- Email: [email protected]
Modeling and finite element simulation of the magneto-mechanical behavior of ferrogels
Journal of Magnetism and Magnetic Materials xx (xxxx) xxxx–xxxx
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Modeling and finite element simulation of the magneto-mechanical behavior of ferrogels ⁎
Abdolhamid Attaran , Jörg Brummund, Thomas Wallmersperger Institut für Festkörpermechanik, Technische Universität Dresden, 01062 Dresden, Germany
A R T I C L E I N F O
A BS T RAC T
Keywords: Ferrogel Modeling Theory of mixtures Magneto-mechanical formulation
In our previous study (see Ref. Attaran et al. (in press) [1]) we formulated a continuum model for ferrogels considering them as multicomponent materials. In the present work a reduced model for ferrogels is presented consisting only of a polymer network (P) and fixed magnetic particles (f). The reduced model is solved using the finite element method where the only degrees of freedom are mechanical displacement and magnetic potential. Elongation and contraction of a ferrogel are observed parallel and perpendicular to the applied magnetic field direction, respectively. These results are in a good qualitative agreement with experimental results. With our modeling approach, we were able to investigate (i) the influence of the magnetic field on the polymer gel containing magnetic particles and (ii) the resulting mechanical deformation of a ferrogel.
1. Introduction Ferrogels are magneto-sensitive materials, the fabrication of which was first reported in the late 90s [2]. They primarily consist of a polymer gel and magnetic filler nanoparticles (e.g. magnetite Fe3O4 [3]). Polymer gels are usually synthesized by chemically cross-linking of an array of polymer networks. By application of an external magnetic field, magnetic particles orient themselves towards the applied field. Since some of the particles are adhered to the polymer network, ferrogels deform in the direction of the applied magnetic field, see Fig. 1. Shortly after their introduction, first theoretical models for ferrogels emerged both at macroscopic and microscopic scales. Notable development in macroscopic modeling are due e.g. to the works of Raikher and Stolbov [4,5], and Zubarev [6]. Hydrodynamics models for ferrogels were developed e.g. by Jarkova et al. [7] for isotropic gels and by Bohlius et al. [8] for uniaxial ferrogels. Development of microscopic models for ferrogels has also been pursued by several researchers over the past years. Worth mentioning here is for example a modeling approach in which the magnetic particles are modeled as hard spheres along with harmonic springs, resembling the elastic matrix [9–11]. Using a microscopic approach some authors used numerical tools e.g. Monte Carlo simulations [12] or Molecular Dynamics (MD) in 2D [13] and 3D [14,15] to simulate the magnetoelastic behavior of ferrogels. For instance Weeber et al. [13] studied deformation mechanism of ferrogels where the particle matrix interaction were examined
⁎
using Néel relaxation (no direct transfer of magnetic torque to the polymer matrix) or Brownian relaxation (direct transfer of magnetic torque to the polymer matrix). Depending on the mechanism, contraction or elongation of ferrogels was reported. Recently few scalebridging models [16,17] have also been proposed to create a bridge over different length scales. The motivation behind this research stems itself from the fact that despite recent developments in modeling the magneto-mechanical behavior of ferrogels, there have hardly been any attempt to propose a macroscopic model based on the multiphase, multicomponent nature of ferrogels. Our ultimate goal is to obtain a magneto-mechanical formulation capable of describing mechanical deformation of a ferrogel placed in a magnetic field using this approach. A comprehensive magneto-mechanical formulation for ferrogels has been developed by the authors in [1] within the framework of continuum mechanics of mixtures. In the current paper a reduced version of this model is presented. The reduced model for ferrogels consists only of polymer network (P) and fixed magnetic particles (f). In this model mechanical displacement and magnetic potential are the degrees of freedom. The finite element method (FEM) is used for the numerical treatment of this model. 2. Coupled magnetomechanical formulation In this section a coupled magnetomechanical formulation is presented for ferrogels including field equations, constitutive relations, and boundary conditions. This formulation is a reduction of the model
Corresponding author. E-mail address: [email protected] (A. Attaran).
