Polyamorphism and Density Anomalies in Network-Forming Fluids

Polyamorphism and density anomalies in network-forming fluids: Zeroth- and first-order approximations C. J. Roberts and Pablo G. Debenedettia) Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 ͑Received 20 February 1996; accepted 4 April 1996͒ A molecular model of network-forming liquids has been formulated in terms of a lattice fluid in which bond formation depends strongly on molecular orientations and local density. The model has been solved in the zeroth- and first-order approximations for molecular and bond geometries similar to water or silica. Results are presented in the form of fluid-phase boundaries, limits of stability, and loci of density extrema. At low temperatures and high pressures the first-order solution shows a liquid–liquid transition with upper and lower consolute temperatures, in addition to vapor–liquid equilibrium; and two loci of density extrema. In contrast to previous models which display a second fluid–fluid transition, the phase behavior of the present model fluid does not result from an a priori imposed dependence of the bonding interaction upon bulk density. The zeroth-order solution shows only one, vapor–liquid, phase transition; and a single, continuous locus of density maxima. The results suggest that low-temperature polyamorphic phase transitions in a pure substance can arise from orientation-dependent interactions; and in particular that a phase transition between two dense fluids can be driven by the greater orientational entropy of high density states which are not fully bonded. The predicted phase behavior is, however, sensitive to the level of approximation used to solve the partition function. © 1996 American Institute of Physics. ͓S0021-9606͑96͒52226-9͔

I. INTRODUCTION structures in the supercooled state. Examples include nega- Network-forming fluids are ubiquitous in nature, water tive thermal expansion coefficients (␣ p) for both silica and water, and increases in isothermal compressibility (␬T) and and silica being the most common examples. They are dis- 4,6 tinguished from ‘‘normal’’ molecular fluids by the presence isobaric heat capacity (c p) for water upon supercooling. of strong, highly directional, intermolecular attractions that Interesting properties of network fluids also occur in 7 8 result in the formation of ‘‘bonds’’ between molecules that their amorphous solid states. Both silica and water are 9 are correctly spaced and oriented with respect to one another. known to exhibit polyamorphism. This term refers to dis- The favorable bonding energy results in the formation of tinct amorphous phases that differ in density and local struc- locally structured regions which, due to geometric con- ture. This phenomenon is especially intriguing in water, as straints on the bonded molecules, have lower density than an apparently first-order, reversible transition occurs between unbonded regions. Such structures in the liquid are local and the two forms of amorphous solid water, LDA and 8,10,11 transient; a fully bonded network exists only in the limit of HDA. Although the amorphous solids are nonergodic low temperatures. For example, transient, ‘‘icelike’’ regions on experimental time scales, the presence of distinct, repro- exist in liquid water, and increase in lifetime and average ducible phases which lack long-range order, and have differ- size as temperature is lowered.1,2 The geometric require- ent densities suggests the existence of an underlying liquid– ments for bonding dictate the spatial distribution of bonds liquid phase separation for a pure fluid. This is quite and the resulting correlations in molecular positions.3,4 We remarkable, as liquid–liquid phase transitions are normally refer collectively to this arrangement of bonds and molecular associated with phases of different composition.12 positions as the topology of the network. In fluids such as A full understanding of the effects of the number, spatial silica, short- and intermediate-range order in the tetrahedrally orientation, and strength of bonds on the global fluid-phase coordinated network is resistant to thermal degradation. Con- behavior of network fluids is still lacking. Particular interest sequently the configurational entropy depends weakly on lies in the low-temperature behavior and the possibility of temperature close to the glass transition. This is the polyamorphic transitions. Because the material properties of ‘‘strong’’ extreme of Angell’s useful classification scheme different amorphous phases may be markedly different ͑e.g., 5 for liquids, the ‘‘fragile’’ limit being typified by one amorphous phase of Si is metallic, while another is a o-terphenyl. In the latter case there are no directional bonds semiconductor13͒, polyamorphism is of interest both as a sci- and no network. As a result, the configurational entropy entific phenomenon and because of its implications for ma- close to the glass transition is sensitive to changes in tem- terials processing. perature. We present here a lattice model of network fluids that Many intriguing properties of liquid silica and water oc- shows two fluid–fluid phase transitions when solved with a cur at low temperatures ͑i.e., well below the equilibrium first-order approximation. This is the first model of its kind freezing point͒ and are related to the presence of network to show this behavior based solely upon the molecular interactions described by the model Hamiltonian, without re- a͒Author to whom correspondence should be addressed. course to imposed density dependences on interaction

