A unifying identity for the work of cluster formation in heterogeneous and homogeneous nucleation theory

Nucleation is a ubiquitous phenomenon that manifests itself wherever there is a thermodynamic barrier to phase change—from the formation of bubbles in superheated liquids to new particle formation from the condensation of supersaturated vapor. Nucleation in the atmosphere is responsible for the formation of new particles that can subsequently grow to exert a significant impact on global climate directly, through the scattering and absorption of solar radiation, and indirectly by serving as sites for cloud droplet condensation.¹ There is evidence that both homogeneous and heterogeneous nucleation pathways contribute to this process.^2,3 Indeed, the interesting two-stage model of Kulmala and co-workers, heterogeneous nucleation of (most likely) oxidized organic vapors on stabilized 1-2 nm neutral cluster condensation sites formed from the gas phase,³ effectively eliminates distinction between heterogeneous and homogeneous nucleation mechanisms as the condensation sites approach molecular size.

The present analysis applies to nucleation models that separate the free energy of cluster formation into volume/number and surface work components, but the general method should be adaptable to inclusion of other free-energy terms such as solute effects in multicomponent nucleation and molecular adsorption. The treatment is not limited to classical nucleation theory (CNT), but homogeneous nucleation in the CNT is used as the model of reference throughout this paper, as components of the true free energy are ratioed to their classical analogs.

The reversible work of cluster formation takes the general form for a single-component condensate,⁴ 

with Δμ = kT ln S, where S is the saturation ratio, n is the number of molecules in the cluster, k is the Boltzmann constant, and T is the temperature. The lead term on the right-hand side is the “volume” work—here the number of molecules present in the nucleus, rather than volume, is used to represent nucleus size. Under the assumption that the cluster is incompressible, these are equivalent. The second term on the right-hand side is the surface work.

The CNT analog to Eq. (1) for homogeneous nucleation has the form

where the asterisk refers to the clusters of critical size and σ_lv is the surface tension between the liquid drop and surrounding vapor phase. The critical radius of curvature, r^*, and the number of molecules present in the nucleus, g^*, are connected under the assumption of incompressibility through the molecular volume of the fluid v_l as $4π(r*)3/3=g*vl$⁠. In the following, g (g^*) will refer to the number of molecules in a non-critical (critical) cluster modeled according to classical homogeneous nucleation theory. n and n^* will be used to distinguish clusters in other cases that will include heterogeneous nucleation on curved substrates and departures from the classical theory. The critical radius, r^*, is given by the Kelvin relation

The focus throughout the remainder of this paper will be on the following three non-dimensional ratios:

The numerators and denominators in Eq. (4) are evaluated at the same chemical potential, Δμ, which then cancels in the last equality [see also Eq. (7)]. The denominators refer to CNT for homogeneous nucleation, which is the reference model.

The main result of this paper is the following identity, henceforth unifying identity, connecting the f-ratios appearing in Eqs. (4):

There is a surprising simple derivation of this result that lends insight into its origin, structure, and generality well beyond classical nucleation theory, where f_W = f_N = f_S = 1 and Eq. (5) reduces to the trivial identity 1 = −2 + 3. The derivation makes use of the following thermodynamic relations for the work of critical droplet formation from a uniform surrounding vapor in the reference model:

Equations (6), which are at the thermodynamic foundation of classical homogeneous nucleation theory, were obtained early on by Gibbs and can be found in his collected studies.⁵ To obtain the main result, simply divide Eq. (1) by $Whomo*$ and then make use of the Gibbs relations to obtain a more explicit form for the general unifying identity,

The second equality brings in the Gibbs relations and the third gives the desired result. Reflection shows that only the classical reference model denominators appearing in Eqs. (4) are involved in the derivation. As long as the application-specific numerators, which appear generally in Eq. (1), refer to a surrounding vapor at the same chemical potential as the reference model, Eqs. (5) and (7) remain valid.

The simplicity of its derivation, which incorporates no other physics beyond the separation of work terms in Eq. (1) and the Gibbs relations themselves, belies the utility of the unifying identity to a broad range of applications, some of which are sampled in Sec. III. This range is limited only by Eq. (1). The great value of the Gibbs relations here lies in determining both the reference denominators and the integer coefficients for the f-ratios appearing in the identity. Before discussing the applications, it is worthwhile to establish the special circumstance under which the Gibbs relations and the unifying identity are equivalent.

