The interface dynamics of many non equilibrium systems arises from the interplay between nonlocal interactions and morphological instabilities. Nonlocal effects can be due to diverse physical mechanisms like diffusive, ballistic, or anomalous transport, as occurs e.g. in combustion fronts or in thin film growth. Often, the dynamics of these interfaces can be cast into an stochastic partial differential equation for the surface height h on a d-dimensional substrate. In this work we study the family of equations (after Fourier transform F) ?thk(t) = (-?k?-?km)hk(t)+(?/2) F[(?xh)2]+?k(t), (1) where ?, m and ? are positive constants with 0?2 and m?2, while ? is Gaussian uncorrelated noise, and the nonlinear term is the celebrated Kardar-Parisi-Zhang (KPZ) nonlinearity. Important examples of nonlocal dispersion relations included in (1) are the Mullins-Sekerka or the Saffman-Taylor instabilities. Also the Darrieus-Landau instability occurring in the propagation of a premixed laminar flame, for which the gas expansion produced by heat induces wrinkles on the flame front. Moreover, in the case of ballistic relaxation, i.e. when ?=1, and for m=2, Eq. (1) becomes a stochastic generalization of the Michelson-Sivashinsky equation, derived for reactive infiltration in porous media. When ?>0 (stable fronts), dimensional analysis correctly predicts the critical exponents obtained through the pseudo-spectral integration of the equations in one and two substrate dimensions. However, when ?<0 (instability), the scaling properties of the surface are nontrivial and an improved analytical argument is needed to calculate the critical behavior of the equations. In this work, we have done a one-loop dynamic renormalization group (DRG) of Eq. (1) for the unstable case and compared with pseudo-spectral numerical integrations for d=1,2. For a wide range of parameters, the asymptotic dynamics is scale invariant with dimension-independent exponents reflecting a hidden Galilean symmetry. The KPZ nonlinearity, albeit irrelevant in that parameter range, plays a key role in the stabilization of the system for intermediate to long times and seems to be responsible for a scaling relation among exponents. In the DRG language, somehow the KPZ nonlinearity renormalizes to zero in infinite RG flow "time".
2009 Materials Research Society Fall Meeting
Publication date: November 2009.
M. Nicoli, R. Cuerno, M. Castro, Unstable nonocal interface dynamics, 2009 Materials Research Society Fall Meeting. Boston, United States of America, 30 November - 04 December 2009