Crystal-liquid interfacial free energy via thermodynamic integration

Theory of Anisotropic Metal Nanostructures

A significant challenge in the development of functional materials is understanding the growth and transformations of anisotropic colloidal metal nanocrystals. Theory and simulations can aid in the development and understanding of anisotropic nanocrystal syntheses. The focus of this review is on how results from first-principles calculations and classical techniques, such as Monte Carlo and molecular dynamics simulations, have been integrated into multiscale theoretical predictions useful in understanding shape-selective nanocrystal syntheses. Also, examples are discussed in which machine learning has been useful in this field. There are many areas at the frontier in condensed matter theory and simulation that are or could be beneficial in this area and these prospects for future progress are discussed.

Direct Calculation of the Interfacial Free Energy between NaCl Crystal and Its Aqueous Solution at the Solubility Limit

Salty water is the most abundant electrolyte aqueous mixture on Earth, however, very little is known about the NaCl-saturated solution interfacial free energy (γ_{s}). Here, we provide the first direct estimation of γ_{s} for several NaCl crystallographic planes by means of the mold integration technique, a highly efficient computational method to evaluate interfacial free energies with anisotropic crystal resolution. Making use of the JC-SPC/E model, one of the most benchmarked force fields for NaCl water solutions, we measure γ_{s} of four different crystal planes, (100), (110), (111), and (112[over ¯]) with the saturated solution at normal conditions. We find high anisotropy between the different crystal orientations with values ranging from 100 to 150  mJ m^{-2}, and the average value of the distinct planes being γ[over ¯]_{s}=137(20)  mJ m^{-2}. This value for the coexistence interfacial free energy is in reasonable agreement with previous extrapolations from nucleation studies. Our Letter represents a milestone in the computational calculation of interfacial free energies between ionic crystals and aqueous solutions.

A General Method for Calculating Solid/Liquid Interfacial Free Energies from Atomistic Simulations: Application to CaSO4.xH2O

We present a general method for computing interfacial free energies from atomistic simulations, which is particularly suitable for solid/liquid interfaces. Our method uses an Einstein crystal as a universal reference state and is more flexible than previous approaches. Surfaces with dipoles, complex reconstructions, and partially dissolved species are all easily accommodated within the framework. It may also be extended to calculating the relative free energies of different phases and other types of defect. We have applied our method to interfaces of bassanite and gypsum with water and obtained interfacial free energies of the order of 0.15 J/m2, of which approximately 50 % is due to entropic contributions. Our calculations of the interfacial free energy of NaCl with water obtained a value of 0.13 J/m2 of which only 19 % is from entropic contributions. We have also predicted equilibrium morphologies for bassanite and gypsum that compare well with experiments and previous calculations.

Cleaving Method for Molecular Crystals and Its Application to Calculation of the Surface Free Energy of Crystalline β-d-Mannitol at Room Temperature

Calculation of the surface free energy (SFE) is an important application of the thermodynamic integration (TI) methodology, which was mainly employed for atomic crystals (such as Lennard–Jones or metals). In this work, we present the calculation of the SFE of a molecular crystal using the cleaving technique which we implemented in the LAMMPS molecular dynamics package. We apply this methodology to a crystal of β-d-mannitol at room temperature and report the results for two types of force fields belonging to the GROMOS family: all atoms and united atoms. The results show strong dependence on the type of force field used, highlighting the need for the development of better force fields to model the surface properties of molecular crystals. In particular, we observe that the united-atoms force field, despite its higher degree of coarse graining compared to the all-atoms force field, produces SFE results in better agreement with the experimental results from inverse gas chromatography measurements.

Crystal growth of bcc titanium from the melt and interfacial properties: A molecular dynamics simulation study

The crystal growth kinetics and interfacial properties of titanium (Ti) are studied using molecular dynamics computer simulation. The interactions between the Ti atoms are modeled via an embedded atom method potential. First, the free solidification method (FSM) is used to determine the melting temperature T_m at zero pressure where the transition from liquid to body-centered cubic crystal occurs. From the simulations with the FSM, the kinetic growth coefficients are also determined for different orientations of the crystal, analyzing how the coupling to the thermostat affects the estimates of the growth coefficients. At T_m, anisotropic interfacial stiffnesses and free energies as well as kinetic growth coefficients are determined from capillary wave fluctuations. The so-obtained growth coefficients from equilibrium fluctuations and without the coupling of the system to a thermostat agree well with those extracted from the FSM calculations.

