^{1}, Alexey O. Ivanov

^{1}and Philip J. Camp

^{2,a)}

### Abstract

Anisotropic pair correlations in ferrofluids exposed to magnetic fields are studied using a combination of statistical-mechanical theory and computer simulations. A simple dipolar hard-sphere model of the magnetic colloidal particles is studied in detail. A virial-expansion theory is constructed for the pair distribution function (PDF) which depends not only on the length of the pair separation vector, but also on its orientation with respect to the field. A detailed comparison is made between the theoretical predictions and accurate simulation data, and it is found that the theory works well for realistic values of the dipolar coupling constant (λ = 1), volume fraction (φ ⩽ 0.1), and magnetic field strength. The structure factor is computed for wavevectors either parallel or perpendicular to the field. The comparison between theory and simulation is generally very good with realistic ferrofluid parameters. For both the PDF and the structure factor, there are some deviations between theory and simulation at uncommonly high dipolar coupling constants, and with very strong magnetic fields. In particular, the theory is less successful at predicting the behavior of the structure factors at very low wavevectors, and perpendicular Gaussian density fluctuations arising from strongly correlated pairs of magnetic particles. Overall, though, the theory provides reliable predictions for the nature and degree of pair correlations in ferrofluids in magnetic fields, and hence should be of use in the design of functional magnetic materials.

E.A.E. and A.O.I. gratefully acknowledge grants from the Ministry of Education and Science of the Russian Federation (N2.609.2011) and the Russian Foundation for Basic Research (N10-02-00034-a). E.A.E. thanks the Ural Federal University for supporting a research visit to the University of Edinburgh where this work was completed.

I. INTRODUCTION

II. MODEL

III. THEORY

A. Virial expansion

B. Zero-field PDF expansion coefficients

C. Field-dependent PDF expansion coefficients

IV. SIMULATIONS

V. RESULTS

A. Pair distribution function

B. Structure factor

VI. CONCLUSIONS

### Key Topics

- Ferrofluids
- 27.0
- Magnetic fields
- 21.0
- Magnetic anisotropy
- 13.0
- Many body problems
- 10.0
- Anisotropy
- 8.0

##### H01F1/44

## Figures

Laboratory frame (*x*, *y*, *z*) and shifted frame (*x* ^{′}, *y* ^{′}, *z* ^{′}) for the calculation of *I* _{3} and *I* _{4} described in Sec. III C. is the separation vector between particles *i* and *j*.

Laboratory frame (*x*, *y*, *z*) and shifted frame (*x* ^{′}, *y* ^{′}, *z* ^{′}) for the calculation of *I* _{3} and *I* _{4} described in Sec. III C. is the separation vector between particles *i* and *j*.

Pair distribution functions calculated in MC simulations of DHS fluids with φ = 0.2: (a) λ = 1 and α = 0; (b) λ = 1 and α = 5; (c) λ = 2 and α = 0; (d) λ = 2 and α = 5. The horizontal axis is *x* _{⊥} = *x* sin θ and the vertical axis is *x* _{‖} = *x* cos θ, where θ is the angle of with respect to the field.

Pair distribution functions calculated in MC simulations of DHS fluids with φ = 0.2: (a) λ = 1 and α = 0; (b) λ = 1 and α = 5; (c) λ = 2 and α = 0; (d) λ = 2 and α = 5. The horizontal axis is *x* _{⊥} = *x* sin θ and the vertical axis is *x* _{‖} = *x* cos θ, where θ is the angle of with respect to the field.

Pair distribution functions *g*(*x*, θ) parallel to the field (θ = 0, shifted along the ordinate by one unit, for clarity) and perpendicular to the field (θ = π/2) in DHS fluids with φ = 0.1, λ = 1, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory.

Pair distribution functions *g*(*x*, θ) parallel to the field (θ = 0, shifted along the ordinate by one unit, for clarity) and perpendicular to the field (θ = π/2) in DHS fluids with φ = 0.1, λ = 1, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory.

Pair distribution functions *g*(*x*, θ) parallel to the field (θ = 0, shifted along the ordinate by one unit, for clarity) and perpendicular to the field (θ = π/2) in DHS fluids with φ = 0.1, λ = 2, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory.

Pair distribution functions *g*(*x*, θ) parallel to the field (θ = 0, shifted along the ordinate by one unit, for clarity) and perpendicular to the field (θ = π/2) in DHS fluids with φ = 0.1, λ = 2, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory.

Pair distribution functions *g*(*x*, θ) parallel to the field (θ = 0, shifted along the ordinate by one unit, for clarity) and perpendicular to the field (θ = π/2) in DHS fluids with φ = 0.2, λ = 1, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory.

Pair distribution functions *g*(*x*, θ) parallel to the field (θ = 0, shifted along the ordinate by one unit, for clarity) and perpendicular to the field (θ = π/2) in DHS fluids with φ = 0.2, λ = 1, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory.

