratification of colloids (Chapter 2) it might be of definite interest to regard a magnetic fluid as an emulsion of strongly magnetic microdrops within a weak magnetizable media [143]. Thus interesting and yet unsolved problems may arise concerning the behaviour of a magnetic drop in the shear flow in the presence of a field (Chapter 5). Interrelation between the extent of deformation of a microdrop and the shear rate can lead to significant consequences regarding the rheological behaviour of an emulsion (for example, occurrence of a negative differential effective viscosity in the presence of a field). Numerical simulation demonstrates that analogous phenomena may take place also in colloids of hard magnetic particles (Chapter 5). Under such a rheological law several hydrodynamic instabilities may occur [327, 595], giving rise to formation of cellular velocity structures. Apart from indirect corroboration regarding coarsedisperse suspensions, phenomena of this class have not yet been verified experimentally. The concept of internal rotation in suspensions from the viewpoint of mechanics of heterogeneous systems is being developed in [107]. Models considering internal rotations offer good perspectives in describing individual hydrodynamic effects in polarizable liquids. With allowance for the electroviscous effect in the electric field of a moving ion a non-monotonic dependence of the ionic mobility upon the crystallographic radius, as well as dependence of the Walden product on the dielectric permeability of a media [129, 132, 301, 302]. These phenomena are being studied intensively by experimental means at present [449]. The model employed to describe these include the relaxation equation of polarisation, as well as an equation equivalent to the magnetisation relaxation equation for magnetic fluids with internal rotation in the limiting case of weak field values. A number of effects pertaining to the hydrodynamics of polarizable heterogeneous media can be described on the basis of models providing for internal rotations. For instance, a polarisation relaxation equation of the suspension characteristic of accumulation of a free charge on interface boundaries turns out to be identical to the magnetisation relaxation equation of a ferrocolloid under stipulation that the Brownian orientation relaxation time of particles is substituted for the Maxwellian relaxation time of the free charge [132]. Within a framework of this model an existence of interesting oscillatory phenomena can be inferred [127] occurring in magnetic colloids in the presence of an electric field [462]. It is worth noting that effects of internal

Introduction

3

rotation determined by accumulation of charges on interphase boundaries can also be observed in various biological suspensions [248, 436J. Let us confine ourselves to this concise juxtaposition of polarizable and magnetizable media. It is worth pointing out that on the basis of existing analogies novel phenomena may be sought and studied. For example, discovery of conditions and a mechanism of development of labyrinth structures of magnetic fluids in flat gaps [162J facilitated the detection of analogue structures also in dielectric liquids [528J. There is a likelihood that analogy between electric and magnetic phenomena can do a service in future in studying natural two-dimensional systems - biological membranes. In neglect of gyromagnetic effects (Chapter 3) the magnetisation of colloidal ferromagnets can occur due to two independent mechanisms [561J. The first one - the Neel process of overcoming energy barriers between equilibrium states in a particle by magnetic moments due to thermal fluctuations known for superparamagnetic hard dispersions [50J. The second one - a magnetization process due to orientation of the magnetic moment jointly with a particle in a viscous medium. The last mechanism may be observed only in magnetic colloids provided the strength of a magnetising field is less than a value of the characteristic strength of the effective field of anisotropy Ha which fixes the magnetic moment along preferential directions in a particle. Since Ha dernding upon the material of a particle by order of magnitude falls within a range of 10 to 103 Oe both the mechanisms shall be taken into consideration. This can be deduced also from kinetic understanding of the magnetization of a colloid in the presence of a weak field. The magnetization relaxation time (may be expressed using the characteristic Neelian ('tN) and Brownian ('tb) relaxation times [561, 536J: 't = 'tN 'ti/('tN +tb). Hence, it may be inferred that transition from one type of magnetic relaxation to another is determined by equality 'tN = 'tb. This condition is satisfied approximately within a size range characteristic of particles in magnetic fluids, thus, in practice, the Neelian, as well as Brownian magnetization mechanism shall be taken into account. Relative contribution of each of these can be determined on the basis of the Cole-Cole plots [396J (Chapter 1). Dependence of magnetic relaxation times upon the field strength is studied in two limiting cases - 'tN >>tb ( hard dipole approximation) [412J and 'tJotC R. the minimum F'I occurs at a non-unifonn f and equals (2.08) 2 aMg / R 2 . Upon selection of a concrete remagnetization mode, let one find the upper limit for min F I and demonstrate that it coincides with (2.08)2aMg / R 2 . And indeed, for the remagnetization mode in the fonn oM =M o(- f(r)y, f(r)x, 0) (twirl mode) due to

oMn = 0 and div oM = 0 , the demagnetizing field oR is identically zero and

f «Vo) + (V&)2)dV F. =------''-:----:--::---1/ 2f (0 + )dV 2

1/ 2aM;

1

2

&

2

The minimal functional value on the right-hand side of the last equality for the given function class equals (2.08)2aMg / R 2 therefore, min quently, the upper limit of min

Pi

Pi :5: (2.08)2 aMt / R 2 .

Conse-

coincides with the lower one and at R > R. ,

(2.08)2aMg / R 2 is the minimal value of a functional Fl. Relationship (1.10) yields

the coercivity of a spherical particle with R > R.

He =

f3Mo- 4~o (1-( ~r)

(1.12)

Relationship (1.12) indicates, that at R> R. it is energetically more favourable to remagnetize a single-domain particle by the mechanism of incoherent magnetization rotation (curling mode). There is experimental evidence [478-480], corroborating relationship (1.12) for Co, Fe and Ni particles. The obtained coercivity values as affected by Co particle radius are plotted on Fig. 1.1 [479].

1400

1200

1000

800 1

-10

-:2 ·10

d

2

3

,cm-

2

4

Fig. 1.1 The coercivity of the hard Co dispersion versus particle size (T = 4.2 K) [479].

18

Magnetic Properties of Colloidal Ferromagnetics

Manifestation of incoherent rotation mechanisms for magnetic spins of small particles is governed by the competition between the exchange interaction energies, growing in the event, that a non-uniform magnetization distribution appears, and the demagnetizing field energy, which may decrease in this case. Upon solidification of a ferromagnetic colloid, the fixation of chain structures, caused by the magnetic dipole interaction of single-domain particles, may occur. The magnetization of the similar structures, provided the crystallographic constant of magnetic anisotropy is sufficiently small (K1 «M 2 ), is determined by the magnetic dipole interaction and is incoherent in nature [309]. Here the incoherence is attributed to the magnetic dipole interaction energy drop owing to occurrence of non-uniform distribution of particle magnetic moments in a chain structure during the magnetization process. Peculiar magnetic properties of these structures permits one to detect their presence in the samples under investigation. The coercivity of single-domain particle chain structure as affected by the angle \jI between directions of chain axes and the magnetizing field strength at K 1 «M2 is calculated in [309] (see problem 1.2). It is demonstrated that, provided the angles \jI between the directions of chain axes and the magnetizing field strength are smaller than a certain limiting \jI 0' the coercivity is determined by the appearance of a fan-like individual particle magnetic moment distribution in a chain. Here, an approximately constant He value is retained, possessing a feebly pronounced maximum at \jI - \jI o. In the range \jI > \jI 0 the remagnetization of an ensemble is governed by the mechanism of coherent rotation of magnetic moments, hence He diminishes, going to zero at \jI = 1t /2.

3

2

10

30

50

70

90

Fig. 1.2 The coercivity of an ensemble of ferroparticle chains as affected by the angle \jI between the chain axes and the magnetizing field. - the coercivity owing to the incoherent remagnetization mechanism for chains of two and infinite number of particles, - - - the same in case of coherent remagnetization.

Magnetization of Single-Domain Particle Dispersions

19

The dependencies of Hc(IV) for chain structures are displayed graphically in Fig. 1.2. The similar dependencies for frozen water-base magnetite and toluene-base cobalt colloids are reported in [117] (Fig. 1.3). Qualitatively distinct data [270] from those, presented in Fig. 1.3, are obtained for textured in a field of 30 kOe ferrite cobalt samples. The dependence of the coercivity upon the angle, texture axis makes with the magnetizing field strength is plotted in Fig. 1.4. The presence of the ensemble coercivity upon magnetization, perpendicular with respect to the texture axis, indicates that the cubic crystallographic anisotropy of a particle plays a remarkable role in the case under consideration. This conclusion is in conformity with the quantitative agreement of cubic crystallographic anisotropy constant value for the sample particles, frozen without a field, and arrived at on account of the saturation magnetization behaviour, with those, listed in table for T = 77 K (see Fig. 1.6). Thus, it is permissible to conclude, that the cobalt ferrite particles possess a crystalline, not an amorphous structure. Quantitative description of data on the He dependence upon IV (Fig. 1.4) require one to allow for the contribution due to superparamagnetic particles, present in an ensemble at T = 77 K. This is pointed out, firstly, by the findings, obtained by using numerical calculations, according to which, the dependence He (IV) for ideally textured ensemble of particles possessing cubic crystallographic symmetry exhibit a minimum, in contrast to the dependence, expressed in Fig. 1.4 at IV ~ 50°. Secondly, the of textured

remanence

samples

at

(Mr / M. = 0.5)

IV = 0°

and

IV = 90°

0.5

o

50

90

Fig. 1.3 The coercivity of textured toluene-base cobalt colloid (1) and water-base magnetite colloid (2) versus the angle between the texture axis and magnetizing magnetic field [I 17].

20

Magnetic Properties of Colloidal Ferromagnetics

HcI s

0.8 -K-

0.6

--. -. - -.-....._-.

-- ..............

-. ..............

0.4

......

.......

so Fig. 1.4 The coercivity of textured in a field 000 kOe ensemble of cobalt ferrite particles as a ftmction of the angle between the texture axis and a magnetizing field [270] (K is the cubic crystallographic anisotropy constant for cobalt ferrite; I. is its saturation magnetization at T = 77 K).

(Mr

/

M.

:=

0.31) is significantly lower than the theory predicts for ideally textured

dispersions ofnon-superparamagnetic dispersions - 1 and 4/ 1t sin(1t /4) [270J. Problem 1.2 Determine the coercivity of an ensemble of spherical ferromagnetic particle chain aggregates with allowance for coherent and incoherent remagnetization mechanisms as affected by the angle 'II between the directions of the magnetizing field strength and axes of chain aggregates ([309]). Regard that upon incoherent remagnetization a fan-like distribution of magnetic moments along a chain aggregate m; = m(sine cosvo; { I; v . (v,O)'t o) =

exp(~(~-I)) I+exp -- (v)) --I . (2av

kT

o

V

o

Since q>\ (v, O)'t o) exhibit a pronounced maximum at v = Vo and runs close to zero at v > Vo and v < Yo, relationship (1.55) for X" yields

X',X"

0.5

-----.Jl..&..&.-.-----.J>~ 10

10

2

10

3

10

4

10

5

..

- --f, Hz

10 6

Fig. 1.22 Dispersion of magnetic susceptibility for a concentrated kerosene-base magnetite colloid. Circles represent the experimental [mdings, reported in [506].

tr

Magnetization Kinetics of Colloidal Ferromagnetics

x" = x. Vof;V,)

j -0)

Hcz

= 1t 1K / D 1M oc.

(1.85)

Combining (1.82) and (1.85) yields the ratio H C1

H cI

1t '"

1

(d)1

8clnIl d D

'

which gives estimate of about 0.01 , corroborating the efficiency of the cooperative remagnetization. In a field of H > H C1 , the magnetization average over the layer thickness of a ferronematic sample reads < M. >= cMo(2E(sin 1jI.. /2) / K(sin IjIm /2) - 1), (1.86) where E is the complete elliptical integral of the second kind. The magnetization curve for a ferronematic is plotted in Fig. 1.33. The sample coercivity is estimated to be about 1.5 H c1 .

2 Structure of Ferromagnetic Colloids 2.1 Stabilization of Colloids In order to prepare stable ferromagnetic colloids, which do not settle when experiencing a gravitational force, or subjected to an uniform or non-uniform magnetic field, it is necessary to stabilize the particles, thus reducing the attraction forces, acting between them. Surfactants may serve for this objective by adsorbing onto the surface of particles thus preventing them from approaching one another. In some cases the presence of repulsion forces during the surfactant adsorption on particles may be related to the appearance of the entropic barrier, forming as a consequence of a decrease in the number of possible conformations of adsorbed molecules when the particles approach each other. The entropic barrier for the adsorption of polymer molecules can be evaluated on the bases of scaling considerations [201]. For the free energy of a polymer molecule, adsorbed in a gap of thickness d is permissible to write F = kTN(a / di f3 , where a is the size of a monomer; N is the number of monomers. Upon bringing two spherical particles of radius R (R»8, 8 is the surface-to-surface separation distance) together, with cr polymer molecules adsorbed per unit area, the integration over the varying gap thickness between the particles, omitting a constant coefficient, yields for the free energy of adsorbed molecules [482] Fa = kTcrRNa5/38-213 . By introducing the molecule size / = N 3f5 a [201] the previous formula may be recast as Fa = kTcrR/ 5/3 /8 213 • The thickness of adsorbed surfactant layers on colloidal ferromagnetic particles are assessed experimentally on the basis of the findings on viscosity [617], magnetoviscous effect [326, 422, 447], kinetics of magnetic birefringence [403, 576], magnetic relaxation [393] etc. The absorption layers of surfactants may disrupt upon dilution of a colloid with pure carrying medium [171] or upon addition of coagulants [208, 485]. Obviously, the stability of colloids may be reduced due to the presence offree surfactant in a carrying medium owing to the occurrence of the displacement flocculation [453] (problem 2.1). The rupture of stability of the toluene-base cobalt colloid, stabilized with sodium dioctyl sulphosuccinate, is reported upon dilution with a carrying medium in [171]. The concentration distribution curves for ferroparticles under gravitational sedimentation as registered at various elapsed time intervals after a 200-fold dilution with pure toluene are plotted in Fig. 2.1. The size of aggregates as determined from the kinetics of the concentration profiles is about 1 fJ.m. The dilution of a system with 10% solution of surfactant in toluene does not give rise to the gravitational separation [171]. As a consequence, the destabilization of a colloid upon dilution may be attributed to the desorption of a surfactant molecule from the particle surface due to their diminishing concentration in an ambient liquid.

