A novel method for evaluating the critical nucleus and the surface tension in systems with first order phase transition

We introduce a novel method for calculating the size of the critical nucleus and the value of the surface tension in systems with first order phase transition. The method is based on classical nucleation theory, and it consists in studying the thermodynamics of a sphere of given radius embedded in a frozen metastable surrounding. The frozen configuration creates a pinning field on the surface of the free sphere. The pinning field forces the sphere to stay in the metastable phase as long as its size is smaller than the critical nucleus. We test our method in two first order systems, both on a two-dimensional lattice: a system where the parameter tuning the transition is the magnetic field, and a second system where the tuning parameter is the temperature. In both cases the results are satisfying. Unlike previous techniques, our method does not require an infinite volume limit to compute the surface tension, and it therefore gives reliable estimates even by using relatively small systems. However, our method cannot be used at, or close to, the critical point, i.e., at coexistence, where the critical nucleus becomes infinitely large.     

Hit and miss of classical nucleation theory as revealed by a molecular simulation study of crystal nucleation in supercooled sulfur hexafluoride

Classical nucleation theory pictures the homogeneous nucleation of a crystal as the formation of a spherical crystalline embryo, possessing the properties of the macroscopic crystal, inside a parent supercooled liquid. In this work we study crystal nucleation in moderately supercooled sulfur hexafluoride by umbrella sampling simulations. The nucleation free energy evolves from 5.2kBT at T=170 K to 39.1kBT at T=195 K. The corresponding critical nucleus size ranges from 40 molecules at T=170 K to 266 molecules at T=195 K. Both nucleation free energy and critical nucleus size are shown to evolve with temperature according to the equations derived from the classical nucleation theory. Inspecting the obtained nuclei we show, however, that they present quite anisotropic shapes in opposition to the spherical assumption of the theory. Moreover, even though the critical nuclei possess the structure of the stable bcc plastic phase, the only mechanically stable crystal phase for SF6 in the temperature range investigated, they are shown to be less ordered than the corresponding macroscopic crystal. Their crystalline order is nevertheless shown to increase regularly with their size. This is confirmed by a study of a nucleus growth from a critical size to a size of the order of 10(4) molecules. Similarly to the fact that it does not affect the temperature dependence of the nucleation free energy and of the critical nucleus size, the ordering of the nucleus with size does not affect the growth rate of the nucleus.     

Test of classical nucleation theory on deeply supercooled high-pressure simulated silica

We test classical nucleation theory (CNT) in the case of simulations of deeply supercooled, high density liquid silica, as modeled by the van Beest-Kramer-van Santen potential. We find that at density rho=4.38 gcm(3), spontaneous nucleation of crystalline stishovite occurs in conventional molecular dynamics simulations at temperature T=3000 K, and we evaluate the nucleation rate J directly at this T via "brute force" sampling of nucleation events in numerous independent runs. We then use parallel, constrained Monte Carlo simulations to evaluate DeltaG(n), the free energy to form a crystalline embryo containing n silicon atoms, at T=3000, 3100, 3200, and 3300 K. By comparing the form of DeltaG(n) to CNT, we test the ability of CNT to reproduce the observed behavior as we approach the regime where spontaneous nucleation occurs on simulation time scales. We find that the prediction of CNT for the n dependence of DeltaG(n) fits reasonably well to the data at all T studied. Deltamu, the chemical potential difference between bulk liquid and stishovite, is evaluated as a fit parameter in our analysis of the form of DeltaG(n). Compared to directly determined values of Deltamu extracted from previous work, the fitted values agree only at T=3300 K; at lower T the fitted values increasingly overestimate Deltamu as T decreases. We find that n(*), the size of the critical nucleus, is approximately ten silicon atoms at T=3300 K. At 3000 K, n(*) decreases to approximately 3, and at such small sizes methodological challenges arise in the evaluation of DeltaG(n) when using standard techniques; indeed even the thermodynamic stability of the supercooled liquid comes into question under these conditions. We therefore present a modified approach that permits an estimation of DeltaG(n) at 3000 K. Finally, we directly evaluate at T=3000 K the kinetic prefactors in the CNT expression for J, and find physically reasonable values; e.g., the diffusion length that Si atoms must travel in order to move from the liquid to the crystal embryo is approximately 0.2 nm. We are thereby able to compare the results for J at 3000 K obtained both directly and based on CNT, and find that they agree within an order of magnitude. In sum, our work quantifies how certain predictions of CNT (e.g., for Deltamu) break down in this deeply supercooled limit, while others [the n dependence of DeltaG(n)] are not as adversely affected.     

