Lindemann Criterion Research Articles

An extension of atomic mean square displacement method for calculating melting temperatures in II-VI compounds

Understanding how solids melt and determining their melting temperatures is of great significance for studying the properties of materials. Based on the main idea of Lindemann's melting criterion and the first-principles calculation of density functional theory, we proposed the atomic mean square displacement method to predict the melting temperature of the material. In this paper, the application range of this method for calculating melting temperature is extended. 8 kinds of Ⅱ-Ⅵ compounds were selected as verification objects. The results show the accuracy of our method in predicting the melting temperature of Ⅱ-Ⅵ compounds.

Investigation of the Lindemann criterion and the melting point in complex alloys

Investigation of the Lindemann criterion and the melting point in complex alloys

Electron-ion relaxation times in 1–100 eV warm dense aluminum and gold

We calculated the electron-ion energy relaxation times in warm dense aluminum and gold. Our calculations span a wide temperature range from a thousand to a million kelvins for both dense aluminum and gold. To determine the temporal evolution of electron and ion temperatures, we employed the two-temperature model with the SESAME equation-of-state tables, and well-known electron-ion coupling factors. Our calculations indicate that the relaxation times for warm dense aluminum were within 5 ps, whereas they were within 20 ps for warm dense gold. The maximum relaxation times were observed at initial electron temperatures of approximately 30,000 K for both aluminum and gold. Our calculated relaxation times using the SESAME database agree very well with the existing theories both in the cold-solid and hot-plasma limits. Furthermore, we found that the melting times calculated using the Lindemann criterion were in good agreement with the melting times for gold from measurements in recent literature.

Lindemann ratio for classical and quantum crystals

Based on the delocalized of melting criterion, a relatively simple analytical (i.e., without computer simulation) method for calculating the Lindemann ratio is proposed. It is shown that for classical single-component solids (in which the melting point (Tm) is greater than the Debye temperature (Θ): Tm/Θ > 1.5), the Lindemann ratio is determined only by the packing coefficient of the structure. Calculations for various structures of classical solids (both crystalline and amorphous) showed good agreement with the estimates of other authors. For quantum single-component crystals (in which Tm/Θ < 0.4), the Lindemann ratio is determined not only by the crystal structure, but also by the Θ/Tm function. Therefore, when passing from the classical to the quantum area, the Tm(Θ) function changes its functional dependence. It was shown that for quantum crystals, the Lindemann ratio decreases with increasing pressure along the melting line. For quantum nanocrystals, the Lindemann ratio increases with an isobaric decrease in a nanocrystal size. At this, the more noticeably the shape of the nanocrystal deviates from the energy-optimal shape, the greater the sized increase in the Lindemann ratio. Therefore, the use of the Lindemann criterion to study the melting of quantum crystals (as they tried to when studying the melting of atomic metallic hydrogen) showed incorrect results.

Revisiting Lindemann’s criterion from a minimal viscosity perspective

A revisiting of the Lindemann criterion under the recently established minimal viscosity formulation is proposed which uncovers intriguing insights into the melting process. The approach suggests that melting involves competition between the electron rest energy and the thermal one caused by temperature increase when considering a defect-free crystal structure. However, when accounting for the role of vacancies in the melting process, the analysis suggests that the key competing factors are the thermal energy and the one needed for vacancy migration. The estimated value of the Lindemann constant, in this case, is close to the reported values and with recent simulations studying the correlation between Born and Lindemann melting criteria.

Glass spectrum, excess wing phenomenon, and master curves in molecular glass formers: A multi-method approach.

