# Gibbs Thermodynamics of the  Renninger-Wilemski Problem

Authors: Victor Kurasov
Journal: Journal of Thermodynamics
 (2015)

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## Abstract

The Renninger-Wilemski problem in nucleation is analyzed. The Gibbs dividing surfaces method with external parameters is used to enrich the initial model. It is shown that both the traditional (Doyle) model and the Renninger-Wilemski model are not complete ones and, namely, the Gibbs dividing surface approach can solve this problem. It is shown that the application of the Gibbs approach also requires some model constructions. The simplified Gibbs model is proposed. It is shown that the simplified Gibbs model gives for the height of activation barrier the same numerical results as the Renninger-Wilemski model.


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## Body

## 1. Introduction

Although the principles of thermodynamics are well grounded, there exist many narrow questions in applications. One of such questions belongs to the new phase embryos formation. Despite the fact that the free energy of the new phase embryo can be calculated in frames of equilibrium thermodynamics, there remain many questions of account of specific conditions for the critical embryo appearance.There is no need to demonstrate the actuality of nucleation taking place practically overall. The importance of nucleation initiates investigations from Wilson’s experiments [1–3] up to modern investigations. Thermodynamic aspects of investigations occupy an important place in this field considering not only the fundamental question of stationarity (see to clarify it, e.g., [4, 5]), but also the problems of smallness of characteristic embryos and corrections appeared because of its sizes and shapes [6]. The results given by thermodynamics immediately lead to quantitative description of kinetics [7] and can be included into consideration of some rather complex aggregates with very specific properties [8]. This consideration is based mainly on thermodynamics of small embryos. Here we will discuss, namely, some important specific features of the small embryos thermodynamics.The bright demonstration of specific features of thermodynamic aspects is the Renninger-Wilemski problem which occupies an important place in the nucleation of multicomponent liquids. An abbreviated version of this problem sounds as follows:(i)
Is it necessary to take the derivative of the surface tension on concentration to determine the characteristics of a critical droplet?From the first glance this problem sounds so simple that the answer seems to be evident. But it took a dozens of years to clarify this problem.This problem arises from the very beginning of investigations of the multicomponent nucleation since the pioneer paper written by Reiss [9], but until the paper of Renninger et al. [10] this problem was somehow in a shadow. The classical theory of nucleation [11, 12] is standing somehow aside of this question. The reason of this shadow is the rather weak dependence of the surface tension on concentration in the intermediate region of concentrations for the dominating majority of mixtures. The physical nature of this weakness is the enrichment of the surface layer with the surface-active molecules. When it is easy to find in the bulk of an embryo a molecule of the surface-active substance then it is profitable to catch it and to put in the surface. Roughly speaking only an entropy of mixing has to be paid here. The energy profit in this situation is a surface with a surface-active substance. So, the surface tension appears to be a relatively smooth function of concentration. Only when it is difficult to find such a molecule in a bulk liquid one has to pay some relatively essential work to find it and the surface tension begins to depend on concentration essentially. But the last situation occurs at small concentrations and this case should be preferably described as a one-component nucleation with some impurities.The mentioned weak dependence allows many authors in 1950–80s to ignore the fact of dependence of the surface tension on concentration, although there exist some investigations like that by Mirabel and Katz [13]. In [13] the correction for the Young-Laplace formula with the derivative of the surface tension on concentration was presented. This correction is rather doubtful.The statement that it is necessary to forbid to differentiate the surface tension was announced by Renninger and coauthors in [10] criticizing mainly Doyle [14] and Mirabel and Katz [13]. Before [10] the situation was the following: there exists the expression for the free energy, it is clear that the free energy of the critical embryo is the free energy at the saddle point, and it is clear that to get the coordinates of the saddle point it is necessary to take the free energy partial derivatives and to put them to zero, but no special attention was paid for the question what to do with the derivative of the surface tension on the solution concentration.It was Doyle [15] who presented contrary to [10] the arguments for the traditional version with the differentiation of the free energy on concentration having justified his initial approach in [14]. Later Wilemski clarified the problem in [16, 17] and one could say after [16, 17] that the phenomenological recipe required that it was necessary to forbid to differentiate the surface tension. This seemed to be the final conclusion in this question. Since that time the problem is called the Renninger-Wilemski (RW) problem and seemed to be the closed problem in thermodynamics.This problem was revitalized in 2003 by Reiss and Reguera [18, 19] who showed by a refinement of some connections for derivatives of chemical potentials in droplet the alternative variant of derivation of equations which are similar to those used by Dole. One can say that technically the publication [18, 19] was the top of analysis of this problem. But the Gibbs approach has not been considered in [18, 19]. It was announced in [18, 19] that the Gibbs approach [20] “is of course the most sophisticated and general, but its application requires more information than is available in macroscopic thermodynamic observables.” Meanwhile the recent investigations on thermodynamics of complex systems successfully operate with the quantities of excesses which is typical for the Gibbs dividing surfaces conception.One has to mention that already after Wilemski in the field of multicomponent nucleation many models use the elements of the Gibbs dividing surface method but in their proper constructions. Instead of application of thermodynamics with some parameters of the model these papers try to build their “true” constructions without any choice of parameters. Here one can mention the approach of Debenedetti [21], the model of Nishioka and Kusaka [22], and the approach of Laaksonen et al. [23] which was analyzed also in [24]. All these models have the features of the Gibbs dividing surface method, but all of them propose some specific relations to close the system of equations. All of them announced the recipe for determination of the critical embryo characteristics as the only “true” one although the Gibbs model is principally a model with external parameters.Here we will analyze both traditional models (the Doyle model and the Renninger-Wilemski model) and show that both of them are not self-consistent. The alternative model as realization of the Gibbs dividing surface method will be presented. In the last part it will be shown how to simplify this approach.


