# Nucleation

Not to be confused with Microtubule nucleation.
When sugar is 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 many thermodynamic phases, such as a crystals and liquids. It is the process that determines how long you have to wait before the new crystal, liquid, etc, appears. If for example, liquid water is cooled below 0 °C then ice is thermodynamically more stable than the water, and so the water will tend to freeze to form ice. However, it is frequently observed that water can remain as a liquid for long periods at temperatures below zero Celsius.[1](this is called supercooling). Then nucleation is said to be slow. If the water is further cooled, ice may form rapidly, and then nucleation is said to be fast. The formation of ice is just an example, all crystals start to form via nucleation, as do most liquids and gases. When the new phase (e.g., ice) is forming is separated from the existing phase (e.g., water) by a first-order phase transition then usually we would expect the new phase to start with nucleation,[2] although under some conditions spinodal decomposition can occur.

The classical theory for nucleation, discussed below in the rate of nucleation section, assumes that what determines whether nucleation is fast or slow, is the rate at which a rare microscopic fluctuation of the new phase forms,.[1][2] For example, the rate of nucleation of liquid water in supersaturated air is then the rate at which a microscopic water droplet of perhaps only a thousand water molecules forms due to a thermal fluctuation. This assumption agrees well with the findings of computer simulation studies for simple models, see below, but these fluctuations are too small and transient to be observed in experiment.[1]

Nucleation normally occurs at nucleation sites on surfaces contacting the liquid or vapor.[1] This is called heterogeneous nucleation. Nucleation that occurs away from any surface is called homogeneous nucleation.

## Examples

• Clouds form when wet air cools (often because the air is rising) and many small water droplets nucleate from the supersaturated air.[1] The amount of water that can exist in air as a vapour decreases on cooling and cooling air can result in there being too much water vapour for the air to hold, driving the nucleation of small water droplets which form a cloud. Nucleation of the droplets of liquid water is heterogeneous, it occurs on particles referred to as cloud condensation nuclei, and cloud formation can be altered by adding artificial cloud condensation nuclei, this is called cloud seeding.
• Nucleation is by definition the first step in crystallisation, and so it determines if a crystal can form. Frequently crystals do not form even when they are thermodynamically the favoured state. For example small droplets of very pure water can remain liquid down to below -30 C although ice is the stable state below 0 C.[1]
Nucleation of carbon dioxide bubbles around a finger.
• Bubbles of carbon dioxide nucleate shortly after the pressure is released from a container of carbonated liquid such Coca-Cola or champagne. Nucleation often occurs more easily at a pre-existing interface (heterogeneous nucleation), as happens on boiling chips and string used to make rock candy. The so-called Diet Coke and Mentos eruption is a dramatic example of this.
• Nucleation in boiling can occur in the bulk liquid if the pressure is reduced so that the liquid becomes 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 nanometer sized materials,[3] and plays an important role in atmospheric processes.
• Nucleation is a key concept in polymer,[4] alloy and ceramic systems.
• In chemistry and biophysics, nucleation is the phaseless formation of multimers, which are intermediates in polymerization processes. This sort of process is believed to be the best model for processes such as crystallization and amyloidogenesis.
• In molecular biology, nucleation is the critical stage in the assembly of a polymeric structure, such as a microfilament, at which a small cluster of monomers aggregates in the correct arrangement to initiate rapid polymerization. For instance, two actin molecules bind weakly, but addition of a third stabilizes the complex. This trimer then adds additional molecules and forms a nucleation site. The nucleation site serves the slow, or lag phase of the polymerization process.
• Some champagne stirrers 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.
• Instruments such as the bubble chamber and the cloud chamber rely on nucleation.

## Mechanics

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 a stochastic process, i.e., there is a random element to it and so in a set of identical droplets nucleation will occur over a spread of times. If the nucleation rate is constant then the fraction of these droplets where nucleation has not occurred should decrease exponentially with time. This is seen for example in the nucleation of ice in supercooled small water droplets.[5] The decay rate of the exponential gives the nucleation rate.

