[Paper Review] Effects of stochastic nucleation in the first order phase transition

Victor Kurasov|arXiv (Cornell University)|Jul 1, 2002

nanoparticles nucleation surface interactions1 references3 citations

TL;DR

This paper re-evaluates stochastic nucleation effects in first-order phase transitions by demonstrating that fluctuations from all droplets—not just early-formed 'main consumers'—must be considered for accuracy. Using an algebraic approach and numerical validation, it shows stochastic corrections are negligible in macroscopic systems due to the universality of averaged nucleation kinetics, invalidating prior models that overemphasized early droplets. The key contribution is a corrected framework for stochastic nucleation based on similarity of nucleation conditions across time intervals.

ABSTRACT

The effects of stochastic apppearence of embryos of a new phase are analyzed analytically. A new approach by the similarity of nucleation conditions is proposed. Corrections for a number of droplets are estimated. A comparison with numerical simulation is given. A good coincidence between theoretical and numerical results can be seen.

Motivation & Objective

• To resolve contradictions between two prior approaches ([1], [2]) on stochastic nucleation effects in first-order phase transitions.
• To justify the use of averaged nucleation kinetics by rigorously quantifying the smallness of stochastic corrections in macroscopic systems.
• To identify the true source of stochastic effects, showing it is not just early-formed droplets but all droplets during the nucleation period.
• To establish a physically consistent definition of the system volume relevant to nucleation, based on diffusion-induced perturbation ranges.
• To develop a corrected analytical and numerical framework for stochastic nucleation that avoids unjustified assumptions from prior models.

Proposed method

• Uses an algebraic approach to demonstrate that stochastic effects scale as (∆N)⁻¹/², confirming their smallness in macroscopic systems.
• Applies the concept of similarity in nucleation conditions—both locally and integrally—across the nucleation period to model collective droplet influence.
• Defines a characteristic volume V₁ = 4π(4Dt₁)³/²/3 based on diffusion range from droplet formation, to determine the effective system size for nucleation.
• Validates analytical results via Monte Carlo simulations with 1000 realizations per parameter set, varying initial supersaturation u₀ from 0 to 1.
• Compares results from averaged kinetics with stochastic corrections, confirming agreement in dispersion and mean shifts.
• Analyzes dynamic conditions and stabilized supersaturation regimes to assess robustness of small stochastic effects across different kinetic scenarios.

Experimental results

Research questions

• RQ1 Why do prior models ([1], [2]) yield numerically close but physically unjustified results despite contradictory assumptions?
• RQ2 What is the true source of stochastic nucleation effects—early droplets or all droplets formed during the nucleation period?
• RQ3 How can the system volume be consistently defined for nucleation kinetics when droplet-induced perturbations vary spatially and temporally?
• RQ4 To what extent are stochastic corrections to the total number of droplets negligible in macroscopic systems?
• RQ5 Can the universality of averaged nucleation kinetics be preserved when accounting for stochastic effects from all droplets?

Key findings

• Stochastic corrections to the total number of droplets are negligible in macroscopic systems, scaling as (∆N)⁻¹/², confirming the validity of averaged nucleation kinetics.
• Numerical simulations confirm analytical predictions: the shift in droplet number is small, and dispersion γ matches theoretical estimates across all values of ln N^(∞).
• The assumption in [1] and [2] that early droplets dominate stochastic effects is incorrect; all droplets contribute significantly to the fluctuation spectrum.
• The effective system volume for nucleation is not the macroscopic container but V₁ = 4π(4Dt₁)³/²/3, based on diffusion range from nucleation.
• The property of similarity in nucleation conditions—both locally and integrally—enables consistent modeling of collective droplet influence across time.
• Even in dynamic conditions with time-dependent supersaturation, stochastic effects remain small due to weak dependence of averaged kinetics on microscopic fluctuations.