Family-Vicsek scaling in classical nucleation theory.

Area of Science:

• Surface science
• Materials science
• Physical chemistry

Background:

• Classical nucleation theory (CNT) in open systems yields double-exponential island-size distributions (ISDs).
• Family-Vicsek scaling functions describe irreversible island growth using scaled natural size.
• A gap exists in understanding the relationship between these two theoretical frameworks.

Purpose of the Study:

• To integrate classical nucleation theory with Family-Vicsek scaling.
• To demonstrate that ISDs in CNT satisfy Family-Vicsek scaling under specific conditions.
• To explore the impact of capture coefficients on island-size distribution scaling.

Main Methods:

• Utilizing the framework of classical nucleation theory for open systems.
• Analyzing island-size distributions with power-law size dependence of attachment rates.
• Converting ISDs to scaled natural size (x=s/〈s〉) for comparison with scaling functions.
• Investigating size-linear and power-law capture coefficients (σ(s)).

Main Results:

• Island-size distributions in CNT, when scaled, satisfy the Family-Vicsek scaling hypothesis in the large time limit.
• For size-linear capture coefficients, analytic scaling functions are monomodal and approach zero at x=0.
• For power-law capture coefficients (α<1), the ISD transforms into a delta function at x=1 for large islands.

Conclusions:

• A significant relationship exists between ISDs in different variables within CNT.
• Confirms the validity of Family-Vicsek scaling within the context of classical nucleation theory.
• Provides valuable insights for modeling the growth kinetics and statistical properties of nano-object ensembles.