The relaxation spectra of glass formers solely displaying an α-peak and excess wing contribution collected by various methods are reanalyzed to pin down their different spectral evolution. We show that master curve construction encompassing both α-peak and emerging excess wing works for depolarized light scattering (DLS) and nuclear magnetic resonance (NMR) relaxometry. It reveals the self-part of the slow dynamics' spectrum. Master curves are to be understood as a result of a more extensive scaling covering all temperatures instead of strict frequency-temperature superposition. DLS and NMR display identical relaxation spectra; yet, comparing different systems, we do not find a generic structural relaxation at variance with recent claims. Dielectric spectroscopy (DS) spectra show particularities, which render master curve construction obsolete. The DS α-peak is enhanced or suppressed with respect to that of DLS or NMR, yet, not correlated to the polarity of the liquid. Attempting to single out the excess wing from the overall spectrum discloses a stronger exponential temperature dependence of its amplitude compared to that below Tg and a link between its exponent and that of the fast dynamics' spectrum. Yet, such a decomposition of α-peak and excess wing appears to be unphysical. Among many different glasses, the amplitude of the excess wing power-law spectrum is found to be identical at Tg, interpreted as a relaxation analog to the Lindemann criterion.

Effect of composition and architecture on the thermodynamic behavior of AuCu nanoparticles.

The chemical and physical properties of nanomaterials ultimately rely on their crystal structures, chemical compositions and distributions. In this paper, a series of AuCu bimetallic nanoparticles with well-defined architectures and variable compositions has been addressed to explore their thermal stability and thermally driven behavior by molecular dynamics simulations. By combination of energy and Lindemann criteria, the solid-liquid transition and its critical temperature were accurately identified. Meanwhile, atomic diffusion, bond order, and particle morphology were examined to shed light on thermodynamic evolution of the particles. Our results reveal that composition-dependent melting point of AuCu nanoparticles significantly departs from the Vegard's law prediction. Especially, chemically disordered (ordered) alloy nanoparticles exhibited markedly low (high) melting points in comparison with their unary counterparts, which should be attributed to enhancing (decreasing) atomic diffusivity in alloys. Furthermore, core-shell structures and heterostructures demonstrated a mode transition between the ordinary melting and the two-stage melting with varying Au content. AuCu alloyed nanoparticles presented the evolution tendency of chemical ordering from disorder to order before melting and then to disorder during melting. Additionally, as the temperature increases, the shape transformation was observed in AuCu nanoparticles with heterostructure or L10 structure owing to the difference in thermal expansion coefficients of elements and/or of crystalline orientations. Our findings advance the fundamental understanding on thermodynamic behavior and stability of metallic nanoparticles, offering theoretical insights for design and application of nanosized particles with tunable properties.

Dynamics of rotlet’s ensemble

We investigate a two-dimensional ensemble of spinners driven by a rotating magnetic field. We demonstrate that in the absence of any interactions except hydrodynamic forces, the ensemble remains disordered. However, the introduction of a repulsive interaction between the spinners induces ordering within the ensemble. Through numerical simulations, we observe that this ordering resembles a freezing phenomenon and can be characterized using a Lindemann criterion. The freezing occurs because during the ordering of the ensemble, the velocity fluctuations caused by the hydrodynamic interactions decay substantially.

Melting of electronic/excitonic crystals in 2D semiconductor moiré patterns: A perspective from the Lindemann criterion

Using the Lindemann criterion, we analyzed the quantum and thermal melting of electronic/excitonic crystals recently discovered in two-dimensional (2D) semiconductor moiré patterns. We show that the finite 2D screening of the atomically thin material can suppress (enhance) the inter-site Coulomb (dipolar) interaction strength, thus inhibits (facilitates) the formation of the electronic (excitonic) crystal. Meanwhile, a strong enough moiré confinement is found to be essential for realizing the crystal phase with a wavelength near 10 nm or shorter. From the calculated Lindemann ratio which quantifies the fluctuation of the site displacement, we estimate that the crystal will melt into a liquid above a critical temperature ranging from several tens Kelvin to above 100 K (depending on the system parameters).

