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Epitaxial growth methods are a key technology used in producing large-areathin films on substrates but as a result of various factors controlling growthprocesses the rational optimization of growth conditions is rather difficult.Mathematical modeling is one approach used in studying the effects ofcontrolling factors on domain growth. The present study is motivated by arecently found scaling relation between the domain radius and time for chemicalvapor deposition of graphene. Mathematically, we need to solve the Stefanproblem; when the boundary moves, its position should be determined separatelyfrom the boundary conditions needed to obtain the spatial profile of diffusingadsorbates. We derive a closed equation for the growth rate constant defined asthe domain area divided by the time duration. We obtain approximate analyticalexpressions for the growth rate; the growth rate constant is expressed as afunction of the two-dimensional diffusion constant and the rate constant forthe attachment of adsorbates to the solid domain. In experiments, the area isdecreased by stopping the source gas flow. The rate of decrease of the area isobtained from theory. The theoretical results presented provide a foundation tostudy controlling factors for domain growth.

Scaling theory for two-dimensional single domain growth driven by attachment of diffusing adsorbates