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Journal Article

Dynamic scaling and stochastic fractal in nucleation and growth processes


Lahiri,  Amit
External Organizations;

Kamrul Hassan,  Md.
External Organizations;

Blasius,  Bernd
External Organizations;


Kurths,  Jürgen
Potsdam Institute for Climate Impact Research;

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Lahiri, A., Kamrul Hassan, M., Blasius, B., Kurths, J. (2022): Dynamic scaling and stochastic fractal in nucleation and growth processes. - Chaos, 32, 9, 093124.

Cite as: https://publications.pik-potsdam.de/pubman/item/item_27944
A class of nucleation and growth models of a stable phase is investigated for various different growth velocities. It is shown that for growth velocities v≈s(t)/t and v≈x/τ(x), where s(t) and τ are the mean domain size of the metastable phase (M-phase) and the mean nucleation time, respectively, the M-phase decays following a power law. Furthermore, snapshots at different time t that are taken to collect data for the distribution function c(x,t) of the domain size x of the M-phase are found to obey dynamic scaling. Using the idea of data-collapse, we show that each snapshot is a self-similar fractal. However, for v=const., such as in the classical Kolmogorov–Johnson–Mehl–Avrami model, and for v≈1/t , the decays of the M-phase are exponential and they are not accompanied by dynamic scaling. We find a perfect agreement between numerical simulation and analytical results. A class of nucleation and growth process is studied analytically by solving an integro-partial differential equation and verified numerically by Monte Carlo simulation. The growth velocity is defined as the ratio of the distance traveled s by the new phase, and the magnitude of time t needed to travel that distance. We first choose constant growth velocity by assuming both s and t equal to constant and reproduce the results of the much studied classical Kolmogorov–Johnson–Mehl–Avrami (KJMA) model. Choosing one of them a constant still makes the exponential decay of the meta-stable phase, such as the KJMA model. However, the growth velocity is so chosen that neither s nor t is constant, and we find that the meta-stable phase decays following a power law. Such a power law is also accompanied by the emergence of fractal. The self-similar property of fractal is verified by showing that the system exhibits dynamic scaling revealing a self-similar symmetry along the continuous time axis. According to Noether’s theorem, there must exist a conserved quantity, and we do find that the dfth moment, where df is the fractal dimension, is always conserved.