Nucleation Free Energy Barrier Research Articles

Nonequilibrium melting and crystallization of a model Lennard-Jones system.

Nonequilibrium melting and crystallization of a model Lennard-Jones system were investigated with molecular dynamics simulations to quantify the maximum superheating/supercooling at fixed pressure, and over-pressurization/over-depressurization at fixed temperature. The temperature and pressure hystereses were found to be equivalent with regard to the Gibbs free energy barrier for nucleation of liquid or solid. These results place upper bounds on hysteretic effects of solidification and melting in high heating- and strain-rate experiments such as shock wave loading and release. The authors also demonstrate that the equilibrium melting temperature at a given pressure can be obtained directly from temperatures at the maximum superheating and supercooling on the temperature hysteresis; this approach, called the hysteresis method, is a conceptually simple and computationally inexpensive alternative to solid-liquid coexistence simulation and thermodynamic integration methods, and should be regarded as a general method. We also found that the extent of maximum superheating/supercooling is weakly pressure dependent, and the solid-liquid interfacial energy increases with pressure. The Lindemann fractional root-mean-squared displacement of solid and liquid at equilibrium and extreme metastable states is quantified, and is predicted to remain constant (0.14) at high pressures for solid at the equilibrium melting temperature.

A molecular dynamics study of homogeneous crystal nucleation in liquid nitrogen

The crystal nucleation of nitrogen is studied by molecular dynamics. Using the umbrella sampling technique, we are able to determine the free energy barrier of nucleation and the structure of the critical nucleus formed under typical experimental supercoolings. In agreement with Ostwald’s ‘step rule’, nucleation is shown to proceed into an orientationally disordered cubic phase instead of the thermodynamically stable hexagonal β phase. We also give an account of the system size dependence of the free energy profiles so obtained.

Nucleation in binary polymer blends: Effects of adding diblock copolymers

The effects of adding AB diblock copolymers to A/B binary blends on the structure and thermodynamics of critical nuclei are studied using the self-consistent field theory. At a fixed ratio of the amount of the two homopolymers, depending on the degree of polymerization and composition of the diblocks, their addition to the blends can either increase or decrease the nucleation free energy barrier relative to the pure A/B blends. The qualitative trend can be deduced from the shift in the coexistence boundary and the spinodal. The distribution of diblock copolymers in critical nuclei depends on the composition of the diblocks and the quench depth. Near the coexistence, symmetric diblocks exhibit surfactant behavior, being highly concentrated on the interface of the critical nuclei. Near the spinodal, they act more like co-solvent with a relatively uniform distribution.

Concentration fluctuation in binary polymer blends: χ parameter, spinodal and Ginzburg criterion

A theory for concentration fluctuations in binary polymer blends is developed using field-theoretic techniques. The theory provides a simple, unified framework for addressing a number of important issues. First, consideration of the fluctuation and correlation effects on different length scales leads to a clarification of three different χ parameters and their interrelationship. By incorporating interaction (modeled by the bare χb) and packing effects up to the polymer size, an effective χe emerges as the natural parameter for characterizing the molecular compatibility of the two polymer species. The measured quantity in small-angle neutron scattering (SANS) experiments is an apparent χa that includes long wavelength critical and spinodal fluctuations, and is related to χe through a self-consistent equation. χa exhibits the typical upward parabolic composition dependence observed in experiments and computer simulations. Second, a unified Ginzburg criterion involving both the composition and temperature (or temperaturelike variable) is derived that is applicable to both the critical and the off-critical spinodal regimes. The common characterization of the Ginzburg criterion in terms of a range of temperature (or temperaturelike variable) alone is generally inadequate. The molecular weight scaling proposed by de Gennes and Binder in the respective critical and off-critical spinodal regimes are recovered as special cases in the limit of large molecular weights. For typical molecular weights used in experiments the Ginzburg region is larger than commonly believed. Finally, the nature of the thermodynamic spinodal is examined. It is shown that a true off-critical thermodynamic spinodal does not exist in spatial dimensions less than 4. In its place, a pseudo-spinodal can be defined where the susceptibility reaches a finite maximum. The pseudo-spinodal precedes the mean-field spinodal but approaches the latter in the limit of infinite molecular weights. The pseudo-spinodal correlates strongly with the free energy barrier for nucleation becoming order kT. Thus it provides a kinetic limit for the physically accessible metastable state, beyond which phase separation may exhibit features characteristic of spinodal decomposition. The calculated location of the pseudo-spinodal for two samples used in a recent experiment of Balsara and co-workers agrees with the onset of spinodal-decomposition-like nucleation observed in the experiement.

