Dislocation lines as the precursor of the melting of crystalline solids observed in Monte Carlo simulations

A statistical analysis of the first stages of freezing and melting of Lennard-Jones particles: Number and size distributions of transient nuclei

The freezing/melting transition is at the heart of many natural and industrial processes. In the classical picture, the transition proceeds via the nucleation of the new phase, which has to overcome a barrier associated with the free energy cost of the growing nucleus. The total nucleation rate is also influenced by a kinetic factor, which somehow depends on the number of attempts to create a nucleus, that translates into a significant density of proto-nuclei in the system. These transient tiny nuclei are not accessible to experiments, but they can be observed in molecular simulations, and their number and size distributions can be acquired and analyzed. The number distributions are carefully characterized as a function of the system size, showing the expected behavior, with limited spurious effects due to the finite simulation box. It is also shown that the proto-nuclei do exist even in the stable phase, in agreement with the fact that the (unfavorable) volume contribution to their free energy is negligible in the first stages of nucleation. Moreover, the number and size distributions evolve continuously between the stable and the metastable phases, in particular when crossing the coexistence temperature. The size distributions associated with any nucleus and with the largest one have also been calculated, and their relationship recently established for bubbles in a liquid [Puibasset, J. Chem. Phys. 157, 191102 (2022)] has been shown to apply here. This is an important relation for free energy barrier calculations with biased molecular simulations.

Finding the differences: Classical nucleation perspective on homogeneous melting and freezing of hard spheres

By employing brute-force molecular dynamics, umbrella sampling, and seeding simulations, we investigate homogeneous nucleation during melting and freezing of hard spheres. We provide insights into these opposing phase transitions from the standpoint of classical nucleation theory. We observe that melting has both a lower driving force and a lower interfacial tension than freezing. The lower driving force arises from the vicinity of a spinodal instability in the solid and from a strain energy. The lower interfacial tension implies that the Tolman lengths associated with melting and freezing have opposite signs, a phenomenon that we interpret with Turnbull's rule. Despite these asymmetries, the nucleation rates for freezing and melting are found to be comparable.

Melting Is Well-Known, but Is It Also Well-Understood?

Contrary to continuous phase transitions, where renormalization group theory provides a general framework, for discontinuous phase transitions such a framework seems to be absent. Although the thermodynamics of the latter type of transitions is well-known and requires input from two phases, for melting a variety of one-phase theories and models based on solids has been proposed, as a generally accepted theory for liquids is (yet) missing. Each theory or model deals with a specific mechanism using typically one of the various defects (vacancies, interstitials, dislocations, interstitialcies) present in solids. Furthermore, recognizing that surfaces are often present, one distinguishes between mechanical or bulk melting and thermodynamic or surface-mediated melting. After providing the necessary preliminaries, we discuss both types of melting in relation to the various defects. Thereafter we deal with the effect of pressure on the melting process, followed by a discussion along the line of type of materials. Subsequently, some other aspects and approaches are dealt with. An attempt to put melting in perspective concludes this review.

Atomic-scale observation of premelting at 2D lattice defects inside oxide crystals

Since two major criteria for melting were proposed by Lindemann and Born in the early 1900s, many simulations and observations have been carried out to elucidate the premelting phenomena largely at the crystal surfaces and grain boundaries below the bulk melting point. Although dislocations and clusters of vacancies and interstitials were predicted as possible origins to trigger the melting, experimental direct observations demonstrating the correlation of premelting with lattice defects inside a crystal remain elusive. Using atomic-column-resolved imaging with scanning transmission electron microscopy in polycrystalline BaCeO₃, here we clarify the initiation of melting at two-dimensional faults inside the crystals below the melting temperature. In particular, melting in a layer-by-layer manner rather than random nucleation at the early stage was identified as a notable finding. Emphasizing the value of direct atomistic observation, our study suggests that lattice defects inside crystals should not be overlooked as preferential nucleation sites for phase transformation including melting.