http://dx.doi.org/10.1016/j.jmmm.2016.09.058 Received 5 July 2016; Received in revised form 31 August 2016; Accepted 11 September 2016 Available online xxxx 0304-8853/ © 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Attaran, A., Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j.jmmm.2016.09.058
Journal of Magnetism and Magnetic Materials xx (xxxx) xxxx–xxxx
A. Attaran et al.
H
Polymer network (P) Fixed magnetic particles (f)
Fig. 1. Ferrogel before application of the magnetic field (left). Deformation of a ferrogel in magnetic field (H) (right).
of Ref. [1]. In the development of this formulation the ferrogel is assumed as a continuum body . in three dimensional Euclidean space, enclosed by boundary ∂. . To obtain the reduced model for ferrogels (i) isothermal and reversible processes are only considered, (ii) mobile magnetic particles are not taken into account, (iii) no liquid is assumed to be present, (iv) small strain theory is used, and (v) mass density of ferrogels remains constant. Corresponding to this reduced model a suitable free energy function is introduced as [1]
ρ0 f =
1 1 ijkl εij εkl + λ ijkl εij mk ml 2 2
in .,
Table 2 Constitutive equations for a magneto-mechanical problem.
bi = −
1 λ ijkl εjk ml in ρ0
(
)
eijk m j k + bk = 0
σij = ijkl εkl +
., in
(8a)
.,
(8b)
1 λ ijkl mk ml + λ ijkl εkq m q ml in .. 2
(8c)
(1)
where f is the specific free energy density, ρ0 the reference mass density, ijkl the fourth order elasticity tensor, λijkl the magnetostrictive tensor, mi the specific magnetization of the gel, and εij is the strain tensor. In Tables 1–3 the corresponding field equations, constitutive equations as well boundary conditions are presented. In Table 1. i is the magnetic induction vector, i the magnetic intensity vector, i = ρ mi the magnetization vector, ρ is the mass density of the ferrogel, ϵijk the Levi-Civita symbol, δij the Kronecker delta, μ0 magnetic permeability of vacuum, Ψ mag scalar magnetic potential, σij Cauchy stress tensor, σijE Maxwell stress tensor, and ui
Table 3 Boundary conditions for a magneto-mechanical problem.
Magnetic field Ψ mag = Ψ¯ mag on
i ni = ¯
∂.,
(9a)
∂.,
(9b)
Mechanical field u i = u¯ i on ∂.,
(10a)
(σ
ji
)
on
+ σ jiE nj = t¯i
on
∂..
(10b)
Table 1 Field equations for a magneto-mechanical problem.
Maxwell's equations i, i = 0 in .
(2a)
ϵijk k, j = 0
(2b)
in
.,
with the relationship i = μ 0 ( i + i ) in
..
(3) 3. Deformation of a ferrogel in magnetic field
Scalar magnetic potential Ψ mag i = −Ψ , mag in ., i
(4a)
Ψ , mag ii
(4b)
= i, i
in
..
Balance of linear momentum (σji + σ jiE ), j = 0 in .,
is the displacement vector. Please note that elastic strain and displacement are those of the polymer network. The fixed magnetic particles have the same displacement field as that of the polymer network and therefore their motion is dependent on the polymer network.1 In Table 2 bi is the local magnetic induction vector. In Table 3 ni is the unit normal vector and ti is the traction vector.
Using FEM, deformation of a ferrogel in magnetic field is investigated in this section. The numerical treatment of the present work is however restricted to 2D. To investigate different geometries, two simulations domains are investigated, see Fig. 2. The simulation domain consists of a ferrogel (in the form of a circle or a strip) which is assumed to be surrounded by a vacuum. Boundaries of the simulation domain are fixed. The ferrogel is also constrained in the middle to avoid any rigid body rotation. The material parameters for the numerical simulation are bulk modulus K = 14.21 kPa , shear modulus N G = 0.87 kPa , magnetic permeability of vacuum μ0 = 1.2566·10−6 2 , as
(5)
where
σ jiE
⎞ δji ⎛ 1 ⎜ k k − 2 q q ⎟. = ji − 2 ⎝ μ0 ⎠
A
kg3
(6)
Small deformation assumption
εij =
1 (u i, j + u j, i ) in 2
..
kg3
well as B0 = −0.30 2 2 5 and B1 = −0.46 2 2 5 as the magnetostrictive As m As m coefficients. Material parameters are adopted from Ref. [3,18]. This means that according to [18] a ferrogel – with about 4% of volume fraction of magnetic particles – was investigated. 1 This description corresponds to the kinematics of a continuum body where the motion is understood as the mapping from the initial configuration to the current configuration. It is therefore no to be mistaken with the dynamics of a thermodynamical system.
