## Abstract

The activation free energy for nucleation of crystals from melts and solutions is described and simulated in a different way from traditional nucleation theory. For molecules at the surface of a nucleus, the unfavourable entropy of fusion outweighs the favourable enthalpy change. The free energy barrier to nucleation in pure melts is simulated from the entropy and enthalpy of fusion. By incorporating the unfavourable entropy of unmixing, nucleation from impure melts and solutions is also simulated. The nature of the metastable zone alters and its size is estimated.

## 1. Introduction

Phase changes involving the melting or dissolution of solids usually occur rapidly once the relevant temperature has been exceeded. The converse is not true, and the existence of supercooled liquids and supersaturated solutions is well known and frequently exploited, for example in using seeds to control the course of crystallization. The existence and persistence of such metastable states is usually accounted for by traditional nucleation theory (Davey & Garside 2000). In this theory, the unfavourable energetic consequences of formation of the interface between the solid and the liquid are used to derive a ‘critical nucleus size’. This size must be exceeded for the nucleus to grow, and this can only happen when the supersaturation reaches a certain value. The key system properties in this analysis are the interfacial tension between the solid and the liquid, and the ‘pre-exponential factor’. The interfacial tension is difficult to measure and calculate.

Homogeneous nucleation from liquid metal droplets has been studied in detail (Turnbull 1950). The ratio between the nucleation temperature *T*_{2} and the melting temperature *T*_{m} was found to be approximately constant for many metals. Values for the interfacial energies were derived from the nucleation temperatures using classical nucleation theory. An empirical ratio between molar interfacial energy and enthalpy of fusion was reported for 14 metals. Four other materials gave lower ratios, and it was suggested that this was related to structural differences.

Early models of nucleation were based on the ‘simple cubic’ lattice in which each molecule is represented by a cube (Dunning 1963; Strickland-Constable 1968). In the ‘nearest neighbour’ bonding approximation, only the binding between neighbouring cube faces is considered. It was shown that in this model, the molar interfacial energy between melt and solid is equal to one-third of the enthalpy of fusion. The link between this ratio and *T*_{2}/*T*_{m} (see table 1 for nomenclature) was discussed briefly.

A different way of describing and simulating nucleation is introduced here. The approach is developed firstly for the one-component system of nucleation from pure melts, and then applied to the two-component systems of impure melts and nucleation from solutions. The main aim is a theory linking nucleation to readily measurable crystal properties. A further aim is to simulate nucleation using spreadsheets rather than intensive computational resources.

## 2. Theory

### (a) The model

In this model, nuclei are assumed to contain two different types of molecules—those that are completely enclosed inside the nucleus and those on the surface. The molecules inside the nucleus are in an identical environment to molecules in the bulk crystal—they are truly ‘solid’. Molecules in the surface are not. They can only form the strong bonds that hold the crystal together with a restricted number of molecules. However, in terms of their entropy, the molecules in the surface are 100% solid; they have lost all conformational, translational and rotational freedom associated with the liquid state.

This sharp distinction between surface and bulk molecules is closely related to the nearest neighbour approximation (Strickland-Constable 1968). The approximation seems reasonable for structures in which bonding is dominated by short-range Van der Waals and hydrogen bonds. Surface molecules may retain some conformational flexibility, but for larger molecules this will be only a small fraction of the total entropy of fusion (Yalkowsky 1979).

In this model, surface molecules experience the ‘entropic pain’ of solid formation without the full benefit of the ‘enthalpic gain’. The barrier to nucleation arises because small nuclei contain a relatively high proportion of surface molecules. The entropic nature of the barrier to nucleation has been discussed elsewhere (Radhakrishnan & Trout 2003) using order parameters and intensive computer simulations. The model developed here is simpler and can be implemented using a spreadsheet.

The model is described first for a general nucleus shape. This model is expressed algebraically, and some of the algebraic manipulations are in appendix A. The model is developed further for the simple cubic nucleus, enabling simple calculations of the free energy barrier to nucleation.

### (b) Pure systems

For a model nucleus, the surface area of the nucleus is directly proportional to *n*^{2}, where *n*^{3} is the number of molecules in the nucleus. The surface enthalpy is directly proportional to the surface area. It follows that(2.1)where *s* is a ‘structure constant’ related to the structure and bonding in the nucleus. One consequence of the nearest neighbour approximation is that *s* is not a function of nucleus size. In the simple cubic nucleus, *s*=1 (§3*a*). As *n* increases, the enthalpy of nucleation approaches the enthalpy of fusion, but for small nuclei the surface term is significant.

Note that the units of enthalpy of nucleation (also entropy, free energy) are per mole of nuclei, as distinct from the units for enthalpy of fusion (also entropy, free energy), which are per mole of molecules.