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strengths. In contrast, the zeroth-order, or mean-field ͑MF͒ solution, shows only a vapor–liquid transition. Both solutions exhibit density anomalies (␣ pϽ0). Our use of the term mean-field requires clarification. In the study of critical phenomena, theories which assume the Helmholtz energy to be analytical at the critical point are called mean-field theories. They all yield the classical critical exponents. In the study of lattice fluids and magnets, the term mean-field denotes the simplest solution technique, in which no account is taken of short-range order. This is also known as the zeroth-order, Bragg-Williams, or molecular field approximation. Throughout the remainder of this paper, we use the term mean-field in the latter sense. The two approximate solution techniques used in this work yield classi- FIG. 1. BCC lattice showing the four sublattices, ᭹, ᭡, ᭺, and ᭝. Bold cal critical exponents. lines indicate bonds in the fully bonded, diamondlike structure. In this struc- Lattice models have been successfully used to study as- ture all ᭹ and ᭡ are occupied, and all ᭺ and ᭝ are unoccupied. pects of the thermodynamics of network fluids. Behavior such as freeze-expansion, density extrema (␣ ϭ0), and p II. MODEL FORMULATION sharp increases in c p and ␬T with decreasing temperature have been reproduced by lattice models for water,14 and by The lattice is composed of body centered cubic unit cells more general models.15,16 Recent models of Borick et al.,15 which can be equivalently represented by four, interpenetrat- Poole et al.,17 and Ponyatovskii et al.18 have shown two ing, simple cubic sublattices ͑denoted with ᭹, ᭺, ᭡, and ᭝ fluid–fluid transitions. However, two of these models15,17 re- in Fig. 1͒. The central site is surrounded by eight nn sites, quire the additional assumption of an imposed dependence of located at the vertices of the cube as shown in Fig. 2͑a͒. Each the bonding interaction on bulk density, while the other18 is occupied nn pair of sites contributes an attractive energy-⑀. based on the assumption that the molecules exist in one of Every molecule has q distinguishable orientations via rota- two ‘‘states’’ within the liquid. Our results show that an tion about the central atom. When a pair of molecules point additional, low temperature phase transition between fluid their bonding arms toward each other, a bond forms and phases results when a better approximation is used to solve contributes an energy-J. The number and relative orienta- the partition function. In particular, there is no need to im- tions of the bonding arms on a molecule is an input to the pose a density-dependent bonding strength, or to assume par- model. Figure 2͑b͒ illustrates the waterlike geometry used in the present work. The central site represents oxygen, and the ticular ‘‘states’’ of molecules in the fluid. molecule has two proton-donor arms H atoms and two Our model is formulated in terms of molecules with wa- ͑ ͒ proton-acceptor arms ͑electron lone pairs on oxygen͒; arrows terlike geometry, on a body-centered cubic ͑BCC͒ lattice. pointing away and toward the central site indicate the donor The molecules interact via nearest-neighbor (nn) attractions and acceptor arms, respectively. A bond forms when a donor and orientation-specific bonds between nn pairs. The open, arm and an acceptor arm point to one another. This restricts fully bonded state is a tetrahedral network with a diamond or the possible orientations of the remaining arms on each of cubic icelike geometry. In addition to molecular orientations the molecules participating in the bond. The formation of which correspond to the diamond-lattice structure, we also two bonds by the same molecule completely specifies its account for molecular orientations of unbonded and partially orientation. bonded molecules. Included in these considerations are con- An open network is promoted in this model by imposing straints imposed upon the orientations of partially bonded an energy penalty for occupied sites which encircle the line molecules by molecular geometry. Open structures are stabi- 15 of centers of an existing bond. This is illustrated in Fig. 2͑b͒, lized by including a bond-weakening interaction which where the three k sites which lie adjacent to the bond formed causes the local energy and density to be negatively corre- between the ij pair weaken that bond by an energy cJ/3 lated, as in liquid water, and also accounts for cooperative (cϾ0) for each occupied k site. All of these sites lie on the effects associated with the formation of local ‘‘icelike’’ or- same sublattice, which is different from both sublattices oc- der. cupied by molecules involved in the bond ͑compare with The paper is organized as follows. The model is formu- Fig. 1͒. The fourth k site in this cell is opposite to the bond, lated in Sec. II. The zeroth- and first-order solutions are dis- and does not weaken it. Thus, a bond surrounded by occu- cussed in Secs. III and IV, respectively. Finally, in Sec. V we pied sites is ‘‘fully weakened’’ and contributes an energy of present calculations on the phase behavior, stability limits, Ϫ(1Ϫc)J, as indicated in Fig. 2͑a͒. and density anomalies predicted by the two solution methods The potential energy of the system is expressed math- for different interaction strengths and degrees of orienta- ematically by the following Hamiltonian: tional entropy. We discuss the differences between the pre- ␹ dictions of the two solutions, and the molecular origins of the * 1 HϭϪ⑀ n n ϪJ n n ␦ ͑1Ϫcn ͒. ͑1͒ ͚ i j ͚ i j ␴i␴j ͚ k second fluid–fluid phase transition in the first-order solution. ͗ij͘ ͗ij͘ k͑ij͒ ␹

