Free-energy landscape of nucleation with an intermediate metastable phase studied using capillarity approximation

Capillarity approximation is used to study the free-energy landscape of nucleation when an intermediate metastable phase exists. The critical nucleus that corresponds to the saddle point of the free-energy landscape as well as the whole free-energy landscape can be studied using this capillarity approximation, and various scenarios of nucleation and growth can be elucidated. In this study, we consider a model in which a stable solid phase nucleates within a metastable vapor phase when an intermediate metastable liquid phase exists. We predict that a composite critical nucleus that consists of a solid core and a liquid wetting layer as well as pure liquid and pure solid critical nuclei can exist depending not only on the supersaturation of the liquid phase relative to that of the vapor phase but also on the wetting behavior of the liquid surrounding the solid. The existence of liquid critical nucleus indicates that the phase transformation from metastable vapor to stable solid occurs via the intermediate metastable liquid phase, which is quite similar to the scenario of nucleation observed in proteins and colloidal systems. By studying the minimum-free-energy path on the free-energy landscape, we can study the evolution of the composition of solid and liquid within nuclei which is not limited to the critical nucleus.     

Pore nucleation in mechanically stretched bilayer membranes

We report a computer-simulation study of the free-energy barrier for the nucleation of pores in the bilayer membrane under constant stretching lateral pressure. We find that incipient pores are hydrophobic but as the lateral size of the pore nucleus becomes comparable with the molecular length, the pore becomes hydrophilic. In agreement with previous investigations, we find that the dynamical process of growth and closure of hydrophilic pores is controlled by the competition between the surface tension of the membrane and the line tension associated with the rim of the pore. We estimate the line tension of a hydrophilic pore from the shape of the computed free-energy barriers. The line tension thus computed is in a good agreement with available experimental data. We also estimate the line tension of hydrophobic pores at both macroscopic and microscopic levels. The comparison of line tensions at these two different levels indicates that the "microscopic" line tension should be carefully distinguished from the "macroscopic" effective line tension used in the theoretical analysis of pore nucleation. The overall shape of the free-energy barrier for pore nucleation shows no indication for the existence of a metastable intermediate during pore nucleation.     

Multicomponent nucleation: thermodynamically consistent description of the nucleation work

A thermodynamically consistent formula is derived for the nucleation work in multicomponent homogeneous nucleation. The derivation relies on the conservative dividing surface which defines the nucleus as having specific surface energy equal to the specific surface energy sigma0 of the interface between the macroscopically large new and old phases at coexistence. Expressions are given for the radius of the nucleus defined by the conservative dividing surface and by the surface of tension. As a side result, the curvature dependence of the surface tension sigmaT of the nucleus defined by the surface of tension is also determined. The analysis is valid for nuclei of any size, i.e., for nucleation in the whole range of conditions between the binodal and the spinodal of the metastable old phase provided the inequality sigmaT < or = sigma0 is satisfied. It is found that under the conditions of validity of the analysis the nucleation rate is higher than the nucleation rate given by the classical nucleation theory. The general results are applied to nucleation of unary liquids or solids in binary gaseous, liquid or solid mixtures.     

Heterogeneous critical nucleation on a completely wettable substrate

Heterogeneous nucleation of a new bulk phase on a flat substrate can be associated with the surface phase transition called wetting transition. When this bulk heterogeneous nucleation occurs on a completely wettable flat substrate with a zero contact angle, the classical nucleation theory predicts that the free-energy barrier of nucleation vanishes. In fact, there always exists a critical nucleus and a free-energy barrier as the first-order prewetting transition will occur even when the contact angle is zero. Furthermore, the critical nucleus changes its character from the critical nucleus of surface phase transition below bulk coexistence (undersaturation) to the critical nucleus of bulk heterogeneous nucleation above the coexistence (oversaturation) when it crosses the coexistence. Recently, Sear [J. Chem. Phys. 129, 164510 (2008)] has shown, by a direct numerical calculation of nucleation rate, that the nucleus does not notice this change when it crosses the coexistence. In our work, the morphology and the work of formation of critical nucleus on a completely wettable substrate are re-examined across the coexistence using the interface-displacement model. Indeed, the morphology and the work of formation changes continuously at the coexistence. Our results support the prediction of Sear and will rekindle the interest on heterogeneous nucleation on a completely wettable substrate.     

