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Hysteresis of ferrogels magnetostriction
Author’s Accepted Manuscript Hysteresis of ferrogels magnetostriction Andrey Zubarev, Dmitry Chirikov, Gennady Stepanov, Dmitry Borin
www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(16)33074-8 http://dx.doi.org/10.1016/j.jmmm.2016.11.069 MAGMA62144
To appear in: Journal of Magnetism and Magnetic Materials Received date: 23 June 2016 Revised date: 12 November 2016 Accepted date: 16 November 2016 Cite this article as: Andrey Zubarev, Dmitry Chirikov, Gennady Stepanov and Dmitry Borin, Hysteresis of ferrogels magnetostriction, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.11.069 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Hysteresis of ferrogels magnetostriction Andrey Zubareva, Dmitry Chirikova, Gennady Stepanovb, Dmitry Borinc a
Urals Federal University, 620000, Ekaterinburg, Russia State Scientific Research Institute of Chemistry and Technology of Organoelement Compounds 105118, Moscow, Russia c Technische Universität Dresden, Magnetofluiddynamics, Measuring and Automation Technology, 01062, Dresden, Germany b
ABSTRACT We propose a theoretical model of magnetostriction hysteresis in soft magnetic gels filled by micronsized magnetizable particles. The hysteresis is explained by unification of the particles into linear chain-like aggregates while the field increasing and rupture of the chains when the field is decreased. Keywords Ferrogels, Magnetodeformation
Composites of nano- or micronsized magnetic particles in a polymeric matrix present new kind of magnetically controlled soft materials, usually named as ferrogels, ferroelasts, magnetically active elastomers, etc. These systems attract considerable interest of researches and engineers due to rich set of physical properties valuable for many industrial and biomedical applications. Experiments demonstrate strong hysteretic effects in ferrogels with soft matrix (see, for example, [1,2]). As, the wide hysteresis loops of the stress – strain dependence have been observed in [1] when the ferrogels were placed in uniform magnetic fields. In contrast, the linear dependence of the stress vs. strain has been observed in [1] for the same samples without the field action. Hysteretic dependence of the gels magnetostriction (elongation under the field action) has been detected in [1,2]. The similar shape of the curves of the samples magnetization vs. applied magnetic field has been observed in [1]. Remarkably that the hysteresis of magnetization has not been observed for the dry powder of the magnetic filler; for the ferrogels the area of the hysteresis loops increased with decrease of the sample elastic modulus. The last results show that the physical cause of the hysteresis effects can lie only in rearrangement, under the field action, of the particles in the soft matrix. Indeed, observations of [1] show that the initially randomly distributed particles unite into chain-like aggregates when the gel is placed in the field. Unification of the particles into the chains and other aggregates in liquid magnetic suspensions is well known phenomenon, which has been studied in many works (see, for example, overviews in [3,4]). However, to the best of our knowledge, the magnetically induced rearrangement of magnetic particles in elastic matrixes, formation of the internal heterogeneous structures as well as effect of these structures on macroscopical properties of the soft magnetic composites have not yet been studied in literature. At the same time, these phenomena present significant interest both from the scientific point of view and viewpoint of practical applications of the mechanically soft magnetic materials. We suggest a simple model of formation of the chain-like structures in a system of magnetizable micronsized particles embedded in soft elastic matrix and estimate dependence of the composite strain on an applied magnetic field. We consider a magnetic gel, consisting of identical spherical micron-sized magnetizable particle embedded in an incompressible gel matrix. We suppose that the matrix was cured without magnetic field; therefore, the particles, initially, are uniformly distributed in the gel. Our first goal is to describe unification of the particles, under magnetic field action, into the linear chain-like aggregates.
1
For maximal simplification of the mathematical part of the problem, we will use the main ideas of the hierarchical model [5] of the chain formation in magnetorheological suspensions, combined with the lattice model, often used in statistical physics of gas and liquid systems [6]. In the framework of the lattice approach, we will suppose that initially the particles are situated in the centers of cells of a cubic lattice (Fig.1). Let one of the lattice axes be directed along the applied magnetic field .
