Nucleation and growth in one dimension.

Inhomogeneous systems where stable and metastable regions coexist are common in nature. Phase separation and coarsening [1], aggregation [2], wetting [3], dendritic growth [4], and growth of breath figures [5] are just a few examples of such systems. Typically, the stable phase grows into the metastable phase according to complicated kinematic rules. However, in certain cases such as adsorption [6], simple ballistic growth rules apply. The KolmogorovAvrami-Johnson-Mehl (KAJM) nucleation-and-growth process is a natural model incorporating nucleation of stable phases with ballistic growth [6-17]. In this work we present exact results for various statistical properties of the KAJM growth model in one dimension. The process depends on the nucleation rate as well as the initial concentration of the growing phase. There are two limiting cases, instantaneous nucleation (subsequent nucleation rate vanishes) and continuous nucleation (vanishing initial concentration). The growth rate of the stable phase may also depend on size of the growing islands. Besides the ordinary KAJM model with size-independent growth rate, the accelerated KAJM model with growth rate linear in size is also solvable as we shall demonstrate in this paper. To solve the KAJM model with linear growth rate, a deeper understanding of the island-size distribution is necessary. Surprisingly, even for the classical KAJM model little is known about the island-size distribution function. Hence, we we first investigate the KAJM model with a constant growth velocity. We introduce the density of islands containing n “seeds” and show that this distribution is not a smooth function of the space variable. As a result, the total island length distribution has spatial discontinuous derivatives at every integer multiple of t. In the continuous nucleation case, we obtain only the inverse Laplace transform of this generalized island distribution. However, an asymptotic analysis shows that the relative fraction of islands containing n seeds decays algebraically in time rather than exponentially as in the case of instantaneous nucleation. In the second part of our study, we introduce an accelerated nucleation-and-growth process where the growth velocity depends on the island size. This growth mechanism is motivated by an accelerated random sequential adsorption (RSA) process [18]. In the ordinary RSA of monomers on a lattice an adsorption attempt on a site is successful only if that site is empty. Unlike ordinary RSA where an attempt to adsorb on an occupied site is rejected, in the accelerated process any attempt is successful – if a monomer is deposited onto an already existing island it diffuses until it reaches an empty site on the island boundary. Hence, islands grow with a rate increasing with island size; if the diffusion time scale is very small compare to the adsorption time scale, the growth rate becomes a linear function of the island length. The continuum version of this model is simply the KAJM nucleation-and-growth process with growth rate linear in the island size. While for the lattice model only an approximate theory exists, we generalize the KAJM theory to the accelerated growth model. Exact results for the island gap distribution show that the system is covered in a finite time. Also, the behavior near complete coverage is robust. It is independent of many details of the growth velocity as well as the nucleation mechanism; it is the same for instantaneous and continuous nucleation. The rest of this paper is organized as follows. In Section II, we consider the ordinary KAJM nucleation-and-growth process. We first review the existing theory, and present a summary of the exact results for both instantaneous and continuous nucleation. We then consider the detailed island gap density and analyze its properties. In Section III, we introduce the accelerated growth model and solve for the exact inter-island gap distribution. Additionally, we analyze the behavior close to complete coverage.