Histogram Monte Carlo simulation on phase transitions in single‐polymer systems

Applying the histogram Monte Carlo simulation method and the bond‐fluctuation model, various phase transitions in single‐polymer systems were investigated. The critical transition temperature (Θ point) in the coil‐globule collapse transition of a macromolecular chain is accurately determined. Finite‐size scaling results near the transition point are verified. The first‐order transition associated with the freezing/crystallization of a polymer at a temperature below the Θ point is also observed. The free energy profiles associated with these two transitions are explicitly computed. Furthermore, the unfolding phase transition associated with stretching a collapsed polymer chain is investigated. The free energy profile associated with the transition is explicitly computed. Results on the energy cumulants and free energy profiles provide direct evidences for the first‐order nature of the unfolding phase transition.     

Excluded volume in the generic van der Waals equation of state and the self-diffusion coefficient of the Lennard-Jones fluid

In the previous papers applying the generic van der Waals equation of state the mean excluded volume was defined with the contact diameter of particles at which the potential energy is equal to zero-the size parameter in the case of the Lennard-Jones potential. This parameter appears as the upper limit of the integral for the generic van der Waals parameter B (mean excluded volume divided by the density) in the generic van der Waals equation of state. Since the choice is not unique, in this paper we reexamine the manner of defining the upper limit and propose another choice for the upper limit. We also propose an interpretation of the free volume overlap factor alpha appearing in the free volume theory of diffusion and a method of estimating it in terms of the intermolecular potential energy only. It is shown that with the so-estimated free volume overlap factor and the new choice of the upper limit of the integral for B the self-diffusion coefficient in the modified free volume theory of diffusion not only acquires a better accuracy than before, but also becomes calculable in terms of only the intermolecular interaction potential without an adjustable parameter. We also assess some of effective diameters of molecules proposed in the literature for their ability to predict the self-diffusion coefficient within the framework of the modified free volume theory of diffusion.     

The influence of pretransitional phenomena on blue phase range

Abstract

The particular phase in which a liquid crystal system will exist is that which has the lowest free energy at a certain temperature. The free energy may depend on variables such as temperature, pressure, chirality, etc. One way in which the stability of a particular thermodynamic phase, relative to its neighbours, would be manifest is in its temperature range. The effect of chirality, in particular, on the temperature range or stability of blue phases has been well-studied both experimentally and theoretically. To date these studies assume that chirality is the only parameter which will influence the existence of blue phases. However, blue phases with relatively low chirality and broad range, which should in principle only show very narrow blue phases, have been reported. This suggests that factors other than chirality are involved in blue phase stability. In this paper we investigate the phase stability of various blue phase mixtures containing equal amounts of a chiral dopant, via their blue phase temperature range. Correlation between blue phase stability and the elastic constants k ₂ and k ₂₂, molecular length of the nematic host, and the order parameter at the blue phase to cholesteric transition is discussed. We have confirmed that for our mixtures the total blue phase temperature range may be related equally to the elastic constant k ₂₂ and the chirality. We also present the first data displaying an odd-even effect in blue phases. Finally, we have found an apparent correlation between the stability of the blue phases and the magnitude of the orientational order parameter of the cholesteric phase at the cholesteric to blue phase transition.     

Colloidal aggregation in polymer blends

We consider here a low-density assembly of colloidal particles immersed in a critical polymer mixture of two chemically incompatible polymers. We assume that, close to the critical point of the free mixture, the colloids prefer to be surrounded by one polymer (critical adsorption). As result, one is assisted to a reversible colloidal aggregation in the nonpreferred phase, due the existence of a long-range attractive Casimir force between particles. This aggregation is a phase transition driving the colloidal system from dilute to dense phases, as the usual gas-liquid transition. We are interested in a quantitative investigation of the phase diagram of the immersed colloids. We suppose that the positions of particles are disordered, and the disorder is quenched and follows a Gaussian distribution. To apprehend the problem, use is made of the standard phi(4) theory, where the field phi represents the composition fluctuation (order parameter), combined with the standard cumulant method. First, we derive the expression of the effective free energy of colloids and show that this is of Flory-Huggins type. Second, we find that the interaction parameter u between colloids is simply a linear combination of the isotherm compressibility and specific heat of the free mixture. Third, with the help of the derived effective free energy, we determine the complete shape of the phase diagram (binodal and spinodal) in the (Psi,u) plane, with Psi as the volume fraction of immersed colloids. The continuous "gas-liquid" transition occurs at some critical point K of coordinates (Psi(c) = 0.5,u(c) = 2). Finally, we emphasize that the present work is a natural extension of that, relative to simple liquid mixtures incorporating colloids.     