Nucleation in the presence of slow microscopic dynamics

Nucleation of a new thermodynamic phase is often a slow process due to the need to overcome a high free-energy barrier. However, there are other sources of slow dynamics; for example, at high densities/low temperatures, the movement of individual molecules or spins may be slow. Here, we study nucleation in a simple phenomenological model that has this type of slow microscopic dynamics. We do this to better understand how the two sources of slow dynamics interact. We find that as nucleation is intrinsically slow, only very slow microscopic dynamics strongly affect how nucleation occurs. The composition of the nucleus at the top of the nucleation barrier is much less sensitive to slow microscopic dynamics than is the composition of the nucleus once it is postcritical. However, slow dynamics affects not only the rate but also the pathway, which no longer goes over the saddle point in the free energy. We also find that the slow microscopic dynamics can cause sampling problems in an algorithm developed to calculate nucleation rates, and so cause it to predict the rate incorrectly.     

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.     

Condensed thymine films at the mercury/water interface: Part V. Phase transitions

The topography of the various phase transitions in a supersaturated thymine solution is described. Most transitions leading from a non-condensed to a condensed film involve nucleation and growth, via a metastable intermediate surface state, as do some transitions between condensed films. The latter have unusually high Avrami slopes and, sometimes, involve multiple intermediate states. Other transitions exhibit exponential transients. Of the three pit regions observed, that at the most positive potentials is present only in super-saturated solutions, is clearly associated with three-dimensional phase formation, and does not lead to a well-defined steady-state interfacial capacitance. Most transitions leading to a non-condensed final state exhibit exponential transients.     

Synthesis of the new metastable skutterudite compound NiSb(3) from modulated elemental reactants

A new metastable binary compound with the skutterudite crystal structure has been synthesized from modulated elemental reactants, through an amorphous intermediate, using a novel low-temperature synthesis technique. The amorphous reaction intermediate undergoes nucleation at 87 degrees C, an extremely low temperature for solid-state reactions. When heated above 350 degrees C, the metastable phase NiSb(3) disproportionates into the thermodynamically stable phases NiSb(2) and Sb. Also, if the sum of the individual elemental layer thicknesses is greater than 30 A, a mixture of different phases forms. Simulation of the high-angle powder X-ray diffraction spectrum confirms that NiSb(3) is isostructural with CoSb(3).     

A note on the nucleation with multiple steps: parallel and series nucleation

Parallel and series nucleation are the basic elements of the complex nucleation process when two saddle points exist on the free-energy landscape. It is pointed out that the nucleation rates follow formulas similar to those of parallel and series connection of resistors or conductors in an electric circuit. Necessary formulas to calculate individual nucleation rates at the saddle points and the total nucleation rate are summarized, and the extension to the more complex nucleation process is suggested.     

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.     

A kinetic approach to the theory of heterogeneous nucleation on soluble particles during the deliquescence stage

Deliquescence is the dissolution of a solid nucleus in a liquid film formed on the nucleus due to vapor condensation. Previously, the kinetics of deliquescence was examined in the framework of the capillarity approximation which involves the thermodynamic interfacial tensions for a thin film and the approximation of uniform density therein. In the present paper we propose a kinetic approach to the theory of deliquescence which avoids the use of the above macroscopic quantities for thin films. The rates of emission of molecules from the liquid film into the vapor and from the solid core into the liquid film are determined through a first passage time analysis whereas the respective rates of absorption are calculated through the gas kinetic theory. The first passage time is obtained by solving the single-molecule master equation for the probability distribution of a "surface" molecule moving in a potential field created by the cluster. Furthermore, the time evolution of the liquid film around the solid core is described by means of two mass balance equations which involve the rates of absorption and emission of molecules by the film at its two interfaces. When the deliquescence of an ensemble of solid particles occurs by means of large fluctuations, the time evolution of the distribution of composite droplets (liquid film+solid core) with respect to the independent variables of state is governed by a Fokker-Planck kinetic equation. When both the vapor and the solid soluble particles are single component, this equation has the form of the kinetic equation of binary nucleation. A steady-state solution for this equation is obtained by the method of separation of variables. The theory is illustrated with numerical calculation regarding the deliquescence of spherical particles in a water vapor with intermolecular interactions of the Lennard-Jones kind. The new approach allows one to qualitatively explain an important feature of experimental data on deliquescence, namely the occurrence of nonsharp deliquescence, a feature that the previous deliquescence theory based on classical thermodynamics could not account for.     

