Brownian motion near a soft surface

TL;DR

This work addresses how Brownian motion of a colloid near a soft boundary is modified by elastohydrodynamic coupling. By deriving a Langevin description from leading-order soft-lubrication theory, the authors reveal a negative, position-dependent added mass and a softness-modified noise, along with a temperature-dependent drift term necessary to maintain Gibbs-Boltzmann equilibrium. The theory is validated numerically for a particle in a harmonic trap, showing enhanced velocity fluctuations due to the added mass while intermediate-time diffusion remains essentially unaffected. The results provide a thermodynamically consistent framework for near-surface Brownian dynamics in soft environments, with potential implications for nanoscale and biological systems.

Abstract

Brownian motion near soft surfaces is a situation widely encountered in nanoscale and biological physics. However, a complete theoretical description is lacking to date. Here, we theoretically investigate the dynamics of a two-dimensional colloid in an arbitrary external potential and near a soft surface. The latter is minimally modelled by a Winkler's foundation, and we restrict the study to the colloidal motion in the direction perpendicular to the surface. We start from deterministic hydrodynamic considerations, by invoking the already-established leading-order soft-lubrication forces acting on the particle. Importantly, a negative softness-induced and position-dependent added mass is identified. We then incorporate thermal fluctuations in the description. In particular, an effective Hamiltonian formulation is introduced and a temperature-dependent generalized potential is constructed in order to ensure equilibrium properties for the colloidal position. From these considerations and the Fokker-Planck equation, we then derive the relevant Langevin equation, which self-consistently allows to recover the deterministic equation of motion at zero temperature. Interestingly, besides an expected multiplicative-noise feature, the noise correlator appears to be modified by the surface softness. Moreover, a softness-induced temperature-dependent spurious drift term has to be incorporated within the Ito prescription. Finally, using numerical simulations with various initial conditions and parameter values, we statistically analyze the trajectories of the particle when placed within a harmonic trap and in presence of the soft surface. This allows us to: i) quantify further the influence of surface softness, through the added mass, which enhances the velocity fluctuations; and ii) show that intermediate-time diffusion is unaffected by softness, within the assumptions of the model.

Figures (3)

• Figure 1: Schematic of the system. A two-dimensional colloidal particle (light grey) of radius $r$ and mass per unit length $M$ is free to move along the vertical direction $\tilde{z}$ and time $\tilde{t}$ under the action of thermal fluctuations, at temperature $\tilde{T}$, and an external potential $\tilde{V}(\tilde{q})$, where $\tilde{q}(\tilde{t})$ is the particle's vertical coordinate with respect to the $\tilde{z}=0$ line. The motion takes place in a Newtonian liquid (blue) of dynamic shear viscosity $\eta$, and remains close to a flat horizontal rigid substrate (dark grey) coated with a soft elastic layer (yellow) of thickness $h_{\textrm{s}}$, and Lamé's coefficients $\mu$ and $\lambda$. We denote by $\tilde{\delta}(\tilde{x},\tilde{t})$ the deformation field of the soft layer.
• Figure 2: Dimensionless mean-square displacement (MSD) of the particle as a function of dimensionless time increment $\tau$, as obtained from statistical analysis of the numerical solutions of Eqs. (\ref{['qfinal']}) and (\ref{['bruit']}), for dimensionless trap stiffnesses $k = 4\times10^3$, $4\times10^4$, $4\times10^5$, $4\times10^6$, $4\times10^7$, and dimensionless surface softnesses $\kappa = 0$, $1\times10^{-5}$, $2.5\times10^{-5}$, $6\times10^{-5}$, $1\times10^{-4}$, $2.5 \times10^{-4}$, $6\times10^{-4}$, $1\times10^{-3}$. The fixed parameters are the dimensionless viscosity $\xi = 1000$, the dimensionless temperature $T = 1$, and the dimensionless rest position in the trap $q_0 = 1.5$. Straight lines indicate ballistic (2) and diffusive (1) exponents, as guides to the eye.
• Figure 3: Equilibrium probability density functions (PDFs) of (a) the rescaled position $q/\langle q^2\rangle^{1/2}$, and (b) the rescaled velocity $\dot{q}/\langle \dot{q}^2\rangle^{1/2}$ of the particle, as obtained from statistical analysis of the numerical solutions of Eqs. (\ref{['qfinal']}) and (\ref{['bruit']}), for dimensionless trap stiffnesses $k = 4\times10^3$, $4\times10^4$, $4\times10^5$, $4\times10^6$, $4\times10^7$, and dimensionless surface softnesses $\kappa = 0$, $1\times10^{-5}$, $2.5\times10^{-5}$, $6\times10^{-5}$, $1\times10^{-4}$, $2.5 \times10^{-4}$, $6\times10^{-4}$, $1\times10^{-3}$. The fixed parameters are the dimensionless viscosity $\xi = 1000$, the dimensionless temperature $T = 1$, and the dimensionless rest position in the trap $q_0 = 1.5$. The dashed lines in the main plots indicate normalized Gaussian distributions. The inset in panel a) shows the variance of the position $q$ as a function of the trap stiffness $k$. The dashed line corresponds to the energy-equipartition prediction $\langle q^2\rangle=T/k$. The inset in panel b) shows the rescaled variance of $\dot q$ as a function of surface softness$\kappa$. The dashed line corresponds to the energy-equipartition prediction $\langle \dot{q}^2\rangle\simeq T/m(q_0)$ in the stiff-trap limit.