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classical nucleation theory now in separate page, as nucleation is more general so two are logically distinct
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{{verbosity|date=October 2014}}
[[File:Rock-Candy-Closeup.jpg|thumb|right|When sugar is [[supersaturation|supersaturated]] in water, nucleation will occur, allowing sugar molecules to stick together and form large crystal structures.]]
'''Nucleation''' is the first step in the formation of either a new thermodynamic phase or a new structure via [[self-assembly]] or [[self-organisation]]. Nucleation is typically defined to be the process that determines how long we have to wait before the new phase or self-organised structure, appears. Nucleation is often found to be very sensitive to impurities in the system. Because of this, it is often important to distinguish between heterogenous nucleation and homogeneous nucleation. Heterogeneous nucleation occurs at ''nucleation sites'' on surfaces in the system.<ref name=pruppacher_book/> Homogenous nucleation occurs away from a surface.
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[[File:Nucleation of the equilibrium phase (red) from a metastable phase (white) in the Ising model.ogg|thumb|Nucleation at a surface (black) in the 2D Ising model. Up spins (particles in lattice-gas terminology) shown in red, down spins shown in white.]]
 
Nucleation is usually a [[stochastic]] process, i.e., there is a random element to it and so even in two identical systems nucleation will occur at different times. This behaviour is similar to [[radioactive decay]]. A common mechanism is illustrated in the animation to the right. This shows nucleation of a new phase (shown in red) in an existing phase (white). In the existing phase microscopic fluctuations of the red phase appear and decay continuously, until an unusually large fluctuation of the new red phase is so large it is more favourable for it to grow than to shrink back to nothing. This nucleus of the red phase then grows and converts the system to this phase. The standard theory that describes this behaviour for the nucleation of a new thermodynamic phase is called classical nucleation theory.<ref name=pruppacher_book/><ref name=sear_review/>
 
For nucleation of a new thermodynamic phase, such as a the formation of ice in water below 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]] as seen in [[radioactive decay]]. This is seen for example in the nucleation of ice in [[supercooled]] small water droplets.<ref>{{cite journal|last=Duft|first=D.|author2=Leisner |title=Laboratory evidence for volume-dominated nucleation of ice in supercooled water microdroplets|journal=Atmospheric Chemistry & Physics|year=2004|volume=4|page=1997|doi=10.5194/acp-4-1997-2004|url=http://www.atmos-chem-phys.net/4/1997/2004/acp-4-1997-2004.html}}</ref> The decay rate of the exponential gives the nucleation rate.
 
It is not just new phases such as liquids and crystals that form via nucleation followed by growth. The self-assembly process that forms objects like the [[amyloid]] aggregates associated with [[Alzheimer's disease]] also starts with nucleation.<ref>{{cite journal|last=Gillam|first=J.E.|last2=MacPhee|first2=C.E. |title=Modelling amyloid fibril formation kinetics: mechanisms of nucleation and growth|journal=J. Phys. Condens. Matter|year=2013|page= 373101 |doi=10.1088/0953-8984/25/37/373101|url=http://iopscience.iop.org/0953-8984/25/37/373101}}</ref> Energy consuming self-organising systems such as the [[microtubules]] in cells also show [[microtubule nucleation|nucleation]] and growth.
 