(7)
2
Journal of Magnetism and Magnetic Materials xx (xxxx) xxxx–xxxx
A. Attaran et al.
y
y
Simulation Domain A
Simulation Domain B
Surrounding Surrounding
Gel
ly, gel = 1 mm
ly = 11 mm
ly = 15 mm
R = 2.5 mm R = 2.5 mm
Gel lx,gel = 5 mm
x
x
lx = 15 mm
lx = 15 mm
Fig. 2. Simulation domains consisting of a circular gel (left) and a strip of ferrogel (right).
Undeformed
y
Deformed without magnetostriction
x
Deformed with magnetostriction
Hy
Hx
Fig. 3. Deformation of a circular gel and a gel strip in applied magnetic field.
Two test cases will be investigated: In the first test case, an external magnetic field is placed in x-direction. In the second test case a magnetic field is applied in y-direction. To apply a magnetic field in x-direction a magnetic potential is prescribed on the left (Ψ¯ mag|left = 1.5 A) and right (Ψ¯ mag|right = −1.5 A) boundaries of the simulation domain. To apply a magnetic field in y-direction a magnetic potential is prescribed on the top (Ψ¯ mag|top = −1.5 A) and bottom (Ψ¯ mag|bottom = 1.5 A) of the simulation domain. The results indicate that the ferrogel elongates in the direction of applied magnetic field and contracts perpendicular to the applied magnetic field, see Fig. 3. The results are in a good qualitative agreement with the experimental results (e.g. Ref. [3]).
This model is a reduction of a more comprehensive one proposed by the authors in Ref. [1]. The deformation of a ferrogel in a magnetic field was numerically investigated in 2D using FEM. It is concluded that the numerical results of presented model are in a good agreement with experimental results, though the model is only valid for small strain regime. For a more detailed comparison with the experimental results, the current modeling approach will be extended to finite and large strains. Moreover the presence of mobile magnetic particles has not been considered in the reduced model. Such particles, in our modeling approach, appear either as trapped within the networks of the polymer or not bound to polymer network junctions. The latter type of mobile magnetic particles can therefore freely diffuse within the polymer. This extension of our reduced model is currently being performed. This enables us to get a better picture of the processes involving the deformation mechanism of different types of ferrogels.
4. Conclusion In the present paper a model for ferrogels was introduced which consisted only of polymer network (P) and fixed magnetic particles (f). 3
Journal of Magnetism and Magnetic Materials xx (xxxx) xxxx–xxxx
A. Attaran et al.
[8] S. Bohlius, H.R. Brand, H. Pleiner, Macroscopic dynamics of uniaxial magnetic gels, Phys. Rev. E 70 (6) (2004) 61411. [9] M.A. Annunziata, A.M. Menzel, H. Löwen, Hardening transition in a one-dimensional model for ferrogels, J. Chem. Phys. 138 (20) (2013) 204906. [10] P.A. Sánchez, J.J. Cerdà, T. Sintes, C. Holm, Effects of the dipolar interaction on the equilibrium morphologies of a single supramolecular magnetic filament in bulk, J. Chem. Phys. 139 (4) (2013) 044904. [11] M. Tarama, P. Cremer, D.Y. Borin, S. Odenbach, H. Löwen, A.M. Menzel, Tunable dynamic response of magnetic gels: impact of structural properties and magnetic fields, Phys. Rev. E 90 (4) (2014) 042311. [12] D.S. Wood, P.J. Camp, Modeling the properties of ferrogels in uniform magnetic fields, Phys. Rev. E 83 (1) (2011) 011402. [13] R. Weeber, S. Kantorovich, C. Holm, Deformation mechanisms in 2D magnetic gels studied by computer simulations, Soft Matter 8 (38) (2012) 9923. [14] R. Weeber, S. Kantorovich, C. Holm, Ferrogels cross-linked by magnetic particles: field-driven deformation and elasticity studied using computer simulations, J. Chem. Phys. 143 (15) (2015) 154901. [15] R. Weeber, S. Kantorovich, C. Holm, Ferrogels cross-linked by magnetic nanoparticles - deformation mechanisms in two and three dimensions studied by means of computer simulations, J. Magn. Magn. Mater. 383 (2015) 262–266. [16] A.M. Menzel, Bridging from particle to macroscopic scales in uniaxial magnetic gels, J. Chem. Phys. 141 (19) (2014) 194907. [17] G. Pessot, R. Weeber, C. Holm, H. Löwen, A.M. Menzel, Towards a scale-bridging description of ferrogels and magnetic elastomers, J. Phys.: Condens. Matter 27 (32) (2015) 325105. [18] C. Gollwitzer, M. Krekhova, G. Lattermann, I. Rehberg, R. Richter, Surface instabilities and magnetic soft matter, Soft Matter 5 (10) (2009) 2093–2100.