## 2. Initial Remarks

To determine the coordinates of the critical embryo one can follow at least two ways.(i)
The first way is to construct some expression (namely, the expression in the capillary approximation) for the total free energy of the critical embryo formation and then to determine the coordinates of the critical embryo as the coordinates of the free energy saddle point by differentiating and putting the derivatives to zero.(ii)
The second way is to use the general relations coming from the differential analysis such as the Laplace-Young formula and the Gibbs-Thomson formula for the equilibrium (critical) droplet and to put into these formulas some concrete models for necessary values.When the expression for the free energy is precise then both approaches have to give one and the same result. The trouble is that everything we can suggest is no more than an approximation. In fact these approaches historically led to two different results and this discrepancy has now the name of the Renninger-Wilemski problem (for such a terminology see [18, 19]). Now it is clear that both results can be derived in both approaches but historically Renninger et al. [10] used the second approach and their opponent Doyle [15] used the first one. Following the history of the question we will speak about RW-model and about D-model correspondingly. Certainly, this classification is rather artificial.The question of thermodynamic derivations of the surface characteristics was the subject of numerous investigations. The thermodynamic definition of the surface tension is discussed in [25]. Thermodynamics of a surface layer is analyzed in [26].At first it is necessary to present the formal thermodynamic derivation in order to see where the hidden suppositions about the structure of the system under consideration have been done. Here we follow the manner of [18, 19] because, namely, that derivation opened the possibility to see different approaches in frames of the differential formalism. Certainly some essential modifications will be introduced.Since in the system where the embryo is born the variablesT (temperature) and V (volume) are supposed to be fixed, one has to use the Helmholtz free energy
(1)F=U-TS.
Really, according to principles of thermodynamics,
(2)dF=-SdT-pdV.
We suppose that the terms ∑i‍μiNi (μi are the chemical potentials, Ni are the numbers of particles, and index i numerates the components) are included into thermodynamic potentials1.One has to splitU and S into contributions from the liquid phase (index l) and the vapor phase (index v)2. Then
(3)U=Ul+Uv,S=Sl+Sv,V=Vl+Vv,dUv=TdSv-pvdVv+∑i‍μvidNvi,dUl=TdSl-pldVl+∑i‍μlidNli.
Since
(4)dVl=-dVv,dNvi=-dNli
we come to
(5)dU=TdS-pl-p^vdVl+∑i‍μli-μvidNli.
Here the symbol ⋯^ indicates that the pressure is the pressure under the curved surface; that is, the Young-Laplace correction 2γ/r has to be added (γ is a surface tension, r is a radius of a droplet). Then
(6)dF=-pl-pv+2γrdVl+∑i‍μli-μvidNli.
In the state of equilibrium we have dF=0 which gives
(7)pl=pv+2γr.
This is the Young-Laplace equation and
(8)μlipv+2γr=μvipv.
Here and later the square brackets show the functional dependence to avoid misreading as multiplication. If the index of variable is absent it means that the dependence takes place over the whole set of variables.Now it is necessary to go in chemical potentials fromp to p+2γ/r explicitly. To do this it is necessary to calculate ∂μli/∂p. To fulfil this transformation we take the Maxwell relations for the Gibbs free energy Gl=Ul-TSl+plVl with a differential
(9)dGl=-SldT+Vldpl+∑i‍μlidNli.
Then
(10)∂μil∂pl=vli,
where vli is the volume per one molecule of component i in a liquid phase. We suppose the liquid to be incompressible. So, the rhs of the previous equation is constant and can be easily integrated. Equation (10) is of great importance since it will lead to the Gibbs-Thomson equation. Then
(11)μlip+2γr=μlip+vli2γr
and (8) looks like
(12)Δμi=vli2γr,
where Δμi is the difference of chemical potentials at the external pressure in the system. The last relation is the so-called Gibbs-Thomson equation well known in thermodynamics.Equation (12) leads to invariant
(13)Δμlivli=Δμljvlj
and allows a clear physical interpretation. Namely, (12) is the basis for RW-model.Here it is necessary to make one very essential comment. Having written equation (9) for the differential of Gl=Ul-TSl+plVl we miss the term γA, that is, the surface tension γ multiplied on the surface area A. We can include this term and write
(14)dGl=-SldT+Vldpl+∑i‍μlidNli+γdA.
This does not change the derivation. It is another moment which is important here. Beside this we have to note that ordinary it is implied that γA is independent of other characteristics presented in (14). But in the droplet there is a connection between A and Nli. This puts a question on the validity of this derivation. It will be discussed below.Equation (6) for dF can be integrated from Nli=0 up to the current size which gives F[Ni]. To fulfil this procedure it is convenient to present the dependence on Nli as the dependence on concentrations defined3 as
(15)ξi=NliN0,
where
(16)N0=∑i‍Nli
is the total number of molecules in a drop.As an extensive variable4 we can take N0, but some other variants of choice are possible and will be presented below.Instead of integration one can suggest to present the model forF and then to check whether it leads to (7), (12) by differentiation. We are interested here, namely, in the last relations.One can easily suggest the following expression forF:
(17)FNli=F0-∑i‍ΔμiNli+γA.
Here and below A is the surface area of the embryo. The term Vl(pl-p) is missed here since it is very small. It corresponds to the difference between the situation when the volume of the system is fixed or the pressure in the system is fixed. Concrete conditions ordinary are not well posed.The condition for the equilibrium is(18)∂F∂Nli=0.
The explicit differentiation has to be done with account of the Gibbs-Duhem equation for the bulk liquid
(19)∑‍Nlidμli+SldT-Vldp=0
which is actually
(20)∑‍Nlidμli=0.
The last relation will be satisfied automatically and inevitably when we use some concrete dependencies for chemical potentials in liquid.The result of differentiation (we take for simplicity a two-component case) on account of(21)A=C∑i‍vliNli2/3C=36π1/3
is
(22)∂FNli∂Nli=-Δμi+2γr+3vm1-ξirdγdξi,
where
(23)vm=∑‍ξivli.
We see that the equilibrium condition
(24)Δμi=2γr+3vm1-ξirdγdξi
differs from the Gibbs-Thomson equation by the last term. This equation is the basis in D-model.In RW-model one can also explain how to come to (12) starting from (17). In the differentiation of F instead of (19) it is necessary to take the surface variant of the Gibbs-Duhem equation
(25)∑‍Nlidμli+SldT-Vldp+Adγ=0
which can be approximately reduced to
(26)∑‍Nlidμli+Adγ=0.
The last term in the previous equation cancels the derivative of the surface tension on concentration and we come to Gibbs-Thomson equations (12).For a long time it seemed that (12) is preferable because there exists a differential method to see (12). But Reiss and Reguera showed that it is possible to get (24) in modification of the way we came here to (12). In integration dμli/dp it is necessary to take into account not only the direct dependence ∂μli/∂p=vli but also the dependence on size of the embryo through the dependence
(27)∂μli∂A=∂γ∂Nli.
The last relation can be found if we start with the Gibbs potential taking into account that in Ul there already exits the surface term γA. Then
(28)dGl=-SldT+Vldpl+∑i‍μlidNli+γdA.
Then the Maxwell relation on variables Nli, A gives (27). Then the integration will lead precisely to (24). This derivation revitalizes D-model.To complete the overview it is necessary to add the approach of Mirabel and Katz [13] who follow the differential way of consideration which led to (12). In order to get (24) Mirabel and Katz modified Young-Laplace equation (7) as
(29)pl-pv-2γr~dγdξi.In [10] this approach was classified as an inappropriate one. We see that this approach is one of explanations how to reconcile the requirement to get the Gibbs-Kelvin equation and the requirement to have the free energy in the capillary model as described above. So, under the mentioned form of the free energy the true alternative is to modify the Gibbs-Kelvin equation or to modify the Young-Laplace equation.


