Three-dimensional rough surface growth: A radial continuum equation and a discrete off-lattice Eden cluster growth model

The study of the propagation of rough surfaces has a rich variety of applications in physics and biology including fluid flow, vapor and electron deposition, tumor and bacterial growth, and epidemiology. A stochastic partial differential equation for three-dimensional surface growth in the radial geometry is proposed. The equation reduces to the KPZ equation in the large radius limit. The equation is analyzed numerically and the growth exponent is estimated. Two distinct scaling regimes are discovered. Further, a three-dimensional, off-lattice Eden-C cluster growth model is proposed. The discrete model grows compact clusters of non-overlapping, contiguous spherical cells. A proof that Eden clusters are compact is extended to off-lattice clusters. Large-scale computer simulations show that the model creates isotropic clusters, with interior volume density constant at approximately 0.43. The clusters have a self-affine surface, which propagates with a growth exponent β ≈ 0.12. These results are used to discuss whether the radial continuum equation and three-dimensional off-lattice Eden growth belong to the KPZ universality class.