Thermodynamic stability of fluid-fluid phase separation in binary athermal mixtures: the role of nonadditivity

We studied the thermodynamic stability of fluid-fluid phase separation in binary nonadditive mixtures of hard-spheres for moderate size ratios. We are interested in elucidating the role played by small amounts of nonadditivity in determining the stability of fluid-fluid phase separation with respect to the fluid-solid phase transition. The demixing curves are built in the framework of the modified-hypernetted chain and of the Rogers-Young integral equation theories through the calculation of the Gibbs free energy. We also evaluated fluid-fluid phase equilibria within a first-order thermodynamic perturbation theory applied to an effective one-component potential obtained by integrating out the degrees of freedom of the small spheres. A qualitative agreement emerges between the two different approaches. We also addressed the determination of the freezing line by applying the first-order thermodynamic perturbation theory to the effective interaction between large spheres. Our results suggest that for intermediate size ratios a modest amount of nonadditivity, smaller than earlier thought, can be sufficient to drive the fluid-fluid critical point into the thermodinamically stable region of the phase diagram. These findings could be significant for rare-gas mixtures in extreme pressure and temperature conditions, where nonadditivity is expected to be rather small.     

Temperature accelerated Monte Carlo (TAMC): a method for sampling the free energy surface of non-analytical collective variables

We introduce a new method to simulate the physics of rare events. The method, an extension of the Temperature Accelerated Molecular Dynamics, comes in use when the collective variables introduced to characterize the rare events are either non-analytical or so complex that computing their derivative is not practical. We illustrate the functioning of the method by studying the homogeneous crystallization in a sample of Lennard-Jones particles. The process is studied by introducing a new collective variable that we call Effective Nucleus Size N. We have computed the free energy barriers and the size of critical nucleus, which result in agreement with data available in the literature. We have also performed simulations in the liquid domain of the phase diagram. We found a free energy curve monotonically growing with the nucleus size, consistent with the liquid domain.     

Landau model of the direct isotropic to smectic-C*A phase transition in antiferroelectric liquid crystals

An orientational order parameter is proposed for the isotropic to smectic C*A phase transition in antiferroelectric liquid crystals. A phenomenological theory is developed to describe the direct isotropic to smectic-C*A phase transition on the basis of a free energy expansion. The free energy is written in terms of the coupled order parameters including the antipolar correlations. We present a detailed analysis of the different phases that can occur and analyze the question: under which conditions a direct isotropic to smectic-C*A phase transition is possible when compared to other phase transitions? The theoretical results are compared with experimental results.     

An analysis of molecular packing and chemical association in liquid water using quasichemical theory

We calculate the hydration free energy of liquid TIP3P water at 298 K and 1 bar using a quasi-chemical theory framework in which interactions between a distinguished water molecule and the surrounding water molecules are partitioned into chemical associations with proximal (inner-shell) waters and classical electrostatic-dispersion interactions with the remaining (outer-shell) waters. The calculated free energy is found to be independent of this partitioning, as expected, and in excellent agreement with values derived from the literature. An analysis of the spatial distribution of inner-shell water molecules as a function of the inner-shell volume reveals that water molecules are preferentially excluded from the interior of large volumes as the occupancy number decreases. The driving force for water exclusion is formulated in terms of a free energy for rearranging inner-shell water molecules under the influence of the field exerted by outer-shell waters in order to accommodate one water molecule at the center. The results indicate a balance between chemical association and molecular packing in liquid water that becomes increasingly important as the inner-shell volume grows in size.     

