Brownian motion with soft constraints in soft matter systems

TL;DR

The paper addresses the challenge of deriving overdamped Brownian dynamics for systems with stiff restraints by treating restraints as soft constraints and taking the stiff limit. It develops a practical, operator-based toolkit of softly constrained dynamics, including extrinsic and intrinsic forms, a projected mobility M_P, and an effective potential U_eff, with a rigorous singular-perturbation derivation. A key advance is the extension to soft-soft constraints where mobility varies on the same scale as the confinement, requiring averaging over the confining degrees of freedom. The framework is demonstrated on diverse soft-matter scenarios such as particles near walls, hydrodynamic couplings, and tethered assemblies, with numerical validations confirming the predicted drifts and mobility corrections. Collectively, the work provides a robust, broadly applicable method to model tethered or confined soft-matter dynamics with correct drift and mobility corrections, and clarifies when and how constrained dynamics are valid for mesoscale simulations.

Abstract

Stiff forces, which bind objects together or otherwise confine motion, are found widely in soft-matter systems - colloids with short range attractions, ligand-receptor contacts, particles in optical traps, fibres that resist stretching, etc. To assess the long-term effect of these stiff forces on dynamics and structure, it is useful to consider the limit where they are treated as constraints, so the system evolves strictly within allowed configurations. Efforts to derive equations involving both constraints, and the stochastic motion appropriate at the scales of soft matter, began around 50 years ago, yet, we are still lacking a straightforward way to extract the projected equations and apply them in modern formulations of mesoscale dynamics. Here, we address this gap with two key contributions: (1) a practical summary of the constrained Brownian dynamics equations with ``soft'' constraints, i.e. constraints imposed by stiff forces, which is illustrated through several representative examples, taking care to highlight the nontrivial effects of the constraints; and (2) a novel derivation using singular perturbation theory, establishing the validity of these equations over timescales exceeding the relaxation of stiffly constrained degrees of freedom. We further extend our approach to ``soft soft'' constraints, where mobility varies on lengthscales comparable to the restraining forces - a scenario typical for particles in fluids experiencing hydrodynamic interactions. We hope our results will be useful for soft matter research, as a robust toolkit for studying tethered or confined systems.

Figures (10)

• Figure 1: Schematic of a softly constrained particle to a curved line in 2D. Here $\mathcal{M}$ is the confined space over which the constraint $c(x,y)=0$, $U^c(x,y)$ the confining potential around $\mathcal{M}$, and $k$ a typical spring constant associated with the confining potential.
• Figure 2: Physical implications of constraining particle motion near a wall. (a) Example of particle (gray) diffusing and confined by gravity near a bottom wall (dark gray). (b) Effective dynamics of the constrained motion would consider only motion along the wall. (c) Definition of different diffusion coefficients near the wall. (d) Analytical predictions of the effective diffusion coefficients $\overline{D_i(z)}$ versus $D_i(\overline{z})$ where $\overline{ z}$ is the mean height for different values of particle-fluid mass imbalance $m$. (e) Associated equilibrium distributions $P(z)$ and formulas for $D_i(z)$ on the same horizontal axis. Note that the $D_i(z)$ are renormalized by their values at $z = \infty$. Numerical parameters: reference mass $m_0 g = 0.0592~\mathrm{pN}$, with $g$ gravity; particle radius $a = 1.395~\mathrm{\mu m}$; viscosity $\eta = 1.4\times 10^{-3}~\mathrm{Pa.s}$ and temperature $T = 300~\mathrm{K}$.
• Figure 3: Particle mobility near another trapped particle. (a) Setup of two particles moving on separated lines and where particle 2 is trapped in a confining potential around $x_2(t) \simeq 0$. (b) Effective long time mobility of particle 1 as a function of its position along the line $x_1$, as given by Eq. \ref{['eq:offdiagproj']}. The colors indicate different separation distances $\Delta y$ between the lines.
• Figure 4: Effective mobility of tethered particles. (a) Setup of two particles moving on a line with a soft constraint between them. (b) Mean-squared displacement of the center of mass of the pair depicted in (a) with $k = 1 \gamma_0/\tau_0$, $\Gamma_1 = \gamma_0$ and $\Gamma_2 = 6 \gamma_0$ such that the effective bound friction coefficient predicted by the theory is $\Gamma = \Gamma_1 + \Gamma_2 = 7 \gamma_0$ and $D_{\infty} = k_B T/\Gamma \simeq 0.14 k_B T/\gamma_0$ close to the measured value. The lines show fits at short (done on the 2 first time points) and long timescales (done over the range where the pink line is plotted). These results correspond to an average of $100$ independent runs of $10^6$ time steps with time step $0.01 \tau_0$. The mean-squared displacement is calculated as in Eq. \ref{['eq:msd']}. In this simulation, $k_BT$, $\tau_0$, $\gamma_0$ are used as units.
• Figure 5: Rotation of a particle tethered by a floppy spring to a wall. Setup of notations used in the main text.