68

Structure ofFerromagnetic Colloids

RELATIVE CONCENTRATION

3

1200h

2 170h

6 Fig. 2.1 Concentration profile for stabilized toluene-base cobalt colloid after 200-fold dilution with pure toluene (171].

Analogous phenomena of forming large conglomerates upon dilution with a carrying medium are reported in [215, 213], where an oleic-acid-stabilized magnetite colloids in kerosene are investigated. It is of interest to discuss the data, obtained by using a technique of IR-spectroscopy. It turns out, that according to the technological circumstances, occurs either physical adsorption of oleic acid onto the particles, or chemosorption. A distinctive character of the binding of a surfactant molecule onto a particle may also explain the failure to observe the loss of stability upon dilution with a carrying medium [551]. Interesting data on the kinetics of the fonnation of ferroparticle aggregates upon dilution of a magnetite colloid in kerosene, stabilized with oleic acid, when a coagulant (butyl alcohol) is added to the system are obtained [485] on the basis of the Procopiu effect [380]. In this case applying a small periodic component H' perpendicular with respect to a constant field, gives rise to oscillations of the magnetization component at a doubled frequency, directed along H o. Provided the period of a field change is substantially smaller than a characteristic magnetic relaxation time of a colloid, the phenomenon can be treated in an equilibrium approximation. For the magnetization according to the Langevin law

= nmL(s)b (S = mJH5 + H,2

/ kT, b = (H o + H') / J H5 + H,2 the magnetization component M., pointing along H o, may be found from the following equation M

ronm3 (SoL'(So) - L(So» H,2 . 2 · M • = --2 2 a sm rot, (k1) 2S o

= H~ sinwt, H~« Ho ~o =mHo / kT. M:, registered by the emf E, induced in a coil, falls at the value So

H'

A peak for = 1,93. A change in E upon the dilution of a colloid with butyl alcohol is depicted in Fig. 2.2. [485]. A shift of the induced emf peak towards lower field values indicates, that the addition of coagulant cause the particles to unite to fonn aggregates, possessing the

Stabilization of Colloids

69

E 30

0-1

"-2 6-3 ·-4 x-5 0-6 ·-7

o

H,Oe 50

100

Fig.2.2 The Procopiu effect in a kerosene-base magnetite colloid (saturation magnetization 45 G [485]): 1 - prior to introduction of a coagulant - butyl alcohol (remaining data are taken after corresponding time lapses, following the introduction of a coagulant in the amount of 30% by volume): t = 0 (2); 15 min (3); 30 min (4); 45 min (5); 2 hrs (6); 24 hrs (7).

magnetic moment nearly 5 times the initial value for the particles. Kinetics of the peak size and position for curves in Fig. 2.2, is indicative of narrowing of the aggregate size distribution with time with respect to the definite mean value, which remains nearly constant. Water-base ferromagnetic colloids can be stabilized also by making use of the electrostatic method without resort to surfactants. A technique for preparing the like systems is described in [416J. The colloidal particles are synthesized by means of a coprecipitation reaction from an aqueous mixture of ferric chloride and ferrous chloride in the presence of an alkali (solution of ammonia) and peptizing the precipitate in an alkali medium (tetramethylammonium hydroxide) or the acidic one (perchloric acid). In an alkali medium the hydroxyl ions determine the potential, whereas ions of N(CH3) / serve as counterions. At a sufficiently high counterion concentration, an effective shielding of the electrostatic repulsion of the particles is furnished and phases separation occurs, resulting in the phases of low and high particle concentration [31 J. The obtained phase diagram for a colloid (the particle concentration v in Fe moles per liter is expressed in terms of the volume fraction by using formula c = IJ.' 10-3 v /3p, IJ. is the molecular mass of a magnetite - 232; P is its density - 5.2 g1cm3) is depicted in Fig. 2.3. It is worth noting that an external magnetic field evokes the phase separation of a colloid, otherwise being stable in the absence of a field [31J. This reveals the role the magnetic

70

Structure of Ferromagnetic Colloids

• • • •

• • • •

•

• •

Q5

• •

•

o

1

2

l

I

10

15

Fig. 2.3 A phase diagram of water-base magnetite colloid, electrostatically stabilized at alkali reaction of a carrying liquid [30].

interactions between the particles may play during a phase separation. Similar phenomena occur in carbohydrate-based colloidal systems [32, 33]. Influence of magnetic interactions between particles upon the stability of magnetic fluids proves the necessity of devising appropriate thermodynamic models. In contrast to conventional systems, the phase diagrams of ferromagnetic colloids apart from the concentration and temperature are affected also by the field strength (see following section). Problem 2.1 Consider the thermodynamic stability of a magnetic colloid with regard to the osmotic force, caused by the presence of free molecules of a surfactant in a carrying liquid. Solution Upon approaching of colloidal particles within a distance less than the size of molecules of a surfactant, the last fail to penetrate into the gap between particles and the resultant force of osmotic pressure, acting upon the particles, assumes values other than zero [453]. The action of this force leads to a mutual attraction of the particles, which, in the event these possess a spherical shape may be computed by the following formula F = PoS, (2.1) where S = 1t[(D p

+ Db ) 2 _1 2 ] I 4; Dp , Db

are the diameters of dispersed particles and

dissolved coils; I is the distance between the centers of particles, and po is the osmotic pressure of a solution. The following potential corresponds to the force (2.1) in the event the osmotic pressure is determined in the ideal solution approximation Po = kTnp

V(/) = -kTn p~[3:.(Dp +DS -/(Dp+Db )2 +.!./l ], Dp < I < Dp+Db ,

4 3 3 as well as yields the following second virial coefficient for a gas of dispersed particles

Stabilization of Colloids

71

B1 =2/Pr'[l-exp[n,1t[~(D, +DS -lCD, +Db)1 +~I)Jf ]]Pdl. The van der Waals interpolation fonnula for the osmotic pressure of a disperse particle gas yields [365]:

nkT

1

p = - - + n kTB1 • I-nv o With allowance for the magnetic interactions within a framework of the mean effective field model [156] the equation of state in tenns of p may be written in the following fonn nkT I p = - - + n1kTB1 --A(nm)1 L1(~.) (2.2) I-nvo 2

Owing to the action of osmotic forces in a solution, the particle interaction displays an attraction

type behaviour, B2 asswnes negative values and a disrupture of the thennodynamic stability of a colloid may occur even in the absence of magneto-dipole forces. The critical concentration for a free surfactant may be obtained from equations

P) =(8 =0. (8anT anT 1

;)

In the absence of the magnetic interactions of particles these yield a ratio for determination of the effective volume fraction of a free surfactant in the fonn

c;

f I

1I(1+~/D,»)

[1- exp(8(l- 3Z I / 3 /2 + z / 2»]dz = -

27

8(1+ Db / Dp )

3'

(2.3)

where

1tD~ 3 8 =n p -6- (l + Dp / Db) . The

c;

values, obtained by means of equation (2.3) equal about 0.5 at Db / D, e[O, 2; I].

Interestingly, the values of the volwne fraction for free surfactant coils, at which a colloid may lose its thennodynamic stability, correspond to the range of semi-diluted solutions [201). Moreover, if the molecule size for an oleic acid is assumed 2 nrn, for the volwne fraction of coils in a 10% solution of free surfactant in kerosene one arrives at the value, close to the aforesaid 0.5. This pennits one to conclude, that the fonnation of microdrop aggregates of ferroparticles, so readily defonning in weak magnetic fields [213, 215], may be attributed to manifestation of the osmotic forces due to an excess surfactant. As a consequence, it follows, that in order to

0.5

CC

P

Fig. 2.4 Critical effective volwne concentration of a solution as affected by the magnetic field strength. y = 1 (1); 2 (2); 3 (3); 4 (4).

72

Structure of Ferromagnetic Colloids

increase the stability of the ferrocolloids it is advisable to remove the free surfactant. The excess presence of a surfactant may be monitored by a JR.-spectroscopy teclmique [551]. Provided the magneto-dipole interaction parameter y =n/..o / 2nvo for ferroparticles assumes sufficiently large values, upon applying a magnetic field the critical volume concentration of a free surfactant diminishes significantly. It is illustrated in Fig. 2.4 for various y values by plotting against the magnetic field strength as calculated on the basis of equation of state

c;

(2.2) and relation (2.3) at an actual oleic acid value Db / Dp =0.2.

2.2 Thennodynamics of Colloids and Magnetic Interactions To describe the structural transformations of magnetic fluids, occurring in an applied field, as well as a change in the particle concentration and temperature, the thermodynamic models are required, which provide for the existence of magnetic particle-particle interactions. An essential characteristic of ferromagnetic colloids, unlike the majority of known magnetic systems, is the translational freedom with respect to the motion of particles, which may serve as a cause for the structural transformations, related to a simultaneous change in a character of magnetic ordering and spatial positioning of the particles. Several thermodynamic models, apt to describe the above effects, are known at the present moment [156, 439, 440, 533, 562]. Among these, a model is devised on the basis of the mean effective field approximation [156]. The fundamentals of this model statistically are substantiated in [533]. The model assumes, that apart from a macroscopic field of strength H, the given dipole experiences a local field HI = AM, proportional to magnetization. A choice of a local field, possessing a similar form, may be based on lattice models for dipole systems [489] or upon cutting the Lorentz sphere out of a uniformly magnetized medium [599], when A takes the value 4m'3. The magnetic part of the free energy 1m per unit volume of a dipole system, allowed for the energy of the magnetic particle-particle interaction and the orientational disorder of dipoles in the mean effective field He = H + AM may be written as

1m =-nH-1I2nHI +kTnj/ ln /d 2 e

(2.4)

the multiplier 1/2 in the second term is introduced to provide that the interaction of the particle under consideration with others is taken into account only once; the distribution function for dipole orientations is essentially a Boltzmann type in an effective field He 1= Z-I exp(mH e ! kT). The effective field value may be determined by satisfying the self-consistency condition (Se = mHe ! kT ):

Se = So + nm 2 AL(Se)! kT.

(2.5)

In this case (2.4) yields 2

2 2

r = -kTn In 41tshSe + A.n m L (Se) Jm

Se

2

(2.6)

Relationships (2.5) and (2.6) indicate the occurrence of the spontaneuos magnetization in a colloid at nm 2 A! kT > 3. Besides, relationship (2.6) describes a phenomenon of the particle aggregation, caused by the magnetic particle-particle interaction. This is opposed by the thermal translational motion of the particles, as well

Thennodynamics of Colloids and Magnetic Interactions

73

as by the existence of an excluded volume owing to the impermeability of particles. In an expression for the free energy of a colloid [156] the above factors are taken into account in a fashion analogous to that, used in the van der Waals gas model [365]. Hence, the free energy per unit volume /vol within a framework of the mean effective field model may be written as nvo sb;e I 2 2 2 (2.7) Ivol =nIO(1)+kTnln---kTnln--+-An m L , I-nv O ~e 2 where Vo = 4v, v is the volume of the ferroparticle possessing a solvate layer, which is considered incompressible. On the basis of relationships (2.5) and (2.7) one may obtain equations of state for the chemical potential of particles cP =

(0On1 )

and the osmotic pressure p = cpn - Ivol: T.H

_ r kTln -nv kT- - kTln sh~e CP-Jo+ -o- + -, I-n~

I-n~

(2.8)

~e

and

kTn 1 2 2 p=----A(nm) L (~e)' (2.9) I-nv o 2 Equation of state for the osmotic pressure (2.9) with the last factor in the second term, depending upon the magnetic field strength, excluded, bears analogy to the equation of state for the van der Waals gas. Increase of the particle attraction with the field strength demonstrates a possibility of the particle condensation when a field is applied. A boundary of the thermodynamic stability region with respect to the spatial stratification is defined by equation

( Ocp)

= O. On T.H The instability of this type occurs, provided the magneto-dipole interaction parameter

y = Am 2 /2v okT exceeds a certain critical value y c' dependent upon the field strength; y c is found from the equations, determining a critical point:

(:) =(02~) =0. On T.H

T.H

r 4

5

2~-------=----------:-,=-

o

5

10

Fig.2.5 The critical value of magneto-dipole interaction parameter as affected by the Langevin parameter.

74

Structure ofFerromagnetic Colloids

10

8

o

5

10

Fig. 2.6 Volwne fraction of colloidal particles at a critical stratification point as a function of the Langevin parameter.

For large

~ one arrives at

'Y c = 2:

(I + t) .

The dependence of 'Y c on the parameter

~ value is shown in Fig. 2.5. The volume concentrations of particles at a critical point for various field strength values are shown in Fig. 2.6. The boundaries of a region for the thermodynamic stability of a colloid (spinodal) for various 'Y values are depicted in Fig. 2.7. It is worth noting, that at 'Y = 4.7 the upper and lower critical field strength values exist, above and below which a colloid is unstable with respect to a spatial phase separation. The occurrence of this effect may be explained quantitatively within a framework of the model under consideration by recalling, that the magnetic susceptibility of the ordered phase near the Curie point (nA o = 3) X. = a / 2(Tk - T) is smaller than the magnetic susceptibility of the

disordered phase X. = a / (T - Tk ) [533]. As a consequence, upon applying a field the chemical potential of the disordered phase suffers a more marked drop than the ordered one, thus leading to a restoration of its stability. Since the critical value of 'Y c' at which the ordered phase experiences an onset of the instability with respect to the

10

5

3

o

20

Fig. 2.7 Thermodynamic instability region for a ferromagnetic colloid as a function of strength of a magnetic field. y == 4.7 (1); 5 (2); 6 (3).

