Thin film instability with thermal noise

We study the effects of stochastic thermal fluctuations on the instability of the free surface of a flat liquid metallic film on a solid substrate. These fluctuations are represented by a stochastic noise term added to the deterministic equation for the film thickness within the long-wave approximation. Unlike the case of polymeric films, we find that this noise, while remaining white in time, must be colored in space, at least in some regimes. The corresponding noise term is characterized by a nonzero correlation length, l(c), which, combined with the size of the system, leads to a dimensionless parameter beta that accounts for the relative importance of the spatial correlation (beta similar to l(c)(-1)). We perform the linear stability analysis (LSA) of the film both with and without the noise term and find that for l(c)larger than some critical value (depending on the system size), the wavelength of the peak of the spectrum is larger than that corresponding to the deterministic case, while for smaller l(c) this peak corresponds to smaller wavelength than the latter. Interestingly, whatever the value of l(c), the peak always approaches the deterministic one for larger times. We compare LSA results with the numerical simulations of the complete nonlinear problem and find a good agreement in the power spectra for early times at different values of l(c). For late times, we find that the stochastic LSA predicts well the position of the dominant wavelength, showing that nonlinear interactions do not modify the trends of the early linear stages. Finally, we fit the theoretical spectra to experimental data from a nanometric laser-melted copper film and find that at later times, the adjustment requires smaller values of beta (larger space correlations).