## 2 Materials and Methods
### CA Model for Particle Movement  
In the CA model, the particles are constrained to move on the regular lattices as fluid particles, and the transport probabilities (\(p_{i}\)) to neighboring nodes depend on the local fluid flow and other external forces. Let \(N(\mathbf{x}_{\text{p}},\mathbf{t})\) be the number of simulation particles at site \(\mathbf{x}_{\text{p}}\) and time t, and \(N(\mathbf{x}_{\text{p}},\mathbf{t})\) can take any nonnegative value. The key idea of CA probabilistic model is that each of the N(\(\mathbf{x},\mathbf{t}\)) particles jumps to a neighboring node at site \(\mathbf{x}_{\text{p}}+\mathbf{c}_{i}\Delta t\) with a probability \(p_{i}\), which is proportional to the projection of the actual displacement of the particle in the direction \(\mathbf{c}_{i}\). Based on the \(D3Q15\) model, the probability of one fluid particle traveling in the six adjacent directions (north, south, west, east, front, and back) can be calculated as:  
\[p_{i}=\max\left(0,\frac{\Delta\mathbf{x}_{\text{p}}\cdot\mathbf{c}_{i}}{ \Delta x}\right)\ (i=1\ \sim 6) \tag{6}\]The final particle position after each time step \(\Delta t\) is determined as follows:  
\[\mathbf{x_{p}^{n+1}}=\mathbf{x_{p}^{n}}+\mu_{1}\mathbf{c_{1}}\Delta t+\mu_{2} \mathbf{c_{2}}\Delta t+\mu_{3}\mathbf{c_{3}}\Delta t+\mu_{4}\mathbf{c_{4}} \Delta t+\mu_{5}\mathbf{c_{5}}\Delta t+\mu_{6}\mathbf{c_{6}}\Delta t \tag{7}\]  
where \(\mu_{1}\) is a Boolean variable that is equal to 1 with a probability \(p_{i}\), and superscripts \(n\) and \(n\)+1 represent the present and next moment, respectively. For example, the three probabilities \(p_{1}\), \(p_{4}\), and \(p_{5}\) are positive. Thus, we create three independent random numbers (\(r_{1}\), \(r_{2}\), and \(r_{3}\)) from a uniform distribution in the interval [0, 1]. If \(p_{1}>r_{1}\), \(p_{4}>r_{2}\), and \(p_{5}>r_{3}\), then \(\mu_{1}\), \(\mu_{2}\), and \(\mu_{3}\) are set to 1, and so the particle will travel to position \(\mathbf{x_{p}}+\mathbf{c_{1}}\Delta t+\mathbf{c_{4}}\Delta t+\mathbf{c_{5}} \Delta t\). The rules for the other possible directions can be obtained using the same principle. Note that the actual displacement of particles \(\Delta\mathbf{x_{p}}\) is still unknown. The particle motion can thus be calculated as:  
\[\frac{d\mathbf{u_{p}}}{dt}=\mathbf{F_{D}}+\mathbf{F_{B}}=\frac{\mathbf{u}- \mathbf{u_{p}}}{\tau_{p}}+\xi\sqrt{\frac{216\mu k_{\mathrm{B}}T}{\pi\rho_{ \mathrm{p}}^{2}d_{\mathrm{p}}^{5}\Delta t}} \tag{8}\]  
\[\frac{d\mathbf{x_{p}}}{dt}=\mathbf{u_{p}} \tag{9}\]  
where \(\mathbf{u_{p}}\) represents the particle velocity, \(\tau_{p}\) is the relaxation time of the particle, \(\tau_{p}=C\ell\rho_{\mathrm{p}}{d_{\mathrm{p}}}^{2}/(18\mu)\), \(Cc\) is the Cunningham slip correction factor, \(\mu\) is the dynamic viscosity of the fluid, \(\mathbf{F_{D}}\) is the drag force, \(\mathbf{F_{B}}\) is the Brownian force, \(\xi\) is a Gaussian random number, \(d_{\mathrm{p}}\) is the particle diameter, \(k_{\mathrm{B}}\) is the Boltzmann constant, and \(T\) is the fluid temperature. In general, the conventional finite difference method is applied to determine the particle displacement, and the particle velocity and displacement are thus calculated by:  
\[\mathbf{u_{p}^{n+1}}=\mathbf{u_{p}^{n}}\cdot\exp{(-\frac{\Delta t}{\tau_{p}}) }+\left(\mathbf{u}+\mathbf{F_{B}}\cdot\tau_{p}\right)\cdot(1-\exp{(-\frac{ \Delta t}{\tau_{p}})}) \tag{10}\]  
\[\mathbf{x_{p}^{n+1}}=\mathbf{x_{p}^{n}}+\left(\mathbf{u_{p}^{n}}-\mathbf{u} \right)\left(1-\exp{\left(-\frac{\Delta t}{\tau_{p}}\right)}\right)\tau_{p}+ \mathbf{u}\Delta t+\left(\Delta t-\left(1-\exp{\left(-\frac{\Delta t}{\tau_{p }}\right)}\right)\tau_{p}\right)\mathbf{F_{B}}\cdot\tau_{p} \tag{11}\]  
Therefore, the actual displacement of particle within \(\Delta t\) is obtained from Equation (11), and the final particle position after each time step \(\Delta t\) is determined from Equation (7).