The fingering instability of growing dry patches in an evaporating film of a polar liquid placed on a solid substrate is investigated. The instability manifests itself as fingering of mobile fronts between growing "dry" (thin) and shrinking "wet" (thick) regions of the film corresponding to two stable states of the evaporating film in contact with its vapor. The boundaries of the fingering instability are found through linear stability analysis of numerical solutions of the nonlinear evolution equation defining the film profile, and the influence of the evaporation rate, polar intermolecular forces, and chemical heterogeneity of the substrate is investigated.

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A theory of the spontaneous formation of nanoscale porous structures in aluminum oxide films growing during aluminum anodization is presented. The main elements of this theory are the Butler-Volmer relation describing the exponential dependence of the current on the overpotential and the dependence of the activation energies of the oxide-electrolyte interfacial reactions on the Laplace pressure and the elastic stress in the oxide layer. Two cases are considered, distinguished by whether the elastic stress dependence is significant or not. In the case when the effect of elastic stress is negligible, a linear stability analysis predicts a long-wave instability resulting from the field-assisted dissolution reaction; its competition with the stabilizing effect of the Laplace pressure due to the surface energy provides the wavelength selection mechanism. A weakly nonlinear analysis near the instability threshold reveals that the nonlinear dynamics of the interface perturbations is governed by the Kuramoto-Sivashinsky equation. The spatiotemporally chaotic solutions of this equation can explain the formation of spatially irregular pore arrays that are observed in experiments. In the case when the effect of elastic stress in the oxide layer is significant we show that the instability can transform from the long-wave type to the shortwave type. A weakly nonlinear analysis of the shortwave instability shows that it leads to the growth of spatially regular, hexagonally ordered pore arrays, as observed experimentally.

A nonlinear evolution equation for surface-diffusion-driven Asaro-Tiller-Grinfeld instability of an epitaxially strained thin solid film on a solid substrate is derived in the case where the film wets the substrate. It is found that the presence of a weak wetting interaction between the film and the substrate can substantially retard the instability and modify its spectrum in such a way that the formation of spatially regular arrays of islands or pits on the film surface becomes possible. It is shown that the self-organization dynamics is significantly affected by the presence of the Goldstone mode associated with the conservation of mass.

In this paper we demonstrate that convective Cahn-Hilliard models, describing phase separation of driven systems (e.g., faceting of growing thermodynamically unstable crystal surfaces), exhibit, with the increase of the driving force, a transition from the usual coarsening regime to a chaotic behavior without coarsening via a pattern-forming state characterized by the formation of various stationary and traveling periodic structures as well as structures with localized oscillations. Relation of this phenomenon to a kinetic roughening of thermodynamically unstable surfaces is discussed.

Consider faceting of a crystal surface caused by strongly anisotropic surface tension, driven by surface diffusion and accompanied by deposition (etching) due to fluxes normal to the surface. Nonlinear evolution equations describing the faceting of 1+1 and 2+1 crystal surfaces are studied analytically, by means of matched asymptotic expansions for small growth rates, and numerically otherwise. Stationary shapes and dynamics of faceted pyramidal structures are found as functions of the growth rate. In the 1+1 case it is shown that a solitary hill as well as periodic hill-and-valley solutions are unique, while solutions in the form of a solitary valley form a one-parameter family. It is found that with the increase of the growth rate, the faceting dynamics exhibits transitions from the power-law coarsening to the formation of pyramidal structures with a fixed average size and finally to spatiotemporally chaotic surfaces resembling the kinetic roughening.

A mechanism for the formation of quantum dots on the surface of thin solid films is proposed, not associated with the Asaro-Tiller-Grinfeld instability caused by epitaxial stresses. This mechanism, free of stress, involves instability of the film surface due to strong anisotropy of the surface energy of the film, coupled to wetting interactions between the film and the substrate. According to the mechanism, the substrate induces the film growth in a certain crystallographic orientation. In the absence of wetting interactions with the substrate, due to a large surface-energy anisotropy, this orientation would be thermodynamically forbidden and the surface would undergo a long-wave faceting (spinodal decomposition) instability. We show that wetting interactions between the film and the substrate can suppress this instability and qualitatively change its spectrum, leading to the damping of long-wave perturbations and the selection of the preferred wavelength at the instability threshold. This creates a possibility for the formation of stable regular arrays of quantum dots even in the absence of epitaxial stresses. This possibility is investigated analytically and numerically, by solving the corresponding nonlinear evolution equation for the film surface profile, and analyzing the stability of patterns with different symmetries. It is shown that, near the instability threshold, the formation of stable hexagonal arrays of quantum dots is possible. With the increase of the supercriticality, a transition to a square array of dots or the formation of spatially localized dots can occur. Different models of wetting interactions between the film and the substrate are considered and the effects of the wetting potential anisotropy are discussed. It is argued that the mechanism can provide a new route for producing self-organized quantum dots.