Kazumi Nishioka

Thermodynamic formulas of liquid phase nucleation from vapor in multicomponent systems

Interfacial thermodynamics is extended to noncritical liquid clusters in vapor for multicomponent systems so as to clarify the uncertainty over whether or not size and composition dependence of interfacial tension must be taken into account in taking the extremity condition of the reversible work of forming a noncritical cluster to derive the size and the composition of a critical nucleus. It is found that the differential of interfacial tension does not arise in the extremity condition due to the Gibbs–Duhem relation derived for a system of a noncritical cluster and the vapor which is in equilibrium under an additional constraint to maintain the number of molecules contained in a cluster.

Reconsideration of the concept of critical nucleus and the Gibbs—Thomson equation

The concept of critical nucleus is reconsidered by taking a single component system as an example and it is found that the size n K of a cluster for which the probabilities of decay and growth balance is not equal to the size n for which the reversible work of cluster formation takes its maximum value. n K is in general smaller than n* when n is treated as a continuous variable. There exist two values for n K , the larger of the two is kinetically unstable but the smaller is stable. The difference between the larger n K and n* increases but the difference between the two values of n K decreases with the degree of supersaturation or supercooling, and in the critical state the two values of n K coincide and it diminishes to 8/27 of n* for three-dimensional homogeneous nucleation and to 1/4 of n* for two-dimensional nucleation on a substrate. Beyond this critical state n K does not exist and for a cluster with any size the probability of growth is higher than that of decay. Whenever the Gibbs-Thomson equation is employed to take into account the curvature effect in describing kinetic processes such as in the BCF theory of spiral growth or in the studies of morphological instability, it must be replaced by the relation between the larger n K and the degree of supersaturation or supercooling.

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.

Interfacial tension for small nuclei in binary nucleation

Abstract Interfacial tension for small nuclei in a binary system is considered by applying the thermodynamics of the interface, and the inhomogeneous regular solution model is employed for the numerical evaluation. It is found that the interfacial tension decreases with curvature, and the radius of the surface of tension, the interfacial tension and the reversible work to form a critical nucleus all tend to vanish as the mean field spinodal composition is approached for the parent phase. However, this does not mean denying the spinodal decomposition, even outside the mean field spinodal, because the mean field spinodal itself does not possess a well-defined physical significance [K. Kaski, K. Binder and J.D. Gunton, J. Phys. A 16 (1983) 1623].

Energy barrier effect on transient nucleation kinetics: Nucleation flux and lag-time calculation

Energy barrier effect on the transient nucleation kinetics is studied. For a high barrier case an advanced interpolation of the “boundary-layer” type is suggested that is valid for the entire time interval. The nucleation kinetics for the “nucleated cluster” size nc is described, the lag time estimate for n=nc is given. For a low barrier case a new “similarity” solution is reported. It is shown that analytical results provide an adequate description of the transient nucleation kinetics (in comparison with numerical solution of governing equation) for both high-barrier and low-barrier conditions.

Transient nucleation in binary vapor of water and sulfuric acid

Transient nucleation is studied for a binary vapor of water and sulfuric acid, in which the acid is assumed to form a hydrate H2SO4⋅2H2O. The rate equation is solved numerically under the condition that only monomers of water and the hydrate exist initially, and it is found that (1) At the first stage the number densities of clusters with single acid molecule but containing more and more water molecules increase very rapidly. (2) Soon the rates of increase in the densities of those clusters slow down, then the densities of clusters with two acid molecules develop from the region around the valley in the contours of the reversible work of cluster formation. These observations seem to support the idea that in the analysis of steady state nucleation the population of clusters may be treated as in equilibrium with respect to exchange with water molecules in the vapor. (3) As for the effect of the difference in the formulas for the reversible work of cluster formation, the conventionally used formula and the one due to Nishioka and Mori, on the transient nucleation behavior, the conventionally used formula may cause significant errors when the critical nucleus obtained from it contains erroneous number of acid molecules. (4) The time lag to reach the steady state is found to be much longer than 1 ms, and it is not possible to continue the computer simulation up to the steady state in the present system.

Interpretation of the atomic formulae for stress and stiffness coefficients

Abstract The atomic formulae for stress {δαβ(i)} and stiffness coefficients {C*αβγλ(i)} are carefully examined particularly with their application to the results of computer simulation for imperfect lattices or amorphous matter. It is found that: (1) The formulae are valid irrespective of whether the force balance condition is satisfied for each atom or not. (2) When it is satisfied, the relation holds, where v(i) denotes the volume of the Wigner-Seitz cell surrounding the ith atom and {fex α(i)} the external forces. The physical meanings of the formulae are investigated, and it is found that: (1) 2v(i)δαβ(i) may be interpreted as the sum of the work done by the counterbalance force (CBF), which is imagined to be exerted on each atom surrounding the ith atom to balance the interaction force exerted by the ith atom, upon the deformation described by the unit value of the strain component ϵαβ under the supposition that the ith atom is fixed. (2) v(i)C*αβγλ(i) may be interpreted as the sum of the work done b...

Thermodynamic formula for the reversible work of forming a noncritical cluster from the vapor in multicomponent systems

A thermodynamic formula is derived for the reversible work of forming a noncritical liquid cluster from the vapor in multicomponent systems. It is found that the commonly used formula may be used but, for a given state of the parent vapor phase, the value of the surface tension for the critical nucleus must be used in the surface term even if the surface tension of a noncritical cluster may differ significantly from that of the critical nucleus due to the differences in the composition and the size.

Thermodynamic analysis of the driving force for forming a critical nucleus in multicomponent nucleation

The Gibbs formula for the reversible work of forming a critical nucleus in multicomponent systems is summarized and the meaning of the bulk term or the driving force for nucleation is discussed. A commonly used formula, in which the bulk term is given in terms of the difference in chemical potentials, is presented as an approximation to the Gibbs formula for systems where the nucleating phase is nearly incompressible. Further, if the composition dependence of the mean molecular volume for the bulk nucleating phase is negligible, the bulk term may be evaluated simply as the difference in chemical potentials for any one of the components. When the bulk of a nucleating phase exists in a narrow composition range such as of a crystal with sharp stoichiometry, the bulk term can be evaluated in terms of the difference in the chemical potentials of the parent phase and of the bulk nucleating phase in the macroscopic two-phase equilibrium.

Growth and perfection of KCl crystals grown from aqueous solution

Abstract KCl crystals were grown from aqueous solution by the evaporation method at the growth temperature of 40°C, the Pb ion concentration was 0–1/2000 in molar fraction, the pH was 2.0–4.0, the evaporation rate 0.4–1.1 g/day and the duration of growth 5–32 days. The results obtained are summarized as follows. (1) Pb ion addition played the most important role for growing larger KCl crystals, since it reduced the generation of parasite crystals. (2) Growth shape changed from cubic to octahedral with increasing Pb ion concentration. (3) Dislocation density in the crystals grown from seeds was always much higher than that in the crystals grown from spontaneous nuclei. Some crystals grown from spontaneous nuclei were nearly free of dislocations. (4) Growth striations were always observed on the etched {100} surfaces of KCl crystals grown in this work. A growth history could be seen from the change of striation pattern with the growth.