An extension of the Landau-Ginzburg-de Gennes theory for liquid crystals

Using angular momentum representation a method is proposed that allows the systematic construction of a generalized Landau-de Gennes elastic free energy of liquid crystals, in powers of a symmetric and traceless tensor order parameter, polarization field, of external fields and all respective derivatives. By this method all linearly independent elastic invariants and surface terms are constructed for nematics and cholesterics up to fourth order terms. In particular it is shown that up to fourth order in the tensor order parameter there are nineteen bulk elastic constants and four surface terms in the free energy of a general, biaxial nematic. In addition, the stability of this expansion is studied in detail. Some special cases of the elastic free energy of liquid crystals, already discussed in the literature, are reexamined and discrepancies with our results are emphasized. Finally, a thermo-dynamically correct way of establishing contact between the generalized de Gennes elastic free energy and other theories, like those of Oseen-Frank or Meyer, is proposed by applying fluctuation theory. Thus, the degeneracy of splay and bend elastic constants is removed even when these are calculated from the standard de Gennes free energy. Restrictions on higher order elastic constants are also obtained by comparing mean field relations and stability conditions with available experimental data.     

Induced smectic phases in phase diagrams of binary nematic liquid crystal mixtures

To elucidate induced smectic A and smectic B phases in binary nematic liquid crystal mixtures, a generalized thermodynamic model has been developed in the framework of a combined Flory-Huggins free energy for isotropic mixing, Maier-Saupe free energy for orientational ordering, McMillan free energy for smectic ordering, Chandrasekhar-Clark free energy for hexagonal ordering, and phase field free energy for crystal solidification. Although nematic constituents have no smectic phase, the complexation between these constituent liquid crystal molecules in their mixture resulted in a more stable ordered phase such as smectic A or B phases. Various phase transitions of crystal-smectic, smectic-nematic, and nematic-isotropic phases have been determined by minimizing the above combined free energies with respect to each order parameter of these mesophases. By changing the strengths of anisotropic interaction and hexagonal interaction parameters, the present model captures the induced smectic A or smectic B phases of the binary nematic mixtures. Of particular importance is the fact that the calculated phase diagrams show remarkable agreement with the experimental phase diagrams of binary nematic liquid crystal mixtures involving induced smectic A or induced smectic B phase.     

Free energy landscapes for homogeneous nucleation of ice for a monatomic water model

We simulate the homogeneous nucleation of ice from supercooled liquid water at 220 K in the isobaric-isothermal ensemble using the MW monatomic water potential. Monte Carlo simulations using umbrella sampling are performed in order to determine the nucleation free energy barrier. We find the Gibbs energy profile to be relatively consistent with that predicted by classical nucleation theory; the free energy barrier to nucleation was determined to be ~18 k(B)T and the critical nucleus comprised ~85 ice particles. Growth from the supercooled liquid gives clusters that are predominantly cubic, whilst starting with a pre-formed subcritical nucleus of cubic or hexagonal ice results in the growth of predominantly that phase of ice only.     

One-dimensional trapped atoms: critical coupling parameter and critical number of particles

In this paper we consider the case of a Bose gas in low dimension in order to illustrate the applicability of a method that allows us to construct analytical relations, valid for a broad range of coupling parameters, for a function which asymptotic expansions are known. The method is well suitable to investigate the problem of stability of a collection of Bose particles trapped in one- dimensional configuration for the case where the scattering length presents a negative value. The eigenvalues for this interacting quantum one-dimensional many particle system become negative when the interactions overcome the trapping energy and, in this case, the system becomes unstable. Here we calculate the critical coupling parameter and apply for the case of Lithium atoms obtaining the critical number of particles for the limit of stability.     

The putative liquid-liquid transition is a liquid-solid transition in atomistic models of water

We use numerical simulation to examine the possibility of a reversible liquid-liquid transition in supercooled water and related systems. In particular, for two atomistic models of water, we have computed free energies as functions of multiple order parameters, where one is density and another distinguishes crystal from liquid. For a range of temperatures and pressures, separate free energy basins for liquid and crystal are found, conditions of phase coexistence between these phases are demonstrated, and time scales for equilibration are determined. We find that at no range of temperatures and pressures is there more than a single liquid basin, even at conditions where amorphous behavior is unstable with respect to the crystal. We find a similar result for a related model of silicon. This result excludes the possibility of the proposed liquid-liquid critical point for the models we have studied. Further, we argue that behaviors others have attributed to a liquid-liquid transition in water and related systems are in fact reflections of transitions between liquid and crystal.     

