Classical and generalized Gibbs’ approaches and the work of critical cluster formation in nucleation theory

Classical and generalized Gibbs’ approaches and the work of criticalcluster formation in nucleation theoryJürn W. P. Schmelzer, Grey Sh. Boltachev, and Vladimir G. Baidakov Citation: J. Chem. Phys. 124, 194503 (2006); doi: 10.1063/1.2196412 View online: http://dx.doi.org/10.1063/1.2196412 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v124/i19 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 124, 194503 �2006�

D

Classical and generalized Gibbs’ approaches and the work of criticalcluster formation in nucleation theory

Jürn W. P. Schmelzera�

Institut für Physik der Universität Rostock; Universitätsplatz, 18051 Rostock, Germany

Grey Sh. BoltachevInstitute of Electrophysics, Ural Branch of the Russian Academy of Sciences, Amundsen Street 106,620016 Ekaterinburg, Russia

Vladimir G. BaidakovInstitute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, Amundsen Street 106,620016 Ekaterinburg, Russia

�Received 20 February 2006; accepted 23 March 2006; published online 16 May 2006�

In the theoretical interpretation of the kinetics of first-order phase transitions, thermodynamicconcepts developed long ago by Gibbs are widely employed giving some basic qualitative insightsinto these processes. However, from a quantitative point of view, the results of such analysis, basedon the classical Gibbs approach and involving in addition the capillarity approximation, are oftennot satisfactory. Some progress can be reached here by the van der Waals and more advanceddensity functional methods of description of thermodynamically heterogeneous systems having,however, its limitations in application to the interpretation of experimental data as well. Moreover,both mentioned theories—Gibbs’ and density functional approaches—lead to partly contradictingeach other’s results. As shown in preceding papers, by generalizing Gibbs’ approach, existingdeficiencies and internal contradictions of these two well-established theories can be removed anda new generally applicable tool for the interpretation of phase formation processes can bedeveloped. In the present analysis, a comparative analysis of the basic assumptions and predictionsof the classical and the generalized Gibbs approaches is given. It is shown, in particular, that—interpreted in terms of the generalized Gibbs approach—the critical cluster as determined via theclassical Gibbs approach corresponds not to a saddle but to a ridge point of the appropriatethermodynamic potential hypersurface. By this reason, the classical Gibbs approach �involving theclassical capillarity approximation� overestimates as a rule the work of critical cluster formation innucleation theory and, in general, considerably. © 2006 American Institute of Physics.�DOI: 10.1063/1.2196412�

I. INTRODUCTION

Nucleation-growth and spinodal decomposition pro-cesses are two basic mechanisms first-order phasetransitions—such as condensation and boiling, segregation insolid and liquid solutions, or crystallization and melting—may proceed. They determine the kinetics of self-structuringprocesses of matter from nanoscale up to galactic dimensionswith a wide spectrum of applications in both fundamentaland applied research �physics, astronomy, chemistry, biology,meteorology, medicine, and materials science� and technol-ogy.

In the interpretation of experimental results on the dy-namics of first-order phase transitions starting from meta-stable �stable with respect to small and unstable with respectto sufficiently large fluctuations exceeding some criticalsizes, the so-called critical cluster sizes� initial states, up tonow predominantly the classical nucleation theory is em-ployed treating the respective processes in terms of clusterformation and growth.1–9 In the specification of the cluster

a�
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properties, thermodynamic methods are intensively em-ployed based on the majority of cases on the thermodynamicdescription of heterogeneous systems developed by Gibbs.10

As one additional simplifying assumption it is assumedhereby frequently that the bulk properties of the clusters arewidely similar to the properties of the newly evolving mac-roscopic phases.

This or similar assumptions, underlying the classical ap-proach to the description of cluster formation and growth,are supported by the results of Gibbs’ classical theory ofheterogeneous systems applied to processes of critical clusterformation. Indeed, following Gibbs’ thermodynamic treat-ment one comes to the conclusion that the critical clustershave bulk properties widely similar to the properties of thenewly evolving macroscopic phases. Provided this is reallythe case, then one can assume that also the interfacial prop-erties of the clusters are similar to the respective parametersfor phase coexistence at planar interfaces. This assumption isdenoted as capillarity approximation. Treating clusters of ar-bitrary sizes as small particles with bulk and interfacial prop-erties of the macroscopic phase, the process of cluster growth
and dissolution is considered then to proceed basically via
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194503-2 Schmelzer, Boltachev, and Baidakov J. Chem. Phys. 124, 194503 �2006�

D

addition or emission of single units �atoms and molecules�6–9

with properties corresponding to those of the newly evolvingmacroscopic phase.

Remaining inside Gibbs’ classical approach, one cannotmake any quantitative predictions concerning the work ofcritical cluster formation without specifying the value of theinterfacial tension or the specific interfacial energy. By thisreason, the capillarity approximation is widely employed inthe theoretical interpretation of nucleation data. However, ina variety of cases, such approach leads to highly unsatisfac-tory results. In order to come to an agreement between ex-perimental and theoretical results on nucleation-growth pro-cesses, this assumption often has to be released byintroducing a curvature dependence of the surface tension.However, such assumption leads to other internal contradic-tions in the theory which cannot be resolved remaining in-side the concepts of Gibbs’ thermodynamic treatment ofcluster properties.11–13 This way, Gibbs’ classical treatmentof surface phenomena is confronted with serious principaldifficulties in application to nucleation. Moreover, Gibbs re-stricted his analysis to “equilibria of heterogeneous sub-stances” �as reflected already by the title of his thermody-namic investigations10 �cf. also Refs. 14–18��. By thisreason, it cannot be applied directly to the description ofcluster growth and dissolution processes without involving atpart serious additional assumptions �cf. Refs. 19 and 20�.

Gibbs employed in his approach a simplified model con-sidering the cluster as a homogeneous body divided from theotherwise homogeneous ambient phase by a sharp interfaceof zero thickness. The thermodynamic characteristics of thesystem under consideration are represented then as the sumof the contributions of both homogeneous phases and correc-tion terms, the so-called superficial quantities, which are at-tributed, at part, formally to the interface. The superficialquantities reflect deviations from additivity and the diffuse-ness of the interface in the framework of Gibbs’ model ap-proach. In contrast to alternative statements21 we believe thatsuch approach is theoretically well founded and absolutelycorrect provided one is able to determine the bulk parametersof the clusters and the superficial quantities in an adequateway.

The alternative continuum’s concept of the thermody-namic description of heterogeneous systems was developedby van der Waals and Kohnstamm.22 It has been applied forthe first time to an analysis of nucleation by Cahn andHilliard.23,24 In application to nucleation-growth processes,Cahn and Hilliard came to the conclusion that the bulk stateparameters of the critical clusters may deviate considerablyfrom the respective values of the evolving macroscopicphases and from the predictions of Gibbs’ theory. Such de-viations occur, in particular, in the vicinity of the classicalspinodal curve dividing thermodynamically metastable andthermodynamically unstable initial states of the systems un-der consideration. These results of the van der Waals ap-proach were reconfirmed later on by more advanced densityfunctional computations.25

Moreover, Cahn and Hilliard developed also the alterna-tive to the nucleation-growth model theoretical description of
spinodal decomposition. According to the common belief

�having again its origin in the classical analysis of Gibbs�,the nucleation-growth model works well for the descriptionof phase formation starting from metastable initial states,while thermodynamically unstable states are believed to de-cay via spinodal decomposition. As one consequence, theproblem arises how one kinetic mode of transition �nucle-ation growth� goes over into the alternative one �spinodaldecomposition� if the state of the ambient phase is changedcontinuously from metastable to unstable states, i.e., how thetransition proceeds in the vicinity of the classical spinodalcurve. The classical Gibbs approach predicts here some kindof singular behavior, which is, however, not confirmed by theCahn-Hilliard description, statistical-mechanical modelanalyses26,27 and experiment.28 From a more general point ofview, we are confronted here with an internal contradictionin the predictions of two well-established theories which hasto be, hopefully, resolved.

The resolution of this contradiction is possible in theframework of a generalization of Gibbs’ classical thermody-namic method developed by us in the recent years. WhileGibbs’ thermodynamic theory is restricted in its applicabilityto equilibrium states exclusively, the generalized Gibbs ap-proach is aimed from the very beginning at a description ofthermodynamic nonequilibrium states consisting of clustersof arbitrary sizes and composition in the otherwise homoge-neous ambient phase.29–31 It was demonstrated that, by de-veloping such generalization of Gibbs’ thermodynamic ap-proach, Gibbs’ and van der Waals’ methods of description ofcritical cluster formation can be reconciled.19,20,30–34 Thegeneralized Gibbs approach was shown to lead for modelsystems to qualitatively and partly even quantitatively simi-lar results as compared with density functional approaches.In particular, it leads to a significant dependence of both thebulk and surface properties of the critical clusters on super-saturation and—in contrast to the classical Gibbs approachwhen the capillarity approximation is employed—to a van-ishing of the work of critical cluster formation for initialstates in the vicinity of the spinodal curve.

In the present paper, we give a general comparison of theclassical and generalized Gibbs approaches with respect totheir basic assumptions and the resulting predictions con-cerning the work of critical cluster formation in nucleationtheory. For the realization of this task, in Sec. II the basicequations are summarized allowing one to determine thecritical cluster properties in both methods of theoretical treat-ment. In Sec. III, it is shown that—similar to the classicalGibbs approach—the critical cluster in the generalized Gibbsmethod corresponds to a saddle point of the characteristicthermodynamic potential, a maximum with respect to varia-tions of the size parameter and a minimum with respect tovariations of the intensive state parameters of the clusterphase. It is shown that—in terms of the generalized Gibbsapproach—the critical cluster in the classical Gibbs treat-ment refers to a particular ridge point in the thermodynamicpotential hypersurface. It follows as a consequence �Sec. IV�that—with minor exceptions—the work of critical clusterformation determined via the classical Gibbs approach andutilizing the capillarity approximation overestimates the cor-
rect value of this quantity and, in general, considerably.

