Dynamic Monte Carlo simulations of inhomogeneous colloidal suspensions

The dynamic Monte Carlo (DMC) method is an established molecular simulation technique for the analysis of the dynamics in colloidal suspensions. An excellent alternative to Brownian dynamics or molecular dynamics simulation, DMC is applicable to systems of spherical and/or anisotropic particles and to equilibrium or out-of-equilibrium processes. In this work, we present a theoretical and methodological framework to extend DMC to the study of heterogeneous systems, where the presence of an interface between coexisting phases introduces an additional element of complexity in determining the dynamic properties. In particular, we simulate a Lennard-Jones fluid at the liquid-vapor equilibrium and determine the diffusion coefficients in the bulk of each phase and across the interface. To test the validity of our DMC results, we also perform Brownian Dynamics simulations and unveil an excellent quantitative agreement between the two simulation techniques.

Figure 1

Schematic representation of a particle performing an $o\to n$ move within the same layer (frames a and b) and to an adjacent layer (frame c). The intervals $2\delta z_i$ and $2\delta z_{ij}$ indicate the allowed range of displacements for in-layer and interlayer movements, respectively. The vertical dashed lines indicate the center of the layer. See text for details.

Figure 2

Typical equilibrium configurations of LJ spheres forming liquid and vapor phases at $T^{*}=0.7$ (a), 0.8 (b), 0.9 (c), and 1.0 (d).

Figure 3

Density profiles along the $z$ direction of the simulation box at $T^{*}=0.7$ ($◯$), 0.8 ($\square$), 0.9 ($◊$), and 1.0 ($▵$). Symbols indicate DMC simulation results, whereas the solid lines are hyperbolic fits of the type given in Eq. (18).

Figure 4

Acceptance rates along the $z$ direction of the simulation box at three different values of the reference MC time step: (a) $\delta t_{\text{MC,ref}}/\tau ={10}^{-3}$, (b) $\delta t_{\text{MC,ref}}/\tau ={10}^{-2}$, and (c) $\delta t_{\text{MC,ref}}/\tau ={10}^{-1}$. The solid lines are hyperbolic fits of the type given in Eq. (18).

Figure 5

MC time step along the $z$ direction of the simulation box for three different values of the reference MC time step: (a) $\delta t_{\text{MC,ref}}/\tau ={10}^{-1}$, (b) $\delta t_{\text{MC,ref}}/\tau ={10}^{-2}$, and (c) $\delta t_{\text{MC,ref}}/\tau ={10}^{-3}$. The solid lines are hyperbolic fits of the type given in Eq. (18).

Figure 6

Maximum displacements along the longitudinal direction of the simulation box for $\delta t_{\text{MC,ref}}/\tau$ equal to (a) ${10}^{-1}$, (b) ${10}^{-2}$, and (c) ${10}^{-3}$. The solid lines are hyperbolic fits of the type given in Eq. (18).

Figure 7

MSDs of LJ particles calculated in liquidlike ($⧫,◊$), vaporlike ($\blacksquare ,\square$), and interface-like ($•,◯$) layers at $T^{*}=0.7$. Solid and empty symbols represent the MSDs obtained from BD and DMC simulations of heterogeneous systems, respectively. Solid and dashed lines refer to MSDs obtained from independent BD and DMC simulations of single phases, respectively. Note the double linear scale of the inset.

Figure 8

Diffusion coefficients in the bulk of liquid ($◯$) and vapor ($\square$) phases. Solid and dashed lines refer to BD and DMC simulations, respectively. Error bars are smaller than the symbols' size.

Figure 9

Diffusion coefficients along the $z$ direction at (a) $T^{*}=0.7$, (b) $T^{*}=0.8$, (c) $T^{*}=0.9$ and (d) $T^{*}=1.0$, for $\delta t_{\text{MC,ref}}/\tau ={10}^{-3}$ ($◯$), ${10}^{-2}$ ($\square$), and ${10}^{-1}$ ($◊$). Solid circles ($•$) refer to the diffusion coefficients obtained by BD simulations. Upper and lower dashed lines indicate the value of the diffusion coefficients calculated, respectively, in the vapor and liquid phase from DMC simulations in single-phase systems. The solid lines are a fit to a linear combination of hyperbolic functions of the type given in Eq. (18).

Figure 10

A particle in layer $i$ can move within $i$ or to the adjacent layer $j$. In the former case, it will move within the interval $[-a_0,\delta z_i]$, in the latter case within the interval $[-\delta z_{ij},-a_0]$.

Figure 11

A particle moving inside the same layer $i$ and changing its nearest adjacent layer from $j$ to $k$.

Figure 12

Representation of a particle moving from an initial layer $i$ to and adjacent layer $j$.