Solid-liquid Interfacial Energy Research Articles

Solid-liquid interfacial free energy out of equilibrium

The properties of the interface between solid and melt are key to solidification and melting, as the interfacial free energy introduces a kinetic barrier to phase transitions. This makes solidification happen below the melting temperature, in out-of-equilibrium conditions at which the interfacial free energy is ill-defined. Here we draw a connection between the atomistic description of a diffuse solid- liquid interface and its thermodynamic characterization. This framework resolves the ambiguities in defining the solid-liquid interfacial free energy above and below the melting temperature. In addition, we introduce a simulation protocol that allows solid-liquid interfaces to be reversibly created and destroyed at conditions relevant for experiments. We directly evaluate the value of the interfacial free energy away from the melting point for a simple but realistic atomic potential, and find a more complex temperature dependence than the constant positive slope that has been generally assumed based on phenomenological considerations and that has been used to interpret experiments. This methodology could be easily extended to the study of other phase transitions, from condensation to precipitation. Our analysis can help reconcile the textbook picture of classical nucleation theory with the growing body of atomistic studies and mesoscale models of solidification.

Calculating the role of composition in the anisotropy of solid-liquid interface energy using phase-field-crystal theory.

This work uses Ginzburg-Landau theory derived from a recent structural phase-field-crystal model of binary alloys developed by the authors to study the roles of concentration, temperature, and pressure on the interfacial energy anisotropy of a solid-liquid front. It is found that the main contribution to the change in anisotropy with concentration arises from a change in preferred crystallographic orientation controlled by solute-dependent changes in the two-point density correlation function of a binary alloy, a mechanism that leads to such phenomena as solute-induced elastic strain and dislocation-assisted solute clustering. Our results are consistent with experimental observations in recent studies by Rappaz et al. [J. Fife, P. Di Napoli, and M. Rappaz, Metall. Mater. Trans. A 44, 5522 (2013)]. This is the first PFC work, to our knowledge, to incorporate temperature, pressure, and density into the thermodynamic description of alloys.

Free-energy landscape and nucleation pathway of polymorphic minerals from solution in a Potts lattice-gas model.

Metastable minerals commonly form during reactions between water and rock. The nucleation mechanism of polymorphic phases from solution are explored here using a two-dimensional Potts model. The model system is composed of a solvent and three polymorphic solid phases. The local state and position of the solid phase are updated by Metropolis dynamics. Below the critical temperature, a large cluster of the least stable solid phase initially forms in the solution before transitioning into more-stable phases following the Ostwald step rule. The free-energy landscape as a function of the modal abundance of each solid phase clearly reveals that before cluster formation, the least stable phase has an energetic advantage because of its low interfacial energy with the solution, and after cluster formation, phase transformation occurs along the valley of the free-energy landscape, which contains several minima for the regions of three phases. Our results indicate that the solid-solid and solid-liquid interfacial energy contribute to the formation of the complex free-energy landscape and nucleation pathways following the Ostwald step rule.

Estimation of solid–liquid interfacial tension using curved surface of a soft solid

Unlike liquids, for crystalline solids the surface tension is known to be different from the surface energy. However, the same cannot be said conclusively for amorphous materials like soft cross-linked elastomers. To resolve this issue we have introduced here a direct method for measuring solid-liquid interfacial tension by using the curved surface of a solid. In essence, we have used the inner surface of tiny cylindrical channels embedded inside a soft elastomeric film for sensing the effect of the interfacial tension. When a liquid is inserted into the channel, because of wetting-induced alteration in interfacial tension, its thin wall deflects considerably; the deflection is measured with an optical profilometer and analyzed using the Föppl-von Kármán equation. We have used several liquids and cross-linked poly(dimethylsiloxane) as the solid to show that the estimated values of the solid-liquid interfacial tension matches with the corresponding solid-liquid interfacial energy reasonably well.

Pattern-Formation During Self-organization in Three-Phase Eutectic Solidification

Eutectic growth is an interesting example for exploring the topic of pattern-formation in multi-phase systems, where the growth of the phases is coupled with the diffusive transport of one or more components in the melt. While in the case of binary alloys, the number of possibilities are limited (lamellae, rods, labyrinth etc.), their number rapidly increases with the number of components and phases. In this paper, we will investigate pattern formation during three-phase eutectic solidification using a state-of-the art phase-field method based on the grand-canonical density formulation. The major aim of the study is to highlight the role of two properties, which are the volume fraction of the solid phases and the solid–liquid interfacial energies, in the self-organization of the solid phases during directional growth. Thereafter, we will show representative phase-field simulations of a micro-structure in a real alloy (Ag–Al–Cu) using an asymmetric phase diagram as well as interfacial properties.