The Gibbs relations and their equivalence to the special case of f-ratio equality: It was shown above that the unifying identity follows from the Gibbs relations. The converse is not generally true as the new result is more general. But in the special case of f-ratio equality, which includes classical homogeneous nucleation theory and heterogeneous nucleation on flat substrates, the Gibbs relations and the unifying identity are equivalent as follows:

Inspection of Eq. (5) shows that if two f-ratios are equal, all three are equal. In the equality f_W = f_N, for example, the denominators are equal from the first Gibbs relation, $Whomo*=g*Δμ/2$⁠, so the same must hold for the numerators, and this can only happen at the critical cluster size to give $W*=n*Δμ/2$⁠. This proves the converse for the first of Eqs. (6). Similarly for f_W = f_S and the second Gibbs relation, $W*=Φ*/3$⁠, again only at the critical size. But all three ratios are equal if any two of them are, so the converse is fully shown: the Gibbs relations and the unifying identity are equivalent in the special case that the f-ratios are equal.

This section demonstrates applications that can benefit from the unifying identity. The identity itself was discovered in connection with heterogeneous nucleation studies, and new results appear in Sec. III A. These include Eq. (10), which connects the rather complicated algebraic expressions for the work of spherical cap formation given in the  Appendix [Eqs. (A1)–(A3)] and verifies the connection through algebraic reduction and numerical calculations summarized in Fig. 2. Sections III B and III C examine (mostly) known results in light of the new identity. Both examples cover ground between Eq. (10) and the much more general result obtained in Sec. II. Section III B is an application to clusters of non-critical size, and Sec. III C is an application to a molecular based extension of classical homogeneous nucleation theory.

Consider, first, the specialization of Eq. (1) to heterogeneous nucleation of a spherical cap on a spherical substrate in the classical Fletcher model.⁶ The geometric parameters of the model are indicated in Fig. 1. The reversible work of critical cap formation takes the following form:

where n^* is the number of liquid condensate molecules present in the critical-size cap. The subscript designates heterogeneous nucleation in the CNT.

The second term on the right-hand side of Eq. (8) is the surface work, which in the Fletcher model is given as⁶ 

Here s, l, and v refer to the solid substrate, liquid condensate, and surrounding vapor phases, respectively. σ_ij and Ω_ij are the surface tension and interfacial area between phases i and j. The second equality uses Young’s relation⁷ σ_sv − σ_sl = σ_lv cos(θ). The factor appearing before Ω_sl includes contributions from molecular interactions between the seed and condensate, parameterized in terms of the contact angle θ.

The f-ratios, as defined according to Eqs. (2) and (8), take the form $fW=Whetero*/Whomo*$⁠, f_N = n^*/g^*, and $fS=Φhetero*/Φhomo*$⁠. These ratios are given explicitly in the  Appendix for the Fletcher model with f_W and f_N from Ref. 8. The surface work ratio, f_S, follows from Eqs. (9) and (2) and expressions for the interfacial areas (see the  Appendix for a derivation of this quantity). Numerators and denominators are again evaluated at the same chemical potential difference, Δμ, which then cancels in the final result to obtain the following expression for the unifying identity in heterogeneous nucleation:

Numerous studies have made use of one or more of the f-ratios for heterogeneous nucleation, going back to Fletcher’s pioneering work sixty years ago.⁶ Despite this activity, the connection between them through Eq. (10) appears to have gone unnoticed until now. The identity, together with the ratios themselves, is displayed in Fig. 2. Equation (10) can be demonstrated to hold for Fletcher’s model by substitution from the detailed mathematical expressions given in Eqs. (A1)–(A3), followed by considerable algebraic reduction. On the other hand, Eq. (10) follows immediately as a specialization of Eqs. (5) and (7). The fact that the parent theory makes no reference to geometric details of the spherical cap model implies a broad applicability for the unifying identity, not only to the Fletcher model but well beyond.

A physical example of f-ratio equality, discussed in Sec. II, occurs in the case of heterogeneous nucleation on a flat substrate. For m = cos θ (see the  Appendix),

The merging of f-ratios with increasing substrate radius (decreasing curvature) is becoming evident when comparing Figs. 2(a) and 2(b). The classical homogeneous nucleation result is recovered from Eq. (11) in the non-wetting limit, m = −1.

