Towards understanding the local structure of liquids

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PHYSICS

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Physics Reports 288 (1997) 409-434

Towards understanding

the local structure of liquids

Alexander Z. Patashinskia,b, Antoni C. Mitusc,d, Mark A. Rafnere ofPhysics and Astronomy and Materials Research Center, Northwestern University,

aDepartment

’ Depurtment

Evanston IL 60201, USA bBudker Institute I$ Nuclear Physics, RAS, Novosibirsk, Russia ‘Institute qf Theoretical Physics, Technicul University, Braunschweig, Germany ‘Institute ofPhysics, Technical University, Wroclaw, Polund qf Chemistry and Materials Research Center, Northwestern University, Evanston, IL 60201, USA

Abstract In this article we discuss the problem of well-defined crystalline patterns of local atomic arrangements in equilibrium liquids, and their statistical mechanics modelling. We present arguments in favor of the existence of local crystalline structures in liquids (local crystal order hypothesis) and discuss a generalized energy landscape picture in the theory of the liquid state. This picture allows a quantification of the hypothesis of local order and offers basic concepts for the statistical mechanics modelling of the melting phase transition. We review recent results of probabilistic-based searches for local structures in various two- and three-dimensional computer-simulated liquids. Next, some statistical-mechanics models of melting and amorphization in terms of structural states of small clusters are proposed. The models, which have only two characteristic energies, that of the orientationally disordered locally crystalline state, and that of completely amorphous state, are studied in a mean-probability approximation. If the amorphization energy is high, the material retains local crystallinity even in the melt; at higher temperatures a crossover to the locally amorphous state occurs. A material that has a low energy non-crystalline local packing exhibits an amorphization melting; the phase transition is from orientationally ordered crystal state to a locally amorphous melt. PACS:

61.20.-p

Keywords;

Melting; Local structure;

Structural invariants

1. Outline of the problem Melting and solidification play an important role in materials technology. These phase transitions (PT) are intensively studied. While melting is really a universal phenomenon, and computer simulations demonstrate clearly that melting will occur even in simple Lennard-Jones systems, understanding of how melting occurs, what changes in structure are associated with the melting phenomenon, and how the nature of the interparticle interaction is responsible for these structural changes remains poor. 0370-1573/‘97/$32.00 Copyright PII so370-I 573(97)00035-5

0 1997 Elsevier Science B.V. All rights reserved

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Fig. I. Phase diagram of a classical one-component respectively.

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system (see text). TP and CP denote triple point and critical point.

A typical pressure-temperature (p-7’) phase diagram of a one-component classical system is shown schematically in Fig. 1. The phase transitions crystal-liquid (melting), crystal-gas (sublimation) and liquid-gas are of a first Ehrenfest order, i.e. they are accompanied by discontinuous changes of density, entropy, and other materials thermodynamic properties. The line of liquid-gas PT terminates at the critical point. All the three phases coexist in the triple point. In the inserts the typical configurations of the atoms in various phases are shown. While for crystalline solids and gases the typical configurations are well-defined (and correspond to ensembles of phonons and weakly non-ideal gas, respectively), the case of liquid is quite different. The atoms perform complicated movements in time (oscillations, diffusion, jumps, etc.) and a configuration cannot be parameterized in the same way as in a crystal. In equilibrium and supercooled liquids, there is no long-range order; the concept of a typical liquid configuration may only refer to the local order in small parts of the liquids. This state of matter has a straightforward consequence for the statistical mechanics treatment of crystalline solids and liquids: they are treated in different ways. In the case of solids, the central object of interest is a configuration of the atoms. The underlying concepts of the solid lattice and its symmetries give a systematical classification of various movements performed by the atoms (usually described in terms of phonon degrees of freedom). The instantaneous positions of the atoms describe a “typical” configuration which gives a non-zero contribution into the partition function 2 = C ee’jH. Here, p stands for inverse temperature 1/T and H denotes the microscopic Hamiltonian of the system. The case of liquids is quite different. Since the details of the space distribution of the atoms are unknown, the parameters used to describe the liquid have to result from some averaging procedure. In this sense, the statistical mechanics of liquids starts from “one level above” the microscopic level. In a standard approach (see, e.g., [1]) liquids are described via pair correlation function gz(v,, y2), i.e. the Boltzmann factor averaged over all the configurations with two atoms fixed at the positions yI, r2. More information about average space distribution could, in principle, be obtained from higher-order distribution functions gk(Y1,. . . ,vk), with k = 3,. . . , which are extremely difficult to calculate and to measure. For this reason, the main “input” into the statistical mechanics theory of liquids is the radial distribution function y2(r) and this theory contains very little information about local (microscopic) structure. In a statistical-mechanics approach to solids and liquids, different ways of treatment of the phases are used to study the melting PT. It would be nice to develop an unified semi-macroscopic approach

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Fig. 2. First mechanism

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of PT (see text)

to liquids and solids. The question if it is possible remains open. There are some hints that this goal can be achieved; to see it, let us examine more carefully the phase diagram discussed above. In spite of the simplicity of the phase diagram, the understanding of the origin of existence of three phases is poor. It is easy to understand that two phases (and thus one line of PT) have to exist. A naive but intuitively clear mechanism for a low-temperature phase coexistence of a condensed, and a dilute and disordered phase is loss of a single atom by the condensed phase, see Fig. 2. The PT occurs when the change of the free energy F = U - TS of the system due to this elementary act is AF = 0. Here, U denotes the energy and S - the entropy, Ai7 > 0 is the energy necessary to remove one atom from the solid and can be treated as constant; LLS is proportional to the logarithm of the number of available states of the atom, i.e. LLS,X ln( V/Y,), where V denotes the volume of the system and V0 denotes a constant with dimension of the volume (I’>> &). In this naive model there exists a line of first order PT between high-density (solid or liquid) and low-density phases; the underlying mechanism can be treated as a prototype of the sublimation PT. The difference in densities of the two phases decreases along the PT line with increase of temperature, so one may expect that the PT line will terminate in a critical point, at T, cc AU. For this PT between a condensed and a dilute phase to exist, the energy AU has to be positive, this assumes attraction at some distance and a binding energy between atoms. The crystalline nature of the dense phase is irrelevant to the PT, although one can find general arguments why the low-temperature dense phase may be expected to be crystalline. These arguments are based on ideas of the lowest packing energy, and one expects a system of hard spheres (no attraction forces) to have a crystalline phase. The one-atom mechanism is naturally complemented by a one-vacancy mechanism of simultaneous dilution of the crystalline phase. The dilution of the crystalline phase by vacancies cannot annihilate the long-range order of the crystal, it can only trigger, or accompany, some mechanism leading to the loss of long-range orientation and translation order. The changes in density by melting are small (usually less than 1OX), and a lattice with a corresponding concentration of vacancies is still a well-connected infinite crystal. It seems then natural to assume that the mechanism responsible for the melting is quite different from the sublimation mechanism. The experimental data imply that this new mechanism might have a collective character. Namely, melting is accompanied by small relative changes of density, internal energy (latent heat is small compared to the vaporization heat), compressibility etc. This could, hypothetically, occur due to the rearrangement (or disappearance) of a relatively small number of

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Fig. 3. Second mechanism

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of PT (see text).

(collective) degrees of freedom, like, e.g., long-wavelength phonons in a solid. Then, solid and liquid would look the same at small distances but differ noticeably from each other at large distances this is one of the formulations of the hypothesis of crystalline local order in liquids. A very naive model for this new mechanism is presented in Fig. 3; an elementary act is a free rotation of a cluster of atoms in the matrix of the solid. The line of the PT can be found from the condition AF = 0. Taking into account that LIU is due to the broken energy bonds and LU is proportional to the surface of the cluster, one finds that the PT takes place at T, #ccR/In(R) for a spherical cluster with radius R. In this naive model the density does not need to change at the PT; the energy and entropy undergo relatively small changes because of a small relative number of degrees of freedom (proportional to R-’ ) responsible for the process. Thus, this second mechanism can be regarded as a hypothetical prototype for the melting PT at least in some materials. An opposite picture is melting as a collective amorphization of the system, with a large ensemble of non-crystalline local arrangements in small volumes of the liquid material. We conclude that the main features of the phase diagram (Fig. 1) can be qualitatively understood if two dzjjhwzt mechanisms of loss of crystalline order exist. The second mechanism is based on the hypothesis of local order in liquids which puts forward a typical configuration of the atoms in a liquid. This, in principle, makes possible a semi-macroscopic statistical mechanics treatment of liquids. The concept of the local order in liquids was long ago quantified on the phenomenological level by the theory of significant structures, paracrystalline models, crystallite hypothesis, etc., see, e.g., [2]. The phenomenological theories differ considerably in their contents but all assume the existence of relatively large clusters of ordered material within the overall disorder. The quantification of the local-order hypothesis on a semi-macroscopic level in terms of local-order parameters [3-51 has offered a starting point for a semi-macroscopic statistical mechanics treatment of liquids. The verification (direct or indirect) of the hypothesis of local order, whatever it might be, in liquids and supercooled liquids or, equivalently, a parameterization of statistically probable configurations of atoms constitutes an important topic in the field of “disordered” condensed matter. The actual study of local structures both in real liquids and in computer-simulated liquids encounters serious problems.