Fig.1. Sketch of the cubic lattice. Dots present the particles situated in the centers of the lattice cells. Segment of the possible positions one of the particles is shown in the left part of the lattice.
We estimate the edge of the cubic cell by using the condition that the particle volume concentration in the lattice model is equal to the concentration of the particles in the real composite. This gives: ( )
(1)
⁄ is the particle volume; is the particle radius. Here Obviously, the condition , which reflects impossibility for the particles to interpenetrate, must be held. We will take into account interaction only between the particles, situated along the axis, parallel to the applied magnetic field. Analysis shows that the direct application of the lattice model, where the particles are situated right in the centers of the cells, leads to very rough description of the chaining, since this approximation does not take into account the statistical character of the particles mutual distribution. In order to approach the lattice model to the real systems with the random spatial distribution of the particles, we will suppose that the center of each particle, with equal probability, can be situated in a linear segment, aligned along the field . Following to the ideas of the hierarchical theory [5], we will consider the process of the chain formation as the unification of the particles into the doublets; then – unification of the doublets into the “quartets” of the particles, etc. Various stages of this unification are illustrated in Fig.2.
Fig.2. Sketch of three first stages of the particles aggregation. The horizontal arrows illustrate evolution of the particles in time. The segments of possible positions of the chains are shown. The segments boundaries are for the poles of the particles at the chains extremities. The single particles and the chains are shown in the centers of the segments of their possible positions.
2
The center of each doublet can take place in each point of a segment with the center right in the middle between the centers of the cells of the single particles. The similar assumption is made for the centers of the fourparticles clusters, and so on Fig.2). In the hierarchical approach each chain consists of particles, where is the number of the stage of the chaining. We determine the number of particles in the chains on the basis of the equality of elastic and magnetic forces between two chains. We estimate the elastic force ( ) on the basis of the linear Hook law of the incompressible matrix deformation: ( )
(
)
(2)
Here is the Young’s modulus of the matrix; is the distance between the centers of the particles at the nearest extremities of the chains. The magnetic force was evaluated on the basis of the following three assumptions. First, we use the approximation of the dipole-dipole interaction between the particles. Second, we propose the uniform magnetization of the particle in its volume. Next, we take into account the mutual magnetization of the particles in the chains by using the Frolich-Kennelly relation [7,8]. These assumptions lead us to the next equation: ( )
( {∑
[
(
) ]
∑
[
(
) ]
) }
∑
(3)
Here is magnetization of the particles which are part of -particle chain. The term corresponds the mutual magnetization of the particles within the same chain; – the mutual magnetization between different chains. Our analysis shows that formation of a chain, when the field is increased, takes place at the higher strengths, than the chain rupture, when the field decreases. As a consequence, the chain length exhibits the hysteretic dependence on the applied field. Let’s consider of the problem of ferrogels deformation. For maximal simplification of mathematical part of the work, we will restrict ourselves by the consideration of incompressible polymer and suppose that the sample of the composite presents an ellipsoid of revolution with the axes . The ellipsoidal shape provides homogeneity of the internal magnetic field inside the sample and allows us to use the well-known explicit expressions [9] for the sample demagnetizing factor. That is very important for analytical study of the magnetomechanical effects. We well suppose that the external field is aligned along the axis of symmetry (axis ) of the ellipsoid and after deformation the sample also has the ellipsoidal shape with another value of the aspect ratio [9]. In fact, the shape of a ferrogels must deviate from the ellipsoid after the magnetostriction [10,11]. However, the deviation effects are not strong and the simplification of the ellipsoidal shape of the deformed sample is quite applicable to estimate order of magnitude of the magnetostriction. The density of the sample free energy can be presented as a sum of the densities of the magnetic and elastic free energies: (4)
Here is the Young’s modulus of the sample; Frankel-Acrivos formula [12]:
is the strain of the sample. To estimation
(5) [
(
) ]
3
we use the
Let us expand the density of magnetic free energy . In the square approximation we get:
in the power series with respect to small deformation |
|
(6)
Here and are coefficients which depend both of the external magnetic field and the shape of the sample. Substituting expression for into equation (4) we get total density of free energy: (7)
Theoretical results of [13,14] demonstrate that magnetostriction of a ferrogel sample is determined by the following two factors. The first one is the demagnetization shape-factor, which tends to stretch the sample along the applied field. The second factor is change of the sample effective magnetic susceptibility as a consequence of change of the particles mutual disposition after the sample deformation. This factor depends of mean size of the chains and usually tends to shrink the sample. Both, the ferrogel effective susceptibility and its change depend on the size of the chains in the composite. Our calculations show that for the spherical sample with the chains, the first factor dominates. The sample relative elongation depends on the size of the chains, therefore experiences hysteresis at increase and further decrease of the magnetic field. The condition of equilibrium is equivalent to the minimum free energy (7). Differentiating this equation with respect to and equating the derivative to zero, we got: (8)
Results of calculation of the composite strain, as well as its comparison with the experiments [2], are shown in Fig.3.