Revisiting the Frenkel-Ladd method to compute the free energy of solids: the Einstein molecule approach

In this paper a new method to evaluate the free energy of solids is proposed. The method can be regarded as a variant of the method proposed by Frenkel and Ladd [J. Chem. Phys. 81, 3188 (1984)]. The main equations of the method can be derived in a simple way. The method can be easily implemented within a Monte Carlo program. We have applied the method to determine the free energy of hard spheres in the solid phase for several system sizes. The obtained free energies agree within the numerical uncertainty with those obtained by Polson et al. [J. Chem. Phys. 112, 5339 (2000)]. The fluid-solid equilibria has been determined for several system sizes and compared to the values published previously by Wilding and Bruce [Phys. Rev. Lett. 85, 5138 (2000)] using the phase switch methodology. It is shown that both the free energies and the coexistence pressures present a strong size dependence and that the results obtained from free energy calculations agree with those obtained using the phase switch method, which constitutes a cross-check of both methodologies. From the results of this work we estimate the coexistence pressure of the fluid-solid transition of hard spheres in the thermodynamic limit to be p*=11.54(4), which is slightly lower than the classical value of Hoover and Ree (p*=11.70) [J. Chem. Phys. 49, 3609 (1968)]. Taking into account the strong size dependence of the free energy of the solid phase, we propose to introduce finite size corrections, which allow us to estimate approximately the free energy of the solid phase in the thermodynamic limit from the known value of the free energy of the solid phase with N molecules. We have also determined the free energy of a Lennard-Jones solid by using both the methodology of this work and the finite size correction. It is shown how a relatively good estimate of the free energy of the system in the thermodynamic limit is obtained even from the free energy of a relatively small system.     

Analytical phase diagrams for colloids and non-adsorbing polymer

We review the free-volume theory (FVT) of Lekkerkerker et al. [Europhys. Lett. 20 (1992) 559] for the phase behavior of colloids in the presence of non-adsorbing polymer and we extend this theory in several aspects: (i) We take the solvent into account as a separate component and show that the natural thermodynamic parameter for the polymer properties is the insertion work Pi(v), where Pi is the osmotic pressure of the (external) polymer solution and v the volume of a colloid particle. (ii) Curvature effects are included along the lines of Aarts et al. [J. Phys.: Condens. Matt. 14 (2002) 7551] but we find accurate simple power laws which simplify the mathematical procedure considerably. (iii) We find analytical forms for the first, second, and third derivatives of the grand potential, needed for the calculation of the colloid chemical potential, the pressure, gas-liquid critical points and the critical endpoint (cep), where the (stable) critical line ends and then coincides with the triple point. This cep determines the boundary condition for a stable liquid. We first apply these modifications to the so-called colloid limit, where the size ratio q(R)=R/a between the radius of gyration R of the polymer and the particle radius a is small. In this limit the binodal polymer concentrations are below overlap: the depletion thickness delta is nearly equal to R, and Pi can be approximated by the ideal (van't Hoff) law Pi=Pi(0)=phi/N, where phi is the polymer volume fraction and N the number of segments per chain. The results are close to those of the original Lekkerkerker theory. However, our analysis enables very simple analytical expressions for the polymer and colloid concentrations in the critical and triple points and along the binodals as a function of q(R). Also the position of the cep is found analytically. In order to make the model applicable to higher size ratio's q(R) (including the so-called protein limit where q(R)>1) further extensions are needed. We introduce the size ratio q=delta/a, where the depletion thickness delta is no longer of order R. In the protein limit the binodal concentrations are above overlap. In such semidilute solutions delta approximately xi, where the De Gennes blob size (correlation length) xi scales as xi approximately phi(-gamma), with gamma=0.77 for good solvents and gamma=1 for a theta solvent. In this limit Pi=Pi(sd) approximately phi(3gamma). We now apply the following additional modifications: With these latter two modifications we obtain again a fully analytical model with simple equations for critical and triple points as a function of q(R). In the protein limit the binodal polymer concentrations scale as q(R)(1/gamma), and phase diagrams phiq(R)(-1/gamma) versus the colloid concentration eta become universal (i.e., independent of the size ratio q(R)). The predictions of this generalized free-volume theory (GFVT) are in excellent agreement with experiment and with computer simulations, not only for the colloid limit but also for the protein limit (and the crossover between these limits). The q(R)(1/gamma) scaling is accurately reproduced by both simulations and other theoretical models. The liquid window is the region between phi(c) (critical point) and phi(t) (triple point). In terms of the ratio phi(t)/phi(c) the liquid window extends from 1 in the cep (here phi(t)-phi(c)=0) to 2.2 in the protein limit. Hence, the liquid window is narrow: it covers at most a factor 2.2 in (external) polymer concentration.