Acknowledgment This research has been financially supported by the Deutsche Forschungsgemeinschaft (DFG) in the framework of Priority Programme SPP 1681 “Field controlled particle matrix interactions: synthesis multiscale modelling and application of magnetic hybrid materials” under the grant number WA 2323/8. References [1] A. Attaran, J. Brummund, T. Wallmersperger, Development of a continuum model for ferrogels, J. Intell. Mater. Syst. Struct. (2016) (in press). [2] M. Zrínyi, L. Barsi, A. Büki, Ferrogel: a new magneto-controlled elastic medium, Polym. Gels Netw. 5 (5) (1997) 415–427. [3] C. Gollwitzer, A. Turanov, M. Krekhova, G. Lattermann, I. Rehberg, R. Richter, Measuring the deformation of a ferrogel sphere in a homogeneous magnetic field, J. Chem. Phys. 128 (16) (2008) 164709. [4] Y.L. Raikher, O.V. Stolbov, Magnetodeformational effect in ferrogel samples, J. Magn. Magn. Mater. 258–259 (2003) 477–479. [5] Y.L. Raikher, O.V. Stolbov, Magnetodeformational effect in ferrogel objects, J. Magn. Magn. Mater. 289 (2005) 62–65. [6] A.Y. Zubarev, On the theory of the magnetic deformation of ferrogels, Soft Matter 8 (11) (2012) 3174. [7] E. Jarkova, H. Pleiner, H.-W. Müller, H.R. Brand, Hydrodynamics of isotropic ferrogels, Phys. Rev. E 68 (4) (2003) 41706.
4
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Modeling and finite element simulation of the magneto-mechanical behavior of ferrogels ⁎
Abdolhamid Attaran , Jörg Brummund, Thomas Wallmersperger Institut für Festkörpermechanik, Technische Universität Dresden, 01062 Dresden, Germany
A R T I C L E I N F O
A BS T RAC T
Keywords: Ferrogel Modeling Theory of mixtures Magneto-mechanical formulation
In our previous study (see Ref. Attaran et al. (in press) [1]) we formulated a continuum model for ferrogels considering them as multicomponent materials. In the present work a reduced model for ferrogels is presented consisting only of a polymer network (P) and fixed magnetic particles (f). The reduced model is solved using the finite element method where the only degrees of freedom are mechanical displacement and magnetic potential. Elongation and contraction of a ferrogel are observed parallel and perpendicular to the applied magnetic field direction, respectively. These results are in a good qualitative agreement with experimental results. With our modeling approach, we were able to investigate (i) the influence of the magnetic field on the polymer gel containing magnetic particles and (ii) the resulting mechanical deformation of a ferrogel.