Predictive Calculation of the Crystallization Tendency of Model Pharmaceuticals in the Supercooled State from Molecular Dynamics Simulations.

Molecular dynamics (MD) simulations were used to perform a comparative study of the crystallization tendency from the melt of two model pharmaceutical compounds: felodipine and nifedipine. Two crystalline polymorphs of nifedipine (N(α), N(β)) and felodipine (FI, FII) have been studied. Calculations were performed on liquid and crystal systems separately in order to determine their main physical properties (diffusivity, density, and enthalpy). A fair agreement was found between the simulation and the known experimental data confirming the ability of the force field GAFF to reproduce accurately the experimental data for both compounds. Simulations of the crystal-liquid interface enabled the determination of the melting temperature and the interfacial free energy of the different polymorphs. Guided by the classical nucleation theory (CNT) predictions and different growth mechanism models (normal, two-dimensional, and screw dislocation), the nucleation and growth rates have been determined. The present investigation particularly raises the very important role of the solid-liquid interfacial free energy and its interplay with the driving force during the crystallization. The origin of the higher crystallization tendency of nifedipine with respect to felodipine is discussed from the present computed kinetic and thermodynamical factors.

Kinetics of reactive wetting of graphite by liquid Al and Cu–Si alloys

In order to reveal the physical essence of the spreading process of reactive wetting, a sort of model of energy to explain the driving force and wetting mechanism was presented. The reactive wetting of molten Al and Cu–Si on graphite was studied by a modified sessile drop method under a vacuum, in which the contact angles were measured by ADSA software. The thermodynamic and kinetic processes of the typical reactive wetting were focused on, the thermodynamic equations of energy relations were derived, the interfacial energy of graphite and solid–liquid interfacial energy versus time at the triple line were calculated, and the dynamics model of interface energy is established. The presented dynamics model is verified by means of experimental results, and it is shown that solid–liquid interfacial energy decreases with time in exponential relationship. It provides a new method for reference to explain the process from the angle of energy.

The Experimental Determination of Interfacial Energies for Solid Zn in Equilibrium with Zn-Al-Sb Liquid

The equilibrated grain boundary groove shapes of solid Zn in equilibrium with Zn-Al-Sb liquid were observed from a quenched sample using a radial heat flow apparatus. The Gibbs–Thomson coefficient, solid–liquid interfacial energy, and grain boundary energy of the solid Zn were determined from the observed grain boundary groove shapes. The thermal conductivity of the eutectic solid phase for Zn-0.4 at. pct Al-0.4 at. pct Sb alloy and the thermal conductivity ratio of the liquid phase to the solid phase for Zn-0.4 at. pct Al-0.4 at. pct Sb alloy at eutectic temperature were also measured with a radial heat flow apparatus and a Bridgman-type growth apparatus, respectively.

A level set method for materials with texturally equilibrated pores

Textural equilibrium controls the distribution of the liquid phase in many naturally occurring porous materials such as partially molten rocks and alloys, salt–brine and ice–water systems. In these materials, pore geometry evolves to minimize the solid–liquid interfacial energy while maintaining a constant dihedral angle, θ, at solid–liquid contact lines. We present a level set method to compute an implicit representation of the liquid–solid interface in textural equilibrium with space-filling tessellations of multiple solid grains in three dimensions. Each grain is represented by a separate level set function and interfacial energy minimization is achieved by evolving the solid–liquid interface under surface diffusion to constant mean curvature surface. The liquid volume and dihedral angle constraints are added to the formulation using virtual convective and normal velocity terms. This results in an initial value problem for a system of non-linear coupled PDEs governing the evolution of the level sets for each grain, using the implicit representation of the solid grains as initial condition. A domain decomposition scheme is devised to restrict the computational domain of each grain to few grid points around the grain. The coupling between the interfaces is achieved in a higher level on the original computational domain. The spatial resolution of the discretization is improved through high-order spatial differentiation schemes and localization of computations through domain composition. Examples of three-dimensional solutions are also obtained for different grain distributions networks that illustrate the geometric flexibility of the method.