Structure and dynamics of the Lennard-Jones fcc-solid focusing on melting precursors

The Lennard-Jones potential is taken as a basis to study the structure and dynamics of the face centered cubic (fcc) solid along an isochore from low temperatures up to the solid/fluid transition. The Z method is applied to estimate the melting point. Molecular dynamics simulations are used to calculate the pair distribution function, numbers of nearest neighbors, and the translational order parameter, analyzing the weakening of the fcc-symmetry due to emerging premelting effects. A range of dynamic properties, such as the mean-squared displacement, non-Gaussian parameter, Debye-Waller factor, and vibrational density of states, is considered for the analysis of the solid state. All of these parameters clearly show that bulk mobility is activated at about 2/3 of the melting temperature, known as the Tammann temperature. This indicates that vibrational motion of atoms is not maintained exclusively in the entire stable solid state and that collective atomic motion constitutes a precursor of the melting process.

Visualization of ultrafast melting initiated from radiation-driven defects in solids

Materials exposed to extreme radiation environments such as fusion reactors or deep spaces accumulate substantial defect populations that alter their properties and subsequently the melting behavior. The quantitative characterization requires visualization with femtosecond temporal resolution on the atomic-scale length through measurements of the pair correlation function. Here, we demonstrate experimentally that electron diffraction at relativistic energies opens a new approach for studies of melting kinetics. Our measurements in radiation-damaged tungsten show that the tungsten target subjected to 10 displacements per atom of damage undergoes a melting transition below the melting temperature. Two-temperature molecular dynamics simulations reveal the crucial role of defect clusters, particularly nanovoids, in driving the ultrafast melting process observed on the time scale of less than 10 ps. These results provide new atomic-level insights into the ultrafast melting processes of materials in extreme environments.

Melting of bcc crystal Ta without the Lindemann criterion

Understanding of melting is deeply rooted in the Lindemann criterion which predicts that the transition occurs when the mean vibrational atomic displacement reaches a universal value. The criterion also finds its way in atomic description of kinetics of various structural phase transitions involving liquid and amorphous phases. Here we show using atomistic modeling in bcc crystal tantalum that neither the universal displacement exists nor melting occurs at the anticipated value from the Lindemann criterion. Instead, before and at melting a series of strongly correlated atomic diffusional motions are set in with the atomic displacement far more complicated than that predicted by Lindemann based on independent atomic vibrations. The displacement leads to formation of new extended atomic configurations composed of lattice chains and loops of Ta atoms still residing on the crystal lattice. It is the proliferation of these lattice chains that leads to melting.

Computational Study on Homogeneous Melting of Benzene Phase I

Molecular-dynamics simulations are used for examining the microscopic details of the homogeneous melting of benzene phase I. The equilibrium melting temperatures of our model were initially determined using the direct-coexistence method. Homogeneous melting at a higher temperature is achieved by heating a defect- and surfacefree crystal. The temperature-dependent potential energy and lattice parameters do not indicate a premelting phase even under superheated conditions. Further, statistical analyses using induction times computed from 200 melting trajectories were conducted, denoting that the homogeneous melting of benzene occurs stochastically, and that there is no intermediate transient state between the crystal and liquid phases. Additionally, the critical nucleus size is estimated using the seeding approach, along with the local bond order parameter. We found that the large diffusive motion arising from defect migration or neighbor-molecule swapping is of little importance during nucleation. Instead, the orientational disorder activated using the flipping motion of the benzene plane results in the melting nucleus.