Thennodynarnics of Colloids and Magnetic mteractions

75

stratification, equals 4.08, the given phenomenon exists in a narrow interval of values ye[4.08; 4.77]. Upon an onset of the thennodynamic instability a colloid separates into phases, corresponding to the thennodynamically stable branches ((: ) T,H >

0)

of the

concentration-dependence of the chemical potential. One of the phases in this case possesses a higher particle concentration and is magnetized even in the absence of a field. At a thennodynamic equilibrium the chemical potentials and osmotic pressures of phases are equal, i.e., denote the averaging over a

distribution of the directions of the magnetic moments of particles in a moving colloid. On the basis of numerical calculations it is revealed [135, 388] that a distribution of particles in a moving colloid in the orientational space may by satisfactorily described in the effective field approximation, when the distribution function for particle orientations j(e) reads I(e) = Q-l exp(mH.I kT). (2.62) In the stationary case when H = He, and 00 = 0oe, the magnitude and direction of the effective field may be determined from the following equations cose = 1;, /1;;

2't BOO

/

(s. = mH. / kT )

I; = (1/1;, - 1/ L(I;,» sine,

(2.63)

where e is an angle the direction of the macroscopic magnetization of a colloid makes with the vector of the magnetic field strength. It is instructive to note, that a certain discrepancy between the effective field approximation and the numerical simulation data occurs at 2't BOO / I; > 1 and may be attributed to a qualitative rearrangement in the Brownian dynamics of a dipole in a moving colloid, consisting in a transition from random jitter of a magnetic dipole near the equilibrium to a stochastic wandering along a family of periodic trajectories of a dipole in an orientational space [280, 387]. Upon an onset of the latter mode an isotropization of the orientational distribution of the particles occur, manifesting in the distribution diagrams of scattered neutrons. According to (2.61) the intensity of scattered neutrons may be determined from relationship Fd(qXb1 + p1(1_ < (eq)1 »). In confonnity with (2.62) in the effective field approximation the following expression applies for the moments

< ee >= L(S.) , l'

;1

O. + (1- L(S. »)h~h'. II'

~C'

,"

H) .

Consider the distribution diagrams of scattered neutrons at kliOo and kll[~ x Introducing an angle cos

2

2

sin2 sin 1, when a distribution of the magnetic moments of the particles becomes isotropic, crevasses in the diagram of scattering disappear (curve 3). This may serve as a criterion of a transition to the second mode of dynamics of a magnetic dipole. If the incident beam points along the vector [0 0 x H] , the anisotropy of a scattering diagram may be determined by the following multiplier 2

1+J;--(l- < b

e; > cos

2

cp-
sin

2

cp).

This diagram for the same parameter values as in Fig. 2.44 is presented in Fig. 2.45. Dashed line in Fig. 2.45 stands for the scattering diagram, obtained at 2"t BOO = 27 and the values of the

e;

e;

moments < >, < >, found by using the method of Brownian dynamics [135]. A discrepancy between this and the dependence (curve 3), determined in the effective actual field approximation, is insignificant. As seen in Fig. 2.45, a crevasse in the scattering diagram disappears with the intensity of the hydrodynamic vortex increasing (analogous to the case, considered in Fig. 2.44). The performed evaluation demonstrates, that the small angle scattering of neutrons permits one to obtain a valuable information about the dynamics of the magnetic moment of a particle in a moving colloid, which is of utmost importance when the models for such media are devised with the inner rotation allowed for. ,...,------~

Fig. 2.45 Indicatrix of scattering of non-polarized thermal neutrons in a moving magnetic colloid. Incident beam is directed along the vector

[no x H].

2"t BO O = 0 (1); 15 (2);

27 (3) (~ = 20). Dashed line represents the dependence, found using the results of the Brownian dynamics [135].

3 Models of Magnetizing Fluids 3.1 General Principles for Developing Models of Magnetizable Media Magnetic fluids or colloidal ferromagnetics belong to the wide class of media - the magnetizing liquids. Therefore, upon developing models of magnetic fluids one should proceed from the general principles of describing the interaction of continuous medium with an electromagnetic field [549, 550J. In general, the interaction should be considered on relativistic terms, though, owing to negligibly small velocities in practice, it is reasonable to confine the analysis to solely a non-relativistic case. Various questions concerning the development of relativistic models of magnetizing media is considered in [184, 204, 649J. Description of the interaction between an electromagnetic field and a medium is based on dividing the full system into 2 subsystems: a field and a medium. As the energy density of a field in a medium it is convenient to select the following quantity H'B' E'D' e =--+-I

87t

87t'

where the field characteristics are computed in an intrinsic coordinate system of a material element of a continuous medium (designated as primed quantities). Since in the absence of the electric charges the strength of an electric field is proportional to vic, in the non-relativistic limit the energy density of an electric field may be neglected. The equation for an influx of the energy from a medium is of fundamental importance for constructing the models of continuous media, which for its material element may be written in the following form [549J p8VdE = O.4 e' +8Qe +8Q'··, where 0.4 e' is the work, performed by external mechanical surface forces; 8Qe is the influx of heat; 8Q'·· is the influx of energy other than work or heat, which for the case in question may include the work, performed, for instance, by surface couples. The field energy influx 8Q;/·· to a medium becomes -(

~: +divjf )8vot;

here the energy flux of an electromagnetic field jf in a laboratory coordinate system with regard to Galilean transformations reads ··0 e v T(n) 1jk =ljk+vk f - ; ik .

Here TJn) is the stresses tensor of the field;

jj = ~[E' x HJ

is the energy flux of a 47t field in an intrinsic coordinate system. It is instructive to note, that the characteristics of a magnetic field in the absence of the electric charges in the non-relativistic limit remain invariant. As a result equation for the energy influx of a magnetizing media and field becomes p8Vd(tJ +l/2MH+H 2 I 87tp +v 2 12) = O.4 e +8Qe +8Q;; +8Q"", (3.1)

130 Models of Magnetizing Fluids

where Me = otfViPiknkdS is work performed by full external surface forces; nd

oQ:t = -otf -=-[E' x H] dS but the influx of heat oQe according to the 2 41t n

law of

thermodynamics may be expressed in the following form OQe = poVTciS -oQ',

where OQ' is the uncompensated heat, describing the irreversible processes in a continuous medium. A model of a continuous medium is defined provided the uncompensated heat is given and the internal energy as a function of the parameters of state of a continuous medium is known. It is exactly in this stage the diversity of potential models of magnetizing media appears. Consideration of various concrete models is deferred until a later section. In the non-relativistic approximation in the absence of the electric charges the field momentum may be neglected. In this case the theorem of kinetic energy and (3.1) permit to obtain the following relationship for the internal energy VI = V + 1/ 2MH of a material element of a continuous medium, viz., equation of the heat influx Ov dM pLlVOV l = - ' (J;kLlVOt+pM7dS1 -oQ' +pH-LlVOt+oQ &k &

00

(3.2)

Here the following identity is taken account of 2

-

dM Ovr T pd-H - - +C- d·tv[E' x H) =-pH -+-Lrk> dt 81tp 41t dt &k

where Tik = 11 4(HrB k -1/ 2H2o rk ),

and 0rk = Prk - Tik according to its definition. In the following in terms of equation for the influx of heat (3.2) the most known models suitable for description of the magnetic fluids are classed. In the majority of cases the energy influx oQ" other than the work, performed by the external surface forces, may be neglected. Let one consider the situations thus evoked. 1 Quasi-equilibrium approximation. In the most simple case it is assumed, that the irreversible processes are caused solely by the viscous friction and thermal conductivity. Thus, oQ' = Ovi 'tikLlVOt, where 'tik is the tensor of the viscous stresses. &k

The internal energy is governed by the following relation: dV I =-pd(1/p)+TdS+HdM.

In this case the processes of the magnetic relaxation are neglected, i.e., it is assumed that MIIH. A given model corresponds to the so-called quasi-equilibrium approximation in ferrohydrodynamics [459]. Phenomena describable within a framework of the above approximation are considered in chapter 4. 2 Processes of magnetic relaxation. A more sophisticated model of magnetic fluids may be developed, if besides the irreversible processes of viscous friction and

General Principles for Developing Models of Magnetizable Media 131

thennal conductivity, a phenomenon of the magnetic relaxation is taken into account. Thus, for the uncompensated heat one may adopt [128] 8Q' Ov i dM 8V8t ='tik Ox - Pat (He - H). (3.3) k

The determining relationship for the internal energy of a magnetizing medium in this event reads dU 1 =TdS-pd(l/p) + HedM. (3.4) Here He with other parameters of state being equal is determined by the magnetization

r

law of a medium at thennodynamic equilibrium M=f(H) in the fonn He = l (M) . In order to obtain a closed system of equations for a continuous medium it is necessary to introduce an additional assumption concerning the absence of the intrinsic angular momentum of a medium and field. Thus, from equation of moments a condition of the absence of an antisymmetric part of the full stresses tensor Pik may be inferred. As a consequence, relationship (3.3) may be recast to give k) dM - +Ov- (p - - [1 -rotvxM] (H -H).

l(Ov; -8Q' - = ' t ' sk 8V8t

I

2 Oxk

Ox;

dt

2

e

(3.5)

The governing relationships for thennodynamic flux, corresponding to (3.5) may be obtained within a framework of the linear thennodynamics of irreversible processes. The phenomenological coefficients, relating the laumanian derivative of the magnetization to the thennodynamic force He - H, due to the presence of the vector parameter of state M may possess anisotropy, i.e.,

(p dM -[00 dt

x M]) = -L ile (M)(H. - H) k' ;

(3.6)

It is worth noting, that defining the uncompensated heat by (3.3) implies a significant assumption, which is to be substantiated by juxtaposing it with kinetic theories. For a single-domain colloid of non-interacting particles the selection of the non-compensated heat in the fonn of (3.3) corresponds to the effective field approximation [371, 412]. The suitability of this approximation, including the situations far from thennodynamic equilibrium, is studied by a method of numerical simulation in [135, 388] (issue of adequacy of the model, based upon the effective field approximation, is discussed in 3.2). 3 Irreversible processes in internal orientational space. In a number of cases a need for a fuller description of irreversible magnetization processes may be fulfilled on the basis of considering the magnetic moment distribution function of particles in an orientational space. The uncompensated heat in this case may be represented in the following fonn [128]

f'

8Q' Ov;- n Je K ell()d - = 'tik e 2 e, 8V8t Oxk where df(e) = -K (. ) dt e J. ,

(3.7)

132 Models of Magnetizing Fluids

and )l(e) is the chemical potential of particles in an orientational space. By assuming the absence of the intrinsic angular momentum for the asymmetric stresses of a medium the following expression holds ('tf = elik't;k)

'tQ

=

-f f(e)K.J.1(e)d e. 2

Relationship (3.7) in accordance with the last identity may be written in the following form:

.!.( Ov; Ov -nf 0. -o.,,/(e))K.)l(e)d

8Q' = 't:k + k) 8V8t 2 Ox k Ox;

2

e,

Determining relationship for the internal energy in this case reads pdU\ = pTdS - pd(l/ p)+pHdM+nf J.1(e)df(e)d 2 e.

The unknown flues may be determined within a framework of the linear thermodynamics of irreversible processes. By defining corresponding equations of state for the chemical potential in an orientational space, the considered model permits to describe also the effects of incomplete freezing-in of the magnetic moment in colloidal particles. Equation of state for the chemical potential of single-domain particles possessing one easy magnetization axis becomes [13 7] J.1(e) = -kTln f exp(S(mh) + cr(me)2 )d 2

m+kTlnf(e) +const.

Correspondingly, for rigid magnetic dipoles it is permissible to write )lee) = -meH+kTlnf(e)+const Within a framework of the given model by defining linear phenomenological equations and the anisotropy, determined by the unit vector e in the direction of the long axis of a prolated particle, taken into account, it is feasible to describe the rheological properties of a suspension in a field with the rotational Brownian motion of particles allowed for (detailed discussion on this subject is deferred until 5.3). By specifying corresponding equation of state for the chemical potential )l(e) effects of the particle-particle orientational interaction may be provided for, in particular the kinetic model of nematic liquid crystals developed [146] and on its basis the dependence of their phenomenological viscosity coefficients upon the orientational ordering parameter etc. may be obtained. 4 Internal angular momentum of medium. In a number of magnetic fluid studies [261, 560, 588] it is suggested to account for the intrinsic angular momentum of colloidal particles. The uncompensated heat, providing for irreversible processes of viscous friction, magnetic relaxation, as well as for an exchange of the angular momentum between particles and a medium, may be expressed for this case in the following form 8Q' 8V8t

='ts ik

.!.(Ov; + Ovk)_[(pdM_[OXM]~(H -H)+'tQ(O-Ou)]. 2 Ox Ox; dt ~.

(3.8)

k

Equation for the internal energy corresponding to (3.8) is given by dUI = TdS-pd(lIp)+H.dM+Od(IO). (3.9) The last term of (3.9) accounts for a change in the internal energy of a medium, attributed to the rotation of the colloidal particles.

