Solid–liquid interfacial free energy of small colloidal hard-sphere crystals

Unravelling the mechanism of columnar-to-equiaxed transition and grain refinement in ultrasonic vibration assisted laser welding of Ti6Al4V titanium alloy

In this study, the microstructural evolution and mechanical properties of Ti6Al4V titanium alloy welded joints subjected to ultrasonic assisted laser welding were scrutinized, while numerical simulations were employed to explicate the grain refinement mechanism. The simulations indicate that the ultrasonic vibration significantly improves the overall fluidity and temperature of the molten pool. Under the identical condition of laser power and welding speed (1500 W, 1.3 m/min), the presence of 0.2A ultrasonic current yields a more uniform refinement of columnar grains, along with a denser arrangement of acicular martensite. The refinement mechanism can be attributed to the small temperature gradient, cavitation effects, and stress induced by ultrasonic vibration. Notably, the welded joint attains a peak tensile strength of 945.2 MPa under the aforementioned 0.2A condition, distinctly demonstrating the characteristics of ductile fracture. This research further reveals the underlying mechanism of grain refinement in Ti6Al4V alloy laser-welded joints induced by ultrasonic vibration, providing valuable references for optimizing process parameters and improving the quality of such welded joints.

Low Interfacial Free Energy Describes the Bulk Ordering Transition in Colloidal Cubes

Many hard faceted nanoparticles are known to undergo disorder-to-order phase transitions following a classical nucleation and growth mechanism. In a previous study [J. Phys. Chem. B 2018, 122, 9264-9273], it was shown that hard cubes undergo a nonclassical phase transition with a bulk character instead of originating from consolidated nuclei. Significantly, an unusually high fraction of ordered particles was observed in the metastable basin of the disordered phase, even for very low degrees of supersaturation. This work aims to substantiate the conjecture that these unique properties originate from a comparatively low interfacial free energy between the disordered and ordered phases for hard cubes relative to other hard particle systems. Using the cleaving wall method to directly measure the interfacial free energy for cubes, it is found that its values are indeed small; e.g., at phase coexistence conditions, it is only one-fifth that for hard spheres. A theoretical nucleation model is used to explore the broader implications of low interfacial tension values and how this could result in a bulk ordering mechanism.

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.

The energetics of prenucleation clusters in lattice solutions

According to classical nucleation theory, nucleation from solution involves the formation of small atomic clusters. Most formulations of classical nucleation use continuum "droplet" approximations to describe the properties of these clusters. However, the discrete atomic nature of very small clusters may cause deviations from these approximations. Here, we present a self-consistent framework for describing the nature of these deviations. We use our framework to investigate the formation of "polycube" atomic clusters on a cubic lattice, for which we have used combinatoric data to calculate the thermodynamic properties of clusters with 17 atoms or less. We show that the classical continuum droplet model emerges as a natural approach to describe the free energy of small clusters, but with a size-dependent surface tension. However, this formulation only arises if an appropriate "site-normalized" definition is adopted for the free energy of formation. These results are independently confirmed through the use of Monte Carlo calculations. Our results show that clusters formed from sparingly soluble materials (μM solubility range) tend to adopt compact configurations that minimize the solvent-solute interaction energy. As a consequence, there are distinct minima in the cluster-size-energy landscape that correspond to especially compact configurations. Conversely, highly soluble materials (1M) form clusters with expanded configurations that maximize configurational entropy. The effective surface tension of these clusters tends to smoothly and systematically decrease as the cluster size increases. However, materials with intermediate solubility (1 mM) are found to have a balanced behavior, with cluster energies that follow the classical "droplet" scaling laws remarkably well.

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.

