^{1,a)}, Andreas M. Menzel

^{1,b)}and Hartmut Löwen

^{1,c)}

### Abstract

We introduce and investigate a coarse-grained model for quasi one-dimensional ferrogels. In our description the magnetic particles are represented by hard spheres with a magnetic dipole moment in their centers. Harmonic springs connecting these spheres mimic the presence of a cross-linked polymer matrix. A special emphasis is put on the coupling of the dipolar orientations to the elastic deformations of the matrix, where a memory effect of the orientations is included. Although the particles are displaced along one spatial direction only, the system already shows rich behavior: as a function of the magnetic dipole moment, we find a phase transition between “soft-elastic” states with finite interparticle separation and finite compressive elastic modulus on the one hand, and “hardened” states with touching particles and therefore diverging compressive elastic modulus on the other hand. Corresponding phase diagrams are derived neglecting thermal fluctuations of the magnetic particles. In addition, we consider a situation in which a spatially homogeneous magnetization is initially imprinted into the material. Depending on the strength of the magneto-mechanical coupling between the dipole orientations and the elastic deformations, the system then relaxes to a uniaxially ferromagnetic, an antiferromagnetic, or a spiral state of magnetization to minimize its energy. One purpose of our work is to provide a largely analytically solvable approach that can provide a benchmark to test future descriptions of higher complexity. From an applied point of view, our results could be exploited, for example, for the construction of novel damping devices of tunable shock absorbance.

The authors thank H. R. Brand and G. K. Auernhammer for helpful discussions and gratefully acknowledge the new environment and support from the recently founded SPP 1681 by the Deutsche Forschungsgemeinschaft.

I. INTRODUCTION

II. THE MODEL

III. NO ORIENTATIONAL MEMORY

IV. ORIENTATIONAL MEMORY

V. CONCLUSIONS

### Key Topics

- Magnetic moments
- 52.0
- Elasticity
- 28.0
- Magnetic fields
- 28.0
- Polymers
- 26.0
- Ferromagnetism
- 25.0

## Figures

Schematic figure of the model: every two neighboring particles i and i + 1 (i = 1, …, N − 1) are connected by a spring of elastic constant k attached to their centers, the center-to-center distance being a i . Each particle is represented by a hard sphere of diameter σ, and each magnetic moment m i (i = 1, …, N) forms the angle θ i with the chain axis .

Schematic figure of the model: every two neighboring particles i and i + 1 (i = 1, …, N − 1) are connected by a spring of elastic constant k attached to their centers, the center-to-center distance being a i . Each particle is represented by a hard sphere of diameter σ, and each magnetic moment m i (i = 1, …, N) forms the angle θ i with the chain axis .

Schematic figure of the model for two neighboring particles i and j (i = 1, …, N − 1, j = i + 1) as seen from along the chain axis ( is oriented towards the reader). For the ith particle, the direction of the projection of the magnetic moment into the plane perpendicular to the chain axis is given by m i × r ij /‖m i × r ij ‖, except for an additional rotation by π/2. The azimuthal orientation of each magnetic moment m i (i = 1, …, N) around the chain axis is measured by the angle ϕ i .

Schematic figure of the model for two neighboring particles i and j (i = 1, …, N − 1, j = i + 1) as seen from along the chain axis ( is oriented towards the reader). For the ith particle, the direction of the projection of the magnetic moment into the plane perpendicular to the chain axis is given by m i × r ij /‖m i × r ij ‖, except for an additional rotation by π/2. The azimuthal orientation of each magnetic moment m i (i = 1, …, N) around the chain axis is measured by the angle ϕ i .

Interparticle distance a/σ and compressive elastic modulus G at equilibrium, as a function of the magnetic dipole moment m/m 0. In (a) and (b), the three different lines correspond to three different values of the rescaled initial particle separation L/σ, whereas the orientation of the external magnetic field with respect to the system axis is kept fixed at θ B = π/4; here L/σ = 1.1 (dotted line), L/σ = (L/σ) c = 5/4 (dashed line), and L/σ = 2 (solid line). In (c) and (d), the three different lines correspond to three different orientations θ B of the external magnetic field with respect to the system axis, whereas the rescaled initial particle separation is kept fixed at L/σ = 2; here θ B = π/4 (solid line), θ B = θ1 ≈ 0.3π (dotted line), and θ B = π/3 (dashed line).