General Principles for Developing Models of Magnetizable Media 133

't

a

In cases when a change in the intrinsic angular momentum may be neglected, =[M x HJ , thus, with regard to linear determining relationships 't a =a(n - 0.0)

(p ~ -[nxM])

=-L;k(M)(H. -Hh; I

equation for the magnetic relaxation [82J reads p ~;

=[0.0 xMl +a-I[rMxHJxML -L;k(H. -H)k·

In [560, 588J the relaxation term in (3.10) is expressed as

-'t-

(3.10) 1

(M-M o), where

M o is the magnetization of a medium in a state of thermodynamic equilibrium. It is not possible to chose between equations (3.6) and (3.10) in order to describe processes of the magnetic relaxation basing solely on phenomenological reasoning. This issue is of principal importance since equation (3.10), as it follows from [563 J at sufficiently great no't and ~ exhibit an hysteresis of the magnetization dependence upon the intensity of an hydrodynamic vortex, whereas relaxation equations do not possess such hysteresis. Numerical experiments dealing with simulation of the rotational Brownian motion indicate that relaxation equations (3.6) offer an adequate description of the magnetic colloids. There is a variety of models of magnetizing media at present with non-zero couple stresses. This class of models deserves a detailed consideration. S Couple stresses under diffusion of internal angular momentum. Upon describing several phenomena observed in the experiment (for instance, the occurrence of macroscopic motion of a ferromagnetic colloid in a rotating field) the models with a non-zero couple stresses 1t ik are resorted to [642, 587]. In the cases considered until now the occurrence of the couple stresses is related to irreversible processes. Dissipative phenomena in the models with the intrinsic angular momentum of a medium, evoking the appearance of the couple stresses may be described by introducing the term 1t ikOQ i / Oxk in equation (3.8) for the uncompensated heat. Equation for the intrinsic angular momentum in this case may be expressed in the following way d J""'r\ a [M X HJ ; +--. Orr. ;k p1;/;; = -'t; +

&

Ox k

Different hydrodynamic problems of the ferromagnetic colloids are solved on the basis of similar models [138, 305, 342, 586, 587, 642J 6 Couple stresses in quasi-equilibrium approximation. A cause for the occurrence of the couple stresses in magnetizable liquids principally differs from that considered in section 5, attributed to the magnetic particle-particle interaction. Here the stresses like in the case of the flat layer breakup considered in 2.3 is related to a non-uniform distribution of the magnetization. Their presence in equation for the heat influx (3.2) is taken into account by defining SQ·· in the corresponding form. By introducing the orientational stresses tensor gik the flux of the energy SQ·· into a material element of the continuous medium may be expressed in the following way

134 Models of Magnetizing Fluids

f'OLMigiknikdS

(3.11)

( 8 L is the Lagrange variation). Upon neglecting the intrinsic angular momentum of a medium the uncompensated heat may be specified in the following form analogous to that of (3.3): 'OQ' Wi dM - = ' t " k - - p - ( H -H). (3.12) 'O~t I Oxk dt e Hence, equation for the heat influx with regard to (3.11) and (3.12) yields the following equation for the internal energy: pdU 1 = pTdS -ppd(l/ p) + (pH' i +Ogile / Ox k )'O L M ; + g;k 'O L M ;k ,. which immediately gives gile = pdU / 8M;ok; pH.; = pdU / 8M; - Ogile / Ox k . (3.13) The stresses tensor for a medium may be expressed in the following form cr iIe = 'tile + cr~ - PO;k' cr~k = -g/kMJj. (3.14) In order to subject actual problems to the analysis based upon the above model there is necessity to specify the internal energy as a function of the magnetization and its gradients. Besides the appropriate boundary conditions are required for the magnetization. To find these one ought to make use of the dependence of the surface energy of a magnetizable medium upon the magnetization. At present this issue still needs to be studied experimentally. Recent developments in building theoretical models of surface phenomena are reported in [261, 265, 595, 596] (see problem 3.1). The couple stresses related to the non-uniform distribution of the magnetization in the case under study when the intrinsic angular momentum of a medium may be neglected, are arrived at on the basis of equation for couples dL 8 p d; = Ox (eilmx/Pmk)+eimnPmn, (3.15) k

hence,

&rr. "k eimnPmn = -~-. Ox A form of the couple stresses is determined according to the condition of invariance for the internal energy and uncompensated heat upon a simultaneous turn of the medium, magnetization and field through the same angle. The first of these (cr?Q = e/ijcr~; IjQ = e/ij1';j) with regard to relationships (3.13) and (3.14) yields

8U]

8 [Mxp~. cr o0 + r =[Mx(H. -H)]+Oxk

8M ok

(3.16)

Since the requirement concerning the absence of the dissipation during a simultaneous rotation of the medium, magnetization and field permits one to express the antisymmetric part of the viscous stresses tensor as 'to = -[M x (H. - H)], it follows from relationships (3.15) and (3.16), that the couple stresses reduces to

nile

BU t ] =[ -M x P 8M ;' ok

(3.17)

General Principles for Developing Models of Magnetizable Media 135

It is necessary to take relationship (3.17) into account when expressing the torque, associated with the magnetizable medium and acting upon a macroscopic body immersed in it. By representing 0LM as 0LM = M / MOLM + [OL x M] relationship (3.11)

permits one to express the energy influx (m=M/M):

SQ**

1

= 0 Lj7t jknkdS

oQ** as the following sum of two terms

+1 0 LMmigiknkdS,

the first term being related to a work done by the couple stresses upon altering the direction of the magnetization ofa medium. 7 Anisotropic magnetizable media - liquid crystals. An important class of the magnetizable media being under detailed study at present is that of anisotropic liquids i.e., the liquid crystals. By introducing the anisotropic ferromagnetic particles into liquid crystals novel media are created - ferronematics, whose structure may be controlled on the part ofa moderate magnetic field (see 1.7) [113, 235]. The anisotropy of liquid crystals may be described by introducing the unit vector n - the director, pointing along the preferred direction of a medium. Since the magnetic characteristics of liquid crystals are determined by the diamagnetic susceptibility of anisotropic molecules, it is permissible to neglect the irreversible processes of magnetic relaxation when describing these. The uncompensated heat associated with the irreversible processes of the viscous friction and motion of a director may be expressed as

-oQ'- ='t~ -1 ( Ov

_I

2 Ox};

o~t

Ov) +'t°([n x n]-[n . +-}; x [~x olD, Oxl

where the second term is attributed to the process of the viscous friction upon rotating the anisotropic molecules with respect to a medium. Besides, the term, describing a work done by the couple stresses upon rotating a director is to be taken into account in the equation for the energy influx,

oQ**

=

f 0 = T-' o(t')dt' . Convergence of the time averages to the o ensemble averages takes place in the probability sense and is associated with the occurrence of the stationary distribution of probability on the unity sphere, as well as with the finiteness of the orientational correlation time ofa dipole [371]. In order to find the random trajectory of a dipole on the unit sphere the procedure giving an accuracy on the order of the square root of a time step At is adopted. In this case, alongside with a stationary Cartesian coordinate system with the Z-axis oriented in the direction of b, a moving system is introduced, coupled to the instantaneous direction of the unit vector 00. 00, e, = [no x b) / [00 x b ~ and e2 = [no x ed serve as the unit vectors for this coordinate system. As to the change of the vector An during the time step At in terms of projections on the unit vectors e, and e2 the system of equations (3.43) yields (the case when Oo.lH is considered, the vector 00 points in a direction along the X-axis) An, = 02At; An2 = -O,At. The angular velocity vector components 0, and O 2 in a moving coordinate system may be expressed in the following form:

o , =0Oy nO/~n02+n02 +~a.-'kT~n02+n02 +Oran. x y x y " ~

0 0 /~n 02 +n 02 + Oran O 2 = 0 onxn 2 . y z x

The statistically independent random rotation angles Or"' At and O~n At on the average equal zero and as a consequence of the action of multiple collisions are 2

distributed according to the Gaussian law with the dispersion < (o~ At) >= 2DAt (D = kT / a. is the rotational diffusion coefficient). Introduction of the characteristic scale a./kT permits one to express An, and An2 in the following form (ro = 0oa. / kT

is the rotational Peelet number):

AnI = roAtn~n~ / ~rn-~2-+-n-~2-+ ,bAty 2;

An2 = -roAtn°y / V'n x02 +ny02

02 2 -~At ~ V'n x +n0y -112At",I>

where YI and Y2 are the statistically independent random quantities, distributed according to the normal law with the average equal to zero and the dispersion equal unity. The components of the vector n upon a termination of the time step in a stationary coordinate system in accordance with the normalizing condition 0 2 = 1 reads (Ani = Anle, + An 2e2):

Selection and Approbation of Magnetic Fluid Models by Statistical Modelling 145

0.5

Fig. 3.1 Moments of magnetic moment components as a fimction of the Langevin parameter ~ stands for theoretical ensemble averages. Length of the averaging interval T = NM (N = lOs, M = 10-2 / ~), curve 1 - < z > ; curve 2 - < >.

n

nx

2 2 2 =JI- &1 .Lx nO + /in Iy nO / ~n02 x +n0 Y + /in 2nOnO x z / ~n02 x +n0 y '.

JI- nO - nO / ~n02 +n02 + =JI- &II n~ - !ln2~n22 +n~2 .

nx = nx

n;

2 &1 .Ly

/in Ix

x

Y

2 /in 2nOnO y z / ~n02 x +n0 y '.

Generation of normally distributed random values of YI and Y2 is carried out by an URAND random-number generator using a method of polar coordinates [240]. The numerical procedure is tested by using the known values of the moments of the distribution function for dipoles in the thermodynamic equilibrium: =< ny >= 0; < nz >= L(~);

== L(~)/~; = 1-2L(~)/~. Some of the values of these moments obtained by averaging over a random trajectory as affected by ~ are displayed in Fig. 3.1. The dependence of the magnetization relaxation time upon the field strength in a numerical simulation experiment is obtained by computing the time correlation functions for the components of the magnetic moment in accordance with the ergodic theorem. These are found by averaging the product of two component values of the magnetic moment, shifted by a fixed time interval t, along the random trajectory of a dipole. Thus

IT

fnO. (r)no. ('t - t)d't::::l N- ~::no. ('t; )no. ('t; - t). T°

< nO. (t)no. >= lim T-+«>

I

N

;=1

146 Models of Magnetizing Fluids

Q 25

------------=-----==-~-~~

o

2.5

5

Fig. 3.2 The time correlation fimction of magnetization < nz(t)nz(O) >, the Langevin parameter ~= 1. Length of the averaging interval N = 2.105. - - - represents the theoretical limit for large time intervals.

If the dipole orientational correlation time "to; is introduced, the time correlation function may be expressed in the following form =< n(l

>~

+( < n; >

p -

< n(l

>~ )exp(-t h(l)'

where < >p denotes the averaging with the distribution function in a state of the thermodynamic equilibrium. The correlation function of the component ~= 1 of the magnetic moment obtained by making use of the above technique in the direction of a field is shown in Fig. 3.2. The magnetization relaxation time is determined by a slope of the line, approximating the findings of a numerical simulation experiment upon plotting these in semi-log coordinates. The magnetization relaxation time as affected by ~ obtained in accordance with the above technique is presented in Fig. 3.3. Dashed lines denote stand for the theoretical dependencies, determined in the effective field approximation [412]. Evidently, the approximation furnishes an adequate representation of the magnetization dynamics near the state of thermodynamic equilibrium. In order to describe the hydrodynamic effects of the internal rotation, caused by a motion of the magnetic colloids in an external field, it is necessary to have the information about the dependence of the magnetization of a medium upon its velocity of motion. For small deviations from a state of the thermodynamic equilibrium when the difference between H. and H is negligible, it is permissible, as is mentioned above, to linearize the relaxation equations (3.37). In this case, with the hydrodynamic disorientation term allowed for, = __ 1 (M-Mo)I,l.' ( pdM_[~XM]l dt ~ILJ.

(3.44)

't1LJ.

Linear relaxation equation (3.44), when the vector of the hydrodynamic vortex is oriented transversally with respect to a field, yields (H = He z ; ~ =.00 e... )

Selection and Approbation of Magnetic Fluid Models by Statistical Modelling 147

I • )( -1

• -2 0-3

0.5

o

10

5

Fig. 3.3 The magnetization relaxation time plotted against the magnetic field strength: J ("t.It B ); 2 ("ty/"t B ); 3 ("tzIt B ) - results of numerical simulation [142]. The relaxation times are obtained by processing 15 values of the time correlation functions set out unifonnly in the interval [0; 1.5 "t B ]. Curve J (longitudinal component), curve 2 (transversal components) - theoretical dependencies (3.41).

M x = 0; My = -'t.l .ooMo I (l + 'tl't.l n~); M z = M o I (l + 'tl't.l n~). (3.45) Non-linear equation (3.37) for the same geometry gives a set of equations, which may be used in order to find the Langevin effective field parameter 1;e and an angle e the magnetization vector makes with the magnetic field strength vector ( MI nm = (0, L(~e)sine, L(~e)cose»

•• • 0.025

·-1 x-2

•

• • • •

x

w

o

5

10

Fig.3.4 Results of statistical modeling of the magnetization in a shear flow (1); (2) represent the data listed in [l1O]. The Langevin parameter ~ = 1.

148 Models of Magnetizing Fluids

a) (n y>. (nz) 0.25

~

-""0

e

01

e e

o

~

e_ e-e_e

2.5

't:t> 9. 0

5

b)

,

Fig.3.5 Components of the magnetization of the colloid of ferromagnetic in shear flow: a - ~=l; b - ~=2. Empty circles - transversal component < ny > - full circles longitudinal component < nz > . times

'tn

and

'tol

theoretical dependencies (3.45), where relaxation

are determined from (3.41); - - - dependencies (3.46) (curve 1 -

< n y > ; 2 - < n z > ). Length of averaging interval 2.105• Time step 10'2 10,2/00 ( 00 ~ 2 ).

cose

= S. / s;

2't~no = (L(~.) -slJ sine.

(00 :'5:

2 ) and

(3.46)

The non-linear relaxation equation (3.42) found within a framework of the internal angular momentum model instead of (3.46) yields the following relationships (M=nm I a certain discrepancy may be observed between these equations and numerical experiments. The parameter 2't BOO / S = aDo / mH represent a ratio of the viscous and magnetic torque. In the case when 't BOO> S/ 2 in the absence of random thermal jolts a

Selection and Approbation of Magnetic Fluid Models by Statistical Modelling 151

1 lOy)

0-1

.-2 x-3 />-4

05

15

Fig. 3.8 Transversal to a field magnetization component of a colloid plotted versus ratio of characteristic magnetic to viscid torques. - represents theoretical dependence (3.55) at

n 1

(stable

stationary state of a dipole in a field of the hydrodynamic vortex). Results of numerical simulation: ~ = IS (1), 20 (2), 25 (3), 30 (4).

qualitative rearrangement of the character of the dipole motion in shear flow takes place [280J. Provided 0.o.LH, the stationary state of a dipole is allowed solely in the region c:xn o / mH < 1. When c:xn o / mH > I there is no asymptotically stable stationary state and a family of periodic trajectories is observed on the unity sphere for the orientations of a dipole [280J. Selection of trajectories from this family is determined by the initial conditions. Since the period averages of the magnetic moment of particles for various trajectories differ, it is necessary to know the distribution function of the particles for a given family of periodic trajectories, if the magnetization of a colloid is to be determined. In practice, there is a natural physical cause responsible for the establishing of the distribution - the rotational Brownian motion. Irrespective of how weak this motion is, it finally results in establishing the distribution of particle trajectories, for which the mean diffusion flux of particles between different trajectories equals zero. Computation of this distribution in the weak Brownian motion limit is presented in problem 3.2. The result ofjuxtaposing the magnetization component of a colloid transversal with respect to a field, found on the basis of the aforementioned distribution and the findings, obtained by numerical simulation using a method of Brownian dynamics is displayed in Fig. 3.8. Evidently, the findings obtained by numerical simulation at sufficiently large aOo / mH and ~ are in a fairly good agreement with the dependence, determined in the weak Brownian motion limit. It is worth noting, that in the presence of random thermal perturbations, a peak of finite magnitude may be observed for the transversal component of the magnetization at c:xno / mH value somewhat less than unity (a maximum of the theoretical dependence in Fig. 3.8 just corresponds to this

152 Models of Magnetizing Fluids 'tx 'ell

\

\ \ \

\ \

\

... \ \

0.5

\

.

