Editing Flash freezing (section)

=== Nucleation ===
{{Main article|Classical nucleation theory}}

Crystal growth or nucleation is the formation of a new [[Thermodynamics|thermodynamic]] phase or a new structure via self-assembly. Nucleation is often found to be very sensitive to impurities in the system. For nucleation of a new thermodynamic phase, such as the formation of ice in water below {{convert|0|C|}}, if the system is not evolving with time and nucleation occurs in one step, then the probability that nucleation has not occurred should undergo [[exponential decay]]. This can also be observed in the nucleation of ice in supercooled small water droplets.<ref>{{Cite book|title=Laboratory evidence for volume-dominated nucleation of ice in supercooled water microdroplets|last=Duft|first=D|publisher=Atmospheric Chemistry and Physics|year=2004}}</ref> The decay rate of the exponential gives the nucleation rate and is given by
:<math>R\ =\ N_S Zj\exp \left( \frac{-\Delta G^*}{k_BT} \right)</math>
where  
* <math>N_S</math> is the number of nucleation sites;
* <math>Z</math> is the probability that a nucleus at the top of the barrier will go on to form the new phase, not dissolve (called the Zeldovich factor);
* <math>j</math> is the rate at which molecules attach to the nucleus, causing it to grow;
* <math>\Delta G^* </math> is the free energy cost of the nucleus at the top of the nucleation barrier;
* <math>k_BT </math> is the [[thermal energy]], where <math>T</math> is the absolute temperature and <math>k_B</math> is the [[Boltzmann constant]].
[[File:Hethomnucdifference.JPG|thumb|Difference in energy barriers. Homogeneous nucleation (blue) has a higher nucleation barrier <math>\Delta G^* </math>at ''r<sub>c</sub>'' than heterogeneous nucleation (red).|255x255px]]
[[Classical nucleation theory]] is a widely used approximate theory for estimating these rates, and how they vary with variables such as temperature. It correctly predicts that the time needed for nucleation decreases extremely rapidly when supersaturated.<ref>{{Cite book|title=Microphysics of Clouds and Precipitation|last=Pruppacher. Klett|first=H.R., J.D.|publisher=Kluwer|year=1997}}</ref><ref>{{Cite book|title=Nucleation: theory and applications to protein solutions and colloidal suspensions|last=Sear|first=R.P.|publisher=Physics Cond. Matt.|year=2007}}</ref>

Nucleation can be divided into homogeneous nucleation and heterogeneous nucleation. Homogeneous nucleation is the rarer, but simpler, case. In homogeneous nucleation, classical nucleation theory assumes that for a microscopic, spherical nucleus of a new phase, the [[Gibbs free energy|free energy]] change of a droplet <math>\Delta G(r) </math> is a function of the size of the nucleus, and can be written as the sum of terms proportional to the nucleus' volume and surface area:
:<math>\Delta G={\frac{4}{3}}\pi r^{3}\Delta g+4\pi r^{2}\sigma </math>
The first term represents volume, and (assuming a spherical nucleus) this is the volume of a sphere of radius <math>r</math>. Here, <math>\Delta g</math> is the difference in free energy per unit volume between the thermodynamic phase in which nucleation is occurring, and the phase that is nucleating. The second term represents the surface area, again assuming a sphere, where <math>\sigma</math> is the [[surface tension]].

At some intermediate value of <math>r</math>, the free energy <math>\Delta G </math> goes through a maximum, and so the probability of formation of a nucleus goes through a minimum. This occurs when <math>\frac{dG}{dr}=0 </math>. This point, <math>\Delta G^* </math>, is called the ''critical nucleus'' and represents the ''nucleation barrier''; it occurs at the critical radius
:<math>r_c=-{\frac{2\sigma}{\Delta g}}</math>
The addition of new molecules to nuclei larger than this critical radius decreases the free energy, so these nuclei are more probable.

Heterogeneous nucleation occurs at a surface or impurity. In this case, part of the nucleus boundary is accommodated by the surface or impurity onto which it is nucleating. This reduces the surface area term in <math>\Delta G </math>, and thus lowers the nucleation barrier <math>\Delta G^* </math>. This lowered barrier is what makes heterogeneous nucleation much more common and faster than homogeneous nucleation.<ref>{{cite journal |last1=Liu |first1=X. Y. |date=31 May 2000 |title=Heterogeneous nucleation or homogeneous nucleation? |journal=The Journal of Chemical Physics |volume=112 |issue=22 |pages=9949–9955 |bibcode=2000JChPh.112.9949L |doi=10.1063/1.481644 |issn=0021-9606}}</ref>