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Hardening transition in a one-dimensional model for ferrogels
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10.1063/1.4807003
/content/aip/journal/jcp/138/20/10.1063/1.4807003
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/20/10.1063/1.4807003
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

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

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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 = π/4. The bottom plane /σ = 1 corresponds to the “hardened” phase, while the upper tilted surface corresponds to the “soft-elastic” phase. For 1 < /σ < 5/4 the transition is continuous, while it is discontinuous for /σ > 5/4.

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

Phase diagram of the system for / ≈ 0.3 and θ = π/4 in the plane of the rescaled rotation parameters 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.

Image of FIG. 8.
FIG. 8.

Phase diagram (a), as well as rescaled interparticle distance /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/ = 1 and / = 5 (large-τ and large- regime). For the chosen parameters, the ground state of the magnetization is always spiral-like.

Image of FIG. 9.
FIG. 9.

Phase diagram (a), as well as rescaled interparticle distance /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/ = 5 × 10 and / = 5 (small-τ and large- regime). The ground state of the magnetization is spiral-like for / ≲ 0.25, for larger values of / it becomes antiferromagnetic.

Image of FIG. 10.
FIG. 10.

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

Image of FIG. 11.
FIG. 11.

Phase diagram (a), as well as rescaled interparticle distance /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/ = 2.5 and / = 0.105 (large-τ and small- regime). The ground state of the magnetization is spiral-like for / ≲ 0.3, for larger values of / it becomes ferromagnetic.

Image of FIG. 12.
FIG. 12.

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

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/content/aip/journal/jcp/138/20/10.1063/1.4807003
2013-05-28
2014-04-21
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Hardening transition in a one-dimensional model for ferrogels
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/20/10.1063/1.4807003
10.1063/1.4807003
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