Hydrodynamic interactions and Brownian forces in colloidal suspensions: Coarse-graining over time and length scales

We describe in detail how to implement a coarse-grained hybrid molecular dynamics and stochastic rotation dynamics simulation technique that captures the combined effects of Brownian and hydrodynamic forces in colloidal suspensions. The importance of carefully tuning the simulation parameters to correctly resolve the multiple time and length scales of this problem is emphasized. We systematically analyze how our coarse-graining scheme resolves dimensionless hydrodynamic numbers such as the Reynolds number Re, which indicates the importance of inertial effects, the Schmidt number Sc, which indicates whether momentum transport is liquidlike or gaslike, the Mach number, which measures compressibility effects, the Knudsen number, which describes the importance of noncontinuum molecular effects, and the Peclet number, which describes the relative effects of convective and diffusive transport. With these dimensionless numbers in the correct regime the many Brownian and hydrodynamic time scales can be telescoped together to maximize computational efficiency while still correctly resolving the physically relevant processes. We also show how to control a number of numerical artifacts, such as finite-size effects and solvent-induced attractive depletion interactions. When all these considerations are properly taken into account, the measured colloidal velocity autocorrelation functions and related self-diffusion and friction coefficients compare quantitatively with theoretical calculations. By contrast, these calculations demonstrate that, notwithstanding its seductive simplicity, the basic Langevin equation does a remarkably poor job of capturing the decay rate of the velocity autocorrelation function in the colloidal regime, strongly underestimating it at short times and strongly overestimating it at long times. Finally, we discuss in detail how to map the parameters of our method onto physical systems and from this extract more general lessons—keeping in mind that there is no such thing as a free lunch—that may be relevant for other coarse-graining schemes such as lattice Boltzmann or dissipative particle dynamics.

Figure 1

Schematic picture depicting how a colloidal dispersion can be coarse-grained by replacing the solvent with a particle-based hydrodynamics solver such as lattice Boltzmann (LB), dissipative particle dynamics (DPD), the Lowe-Anderson thermostat, or stochastic rotation dynamics (SRD). Each method introduces an effective coarse-graining length scale that is chosen to be smaller than those of the mesoscopic colloids but much larger than the natural length scales of a microscopic solvent. By obeying local momentum conservation, these methods reproduce Navier-Stokes hydrodynamics on larger length scales. For LB, thermal fluctuations must be added in separately, but these emerge naturally for the other three methods. In this paper we focus on SRD, but many of the lessons learned should also apply to other particle based coarse-graining methods.

Figure 2

Schematic picture depicting how a fluid molecule interacts with a colloid, imparting both linear and angular momentum. Near the colloidal surface, here represented by the shaded region, there may be a steric stabilization layer or a double-layer made up of co- and counter-ions. In SRD, the detailed manner in which a fluid particle interacts with this boundary layer is represented by a coarse-grained stick or slip boundary condition.

Figure 3

(Color online) Effective depletion potentials induced between two colloids by the SRD fluid particles, for HS fluid-colloid (solid line) and WCA fluid-colloid (dashed line) interactions taken from Eq. (17). The interparticle distance is measured in units of ${\sigma }_{\mathit{cf}}$. Whether these attractive potentials have important effects or not depends on the choice of diameter ${\sigma }_{\mathit{cc}}$ in the bare colloid-colloid potential (16).

Figure 4

(Color online) Colloid radial distribution functions $g\left(r\right)$ for various colloid volume fractions ${\varphi }_c$. The simulations were performed with BD (black lines) and SRD (red lines) simulations, and are virtually indistinguishable within statistical errors.

Figure 5

Finite-system-size scaling of the Stokes friction ${\xi }_S$ on a sphere of size ${\sigma }_{cf}=2$ in a flow of velocity $v_s=0.01a_0∕t_0$. The correction factor $f$ defined in Eq. (42) can be extracted from measured friction $\xi$ (drag force/velocity) and estimated Enskog friction ${\xi }_E$ defined in Eq. (40). It compares well to theory, Eq. (43).