Approach to hyperuniformity in a metallic glass-forming material exhibiting a fragile to strong glass transition

We investigate a metallic glass-forming (GF) material (Al90Sm10) exhibiting a fragile-strong (FS) glass-formation by molecular dynamics simulation to better understand this highly distinctive pattern of glass-formation in which many of the usual phenomenological relations describing relaxation times and diffusion of ordinary GF liquids no longer apply, and where instead genuine thermodynamic features are observed in response functions and little thermodynamic signature is exhibited at the glass transition temperature, Tg. Given the many unexpected similarities between the thermodynamics and dynamics of this metallic GF material with water, we first focus on the anomalous static scattering in this liquid, following recent studies on water, silicon and other FS GF liquids. We quantify the “hyperuniformity index” H of our liquid, which provides a quantitative measure of molecular “jamming”. To gain insight into the T-dependence and magnitude of H, we also estimate another more familiar measure of particle localization, the Debye–Waller parameter 〈u2〉 describing the mean-square particle displacement on a timescale on the order of the fast relaxation time, and we also calculate H and 〈u2〉 for heated crystalline Cu. This comparative analysis between H and 〈u2〉 for crystalline and metallic glass materials allows us to understand the critical value of H on the order of 10–3 as being analogous to the Lindemann criterion for both the melting of crystals and the “softening” of glasses. We further interpret the emergence of FS GF and liquid–liquid phase separation in this class of liquids to arise from a cooperative self-assembly process in the GF liquid.Graphical abstract

The influence of boundary scattering phase shift on melting behavior of nanoparticles

Based on the Lindemann melting criterion and the lattice dynamical approach, we analytically estimated the melting temperature of nanoparticles with emphasis on the influence of the boundary scattering phase shift parameter α . With the decrease of size, the estimated relation shows that the melting temperature of free standing like nanoparticles will decrease, and the melting temperature of embedded nanoparticles will increase. The increase or decrease of the melting temperature depends on the boundary scattering phase shift parameter. The gradual change of parameter α from − 1 to 1 corresponds to the transition of the surface or environment of nanoparticles from the free boundary condition to the fixed boundary condition. Our analytical estimation qualitatively explains the experimental data, and its predictive capability will help the design of nanoparticles. • Based on the Lindemann criterion, An analytical estimation for size-dependent melting temperature of nanoparticles by lattice dynamical method will be given. • Our theoretical result can fit experimental data of nanoparticles in different environments qualitatively. • A simple boundary scattering phase shift is used to characterize the environment of nanoparticles.

Thermal expansion and the glass transition

Melting is well understood in terms of the Lindemann criterion, which essentially states that crystalline materials melt when the thermal vibrations of their atoms become so vigorous that they shake themselves free of the binding forces. This picture does not necessarily have to hold for glasses, where the nature of the solid–liquid cross-over is highly debated. The Lindemann criterion implies that the thermal expansion coefficients of crystals are inversely proportional to their melting temperatures. Here we find that, in contrast, the thermal expansion coefficient of glasses decreases more strongly with increasing glass temperature, which marks the liquid–solid cross-over in this material class. However, this proportionality returns when the thermal expansion coefficient is scaled by the fragility, a measure of particle cooperativity. Therefore, for a glass to become liquid, it is not sufficient to simply overcome the interparticle binding energies. Instead, more energy must be invested to break up the typical cooperative particle network that is common to glassy materials. The thermal expansion coefficient of the liquid phase reveals similar anomalous behaviour and is universally enhanced by a constant factor of approximately 3. These universalities allow the estimation of glass temperatures from thermal expansion and vice versa.

Ab initio calculation of hafnium and zirconium melting curves via the Lindemann criterion

In this work we present the melting curves of hafnium and zirconium obtained using quantum molecular dynamics calculations. The mean-square displacements computed during ab initio simulations of a crystal phase are used to reconstruct the melting curve according to the Lindemann criterion. The resulting Zr melting curve shows a steeper slope in the low-pressure region compared to some recent diamond-anvil cell experiments but agrees with our previous estimate via the Clausius-Clapeyron relation. The slope for higher pressures is consistent with the experimental one. As for Zr, the Hf melting curve also has a similar steep initial slope being in agreement with our estimate from the Clausius-Clapeyron relation. Currently there are no data on the melting of Hf above atmospheric pressure, so we demonstrate the first ab initio estimate of the melting curve of Hf up to 225 GPa.

