Classical nucleation theory

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. Classical nucleation theory (CNT) is the most common theoretical model used to understand why nucleation may take hours or years, or in effect never happen.^[1][2][3]

Outline of 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 CNT 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 0.1m³ in 1s, then the rate R=1000/s. The description here follows modern standard CNT.^[3] The prediction for the rate R is

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

where

• $\Delta G^*$ is the free energy cost of the nucleus at the top of the nucleation barrier, and k_BT is the thermal energy with T the absolute temperature and k_B is the Boltzmann constant.
• $N_S$ 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, $N_S \exp \left( -\Delta G^*/k_BT \right)$, is the number of nucleation sites multiplied by the probability that a nucleus of critical size has grown around it. It can be interpreted as the average, instantaneous number of nuclei at the top of the nucleation barrier. Free energies and probabilities are closely related in general, by definition.^[4] The probability of a nucleus forming at a site is proportional to $\exp[-\Delta G^*/kT]$. So if $\Delta G^*$ 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, $Zj$. Here, $j$ expresses the rate of incoming matter and $Z$ is the probability that a nucleus of critical size (at the maxiumum of the energy barrier) will continue to grow and not dissolve. The Zeldovich factor is derived by assuming that the nuclei near the top of the barrier are effectively diffusing along the radial axis. By statistical fluctuations, a nucleus at the top of the barrier can grow diffusively into a larger nucleus that will grow into a new phase, or it can lose molecules and shrink back to nothing. The probability that a given nucleus goes forward is $Z$.

To see how this works in practice we can look at an example. Sanz and coworkers^[5] 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., 19.5 °C 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 10⁸³s (10⁷⁶ years) 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—in experiments we cannot grow nuclei of 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.

Homogeneous nucleation

Homogeneous nucleation is much rarer than heterogeneous nucleation.^[1][6] However, homogeneous nucleation is simpler and 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 CNT calculation for the homogeneous nucleation barrier $\Delta G^*$.

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, $\Delta G^*$, 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.

To understand if nucleation is fast or slow, $\Delta G(r)$ needs to be calculated. The classical theory^[7] 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 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. $\sigma$ 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 $\Delta G$ where

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

This is called the critical nucleus and occurs at a critical nucleus radius

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

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

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

This is the free energy barrier needed in the CNT expression for $R$ above.

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 $4\pi r^2$. This geometrical factor reduces the interfacial area and so the interfacial free energy, which in turn reduces the nucleation barrier.^[3] 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.^[8] 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

Comparison between CNT and computer simulation

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, but only 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.^[9] 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.

References

1. H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, Kluwer (1997)
2. P.G. Debenedetti, Metastable Liquids: Concepts and Principles, Princeton University Press (1997)
3. 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.
4. Frenkell, Daan; Smit, Berent (2001). Understanding Molecular Simulation, Second Edition: From Algorithms to Applications. p. Academic Press. ISBN 0122673514.
5. 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 of the American Chemical Society. 135 (40): 15008–15017. arXiv:1312.0822. doi:10.1021/ja4028814.
6. Sear, Richard P. (2014). "Quantitative Studies of Crystal Nucleation at Constant Supersaturation: Experimental Data and Models" (PDF). CrystEngComm. 16 (29): 6506–6522. doi:10.1039/C4CE00344F.
7. F. F. Abraham (1974) Homogeneous nucleation theory (Academic Press, NY).
8. Sholl, C. A.; N. H. Fletcher (1970). "Decoration criteria for surface steps". Acta Metall. 18 (10): 1083–1086. doi:10.1016/0001-6160(70)90006-4.
9. Auer, S.; D. Frenkel (2004). "Numerical prediction of absolute crystallization rates in hard-sphere colloids". J. Chem. Phys. 120 (6): 3015. Bibcode:2004JChPh.120.3015A. doi:10.1063/1.1638740.