Interfacial free energy of a hard-sphere fluid in contact with curved hard surfaces

Using molecular-dynamics simulation, we have calculated the interfacial free energy γ between a hard-sphere fluid and hard spherical and cylindrical colloidal particles, as functions of the particle radius R and the fluid packing fraction η=ρσ(3)/6, where ρ and σ are the number density and hard-sphere diameter, respectively. These results verify that Hadwiger's theorem from integral geometry, which predicts that γ for a fluid at a surface, with certain restrictions, should be a linear combination of the average mean and Gaussian surface curvatures, is valid within the precision of the calculation for spherical and cylindrical surfaces up to η ≈ 0.42. In addition, earlier results for γ for this system [Bryk et al., Phys. Rev. E 68, 031602 (2003)] using a geometrically based classical density functional theory are in excellent agreement with the current simulation results for packing fractions in the range where Hadwiger's theorem is valid. However, above η ≈ 0.42, γ(R) shows significant deviations from the Hadwiger form indicating limitations to its use for high-density hard-sphere fluids. Using the results of this study together with Hadwiger's theorem allows one, in principle, to determine γ for any sufficiently smooth surface immersed in a hard-sphere fluid.

Equilibrium fluid-solid coexistence of hard spheres

We present a tethered Monte Carlo simulation of the crystallization of hard spheres. Our method boosts the traditional umbrella sampling to the point of making practical the study of constrained Gibbs' free energies depending on several crystalline order parameters. We obtain high-accuracy estimates of the fluid-crystal coexistence pressure for up to 2916 particles (enough to accommodate fluid-solid interfaces). We are able to extrapolate to infinite volume the coexistence pressure [p(co)=11.5727(10)k(B)T/σ(3)] and the interfacial free energy [γ({100})=0.636(11)k(B)T/σ(2)].

Colloids as model systems for metals and alloys: a case study of crystallization

Metallic systems are widely used as materials in daily human life. Their properties depend very much on the production route. In order to improve the production process and even develop novel materials a detailed knowledge of all physical processes involved in crystallization is mandatory. Atomic systems like metals are characterized by very high relaxation rates, which make direct investigations of crystallization very difficult and in some cases impossible. In contrast, phase transitions in colloidal systems are very sluggish and colloidal suspensions are optically transparent. Therefore, colloidal systems are often discussed as model systems for metals. In the present work, we study the process of crystallization of charged colloidal systems from the very beginning. Charged colloids offer the advantage that the interaction potential can be systematically tuned by a variation of the particle number density and the salt concentration. We use light scattering and ultra-small angle x-ray scattering to investigate the formation of short-range order in the liquid state even far from equilibrium, crystal nucleation and crystal growth. The results are compared with those of equivalent studies on metallic systems. They are critically assessed as regards similarities and differences.

Hard-sphere crystallization gets rarer with increasing dimension

We recently found that crystallization of monodisperse hard spheres from the bulk fluid faces a much higher free-energy barrier in four than in three dimensions at equivalent supersaturation, due to the increased geometrical frustration between the simplex-based fluid order and the crystal [J. A. van Meel, D. Frenkel, and P. Charbonneau, Phys. Rev. E 79, 030201(R) (2009)]. Here, we analyze the microscopic contributions to the fluid-crystal interfacial free energy to understand how the barrier to crystallization changes with dimension. We find the barrier to grow with dimension and we identify the role of polydispersity in preventing crystal formation. The increased fluid stability allows us to study the jamming behavior in four, five, and six dimensions and to compare our observations with two recent theories [C. Song, P. Wang, and H. A. Makse, Nature (London) 453, 629 (2008); G. Parisi and F. Zamponi, Rev. Mod. Phys. (to be published)].

Homogeneous nucleation in thermal dust-electron plasmas

The homogeneous nucleation in the dust-electron plasma, which is formed in the zone of metal powder combustion products in the premixed laminar two-phase flame, has been studied. The classical nucleation theory has been used to determine the free energy and the critical radius of the nucleus. The influence of nucleus charging as a result of interaction between the nucleus and the electronic component of the plasma on the free energy has been determined. The dependence of the nucleus' critical radius on the plasma temperature and number density of the plasma's dust component has been determined. The proposed theoretical model shows that nucleation in the thermal dust-electron plasma is a self-consistent process, which is opposed to changing of the plasma's disperse structure.

