From First-Order to Two Continuous Melting Transitions: Monte Carlo Study of a New 2D Lattice-Defect Model

Abstract

We present a Monte Carlo study of a new lattice model describing defects and their elastic interactions, including a term that accounts for rotational stiffness over a length scale /. We show that, when going from small / (as in most atomic crystals) to large / (as in many molecular crystals), the melting process changes from a single first-order to two successive Kosterlitz-Thouless transitions.PACS numbers: 64.70.DvThe Kosterlitz-Thouless-Halperin-Nelson-Young suggestion l that two-dimensional melting should proceed via two successive continuous transitions of the Kosterlitz-Thouless type seems to be applicable only to layers of liquid crystals. 2 Experiments on atomic layers loosely absorbed on surfaces, 3 or molecular-dynamic simulations of two-dimensional Lennard-Jones systems, 4 have always been found to undergo only a single first-order melting transition. For high coverage, it may sometimes become continuous, 5 but it has never been seen to split into two successive transitions. The same has been found for all lattice models which simulate an ensemble of crystal defects with long-range elastic interactions of linear elasticity 6 £ c V ) -Jrf 2 «[W4)0,-tty + 8^/) 2 + a/2)(8 I -M I -) 2 ], J where w, are the lattice displacements and n,X are the elastic moduli.Recently, 7 it has been shown that the characteristic parameter which specifies the difference between atomic crystals and those formed by liquid crystals, namely, the length scale of rotational stiffness, /, is responsible for the different melting processes. It appears in the second gradient elastic interaction E%r = 2fj,l 2 fd 2 %(di(a) 2 , where <o= j 67/d/Wy. If the combined energy is placed on a lattice of spacing a, with the gradients d,-substituted by \/a times lattice differencesif, further, displacements «,E ( -a/2,a/2) are scaled to yi-lnuifa E ( -n,n) and co to j €tj\iYj, an d if discrete-valued plastic distortions are inserted to supply the Volterra jumping surfaces of dislocations and disclinations, the ensuing model energy reads (in natural units in which fia 2 /4x 2 = \, and lattice spacing a = 1) £-TX(V/yy+V yy/ -4^) 2 + ^Z x,/y 2/i-Z(V I -r < -2CT/ < )l 2 + 2/ 2 £{V,CB-«[2/H, + V/0ti2-B2l)H 2 .(2) summed over all integer nij [nfj = (njj + nji)/2] and m,, was shown 7 to possess, for very small /, a single first-order melting transition and, for very large /, two successive Kosterlitz-Thouless transitions (at least for v = l). A few manipulations 7 on the partition function lead to two alternative representations. One involves the energy ££ r =(l//?)£ x (iUl/(l + v^ (3)with v =ss X/(2fi+X) ^Poisson ratio, and h,Ai being integer-valued stress gauge fields (to be summed over in the partition function). The other, dual representation is 0£j=j3J4;r 2 (l + v)X x 77(V-V)-2^ (4)where b\ and 0 are integer-valued dislocation and disclination densities, and T] = €ijVibj + 0 is the so-called defect density. The partition function involving f3E r is a generalization of the Laplacian roughening (LR) model 8 0£LR = n/4/3(l + v)]Xx(V*V/j) 2 to which ...

Tetratic Phase in the Planar Hard Square System?

“…It may be interesting to perform analogous studies for the system of hard triangles. If the tetratic phase of the hard squares is induced by the molecular anisotropy, as it is suggested by the model discussed by Kleinert and Janke [33,34], then the triangle system (as consisted of more anisotropic particles) should exhibit a hexatic phase. Otherwise, e.g.…”

“…In the free energy expansion he considered an additional term which he related to molecular anisotropy [33]. Janke and Kleinert [34] studied this model and observed either a single first-order transition or two continuous transitions, depending on the value of a coefficient at the new term. It is not obvious, however, if and when the expansion proposed by Kleinert is applicable to real systems.…”

Nematic world crystal model of gravity explaining absence of torsion in spacetime

“…[12], without a simultaneous condensation of disclinations in a first-order melting transition, the model requires a modification by an additional highergradient rotational energy. This was shown in [13] and verified by Monte Carlo simulations in [14]. The threedimensional extension of the extended model is described in [6].…”

Melting of Wigner-like lattice of parallel polarized dipoles

“…For high ℓ, on the other hand, beginning about with the lattice spacing a 0 , the disclinations could be suppressed with the consequence that the transition based on disclination unbinding would occur later than that of dislocation unbinding. The precise location of the critical ℓ was found by computer simulations [6], and is plotted in Fig. 1.…”

“…For small ℓ 2 , the transition is of first order, for ℓ 2 1 it splits into two Kosterlitz-Thouless transitions. The simulation data are from Ref [6]…”