194503-3 Critical cluster formation J. Chem. Phys. 124, 194503 �2006�

D

II. GENERALIZED GIBBS’ APPROACH IN THEDESCRIPTION OF CLUSTER FORMATION

A. Basic equations and general results

The change �U of the internal energy U in cluster for-mation in an isolated system of constant volume V can beexpressed according to the generalized Gibbs approach30,31

as

�U = S��T� − T�� + �p� − p��V� + �A + �i=1

k

ni��i� − �i�� .

�1�

Here S denotes the entropy, T the absolute temperature, p thepressure, � the surface tension, and A the surface area of thecluster; ni are the mole or particle numbers of the k differentcomponents and �i their chemical potentials. With a sub-script �, the parameters of the cluster phase are specifiedwhile the subscript � refers to the parameters of the ambientphase.

Equation �1� describes the change of the internal energyof a system if a cluster of arbitrary size and composition isformed in the initially homogeneous ambient phase. It is notrequired in the derivation that the cluster has to be in equi-librium with the surrounding ambient phase. This relation isvalid, moreover, independent of the choice of the state pa-rameters of the reference phase for the description of thebulk properties of the cluster and independent of the choiceof the dividing surface. In the following considerations, wewill use the �generalized� surface of tension as the dividingsurface and assume the clusters to be of spherical sizes witha radius R.

The generalization of Gibbs’ fundamental equation forthe superficial parameters �specified by the subscript �� readsthen as30,31

dU� = T� dS� + �i=1

k

�i� dni� + �dA + �j=1

k+1

� j�d� j�, �2�

with

T� = T�, �i� = �i� for i = 1,2, . . . ,k . �3�

In Eq. �2�, �� j�� is a complete set of intensive variables,specifying the bulk state of the cluster phase. We will iden-tify them here with the volume densities of particles �i� ofthe different components and the volume density of the en-tropy s�,

�i� = �i� for i = 1,2, . . . ,k, �k+1 = s�. �4�

The term � j=1k+1� j� d� j� describes changes of the superficial

internal energy due to variations of the state parameters ofthe cluster for the general case when the cluster is not inequilibrium with the ambient phase. Hereby it is assumedthat irreversible flow processes are inhibited by certain well-defined constraints �cf. also Refs. 17 and 18�.

The introduction of the term � j=1k+1� j�d� j� into Eq. �2�

was motivated by the requirement that the surface tension ofa cluster being not in equilibrium with the ambient phase
depends, in general, on the bulk state parameters of both the

ambient and the cluster phases. This requirement is reflectedthen in the generalized Gibbs adsorption equation which getsthe form30,31

S� dT� + Ad� + �i=1

k

ni�d�i� = �j=1

k+1

� j�d� j�. �5�

This result—the dependence of the surface tension on thestate parameters of both ambient and cluster phases—is thebasic difference between the classical and generalized Gibbsapproaches as discussed here.

Denoting similarly by �� j�� the set of independent inten-sive variables for the description of the state of the ambientphase, we have

� j� = A� ��

��i�

��6�

as the expression for the determination of the coefficients � j�

in Eq. �5�. It follows as a consequence that Gibbs’ classicaltheory is retained in the generalized Gibbs approach as alimiting case if, at given values of the intensive state param-eters of the ambient phase, the derivatives of the surfacetension with respect to the state parameters of the clusterphase are set equal to zero.

The general thermodynamic equilibrium conditions10

�dU�V,S,�i�= 0 �7�

lead in the generalization of Gibbs’ approach to theexpressions30,31

�T� − T��s� + �p� − p�� +2�

R+ �

i=1

k

�i��i� − �i�� = 0,

�8�

��i� − �i�� =3

R� ��

��i�

��for i = 1,2, . . . ,k , �9�

�T� − T�� =3

R� ��

�s�

��. �10�

Note that the classical Gibbs equilibrium conditions are ob-tained not only in the case that the derivatives of the surfacetension with respect to the parameters of the cluster phase areset equal to zero but also in the limit of large critical clustersizes, i.e., if the radius of the generalized surface of tensiontends to infinity. This way, the generalized Gibbs approachleads to identical equilibrium conditions for the case of phasecoexistence at planar interfaces as derived by Gibbs in hisclassical treatment.

A substitution of the mechanical equilibrium conditions,Eq. �8�, into the general expression for the change of theinternal energy in cluster formation, Eq. �1�, yields theconventional expression for the work of critical cluster
formation


D

�U = 13�A, A = 4R2, �11�

known from Gibbs’ classical approach. However, in the gen-eralized Gibbs approach, the radius of the clusters is deter-mined by

R =2�

�T� − T��s� + �p� − p�� − �i=1

k�i��i� − �i��

. �12�

It follows that �U and R in Eqs. �11� and �12� are functionsof the intensive state parameters of the cluster phase forgiven values of the state parameters of the ambient phase.They refer to the critical cluster only as far as the intensivestate parameters of the cluster are determined appropriately.In the generalized Gibbs approach, this procedure is realizedvia Eqs. �9� and �10�.

Introducing the notations

��i = �i� − �i�, i = 1,2, . . . ,k, ��k+1 = T� − T�,

�13�

we can formulate the necessary thermodynamic equilibriumconditions, Eqs. �9� and �10�, in the compact form

��i =3

R� ��

��i�

��, i = 1,2, . . . ,k + 1. �14�

This expression will be employed in the further computa-tions. The classical Gibbs limiting result we obtain then bysetting ��i equals to zero.

The expressions for the determination of the parametersof the critical clusters and the work of critical cluster forma-tion are independent on the thermodynamic boundary condi-tions. Indeed, if we assume, for example, constancy of exter-nal pressure �p= p�=const� and constancy of externaltemperature �T=T�=const�, then we get with the definitionG=U−T�S+ p�V the equations

�G = S��T� − T�� + �p� − p��V� + �A

+ �i=1

k

ni��i� − �i�� , �15�

dG = �T� − T��dS� + �p� − p��dV� + �dA

+ �i=1

k

��i� − �i��dni� + �i=1

k+1

�i�d�i�, �16�

resulting into relations equivalent �in Eq. �11�, �U has to bereplaced by �G� or identical to Eqs. �8�–�10�, again.

B. Classical and self-consistent capillarityapproximations

Equations �8�–�10� are the general relations allowing oneto determine the state parameters of the critical clusters andthe change of the characteristic thermodynamic potentials incritical cluster formation as derived in the generalized Gibbsapproach. They contain the classical Gibbs result as a limit-ing case �for ��i=0, i=1,2 , . . . ,k+1�. In order to employthese relations, in addition to the knowledge of the bulk
properties of both macrophases in dependence on the respec-

tive state parameters, a similar expression for the dependenceof the surface tension on the state parameters of both coex-isting phases has to be known.

According to the general result, Eq. �5�, both in the clas-sical and generalized Gibbs approaches the interfacial ten-sion depends primarily on the bulk state parameters of thecoexisting phases as far as the surface of tension is chosen asthe dividing surface. A curvature dependence of the surfacetension is connected then exclusively with the changes ofthese bulk properties in dependence on the size of the clus-ters. Taking into account these general results, the depen-dence of the surface tension on the state parameters of thecoexisting phases can be expressed in the simplest formas30,31

� = �i,j=1

k+1

�ij��i� − �i�� j� − � j��, with �ij = � ji. �17�

In the derivation of Eq. �17� it is merely assumed that thesurface tension has to depend basically on the difference ofthe state parameters of both coexisting phases, i.e., it is as-sumed that the dependence of the surface tension on the stateparameters can be expressed as �� , ��=��

� �� , ��i�−�i��. The first nonvanishing term in a Tay-lor expansion with respect to the differences ��i�−�i�� re-sults then in Eq. �17�. This relation we will employ hereleaving the analysis of the effect of higher-order terms ontothe results to a future study. Note that due to symmetry con-siderations only even terms have to be retained in the expan-sion. Moreover, the surface tension becomes equal to zero ifthe parameters of both the ambient and the newly evolvingphases become identical �here we get the limit of a one-phasestate and interfacial contributions have to vanish�.

The coefficients of the expansion �ij have to obey theconditions that the quadratic form as given by Eq. �17� ispositive semidefinite. In application to the interpretation ofphase formation in real or model systems, the values of thecoefficients �ij in the expansion, Eq. �17�, can be determinedeither from measurements or from statistical-mechanicalmodel computations of the values of the surface tension forequilibrium phase coexistence at planar interfaces knowingthe values of the state parameters of the coexisting phasesand the value of the surface tension for these particularstates. By this reason, the coefficients �ij may depend on thestate parameters of the ambient phase coexisting in equilib-rium with the newly evolving phase at a planar interface, i.e.,�ij =�ij��

� ��.Expressions for the surface tension of the type as given

by Eq. �17�, we denote as self-consistent capillarity approxi-mation. It is assumed in agreement with the general theoret-ical result, Eq. �5�, that the bulk properties of the ambientand cluster phases determine the value of the interfacial ten-sion. It follows that Eq. �17�, being consistent with Eq. �5�,has to be considered as an approximation only as far ashigher-order terms in the expansion with respect to the dif-ferences of the intensive state parameters of both coexistingphases are omitted. In contrast, employing the classical cap-illarity approximation it is supposed that the value of the
interfacial tension is determined by the bulk properties of the


D

coexisting phases they have for the particular state of anequilibrium coexistence of the respective phases at planarinterfaces. Since the properties of the critical clusters and theambient phase change with supersaturation, latter approxi-mation has to be considered as theoretically not consistentwith the basic equation, Eq. �5�, both in the classical andgeneralized Gibbs approaches. Consequently, employing theself-consistent capillarity approximation in the classicalGibbs approach, one can expect to get a better �although notsufficiently correct� description of the critical cluster proper-ties as compared with the case that the classical capillarityapproximation is employed. We will demonstrate the validityof such statement in Sec. IV C.

