[Paper Review] General trends of the late period of evolution in the quasichemical model of nucleation

Victor Kurasov|arXiv (Cornell University)|Jul 28, 2006

nanoparticles nucleation surface interactions3 citations

TL;DR

This paper develops an analytical framework for the late-stage evolution of nucleation in the quasichemical model, focusing on post-nucleation growth and diffusion-driven spectral erosion. It models droplet size distribution dynamics through regular growth, relaxation, and diffusion processes, showing that the spectrum collapses rapidly due to diffusion and that the system evolves through self-sustaining loops of spectral decay and reorganization, with key transitions governed by algebraic and diffusion equations.

ABSTRACT

The periods after the end of the "primary" nucleation are considered. The approximate analytical description is given. The process is split into several periods which form the loop of evolution.

Motivation & Objective

• To describe the evolution of droplet size distributions after primary nucleation ends in the quasichemical model of nucleation.
• To model the transition from regular growth to diffusion-dominated spectral erosion in the late period of nucleation.
• To analyze the hierarchical structure of the size spectrum and its temporal evolution using analytical approximations.
• To identify conditions under which the size distribution collapses and reorganizes into new quasi-stationary states.
• To close the loop of nucleation evolution by incorporating diffusion effects and re-initiating the process in a self-consistent manner.

Proposed method

• Uses a natural size coordinate $ \rho = \nu^{1/3} $, where $ \nu $ is the number of molecules in a droplet, to simplify growth dynamics.
• Applies a first-order differential equation for $ z $, the size coordinate of the droplet spectrum, based on vapor consumption and momentums of the distribution function.
• Introduces a regular relaxation model governed by $ \tau \frac{dz}{dt} = \Phi_* - \sum_i \mu_i^\infty z^{3-i} C_3^i $, with $ \mu_i^\infty $ as full momentums of the distribution.
• Models diffusion erosion using a Fokker-Planck-type equation in $ \nu $-space, transformed into $ S = \nu^{2/3} $-space for analytical tractability.
• Uses Green’s function solutions for the diffusion process, with initial conditions approximated as $ \delta $-functions or flat tails.
• Closes the loop by reapplying the same spectral evolution procedure after diffusion erases the initial form, leading to repeated cycles of spectral decay and reorganization.

Experimental results

Research questions

• RQ1 How does the droplet size distribution evolve after the end of primary nucleation in the quasichemical model?
• RQ2 What determines the characteristic timescale for the collapse of the size spectrum due to diffusion?
• RQ3 How can the transition from regular growth to diffusion-dominated erosion be modeled analytically?
• RQ4 What conditions close the loop of nucleation evolution, and how are successive cycles of spectral reorganization initiated?
• RQ5 How does the number of droplets change across successive loops, and what limits the observability of these cycles?

Key findings

• The size spectrum collapses rapidly due to diffusion, with the half-width of the spectrum in $ S $-space growing as $ \Delta_2 S \simeq 2 \Delta_{1}x z $, leading to loss of initial form on a timescale $ t_2 \sim (\Delta_2 S)^2 / (4D) $.
• The end of the regular relaxation phase is marked when $ z \approx (2\div3)\rho_c $, corresponding to a critical condition on the supersaturation and droplet size.
• The diffusion process leads to a Gaussian-like tail over time, especially when $ t \gg S_k^2 / (4D) $, indicating a universal late-stage behavior.
• The number of droplets decreases significantly over each loop, with $ N_{\text{ENR}} \sim N_{\text{total}} - N_{\text{ENR}} $ marking the end of a cycle, implying exponential decay in droplet count across loops.
• The system evolves through self-sustaining loops of spectral decay and reorganization, with each loop governed by a closed differential equation involving $ z_c(z) $, the critical size as a function of droplet size.
• The model shows that even with constant tail height, the system can be solved analytically via integration of $ t = \tau \int \frac{dz}{(2a)/(3(z - x_c(z)))} $, demonstrating the feasibility of closed-form solutions under simplifying assumptions.