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A direct observation of local structure in liquids still remains a challenge. The scattering experiments yield a time-averaged description of local structures in terms of radial distribution function y(r) and dynamic structure factor S(q, o) [l] which means a loss of detailed information about microscopic arrangements of the atoms. This information is, on the contrary, in principle available via computer simulations using Monte-Carlo or molecular dynamic techniques. The question how to analyze this information becomes of the first importance for the understanding of local structure of condensed matter. On the other hand, indirect support for the local-order hypothesis might result from study of various models of locally ordered condensed matter and their comparison with experiments. The aim of this review is a survey of various topics closely related to the statistical mechanics description of locally ordered condensed matter. On the one hand, we present the recent results on the local structure in computer-simulated liquids. On the other hand, we review several theoretical concepts, like e.g. the energy landscape formulation of local-order hypothesis or the statistical mechanics model of condensed matter in terms of local states. This model is then studied in a mean field approximation leading to a variety of experimentally confirmed and also hypothetical physical effects. The paper is organized as follows. In the next section we present the energy landscape picture of liquid and glassy state of matter. In Section 3 we report the recent results of a probabilistic analysis of local structures in various two- and three-dimensional liquids. The next section is devoted to the modelling of melting/amorphization of solids in terms of the local state model of condensed matter. Some conclusions are presented in the last section.

2. The energy landscape picture in the theory of liquid state The nature of difficulties encountered by the theory of liquid state, and the assumed structure of the liquid, may be illustrated in the energy landscape picture. One considers a system of N classical particles in 3D space, A configuration of the system is characterized by 3N coordinates of constituent particles and may be seen as a point in the 3N-dimensional configuration space. If a more detailed description of the configuration space is needed, one defines a 3N-dimensional Euclidean space as a direct product of 3D Euclidean spaces of each particle. The configuration Hamiltonian H of the system assigns a potential energy to each point of the configuration space. The crystalline lattice corresponding to the low-temperature stable phase is then an absolute minimum of potential energy H. The near vicinity of this absolute minimum of energy point represents small displacements of particles from the lattice sites; it may be characterized in terms of normal modes and phonons in the ideal crystal. In the 3N-dimensional configuration space the energy in this nearest vicinity to the ideal lattice configuration may be approximated as a positive quadratic form. By allowing a finite density of defects one arrives at a broader vicinity to the ideal crystal state. For the defect density not exceeding some threshold, one still may treat a small part of the system as a part of a crystal. The maximal characteristic size of such a small part depends on the defect density. In this way, it is possible to construct a set of configurations having the energy H higher than those near the ideal lattice. The local orientation of the axes of the crystalline anisotropy becomes an important component of the configuration. The statistical mechanics of a system confined to this set of states constitutes the theory of locally crystalline materials. It was proposed and developed for 2D systems, for which the configuration space is 2N-dimensional (KTNHY theory; see [6&S, lo] and

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the review paper of Strandburg [9]) w h ere the existence of the local order in liquids is supported by theoretical reasoning and experimental observations. In 3D-systems, the theory was formulated [ 11-l 31 but quantitative analysis is still a challenging problem. In statistical mechanics, the possibility to limit the consideration to a chosen set of configurations depends on the statistical weight of this set. The main assumption in the theory based on local crystallinity is that some physical mechanism (repulsion of defects?) prevents the defect density from increasing up to a level where the concept of local crystalline order becomes ambiguous, and a classification of local (in small part of the system) arrangements based on the crystalline order becomes useless. We will refer to the local arrangements belonging to a wide set of noncrystalline packing, in which it is impossible to recognize the crystalline or any other geometrically defined unique local structure (e.g. icosahedral), as amorphous. The energy of configurations with amorphous local arrangements may be higher than the energy of locally crystalline states, but one expects the number of amorphous states (and correspondingly the entropy) to be large. The part of the configuration space where the probability to find the system is overwhelmingly large corresponds to the minimum of the free energy F = U - ES. The increase of energy U may be compensated by increase of the entropy S. We suggest that the balance between the energy and entropy in the liquid near the melting point is not universal, and depends on the interaction potential between particles, while some materials may conserve the local crystallinity by melting, other materials may become amorphous. The local crystallinity in the liquid is increased if the microscopic interaction favors angles and distances characteristic for the lattice. That may be expected in materials with tetrahedral organization (SiOz, H20) and covalent or hydrophobic interactions, or (for a liquid crystal order) in molecular species with highly anisotropic molecules. The above general considerations say nothing about the energy landscape in the part of the configuration space corresponding to non-crystalline states. An additional assumption is that this landscape is characterized by many relative minima of the potential energy, and energy barriers intervening between those minima. This picture of the energy landscape underlies the contemporary understanding of the glassy state [ 141. One can then divide the configuration space into cells (domains, “basins”) assigned each to a corresponding minimum of energy; if one postulates a pure friction dynamics, a basin consists of all initial points from which trajectories lead to the corresponding energy minimum. That idea was used [ 15, 161 to demonstrate the existence of other than crystalline energy minima in a system of an order of lo3 particles with a two particle interaction potential. The division of the configuration space into domains assigned to energy minima provides a basis for classification of configurations under discussion in terms of local order. Namely, we will always consider the physical configurations belonging to a basin as fluctuations of atoms from the corresponding minimum energy configuration. Physically, the minimum energy configuration is related (but not identical) to a time averaged configuration of atoms over the vibrations in local cages. The possible escape of the system from the basin during the averaging period makes the definition of the time averaged structure ambiguous. In glasses and supercooled liquids, the system is, with a high probability, close to one of the minimum energy configurations, and one may neglect the difference between the physical configurations and the minimum energy configurations. At higher temperatures, the fluctuations are larger. The division of the configuration space into domains represented by minimum energy structures allows one to reduce the discussion of the local and global order to the discussion of the order in these configurations. We will refer the probability w to find the system in the corresponding domain

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of the phase space as the probability of the minimum energy configuration. tive Hamiltonian &(a), where a labels the minimum energy configuration, domain in the configuration state: 4~)

= exp[-Kf(a)l(knT)l

Z = C exp[-H/(&T)]

=Z-I?

41.5

This defines the effecand the corresponding

exp[-H/(kBT)l,

.

(1)

In this formula, H is the microscopic Hamiltonian of the system, kB denotes the Boltzmann constant, the sum 1’ is over the states in the basin, and the sum C is over all states. We will discuss below in more detail the state label a, and the form of Hef(cc). The 3N-dimensional energy landscape picture is merely an abstraction because of very large N in a macroscopic system. In such a picture, the “boundary” dividing two “basins” is a (3N1)-dimensional hyperspace having its own geometry. A much more physical picture emerges if the general idea of “basins” is combined [ 171 with the idea of local inherent structures. The idea emerges from the empirical observation of the local rearrangement kinetics in glassy materials and supercooled liquids, and postulates a local mechanism of hopping in the configuration space. It is assumed that to get from one basin to others, a series of local rearrangements is needed. Here, local means that the substantial changes (mutual exchange of atomic positions and, for molecular liquids, molecular reorientation) take place in a small part of the system referred below as cluster, while the particles in the rest of the material have small adjustment displacements. An experimental observation is that in polymeric glasses the number of repeating units that rearrange is of the order of 10; this is qualitatively in correspondence to the computational observation of [ 15, 161. Then, the energy barriers between nearest basins are identical with the rearrangement barriers in small clusters. The idea of local rearrangements naturally leads to a more detailed description of the state a corresponding to a basin: this state may be described in terms of structural states of small parts of the system having the typical size of a rearranging cluster. Idealizing the situation, we divide the volume of the system in small parts, and consider each such small part as a cluster that may acquire one of a set of local structural states. Depending on the chosen set of states, we will obtain different models of global ordering and disordering of the system. 3. Probabilistic

analysis of solid-like order in computer-simulated

atomic liquids

The full information about local arrangements of the atoms is given by instantaneous configurations. At present, these configurations cannot be observed directly but can be obtained from computer simulations (see, e.g., in [ 181). In this section we present recent results of an analysis of local structures in atomic 2D- and 3D-liquids. We study the case when the set of states introduced above corresponds to different orientations of a small crystallographic cluster. The concepts of symmetry can then be helpful in the search for the parameters that describe its structure. 3.1. The method The task of retrieving information about local structures from instantaneous configurations of the atoms belongs to the field of pattern recognition. Below, we present the main points of the method; more details can be found in the literature cited in the text.