Computation of solid-fluid interfacial free energy in molecular systems using thermodynamic integration

In this article, we present two methods based on thermodynamic integration for computing solid-fluid interfacial free energy for a molecular system. As a representative system, we choose two crystal polymorphs of orcinol (5-methylbenzene-1,3-diol) as the solid phase and chloroform and nitromethane as the liquid phase. The computed values of the interfacial free energy are then used in combination with the classical nucleation theory to predict solvent induced polymorph selectivity during crystallization of orcinol from solution.

Interfacial free energy of a liquid-solid interface: Its change with curvature

We analyze the changes in the interfacial free energy between a spherical solid cluster and a fluid due to the change of the radius of the solid. Interfacial free energies from nucleation studies using the seeding technique for four different systems, being hard spheres, Lennard-Jones, and two models of water (mW and TIP4P/ICE), were plotted as a function of the inverse of the radius of the solid cluster. In all cases, the interfacial free energy was a linear function of the inverse of the radius of the solid cluster and this is consistent with Tolman's equation. This linear behavior is shown not only in isotherms but also along isobars. The effect of curvature on the interfacial free energy is more pronounced in water, followed by hard spheres, and smaller for Lennard-Jones particles. We show that it is possible to estimate nucleation rates of Lennard-Jones particles at different pressures by using information from simple NpT simulations and taking into account the variation of the interfacial free energy with the radius of the solid cluster. Neglecting the effects of the radius on the interfacial free energy (capillarity approximation) leads to incorrect values of the nucleation rate. For the Lennard-Jones system, the homogeneous nucleation curve is not parallel to the melting curve as was found for water in previous work. This is due to the increase in the interfacial free energy along the coexistence curve as the pressure increases. This work presents a simple and relatively straightforward way to approximately estimate nucleation rates.

Communication: Computing the Tolman length for solid-liquid interfaces

The curvature dependence of interfacial free energy, which is crucial in quantitatively predicting nucleation kinetics and the stability of bubbles and droplets, is quantified by the Tolman length δ. For solid-liquid interfaces, however, δ has never been computed directly due to various theoretical and practical challenges. Here we perform a direct evaluation of the Tolman length from atomistic simulations of a solid-liquid planar interface in out-of-equilibrium conditions, by first computing the surface tension from the amplitude of thermal capillary fluctuations of a localized version of the Gibbs dividing surface and by then calculating how much the surface energy changes when it is defined relative to the equimolar dividing surface. We computed δ for a model potential, and found a good agreement with the values indirectly inferred from nucleation simulations. The agreement not only validates our approach but also suggests that the nucleation free energy of the system can be perfectly described using classical nucleation theory if the Tolman length is taken into account.

Extraction of effective solid-liquid interfacial free energies for full 3D solid crystallites from equilibrium MD simulations

Molecular dynamics simulations of an embedded atom copper system in the isobaric-isenthalpic ensemble are used to study the effective solid-liquid interfacial free energy of quasi-spherical solid crystals within a liquid. This is within the larger context of molecular dynamics simulations of this system undergoing solidification, where single individually prepared crystallites of different sizes grow until they reach a thermodynamically stable final state. The resulting equilibrium shapes possess the full structural details expected for solids with weakly anisotropic surface free energies (in these cases, ∼5% radial flattening and rounded [111] octahedral faces). The simplifying assumption of sphericity and perfect isotropy leads to an effective interfacial free energy as appearing in the Gibbs-Thomson equation, which we determine to be ∼177 erg/cm², roughly independent of crystal size for radii in the 50-250 Å range. This quantity may be used in atomistically informed models of solidification kinetics for this system.