III. GENERALIZED OSTWALD’S RULE OF STAGESIN NUCLEATION

A. Necessary conditions of validity of the generalizedOstwald’s rule of stages

In determining the state parameters of the critical clus-ters it was demanded originally in developing the general-ized Gibbs approach in application to nucleation32 that wehave to search—at fixed parameters of the ambient phase—for those values of the intensive state parameters of the clus-ter phase at which the expression for the internal energychange �or some equivalent thermodynamic potential differ-ence� has a minimum. This statement we denoted as gener-alized Ostwald’s rule of stages in application to nucleationand formulated it in the following way: In phase transforma-tion processes, the structure and properties of the criticalnucleus may differ qualitatively from the properties both ofthe ambient and newly evolving macrophases. Those classesof critical clusters determine the process of the transforma-tion, which correspond to a minimum of the work of criticalcluster formation �as compared with all other possible alter-native structures and compositions, which may be formed atthe given thermodynamic constraints�.

In such an approach, the state parameters of the criticalclusters are determined via

� ��U

�� j�

�i�=const,i�j;��= 0, j = 1,2, . . . ,k + 1 �18�

or

��U

�� j�=

4

3� ��

�� j�R2 + 2�R

�R

�� j�� = 0. �19�

By a derivation of Eq. �8� or, equivalently, Eq. �12� withrespect to the intensive parameters of the cluster phase, weobtain

�

�� j��T� − T��s� + �p� − p�� +

2�

R

+ �i=1

k

�i��i� − �i�� = 0 �20�

or


�T� − T��s�

�� j�+ �

i=1

k � ��i�

�� j��i� − �i�� +

2

R

��

�� j�

−2�

R2

�R

�� j�+ s�

�T�

�� j�−

�p�

�� j�+ �

i=1

k ��i��i�

�� j�� = 0.

�21�

Due to the Gibbs-Duhem equation for the cluster bulk phase,the second line in the above equation is equal to zero and wearrive straightforwardly at Eqs. �22� and �23�, i.e.,

2

R

��

��i�−

2�

R2

�R

��i�+ ��i� − �i�� = 0, i = 1,2, . . . ,k , �22�

2

R

��

�s�

−2�

R2

�R

�s�

+ �T� − T�� = 0. �23�

With the notations, Eq. �13�, Eqs. �22� and �23� can be trans-formed into the compact form

2

R

��

�� j�−

2�

R2

�R

�� j�= �� j, j = 1,2, . . . ,k + 1. �24�

The above equations are generally valid as long as Eqs. �8�and �12� are supposed to hold. Provided that the thermody-namic equilibrium conditions, Eqs. �14�, are fulfilled, thenEqs. �22� and �23� �or Eqs. �24�� are reduced to

��U

�� j�=

4

3� ��

�� j�R2 + 2�R

�R

�� j�� = 0. �25�

Vice versa, Eqs. �22� and �23� yield directly the equilibriumconditions for the state parameters of the critical clusters,i.e., the conditions given by Eqs. �25� and the thermody-namic equilibrium conditions, Eqs. �9� and �10�, lead to iden-tical results. In other words, critical clusters are characterizedin the generalized Gibbs approach by the set of conditions

�

�� j��4

3�R2� = 0, j = 1,2, . . . ,k + 1, �26�

where the radius of the generalized Gibbs surface of tensionis defined by Eq. �12�.

B. Sufficient conditions of validity of the generalizedOstwald’s rule of stages

So far, we have shown the equivalence of the equilib-rium conditions, Eqs. �8�–�10�, and the generalized Ostwaldrule �Eqs. �11�, �12�, and �26��. In order to prove that theabove conditions indeed refer to a minimum of the internalenergy with respect to variations of the intensive state pa-rameters of the cluster phase, we have to show that the qua-dratic form

� = �i=1

k+1

�j=1

k+1 �2�U

��i�� j�

�c��i� − �i�

�c�� j� − � j��c�� 27�

is positive semidefinite. Here by the superscript �c� the pa-rameters of the critical clusters are specified and the deriva-tives have to be taken at the critical cluster coordinates as
well.


D

In order to compute the coefficients ��2�U /��i�� j�� c�, we start with a slightly modified version of Eq.�25�, writing it as

��U

�� j�=

4

3R� ��

�� j�R3 + 2�R2 �R

�� ja� , �28�

and, with Eq. �25�, we arrive at

�2�U

��i�� j�=

4

3R

�

��i��

�� j�R3 + 2�R2 �R

�� j�� . �29�

Taking into account Eq. �24�, we obtain further

�2�U

��i�� j�=

4

3R

�

��i��3R3 ��

�� j�− R4�� j� . �30�

This equation is equivalent to

�2�U

��i�� j�=

4R3

3

�

��i�� 3

R

��

�� j�− �� j� �31�

resulting in

�2�U

��i�� j�=

4R3

3�−

3

R2

�R

��i�

��

�� j�+

3

R

�2�

��i�� j�−

�� j

��i�� .

�32�

Now, let us consider separately the different contribu-tions to the quadratic form � given by Eqs. �27� and �32�.For the first term �1, we get with Eq. �25�

�1 = − 4R�i=1

k+1

�j=1

k+1�R

��i�

��

�� j��i� − �i�

�c�� j� − � j��c��

=8R2

��i=1

k+1

�j=1

k+1��

��i��i� − �i�

�c��

�� j�� j� − � j�

�c��

=8R2

��

i=1

k+1��

��i��i� − �i�

�c��2

� 0. �33�

The second term in Eq. �32� is determined by the quantities��2� /��i�� j��. According to Eq. �17�, the identity

�ij =�2�

��i�� j��34�

holds. It follows from the remarks made with respect to thesurface tension � that the second term

�2 = 4R2�l=1

k+1

�j=1

k+1�2�

��i�� j��i� − �i�

�c�� j� − � j��c�� 0

�35�

is a positive semidefinite quadratic form as well. Finally,taking into account the definitions, Eq. �13�, of the quantities��i, we can conclude that the condition

�3 = −4

3R3�

i=1

k+1

�j=1

k+1�� j

��i��i� − �i�

�c�� j� − � j��c�� 0

�36�

is a direct consequence from the assumed internal stability of35,36
the bulk properties of the cluster phase. Since all three

terms in Eq. �27� result in positive contributions the inequal-ity ��0 holds true. It follows that the state parameters of thecritical cluster, defined by Eqs. �9� and �10�, correspond in-deed to a minimum of the internal energy in the subspace ofthe thermodynamic space defined by Eq. �8� or �12�, i.e., inthe considered subspace of the intensive state variables of thecluster phase.

On the other hand, at fixed values of the state parametersof the cluster phase, corresponding to the critical cluster,slight deviations of the value of the radius result in condi-tions favoring either a spontaneous increase �R=Rc+�R� or afurther decrease �R=Rc−�R� of the radius. Both these spon-taneous processes are possible only if they are accompaniedby a decrease of the thermodynamic potential. Indeed, takingthe derivative of �U �cf. Eq. �1��,

�U =4R3

3 �s��T� − T�� + �p� − p�� +3�

R

+ �i=1

k

�i��i� − �i�� , �37�

with respect to the cluster size R, assuming constancy of theintensive state parameters of both ambient and clusterphases, we get with Eq. �8� or �12�,

��U

�R

�c�= 0, �2�U

�R2 �c�

= − 8� � 0. �38�

With respect to variations of the size variable, the criticalcluster refers to a maximum of the thermodynamic potential.In general, the critical cluster in the generalized Gibbs ap-proach corresponds, consequently, to a saddle point.

Note that Eqs. �37� and �38� hold not only for the saddlepoint but always �at assumed constancy of the cluster bulkstate parameters independent on their actual values� if Eq. �8�or �12� is fulfilled. This way, latter equations determine theridge path in the thermodynamic potential surface in the gen-eralized Gibbs approach.

The results, outlined in the present section, are illustratedin Fig. 1. Here the change of the internal energy is given dueto the formation of the cluster with the state parameters ��in the initially homogeneous ambient phase �here for the caseif the bulk properties of the cluster are completely describedby two such state parameters�. The ridge path in the hyper-surface of the internal energy is specified by a dashed curve.The critical cluster parameters in the generalized Gibbs ap-proach correspond to a minimum with respect to the varia-tion of the state parameters of the cluster phase along theridge path. With respect to variations of the cluster size at
fixed values of the state parameters of the cluster phase, the


D

ridge path corresponds to maxima of the thermodynamic po-tential both for the critical �black curve� and the ridge �whitecurve� clusters.

IV. CRITICAL CLUSTER SIZE AND WORKOF CRITICAL CLUSTER FORMATION: COMPARISONOF CLASSICAL AND GENERALIZED GIBBS’APPROACHES

A. Behavior of the size parameter R of the ridgeof the thermodynamic potential surface in the vicinityof the critical cluster

According to the results, outlined in the previous section,the radius of the cluster at any state along the ridge of thehypersurface of the characteristic thermodynamic potential�here the internal energy� is determined by Eq. �12�. Theintensive bulk state parameters of the critical cluster aregiven further via Eq. �25�. Employing the generalized Gibbsapproach, the partial derivatives �� /��i�� are different fromzero, in general. It follows that the cluster size R does notobey any peculiar features �e.g., extrema� at the critical clus-ter composition.