==Examples==
* [[Clouds]] form when wet air cools (often because the air is rising) and many small water droplets nucleate from the supersaturated air.<ref name="pruppacher_book">H. R. Pruppacher and J. D. Klett, ''Microphysics of Clouds and Precipitation'', Kluwer (1997).</ref> The amount of water vapor that air can carry decreases with lower temperatures. The excess vapor begins to nucleate and form small water droplets which form a cloud. Nucleation of the droplets of liquid water is heterogeneous, occurring on particles referred to as [[cloud condensation nuclei]]. [[Cloud seeding]] is the process of adding artificial condensation nuclei to quicken the formation of clouds.
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* Nucleation in [[boiling]] can occur in the bulk [[liquid]] if the pressure is reduced so that the liquid becomes [[superheating|superheated]] with respect to the pressure-dependent boiling point. More often, nucleation occurs on the heating surface, at ''nucleation sites''. Typically, nucleation sites are tiny crevices where free gas-liquid surface is maintained or spots on the heating surface with lower [[wetting]] properties. Substantial superheating of a liquid can be achieved after the liquid is de-gassed and if the heating surfaces are clean, smooth and made of materials well wetted by the liquid.
* Nucleation is relevant in the process of crystallization of [[Nanoparticle|nanometer sized materials]],<ref>{{cite journal|arxiv=cond-mat/0702605|doi=10.1103/PhysRevLett.98.185503|title=Surface Nucleation in the Freezing of Gold Nanoparticles|year=2007|last1=Mendez-Villuendas|first1=Eduardo|last2=Bowles|first2=Richard|journal=Physical Review Letters|volume=98|issue=18|bibcode = 2007PhRvL..98r5503M }}</ref> and plays an important role in atmospheric processes.
* Nucleation is a key concept in [[polymer]],<ref>Young, R. J. (1981) ''Introduction to Polymers'' (CRC Press, NY) ISBN 0-412-22170-5.</ref> [[alloy]] and [[ceramic]] systems.
* Some [[champagne stirrer]]s operate by providing many nucleation sites via high surface area and sharp corners, speeding the release of bubbles and removing carbonation from the wine.
* [[Sodium acetate]] heating pads use cavitation voids caused by the deflection of a metal disk as nucleation centres for exothermic crystallization.
* [[Microtubule nucleation]] is the nucleation of [[microtubules]], large rod-like structures found in [[eukaryote]] cells. This nucleation is not of a thermodynamic phase, and like other aspects of microtubule dynamics, it is an energy consuming process.
 
== Classical nucleation theory ==
{{Main|Classical nucleation theory}}
 
This is the standard simple theory for nucleation of a new thermodynamic phase, such as a liquid or a crystal. It should be borne in mind that it is approximate. The basic classical nucleation theory for nucleation of a new phase provides an approximate but physically reasonable
prediction for the rate at which nuclei of a new phase form, via nucleation on a set of identical nucleation sites. This rate, ''R'' is the number of, for example, water droplets nucleating
in a uniform volume of air supersaturated with water vapour, per unit time. So if a 100 droplets
nucleate in a volume of 0.1m<sup>3</sup> in 1 s, then the rate ''R''=1000/s. The description here follows modern
standard classical nucleation theory.<ref name=sear_review/>
The prediction for the rate ''R'' is
 
::<math>R\ =\ N_S Zj\exp \left( \frac{-\Delta G^*}{k_BT} \right)</math>
 
where
:*<math>\Delta G^*</math> is the free energy cost of the nucleus at the top of the nucleation barrier, and ''k''<sub>B</sub>T is the thermal energy with ''T'' the absolute temperature and ''k''<sub>B</sub> is the [[Boltzmann constant]].
:*<math>N_S</math> is the number of nucleation sites.
:* ''j'' is the rate at which molecules attach to the nucleus causing it to grow.
:* ''Z'' is what is called the Zeldovich factor Z. Essentially the Zeldovich factor is the probability that a nucleus at the top of the barrier will go on to form the new phase, not dissolve.
 
This expression for the rate can be thought of as a product of two factors. The first is the
 
::average instantaneous number of fluctuations that at the top of the nucleation barrier <math> =N_S \exp \left( -\Delta G^*/k_BT \right)</math>
 
This is the
number of nucleation sites, <math>N_S</math>, multiplied by the probability
that a nucleus of critical size has grown around it, <math> \exp \left(-\Delta G^*/k_BT\right)</math>.
Free energies and probabilities are closely related in general, by definition <ref name=frenkel_book>{{cite book |last= Frenkell|first=Daan |last2=Smit | first2=Berent |date=2001 |title=Understanding Molecular Simulation, Second Edition: From Algorithms to Applications |publisher= |page=Academic Press |isbn=0122673514 }}</ref> probability of a nucleus forming at a site <math> \propto \exp[-\Delta G^*/kT]</math>. So if <math>\Delta G^*</math> is large and positive the probability of forming a nucleus is very low and nucleation will be slow. Then the average number will be much less than one, i.e., it is likely that at any given time none of the sites has a nucleus.
 