Theory of the thermodynamic influence of solution-phase additives in shape-controlled nanocrystal synthesis

Though many experimental studies have documented that certain solution-phase additives can play a key role in the shape-selective synthesis of metal nanocrystals, the origins and mechanisms of this shape selectivity are still unclear. One possible role of such molecules is to thermodynamically induce the equilibrium shape of a nanocrystal by altering the interfacial free energies of the facets. Using a multi-scheme thermodynamic integration method that we recently developed [J. Chem. Phys., 2016, 145, 194108], we calculate the solid-liquid interfacial free energies γ_sl and investigate the propensity to achieve equilibrium shapes in such syntheses. We first apply this method to Ag(100) and Ag(111) facets in ethylene glycol solution containing polyvinylpyrrolidone (PVP), to mimic the environment in polyol synthesis of Ag nanocrystals. We find that although PVP has a preferred binding to Ag(100), its selectivity is not sufficient to induce a thermodynamic preference for {100}-faceted nanocubes, as has been observed experimentally. This indicates that PVP promotes Ag nanocube formation kinetically rather than thermodynamically. We further quantify the thermodynamic influence of adsorbed solution-phase additives for generic molecules, by building a γ_sl ratio/nanocrystal shape map as a function of zero-temperature binding energies. This map can be used to gauge the efficacy of candidate additive molecules for producing targeted thermodynamic nanocrystal shapes. The results indicate that only additives with a strong facet selectivity can impart significant thermodynamic-shape change. Therefore, many of the nanocrystals observed in experiments are likely kinetic products.

Studying of crystal growth and overall crystallization of naproxen from binary mixtures

Broadband dielectric spectroscopy (BDS) and differential scanning calorimetry (DSC) were applied to investigate the molecular dynamics and phase transitions in binary mixtures composed of naproxen (NAP) and acetylated saccharides: maltose (acMAL) and sucrose (acSUC). Moreover, the application of BDS method and optical microscopy enabled us to study both crystallization kinetics and crystal growth of naproxen from the solid dispersions with the highest content of modified carbohydrates (1:5wt ratio). It was found that the activation barriers of crystallization estimated from dielectric measurements are completely different for both studied herein mixtures. Much higher E_a (=205kJ/mol) was obtained for NAP-acMAL solid dispersion. It is probably due to simultaneous crystallization of both components of the mixture. On the other hand, lower value of E_a in the case of NAP-acSUC solid dispersion (81kJ/mol) indicated, that naproxen is the only crystallizing compound. This hypothesis was confirmed by X-ray diffraction studies. We also suggested that specific intermolecular dipole-dipole interactions between active substance and excipient may be an alternative explanation for the difference between activation barrier obtained for NAP-acMAL and NAP-acSUC binary mixtures. Furthermore, optical measurements showed that the activation energy for crystal growth of naproxen increases in binary mixtures. They also revealed that both excipients: acMAL and acSUC move the temperature of the maximum of crystal growth towards lower temperatures. Interestingly, this maximum occurs for nearly the same structural relaxation time, which is a good approximation of viscosity, for all samples. Finally, it was also noticed that although naproxen crystallizes to the same polymorphic form in both systems, there are some differences in morphology of obtained crystals. Thus, the observed behavior may have a significant impact on the bioavailability and dissolution rate of API produced in that way.

Silicon-wall interfacial free energy via thermodynamics integration

We compute the interfacial free energy of a silicon system in contact with flat and structured walls by molecular dynamics simulation. The thermodynamics integration method, previously applied to Lennard-Jones potentials [R. Benjamin and J. Horbach, J. Chem. Phys. 137, 044707 (2012)], has been extended and implemented in Tersoff potentials with two-body and three-body interactions taken into consideration. The thermodynamic integration scheme includes two steps. In the first step, the bulk Tersoff system is reversibly transformed to a state where it interacts with a structureless flat wall, and in a second step, the flat structureless wall is reversibly transformed into an atomistic SiO₂ wall. Interfacial energies for liquid silicon-wall interfaces and crystal silicon-wall interfaces have been calculated. The calculated interfacial energies have been employed to predict the nucleation mechanisms in a slab of liquid silicon confined by two walls and compared with MD simulation results.

Seeding approach to crystal nucleation

We present a study of homogeneous crystal nucleation from metastable fluids via the seeding technique for four different systems: mW water, Tosi-Fumi NaCl, Lennard-Jones, and Hard Spheres. Combining simulations of spherical crystal seeds embedded in the metastable fluid with classical nucleation theory, we are able to successfully describe the nucleation rate for all systems in a wide range of metastability. The crystal-fluid interfacial free energy extrapolated to coexistence conditions is also in good agreement with direct calculations of such parameter. Our results show that seeding is a powerful technique to investigate crystal nucleation.