'\.

"......... " " .-

-.

--e--e_e_ _

5e

o

10

20

Fig. 3.9 Relaxation time of the magnetization component longitudinal with respect to the hydrodynamic vortex (~ = 20).

value), being indicative of the influence the thermal agitation exerts upon the rearrangement of the motion of a dipole at a..Q o / mH == 1. The discordance between the findings, obtained by numerical simulation and the theoretical dependencies according to the effective field model, shown in Figs. 3.6 and 3.7 is connected with a qualitative rearrangement of the character of the motion of a dipole in a shear flow and this link may be further traced by numerical values of the time correlation functions of a magnetic moment. The relaxation times for various magnetization components, determined on the basis of these findings are presented in Fig. 3.9 and 3.10. It is informative to note the presence of peaks in 'tl. and 'til curves (Fig. 3.10) at a.rlo / mH 0;::: 1, i. g., in the region the dipole trajectory rearrangement takes place. The growth of the relaxation times is in a qualitative agreement with the fact, that the period of the dipole reversal by a periodic trajectory at the moment of its birth is large. The theoretical dependence of the relaxation time in the event the perturbations of the magnetization component longitudinal with respect to the hydrodynamic vortex are small upon the effective field parameter ~ e : 't = J(

2't B L(~.) (~.

_

L(~.»

(3.48)

is presented in Fig. 3.9 by a dashed line. It is worth noting, that (3.48) checks with the dependence of the relaxation time for the magnetization component transversal with respect to a field (3.41) upon substituting ~ for ~ e' Fig. 3.9 indicates that a departure of the numerical simulation

Selection and Approbation of Magnetic Fluid Models by Statistical Modelling 153

0.1

0

0- - - - - - - - - - - - - - ~ 'ttl

o 10

20

Fig. 3.10 Relaxation time of the magnetization component transversal to the hydrodynamic vortex = 20).

(s

findings from theoretical dependence (3.48) occurs at

Se

values, corresponding to the

region where 2't BnO / S > 1. Thus, numerical simulation shows, that the model, based upon the effective field approximation (see 3.1, section 2) furnishes a closest fit in the event a description of the magnetization processes of ferromagnetic colloids in shear flows and what being equivalent in rotating magnetic fields. For large hydrodynamic vortex intensities and magnetic field strength conclusions drawn from this model are to be subjected to a quantitative refinement. In order to obtain quantitatively reliable data in the region under consideration at the present moment the model should be applied, where the distribution function of particles in the orientational space is used in the capacity of the parameter of an internal state (see 3.1, section 3). Equation of the magnetization relaxation for the internal angular momentum model (3.42) at small values of the intensity of the hydrodynamic vortex and magnetic field strength leads to even qualitatively wrong conclusions, since the dependence of the magnetization upon the hydrodynamic vortex intensity does not exhibit any hysteresis. For moderate hydrodynamic vortex intensity, when a deviation from a state of the thermodynamic equilibrium is negligible, linear relaxation equations (3.44) apply. Problem 3.2 [293,388]. In the limiting case of weak Brownian motion consider the distribution function for orientations of the spherical single-domain particles of a colloid under shear flow in a field ( no1..8 ). Examine the case, when an o / mH > 1. Solution Let the X-axis points along H, whereas the Z-axis along no' Equation of motion for the unit vector 0 = (x, y, z), orientated along the magnetic moment of a particle reads it = [00 x oj +a -lmH[[o x h] x oj. (3.49)

154 Models of Magnetizing Fluids Introducing dimensionless time t = 00t equation (3.49) in the components on the axes of a Cartesian coordinate system yields ( 0 =mH I aOo ) 2 x= -Oxz; y= -z-Oyz; z= y+0(I-z ). It follows from the first two equations of a set, that apart from x 2 + y 2 +Z2 it involves also integral F= __X_ _ (3.50) 1/0+y At 0 1 two stationary states of a dipole exist, only one of them being stable. Distribution of dipoles with respect to a family of periodic trajectories at 0 and < 0 >, in the linear with respect to Am and A0 approximation yield the following relaxation equations for the magnetization and the mean angular moment of a colloidal particle:

Models of Magnetizable Fluids with Regard to Gyrornagnetic Effects 159

( d -m o)lI; dt 'trY

(3.67)

dJ. 1 1 =-a,«o>_n) --«m>-m) , dt ""'0 J. 't Y 0 J. J.

where indices II and .1 designate the components longitudinal and transversal with respect to the field strength vector; the relaxation times 'tJl and 't J. are determined by the following dependencies 'tt

= ~'(I;) /2kTL(I;);

't.l

=8L(I;) / kT(1; -

L(I;».

(3.68)

The equilibrium magnetization mo with the Barnett effect allowed for in the linear with respect to 0 approximation may be expressed in the following form m

011

=mL(l:)+~L'(l:)«O»' .."

kT

.."

Y

II '

m

OJ.

=~ L(I;) «0» kT

I;

y

J.

.

It is readily understandable, that the set of relaxation equations (3.66), (3.67) yields the following equations of balance of the full internal moment:

!!...[< m > +1 ] = [< m > xH]-a,«O > -00). dt Y The performed analysis permits one to determine the coefficients for the linear phenomenological laws (3.59) - (3.61). In the first place, the set (3.66), (3,67) indicate, that there are no cross effects between the magnetization relaxation and the antisymmetric part of the viscous stress tensor in a diluted colloid. In the second place, in the low field limit the relaxation time 't = 8 /2kT in equation (3.63) may be determined through the coefficient 8, appearing in the Landau-Lifshitz equation for the magnetic moment of a ferromagnetic. The magnitude of 8 may be judged by the line width in ferromagnetic resonance. Applying a rotating field evokes the appearance of forced precession of the magnetic moment of a immobile ferromagnetic particle in the stationary field H transversal with respect to the magnetic moment. The Landau-Lifshitz equation for the energy dissipation occurring in this case gives . m2 oo 2h 2 Q=

8[(00 +yH)2 +(mH /8)2]"

Obviously, the ratio of the line width of ferromagnetic resonance to the resonance frequency Iioo /00 0 equals ~ =m / y8 . For the majority of ferromagnetics the ~ is about 0.01 [561]. In particular, the experimental research work on the domain wall displacement velocity [252] give the parameter ~ = A. / M s y value of 4.10-2 (A. = 3.5.108 , y = 1.7.107 , M s = 500 G).

160 Models of Magnetizing Fluids

Upon reducing the size of ferromagnetic the thermal fluctuations gain in their importance. According to [256], l:i.ro 0 / ro 0 in this case equals 2/ 't .1 yH, where 't.l may be determined from formula (3.68). Hence, l:i.ro / roo = 2f3(~ - L) / ~, i. e., upon a colloidal particle reducing in size (~ ~ 0) the ferromagnetic resonance line widens. The experimental study of the ferromagnetic resonance in Ni, Co and Fe304 colloids is reported in [11, 12, 552]. A feasibility of explaining the increase of the effective viscosity observed in certain experiments upon applying a field [422, 447] by the action of gyromagnetic effects is discussed in [648, 649]. The presented evaluation permits one to subject this assumption to critical analysis. Consider the low field limit, when the relaxation equation for the magnetization appears as (3.63). Then, in the case of a stationary flat shear flow (0, vy(z), 0) equation (3.63) for the magnetization components in a field ofR = (0, 0, If) yields M =[

X(l/2)dvy / dz;

X'tH(1/2)dvy / dz;

y[1+('tYH)2]

1+ ('tyH)2

0].

Equation for the moments (3.56) permits to compute the antisymmetrical part of the viscous stress tensor 'to, consequently, the effective viscosity increment of a colloid I:i.TJ , associated with the gyromagnetic effects reads

=.!.

X ('tYH)2 (3.69) 4 y 2't 1+ ('tYH)2 . Let juxtapose relationship (3.69) and the limiting values of the increment of the effective viscosity of a colloid I:i.TJoo = ex. /4 appearing upon a complete colloidal particle rotation slowdown: I:i.TJ

~ _3.(!!!-.) 2 _ 0

I:i.TJoo 3 8y 6Vr TJ It follows from here, that the contribution of the gyromagnetic effects to the increase of

the effective viscosity is proportional to 13 2, i.e., to a ratio of the rotational friction coefficients of the magnetic moment and a particle. By expressing 0 as m/yf3 a colloidal magnetite (Mo = 480 G) gives I:i.TJ / I:i.TJoo = 3·10'6 in a carrying liquid with the viscosity of 10'2 P (13 = 10,2 , Vm is the volume of the magnetic core of a particle, Vr is the volume of a particle together with a solvate layer of the surfactant). The presented estimate indicates, that the gyromagnetic effects lead to the effective viscosity increment values of a colloid in a field, which are at least by six orders of magnitude smaller than observed in the experiment. This serves as a reasonable justification for neglecting the gyromagnetic effects when the motion of ferromagnetic colloids is to be described. The gyromagnetic effects may occur solely in molecular liquid paramagnetics and gases [558, 566]. The effects related to the magnetic anisotropy of particles turn out to be much more important if colloidal solutions of ferromagnetics are considered. The magnetic anisotropy may be crystallographic in nature, as well as, by virtue of the non-ideality of the particles, may be attributed to the shape anisotropy. For actual colloids the time interval 't s = I / ex. during which the rotational inertia plays a significant role is of order 10,11 - 10,12 s, being even less than the Larmor

Models of Magnetizable Fluids with Regard to Gyromagnetic Effects 161

precession period of the magnetic moment in a field of H = 1 kOe. Thus, it would be reasonable to assume the rotational inertia of the particles negligible. Therefore, the following equation holds for the mean magnetic moment of a colloidal particle [150] d a. 1 ---= +y + dt a. +a. m &y a. a. +Y + (3.70) a.+a. m (a.+a.m)o -2kTa. < m > lo(a. +a. m), where < > stands for the averaging over the magnetic moment (e=mlm) and the magnetic anisotropy axis n of a particle by the distribution function P(e, D, t), a. m =m 2 1&y2. The mean angular rotation velocity of the colloidal particles, determining the antisyrnmetric viscous stresses, in this case reads < 0> -00 = a.-I < [m x H] > _(a.y)-l d < m > Idt. (3.71) The last tenn in relationship (3.71) describes the Einstein-de Haas effect in a colloidal particle of ferromagnetic, a. m I a. is proportional to L\ll I L\lloo and, in confonnity with the above estimate may be safely neglected. Thus, upon substitution of equation (3.70) into relationship (3.71) and by dropping the tenns of P order and neglecting a. m I a. at the magnetic anisotropy field of > kTlm one can rearrange the relationship for the rotational slip of the colloidal particles to read: -00 =_a.- 1 . (3.72) Relationship (3.72) in the non-inertial approximation describes the rotational motion of particles, associated with an interaction of the magnetic moment with its axis of the magnetic anisotropy. For a concrete computation of the torque in the right-hand side of relationship (3.72) there is a need for the infonnation about the probability density function of combined distribution of the m and n (see 1.3 for a detailed discussion). In the limiting case, when H a »H and the time required for the magnetic moment of a particle to attain the equilibrium is negligible if compared with a characteristic time of its motion in a viscous medium, the following relationship applies =- and -00 = a.-I . Here the magnetic moment is assumed directed along an axis of the magnetic anisotropy D of a particle (a case of the rigid magnetic moment, its constitutive relation is subjected to a detailed analysis by the technique of statistical modelling in the previous section). The evaluation indicates, that in actual ferromagnetic colloids the gyromagnetic effects, related to the internal spin angular momentum of electrons, are negligibly small in comparison with the effects, attributed to the finite magnetic anisotropy of the particles.

4 Quasi-Equilibriwn Hydrodynamics of Magnetic Fluids 4.1 Basic Equations In those cases when it is permissible to neglect the magnetization relaxation processes in the magnetic fluids, the quasi-equilibrium model may be applied (see 3.1, section 1). In accordance with this model the equation of motion reads dv dt

p - = -Vp

+ l1L'1v + MV'H; divv =0,

where the magnetic field strength in a medium is determined by the equations of magnetostatics divB = 0; rotH = 0

under the boundary conditions of continuity across the surfaces of discontinuity of the normal field induction B component and the tangential field strength H component. The following dynamic conditions are fulfilled on the free surface of a magnetic fluid [( -P~;t +t /t + 1;'t)nt ] = 0"(1 / ~ + 11 ~)nl

(4.1)

where [ ] designate an abrupt change upon transition across the discontinuity surface, 1/ R1 + 1/ R2 is the sum of the principal radii of its curvature. If at rest, the boundary condition (4.1) yields the Laplace law for a magnetizable media PI - P2

= 0"(11 RI +11 ~)-27rM~.

There is an extensive class of the magnetic fluid flows, when the pondermotive force MVH is potential. For instance, the isothermal flows under a constant concentration of the ferroparticles etc. represent this trend. In the case of a potential flow of the ideal magnetic fluid the modified theorem of Bernoulli applies, yielding in the stationary case accounted for an action of the gravity force [459] H

p+pv 2 /2+pgz- fMdH = const.