However, the situation is different in the classical Gibbsapproach. Here �� /��i��=0 is assumed to be fulfilled and

FIG. 1. Hypersurface of the thermodynamic potential difference �U in de-pendence on the state parameters of the cluster phase. The ridge path in thehypersurface of the internal energy change is specified by a dashed curveand determined by Eq. �8� or �12�. The critical cluster parameters in thegeneralized Gibbs approach correspond to a minimum with respect to thevariations of the state parameters of the cluster phase along the ridge path�determined via Eq. �14� or �26��. With respect to variations of the clustersize at fixed values of the state parameters of the cluster phase, the ridgepath corresponds to maxima of the thermodynamic potential both for thecritical �black curve� and the ridge �white curve� clusters �cf. Eqs. �37� and�38��. In terms of the generalized Gibbs approach, the classical Gibbs equi-librium conditions �Eq. �42�� correspond to a point along the ridge of theinternal energy hypersurface, i.e., we have a maximum with respect to varia-tions of the size parameter but not a minimum with respect to the variationsof the intensive bulk state parameters of the cluster phase �cf. Eqs. �44� and�45��. The minimum is found for those state parameters which are deter-mined via the generalized Gibbs approach. By mentioned reasons, the clas-sical Gibbs approach overestimates commonly the value of the work ofcritical cluster formation.

Eq. �25� leads to


�R

�� j�

�c�= 0. �39�

Taking into account both these conditions, Eqs. �27� and �29�yield

� =8

3��R��

i=1

k+1

�j=1

k+1 �2R

��i�� j�

�c��i� − �i�

�c�� j� − � j��c�� 0.

�40�

Since this quadratic form is positive semidefinite, the sizeparameter R, corresponding to the ridge of the potential sur-face, has—in the classical Gibbs approach—a minimum atthe critical cluster composition. Consequently, in the classi-cal Gibbs approach the saddle point is characterized not onlyby the lowest value of �U but also by the lowest value of thesize parameter as compared with neighboring states alongthe ridge. By this reason, it is very hard to understand, in theframework of the classical Gibbs theory of capillarity, whyridge crossing could be under certain conditions a more ap-propriate path of evolution to the new phase since ridgecrossing always results here in both larger activation energiesand in larger sizes of the ridge clusters as compared with thesaddle point. Such problem does not occur in the generalizedGibbs method since here it is possible, in general, to passridge clusters having higher activation energies but smallercluster sizes as compared with the respective saddle point.This way, here ridge crossing may be favored by the kineticsof cluster growth even if ridge crossing is energetically lessfavorite �cf. also Ref. 20�.

B. Work of critical cluster formation

1. Some general conclusions

Employing the classical Gibbs method, we have to set

� ��

�� j�

��= 0, j = 1,2, . . . ,k + 1. �41�

The necessary equilibrium conditions �cf. Eqs. �9� and �10�or Eqs. �13� and �14�� are reduced then to the well-knownrelations

�� j = 0, j = 1,2, . . . ,k + 1. �42�

Instead of Eq. �32�, we have then

�2�U

��i�� j�

�c�= −

4R3

3� �� j

��i�� . �43�

Consequently, �=�3�0 holds and the critical cluster corre-sponds also to a minimum in the space of the intensive statevariables of the cluster phase and to a maximum with respectto variations of the size parameter. In this respect, the prop-erties of the critical clusters are similar in the two differenttheoretical schemes employed.

However, in Gibbs’ classical approach, the search for thesaddle is restricted to a more confined space since the surfacetension is assumed to depend on the properties of one of thecoexisting phases merely. From the point of view of the gen-eralization of Gibbs’ approach, employed in our analysis, the
conditions, Eq. �42�, do not refer to the minimum of the


D

work of critical cluster formation in dependence on the in-tensive state parameters of the ambient phase but to highervalues of the change of the internal energy: In the general-ized Gibbs approach, the equilibrium conditions are given byEq. �24� and they refer to a minimum with respect to varia-tions of the intensive state parameters of both coexistingphases. On the other hand, if the classical Gibbs equilibriumconditions, Eq. �42�, hold then Eq. �8� is transformed into theYoung-Laplace equation

�p� − p�� +2�

R= 0. �44�

Determining the bulk parameters of the clusters �followingthe classical Gibbs results� via Eq. �42�, Eq. �37� reads

�U =4R3

3��p� − p�� +

3�

R� . �45�

Finally, substituting into Eq. �45� the conditions for pressureequilibrium in the classical Gibbs approach, Eq. �44�, wearrive at Eq. �11�, again.

Further, taking the derivatives of �U �given by Eq. �45��with respect to the cluster radius at constant values of thebulk parameters of the coexisting phases and employing thecondition for pressure equilibrium, Eq. �44�, we arrive at Eq.�38�, again. With respect to changes of the size parameter,the state given by Eq. �42� corresponds to a maximum, again.Slight deviations of the radius at fixed by Eq. �42� state pa-rameters of the cluster phase result in conditions favoringeither a spontaneous increase �R=Rc+�R� or a further de-crease �R=Rc−�R� of the radius. Both these spontaneousprocesses are possible only if they are accompanied by adecrease of the thermodynamic potential. Consequently, theclassical Gibbs solution corresponds—in terms of the gener-alized Gibbs approach—to a maximum with respect to varia-tions of the size parameter but not to a minimum with respectto variations of the intensive state parameters of the clusterphase; i.e., from the point of view of the generalized Gibbsapproach, Gibbs’ classical solution corresponds to a ridgepoint of the characteristic thermodynamic potential. Conse-quently, since any of the ridge points in the generalizedGibbs approach corresponds to higher values of the changeof the internal energy �or any other appropriate thermody-namic potential for different boundary conditions�, the workof critical cluster formation, determined via the classicalGibbs approach and employing the self-consistent capillarityapproximation, is larger as compared with the respectivevalue obtained via the generalized Gibbs approach.

These results are illustrated in Fig. 1 as well. In terms ofthe generalized Gibbs approach, the classical Gibbs equilib-rium conditions correspond to a point along the ridge of theinternal energy hypersurface, i.e., we have a maximum withrespect to variations of the size parameter but not a minimumwith respect to the variations of the intensive bulk state pa-rameters of the cluster phase. The minimum is found forthose state parameters which are determined via the general-
ized Gibbs approach.

2. Relation between the values of the surface tensionin the classical and generalized Gibbsapproaches

The change of the internal energy, determined via Eq.�11�, is a function of the intensive state parameters of thecluster phase. Specifying the state parameters, determinedvia the classical Gibbs approach by the superscript �CG�, wemay express the work of critical cluster formation in thegeneralized Gibbs approach �specified by a superscript �GG��as

�Uc�GG� = �Uc

�CG� + �j=1

k+1 ��U

�� j�

CG�� j�

�GG� − � j��CG��

= �Uc�CG� + �

j=1

k+1 4

3� ��

�� j�R2 + 2�R

�R

�� j��

CG

�� j��GG� − � j�

�CG�� . �46�

With Eqs. �24� and �42�, we arrive at


�CG� + �j=1

k+1 �4

3R2 ��

�� j��

CG�� j�

�GG� − � j��CG�� .

�47�

Writing the surface tension in the form

��GG� = ��CG� + �j=1

k+1 ��

�� j�

CG�� j�

�GG� − � j��CG�� , �48�

we can rewrite Eq. �47� as


�CG� + 4RCG2 ��GG� − ��CG�� . �49�

In the considered linear approximation, the difference in thework of critical cluster formation, computed via the classicaland generalized Gibbs approaches, respectively, is equal tothe change in the surface contributions to the cluster with asize as determined via Gibbs’ classical method employingthe self-consistent capillarity approximation. Alternatively,we may write

�Uc�GG�

�Uc�CG� = 1 + 3��GG�

��CG� − 1 . �50�

Since the classical Gibbs approach �employing the self-consistent capillarity approximation� corresponds—interpreted in terms of the generalized Gibbs approach—to aridge point of the thermodynamic potential, the relations

�Uc�GG�

�Uc�CG� � 1,

��GG�

��CG� � 1 �51�

hold. The surface tension for the critical clusters, as deter-mined via the generalized Gibbs approach, cannot be largercompared with the respective values as obtained in the clas-sical Gibbs approach, when in latter one the self-consistent
capillarity approximation is employed.


D

3. Classical and self-consistent capillarityapproximations and the work of critical clusterformation

In deriving the above made conclusions concerning therelation between the estimates for the work of critical clusterformation, we have employed the following prescription forthe determination of the value of the surface tension. Weassumed that the surface tension is determined by relationssuch as Eq. �17� with the difference that the bulk parametersof the cluster phase are determined either by Gibbs’ classicalequilibrium conditions, resulting in

��CG� = ��,��CG�� , �52�

or, alternatively, by the equilibrium conditions obtained inthe generalized Gibbs approach, i.e.,

��GG� = ��,��GG�� . �53�

However, in the classical understanding of the capillarity ap-proximation, the procedure employed in the determination ofthe interfacial tension is a slightly different one. Here � istaken to be equal to � ,

� = �� ,��

� �� , �54�

i.e., equal to the value of the surface tension for a stableequilibrium coexistence of both phases at planar interfaceswith the intensive state parameters, ��

� �� and �� , of the

both coexisting phases. The question, now, is the following:If the surface tension is determined is such a way indepen-dent of supersaturation �resulting in a value of the work ofcritical cluster formation equal to �Uc

� ��, what is the relationbetween �Uc

� � and �Uc�GG�?

