Binary nucleation kinetics. III. Transient behavior and time lags

Communication: On the diffusion tensor in macroscopic theory of cavitation

The classical description of nucleation of cavities in a stretched fluid relies on a one-dimensional Fokker-Planck equation (FPE) in the space of their sizes r, with the diffusion coefficient D(r) constructed for all r from macroscopic hydrodynamics and thermodynamics, as shown by Zeldovich. When additional variables (e.g., vapor pressure) are required to describe the state of a bubble, a similar approach to construct a diffusion tensor D^ generally works only in the direct vicinity of the thermodynamic saddle point corresponding to the critical nucleus. It is shown, nevertheless, that "proper" kinetic variables to describe a cavity can be selected, allowing to introduce D^ in the entire domain of parameters. In this way, for the first time, complete FPE's are constructed for viscous volatile and inertial fluids. In the former case, the FPE with symmetric D^ is solved numerically. Alternatively, in the case of an inertial fluid, an equivalent Langevin equation is considered; results are compared with analytics. The suggested approach is quite general and can be applied beyond the cavitation problem.

Kinetic theory of diffusion-limited nucleation

We examine binary nucleation in the size and composition space {R,c} using the formalism of the multivariable theory [N. V. Alekseechkin, J. Chem. Phys. 124, 124512 (2006)]. We show that the variable c drops out of consideration for very large curvature of the new phase Gibbs energy with composition. Consequently nuclei around the critical size have the critical composition, which is derived from the condition of criticality for the canonical variables and is found not to depend on surface tension. In this case, nucleation kinetics can be investigated in the size space only. Using macroscopic kinetics, we determine the general expression for the condensation rate when growth is limited by bulk diffusion, which accounts for both diffusion and capillarity and exhibits a different dependence with the critical size, as compared with the interface-limited regime. This new expression of the condensation rate for bulk diffusion-limited nucleation is the counterpart of the classical interface-limited result. We then extend our analysis to multicomponent solutions.

Thermodynamics and kinetics of binary nucleation in ideal-gas mixtures

The nonisothermal single-component theory of droplet nucleation [N. V. Alekseechkin, Physica A 412, 186 (2014)] is extended to binary case; the droplet volume V, composition x, and temperature T are the variables of the theory. An approach based on macroscopic kinetics (in contrast to the standard microscopic model of nucleation operating with the probabilities of monomer attachment and detachment) is developed for the droplet evolution and results in the derived droplet motion equations in the space (V, x, T)—equations for V̇≡dV/dt, ẋ, and Ṫ. The work W(V, x, T) of the droplet formation is obtained in the vicinity of the saddle point as a quadratic form with diagonal matrix. Also, the problem of generalizing the single-component Kelvin equation for the equilibrium vapor pressure to binary case is solved; it is presented here as a problem of integrability of a Pfaffian equation. The equation for Ṫ is shown to be the first law of thermodynamics for the droplet, which is a consequence of Onsager's reciprocal relations and the linked-fluxes concept. As an example of ideal solution for demonstrative numerical calculations, the o-xylene-m-xylene system is employed. Both nonisothermal and enrichment effects are shown to exist; the mean steady-state overheat of droplets and their mean steady-state enrichment are calculated with the help of the 3D distribution function. Some qualitative peculiarities of the nucleation thermodynamics and kinetics in the water-sulfuric acid system are considered in the model of regular solution. It is shown that there is a small kinetic parameter in the theory due to the small amount of the acid in the vapor and, as a consequence, the nucleation process is isothermal.

Nucleation pathway of core-shell composite nucleus in size and composition space and in component space

The kinetics of nucleation of a core-shell composite nucleus that consists of a core of stable final phase surrounded by a wetting layer of intermediate metastable phase is studied using the kinetic theory of binary nucleation not only in the size and composition space but also in the component space. The steady-state solution of the Fokker-Planck equation is considered. Various formulas for the critical nucleus at the saddle point as well as for the postcritical nucleus are derived. The kinetics of nucleation at the saddle point is more appropriately characterized in the size and composition space, while the kinetics of the postcritical nucleus is more appropriately described in the component space. Although both the free-energy landscape and the reaction rates play decisive role to determine the kinetics of nucleation at the saddle point, the details of the free-energy landscape are irrelevant to the kinetics of the postcritical nucleus.

