Interaction potentials for mutually induced dipoles in uniform fields

TL;DR

The paper derives a force-based interacting potential for systems of mutually induced dipoles in a uniform external field, showing that mutual induction adds two- and three-body corrections atop the classical fixed-dipole term. It provides explicit expressions for pair and many-body potentials, demonstrates significant errors in simplified DM models (especially for anisotropic structures and certain field orientations), and introduces an $O(N^2)$ iterative scheme to compute mutual magnetization efficiently. The work highlights the critical role of mutual magnetization in predicting structure and dynamics and offers a practical computational method for large-scale simulations. The approach is applicable to polarizable particles under electric fields and, more broadly, to magnetostatic problems in soft matter and related systems.

Abstract

Dipolar interactions govern the structure and dynamics of many soft-matter systems, from molecular to colloids assemblies. When dipole moments are induced by an external field, mutual interactions lead to a many-body magnetization response that cannot be described by fixed-dipole models. Here, we derive the interaction potential for a system of mutually interacting induced dipoles in a uniform external field using a force-based approach. By accounting for the displacement-induced variation of the dipole moments, we obtain an interaction potential consisting of the classical dipole-dipole term supplemented by two- and three-body corrections arising from mutual induction. Comparisons with simplified models that neglect mutual magnetization reveal significant errors in the interaction potential, particularly in anisotropic particle assemblies. We also discuss an efficient $\mathcal{O}(N^2)$ iterative scheme for computing the mutual magnetization, enabling accurate simulations of large dipolar systems.

Figures (5)

• Figure 1: Sketch of a pair of mutually interacting magnetic dipoles separated by a distance $\textbf{r}$ subjected to an external magnetic field $\textbf{H}_0$. The dashed lines represent the field induced by each dipole.
• Figure 2: a) and b) present the pair potential as a function of $r$ for $\textbf{H}_0$ applied perpendicular to and parallel to $\textbf{r}$, respectively. c) and d) present the interparticle force as a function of $r$ for $\textbf{H}_0$ applied perpendicular to and parallel to $\textbf{r}$, respectively. Negative values correspond to attractive forces while positive corresponds to repulsive ones. The solid lines corresponds to the MDM model and the dashed lines corresponds to the DM model. Probed values of $\chi_{eff}$ are equal to 0.5 (black), 1.0 (red), 1.5 (green), and 2.0 (blue). The arrow perpendicular to the curves indicated increasing $\chi_{eff}$. Each graph presents an inset with the relative error in percentage of the DM model relative to the MDM model.
• Figure 3: Pair potential for a probe paramagnetic particle placed at $\textbf{x} = x\,\hat{\textbf{e}}_x + y\,\hat{\textbf{e}}_y$ around another fixed at $\textbf{x} = \textbf{0}$, for $\textbf{H}_0 = H_0\,\hat{\textbf{e}}_x$ and $\chi_{eff}=2$. Above the dashed line $y/a>0$, the interacting potential corresponds to the MDM model. Below the dashed line $y/a<0$, the interacting potential corresponds to the DM model. The white region refers to the non-physical overlap between the two particles.
• Figure 4: a) and b) total interacting potential for a cubic cluster of $5^3$ particles as a function of $\textbf{x}$ for $\textbf{H}_0$ applied perpendicular to and parallel to $\textbf{x}$, respectively. c) and d) total interacting potential for a linear chain of six particles as a function of $\textbf{x}$ for $\textbf{H}_0$ applied perpendicular to and parallel to $\textbf{x}$, respectively. The solid lines corresponds to the MDM model and the dashed lines corresponds to the DM model. Probed values of $\chi_{eff}$ are equal to 0.5 (black), 1.0 (red), 1.5 (green), and 2.0 (blue).
• Figure 5: Computational time cost given in wall clock time solving the linear system by LU factorization and the iterative approach based on Eq. \ref{['eq:recursive_2']} as a function of the number of particles $N$ in cubic clusters for $\chi_{eff}=2$. The dashed lines in black serve as a guide to the eyes.