Capillary Leveling of Freestanding Liquid Nanofilms

We report on the capillary-driven leveling of a topographical perturbation at the surface of a freestanding liquid nanofilm. The width of a stepped surface profile is found to evolve as the square root of time. The hydrodynamic model is in excellent agreement with the experimental data. In addition to exhibiting an analogy with diffusive processes, this novel system serves as a precise nanoprobe for the rheology of liquids at interfaces in a configuration that avoids substrate effects.

Figure 1

(a) Schematic of a freestanding stepped film. (b) In the melt state, the profile quickly symmetrizes with respect to the central plane (dotted line) at $z=0$. (c) The two mirrored steps broaden horizontally over time. The half-height profile $h(x,t)$ is recorded with AFM. (d) AFM profiles of a freestanding stepped film with $h_1=h_2=176\mathrm{nm}$, for different times $t$ spent above $T_{\mathrm{g}}$. As sketched in the inset (arbitrary units), the profile width $w$ is defined from the tangent (tilted dashed) line in the middle of the step and the film thickness.

Figure 2

Profile width $w$ [Fig. 1, inset] as a function of the square root of time, $t^{1/2}$, for freestanding stepped films with $h_1=h_2$. The open symbols are the measured widths for three different film thicknesses, as indicated. The solid lines are fits to $w=(Mt{)}^{1/2}$, where the mobility $M$ is the fitting parameter. Inset: mobility as a function of thickness $h_2$, when $h_1=h_2$. The dashed line is a linear fit.

Figure 3

Normalized long-time experimental height profiles as a function of the self-similar variable $x/t^{1/2}$, at several times and for two different geometries as indicated. The light (yellow) dashed line represents a fit using Eq. (6). The dark (green) dashed line indicates a fit using the numerical solution of Eqs. (7) and (8) for the $h_1=h_2$ case.

Figure 4

(a) Normalized long-time experimental height profiles (solid lines) of supported PS stepped films ($55\mathrm{kg}/\mathrm{mol}$, $h_1=h_2=390\mathrm{nm}$) as a function of the self-similar variable $x/t^{1/4}$ of Eq. (1), for ten samples and two different times. The numerical solution [57] is fit (dashed line) to the experimental profiles with $v_{\mathrm{c}}$ as the free parameter. (b) Normalized long-time experimental height profiles (solid lines) of freestanding PS stepped films ($55\mathrm{kg}/\mathrm{mol}$, $h_1=h_2=390\mathrm{nm}$) as a function of the self-similar variable $x/t^{1/2}$ of Eq. (2), for 12 samples and two different times. The numerical solution of Eqs. (7) and (8), for the $h_1=h_2$ case, is fit (dashed line) to the experimental profiles with $v_{\mathrm{c}}$ as the free parameter.