Transient kinetics of nucleation

Transient nucleation driven by solvent evaporation

We theoretically investigate homogeneous crystal nucleation in a solution containing a solute and a volatile solvent. The solvent evaporates from the solution, thereby continuously increasing the concentration of the solute. We view it as an idealized model for the far-out-of-equilibrium conditions present during the liquid-state manufacturing of organic electronic devices. Our model is based on classical nucleation theory, taking the solvent to be a source of the transient conditions in which the solute drops out of the solution. Other than that, the solvent is not directly involved in the nucleation process itself. We approximately solve the kinetic master equations using a combination of Laplace transforms and singular perturbation theory, providing an analytical expression for the nucleation flux. Our results predict that (i) the nucleation flux lags slightly behind a commonly used quasi-steady-state approximation. This effect is governed by two counteracting effects originating from solvent evaporation: while a faster evaporation rate results in an increasingly larger influence of the lag time on the nucleation flux, this lag time itself is found to decrease with increasing evaporation rate. Moreover, we find that (ii) the nucleation flux and the quasi-steady-state nucleation flux are never identical, except trivially in the stationary limit, and (iii) the initial induction period of the nucleation flux, which we characterize as a generalized induction time, decreases weakly with the evaporation rate. This indicates that the relevant time scale for nucleation also decreases with an increasing evaporation rate. Our analytical theory compares favorably with results from a numerical evaluation of the governing kinetic equations.

Classical nucleation theory in the phase-field crystal model

A full understanding of polycrystalline materials requires studying the process of nucleation, a thermally activated phase transition that typically occurs at atomistic scales. The numerical modeling of this process is problematic for traditional numerical techniques: commonly used phase-field methods' resolution does not extend to the atomic scales at which nucleation takes places, while atomistic methods such as molecular dynamics are incapable of scaling to the mesoscale regime where late-stage growth and structure formation takes place following earlier nucleation. Consequently, it is of interest to examine nucleation in the more recently proposed phase-field crystal (PFC) model, which attempts to bridge the atomic and mesoscale regimes in microstructure simulations. In this work, we numerically calculate homogeneous liquid-to-solid nucleation rates and incubation times in the simplest version of the PFC model, for various parameter choices. We show that the model naturally exhibits qualitative agreement with the predictions of classical nucleation theory (CNT) despite a lack of some explicit atomistic features presumed in CNT. We also examine the early appearance of lattice structure in nucleating grains, finding disagreement with some basic assumptions of CNT. We then argue that a quantitatively correct nucleation theory for the PFC model would require extending CNT to a multivariable theory.

Theoretical model of ice nucleation induced by inertial acoustic cavitation. Part 2: Number of ice nuclei generated by a single bubble

In the preceding paper (part 1), the pressure and temperature fields close to a bubble undergoing inertial acoustic cavitation were presented. It was shown that extremely high liquid water pressures but quite moderate temperatures were attained near the bubble wall just after the collapse providing the necessary conditions for ice nucleation. In this paper (part 2), the nucleation rate and the nuclei number generated by a single collapsing bubble were determined. The calculations were performed for different driving acoustic pressures, liquid ambient temperatures and bubble initial radius. An optimal acoustic pressure range and a nucleation temperature threshold as function of bubble radius were determined. The capability of moderate power ultrasound to trigger ice nucleation at low undercooling level and for a wide distribution of bubble sizes has thus been assessed on the theoretical ground.

Comparing crystal-melt interfacial free energies through homogeneous nucleation rates

In this work, we compared several available crystal-melt interfacial free energies via homogeneous nucleation rates in a pure Lennard-Jones model system using both model fitting and numerical methods. We examined the homogeneous nucleation temperature obtained from the classical nucleation theory using the available interfacial free energies from three different methods as inputs, i.e. the free energy integration method, the interface fluctuation method and the classical nucleation theory based method. We found that the critical temperature obtained by using the interfacial free energy calculated recently (Bai and Li 2006 J. Chem. Phys. 124 124707) is in better agreement with that obtained from spontaneous crystallization in an independent molecular dynamics simulation. The discrepancies among the interface energies are discussed in light of these results.

Quantitative parameter-free prediction of simulated crystal-nucleation times

We present direct comparisons between simulated crystal-nucleation times and theoretical predictions using a model of aluminum, and demonstrate that a quantitative prediction can be made. All relevant thermodynamic properties of the system are known, making the agreement of our simulation data with nucleation theories free of any adjustable parameters. The role of transient nucleation is included in the classical nucleation theory approach, and shown to be necessary to understand the observed nucleation times. The calculations provide an explanation on why nucleation is difficult to observe in simulations at moderate undercoolings. Even when the simulations are significantly larger than the critical nucleus, and when simulation times are sufficiently long, at moderate undercoolings the small concentration of critical nuclei makes the probability of the nucleation low in molecular dynamics simulations.

Igniting homogeneous nucleation

Transient homogeneous nucleation is studied in the limit of large critical sizes. Starting from pure monomers, three eras of transient nucleation are characterized in the classic Becker-Döring kinetic equations with two different models of discrete diffusivity: the classic Turnbull-Fisher formula and an expression describing thermally driven growth of the nucleus. The latter diffusivity yields time lags for nucleation which are much closer to values measured in experiments with disilicate glasses. After an initial stage in which the number of monomers decreases, many clusters of small size are produced and a continuous size distribution is created. During the second era, nucleii are increasing steadily in size in such a way that their distribution appears as a wave front advancing towards the critical size for steady nucleation. The nucleation rate at critical size is negligible during this era. After the wave front reaches critical size, it ignites the creation of supercritical clusters at a rate that increases monotonically until its steady value is reached. Analytical formulas for the transient nucleation rate and the time lag are obtained that improve classical ones and compare very well with direct numerical solutions.

Nucleation rates and induction times during colloidal crystallization: links between models and experiments

A kinetic model for the evolution of the cluster size distribution during crystal nucleation and growth is presented. The model allows one to establish precise links between model parameters and experimental measures of nucleation kinetics. This approach demonstrates the significance of several processes not accounted for in classical nucleation theories. Chief among these is that the driving force for crystal nucleation decreases rapidly due to a reduction of the background monomer concentration as crystallization progresses, resulting in a reduction of nucleation rates. This, coupled with the disparities in the definitions of measured and predicted quantities, leads to significant discrepancies between predictions of extant models and experimental estimates of nucleation rates. Accounting for these effects, calculations of the kinetic model are shown to be in good agreement with experimental estimates of nucleation rates, crystal growth velocities, and induction times during the crystallization of hard sphere colloidal suspensions.

Analytical approach to time lag in binary nucleation

We present an analytical formula for the time required to establish steady state in a nucleating binary system. To test our solution, we evaluate the time lag for a range of activities of both components at the vapor-liquid transition, and show that our result is in much better agreement with a purely numerical simulation than other available analytical formulas, which overestimate the time lag by factors of from 2 to 200.