Direct Atomic-Scale Observation of Intermediate Pathways of Melting and Crystallization in Supported Bi Nanoparticles

Uncovering the evolutional pathways of melting and crystallization atomically is critical to understanding complex microscopic mechanism of first-order phase transformation. We conduct in situ atomic-scale observations of melting and crystallization in supported Bi nanoparticles under heating and cooling within an aberration-corrected TEM. We provide direct evidence of the multiple intermediate state events in melting and crystallization. The melting of the supported nanocrystal involves the formation and migration of domain boundaries and dislocations due to the atomic rearrangement under heating, which occurs through a size-dependent multiple intermediate state. A critical size, which is key to inducing the transition pathway in melting from two to four barriers, is identified for the nanocrystal. In contrast, crystallization of a Bi droplet involves three stages. These findings demonstrate that the phase transformations cannot be viewed as a simple single barrier-crossing event but as a complex multiple intermediate state phenomenon, highlighting the importance of nonlocal behaviors.

Diffusive and martensitic nucleation kinetics in solid-solid transitions of colloidal crystals

Solid–solid transitions between crystals follow diffusive nucleation, or various diffusionless transitions, but these kinetics are difficult to predict and observe. Here we observed the rich kinetics of transitions from square lattices to triangular lattices in tunable colloidal thin films with single-particle dynamics by video microscopy. Applying a small pressure gradient in defect-free regions or near dislocations markedly transform the diffusive nucleation with an intermediate-stage liquid into a martensitic generation and oscillation of dislocation pairs followed by a diffusive nucleus growth. This transformation is neither purely diffusive nor purely martensitic as conventionally assumed but a combination thereof, and thus presents new challenges to both theory and the empirical criterion of martensitic transformations. We studied how pressure, density, grain boundary, triple junction and interface coherency affect the nucleus growth, shape and kinetic pathways. These novel microscopic kinetics cast new light on control solid–solid transitions and microstructural evolutions in polycrystals.

Solid-solid transitions between different crystalline structures have broad implications in earth science, steel and ceramic materials. Peng et al. show a transformation pathway that starts off as being martensitic then switches to diffusive at the single particle level in a colloidal system under pressure.

Phase transition and surface sublimation of a mobile Potts model

We study in this paper the phase transition in a mobile Potts model by the use of Monte Carlo simulation. The mobile Potts model is related to a diluted Potts model, which is also studied here by a mean-field approximation. We consider a lattice where each site is either vacant or occupied by a q-state Potts spin. The Potts spin can move from one site to a nearby vacant site. In order to study the surface sublimation, we consider a system of Potts spins contained in a recipient with a concentration c defined as the ratio of the number of Potts spins N(s) to the total number of lattice sites N(L)=N(x)×N(y)×N(z). Taking into account the attractive interaction between the nearest-neighboring Potts spins, we study the phase transitions as functions of various physical parameters such as the temperature, the shape of the recipient, and the spin concentration. We show that as the temperature increases, surface spins are detached from the solid phase to form a gas in the empty space. Surface order parameters indicate different behaviors depending on the distance to the surface. At high temperatures, if the concentration is high enough, the interior spins undergo a first-order phase transition to an orientationally disordered phase. The mean-field results are shown as functions of temperature, pressure, and chemical potential, which confirm in particular the first-order character of the transition.

Direct observation of liquid nucleus growth in homogeneous melting of colloidal crystals

The growth behaviour of liquid nucleus is crucial for crystal melting, but its kinetics is difficult to predict and remains challenging in experiment. Here we directly observed the growth of individual liquid nuclei in homogeneous melting of three-dimensional superheated colloidal crystals with single-particle dynamics by video microscopy. The growth rate of nucleus at weak superheating is well fitted by generalizing the Wilson–Frenkel law of crystallization to melting and including the surface tension effects and non-spherical-shape effects. As the degree of superheating increases, the growth rate is enhanced by nucleus shape fluctuation, nuclei coalescence and multimer attachment. The results provide new guidance for the refinement of nucleation theory, especially for the poorly understood strong-superheating regime. The universal Lindemann parameter observed at the superheat limit and solid–liquid interfaces indicates a connection between homogeneous and heterogeneous melting.

Monitoring crystal melting at a single-atom level is challenging because of the small spatial and temporal scales involved, especially for a bulk process. Wang et al. report the melting dynamics of a colloidal crystal with emphasis on the growth of critical nuclei upon different degrees of superheating.

