Classical nucleation theory

Classical nucleation theory (CNT) is the most common theoretical model used to quantitatively study the kinetics of nucleation.[1][2][3]

Nucleation is the first step in the spontaneous formation of a new thermodynamic phase or a new structure, starting from a state of metastability. The kinetics of formation of the new phase is frequently dominated by nucleation, such that the time to nucleate determines how long it will take for the new phase to appear. The time to nucleate can vary by orders of magnitude, from negligible to exceedingly large, far beyond reach of experimental timescales. One of the key achievements of classical nucleation theory is to explain and quantify this immense variation.[4]

DescriptionEdit

The central result of classical nucleation theory is a prediction for the rate of nucleation  , in units of (number of events)/(volume·time). For instance, a rate   in a supersaturated vapor would correspond to an average of 1000 droplets nucleating in a volume of 1 cubic meter in 1 second.

The CNT prediction for   is[3]

 

where

  •   is the free energy cost of the nucleus at the top of the nucleation barrier, and   is the average thermal energy with   the absolute temperature and   the Boltzmann constant.
  •   is the number of nucleation sites.
  •   is the rate at which molecules attach to the nucleus.
  •   is the Zeldovich factor, which gives the probability that a nucleus at the top of the barrier will go on to form the new phase, rather than dissolve.

This expression for the rate can be thought of as a product of two factors: The first,  , 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 by definition.[5] The probability of a nucleus forming at a site is proportional to  . So if   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,  . Here,   expresses the rate of incoming matter and   is the probability that a nucleus of critical size (at the maximum 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  .

To see how this works in practice we can look at an example. Sanz and coworkers[6] 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 TIP4P/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  . They also estimate a rate of addition of water molecules to an ice nucleus near the top of the barrier of   and a Zeldovich factor  . The number of water molecules in 1 m3 of water is approximately 1028. These leads to the prediction  , which means that on average one 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 one cannot grow nuclei of water and so cannot directly determine the values of the barrier  , or the dynamic parameters such as  , 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 water freezes at temperatures of -20 °C and above this is due to heterogeneous nucleation, i.e., the ice nucleates in contact with a surface.

Homogeneous nucleationEdit

Homogeneous nucleation is much rarer than heterogeneous nucleation.[1][7] 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  .

 
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,  , and radius at the top of the barrier,  . 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,   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   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

 

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  .   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   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,   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.   is the surface tension of the interface between the nucleus and its surroundings, which is always positive.

For small   the second surface term dominates and  . The free energy is the sum of an   and   terms. Now the   terms varies more rapidly with   than the   term, so as small   the   term dominates and the free energy is positive while for large  , the   term dominates and the free energy is negative. This shown in the figure to the right. Thus at some intermediate value of  , 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 size, i.e., the one with the highest value of   where

 

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  , which is

 

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

From an experimental standpoint, this theory grants tuning of the critical radius through the dependence of   on temperature. The variable  , described above, can be expressed as

 

where   is the melting point and   is the enthalpy of formation for the material. Furthermore, the critical radius can be expressed as

 

revealing a dependence of reaction temperature. Thus as you increase the temperature near  , the critical radius will increase.

Heterogeneous nucleationEdit

 
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.
 
A diagram featuring all of the factors that affect heterogeneous nucleation

Unlike homogeneous nucleation, heterogeneous nucleation occurs on a surface or impurity. It is much more common than homogeneous nucleation. This is because the nucleation barrier for heterogeneous nucleation is much lower than for homogeneous nucleation. To see this, note that the nucleation barrier is determined by the positive term in the free energy  , which is proportional to the total exposed surface area of a nucleus. For homogeneous nucleation the surface area is simply that of a sphere. For heterogeneous nucleation, however, the surface area is smaller since part of the nucleus boundary is accommodated by the surface or impurity onto which it is nucleating.[9]

There are several factors which determine the precise reduction in the exposed surface area.[9] As shown in a diagram on the left, these factors include the size of the droplet, the contact angle between the droplet and surface, and the interactions at the three phase interfaces: liquid-solid, solid-vapor, and liquid-vapor.

The schematic to the right illustrates the decrease in the exposed surface area of the droplet as the contact angle decreases. Deviations from a flat interface decrease the exposed surface even further: there exist expressions for this reduction for simple surface geometries.[10] In practice, this means that nucleation will tend to occur on surface imperfections.