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Fig. 4. Delineation of the space of parameters (c) high fluctuations); see text.

describing

local structures

((a) no fluctuations;

(b) small fluctuations

and

The general strategy is as follows. The object of the analysis is a cluster of atoms, consisting of a central atom and its N metrically defined nearest neighbors, i.e., the N atoms which have the smallest distances from the central one. The atoms are treated as material points. Next, one defines (experimentally or computationally) a set of patterns c representing various types of local structures. For example, in three dimensions (3D) some local structures with dense packing are represented by 13-atom clusters: the nuclei of face-centered-cubic (fee) and of hexagonally closepacked (hcp) lattices and an icosahedron. In 2D, the only candidate for solid-like dense packing is a 7-atom hexagon. The recognition of a type of a trial N-atom cluster proceeds via an algorithm which chooses one of the patterns rj as the “most similar” to the trial one. To this end, one introduces parameter(s) Q which describe the structure of an N-atom cluster. In the space of these parameters (the so-called feature space _S) various local structures c correspond to different points as shown in Fig. 4(a). In the presence of the (thermal) fluctuations, an ideal pattern ri gives rise to a (continuous) family of “excited” structures, which are represented in the feature space by a cloud of points. When the fluctuations are small, the clouds corresponding to different r; do not intersect each other (Fig. 4(b)), while for higher fluctuations the intersection occurs (Fig. 4(c)). In the former case, the delineation of the feature space has a deterministic character: a point Q E 2 corresponds to a fluctuation of one and only one (or none) of the patterns c. At higher levels of fluctuations, this one-to-one correspondence is lost and the delineation of the feature space has a probabilistic character: a point Q E 2 can correspond to a fluctuation of more than one pattern c. The fluctuations “blur” any information about the structure and this fact has to be properly accounted for in a reliable analysis [ 19-231. In particular, it implies the necessity of using geometrical concepts together with probabilistic concepts, or, in other words, using the concepts of probabilistic geometry. In what follows, we restrict ourselves to the case of 1D feature space; the case of higher-dimensional feature space is discussed in [20,22]. A fluctuating cluster r is characterized by a distribution of the random variable Q, described by a probability density function (PDF) p(Q, c), where < is a parameter which characterizes the strength of the fluctuations. For example, if the atoms are assumed to fluctuate independently of each other and the fluctuations are normally distributed, then < is proportional to the root-mean-square fluctuation of the atom’s displacement from its equilibrium position [20, 221. The PDFs are the main objects of interest in our probabilistic approach to the recognition of local structures. When no fluctuations are present (5 = 0), the PDF reduces to the Dirac delta function: p(Q, < = 0) #CC S(Q-Qid), where Qld denotes the value of invariant Q for the ideal (non-fluctuating) cluster. The shape of PDF

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PDF

(4

PDF r/l,

Fig. 5. Plot of the probability density functions pr,(Q, 0, fluctuations: (b) small fluctuations and (c) high fluctuations).

changes with increasing

Edt) =

(,)~PDF

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(4

r 1'1

prz(Q, 0

f or various values of i_ (schematically)

((a) no

[; see Fig. 5. The overlap of the two PDFs

smin

h(Q, 0, Pr2(Q, 5)) dQ

(2)

is a measure of the structural similarity of two jIuctuatiny clusters. Let us now consider two fluctuating clusters r,, r,, and let Q; denote a value of parameter Q calculated from an instant configuration of one of the clusters (which one is not specified) undergoing fluctuations of the atoms characterized by <. Intuitively, one classifies this cluster as being a fluctuation of r, if pfi(Qz, 4) > pr,(Qz, [); in the opposite case the cluster is classified as a fluctuation of I’,. Then, the parameter Et2([) gives the total probability of false classification: a fluctuation of structure r, as fluctuation of r, and vice versa. The parameter EIZ(t) has the following properties. For 4 = 0, Et2([) = 0, and the classification of the unknown structure has deterministic character. At higher values of [, the PDFs starts to overlap each other. When the overlap (given by Et2(<)) is small (see Fig. 5) then the probability of a false classification is small and both fluctuating clusters have well-defined structures. With increasing [, the overlap Er2(t) increases, the information about the structures is gradually lost. As observed in computer simulations, the overlap is small for r < <,, and sharply increases at larger 5. At high enough r’s, Ej2(<) 2 1 ( see Fig. 5) the two types of structures cannot be distinguished. Thus, the smallness of E,,(t) is a necessary condition for reliable analysis of the structure. The above discussion shows how single fluctuating clusters can be classified. The search for local structures in a physical system is done in the following way [24, 251. For an instantaneous configuration of the system, one considers all N-atom clusters that may be defined in the material; the parameter Q is calculated for each such cluster. This gives the distribution p(Q, T*, p*) as a space average. Here, T’ and p* denote reduced temperature and density. For a Lennard-Jones potential V(u) = 4ao((r/o)-t2 - (Y/o)-~), T* = kgT/cO and p* = ~8, where D denotes the spatial dimension of the system. For a finite (and not very large) system in thermal equilibrium at given T*, p* one may increase the ensemble by studying subsequent configurations of the system. The simplest ansatz for an analysis of those unknown structures is to assume that this distribution is a linear superposition of the distributions for patterns c (see Fig. 6):

P(Q, T*, P*>= CC~P~(Q, i”i>>

(3)

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of PDF (thick line) into patterns of local structure, with cl = 0.4 and c2 = 0.6 (see text).

where Ci, 5; are unknown fit parameters, and Cici = 1. The quantities ci( T’, p*) measure the weight of the structure represented by ideal pattern c. The “best” decomposition of p(Q, T*, p*) is calculated by maximizing with respect to the parameters cl, & the significance level calculated from x2-test verification of the hypothesis (see, e.g., [26]) which states that the data corresponding to the 1.h.s and r.h.s of Eq. (3) are drawn from the same distribution. When CIis too small [26] the decomposition is not reliable. The structural identity of found local structures has still to be checked by calculation of the errors I!?;~(&,tj) (being a straightforward generalization of formula (2)) as discussed above. The quantities we use to describe local structures, which effectively differentiate between various symmetrical structures of fluctuating clusters of the atoms, are constructed from local-order parameters which are the multipole moments of the density of the matter. Those parameters may take the form either of Cartesian tensors [3,5], or, equivalently, of bond-order parameters Qlm [4,27,28,20]:

The summation extends over all the N neighbors r (‘) of the central atom in a cluster, 8, stand for spherical harmonics and Szco) denotes the polar and azimuthal angles O(“),@‘) corresponding to Y(“) (in a fixed coordinate frame). Parameters Qlm which depend on the choice of the coordinate system contain the information about (i), the structure, which is independent of the coordinate system and (ii), the orientation of the cluster in space. For the atomic liquids considered in this paper, we are interested in the structure of the cluster only; the question of an orientation analysis is discussed in [31]. The information about the structure of the cluster is contained in geometrical objects which are SO(3) or SO(2) scalars (for 3D and 2D systems, respectively), constructed from bond-order parameters. We call those scalars structural invariants. The nth-order structural invariants (e.g., for n = 2, Qi and for n = 3, Q~,/,I,) are calculated by the “angular momentum” vector addition of II bond-order parameters ([27], see also [22,28]):

