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Macroscopic effects of an anisotropic Gaussian-type repulsive potential: nematic alignment and spatial effects

Sara Merino-Aceituno, Steffen Plunder, Claudia Wytrzens, Havva Yoldaş

TL;DR

This work derives nematic alignment and spatial structure from volume exclusion by an anisotropic Gaussian-type repulsive potential, without imposing alignment a priori. By formulating a discrete particle model and passing to a mean-field kinetic equation, the authors perform nondimensionalization and a scaling analysis to obtain macroscopic equations for the mass density ρ and mean-nematic direction Ω via Generalized Collision Invariants. The main result is a rigorous macroscopic limit in which f^ε converges to ρ G_{η(ρ) A_Ω}, yielding a diffusion-modulated density equation and a transport-diffusion Ω-equation, with a special Π_3 term at a=2 that distinguishes oblate from prolate shapes. Numerical particle simulations validate the theory, showing alignment only in the well-posed σ<ν regime and illustrating the impact of anisotropy and density; the Berne-Pechukas potential fails to produce alignment, aligning with the theoretical predictions. The framework provides insights into how particle shape and repulsive interactions influence both the collective orientation and the spatial distribution, with potential applications to dense suspensions of anisotropic particles and active matter systems.

Abstract

Elongated particles in dense systems often exhibit alignment due to volume exclusion interactions, leading to packing configurations. Traditional models of collective dynamics typically impose this alignment phenomenologically, neglecting the influence of volume exclusion on particle positions. In this paper, we derive nematic alignment from an anisotropic repulsive potential, focusing on a Gaussian-type potential and first-order dynamics for the particles. By analyzing larger particle systems and performing a hydrodynamic limit, we uncover the effects of anisotropy on both particle density and direction. Our findings reveal that while particle density evolves independently of direction, anisotropy slows down nonlinear diffusion. The direction dynamics are affected by the particles' position and involve complex transport and diffusion processes, with different behaviors for oblate and prolate particles. The key to obtaining these results lies in recent advancements in Generalized Collision Invariants offered by Degond, Frouvelle and Liu (KRM 2022).

Macroscopic effects of an anisotropic Gaussian-type repulsive potential: nematic alignment and spatial effects

TL;DR

This work derives nematic alignment and spatial structure from volume exclusion by an anisotropic Gaussian-type repulsive potential, without imposing alignment a priori. By formulating a discrete particle model and passing to a mean-field kinetic equation, the authors perform nondimensionalization and a scaling analysis to obtain macroscopic equations for the mass density ρ and mean-nematic direction Ω via Generalized Collision Invariants. The main result is a rigorous macroscopic limit in which f^ε converges to ρ G_{η(ρ) A_Ω}, yielding a diffusion-modulated density equation and a transport-diffusion Ω-equation, with a special Π_3 term at a=2 that distinguishes oblate from prolate shapes. Numerical particle simulations validate the theory, showing alignment only in the well-posed σ<ν regime and illustrating the impact of anisotropy and density; the Berne-Pechukas potential fails to produce alignment, aligning with the theoretical predictions. The framework provides insights into how particle shape and repulsive interactions influence both the collective orientation and the spatial distribution, with potential applications to dense suspensions of anisotropic particles and active matter systems.

Abstract

Elongated particles in dense systems often exhibit alignment due to volume exclusion interactions, leading to packing configurations. Traditional models of collective dynamics typically impose this alignment phenomenologically, neglecting the influence of volume exclusion on particle positions. In this paper, we derive nematic alignment from an anisotropic repulsive potential, focusing on a Gaussian-type potential and first-order dynamics for the particles. By analyzing larger particle systems and performing a hydrodynamic limit, we uncover the effects of anisotropy on both particle density and direction. Our findings reveal that while particle density evolves independently of direction, anisotropy slows down nonlinear diffusion. The direction dynamics are affected by the particles' position and involve complex transport and diffusion processes, with different behaviors for oblate and prolate particles. The key to obtaining these results lies in recent advancements in Generalized Collision Invariants offered by Degond, Frouvelle and Liu (KRM 2022).