A characteristic of systems with f-ratio equality is that nucleus shape is independent of nucleus size. In addition to homogeneous nucleation in the CNT (where f_W = f_N = f_S = 1), shape preservation is seen to apply to heterogeneous nucleation on a flat substrate in the Fletcher model, in which case cap shape, including contact angle θ, is preserved with size. Note that line tension can cause the contact angle to become size dependent,⁹ breaking the size-independent scaling described here. But in the Fletcher model (with flat substrate), the f-ratios follow Eq. (11), and the surface and volume numerators scale as g^2/3 and g, respectively. Such systems can be described by a further specialization of Eq. (1), with size-independent shape factor α,

The general form of Eq. (1) forces no requirement on the numerators of the unifying identity that they apply only at critical size. With extension to clusters of non-critical size, the f-ratios in accord with the scaling exhibited in Eq. (12) can be written immediately as

Substitution into Eq. (7) gives a convenient scaling result valid for clusters of any size,

Equation (14) is a known result showing, for a broad class of systems satisfying Eq. (12), that the nucleation barrier has the same overall shape independent of substance and saturation ratio. See, for example, Ref. 10 for a conventional derivation of this result that first locates the critical cluster size and barrier height, W_CNT(g^*), from Eq. (12), and uses these to eliminate the substance- and saturation ratio-dependent parameters α and Δμ.

A major limitation of classical homogeneous nucleation theory is its failure to predict a spinodal, the locus of states at the boundary between the metastable (nucleation) and unstable (spontaneous spinodal decomposition) regions of the phase diagram. Talanquer¹¹ obtained the following expression for nucleation barrier height, accommodating a spinodal while significantly improving agreement with cloud chamber studies of the temperature dependence of nucleation rates closer to the coexistence curve:

Δμ_S is the temperature-dependent driving free energy for nucleation at the spinodal, at which location the barrier height W^* vanishes. The molecular features of the underlying (nonclassical) theory¹² are based on its use of the nucleation theorem,⁴ 

which has been shown from a molecular-based derivation to follow as a direct consequence of the law of mass action.¹³ The approximate equality neglects subtracting the small number of molecules present in a volume, equal to the nucleus volume, of displaced vapor phase.⁴ 

A key assumption of the underlying theory is the validity of the Kelvin relation to obtain the molecular content of the critical nucleus, n^*.¹² In other words, the assumption is made that n^* = g^* where g^* is given by the Kelvin relation,

which is consistent with the last of Eqs. (4) for f_N = 1. C(T) is a substance dependent prefactor that is otherwise a function of temperature alone. In this context, it is noteworthy that the Kelvin relation has been shown experimentally to hold for homogeneous and heterogeneous nucleation down to a few nanometers.^14,15

The objective here is to derive Eq. (15) from the unifying identity to demonstrate the latter as an analysis tool. With the Kelvin assumption expressed compactly as f_N = n^*/g^* = 1, the identity becomes

At the spinodal, f_W = 0, as the nucleation barrier vanishes there, and f_S = 2/3. Multiplication of Eq. (18) by g^*Δμ/2 gives

To obtain f_S, differentiate both sides of Eq. (19) with respect to Δμ using the nucleation theorem [Eq. (16)] on the left-hand side, followed by replacement of n^* with g^* (per Kelvin assumption). Next use the Kelvin relation—in the form given by Eq. (17), from which follows the identity dg^*/dΔμ = −3g^*/Δμ. Collecting terms yields the following differential equation for f_S:

Integration of Eq. (20) using the boundary condition f_S = 2/3 at the spinodal, where Δμ = Δμ_S, gives

Substituting this new expression into Eq. (18) recovers Talanquer’s result. While the connection between Eqs. (15) and (21) is not immediately apparent otherwise, they are immediately connected through the new identity.

We conclude this section with an application of the unifying identity to surface excess quantities using this same model.