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and (6)

where

(

11

12

13

MI

m2

m3

1

denotes the Wigner 3j symbol (see, e.g., in [29]). The choice of the informative invariants depends on the number of spatial dimensions of the system and on the local symmetry looked for and is discussed in the next sections. 3.2. Results: 20 liquids In 2D the atoms lie in a plane which we choose to be perpendicular to the Z-axis of the coordinate frame; all 0; in Eq. (4) are equal to 7t/2. The local crystallograpic 2D close-packed structure, represented by a 7-atom hexagon, can be effectively differentiated from other types of local (7atom) structures via invariant Q, with I = 6 [24]. For convenience, we omit from now on the subscript 1 and use Q = Q6. Contrary to other approaches [30], our analysis is based directly on the probability density function p(Q). Intensive studies of local structures in 2D Lennard-Jones computer-simulated liquids [24, 311 have shown that in the wide interval of densities below the solid-liquid coexistence line the liquid (melt) contains a noticeable concentration cg( T*, p*) of local solid-like hexagons, with cg z 0.5 in the vicinity of the liquidus line. The concentration cg obeys a kind of a law of corresponding state [3 I]: cg as a function of scaled density psc = p*/&(T), where p&(T) corresponds to the liquidus line, is temperature-independent. This universal function, which can be satisfactorily approximated by quadratic dependence, cg cx (psc - const)2, is shown in Fig. 7. The solid-like clusters have a tendency to group together into bigger solid-like clusters. A typical configuration of the atoms of a 2500-atom LJ liquid in the vicinity of the liquidus line (psc = 0.95) is shown in Fig. 8. The concentration of solid-like clusters is cg = 0.43. The bigger aggregations of solid-like clusters are very short-lived: after the typical time of lo-r2 s the subsequent configuration displays a totally different distribution of solid-like matter; nevertheless, its concentration and the distribution of the sizes of solid-like clusters remain, on the average, conserved. Thus, those clusters are not nuclei of a new thermodynamic phase (solid). Such a dynamical physical picture of structure of liquids was put forward years ago by Frenkel [32]. Similar results were found for other types of 2D liquids. The concentration of solid-like matter in a liquid with quantum degrees of freedom [25] and in the liquid of hard discs [33] may be satisfactorily described by the universal curve found for LJ liquid. Also, the typical configurations look like those in the LJ liquid. To summarize, we find that 2D atomic liquids are locally solid-like ordered. This offers a possibility of a parameterization of a typical liquid configuration in the vicinity of the liquid-solid coexistence region; the key to it is the observed existence of high concentration of local solid-like structures. In particular, the physical picture of a locally ordered liquid may cast new light on the mechanism of freezing of purely 2D liquids, which is mainly of theoretical interest, but also onto

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+ 0.4

Fig. 7. Law of the corresponding in 2D LJ liquid (see text).

states: dependence

Fig. 8. A typical configuration of 2500-atom denote the centers of solid-like hexagons.

the closely related (and important thin films.

of the concentration

c6 of local solid-like structures on scaled density

LJ liquid in the vicinity of the liquidus line (pSC= 0.95). The small circles

for material sciences)

problems

of freezing of intercalates

and of

3.3. Results: 30 liquids The search of close-packed local structures in 3D systems is much more complicated than in 2D systems. The reason is as follows. In 3D, there exist three close-packed local atomic arrangements, namely the nuclei of fee and hcp crystallographic lattices and an icosahedron (see Section 3.1). Contrary to the 2D case, the point symmetry now plays an important role. To differentiate effectively between those three local structures one has to use higher-rank tensors (or spherical harmonics, Eq. (4)) because they are more sensitive to small changes of the spatial arrangements of the atoms (recall, e.g., that 13-atom fee and hcp clusters differ by a rc/3-rotation of three properly chosen atoms). On the other hand, higher-order local order-parameters become non-informative (lose the ability to differentiate between fluctuating structures) at lower fluctuations than lower-order parameters [20]. Those two facts make the search for informative parameters a very delicate and tedious task; the study of the parameters like E,(t) is then obligatory. In computer simulations of small systems the fluctuation of the atoms in the solid phase can be quite high, with c 5 0.18 (see, e.g., [28, 341). Thus, the parameters to describe the local structure of liquids should be informative at least at such level of fluctuations. This requirement is very hard to fulfill. The widely used method of Voronoi construction with small edge reduction [35] correctly identifies fluctuating fee clusters for 5 = 0.1 and is acceptable up to 5 = 0.14; at higher fluctuations the error of the identification grows rapidly. It may happen [36] that this method is incapable of a correct recognition of an fee solid in the vicinity of its melting line. Similar limitations hold for local hcp structures.

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1000 0.3

i 500

0.00

0.05

0.10

0.15

0.20

0.25

Fig. 9. Plot of the probability Fig. IO. The PDF of invariant (3446 calculated

0 -0.002

-0.001

0.000

0.001

0.C 02

EFCC,HcPas a function of (. from LJ liquid at p* = 0.95 and T” = 1.2.

The use of structural invariants introduced above makes possible a systematical search for the most informative invariants. In particular, one of the most informative invariants for the 13-atom fee structure is a third-order invariant Q 446 [22], calculated by vector addition of three bond-order parameters with I= 4,4,6; cf. Eq. (6). The measure of reliability of the recognition of fluctuating 13-atom fee clusters is given by the error probability E FCC,HCP(<).In Fig. 9 we show the plot of the error EFCC.nCP(<) as a function of < calculated in the independent Gaussian fluctuations ensemble, i.e. when the fluctuations of the atoms are independent normally distributed random variables with r.m.s. displacements equal to [ (because of somewhat different definitions of t used here and in Ref. [22], Fig. 9 is not identical with an analogous figure in Ref. [22]). The range of application of invariant Qdd6 for differentiation between fluctuating FCC and other close-packed 13-atom clusters is c <0.1550.18: E(0.15)~O.l and E(O.18)=0.35. In the latter case, Q 446offers rather poor distinction between fee and other structures, but still can be used as a first try. A search of local close-packed structures in 3D LJ liquids using the parameter Qdb6 revealed no existence of local fee arrangements [37]. In Fig. 10 we show the probability density function (PDF) of invariant Q 44h calculated from a 500-atom LJ liquid at p* = 0.95 and T* = 1.2. The PDF is centered around the value Q 446= 0, which corresponds to non-fee local arrangements. We find no traces of a peak centered around Qeh6zz 0.0027 which corresponds to crystalline fee structure. An alternative possibility for the existence of local fee arrangements in liquids follows from the observation [37] that strong fluctuations (5 > 0.18) lead to various possibilities for defining a cluster of the atoms. In particular, efforts were made [37] to project the local structures in LJ liquid onto the local structures present in a model system used for studies of hypothetical high-temperature solids, the so-called hot-solid (see, e.g., [38]). Al so in this case the calculated PDF did not possess any fee-like peaks. To summarize, the use of structural invariants and of alternative concepts about the choice of clusters did not reveal any traces of local fee structures in 3D LJ liquids. To search for local icosahedral arrangements in 3D LJ liquid (500 atoms, p” = 0.95, T” = 1.4) the structural invariant &be proportional to Q 666 (introduced in papers [27, ZS]), which is very informative for a single fluctuating 13-atom icosahedron, was used [37]. As in the previous case, no traces of icosahedral clusters were found. The search for local hcp structures failed because of lack of informative invariants at high enough fluctuations.

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To summarize, no traces of local close-packed structures were found in equilibrium 3D LJ liquids. Similar studies were done for an undercooled 3D LJ liquid, with 4000 atoms at p* = 0.95 and T’ = 0.8 [37]. The PDFs calculated for this system were much like those calculated from an equilibrium liquid and did not show any traces of local close-packed structures. The absence of local solid-like order is probably the effect of the two-particle form of the interaction potential. Other types of interatomic interactions, see below, can probably force well-defined local (solid-like) order in an undercooled liquid, or even in an equilibrium liquid. This was shown for a case of a slightly unusual potential [39] which imitates the interaction in a two-component system. This potential has a local maximum which suppresses the local solid-like structures and forces local icosahedral order. Preliminary studies of the equilibrium liquid phase in this system [37] (500 atoms, p* = 0.88 and T* = 1.6) did not show any differences with the LJ liquid. However, the situation changes dramatically for undercooled liquids. In a system of 4000 atoms, at p* = 0.88 and T* = 0.5 a part of the PDF of invariant Oh66 corresponds clearly to the existence of local icosahedral order. It thus seems plausible that more complicated potentials, e.g. interactions that are not of a twobody type, non-spherical particles, hydrogen bonding, steric interactions, weak covalent interactions, might induce well-defined types of local structures (probably not close-packed) even in equilibrium liquids. We are not aware of any systematic studies of such kind. In quenched systems, as a rule, crystallization into one or more types of crystallographic structures takes place. When the nuclei of solid phase are big enough, their structure can be more easily studied because the fluctuations are noticeably smaller. The details of the kinetics of nucleation and growth are of primary interest for material sciences. This topic does not belong to the scope of the present paper; a review of the methods and recent results can be found in the papers [23,40].

