Diffusion-Limited Aggregation on Curved Surfaces

We develop a general theory of transport-limited aggregation phenomena occurring on curved surfaces, based on stochastic iterated conformal maps and conformal projections to the complex plane. To illustrate the theory, we use stereographic projections to simulate diffusionlimited-aggregation (DLA) on surfaces of constant Gaussian curvature, including the sphere (K > 0) and pseudo-sphere (K < 0), which approximate “bumps” and “saddles” in smooth surfaces, respectively. Although curvature affects the global morphology of the aggregates, the fractal dimension (in the curved metric) is remarkably insensitive to curvature, as long as the particle size is much smaller than the radius of curvature. We conjecture that all aggregates grown by conformally invariant transport on curved surfaces have the same fractal dimension as DLA in the plane. Our simulations suggest, however, that the multifractal dimensions increase from hyperbolic (K < 0) to elliptic (K > 0) geometry, which we attribute to curvature-dependent screening of tip branching. Introduction.— The Laplacian growth model and its stochastic analogue, diffusion-limited aggregation (DLA) [3], describe the essential physics of nonequilibrium pattern formation in diverse situations [2]. Examples include viscous fingering [4], dendrite solidification [5], dielectric breakdown [6], and dissolution [11], depending on conditions at the moving, free boundary. Some extensions to non-Laplacian growth phenomena, such as advection-diffusion-limited aggregation [13, 16] (ADLA) and brittle fracture [12], are also available, which exploit mathematical similarities with the DLA model. Almost all prior work with these growth models has assumed flat Euclidean space, typically a two-dimensional plane, but real aggregates, such as mineral dendrites [17], cell colonies [18], and cancerous tumors [19], often grow on curved or rough surfaces. In principle, surface curvature should affect the morphology of stochastic aggregates, but we are not aware of any prior work, except for simulations of Eden-like clusters on spheres [18], which lack the longrange interactions of DLA and related models through the evolving concentration field. In the case of continuous interfacial motion, there have been several mathematical studies of viscous fingering (without surface tension) on spheres [20, 21], but we will show that stochastic aggregation of discrete particles is rather different, due to the physical constraint of fixed particle size in the curved space. In this Letter, we extend transport-limited growth models to curved two-dimensional surfaces via conformal projections from the plane. Time-dependent conformal maps are widely used in physics [1] and materials science [14] to describe interfacial dynamics in two dimensions. Continuous conformal maps have long been applied to viscous fingering [4,7], and more recently, Hastings and Levitov introduced stochastic, iterated conformal maps for DLA [24]. Both continuous and stochastic conformal-map dynamics have also been extended to other conformally invariant (but non-Laplacian and nonlinear) gradient-driven transport processes [9, 13, 16], such as advection-diffusion in a potential flow [8,10] or electrochemical transport in a quasi-neutral solution [9]. Indeed, there is nothing special about harmonic functions (solutions to Laplace’s equation) in the plane, aside from the direct connection to analytic functions of a complex variable (real or imaginary part). The key property of conformal invariance is shared by other equations [9,14] and, as note here, applies equally well conformal (i.e. angle preserving) transformations between curved surfaces. We formulate continuous and discrete conformal-map dynamics for surfaces of constant Gaussian curvature, not only the sphere with positive curvature, but also the pseudosphere, with negative curvature. We use the approach to study the fractal and