Crystal-melt Interfacial Free Energy Research Articles

Crystal Growth Rates from Molecular Liquids: The Kinetics of Entropy Loss

It has been established empirically that the rate of addition of molecules to the crystal during crystal growth from the melt is proportional to exp(-|ΔSfus|/R), where ΔSfus is the entropy of fusion. Here we show that this entropic slowdown arises directly from the separation of the entropy loss and energy loss processes associated with the freezing of the liquid. We present a theoretical treatment of the kinetics based on a model flat energy landscape and derive an explicit expression for the coupling magnitude in terms of the crystal-melt interfacial free energy. The implications of our work for nucleation kinetics are also discussed.

“Plunger” Method for Simulating Crystal–Melt Interfacial Free Energies

Crystal–melt interfacial free energies are important ingredients in predicting the nucleation barrier for polymers to crystallize. Experimental measurement of polymer crystal–melt interfacial tensions is challenging, particularly for crystal facets other than the lamellar surfaces. We propose a simple way to obtain the interfacial free energy for any polymer crystal surface and melt using molecular dynamics simulation. We measure the force on a simulated nanoscale “plunger”, that restrains a melt from flowing into the gap between two crystals cleaved along the (100) plane. This gives the difference between the crystal–vacuum and crystal–melt interfacial free energies. Separately, the crystal–vacuum interfacial free energy is obtained by measuring the force required to hold two crystals apart and integrating the force with respect to distance. We obtain the crystal–melt interfacial free energy by subtracting the above values. Results from this method applied to n-alkanes can be compared to measurements of ...

Interfacial free energy and stiffness of aluminum during rapid solidification

Using molecular dynamics simulations and the capillary fluctuation method, we have calculated the anisotropic crystal-melt interfacial free energy and stiffness of aluminum in a rapid solidification system where a temperature gradient is applied to enforce thermal non-equilibrium. To calculate these material properties, the standard capillary fluctuation method typically used for systems in equilibrium has been modified to incorporate a second-order Taylor expansion of the interfacial free energy term. The result is a robust method for calculating interfacial energy, stiffness and anisotropy as a function of temperature gradient using the fluctuations in the defined interface height. This work includes the calculation of interface characteristics for temperature gradients ranging from 11 to 34 K/nm. The captured results are compared to a thermal equilibrium case using the same model and simulation technique with a zero gradient definition. We define the temperature gradient as the change in temperature over height perpendicular to the crystal-melt interface. The gradients are applied in MD simulations using defined thermostat regions on a stable solid-liquid interface initially in thermal equilibrium. The results of this work show that the interfacial stiffness and free energy for aluminum are dependent on the magnitude of the temperature gradient, however the anisotropic parameters remain independent of the non-equilibrium conditions applied in this analysis. The relationships of the interfacial free energy/stiffness are determined to be linearly related to the thermal gradient, and can be interpolated to find material characteristics at additional temperature gradients.

Anisotropic crystal–melt interfacial energy and stiffness of aluminum

Abstract

Calculation of crystal-melt interfacial free energy of Cu by molecular dynamics simulation

The growing and melting of crystal nuclei in liquid Cu are investigated by molecular dynamics simulation. The critical undercooling is proportional to the reciprocle of the nanoparticle radius. The Gibbs-Thomson coefficient of Cu is 1.12× 10-7 K·m. Then the crystal-melt interfacial free energy of Cu is 0.146 J/m2 estimated from the Gibbs-Thomson coefficient, and the Turnbull coefficient of Cu is 0.416. All the values by simulation are consistent with the experimental results of Turnbull.

A molecular dynamics simulation study of the crystal–melt interfacial free energy and its anisotropy in the Cu–Ag–Au ternary system

The crystal–melt interfacial free energy and its small anisotropy were computed using molecular dynamics simulations and the capillary fluctuation method applied to an embedded atom description of the ternary Cu–Ag–Au alloy system. The use of a ternary system allows for an assessment of the variation in anisotropy as one solute species (Au) is replaced by a nearly equal atomic size species (Ag) at a constant temperature, thereby isolating the effect of solute–solvent binding characteristics. Both the four-fold anisotropy parameter, ε 1 , and the six fold term, ε 2 , were computed for five compositions within the Gibbs triangle at T = 1275 K . The results suggest that solute species exhibiting a large enthalpy of mixing will tend to promote a transition of the dendrite crystallographic growth directions from 〈 1 0 0 〉 to 〈 1 1 0 〉 .

Crystal-melt interfacial free energy of binary hard spheres from capillary fluctuations

This is the publisher's version, also available electronically from http://journals.aps.org/prb/abstract/10.1103/PhysRevB.78.144112.