Enhanced protein crystallization around the metastable critical point

We report on a computer-simulation study of homogeneous crystal nucleation in a model for globular proteins. We find that the presence of a metastable vapour-liquid critical point drastically changes the pathway for the formation of a critical nucleus. But what is more important, the large density fluctuations near the critical point also lowers the free-energy barrier to nucleation and hence increases the nucleation rate. As␣the location of the vapour-liquid critical point can be controlled by changing the solvent conditions, our simulation results suggest a guided approach to protein crystallization. Received: 4 June 1998/Accepted: 3 September 1998 / Published online: 10 December 1998     

Nucleation of bulk phases in the HCl/H2O system

We report experimental results on the low-temperature uptake of HCl on H(2)O ice (ice). HCl was deposited on the surface at greater than monolayer amounts at 85 K, and the ice substrate was heated. The temperature dependence of the HCl vapor pressure from this phase was measured from 110 to 150 K, with the nucleation of a bulk hydrate phase observed at 150 K. Measurements were conducted in a closed system by simultaneous application of gas phase mass spectrometry and surface spectroscopy to characterize vapor/solid equilibrium and the nucleation of bulk hydrate phases. Combining the nucleation data reported here with data we reported previously (180 to 200 K) and data from two other laboratories (165 and 170 K), the thermodynamic boundaries for the nucleation of both the metastable bulk solution and bulk hydrate phases subsequent to monolayer adsorption of HCl have been determined. The nucleation of the metastable bulk solution phase occurs promptly at monolayer coverage at the ice/liquid coexistence boundary on the binary bulk phase diagram. The nucleation of the bulk hexahydrate occurs from this metastable solution along a locus of points defining a state of constant solution free energy. This measured free energy is -51.2 +/- 0.9 kJ/mol. Finally, the temperature dependence of the HCl vapor pressure from the low-temperature phase is reported here for the first time and is consistent with that of the metastable solution predicted by this thermodynamic model of uptake, extending the range of validity of this model of adsorption followed by bulk solution and hydrate nucleation to a lower bound in temperature of 110 K.     

Photochemically-induced crystallization of protein

I demonstrate photochemically induced crystallization of metastable hen egg-white lysozyme solution by weak UV irradiation for several tens seconds. The most effective irradiation time range is 10–60 s, and in this range the enzyme activity is maintained. Intermediates, neutral radicals at tryptophan residual produced by one-photon absorption, enhance nucleation. When the intermediate is selectively excited by visible light, the intermediate is denatured. At that time the light-induced nucleation is inhibited. This result indicates the intermediate induces nucleation. The radical forms lysozyme dimer that is detected by an SDS-PAGE electrophoresis experiment. An addition of polyethylene glycol (PEG) greatly enhances light-induced nucleation. PEG affects to shorten the intermediate radical lifetime, which suggests that PEG assists to form dimer. We consider that the photochemical dimer behaves as smallest cluster to grow critical nucleus. The smallest cluster formation is the rate determining step in classical nucleation theory due to surface energy disadvantage. The photochemical dimer is formed by a covalent bond, and the nucleation is initiated from stable dimer. The nucleation enhancement is reasonably explained. The present researches results point out the development of a new method for controlling nucleation and growth that could be applied for structural genomics and pharmaceutical industry for instance.     

Thermodynamics of crystalline nanonucleus formation from liquid phase

The energy of crystal nucleation from liquid phase was considered, with the following two stages taken into account: (1) the formation of metastable supercooled melt (solution), containing pre-nuclei with intermediate amorphous (quasicrystalline) structure, and (2) the transformation of amorphous clusters into solid crystalline nuclei having different structures. With growth of a nucleus the nucleation energy profile manifests 2–3 maxima corresponding to these stages, and the kinetics of the non-stationary nucleation has five characteristic variations.     