Kinetics of phase transformations in a model with metastable fluid–fluid separation: A molecular dynamics study

By molecular dynamics (MD) simulations we study the crystallization process in a model system whose particles interact by a spherical pair potential with a narrow and deep attractive well adjacent to a hard repulsive core. The phase diagram of the model displays a solid–fluid equilibrium, with a metastable fluid–fluid separation. Our computations are restricted to fairly small systems (from 2592 to 10368 particles) and cover long simulation times, with constant energy trajectories extending up to 76×106 MD steps. By progressively reducing the system temperature below the solid–fluid line, we first observe the metastable fluid–fluid separation, occurring readily and almost reversibly upon crossing the corresponding line in the phase diagram. The nucleation of the crystal phase takes place when the system is in the two-fluid metastable region. Analysis of the temperature dependence of the nucleation time allows us to estimate directly the nucleation free energy barrier. The results are compared with the predictions of classical nucleation theory. The critical nucleus is identified, and its structure is found to be predominantly fcc. Following nucleation, the solid phase grows steadily across the system, incorporating a large number of localized and extended defects. We discuss the relaxation processes taking place both during and after the crystallization stage. The relevance of our simulation for the kinetics of protein crystallization under normal experimental conditions is discussed.

Dynamics of the Primary Processes of Protein Folding: Helix Nucleation

We report on the folding and unfolding dynamics of α-helix nucleation in two model helical peptides. One of the peptides studied unfolds at lower temperatures (cold denatures) in solutions of 9% hexafluoroisopropanol. Laser-induced temperature jumps were used to rapidly perturb the helix/coil equilibrium in this peptide from a predominantly unfolded to a more folded ensemble. The peptide conformation was monitored through time-resolved absorption of the amide I‘ band. These experiments directly probe α-helix nucleation, as a majority of α-helices formed must start from a completely random coil conformation. In another α-helical peptide, the unfolding and folding kinetics of specific residues were monitored through the use of isotopically (13CO) labeled amino acids. By selectively measuring the unfolding kinetics of the middle of the helix, one can minimize the contribution of end-fraying effects and explicitly probe the crossing of a nucleation free energy barrier in the helix to coil direction. The resul...

Vacancy-enhanced submonolayer nucleation of Si on Si( [formula omitted

Scanning tunneling microscopy (STM) and high-resolution low-energy electron diffraction techniques are used to determine the characteristics of the two-dimensional Si island distribution on defect-free and imperfect Si(1 1 1) surfaces in real space and reciprocal space. The surface power spectra of STM and modeling of diffraction profiles give consistent island separation. Unlike the submonolayer nucleation of silicon islands on a defect-free Si(1 1 1) surface, it is shown that the island size distribution of Si, nucleated on a Si(1 1 1) surface having a high density of vacancy defects, does not exhibit a preferred characteristic island size. The process is interpreted as defect-induced nucleation with the critical island size equal to zero, for which there is no free energy barrier for nucleation. The results are qualitatively consistent with recent computer simulations, and are compared with other recent experimental work.

Polymorph screening: Influence of solvents on the rate of solvent-mediated polymorphic transformation

Solvent-mediated polymorphic transformation is an efficient technique to obtain the most stable polymorph. The rate of solvent-mediated polymorphic transformation of sulfamerazine at 24°C in various solvents and solvent mixtures is controlled by the nucleation rate of the more stable Form II. The transformation rate is generally higher in the solvent giving a higher solubility and is low in the solvent giving a low solubility (8 mmol/L). In these solvents, because of a high interfacial energy, the metastable zone may be wider than the solubility difference between two polymorphs, such that the critical free energy barrier for nucleation cannot be overcome. In addition to the solubility, the strength of the solvent-solute interactions is also important in determining the transformation rate. For sulfamerazine, the transformation rate is lower in the solvent with a stronger hydrogen bond acceptor propensity. Because solubility is higher in the solvent with stronger hydrogen bond acceptor propensity, the balance of solubility and strength of hydrogen bonding interactions between the solute and solvent molecules determines the polymorphic transformation rate. Degree of agitation and temperature also change the polymorphic transformation rate by influencing the crystallization kinetics of the more stable polymorph. © 2001 Wiley-Liss, Inc. and the American Pharmaceutical Association J Pharm Sci 90:1878–1890, 2001

A computational study of homogeneous liquid–vapor nucleation in the Lennard-Jones fluid

Umbrella sampling Monte Carlo simulations are used to calculate free energy barriers to homogeneous liquid–vapor nucleation in the superheated Lennard-Jones fluid. The calculated free energy barriers decrease with increased superheating and vanish at the spinodal curve. A statistical geometric analysis reveals the existence of two types of voids: Small interstitial cavities, which are present even in the equilibrium liquid, and much larger cavities that develop as the system climbs the nucleation free energy barrier. The geometric analysis also shows that the average cavity size within the superheated liquid is a function of density but not of temperature. The critical nucleus for the liquid–vapor transition is found to be a large system-spanning cavity that grows as the free energy barrier is traversed. The weblike cavity is nonspherical at all superheatings studied here, suggesting a phenomenological picture quite different from that of classical nucleation theory.