### Classical nucleation theory

#### Homogeneous nucleation

Homogeneous nucleation is much rarer than heterogeneous nucleation.[1][6] 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 for homogeneous nucleation.

probability of a nucleus of radius r forming $\propto \exp[-\Delta G(r)/kT]$

The probability of any fluctuation occurring is proportional to the exponential of minus its free energy $\Delta G$ divided by the thermal energy $kT$. This is a general result.[7] Here we apply it to the case of a microscopic droplet or crystallite of the new phase, of radius $r$. If $\Delta G$ is large and positive the probability of forming a nucleus is very low and nucleation will be slow.

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, Δ G*, and radius at the top of the barrier, r*. 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.

So, now to understand if nucleation is fast or slow, $\Delta G(r)$ needs to be calculated. The classical theory[8] assumes that even for a microscopic nucleus of the new phase, we can write the free energy of a droplet $\Delta G$ 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

$\Delta G = -\frac{4}{3} \pi r^3 \Delta g + 4 \pi r^2 \sigma$

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 $r$. $\Delta g$ 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 $\Delta g$ 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, $\Delta g$ is always positive. 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. $\sigma$ is the surface tension of the interface between the nucleus and its surroundings, which is always positive.

For small $r$ the second surface term dominates and $\Delta G(r)>0$. The free energy is the sum of an $r^2$ and $r^3$ terms. Now the $r^3$ terms varies more rapidly with $r$ than the $r^2$ term, so as small $r$ the $r^2$ term dominates and the free energy is positive while for large $r$, the $r^3$ term dominates and the free energy is negative. This shown in the figure to the right. Thus at some intermediate value of $r$, 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 $\Delta G$. This is called the critical nucleus and occurs at a critical nucleus radius

$r^* = \frac{2 \sigma}{\Delta g}$

where

$\frac{dG}{dr} = 0$

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 $\Delta G^*$, which is

$\Delta G^* = \frac{16 \pi \sigma ^3}{3(\Delta g)^2}$

For nucleation of the crystal in supercooled liquids, the term $\Delta G_v$ can be related to the equilibrium temperature, enthalpy of fusion ($\Delta H_v$), and the degree of undercooling ($\Delta T$) as follows,

$\Delta G_v=\Delta H_v - T \Delta S_v$

By evaluating this equation at the equilibrium point ($\Delta G_v = 0$) at the melting temperature $T_m$ we achieve,

$\Delta S_v=\frac{\Delta H_v}{T_m}$

Substitution of $\Delta S_v$ into the first equation leads to

$\Delta G_v=\Delta H_v - T(\frac{\Delta H_v}{T_m})$

Which by using common denominators and the definition of $\Delta T = T_m-T$ provides

$\Delta G_v=\frac{\Delta H_v}{T_m}\Delta T$

As the phase transformation becomes more and more favorable, the formation of a given volume of nucleus frees enough energy to form an increasingly large surface, allowing progressively smaller nuclei to become viable. Eventually, thermal activation will provide enough energy to form stable nuclei. These can then grow until thermodynamic equilibrium is restored.

A greater degree of supercooling favors phase transformation, and we can relate $\Delta G^*$ to supercooling and find r* and $\Delta G^*$ as a function of $\Delta T$ by the substitution of $\Delta G_v$

$r^* = -\frac{2 \sigma T_m}{\Delta H_v} \frac{1}{\Delta T}$

and

$\Delta G^* = \frac{16 \pi \sigma ^3 T_m^2}{3\Delta H_v^2} \frac{1}{(\Delta T)^2}$

The greater the supercooling, the smaller the critical radius and the less energy needed to form it.

#### Heterogeneous nucleation

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 ΔG* is much lower at a surface. This is because the nucleation barrier comes from the positive term in the free energy ΔG, 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, 4πr2, times the surface tension σ. 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 4πr2. This geometrical factor reduces the interfacial area and so the interfacial free energy, which in turn reduces the nucleation barrier.[2] 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.[9] 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.

Difference in energy barriers

Similar effects can cause precipitate particles to form at the grain boundaries of a solid. This can interfere with precipitation strengthening, which relies on homogeneous nucleation to produce a uniform distribution of precipitate particles.

