Spreading fronts of wetting liquid droplets: Microscopic simulations and universal fluctuations

We have used kinetic Monte Carlo (kMC) simulations of a lattice gas to study front fluctuations in the spreading of a nonvolatile liquid droplet onto a solid substrate. Our results are consistent with a diffusive growth law for the radius of the precursor layer, $R\sim t^{\delta }$, with $\delta \approx 1/2$ in all the conditions considered for temperature and substrate wettability, in good agreement with previous studies. The fluctuations of the front exhibit kinetic roughening properties with exponent values which depend on temperature $T$, but become $T$ independent for sufficiently high $T$. Moreover, strong evidence of intrinsic anomalous scaling has been found, characterized by different values of the roughness exponent at short and large length scales. Although such a behavior differs from the scaling properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) universality class, the front covariance and the probability distribution function of front fluctuations found in our kMC simulations do display KPZ behavior, agreeing with simulations of a continuum height equation proposed in this context. However, this equation does not feature intrinsic anomalous scaling, at variance with the discrete model.

Figure 1

Top views of three snapshots of the lattice gas model for increasing values of the Hamaker constant $A$, left to right. Occupied cells in the precursor and supernatant layers are in gray and black, respectively, with the red and green lines delimiting the corresponding fronts; empty cells are uncolored. Parameters used are $J=1, T=1, L_y=100, L_x=50$, and (a) $A=0.1$, (b) $A=1$, and (c) $A=10$. The three snapshots were taken at the same simulation time. All units are arbitrary.

Figure 2

Same as in Fig. 1 but for decreasing values of the temperature $T$, left to right. Specifically, $J=1, A=10, L_y=100, L_x=50$, and (a) $T=3$, (b) $T=1$, and (c) $T=1/3$. All units are arbitrary.

Figure 3

Average front position $\langle \overline{h}\left(t\right)\rangle$ as a function of time for $J=1, T=1, L_x=1000, L_y=256$, and several values of $A$. The solid black line corresponds to the reference scaling $\langle \overline{h}\left(t\right)\rangle \sim t^{1/2}$. All units are arbitrary.

Figure 4

Squared roughness $w^2\left(t\right)$ as a function of time, as obtained for $J=1, T=3, L_x=1000, L_y=256$, and several values of $A$. The corresponding values of $2\beta$ are given in Table 3. As a visual reference, the solid black line corresponds to $w^2\left(t\right)\sim t^{1/2}$. All units are arbitrary.

Figure 5

Values of $\alpha$ (top) and $\beta$ (bottom) taken from Table 5 and Table 3 vs $T$ for $J=1$ and for $A=0.01$ (circles), $A=1$ (squares), and $A=10$ (triangles). Lines are guides to the eye. All units are arbitrary.

Figure 6

Estimates ${\xi }_{0.8}\left(t\right)$ and ${\xi }_{0.9}\left(t\right)$ as functions of time, obtained for $J=1, T=1, A=1$, and $L_x=1000$. As a visual reference, the solid black line corresponds to $\xi \left(t\right)\sim t^{0.3}$. All units are arbitrary.

Figure 7

Height-difference correlation function $C_2(L_y/2,t)$ versus ${\xi }_{0.8}\left(t\right)$ and ${\xi }_{0.9}\left(t\right)$ at different times. Conditions are $J=1, T=1, A=1, L_x=1000$, and $L_y=256$. As a visual reference, the solid black line corresponds to $C_2(L_y/2,t)\sim t^{1.75}$. All units are arbitrary.

Figure 8

Data collapse of the height-difference correlation function obtained for different values of time, for $J=1, T=1, A=1, L_x=1000$, and $L_y=256$, using $\alpha =0.9$. The curve onto which collapse occurs is the function $g(r/\xi \left(t\right))$ of Eq. (11), with the solid black line representing the theoretical behavior for large $u, g\left(u\right)\sim u^{-2\alpha }$, and the solid gray line representing the behavior for small $u, g\left(u\right)\sim u^{-2{\alpha }^{\prime }}$ (see Tables 4 and 6 in the Appendix). All units are arbitrary. Inset: height-difference correlation function as a function of $r$ for times increasing from 20 to 100 bottom to top at regular intervals.

Figure 9

Structure factor calculated for the precursor layer at (a) $J=1, T=0.5, A=0.1$ and (b) $J=1, T=3, A=1$, for times increasing bottom to top in both panels. The scaling behavior at fixed time is $S(k,t)\sim {\left|k\right|}^{-(2{\alpha }_{\mathrm{loc}}+1)}$, where ${\alpha }_{\mathrm{loc}}$ has been evaluated as $\alpha -{\alpha }^{\prime }$; see Tables 4 and 6 in the Appendix. The power laws represented by the solid lines are indicated in the corresponding legends. All units are arbitrary.

Figure 10

Fluctuation histograms calculated according to Eq. (14) for $J=1, A=1, T=1$, and several system sizes, as indicated in the legend. The solid line corresponds to the GOE Tracy-Widom distribution. All units are arbitrary.

Figure 11

$R(\tilde{x},t)\equiv \frac{C_1(\tilde{x}{t}^{1/z}/a_2)}{a_1{t}^{2\beta }}$ versus $\tilde{x}\equiv a_{2}r/t^{1/z}$ for $t=60$, 80, and 100, calculated for parameters as in Fig. 10, using $1/z=0.32, 2\beta =0.544, a_1=1.834\times {10}^{-4}$, and $a_2=8.985\times {10}^{-3}$. The solid line corresponds to the exact ${\mathrm{Airy}}_1\left(\tilde{x}\right)$ function. All units are arbitrary.

Figure 12

Average front position (left vertical axis, circles) and squared roughness (right vertical axis, triangles) as functions of $t$, as obtained from numerical simulations of Eq. (16) for $\nu =\lambda =1$. The solid red line corresponds to power law $\langle \overline{h}\left(t\right)\rangle \sim t^{\delta }$ and the dashed green line to $w^2\left(t\right)\sim t^{2\beta }$ with exponent values as in the legend. All units are arbitrary.

Figure 13

(a) Structure factor of the front, $S(k,t)$, as a function of wave-vector modulus $k$ for increasing times as indicated by the color scale, from numerical simulations of Eq. (16) using parameters as in Fig. 12. (b) Collapse of the data of panel (a) following Eq. (12). The $u$-independent behavior at large $u=k{t}^{1/z}$ indicates standard FV scaling with $\alpha ={\alpha }_{\mathrm{loc}}\simeq 1$. All units are arbitrary.

Figure 14

(a) PDF of front fluctuations as obtained from numerical simulations of Eq. (16) using parameters as in Fig. 12, for $t={10}^{-2}$ (green points) and for $t=2621.44=2^{18}\times {10}^{-2}$ (red points). The dashed line shows a Gaussian distribution, while the solid line corresponds to the exact TW-GOE form. Inset: time evolution of the skewness and excess kurtosis corresponding to the same set of simulations as the main panel. The TW-GOE values are the abscissas of the corresponding horizontal lines and are shown for reference. (b) Data collapse according to Eq. (15) for the same simulations as in panel (a). For reference, the exact ${\mathrm{Airy}}_1$ covariance is shown as a dashed line. All units are arbitrary.