Fig.3. Hysteresis of strain of a ferrogel vs. applied external magnetic field . System parameters: volume concentration of the particles = 0.30; saturated magnetization of the particle material = 1650 kA/m; the aspect ratio of non deformed sample = 1.03; initial susceptibility of the particles material = 100; elastic modulus of the matrix = 140 kPa; 1 – magnetic field increases; 2 – magnetic field decreases; dots – experiment; solid lines – theory.
In spite of the model simplicity, the theoretical curves, presented in Fig.3, reproduce the main physical feature of the experimental results – the hysteretic behavior of the strain as a function of the applied field, as well as the tendency of the elongation to saturation while the field increasing. In the order of magnitude our results are in agreement with the experimental one (no fit parameters have been used). The quantitative discrepancy can be explained by the fact that we, for maximal simplification of calculations, have ignored dependence of the 4
composite elastic modulus on the applied field, whereas in experiment this dependence can be quite significant [1]. Theoretical analysis of this dependence can be a natural continuation of this work. In Conclusion, we present a theoretical model of the hysteretic dependence of a ferrogel magnetostriction on the applied magnetic field. This hysteresis is explained by the unification of the particles into linear chainlike aggregates when the field is increased and by the rupture of the chains when the field is decreased. Acknowledgments This work has been done under the financial support of the Russian Fund for Basic Research, projects 1408-00283, 16-58-12003, 16-32-00019 mol a. and the Program of Russian Ministry of Science and Education, project 3.12.2014/K. One of us (D.B.) would like to acknowledge the financial support of Deutsche Forschungsgemeinschaft (DFG) under Grant Bo 3343/1-1 within PAK 907 References [1] G. Stepanov, D. Borin, Yu. Raikher, P. Melenev and N.S. Perov, J. Phys.: Condens. Matter, 20 (2008) 204121. [2] G. Diguet, E. Beaugnon and J.Y. Cavaille, J. Magnetism and Magnetic Materials 322 (2010) 3337. [3] G. Bossis, O. Volkova, S. Lacis, A. Meunier, in Ferrofluids, Magnetically Controllable Fluids and Their Application, edited by S. Odenbach (Springer, Berlin, 2002). [4] P. Ilg, S. Odenbach, in Colloidal Magnetic Fluids. Basics, Development and Application of Ferrofluids (Ed. S. Odenbach), Lectures Notes in Physics, Springer, 2009, P.249. [5] H. See and M. Doi, J. Phys. Soc. Jpn. 60 (1991) 2778. [6] T.L. Hill, Statistical Mechanics – Principles and Selected Applications, Courier Corporatiopn, 2013. [7] Jiles, D.: Introduction to Magnetism and Magnetic Materials (Chapman & Hill, London,1991). [8] R.M. Bozorth, Ferromagnetism (Wiley, New York, 1993). [9] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuum Media, Pergamon Press, London, 1960. [10] Yu.L. Raikher, O.V. Stolbov, J. Magn. Magn. Materials 477 (2003) 258–259. [11] K. Morozov, M. Shliomis, H. Yamaguchi, Physical Review E 79 (2009) 040801. [12] N.A. Frankel, A. Acrivos, Chem. Eng. Sci. v. 22 (1967) 847. [13] A. Zubarev, Physica A 392 (2013) 4824–4836. [14] A.Yu. Zubarev, A.S. Elkady, Physica A 413 (2014) 400–408.