Pair distribution functions *g*(*x*, θ) parallel to the field (θ = 0, shifted along the ordinate by one unit, for clarity) and perpendicular to the field (θ = π/2) in DHS fluids with φ = 0.2, λ = 2, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory.

Pair distribution functions *g*(*x*, θ) parallel to the field (θ = 0, shifted along the ordinate by one unit, for clarity) and perpendicular to the field (θ = π/2) in DHS fluids with φ = 0.2, λ = 2, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory.

Structure factors parallel to the field [*S*(*k* _{‖}), shifted up one unit for clarity] and perpendicular to the field [*S*(*k* _{⊥})] in DHS fluids with φ = 0.1 and λ = 1, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory with the unphysical regions highlighted with dashed lines.

Structure factors parallel to the field [*S*(*k* _{‖}), shifted up one unit for clarity] and perpendicular to the field [*S*(*k* _{⊥})] in DHS fluids with φ = 0.1 and λ = 1, and (a) α = 0, (b) α = 1, (c) α = 2, and (d) α = 5. The points are from simulations, and the lines are from theory with the unphysical regions highlighted with dashed lines.

Structure factors parallel to the field [*S*(*k* _{‖}), top row] and perpendicular to the field [*S*(*k* _{⊥}), bottom row] in DHS fluids with φ = 0.01–0.2, λ = 1, and α = 0 (left column) and α = 5 (right column). The points are from simulations, and the lines are from theory: (black circles and lines) φ = 0.01; (red squares and lines) φ = 0.02; (green diamonds and lines) φ = 0.05; (blue up triangles and lines) φ = 0.1; (magenta left triangles and lines) φ = 0.2.

Structure factors parallel to the field [*S*(*k* _{‖}), top row] and perpendicular to the field [*S*(*k* _{⊥}), bottom row] in DHS fluids with φ = 0.01–0.2, λ = 1, and α = 0 (left column) and α = 5 (right column). The points are from simulations, and the lines are from theory: (black circles and lines) φ = 0.01; (red squares and lines) φ = 0.02; (green diamonds and lines) φ = 0.05; (blue up triangles and lines) φ = 0.1; (magenta left triangles and lines) φ = 0.2.

Structure factors parallel to the field [*S*(*k* _{‖}), top row] and perpendicular to the field [*S*(*k* _{⊥}), bottom row] in DHS fluids with φ = 0.01–0.2, λ = 2, and α = 0 (left column) and α = 5 (right column). The points are from simulations, and the lines are from theory: (black circles and lines) φ = 0.01; (red squares and lines) φ = 0.02; (green diamonds and lines) φ = 0.05; (blue up triangles and lines) φ = 0.1; (magenta left triangles and lines) φ = 0.2.

Structure factors parallel to the field [*S*(*k* _{‖}), top row] and perpendicular to the field [*S*(*k* _{⊥}), bottom row] in DHS fluids with φ = 0.01–0.2, λ = 2, and α = 0 (left column) and α = 5 (right column). The points are from simulations, and the lines are from theory: (black circles and lines) φ = 0.01; (red squares and lines) φ = 0.02; (green diamonds and lines) φ = 0.05; (blue up triangles and lines) φ = 0.1; (magenta left triangles and lines) φ = 0.2.

Structure factor perpendicular to the field [*S*(*k* _{⊥})] in DHS fluids with φ = 0.01–0.1, λ = 2, and α = 5. The points are from simulations, and the lines are fits to the function : (black circles and line) φ = 0.01; (red squares and line) φ = 0.02; (green diamonds and line) φ = 0.05; (blue up triangles and line) φ = 0.1.

Structure factor perpendicular to the field [*S*(*k* _{⊥})] in DHS fluids with φ = 0.01–0.1, λ = 2, and α = 5. The points are from simulations, and the lines are fits to the function : (black circles and line) φ = 0.01; (red squares and line) φ = 0.02; (green diamonds and line) φ = 0.05; (blue up triangles and line) φ = 0.1.

## Tables

Expansion coefficients β_{ kl }(*x*) for the case of zero field (from Refs. 19 and 35), and functions γ_{ kl }(*x*) for the case of non-zero field. Each of the functions is equal to zero for *x* < 1.

Expansion coefficients β_{ kl }(*x*) for the case of zero field (from Refs. 19 and 35), and functions γ_{ kl }(*x*) for the case of non-zero field. Each of the functions is equal to zero for *x* < 1.

Expansion coefficients *h* _{ i }(*x*) and *h* _{ ij }(*x*) for the calculation of β_{12}(*x*, θ) in non-zero field. Each of the functions is equal to zero for *x* < 1.

Expansion coefficients *h* _{ i }(*x*) and *h* _{ ij }(*x*) for the calculation of β_{12}(*x*, θ) in non-zero field. Each of the functions is equal to zero for *x* < 1.

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