Interparticle distance a/σ and compressive elastic modulus G at equilibrium, as a function of the magnetic dipole moment m/m 0. In (a) and (b), the three different lines correspond to three different values of the rescaled initial particle separation L/σ, whereas the orientation of the external magnetic field with respect to the system axis is kept fixed at θ B = π/4; here L/σ = 1.1 (dotted line), L/σ = (L/σ) c = 5/4 (dashed line), and L/σ = 2 (solid line). In (c) and (d), the three different lines correspond to three different orientations θ B of the external magnetic field with respect to the system axis, whereas the rescaled initial particle separation is kept fixed at L/σ = 2; here θ B = π/4 (solid line), θ B = θ1 ≈ 0.3π (dotted line), and θ B = π/3 (dashed line).

Energy per particle E 1/N as a function of the rescaled interparticle distance a/σ, for m/m 0 = 0.5 and θ B = π/4. The black points are equal-energy minima of E 1/N: at the point a/σ = 1 the hard spheres touch each other and the system is in a “hardened” state, while at the point a/σ = a*/σ ≈ 2.45 the system is still “soft-elastic”.

Energy per particle E 1/N as a function of the rescaled interparticle distance a/σ, for m/m 0 = 0.5 and θ B = π/4. The black points are equal-energy minima of E 1/N: at the point a/σ = 1 the hard spheres touch each other and the system is in a “hardened” state, while at the point a/σ = a*/σ ≈ 2.45 the system is still “soft-elastic”.

Phase diagram for a quasi-1D ferrogel system without orientational memory in the presence of a strong external magnetic field that is tilted with respect to the system axis by the angle θ B = π/4. The bottom plane a/σ = 1 corresponds to the “hardened” phase, while the upper tilted surface corresponds to the “soft-elastic” phase. For 1 < L/σ < 5/4 the transition is continuous, while it is discontinuous for L/σ > 5/4.

Phase diagram for a quasi-1D ferrogel system without orientational memory in the presence of a strong external magnetic field that is tilted with respect to the system axis by the angle θ B = π/4. The bottom plane a/σ = 1 corresponds to the “hardened” phase, while the upper tilted surface corresponds to the “soft-elastic” phase. For 1 < L/σ < 5/4 the transition is continuous, while it is discontinuous for L/σ > 5/4.

Antiferromagnetic (left) and spiral-like (right) configurations of the magnetic moments when seen from along the chain axis. In the antiferromagnetic case we plot two neighboring magnetic moments, in the spiral case we plot three neighboring magnetic moments. The antiferromagnetic case can be viewed as a degenerate spiral of Δ = π.

Antiferromagnetic (left) and spiral-like (right) configurations of the magnetic moments when seen from along the chain axis. In the antiferromagnetic case we plot two neighboring magnetic moments, in the spiral case we plot three neighboring magnetic moments. The antiferromagnetic case can be viewed as a degenerate spiral of Δ = π.

Phase diagram of the system for m/m 0 ≈ 0.3 and θ(0) = π/4 in the plane of the rescaled rotation parameters D and τ. We show the location of the three states “FERRO” (uniaxially ferromagnetic), “SPIRAL” (spirally magnetized), and “AF” (antiferromagnetic). The dashed-dotted, dashed, and dotted lines correspond to the “FERRO”–“SPIRAL,” “FERRO”–“AF,” and “SPIRAL”–“AF” phase boundaries, respectively.

Phase diagram of the system for m/m 0 ≈ 0.3 and θ(0) = π/4 in the plane of the rescaled rotation parameters D and τ. We show the location of the three states “FERRO” (uniaxially ferromagnetic), “SPIRAL” (spirally magnetized), and “AF” (antiferromagnetic). The dashed-dotted, dashed, and dotted lines correspond to the “FERRO”–“SPIRAL,” “FERRO”–“AF,” and “SPIRAL”–“AF” phase boundaries, respectively.