## 3. Physical Model

Ideas to find a tool outside the nucleation theory to solve the problem formulated above seem to be very attractive. But unfortunately there is no such tool. Speaking about the possibility of the direct measurement of the free energy of embryos one has to confess that except for some specific cases (see [27]) there is no direct mechanism to realize this possibility.Certainly, practically every reliable model can be expressed in terms of statistic-mechanical approach (see [28]), but this can not be a tool to decide whether one can cancel the derivative of the surface tension. Here also both approaches can be justified on statistic-mechanical basis, the matter is which reference state of equilibrium is chosen.One has to stress that the density functional theory cannot be the instrument to decide to cancel the derivatives [29, 30] because here the model form of the free energy is simply postulated at the level of the elementary hydrodynamic subsystems.Unfortunately, the comparison with the 2D Ising model as it is proposed in [31] to solve the question of the necessity of cancelation cannot be fulfilled because one cannot generalize this model to the multicomponent case.So, we have to formulate some physical model and discuss it.


### 3.1. Different Meanings of the Surface Tension

One cannot argue that some excesses initiated by the interface have to be put into the theory. The most common one is the excess of the free energy, that is, the surface tension.The answer to the question what approach is the most appropriate one lies in the method of construction of the capillary approximation and the expression for a free energy in this approximation. We have to reformulate a question: what is the surface tension which we will put in the surface term for the free energy of the embryo? Several answers are possible.(i)
This is the surface tension of an embryo.Then the nucleation theory is the absolutely self-closed theory and it cannot be checked. At the modern level of experimental devices there is no way to measure the surface tension of droplets otherwise than in the nucleation rate experiments. Everything which is measured in experiments is attributed to the value of surface tension according to theoretical dependencies derived in frames of an approach to calculate the nucleation rate.(ii)
This is the surface tensionγt observed in a capillary tube (index t) of the radius equal to the radius of the embryo (with corresponding wetting).Here one can perform direct experiments to measure the surface tension. This is the way to construct a theory with the predicting force contrast to the previous case.(iii)
The surface tension is the plane surface tensionγp.Certainly, this is a rather rough approximation, but ordinary, namely, the flat surface tension is known. By application of the methods of thermodynamics of small objects it is possible to construct some approximation for the surface tension in a tube and to reduce this case to the previous one. Also one has to mention that in some sense this case can be preferable because the surface excesses are mainly known in the case of the flat surface.(iv)
One can propose the conventional variant:γ is the surface tension of a flat surface but at the surplus pressure (plus the Laplace-Young addition); this value can be found by interpolation from P,T coexistence curve to surplus temperature or pressure.There are no principal differences between this variant and the previous one. So, no special attention will be paid to this variant.To go further one has to note that the capillary tube and the flat liquid are not a closed object as the droplet is. This is the inevitable difference and this difference will be important. The question is now the following:(i)
is it enough to account the compactness of the embryo only at the level of expression forAas a function of the number of molecules inside the embryo or is it necessary to rewrite the analog of the Gibbs-Duhem equation which will be different from (25)?Our conclusion is that there is no direct answer on this question in frames of macroscopic thermodynamics, but the principle of correspondence formulated below is rather reliable and solves this question.


