Force calculation on walls and embedded particles in multiparticle-collision-dynamics simulations

Colloidal solutions posses a wide range of time and length scales so that it is unfeasible to keep track of all of them within a single simulation. As a consequence, some form of coarse graining must be applied. In this work we use the multiparticle collision dynamics scheme. We describe a particular implementation of no-slip boundary conditions upon a solid surface, capable of providing correct forces on the solid bypassing the calculation of the velocity profile or the stress tensor in the fluid near the surface. As an application we measure the friction on a spherical particle when it is placed in a bulk fluid and when it is confined in a slit. We show that the implementation of the no-slip boundary conditions leads to an enhanced Enskog friction, which can be understood analytically. Because of the long-range nature of hydrodynamic interactions, the Stokes friction obtained from the simulations is sensitive of the simulation box size. We address this topic for the slit geometry, showing that the dependence on the system size differs very much from what is expected in a three-dimensional system where periodic boundary conditions are used in all directions.

Figure 1

Velocity profile across a slit of width $L_x$. Simulations are performed in a box of sides $L_x=L_y=L_z=64a$. The external applied field is ${\mathit{F}}^{\text{ext}}={10}^{-4}{\mathit{f}}_0\mathit{ẑ}$. The inset shows a magnification of the profile near the wall at $x=0$.

Figure 2

Average force exerted by the fluid upon each wall of a slit of width $L_x=64a$, versus time. The total expected value is ${\mathit{f}}^{\text{exp}}=16{\mathit{f}}_0$; ${\mathit{f}}_s$ and ${\mathit{f}}_c$ are the force exerted during the streaming and the collision steps.

Figure 3

Running integral of the autocorrelation functions of the constraint force. The off-diagonal terms of Eq. (8), which are not shown here, go quickly to zero.

Figure 4

FSEs corrections for the friction upon a spherical particle placed in the mid-plane of a slit geometry. The lateral size of the square walls is $L$, while the wall separation is $L_z=32a$. Circles: simulation results. Solid line: the corrective terms for a cubic box with periodic BC in all three directions [Eq. (10)].

Figure 5

FSEs for $L>20$ in comparison with two different fit functions. Red (gray) lines are based on a function of the type as in Eq. (10); black lines are based upon the function in Eq. (27). Dotted lines are for the perpendicular friction, dashed lines for the parallel one.