Thermal conductivity and interfacial energy of solid Bi in the Bi–Ag eutectic system

The equilibrated grain boundary groove shapes for solid Bi (Bi–2.87 at.%Ag) in equilibrium with Bi–Ag eutectic liquid have been observed from quenched sample with a radial heat flow apparatus. The Gibbs–Thomson coefficient, solid–liquid interfacial energy and grain boundary energy of solid Bi have been determined from the observed grain boundary groove shapes. The variation of thermal conductivity with temperature for eutectic solid phase (Bi–4.7 at.%Ag) has been measured. The ratio of thermal conductivity of equilibrated eutectic liquid phase to eutectic solid phase has also been measured with a Bridgman-type growth apparatus at the melting temperature. The Gibbs–Thomson coefficient, solid–liquid interfacial energy and grain boundary energy of solid Bi in equilibrium with Bi–Ag eutectic liquid were determined to be (9.2 ± 0.6) × 10–8 K m, (52.7 ± 6.3) × 10–3 J m–2 and (102.4 ± 13.3) × 10–3 J m–2, respectively, from observed grain boundary groove shapes.

Phase-field crystal model for a diamond-cubic structure.

We present a structural phase-field crystal model [M. Greenwood et al., Phys. Rev. Lett. 105, 045702 (2010)] that yields a stable dc structure. The stabilization of a dc structure is accomplished by constructing a two-body direct correlation function (DCF) approximated by a combination of two Gaussian functions in Fourier space. A phase diagram containing a dc-liquid phase coexistence region is calculated for this model. We examine the energies of solid-liquid interfaces with normals along the [100], [110], and [111] directions. The dependence of the interfacial energy on a temperature parameter, which controls the heights of the peaks in the two-body DCF, is described by a Gaussian function. Furthermore, the dependence of the interfacial energy on the peak widths of the two-body DCF, which controls the excess energy associated with interfaces, defects, and strain, is described by an inverse power law. These relationships can be used to parametrize the phase-field crystal model for the dc structure to match solid-liquid interfacial energies to those measured experimentally or calculated from atomistic simulations.

Experimental determination of interfacial energies for solid Sn in equilibrium with Sn-Mg-Zn liquid

The equilibrated grain boundary groove shapes of solid Sn in equilibrium with Sn-Mg-Zn liquid were observed from a quenched sample by using a radial heat flow apparatus. The Gibbs-Thomson coefficient, solid-liquid interfacial energy and grain boundary energy of solid Sn were determined from the observed grain boundary groove shapes. The thermal conductivity of the eutectic solid phase for Sn-8.12 at% Mg-4.97 at% Zn alloy and the thermal conductivity ratio of the liquid phase to the solid phase for Sn-8.12 at% Mg-4.97 at% Zn alloy at eutectic temperature were also measured with a radial heat flow apparatus and a Bridgman-type growth apparatus, respectively. The Gibbs-Thomson coefficient, solid-liquid interfacial energy and grain boundary energy of solid Sn in equilibrium with Sn-Mg-Zn liquid were determined to be (8.3 ± 0.6)×10-8 Km, (118.5 ± 14.2)×10-3 Jm-2 and (225.1 ± 29.3)×10-3 J m-2 respectively from observed grain boundary groove shapes. A comparison of present results for solid Sn in the Sn-8.12 at% Mg-4.97 at% Zn alloy with the results obtained in previous works for similar solid Sn in equilibrium with different binary or ternary liquid was made.