Thermodynamic expansion of nucleation free-energy barrier and size of critical nucleus near the vapor-liquid coexistence

Nucleation free-energy barrier height and size of the critical nucleus are expanded in powers of the chemical potential difference between the supersaturated vapor (or expanded liquid) in the metastable state and the saturated vapor-liquid system in the stable equilibrium state at the same temperature. The coefficients in the expansions are expressed in terms of the thermodynamic properties at the stable equilibrium state. Comparisons with the results obtained from the density-functional calculation for nucleation of the Lennard–Jones fluid show that systematic improvement in predicting properties of the critical nucleus, either liquid droplet or vapor cavity, is achieved by adding the higher order terms in the expansions. The scaling relations proposed by McGraw and Laaksonen are found to be good approximations to the general expansion; in particular, the barrier height displacement appearing in these scaling relations is naturally given as the second order coefficient in the expansion of the barrier height.

Surface Topology of the Ion-Induced Vapor Nucleation Rate

ABSTRACT Ion-induced nucleation involves additional electrostatic interactions between vapor molecules (atoms) and ions. The electrostatic force helps to form the ion stabilized prenucleation embryos and lowers the free energy barrier for nucleation. The free energy for ion-induced critical embryos formation has been calculated using the Thomson's term in the framework of the classical theory of nucleation. Nucleation theory has made obvious progress, but the understanding of the nucleation phenomenon is far from complete. Some ideas for the identification of possible new directions to improve both homogeneous and heterogeneous nucleation theories can be found by analyzing the topology of nucleation rate surfaces. The creation of the ion-induced nucleation rate surfaces is based on the knowledge of phase state diagrams, experimental results on ion-induced nucleation, and a few plausible assumptions. In this article the surfaces of the ion-induced nucleation rates for metastable vapor nucleation will be constructed. Semiempirical rules are formulated for the ion-induced vapor nucleation based on both known experimental and theoretical results. By using surface topology analysis, problems in heterogeneous nucleation theory can be formulated more clearly, and future directions for improvements can be discussed. A point on the spinodal line was found by extrapolation of the logarithm of the actual vapor pressure for ion-induced and homogeneous nucleation at a constant nucleation rate as a function of inverse temperature. Nucleation rate scaling of the surface should yield a quantitative scale for the ion-induced nucleation rates. Phase diagrams need to be incorporated into the interpretation of experimental and theoretical results on ion-induced nucleation.

Laser temperature jump study of the helixcoil kinetics of an alanine peptide interpreted with a 'kinetic zipper' model.

The kinetics of the helix<==>coil transition of an alanine-based peptide following a laser-induced temperature jump were monitored by the fluorescence of an N-terminal probe, 4-(methylamino)benzoic acid (MABA). This probe forms a peptide hydrogen bond to the helix backbone, which changes its fluorescence quantum yield. The MABA fluorescence intensity decreases in a single exponential relaxation, with relaxation times that are weakly temperature dependent, exhibiting a maximum value of approximately 20 ns near the midpoint of the melting transition. We have developed a new model, the kinetic version of the equilibrium 'zipper' model for helix<==>coil transitions to explain these results. In this 'kinetic zipper' model, an enormous reduction in the number of possible species results from the assumption that each molecule contains either no helical residues or a single contiguous region of helix (the single-sequence approximation). The decay of the fraction of N-terminal residues that are helical, calculated from numerical solutions of the kinetic equations which describe the model, can be approximately described by two exponential relaxations having comparable amplitudes. The shorter relaxation time results from rapid unzipping (and zipping) of the helix ends in response to the temperature jump, while the longer relaxation time results from equilibration of helix-containing and non-helix-containing structures by passage over the nucleation free energy barrier. The decay of the average helix content is dominated by the slower process. The model therefore explains the experimental observation that relaxation for the N-terminal fluorescent probe is approximately 8-fold faster than that for the infrared probe of Williams et al. [(1996) Biochemistry 35, 691-697], which measures the average helix content, but does not account for the absence of observable amplitude for the slow relaxation in the fluorescence experiments (<10% slow phase). If we assume that the activation barrier for the coil-->helix rate is purely entropic, the model can also explain the maximum in the temperature dependence of the relaxation time for the fluorescent probe. Parameters that best reproduce the melting curves and the ratio of relaxation times predict a value of the cooperativity parameter sigma which is approximately 3-fold larger than previously reported values obtained from fitting equilibrium data only. The helix growth rate of approximately 10(8) s-1 that reproduces the experimental relaxation times is approximately 100-fold slower than those observed in molecular dynamics simulations. These parameters can be used to simulate the kinetically cooperative formation of a helix from the all-coil state.

