G. M. Pound

Surface structure and surface tension: Perturbation theory and Monte Carlo calculation

A Monte Carlo computation of the surface structure and surface tension of a liquid with molecules interacting according to the Lennard‐Jones 12–6 potential is described. The computation demonstrates the presence of a well‐developed layer structure at the surface extending to a depth of about eight molecular layers. The calculated surface tension agrees well with that predicted by Toxvaerds extension of the Barker‐Henderson perturbation theory to nonuniform fluids. The latter theory also predicts well the averaged behavior of the surface density profile (though not the development of the layered structure). This theory is used with accurate pair and triplet potentials to calculate values of the surface tension of liquid argon that agree quantitatively with experiment over a wide range of temperatures.

A new Monte Carlo method for calculating surface tension

A new Monte Carlo method for calculating the surface tension of a liquid is described. The method is based on a direct evaluation of the free energy required to create a surface, unlike earlier Monte Carlo calculations which evaluated the surface stress. It is applied to the 6:12 fluid in conditions close to the triple point for argon. The calculated surface tension agrees within statistical uncertainty with previous Monte Carlo estimates, but the statistical uncertainty of the present method is much lower. Agreement with experimental data for argon is not good, as should be expected; estimates of the effects of using a correct pair potential and particularly of including three‐body interactions indicate that they would lead to good agreement.

Homogeneous nucleation of crystalline gallium from liquid gallium

Abstract The nucleation rate of crystalline gallium from supercooled liquid gallium was measured using a dilatometric technique. The results appear to represent homogeneous nucleation. They may be explained by ‘classical’ nucleation theory upon assumption of a negative interfacial entropy of formation.

Effect of Concentration‐Dependent Diffusion Coefficient on the Migration of Interphase Boundaries

The diffusion‐controlled interphase‐boundary migration problem is defined by a system of equations in which the boundary conditions for the diffusion equation are determined by capillarity. Numerical computer solutions to these equations were obtained for steady‐state Widmanstatten ferrite plate growth. The effect of a concentration‐dependent diffusion coefficient on the migration rate of some simple interphase boundary shapes was investigated. It was found that a constant diffusion coefficient corresponding to the weighted average coefficient in the matrix gave an exact growth rate.

Replacement Partition Function for Small Crystals in Homogeneous Nucleation Theory

The replacement partition function was calculated for small fcc crystals by numerical normal mode analysis using exactly the same model, which considers nearest‐neighbor interactions only and neglects the surface relaxation and thermal expansion, to compute all the necessary quantities. A method of calculating the surface free energy of a small crystal from data on the macroscopic surface was introduced. The numerical value of the replacement partition function thus computed was, for example, 108 at T=2θ and for n=87. This is considerably larger than the value estimated for a crystal so far, which is 102‐−04 at T=2θ and for n ≅ 100. This result suggests the need of further work both in numerical calculation and in theoretical studies.

Growth Kinetics of Plate‐Like Precipitates

Different models for the growth kinetics of plate‐like precipitates have been critically reviewed. It is shown that the Zener‐Hillert equation does not properly take into consideration the capillarity effect at the advancing interphase boundary. A mathematical treatment of the diffusion controlled steady‐state growth of plate‐like precipitate is presented for the case in which the variation of concentration along the interface is determined from the theory of capillarity. This variation is assumed to be small so that the steady‐state shape does not deviate appreciably from a parabola. Applicability of the present treatment to the growth of a group of plates is then discussed.