Nicotinamide–Indole Binary Drug System: Thermodynamic and Interfacial Studies

Crystal engineering, which can potentially be applied to a wide range of crystalline materials, offer an alternative and potentially fruitful method for improving pharmaceutical properties of drugs. The molecular mobility inherent to amorphous/crystalline phases can lead to molecular associations between different components, such that a single crystalline phase of multiple components is formed. The present investigation measures the solid–liquid equilibria (SLE) of nicotinamide (NA)-indole (IN) binary drug system. The solid–liquid equilibrium phase diagram of NA-IN system was determined by the thaw-melt method in the form of a temperature-composition curve. It shows the formation of a simple eutectic at a 0.953 mole fraction of indole, at 48°C. The excess thermodynamic quantities (gE, hE, and sE) have been estimated by computing heat of fusion data and activity coefficient of the component in a binary mix. These values highlight the ordering, stability, and structure of eutectic and non-eutectic alloys. The thermodynamic mixing functions provide information about the nature of mixing of the components during alloying. The negative value of the integral mixing function, ΔGM for eutectic and some non-eutectic alloys indicates spontaneous mixing. The driving force of nucleation during solidification (ΔGv) and the critical free energy of nucleation (ΔG*) obtained at different undercoolings have also been highlighted. The values of radius of critical nucleus (r*) of alloys lies within the nm scale, which suggests new dimensions of solidification processes for nano solid drug dispersions. The solid–liquid interfacial energy (σ), the grain boundary energy (σgb), and the Gibbs–Thomson coefficient (τ) of the drug alloys have been discussed. Interface morphology of the alloys follows the Jackson's surface roughness (α) theory and predicts that faceted growth (α > 2) proceeds in all cases.Supplemental data for this article can be accessed at www.tandfonline.com/gmcl

Solid–liquid interfacial energy of solid succinonitrile in equilibrium with succinonitrile-1,4-diiodobenzene eutectic liquid

The grain boundary groove shapes for equilibrated solid SCN in equilibrium with the SCN-0.5 mol% DIB eutectic liquid have been directly observed by using a horizontal linear temperature gradient apparatus. The ratio of thermal conductivity of equilibrated liquid to thermal conductivity of solid SCN has also been determined to be 0.90. From the observed grain boundary groove shapes and measured thermal conductivity ratio, the Gibbs–Thomson coefficient (Г) of solid SCN has been determined to be (5.46 ± 0.55) × 10−8 K m. The solid–liquid interfacial energy (σ SL) and the grain boundary energy of solid SCN have also been determined to be (8.25 ± 1.24) × 10−3 J m−2 and (15.84 ± 2.53) × 10−3 J m−2, respectively.

Measuring Solid Liquid Interfacial Energy by Grain Boundary Groove Profile Method (GBG)

The solid liquid interfacial energy plays important role in the liquid solid transformation, thus it is very important to have quantitative values of the interfacial energy as this will help gain more knowledge into the structural nature of the interface. It will assist in gaining fundamental knowledge into the physics of interfaces and also in improving the technology in the crystal growth and foundry. Gündüz designed a radial heat flow apparatus in applied temperature gradient to produce grain boundary groove profile in 1984[2]. Based on the design by Gündüz, a novel radial heat flow apparatus is designed. To obtain the solid liquid interfacial energy, Gibbs Thomson coefficient value is required. To obtain accurate values of Gibbs Thomson coefficient, the interface groove profile, temperature gradient in solid and thermal conductivity values for the solid and liquid must be known.

Molecular dynamics simulation on the wetting characteristic of micro-droplet on surfaces with different free energies

The wetting characteristic of micro-droplets on surfaces with different free energies is crucial to heterogeneous nucleation theory and the growth mechanism of micro-droplets during vapor condensation. In this paper, the spreading processes and wetting characteristics of nanoscale water droplets on various surfaces are explored by molecular dynamics simulation method. The surfaces are constructed from face centered cubic copper-like atoms with different Lennard-Jones potential parameters. The Lennard-Jones interaction energy well-depth of the surface atoms is adjusted to acquire different surface free energies, and the ratio of surface-water interaction energy well-depth to the water-water interaction energy well-depth is defined as the interaction intensity. In the present study, the relationship between interfacial free energies and solid-liquid interaction intensities is evaluated using molecular dynamics simulations. The wetting characteristics of TIP4P/2005 water droplets on surfaces with various free energies are simulated and analyzed as well, using molecular dynamics simulations under an NVT ensemble. Results indicate that the solid-liquid interfacial free energy increases as the solid-liquid interaction intensity increases, with different spreading processes and wetting characteristics achieved for the water droplets on these surfaces. For the surfaces with lower interaction intensities, water cannot spread on the solid surfaces and hydrophobic surfaces are obtained when the interaction intensity ratio between surface atoms and water molecules is lower than 1.6. As the interaction intensity increases, the surface translates from hydrophobic into hydrophilic, and finally into a complete wetting state as the interaction intensity reaches up to 3.5. Due to the limitation of nanoscale dimensions, the forces that exert on droplet surface are non-continuous and asymmetric. As a result, significant fluctuations of liquid-vapor interface and local solid-liquid contact line can be observed for the droplet in nanoscale. The transient contact angle of nano-droplets is also fluctuating within a certain range, which is different from that observed for macro-droplets. From the viewpoint of statistics, an apparent contact angle can be obtained for the droplet on each surface. The contact angle decreases with solid-liquid interaction intensities linearly, which is in accordance with the calculated results of classic Young's theory using the interfacial free energies obtained from molecular dynamics simulations. The fact that an apparent contact angle is already established for a droplet in nanoscale, supporting the capillary assumption that is widely adopted in classic nucleation theory. The fluctuation of liquid-vapor interface and contact angle also provides a qualitative explanation for the discrepancy between experimental nucleation rates and predictions in classic nucleation theory.

