The well-known thermal capillary wave theory, which describes the capillary spectrum of the free surface of a liquid film, does not reveal the transient dynamics of surface waves, e.g., the process through which a smooth surface becomes rough. Here, a Langevin model is proposed that can capture these dynamics, goes beyond the long-wave paradigm which can be inaccurate at the nanoscale, and is validated using molecular dynamics simulations for nanoscale films on both planar and cylindrical substrates. We show that a scaling relation exists for surface roughening of a planar film and the scaling exponents belong to a specific universality class. The capillary spectra of planar films are found to advance towards a static spectrum, with the roughness of the surface $W$ increasing as a power law of time $W\sim t^{1/8}$ before saturation. However, the spectra of an annular film (with outer radius $h_0$) are unbounded for dimensionless wavenumber $qh_0<1$ due to the Rayleigh-Plateau instability.
Interfacial roughening in non-ideal fluids: Dynamic scaling in the weak- and strong-damping regime
界面粗糙化是指初始平坦界面达到其平衡态的非平衡过程，其特征在于热激发毛细波的存在。流体界面的粗糙化首先由Flekkoy和Rothman [Phys. Rev. Lett. 75，260（1995）]，其中在二维弱阻尼情况下的动态标度指数被发现符合Kardar-Parisi-Zhang普适类。我们扩展这项工作，同时考虑到强阻尼制度，并进行广泛的波动流体力学模拟在两个维度上使用格子玻尔兹曼方法。我们发现，在弱阻尼和强阻尼的情况下，动力学标度行为是不同的。
Capillary rise dynamics of liquid hydrocarbons in mesoporous silica as explored by gravimetry, optical and neutron imaging: Nano-rheology and determination of pore size distributions from the shape of imbibition fronts
