A model for front evolution with a nonlocal growth rate
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RAPID COMMUNICATIONS The purpose of this Rapid Communications section is to provide accelerated publication of important new results in the fields regularly covered by Journal of Materials Research. Rapid Communications cannot exceed four printed pages in length, including space allowed for title, figures, tables, references, and an abstract limited to about 100 words.
A model for front evolution with a nonlocal growth rate Shi Jina) and Xuelei Wangb) School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Thomas L. Starrc) Department of Chemical Engineering, University of Louisville, Louisville, Kentucky 40292 (Received 8 January 1999; accepted 4 June 1999)
In this paper we provide a new mathematical model for front propagation with a nonlocal growth law in any space dimension. Such a problem arises in composite fabrication using the vapor infiltration process and in other physical problems involving transport and reaction. Our model, based on the level set equation coupled with a boundary value problem of the Laplace equation, is an Eulerian formulation, which allows robust treatment for topological changes such as merging and formation of pores without artificially tracking them. When applied to the fabrication of continuous filament ceramic matrix composites using chemical vapor infiltration, this model accurately predicts not only residual porosity but also the precise locations and shapes of all pores.
The evolution of material microstructure during processing or reaction is a general problem of interest in many materials technologies. Mathematical modeling of this evolution has a long history. Many analyses are based on the early work of Kolmogorov,6 Johnson and Mehl,5 and Avrami1 for growth of discrete nuclei of one phase in another continuous phase. This solution assumes a random distribution of nuclei and a continuous, constant growth rate until transformation is complete and results in an analytical expression for the fraction transformed versus time. This model has been applied to metal solidification, polymer crystallization, and second phase precipitation with various extensions and elaborations. Fundamental features of these models are (i) local control of the growth rate, i.e., constant or dependent only on local characteristics such as curvature, and (ii) ideally random distribution of initial nuclei (size and location). A variation on this problem arises where the growth rate of the discrete phase depends on a nonlocal phenomenon such as diffusion of a reacting species to the growing surface from a remote source. Such problems often involve gas–solid reactions where a particulate solid phase grows by reaction with a mobile, gaseous reactant
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e-mail: [email protected] e-mail: [email protected] c) e-mail: [email protected] b)
J. Mater. Res., Vol. 14, No. 10, Oct 1999
that diffuses from the surroundings. In this case, the growth rate at a particular location depends on nonlocal transport. In one limiting case,
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