1. Introduction Ferrogels are magneto-sensitive materials, the fabrication of which was first reported in the late 90s [2]. They primarily consist of a polymer gel and magnetic filler nanoparticles (e.g. magnetite Fe3O4 [3]). Polymer gels are usually synthesized by chemically cross-linking of an array of polymer networks. By application of an external magnetic field, magnetic particles orient themselves towards the applied field. Since some of the particles are adhered to the polymer network, ferrogels deform in the direction of the applied magnetic field, see Fig. 1. Shortly after their introduction, first theoretical models for ferrogels emerged both at macroscopic and microscopic scales. Notable development in macroscopic modeling are due e.g. to the works of Raikher and Stolbov [4,5], and Zubarev [6]. Hydrodynamics models for ferrogels were developed e.g. by Jarkova et al. [7] for isotropic gels and by Bohlius et al. [8] for uniaxial ferrogels. Development of microscopic models for ferrogels has also been pursued by several researchers over the past years. Worth mentioning here is for example a modeling approach in which the magnetic particles are modeled as hard spheres along with harmonic springs, resembling the elastic matrix [9–11]. Using a microscopic approach some authors used numerical tools e.g. Monte Carlo simulations [12] or Molecular Dynamics (MD) in 2D [13] and 3D [14,15] to simulate the magnetoelastic behavior of ferrogels. For instance Weeber et al. [13] studied deformation mechanism of ferrogels where the particle matrix interaction were examined
⁎
using Néel relaxation (no direct transfer of magnetic torque to the polymer matrix) or Brownian relaxation (direct transfer of magnetic torque to the polymer matrix). Depending on the mechanism, contraction or elongation of ferrogels was reported. Recently few scalebridging models [16,17] have also been proposed to create a bridge over different length scales. The motivation behind this research stems itself from the fact that despite recent developments in modeling the magneto-mechanical behavior of ferrogels, there have hardly been any attempt to propose a macroscopic model based on the multiphase, multicomponent nature of ferrogels. Our ultimate goal is to obtain a magneto-mechanical formulation capable of describing mechanical deformation of a ferrogel placed in a magnetic field using this approach. A comprehensive magneto-mechanical formulation for ferrogels has been developed by the authors in [1] within the framework of continuum mechanics of mixtures. In the current paper a reduced version of this model is presented. The reduced model for ferrogels consists only of polymer network (P) and fixed magnetic particles (f). In this model mechanical displacement and magnetic potential are the degrees of freedom. The finite element method (FEM) is used for the numerical treatment of this model. 2. Coupled magnetomechanical formulation In this section a coupled magnetomechanical formulation is presented for ferrogels including field equations, constitutive relations, and boundary conditions. This formulation is a reduction of the model
Corresponding author. E-mail address: [email protected] (A. Attaran).
http://dx.doi.org/10.1016/j.jmmm.2016.09.058 Received 5 July 2016; Received in revised form 31 August 2016; Accepted 11 September 2016 Available online xxxx 0304-8853/ © 2016 Elsevier B.V. All rights reserved.
Please cite this article as: Attaran, A., Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/j.jmmm.2016.09.058
Journal of Magnetism and Magnetic Materials xx (xxxx) xxxx–xxxx
A. Attaran et al.
H
Polymer network (P) Fixed magnetic particles (f)
Fig. 1. Ferrogel before application of the magnetic field (left). Deformation of a ferrogel in magnetic field (H) (right).
of Ref. [1]. In the development of this formulation the ferrogel is assumed as a continuum body . in three dimensional Euclidean space, enclosed by boundary ∂. . To obtain the reduced model for ferrogels (i) isothermal and reversible processes are only considered, (ii) mobile magnetic particles are not taken into account, (iii) no liquid is assumed to be present, (iv) small strain theory is used, and (v) mass density of ferrogels remains constant. Corresponding to this reduced model a suitable free energy function is introduced as [1]
ρ0 f =
1 1 ijkl εij εkl + λ ijkl εij mk ml 2 2
in .,
Table 2 Constitutive equations for a magneto-mechanical problem.
bi = −
1 λ ijkl εjk ml in ρ0
(
)
eijk m j k + bk = 0
σij = ijkl εkl +
., in
(8a)
.,
(8b)
1 λ ijkl mk ml + λ ijkl εkq m q ml in .. 2
(8c)
(1)
where f is the specific free energy density, ρ0 the reference mass density, ijkl the fourth order elasticity tensor, λijkl the magnetostrictive tensor, mi the specific magnetization of the gel, and εij is the strain tensor. In Tables 1–3 the corresponding field equations, constitutive equations as well boundary conditions are presented. In Table 1. i is the magnetic induction vector, i the magnetic intensity vector, i = ρ mi the magnetization vector, ρ is the mass density of the ferrogel, ϵijk the Levi-Civita symbol, δij the Kronecker delta, μ0 magnetic permeability of vacuum, Ψ mag scalar magnetic potential, σij Cauchy stress tensor, σijE Maxwell stress tensor, and ui
Table 3 Boundary conditions for a magneto-mechanical problem.
Magnetic field Ψ mag = Ψ¯ mag on
i ni = ¯
∂.,
(9a)
∂.,
(9b)
Mechanical field u i = u¯ i on ∂.,
(10a)
(σ
ji
)
on
+ σ jiE nj = t¯i
on
∂..