(4.2) o On the basis of relationship (4.2) it is permissible to consider such issues as the change in the cross-section of the jet of a magnetic fluid upon its passage through a region of non-homogeneous field [522] etc. The conclusion may be drawn from relationship (4.2) concerning the modification of the Archimedes' law determining the buoyancy force, acting on a non-magnetic body, immersed in a magnetic fluid. Here the effective density of a magnetic fluid in the external field, possessing the gradient VH increases with respect to 1+ MjVHlpg . Therefore, the pronounced magnetic properties of ferrocolloids permits one to perform the flotation of nonmagnetic, high density bodies, having broad utility in designing the magnetic-fluid separators [581]. Provided the pondermotive forces are potential the characteristic features of hydrodynamic problems of magnetizable liquids reveal in the case when there are free boundaries. Owing to the action of the pondermotive force of a field, these may experience deformation, leading to the appearance of complex and fascinating structures. The main complexity of the problems under consideration lies in the fact, that the pondermotive force in the presence of the internal field of a fluid is affected by the shape, the fluid occupies in space. Thus, an extensive class of non-linear problems arises, whereas only few approximate solutions are provided at present. Some of these

164 Quasi-Equilibrium Hydrodynamics of Magnetic Fluids

solutions, based upon a behaviour of the droplet and the free surface of a magnetic fluid are considered in the following. Besides this case, when the forces associated with the intrinsic field of a fluid prevail, there is a class of phenomena, which may be considered in the noninductive approximation. Versatility of eventual distribution of non-uniform field, generated by external sources, provide plenty of opportunities to execute a control over the magnetic fluids. Thus, by making use of a non-uniform magnetic field, furnished by a cylindrical, current-carrying conductor it is feasible to steady the capillary instability of a magnetic fluid column (problem 4.1). In a non-uniform field of a horizontal currentcarrying linear conductor one may observe formation of multiple-connected equilibrium pattern, besides this, depending upon the current strength in a conductor, this phenomenon may exhibit hysteresis (problem 4.2). The equilibrium pattern, establishing upon passing a current through a vertical rod running through a pool of magnetic fluid, is among the first experimental illustrations of the exceptionally strong magnetic properties the ferromagnetic colloids possess [459]. Later on the shape of a magnetic fluid, governed by the balance of the attraction force, acting in a direction towards a cylindrical conductor, and the gravity force, has been calculated by several authors in [40, 355]. The self-levitation of bodies in the magnetic fluid belong to the class of phenomena, for which the non-inductive approximation may be applied in certain cases [35, 134, 518,523]. This class of phenomena, showing promise what concerns various technical applications, is discussed in the following section. Problem 4.1 [16, 57] Analyze the possibility of stabilizing an infinite, cylindrical column of a magnetic fluid, surroWlding a linear, current-carrying conductor. Consider the dynamics of perturbations, developing in a column. The action of the gravity force may be neglected (since the column of a magnetic fluid occurs in an inuniscible liquid of equal density). Solution Provided the case the magnetic fluid configuration is that of azimuthal symmetry, the non-inductive approximation affords exact solution. The energy fimctional of a column, comprising the boWldary surface energy, as well as the potential energy of a current-carrying rod in a non-Wliform field may be expressed in the following form (p = R +~(z) is the column boWldary equation in a cylindrical coordinate system) R+t;(z)

F = C1 f (R +~)Jl +~~dq>dz - f dz f

H(p)

pdpdlp f M(H')dH'.

(4.3)

o

Should the cylindrical shape of a liquid column surroWlding a conductor be stable, it is required, that the second variation of the energy functional with respect to the state R + ~ = 1\ = const is positive. By virtue of the incompressibility of a liquid its volume

V=1/2fd0. (4.5) Relationship (4.5) demonstrates, that for the magnetic Bond numbers larger than unity it is feasible to attain a complete stabilization of the capillary instability, related to the tendency of surface energy to decrease. The growth of perturbations of the wavelength, exceeding a certain critical value, may occur at Bm < 1. The establishing structure is determined in this case by dynamics of perturbation development. It would be sound to assume, that such structure develops, the increment of which, arrived at within a framework of the linear stability theory, is the greatest. In the case of sufficiently viscous and thin films of a magnetic fluid enveloping a conductor, the increment of small perturbations may be calculated in the approximation of lubrication theory [45]. In this event,

upon introducing the effective pressure

p= p

H

- f M(B')dH' , the Y-coordinate, perpendicular to o

a film and neglecting the film curvature, the equation for the creeping motion of a liquid reads

Oft 8 2v, = 0: --+"8z 8y2 '

Oft

8y

= O.

(4.6)

The above equations are to be considered under the no-slip boundary conditions on the conductor v,(O) =0 and vanishing tangential stress on the free surface of a film (the viscosity of a surrounding medium is neglected)

dv, (5) =o. dy As indicates the second equation in (4.6), Ii acts constantly perpendicularly with respect to a film and may be determined by the pondermotive force of a field and the capillary pressure, yielding for small perturbations t::, of the free surface of a film p = -crt::, / R; - crt::,,,, + MIVBIt::,· A profile of the flow velocity of a liquid along the boundary of a film may be obtained from (4.6) and expressed in the following form 1 OF 2 V, = - - ( y - 25y). (4.7)

2" 8z

166 Quasi-Equilibriwn Hydrodynamics of Magnetic Fluids The velocity component transversal to the film boundaries be found from the continuity equation in the following way: v (3) =

-I az

2

Ovz ely'" _ 1 a v z

2 azo/

0

y

I

Vy ,

determining its deformation may

32 •

y=o

As a result, by making use of equation (4.7) and in accordance with the kinematic condition

at:, =v/3) one obtains the differential equation for small free boundary perturbations

at

3 ap 3 at = 211 az2 =- 2~ «1- Bm)Szz + ~s=),

at:,

2

2

3

(1

2

At»

giving as the increment A. dependence upon the wave nwnber (s(t) = So exp(ikz + the following expression 3 A. = 3 (1. (kRo)2[l- (kRo/ - Bm). (4.8) 211~

A similar dependence of the increment for small perturbations, though with different scaling factor may be arrived at within a framework of the ideal fluid model [16], which however is not adequate for thin films of viscous liquid. Relationship (4.8) indicates, that at Bm ~ 1, there exists the perturbation wavelength, for which the increment reaches maximwn. It may be found from the following relationship (kRS =1/ 2(1- Bm), which for the developing structure of the dimensionless period A. = L / 2xRo yields the root law [16] J2I(l-Bm) = A •. The increment of these perturbations may be expressed as

..

o

3

...

2.5

/t

/

/

/

/0

2

../

0

/

(4.9)

/

/

/

• -1

0-2

•••0.0 01/ / /

~8°/

1.5

1

/

/

1 y1-Bm

1.5

2

2.5

Fig. 4.1 Wavelength of the structure, evolving on a cylindrical column as affected by the magnetic Bond nwnber [16]: 1 - column length of magnetic fluid equals 18 cm; 2 - 50 cm. - - - represents theoretical dependence (4.9).

Basic Equations 167 Sl(J

,

A:: 2~ (i-Bm) .

(4.10)

Relations (4.9) and (4.10) offer a good consonance with the experimental values. It is established [16, 57], that depending upon the reduced current intensity in a conductor, the column of a magnetic fluid breaks down into varying number of drops. The determined

wavelength of the developing structure as affected by 1/ Jl- Bm is shown in Fig. 4.1. As seen in the figure, the fmdings, obtained for a longer column are in good conformation with the theoretical dependence (4.9). The stepwise change in the period of a structure as Bm increases indicates, that for a shorter cylinder the fringe effects, not accounted for in relationship (4.9), may play a significant role. The developing perturbation increments for the columns of a magnetic fluid of various diameters versus the Bm values are depicted in Fig. 4.2. Evidently, the experimental results are in a good agreement with the linear relationship A:: /3(1- Bm)' . The linear dependence coefficient values, obtained from the slope of the curves, displayed in Fig. 4.2. by a dashed line, for mentioned above cases equal 1.4 c· l (Ro:: 2.8 mm) and 2.35 c') (Ro:: 2.1 mm) respectively. By making use of values of the magnetic fluid parameters.., :: 6 cP, (J:: 10 erg/cm2 reported in [16], as well as of the radius value ofa conductor r:: I mm, one arrives at corresponding values of 20 and 14. Such a discrepancy of almost an order permits one to infer, that upon describing the dynamics of a developing instability it is necessary to take into account the friction in a surrounding medium (a glycerol-water solution having the viscosity of 1.7 Pis reported in [16]). It would be permissible to resort to relationship (4.10) in the event, when the friction in a surrounding medium is negligible. Problem 4.2 [341] Consider the figures of equilibrium in the non-inductive approximation, establishing upon passing a current through a straight conductor, positioned at a distance h parallel with respect to the free surface of a magnetic fluid. The surface tension forces may be neglected.

:1

J1., S

-1

•

2

1 • -1

0-2

(1-Bm)

o

2

0.5

Fig.4.2 Increment of cylindrical column perturbations as a function of the magnetic Bond number [16]: 1 - magnetic fluid column radius equals 2.1 mm; 2 - 2.8 mm.

168 Quasi-Equilibriwn Hydrodynamics of Magnetic Fluids

Solution The equilibriwn condition for a magnetic fluid experiencing the gravitational force and the magnetic field force may be written as l _'Vp+ll- 'VH 2 +pg=0, (4.11) 81t where H in the non-inductive approximation stands for the strength of the magnetic field, generated by a straight conductor carrying current [- H =2I 1cr. Since the pressure in the absence of the surface tension forces to an accuracy of the 2nd-order infInitesimal terms with

respect to (ll-l)2 is continuous, the equilibriwn condition permits one to obtain the boundary equation in the following form: Il - I H 2 + pgr 81t

=const.

Then, by introducing a Cartesian coordinate system with the Y-axis pomtmg along a conductor, whereas the Z-axis is directed parallel to a conductor and the unperturbed free surface of a liquid, as well as the dimensionless parameter Mi = (Il - 1)[2 1 21tpgh 3 , representing a ratio of the pondermotive force to the gravitational force, and using the characteristic length scale h yields the free surface equation in an implicit form: Mi Z=-2--2+C. (4.12) x +z Equation (4.12) admits solutions for two shapes of the free surface: 1) the surface of a liquid approaching the flat, unbounded shape far away from a conductor; 2) closed one, corresponding to the magnetic fluid shape, enveloping a current-carrying conductor. In order to solve the first of these types the C value may be found from the boundary conditions z( ±oo) =-1 . Equation for the shape of a surface may be expressed in the following way Mi Z=-2--2 -1. (4.13) x +z As the distance from the surface of a liquid to a conductor ho equation (4.13) yields the following relationship

l=ho+Milh;,

(4.14)

a positive solution to which is admitted only at Mi < 4/27. At greater Mi values there are no solutions with all the points located below a conductor. In this region the MS liquid rests also around a conductor. Let one consider the free surface shape of the second type - the closed one, enveloping a conductor. By designating, the same way as above, the distance from the lowest point of the surface of a liquid to a conductor through ho, solution (4.12) for the closed boundary shape yields

Z+ho=Mi(~--\-). x +z h

(4.15)

o

In order to obtain the required solution it is worth noting, that there is a peak volume of a liquid for each Mi value, which may be retained in the proximity of a conductor. Since the volwne per unit length of a conductor may be expressed in the following form V

=2

z(ho)

f x(z)dz (x ~ 0),

x(z(ho

»=x(-ho) =0,

-no

then the derivative dV 1 dho with regard to equation (4.15) reads dV Z(fho) dx I Z(fllo) 3· 2 2 =2 - d z =-. (2Mi / ho - IXx + z )dz / x dho -no dho Ml -no

Hence, the maximwn volwne value is reached at

Levitation in Magnetic Fluids 169

z

-1

x

Fig.4.3 Magnetic fluid figures of equilibriwn in the proximity of a current-carrying conductor (Mi = 4/27 (1); 6/27 (2); 8/27 (3). Equilibriwn pattern 1 consists of two branches: one is situated entirely below conductor, whereas another surrounds it.

hD

= (2Mii'3.

(4.16)

It is worth noting, that the index value 1/3 in relationship (4.16) may be obtained also from scaling considerations, since the volwne of the closed region of a liquid, which may be retained in the proximity of a current-carrying conductor, in the non-inductive approximation should not depend upon the distance from a pool of ferrofluid. On the basis of relationships (4.14) and (4.16) it is feasible to imagine a qualitative picture of how a shape of the free surface evolves as the current intensity in a conductor is increased or reduced. In the region of small current intensity values a bulge of liquid forms below a conductor. Upon achieving the critical value Mi = 4/27, at which the conductor is situated at a distance hD = 2/3 from the surface of a liquid, a state of the liquid undergoes a stepwise change,

thus, assuming the shape, enveloping the surface of a conductor. The peak height magnetic fluid reaches above the conductor for the given Mi value equals 1/3. At Mi = 8/27 according to analysis of equation (4.13) the neck disappears and the bulge of magnetic fluid surrounds a conductor. Evolution of the equilibriwn patterns is displayed in Fig. 4.3. As the current intensity is reduced, owing to the existence of closed solutions for the shape of the free surface, the hysteresis phenomena may occur. Upon reaching the value of Mi = 8/27 one may observe the initiation of a neck of magnetic fluid, which a breakdown at Mi = 4/27, leading to the formation of a pendent drop on the conductor. As a result, the equilibriwn pattern of a liquid transforms from a single-connected into a double-connected one. A subsequent reduction of Mi causes in accordance with relationship (4.16) the diminishing of the volwne of a pendent drop. Analogous phenomena of formation of the rnulti-connected patterns may be observed for more complicated configurations of an external field, as well. Thus, it is reported in [341], that upon passing a current in the same direction through two parallel conductors, positioned on the surface of a magnetic fluid, a cylindrical cavity may form in bulk of a liquid.