#### Rate of nucleation

The basic classical nucleation theory for nucleation provides an approximate but physically reasonable prediction for the rate at which nuclei of a new phase form, via homogeneous nucleation in a uniform volume V. 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 0.1m3 in 1 s, then the rate R=1000/s. The description here follows modern standard classical nucleation theory.[2] It should be borne in mind that it is approximate. The prediction for the rate R is

$R\ =\ N Zj\exp \left( \frac{-\Delta G^*}{k_BT} \right)$

where

• ΔG* is the free energy cost of the nucleus at the top of the nucleation barrier, and kBT is the thermal energy with T the absolute temperature and kB is the Boltzmann constant.
• N is the number of molecules in the volume V.
• 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 number of fluctuations that at the top of the nucleation barrier, this is: $N \exp \left( -\Delta G^*/k_BT \right)$, which is the number of nucleation sites, which here is just the number of molecules, N, times the probability that a nucleus of critical size has grown around it, $\exp \left(-\Delta G^*/k_BT\right)$. The second is the dynamic part, which also has two parts. The first is the rate at which molecules add to the nucleus, increasing its size, $j$, 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 [10] 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 $\Delta G^*=275k_BT$. They also estimate a rate of addition of water molecules to an ice nucleus near the top of the barrier of j = 1011/s and a Zeldovich factor Z = 10−3 (note that this factor is dimensionless because it is basically a probability). The number of water molecules in 1 m3 of water is approximately 1028. Putting all these numbers into the formula we get a nucleation rate of approximately 10−83/s. This means that on average we would have to wait 1083 s (1076years) to see a single ice nucleus forming in 1 m3 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 ΔG*, 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.

#### 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 σ. 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.[11] 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

Phase transition processes can also be explained in terms of spinodal decomposition, where phase separation is delayed until the system enters the unstable region where a small perturbation in composition leads to a decrease in energy and, thus, spontaneous growth of the perturbation.[12] This region of a phase diagram is known as the spinodal region and the phase separation process is known as spinodal decomposition and may be governed by the Cahn–Hilliard equation.

## Modern technology

Nucleation is a topic of wide interest in many scientific studies and technological processes. It is used heavily in the chemical industry for cases such as in the preparation of metallic ultradispersed powders that can serve as catalysts. For example, platinum deposited onto TiO2 nanoparticles catalyses the liberation of hydrogen from water.[13] It is an important factor in the semiconductor industry, as the gap width in semiconductors is influenced by the size of metal nanoclusters.[14] As another example, understanding calcium carbonate nucleation could help scientists control its formation to keep carbon dioxide from getting into the atmosphere.[15]

## Experiment

It is typically difficult to experimentally study nucleation. The nucleus is microscopic (its size is r* above) so the nucleus is too small to be directly observed. In large liquid volumes there are typically multiple nucleation events and it is difficult to disentangle the effects of nucleation from those of growth of the nucleated phase. These problems can be overcome by working with small droplets. As nucleation is stochastic, many droplets are needed so that statistics for the nucleation events can be obtained.

The black triangles are the fraction of a large set of small supercooled liquid tin droplets that are still liquid, i.e., where the crystal state has not nucleated, as a function of time. The data is from Pound and La Mer (1952). The red curve is a fit of a function of the Gompertz form to this data.

To the right is shown an example set of nucleation data. It is for the nucleation at constant temperature and hence supersaturation of the crystal phase in small droplets of supercooled liquid tin; this is the work of Pound and La Mer.[16]

Nucleation occurs in different droplets at different times, hence the fraction is not a simple step function that drops sharply from one to zero at one particular time. The red curve is a fit of a Gompertz function to the data. This is a simplified version of the model Pound and La Mer used to model their data.[16] The model assumes that nucleation occurs due to impurity particles in the liquid tin droplets, and it makes the simplifying assumption that all impurity particles produce nucleation at the same rate. It also assumes that these particles are Poisson distributed among the liquid tin droplets. The fit values are that the nucleation rate due to a single impurity particle is 0.02/s, and the average number of impurity particles per droplet is 1.2. Note that about 30% of the tin droplets never freeze; the data plateaus at a fraction of about 0.3. Within the model this is assumed to be because, by chance, these droplets do not have even one impurity particle and so there is no heterogeneous nucleation. Homogeneous nucleation is assumed to be negligible on the timescale of this experiment. The remaining droplets freeze in a stochastic way, at rates 0.02/s if they have one impurity particle, 0.04/s if they have two, and so on.