Surface excess quantities in the Kelvin scaling model: Comparison of Eqs. (15) and (21) reveals that f_W and f_S have different forms consistent with the identity and f_N = 1. The latter equality is just the Kelvin scaling assumption¹² that underlies the spinodal-compatible nucleation model.¹¹ Interpretation of the Kelvin assumption and its allowance of departure from CNT were also developed in Ref. 12 using the Gibbs concept of superficial density.⁴ Departure from CNT is described by the following equation:

where D is a temperature-dependent parameter that describes departure from the classical theory. Two relationships involving surface excess quantities were also obtained in Ref. 12. These are

and

where $nL*$ is defined as the number of molecules contained within the volume bounded by the surface of tension (of radius R_S) for uniform interior density of the drop and $nS*$ divided by $4πRS2$ is the superficial density. Thus Eq. (24) frames the departure from CNT in terms of the superficial density. Neglecting the displaced vapor component [cf. Eq. (16)], the molecular occupation numbers are related as

where the first equality expresses the Kelvin scaling assumption, f_N = 1. From Eqs. (23) and (25), and the first of the Gibbs relations [Eq. (6)], $fW=nL*/g*$ and two of the f-ratios are determined. The new identity [Eq. (5)] can be used immediately to obtain the third ratio,

Combine the Gibbs relations to obtain $ΦCNT*=(3/2)g*Δμ$⁠, and use Eq. (26) to obtain a new result for the surface free energy component in the model,

Although inspired by the new identity, Eq. (27) can of course be validated by other methods. This is particularly easy to do here: returning to the general expression, Eq. (1), and making substitutions from Eqs. (23) and (25) yield the desired result.

A unifying identity connecting the surface and volume work components of the reversible work of general cluster formation has been derived using a fundamental result, proposed by Gibbs, for the work of forming a spherical liquid drop from vapor in classical homogeneous nucleation theory. The new result yields productive insights into much broader applications that include Fletcher’s model of heterogeneous nucleation, extension to clusters of non-critical size, and molecular-based extensions of classical nucleation theory. It can be expected that the new identity will hold for extensions of the Fletcher model that include line tension and microscopic contact angle⁹ and molecular adsorption.¹⁶ Moreover, its power as an analysis tool that enables heterogeneous and homogeneous nucleation to be examined within the same unified theoretical framework has been demonstrated, as has its potential to serve as a consistency check on mathematical formulas and numerical calculations and as a guide to the interpretation of theory and measurements.

The examples provided in Sec. III demonstrate versatility in how the new unifying identity can be used. While its denominators are referenced to classical expressions, its numerators are adaptable to any application that is compatible with the generic cluster model of Eq. (1). New fundamental insights can be expected on this basis into a range of current active topics including nucleation on ions and ionic clusters,¹⁵ ice nucleation,¹⁷ and Köhler/nano-Köhler theory describing the processes by which water vapor condenses on cloud condensation nuclei (CCN) and forms cloud droplets¹⁸ or organic vapors condense on nano-clusters that can subsequently grow to form CCN. The only requirements are (1) that the work of cluster formation should be separable into surface and volume components and (2) that the parent phase chemical potential should be the same in the application and reference models.

R.L.M. acknowledges support by the Atmospheric Systems Research (ASR) Program of the U.S. Department of Energy (Grant No. DE-SC0012704). P.M.W. and P.E.W. acknowledge support by the European Research Council under the European Community’s Seventh Framework Programme (No. FP7/2007–2013)/ERC Grant Agreement No. 616075 and the Austrian Science Fund (FWF) (Project Nos. P19546 and L593).

With reference to Fig. 1, define x = R/r^*, where R is the seed radius, r^* is the critical radius of curvature of the liquid-vapor interface of the cap, and m = cos(θ) for contact angle θ. In terms of the additional non-dimensional quantity,

the f-ratios take the following forms:

The expressions in (A1) and (A2) follow the work of Vehkamäki et al.⁸ A derivation of (A3) is given below. In the limit that R → ∞, x → ∞, with l’Hôpital’s rule used to evaluate indeterminate products involving powers of x, Eqs. (A1)–(A3) reduce to

which is the result for cap formation on a flat substrate, discussed as a special case in the text. For R = x = 0, or for m = −1, the homogeneous nucleus is recovered and f_W = f_N = f_S = 1. In this case, Eq. (5) becomes a trivial identity: 1 = −2 + 3.

The interfacial areas appearing in Eq. (9) take the explicit forms⁶ 

and

where the equalities (see Fig. 1 for relevant angles), cos ϕ = (R − r^*m)/d = (x − m)/g_F and $cos⁡ψ=−(r*−Rm)/d=−(1−mx)/gF$⁠, have been used. Substitution using the last equality of Eq. (9) gives