4. Statistical mechanics of melting For a conventional statistical mechanics description, one can start with a particle Hamiltonian of the material, and obtain the properties by computer simulations. This approach gives, however, little insight into the nature of the melting process. On the other hand, the results presented in the last section indicate clearly that typical configurations of 2D liquids can be described in terms of local solid-like arrangements of the atoms. In 3D LJ liquids no traces of close-packed local structures were found; however, it is highly probable that more complicated interactions may lead to the local-order structure. This assumption was discussed earlier in Section 2 in terms of the energy landscape concept. It is the presence of these clusters, and their interactions and rearrangements, that constitute the heart of the melting picture to be discussed here. It focuses not on the particles (atoms or molecules) themselves, but rather on the local states that the clusters can adopt. The model was developed [41] to deal with melting in conceptual, rather than purely computational terms. 4.1. The local state model of a condensed material In this section, we propose and discuss an idealized mode1 of a material having many local states. The material is treated as a system of small clusters of equal size, each cluster being in one of a set

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Fig. 11. An idealized model of a material having many local states, see text.

of N, local structural states. The same set of local structures is assumed for all clusters; a number N, is assigned to each of the structural states of this set, see Fig. 11. A structure of the i=l,..., whole material is then described by the field i(v) which denotes the structural state at the point Y. The field i(r) plays the role of the structural order parameter of the system. To proceed with a statistical theory of the local and global order, one has to find the form of the non-equilibrium free energy (effective Hamiltonian [42,43]) H{~‘(Y)} as a functional of the field i(v). In addition to the order parameter, the material has many other fluctuating degrees of freedom. To calculate H{~(Y)} the average over these degrees of freedom has to be taken, as explained in the fluctuation theory of phase transitions [42,43]. The free energy H{ i(v)} d e fi nes the probability w{i(v)) of a configuration i(v):

4(r)) =

&) exp(-“L:;“),

Z(T) = Cc)1 exp

(

-

H;:;)])

.

Here, kiS denotes the Boltzmann constant, which for the sake of simplicity will be omitted in the subsequent formulas, and Z(T) denotes the partition function being a sum over all configurations of local states. The ensemble (7) describes the statistics of local order in a state of thermodynamic equilibrium. The formula (7) may be seen as a definition of the effective Hamiltonian. A model of the local and global ordering is then specified by the set {i} of local states and the form of H { i( r)}. Quite generally, the effective Hamiltonian should include independent contributions h{ i( v)} from different clusters, and a part Hint that depends on the states of more than one cluster: H{i(r)}

= C h{i(~)} + H,,, . r

(8)

The relative significance of these two parts depends on the size R of the clusters. If a large R >a is chosen, where c1 denotes a mean interatomic distance, then H,,, becomes relatively less important, but the internal order in such a large cluster becomes almost as complex as that of the whole material. For small R, the interaction becomes the dominant part of the effective Hamiltonian, and one returns to the exact but mathematically hard microscopic theory. The optimal choice of R would allow one to classify the structural states of the clusters as corresponding to minima of potential energy of the cluster by some special configuration (e.g. crystalline) of the embedding matrix; the change of the energy of a given structural state when the surrounding matrix is changed from the standard one to the actual structural state of neighboring clusters may then be described in terms of the interaction of local orders. A cluster of optimal size has to include subunits (atoms or molecules)

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that change relative positions in a local rearrangement. We suggest that for the optimal choice of the cluster size the local and the interaction energies are of a comparable order. Such a cluster includes a few coordination shells of the central atom, containing cxlO’-1 O3 subunits. For non-polar systems, the typical radius of the inter-particle potential is the interparticle distance. The interaction between non-neighboring clusters may be then neglected if the size of the cluster is larger than the interaction radius; the energy of the system includes the interaction term depending on the states i and i’ of neighboring clusters. The two-cluster interaction may be written in the general form

A special local state is the crystalline one. The local crystallinity means that the cluster may be treated as a small crystal. Some number of defects is allowed, but this number has to be small to allow the recognition of the crystalline order [ 121. The fluctuating characteristic of a crystalline cluster is the orientation of local crystalline axes. The ideal crystal is then characterized by parallel orientations of corresponding local crystalline axes in all clusters. In a locally crystalline material, the phase transition between the ordered crystalline and the disordered liquid state may be described in terms of local orientations, or in terms of the distribution of topological defects. The theory of 2D defect mediated melting [6-lo] was successful in explaining the nature of phase transitions in such systems. The results of Section 3 show a high concentration of non-crystalline clusters at the melting temperature in computer simulations. It implies that the core energy of a defect is strongly renormalized near the melting temperature, and one has to take into account the interaction between the orientation order and the local amorphization. A microscopic theory of defect melting was formulated for 3D systems in [ 12, 131; the study of melting in this theory is a difficult mathematical problem. A much simpler phenomenological model of melting, formulated in terms of rotations of local crystalline clusters [5, 44, 451 allows a detailed study of the statistical properties of the local orientation order. Unlike the 2D case, the melting transition in 3D is of first Ehrenfest order (discontinuous). The phenomenological model deals with a simple lattice version of orientation order-disorder similar to those known in the theory of magnetism; the basic difference is the symmetry of the order parameter. An absolutely symmetric rank four tensor (“cubic nonor”) plays the role of the order parameter in the case of local fee lattice. For other than fee symmetries of the local (“tangent” [ 121) lattice the number of the tensor order parameter fields and the rank of the tensors may differ from the cubic case. These models explain some important common features of melting [3, 5, 44, 461: (i), the phase transition is, in 3D, of first order (discontinuous), (ii), the absolute instability of the crystal state is close to the equilibrium melting temperature while the absolute instability of the liquid state extends over much bigger temperature interval, (iii) the existence of premelting phenomena in heat capacity, found later experimentally [47], and (iv) the existence of surface melting, confirmed later experimentally [48]. They also predict new effects like the (discontinuous) liquid-liquid phase transition [49] reported later in experiments [51, 521. However, the correlation radius of the orientation order is, both in the defect melting and in the orientation melting models with local crystallinity, of the order of an elementary length of the model. This length depends on the interparticle interaction; for a system with ordered clusters of the interparticle distance size, the structure of resulting disordered state is hardly compatible with the assumption of the local crystallinity. To study the general situation, one has to consider the possibility for a cluster to be in the crystalline, or in an amorphous local

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state. In such a generalized theory, the amount of crystallinity in the equilibrium and supercooled liquid is determined, in the framework of the statistical mechanics theory, by the Hamiltonian of the system. 4.2. A locally uystalline liquid The physical picture of a locally crystalline liquid is that each cluster may be treated as a cluster of a matching crystal [ 121; there are topological defects that allow the orientation Q of local crystalline axes in the liquid to vary randomly along the system. The defect density has to be not too high for the local order and its orientation in space to be well defined. In models of orientation melting, one studies the statistics of orientations Q(r) [5, 441. This orientation at point r is defined, e.g., by three Euler angles in 3D, and one rotation angle in 2D space. The probability ~~{Q(r)} of an orientation configuration Q(r) is determined by the effective Hamiltonian H{Q(v)} as described in formula (7); the orientation B plays the role of a multidimensional “index” of a local state. For the orientation of two crystalline clusters sharing a common boundary to differ, a minimal amount of topological defects, and a minimal energy is necessary. This energy may be characterized by a function ,f(Q(r), a(~‘)), where r, 1” denote the coordinates of the clusters. The three-cluster, fourcluster, etc., interactions may be written as well; we limit ourselves to two-cluster interactions for which the effective Hamiltonian has the form

Since the effective Hamiltonian results from averaging over small-scale molecular vibrations and variations of defect density, it may depend on thermodynamical parameters like temperature and pressure. We neglect this dependence in a narrow range of temperatures near the melting temperature 7;,. The single-crystal state is the ground state of the system, so that the function ,f’(Q(v), Q(r’)) has a sharp minimum at Q(u) = B(v’). Models of orientation melting [4, 5, 441 having different functions ,f‘(Q(r), G?(r’)) show similar melting behavior. The orientationally ordered phase (crystal) occupies the low temperature part of the thermodynamic plane. A first-order phase transition leads to the orientationally disordered (liquid) phase. Let us divide the orientation space in equal cells and use a coarsened description of orientations in terms of orientation cells. We label the cells with the number Y = 0, I,. . . , No,; the orientation of a cluster centered at point Y is described as X(Y). The size of the orientation cell has to be chosen as to comprise the orientational attraction region, where the interaction energy is close to its minimum. The simple model of orientation ordering is then given by the effective Hamiltonian of the form

The first term of the sum in (11) is the energy of parallel orientations in corresponding clusters, while the second term describes the increase of the energy when the orientations belong to different cells (we assume Z,,Iz > 0). The Hamiltonian ( 11) is that of a Potts model [5]. All (N,, + 1) orientation states are equivalent, so the model has a high internal symmetry. In the symmetric hightemperature phase (“liquid”), the probability to find a given cluster in any state is IV, = l/(N,, + 1).

