Lindemann Criterion Research Articles

Use of polydispersity index as control parameter to study melting/freezing of Lennard-Jones system: Comparison among predictions of bifurcation theory with Lindemann criterion, inherent structure analysis and Hansen-Verlet rule

Using polydispersity index as an additional order parameter we investigate freezing/melting transition of Lennard-Jones polydisperse systems (with Gaussian polydispersity in size), especially to gain insight into the origin of the terminal polydispersity. The average inherent structure (IS) energy and root mean square displacement (RMSD) of the solid before melting both exhibit quite similar polydispersity dependence including a discontinuity at solid-liquid transition point. Lindemann ratio, obtained from RMSD, is found to be dependent on temperature. At a given number density, there exists a value of polydispersity index (δ P) above which no crystalline solid is stable. This transition value of polydispersity(termed as transition polydispersity, δ P ) is found to depend strongly on temperature, a feature missed in hard sphere model systems. Additionally, for a particular temperature when number density is increased, δ P shifts to higher values. This temperature and number density dependent value of δ P saturates surprisingly to a value which is found to be nearly the same for all temperatures, known as terminal polydispersity (δ TP). This value (δ TP∼ 0.11) is in excellent agreement with the experimental value of 0.12, but differs from hard sphere transition where this limiting value is only 0.048. Terminal polydispersity (δ TP) thus has a quasiuniversal character. Interestingly, the bifurcation diagram obtained from non-linear integral equation theories of freezing seems to provide an explanation of the existence of unique terminal polydispersity in polydisperse systems. Global bond orientational order parameter is calculated to obtain further insights into mechanism for melting.

Unified Criterion for Temperature-Induced and Strain-Driven Glass Transitions in Metallic Glass.

In a model metallic glass, we study the relaxation dynamics in both the linear and the nonlinear response regimes by numerical simulations of dynamical mechanical spectroscopy and analyze the atomic displacement statistics. We find that the primary (α) relaxation always takes place when the most probable atomic displacement reaches a critical fraction (~20%) of the average interatomic distance, irrespective of whether the relaxation is induced by temperature (linear response) or by mechanical strain (nonlinear response). Such a unified scenario, analogous to the well-known Lindemann criterion for crystal melting, provides insight into the structural origin of the strain-induced glass-liquid transition.

Volume and pressure-dependent thermodynamic properties of sodium

In this work, the volume and pressure effects on thermodynamic quantities including Grüneisen parameter and melting temperature of sodium have been studied of to 65 GPa. The volume dependence of Grüneisen parameter of sodium has been firstly analyzed. Based on this result and the Lindemann criterion of melting we derived the analytical expression of volume-dependent melting temperature. Combining with Vinet equation-of-state, the melting curve of sodium at high pressure has been proposed. Numerical calculations were performed for solid sodium up to pressure 65 GPa. Theoretical calculations are compared with available experimental data showing the good and reasonable agreements in the pressure range 0–30 GPa. This research proposes the good volume- and pressure-dependent Grüneisen parameter as well as the potential of Lindemann criterion approach on predicting high-pressure melting of materials.

Mode couplings and resonance instabilities in finite dust chains.

Employing a numerical simulation, the normal modes are investigated for finite, one-dimensional horizontal dust chains in complex plasma. Mode couplings induced by the ion flow within the sheath are identified in the mode spectra and the coupling rules are determined. Two types of resonance-induced instabilities are observed, one bidirectional and one unidirectional. Bidirectional instability is found to cause melting of the chain with the melting proceeding via a two-step process which obeys the Lindemann criterion. The relationship between the normal mode spectra observed in finite systems and the wave dispersion relations seen in larger systems was also examined using a dust chain model. For this case, the dispersion relation was obtained through multiplication of the mode spectra matrix by a transition matrix. The resulting dispersion relations exhibit both the general features observed in larger crystals as well as several characteristics unique to finite systems, such as discontinuities and strong energy-density fluctuations.