A physical analogue of the Schelling model

We present a mathematical link between Schelling's socio-economic model of segregation and the physics of clustering. We replace the economic concept of "utility" by the physics concept of a particle's internal energy. As a result cluster dynamics is driven by the "surface tension" force. The resultant segregated areas can be very large and can behave like spherical "liquid" droplets or as a collection of static clusters in "frozen" form. This model will hopefully provide a useful framework for studying many spatial economic phenomena that involve individuals making location choices as a function of the characteristics and choices of their neighbors.

The anisotropic hard-sphere crystal-melt interfacial free energy from fluctuations

We have calculated the interfacial free energy for the hard-sphere system, as a function of crystal interface orientation, using a method that examines the fluctuations in the height of the interface during molecular dynamics simulations. The approach is particularly sensitive for the anisotropy of the interfacial free energy. We find an average interfacial free energy of gamma=0.56+/-0.02k(B)Tsigma(-2). This value is lower than earlier results based upon direct calculations of the free energy [R. L. Davidchack and B. B. Laird, Phys. Rev. Lett. 85, 4751 (2000)]. However, both the average value and the anisotropy agree with the recent values obtained by extrapolation from direct calculations for a series of the inverse-power potentials [R. L. Davidchack and B. B. Laird, Phys. Rev. Lett. 94, 086102 (2005)].

Calculation of solid-liquid interfacial free energy: A classical nucleation theory based approach

We present a simple approach to calculate the solid-liquid interfacial free energy. This new method is based on the classical nucleation theory. Using the molecular dynamics simulation, we employ spherical crystal nuclei embedded in the supercooled liquids to create an ideal model of a homogeneous nucleation. The interfacial free energy is extracted by fitting the relation between the critical nucleus size and the reciprocal of the critical undercooling temperature. The orientationally averaged interfacial free energy is found to be 0.302+/-0.002 (in standard LJ unit). The temperature dependence of the interfacial free energy is also obtained in this work. We find that the interfacial free energy increases slightly with increasing temperature. The positive temperature coefficient of the interfacial free energy is in qualitative agreement with Spaepen's analysis [Solid State Phys. 47, FS181 (1994)] and Turnbull's empirical estimation [J. Appl. Phys. 21, 1022 (1950)].

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.

Breakdown of classical nucleation theory near isostructural phase transitions

We report simulations of crystal nucleation in binary mixtures of hard spherical colloids with a size ratio of 1:10. The stable crystal phase of this system can be either dense or expanded. We find that, in the vicinity of the solid-solid critical point where the crystallites are highly compressible, small crystal nuclei are less dense than large nuclei. This phenomenon cannot be accounted for by either classical nucleation theory or by the Gibbsian droplet model. We argue that the observed behavior is due to the surface stress of the crystal nuclei. The observed effect highlights a general deficiency of the most frequently used thermodynamic theories for crystal nucleation. Surface stress should lead to an experimentally observable expansion of crystal nuclei of colloids with short-ranged attraction and of globular proteins.

QUANTITATIVE PREDICTION OF CRYSTAL-NUCLEATION RATES FOR SPHERICAL COLLOIDS: A Computational Approach

This review discusses the recent progress that has been made in the application of computer simulations to study crystal nucleation in colloidal systems. We discuss the concept and the numerical methods that allow for a quantitative prediction of crystal-nucleation rates. The computed nucleation rates are predicted from first principles and can be directly compared with experiments. These techniques have been applied to study crystal nucleation in hard-sphere colloids, polydisperse hard-sphere colloids, weakly charged or slightly soft colloids, and hard-sphere colloids that are confined between two-plane hard walls.

Numerical prediction of absolute crystallization rates in hard-sphere colloids

Special computational techniques are required to compute absolute crystal nucleation rates of colloidal suspensions. Using crystal nucleation of hard-sphere colloids as an example, we describe in some detail the novel computational tools that are needed to perform such calculations. In particular, we focus on the definition of appropriate order parameters that distinguish liquid from crystal, and on techniques to compute the kinetic prefactor that enters in the expression for the nucleation rate. In addition, we discuss the relation between simulation results and theoretical predictions based on classical nucleation theory.