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The symmetry is broken in the ordered phase where one of the states (labelled by a = 0) is occupied with higher probability w(0) - w. = w than the other N,, states that have the probability of occupation wl = w2 = ’ ’ . = (I - wo)/No,. We use a mean-probability approximation [5,44,53] to study the phase transition in the model. This approximation may be formally derived from the initial Hamiltonian (see Ref. [53]); here we use intuitive arguments to obtain the equations. Consider a representative cluster at point Y= 0. Suppose that all other clusters are independently occupying orientational states with a probability w,. The mean energy H MFA(a) of the representative cluster depends on its state. From ( 11) one obtains Hhj)T*(!x) = --cw, + v,

t: =

s

(/I(Y) +4(~.)>dK

I!=

s

I*(v)dV.

(12)

In (12) the relation wI =w,=...=(l -wo)/N,, was taken into account. The shift of the origin of the energy scale does not change the probability of states. We choose the orientationally disordered but locally crystalline state as the origin of the energy scale; that yields v = 0. Then, we obtain the self-consistent equation for w in the form 1 w=-exp zx

ew , ( T 1

Z,,-exp(y)(l

+NO,exp(+*(wi

-w))).

(13)

The solutions of this equation for N,, = 3, c = 1 are shown in Fig. 12. The symmetric solution it’/ = i exists for all temperatures while the non-symmetric solution exists only for T < T*, where T* cx t: denotes the spinodal temperature for the crystalline state. As is usual in mean-field approximations, there is a branch of the broken symmetry solution that corresponds to an unstable homogeneous state (but may play some role in non-homogeneous states of the system [46, 541). The equilibrium-phase transition temperature is T,, < T*. Numerical values for the temperature T” of the crystal state spinodal and for w* = w(T*) are T*/c = 0.31, w* = 0.65 for N,, = 3 [5], and T*/t: = 0,214, w* = 0.73 for N,, = 10. The melting temperature T, is determined by the condition that the free energy of the ordered phase equals that of the disordered. Eq. (13) may be obtained as the minimum condition of the non-equilibrium free energy F(M), T) of the form F(w,T)=;

w2+

&(l -w)’

)-

Tln(exp(~)+NO~exp(“‘l

iw))).

(14)

or

In Fig. 13 we present the plots of F(w, T) (No, = 3, c = 1) for various temperatures. In the temperature range T < T* F(w, T) has two minima, w, and w/ (at low temperatures w/ is a maximum). The stable phase corresponds to the smaller value of F(w, T), and the condition F(w/, T,) = F(wc, T,,) determines the temperature of orientation melting. The ratio (T* - T,,)/T* ry 0.01 for N,, = 3, and it nears 1 as N,, becomes large. The orientation partition function Z,,(T) determines the orientation free energy F,, = - T In Z,,( T). From (13) one obtains 9

&,/

=

(15)

Here, &.t stands for the symmetric (“liquid”) solution. In Fig. 14 we show the free energy calculated from formula (15) (N,, = 3, c = 1). At T > T,, the liquid solution describes the stable orientationally disordered high-temperature phase (liquid). This solution becomes metastable at T,,

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421

I

I------, &-/,: T~

0.5

/

/

/

‘I- ,

o..

0.2

0.1 Fig. 12. Various solutions of MFA equations

T’,

0.3

for N,, = 3: stable (and metastable)

F

0.4 (solid line) and unstable (dashed line).

(a> I

\ /

\ \

/

---

\

_’

w / 3.0

,

1

I

free energy F(w, T) for N,, = 3: case (a), 7’~ T,,, and T > r,, (dashed line), T < T,,

For

-0.6

-1.0

.O

0.5I

WI

Fig. 13. Plot of the non-equilibrium (solid line); case (b), T>>T*.

/

\

\

ai:---- ,

0.1

Fig. 14. Plot of the free energy F,,(T)

0.2

calculated

T*,

0.3

0.4

for N,, = 3: crystal (solid line) and liquid (dashed line).

and unstable at some temperature r:. The low-temperature ordered phase, identified with the crystal state, is stable at T < T,, and metastable at T,, < T < T*; this solution is characterized by IV(T) MI/ r” w( T ) > 0. 4.3. Anmphous

loud states

A more general assumption is that the local arrangement in a small cluster may be crystalline or amorphous. To account for the amorphous local states, let us assign a number i = 1,. . . , N,,,, to each of these states. The number i= 0 is reserved for the crystalline state of the cluster. A given amorphous local arrangement may occur in different orientations; we count those states as different amorphous states, so N,,, has the meaning of the total phase volume of amorphous states of a cluster. Following the objective to study the most simple schematic models, we consider all N,,,, amorphous states having equal internal energies, and neglect the inter-cluster part of the energy of amorphous clusters. The general form of the Hamiltonian is then

(16) The second term in ( 16) represents the orientation interaction in the crystalline (i = 0) part of the material, while the first one models the amorphization energy of a cluster. As will be discussed below, /l may be positive or negative. The sums are over all clusters in the material. The orientation interaction is taken in the Potts-model form of Section 4.2; the resulting Hamiltonian is then

with the energy p being the only characteristic of an amorphous cluster. A more general model may be formulated in terms of local states by a more realistic orientation interaction and taking into account differences in energy and interactions of amorphous states. This improvement may be important for modelling the supercooled liquid state of the system. The simple model considered in this paper is sufficient to describe the differences in the liquid states of different liquids near their crystallization temperature. To study the statistical mechanics of the model (17), we apply a mean-probability approximation (MPA) [53] similar to that used in the previous section. Let n(O) E II denote the probability to find a given cluster in the crystalline state. Due to the homogeneity of the system, M has the same value at all points r, and gives the fraction of crystalline clusters in the system. All amorphous states (i= 1 , . . . , N;,,) are statistically equivalent, so the probability n(i), i = 1,. . . , N, ,,,, to find a given cluster in the ith amorphous state is n(i) = (1 - n)/N,,,. A crystalline cluster (i = 0) is additionally characterized by its orientation state x(. The probability for a crystalline cluster to have the orientation state r is I$‘,. The probability to find a given cluster in the crystalline state (i = 0) and in the orientational state r is p(x) = ~IV,; the normalization conditions for p and r are C M;‘,= 1, C p(r) = n. We find the MPA Hamiltonian H(i,&) in a similar way as in previous section. The mean energy HMpA(i,z) of the trial cluster (at v=O) depends on the state i of this cluster. If the cluster is amorphous (i > 0) then 14MPA(i)= I_I; if the cluster is crystalline, there is an additional

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characteristic r of the state. For a given cluster, the probability to be crystalline regardless of orientation U (i = 0, a = 0,. ..,N,,) is n(~)=n, and that to be amorphous (i=l,...,N,,) is (1 -n). From the model Hamiltonian ( 17), one obtains then the MPA Hamiltonian H,,,,p4 in the form HMp*(i,CI) = p( 1 - (5,,) - a?Zbi,o6,.”. The self-consistency

conditions

give the equations

(18)

for w and ~1:

-G = exp(c-:w/O) + N,, exp(c( 1 - w)/N,,O),

n = Z,,,( 0)/Z

= exp(Ew/O)/wZ,

Z = Namexp( -p/T)

+ Z,,,( 0) .

0 = T/n, (19)

The equation for the orientation probability of crystalline clusters coincides with that for a pure orientational model discussed in previous subsection, with 0 playing the role of the temperature. There are two solutions (w,, w/ ) of the self-consistency conditions in these models, and correspondingly two different orientational states. As explained in previous subsection, the stable phase corresponds to the smaller value of non-equilibrium free-energy F(w, O), formula (14) and the condition F(ws, 0,) = F( w/, 0,) determines the temperature of orientation melting. It is convenient to write the non-equilibrium thermodynamic potential F of the system using as independent variables the probabilities n and p = nw: F(n,p,T)=~c(p2+(l/N0,)(n-p)2)-TlnZ

(20)

where Z was defined in formula (19); Eqs. (19) are the minimum conditions for the function F(n, p, T ). There are two different orientation phases at 0 < O*, where O* = K*t: is the spinodal of the orientationally ordered phase. Here, K* denotes a number dependent only on N,,, so there are at least two solutions of Eqs. (19). If for some temperatures there exists more than one solution of (19), the stable phase corresponds to the lowest value of F(n, p). A detailed study of the solutions of the self-consistency conditions (19), (20) for different relations of characteristic energy goes beyond the scope of the present paper and will be published elsewhere [41]; some asymptotic cases are discussed in the next subsection. 4.4. Amorphisation

vs. orientational

meltiny

One obtains from (19) the equation for n(T) in the form n(T) = (1 + wN~,,,exp( -(/l + nr-:w)/T))P’.

