Model of Cluster Growth and Phase Separation : Exact Results in One Dimension

We present exact results for a lattice model of cluster growth in 1D. The growth mechanism involves interface hopping and pairwise annihilation supplemented by spontaneous creation of the stable-phase, +1, regions by overturning the unstable-phase, −1, spins with probability p. For cluster coarsening at phase coexistence, p = 0, the conventional structure-factor scaling applies. In this limit our model falls in the class of diffusion-limited reactions A+A→inert. The +1 cluster size grows diffusively, ∼ √ t, and the two-point correlation function obeys scaling. However, for p > 0, i.e., for the dynamics of formation of stable phase from unstable phase, we find that structure-factor scaling breaks down; the length scale associated with the size of the growing +1 clusters reflects only the short-distance properties of the two-point correlations. 1. INTRODUCTION Lattice cellular automaton-type models with local tendency for ordering, termed voter models, can be used to study phase segregation and cluster coarsening reminiscent of spin-odal decomposition, (1,2) at least in low dimensions. Both the cluster-size (3) and structure-factor (4) scaling at phase separation have been subjects of numerous investigations. However , most of the available results for realistic 2D and 3D dynamical models are numerical. We distinguish between the two " scaling " terms as follows. By cluster-size scaling we mean scaling properties of the cluster size distribution. The term structure-factor scaling is reserved for the scaling properties of the two-point order parameter correlation function. The latter is accessible to scattering experiments.