 
Difference in energy barriers

Statistical mechanical treatmentEdit

The classical nucleation theory hypothesis for the form of   can be examined more rigorously using the tools of statistical mechanics.[11] Specifically, the system is modeled as a gas of non-interacting clusters in the grand canonical ensemble. A state of metastable equilibrium is assumed, such that the methods of statistical mechanics hold at least approximately.[12] The grand partition function is[13]

 

Here the inner summation is over all microstates   which contain exactly   particles. It can be decomposed into contributions from each possible combination of clusters which results in   total particles.[14] For instance,

 

where   is the configuration integral of a cluster with   particles and potential energy  :

 

(The quantity   is the thermal de Broglie wavelength of the particle, which enters due to the integration over the   momentum degrees of freedom.) Note that the inverse factorials are included in the above expressions to compensate for overcounting, since particles and clusters alike are assumed indistinguishable.

More compactly,

 .

Then, by expanding   in powers of  , one can check that the probability   of finding exactly   clusters which each has   particles is

 

The number density   of  -clusters can therefore be calculated as

 

This is also called the cluster size distribution.

The grand potential   is equal to  , which, using the thermodynamic relationship  , leads to the following expansion for the pressure:

 

If one defines the right hand side of the above equation as the function  , then various other thermodynamic quantities can be calculated in terms of derivatives of   with respect to  .[15]

The connection with the simple version of the theory is made by assuming perfectly spherical clusters, in which case   depends only on  , in the form

 

where   is the binding energy of a single particle in the interior of a cluster, and   is the excess energy per unit area of the cluster surface. Then,  , and the cluster size distribution is

 

which implies an effective free energy landscape  , in agreement with the form proposed by the simple theory.

On the other hand, this derivation reveals the significant approximation in assuming spherical clusters with  . In reality, the configuration integral   contains contributions from the full set of particle coordinates  , thus including deviations from spherical shape as well as cluster degrees of freedom such as translation, vibration, and rotation. Various attempts have been made to include these effects in the calculation of  , although the interpretation and application of these extended theories has been debated.[4][16][17] A common feature is the addition of a logarithmic correction   to  , which plays an important role near the critical point of the fluid.[18]

LimitationsEdit

Classical nucleation theory makes a number of assumptions which limit its applicability. Most fundamentally, in the so-called capillarity approximation it treats the nucleus interior as a bulk, incompressible fluid and ascribes to the nucleus surface the macroscopic interfacial tension  , even though it is not obvious that such macroscopic equilibrium properties apply to a typical nucleus of, say, 50 molecules across.[19][20] In fact, it has been shown that the effective surface tension of small droplets is smaller than that of the bulk liquid.[21]

In addition, the classical theory places restrictions on the kinetic pathways by which nucleation occurs, assuming clusters grow or shrink only by single particle adsorption/emission. In reality, merging and fragmentation of entire clusters cannot be excluded as important kinetic pathways in some systems. Particularly in dense systems or near the critical point – where clusters acquire an extended and ramified structure – such kinetic pathways are expected to contribute significantly.[21] The behavior near the critical point also suggests the inadequacy, at least in some cases, of treating clusters as purely spherical.[22]

Various attempts have been made to remedy these limitations and others by explicitly accounting for the microscopic properties of clusters. However, the validity of such extended models is debated. One difficulty is the exquisite sensitivity of the nucleation rate   to the free energy  : even small discrepancies in the microscopic parameters can lead to enormous changes in the predicted nucleation rate. This fact makes first-principles predictions nearly impossible. Instead, models must be fit directly to experimental data, which limits the ability to test their fundamental validity.[23]

Comparison with simulation and experimentEdit

For simple model systems, modern computers are powerful enough to calculate numerically exact nucleation rates. One such example is the nucleation of the crystal phase in the model of hard spheres. This is a simple model of some colloids consisting of perfectly hard spheres in thermal motion. The agreement of CNT with the calculated rates for this system confirms that the classical theory is a very reasonable approximate theory.[24] For the simple models CNT works quite well, however it is unclear if it describes complex (e.g. molecular) systems equally well. Jones et al. computationally explored the nucleation of small Water cluster using classical water model. It was found that CNT could describe the nucleation of clusters of 8-50 water molecules well, but failed to describe smaller clusters.[25] Corrections to CNT, obtained from higher accuracy methods such as quantum chemical calculations, can provide necessary interactions for accurate nucleation rates.[26] However, the CNT fails in describing experimental results of vapour to liquid nucleation even for model substances like Argon by several orders of magnitude.[27]

