Condensed Matter Seminar

Growing interfaces appear commonly in the natural world. Some of them are as important as the technologically motivated thin film deposition, medically relevant as bacterial colonies development, and physically interesting as fluid flow in porous media. Despite their diverse origin, all these phenomena have been studied within the common formalism of scaling analysis. This methodology has been successfully applied to a broad range of situations, but however, there are a number of restrictions that limit scaling analysis as we know it. One of them is the assumption that the interface must be planar and the substrate size constant for all times. Despite the usefulness of this representation in some cases, there are many growth profiles that can not be described according to it. Physical settings such as adatom and vacancy islands on crystal surfaces present interfaces that violate the hypothesis of the Euclidean representation. Biological systems are also characterized by an approximate spherical symmetry: bacterial colonies, plant calli, and tumors develop rough surfaces which are not describable from a planar reference frame. A particularly important example of spherical growth is tumor development. Standard scaling analysis applied to this case suggested a behavior compatible with molecular beam epitaxy surface dynamics, and it served as the basis of a therapeutic strategy targeted in stopping tumor growth. However, the results obtained in these works enter in contradiction with others present in the medical literature, raising the problem of understanding the dynamics of curved interfaces to one important open question in scaling analysis. Due to the disparity of experimental and numerical results obtained so far, it seems necessary to build a theory based on analytical progress in order to clarify curved interface dynamics. Such a theoretical framework started with the introduction of the radial Kardar-Parisi-Zhang and Mullins-Herring equations. The analysis of these equations revealed that the properties of spherical interfaces are totally different from their planar counterparts. We will comment the analysis of growth phenomena on curved interfaces performed so far, pointing out the profound differences in morphological and dynamical properties that they present. Surfaces of a two or higher dimensional nature become flat in the long time limit, due to noise irrelevance in such cases. However, some residual roughness is developed in the first stages of growth, and it can be the source of spurious results in numerical simulations initialized with small cluster sizes. The one dimensional situation is different, as fluctuations, despite being marginal and only of a logarithmic amplitude, are not irrelevant. In this case, flat interfaces correspond to models showing a super-ballistic diffusivity, which is able to erase the effect of early fluctuations asymptotically in time. Sub-ballistic diffusivity plays actually no role in the large scale dynamics and this type of equations reduce to the radial random deposition model, characterized by an uncorrelated interface of marginal logarithmic amplitude, in the long time limit. The critical situation, in which correlations propagate ballistically, has again a logarithmic width interface, due to the marginal intensity of the noise term, but the correlation function is substantially different. The analysis of this function for long times and short spatial scales reveals the appearance of the new local dynamical exponent, which is nonuniversal. These facts have a strong impact on the analysis of growing interfaces, and imply the necessity of reconsidering some of the experimental results obtained so far.