Exploration of non-trivial relations for the non-steady state nucleation rate: usefulness of the elliptic theta functions for its experimental estimations

Abstract

Abstract It is known that the non-steady state nucleation rate can be expressed as the product of the steady state nucleation rate and an elliptic theta function. However, there has been no research where rich mathematical properties of the elliptic theta functions were utilized to analyze the non-steady state nucleation processes. In this paper, we achieved two objectives by using the properties. The first one is to derive non-trivial relations for the non-steady state nucleation rate by using the rich mathematical properties of the elliptic theta functions, which could not be discovered in the conventional classical nucleation theory because of its complexities. The second one is to find solutions to two problems by solving a difference equation for the non-steady state nucleation rate: (i) it requires a large amount of effort and cost to estimate the time evolution of the non-steady state nucleation rate under some special conditions, and (ii) it is impossible to measure the non-steady state nucleation rate under some special cases. It is shown that our result help reduce the estimation cost of the non-steady state nucleation rate and makes mechanical estimation of the non-steady state nucleation rate possible from the past to the future. Article highlights Non-trivial relations for the non-steady state nucleation rate are found from the mathematical properties of the elliptic theta functions. It is shown that our result help reduce the estimation cost of the non-steady state nucleation rate. It is shown that our result makes mechanical estimation of the non-steady state nucleation rate possible from the past to the future.

Keywords

• Classical nucleation theory
• Non-steady state nucleation rate
• Elliptic theta function
• Difference equation