Dynamic regimes of fluids simulated by multiparticle-collision dynamics

We investigate the hydrodynamic properties of a fluid simulated with a mesoscopic solvent model. Two distinct regimes are identified, the “particle regime” in which the dynamics is gaslike and the “collective regime” where the dynamics is fluidlike. This behavior can be characterized by the Schmidt number, which measures the ratio between viscous and diffusive transport. Analytical expressions for the tracer diffusion coefficient, which have been derived on the basis of a molecular-chaos assumption, are found to describe the simulation data very well in the particle regime, but important deviations are found in the collective regime. These deviations are due to hydrodynamic correlations. The model is then extended in order to investigate self-diffusion in colloidal dispersions. We study first the transport properties of heavy pointlike particles in the mesoscopic solvent, as a function of their mass and number density. Second, we introduce excluded-volume interactions among the colloidal particles and determine the dependence of the diffusion coefficient on the colloidal volume fraction for different solvent mean-free paths. In the collective regime, the results are found to be in good agreement with previous theoretical predictions based on Stokes hydrodynamics and the Smoluchowski equation.

Figure 1

Dimensionless kinematic viscosity for the simple fluid in MPCD. The symbols are the simulation results, the solid line is the total theoretical prediction, the dotted line is the collisional contribution, and the dashed line the kinetic contribution. In both cases the system size is $L∕a=20$. (a) $\alpha$ dependence with $\lambda =0.2$ and $\rho =10$. (b) $\lambda$ dependence with $\alpha =130$ and $\rho =5$.

Figure 2

Theoretical Schmidt number versus collision time. The $\alpha$ and $\rho$ parameters are specified in the plot.

Figure 3

Normalized velocity autocorrelation function as a function of the dimensionless time for mean free paths $\lambda =1$ and $\lambda =0.1$. Dashed lines correspond to the exponential decays in Eq. (9). In both cases the number density is $\rho =5$, the rotation angle $\alpha =130$, and the system size $L∕a=20$.

Figure 4

Time dependence of the normalized velocity autocorrelation function. The parameters are the same as in Fig. 3. The data are compared with long-time tail prediction $t^{-1∕3}$. The amplitude predicted in Eq. (11) is (within the statistical error) exact for $\lambda =1$ and about 10% larger for $\lambda =0.1$.

Figure 5

Number of remaining neighbors in a collision box after $n$ collisions as a function of the root-mean-square displacement. Solid symbols correspond to $\rho =5$, $\alpha =130$ with mean free path specified in the legend. Open symbols correspond to $\rho =10$, $\alpha =110$ with $\lambda =0.6$ (▵) and $\lambda =1$ (◻).

Figure 6

Time dependence of the normalized velocity autocorrelation function. The dashed line is the exponential decay in Eq. (9), crosses $(\times )$ are the simulation results and pluses $(+)$ are the predicted values obtained by employing the ${\zeta }_n$ numbers, as indicated in Eq. (15). The parameters of the simulation are $\rho =10$, $\alpha =110$, $\lambda =0.1$, and $L∕a=20$. The inset is a zoom into the regime of the first few collisions.

Figure 7

Relative deviation $\Delta D=(D_{\mathrm{sim}}-D_0)∕D_0$ of the simulated diffusion coefficient from the Brownian approximation, as a function of the scaled mean free path $\lambda$. Full circles are simulation results, the solid line represents the analytical expression of $D_0$ in Eq. (17). Simulation parameters are $\alpha =130$, $\rho =5$, and $L∕a=20$.

Figure 8

Time dependence of the normalized velocity autocorrelation function for different heavy particles. Simulation parameters are $\lambda =0.1$, $\alpha =130$, $L∕a=20$, and $\rho =M∕m$. Dashed lines are simulation results and the solid line is the molecular-chaos approximation (24).

Figure 9

Time dependence of the normalized velocity autocorrelation function of a heavy particle of mass $M=5m$ for mean free path $\lambda =1$ and $\lambda =0.1$. Dashed lines correspond to the exponential decays in Eq. (24). In both cases the number density is $\rho =5$ and the rotation angle $\alpha =130$. Compare with Fig. 3.

Figure 10

Relative deviation of the simulated diffusion coefficient $D_M$ from the Brownian approximation $D_0$, in Eq. (26), as a function of the heavy particle mass for $\rho =5$. These deviations represent the hydrodynamic contribution to the diffusion coefficient $D_H=D_M-D_0$ in units of the Brownian contribution. Symbols are simulation results and the dashed line is a guide to the eye which represents a 75% enhancement of the hydrodynamic term over the Brownian one.

Figure 11

Hydrodynamic contribution to the diffusion coefficient in units of the Brownian contribution as a function of the scaled mean free path $\lambda$. Symbols are simulation measurements, and the ordinate zero axis represents perfect agreement with the analytical expression $D_0$. Simulation parameters are $\alpha =130$, $\rho =5$, and $L∕a=20$. Compare with Fig. 7.

Figure 12

Diffusion coefficients for a heavy particle as a function of the concentration $\varphi =N_M{(a∕L)}^3$, normalized with the diffusion coefficient $D_M\left(0\right)$ of heavy particles at zero number density. The dashed line is the analytical approximation from Eqs. (26, 29), symbols correspond to the simulation data with $\lambda$ specified in the legend. The other parameters are $M∕m=\rho =5$, $\alpha =130$, and $L∕a=20$.

Figure 13

Diffusion coefficient for colloidal particles without solvent as a function of the volume fraction $\phi$. Symbols are simulation results, the dashed line corresponds to the analytical prediction (32). The inset is a zoom over the small values of the diffusion coefficient, and the solid line is a linear extrapolation for large values of $\phi$.

Figure 14

Diffusion coefficient as a function of the volume fraction of colloidal particles dispersions interacting with a solvent represented with MPCD at different collision times. For comparison, the MD results of Fig. 13 are also plotted.

Figure 15

Dependence of the normalized diffusion coefficient on the volume fraction $\phi$ of colloidal particles. The same data are shown as in Fig. 14. The normalization factor $D\left(0\right)$ is obtained by extrapolation of the data to zero volume fraction. The solid line corresponds to the hydrodynamic prediction in Eq. (33).