### 3.2. Mean Chemical Potentials in Different Models

The object of further investigation will be the cluster of a liquid phase which is dense and compact contrast to a vapor phase. So, we will omit the indexl, use νi instead of Nli, and speak about chemical potentials instead of differences in chemical potentials5.We will formulate the principle of correspondence.(i)
It is necessary to use the Gibbs-Duhem equation from the system which provides the surface tension used in the model. If we use the surface tension from a flat surface, we write the Gibbs-Duhem equation for a flat surface. If we use the surface tension from a capillary tube, we write the Gibbs-Duhem equation for a capillary tube.Certainly, the most similar system with the explicitly measured surface tension for the situation of the embryo is the capillary tube. We take the surface tension from the capillary tube and have to write the Gibbs-Duhem equation for the capillary tube. This leads (as it will be shown later) to the Renninger-Wilemski recipe. But one has to stress that the principle of correspondence is no more than a supposition. Also it is necessary to add that one can act in a manner of the explicit attribution of the surface excesses which is typical for the Gibbs dividing surface approach.In capillary models one has to use the surface tension from the situation with a capillary tube. It leads to the following consequence discussed below.(i)
In the droplet there is a strong connection between the radiusr and the number of molecules νi; this connection is absent in the capillary tube.The last connection can be written in a most simple version of the theory (without surface excesses taken into account) as
(30)4πr33=∑i‍νivli.
If we use the Gibbs formalism there are surface excesses Ψi of the substance (index s marks the surface) and then
(31)4πr33=∑i‍νi-Ψivli.
We attribute in frames of conception of a phase in the Gibbs method all other molecules to the bulk phase
(32)νbi=νi-Ψi
(index b marks the bulk region).ApproximatelyΨi are proportional to the surface area 4πr2:
(33)Ψi=4πr2ρi,
where ρi is the density of excess6.As the main dividing surface it is preferable to take the surface of tension where the Young-Laplace equation is precise.Then(34)4πr33=∑i‍νi-4πr2ρivli
is an equation on r as a function of νi.The concentrations are defined as(35)ξi=νbi∑jνbj.
It is a principal requirement that in the definition of concentration there stand the bulk numbers of molecules.Now we will present constructions on the base of a capillary tube as a corresponding system for the free energy in capillary approximation. Until the place where the Gibbs model is formulatedwe forget about the excesses at the dividing surface. In the capillary tube there is no such balance relation as νbi=νi-Ψi. If one constructs the free energy of a capillary tube one gets
(36)Ft=∑i‍Niμi+γtA
(index t marks that we deal with a tube; Ni are the numbers of molecules in a tube). Since the numbers Ni are not fixed in principle (the tube is connected with a bath) one has to express the last relation through concentrations. This gives
(37)Ft=∑‍NiBbξ+γA,
where Bb is
(38)Bb=∑‍ξiμiξ.
The value of Bb has the sense of the averaged chemical potential. Consider
(39)Bb=μ.The condition for the equilibrium will be(40)dBbdξ=0.
Since Gibbs-Duhem equation (19) takes place the last condition means
(41)μi=μj
with a clear physical sense.The same picture will be in a situation with a plane surface. The difference is that the surplus chemical potentials are counted here from the coexistence line, while in the case of a tube they are counted from the potentials in vapor phase under the Laplace pressure.The termγA does not influence the concentration equilibrium when we suppose that the contact angle in a capillary tube is constant (namely, this situation is considered). Certainly, there exists a dependence on concentration through the increase of pressure, but we suppose that the dependence of activity coefficients, that is, the functions fi staying in standard expressions
(42)μi=ln⁡ppeq-ln⁡ξi-ln⁡fiξi
on pressure, is small. Here peq is the pressure over a pure plane liquid phase and ln⁡(p/peq) is a logarithm of the supersaturation plus one.The situation in a droplet is another one. The surface areaA depends on Ni and we have to take this dependence into account. There are several ways how to do it.To see the preferable way we will analyze the structure of the free energy in the embryo. We presentF from (17) in the form like that in the one-dimensional case:
(43)F=-BdΩ+Ω2/3.
Here Ω is defined in order that Ω2/3 is the surface surplus input into the droplet free energy
(44)Ω2/3=Aγ.ForBd one can get on the base of (17):
(45)Bd=C-3/2γ-3/2Bbvm,
where vm is the mean volume of a molecule in a droplet.The value ofBd allows the following interpretation:
(46)Bd~μγ3/2v
and one can consider the denominator in the last expression as the mean surface tension for one molecule in a power 3/2 (certainly, one molecule has no surface tension).Condition for an equilibrium concentrationξc will be
(47)dBddξi=0.
Condition for an equilibrium when the derivative of γ is put to zero will be
(48)dBb/vmdξi=0.
Here one can also get the interpretation for Bb/vm as the ratio of two mean values μ=μm and v=vm.Both conditions are independent of the value ofΩ and it is important. When we choose as a variable which is supplementary to ξ any other variable we can come to the dependence of ξc on Ω but at the saddle point the result will be the same.We will callBd the mean chemical potential in the droplet in D-model and BRW~Bb/vm the mean chemical potential in the droplet in RW-model.The property of independence onγ of the minimum of Bb/vm is the serious argument for RW-model and against D-model.(A1) In the capillary tube which gives the tension used in the capillary approximation forF there exists the same independence; the concentration is the argument which provides the minimum of Bb. The position of this minimum is independent of γ.Immediately there arise two arguments for D-model against RW-model.(A2) The number of molecules and the surface termγA (γ is the surface tension, A is the surface area) are not independent in droplet.In the tube the situation is opposite. They are independent.When the molecule comes to the droplet it enlarges the volume and, thus, the surface termγA. In the tube such effect is absent.So, here appears a question why we will use the model of a tube7.Even when the molecule does not change the volume of an embryo (e.g., this molecule is changed by the molecule of another type but with the same volume) the surface tension will be changed. We come to the following conclusion.(A3) The composition of the surface layer has to be chosen not only to minimize the surface tension as it is in the tube but to minimize the whole surface termγA; that is, the composition of the bulk region also acts on A and, thus, on the surface term in a way different from the case of a tube.This argument strengthens the conviction not to use the tube as the source of approximation forF but to use directly D-model.But the most important argument comes from the analysis of the Gibbs-Duhem relation. It is formulated below. The problem staying there is that the Gibbs-Duhem relation has to be used inevitably in any model, but to write it one has to adopt also some model. This model has the direct influence on the result.


### 3.3. Interpretation of the Gibbs-Duhem Relation

The principle of correspondence states that since we use the surface tension from the measurements with a capillary tube we have to use the Gibbs-Duhem relation for the surface of a tube. The Gibbs-Duhem relation for the surface in a tube has the following form8:
(49)dγ+∑‍ρidμiξbi=0.One can easily note that if(50)ρi∑jρj=ξbi
then
(51)∑i‍ρidμiξbi≡0
and the derivative of the surface tension cannot be compensated9.So, the physical reason of the compensation of the derivative of the surface tension on the concentration is the difference of the concentration in the surface layer from the bulk concentration. Surface enrichment as a complication for consideration was noticed in [32], but there the different possibilities to formulate the reference system were not even mentioned.Really, in any surface layer (let it be the layer of an imaginary thicknessl) we see that the volume of the layer is approximately lA and there are
(52)ξbilAvm+ρiA
molecules of a sort i. So the concentration ξsi in a surface layer will be
(53)ξsi=ξbil/vm+ρi∑jξbjl/vm+ρj
and it does not coincide with ξbi. The difference between ξbi and ξsi explains why segregation is size dependent in diluted solutions, as in droplets of small size there is no sufficient number of atoms. Certainly, l is a parameter and ξsi cannot be calculated explicitly. But under any l the value ξsi does not coincide with ξbi.We see that D-model is the model with uniform liquid and RW-model is the model with nonuniform liquid. The account of nonuniformity of liquid in frames of RW-model is the simplest one but we see that this account is very important and makes the model more complete. It is rather difficult to realize that RW-model is the model with nonuniform liquid because the region of enrichment is not described explicitly. Also there are no explicit values of excesses which also can show the nonuniformity. The Gibbs-Duhem equation is the only trace of such nonuniformity. It is simply a necessary condition of enrichment in the model.We come to the following argument.(A4) Having usedγ from the situation with capillary tube of flat surface where the surface tension region is enriched with the surface-active component it is reasonable to take the model where the surface region is enriched according to the same rule, that is, according to Gibbs-Duhem equation (49). This argument is the decisive argument in a choice favorable for RW-model.Despite the choice of RW-model as a more adequate model we see that both models are not free from objections. For the RW-model the arguments A2 and A3 are the features ignored by this model. For the D-model the arguments A1 and A4 are the unresolved problems. Below the Gibbs dividing surface method (G-model) will be presented and this method is more sophisticated than RW- and D-models. But still the Gibbs dividing surface method (at least at the level of the several first terms of decomposition) is no more than a model because the arguments A2 and A3 are not fully resolved. But it is the best model one can propose at the relatively simple level.The terminology “Gibbs model” or “Gibbs dividing surface method” means nothing without any concrete recipe of approximations. Certainly, if we can get the characteristics of the curved surface of the closed confined droplet then the description becomes precise. The problem is that we do not know these characteristics; the optimal thing we can do is to use the characteristics of the curved surface over the unbounded system like the capillary tube is. The transmission of these characteristics to the situation of a droplet means the usage of the model. Evidently the capillary model is rather accurate and natural. But it is still a model. Here lies the source of discrepancy between theory and experiments on nucleation which tortures this field more than half a century. Certainly, this difficulty takes place in one-component case also.