The entropy of formation of the nucleus is given by(2.2)The free energy of nucleation is(2.3)Substituting (2.1) and (2.2) into (2.3)(2.4)Rearranging(2.5)This expression is very similar to the traditional equation for free energy of nucleation for a spherical nucleus (Davey & Garside 2000)(2.6)The difference is that the ‘surface energy’ term is now related to the molar enthalpy of fusion rather than interfacial free energy per unit area. This has the advantage that the free energy of nucleation as a function of *n* can now be simulated from the melting properties of the crystal, without knowing or estimating *γ*. There is no explicit length-scale in the simulation—the implied scale is the size of the molecule.

At the critical nucleus size (*n*=*n*_{c}), Δ*G*_{nucleation} is at a maximum, and the derivative with respect to *n* will be zero. Differentiation of equation (2.4), after some mathematical manipulation (appendix A*a*), gives(2.7)This is a very different result from classical nucleation theory. The size of the critical nucleus is a function of *s*, *T*_{m} and the degree of under-cooling only, and is not dependent on Δ*H*_{f} or Δ*S*_{f}.

Substituting back into equation (2.4), with some algebraic manipulation (appendix A*b*), gives(2.8)For compounds with similar melting points (and similar *s*) higher entropies (and enthalpies) of fusion are predicted to correlate linearly with greater barriers to nucleation.

### (c) Impure systems

Here it is assumed that the effect of impurity can be described completely by defining *x* as the mole fraction of the crystallizing material in the system, where *x*<1. This is equivalent to assuming that there are no preferential interactions between impurity or solvent and the crystallizing molecule. Δ*H*_{unmix}=0 for any mixing process, and Δ*S*_{unmix} is determined solely by configurational entropy.

Two types of impure system are of interest. The first is where the crystallizing material is the major component in the system (*x*>0.75). The minor component could be residual solvent or an additive. The second is crystallization from solution where the crystallizing material (the solute) is the minor component (*x*<0.1). In both cases, it is assumed that the impurity is not incorporated into the solid. Here the same theoretical treatment is used to describe both systems.

The presence of impurity/solvent affects both the thermodynamics and the kinetics of crystallization. The effect on the thermodynamics is well known as the ‘depression of freezing point’ or the ‘ideal solubility’ (Atkins & de Paula 2002) and may be simulated using the van't Hoff equation(2.9)Defining *j* as a dimensionless melting constant such that(2.10)where *j* is always positive and approaches 0 as *x* approaches 1, equation (2.9) can be simplified to(2.11)As *x* decreases, *j* increases and *T*′ decreases. The effect of impurity is predicted to be greater for compounds with lower entropies of melting.

Impurity/solvent will have an additional and completely distinct effect on the kinetics of crystallization, if it is assumed that a nucleus cannot contain any impurity or solvent. It follows that the liquid phase must segregate (unmix) prior to nucleation. The entropy of nucleation is then given by(2.12)where Δ*S*_{unmix} is given by (Bourne & Davey 1976)(2.13)Combining equations (2.4), (2.12) and (2.13) gives the following expression for the free energy of nucleation in an impure system:(2.14)Now, by defining a dimensionless unmixing factor *k* such that(2.15)where *k* is always positive and approaches 0 as *x* approaches 1, equation (2.14) can be simplified to(2.16)This reduces to equation (2.4) when *k*=0.

Now proceeding as above for pure melts, by differentiating with respect to *n*, and after some manipulation (appendix B)(2.17)This agrees with equation (2.7) when *k*=0.

Now for any positive value of *k*, there will be a critical value of *T*=*T*″, such that(2.18)If *T*>*T*″, then there will be no solution to equation (2.17). This implies that there is a temperature *T*″ above which nucleation is not possible.

Equation (2.18) is similar, but not identical to equation (2.11) which defines the temperature *T*′ of equilibrium in the impure system. It is possible to compare *T*′ and *T*″ qualitatively by comparing *j* (equation (2.10)) and *k* (equation (2.15)); the equations are repeated here to emphasize the similarities.

As 0<*x*<1, it follows that *k*>*j* and therefore *T*″<*T*′.

At temperatures in between *T*′ and *T*″ the impure system is supersaturated, but nucleation cannot occur. This corresponds to the metastable zone. In contrast to the metastable zone in pure systems, or in conventional nucleation theory, its existence and size are not a function of time, scale or cooling rate. The size of this metastable zone is a function of *T*_{m}, Δ*S*_{f} and *x* only.

## 3. Calculations

In order to carry out calculations, it is necessary to assign meaningful numerical values to Δ*H*_{f}, Δ*S*_{f}, *T*_{m} and *s*. This is achieved using a simple cubic nucleus as a model, with the expectation that the main conclusions will be more generally valid.