Phase diagram (a), as well as rescaled interparticle distance a/σ (b), angle θ formed by the magnetic moments with the chain axis (c), and relative azimuthal angle Δ between neighboring magnetic moments (d), for states of coexistence. This coexistence is stressed by the tie lines. The data curves are obtained for a case of orientational memory that is characterized by the parameter values τ/kL 2 = 1 and D/kL 2 = 5 (large-τ and large-D regime). For the chosen parameters, the ground state of the magnetization is always spiral-like.

Phase diagram (a), as well as rescaled interparticle distance a/σ (b), angle θ formed by the magnetic moments with the chain axis (c), and relative azimuthal angle Δ between neighboring magnetic moments (d), for states of coexistence. This coexistence is stressed by the tie lines. The data curves are obtained for a case of orientational memory that is characterized by the parameter values τ/kL 2 = 1 and D/kL 2 = 5 (large-τ and large-D regime). For the chosen parameters, the ground state of the magnetization is always spiral-like.

Phase diagram (a), as well as rescaled interparticle distance a/σ (b), angle θ formed by the magnetic moments with the chain axis (c), and relative azimuthal angle Δ between neighboring magnetic moments (d), for states of coexistence. This coexistence is stressed by the tie lines. The data curves are obtained for a case of orientational memory that is characterized by the parameter values τ/kL 2 = 5 × 10−4 and D/kL 2 = 5 (small-τ and large-D regime). The ground state of the magnetization is spiral-like for m/m 0 ≲ 0.25, for larger values of m/m 0 it becomes antiferromagnetic.

Phase diagram (a), as well as rescaled interparticle distance a/σ (b), angle θ formed by the magnetic moments with the chain axis (c), and relative azimuthal angle Δ between neighboring magnetic moments (d), for states of coexistence. This coexistence is stressed by the tie lines. The data curves are obtained for a case of orientational memory that is characterized by the parameter values τ/kL 2 = 5 × 10−4 and D/kL 2 = 5 (small-τ and large-D regime). The ground state of the magnetization is spiral-like for m/m 0 ≲ 0.25, for larger values of m/m 0 it becomes antiferromagnetic.

Magnification of the upper branches of the a/σ and θ coexistence curves in Figs. 9(b) and 9(c) . The characteristics of a second-order phase transition are visible in the a/σ and θ variables at the point above which only antiferromagnetic states are found (m/m 0 ≈ 0.25). In the curve for θ the discontinuity in the first derivative is more evident.

Magnification of the upper branches of the a/σ and θ coexistence curves in Figs. 9(b) and 9(c) . The characteristics of a second-order phase transition are visible in the a/σ and θ variables at the point above which only antiferromagnetic states are found (m/m 0 ≈ 0.25). In the curve for θ the discontinuity in the first derivative is more evident.

Phase diagram (a), as well as rescaled interparticle distance a/σ (b), angle θ formed by the magnetic moments with the chain axis (c), and relative azimuthal angle Δ between neighboring magnetic moments (d), for states of coexistence. This coexistence is stressed by the tie lines. The data curves are obtained for a case of orientational memory that is characterized by the parameter values τ/kL 2 = 2.5 and D/kL 2 = 0.105 (large-τ and small-D regime). The ground state of the magnetization is spiral-like for m/m 0 ≲ 0.3, for larger values of m/m 0 it becomes ferromagnetic.

Phase diagram (a), as well as rescaled interparticle distance a/σ (b), angle θ formed by the magnetic moments with the chain axis (c), and relative azimuthal angle Δ between neighboring magnetic moments (d), for states of coexistence. This coexistence is stressed by the tie lines. The data curves are obtained for a case of orientational memory that is characterized by the parameter values τ/kL 2 = 2.5 and D/kL 2 = 0.105 (large-τ and small-D regime). The ground state of the magnetization is spiral-like for m/m 0 ≲ 0.3, for larger values of m/m 0 it becomes ferromagnetic.

Magnification of the upper branch of the a/σ coexistence curve in Fig. 11(b) . The characteristics of a second-order phase transition are visible in the a/σ variable at the point above which only uniaxial ferromagnetic states are found (m/m 0 ≈ 0.3).

Magnification of the upper branch of the a/σ coexistence curve in Fig. 11(b) . The characteristics of a second-order phase transition are visible in the a/σ variable at the point above which only uniaxial ferromagnetic states are found (m/m 0 ≈ 0.3).

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