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

Abstract

Epitaxial growth methods are a key technology used in producing large-area thin films on substrates but as a result of various factors controlling growth processes the rational optimization of growth conditions is rather difficult. Mathematical modeling is one approach used in studying the effects of controlling factors on domain growth. The present study is motivated by a recently found scaling relation between the domain radius and time for chemical vapor deposition of graphene. Mathematically, we need to solve the Stefan problem; when the boundary moves, its position should be determined separately from the boundary conditions needed to obtain the spatial profile of diffusing adsorbates. We derive a closed equation for the growth rate constant defined as the domain area divided by the time duration. We obtain approximate analytical expressions for the growth rate; the growth rate constant is expressed as a function of the two-dimensional diffusion constant and the rate constant for the attachment of adsorbates to the solid domain. In experiments, the area is decreased by stopping the source gas flow. The rate of decrease of the area is obtained from theory. The theoretical results presented provide a foundation to study controlling factors for domain growth.

Domain Size Regulation in Phospholipid Model Membranes Using Oil Molecules and Hybrid Lipids

“…It is important to note that in phase transition processes near critical phenomena the exponent of the power law describing the time evolution of the average domain size depends only on the dominating mechanism, i.e., diffusion, and on the dimensionality of the system, but it does not depend on the microscopic details . Within the experimental precision (±20%), our experimental data for hexadecane–squalene mixtures are all compatible with a slope 1 (denoting a purely diffusive 2D growth), despite there certainly being some residual oil in the membrane. This is particularly remarkable, given the branched structure of squalene and its size.…”

Transition from Reaction- to Diffusion-Limited Growth of Graphene by Chemical Vapor Deposition

“…Here, we present theoretical results on single domain growth by ignoring the influence from the other domains. 23,24 The radial axis is introduced, where the origin is set at the center of the domain. We ignore shape changes during the domain growth and assume an axisymmetric shape of the domain.…”

“…This is the nonlinear equation for determining α; the domain growth law can be obtained by substituting α into eq 5 and by solving the resultant equation for R. The resultant equation was numerically solved for κ i → ∞. 23 Kummer's confluent hypergeometric function of the second kind can be approximated as 34…”

“…We can show that eq 17 can be also derived from the expression of the radial velocity in terms of the modified Bessel functions in the BCF model. 24,33 The relation R(t) 2 ∝ Dt in the diffusion-controlled limit naturally arises in considering two-dimensional growth of isotropic domain by diffusion, 23,24,32 while the relation R(t) ∝ κ i t has been well-recognized for reaction-controlled growth. 24 ■ BACK-STRESS EFFECT: INTERFERENCE OF…”

“…21,22 If an isolated domain far away from the other domains is selected to measure the domain size growth rate, the back-stress effect could be minimized. More recently, it is theoretically shown that the growth can be intrinsically suppressed even for a single domain growth by increasing the periphery length; 23,24 the initial (incorporation) reaction-limited domain growth transforms into diffusion-limited growth as the domain size increases.…”