### (a) The simple cubic nucleus

In a simple cubic nucleus, each molecule occupies a cubic box with six equivalent faces. The cubic nucleus is made up of *n*×*n*×*n* identical molecules packed in a simple cubic structure. In the nearest neighbour approximation, binding only occurs between faces that are in contact, so each molecule (apart from the surface molecules) is bound to six neighbouring molecules only. For a cubic cluster of *n*×*n*×*n* molecules within an infinite structure, the enthalpy of fusion per mole of clusters is given by *n*^{3}×Δ*H*_{f}. Δ*H*_{nucleation} may be calculated by considering the difference between this value and the enthalpy of formation of a nucleus of the same size. The difference is that a nucleus has a surface area determined by simple geometry to be 6*n*^{2}. For each exposed face on the surface of the nucleus, an enthalpy of one-sixth Δ*H*_{f} is ‘missing’. Hence, the total missing enthalpy is *n*^{2}×Δ*H*_{f}. By comparison with equation (2.1), *s*=1 for a simple cubic nucleus.

### (b) Pure system

Setting *s*=1 simplifies equations (2.4), (2.7) and (2.8)(3.1)(3.2)(3.3)The variation of *n*_{c} with under-cooling can be plotted directly from equation (3.2) for different values of *T*_{m} (figure 1).

The variation with *T*_{m} is small, at least over the range 400–500 K, which includes many molecular crystals. The form of this graph is more generally valid—for other structures (s not equal to 1) *n*_{c}/*s* replaces *n*_{c}.

The smallest possible nucleus has *n*=2. Substituting *n*_{c} =2 in equation (3.2) gives(3.4)where *T*_{2} is the temperature below which nucleation is predicted to be inevitable and instantaneous. This suggests that if *T*_{g}/*T*_{m} <2/3, where *T*_{g} is the glass transition temperature, then the amorphous state will not be kinetically stable.

For several metals, the measured value of *T*_{2}/*T*_{m} is in the range 0.80–0.85 (Strickland-Constable 1968), higher than the value of 0.67 predicted here. The difference may be related to the different structures involved and the breakdown of the nearest neighbour approximation.

Equation (3.1) simulates the free energy barrier to nucleation as a function of *n*, for a simple cubic nucleus, at different values of under-cooling (Δ*T*=*T*_{m}−*T*).

The following material properties have been used here:These are typical values for an organic molecular solid with molecular weight approximately 500. The simulation results are given in figure 2

The plots show the same features as seen in conventional nucleation theory. With increasing nucleus size, the free energy increases initially to a maximum value and then decreases (Davey & Garside 2000). At this maximum value, Δ*G*_{nucleation} is Δ*G*_{nucleation}^{*}, the activation free energy for nucleation. The corresponding value of *n* is *n*_{c}, the critical nucleus size or radius. Both Δ*G*_{nucleation}^{*} and *n*_{c} decrease as under-cooling increases.

The variation of activation free energy of nucleation, Δ*G*_{nucleation}^{*} with under-cooling is plotted using equation (3.3) in figure 3. As expected, at small under-cooling, Δ*G*_{nucleation}^{*} is very large and nucleation is very unlikely. As the under-cooling (supersaturation) increases, Δ*G*_{nucleation}^{*} falls rapidly, and a corresponding increase in nucleation rate is expected. These results are qualitatively very similar to traditional nucleation theory (Davey & Garside 2000).

### (c) Nucleation of polymorphs from pure melts

The free energy of nucleation is strongly dependent on the amount of under-cooling. Thus, the polymorph with the higher melting point will generally be favoured in this model. There is a smaller effect favouring the polymorph with the lower enthalpy of melting. Hence, it is possible to simulate the case where the metastable polymorph has a lower free energy of nucleation. One such simulation is shown belowThe nucleation temperature was set at 460 K, i.e. an under-cooling of 20° for form 1 and 18° for form 2. The results are shown in figure 4.

Form 1, which is more thermodynamically stable at all temperatures (monotropic), has a higher free energy of nucleation. This simulation is consistent with ‘Ostwald's Law of Stages’ (Ostwald 1897). However, there are many other combinations of melting temperature and enthalpy of melting for which the more stable polymorph would have a lower free energy of nucleation, in which case the ‘metastable’ form will never appear. This simulation uses a very small (2°C) difference in melting point and a very large (more than 20%) difference in melting enthalpy. Therefore, these simulations provide no support for the generality of Ostwald's Law. They are more consistent with the observation that only a small fraction of the possible polymorphs generated by crystal structure prediction methods is actually observed (Day *et al.* 2005). This suggests a method for ‘screening’ trial structures from a polymorph prediction, via calculations of nucleation free energies.