## 3.1. Different Meanings of the Surface Tension

One cannot argue that some excesses initiated by the interface have to be put into the theory. The most common one is the excess of the free energy, that is, the surface tension.The answer to the question what approach is the most appropriate one lies in the method of construction of the capillary approximation and the expression for a free energy in this approximation. We have to reformulate a question: what is the surface tension which we will put in the surface term for the free energy of the embryo? Several answers are possible.(i)
This is the surface tension of an embryo.Then the nucleation theory is the absolutely self-closed theory and it cannot be checked. At the modern level of experimental devices there is no way to measure the surface tension of droplets otherwise than in the nucleation rate experiments. Everything which is measured in experiments is attributed to the value of surface tension according to theoretical dependencies derived in frames of an approach to calculate the nucleation rate.(ii)
This is the surface tensionγt observed in a capillary tube (index t) of the radius equal to the radius of the embryo (with corresponding wetting).Here one can perform direct experiments to measure the surface tension. This is the way to construct a theory with the predicting force contrast to the previous case.(iii)
The surface tension is the plane surface tensionγp.Certainly, this is a rather rough approximation, but ordinary, namely, the flat surface tension is known. By application of the methods of thermodynamics of small objects it is possible to construct some approximation for the surface tension in a tube and to reduce this case to the previous one. Also one has to mention that in some sense this case can be preferable because the surface excesses are mainly known in the case of the flat surface.(iv)
One can propose the conventional variant:γ is the surface tension of a flat surface but at the surplus pressure (plus the Laplace-Young addition); this value can be found by interpolation from P,T coexistence curve to surplus temperature or pressure.There are no principal differences between this variant and the previous one. So, no special attention will be paid to this variant.To go further one has to note that the capillary tube and the flat liquid are not a closed object as the droplet is. This is the inevitable difference and this difference will be important. The question is now the following:(i)
is it enough to account the compactness of the embryo only at the level of expression forAas a function of the number of molecules inside the embryo or is it necessary to rewrite the analog of the Gibbs-Duhem equation which will be different from (25)?Our conclusion is that there is no direct answer on this question in frames of macroscopic thermodynamics, but the principle of correspondence formulated below is rather reliable and solves this question.