The intensive state parameters of the cluster phase, asdetermined via the classical Gibbs approach, have in mostcases values widely corresponding to the parameters of thenewly evolving macrophase in equilibrium coexistence with

FIG. 2. In the limit of large critical cluster sizes, the equilibrium conditionresults. The locations of the saddle �corresponding to the generalized Gibbs ahere the self-consistent capillarity approximation is employed� are close to ecritical clusters, determined via the generalized Gibbs approach, differs, asmethod. In such cases, the work of critical cluster formation, determinedgeneralized Gibbs approach and, as a rule, significantly �cf. Fig. 2�b��, i.e., tpoints, which are not located on the ridge path, correspond to the classical Gpoints are located near to the ridge points but may not coincide with them.classical Gibbs approach involving the classical capillarity approximation,approach; however, for small supersaturations exceptions from this rule ma

the ambient phase at planar interfaces. By this reason, the


relation �Uc� ��Uc

�CG� holds often in a good approximationand the work of critical cluster formation �Uc

�GG�, derived viathe generalized Gibbs approach, in such cases is always lessas compared with the classical Gibbs approach involving theclassical capillarity approximation. This is the case one cantypically expect in application to phase formation in first-order phase transitions as it is exemplified also in the analy-sis of a variety of special cases �condensation and boiling inone-component van der Waals’34 and boiling in binaryfluids30 and segregation processes in solutions33�.

The general situation is illustrated in Fig. 2. Accordingto Eqs. �8�–�10�, in the limit of small supersaturations corre-sponding to large critical cluster sizes, both sets of equilib-rium conditions—derived in the classical and generalizedGibbs approaches—lead to identical results. The locations ofthe saddle �corresponding to the generalized Gibbs approach�and the ridge points �corresponding to the classical Gibbsmethod when the self-consistent capillarity approximation isemployed� are close to each other �cf. Fig. 2�a��. With in-creasing supersaturation, the composition of the critical clus-ters, determined via the generalized Gibbs approach, differs,as a rule, significantly from the compositions as obtained viathe classical Gibbs approach. In such cases, the work of criti-cal cluster formation, determined via the classical Gibbs ap-proach, exceeds the respective value found via the general-ized Gibbs approach as a rule significantly �cf. Fig. 2�b��,i.e., the distance between saddle and ridge points becomeslarge.

The positions of the points in Fig. 2 which are not lo-cated on the ridge correspond to possible values of the workof critical cluster formation computed via the classical Gibbsapproach, when there the classical capillarity approximationis employed. Since, by the above mentioned reasons, �� hascommonly values near to ��CG�, these points are located nearto the ridge points but may not coincide with them. Again,

rived in the classical and generalized Gibbs approaches—lead to identicalch� and the ridge points �corresponding to the classical Gibbs method whenther �cf. Fig. 2�a��. With increasing supersaturation, the composition of the

le, significantly from the compositions as obtained via the classical Gibbsthe classical Gibbs approach, exceeds the respective value found via thestance between saddle and ridge points becomes large. The positions of theapproach, when the classical capillarity approximation is employed. These

n, we conclude that the work of critical cluster formation, obtained via thea rule larger then the respective value obtained via the generalized Gibbs

found.

s—depproaach oa ruvia

he diibbs

Agaiis asy be

we conclude that the work of critical cluster formation, ob-



D

tained via the classical Gibbs approach involving the classi-cal capillarity approximation, is as a rule larger then the re-spective value obtained via the generalized Gibbs approach.As illustrated in Fig. 2, exceptions from this rule may beexpected in cases when saddle point �corresponding to thework of critical cluster formation in the generalized Gibbsapproach� and ridge point �corresponding to the classical ap-proach involving the self-consistent capillarity approxima-tion� are located near to each other, i.e., for very small su-persaturations.

In the limit of very small supersaturations, the men-tioned different approaches lead to identical characteristicparameters of the critical clusters, i.e.,

of the ambient and cluster phases. The change of the param-


�� , ��

� ��, � � ��CG� � ��GG�,

�55�

and the ratios of the work of critical cluster formation deter-mined in the different ways are equal to 1, i.e.,

�Uc�GG�

�Uc� � �

�Uc�CG�

�Uc� � = 1. �56�

Employing Eqs. �8�–�12�, we can express this ratio �formoderate supersaturations, when the difference �p�− p�� hasnearly the same values in both considered methods of deter-mination, the classical and generalized Gibbs approaches�either in the form

�Uc�GG�

�Uc� � = � �

� 3� �p� − p��2

��T� − T��s� + �p� − p�� − �i=1

k�i��i� − �i��2�

�c�

�57�

or

�Uc�GG�

�Uc� � = � �

� 3� 1

1 + �3/R�p� − p��/�s��s� + �i=1

k��/��i��i��2

�c�

. �58�

In the limit of vanishing supersaturations, the ratio��Uc

�GG� /�Uc� �� tends to 1. Consequently, the second term in

the denominator in the above relation tends to zero and it isa small parameter at small supersaturations. Taking into ac-count these considerations, the change of the ratio��Uc

�GG� /�Uc� �� due to an increase of the supersaturation

�starting with an initial state on the coexistence curve� isgiven in a first approximation by

d��Uc�GG�

�Uc� �

Rc→

= � 3d�

�

− 2d� 3

R�p� − p��

�s�s�

+ �i=1

k � ��

��i��i��

Rc→

�59�

or, employing the notations, Eqs. �4� and �13�, by

d��Uc�GG�

�Uc� �

Rc→

= � 3d�

�

− 2d� 1

�p� − p��i=1

k+1

�i��i��Rc→

. �60�

The kind of variation �increase or decrease� of the ratio��Uc

�GG� /�Uc� �� is consequently determined in such linear

approximation by the variation of both the state parameters

eters of the ambient phase is determined externally, the re-sulting change in the parameters of the cluster phase can becomputed then via Eqs. �9� and �10� employing the classicallimit Rc→ .

Equations �59� and �60� are generally valid both for thegeneralized and classical �here the derivatives of the surfacetension with respect to the state parameters of the clusterphase or the quantities ��i have to be set equal to zero�Gibbs approaches when in latter one the self-consistent cap-illarity approximation is employed. In the generalized Gibbsapproach, in addition to changes of the surface tension, alsodeviations of the changes of the bulk state parameters ascompared with the classical Gibbs approach may affect theresponse of the ratio ��Uc

�GG� /�Uc� �� to variations of the

state parameters of the ambient phase. We will illustrate nowdifferent possibilities for special cases of condensation andboiling in one-component systems and segregation in binarysolutions giving an interpretation both in terms of the classi-cal and generalized Gibbs approaches.

C. Curvature dependence of the surface tensionfor small supersaturations: Classical Gibbs’ approach

Employing the classical Gibbs approach, the change ofthe ratio of the work of critical cluster formation divided bythe respective value obtained by employing the capillarityapproximation is given in the limit of small supersaturations
by


D

d��Uc�CG�

�Uc� �

Rc→

= 3d�

�

Rc→

. �61�

Consequently, in such treatment the behavior of the ratio ��Uc

�CG� /�Uc� �� Rc→ is uniquely determined by the type of

change of the surface tension due to an increase of the initialsupersaturation. The consequences we will study below fortwo special cases.

1. A first special case: Condensation and boilingin one-component fluids at isothermal conditions

Considering as an example processes of condensationand boiling of one-component fluids at isothermal condi-tions, we have34,37,38

� = ��T�� − ��, �62�

where, according to Eq. �17�, we set here �=2. Varying thesupersaturation by changes in ��, we get in the consideredlimit of vanishing supersaturations,

d� = 2�� − ��

� ��d��

d��

− 1d��, � ��

��d��

= � ��

��d��, �63�

where the change of the density of the cluster phase is deter-mined via the equilibrium condition, �� ,T�=�� ,T�.Equation �63� yields generally

d� = 2�� − ��

� �� /��/��

R→

− 1d��. �64�

In the case of boiling �formation of bubbles, ��=�gas,��=�liquid�, we have �� and the degree of metastabilityis increased by a decrease of the density of the liquid �d��

�0�. Equation �64� yields then

d� = 2��gas� � − �liquid

� � �

�� liquid/��liquid��gas/��gas�

R→

− 1d�liquid, d�liquid � 0.

�65�

In the opposite case of condensation of gases �formation ofdroplets, ��=�liquid, ��=�gas�, we have instead �� andthe degree of metastability is increased by an increase of thedensity of the gas, i.e., d��0. Equation �64� yields here

d� = 2��liquid� � − �gas

� ��

�� gas/��gas��liquid/��liquid�

R→

− 1d�gas, d�gas � 0. �66�

Since the sign of the product �� −��

� ��d�� is the same forboth considered cases �it is positive�, the surface tension be-haves in opposite ways for condensation and boiling, onetime increasing and one time decreasing with increasing su-persaturation. The way of change is determined hereby bythe value of the ratio of the partial derivatives of the chemi-cal potentials with respect to the densities. Due to the condi-
tions of internal stability of both bulk phases �� /��T

�0�, this ratio is always greater than zero. Its value may bedifferent in dependence on the equation of state of the fluidunder consideration.

For a van der Waals fluid,34,37,38 we have, for example,

�� +3

�2�3� − 1� = 8� , �67�

with

� =p

pc, � =

�

�c, � =

T

Tc. �68�

By pc, �c, and Tc the critical values of pressure, volume, andtemperature are denoted.

The expressions for the chemical potential of a van derWaals fluid and its derivative with respect to � are generallyof the form37

��,��pc�c

= −8�

3ln�3� − 1� +

8��

3� − 1−

6

�+ �� , �69�

�

��

pc�c = − 6�� 4�

�3� − 1�2 −1

�3� . �70�

Here �� is some well-defined function only of tempera-ture. The knowledge of its particular form is not required forthe further derivations.

