Slip length of confined liquid with small roughness of solid-liquid interfaces

We have studied the slip length of confined liquid with small roughness of solid-liquid interfaces. Dyadic Green function and perturbation expansion have been applied to get the slip length quantitatively. In the slip length, both the effects of the roughness of the interfaces and the chemical interaction between the liquid and the solid surface are involved. For the numerical calculation, Monte Carlo method has been used to simulate the rough interfaces and the physical quantities are obtained statistically over the interfaces. Results show that the total slip length of the system is linearly proportional to the slip length contributed from the chemical interaction. And the roughness of the interfaces plays its role as the proportionality factor. For the roughness, the variance of the roughness decreases the total slip length while the correlation length of the roughness can enhance the slip length dramatically to a saturation value.

Figure 1

Statistical average of $N$ over interfaces, noted by $\langle N\rangle$, as a function of the correlation length $l$ with various roughness variances $\sigma$. The $N$ has been defined in the text by Eq. (21). And the $\langle N\rangle$ is plotted in the logarithmic scale.

Figure 2

Slip lengths as functions of the correlation length $l$ with various roughness variances $\sigma$. (a) and (b) are for the total slip length $b$ of the system while the (c) and (d) are for the slip length $b_r$ contributed from the roughness of interfaces. In the plot, the slip lengths and $\sigma$ are all renormalized by the slip length $b_w$ contributed from the chemical interaction of the interfaces.

Figure 3

(a) Proportion of $b$ to $b_w$. (b) Proportion of $b_r$ to $b_w$. In the plot, the $b$ and $b_r$ have been renormalized by the $b_w$ for clarity.