Size-dependent freezing temperature of metallic and semi-metallic nanoparticles

The influence of size on freezing temperature of metallic and semi-metallic nanoparticles is investigated according to the classical nucleation theory by considering size and temperature effect on solid–liquid interface energy. The advantage of this improvised model is the ability to predict the freezing temperature for low dimensional materials, literally free from any arbitrarily adjustable parameters. It predicts that freezing temperature decreases almost exponentially with decreasing size due to the increasing surface/volume ratio. In addition, the critical radius at which the difference between the melting and freezing temperature disappears has been predicted. The model predictions agree well with available experimental results of Pb and Bi nanoparticles, which supply an easy way to understand freezing behavior of nanomaterials.

Effect of ultrasound and stabilizers on nucleation kinetics of curcumin during liquid antisolvent precipitation

Nucleation kinetics of liquid antisolvent precipitation of a poorly water soluble drug curcumin in presence of ultrasound and surfactants have been estimated. Ultrasound and stabilizers were found to have opposing effects on induction time (τind), metastable zone width (MSZW) and nucleation rates (J) of curcumin during antisolvent precipitation. The use of ultrasound (in presence or absence of stabilizers) was found to decrease τind and MSZW drastically while the values of nucleation rates were found to increase. In contrast to these observations, use of stabilizers (in presence or absence of ultrasound) were found to increase MSZW, increase τind and lower the nucleation rates (J) of curcumin. The solid–liquid interfacial energies (γSL) for curcumin in aqueous ethanolic solutions (with and without stabilizers) have also been calculated using experimentally estimated induction time (τind) and supersaturation data. The values of solid–liquid interfacial energies were found to be in the range of 1.5–3.5mJ/m2. In comparison to these values, the values of γSL predicted by Mersmann equation and equation proposed by Bennema and Sohnel were found to be significantly higher and were in the range of 10–30mJ/m2.

Phase equilibrium and mechanisms of crystallization in liquid–liquid phase separating system

The influence of liquid–liquid phase separation (LLPS) in a crystallizing system is examined, and the liquid–liquid tie lines are determined in the ternary system of butyl paraben, water and ethanol. Crystallization by evaporation in droplets (solution of butyl paraben in ethanol and water mixtures) of homogeneous solution and two-phase liquid mixtures on plates of different materials: glass, plastic, steel, and tin were investigated by microscopy. Different crystallization and liquid–liquid phase separation behaviour are observed depending on initial droplet composition and the surface material. This work reveals that crystallization in the LLPS solution of butyl paraben in ethanol–water mixtures starts in the butyl paraben rich phase as can be explained by a higher butyl paraben concentration, and a lower solid–liquid interfacial energy.

Melting curves and entropy of melting of iron under Earth’s core conditions

The melting curves of iron are determined up to 365GPa via molecular dynamic (MD) simulations combining with the embedded atom model (EAM) potential developed by Ackland et al. We simulated the melting with three approaches, the hysteresis, two-phase and recently modified Z methods. All three techniques can produce satisfying results, consistent well with most of static compression measurements and shock experiments. Hence, we recommend that these three techniques and this EAM potential are reliable techniques and potential for simulating melting properties of iron. Fitting the well-known Simon equation to our two-phase data we yield the analytical melting curve for iron: 1825(1+P/57.723)0.654, which gives a melting point at the inner core boundary of 6345K, very close to the recent diamond anvil cell (DAC) extrapolated value and other ab initio calculations. Furthermore, the analyses of our entropy of melting and solid–liquid interfacial energy γsl indicate that at high pressure, the entropy of fusion shows weak pressure effect. The γsl increases monotonically with pressure, and can be described as a second-order polynomial relation.