In this chapter, we describe a few models that have had a key impact on our knowledge about specific aspects of interface roughening. Due to intractable mathematical difficulties, numerical methods are commonly used to determine the scaling exponents for systems with d > 1. Most growth models originate from specific physical or biological problems, and only recently have been investigated using the methods described in this book.

Ballistic deposition

The ballistic deposition model introduced in Chapter 2 is the simplest version – termed the nearest-neighbor (NN) model because falling particles stick to the first nearest neighbor on the aggregate. If we allow particles to stick to a diagonal neighbor as well, we have the next-nearest neighbor (NNN) model (Fig. 8.1). Since the nonlinear term is present for both models (λ ≠ 0), the scaling properties for both models are described by the nonlinear theory. These two models therefore belong to the same universality class, since they share the same set of scaling exponents, α, β, and z. Their non-universal parameters, however, are different. For example, for the velocity V0 (see (A.13)), we find v0 = 2.14, 4.26 for the NN and NNN models, respectively. The coefficient λ of the nonlinear term differs as well, with λ = 1.30, 1.36, respectively.

The origin of the nonlinear term in the model is the lateral sticking rule, leading to the presence of voids.