Highlights
- A theoretical model of magnetostriction hysteresis in magnetic gels is proposed.
5
www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(16)33074-8 http://dx.doi.org/10.1016/j.jmmm.2016.11.069 MAGMA62144
To appear in: Journal of Magnetism and Magnetic Materials Received date: 23 June 2016 Revised date: 12 November 2016 Accepted date: 16 November 2016 Cite this article as: Andrey Zubarev, Dmitry Chirikov, Gennady Stepanov and Dmitry Borin, Hysteresis of ferrogels magnetostriction, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.11.069 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Hysteresis of ferrogels magnetostriction Andrey Zubareva, Dmitry Chirikova, Gennady Stepanovb, Dmitry Borinc a
Urals Federal University, 620000, Ekaterinburg, Russia State Scientific Research Institute of Chemistry and Technology of Organoelement Compounds 105118, Moscow, Russia c Technische Universität Dresden, Magnetofluiddynamics, Measuring and Automation Technology, 01062, Dresden, Germany b
ABSTRACT We propose a theoretical model of magnetostriction hysteresis in soft magnetic gels filled by micronsized magnetizable particles. The hysteresis is explained by unification of the particles into linear chain-like aggregates while the field increasing and rupture of the chains when the field is decreased. Keywords Ferrogels, Magnetodeformation
Composites of nano- or micronsized magnetic particles in a polymeric matrix present new kind of magnetically controlled soft materials, usually named as ferrogels, ferroelasts, magnetically active elastomers, etc. These systems attract considerable interest of researches and engineers due to rich set of physical properties valuable for many industrial and biomedical applications. Experiments demonstrate strong hysteretic effects in ferrogels with soft matrix (see, for example, [1,2]). As, the wide hysteresis loops of the stress – strain dependence have been observed in [1] when the ferrogels were placed in uniform magnetic fields. In contrast, the linear dependence of the stress vs. strain has been observed in [1] for the same samples without the field action. Hysteretic dependence of the gels magnetostriction (elongation under the field action) has been detected in [1,2]. The similar shape of the curves of the samples magnetization vs. applied magnetic field has been observed in [1]. Remarkably that the hysteresis of magnetization has not been observed for the dry powder of the magnetic filler; for the ferrogels the area of the hysteresis loops increased with decrease of the sample elastic modulus. The last results show that the physical cause of the hysteresis effects can lie only in rearrangement, under the field action, of the particles in the soft matrix. Indeed, observations of [1] show that the initially randomly distributed particles unite into chain-like aggregates when the gel is placed in the field. Unification of the particles into the chains and other aggregates in liquid magnetic suspensions is well known phenomenon, which has been studied in many works (see, for example, overviews in [3,4]). However, to the best of our knowledge, the magnetically induced rearrangement of magnetic particles in elastic matrixes, formation of the internal heterogeneous structures as well as effect of these structures on macroscopical properties of the soft magnetic composites have not yet been studied in literature. At the same time, these phenomena present significant interest both from the scientific point of view and viewpoint of practical applications of the mechanically soft magnetic materials. We suggest a simple model of formation of the chain-like structures in a system of magnetizable micronsized particles embedded in soft elastic matrix and estimate dependence of the composite strain on an applied magnetic field. We consider a magnetic gel, consisting of identical spherical micron-sized magnetizable particle embedded in an incompressible gel matrix. We suppose that the matrix was cured without magnetic field; therefore, the particles, initially, are uniformly distributed in the gel. Our first goal is to describe unification of the particles, under magnetic field action, into the linear chain-like aggregates.
1
For maximal simplification of the mathematical part of the problem, we will use the main ideas of the hierarchical model [5] of the chain formation in magnetorheological suspensions, combined with the lattice model, often used in statistical physics of gas and liquid systems [6]. In the framework of the lattice approach, we will suppose that initially the particles are situated in the centers of cells of a cubic lattice (Fig.1). Let one of the lattice axes be directed along the applied magnetic field .
Fig.1. Sketch of the cubic lattice. Dots present the particles situated in the centers of the lattice cells. Segment of the possible positions one of the particles is shown in the left part of the lattice.
