Triple-line kinetics for solid films

We present a derivation of the triple-line kinetic boundary conditions for a solid film in contact with a solid substrate for both nonconserved (evaporation-condensation) and conserved (surface diffusion) dynamics. The result is obtained via a matched asymptotic expansion from a mesoscopic model with a thickness-dependent wetting potential (or disjoining pressure) and mobility. In the nonconserved case, we obtain a single boundary condition, which relates the triple-line velocity with the deviation of the contact angle from its equilibrium value. In the conserved case, two kinetic boundary conditions are needed. They relate the velocity and mass flux at the triple line to the contact angle deviation and discontinuity of the chemical potential. These linear relations are described by three kinetic coefficients. The conditions under which the kinetic coefficients remain finite are obtained. We find, for example, that some kinetic coefficients diverge within the conserved model in the presence of van der Waals interaction.

Figure 1

Wetting potential as a function of film thickness in the case of partial wetting. By convention, the origin of heights ($h=0$) corresponds to the substrate equilibrium height at the minimum of $w$.

Figure 2

Schematic showing the division of the full film profile into substrate, triple-line, and island regions. The dashed line indicates the equilibrium substrate height.

Figure 3

Surface mobility as a function of film thickness for the four cases discussed in the text. The values of the parameters $p_1$ and $p_2$ for cases 3 and 4 are reported in Table 1.

Figure 4

Numerical simulation of the nonconserved model for ${\mu }_c=0$. (a) Time evolution of the film profile in the constant mobility case (case 1). (b) Schematic of the fitting procedure to find the triple-line position. The black solid curve represents the film profile. Blue and red curves are the functions in the island and substrate regions obtained by the fitting of the film profile in the fitting intervals FitS and FitF, respectively. Also shown is the time evolution of (c) the triple-line position $x_{\mathrm{TL}}$, (d) the triple-line velocity $v$, and (e) the thermodynamic force ${\left[u\right]}_{\mathrm{TL}}$. The four different cases correspond to four different forms of the mobility defined in the text. (f) Variation of the kinetic coefficients as a function of the center of the fitting region on the film side. The kinetic coefficients are normalized with respect to the theoretical prediction.

Figure 5

Numerical results for the conserved model. (a) Evolution of the film profile starting from an step initial condition for constant mobility (case 1). (b) Time evolution of triple-line position for four different cases of surface mobility. Here symbols show the numerical data and corresponding black lines are the fit to the data according Eq. (110). Also shown is the temporal evolution of (c) the triple-line velocity $v$, (d) the mass flux $j_{\mathrm{TL}}$ approaching from the substrate side, (e) the thermodynamic force ${\left[u\right]}_{\mathrm{TL}}$, and (f) the difference in chemical potential ${\left[\mu \right]}_{\mathrm{TL}}$.

Figure 6

Variation of ${\left[j\right]}_{\mathrm{TL}}/j_{\mathrm{TL}}$ with time.

Figure 7

Numerical verification of the kinetic boundary conditions. Here $j_{\mathrm{TL}}$ is the value obtained from the extrapolation of the mass flux from the substrate side. Closed and open symbols denote the data for the initial film thicknesses $\overline{h}=2$ and $\overline{h}=5$, respectively. In addition, ${\mathcal{E}}_1, {\mathcal{E}}_2$, and $\mathcal{E}$ are the time-averaged errors defined in the text. Averages are performed in the intervals $I_1$ with $2\times {10}^4<t<2\times {10}^6$ and $I_2$ with $2\times {10}^5<t<2\times {10}^6$. The legend for each column is shown on top. The dashed black line in each plot corresponds to the analytical prediction reported in Table 2. As discussed in the text, the four panels in each column correspond to the four different mobility cases.

Figure 8

Convergence of the numerical procedure to extract ${\mathcal{L}}_{2v}$ for case 2. Fit of numerical data for ${\mathcal{L}}_{2v}$, using linear (solid red line) and quadratic (dashed blue curve) functions described in the text. The horizontal dashed line shows the predicted value. The vertical dashed line shows the position of the triple line.

Figure 9

Relative deviation of the contact angle and mass flux balance during dewetting. The relative deviation of the dynamic contact angle from its equilibrium value is shown for (a) the nonconserved and (b) the conserved models. (c) Comparison of the mass flux through the triple line to the mass flux corresponding to the increase of the volume of the rim resulting from its motion at velocity $v$.

Figure 10

Decomposition of ${\left[u\right]}_{\mathrm{TL}}$ following Eq. (113) for the nonconserved model. The simulation was performed with a vapor chemical potential ${\mu }_c=0.02$.

Figure 11

Numerical verification of the kinetic boundary conditions. Here $j_{\mathrm{TL}}$ is the value obtained from the extrapolation of the mass flux from the island side. The notation and symbols are similar to those in Fig. 7.