This data is just one example but it does illustrate common features of the nucleation of crystals in that there is clear evidence for heterogeneous nucleation, and that nucleation is clearly stochastic.

## References

1. H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, Kluwer (1997).
2. ^ a b c d Sear, R. P. (2007). "Nucleation: theory and applications to protein solutions and colloidal suspensions". J. Physics Cond. Matt. 19 (3): 033101. Bibcode:2007JPCM...19c3101S. doi:10.1088/0953-8984/19/3/033101.
3. ^ Mendez-Villuendas, Eduardo; Bowles, Richard (2007). "Surface Nucleation in the Freezing of Gold Nanoparticles". Physical Review Letters 98 (18). arXiv:cond-mat/0702605. Bibcode:2007PhRvL..98r5503M. doi:10.1103/PhysRevLett.98.185503.
4. ^ Young, R. J. (1981) Introduction to Polymers (CRC Press, NY) ISBN 0-412-22170-5.
5. ^ Duft, D.; Leisner (2004). "Laboratory evidence for volume-dominated nucleation of ice in supercooled water microdroplets". Atmospheric Chemistry & Physics 4: 1997. doi:10.5194/acp-4-1997-2004.
6. ^ Sear, Richard P. (2014). "Quantitative Studies of Crystal Nucleation at Constant Supersaturation: Experimental Data and Models" (PDF). CrystEngComm 16: 6506. doi:10.1039/C4CE00344F.
7. ^ a b Frenkell, Daan; Smit, Berent (2001). Understanding Molecular Simulation, Second Edition: From Algorithms to Applications. p. Academic Press. ISBN 0122673514.
8. ^ F. F. Abraham (1974) Homogeneous nucleation theory (Academic Press, NY).
9. ^ Sholl, C. A.; N. H. Fletcher (1970). "Decoration criteria for surface steps". Acta Metall. 18: 1083. doi:10.1016/0001-6160(70)90006-4.
10. ^ Sanz, Eduardo; Vega, Carlos; Espinosa, J. R.; Cabellero-Bernal, R.; Abascal, J. L. F.; Valeriani, Chantal (2013). "Homogeneous Ice Nucleation at Moderate Supercooling from Molecular Simulation". Journal American Chemical Society 135: 15008. arXiv:1312.0822. doi:10.1021/ja4028814.
11. ^ Auer, S.; D. Frenkel (2004). "Numerical prediction of absolute crystallization rates in hard-sphere colloids". J. Chem. Phys. 120: 3015. Bibcode:2004JChPh.120.3015A. doi:10.1063/1.1638740.
12. ^ Mendez-Villuendas, Eduardo; Saika-Voivod, Ivan; Bowles, Richard K. (2007). "A limit of stability in supercooled liquid clusters". The Journal of Chemical Physics 127 (15): 154703. Bibcode:2007JChPh.127o4703M. doi:10.1063/1.2779875. PMID 17949187.
13. ^ Palmans, Roger; Frank, Arthur J. (1991). "A molecular water-reduction catalyst: Surface derivatization of titania colloids and suspensions with a platinum complex". The Journal of Physical Chemistry 95 (23): 9438. doi:10.1021/j100176a075.
14. ^ Rajh, Tijana; Micic, Olga I.; Nozik, Arthur J. (1993). "Synthesis and characterization of surface-modified colloidal cadmium telluride quantum dots". The Journal of Physical Chemistry 97 (46): 11999. doi:10.1021/j100148a026.
15. ^ Nielsen, Michael; Aloni, Shaul; De Yoreo, James (September 4, 2014). "In Situ TEM Imaging of CaCO3 Nucleation Reveals Coexistence of Direct and Indirect Pathways". Science 345 (6201): 1158–1162. doi:10.1126/science.1254051.
16. ^ a b Pound, Guy M.; V. K. La Mer (1952). "Kinetics of Crystalline Nucleus Formation in Supercooled Liquid Tin". J. American Chemical Society 74: 2323. doi:10.1021/ja01129a044.