We estimate the edge of the cubic cell by using the condition that the particle volume concentration in the lattice model is equal to the concentration of the particles in the real composite. This gives: ( )
(1)
⁄ is the particle volume; is the particle radius. Here Obviously, the condition , which reflects impossibility for the particles to interpenetrate, must be held. We will take into account interaction only between the particles, situated along the axis, parallel to the applied magnetic field. Analysis shows that the direct application of the lattice model, where the particles are situated right in the centers of the cells, leads to very rough description of the chaining, since this approximation does not take into account the statistical character of the particles mutual distribution. In order to approach the lattice model to the real systems with the random spatial distribution of the particles, we will suppose that the center of each particle, with equal probability, can be situated in a linear segment, aligned along the field . Following to the ideas of the hierarchical theory [5], we will consider the process of the chain formation as the unification of the particles into the doublets; then – unification of the doublets into the “quartets” of the particles, etc. Various stages of this unification are illustrated in Fig.2.
Fig.2. Sketch of three first stages of the particles aggregation. The horizontal arrows illustrate evolution of the particles in time. The segments of possible positions of the chains are shown. The segments boundaries are for the poles of the particles at the chains extremities. The single particles and the chains are shown in the centers of the segments of their possible positions.
2
The center of each doublet can take place in each point of a segment with the center right in the middle between the centers of the cells of the single particles. The similar assumption is made for the centers of the fourparticles clusters, and so on Fig.2). In the hierarchical approach each chain consists of particles, where is the number of the stage of the chaining. We determine the number of particles in the chains on the basis of the equality of elastic and magnetic forces between two chains. We estimate the elastic force ( ) on the basis of the linear Hook law of the incompressible matrix deformation: ( )
(
)
(2)
Here is the Young’s modulus of the matrix; is the distance between the centers of the particles at the nearest extremities of the chains. The magnetic force was evaluated on the basis of the following three assumptions. First, we use the approximation of the dipole-dipole interaction between the particles. Second, we propose the uniform magnetization of the particle in its volume. Next, we take into account the mutual magnetization of the particles in the chains by using the Frolich-Kennelly relation [7,8]. These assumptions lead us to the next equation: ( )
( {∑
[
(
) ]
∑
[
(
) ]
) }
∑
(3)
Here is magnetization of the particles which are part of -particle chain. The term corresponds the mutual magnetization of the particles within the same chain; – the mutual magnetization between different chains. Our analysis shows that formation of a chain, when the field is increased, takes place at the higher strengths, than the chain rupture, when the field decreases. As a consequence, the chain length exhibits the hysteretic dependence on the applied field. Let’s consider of the problem of ferrogels deformation. For maximal simplification of mathematical part of the work, we will restrict ourselves by the consideration of incompressible polymer and suppose that the sample of the composite presents an ellipsoid of revolution with the axes . The ellipsoidal shape provides homogeneity of the internal magnetic field inside the sample and allows us to use the well-known explicit expressions [9] for the sample demagnetizing factor. That is very important for analytical study of the magnetomechanical effects. We well suppose that the external field is aligned along the axis of symmetry (axis ) of the ellipsoid and after deformation the sample also has the ellipsoidal shape with another value of the aspect ratio [9]. In fact, the shape of a ferrogels must deviate from the ellipsoid after the magnetostriction [10,11]. However, the deviation effects are not strong and the simplification of the ellipsoidal shape of the deformed sample is quite applicable to estimate order of magnitude of the magnetostriction. The density of the sample free energy can be presented as a sum of the densities of the magnetic and elastic free energies: (4)
Here is the Young’s modulus of the sample; Frankel-Acrivos formula [12]:
is the strain of the sample. To estimation
(5) [
(
) ]
3
we use the
Let us expand the density of magnetic free energy . In the square approximation we get:
in the power series with respect to small deformation |
|
(6)
Here and are coefficients which depend both of the external magnetic field and the shape of the sample. Substituting expression for into equation (4) we get total density of free energy: (7)
Theoretical results of [13,14] demonstrate that magnetostriction of a ferrogel sample is determined by the following two factors. The first one is the demagnetization shape-factor, which tends to stretch the sample along the applied field. The second factor is change of the sample effective magnetic susceptibility as a consequence of change of the particles mutual disposition after the sample deformation. This factor depends of mean size of the chains and usually tends to shrink the sample. Both, the ferrogel effective susceptibility and its change depend on the size of the chains in the composite. Our calculations show that for the spherical sample with the chains, the first factor dominates. The sample relative elongation depends on the size of the chains, therefore experiences hysteresis at increase and further decrease of the magnetic field. The condition of equilibrium is equivalent to the minimum free energy (7). Differentiating this equation with respect to and equating the derivative to zero, we got: (8)
Results of calculation of the composite strain, as well as its comparison with the experiments [2], are shown in Fig.3.
