National Research Centre "Kurchatov Institute", Kurchatov Sq. 1, 123182 Moscow, Russia.
National Science Center "Kharkov Institute of Physics and Technology", Akademicheskaya St. 1, 61108 Kharkov, Ukraine.
Phys Rev E. 2017 Jan;95(1-1):012801. doi: 10.1103/PhysRevE.95.012801. Epub 2017 Jan 6.
The nucleation of islands in a supersaturated solution of surface adatoms is considered taking into account the possibility of diffusion profile formation in the island vicinity. It is shown that the treatment of diffusion-controlled cluster growth in terms of the Fokker-Planck equation is justified only provided certain restrictions are satisfied. First of all, the standard requirement that diffusion profiles of adatoms quickly adjust themselves to the actual island sizes (adiabatic principle) can be realized only for sufficiently high island concentration. The adiabatic principle is essential for the probabilities of adatom attachment to and detachment from island edges to be independent of the adatom diffusion profile establishment kinetics, justifying the island nucleation treatment as the Markovian stochastic process. Second, it is shown that the commonly used definition of the "diffusion" coefficient in the Fokker-Planck equation in terms of adatom attachment and detachment rates is justified only provided the attachment and detachment are statistically independent, which is generally not the case for the diffusion-limited growth of islands. We suggest a particular way to define the attachment and detachment rates that allows us to satisfy this requirement as well. When applied to the problem of surface island nucleation, our treatment predicts the steady-state nucleation barrier, which coincides with the conventional thermodynamic expression, even though no thermodynamic equilibrium is assumed and the adatom diffusion is treated explicitly. The effect of adatom diffusional profiles on the nucleation rate preexponential factor is also discussed. Monte Carlo simulation is employed to analyze the applicability domain of the Fokker-Planck equation and the diffusion effect beyond it. It is demonstrated that a diffusional cloud is slowing down the nucleation process for a given monomer interaction with the nucleus edge.
考虑到在岛附近形成扩散分布的可能性，研究了过饱和表面原子吸附物溶液中岛的成核。结果表明，只有在满足一定限制的情况下，用福克-普朗克方程处理扩散控制的团簇生长才是合理的。首先，原子吸附物扩散分布能够快速适应实际岛尺寸的标准要求（绝热原理），仅当岛的浓度足够高时才能实现。对于原子吸附物在岛边缘的附着和脱附概率与原子吸附物扩散分布建立动力学无关，绝热原理对于岛成核处理为马尔可夫随机过程是至关重要的。其次，结果表明，在福克-普朗克方程中，通常使用原子吸附物附着和脱附速率来定义“扩散”系数的方法，仅当附着和脱附在统计上是独立的情况下才是合理的，而对于岛的扩散限制生长，这种情况通常并不成立。我们提出了一种特定的方法来定义附着和脱附速率，使其能够满足这一要求。当应用于表面岛成核问题时，我们的处理方法预测了稳态成核势垒，该势垒与传统的热力学表达式一致，尽管没有假设热力学平衡，并且明确地处理了原子吸附物的扩散。还讨论了原子吸附物扩散分布对成核率指数前因子的影响。采用蒙特卡罗模拟来分析福克-普朗克方程及其以外的扩散效应的适用范围。结果表明，对于给定的单体与核边缘的相互作用，扩散云会减缓成核过程。