The location of the binodal curve may be determinedfrom the necessary thermodynamic equilibrium conditions�for planar interfaces�—equality of pressure and chemicalpotentials—via the solution of the set of equations

�liquid��liquid� � ,�� = �gas��gas

� ��, �liquid��liquid� � ,��

= �gas��gas� �,�� . �71�

With �=1/�, we may write

��

��= − �2��

��, �72�

resulting in

� = ��liquid/��liquid��gas/��gas�

R→

= �4��liquid3 /�3�liquid − 1�2� − 1

�4��gas3 /�3�gas − 1�2� − 1

�=��

. �73�

The value of this parameter � as a function of the reducedtemperature � is shown in Fig. 3. It is less then 1 in thetemperature range ��1 below the critical point. At the criti-cal point it approaches the value �=1. Consequently, for avan der Waals fluid, the surface tension and the ratio of theworks of critical cluster formation �cf. Eq. �59�� of bubbles inthe liquid decrease with increasing supersaturation startingwith states at the binodal curve. In contrast, the surface ten-sion of droplets of critical sizes first increases as comparedwith the value given by Gibbs’ classical theory and employ-ing the classical capillarity approximation.

The results of this analysis are illustrated in Fig. 4. InFigs. 4�a� and 4�b�, the ratio �� /� � is shown in dependence
on supersaturation for small supersaturations �Fig. 4�a�� and


D

for the whole range of metastable states of the ambient phase�Fig. 4�b�� both for condensation and boiling. Here the bulkstate parameters of the critical clusters are determined by theclassical Gibbs approach and the surface tension � via theself-consistent capillarity approximation. By � , the value ofthe surface tension according to the classical capillarity ap-proximation is specified, again. Since in the classical Gibbsapproach, the relation

�Uc�CG�

�Uc� � = � �

� 3

�74�

holds �cf. Eq. �58� and remember that in the classical Gibbsapproach the derivatives of the surface tension with respectto the bulk state parameters of the cluster phase have to beset equal to zero� the ratio of the values of the surface tensiongives a qualitative impression also of the ratio of the work ofcritical cluster formation computed in the classical Gibbsapproach and employing the classical and the self-consistentcapillarity approximations, respectively.

This result is in agreement with van der Waals squaregradient computations of the work of critical cluster forma-

FIG. 3. Dependence of the parameter � �cf. Eq. �73�� on the reduced tem-perature, �=T /Tc, for a van der Waals fluid in the range 0.3��1.

FIG. 4. Ratio �� /� � in dependence on supersaturation of the ambient phaseboth for condensation �upper curve� and boiling �lower curve�. Here thebulk state parameters of the critical clusters are determined by the classicalGibbs approach and the surface tension � via the self-consistent capillarityapproximation. By � , the value of the surface tension according to the
classical capillarity approximation is specified, again.

tion and the value of the surface tension for bubbles anddroplets of critical sizes in a van der Waals fluid interpretedin terms of the classical Gibbs approach �cf. Refs. 39 and40�. The observed for droplets increase of the surface tensionis, however, restricted to states in the immediate vicinity ofthe binodal curve. Starting with some sufficiently large su-persaturation, the work of critical cluster formation deter-mined via the classical Gibbs approach and employing theself-consistent capillarity approximation becomes alwayssmaller as compared to the respective value when the classi-cal capillarity approximation is employed.

Similar considerations as made above for the example ofa van der Waals fluid can be easily repeated for any moreappropriate equations of state of the fluids, provided that—similarly to Eq. �69�—the dependence of the chemical poten-tials on the state parameters of the bulk phases is known. Theresults remain qualitatively the same even in cases when theexponent in Eq. �62� is not set equal to 2 �cf. Refs. 34, 37,and 38�. Indeed, generally we obtain for the case of dropletformation in vapors

d� = ��liquid� � − �gas

� ��−1�d�liquid

d�gas− 1d�gas. �75�

In the opposite case of formation of bubbles in the liquid, wehave similarly

d� = ��liquid� � − �gas

� ��−1 d�gas

d�liquid�d�liquid

d�gas− 1d�liquid.

�76�

In both cases, we can employ again the relation

� ��liquid

��liquidd�liquid = � ��gas

��gasd�gas. �77�

The condition of internal thermal stability of both bulkphases, �� /��T�0, yields �d�liquid /d�gas�T�0. Conse-quently, also in such more general cases, the surface tensionand thus the work of critical cluster formation behave differ-ently in condensation and boiling in dependence on super-saturation for initial states near the respective binodal curves.

2. A second special case: Segregationin solutions

Let us consider as a second special case segregation pro-cesses in binary solutions, when the states of the ambient andthe newly evolving phases can be described completely viathe molar fraction x of one �here the second� of thecomponents.4,32,33 Similar to Eq. �62�, the surface tension isgiven then as

� = ��T��x� − x��2. �78�

The change of the surface tension can be expressed similarlyto Eq. �63� as

d� = 2��x�� − x�

� ��dx�

dx�

− 1dx�. �79�

In the limit of vanishing supersaturations �initial states near
the binodal curves�, the change of the composition of the


D

critical cluster dx� in dependence on the change of the stateof the ambient phase is given generally by19

��1��x�� − �2��x�� − ��1��x�� − �2��x�� = 0. �80�

Equations �79� and �80� yield

d� = 2��x�� − x�

� ��

�� 1�/�x�� − ��2�/�x��1�/�x�� − ��2�/�x��

R→

− 1dx�. �81�

To be definite, let us consider metastable initial states ofthe ambient phase located near the left branch of the binodalcurve �cf. Fig. 5�. In such cases, we have x�

� �=xbinodal�left� and

x�� =xbinodal

�right� and the increase of the supersaturation isachieved by increasing the molar fraction of the segregatingcomponent in the ambient phase, i.e., dx��0. Consequently,the product �x�

� �−x�� dx� is a positive quantity. Obviously,

the same conclusion can be drawn in the case of initial statesnear the right branch of the binodal curve. In both cases, thesign of the change of the surface tension �and the behavior ofthe ratio of the work of critical cluster formation �cf. Eq.�59�� is determined by the sign of the ratio

�̃ = ��1�/�x�� − ��2�/�x��1�/�x�� − ��2�/�x��

R→

. �82�

For the case of a regular solution, we have

�1�p,T,x� = �1*�p,T� + kBT ln�1 − x� + �x2, �83�

�2�p,T,x� = �2*�p,T� + kBT ln�x� + ��1 − x�2. �84�

The parameter � is connected here with the critical tempera-ture of the solution via Tc=� / �2kB� and the location of thebinodal curve is given by

ln�1 − x

x = 2�Tc

T�1 − 2x� . �85�

FIG. 5. Phase diagram of a binary regular solution. The binodal curve,which separates metastable from stable homogeneous initial states of theambient phase, is given by a full line. The spinodal curve, separating ther-modynamically metastable from thermodynamically unstable initial states,is given by a dashed curve. In the analysis, we consider in detail phaseseparation for initial states of the ambient phase near the left branch of thebinodal curve, x��xbinodal

�left� . In such case, the composition of the newlyevolving phase in stable equilibrium with the ambient phase at planar inter-faces is given by x��xbinodal

�right� .

With the above notations, we get


��1

�x−

��2

�x= 2kBTc�1 −

T

4Tc� 1

x�1 − x�� 0. �86�

Since the position of the spinodal curve is determined for aregular solution via the relation19

x�1 − x� =T

4Tc, �87�

the difference of the derivatives of the chemical potentials inEq. �86� is not equal to zero at the binodal curve. Conse-quently, we may rewrite Eq. �82� in the form

�̃ = �1 − �T/4Tc��1/�x��1 − x��1 − �T/4Tc��1/�x��1 − x��

R→

. �88�

In the above relation, x� has values near to one of thebranches of the binodal curve �e.g., x�=xbinodal

�left� � and x� isequal to the alternative value �x�=xbinodal

�right� �. From Eq. �85�, forany temperature we have

xbinodal�left� �1 − xbinodal

�left� � = xbinodal�right� �1 − xbinodal

�right� � �89�

and the ratio �̃ turns out to be equal to 1.In the considered first-order approximation, the value of

the ratios of the work of critical cluster formation and thevalue of the surface tension of critical clusters do notdepend—due to the symmetry in the phase diagram �cf. Fig.5�—near the binodal curves on changes of the supersatura-tion. This result is in agreement with conclusions of Fisherand Wortis41 and previous own investigations33,39,40 of thecurvature dependence of the surface tension of critical clus-ters in regular solutions. According to these results—again,due to the symmetry of regular solutions—the first term inthe expansion of the surface tension with respect to �1/R�,

��R� = � �1 +�1

R+

�2

R2 + ¯ , �90�

is equal to zero ��1=0�. As a consequence, the work of criti-cal cluster formation for phase formation in regular solutionsis always equal or smaller than the respective value obtainedvia the classical approach. For condensation and boiling invan der Waals’ fluids, the first-order terms are nonequal tozero resulting in the discussed already differences. A similardifference in the behavior of the surface tension of dropletsand bubbles of critical sizes has to be expected, in general,for phase separation in solutions not obeying the symmetryof regular solutions and, for example, also for condensationand boiling in multicomponent fluids �see, e.g., Refs. 30, 42,and 43�.

Summarizing the results of the above given analysis, wecan conclude: employing the classical Gibbs theory of cap-illarity and the self-consistent capillarity approximation, wecan, as shown, reproduce results of the van der Waals squaregradient approximation in the limit of small supersaturations.This is an additional indication that the choice of the expres-sion for the surface tension, Eq. �17�, is an appropriate ap-proximation giving already in the framework of the classicalGibbs approach the possibility to get more detailed insightsinto the critical cluster properties as compared with the case
when the classical capillarity approximation is used. In the


D

next section, we will compare the predictions of the classicalwith those of the generalized Gibbs approach.