Fig.3. Hysteresis of strain of a ferrogel vs. applied external magnetic field . System parameters: volume concentration of the particles = 0.30; saturated magnetization of the particle material = 1650 kA/m; the aspect ratio of non deformed sample = 1.03; initial susceptibility of the particles material = 100; elastic modulus of the matrix = 140 kPa; 1 – magnetic field increases; 2 – magnetic field decreases; dots – experiment; solid lines – theory.
In spite of the model simplicity, the theoretical curves, presented in Fig.3, reproduce the main physical feature of the experimental results – the hysteretic behavior of the strain as a function of the applied field, as well as the tendency of the elongation to saturation while the field increasing. In the order of magnitude our results are in agreement with the experimental one (no fit parameters have been used). The quantitative discrepancy can be explained by the fact that we, for maximal simplification of calculations, have ignored dependence of the 4
composite elastic modulus on the applied field, whereas in experiment this dependence can be quite significant [1]. Theoretical analysis of this dependence can be a natural continuation of this work. In Conclusion, we present a theoretical model of the hysteretic dependence of a ferrogel magnetostriction on the applied magnetic field. This hysteresis is explained by the unification of the particles into linear chainlike aggregates when the field is increased and by the rupture of the chains when the field is decreased. Acknowledgments This work has been done under the financial support of the Russian Fund for Basic Research, projects 1408-00283, 16-58-12003, 16-32-00019 mol a. and the Program of Russian Ministry of Science and Education, project 3.12.2014/K. One of us (D.B.) would like to acknowledge the financial support of Deutsche Forschungsgemeinschaft (DFG) under Grant Bo 3343/1-1 within PAK 907 References [1] G. Stepanov, D. Borin, Yu. Raikher, P. Melenev and N.S. Perov, J. Phys.: Condens. Matter, 20 (2008) 204121. [2] G. Diguet, E. Beaugnon and J.Y. Cavaille, J. Magnetism and Magnetic Materials 322 (2010) 3337. [3] G. Bossis, O. Volkova, S. Lacis, A. Meunier, in Ferrofluids, Magnetically Controllable Fluids and Their Application, edited by S. Odenbach (Springer, Berlin, 2002). [4] P. Ilg, S. Odenbach, in Colloidal Magnetic Fluids. Basics, Development and Application of Ferrofluids (Ed. S. Odenbach), Lectures Notes in Physics, Springer, 2009, P.249. [5] H. See and M. Doi, J. Phys. Soc. Jpn. 60 (1991) 2778. [6] T.L. Hill, Statistical Mechanics – Principles and Selected Applications, Courier Corporatiopn, 2013. [7] Jiles, D.: Introduction to Magnetism and Magnetic Materials (Chapman & Hill, London,1991). [8] R.M. Bozorth, Ferromagnetism (Wiley, New York, 1993). [9] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuum Media, Pergamon Press, London, 1960. [10] Yu.L. Raikher, O.V. Stolbov, J. Magn. Magn. Materials 477 (2003) 258–259. [11] K. Morozov, M. Shliomis, H. Yamaguchi, Physical Review E 79 (2009) 040801. [12] N.A. Frankel, A. Acrivos, Chem. Eng. Sci. v. 22 (1967) 847. [13] A. Zubarev, Physica A 392 (2013) 4824–4836. [14] A.Yu. Zubarev, A.S. Elkady, Physica A 413 (2014) 400–408.
Highlights
- A theoretical model of magnetostriction hysteresis in magnetic gels is proposed.
5