Microstructure Evolution of Amorphous Si1-xGex Thin Films

ABSTRACTThe composition dependence of the nucleation free energy barrier W* in amorphous Si1-xGex thin films is investigated. Within the composition range of x = 0.25 ∼ 0.52, the nucleation free energy barrier exhibits a maximum, which is in a good agreement with our theoretical analysis. The results are significant for processing polycrystalline SiGe thin films with desirable microstructures for thin film transistor applications. In addition, the incubation time of crystallization of amorphous Si1-xGex (x=0.5) thin films is investigated as a function of temperature.

Crystallization of amorphous SiGe thin films

The rate of nucleation and the rate of growth of crystallites in amorphous Si 50Ge 50 thin films was determined using transmission electron microscopy. With the data of nucleation and growth kinetics, the temperature dependence of the nucleation free energy barrier W ∗ was obtained using a recently developed method. Within the temperature range of 525 ∼ 575 °C, the nucleation free energy barrier shows a linear increase with temperature and the mean value of W ∗ is 1.92 eV. The theoretical relationship, developed by the authors, between the nucleation free energy barrier and the average grain size in the fully crystallized Si 50Ge 50 thin films was examined in the light of the present data.

Direct measurement of the free-energy barrier to nucleation from the size distribution of dendritic crystallites in a-Si thin films.

We introduce a method to determine the free-energy barrier to nucleation of crystallites, independent of any model for the nucleation free-energy barrier, and independent of the energy barrier to the growth. This model-independent method is developed based on the dynamic scaling of the cluster size distribution and other universal kinetics in the early stages of nucleation and growth. The method is applied to determine the free-energy barrier to nucleation of the amorphous-to-crystalline transformation in a-Si thin films. After considering the dendritic nature of Si crystallites, we obtain the free-energy barrier to nucleation ${\mathit{W}}_{\mathrm{*}}$\ensuremath{\approxeq}2.15--2.18 eV, the enthalpic (i.e., the activation energy) barrier \ensuremath{\Delta}${\mathit{H}}_{\mathrm{*}}$=1.32 eV, and the entropic barrier \ensuremath{\Delta}${\mathit{S}}_{\mathrm{*}}$=-9.57\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$ eV ${\mathrm{K}}^{\mathrm{\ensuremath{-}}1}$, within the temperature range of T=863--893 K. The entropic contribution to ${\mathit{W}}_{\mathrm{*}}$ is found to be considerably large. We also show that the magnitude of the measured ${\mathit{W}}_{\mathrm{*}}$ could not be accounted for by the classical expression for the free-energy barrier with the previous suggested values for its parameters.

Effective cross‐sections for molecular aggregation in external orienting fields

AbstractEffects of an external orienting field and orientational cross‐section for molecular association on the kinetics of crystal nucleation and critical temperature of crystallization is discussed. Dipole and quadrupole effects of an electric (magnetic) field and quadrupole effects of hydrodynamic potential of uniaxial flow are considered. Critical temperature of crystallization, free energy barrier of nucleation and nucleation rates are orientation‐dependent. Smaller orientational cross‐sections result in stronger anisotropy of the nucleation rates.

Role of Small Clusters in Nucleation about Ions

A consideration of the molecular aspects of ion induced nucleation shows that, in contrast to accepted theory, small clusters are a segment of the over-all nucleation size spectrum. No direct relationship exists between ion solvation energy and the free energy barrier for nucleation. Instead, it is established that the properties of the small clusters govern the free energy functions in the neighborhood of the peak cluster concentration, and that these in turn affect the over-all height of the free energy barrier for nucleation. The fact that ion-molecule reactions are taking place during the course of the usual ion induced nucleation experiment is clearly demonstrated. Assumptions that nucleation generally occurs directly about ions of the carrier gas are disproven.

Heterogeneous nucleation at macroscopic steps

Macroscopic steps on a substrate surface are known to affect condensation characteristics of a substance from its vapor. A simple model is presented, for the case of isotropic interfacial free energy, to explain the kinetics of nucleation on a macroscopic step in terms of capillarity theory. A critical contact angle θ c is shown to exist above which a macroscopic step on a substrate will have no effect on nucleation. The model also predicts a contact angle of 45° below which the free energy barrier for nucleation at a step vanishes. Between 50° and θ c nucleation occurs preferentially at steps. Such nucleation is thought to give rise to a characteristic decoration pattern in electron-microscope replicas of the substrate surface. Experiments in which cadmium and gold were condensed from the vapor onto cleaved sodium chloride and lithium fluoride crystals tend to support the theory. However one notes a reservation in regard to application of the present thermodynamic theory to these small nuclei which are of the order of only several atoms.