Instability of insulating states in optical lattices due to collective phonon excitations

The role of collective phonon excitations on the properties of cold atoms in optical lattices is investigated. These phonon excitations are collective excitations, whose appearance is caused by intersite atomic interactions correlating the atoms, and they do not arise without such interactions. These collective excitations should not be confused with lattice vibrations produced by an external force. No such a force is assumed. But the considered phonons are purely self-organized collective excitations, characterizing atomic oscillations around lattice sites, due to intersite atomic interactions. It is shown that these excitations can essentially influence the possibility of atoms to be localized. The states that would be insulating in the absence of phonon excitations can become delocalized when these excitations are taken into account. This concerns long-range as well as local atomic interactions. To characterize the region of stability, the Lindemann criterion is used.

Dynamical properties and melting of binary two-dimensional colloidal alloys.

A two-dimensional (2D) binary colloidal system consisting of interacting dipoles is investigated using an analytical approach. Within the harmonic approximation we obtain the phonon spectrum of the system as a function of the composition, dipole-moment ratio, and mass ratio between the small and big particles. Through a systematic analysis of the phonon spectra we are able to determine the stability region of the different lattice structures of the colloidal alloys. The gaps in the phonon frequency spectrum, the optical frequencies in the long-wavelength limit, and the sound velocity are discussed as well. Using the modified Lindemann criterion and within the harmonic approximation we estimate the melting temperature of the sublattice generated by the big particles.

Molecular Dynamics Simulation on Liquid Nucleation in Ideal Silicon Crystal upon Melting

Liquid nucleation in an ideal silicon crystal is studied in the framework of Lindemann criteria by using molecular dynamics simulations. Lindemann parameter is traced during the transition. It is found that when Lindemann parameter reaches a critical value δL*=0.328, liquid emerges, and when 9.4% atoms in the crystal exceeds the critical value, a complete melting is achieved.

Systematic prediction of high-pressure melting curves of transition metals

The pressure effects on melting temperatures of transition metals have been studied based on the combination of the modified Lindemann criterion with statistical moment method in quantum statistical mechanics. Numerical calculations have been performed for five transition metals including Cu, Pd, Pt, Ni, and Mn up to pressure 100 GPa. Our results are in good and reasonable agreements with available experimental data. This approach gives us a relatively simple method for qualitatively calculating high-pressure melting temperature. Moreover, it can be used to verify future experimental and theoretical works. This research proposes the potential of the combination of statistical moment method and the modified Lindemann criterion on predicting high-pressure melting of materials.

Melting of solids under high pressure

The pressure effects on melting curves of solids have been studied by combining Lindemann criterion of melting with pressure-dependent Grüneisen parameter. The analytical expression of melting temperature Tm as a function of pressure has been derived. Numerical calculations were performed for some solids including ε-Fe up to pressure 350 GPa and Cu up to pressure 100 GPa. Theoretical calculations are compared with available experimental data showing the good and reasonable agreements. This research proposes the potential of Lindemann criterion approach on predicting high-pressure melting of materials.

Quantum hexatic order in two-dimensional dipolar and charged fluids

Recent advances in cold atom experimentation suggest that studies of quantum two-dimensional melting of dipolar molecules, with dipoles aligned perpendicular to ordering plane, may be on the horizon. An intriguing aspect of this problem is that two-dimensional classical aligned dipoles (already studied in great detail in soft matter experiments on magnetic colloids) are known to melt via a two-stage process, with an intermediate hexatic phase separating the usual crystal and isotropic fluid phases. We estimate here the effect of quantum fluctuations on this hexatic phase, for both dipolar systems and charged Wigner crystals. Our approximate phase diagrams rely on a pair of Lindemann criteria, suitably adapted to deal with the effects of thermal fluctuations in two dimensions. As part of our analysis, we determine the phonon spectra of quantum particles on a triangular lattice interacting with repulsive $1/{r}^{3}$ and $1/r$ potentials. A large softening of the transverse and longitudinal phonon frequencies, due to both lattice effects and quantum fluctuations, plays a significant role in our analysis. The hexatic phase is predicted to survive down to very low temperatures.