## 3.2. Mean Chemical Potentials in Different Models

The object of further investigation will be the cluster of a liquid phase which is dense and compact contrast to a vapor phase. So, we will omit the indexl, use νi instead of Nli, and speak about chemical potentials instead of differences in chemical potentials5.We will formulate the principle of correspondence.(i)
It is necessary to use the Gibbs-Duhem equation from the system which provides the surface tension used in the model. If we use the surface tension from a flat surface, we write the Gibbs-Duhem equation for a flat surface. If we use the surface tension from a capillary tube, we write the Gibbs-Duhem equation for a capillary tube.Certainly, the most similar system with the explicitly measured surface tension for the situation of the embryo is the capillary tube. We take the surface tension from the capillary tube and have to write the Gibbs-Duhem equation for the capillary tube. This leads (as it will be shown later) to the Renninger-Wilemski recipe. But one has to stress that the principle of correspondence is no more than a supposition. Also it is necessary to add that one can act in a manner of the explicit attribution of the surface excesses which is typical for the Gibbs dividing surface approach.In capillary models one has to use the surface tension from the situation with a capillary tube. It leads to the following consequence discussed below.(i)
In the droplet there is a strong connection between the radiusr and the number of molecules νi; this connection is absent in the capillary tube.The last connection can be written in a most simple version of the theory (without surface excesses taken into account) as
(30)4πr33=∑i‍νivli.
If we use the Gibbs formalism there are surface excesses Ψi of the substance (index s marks the surface) and then
(31)4πr33=∑i‍νi-Ψivli.
We attribute in frames of conception of a phase in the Gibbs method all other molecules to the bulk phase
(32)νbi=νi-Ψi
(index b marks the bulk region).ApproximatelyΨi are proportional to the surface area 4πr2:
(33)Ψi=4πr2ρi,
where ρi is the density of excess6.As the main dividing surface it is preferable to take the surface of tension where the Young-Laplace equation is precise.Then(34)4πr33=∑i‍νi-4πr2ρivli
is an equation on r as a function of νi.The concentrations are defined as(35)ξi=νbi∑jνbj.
It is a principal requirement that in the definition of concentration there stand the bulk numbers of molecules.Now we will present constructions on the base of a capillary tube as a corresponding system for the free energy in capillary approximation. Until the place where the Gibbs model is formulatedwe forget about the excesses at the dividing surface. In the capillary tube there is no such balance relation as νbi=νi-Ψi. If one constructs the free energy of a capillary tube one gets
(36)Ft=∑i‍Niμi+γtA
(index t marks that we deal with a tube; Ni are the numbers of molecules in a tube). Since the numbers Ni are not fixed in principle (the tube is connected with a bath) one has to express the last relation through concentrations. This gives
(37)Ft=∑‍NiBbξ+γA,
where Bb is
(38)Bb=∑‍ξiμiξ.
The value of Bb has the sense of the averaged chemical potential. Consider
(39)Bb=μ.The condition for the equilibrium will be(40)dBbdξ=0.
Since Gibbs-Duhem equation (19) takes place the last condition means
(41)μi=μj
with a clear physical sense.The same picture will be in a situation with a plane surface. The difference is that the surplus chemical potentials are counted here from the coexistence line, while in the case of a tube they are counted from the potentials in vapor phase under the Laplace pressure.The termγA does not influence the concentration equilibrium when we suppose that the contact angle in a capillary tube is constant (namely, this situation is considered). Certainly, there exists a dependence on concentration through the increase of pressure, but we suppose that the dependence of activity coefficients, that is, the functions fi staying in standard expressions
(42)μi=ln⁡ppeq-ln⁡ξi-ln⁡fiξi
on pressure, is small. Here peq is the pressure over a pure plane liquid phase and ln⁡(p/peq) is a logarithm of the supersaturation plus one.The situation in a droplet is another one. The surface areaA depends on Ni and we have to take this dependence into account. There are several ways how to do it.To see the preferable way we will analyze the structure of the free energy in the embryo. We presentF from (17) in the form like that in the one-dimensional case:
(43)F=-BdΩ+Ω2/3.
Here Ω is defined in order that Ω2/3 is the surface surplus input into the droplet free energy
(44)Ω2/3=Aγ.ForBd one can get on the base of (17):
(45)Bd=C-3/2γ-3/2Bbvm,
where vm is the mean volume of a molecule in a droplet.The value ofBd allows the following interpretation:
(46)Bd~μγ3/2v
and one can consider the denominator in the last expression as the mean surface tension for one molecule in a power 3/2 (certainly, one molecule has no surface tension).Condition for an equilibrium concentrationξc will be
(47)dBddξi=0.
Condition for an equilibrium when the derivative of γ is put to zero will be
(48)dBb/vmdξi=0.
Here one can also get the interpretation for Bb/vm as the ratio of two mean values μ=μm and v=vm.Both conditions are independent of the value ofΩ and it is important. When we choose as a variable which is supplementary to ξ any other variable we can come to the dependence of ξc on Ω but at the saddle point the result will be the same.We will callBd the mean chemical potential in the droplet in D-model and BRW~Bb/vm the mean chemical potential in the droplet in RW-model.The property of independence onγ of the minimum of Bb/vm is the serious argument for RW-model and against D-model.(A1) In the capillary tube which gives the tension used in the capillary approximation forF there exists the same independence; the concentration is the argument which provides the minimum of Bb. The position of this minimum is independent of γ.Immediately there arise two arguments for D-model against RW-model.(A2) The number of molecules and the surface termγA (γ is the surface tension, A is the surface area) are not independent in droplet.In the tube the situation is opposite. They are independent.When the molecule comes to the droplet it enlarges the volume and, thus, the surface termγA. In the tube such effect is absent.So, here appears a question why we will use the model of a tube7.Even when the molecule does not change the volume of an embryo (e.g., this molecule is changed by the molecule of another type but with the same volume) the surface tension will be changed. We come to the following conclusion.(A3) The composition of the surface layer has to be chosen not only to minimize the surface tension as it is in the tube but to minimize the whole surface termγA; that is, the composition of the bulk region also acts on A and, thus, on the surface term in a way different from the case of a tube.This argument strengthens the conviction not to use the tube as the source of approximation forF but to use directly D-model.But the most important argument comes from the analysis of the Gibbs-Duhem relation. It is formulated below. The problem staying there is that the Gibbs-Duhem relation has to be used inevitably in any model, but to write it one has to adopt also some model. This model has the direct influence on the result.


## 3.3. Interpretation of the Gibbs-Duhem Relation

The principle of correspondence states that since we use the surface tension from the measurements with a capillary tube we have to use the Gibbs-Duhem relation for the surface of a tube. The Gibbs-Duhem relation for the surface in a tube has the following form8:
(49)dγ+∑‍ρidμiξbi=0.One can easily note that if(50)ρi∑jρj=ξbi
then
(51)∑i‍ρidμiξbi≡0
and the derivative of the surface tension cannot be compensated9.So, the physical reason of the compensation of the derivative of the surface tension on the concentration is the difference of the concentration in the surface layer from the bulk concentration. Surface enrichment as a complication for consideration was noticed in [32], but there the different possibilities to formulate the reference system were not even mentioned.Really, in any surface layer (let it be the layer of an imaginary thicknessl) we see that the volume of the layer is approximately lA and there are
(52)ξbilAvm+ρiA
molecules of a sort i. So the concentration ξsi in a surface layer will be
(53)ξsi=ξbil/vm+ρi∑jξbjl/vm+ρj
and it does not coincide with ξbi. The difference between ξbi and ξsi explains why segregation is size dependent in diluted solutions, as in droplets of small size there is no sufficient number of atoms. Certainly, l is a parameter and ξsi cannot be calculated explicitly. But under any l the value ξsi does not coincide with ξbi.We see that D-model is the model with uniform liquid and RW-model is the model with nonuniform liquid. The account of nonuniformity of liquid in frames of RW-model is the simplest one but we see that this account is very important and makes the model more complete. It is rather difficult to realize that RW-model is the model with nonuniform liquid because the region of enrichment is not described explicitly. Also there are no explicit values of excesses which also can show the nonuniformity. The Gibbs-Duhem equation is the only trace of such nonuniformity. It is simply a necessary condition of enrichment in the model.We come to the following argument.(A4) Having usedγ from the situation with capillary tube of flat surface where the surface tension region is enriched with the surface-active component it is reasonable to take the model where the surface region is enriched according to the same rule, that is, according to Gibbs-Duhem equation (49). This argument is the decisive argument in a choice favorable for RW-model.Despite the choice of RW-model as a more adequate model we see that both models are not free from objections. For the RW-model the arguments A2 and A3 are the features ignored by this model. For the D-model the arguments A1 and A4 are the unresolved problems. Below the Gibbs dividing surface method (G-model) will be presented and this method is more sophisticated than RW- and D-models. But still the Gibbs dividing surface method (at least at the level of the several first terms of decomposition) is no more than a model because the arguments A2 and A3 are not fully resolved. But it is the best model one can propose at the relatively simple level.The terminology “Gibbs model” or “Gibbs dividing surface method” means nothing without any concrete recipe of approximations. Certainly, if we can get the characteristics of the curved surface of the closed confined droplet then the description becomes precise. The problem is that we do not know these characteristics; the optimal thing we can do is to use the characteristics of the curved surface over the unbounded system like the capillary tube is. The transmission of these characteristics to the situation of a droplet means the usage of the model. Evidently the capillary model is rather accurate and natural. But it is still a model. Here lies the source of discrepancy between theory and experiments on nucleation which tortures this field more than half a century. Certainly, this difficulty takes place in one-component case also.


