Development of a continuum model for ferrogels

Abstract

A systematic development of a continuum model is presented, which is capable of describing the magneto-mechanical behavior of magnetic polymer gels commonly referred to as “ferrogels”. In the present research, ferrogels are treated as multicomponent, multiphase materials. They consist of a polymer network (P), fixed magnetic particles (f), mobile magnetic particles (m), and liquid (L). By considering ferrogels as multicomponent materials, interaction among constituents of ferrogels can be captured. This helps in understanding the process occurring inside ferrogels under the influence of external stimuli, such as magnetic fields. In our modeling approach, the field equations of ferrogels are derived within the framework of the theory of mixtures. The basic equations include Maxwell’s equations, balance of mass, linear momentum, angular momentum, energy, and entropy. In the framework of the theory of mixtures, balance relations are first presented at the constituent level also referred to as partial balance relations. By summing partial balance relations over all constituents and imposing the restrictions of theory of mixtures, balance relations of mixture (for the ferrogel) are obtained. In the current work the specific magnetization (magnetization per density) is considered as an evolving variable. It is demonstrated that balance of angular momentum is satisfied using the evolution equation of specific magnetization and constitutive laws. In the process of modeling, a suitable free energy function is introduced and thermodynamically consistent constitutive laws are formulated. Introducing certain assumptions, a reduced model of the ferrogel, a coupled magneto-mechanical formulation, is subsequently presented. The reduced model consists only of a polymer network (P) and fixed magnetic particles (f). It is concluded that the reduced model compares well to the existing ones in the literature. The magneto-mechanical problem based on the reduced model is solved in 2D using the finite element method. The only unknowns for the finite element method implementation are mechanical displacement and magnetic potential. Deformation of a ferrogel in a magnetic field is subsequently investigated. Elongation and contraction of a ferrogel are observed when a magnetic field is applied in the x- and y-directions, respectively. The numerical results were compared with existing experimental work in the literature. A good qualitative agreement was found between numerical and experimental results.

Review on the advancements of magnetic gels: towards multifunctional magnetic liposome-hydrogel composites for biomedical applications

“…Concerning the requirement of understanding the behaviour of these materials to tailor with the desired properties, various theoretical studies have been carried out, including the use of molecular dynamics [197][198][199][200][201][202][203][204][205][206]. For instance, Lopez-Lopez et al [207] evaluated the effect of the magnetic field-induced nanoparticle rearrangement on the shear elasticity of isotropic magnetic gels.…”

“…Different simulation approaches have been used to study the magnetic gels properties at different scales (from micro-to macroscopic) [212], such as full continuum-mechanical approaches [213], an elastic background continuum (particles are explicitly modelled) [214] or elastic spring matrix [215,216], an explicit approach [171], and density functional approach [200]. For instance, molecular dynamics has helped the understanding of magnetic field-induced deformation experimentally observed, which requires a large concentration of nanoparticles [171,218,219].…”

Magnetic susceptibility, nanorheology, and magnetoviscosity of magnetic nanoparticles in viscoelastic environments

“…Despite a number of recent efforts, the dynamics of magnetic nanocolloids in viscoelastic environments remains considerably less well-studied than ferrofluids. Macroscopic approachs in terms of nonequilibrium thermodynamics established the hydrodynamic equation of ferrogels [20,21]. Some first steps in coarse graining a molecular model of polymer chains permanently attached to magnetic particles are presented in [22].…”

Multiphysics modeling and simulation of fluid‐saturated porous ferrogels at finite strains

“…Porous ferrogels are a class of these materials consisting of a porous polymeric matrix with dispersed micro-or nano-sized ferromagnetic particles [1][2][3]. Porous ferrogels are a class of these materials consisting of a porous polymeric matrix with dispersed micro-or nano-sized ferromagnetic particles [1][2][3].…”

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Over the last years there has been a growing interest in the study of the behavior of field-responsive or so called smart materials. Porous ferrogels are a class of these materials consisting of a porous polymeric matrix with dispersed micro-or nano-sized ferromagnetic particles [1][2][3]. Due to their ability to exhibit large deformations and alter their effective material characteristics upon external magnetic stimulation, these materials are interesting for a wide range of applications in biomedical engineering, microfluidics and other innovative fields of research.

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