Statistical mechanics of homogeneous nucleation in vapor

Abstract The thermodynamics and statistical mechanics of fluctuations in nucleation are considered in Sec. I-C, II-A, II-D, and II-E of this chapter. A prescription is given for defining a liquid microcluster in vapor in terms of a Gibbs dividing surface which is valid even for small microclusters where the density is not homogeneous and does not reach the density of the bulk liquid even at the center of the microcluster. This definition can be given only in 3n - 6 degrees of freedom, that is, translation and rotation of the microcluster cannot be included in the corresponding definition of the surface tension. Thus, this definition of the microcluster in vapor can be used with the Lothe-Pound definition of the surface tension of microclusters but not with the Kikuchi definition. Sec. II-D treats the statistical mechanics of the capillarity approximation in terms of the Lothe-Pound theory. The description is developed in some detail for the case of a crystal. A calculation is made to determine the extent to which the contributions from free translation and rotation of the microcluster in vapor are already contained in the volume free energy of bulk crystal. This effect is described by the replacement partition function, which gives the free energy a microcluster in vapor does not have because it is no longer a part of the bulk phase. It is concluded that the replacement partition function is due to the six translational and rotational oscillations of the mathematical microcluster in the bulk crystal for which all atoms are fixed at their local lattice sites. The numerical value of the replacement partition function for a crystalline microcluster of n = 100 atoms is estimated to be only about 104. In other words, very little of the contributions from free translation and rotation of the microcluster in vapor, corresponding to a partition function of typically 1020, are already contained in the volume free energy of the bulk crystal. Furthermore, the Lothe-Pound theory assumes that these contributions are not already contained in the macroscopic surface tension of the bulk crystal either. In other words, it is assumed that there is no release of correlation on forming macroscopic surface on a bulk crystal. Accordingly, the contributions from free translation and rotation of the crystalline microcluster in vapor must be added to the classical prescription. In the Lothe-Pound picture, a similar state of affairs is assumed to be valid also for liquids. Sec. II-E discusses the statistical mechanics of the capillarity approximation in terms of the Kikuchi theory, which is specifically for liquids. Secs. II-F and II-G discuss the difference and similarities between the crystal and liquid cases in terms of correlation distances, microcluster size and surface reconstruction in the bulk phase. Qualitative interpretations of the replacement partition function and the release of correlation on forming macroscopic surfaces are given. Also, the effects of change in properties of the liquid between the triple point and the critical point are considered. It is shown that the replacement partition function, which describes the extent to which the contributions from free translation and rotation of the microcluster in vapor are already contained in the volume free energy of the bulk phase, should be small for crystals. Furthermore, even for liquids, the replacement partition function may not be large. Rather, it is a function of the correlation distance in the liquid, being smaller for larger correlation distances. For example, it cannot be assumed that a microcluster in bulk liquid already has its entire contribution for free rotation simply because all its rotated configurations are equally probable. The replacement partition function for a liquid remains to be calculated. Secs. II-F and II-G conclude that the Lothe-Pound theory is correct for crystals. In the case of liquids, the Lothe-Pound theory assumes that none of the contributions from free translations and rotation of the microcluster in vapor are contained in the surface tension of the bulk phase, that is, there is no release of translational or rotational correlation on forming macroscopic surface on a bulk phase. Longer correlation distances and surface reconstruction of the macroscopic surface favor the Lothe-Pound assumption. The true state of affairs in any given case must be determined by calculation, but this has not yet been done. Secs. II-F and II-G emphasize that any serious analysis of nucleation in relation to the capillarity approximation must relate the properties of the microcluster to the volume free energy and to the surface free energy of the bulk phase. Specifically, one should not assume a priori that the contributions from free translation and rotation of an isolated liquid nucleus in vapor are already contained in the volume free energy or in the surface tension of the bulk liquid. Neither should it be assumed a priori that these contributions are not already contained in the macroscopic terms. The points at issue are strongly dependent on the structure of the bulk liquid and its surface . The necessary calculations, that is, of the replacement partition function and of the release of correlation on forming macroscopic surface, have yet to be carried out for bulk liquids. One notes that the above considerations have nothing to do with another fundamental difficulty of the capillarity approximation, namely, in the usually expected situation where the structure of the microcluster does not correspond to that of the bulk phase in terms of either the macroscopic volume or surface. Sec. III deals with the results of computer calculations in relation to the capillarity approximation. Normal mode, Monte Carlo, and molecular dynamics methods are being applied to calculate the free energy of microclusters in vapor, the volume free energy of the bulk condensed phase, the surface free energy of the bulk condensed phase, and the replacement free energy. A principal object of these model calculations is to evaluate the free energy of formation of microclusters and compare this “exact” free energy with the results obtained from the capillarity approximation with and without the corrections for free translation, rotation, and the replacement partition function. From the normal mode work on crystals, it appears that the uncorrected capillarity approximation underestimates both the potential energy and entropy of the crystalline microcluster. In some cases this results in fortuitous agreement of the uncorrected capillarity approximation with the “exact” results. In general, it seems that the capillarity approximation grossly underestimates the entropy of the crystalline microcluster and that the corrections for free translation and rotation must be added to obtain agreement with the “exact” entropy. This is to be expected from the classical phase integral treatments of sec. II. The present computer evidence indicates that the replacement entropy for a microcrystallite containing roughly 100 atoms is 18 this corresponds to a replacement partition function of about 108. The Monte Carlo and molecular dynamics results for liquid or solid argon microclusters are in fair agreement and may be almost precise enough to describe experimental results on the homogeneous nucleation of argon. However, the present Monte Carlo calculation for the surface tension of bulk liquid argon does not agree with experiment and is itself too imprecise to afford a comparison of the capillarity approximation with the “exact” results. It seems worthy of note that in all the examples of “exact” calculations, as in the examples of the capillarity approximation, the contributions to the microcluster free energy which are most precisely known are those of free translation and rotation. The experimental critical supersaturations for liquids generally fall into one of two categories, that is, those that approximately follow the classical theory and those which approximately follow the Lothe-Pound theory. One notes that the predicted critical supersaturations are typically 15% lower in the case of the Lothe-Pound theory. A 15% decrease in the surface tension, would give about the same reduction in critical supersaturation in the classical theory. In view of the rudity of the capillarity approximation, it is surprising that the experimental results for critical supersaturations in homogeneous nucleation from the vapor should follow either theory.

