Collective colloid diffusion under soft two-dimensional confinement

Abstract

This work presents a numerical and theoretical investigation of the collective dynamics of colloids in an unbounded solution but trapped in a harmonic potential. Under strict two-dimensional confinement (infinitely stiff trap) the collective colloidal diffusion is enhanced and diverges at zero wave number (like k^{-1}), due to the hydrodynamic propagation of the confining force across the layer. The analytic solution for the collective diffusion of colloids under a Gaussian trap of width δ still shows enhanced diffusion for large wavelengths kδ<1, while a gradual transition to normal diffusion for kδ>1. At intermediate and short wavelengths, we illustrate to what extent the hydrodynamic enhancement of diffusion is masked by the conservative forces between colloids. At very large wavelengths, the collective diffusion becomes faster than the solvent momentum transport and a transition from Stokesian dynamics to inertial dynamics takes place. Using our inertial coupling method code (resolving fluid inertia), we study this transition by performing simulations at small Schmidt number. Simulations confirm theoretical predictions for the k→0 limit [Phys. Rev. E 90, 062314 (2014)PLEEE81539-375510.1103/PhysRevE.90.062314] showing negative density-density time correlations. However, at finite k simulations show deviations from the theory.

Solvent Hydrodynamics Enhances the Collective Diffusion of Membrane Lipids

“…As a consequence, the collective diffusion increases without bounds with the disturbance wavelength. This type of anomalous collective diffusion is in quantitative agreement with the quasi-2D hydrodynamics observed in colloids confined in a plane and surrounded by solvent [21], recently analyzed in several works [22][23][24]. In membranes, this phenomena introduces a so far unexplored intermediate dynamic regime which significantly enhances the collective diffusion at short times (over about 100ns) and affects a wide range of scales, from nanometers to microns.…”

“…This central result shows that hydrodynamic interactions between the membrane and the ambient fluid leads to a significant enhancement of collective lipid diffusion compatible to that observed in colloidal q2D dynamics [22]. While in colloids, the confinement arise from an external force field [24], in membranes, it arises from the internal elastic forces acting in normal direction to the plane. These forces transfer normal momentum to the surrounding fluid which spreads tangentially over the membrane.…”

“…The solvent is incompressible so ∇ · M 3D = 0, but a non-zero divergence appears when evaluated in the plane (z = 0 and r = r ) simply because ∇ r · M 3D = −∂ z M 3D = 0. Particles confined to move in the plane are thus exposed to a repulsive hydrodynamic force f q2D = k B T ∇ r · M 3D (r , z = 0) which is long ranged f q2D ∝ 1/r 2 because M 3D ∝ 1/r [24,36,37].…”

Hydrodynamic fluctuations in quasi-two dimensional diffusion

“…In this paper we consider the limit of infinitely strong confining forces. In this limit, it is not difficult to prove [13] that the effective hydrodynamic drag is proportional to the divergence of the hydrodynamic mobility evaluated in the plane. To see this, let us consider non-interacting particles of radius a confined to the x − y plane by a quadratic potential U (z) = (k s /2)z 2 .…”

“…This shows that the spectrum of the random velocity field w (r, t) decays like 1/k and that the field is compressible. Specifically, for the divergence we get k ·R k · k = k/4η, which allows us to write (13) in the form…”