Physica A 205 (1994) 738-746 I North-Holland SSDI 0378-4371(93)E0490-6 Repulsive potentials, clumps and the metastable glass phase W. Klein a, Harvey Gould b, Raphael A. Ramos", I. Clejan a and Andrew I. Mel'cuk b aDepartment of Physics, Boston University, Boston, MA 02215, USA bDepartrnent of Physics, Clark University, Worcester, MA 01610, USA Received 15 September 1993 Revised manuscript received 5 October 1993 We present the results of simulations and theoretical investigations of a system of particles with a weak, long-range repulsive interaction. At fixed density the fluid phase, which is stable at high temperatures, becomes metastable and then unstable as the temperature is lowered. For low temperatures the particles form approximately equal size clumps that interact weakly with neighboring clumps. The global free energy minimum at low temperatures corresponds to a crystalline lattice clump structure. However the free energy surface has many minima associated with metastable amorphous phases. Quenches from the fluid phase to low temperatures almost always will result in an amorphous clump structure with the number of clumps dependent on quench history. The association of the amorphous phase with a large number of metastable minima is similar to a mean-field spin glass. The understanding of the processes of crystallization and glass formation has proven to be a formidable task [1,2]. A high degree of non-linearity and evolution on several length and time scales makes these processes difficult to attack theoretically, experimentally, and by simulations. In addition, glass formation and the crystallization processes often compete with each other, further complicating our ability to understand them [3]. To gain more insight into these processes, we have begun a study of systems with a weak, long-range repulsive interaction, i.e., systems in the mean-field limit. The advantage of studying mean-field systems is that several aspects of the problem can be addressed theoretically. In this paper we report our first results which strongly indicate the presence of a metastable glass phase and a new phase of matter that we call the "clump" phase. We will see that the clumps play a significant role in the formation of the glass phase in these models. We consider a pairwise, radially symmetric Kac potential [4] of the form 0378-4371/94/$07.00 © 1994-Elsevier Science B.V. All rights reservedW. Klein et al. / Repulsive potentials and the metastable glass phase 739 V(r) = ydr(rr) >1 o, (1) where r = Irl, y is a parameter related to the strength and range of the potential, and d is the spatial dimension. In the y ~ 0 limit, it is known [5] that the system can be described exactly by a mean-field theory in which the single particle distribution fu:~tion p~(x) satisfies the equation p,(x) = z exp(-13 f ¢k(lx- x'D o,(x') ar') , (2) where z is the activity, /3-~= kBT, and x = yr. For high temperatures (and z fixed), the solution to (2) is a unique spatial constant [5]. For potentials whose Fourier transform ~(q) is negative for some value of q = q0 (the dimensionless quantity q = k/y), the spatially constant solution of (2) becomes unstable for the smallest value of/3p such that [5] 1 + flpqb(qo) = 0, (3) where ~(q0) is the minimum of q~(q). The number density p is the spatially constant solution of (2). It has been shown [5] that at the critical value /3p = (/3p) o found from the solution of (3), the structure function diverges at q = q0, and hence (3) defines the location of the spinodal. Unless otherwise noted, we will consider the step potential ya x~l YdS'(X) = 0, X>I. (4) It is easy to verify that there is a range of q for which ~s < 0 and that the first minimum of as(q) occurs for q0 = 5.76. Hence a system with the step potential (4) exhibits a spinodal in the mean-field limit y---~0. Our main interest is the structure of the system for the step potential in (4) for/3p > (/3P)0, that is, for temperatures below the spinodal at fixed density. The interaction energy per particle for d = 3 when the system is in a uniform state with density p is E u -- ~'rrp.14 (The factor of ½ accounts for the sharing of the energy of interaction between two particles). Now consider a nonuniform configuration where the particles are grouped into "clumps" with an equal number of particles in each clump. Each particle interacts with all of the particles in the same clump. If there is no interaction between particles in different clumps, the energy of interaction per particle is E c = ½"y3N/Nc, where N is the number of particles in the system, N c is the number of clumps, and N/N c is the mean number of particles in a clump. To estimate No, we calculate the number of clumps that just fit in a box of volume L 3 in a simple cubic (sc)740 W. Klein et al. / Repulsive potentials and the metastable glass phase arrangement. If we take the mean distance between the clumps, (L3/N ]1/3 to be the interaction range y -1, we have N c = L3/T -3, and hence Esc = 1 p, which implies that Esc < E u. The energy per particle for fcc, bcc, and random close packing (rcp) arrangements of the clumps is given by Ef¢¢ = p/2V2~O.35p, Ebc c ~- 2p/(3/V~) ~ 0.38p, and Erc p ~ 0.41p respectively. These qualitative considerations suggest that as T---~ 0, the arrangement of particles into non-interacting clumps will have a lower energy and hence lower free energy than the uniform distribution. This result appears to be true whether the centers of mass of the clumps are arranged in a crystalline configuration or are random. Our conclusion is that the clumps will determine the structure of the system for large ]3p. To test this hypothesis we have performed Monte Carlo simulations on a system of N = 8000 and N = 1000 particles with the step potential (4) with -1 T = 3. The density is fixed at p = 1.95 and periodic boundary conditions are used. For this density, it is easy to show that the spinodal temperature T O given by the solution
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