A new and effective method for thermostatting confined fluids

We present a simple thermostatting method suitable for nanoconfined fluid systems. Two conventional strategies involve thermostatting the fluid directly or employing a thermal wall that couples only the wall atoms with the thermostat. When only a thermal wall is implemented, the temperature control of the fluid is true to the actual experiment and the heat is transferred from the fluid to the walls. However, for large or complex systems it can often be computationally prohibitive to employ thermal walls. To overcome this limitation many researchers choose to freeze wall atoms and instead apply a synthetic thermostat to the fluid directly through the equations of motion. This, however, can have serious consequences for the mechanical, thermodynamic, and dynamical properties of the fluid by introducing unphysical behaviour into the system [Bernardi et al., J. Chem. Phys. 132, 244706 (2010)]. In this paper, we propose a simple scheme which enables working with both frozen walls and naturally thermostatted liquids. This is done by superimposing the walls with oscillating particles, which vibrate on the edge of the fluid control volume. These particles exchange energy with the fluid molecules, but do not interact with wall atoms or each other, thus behaving as virtual particles. Their displacements violate the Lindemann criterion for melting, in such a way that the net effect would not amount to an additional confining surface. One advantage over standard techniques is the reduced computational cost, particularly for large walls, since they can be kept rigid. Another advantage over accepted strategies is the opportunity to freeze complex charged walls such as β-cristobalite. The method furthermore overcomes the problem with polar fluids such as water, as thermalized charged surfaces require higher spring constants to preserve structural stability, due to the effects of strong Coulomb interactions, thus inevitably degrading the thermostatting efficiency.

A criterion for the glass–liquid transition

Criterion for softening of glassy solids, analogous to the Lindemann criterion of melting, is proposed. At glass softening the temperature ratio of root mean square atom displacement to mean interatomic distance is practically of constant value for different glasses. It is assumed that, when the root mean square amplitude of the thermal vibrations of an atom reaches a certain fraction of an interatomic distance, the excited atom loses its stability and the glass–liquid transition begins. The softening process in glasses as in crystals is due to the emergence of vibrational instability of atoms located in the lattice sites.

On the behavior of indicators of melting: Lennard-Jones system in the vicinity of the phase transition

Various indicators of melting for a system of particles whose pair interaction is described by the Lennard-Jones potential have been considered. The behavior of the radial distribution function g(r) and the associated criteria of melting, modified Lindemann criterion, and criteria based on the properties of short-range orientational order (rotational invariants ql and wl of various orders l) has been analyzed in detail in the vicinity of the melting phase transition. A parameter based on the loss of the nearest neighbors of an atom/particle has been proposed to characterize the melting transition. All considered indicators of melting for the Lennard-Jones system have been compared. It has been shown that the indicators of melting derived from the properties of the short-range orientational order are much more sensitive to the melting phase transition and can be used to construct new phenomenological criteria of melting similar to the Lindemann criterion. An additional important advantage of such indicators is the relatively small number of configurations of the system necessary for their calculation.

Mode couplings and resonance instabilities in dust clusters

The normal modes for three to seven particle two-dimensional (2D) dust clusters in a complex plasma are investigated using an N-body simulation. The ion wakefield downstream of each particle is shown to induce coupling between horizontal and vertical modes. The rules of mode coupling are investigated by classifying the mode eigenvectors employing the Bessel and trigonometric functions indexed by order integers (m, n). It is shown that coupling only occurs between two modes with the same m and that horizontal modes having a higher shear contribution exhibit weaker coupling. Three types of resonances are shown to occur when two coupled modes have the same frequency. Discrete instabilities caused by both the first and third type of resonances are verified and instabilities caused by the third type of resonance are found to induce melting. The melting procedure is observed to go through a two-step process with the solid-liquid transition closely obeying the Lindemann criterion.

Specific heat capacity of nanoporous Al2O3

Based on Lindemann's criterion, a specific heat capacity model for nanoporous material was proposed by defining the surface-atom layer, to take the surface atoms and the volume atoms separately into account. The height of the surface-atom layer was determined from the experiment, and results show that only the first layer atoms on the surface should be separately considered for nanoporous Al2O3. The shape factor of the pore was also introduced in the model with values between 2 (for cylindrical pore) and 3 (for spherical pore) to characterize the morphology of the pore. It turns out experimentally that the specific heat capacity of the analyzed nanoporous Al2O3 is much larger than that of the bulk, which can be interpreted as due to the fact that the surface atom plays a more important role than the volume one. And the smaller the radius and/or the larger the porosity, which lead to a larger surface-volume ratio, the larger the specific heat capacity becomes. The nanoporous material could be a better heat storage medium than the corresponding bulk with a much lighter weight, smaller volume but higher heat storage capacity.