Kinetics of droplet condensation through a double free-energy barrier

Results are presented for the kinetics of nucleation of liquid droplets from a one-component vapor phase on a planar lyophobic substrate patterned with a large number of easily wettable (lyophilic) circular domains. If the wettability of these lyophilic domains is characterized by a contact angle smaller than pi2, for intermediate values of the supersaturation, the condensation of a droplet on a lyophilic domain occurs through a free-energy barrier with two maxima, that is, through a double barrier. A simple model is proposed for the kinetics of droplet condensation through a double barrier that combines Kramers's [Physica (Utrecht) 7, 284 (1940)] transition rate theory with known results of nucleation theory. In the framework of this model, the solution is derived for the steady-state limit of the nucleation process. The number of lyophilic domains available for droplet condensation reduces with time as domains are occupied by droplets. The problem of droplet condensation through a double barrier is solved taking into account the effect of the time-dependent depletion in the number of available lyophilic domains.     

Macromolecular nucleation

The rate-determining step of crystallization of macromolecules from melt and solution is shown to be molecular nucleation. Molecular nucleation is described by using classical nucleation theory. Good agreement is found between the molecular nucleus size and the critical length of the rejected species. Molecular nucleation has a higher free-energy barrier than secondary nucleation; it might thus lead initially to independent patches of each molecule on the crystal surface. A possible reason for secondary nucleation on extended surfaces is suggested.     

STOCHASTIC MODELING OF LIGHT ENERGY CONVERSION IN PHOTOSYNTHESIS

Abstract— Photosynthetic quantum conversion and early electron transport is modeled as a stochastic process on a digital computer to determine what free-energy losses are a necessary consequence of specific assumptions about the reaction structure, kinetics, and thermodynamics of the participating molecules. Maximal free-energy yield requires that all dark reactions be near equilibrium, so the potentials of all half-cells on each side of the light act are nearly the same. This near equilibrium requires that all forward rate constants be at least 10² times the rate of light absorption, and that all reverse rate constants be at least the rate of light absorption. The behavior of model systems with one primary donor and one primary acceptor is comparatively independent whether there is one or an infinite number of secondary electron donors and acceptors. A system having no metastable (e.g. triplet) state of the light-activated donor can convert light energy with nearly ideal efficiency, provided that the standard electrode potentials of the primary donor and primary acceptor half-cells are precisely located with respect to one another and to the potentials of the ultimate donor and acceptor. While not necessary for near maximal free-energy yield, a metastable intermediate allows a flexibility in the choice of half-cell potentials which is not possible in the absence of such an intermediate.     

Nucleation of ordered solid phases of proteins via a disordered high-density state: phenomenological approach

Nucleation of ordered solid phases of proteins triggers numerous phenomena in laboratory, industry, and in healthy and sick organisms. Recent simulations and experiments with protein crystals suggest that the formation of an ordered crystalline nucleus is preceded by a disordered high-density cluster, akin to a droplet of high-density liquid that has been observed with some proteins; this mechanism allowed a qualitative explanation of recorded complex nucleation kinetics curves. Here, we present a simple phenomenological theory that takes into account intermediate high-density metastable states in the nucleation process. Nucleation rate data at varying temperature and protein concentration are reproduced with high fidelity using literature values of the thermodynamic and kinetic parameters of the system. Our calculations show that the growth rate of the near-critical and supercritical ordered clusters within the dense intermediate is a major factor for the overall nucleation rate. This highlights the role of viscosity within the dense intermediate for the formation of the ordered nucleus. The model provides an understanding of the action of additives that delay or accelerate nucleation and presents a framework within which the nucleation of other ordered protein solid phases, e.g., the sickle cell hemoglobin polymers, can be analyzed.     

Thermodynamics and kinetics of bubble nucleation: Simulation methodology

The simulation of homogeneous liquid to vapor nucleation is investigated using three rare-event algorithms, boxed molecular dynamics, hybrid umbrella sampling Monte Carlo, and forward flux sampling. Using novel implementations of these methods for efficient use in the isothermal-isobaric ensemble, the free energy barrier to nucleation and the kinetic rate are obtained for a Lennard-Jones fluid at stretched and at superheated conditions. From the free energy surface mapped as a function of two order parameters, the global density and largest bubble volume, we find that the free energy barrier height is larger when projected over bubble volume. Using a regression analysis of forward flux sampling results, we show that bubble volume is a more ideal reaction coordinate than global density to quantify the progression of the metastable liquid toward the stable vapor phase and the intervening free energy barrier. Contrary to the assumptions of theoretical approaches, we find that the bubble takes on cohesive non-spherical shapes with irregular and (sometimes highly) undulating surfaces. Overall, the resulting free energy barriers and rates agree well between the methods, providing a set of complementary algorithms useful for studies of different types of nucleation events.