Comments on Homogeneous Nucleation Theory

Reiss, Katz, and Cohens theory for homogeneous nucleation of liquid from vapor is reformulated by using the criticism of Lothe and Pound. It is concluded that P(0) in Reiss, Katz, and Cohens result has to be replaced by 1/vcell. This gives a metastable equilibrium concentration of critical clusters 102–105 times larger than that predicted by Reiss, Katz, and Cohen for a typical case. The effect described has to do only with translation; the contribution from rotation of the cluster is still wholly ignored.

The Symmetry Number in Classical Statistical Mechanics

The quantum statistical treatment of the symmetry number is briefly reviewed in the Introduction for comparison with the classical statistical treatment. The concept of a localized system is discussed as a convenient method to calculate the classical partition function. The symmetry number in classical statistical mechanics is then defined, and it is applied to some examples. The classical phase integral is discussed as a limiting case of the canonical partition function in quantum statistical mechanics in the Appendix in order to make the meaning of the classical phase integral clear with respect to identity of particles, which is needed for the present purpose.

Theory of the replacement partition function for crystals in homogeneous nucleation

Abstract A replacement partition function was formulated for crystals by dividing the bulk crystal into mathematical “clusters” and expressing the total partition function in terms of the external and the internal variables of the “clusters”. In order to separate the total partition function into external and internal parts, a thermodynamic perturbation theory was employed by treating the coupling potential energy as a perturbation. The first order correction term vanishes and the second order term was shown to be negligibly small compared with the external free energy per “cluster”. Thus the external and the internal partition functions were shown to be separable and the replacement partition function may be defined as the external partition function per “cluster”. It is concluded that: (1) The replacement partition function is a well-defined fundamental quantity of the bulk crystal. (2) It is due to the translational and the rotational motions of the “clusters” in the bulk crystal for which all the atoms are fixed at their local lattice sites defined for each “cluster”, which confirms the Lothe-Pound assertion.