Photoexcitation-Induced Nonthermal Ultrafast Loss of Long-Range Order in GeTe.

Nonadiabatic quantum molecular dynamics is used to investigate the evolution of GeTe photoexcited states. Results reveal a photoexcitation-induced picosecond nonthermal path for the loss of long-range order. A valence electron excitation threshold of 4% is found to trigger local disorder by switching Ge atoms from octahedral to tetrahedral sites and promoting Ge-Ge bonding. The resulting loss of long-range order for a higher valence electron excitation fraction is achieved without fulfilling the Lindemann criterion for melting, therefore utilizing a nonthermal path. The photoexcitation-induced structural disorder is accompanied by charge transfer from Te to Ge, Ge-Te bonding-to-antibonding, and Ge-Ge antibonding-to-bonding change, triggering Ge-Te bond breaking and promoting the formation of Ge-Ge wrong bonds. These results provide an electronic-structure basis to understand the photoexcitation-induced ultrafast changes in the structure and properties of GeTe and other phase-change materials.

Revisiting the melting curves of vanadium and niobium metals under pressure

The Lindemann criterion is adjusted to revisit melting curves of vanadium and niobium metals under pressure thanks to the statistical moment method (SMM) in statistical mechanics. The equation-of-state and atomic mean-square displacement in SMM approach are applied to implement numerical calculations for vanadium and niobium metals up to 250 GPa and 150 GPa, respectively. The obtained theoretical melting curves are compared with results of previous experiments and molecular dynamics (MD) simulations where possible to check the theory. We expose that our theoretical melting line of vanadium is overall in good agreement with previous results determined by in situ X-ray diffraction (XRD) and diffuse scattering signals using synchrotron XRD together with shock experiments up to pressure 250GPa. For niobium, our melting curve follows very well with those of recent experimental XRD data as well as ab initio Z method up to 124 GPa. However, we also observe the difference of classical MD simulation results from SMM calculations at pressure below 2.5 GPa. This research enriches the high-pressure melting point database as well as provides a relatively simple approach with low computational cost to estimate high-pressure melting points of materials.

Theoretical prediction of melting curves of gold and silver up to pressure 150 GPa

We investigate melting curves of gold and silver thanks to the statistical moment method in quantum statistical mechanics and the modified Lindemann melting criterion. Numerical calculations have been performed for face-centered cubic (fcc) gold and silver metals up to pressure 150 GPa. Our results are compared with available experimental data and ab initio molecular dynamics simulations where possible. We show that theoretical melting line of gold goes along with recent X-ray diffraction measurements reported by Weck et al. up to pressure more than 110GPa. For silver, our work virtually coincides with the quantum molecular dynamics melting curve which is a combination of fcc and body-centered cubic (bcc) segments freshly implemented by Baty et al. At the pressure 94.2 GPa of the predicted fcc–bcc-liquid triple point of silver, our calculated melting temperature is 4302 K that is only about 2% difference from the prediction of Baty et al. This research proposes a relatively simple approach made from the modified Lindemann criterion and the moment method for predicting melting points of materials at high pressure.