Nucleation theory in Langevin's approach and lifetime of a Brownian particle in potential wells

The multivariable theory of nucleation suggested by Alekseechkin [J. Chem. Phys. 124, 124512 (2006)] is further developed in the context of Langevin's approach. The use of this approach essentially enhances the capability of the nucleation theory, because it makes possible to consider the cases of small friction which are not taken into account by the classical Zel'dovich-Frenkel theory and its multivariable extensions. The procedure for the phenomenological determination of the nucleation parameters is described. Using the similarity of the Kramers model with that of nucleation, the lifetime of a Brownian particle in potential wells in various dimensionalities is calculated with the help of the expression for the steady state nucleation rate.

Transient nucleation near the mean-field spinodal

Nucleation is considered near the pseudo-spinodal in a one-dimensional phi{4} model with a nonconserved order parameter and long-range interactions. For a sufficiently large system or a system with slow relaxation to metastable equilibrium, there is a non-negligible probability of nucleation occurring before reaching metastable equilibrium. This process is referred to as transient nucleation. The critical droplet is defined to be the configuration of maximum likelihood that is dynamically balanced between the metastable and stable wells. Time-dependent droplet profiles and nucleation rates are derived, and theoretical results are compared to computer simulations. The analysis reveals a distribution of nucleation times with a distinct peak characteristic of a nonstationary nucleation rate. Under the quench conditions employed, transient critical droplets are more compact than the droplets found in metastable equilibrium simulations and theoretical predictions.

Multivariable kinetic theory of the first order phase transitions

The problem of calculation of the steady state homogeneous nucleation rate in the multidimensional space of the variables describing a nucleus is considered. Within the framework of the theory proposed, expressions for the nucleation rate and the steady state distribution function of nuclei are derived. The expression for the nucleation rate is invariant with respect to the space dimensionality and, in particular, involves the result of the one-dimensional theory. The distribution function is obtained in the initial, physical variables. In connection with the analysis of restrictions on the current direction, the question of symmetry of the matrix of diffusivities is considered; on the basis of the detailed balance principle it is shown that this matrix is symmetric. The question of normalizing the equilibrium distribution functions is investigated and the physical picture of the equilibrium state is described. The procedure of reducing the multidimensional theory to the one-dimensional one is described.

Kinetic theory of binary nucleation based on a first passage time analysis

The binary classical nucleation theory (BCNT) is based on the Gibbsian thermodynamics and applies the macroscopic concept of surface tension to nanosize clusters. This leads to severe inconsistencies and large discrepancies between theoretical predictions and experimental results regarding the nucleation rate. We present an alternative approach to the kinetics of binary nucleation which avoids the use of classical thermodynamics for clusters. The new approach is an extension to binary mixtures of the kinetic theory previously developed by Narsimhan and Ruckenstein and Ruckenstein and Nowakowski [J. Colloid Interface Sci. 128, 549 (1989); 137, 583 (1990)] for unary nucleation which is based on molecular interactions and in which the rate of emission of molecules from a cluster is determined via a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution of a "surface" molecule moving in a potential field created by the cluster. The starting master equation is a Fokker-Planck equation for the probability distribution of a surface molecule with respect to its phase coordinates. Owing to the hierarchy of characteristic time scales in the evolution of the molecule, this equation can be reduced to the Smoluchowski equation for the distribution function involving only the spatial coordinates. The new theory is combined with density functional theory methods to determine the density profiles. This is essential for nucleation in binary systems particularly when one of the components is surface active. Knowing these profiles, one can determine the potential fields created by the cluster, its rate of emission of molecules, and the nucleation rate more accurately than by using the uniform density approximation. The new theory is illustrated by numerical calculations for a model binary mixture of Lennard-Jones monomers and rigidly bonded dimers of Lennard-Jones atoms. The amphiphilic character of the dimer component (i.e., its surface activity) is induced by the asymmetry in the interaction between a monomer and the two different sites of a dimer. The inconsistencies of the BCNT are avoided in the new theory.