Anisotropy of crystal-melt interfacial free energy of silicon by simulation

We extend the cleaving wall method to a nonpairwise additive potential. Using this method, we compute the anisotropy of crystal-melt interfacial free energy γ for Stillinger–Weber potential of silicon [F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985)]. The calculated γ for (100), (111), and (110) orientations are 0.42±0.02, 0.34±0.02, and 0.35±0.03J∕m2, respectively. The anisotropy in γ we found is consistent with the experimental observation that Si(100)-melt interface develops (111) facets and also helps in explaining a higher undercooling observed for Si(111)-melt interface in Czochralski method.

Atomistic Underpinnings for Orientation Selection in Alloy Dendritic Growth

In dendritic solidification, growth morphologies often display a pronounced sensitivity to small changes in composition. To gain insight into the origins of this phenomenon, we undertake an atomistic calculation of the magnitude and anisotropy of the crystal-melt interfacial free energy in a model alloy system featuring no atomic size mismatch and relatively ideal solution thermodynamics. By comparing the results of these calculations with predictions from recent phase-field calculations, we demonstrate that alloying gives rise to changes in free-energy anisotropies that are substantial on the scale required to induce changes in growth orientations.

Molecular dynamics simulations of the crystal–melt interfacial free energy and mobility in Mo and V

Molecular dynamics simulations, based on embedded-atom method potentials, have been used to compute thermodynamic and kinetic properties of crystal–melt interfaces in the bcc metals Mo and V. The interfacial free energy and its associated crystalline anisotropy have been obtained with the capillary fluctuation method and for both metals the anisotropy and the value of the Turnbull coefficient are found to be significantly lower than for the case of fcc materials. The interface mobility, or kinetic coefficient, which relates the isothermal crystallization rate to interface undercooling, was computed by non-equilibrium molecular dynamics simulations. Mobilities in the range 9-16 cm s−1K−1 are obtained. For Mo the mobility in the (110) crystallographic growth direction is larger than in the (100) and (111) directions, whereas for V the growth is found to be isotropic within numerical uncertainty. The kinetic-coefficient results are discussed within the framework of a density-functional-based theory of crystal growth.

Calculation of the crystal-melt interfacial free energy of succinonitrile from molecular simulation

The crystal-metal interfacial free energy for a six-site model of succinonitrile [N triple bond C-(CH(2))(2)-C triple bond N] has been calculated using molecular-dynamics simulation from the power spectrum of capillary fluctuations in interface position. The orientationally averaged magnitude of the interfacial free energy is determined to be (7.0+/-0.4)x10(-3) J m(-2). This value is in agreement (within the error bars) with the experimental value [(7.9+/-0.8)x10(-3) J m(-2)] of Marasli et al. [J. Cryst. Growth 247, 613 (2003)], but is about 20% lower than the earlier experimental value [(8.9+/-0.5)x10(-3) J m(-2)] obtained by Schaefer et al. [Philos. Mag. 32, 725 (1975)]. In agreement with the experiment, the calculated anisotropy of the interfacial free energy of this body-centered-cubic material is small. In addition, the Turnbull coefficient from our simulation is also in agreement with the experiment. This work demonstrates that the capillary fluctuation method of Hoyt et al. [Phys. Rev. Lett. 86, 5530 (2001)] can be successfully applied to determine the crystal-melt interfacial free energy of molecular materials.

Crystal Structure and Interaction Dependence of the Crystal-Melt Interfacial Free Energy

We examine via molecular simulation the dependence of the crystal-melt interfacial free energy gamma on molecular interaction and crystal structure (fcc vs bcc) for systems interacting with inverse-power repulsive potentials, u(r)=epsilon(sigma/r)(n), 6< or =n< or =100. Both the magnitude and anisotropy of gamma are found to increase as the range of the potential increases. Also we find that gamma(bcc)<gamma(fcc), consistent with recent observations that some fcc forming fluids nucleate via formation of metastable bcc nuclei. The anisotropy in gamma is also seen to be smaller in the bcc systems. By extrapolation, we also obtain an improved estimate of gamma for hard spheres.

Anisotropic Interfacial Free Energies of the Hard-Sphere Crystal−Melt Interfaces

We present a reliable method to define the interfacial particles for determining the crystal-melt interface position, which is the key step for the crystal-melt interfacial free energy calculations using capillary wave approach. Using this method, we have calculated the free energies gamma of the fcc crystal-melt interfaces for the hard-sphere system as a function of crystal orientations by examining the height fluctuations of the interface using Monte Carlo simulations. We find that the average interfacial free energy gamma(0) = 0.62 +/- 0.02k(B)T/sigma(2) and the anisotropy of the interfacial free energies are weak, gamma(100) = 0.64 +/- 0.02, gamma(110) = 0.62 +/- 0.02, gamma(111) = 0.61 +/- 0.02k(B)T/sigma(2). The results are in good agreement with previous simulation results based on the calculations of the reversible work required to create the interfaces (Davidchack and Laird, Phys. Rev. Lett. 2000, 85, 4571). In addition, our results indicate gamma(100) > gamma(110) > gamma(111) for the hard-sphere system, similar to the results of the Lennard-Jones system.