(21)

The temperature-dependent quantity w is w = w/ = (1 + N,,))’ in the symmetric orientation state, and w = w, = 1 in the non-symmetric one. The quantities p and E are materials characteristics, and may differ substantially for different materials. Let us consider a material that has a very large amorphization energy p> c. The orientation melting will then take place at the temperature T, = To,, x c<< p, and at this point one has n r” 1 - wN~, exp(-(p

+ cw)/T,)

.

(22)

The last term in (22) determines the small jump in the number of amorphous clusters at melting, when w jumps from the ordered phase value w, to the w,. The liquid state for that class of materials

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has a predominantly crystalline structure, as described in Refs. [55,4, 5, 44, 121. The materials in that class are those with interatomic interactions favoring crystalline packing. If there are competing crystalline structures, the material may exhibit a liquid-liquid-phase transition, in which the local crystalline structure is changed (see Section 4.1). The amorphization of materials with large P>>E takes place in the liquid state, at T cc ,u(>>T,,,,. Because of the crystalline structure of the liquid, the flow mechanism involves defects. One expects then the viscosity of the melt to be high at T,. Let us now consider a material in which melting is amorphization-dominated. For that class of materials, the perfect crystalline state does not substantially lower the local energy, but gives the lowest global energy due to the continuity (in absence of boundaries) of the crystalline lattice. For such materials (e.g. those with soft sphere or LJ interactions) the global energy of the material in the orientationally disordered phase is larger than that of the amorphous state. For small clusters of such materials the icosahedral local packing was shown to have a lower energy than any crystalline packing. In the model considered in this paper, the energy per cluster of the amorphous state is E,,, = ,u, the energy of a completely crystalline and ordered material is EC, = - I, and the energy of a locally crystalline liquid is EC,,, = - c/( 1 + N,,), so the expected conditions for the amorphization melting are EC, < E,, < EC,,/, or 2 > - ,u > c/( 1 +N,,). As was mentioned in Section 4.2, the theory allows arbitrary choice of the origin of the energy scale. Here, this scale is chosen to have the constant v = 0, otherwise E,,, EC,,/ and EC, would add the term +v. By the choice v = 0 one must have ~1< 0 to yield the condition of E,, < Ecr,/, so the condition reads t‘ > 1~1> I-:/(1 + N,,,). The melting transition is expected to lead from a well-ordered crystalline state (n z 1, p r” 1) to an amorphous state (n = 0, p ~0); the melting temperature is expected to be T,,
The condition temperature

F( cx ic + ,u - T In N,, . of the phase equilibrium

T,, = (e + ,u)/ln N,, .

(23) F,,(T,)

= F/( T,) gives in this approximation

the melting

(24)

The large number lnN,, in the denominator and a negative .P with eN,,,/( 1 + N,,) > t’ + ,u > 0 shows that T,, -cc. In the next approximation, one calculates the density n of crystalline clusters at T, in the crystal and in the liquid: (25) One sees from formula (25) that the assumption of an amorphization melting that leads from an almost perfect crystal (n - 1 < 1, w - 1<
oj’ the results

Our model operates with only two energy characteristics of a material, one being the energy of the crystalline orientationally ordered state proportional to e, and the other the energy of the

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completely amorphous state, proportional to ,u; the energy of locally crystalline but orientationally disordered state is chosen as the origin of the energy scale. Other important characteristics are the phase volumes (the numbers of states) N,, and N,, of the crystalline and the amorphous states, respectively. The two mechanisms of melting, orientation melting for materials with &$E, and the amorphisation melting for materials having negative ,U and 1,~ CC, are asymptotic cases. A real material may be expected to be closer to one or other mechanism of melting depending on the value of the ratio A = p/c; the materials characterized by a larger A have more of the crystalline component in the liquid. The local structures found in computer-simulated 2D and 3D LennarddJones and other liquids were discussed in Section 3. In 2D systems at the melting point about i of small (7-atom) clusters are non-crystalline; the phase transition is then dominated by the depercolation of the crystalline part of the material. The mean-field approximation used in previous sections of the current paper does not describe the defect molecule dissociation transition (Kosterlitz-Thouless) in 2D systems. The proliferation of non-crystalline clusters does not necessarily contradict the ideas of the dislocation melting theory, but may strongly renormalize the shear modulus that becomes a sharp function of the temperature due to the amorphisation statistics. For the LJ potential and other two-particle interaction potentials studied, ,U and a may be assumed to have the same order of magnitude. In 2D systems, the triangle lattice is unique, having the lowest energy, and simultaneously the highest density. From the point of view of the proposed model, one has then an intermediate case, in which both orientation and amorphisation are important at melting, so the liquid contains a significant concentration of both crystalline and amorphous clusters. In 3D materials, there are two crystalline dense packings corresponding to small (13 atom) clusters of fee and hcp lattices; a non-crystalline competing local structure is icosahedral; the last configuration has a strong energetic preference for twelve-particle coordination shells for most simple pair potentials [50]. One may try to understand the melt starting from a configuration in which these three types of local packing are present with comparable probabilities. Given the level of thermal fluctuations at melting temperatures, and the mutual distortion of atomic positions in neighboring clusters of different packing, one can then hardly expect any definite local order in 3D systems with two-particle interactions of LJ type, in agreement with the results of Section 3.3. In a substantially supercooled computer liquid, small fee clusters nucleate following the temperature quench [23,40]. The idea that locally crystalline behavior can persist above the melting temperature thus seems clearest in materials whose potentials are much more structured and complex than the simple LJ potentials appropriate for atomic liquids. A three (and more) particle potential that favors special bond angles may substantially contribute to the ratio E/F; if these preferred angles correspond to a crystal, the local crystallinity in the melt will be increased. Experimentally observed phase transitions in some melts (Se, Te, Bi) [51, 521 may be interpreted as a change of one local order for another. In these materials, the interaction is more complex than LJ interaction, and there are polymorphic phase transitions in the crystal state between ordered crystalline phases. In addition to the chalcogens and Bi discussed above, spectacular examples occur in hydrogen bonding situations. In carboxylic acids, for example, dimerization occurs even in the vapor phase; such dimers are locally ordered states, and can clearly persist through the phase transition. In the most important liquid, water, there has been a long debate about the structure above the melting point; discussion in terms of flickering clusters and disordered ice islands have been given (for a recent review see [56]). Simulations clearly show that the hydrogen bonding cluster structures remain important in liquid water [57,5X],

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so that the idea discussed here of crystalline structure in the liquid followed by higher-temperature amorphization seems appropriate for liquid water [59-611. More generally, the local crystallinity in the melt should occur in materials with strong covalent, steric, or hydrogen bonding, having welldefined orientationally preferred structures commensurate with some type of crystalline order. An obvious example is any liquid-crystal molecule. Hydrogen-bonded and covalent examples include water (ice), and silicon dioxide. The random bond network model [62], designed to describe the melt structure in these materials may be generalized to include networks with a variable randomness. As a measure of order in such a network, the number of sites having the crystalline arrangement of nearest neighbors may be proposed; the definition of a crystalline arrangement in a small cluster can be given in terms of pattern recognition methods discussed briefly in Section 3.1. The simple model studied above allows many straightfotward generalizations, in particular an introduction of a band of amorphous states, and of an interaction energy between amorphous clusters. We hope that further computer simulation and experimental study will prove sufficient information to test the adequacy of local state models, and give the values of important parameters for the materials.

5. Conclusions In spite of many theoretical studies, the problem of the nature of the arrangements of the atoms in liquids (in particular in the vicinity of the melting line) remains open. The results of computer simulations discussed above indicate that the presence and type(s) of local structures in liquids depend on the kind of interatomic potential and dimensionality of the system. We have tried to discuss the physical picture of liquids as systems in which the competition of various types of local structures leads to different types of liquids. Simple statistical models of this competition studied above show that both locally crystalline and fully amorphous liquids may exist, depending on the type of inter-unit interactions. Simple two-atom interactions, like, e.g. Lennard-Jones interactions, seem to favor (in 3D) amorphous liquids, while more complicated ones may force well-defined local order in the liquid phase. In sum, the statistical mechanics model of melting phase transition introduced in this paper offers a promising starting point for more realistic studies of the melting phase transition, in which both the phases (crystal and liquid) are described by the same structure-based meta-Hamiltonian - this is a goal physicists have sought nearly from the beginning of our century.