Paper Structure

This paper contains 46 sections, 16 theorems, 159 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Volume exclusion and nematic alignment
  3. A discrete model for anisotropic particles
  4. Kinetic equation and macroscopic quantities
  5. Structure of the paper.
  6. Continuum equations
  7. Non-dimensionalization.
  8. Scaling.
  9. Properties of the operator C
  10. The macroscopic limit
  11. Comments on the continuum equations
  12. The equation for the mass density
  13. Effect of particle anisotropy.
  14. The equation for the mean-nematic direction
  15. The case $a\in(0,2)$.
  16. ...and 31 more sections

Key Result

Lemma 2.1

Considering $\ell = \varepsilon \ell'$ and $d = \varepsilon d'$ we have the following expansion $V_f^\varepsilon$ of the scaled weighted Gaussian potential $V_f$ (defined by using eq:weighted_Gaussian_potential): where $\nabla^2_x$ is the Hessian, i.e., a $n\! \times\! n$-matrix with components $(\nabla^2_x)_{ij}=\partial_{x_i}\partial_{x_j}$, '$:$' denotes the double contraction, i.e., $A : B :=

Figures (10)

  • Figure 1: Spheroids are obtained by rotating an ellipse, shown in (a), around one of its principal axes. If the revolution is around the major axis, the spheroid is called prolate (b); if it is around the minor axis, it is called oblate (c).
  • Figure 2:
  • Figure 3: (a) Time evolution of the diffusion coefficient $K(\eta)$ with respect to $\eta$ with different $\chi$ values. The dotted lines and the dashed lines correspond to $1-\chi^2/n$ (upper bound for $K$) and $1-\chi^2$ (lower bound for $K$), respectively, for the matching color $\chi$ values. (b) The range of $K(\eta)$ for $\chi \in (0,1)$ and $\sigma \in [2^{-8}, 2^{8}]$. The dotted and the dashed lines show the upper and the (numerical) lower bound for $K$, respectively.
  • Figure 4: Parameter study for $(D_x, D_u)$. Trajectories of the order parameter $\gamma_f$ are plotted for fixed $D_u=0$ and varying $D_x$ in Figure \ref{['fig:vary_Dx']}; for fixed $D_x=2^{-4}$ and varying $D_u$ in Figure \ref{['fig:vary_Du']}. Figure \ref{['fig:heatmap_DxDu']} displays a heatmap of $\gamma_f$ at time $t_{\text{end}}=1.5 \times 10^5$ while both $D_u$ and $D_x$ vary simultaneously. The red line depicts where $\nu=\sigma$.
  • Figure 5: Parameter study for $(\lambda, \mu)$. Trajectories of the order parameter $\gamma_f$ are plotted for fixed $\mu=2^{13}$ and varying $\lambda$ in Figure \ref{['fig:vary_lambda']}; for $\lambda=2^8$ and varying $\mu$ in Figure \ref{['fig:vary_chi']}. Figure \ref{['fig:heatmap_lambda_mu']} displays the heatmap of $\gamma_f$ at time $t_{\text{end}}=1.5 \times 10^5$ while both $\lambda$ and $\mu$ vary simultaneously. The red line depicts where $\nu=\sigma$.
  • ...and 5 more figures

Theorems & Definitions (41)

  • Remark 1.1
  • Lemma 2.1: Expansion of the potential
  • proof
  • Remark 2.2: Motivation for the scaling factor $a$ and scaling choices
  • Remark 2.3
  • Definition 2.4: Uniaxial tensor
  • Definition 2.5: Gibbs distribution of an uniaxial tensor
  • Definition 2.6: Order parameter
  • Remark 2.7
  • Lemma 2.8: Proposition 2 in DFL22
  • ...and 31 more