Crystallization --> Nucleation

### Nucleation from own melt.

by Leonid Sakharov

A phenomena of spontaneous appearance of crystal phase inside of liquid and then self sustained growth up to the point of complete transformation of liquid into crystal is the very commonly observed for producing ice in refrigerator and most paradoxical from point of clear demonstration of limitations of direct application of thermodynamic laws in case when macro effects are seed out from micro scale incident when statistical fluctuations play major role.

The paradox of nucleation comes from the fact that there is no way to construct a thermodynamically allowed path of transformation some part of area in melt from zero volume to big enough to allow self sustained growth. A change of Gibbs thermodynamic potential for sphere shape crystal inside of melt of the same content can be calculated by formula:

ΔG = (4/3)×r3×ΔGv + 4× r2×σ                         (1),

where r - radius of nucleus (the presumption of the sphere shape of nucleus is based on the fact that this form provides minimum addition of surface for three dimensional form) ΔGv - specific change of free energy per volume, σ - surface energy.

The dependence of change of free energy has maximum on the curve of displaying dependence from size of nucleus as seen on the following picture:

Change of free Gibbs energy depend on radius of nucleus for following parameters:  ΔHv = -2×10-28 J/nm3, temperature of liquidus Tl=1500o C, temperature T=1475o C, surface energy σ = 2×10-20 J/nm2.

As seen from the chart for size of sphere of nucleus less than rcr (critical radius of nucleus) a change of free energy with increasing its size is positive that thermodynamically prohibited. If molecules randomly gathered into nucleolus with size above critical they are able for stable expansion. To find value of critical nucleus the value of deviation of ΔG by radius must be zero:

 d ΔG = 4dπ×r2×ΔGv + 8πd×r×σ = 0 (2) d r

resolving equation (2) toward r an analytical expression can be found:

 rcr = - 2×σ (3) ΔGv

and change of free energy to form critical nucleus is:

 ΔGcr = 16π×σ3 (4) 3×ΔGv2

Important to point out that ΔGv ( specific change of free energy per volume) has negative sign for temperatures below of temperature of equilibrium that makes critical radius positive and energy of formation of critical nucleus is positive as well due the fact that square of negative value is positive. So there are no thermodynamically allowed patch for transformation from melt into crystalline phase. But experimentally we are seeing such phenomenon. There are some discussion about purity of melt where spontaneous nucleation had happened with the suspicion that actually the formation of initial crystal seed are happen on hard alien particles that can diminish barrier for formation of critical nuclei. But even if it could takes place it will not change the fact of necessity to dodge of prohibition of thermodynamic for system to increase potential of Gibbs free energy.

The paradox can be resolved by understanding that laws of thermodynamics are applied for statistically big systems as a whole working with an average values characterizing behavior of molecules in the system. If one could isolate small areas in big system and calculate average values the distribution for them will be found. Depend on size of such subsystems deviations of parameters from average for the whole systems will be the large the smaller subsystems will be. Very be deviations are called as fluctuations. Particularly for our case concerning to the crystallization in the melt every small area in melt can be characterized as having structure of crystal or not. As a result of chaotic movements of atoms there are continuous reversible transformation into and out of crystal structure. If by chance many enough areas with crystal structure are happened to be in direct touch to each other inside a region bigger than radius of critical nucleus it will able to growth stable in accordance with thermodynamics.

Average number of molecules to produce one such fluctuation at one moment of the time can be calculated by formula:

 N = e( ΔGcr ) (5) kT

where k - Boltzmann's constant  (1.380 6504(24) × 10−23 J K−1 ), T - absolute (Kelvin scale) temperature.

The frequency of appearance one over-critical nucleus in volume of melt can be calculated using the parameter of an average volume of one molecule vm and frequency the molecule can change the status from melt to crystal that can be taken from the thermal activated growth model:

 In = (1/vm)× e(-ΔGcr/kT)×γo × e(-Ea in/kT) (6) ,

where γo - own frequency of thermal vibration of molecule, Ea - energy of activation barrier to regroup atoms in area represented one molecule from melt to crystal phase.

Close analogues to the model of nucleation presented above is a lottery drawing when number of tickets for guarantee win is represented by formula 5 and frequency of drawing is defined as frequency of molecules to jump back and forth over activation barrier. The idea is that if one will buy the ticket for every drawing sometimes in distant future the investment will pay off. The interesting pith concerning rate of nucleation is that both components of formula 6 can vary their values in extraordinary broad diapason of values explaining variety of experimentally observed behavior of different materials from almost instant crystalline crystallization for metals till practical absence of any noticeable crystal formation for centuries for glassware.

The principal limitation of the presented model is the contradiction between the premise of established uniformity of temperature as well of distribution of crystalline clusters with size under critical.  At practice such conditions can be reached, at least with reasonable adjacency, only for specific situations such for example in case of very small liquid drops these have small thermo inertia that permits to achieve thermal uniformity in very short time. Other complication is related to the fact that as soon first overcritical nucleus starts to expand into neighborhood conditions for formation others are changing. The most direct effect of growing crystal on nucleation rate will be diminishing a volume of melt available for formation next overcritical nuclei.

More detail discussion about experimental research and advanced theoretical models of nucleation are presented in following articles.

Submitted at Jun. 11, 2010; 18:22