Figure 6

Drag force $F_d$ divided by $4\pi \eta v$ for various colloid sizes and Reynolds numbers (in all cases the box size $L=16{\sigma }_{cf}$). For small Reynolds numbers this ratio converges to the effective hydrodynamic radius $a$. Dashed lines are theoretical predictions from combining Stokes and Enskog frictions in Eq. (41).

Figure 7

Magnified difference fields $10[\mathbf{v}\left(\mathbf{r}\right)-{\mathbf{v}}_{\mathrm{St}}\left(\mathbf{r}\right)]∕v_{\infty }$ for different colloidal sizes (the axes are scaled by ${\sigma }_{cf}$). In all cases $\mathrm{Re}\approx 0.1$ and box size $L=16{\sigma }_{cf}$.

Figure 8

(a) Fluid flow field around a fixed colloid with ${\sigma }_{cf}=2$ and $L=64a_0$ for $\mathrm{Re}=0.08$. (b)–(d) Difference field $10[\mathbf{v}\left(\mathbf{r}\right)-{\mathbf{v}}_{\mathrm{St}}\left(\mathbf{r}\right)]∕v_{\infty }$ for $\mathrm{Re}=0.08$, 0.4, and 2, respectively.

Figure 9

(Color online) Schematic depiction of our strategy for coarse-graining across the hierarchy of time scales for a colloid (here the example taken is for a colloid of radius $1\mu \mathrm{m}$ in ${\mathrm{H}}_2\mathrm{O}$). We first identify the relevant time scales and then telescope them down to a hierarchy which is compacted to maximize simulation efficiency, but sufficiently separated to correctly resolve the underlying physical behavior.

Figure 10

(Color online) Normalized VACF for a single colloid with ${\sigma }_{cf}=2a_0$ and $L=32a_0$: measured (solid line), Enskog short-time prediction (dot-dashed line), and Brownian approximation (dashed line). The normalized force-force auto correlation $C_{FF}\left(t\right)$ (dotted line) decays on a much faster time scale. The inset shows a magnification of the short-time regime. Note that the colloid exhibits ballistic motion on a time scale $\sim {\tau }_f$. The time scales ${\tau }_{\mathit{FP}}\approx 0.09$, ${\tau }_B\approx 2.5$, and ${\tau }_D\approx 200$ are clearly separated by at least an order of magnitude, as required.

Figure 11

Time-dependent self-diffusion coefficient $D\left(t\right)$ for different volume fractions $\varphi$, achieved by varying the number of colloids in a box of fixed volume. The inset shows the self-diffusion coefficient $(D=\lim t\to \infty D\left(t\right)$) as a function of volume fraction, together with an analytical prediction $D=D_0(1-1.1795\varphi )$ [90] valid in the low-density limit.

Figure 12

Mean-square-displacement of a ${\sigma }_{cf}=2a_0$ colloid (solid line). The dotted line is a fit to the long-time limit. The inset shows a magnification of the short-time regime, highlighting the initial ballistic regime [dashed line is ${⟨V\left(0\right)⟩}^2{t}^2$], which rapidly turns over to the diffusive regime.

Figure 13

(Color online) Normalized VACF for $a=2a_0$ and stick boundary conditions (from [58]), compared to the short-time Enskog result (B11), and the full hydrodynamic VACF (B5) by itself, and with the VACF (B9) from sound added. We also show the asymptotic long-time hydrodynamic tail from mode-coupling theory.

Figure 14

(Color online) Integral $D\left(t\right)$ of the normalized VACF from Fig. 13 (solid line), compared to the integral of Eq. (B5) (dashed line) and including the contribution from sound as in Eq. (B14) (dotted line). The simple Langevin form from the integral of Eq. (B3) (dash-dotted line) is calculated with $\xi =6\pi \eta {\sigma }_{cf}$ in order to have the same asymptotic value as $D^H\left(t\right)$. Because in the plot we integrate the normalized VACF, the $t\to \infty$ limit of $D\left(t\right)$ for the two theoretical approaches is $\frac{2}{9}{t}_0$ in our units, independent of colloid size. The measured $D\left(t\right)$ has a larger asymptotic value due to the initial effect of the Enskog friction.