Acknowledgements This study of amorphous systems from the local order perspective was inspired by discussions, in the 70s of one of the authors (A.Z.P) with Ilya Michailovich Lifshitz, and especially by Ilya Michailovich’s work on relaxation in polycrystals [63]. Helpful discussions with Boris 1. Shumilo, Michael E. Chertkov, and Professor Harro Hahn are greatly appreciated. ACM thanks Alexander von Humboldt Stifmng for the support of his stay in Germany in 1989/1990, when some of the methods and results presented in this paper were developed and calculated. MR and AP are grateful to the Materials Division of the NSF, for

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support through the Northwestern University MRC (grant #DMR 9120521). tially supported by the NASA grant # NAG3-1932.

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References [l] [2] [3] [4] [5] [6] [7] [8] [9]

[lo] [I l] [12] [13] [ 141 [15]

[ 161 [I71 [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [3 l] [32] [33] [34] [35] [36] [37] [38]

J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, London, 1986. J.M. Ziman, Models of Disorder, Cambridge Univ. Press, Cambridge, 1979. S. Hess, Z. Naturforsch. 35A (1980) 69. D.R. Nelson, J. Toner, Phys. Rev. B 24 (1981) 363. A.C. Mims, A.Z. Patashinskii, Sov. Phys. JETP 53 (1981) 798; Phys. Lett. A 87 (1982) 79. J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6 (1973) 1181. D.R. Nelson, B.I. Halperin, Phys. Rev. B 19 (1979) 2457. A.P. Young, Phys. Rev. B 19 (1979) 1855. K.J. Strandburg, Rev. Mod. Phys. 60 (1988) 161. H. Kleinert, Gauge Fields in Condensed Matter, vol. II, Part III, Ch. 14, World Scientific, Singapore, 1989. F.R.N. Nabarro, Theory of Crystal Dislocations, Clarendon, Oxford, 1967. A.Z. Patashinskii, B.I. Shumilo, Sov. Phys. JETP 62 (1985) 177. A.Z. Patashinskii, L.D. Son, Sov. Phys. JETP 76 (1993) 534. M. Goldstein, J. Chem. Phys. 51 (1969) 3728; ibid. 67 (1977) 2246. F.H. Stillinger, T.A. Weber, Science 225 (1984) 983; J. Chem. Phys. 81 (1984) 5095; ibid. 83 (1985) 4767; ibid. 87 (1983) 2833; Phys. Rev. A 25 (1982) 978; ibid. A28 (1983) 2408; Phys. Rev. B 32 (1985) 5402; T.A. Weber, F.H. Stillinger, J. Chem. Phys. 80 (1984) 438; Phys. Rev. B32 (1985) 5402. R.A. LaViolette, F.H. Stillinger, J. Chem. Phys. 83 (1985) 4079. A.Z. Patashinski, M. Ratner, J. Chem. Phys. 103 (24) (1995) 10779. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987. J. Michalski, A.C. Mitus, A.Z. Patashinskii, Phys. Lett. A 123 (1987) 293. A.C. Mitus, A.Z. Patashinskii, Physica A 150 (1988) 371, 383. A.C. Mitus, A.Z. Patashinskii, in: D. Walgraef, N.M. Ghoniem (Eds.), Patterns, Defects and Materials Instabilities, Khmer, Amsterdam, 1990, p. 185. H. Gades, A.C. Mitus, Physica A 176 (1991) 297. A.C. Mitus, A.Z. Patashinskii, F. Smolej, H. Hahn, Physica A (1966) in press. A.C. Mitus, A.Z. Patashinskii, S. Sokolowski, Physica A 174 (1991) 244. A.C. Mitus, D. Marx, S. Sengupta, P. Nielaba, A.Z. Patashinskii, H. Hahn, J. Phys.: Condens. Matter 5 (1993) 8509. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1986. J.P. Steinhardt, D.R. Nelson, M. Ronchetti, Phys. Rev. B 28 (1983) 784; S. Nose, F. Yonezawa, J. Chem. Phys. 84 (3) (1986) 1803; Solid State Commun. 56 (1985) 1005, 1009. L.D. Landau, E.M. Lifshitz, Quantum Mechanics-Monrelativistic Theory: Course in Theoretical Physics, vol. 3, Pergamon, Oxford, 1962. J.S. van Duijneveldt, D. Frenkel, J. Chem. Phys. 96 (1992) 4655. A.C. Mitus. I. Stolpe. A.Z. Patashinskii, H. Hahn, in preparation; I. Stolpe, Master Thesis, Technical University, Braunschweig, 1995. J. Frenkel, Kinetic Theory of Liquids, Oxford, Clarendon Press, 1946. A.C. Mitus, D. Marx, H. Weber, in preparation. E.J. Meier, D. Frenkel, J. Chem. Phys. 94 (1991) 2269. C.S. Hsu, A. Rahman, J. Chem. Phys. 71 (1979) 4974. W. Swope, H.C. Andersen, Phys. Rev. B 41 (1990) 7042. F. Smolej, Ph.D. Thesis, Cuvilier Verlag, Goettingen, 1955; A.C. Mitus, F. Smolej, A.Z. Patashinski, H. Hahn, in preparation. A. Rahman, J. Chem. Phys. 45 (1966) 2585.

434 [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63]

A. Z. Patashinski

et al. I Physics Reports 288 (1997)

409434

M. Dzugutov, Phys. Rev. A 46 (1992) 2084. A.C. Mitus, F. Smolej, H. Hahn, A.Z. Patashinski, Europhys. Lett. 32 (9) (1995) 777. A.Z. Patashinski, M. Ratner, in preparation. KG. Wilson, J. Kogut, Phys. Rep. 12C (1974) 75. A.Z. Patashinski, V.L. Pokrovskii, Fluctuation Theory of Phase Transitions, 2nd ed., Pergamon Press, Oxford, (in Russian), Nauka, Moscow, 1993. A.C. Mitus, A.Z. Patashinski, Sov. Solid State Phys. 24 (1982) 1633; ibid, 1900. D.K. Remler, A.D.J. Haymet, Phys. Rev. B 35 (1987) 245. R.T. Lyzhva, A.C. Mitus, A.Z. Patashinskii, Sov. Phys. JETP 54 (1981) 1168. E.B. Amitin, Yu.F. Minenkov, O.A. Nabutovskaya, I.E. Paukov, Sov. Phys. JETP 62 (1985) 1207. J.W.M. Frenken, J.F. van der Veen, Phys. Rev. Lett. 54 (1985) 134. A.C. Mitus, A.Z. Patashinskii, B.I. Shumilo, Phys. Lett. A 113 (1985) 41; A.C. Mitus, A.Z. Patashinskii, Acta Pol. A 74 (1988) 779. F.C. Frank, Proc. Roy. Sot. 215 (1952) 43. V.V. Brazhkin, R.N. Voloshin, S.V. Popova, Sov. Phys. JETP Lett. 50 (1989) 424; Phys. Lett. A 166 (1992) V.V. Brazhkin, R.N. Voloshin, S.V. Popova, A.G. Umnov, J. Phys.: Condens. Matter 4 (1992) 1419; Phys. A 154 (1991) 413. A.Z. Patashinski, M.E. Chertkov, preprint INF, Novosibirsk, 1992. R. Lyzwa, A.Z. Patashinski, Acta Phys. Polonica A 64 (1983) 413. A.R. Ubbellohde, The Molten State of Matter, Wiley, New York, 1978. K. Liu, J.D. Cruzan, R.J. Saykally, Science 271 (1996) 929. K. Liu et al., J. Am. Chem. Sot. 116 (1994) 3507; Faraday Discuss. 97 (1994) 35. N. Pugliano, R.J. Saykally, Science 257 (1992) 1937. G. Corongui, E. Clementi, Chem. Phys. Lett. 214 (1993) 367. P.-O. Astrand, A. Wallqvist, G. Karlstrom, J. Chem. Phys. 100 (1994) 1262. F. Franks, Water, Royal Sot. of Chemistry, London, 1983. W.H. Zachariasen, J. Am. Chem. Sot. 54 (1932) 3841. I.M. Lifshitz, JETP 44 (4) (1963) 1349.

1979;

Phys.

383. Lett.