Numerical Simulation and the Universality Class of the KPZ Equation for Curved Substrates

TL;DR

This study uses extensive numerical simulations to analyze the KPZ equation on curved substrates, revealing that interface width grows logarithmically over time and the transition between universality classes is gradual, contrasting flat substrate behavior.

Contribution

It provides new insights into the long-time behavior of the radial KPZ equation, showing logarithmic growth and a non-sharp transition between universality classes in (1+1)-dimensions.

Findings

Interface width grows logarithmically with time.

Transition between universality classes is gradual.

Evaporation can dominate growth when nonlinear coefficient is small.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzed for over three decades. Some experiments indicated the power law for the interface width, $w(t)\sim t^\beta$, remains the same as in growth on planar surfaces. Escudero (Phys. Rev. Lett. {\bf 100}, 116101, 2008) argued, however, that for the radial KPZ equations in (1+1)-dimension $w(t)$ should increase as $w(t)\sim [\ln(t)]^{1/2}$ in the long-time limit. Krug (Phys. Rev. Lett. {\bf 102}, 139601, 2009) argued, however, that the dynamics of the interface must remain unchanged with a change in the geometry. Other studies indicated that for radial growth the exponent $\beta$ should remain the same as that of the planar case, regardless of whether the growth is linear or nonlinear, but that the saturation regime will not be reached anymore. We present the results of extensive numerical simulations in (1+1)-dimensions of the radial KPZ equation, starting from an initial circular substrate. We find that unlike the KPZ equation for flat substrates, the transition from linear to nonlinear universality classes is not sharp. Moreover, in the long-time limit the interface width exhibits logarithmic growth with the time, instead of saturation. We also find that evaporation dominates the growth process when the coefficient of the nonlinear term in the KPZ equation is small, and that the average radius of the interface decreases with time and reaches a minimum but not zero value.