Effective model hierarchies for dynamic and static classical density functional theories.

The origin and methodology of deriving effective model hierarchies are presented with applications to solidification of crystalline solids. In particular, it is discussed how the form of the equations of motion and the effective parameters on larger scales can be obtained from the more microscopic models. It will be shown that tying together the dynamic structure of the projection operator formalism with static classical density functional theories can lead to incomplete (mass) transport properties even though the linearized hydrodynamics on large scales is correctly reproduced. To facilitate a more natural way of binding together the dynamics of the macrovariables and classical density functional theory, a dynamic generalization of density functional theory based on the nonequilibrium generating functional is suggested.

Interface equations for capillary rise in random environment.

We consider the influence of quenched noise upon interface dynamics in two-dimensional (2D) and 3D capillary rise with rough walls by using a phase-field approach, where the local conservation of mass in the bulk is explicitly included. In the 2D case, the disorder is assumed to be in the effective mobility coefficient, while in the 3D case we explicitly consider the influence of locally fluctuating geometry along a solid wall using a generalized curvilinear coordinate transformation. To obtain the equations of motion for meniscus and contact lines, we develop a systematic projection formalism that allows inclusion of disorder. Using this formalism, we derive linearized equations of motion for the meniscus and contact line variables, which become local in the Fourier space representation. These dispersion relations contain effective noise that is linearly proportional to the velocity. The deterministic parts of our dispersion relations agree with results obtained from other similar studies in the proper limits. However, the forms of the noise terms derived here are quantitatively different from the other studies.

Interface dynamics and kinetic roughening in fractals.

We consider the dynamics and kinetic roughening of single-valued interfaces in two-dimensional fractal media. Assuming that the local height difference distribution function of the fronts obeys Levý statistics with a well-defined power-law decay exponent, we derive analytic expressions for the local scaling exponents. We also show that the kinetic roughening of the interfaces displays anomalous scaling and multiscaling in the relevant correlation functions. For invasion percolation models, the exponents can be obtained from the fractal geometry of percolation clusters. Our predictions are in excellent agreement with numerical simulations.

Interface pinning in spontaneous imbibition.

Evaporation and gravity induced pinning in spontaneous imbibition are examined within a phase field formalism. Evaporation is introduced via a nonconserving term and gravity through a convective term that constrains the influx of liquid. Their effects are described by dimensionless coupling constants epsilon and g, respectively. From liquid conservation, the early time behavior of the average interface position follows H(t) approximately t1/2 until a crossover time t*(g,epsilon). After that the pinning height Hp(g,epsilon) is approached exponentially in time, in accordance with mean field theory. The statistical roughness of the interface is described by an exponent chi approximately 1.25 at all stages of the rise, but the dynamic length scale controlling roughness crosses over from xi(x) approximately H1/2 to a time independent pinning length scale xi(p)(epsilon,g).