The second factor in the expression for the rate is the dynamic part, which also has two parts. The first is the rate at which molecules add to the nucleus, increasing its size, <math> j</math>, and the second part is derived by assuming that near the top of the barrier the nucleus
is effectively diffusing along the ''r'' axis, and so nuclei at the top of the barrier can grow diffusively into larger nuclei that will grow into a new phase, or they can lose molecules and shrink back to nothing. The probability that they go forward is ''Z''.
 
To see how this works in practice we can look at an example. Sanz and coworkers <ref>{{cite journal|arxiv=1312.0822|doi=10.1021/ja4028814|title=Homogeneous Ice Nucleation at Moderate Supercooling from Molecular Simulation|year=2013|last1=Sanz|first1=Eduardo|last2=Vega|first2=Carlos|first3=J. R.|last3=Espinosa|first4=R.|last4=Cabellero-Bernal|first5=J. L. F.|last5=Abascal|first6=Chantal|last6=Valeriani|journal=Journal American Chemical Society|volume=135|page=15008}}</ref>
have used computer simulation to estimate all the quantities in the above equation, for the nucleation of ice in liquid water. They did this for a simple but approximate model of water called TIP5P/2005. At a supercooling of 19.5 C, i.e., 19.5C below the freezing point of water in their model, they estimate a free energy barrier to nucleation of ice of <math> \Delta G^*=275k_BT</math>. They also estimate a rate of addition of water molecules to an ice nucleus near the top of the barrier of ''j'' = 10<sup>11</sup>/s and a Zeldovich factor ''Z'' = 10<sup>−3</sup> (note that this factor is dimensionless because it is basically a probability). The number of water molecules in 1 m<sup>3</sup> of water is approximately 10<sup>28</sup>. Putting all these numbers into the formula we get a nucleation rate of approximately 10<sup>−83</sup>/s. This means that on average we would have to wait 10<sup>83</sup> s (10<sup>76</sup>years) to see a single ice nucleus forming in 1 m<sup>3</sup> of water
at -20 C!
 
This is a rate of homogeneous nucleation estimated for a model of water, not real water—we cannot growing nuclei of real water and so cannot directly determine the values of the barrier ''&Delta;G*'', or the dynamic parameters such as ''j'', for real water. However, it may be that indeed the homogeneous nucleation of ice at temperatures near - 20 C and above is ''extremely'' slow and so that whenever we see water freezing temperatures of - 20 C and above this is due to heterogeneous nucleation, i.e., the ice nucleates in contact with a surface.
 
=== Homogeneous nucleation ===
{{Technical|section|date=February 2014}}
Homogeneous nucleation is much rarer than heterogeneous nucleation.<ref name=pruppacher_book/><ref name=sear_review14>{{cite journal |last=Sear |first=Richard P. |year=2014 |title=Quantitative Studies of Crystal Nucleation at Constant Supersaturation: Experimental Data and Models|url=http://personal.ph.surrey.ac.uk/~phs1rs/crys_eng_review.pdf |format=PDF |journal=CrystEngComm |volume=16 |pages=6506 |doi=10.1039/C4CE00344F}}</ref> However, homogeneous nucleation is simpler and so easier to understand than heterogeneous nucleation, so the easiest way to understand heterogeneous nucleation is to start with homogeneous nucleation. So we will outline the classical nucleation theory calculation for the homogeneous nucleation barrier <math>\Delta G^*</math>.
 
[[File:Plot of classical nucleation theory prediction for the free energy of a nucleus as a function of radius.png|thumb|The green curve is the total (Gibbs if this is at constant pressure) free energy as a function of radius. Shown is the free energy barrier, <math>\Delta G^*</math>, and radius at the top of the barrier,
<math>r*</math>. This total free energy is a sum of two terms. The first is a bulk term, which is plotted in red. This scales with volume and is always negative. The second term is an interfacial term, which is plotted in black. This is the origin of the barrier. It is always positive and scales with surface area.]]
 