D. Curvature dependence of the surface tensionfor small supersaturations: GeneralizedGibbs’ approach

1. Condensation and boiling in one-component fluidsat isothermal conditions

Employing the generalized Gibbs approach, the changeof the ratio of the work of critical cluster formation, dividedby the respective value obtained by employing the classicalGibbs theory and the classical capillarity approximation, canbe expressed, for the case of condensation and boiling ofone-component fluids at isothermal conditions and in thelimit of small supersaturations, as �cf. Eq. �60��

d��Uc�GG�

�Uc� �

Rc→

= �3d�

�

− 2d�� − ��p� − p�

�Rc→

. �91�

From the equilibrium conditions in the generalized Gibbsapproach, Eqs. �8�–�10�, we obtain in the considered case

�� − �� = �p� − p��, � =1 − �3��/2��/��

�3��/2��/��.

�92�

Consequently, variations of the state parameters of ambientand critical cluster phases are coupled generally by the rela-tion

��d�� − d�� + �� − ��d�� = �dp� − dp�� .

�93�

Equations �91�–�93� yield

d��Uc�GG�

�Uc� �

Rc→

= �3d�

�

+2

�2d��Rc→

,

d�

�= −

d��

��

. �94�

Due to the structure of the expression for the surface tension,Eq. �17�, the symmetry relation �� /��=−�� /�� holdsand Eq. �94� can be transformed into

d��Uc�GG�

�Uc� �

Rc→

= − � 3

�

��

��1 +

1

�

d��

d��

Rc→

d��. �95�

In the limit of large critical cluster sizes �Rc→ resulting in��=�� and taking into account the consequences from theGibbs-Duhem equations for the bulk phases �dp=�d��, we
get

d��

d��

= � �� + ��

�� + 1� ��/��

��/��

. �96�

Employing Eq. �64� for the determination of the surface ten-sion, we arrive at

1

�

��

��

=�

�� − ��

, � = −��3� − 2� + 2��

3��,

�� + ��

�� + 1�= −

3� − 2

2, �97�

resulting with Eq. �96� in

d��Uc�GG�

�Uc� �

Rc→

= − 3��1 +3��3� − 2�

2��3� − 2� + 2��

��/��

��/��

Rc→

d��

�� − ��

� ��. �98�

Since the ratio d�� / �� −��

� �� is larger than zero both forcondensation and boiling, the work of critical clusterformation—as computed via the generalized Gibbsapproach—is also for small supersaturations always less ascompared with the result obtained in the classical Gibbs ap-proach if one employs there the classical capillarity approxi-mation.

We come to the conclusion that the results of the gener-alized Gibbs approach concerning the work of critical clusterformation for very small supersaturations are partly in con-tradiction to the predictions of both the classical Gibbsmethod �when here the self-consistent capillarity approxima-tion is employed� and the van der Waals square gradientapproach. On the other hand, when the whole range of initialsupersaturations is considered, the generalized Gibbs ap-proach is in excellent agreement with the predictions of thevan der Waals square gradient approach34,37,38 including theprediction of the vanishing of the work of critical clusterformation near the classical spinodal curve. Consequently, anindependent on the discussed here methods of determinationof the work of critical cluster formation for condensation andboiling at very small supersaturations would be highly desir-able. Such methods exists �cf., e.g., Ref. 44�, however, thediscussed effects cannot be observed in general in computersimulations since the sizes of the critical clusters, required todemonstrate such kind of behavior, exceed commonly thesize of the model systems under consideration.

The work of formation of aggregates of critical sizes in avan der Waals fluid is shown in dependence on supersatura-tion in Fig. 6 for condensation and in Fig. 7 for boiling. As ameasure of the supersaturation, here the deviation of the mo-lar volume of the fluid from the value at the binodal curve istaken divided by the difference of this parameter at the spin-odal and binodal curves. In addition, the ratio �Fig. 8; cf. Eq.�56�� and difference �Fig. 9� of the work of critical clusterformation in condensation and boiling are shown computedvia the classical Gibbs approach involving the self-consistent
capillarity approximation and the generalized Gibbs ap-


D

proach, respectively. In particular, the results confirm tosome extent the suggestion of McGraw and Laaksonen45,46

of constant values of the difference �cf. Fig. 9�, however, forthe system under consideration such relations hold for meta-stable initial states corresponding to sufficiently large super-saturations, only.

In the generalized Gibbs approach, the density differ-ences between ambient and newly evolving phases behavedifferently as compared with the predictions of the classicalGibbs approach �the sign of the changes is different� withincreasing supersaturation �cf. Eqs. �96� and �97�� already atinitial states near to the respective binodal curves. As theresult, the density differences decrease and the surface ten-sion, referred to the generalized surface of tension as definedin the generalized Gibbs approach, decreases with increasingsupersaturation both for condensation and boiling. As a con-sequence, the surface tension decreases with increasing su-persaturation in both cases. This result is illustrated in

FIG. 6. Work of formation of droplets of critical sizes in a van der Waalsfluid as obtained via the generalized Gibbs approach �full curve� and theclassical Gibbs approach involving the capillarity approximation �dashedcurve�. The computations have been performed employing Eq. �62� with avalue of the parameter � equal to �=2.

Fig. 10.


2. Segregation in binary regular solutionsFor the case of segregation in a binary regular solution,

the work of critical cluster formation computed via the clas-sical Gibbs approach and involving the self-consistent capil-larity approximation is always less as compared with theresults of the classical Gibbs approach when here the classi-cal capillarity approximation is employed. Taking into ac-count the earlier obtained general result—the generalizedGibbs approach leads to values for the work of critical clus-ter formation less than those obtained via the classical Gibbsmethod involving the self-consistent capillarityapproximation—we conclude that in this case the general-ized Gibbs approach always leads to values of the work ofcritical cluster formation less as compared with the classicalcapillarity approximation in Gibbs’ classical method. Conse-quently, a separate analysis is not required here.

3. General caseCompleting the analysis, we would like to give the basic

FIG. 7. Work of formation of bubbles of critical sizes in a van der Waalsfluid as obtained via the generalized Gibbs approach �full curve� and theclassical Gibbs approach involving the capillarity approximation �dashedcurve�. The computations have been performed employing Eq. �62� with avalue of the parameter � equal to �=2.

equations allowing one to analyze the behavior of the work



D

of critical cluster formation for small supersaturations for thegeneral case of multicomponent systems not demanding fur-ther that the process has to proceed under isothermal condi-tions. In this analysis, we may start with Eq. �60�. Employ-ing, again, the notations, Eqs. �4� and �13�, we canreformulate the equilibrium conditions, Eqs. �8�–�10�, as

2�

Rc= �p� − p�� + �

i=1

k+1

�i��i, ��i =3

Rc� ��

��i�

��

. �99�

Similar to the one-component case, the change of the stateparameters of the critical clusters due to variations of thedegree of supersaturation is given then by

��i=1

k+1

�i��i = �p� − p�� ,

�100�

� =1 − �i=1

k+1�3�i�/2��/��i��

�i=1

k+1�3�i�/2��/��i��

.

Equation �60� gets then the form

d��Uc�GG�

�Uc� �

Rc→

= �3d�

�

+2

�2d��Rc→

. �101�

This equation can be evaluated easily for any given path ofpenetration into the metastable region starting from initialstates at the respective binodal curves. In order to performthe respective computations, the expression for the surfacetension and the values of the intensive state parameters ofboth phases for an equilibrium coexistence at planar inter-faces have to be known.

V. DISCUSSION

The results of the present analysis are based on the fol-

FIG. 8. Ratio of the work of critical cluster formation, ��Uc�CG� /�Uc

�GG��, forcondensation and boiling of a van der Waals fluid computed via the classical�involving the self-consistent capillarity approximation� and generalizedGibbs approaches.

lowing basic considerations. First, it is realized that, in order


to determine singular points of thermodynamic potential sur-faces, one has to formulate as the starting point the expres-sions for the respective thermodynamic quantities for anythermodynamically well-defined nonequilibrium states con-sisting of a cluster or ensembles of clusters in the otherwisehomogeneous ambient phase. Such task has not been per-formed by Gibbs who restricted his analysis to equilibriumstates of heterogeneous substances,10 exclusively. In gener-alizing Gibbs’ classical approach to the description of theconsidered nonequilibrium states, the surface tension of acluster has to be considered as a function both of the stateparameters of the ambient and the newly evolving phases.This essential new feature is introduced, as a second basicingredient of the theory employed, via the generalization ofGibbs’ fundamental equation for the superficial quantities,Eq. �2�, leading to a generalization of Gibbs’ adsorptionequation, Eq. �5�, with the desired properties. Developing the

FIG. 9. Difference of the work of critical cluster formation, ��Uc�CG�

−�Uc�GG�� /kBT, for condensation and boiling of a van der Waals fluid com-

puted via the classical �involving the self-consistent capillarity approxima-tion� and generalized Gibbs approaches for different ranges of the initialsupersaturation.

theory, the expressions for the thermodynamic potentials re-



D

tain the same form as in conventional—not involving modi-fications of Gibbs’ fundamental equation—generalizations ofGibbs’ classical theory to nonequilibrium states,15–18 how-ever, the expressions for the parameters of the critical clus-ters, Eqs. �8�–�10�, become different as compared withGibbs’ classical results.

