Mesoscale modeling of colloidal suspensions with adsorbing solutes

We construct a mesoscale model of colloidal suspensions that contain solutes reversibly adsorbing onto the colloidal particle surfaces. The present model describes the coupled dynamics of the colloidal particles, the host fluid, and the solutes through the Newton-Euler equations of motion, the hydrodynamic equations, and the advection-diffusion equation, respectively. The solute adsorption is modeled through a square-well potential, which represents a short-range attractive interaction between a particle and a solute molecule. The present model is formulated to be solved through direct numerical simulations. Some numerical results are presented to validate the simulations. The present model enables investigations of solute adsorption effects in the presence of a fluid flow and an inhomogeneous solute concentration distribution.

Figure 1

Interaction potential between a particle and a solute molecule. The potential $u$ is described by a thick solid line and consists of the hard-core exclusion potential $u^{\mathrm{ex}}$ and the adsorption potential $u^{\mathrm{ad}}$, which potentials are described by broken lines.

Figure 2

Virtual solute concentration field $c^{*}$ around a particle with the imposed concentration gradient $\mathit{K}=-0.1(c_0/a)\widehat{\mathit{x}}$. The adsorption layer thickness is $w/a=0.5$. (a) Contour lines of the concentration field at the plane of $z=0$ for $\beta \varepsilon =0.5$. The color lines represent the analytical solution, while the black dashed-two dotted lines represent the simulation result given in Sec. 4a. The surfaces of the particle and the adsorption layer are described by thick and thin broken lines, respectively. (b) Concentration distribution on the line of $y=z=0$. The adsorption energy takes the values of $\beta \varepsilon =0$, $\pm 0.5$. The color lines represent analytical solutions. The markers represent the simulation result given in Sec. 4a. The dashed double-dotted line represents the concentration distribution in the absence of the particle, $c^{*}=\left|\mathit{K}\right|x$. The thick and thin broken lines represent the positions where surfaces of the particle and the adsorption layer are present, respectively.

Figure 3

Configuration of two particles. The particles are situated in a fluid with a center-to-center distance $R=\left|\mathit{R}\right|$. (a) The adsorption layers overlap each other without one particle facing to the adsorption layer of the other as $a+b\le R<2b$. (b) One particle faces to the adsorption layer of the other as $R\le a+b$.

Figure 4

Force acting on a particle, whose adsorption layer overlaps with that of another particle, as a function of the center-to-center distance between the particles for various adsorption energies $\beta \varepsilon$. The adsorption layer thickness is set as $w/a=0.5$. The negative and positive values correspond to attractive and repulsive forces, respectively.

Figure 5

Simulation estimation of force acting on a particle induced by the adsorption layer overlapping with various grid resolutions of the particle $a/\mathrm{\Delta }$. The adsorption layer thickness and adsorption energy are $w/a=0.5$ and $\beta \varepsilon =0.5$, respectively.

Figure 6

Sedimentation velocity of a particle with different adsorption energies $\beta \varepsilon$ as a function of the adsorption layer thickness $w/a$. The velocity is normalized by the sedimentation velocity with no solute adsorption ($\beta \varepsilon =0$), which is given by $V_0$. The adsorption energy takes the values of $\beta \varepsilon =\pm 0.25,\pm 0.5$.

Figure 7

Deviation of virtual solute concentration $c^{*}-c_0$ and velocity field $\mathit{v}$ around a particle, with the adsorption layer thickness $w/a=0.5$. The deviation of the virtual concentration is scaled by the bulk concentration $c_0$. The adsorption energy takes the values of (a) $\beta \varepsilon =0.5$ and (b) $\beta \varepsilon =-0.5$. The cross sections presented here are parallel to the particle motion direction and include the particle center. The particle is represented by a black circle. The direction of the particle velocity is to the downward in the pictures. The deviation of the virtual solute concentration is described by a color scale, which goes from negative (darker) to positive (lighter) deviation.