To understand if nucleation is fast or slow, <math>\Delta G(r)</math> needs to be calculated. The classical theory<ref>F. F. Abraham (1974) ''Homogeneous nucleation theory'' (Academic Press, NY).</ref> assumes that even for a microscopic nucleus of the new phase, we can write the free energy of a droplet <math> \Delta G</math> as the sum of a bulk term that is proportional to the volume of the nucleus, and a surface term, that is proportional to its surface area
 
:<math>\Delta G = \frac{4}{3} \pi r^3 \Delta g + 4 \pi r^2 \sigma</math>
 
The first term is the volume term, and as we are assuming that the nucleus is spherical, this is the volume of a sphere of radius <math>r</math>.
<math>\Delta g</math> is the difference in free energy per unit volume between the thermodynamic phase nucleation is occurring in, and the phase that is nucleating. For example, if water is nucleating in supersaturated air, then <math>\Delta g</math> is the free energy per unit volume of the supersaturated air minus that of water at the same pressure. As nucleation only occurs when the air is supersaturated, <math>\Delta g</math> is always negative. The second term comes from the interface at surface of the nucleus, which is why it is proportional to the surface area of a sphere. <math>\sigma</math> is the [[surface tension]] of the interface between the nucleus and its surroundings, which is always positive.
 
For small <math>r</math> the second surface term dominates and <math>\Delta G(r)>0</math>. The free energy is the sum of an <math>r^2</math> and <math>r^3</math> terms. Now the <math>r^3</math> terms varies more rapidly with <math>r</math> than the <math>r^2</math> term, so
as small <math>r</math> the <math>r^2</math> term dominates and the free energy is positive while for large <math>r</math>, the <math>r^3</math> term dominates and the free energy is negative. This shown in the figure to the right. Thus at some intermediate value of <math>r</math>, the free energy goes through a maximum, and so the probability of formation of a nucleus goes through a minimum.
There is a least-probable nucleus occurs, i.e., the one with the highest value of <math>\Delta G</math>
where
:<math> \frac{dG}{dr} = 0 </math>
This is called the critical nucleus and occurs at a critical nucleus radius
:<math>r^* = -\frac{2 \sigma}{\Delta g}</math>
 
Addition of new molecules to nuclei larger than this [[critical radius]] decreases the free energy, so these nuclei are more probable. The rate at which nucleation occurs is then limited by, i.e., determined by the probability, of forming the critical nucleus. This is just the exponential of minus the free energy of the critical nucleus <math>\Delta G^*</math>, which is
::<math>\Delta G^* = \frac{16 \pi \sigma ^3}{3(\Delta g)^2}</math>
This is the free energy barrier needed in the {{Sectionlink|classical nucleation theory}} expression for <math>R</math> above.
 
=== Heterogeneous nucleation ===
 
[[Image:Surface tension.svg|thumb|right|Three droplets on a surface, illustrating decreasing contact angles. The [[contact angle]] the droplet surface makes with the solid horizontal surface decreases from left to right.]]
 
Heterogeneous nucleation, nucleation with the nucleus at a surface, is much more common than homogeneous nucleation. Heterogeneous nucleation is typically much faster than homogeneous nucleation because the nucleation barrier &Delta;G* is much lower at a surface. This is because the nucleation barrier comes from the positive term in the free energy &Delta;G, which is the surface term. For homogeneous nucleation the nucleus is approximated by a sphere and so has a free energy equal to the surface area of a sphere, 4πr<sup>2</sup>, times the [[surface tension]] &sigma;. However, as we can see in the schematic of macroscopic droplets to the right, droplets on surfaces are not complete spheres and so the area of the interface between the droplet and the surrounding fluid is less than <math> 4\pi r^2 </math>. This geometrical factor reduces the interfacial area and so the interfacial free energy, which in turn reduces the nucleation barrier.<ref name = sear_review>{{cite journal|last=Sear|first=R. P.|title=Nucleation: theory and applications to protein solutions and colloidal suspensions|journal=J. Physics Cond. Matt.|year=2007|volume=19|page=033101|doi=10.1088/0953-8984/19/3/033101|url=http://iopscience.iop.org/0953-8984/19/3/033101|url=http://personal.ph.surrey.ac.uk/~phs1rs/review.pdf|issue=3|bibcode = 2007JPCM...19c3101S }}</ref> Note that this simple theory treats the microscopic nucleus just as if it is a macroscopic droplet.▼
 