Equilibrium fluid-crystal interfacial free energy of bcc-crystallizing aqueous suspensions of polydisperse charged spheres

The interfacial free energy is a central quantity in crystallization from the metastable melt. In suspensions of charged colloidal spheres, nucleation and growth kinetics can be accurately measured from optical experiments. In previous work, from these data effective nonequilibrium values for the interfacial free energy between the emerging bcc nuclei and the adjacent melt in dependence on the chemical potential difference between melt phase and crystal phase were derived using classical nucleation theory (CNT). A strictly linear increase of the interfacial free energy was observed as a function of increased metastability. Here, we further analyze these data for five aqueous suspensions of charged spheres and one binary mixture. We utilize a simple extrapolation scheme and interpret our findings in view of Turnbull's empirical rule. This enables us to present the first systematic experimental estimates for a reduced interfacial free energy, σ(0,bcc), between the bcc-crystal phase and the coexisting equilibrium fluid. Values obtained for σ(0,bcc) are on the order of a few k(B)T. Their values are not correlated to any of the electrostatic interaction parameters but rather show a systematic decrease with increasing size polydispersity and a lower value for the mixture as compared to the pure components. At the same time, σ(0) also shows an approximately linear correlation to the entropy of freezing. The equilibrium interfacial free energy of strictly monodisperse charged spheres may therefore be still greater.

Free energy cost of forming an interface between a crystal and its frozen version

Using a thermodynamic integration scheme, we compute the free energy cost per unit area, γ, of forming an interface between a crystal and a frozen structured wall, formed by particles frozen into the same equilibrium structure as the crystal. Even though the structure and potential energy of the crystalline phase in the vicinity of the wall is the same as in the bulk, γ has a nonzero value and increases with increasing density of the crystal and the wall. Investigating the effect of several interaction potentials between the particles, we observe a positive γ at all crystalline densities if the potential is purely repulsive. For models with attractive interactions, such as the Lennard-Jones potential, a negative value for γ is obtained at low densities. A qualitative explanation for the change of sign of γ when going from repulsive to attractive interactions, at low crystal densities, is suggested.

The mold integration method for the calculation of the crystal-fluid interfacial free energy from simulations

The interfacial free energy between a crystal and a fluid, γcf, is a highly relevant parameter in phenomena such as wetting or crystal nucleation and growth. Due to the difficulty of measuring γcf experimentally, computer simulations are often used to study the crystal-fluid interface. Here, we present a novel simulation methodology for the calculation of γcf. The methodology consists in using a mold composed of potential energy wells to induce the formation of a crystal slab in the fluid at coexistence conditions. This induction is done along a reversible pathway along which the free energy difference between the initial and the final states is obtained by means of thermodynamic integration. The structure of the mold is given by that of the crystal lattice planes, which allows to easily obtain the free energy for different crystal orientations. The method is validated by calculating γcf for previously studied systems, namely, the hard spheres and the Lennard-Jones systems. Our results for the latter show that the method is accurate enough to deal with the anisotropy of γcf with respect to the crystal orientation. We also calculate γcf for a recently proposed continuous version of the hard sphere potential and obtain the same γcf as for the pure hard sphere system. The method can be implemented both in Monte Carlo and Molecular Dynamics. In fact, we show that it can be easily used in combination with the popular Molecular Dynamics package GROMACS.

Crystal-liquid interfacial free energy of hard spheres via a thermodynamic integration scheme

The hard-sphere crystal-liquid interfacial free energy γcl is determined from molecular dynamics simulations using a thermodynamic integration (TI) scheme. The advantage of this TI scheme compared to previous methods is to successfully circumvent hysteresis effects due to the movement of the crystal-liquid interface. This is accomplished by the use of extremely-short-range and impenetrable Gaussian flat walls that prevent the drift of the interface while imposing a negligible free-energy penalty. We find that it is crucial to analyze finite-size effects in order to obtain reliable estimates of γcl in the thermodynamic limit.

Excess free energy of supercooled liquids at disordered walls

Using a thermodynamic integration scheme, we compute the excess free energy, γ, of a glass-forming, binary Lennard-Jones liquid in contact with a frozen amorphous wall, formed by particles frozen into a similar structure as the liquid. We find that γ is nonzero, becoming negative at low temperature. This indicates that the thermodynamics of the system is perturbed by the effect of the amorphous wall.