Vortex matter in the type-II superconductor La3Ni2B2N3−x in the light of NMR

A quantitative analysis of spin-echo NMR measurements of the local field distribution and the spin–spin relaxation time T2 in the field-cooled superconducting state of the type-II superconductor La3Ni2B2N3−x shows evidence for two fluctuation modes of the weakly pinned vortex matter dominating T2 at different temperatures: long wavelength diffusive modes in the vortex solid at low temperature, and plastic deformation of the viscous liquid close to the critical temperature Tc, with a broad crossover region in between where a more complex theory is required. We determine the melting temperature Tm = 11.5 K of the vortex matter as the temperature where the characteristic time of the plastic deformation slows down to the time scale set by the spin-echo experiment. This is preferred to the conventional estimates from a Lindemann criterion for the melting vortex solid since the quantitative description of T2 near Tm by the viscous vortex liquid appears to be more consistent. It indicates that the plastic deformation mode might be important for T2 in the mixed state of other type-II superconductors in the weak pinning regime near Tc as well.

Melting of graphene clusters

Density-functional tight-binding and classical molecular dynamics simulations are used to investigate the structural deformations and melting of planar carbon nanoclusters C${}_{N}$ with $N=2$--55. The minimum-energy configurations for different clusters are used as starting configurations for the study of the temperature effects on the bond breaking and rotation in carbon lines ($N<6$), carbon rings ($5<N<19$), and graphene nanoflakes. The larger the rings (graphene nanoflakes) the higher the transition temperature (melting point) with ring-to-line (perfect-to-defective) transition structures. The melting point was obtained by using the bond energy, the Lindemann criteria, and the specific heat. We found that hydrogen-passivated graphene nanoflakes (C${}_{N}$H${}_{M}$) have a larger melting temperature with a much smaller dependence on size. The edges in the graphene nanoflakes exhibit several different metastable configurations (isomers) during heating before melting occurs.

Anomalous Temperature Dependence of Crystalline-to-Amorphous Transformation Induced by High-Pressure Torsion in Zr<sub>50</sub>(Cu,Al)<sub>50</sub>

Martensite phases of Zr50Cu50 and Zr50Cu44Al6 were deformed by high-pressure torsion (HPT) at different temperatures ranging from 253 to 473K. Microstructural observations revealed that the crystalline-to-amorphous transformation induced by HPT showed remarkable temperature dependence: an increase in deformation temperature up to 473K significantly increased the fraction of the amorphous structure. This result suggests that the lattice instability near martensitic transformation temperature promotes crystalline-to-amorphous transformation, which is coherent to the generalized Lindemann criterion of defect-induced amorphization. [doi:10.2320/matertrans.M2013049]

Monte Carlo Simulation of Polymer–Nanoparticle Composites

Abstract The structural formation of polymer–nanoparticle composites (PNC), which are also called polymer spherical brushes, is investigated by Monte Carlo simulations. Based on an osmotic model, the solid–liquid transition of PNC systems due to the steric effect between PNCs is studied by employing the Lindemann criterion and the Hansen–Verlet criterion. The phase diagram of PNC systems is also clarified by the free-energy analysis of the system. These results indicate the existence of PNC crystals with sufficiently high density and the large number of polymer chains per each particle.

A Lindemann criterion for the atomization of metals

A thermodynamic variation of the Lindemann criterion for the vaporization of metals is proposed. It is shown that the critical amplitude of vibrations of atoms at the boiling point averages 1.42 bond lengths. Close values of interatomic distances result from the Vinet universal equation for the atomization of metals under the action of high temperatures (1.48) and negative pressures (1.50). The last value corresponds to the Van der Waals distances between metal atoms.