From a discrete to continuous description of two-dimensional curved and homogeneous clusters: Some kinetic approach

Starting with a discrete picture of the self-avoiding polygon embeddable in the square lattice, and utilizing both scaling arguments as well as a Stei nhaus rule for evaluating the polygon's area, we are able, by imposing a di screte time-dynamics and making use of the concept of quasi-static approxim ation, to arrive at some evolution rules for the surface fractal. The proce ss is highly curvature-driven, which is very characteristic of many phenome na of biological interest, like crystallization, wetting, formation of biom embranes and interfaces. In a discrete regime, the number of subunits const ituting the cluster is a nonlinear function of the number of the perimeter sites active for the growth. A change of the number of subunits in time is essentially determined by a change in the curvature in course of time, give n explicitly by a difference operator. In a continuous limit, the process i s assumed to proceed in time in a self-similar manner, and its description is generally offered in terms of a nonlinear dynamical system, even for the homogeneous clusters. For a sufficiently mature stage of the growing proce ss, and when linearization of the dynamical system is realized, one may get some generalization of Mullins-Sekerka instability concept, where the func tion perturbing the circle is assumed to be everywhere continuous but not n ecessarily differentiable, like e.g., the Weierstrass function. Moreover, a time-dependent prefactor appears in the simplified dynamical system.