## 4. Gibbs Method

The more adequate model than D- and RW-models can be constructed on the base of surface excesses and it will be done in this section.First of all it is necessary to stress that we speak about the simplest vulgar version of the Gibbs dividing surfaces method. Namely, such a version is necessary to be installed instead of D-model or RW-model because the information about excesses even in a flat case is rather poor.The structure of the Gibbs dividing surface method in the simplest (zero) approximation requires to consider the surface (it is more convenient to consider the surface of tension, i.e., the surface where the Young-Laplace formula takes place precisely) and the surface excesses of all componentsN~si which will be marked by Ψi. Then the surface area A will be calculated on the base of
(54)A=C∑i‍vliνi-Ψi2/3
which forms an equation on A:
(55)A=C∑i‍vliνi-Aρi2/3.
The first iteration is sufficient:
(56)A=C∑i‍vliνi-C∑i‍vliνi2/3ρi2/3.We again recall that here and later we suppose the liquid to be incompressible and the volumesvli to be independent of concentration.From the first glance it seems that the free energyF has to be approximately transformed from F=∑i‍μiνi+γA to
(57)F=∑i‍μiνi+γC∑i‍vliνi-C∑i‍vliνi2/3ρi2/3.
But actually this transformation can be avoided by use of special variables as it will be clear below.One can fulfil the same analysis as above but instead ofΩ one has to choose
(58)Ω⟶γ3/2C∑i‍vliνi-C∑i‍vliνi2/3ρi2/33/2.Then(59)F=-∑j‍λjμj-∑j‍ϱjμjΩ2/3γξ+Ω2/3
with
(60)λi=νi-Ψi.
Certainly,
(61)λiλj=ξiξj.One can choose instead ofΩ the variable κ by the following formula:
(62)κ=A3/2γ-∑i‍ϱiμi3/2.
In these variables the free energy F has the form
(63)F=-κbgξ+κ2/3
with the generalized chemical potential
(64)bg=∑iλiμiκ
or
(65)bg=∑i‍ξiμi∑jλjκ.
One has to show that bg does not depend on κ. To fulfill this derivation one can come to
(66)bg=∑i‍ξiμi∑jλjA3/2γ-∑kϱkμk3/2
or10(67)bg=∑i‍ξiμi∑jλjγ-∑kϱkμk3/2∑lvlλl.
It can be also presented as
(68)bg=∑i‍ξiμi1γ-∑kϱkμk3/2∑lvlξl.
The last relation evidently shows that bg is really a function of ξ. The dependence on κ is absent.Formula (63) is valid for the arbitrary Gibbs model. This form will be called the canonic representation. Now we see that it takes place for the arbitrary model in frames of the Gibbs dividing surface method. This result is of great importance. Directly from this result follows the Gibbs rule. It means that the free energy of the critical embryo is one-third of the surface energy. Now this rule is shown for the arbitrary model from the Gibbs dividing surface method.One can use expression (68) to clarify the Renninger-Wilemski problem. According to the Gibbs absorption relation11(69)dγ=∑j‍ϱidμi
the derivative of the surface tension on concentration is canceled by the corresponding derivatives of ϱi on ξ. So, if we write bg without surface excesses as
(70)bg=∑i‍ξi1γ3/2∑jvjξj
we have to forbid the differentiation of γ on concentration. We see that in frames of the Gibbs approach (G-model) the Renninger-Wilemski problem is explained. Here there are no artificial requirements not to differentiate the surface tension on concentration. It is necessary to stress that the reason is not the formal Gibbs absorption equation, but the difference of concentrations in the bulk solution from the integral values.One has also to recall that in frames of G-model there is one undefined parameter because in theL component case there are only L-1 independent equations:
(71)dγdξbi=∑jρjdμjdξbi.One additional parameter in G-model is inevitable. Ordinary it is the sum of excesses with some weights.


## 5. The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model. It is important that G-model is the self-consistent one. But there exists one additional parameter which cannot be calculated. Namely, for two excessesρ1 and ρ2 there exists only one equation:
(72)ρ1dμ1dξb+ρ2dμ2dξb=dγdξb.
For L-component case there exist L-1 equations:
(73)∑jρjdμjξbdξbi=dγdξbi
and the last Lth equation
(74)∑jρjdμjξbdξbL=dγdξbL
is the superposition of the previous L-1 equations. So, it is necessary to add one equation. When one knows the distance l between the surface of tension and the total equimolecular surface determined as
(75)∑j‍νjvlj=4π3r3
(certainly, it is possible to account the density in a vapor phase) then (l≪r)
(76)∑‍Aρivli=Al.The most simple variant is to say that the total equimolecular surface coincides with the surface of tension. This corresponds to initial relation (30). It gives
(77)∑‍ρivli=0.
Certainly, this is artificial requirement and generally speaking it is wrong. The only justification is that this requirement is the simplest one and that it corresponds to the recipe (30) of the simplest and the oldest capillary approximation which is used since [9]. In [23, 24] this connection was announced to be the only true one. One has to stress that according to the Gibbs dividing surface method this connection has no preferences.The main conclusion for the SG-model (simplified Gibbs model) is that the results for the free energy of the critical embryo, for concentrationξb and for the variable Ω (here κ), coincide with results of RW-model.The last statement can be easily checked by comparison of corresponding formulas.The coincidence of the free energy of the critical embryo, the concentration and the surface of the critical embryo in both (RW and SG) models, does not mean that all characteristics coincide. In RW-model one can get(78)νi=ξbiΩvmγ3/2C3/2=Nbi.Certainly, to see the critical numbersνci of the molecules of ith component in SG-model one has to solve the system of equations and add ρi4πr2 to
(79)νbi=ξbiΩvmγ3/2C3/2.Then(80)νi=ξbiΩvmγ3/2C3/2+ρi4πr2
or
(81)νi=ξbiΩvmγ3/2C3/2+ρiΩ2/3γ-1
and the result slightly differs from RW-model.But this difference means absolutely nothing for the height of activation barrier. For the rate of nucleation since the change of the Zeldovich factor is negligible the error is negligible also.We come to the same functional dependencies but with slightly changed values of coefficients. Such simple dependencies are probably the source for the simple scaling factors like it is observed in [33]. One can also mention that the applicability of the scaling approach to the free energy like the Fisher droplet model and analogous ideas by Stauffer [34] and later by Bauchspiess and Stauffer [35] can be also interpreted as the result of the simple scaling observed here. They can be also included in this picture by some moderate transformations.


