Thermal capillary waves on bounded nanoscale thin films

The effect of confining walls on the fluctuation of a nanoscale thin film's free surface is studied using stochastic thin-film equations (STFEs). Two canonical boundary conditions are employed to reveal the influence of the confinement: (1) an imposed contact angle and (2) a pinned contact line. A linear stability analysis provides the wave eigenmodes, after which thermal-capillary-wave theory predicts the wave fluctuation amplitudes. Molecular dynamics (MD) simulations are performed to test the predictions, and a Langevin diffusion model is proposed to capture oscillations of the contact lines observed in MD simulations. Good agreement between the theoretical predictions and the MD simulation results is recovered, and it is discovered that confinement can influence the entire film. Notably, a constraint on the length scale of wave modes is found to affect fluctuation amplitudes from our theoretical model, especially for 3D films. This opens up challenges and future lines of inquiry.

Figure 1

An illustration of the geometry of the quasi-2D thin-film problem (top) and a snapshot of a representative MD simulation for a thin film with ${90}^{\circ }$ contact angle (below); yellow particles denote liquid argon and red particles denote platinum solid.

Figure 2

Standard deviation of the fluctuations of films with ${90}^{\circ }$ contact angle (black: $L_x=13.04$ nm, blue: $L_x=25.99$, green: $L_x=51.29$ nm). MD results (dashed lines with circles) are compared to our theory, Eq. (14) (solid lines). Results are normalized by the thermal length scale $l_T$. The dashed-and-dotted horizontal lines are fluctuation amplitudes predicted by Eq. (4) for a periodic (unbounded) film.

Figure 3

Wave modes ${\phi }_n\left(x\right)$ for a film with perfectly pinned contact lines.

Figure 4

(a)–(c) MD snapshots in a region near the contact line and the chemical heterogeneity of the solid wall, showing the position of the contact line below, on, and above the proposed pinning point respectively. Blue particles denote hydrophilic wall atoms and red particles denote the hydrophobic wall atoms. Liquid argon atoms are in yellow. (d) A histogram of contact line position extracted from a single realization of MD simulations.

Figure 5

Standard deviation of fluctuations for the films with partially pinned contact lines (black: $L_x=13.04$ nm, blue: $L_x=25.99$ nm, green: $L_x=51.29$ nm, red: $L_x=102.30$ nm). (a) A comparison of MD (dashed lines with circles), theory via Eq. (41) (solid lines), and the classic thermal-capillary-wave theory for periodic films (dashed lines) predicted by Eq. (4). (b) The decomposition of theory (41), where the dashed lines represent the amplitudes of fluctuations caused by thermal noise in bulk, the dotted lines represent the amplitudes of fluctuations caused by noise on the contact lines, and the solid lines represent the full fluctuation amplitudes.

Figure 6

An illustration of the geometry of the 3D circular thin films (half to show cross section).

Figure 7

Wave modes ${\chi }_{n,\alpha }\left(r\right)$ for the circular-bounded 3D film with ${90}^{\circ }$ contact angle. $n$ is the wave number in $\theta$, and $\alpha$ is the wave number in $r$. $\alpha =2$ for black lines, and $\alpha =5$ for red lines.

Figure 8

Snapshots of MD simulations for a 3D circular film. (a) The cross section. (b) The top view with the circular mesh used for extracting the free surface position.

Figure 9

Standard deviation of fluctuations for ${90}^{\circ }$ contact-angle 3D circular films with two different radii (black: $a=11.81$ nm, red: $a=23.39$ nm). MD simulation results (dashed lines with circles) and theory (63) (solid lines) are normalized by the thermal scale $l_T$.

Figure 10

Wave modes ${\psi }_{n,\alpha }\left(r\right)$ for the circular bounded 3D film with a pinned contact line. $n$ is the wave number in $\theta$, and $\alpha$ is the wave number in $r$. $\alpha =2$ for black lines, and $\alpha =5$ for red lines.

Figure 11

Standard deviation of fluctuations for pinned contact-line 3D circular films with two different radii (black: $a=11.81$ nm, red: $a=23.39$ nm). MD simulation results (dashed lines with circles) and theory (78) (solid lines) are normalized by thermal scale $l_T$.

Figure 12

Standard deviation of fluctuation of 3D circular film with radius $a=23.39$ nm and different boundary conditions: (a) ${90}^{\circ }$ contact angle and (b) a pinned contact line. Theoretical predictions (63) and (78) with different cutoff length scales ${\ell }_c={\sigma }_{ff}$, $2{\sigma }_{ff}$, $3{\sigma }_{ff}$ are given by solid lines. The circled lines are MD results obtained using different bin sizes $L_m=1.77{\ell }_c$.

Figure 13

Comparison of fluctuation amplitudes for quasi-2D thin films extracted from MD (circled lines), theoretical predictions considering a cutoff (crossed lines) and without a cutoff (solid lines). (a) ${90}^{\circ }$ contact angle, (b) partially pinned contact lines. Black: $L_x=13.04$ nm, blue: $L_x=25.99$ nm, green: $L_x=51.29$ nm, red: $L_x=102.30$ nm.