The phase diagrams of beryllium and magnesium oxide at megabar pressures

We perform ab initio simulations of beryllium (Be) and magnesium oxide (MgO) at megabar pressures and compare their structural and thermodynamic properties. We make a detailed comparison of our two recently derived phase diagrams of Be (Wu et al 2021 Phys. Rev. B 104 014103) and MgO (Soubiran and Militzer 2020 Phys. Rev. Lett. 125 175701) using the thermodynamic integration technique, as they exhibit striking similarities regarding their shape. We explore whether the Lindemann criterion can explain the melting temperatures of these materials through the calculation of the Debye temperature at high pressure. From our free energy calculations, we find that the melting line of both materials is well represented by the Simon–Glazel fit T m(P) = T 0(1 + P/a)1/c , where T 0 = 1564 K, a = 15.8037 GPa and c = 2.4154 for Be, while T 0 = 3010 K, a = 10.5797 GPa and c = 2.8683 for the MgO in the B1. For the B2 phase, we use the values a = 26.1163 GPa and c = 2.2426. Both materials exhibit negative Clapeyron slopes on the boundaries between the two solid phases that are strongly affected by anharmonic effects, which also influence the location of the solid–solid–liquid triple point. We find that the quasi-harmonic approximation underestimates the stability range of the low-pressure phases, namely hcp for Be and B1 for MgO. We also compute the phonon dispersion relations at low and high pressure for each of the phases of these materials, and also explore how the phonon density of states is modified by temperature. Finally, we derive secondary shock Hugoniot curves in addition to the principal Hugoniot curve for both materials, and study their offsets in pressure between solid and liquid branches.

Optically and thermally driven huge lattice orbital and spin angular momenta from spinning fullerenes

Lattice vibration in solids may carry angular momentum. But unlike the intrinsic spin of electrons, the lattice vibration is rarely rotational. To induce angular momentum, one needs to find a material that can accommodate a twisted normal mode, two orthogonal modes, or excitation of magnons. If excitation is too strong, one may exceed the Lindemann limit, so the material melts. Therefore these methods are not ideal. Here, we theoretically propose a route to phonon angular momentum in a molecular crystal ${\mathrm{C}}_{60}$. We find that a single laser pulse is able to inject a significant amount of angular momentum to ${\mathrm{C}}_{60}$, and the momentum transfer is helicity dependent. Changing from right-circularly polarized light to left-circularly polarized light switches the direction of phonon angular momentum. On the ultrafast timescale, the orbital angular momentum change closely resembles the displacive excitation of coherent phonons, with a cosine-function dependence on time, different from the spin counterpart. Atomic displacements, even under strong laser excitation, remain far below the Lindemann criterion. Under thermal excitation, spinning ${\mathrm{C}}_{60}$ even at room temperature generates a huge angular momentum close to several hundred $\ensuremath{\hbar}$. Our finding opens the door to a large group of fullerenes, from ${\mathrm{C}}_{60},\phantom{\rule{0.16em}{0ex}}{\mathrm{C}}_{70}$ to their endohedral derivatives, where angular momentum can be generated through light or temperature. This paves the way to the phononic control electronic spin and harvesting thermal energy through phonon angular momentum.

The melting curve of gold up to 1500 kbar

The melting temperature Tm of gold has been determined from ambient pressure to 1500 kbar using statistical moment method (SMM) and the Lindemann criterion. The equation of the melting curve obtained is a quadratic polynomial of the melting temperature Tm, with coefficients that are explicitly dependent on pressure P. Simple number calculation and easily verify. Numerical results for the melting temperature of gold up to 1500 kbar are in good agreement with the experimental data and the theoretical results of other authors.

Evolution of the Graphene Layer in Hybrid Graphene/Silicon Carbide Heterostructures upon Heating

The hybrid graphene/SiC model is studied via molecular dynamics simulation to observe the evolution of the graphene layer upon heating. A two-layer model containing 10,000 graphene atoms and 7000 SiC atoms is heated from 50 K to 6000 K via Tersoff and Lennard-Jones potentials. The melting point zone is defined as the temperature range from 4400 K to 4600 K, which is close to the melting zone of graphite in an experiment. The Lindemann criterion for the 2D case is calculated and used to observe the appearance of liquid-like atoms. The evolution upon heating is analyzed on the basis of the occurrence/growth of liquid-like atoms, the radial distribution functions, and the formation of clusters. The liquid-like atoms tend to form clusters, and the largest cluster increases in size slightly to form a single largest cluster of liquid-like atoms.