(10b)
Table 1 Field equations for a magneto-mechanical problem.
Maxwell's equations i, i = 0 in .
(2a)
ϵijk k, j = 0
(2b)
in
.,
with the relationship i = μ 0 ( i + i ) in
..
(3) 3. Deformation of a ferrogel in magnetic field
Scalar magnetic potential Ψ mag i = −Ψ , mag in ., i
(4a)
Ψ , mag ii
(4b)
= i, i
in
..
Balance of linear momentum (σji + σ jiE ), j = 0 in .,
is the displacement vector. Please note that elastic strain and displacement are those of the polymer network. The fixed magnetic particles have the same displacement field as that of the polymer network and therefore their motion is dependent on the polymer network.1 In Table 2 bi is the local magnetic induction vector. In Table 3 ni is the unit normal vector and ti is the traction vector.
Using FEM, deformation of a ferrogel in magnetic field is investigated in this section. The numerical treatment of the present work is however restricted to 2D. To investigate different geometries, two simulations domains are investigated, see Fig. 2. The simulation domain consists of a ferrogel (in the form of a circle or a strip) which is assumed to be surrounded by a vacuum. Boundaries of the simulation domain are fixed. The ferrogel is also constrained in the middle to avoid any rigid body rotation. The material parameters for the numerical simulation are bulk modulus K = 14.21 kPa , shear modulus N G = 0.87 kPa , magnetic permeability of vacuum μ0 = 1.2566·10−6 2 , as
(5)
where
σ jiE
⎞ δji ⎛ 1 ⎜ k k − 2 q q ⎟. = ji − 2 ⎝ μ0 ⎠
A
kg3
(6)
Small deformation assumption
εij =
1 (u i, j + u j, i ) in 2
..
kg3
well as B0 = −0.30 2 2 5 and B1 = −0.46 2 2 5 as the magnetostrictive As m As m coefficients. Material parameters are adopted from Ref. [3,18]. This means that according to [18] a ferrogel – with about 4% of volume fraction of magnetic particles – was investigated. 1 This description corresponds to the kinematics of a continuum body where the motion is understood as the mapping from the initial configuration to the current configuration. It is therefore no to be mistaken with the dynamics of a thermodynamical system.
(7)
2
Journal of Magnetism and Magnetic Materials xx (xxxx) xxxx–xxxx
A. Attaran et al.
y
y
Simulation Domain A
Simulation Domain B
Surrounding Surrounding
Gel
ly, gel = 1 mm
ly = 11 mm
ly = 15 mm
R = 2.5 mm R = 2.5 mm
Gel lx,gel = 5 mm
x
x
lx = 15 mm
lx = 15 mm
Fig. 2. Simulation domains consisting of a circular gel (left) and a strip of ferrogel (right).
Undeformed
y
Deformed without magnetostriction
x
Deformed with magnetostriction
Hy
Hx
Fig. 3. Deformation of a circular gel and a gel strip in applied magnetic field.
Two test cases will be investigated: In the first test case, an external magnetic field is placed in x-direction. In the second test case a magnetic field is applied in y-direction. To apply a magnetic field in x-direction a magnetic potential is prescribed on the left (Ψ¯ mag|left = 1.5 A) and right (Ψ¯ mag|right = −1.5 A) boundaries of the simulation domain. To apply a magnetic field in y-direction a magnetic potential is prescribed on the top (Ψ¯ mag|top = −1.5 A) and bottom (Ψ¯ mag|bottom = 1.5 A) of the simulation domain. The results indicate that the ferrogel elongates in the direction of applied magnetic field and contracts perpendicular to the applied magnetic field, see Fig. 3. The results are in a good qualitative agreement with the experimental results (e.g. Ref. [3]).
This model is a reduction of a more comprehensive one proposed by the authors in Ref. [1]. The deformation of a ferrogel in a magnetic field was numerically investigated in 2D using FEM. It is concluded that the numerical results of presented model are in a good agreement with experimental results, though the model is only valid for small strain regime. For a more detailed comparison with the experimental results, the current modeling approach will be extended to finite and large strains. Moreover the presence of mobile magnetic particles has not been considered in the reduced model. Such particles, in our modeling approach, appear either as trapped within the networks of the polymer or not bound to polymer network junctions. The latter type of mobile magnetic particles can therefore freely diffuse within the polymer. This extension of our reduced model is currently being performed. This enables us to get a better picture of the processes involving the deformation mechanism of different types of ferrogels.