Imaging the homogeneous nucleation during the melting of superheated colloidal crystals

The nucleation process is crucial to many phase transitions, but its kinetics are difficult to predict and measure. We superheated and melted the interior of thermal-sensitive colloidal crystals and investigated by means of video microscopy the homogeneous melting at single-particle resolution. The observed nucleation precursor was local particle-exchange loops surrounded by particles with large displacement amplitudes rather than any defects. The critical size, incubation time, and shape and size evolutions of the nucleus were measured. They deviate from the classical nucleation theory under strong superheating, mainly because of the coalescence of nuclei. The superheat limit agrees with the measured Born and Lindemann instabilities.

Materials science. Melting colloidal crystals from the inside out

How crystals melt can be seen directly by studying size-tunable colloidal particles.

Equation of state from the Potts-percolation model of a solid

We expand the Potts-percolation model of a solid to include stress and strain. Neighboring atoms are connected by bonds. We set the energy of a bond to be given by the Lennard-Jones potential. If the energy is larger than a threshold the bond is more likely to fail, whereas if the energy is lower than the threshold, the bond is more likely to be alive. In two dimensions we compute the equation of state: stress as a function of interatomic distance and temperature by using renormalization-group and Monte Carlo simulations. The phase diagram, the equation of state, and the isothermal modulus are determined. When the Potts heat capacity is divergent the continuous transition is replaced by a weak first-order transition through the van der Waals loop mechanism. When the Potts transition is first order the stress exhibits a large discontinuity as a function of the interatomic distance.

Extended defects in the Potts-percolation model of a solid: renormalization group and Monte Carlo analysis

We extend the model of a 2d solid to include a line of defects. Neighboring atoms on the defect line are connected by springs of different strength and different cohesive energy with respect to the rest of the system. Using the Migdal-Kadanoff renormalization group we show that the elastic energy is an irrelevant field at the bulk critical point. For zero elastic energy this model reduces to the Potts model. By using Monte Carlo simulations of the three- and four-state Potts model on a square lattice with a line of defects, we confirm the renormalization-group prediction that for a defect interaction larger than the bulk interaction the order parameter of the defect line changes discontinuously while the defect energy varies continuously as a function of temperature at the bulk critical temperature.

A numerical study of the growth process of Au nanometre-sized particles in liquid phases

Molecular dynamics simulations have been employed to study the growth of faceted and spherical gold (Au) nanometre-sized particles in undercooled Au melts and supersaturated Kr-, Xe- and Rn-based liquid solutions at different degrees of undercooling and supersaturation. Different mechanisms have been observed depending on the chemical environment and temperature. At relatively high temperatures, surface adsorption is shown to critically depend on the dynamics of surface species with low coordination number. At low temperatures, adsorption occurs instead with no selective feature. Dendritic structures are formed at the particle surface at high adsorption rates.

TWO-STATE STRUCTURE OF NANOMETER-SIZED Cu PARTICLES

Molecular dynamics simulations have been employed to investigate the static and dynamic properties of unsupported spherical Cu particles with size ranging between 1 and 10 nm. The potential energy, the structural arrangement, and the mobility of atomic species were studied for each nanometer-sized particle within the temperature range between 300 K and the melting point. Two distinct regions, namely an internal domain and a surface layer, can be identified within each nanoparticle on the basis of the radial profiles of such quantities. The atomic species belonging to the interior of the particle display a bulk-like behavior. By contrast, the surface layer is characterized by an excess potential energy associated with a disordered structure. At relatively low temperatures, the surface atoms possess structural and energetic features intermediate between the ones of a superheated bulk solid and of an undercooled bulk liquid. Pre-melting processes at the surface are also evident at temperatures close to the melting point. The nanometer-sized particles can be thus regarded as heterogeneous two-state systems consisting of roughly distinguishable bulk-like and surface regions.

