Dynamical Ising Model Simulations of Nucleation and Growth in Copper-Cobalt Alloys
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DYNAMICAL ISING MODEL SIMULATIONS OF NUCLEATION AND GROWTH IN COPPER-COBALT ALLOYS ALFRED CEREZO*, JONATHAN M. HYDE*, MICHAEL K. MILLER**, ROHAN P. SETNA* AND GEORGE D. W. SMITH* *Department of Materials, University of Oxford, Parks Road, Oxford OXI 3PH, U.K. "**Metalsand Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6436.
ABSTRACT A simple dynamical Ising model on a fixed lattice with a single bond energy parameter has been used to simulate the kinetics of diffusion during solid-state phase transformations in binary metallic alloys. The results of these simulations are compared with direct real-space measurements of the atomic distributions of elements in alloys, obtained with the positionsensitive atom probe. Despite the simplicity of the model, there is good quantitative agreement between the development of microstructure in the simulation and the nucleation and growth of cobalt-rich precipitates in copper-cobalt alloys. INTRODUCTION When an alloy is aged at temperatures below the solvus, it is possible for phase separation to occur by a process of diffusion. Conventional theories of diffusional phase transformations are based on continuum models, an approach which appears reasonable given the number of atoms in even a modest amount of material.[1, 2] However, advances in microscopy and microanalysis have extended the study of phase decompositions to the earliest stages, when composition fluctuations are on the scale of only a few atomic spacings. In many engineering alloys, important properties result from microstructural features on the nanometre scale, generated by phase separation during heat treatment. Understanding the early stages of phase decomposition is therefore important not only at a fundamental level, but also for predicting the long-term stability of engineering alloys and in the design of new materials. At these size scales, continuum representations of the composition variations are no longer valid, and this leads to a breakdown of conventional models in predicting the kinetics of the reactions. This paper describes a statistical, atomic-scale simulation of the ageing of materials, using a dynamical Ising model. The modelling work is being performed in conjunction with an experimental programme of atomic-scale studies of the early stages of phase transformations, using ultrahigh spatial resolution atom probe field-ion microscope (APFIM) techniques [3, 4]. With APFIM microanalysis, composition-depth profiles can be obtained with 1-2nm lateral resolution, and a depth resolution of a single atomic layer. In a more recent variant of the technique, the position-sensitive atom probe (PoSAP) [5], this microanalysis capability is extended to a full three-dimensional reconstruction of the distribution of atoms originally present in a volume approximately 20nm x 20nm x 20nm in the solid, with sub-nanometre spatial resolution. It is now possible to make direct comparisons between the results of simulations and the atomic-scale chemical measurements. DYNAMICAL ISING MODEL The simula
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