ReferencesEdit

  1. ^ a b 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. ^ a b Sear, R. P. (2007). "Nucleation: theory and applications to protein solutions and colloidal suspensions". J. Phys.: Condens. Matter. 19 (3): 033101. Bibcode:2007JPCM...19c3101S. CiteSeerX 10.1.1.605.2550. doi:10.1088/0953-8984/19/3/033101.
  4. ^ a b Oxtoby, David W. (1992), "Homogeneous nucleation: theory and experiment", Journal of Physics: Condensed Matter, 4 (38): 7627–7650, doi:10.1088/0953-8984/4/38/001
  5. ^ Frenkel, Daan; Smit, Berent (2001). Understanding Molecular Simulation, Second Edition: From Algorithms to Applications. p. Academic Press. ISBN 978-0122673511.
  6. ^ 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. Bibcode:2013arXiv1312.0822S. doi:10.1021/ja4028814. PMID 24010583.
  7. ^ 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. Archived from the original (PDF) on 2014-10-25. Retrieved 2014-12-26.
  8. ^ F. F. Abraham (1974) Homogeneous nucleation theory (Academic Press, NY).
  9. ^ a b Liu, X. Y. (31 May 2000). "Heterogeneous nucleation or homogeneous nucleation?". The Journal of Chemical Physics. 112 (22): 9949–9955. Bibcode:2000JChPh.112.9949L. doi:10.1063/1.481644. ISSN 0021-9606.
  10. ^ 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.
  11. ^ The discussion which follows draws from Kalikmanov (2001), unless noted otherwise.
  12. ^ Kalikmanov, V.I. (2013), "Nucleation Theory", Lecture Notes in Physics LNP, Lecture Notes in Physics, Springer Netherlands, 860: 17–19, doi:10.1007/978-90-481-3643-8, ISBN 978-90-481-3643-8, ISSN 0075-8450
  13. ^ Kardar, Mehran (2007), Statistical Physics of Particles, Cambridge University Press, p. 118, ISBN 978-0-521-87342-0
  14. ^ Kalikmanov, V.I. (2001), Statistical Physics of Fluids: Basic Concepts and Applications, Springer-Verlag, pp. 170–172, ISBN 978-3-540-417-47-7, ISSN 0172-5998
  15. ^ Kalikmanov (2001) pp. 172-173
  16. ^ Kiang, C. S. and Stauffer, D. and Walker, G. H. and Puri, O. P. and Wise, J. D. and Patterson, E. M., C.S.; Stauffer, D.; Walker, G. H.; Puri, O.P.; Wise, J.D.; Patterson, E.M. (1971), "A Reexamination of Homogeneous Nucleation Theory", Journal of the Atmospheric Sciences, 28 (7): 1222–1232, doi:10.1175/1520-0469(1971)028<1222:AROHNT>2.0.CO;2CS1 maint: multiple names: authors list (link)
  17. ^ Reguera, D.; Rubı́, J.M. (2001), "Nonequilibrium translational-rotational effects in nucleation", The Journal of Chemical Physics, 115 (15): 7100–7106, arXiv:cond-mat/0109270, doi:10.1063/1.1405122
  18. ^ Sator, N. (2003), "Clusters in simple fluids", Physics Reports, 376 (1): 1–39, arXiv:cond-mat/0210566, doi:10.1016/S0370-1573(02)00583-5, ISSN 0370-1573
  19. ^ Kalikmanov (2013), p. 21
  20. ^ Oxtoby (1992), p. 7631
  21. ^ a b Kiang, et al (1971)
  22. ^ Sator (2003)
  23. ^ Oxtoby (1992), p. 7638–7640
  24. ^ Auer, S.; D. Frenkel (2004). "Numerical prediction of absolute crystallization rates in hard-sphere colloids" (PDF). J. Chem. Phys. 120 (6): 3015–29. Bibcode:2004JChPh.120.3015A. doi:10.1063/1.1638740. hdl:1874/12074. PMID 15268449.
  25. ^ Merikanto, Joonas; Zapadinsky, Evgeni; Lauri, Antti; Vehkamäki, Hanna (4 April 2007). "Origin of the Failure of Classical Nucleation Theory: Incorrect Description of the Smallest Clusters". Physical Review Letters. 98 (14): 145702. Bibcode:2007PhRvL..98n5702M. doi:10.1103/PhysRevLett.98.145702. PMID 17501289.
  26. ^ Temelso, Berhane; Morrell, Thomas E.; Shields, Robert M.; Allodi, Marco A.; Wood, Elena K.; Kirschner, Karl N.; Castonguay, Thomas C.; Archer, Kaye A.; Shields, George C. (22 February 2012). "Quantum Mechanical Study of Sulfuric Acid Hydration: Atmospheric Implications". The Journal of Physical Chemistry A. 116 (9): 2209–2224. Bibcode:2012JPCA..116.2209T. doi:10.1021/jp2119026. PMID 22296037.
  27. ^ A. Fladerer, R. Strey: „Homogeneous nucleation and droplet growth in supersaturated argon vapor: The cryogenic nucleation pulse chamber.“ in: The Journal of Chemical Physics 124(16), 164710 (2006). (Online)