Three-dimensional crystal-melt Wulff-shape and interfacial stiffness in the Al-Sn binary system

The quantitative determination of the three-dimensional Wulff shape for a metallic crystal-melt system is reported here. The anisotropy of crystal-melt interfacial free energy is experimentally measured for the $\mathrm{Al}--\mathrm{Sn}$ binary system at temperatures of 300 and $500\phantom{\rule{0.2em}{0ex}}\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$. Equilibrium shapes of liquid droplets entrained within the crystalline phase are measured experimentally on sequential two-dimensional sections, and the three-dimensional Wulff plot is reconstructed. For this system, it is found that a single-parameter description of anisotropy is not sufficient, and the anisotropy is reported using the leading terms of the relevant cubic harmonics. Accordingly, the anisotropy coefficients are determined to be ${\ensuremath{\epsilon}}_{1}=(1.81\ifmmode\pm\else\textpm\fi{}0.36)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ and ${\ensuremath{\epsilon}}_{2}=(\ensuremath{-}1.12\ifmmode\pm\else\textpm\fi{}0.13)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$. In addition, the corresponding normal stiffness components as well as a generalized stiffness are quantified and compared with available predictions from atomistic simulations.

Semi-empirical model for prediction of crystal–melt interfacial tension

A simple formula for the crystal–melt interfacial free energy γ sl is derived with the aid of the concept of an interplanar adhesion potential function. It is found that γ sl scales with the melting point T m and the atomic volume Ω sm 2/3 of the solid at the melting point in conformity with the law of corresponding states and with molecular dynamics simulations. The new expression correctly describes the γ sl data obtained in supercooling experiments and explains the nature of Turnbull's empirical rule. It predicts γ sl for a wide variety of elements including metals and semimetals, and of organic and inorganic substances, in excellent agreement with available experimental data.

Complete mapping of the anisotropic free energy of the crystal-melt interface in Al

We have calculated the complete anisotropic crystal-melt interfacial free energy of aluminum, using molecular dynamics simulations of the interfaces in equilibrium. The interfacial free energy, $\ensuremath{\gamma}(\mathbf{n}^),$ can be expressed in terms of two anisotropic parameters, $\ensuremath{\epsilon}=1.2$% and $\ensuremath{\delta}=\ensuremath{-}1.2%,$ as well as an average free energy of ${\ensuremath{\gamma}}_{0}=149{\mathrm{m}\mathrm{J}/\mathrm{m}}^{2}$ in reasonable agreement with current experimental results. The expansion of the free energy in terms of these parameters is consistent with six different orientations, including the (111) interfacial plane, which is found to be rough despite its large stiffness.

The solid–liquid interfacial free energy of close-packed metals: Hard-spheres and the Turnbull coefficient

Largely due to its role in nucleation and crystal-growth, the free energy of the crystal-melt interfacial free energy is an object of considerable interest across a number of scientific disciplines, especially in the materials-, colloid-, and atmospheric sciences. Over 50 years ago, Turnbull observed that the interfacial free energies (scaled by the mean interfacial area per particle) of a variety of metallic elements exhibit a linear correlation with the enthalpy of fusion. This correlation provides an important empirical “rule-of-thumb” for estimating interfacial free energies, but lacks a compelling physical explanation. In this work we show that the interfacial free energies for close-packed metals are linearly correlated with the melting temperature and are therefore primarily entropic in origin. We also show that the slope of this linear relationship can be determined with quantitative accuracy using a hard-sphere model, and that the correlation with the enthalpy of fusion reported by Turnbull follows as a consequence of the fact that the entropy of fusion for close-packed metals is relatively constant.

Calculation of the Crystal-Melt Interfacial Free Energy from Experimental Radial Distribution Function Data

Calculations of the crystal-melt interfacial free energy at the melting temperature have been carried out using a method based upon Ewing’s model in which the interfacial entropy is evaluated from the liquid radial distribution function. A simple expression is proposed for the energetic contribution to the interfacial free energy for the case of average, high-index crystallographic orientations exposed to the melt, permitting the determination of the interfacial free energy for any metal for which a dependable experimental radial distribution function is available. Calculations have been made for 37 metals. The results are found to agree closely with previous experimental determinations of the interfacial free energy from both homogeneous nucleation data (corrected to the melting point) and dihedral angle studies.