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We consider its low-temperature stability relative to the close-packed, ordered structure formed when the four sublattices are ﬁlled and fully bonded to each other ͑e.g., ᭹ with ᭡, and ᭺ with ᭝͒. This is the analogue of ice VII in our model. For the solid with an open, diamondlike structure, the number of molecules, M, and sites, N, are related by MϭN/2, and all molecules lie on the appropriate sublattices and are properly oriented for bonding ͑see Fig. 1͒. Therefore no bonds are weakened, and Eq. ͑1͒ becomes

EopenϭϪ͑2M⑀ϩ2MJ͒ϭϪN͑⑀ϩJ͒, ͑2͒ from which we can write the enthalpy

Hopenϭ2M͑Pv0Ϫ⑀ϪJ͒, ͑3͒

where v0 is the volume per site. A similar procedure for the close-packed, fully occupied (MϭN) and bonded lattice gives

HclosedϭM͑Pv0Ϫ4⑀Ϫ2Jϩ2cJ͒. ͑4͒ At 0 K, the Gibbs free energy is equal to the enthalpy, so the equilibrium pressure for the transition between these two FIG. 2. ͑a͒ Illustration of the relative positions of interacting tetrahedral structures is found by equating specific enthalpies, from molecules and the associated energies; dotted lines indicate nonbonding pairs, bold full lines indicate bonded pairs. All sites are assumed occupied which we find and with molecules in the correct orientations for bonding if applicable. Pv ϭ2 cJϪ⑀͒. ͑5͒ Bonds are weakened by occupied sites surrounding the line of centers of the 0 ͑ bonding pair. See text for notation of interaction energies. ͑b͒ Effects of Thus, if cϾ⑀/J there exists a range of positive pressures for molecular orientations for bond formation; the illustration is for a partially bonded tetrahedral molecule with two proton-donor arms ͑arrows point which the ground state is open and fully bonded. All values away from central molecule͒ and two proton-acceptor arms ͑arrows point of model parameters used in this work satisfy this condition. toward central molecule͒. The bond is indicated by the filled arrow from the i to the j site. Note the possible bond-weakening sites ͑three of four k sites͒, and possible choices for further bonding ͑indicated by the dashed arrows͒. III. MEAN-FIELD SOLUTION We begin with the grand canonical partition function, In Eq. ͑1͒, n is an occupation variable for site i; n is1ifthe i i ⌶͑␮,V,T͒ϭ e␤␮͚jnjeϪ␤H, ͑6͒ site is occupied, 0 otherwise. Each molecule is considered to ͚ ͚ ͕ni͖ ͕␴i͖ have q distinguishable orientations. The different orientations of the molecule on site i are denoted by a Potts variable where ͕ni͖ and ͕␴i͖ indicate summations over all possible ␴ (␴ ϭ1,2,...,q). The symbol ␦ is given the value 1 if distributions of occupation numbers and molecular orienta- i i ␴i␴ j tions, respectively. In Eq. ͑6͒, ␮ is the chemical potential, V the orientations of molecules i and j are correct for bonding, is the total volume, and T is the temperature. Using Eq. ͑1͒, 0 otherwise. The notation ͗ij͘ specifies that the summation is over nearest-neighbor pairs. The superscript * on the sec- ␤␮͚jnj ond summation indicates that restrictions imposed by va- ⌶͑␮,V,T͒ϭ͚ ͚ e QnbQb , ͑7͒ ͕ni͖ ͕␴i͖ lency and bond geometry are to be included when determining which nn pairs are allowed to be simultaneously bonded. where The third summation is over the bond weakening (k) sites of ␤⑀͚ ij ninj Qnbϭe ͗ ͘ ͑8͒ a given bonding pair (ij). The number of potential bond weakeners ͑␹͒ depends upon the molecular geometry. As and 3 noted above, ␹ is 3 for the present treatment. ␤͑J/3͚͒* n n ␦ ͚ ͑1Ϫcn ͒ Q ϭe ͗ij͘ i j ij k͑ij͒ k . ͑9͒ Ordered, solid phases, which are not included in this b work, arise via preferential occupation of one or more sub- In writing the above expressions we have used ␤ϭ1/kT, k is Boltzmann’s constant, ␦ ϵ ␦ , and have let subscripts nb lattices. It is useful, however, to consider the relative stabili- ij ␴i␴j ties of open- and close-packed ordered phases so as to and b denote nonbonding and bonding contributions to the choose values of the model parameters that will promote partition function. open, bonded structures in the fluid over a range of positive We then average over the distribution of orientation vari- pressures. An open, ordered phase similar to that of cubic ice ables, using a single-graph approximation and sphericalliza- occurs if two sublattices are filled and fully bonded, while tion technique.16,19 This reduces the partition function to the remaining two sublattices are vacant, as shown in Fig. 1. summations over occupation variables, and replaces J with a

