Thermodynamics, gas-liquid nucleation, and size-dependent surface tension

– Phenomenological nucleation theories are considered from the viewpoint of Gibbs’ surface thermodynamics. We point out, in defining the critical nucleus, that it is important to make a distinction between the number of molecules enclosed by the surface of tension and the excess number of molecules over the uniform vapor phase. We show that the Kelvin equation should be employed in determining the size of the critical nucleus even if the nucleus free energy contains a size-dependent surface energy term. Furthermore, we make use of the fact that the classical form of Kelvin equation (containing the surface tension of a flat interface) predicts the equimolar radius of the critical nucleus well down to nuclei of about 40 molecules, and derive a new equation for the size-dependent surface tension that differs from the Tolman relation. Density functional calculations support the new formula. Phenomenological nucleation theories have become increasingly popular in the recent years. Models incorporating size-dependent surface tensions and/or parameters that are determined using critical properties of the fluids in question [1]-[3] sometimes succeed in predicting the nucleation behavior more accurately than the classical nucleation theory (CNT) does. However, in spite of the original enthusiasm inspired by the Dillmann-Meier (DM) theory [1], it has become evident that overall the phenomenological theories do not perform much better than the CNT [2], and when they do, the success is more or less accidental [4]. Our aim is to take a critical look at the phenomenological theories starting from fundamental thermodynamical principles. We will show that certain assumptions that are often made in these theories are thermodynamically inconsistent. A recent development in nucleation studies is the proof of the so-called nucleation theorem [5]-[7] that allows for the measurement of the molecular content of critical nuclei. Making use of the nucleation theorem, it has been shown experimentally [6], [8] that the classical Kelvin equation predicts the number of molecules in critical nuclei, g∗, surprisingly accurately for clusters that are larger than about 40 molecules. The Kelvin equation works well also when compared with results produced using the density functional (DF) theory of nucleation [4]. Below, we will show that, together with a thermodynamically consistent formulation of nucleation phenomenology, this discovery leads to a new form for size-dependent surface tension. We will also show that the new form describes surface tensions calculated using the DF theory better than the expression derived originally by Tolman [9]. c © Les Editions de Physique 368 EUROPHYSICS LETTERS Fig. 1. – A schematic figure of the density profile at the gas-liquid interface illustrating the calculation of the various molecular numbers mentioned in the text. The area between the density profile and the uniform vapor density corresponds to g. Note that the actual density of the nucleus does not have to reach bulk liquid density even at the center. We consider a spherical cluster with a volume V and excess number of molecules g over the uniform vapor phase. At this point we do not fix the volume in any way, i.e. V and g are independent variables. We can then write g = nl − nv + ns, where nl = V ρl and nv = V ρv, with ρl and ρv densities of the uniform liquid and vapor phases, respectively, and ns is the surface excess number of molecules that corrects for the difference between the step profile and the actual interfacial density profile (see fig. 1). The free-energy change to create the cluster can now be expressed as [7] ∆G = (Pv − Pl)V + (μl − μv)nl + (μs − μv)ns + Φ(g, V ). (1) Here the P ’s are the pressures and the μ’s the chemical potentials of the uniform liquid and vapor phases, and Φ is an excess energy term dependent on both V and g that includes the surface free energy and all other possible energetical contributions. The critical nucleus, denoted by an asterisk, is in unstable equilibrium with the environment. Thus we can set the partial derivatives of ∆G with respect to V , nl, and ns zero and obtain the following conditions: μl = μ ∗ s = μv; and ∆P ∗ = P ∗ l − Pv = ∂Φ∗/∂V ∗. The work of nucleus formation is then W ∗ = ∆G∗ = −∆P ∗V ∗ + Φ∗. The above development is completely general. We now proceed to a somewhat more specialized direction by assuming that the excess free energy can be divided into two parts: Φ∗(g∗, V ∗) = A∗σ∗(g∗, V ∗) + F (g∗), where A∗ is the surface area, σ∗(g∗, V ∗) is the surface tension, and F is an arbitrary function that is assumed to depend on g∗ only and not on the location of the dividing surface (as an example, we refer to the τ ln(g∗)-term of the Fisher droplet model [10] present in several phenomenological nucleation theories [1]-[3]). We fix the dividing surface to be the surface of tension (∂σ∗(g∗, V ∗)/∂V ∗ = 0), and denote the corresponding volume, surface area, and radius with a subscript s. The surface tension is now a function of g∗ only, and we have for ∆P ∗ and W ∗ ∆P ∗ = 2σ∗(g∗)/R∗ s , (2) W ∗ = −∆P ∗V ∗ + F (g∗) +Asσ(g) . (3) Equation (2) is the Laplace relation and R∗ s denotes the nucleus radius at the surface of tension. Assuming incompressibility of the liquid phase (∆P ∗V ∗ = nl ∆μ ∗ = nl (μv(Pv) − μl(Pv)) ' A. LAAKSONEN et al.: THERMODYNAMICS, GAS-LIQUID NUCLEATION, ETC. 369 nl (μv(Pv)− μcoex)), these become ∆μ∗ = 2σ∗(g∗)v/R∗ s , (4) W ∗ = −∆μnl + F (g∗) +Asσ(g) . (5) Here the first equality is the Kelvin relation and v denotes the liquid-phase molecular volume. The classical nucleation theory is obtained by assuming that F = 0, and that the surface of tension is located at the equimolar surface, whence g∗ becomes equal to nl , and the surface tension becomes equal to the bulk surface tension (σ = γ∞) [11]. Two points are worth noting. First, the number of molecules nl appearing in eqs. (4) and (5) is not the same (unless the surface of tension happens to coincide with the equimolar surface) as the number given by the nucleation theorem, g∗, which is accessible to measurement. Thus, experimental determinations of g∗ should not without reservation be compared with estimates of the molecular content of the critical nucleus derived from phenomenological nucleation theories that assume size-dependent surface tension. Secondly, the correct theoretical equation to determine the radius of the critical nucleus is the Kelvin relation even if the theory contains a nonzero F (g∗) and a size-dependent σ. The cubic equation for determining the critical nucleus size that appears in some phenomenological theories [1], [3] results from confused treatment of V ∗, g∗, and nl , and it is not thermodynamically correct. Next, we note that eqs. (4) and (5) can be combined to give W ∗ = ∆μnl /2 + F ∗ . (6) On the other hand, following Rowlinson and Widom [11], we have