Molecular Hydrodynamics from Memory Kernels

Abstract

The memory kernel for a tagged particle in a fluid, computed from molecular dynamics simulations, decays algebraically as t −3/2 . We show how the hydrodynamic Basset-Boussinesq force naturally emerges from this long-time tail and generalize the concept of hydrodynamic added mass. This mass term is negative in the present case of a molecular solute, at odds with incompressible hydrodynamics predictions. We finally discuss the various contributions to the friction, the associated time scales and the cross-over between the molecular and hydrodynamic regimes upon increasing the solute radius.The Brownian motion of a particle in a fluid finds its origin in the fluctuating force exerted by the solvent molecules on the solute. It has long been known that the canonical description of this random force by a Gaussian Markov process is only valid in limiting cases. Even in the limit where the solute is much heavier than the solvent particles, for which multiple time-scale analysis allows to recover the Smoluchowski equation for diffusion [1], nonMarkovian effects are expected when the mass density ratio is close to unity [2] -a situation which is rather the rule than the exception e.g. in colloidal suspensions. These non-Markovian effects arise because of momentum conservation, leading to slow hydrodynamic modes that manifest themselves as long-time tails in the velocity autocorrelation function (VACF) [3][4][5][6]. Recent experiments have demonstrated that the force exerted by the bath includes a deterministic component [7], well described for large colloidal spheres by the Basset-Boussinesq (BB) hydrodynamic force [8,9]:where R is the sphere radius, η the solvent viscosity and ρ 0 its mass density. The first term is the usual Stokes friction. The other two account for the inertia of the displaced fluid and involve a finite added mass m BB 0 = 2 3 πR 3 ρ 0 and a viscosity-dependent retarded component describing the transient effects of momentum diffusion in the solvent. While continuous descriptions of steady-state flows appear to hold down to the nanoscale [10][11][12], possibly at the price of adapting the hydrodynamic radius R or the boundary conditions [13], their validity for the transient regimes should be questioned. The implicit assumption of a separation of time scales between the solvent and solute dynamics, which holds a priori for colloidal particles [14], is expected to break down with smaller solutes such as nanoparticles or biomolecules.Here we address the fundamental questions that arise when approaching the regime of molecular solutes by computing directly from Molecular Dynamics (MD) simulations the memory kernel and the random noise of the Generalized Langevin Equation (GLE). A novel algorithm based on the Mori-Zwanzig formalism with high numerical stability allows us to explore long time scales for the first time. We consider the extreme case of a tagged particle (identical masses and sizes) in a pure supercritical fluid.By examining the long-time behaviour of the memory kernel, we demonstrate the gen...

Generalized Langevin dynamics: construction and numerical integration of non-Markovian particle-based models

“…This was due to accumulating errors when applying the inverse Volterra technique, which became most pronounced at large correlation times. In our applications here, we found the IMRV method to be more robust than other non-iterative memory reconstruction techniques [34][35][36][37][38][39] . Fig.…”

111 years of Brownian motion

“…120,121 From the ratio of the added mass and the particle mass $\frac{M}{2m}=\frac{\rho }{2{\rho }_{\mathrm{B}}}$, it is simple to see that for a lighter fluid the compressibility becomes less important for the particle dynamics.…”

Transient hydrodynamic finite-size effects in simulations under periodic boundary conditions

“…Such slow hydrodynamic modes also manifest themselves in the non-Markovian dynamics of solutes, which includes a deterministic component of the force exerted by the suspending fluid, well described for colloidal spheres by the Basset-Boussinesq force [20,21]. Simulations displaying such a hydrodynamic memory, either on a coarse-grained [22] or molecular [23] scale, may therefore suffer from artefacts associated with the use of PBC, at least on long time scales. This was already recognized by Alder and Wainwright in their seminal paper where they reported their results "up to the time where serious interference between neighbouring periodically repeated systems is indicated" [24].…”

“…Similarly, it has been recently shown that the diffusion coefficient of lipids and carbon nanotubes embedded in a membrane diverges logarithmically with system size [11] and one should also observe the impact of PBC on the transient dynamics. This may also prove important for extracting from finite size simulations other dynamical properties for which hydrodynamics play an important role, such as memory kernels [23].…”