## 6. Self-Consistency of Thermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange; there are no evident errors in both D- and RW-models, but at least one result is wrong. Does it mean that in construction of thermodynamic theories it is possible to make an “invisible error”? Fortunately it will be shown below that the difference in results for the free energy of the critical embryo obtained in different models has the order of correction.To see this smallness one can redefineκ as A3/2 and forget about excesses. The value of κ is the big parameter of the theory. Really, the thermodynamic description is valid only at great number of molecules in the embryo which requires a big value of κ.ThenF=-BAκ-γκ2/3 and it is important that BA~Bb/vm does not contain γ. Really, from equation(82)∂F∂ξκ=fixed=∂γ∂ξκ2/3-∂BA∂ξκ
it is seen that the first term (∂γ/∂ξ)κ2/3 with the derivative ∂γ/∂ξ has the correction order κ2/3:
(83)∂γ∂ξκ2/3~κ2/3
in comparison with the second term (∂BA/∂ξ)κ having the order κ:
(84)∂BA∂ξκ~κ.
It means that the correction will be small.We extract this result because of its importance for the reconstruction of the logical self-consistency of thermodynamics. Only the correcting order of the term with the derivative of the surface tension allows to ignore it in the main order and to return the leading role of the ordinary capillary approximation.Certainly, in situations of practical realization in experiments the difference between the orderκ and κ2/3 can be hardly seen.Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation ofγ on concentration then the equation on concentration will be different. It would cause the impression that there is a shift in a leading term. The true answer is that this result causes the shift in Fc which has a correction order as it follows from the last equation.One can treat the surface tension as a coefficient in the first correction term proportional to the surface of the embryo. The coefficients atκ1/3, ln⁡κ, κ-1/3, and so forth depend on intensive variables (concentration is one example). Their derivatives will be canceled by derivatives of corresponding excesses. The structure will resemble the Renninger-Wilemski problem. But here the dimension of “surface” will be κ1/3, ln⁡κ, κ-1/3, and so forth. This effect will be called the “generalized cancelation of derivatives on intensive variables.” The considerations to justify the cancelation are practically the same as those presented here for the leading correction.The necessity to develop the theory with surface excesses is evident because the surface excesses will essentially shift the position of the near-critical region in comparison with the size of the near-critical region. In RW-model the form of the near-critical region can not be explicitly described. But this region is extremely important for solution of kinetic equation to get the rate of nucleation.The Gibbs method allows to get the form of the near-critical region. Rigorously speaking to determine the form of the near-critical region one has to take the expression forF with correction terms up to the order which causes the shift of position of the near-critical region. Now it is clear that the effect results mainly in the shift of the near-critical region while the form of the near-critical region remains approximately the same.


## 7. Results and Conclusions

The main new results of the current analysis are the following:(i)
It is shown that both D-model and RW-model are not complete ones.(ii)
The principle of correspondence is formulated. It is shown that this postulate leads to the RW-model.(iii)
It is shown that canonic representation takes place in the arbitrary Gibbs model.(iv)
The Gibbs rule is shown for the arbitrary model in the class of the Gibbs dividing surface method.(v)
The driving physical reason of RW problem is shown. It is the difference of concentration in a surface layer from the concentration in a bulk liquid phase.(vi)
The self-consistency of thermodynamic models is shown.Does the performance of this analysis mean that the free energy of the critical embryo in nucleation has been found? To comment on this question one has to note that the conditions of applicability of thermodynamic approach to solve the RW problem are very strong. Namely, ordinary it is required that(85)νi≫1.
But also the surface layer has to be the uniform media which requires
(86)Ψi=Aρi≫1.
The last inequality is very hard to satisfy. The difficulty occurs not only because the surface layer has the small volume (we mean the real surface layer which has the finite thickness and it is only reflected in Ψi in Gibbs model) but also because there exists an effect of enrichment of the surface layer by the so-called surface active substances. Then there are only few molecules of the surface passive substance. This breaks the last inequality for the surface-passive components for rather big critical embryos. Otherwise we have to construct the theory with a nonuniform surface layer, which is extremely difficult to do, even approximately.The theoretical constructions presented above can be generalized. In fact there exist several model methods to improve the construction of the free energy, for example, the Debenedetti Reiss method, discussed in [36]. These refinements can be inserted into the theory presented here without radical rearrangements.One can refine the theory by inserting the dependence of the surface tension on radius (see, e.g., [37, 38]). The special effects which appeared in the constrained systems (see [39]) are not taken into account here but they can be easily introduced into this theory. One can add that the possibility of deformation of the center of the embryo can be also taken into account. This effect is quite analogous to the deformation of heterogeneous centers and it was considered in [40].We keep aside the possibility of fluctuations widely investigated theoretically and observed experimentally [41]. They can be also included into consideration without essential transformation of the theory. One can insert here the Tolman-like corrections as well as the scaled expansions introduced in [42]. One has to note that any refinements like [43, 44] of the classical nucleation theory can be included here directly without any interference with results of this consideration. Refinements have to be put instead of the classical expression.However, in this analysis we do not analyze the possibility that the bulk structure of the embryo can differ from the bulk structure of the reference bulk system in a capillary tube (for the principal possibility of such an effect see [45]). We consider the bulk of the embryos as the homogeneous media; for nonuniform density one can see [46]. The theory can be transformed to grasp this case also.We do not pretend to go close to the critical point although the difference between concentrations in bulk and in surface layers can be interpreted in analogous manner. It seems that any extensions of the classical approach like [47] should be replaced by theoretical constructions based on the renormalized group approach.



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