4.2. Levitation in Magnetic Fluids The familiar Earnshaw's theorem asserts [599], that a charged conducting body placed in a field of external electric charges cannot be maintained in stable equilibrium under the influence of electric forces alone. The validity of the theorem is attributed to the

170 Quasi-Equilibrium Hydrodynamics of Magnetic Fluids

properties of harmonic functions, for which internal points of the region cannot be the extremum points. As a consequence, an analogous theorem holds for a system of pennanent magnets. This theorem though, does not rule out the possibility to maintain magnetic bodies in stable equilibrium in the presence of extraneous forces. A similar situation occurs in the event, a pennanent magnet experiences selflevitation in a beaker of magnetic fluid. This phenomenon is demonstrated for the first time by Rosensweig [518], who succeeded in self-levitating a pennanent ceramic disk magnet of density 4.7 glcm3 possessing the saturation magnetization of 395 Gs in the magnetic fluid with j.I. ~ 2. A phenomenon of self-levitation is associated with the repulsion of poles of a pennanent magnet from the interface between the magnetic and nonmagnetic media. An analogous force for a point charge q, located at a distance I from the boundary of dielectric with the dielectric penneability & equals q2 (& - 1) / 4P (1 + &) and reaches its peak value at & = 1+.J2 ~ 2.4 . Evaluation of the magnetic levitation force for a pennanent magnet of an arbitrary shape is a tedious task. An exact analytical solution may be arrived at in the model case of a pennanent cylindrical magnet, magnetized across its axis and placed in a cylindrical beaker of magnetic fluid [134]. Analysis of this solution permits one to get infonnation about the magnitude of the magnetic levitation forces, as well as, to fonnulate several useful relationships to find these. According to geometrical considerations of the above model case it would be natural to resort to the bipolar coordinates a, ~, related to a Cartesian coordinate system by the following relationships x =asha/ (cha +cos~); y =a sin~/ (cha +cos~). The coordinate surfaces a = a l and a = a2 coincide with the surfaces of a magnet and beaker, respectively (the X-axis runs along the center line of a beaker and magnet, positioned at a distance of I between each other). Then Ri-R2_J2 Ri-R2+J2 cha l = coo 2 = 2RI , ; a = R.sha l ·

2lh ; •

2

The magnetostatic potential 'V (H = V''V ) in the regions 1, 2, 3 is found as a solution to the two-dimensional Laplace equation (1 - area of a pennanent magnet, 2 - magnetic fluid, 3 - non-magnetic medium outside a beaker): 02'V 02'V --+--=0 00 2 a~2 under the matching boundary conditions 'VI = 'V2ICLl; 'V2 = 'V3ICL2;

Bnl = Bn2 ICLI; Bn2 = Bn3 1CL2 ,

as well as the restraint conditions 'V I ,'V 3 < 00. The induction vector of a magnetic field in the regions under consideration reads: B I = HI + 41tM; B 2 = j.1.112; B 3 = H 3 , where M is the magnetization of a pennanent magnet. By making use of the relation between the unit vectors of a bipolar and Cartesian coordinate systems, the boundary conditions may be written in the following fonn:

Levitation in Magnetic Fluids 171

Owl I + 47tM,a(l+cha1cos13)

aa.

a,

47tMy asha l sinp

(chal +COS13)1

'1'1 la, = 'l'1Ia,; J..L

(cha l +cos13)

=J..L Ow1 1 ; aa. a,

~ I = ~ I ; '1'11.., = '1'31..,. (I.:

(1.2

By taking a Fourier series expansion of functions (l+cha\ cosP)/(cha.\ +COSp)2 and sha l sin 13 / (cha l + cos13)1 in the interval [0, 27t] of the variable p, solutions for '1'1' '1'1' and may be expressed as the following series

"'3

. .. . .. '1'1 = L(Boe-oa +C.e"a)cosn13 +L(F"e-oa +Goe .. na .. na '1'1 = LA"e-"'cosn13+LEoe-oasinn13; "=0

n=O

na

11=0

)sinn13;

(4.17)

":::0

'1'3 = LDoe cosn13 +LHoe sinn13, ,,=0

11=0

where the undefined coefficients An' .... may be determined as solutions to a linear non-uniform set of equations. The coefficients requisite for computation of the magnetic levitation force, thus becomes B = (J..L_l)41tM"p2(_1)0+l e-10(a,-a,) . o (J..L+l)1[1_(J..L_l)1 e-1o(a,-a,) /(J..L+V]' C =

47tMp2(-I)"+l e-1"',

.

o (J..L + 1)[1- (J..L _1)1 e-10(a,-a,) / (J..L +1)1]' (4.18) (J..L -1)47tMy2a( _1)0+l e -10(a,-a,) . F= o (J..L +1)1[1_ (J..L - Ve 10(«,- of an anisotropic dispersion reads (J.L2 I J.Ll -1) < J.L > IJ.Ll -1 =q> 1 + « J.L> IJ.LI -I)N 141t 1 + (J.L2 I J.Ll -I)N I 41t

Hydrostatics of a Magnetizable Fluid Drop 185

o

250

Fig. 4.14 Relative deformation of a magnetic and non-magnetic drop versus magnetic field strength (Il = 2.5). theoretical dependence (4.31); - - - theoretical dependence (4.35); ... theoretical dependence (4.32). Dependence (4.32) for a non-magnetic drop may not be discerned in practice from the dependence derived for this case from (4.35). Circles stand for experimental fmdings [211] (full circles - a magnetic-fluid drop with a radius of 6.7 rom in glycerol, empty circles represent a glycerol drop with a radius 6.7 rom in magnetic fluid).

Here 112 is the magnetic permeability of an inclusion; N, cP - their demagnetizing factor and volume fraction, III is the magnetic permeability of a surrounding medium. The initial value < Il > in the case when 112 »Ill is determined solely by the demagnetizing factor of microdrops and equals III (l + 4ncp / N (1- cp». Upon stretching the microdrops in a constant field the demagnetizing factor diminishes and the magnetic permeability of a dispersion grows reaching the constant value III (1 + CP(1l2 / III -1» at 4n / N(1l2 / III -1) ~ 1. The presence of a < 1.1 > maximum in this case may be associated with falling off 1.11 as the biasing field is augmented. The quoted relationships indicate, that a maximum of the magnetic differential permeability is expected in the field strength range, where an extent of deformation and, consequently, the demagnetizing factor of a drop suffers the most pronounced change. Since in accordance with the results reported in [208] the given concrete dispersion possess ll2 = 15, with regard to theoretical dependencies (see Fig. 4.11) this may take place in the range of the field strength where HgR / ( j ~ 6. The last evaluation with the findings shown in Fig. 4.15, as well as, a characteristic size of microdrops about 5 llm [209] allowed for as the coefficient of their surface tension yields the value of 2.10-3 erg/cm3 . The quoted value quite satisfactorily corresponds to the results, arrived at when the microdrop deformation has been studied in a field [213]. Kinetics of establishing equilibrium configurations. On the basis of a virial technique it is feasible also to get information about the kinetics of establishing equilibrium configurations. Assume that the deformation of a drop in a field is affine in

186 Quasi-Equilibrium Hydrodynamics of Magnetic Fluids

.".'"

.-.....

./

0. 26

•

0.25

H,Oe

•

5

10

Fig. 4.15 Real part of magnetic susceptibility of a colloid possessing the microdrop structure as affected by a constant field. Circles represent experimental data.

nature, thus, the Lagrange displacements of material points may be expressed in the following fonn 1;3 == L 33 x 3 ; 1;\ == -l/2L 33 X\; 1;3 == -1/2L n x2 · Hence, virial equation (4.28) after eliminating the pressure yields differential equation [157] 2 1 [ 27t(I-e2 )(3-e ) I n l+e- - 3) L -- Bm33 't o 2e5 l-e e4 (4.37) 2 2 _.!.(3-2e ) (3-4e ) arcsine )(I_e2 )1/2] ==0. 2 e2 e3 (l_e 2 )1/2 Here

't 0

== 2,,(3V I 47t) 1/3 I cr is a characteristic time scale for the drop defonnation of a

given type. At ,,== 0.5 Pa and cr == 6.4.10-4 dyn/cm for a drop of 15 ~m in radius 'to amounts to about 2 s. Upon approaching the conditions of loss of the hydrodynamic instability the drop defonnation rate diminishes. Variation of the virial ratio with respect to minute affine defonnations of the equilibrium configuration permits to obtain in the noninertial approximation the linear relaxation equation for a pulsed mode [157] • -1 L 33 + 't p L 33 == 0, where

Hydrostatics of a Magnetizable Fluid Drop 187

~=_91t 't p

Bm(~i-l)aL/~e(1-e2)2

2 (1-(A.-N)3(~i-l)aL/~e/41t)

(e2-3)Inl+e+~)2_ e5

l-e

e4

4 2 2 _31tBm(l_e 2 )((15-7e ) _ (e -12e + 15) In 1+ e) _ e6 2e 7 l-e 4 2 _(I_e2)1/3(27-24e -4e ) 4 4e

2 2 (8e -9)(2e -3)arcSine) 4(1- e 2)1/2 e 5 '

in the small field strength limit ( aL / a; e = 1/ 3 ) yields 'to

~=

31t - 2 (1

2 (e -3)II+e N(~ -1) /41t) eS n 1- e

Bm(~-I)

+

4 2 (e -12e +15)ln 1+ e )_ 2e 7 l-e

2 _31tBm(1_e 2)(15-7e ) e6 -(1- e2)1/3(27 - 24e

4e

2

-

4e

6)2

+;, -

4

) _

4

2

2

(8e - 9)(2e - 3)arCSine) 4(1-e 2 )1/2 e S '

Dependence of 'to / 't p upon the stretching extent of a drop for several shown in Fig. 4.16. The negative range of 't p hydrodynamically unstable states. At

(4.38)

bla~1

~

values is

values corresponds to the

a ratio 't p tends to the value 1.6

corresponding to a decrement of small elliptical deformations of a viscid drop [214]. The deformation dynamics of a microdrop of the concentrated phase ferromagnetic colloid is studied experimentally in [25] , where it is demonstrated that at the critical field strength value with respect to an onset of the hydrodynamic instability, an

2

1

b

a

Of---~=--+---J'------+----

Fig. 4.16 Relaxation time pulsed mode plotted against stretching degree of a drop: 20 (2);40 O.

~

== 10 (1);

188 Quasi-Equilibrium Hydrodynamics of Magnetic Fluids

increment for the elliptical deformations is small (see Fig. 4.16). Thus, differential equation (4.37) with the ratio i 33 =2ee/3(I-e 2 ) accounted for, permits to arrive at the following equation of motion, provided the eccentricity deviation of a drop from the critical value ee, corresponding to the point of loss of its hydrodynamic instability, is small

't oe=Ah+B(e-ee)2; h=H/He-l. (4.39) The numerical values of the constants A and B as affected by the magnetic permeability of a drop f.I. may be found on the basis ofrelationships (4.31) and 4.38). At f.I.= 40 these take the following values: A = 0.52, d / de('t o / 't p)le=e = 5.98 (ee = 0.88,

c

H;Ro / (j = 3.36).

Solution to equation (4.39) under the initial condition e =ec reads

e - ec =

~~ tg(JABht !'to)'

A similar dependence may be used to describe the initial stage of the deformation of a drop when He is exceeded [651]. The experimental values of't reported in [651] for a microdrop with R = 10.5 f.l.m and (j = 6.1·10,4 dyn/cm comply well with the root dependence 't- I = 't OI JABh (Fig. 4.17.). The coefficient 'to proves to equal

.JAB/

0.9 ± 0.1 c-

I

.

For the mentioned A and B values this yields as the viscosity of a

't, s 100

............

......

.................... .............

.........

.......

.........

...............

.......~

10

......

'e.•

....~

.~.......

h

Fig. 4.17 Characteristic time of initial stage of stretching of a microdrop as a function of supercritical field strength values in log coordinates. Line represents exponent dependence with an index of 0.52, circles - experimental results [25].

Hydrostatics of a Magnetizable Fluid Drop 189

microdrop the value of 0.57 Pa, which is consistent with representation of a drop as an inclusion of the concentrated phase of ferromagnetic colloid. Stability of a sessile drop. A virial method may be made use of in order to study also a number of more sophisticated phenomena [157], including the halving of a magnetic fluid drop, resting on a non-magnetic backing, revealed in [34]. A less complicated situation occurs when a magnetic fluid drop rests upon a magnetic backing possessing great magnetic permeability and the contact angle of wetting is equal to 90°. This case is studied experimentally in [57, 58]. On the basis of energy reasoning upon the assumption that a drop possesses a shape of a half of a prolate ellipsoid of revolution its relative stretching as affected by the magnetic Bond number is calculated in [60]. From comparing the energy of a whole drop to that of two half-volume drops the instability of a drop against division is inferred. A similar consideration shows the possibility of division provided finite perturbations of a singleconnected shape of the drop are present. Here, the magnitude of these perturbations, as well as the range where a stable, single-connected equilibrium pattern exists remain unknown. The last issue may be solved on the basis of the virial method [157] (see problem 4.3). It is worth noting that in accordance with the aforesaid, the energy and virial approaches in terms of determining the equilibrium patterns of freely suspended ellipsoidally shaped drops prove equivalent. In the given case under the condition of symmetry it is readily understandable that the field strength Hi within a drop is uniform, though depends upon the demagnetization factor N ik :Hi = H iO - NikMk . The ellipsoid parameters provided for an action of the gravitational forces may be determined by the gravitational (Bg = pg(3V / 4n) 2/3 2cr, V = 2Vk , Vk is the volume of a drop) and magnetic Bond numbers (Bm = M 2 (3V /4n)2/3 / cr,). At a low field strength by virtue of an action of the gravitational forces a drop possesses an oblate

2

o

Bm 5

10

Fig. 4.18 Relative deformation «4.43) and (4.44») of a drop resting on ferromagnetic backing plotted versus the magnetic Bond number (-). - - - designates a neutral curve of instability against deformation into a triaxial ellipsoid «4.46) and (4.47)). Bg = 5 (1); 7.5 (2); 8.5 (3).