Comparative study of microstructural evolution during melting and crystallization

Molecular dynamics simulations, with the interaction between atoms described by a modified analytic embedded atom method, have been performed to obtain the atomic-scale details of isothermal melting in nanocrystalline Ag and crystallization from supercooled liquid. The radial distribution function and common neighbor analysis provide a visible scenario of structural evolution in the process of phase transition. The results indicate that melting at a fixed temperature in nanocrystalline materials is a continuous process, which originates from the grain boundary network. With the melting developing, the characteristic bond pairs (555), (433), and (544), existing in liquid or liquidlike phase, increase approximately linearly till completely melted. The crystallization from supercooled liquid is characterized by three characteristic stages: nucleation, rapid growth of nucleus, and slow structural relaxation. The homogeneous nucleation occurs at a larger supercooling temperature, which has an important effect on the process of crystallization and the subsequent crystalline texture. The kinetics of transition from liquid to solid is well described by the Johnson-Mehl-Avrami equation.

Mechanistic Aspects of Homogeneous and Heterogeneous Melting Processes

Molecular dynamics simulations have been employed to explore the response of crystalline Ar systems with and without a free surface to a gradual temperature rise. The surface-free crystalline bulk undergoes a homogeneous melting process at the limit of superheating, whereas the semicrystal terminating with a free plane surface melts with a heterogeneous mechanism at a temperature corresponding to the equilibrium melting point. Numerical findings suggest that the gradual disordering of the crystalline lattice as well as the homogeneous and heterogeneous melting processes are mediated by atoms with defective coordination. Their concentration in the regions close to the semicrystal surface at the equilibrium melting point is found to be approximately the same as in the surface-free bulk at the limit of superheating.

Cooperative Atomic Displacements and Melting at the Limit of Superheating

This paper shows how the melting of superheated crystals originates from the localization of thermal disorder in excited regions of the crystalline structure. Within such regions, disordered thermal motion is found to induce the formation of bulk topological defects. These consist of atoms with a number of nearest neighbors different from the equilibrium one. Such defectively coordinated atoms arrange according to pseudolinear clusters, the number and size of which depend on temperature. Characterized by high mobility, defective atoms and their nearest neighbors are seen to undergo a cooperative dynamics that can result in net atom displacements between equilibrium lattice sites.

Defect-Mediated Melting in Superheated Noble Gas Crystals

Molecular dynamics simulations have been used to investigate the mechanisms governing the homogeneous melting of pure noble gases at the limit of superheating. For each chemical species considered, the heterogeneous melting point was estimated by monitoring the thermal behavior of crystalline systems containing a high-angle grain boundary. To determine the limit to superheating, calculations were instead carried out on a perfect crystalline bulk. The temperature was gradually increased to bring the systems within the metastable region above the equilibrium melting point. The static order parameter was employed to monitor the structural disordering during the slow temperature increase and to determine the temperature at which the crystalline lattice collapses to a liquid. Structural disorder was further characterized by studying the appearance of atoms with defective coordination. Their relative number and spatial correlation appeared to play a fundamental role in destabilizing the crystalline lattice bulk and triggering the homogeneous melting. The fraction of atoms with defective coordination and the total length of the stringlike clusters they form in the vicinity of the homogeneous melting point were found to be approximately the same for all of the chemical species considered. These findings have been compared with theoretical predictions.

Cooperative Dynamics and Self-Diffusion in Superheated Crystals

Molecular dynamics simulations have been used to study the atomistic scale dynamics of superheated crystals under different temperature and pressure conditions. The limit of superheating was determined by monitoring a suitable order parameter. The occurrence of homogeneous melting was related to the generation of structural defects characterized by the presence of pairs of particles having defective coordination. At temperatures close to the homogeneous melting point such particles formed extended stringlike clusters. Particles involved in clusters change continuously as a result of local structural rearrangements. These can result in the displacement of particles from one lattice site to another, thus providing a mechanism for self-diffusion.