4. Conclusion In the present paper a model for ferrogels was introduced which consisted only of polymer network (P) and fixed magnetic particles (f). 3
Journal of Magnetism and Magnetic Materials xx (xxxx) xxxx–xxxx
A. Attaran et al.
[8] S. Bohlius, H.R. Brand, H. Pleiner, Macroscopic dynamics of uniaxial magnetic gels, Phys. Rev. E 70 (6) (2004) 61411. [9] M.A. Annunziata, A.M. Menzel, H. Löwen, Hardening transition in a one-dimensional model for ferrogels, J. Chem. Phys. 138 (20) (2013) 204906. [10] P.A. Sánchez, J.J. Cerdà, T. Sintes, C. Holm, Effects of the dipolar interaction on the equilibrium morphologies of a single supramolecular magnetic filament in bulk, J. Chem. Phys. 139 (4) (2013) 044904. [11] M. Tarama, P. Cremer, D.Y. Borin, S. Odenbach, H. Löwen, A.M. Menzel, Tunable dynamic response of magnetic gels: impact of structural properties and magnetic fields, Phys. Rev. E 90 (4) (2014) 042311. [12] D.S. Wood, P.J. Camp, Modeling the properties of ferrogels in uniform magnetic fields, Phys. Rev. E 83 (1) (2011) 011402. [13] R. Weeber, S. Kantorovich, C. Holm, Deformation mechanisms in 2D magnetic gels studied by computer simulations, Soft Matter 8 (38) (2012) 9923. [14] R. Weeber, S. Kantorovich, C. Holm, Ferrogels cross-linked by magnetic particles: field-driven deformation and elasticity studied using computer simulations, J. Chem. Phys. 143 (15) (2015) 154901. [15] R. Weeber, S. Kantorovich, C. Holm, Ferrogels cross-linked by magnetic nanoparticles - deformation mechanisms in two and three dimensions studied by means of computer simulations, J. Magn. Magn. Mater. 383 (2015) 262–266. [16] A.M. Menzel, Bridging from particle to macroscopic scales in uniaxial magnetic gels, J. Chem. Phys. 141 (19) (2014) 194907. [17] G. Pessot, R. Weeber, C. Holm, H. Löwen, A.M. Menzel, Towards a scale-bridging description of ferrogels and magnetic elastomers, J. Phys.: Condens. Matter 27 (32) (2015) 325105. [18] C. Gollwitzer, M. Krekhova, G. Lattermann, I. Rehberg, R. Richter, Surface instabilities and magnetic soft matter, Soft Matter 5 (10) (2009) 2093–2100.
Acknowledgment This research has been financially supported by the Deutsche Forschungsgemeinschaft (DFG) in the framework of Priority Programme SPP 1681 “Field controlled particle matrix interactions: synthesis multiscale modelling and application of magnetic hybrid materials” under the grant number WA 2323/8. References [1] A. Attaran, J. Brummund, T. Wallmersperger, Development of a continuum model for ferrogels, J. Intell. Mater. Syst. Struct. (2016) (in press). [2] M. Zrínyi, L. Barsi, A. Büki, Ferrogel: a new magneto-controlled elastic medium, Polym. Gels Netw. 5 (5) (1997) 415–427. [3] C. Gollwitzer, A. Turanov, M. Krekhova, G. Lattermann, I. Rehberg, R. Richter, Measuring the deformation of a ferrogel sphere in a homogeneous magnetic field, J. Chem. Phys. 128 (16) (2008) 164709. [4] Y.L. Raikher, O.V. Stolbov, Magnetodeformational effect in ferrogel samples, J. Magn. Magn. Mater. 258–259 (2003) 477–479. [5] Y.L. Raikher, O.V. Stolbov, Magnetodeformational effect in ferrogel objects, J. Magn. Magn. Mater. 289 (2005) 62–65. [6] A.Y. Zubarev, On the theory of the magnetic deformation of ferrogels, Soft Matter 8 (11) (2012) 3174. [7] E. Jarkova, H. Pleiner, H.-W. Müller, H.R. Brand, Hydrodynamics of isotropic ferrogels, Phys. Rev. E 68 (4) (2003) 41706.
4