Nonlocality in Kinetic Roughening

“Suppose that we take a bin and gently and uniformly pour in granular material. As the material in the bin builds up we can identify a surface and ask the question, ‘What is the magnitude of the fluctuation in the height of surface (measured from the base of the bin)?’ Also of interest is the length scale of the surface fluctuations and how they behave dynamically as more material is added” [1]. And thus was born the Edwards-Wilkinson (EW) model for surface growth—a solvable linear model at the heart of our current understanding of numerous growth processes. A relevant nonlinear term, added to this by Kardar-Parisi-Zhang (KPZ) [2–4], brought to light the nuances of growth phenomena to the extent that the KPZ equation very soon became a paradigm, in particular for dynamic phase transitions. The applicability of the KPZ equation seems to encompass length scales from an atomic level to macroscopic phenomena of everyday life, but still a specter is haunting the field: Why is the KPZ behavior not observed [3]? Many of the experimental situations, however, involve complex processes which beg to go beyond the idealization, as pouring of noninteracting particles. This is especially true if medium or fluctuation induced interactions interfere with the process as, for example, in the several recently studied systems involving proteins, colloids or latex particles [5–8], or in sedimentation. The major interaction one has to reckon with, as detailed numerical computations suggest [9,10], is the long ranged hydrodynamic interaction. Are such long range interactions relevant for the roughness of the surface? This question, the absence of a formalism to handle such interactions in the growth process, and the elusiveness of the KPZ behavior, led us to propose a simple phenomenological model by focusing on the long range nature of the extra force. We developed a Langevin equation-type description, where long range aspects can be simulated by a force at each point of the growing surface exerted by the particles away from it—a hint to go beyond a strict local description. In the linear EW model, the growth is along the global normal to the surface without any overhang. The height hsr, td, at point r and time t, satisfies the diffusion