Lindemann criterion

Finally, if the temperature increases, becomes larger until the crystal melts. The Lindemann criterion predicts that melting sets in when becomes about 0.25 a2, where a is the interatomic distance of the metal. Because the mean squared displacements of surface atoms is higher we expect that the surface melts at lower temperatures than the bulk does [2]. Indeed, evidence has been presented that the (110) surface of lead starts to melt at 560 K whereas the bulk melting temperature is about 600 K [13]. 

A theoretical analysis of the stability of such colloidal crystals of spherical latex particles has been carried out by Marcel ja et al (28.). They employ the Lindemann criterion that a crystal will be stable if the rms thermal displacement of the particles about their equilibrium positions is a small fraction f of the lattice spacing R. Comparison with Monte Carlo simulations shows that f is about 0.1 for "hard crystals, and 0.08 for "soft crystals stabilized by long-ranged electrostatic forces. This latter criterion translates into a critical ratio... 

The constraint in Eq. (38) that enables the direct computation of Tg is obtained by the extension of the Lindemann criterion to the softening transformation in glass-forming liquids [42, 56, 129, 130], and the details of this relation are explained in Section VI. Within the schematic model for glass formation (with specified e, Es, and monomer structure), all calculated thermodynamic properties depend only on temperamre T, on pressure P, and on molar mass Mmoi (which is proportional to the number M of united atom groups in single chains). The present section summarizes the calculations for To, Tg, Ti, and Ta as functions of M for a constant pressure of P = 0.101325 MPa (1 atm). 

Our identification of the Buchenau relation in Eq. (64) with the AG equation for X further implies that the characteristic temperatures of glass formation (Tq, Tg, T, Ta) have their counterparts in characteristic values of (m ) and that the Lindemann criterion (see Section VI) can be generalized to describe successive stages of glass formation. Dividing the arguments of the exponentials defining... 

Zhou YQ, Vitkuo D, Karplus M Native proteins are surface-molten solids Application of the lindemann criterion for the solid versus hquid state. J. Mol. Biol. 1999, 285 1371-1375. 

The critical amplitudes of atom oscillations in condensed helium are close to those in other rare gases. It follows that the Lindemann criterion (see Chap. 6) cannot be used to explain the special features of helium solidification. In this context, note a remarkable similarity of the changes in the properties of MX and Hc2 caused by the transition from molecular to the soUd state along with an increase in we... 

In this section the results obtained from AIMD-B-LYP simulations by energizing three isomers of Li9" clusters will be analyzed [39]. For this purpose three types of quantities will be used. One of them is the root-mean-square (RMS) bond-length fluctuation 8 [25], which is calculated at the end of trajectories. A sharp increase of the 8 value is known as the Lindemann criterion for bulk melting, while in the context of finite-size clusters it can be taken as an indication of transition from solid-like to liquid-like state. We find it particularly instructive to analyze the trajectories in terms of atomic equivalence indexes ,... 

Fig. 4. Averaged relative root mean square bond length fluctuations for the Ar-Ar and Ar-SF distances for certain SF -(Ax) clusters. The shaded band identifies the 8 values chosen to define the Lindemann criterion for melting.

Each of the walls confining the film contains two (111) layers of an fee crystal. The axes are chosen so that the layers lie in the x-y plane and the film thickness is measured along the z-axis. Periodic boundary conditions are imposed in the x-y plane. Each wall atom is tied to a lattice site with a spring. These are stiff enough that thermal displacements are less than the Lindemann criterion for melting (20,27). 

For a simple calculation based on the Lindemann criterion and comparison with experiment see for example Tabor, D. Gases, Liquids and Solids, 1969, Penguin Books, Baltimore, p. 207. The calculation assumes harmonic vibration of the particles, against Young s modulus. Melting temperatures are calculated with reasonable accuracy for metals and even for quartz, but not for organic molecules. 

FIGURE 4.14 (a) Result of the scaling procedure described in the text, (b) Volume fraction dependence of the localization length in units of mean interparticle distances. Dashed horizontal line corresponds to the Lindemann criterion, = 0.1. 

The two glass states differ in their local packing and consequently stiffness. While repulsion localizes particles according to Lindemann criterion, bond formation allows particles less local free volume. Figure 7.8 shows the localization lengths (left panel), calculated from the MSD at infinite time, along the transition lines in Figure 7.7. 

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