### (d) Impure melts

Assigning values of *T*_{m} =480 K and Δ*S*_{f} =−100 J mol^{−1} K^{−1} (as above) allows *T*′ and *T*″ to be plotted using equations (2.11) and (2.18) (figure 5). At high mole fractions, *T*′ is the ideal depression of melting point. At low mole fractions, *T*′ is the ideal solubility curve depicted in an unconventional way (compare Black *et al.* 2006). The ‘metastable zone’ is the region between the two curves, quantified as(3.5)In this region, nucleation cannot occur, although seeds will grow. The figure describes the metastable zone for impure melts (*x*>0.75) and solutions (*x*<0.1).

In the pure melt, MZ=0. As *x* decreases, MZ increases rapidly at first in comparison with the thermodynamic lowering of melting point. For *x*<0.7, MZ is remarkably constant, reaching a maximum of 26°C at *x*=0.2. As *x* decreases further, reaching values corresponding to crystallization from solution, MZ decreases gradually. For example, MZ=21°C at *x*=0.017 corresponding to 100 g l^{−1} for a solute with MW one-sixth of the solvent. This implies a general justification for the widespread use of seeding in the crystallization of molecular solids (Beckmann 2002).

Note that this simulation was carried out by assuming values for *T*_{m} and Δ*S*_{f}, but not for *s*. For a simple cubic nucleus with *s*=1, equation (2.16) becomes(3.6)Δ*G*_{nucleation} may be plotted as a function of *n* for a given *x* (hence *k*) *k* and *T*, giving plots similar to figure 2 . The appearance and scale of the plots depend entirely on the difference between *T* and *T*″.

As for pure melts, it seems reasonable to assume that the smallest possible nucleus is when *n*_{c}=2. Substituting *n*_{c}=2 and *s*=1 into equation (2.17) gives(3.7)Hence, by comparison with equation (2.18)(3.8)This is analogous to equation (3.4) for the pure system. As *x* decreases, *k* increases and *T*_{2}, the temperature below which nucleation is predicted to be inevitable and instantaneous, decreases. The variation of *T*′, *T*″ and *T*_{2} with *x* is shown in figure 6

Note the similarity with figure 5, but the larger temperature scale. The temperature-composition space is now divided into four regions—from top to bottom:

region 1, nuclei dissolve;

region 2, nuclei cannot form but seeds will grow;

region 3, nuclei will form over time; and

region 4, nucleation is inevitable and instantaneous.

Only in region 3 is nucleation is expected to be a function of time, cooling rate and scale.

## 4. Conclusions

There is a fundamental difference between this method for simulation of nucleation and traditional nucleation theory. In this method, the unfavourable entropy of nucleation is considered explicitly to derive an expression for the free energy of nucleation that is related to the enthalpy and entropy of fusion. These quantities can be measured easily, and Δ*G*_{nucleation} can then be calculated using a spreadsheet.

The model replicates the variation of nucleation free energy with nucleus size that is characteristic of traditional nucleation theory. There is no explicit length-scale, with the emphasis on the molecule as the definition of scale. One area for future work is the adaptation of this approach to non-molecular solids.

The method suggests some features of nucleation that are not predicted by classical theory. In pure systems, the size of the critical nucleus depends on *T*_{m}, under-cooling and the structure constant only. A connection with the kinetic stability of amorphous molecular materials is suggested.

Nucleation in impure systems is described by incorporating an explicit term for the entropy of unmixing. The result is a metastable zone that is qualitatively different from that in pure systems, being independent of time, scale and cooling rate. This description of the metastable zone spans impure melts and crystallization from solution. The variation in metastable zone across this range is surprisingly small.

A further feature of the model is the distinction between instantaneous and slow nucleation. The conditions for either process are quantified in terms of mole fraction and under-cooling. Where nucleation is either desired (e.g. in polymorph screens) or unwanted (e.g. in stabilized amorphous dispersion; Konno & Taylor 2006), this may be a relevant concept.

It is hoped that this model will stimulate experimental studies to compare nucleation of the same materials in pure melts, impure melts and from solution. This simple method for describing the free energy of nucleation explicitly may also be useful in molecular mechanics studies of nucleation and in polymorph prediction.

## Acknowledgments

The author is grateful to Roger Davey, Frans Muller, Ron Roberts and Brian Cox for their helpful discussions. The reviewers of this paper are also thanked for their helpful comments.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.0008 or via http://www.journals.royalsoc.ac.uk.

- Received April 28, 2007.
- Accepted July 3, 2007.

- © 2007 The Royal Society