In the schematic to the right the [[contact angle]] between the droplet surface and the surface decreases from left to right (A to C), and we see that the surface area of the droplet decreases as the contact angle decreases. This geometrical effect reduces the barrier and hence results in faster nucleation on surfaces with smaller contact angles. Also, if instead of the surface being flat it curves towards fluid, then this also reduces the interfacial area and so the nucleation barrier. There are expressions for this reduction for simple surface geometries.<ref>{{cite journal|last=Sholl|first=C. A.|author2=N. H. Fletcher |title=Decoration criteria for surface steps|journal=Acta Metall.|year=1970|volume=18|page=1083|url=http://www.sciencedirect.com/science/article/pii/0001616070900064|doi=10.1016/0001-6160(70)90006-4}}</ref> In practice, this means we expect nucleation to be fastest on pits or cracks in surfaces made of material such that the nucleus forms a small contact angle on its surface.
 
[[Image:Hethomnucdifference.JPG|thumb|right|Difference in energy barriers]]
▲Heterogeneous nucleation, nucleation with the nucleus at a surface, is much more common than homogeneous nucleation. Heterogeneous nucleation is typically much faster than homogeneous nucleation because the nucleation barrier &Delta;G* is much lower at a surface. This is because the nucleation barrier comes from the positive term in the free energy &Delta;G, which is the surface term. For homogeneous nucleation the nucleus is approximated by a sphere and so has a free energy equal to the surface area of a sphere, 4πr<sup>2</sup>, times the [[surface tension]] &sigma;. However, as we can see in the schematic of macroscopic droplets to the right, droplets on surfaces are not complete spheres and so the area of the interface between the droplet and the surrounding fluid is less than <math> 4\pi r^2 </math>. This geometrical factor reduces the interfacial area and so the interfacial free energy, which in turn reduces the nucleation barrier.<ref name = sear_review>{{cite journal|last=Sear|first=R. P.|title=Nucleation: theory and applications to protein solutions and colloidal suspensions|journal=J. Physics Cond. Matt.|year=2007|volume=19|page=033101|doi=10.1088/0953-8984/19/3/033101|url=http://iopscience.iop.org/0953-8984/19/3/033101|url=http://personal.ph.surrey.ac.uk/~phs1rs/review.pdf|issue=3|bibcode = 2007JPCM...19c3101S }}</ref> Note that this simple theory treats the microscopic nucleus just as if it is a macroscopic droplet.
 
Similar effects can cause [[Precipitation (chemistry)|precipitate]] particles to form at the [[crystallite|grain boundaries]] of a solid. This can interfere with [[precipitation strengthening]], which relies on homogeneous nucleation to produce a uniform distribution of precipitate particles.
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===Computer simulation studies of simple models===
 
The classical nucleation theory makes a number of assumptions, for example it treats a microscopic nucleus as if it is a macroscopic droplet with a well defined surface whose free energy is estimated using an equilibrium property: the interfacial tension &sigma;. For a nucleus that may be only of order ten molecules across it is not always clear that we can treat something so small as a volume plus a surface. Also nucleation is an inherently out of [[thermodynamic equilibrium]] phenomenon so it is not always obvious that its rate can estimated using equilibrium properties.
 
However, modern computers are powerful enough to calculate essentially exact nucleation rates for simple models. These have been compared with the classical theory, for example for the case of nucleation of the crystal phase in the model of hard spheres. This is a model of perfectly hard spheres in thermal motion, and is a simple model of some [[colloids]]. For the crystallization of hard spheres the classical theory is a very reasonable approximate theory.<ref name=auer04>{{cite journal|last=Auer|first=S.|author2=D. Frenkel |title=Numerical prediction of absolute crystallization rates in hard-sphere colloids|journal=J. Chem. Phys.|year=2004|volume=120|page=3015|doi=10.1063/1.1638740|url=http://scitation.aip.org/content/aip/journal/jcp/120/6/10.1063/1.1638740|bibcode = 2004JChPh.120.3015A }}</ref> So for the simple models we can study CNT works quite well, but we do not know if it works equally well for say complex molecules crystallising out of solution.
 
=== The spinodal region ===
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