In order to apply the theory to the determination of theparameters of the critical clusters, the bulk properties of thephases under consideration and the dependence of the sur-face tension on the intensive state parameters of the clusterand the ambient phases have to be known. Employing onlyvery general arguments, the dependence of the surface ten-sion on the state parameters of the coexisting phases is de-rived and employed here in the form as given by Eq. �17�.This is the third basic assumption of the analysis performedin the present paper. However, although the dependence asgiven by Eq. �17� is believed to be of very general nature, itis not a direct result of the generalized Gibbs approach but ofadditional considerations. By this reason, it can be modified,if required. In particular, it would be of interest to checkwhat modifications of the results outlined can be found if theexpansion in Eq. �17� is extended to include �even� higher-order terms in the differences of the intensive state param-eters of both phases.

Employing the above summarized basic ideas, the basicresults of the present analysis can be summarized as follows:�i� The critical cluster in the generalized Gibbs approach cor-responds �similar to Gibbs’ classical method� to a saddlepoint of the appropriate thermodynamic potential surface, amaximum with respect to variations of the size parameterand a minimum with respect to variations of the intensivestate parameters of the cluster phase; �ii� the critical cluster,as determined via the classical Gibbs approach and involvingthe self-consistent capillarity approximation, corresponds—interpreted in terms of the generalized Gibbs approach—to a ridge point of the thermodynamic potential surface; �iii�as a direct consequence, the work of critical cluster

FIG. 10. Dependence on supersaturation of the value of the surface tension�referred to the surface of tension� as defined in the generalized Gibbs ap-proach. The results are shown for bubbles or drops of critical sizes in con-densation and boiling of a van der Waals fluid �see also Figs. 6 and 7�.

formation—as obtained via the classical Gibbs approach and


involving the classical capillarity approximation—overestimates, as a rule, its real value and, in general, con-siderably.

From a physical point of view, these results are conse-quences from the ability of the generalized Gibbs approachto determine appropriately and in a theoretically well-founded way the bulk properties of the critical clusters independence on supersaturation and/or the size of the criticalclusters. In the classical Gibbs approach, the bulk propertiesof the clusters of the newly evolving phase turn out to besimilar to the bulk properties of the newly evolving macro-scopic phases. If, in addition, the classical capillarity ap-proximation is employed, deviations between theoretical andexperimental data for the steady-state nucleation rate mayreach the order 1020–1030 for condensation of one-component vapors47,48 and even 10100 and more for phaseformation in glass-forming melts.11 Corrections are intro-duced then by assuming some kind �often ad hoc assump-tions are used� of curvature dependence of the surface ten-sion. The generalized Gibbs approach allows us �asdemonstrated here for model systems� to determine both thechanges in the bulk properties and the resulting variations inthe surface properties in a well-defined way utilizing exclu-sively data on the bulk properties of the phases under con-sideration and the dependence of the surface or interfacialtension �or specific surface energies� for phase coexistence atplanar interfaces as the starting point of the analysis. A firstapplication of this method to the analysis of experimentalresults12,13,19,20,49 has shown its ability to resolve a variety ofproblems in the theoretical interpretation of experimentaldata which have not found a satisfactory solution till thattime. The further implementation of this method to theinterpretation of experimental results is the aim of futureanalyses.

ACKNOWLEDGMENT

The authors express their gratitude to the DeutscheForschungsgemeinschaft for financial support �DFG GrantNo. 436 RUS 113/705/0-2�.

1 M. Volmer, Kinetik der Phasenbildung �Th. Steinkopff, Dresden, Leipzig,1939�.

2 J. P. Hirth and G. M. Pound, Condensation and Evaporation �Pergamon,London, 1963�.

3 V. P. Skripov, Metastable Liquids �Wiley, New York, 1974�.4 A. E. Nielsen, Kinetics of Precipitation �Pergamon, Oxford, 1964�.5 J. W. Christian, The Theory of Transformations in Metals and Alloys�Oxford University Press, Oxford, 1975�.

6 I. Gutzow and J. Schmelzer, The Vitreous State: Thermodynamics, Struc-ture, Rheology, and Crystallization �Springer, Berlin, 1995�.

7 H. A. Stanley, Introduction to Phase Transitions and Critical Phenomena�Clarendon, Oxford, 1971�.

8 J. W. P. Schmelzer, in Encyclopedia of Surface and Colloid Science,edited by A. Hubbard �Marcel Dekker, New York, 2002�, pp. 4017–4029.

9 D. T. Wu, Solid State Physics: Advances in Research and Applications,Nucleation Theory Vol. 50, edited by H. Ehrenreich, and F. Spaepen�Academic, New York, 1997�.

10 J. W. Gibbs, The Collected Works, Thermodynamics Vol. 1 �LongmansGreen, New York, 1928�.

11 E. D. Zanotto and V. M. Fokin, Philos. Trans. R. Soc. London, Ser. A361, 591 �2003�.

12
V. M. Fokin, N. S. Yuritsyn, and E. D. Zanotto, in Nucleation Theory and


D

Applications, edited by J. W. P. Schmelzer �Wiley-VCH, Berlin, 2005�,pp. 74–125.

13 V. M. Fokin, E. D. Zanotto, N. S. Yuritsyn, and J. W. P. Schmelzer, J.Non-Cryst. Solids �in press�.

14 E. A. Guggenheim, Thermodynamics: An Advanced Treatment for Chem-ists and Physicists �North-Holland, Amsterdam, 1950�.

15 A. I. Rusanov, Phasengleichgewichte und Grenflächenerscheinungen�Akademie-Verlag, Berlin, 1978�.

16 H. Ulbricht, J. Schmelzer, R. Mahnke, and F. Schweitzer, Thermodynam-ics of Finite Systems and the Kinetics of First-Order Phase Transitions�Teubner, Leipzig, 1988�.

17 K. Nishioka and I. Kusaka, J. Chem. Phys. 96, 5370 �1992�.18 P. G. Debenedetti and H. Reiss, J. Chem. Phys. 108, 5498 �1998�.19 J. W. P. Schmelzer, A. R. Gokhman, and V. M. Fokin, J. Colloid Interface

Sci. 272, 109 �2004�.20 J. W. P. Schmelzer, A. S. Abyzov, and J. Möller, J. Chem. Phys. 121,

6900 �2004�.21 L. Granasy and P. James, J. Non-Cryst. Solids 253, 210 �1999�.22 J. D. van der Waals and Ph. Kohnstamm, Lehrbuch der Thermodynamik

�Johann-Ambrosius-Barth, Leipzig, 1908�.23 J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 �1958�.24 J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 688 �1959�.25 D. W. Oxtoby, Acc. Chem. Res. 31, 91 �1998�.26 J. D. Gunton, M. San Miguel, and P. S. Sahni, in Phase Transitions and

Critical Phenomena, The Dynamics of First-Order Phase Transition Vol.8, edited by C. Domb and J. L. Lebowitz �Academic, London, 1983�.

27 K. Binder, Rep. Prog. Phys. 50, 783 �1987�.28 M. Shah, O. Galkin, and P. Vekilov, J. Chem. Phys. 121, 7505 �2004�.29 J. W. P. Schmelzer, Phys. Chem. Glasses 45, 116 �2004�.30 J. W. P. Schmelzer, V. G. Baidakov, and G. Sh. Boltachev, J. Chem. Phys.

119, 6166 �2003�.31 J. W. P. Schmelzer, G. Sh. Boltachev, and V. G. Baidakov, in Nucleation


Theory and Applications, edited by J. W. P. Schmelzer �Wiley-VCH,Berlin, 2005�, pp. 418–446.

32 J. W. P. Schmelzer, J. Schmelzer, Jr., and I. Gutzow, J. Chem. Phys. 112,3820 �2000�.

33 V. G. Baidakov, G. Sh. Boltachev, and J. W. P. Schmelzer, J. ColloidInterface Sci. 231, 312 �2000�.

34 J. W. P. Schmelzer and V. G. Baidakov, J. Phys. Chem. B 105, 11595�2001�.

35 R. Kubo, Thermodynamics �North-Holland, Amsterdam, 1968�.36 J. Schmelzer and F. Schweitzer, Wissenschaftliche Zeitschrift der

Wilhelm-Pieck-Universität Rostock, Mathematisch-Naturwissenschaftliche Reihe 36, 83 �1987�.

37 J. W. P. Schmelzer and J. Schmelzer, Jr., J. Chem. Phys. 114, 5180�2001�.

38 J. W. P. Schmelzer and J. Schmelzer, Jr., Atmos. Res. 65, 303 �2003�.39 V. G. Baidakov and G. Sh. Boltachev, Phys. Rev. E 59, 469 �1999�.40 V. G. Baidakov and G. Sh. Boltachev, Dokl. Akad. Nauk 363, 753

�1998�.41 M. P. A. Fisher and M. Wortis, Phys. Rev. 29, 6252 �1984�.42 V. G. Baidakov, in Nucleation Theory and Applications, edited by J. W. P.

Schmelzer �Wiley-VCH, Berlin, 2005, pp. 126–177�.43 V. G. Baidakov, Explosive Boiling of Cryogenic Liquids �Wiley-VCH,

Berlin, 2006�.44 A. V. Neimark and A. Vishnyakov, J. Chem. Phys. 122, 174508 �2005�.45 R. McGraw and A. Laaksonen, Phys. Rev. Lett. 76, 2754 �1996�.46 R. McGraw and A. Laaksonen, J. Chem. Phys. 106, 5284 �1997�.47 J. Wölk, R. Strey, C. H. Heath, and B. E. Wyslouzil, J. Chem. Phys. 117,

4954 �2002�.48 K. Iland, Ph.D. thesis, University Cologne, 2004.49 V. M. Fokin, E. D. Zanotto, J. W. P. Schmelzer, and O. V. Potapov, J.

Non-Cryst. Solids 351, 1491 �2005�.


Documents

Classical and generalized Gibbs’ approaches and the work of critical cluster formation in nucleation theory