190 Quasi-Equilibrium Hydrodynamics of Magnetic Fluids

shape. A virial method as the dependence of the eccentricity upon the magnetic Bond number in this case yields relationship (4.44). When Bm exceeds a certain critical value, dependent upon Bg an ellipsoid becomes prolate. In the given case the equilibrium pattern is determined by relationship (4.43) (see Problem 4.3). Dependence of the droplet aspect (semi-axes) ratio bla = j(Bm) evaluated in accordance with relationships (4.43) and (4.44) for several Bg values is plotted in Fig. 4.18. A sufficient instability condition for a drop with respect to its disintegration (breakdown) may lx! derived from the virial relationship (4.28) for small deformations of a drop resulting in a triaxial ellipsoid [157]. The critical Bm values may be arrived at by making use ofrelationships (4.46) and (4.47) (see Fig. 4.18, problem 4.3) and are independent upon the gravitational Bond number. As seen in the figure at sufficiently large Bg values , when the elongation of a drop along the direction of a field is hindered, there exists a limited range of Bm values and, consequently, of the field strength values, where a drop exhibits instability against the deformation into a triaxial ellipsoid. Development of such perturbations in accordance with the experimental findings reported in [57, 58] may bring about the division of a drop. Conclusion concerning the existence of a range of the field strength values bounded from below and from above, where the instability of a drop is allowed, is consistent with experiment [57, 58]. The experimental values of the critical field strength as a function of the volume ofa drop Vd (MH =50 Gs, P = 1.516 gicm3 , cr = 27 dyn/cm) are displayed in Fig. 4.19. As seen in the figure, drops having a volume less than 0.3 cm3 are stable with respect to the deformation into a triaxial ellipsoid throughout the whole investigated range of H values. In conformity with the theoretical dependencies plotted at which instability may still occur, in Fig. 4.18, the minimal volume of a drop

V;

V;,

approximately corresponds to the value Bm = 7.5. This value yields 0.3 as thus offering a good agreement with the value obtained in experiment. In order to effect a more detailed quantitative juxtaposition of the experimental finds and theoretical dependencies (4.46) and (4.47) information about the magnetization curve of a magnetic fluid is required. However, it is worth mentioning that the range of the dimensionless parameters Bg and Bm, at which the findings shown in Fig. 4.19 are obtained, is about [7.5; 9] and [0; 50] thus offering an agreement with the Bg and Bm values of the presented theoretical dependencies (see Fig. 4.18).

0.5

•

•

H.Oe

o

500

1000

1500

Fig. 4.19 Critical field strength values as a function of volume of a drop resting upon a magnetic backing.

Hydrostatics of a Magnetizable Fluid Drop 191

Description of a drop dividing on a non-magnetic basing is more complex. Experimental study of this situation is reported in [34, 56, 57, 58]. The field strength inside a drop in this event is essentially non-uniform and the pondermotive forces thus evoked serve as an additional destabilization factor of the simply connected configuration of a magnetic fluid. Particularly, there is an experimental evidence in [34, 57] demonstrating that the division of a drop initiates by formation of a neck at its base. The observed sausage-like configurations at the base of a drop are essentially similar to the patterns found by means of numerical simulation [163] of the flat layer of the magnetic fluid when the magnetic surface stress is absent. Problem 4.4 On the basis of virial relationships evaluate the extent of stretching of a magneticfluid drop resting upon a ferromagnetic backing of large magnetic permeability (J.1~oo). Presume that the contact angle of a liquid wetting a backing equals 90° and a drop as the part of an ellipsoid of revolution. Consider the stability of a drop with respect to nonaxisynunetrical perturbations. Solution The field strength inside a drop equals H~;) = H IN

NikM (N is the coefficient k ik tensor of the demagnetizing field of an ellipsoid). Equation of hydrostatics accounted for the uniformity of a field inside a drop may be expressed in the following form ( p = p - pgr ) -

-Vji = O. (4.40) Multiplying (4.40) by Xk and carrying out integration over a whole volume of a drop provided for the boundary condition on its surface - p = -0'(1/ R, + 11 ~) + 21t(Mn)2 + pgr gives viria1 relationship of the following form (the X3 axis of a Cartesian coordinate system rooted in the center of an ellipsoid is directed vertically):

Bllf pdV +V;t = B;d pdV +21t f x.n;(Mn)2dSv..

f

0' (oll-nlnt)dS s.

s.

(4.41 )

+Pf xlnl(gr)dS = O. s.

Upon obtaining (4.41) allowance is made for the integral

f xtn;pdS over the area of a backing,

wetted by a liquid being equal to zero owing to the synunetry condition and selection of the origin of a Cartesian coordinate system. Since under the condition of the problem a drop represents a semi-ellipsoid, the second and third integrals in relationship (4.41) may be transformed into integrals over the whole surface area of an ellipsoid. Provided the term O;t

f pdV

is eliminated, relationship (4.41) allows to write the equilibrium condition for a drop

in the following form ~l -1/ 2(V.l +V22 ) = O. (4.42) Hence, by making use of the following moment values for the case of a prolate ellipsoid

-1

f(xlnl-1/2(xlnl+x2n2))nldS=---3V (l-e )(3-e l+e 6)., --s-1o--4 2

2

2

21t s.

f

)

41t

_1 (3n: - 1) dS =(3V)21l 1 21t s. 2 41t 2(1- e 2)1/6

2

1- e

e

2 2 2 (3 - 4e )arcsine _ (1- e )112(3 - e )). el e2 '

1 1 (3V)4/l -f(xlnl-l/2(xlnl+x2n2))xldS=- 21t S.

e

8 41t

1 2 Ill' (I-e)

192 Quasi-Equilibrium Hydrodynamics of Magnetic Fluids

-/1 -

a 2 1b 2 its eccentricity; Vk is the volume where V = 2Vk is the volume of an ellipsoid; e = of a drop obtained under the equilibrium condition (4.42) one arrives at (b is the major semiaxis) ((3-2e 2)(1- e2/ 12 1e2 _ (3-4e 2) arcsine 1e3 + Bg 1(I-e2)11 6) Bm=

2;{I.-e 2)

7/6

4 ((3-e 2)ln((l+e)/(I-e»le5 -6Ie )

'

(4.43)

where Bm = M 2(3V 14X)1/3 1 cr is the magnetic Bond number, Bg = pg(3V 14x)1I3 . It follows from relationship (4.43), that at fInite magnetic Bond numbers owing to the action of the gravitational forces a drop possesses the shape of an hemisphere. For smaller Bm values a drop takes the shape of an oblate ellipsoid of revolution. The moments relevant to calculation of the equilibrium shape of a drop may be expressed in the following form 2 _I f(3n;-I)dS=(3V)213 1 (3-e _(I-e2X3+e2)lnl+e); 2 2 2x s. . 2 4x 2(1- e )113 e 2e3 1- e

2 2 I (3V)(3(e I) + (I - e3 ),,2 (3-2e 2 )arctg r:--;-; e ) -f(X~3-1I2(xl"t+x2n2»n;dS= 42x s• 4x e e vl-e 2 1 I (3V)4/3 2 1/3 -f(x3n3-1I2(x1"t +x2n2»x3dS =- (l-e), 2x s. 8 4x where e = ~I- b2 1a 2 is the eccentricity of an oblate ellipsoid of revolution (b is its minor semi-axis). The equilibrium condition (4.42) yields 1 ((3 +e 2)(1_ e2)ln«1 +e) I (1- e» 12e3-(3-e 2)1 e2+ Bg(l- e2( ) ~=

4X(I_e 2),,3

.

((1-e 2)"2(3-e2)arctg(el ~1-e2)1 e3 -3(I-e 2)1 e4)

~~

On the basis of virial relationships let consider the stability of a drop against the nonaxisymmetrical perturbations. At their neutrality point the following relation applies

OlkOLfpdV +Oi~k =0

(4.45)

where 0 L denotes the Legendre variation. Introducing the Lagrangre displacement of the material points of a liquid ~ - x' = x + ~(r) as the deformation mode into a triaxial ellipsoid relationship (4.45) after ~lirnination of the pressure 0 L(V;. - Vn ) = 0 gives 1 f (x.n 1 -x2n2Xn.2 -n22)n23dS=O. 2x s.

1 f (l-n 2 )dS+cr1 f (n.2 -n 22 2 -2cr3 2) dS+4m\1 -

2x s. 2x s. If the following moment values

Structure ofMagnetic Fluid Surface in a Vertical Magnetic Field 193

_1_J(I-n )dS = (3V)2/3 1 (e2-1+(l-e2Xl+e2)lnl+e)ba, l-e 2e 7

e = Jr-i-_-a2-/-b2. are used the last relation pennits to present the neutrality condition with respect to the perturbations of a given type in the following form: 2 Bm= 1 3(1+e )]/ 41t(1- e 2 ) 4e 1- e 2 (4.46)

l/l.[(3e4+2e2+3)ln~_

I[

(S-4e2) 2e l(l- e2r'l

_

Bm -

1

41t(1-e 2)

arctg_e_+.!._~] 2 2

7/6

Jl- e

3

2e

b'); o 0 0 0 0 here G(r,q>; r' ,q>') = (r 2 - 2rr' cos(

') +r,2)-1I2 _(r 2 -2rr'cos(') is odd function. Since the integral over r in

+2P +112(1; - P(l- A» < > +

a

where

+112(1; + P(l + A» < the characteristic

n: > +11

n:n;

n;

rot ,

rotational

viscosity

may

be

expressed

as

Rheology of Magnetic Colloids in a Field 285

llrot

= -13~ < n y » / 200 . In this case the first and third differences of the normal stress

reads where F. = «X -2"-13) < (n; -n;)nyn > +(l;+13(1+A.» +213 < n yn z » / 200.

Relationships (5.60) for generation of the random trajectory of an anisotropic particle here recast to give the following form: M l = ooM(l- A.)n~n~/ ~r-n-~2-+-n-~-2 +.J2My 2; M 2 =-~M~n~2

+n~2 -oolitn~(1-A.(1-n~2»/~n~2 +n~2

+.J2MYI'

The characteristic viscosity values plotted versus the shear rate are displayed in Fig. 5.27. For sufficiently small ~ in the initial region of the flow curve the obtained results fit the theoretical relationship 11. = 110. + 0.425~ 2 [490]. It is obvious from Fig. 5.27 that when a field is applied longitudinally with respect to the flow the non-Newtonian behaviour of a suspension is considerably less pronounced than in the case of a transversally applied field. It is shown in Fig. 5.28 where at ~ = 2 the characteristic viscosity is plotted against the shear rate in a longitudinal and transversal fields. A similar anisotropy of the rheological behaviour may be observed also experimentally as it is evidenced in Fig. 5.23. The first and third difference of the normal stress drop when a longitudinal field is applied are decreasing. A method of Brownian dynamics may furnish also information about other interesting phenomena in ellipsoidal ferromagnetic particle suspensions, especially, when the dynamo-optical effects are considered. As an illustration, the extinction angle

o

10

20

Fig. 5.27 Characteristic viscosity of an ellipsoidal particle suspension plotted versus shear rate in the presence of a magnetic field parallel to velocity vector. - - - present theoretical values of characteristic viscosity for a given field orientation in the limiting case of small shear rates 11. = 11-0 +0.425;2 [331]: ~ = 0.5 (1); 1.0 (2); 1.5 (3); 2.0 (4). Circles represent numerical simulation values.

286 Internal Rotations in Hydrodynamics of Magnetic Colloids o

\ \

'1-

\o

\

5

o

\

\

0, \

1

" 0" 1\( "

,

0-

2/

o

"

'0,'"''''0........ .....0.....

...... 0_ . . . -.0-

--0__ --0-_=8::-=-0-_8

10

20

-0-__

Fig. 5.28 Anisotropy of viscosity of an ellipsoidal particle suspension in the presence of a magnetic field. Langevin parameter ~ = 2. Curve 1 - field strength vector parallel to velocity gradient; 2 - parallel to velocity. Circles represent numerical simulation values.

of the light passing perpendicular to the shear plane through a ferrosuspension residing between crossed polarizers, is depicted in Fig. 5.29. The angle 0 . Mixture magnetization M is proportional to magnetic

in a non-magnetic carrier

c carrier, ~ > V; , hence phase concentration, therefore the coefficient Pm also is positive. Consequently, the diffusion-eonvective instability appears under the condition, that the particle concentration diminishes in the direction of sum buoyancy. It should be noted, that convective instability in non uniform magnetic colloids and suspensions is markedly more significant than thermoconvective instability. Since in colloidal solutions the coefficients Pnand f3 m exceed the thermal analogues by several orders of magnitude and the Schmidt number is considerably larger then the Prandtl number, then the concentration Rayleigh numbers by 4 to 6 orders of magnitude exceed the thermal ones (see evaluations in 6.3). A more significant role here plays local field of a magnetic fluid. For instance, in transversal uniform external field, in the event, that in a layer dH I dz = -41tPmM(1+41toM I OH)-l dn; I dz,

(A

41tP~M2 dn;) dn; /4 • Ran = I-'npg+ --, 1+41t'X. dz dz pvD;

(6.50)

a situation comes to an effect, under which magnetic force largely exceeds gravitational one, and irrespective of Vn; orientation to g, the Ra: criterion exceeds the critical value, determined by relationship (6.13). Convective instability of diffusion layer of colloidal particles on the interface of a magnetic fluid and non-magnetic carrier under gravitationally stable stratification of liquid dn; I dz < 0 is studied in [402]. The external magnetic field B o is uniform and

Magnetodiffusion Convection 333

directed normal to an interface. Initial concentration distribution in a diffusion layer is assumed linear. As boundary conditions on velocity and magnetic field, the conditions, typical of a layer with free boundaries, are used. When considering development of convection, a magnetic phase is assumed passive, Le., not only magnetophoresis of colloidal particles is neglected, but their diffusion as well. Since, at D; ~ 0 the Rayleigh number grows without limit, it is readily understandable, that convection is thresholdfree in this approximation. Convection appears under the stipulation, that the sum in parenthesis (6.50) becomes negative (let recall, that dn; / dz < 0), i.e., if G=

f3 n Pg(I+41t OM) oH

41tf3~M2 (- ~;)