Recent developments in the kinetic theory of nucleation

A review of recent progress in the kinetics of nucleation is presented. In the conventional approach to the kinetic theory of nucleation, it is necessary to know the free energy of formation of a new-phase particle as a function of its independent variables at least for near-critical particles. Thus the conventional kinetic theory of nucleation is based on the thermodynamics of the process. The thermodynamics of nucleation can be examined by using various approaches, such as the capillarity approximation, density functional theory, and molecular simulation, each of which has its own advantages and drawbacks. Relatively recently a new approach to the kinetics of nucleation was proposed [Ruckenstein E, Nowakowski B. J Colloid Interface Sci 1990;137:583; Nowakowski B, Ruckenstein E. J Chem Phys 1991;94:8487], which is based on molecular interactions and does not employ the traditional thermodynamics, thus avoiding such a controversial notion as the surface tension of tiny clusters involved in nucleation. In the new kinetic theory the rate of emission of molecules by a new-phase particle is determined with the help of a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution function of a surface layer molecule moving in a potential field created by the rest of the cluster. The new theory was developed for both liquid-to-solid and vapor-to-liquid phase transitions. In the former case the single-molecule master equation is the Fokker-Planck equation in the phase space which can be reduced to the Smoluchowski equation owing to the hierarchy of characteristic time scales. In the latter case, the starting master equation is a Fokker-Planck equation for the probability distribution function of a surface layer molecule with respect to both its energy and phase coordinates. Unlike the case of liquid-to-solid nucleation, this Fokker-Planck equation cannot be reduced to the Smoluchowski equation, but the hierarchy of time scales does allow one to reduce it to the Fokker-Plank equation in the energy space. The new theory provides an equation for the critical radius of a new-phase particle which in the limit of large clusters (low supersaturations) yields the Kelvin equation and hence an expression for the macroscopic surface tension. The theory was illustrated with numerical calculations for a molecular pair interaction potential combining the dispersive attraction with the hard-sphere repulsion. The results for the liquid-to-solid nucleation clearly show that at given supersaturation the nucleation rate depends on the cluster structure (for three cluster structures considered-amorphous, fcc, and icosahedral). For both the liquid-to-solid and vapor-to-liquid nucleation, the predictions of the theory are consistent with the results of classical nucleation theory (CNT) in the limit of large critical clusters (low supersaturations). For small critical clusters the new theory provides higher nucleation rates than CNT. This can be accounted for by the fact that CNT uses the macroscopic interfacial tension which presumably overpredicts the surface tension of small clusters, and hence underpredicts nucleation rates.

Kinetics of binary nucleation of vapors in size and composition space

We reformulate the kinetic description of binary nucleation in the gas phase using two natural independent variables: the total number of molecules g and the molar composition x of the cluster. The resulting kinetic equation can be viewed as a two-dimensional Fokker-Planck equation describing the simultaneous Brownian motion of the clusters in size and composition space. Explicit expressions for the Brownian diffusion coefficients in cluster size and composition space are obtained. For characterization of binary nucleation in gases three criteria are established. These criteria establish the relative importance of the rate processes in cluster size and composition space for different gas phase conditions and types of liquid mixtures. The equilibrium distribution function of the clusters is determined in terms of the variables g and x. We obtain an approximate analytical solution for the steady-state binary nucleation rate that has the correct limit in the transition to unary nucleation. To further illustrate our description, the nonequilibrium steady-state cluster concentrations are found by numerically solving the reformulated kinetic equation. For the reformulated transient problem, the relaxation or induction time for binary nucleation was calculated using Galerkin's method. This relaxation time is affected by processes in both size and composition space, but the contributions from each process can be separated only approximately.

Genuine saddle point and nucleation potential for binary systems

A generalized nucleation potential is constructed for binary systems. The potential consists of the reversible work of cluster formation plus additional terms arising from various kinetic effects. We show that the major nucleation flux passes through the saddle point (termed the genuine saddle point) of this generalized nucleation potential. The generalized nucleation potential reduces to the kinetic potential of a unary system when one component vanishes. The genuine saddle point concept provides a convenient way to identify systems and conditions for which the ridge crossing phenomenon occurs. Our theory agrees approximately with exact numerical results.

Analytical approach to time lag in binary nucleation

We present an analytical formula for the time required to establish steady state in a nucleating binary system. To test our solution, we evaluate the time lag for a range of activities of both components at the vapor-liquid transition, and show that our result is in much better agreement with a purely numerical simulation than other available analytical formulas, which overestimate the time lag by factors of from 2 to 200.