J. Chem. Phys., Vol. 105, No. 2, 8 July 1996

temperature dependent, effective bonding interaction param- where we must include a factor of 2 in pb since we can eter ␦J. Using the techniques detailed in Appendix B, Eq. ͑7͒ distinguish between bonding as an A – D pair and bonding as can then be written a D – A pair, for a selected molecule. We next apply the mean-field approximation to the occupation variables.15,16 We denote the fractional occupation ,V,T e͑␤␮ϩln q͚͒ jn j ⌶͑␮ ͒Ϸ͚ of each sublattice by ␳ (iϭ1,...,4), and the overall fraction ͕ni͖ i of occupied sites by ␳(ϭM/N). The equation of state then ␤⑀͚ n n ␤␦J͚* n n ͚ ͑1Ϫc*n ͒ ϫe ͗ij͘ i je ͗ij͘ i j k͑ij͒ k . ͑10͒ follows from PNv0ϭkT ln ⌶͑using the maximum term of ⌶ and the fact that ␳1ϭ␳2ϭ␳3ϭ␳4ϭ␳ for fluid phases͒ as In Eq. ͑10͒, c* is formally an effective bond-weakening pa- P*ϭ͑␮*ϩT* ln q͒␳ϩ4x␳2ϪT*͑␳ ln ␳ϩ͑1Ϫ␳͒ rameter; expressions for c* and ␦J are given in Appendix B. 2 These quantities require knowledge of pb , the probability of ϫln͑1Ϫ␳͒͒ϩ12␦J*␳ ͑1Ϫc*␳͒. ͑13͒ a bond being formed between a nn pair when orientations of The following reduced variables are used in the above the molecules are distributed randomly. We derive an ex- expression: P*ϵPv /J, ␮*ϵ␮/J, xϵ⑀/J, T*ϵkT/J, and pression for p as follows. 0 b ␦J*ϵ␦J/J. We can eliminate ␮* from Eq. ͑13͒ by realizing Consider a pair of molecules situated on nn sites, each that at equilibrium ln ⌶(␮,V,T) must satisfy with q distinguishable orientations. For a tetrahedral mol- ␳͒ ϭ0 and thusץ/ln ⌶ ץ͑ ecule with two donor (D) arms and two acceptor (A) arms, ␮,V,T P* ␳ץ a bond is formed when a D(A) arm is in line with a nn ϭ0ϭ␮*ϩT* ln qϩ8x␳ϪT* ln ␳ ͪ ͩ 1Ϫ␳ͪץ ͩ -A(D) arm. The MF estimate for the bonding probability fol lows from the assumption that bond formation between a 2 given nn pair is independent of bonds formed by that pair ϩ12␦J*͑2␳Ϫ3c*␳ ͒. ͑14͒ with other molecules. However, it should be realized that we Rearranging for ͑␮*ϩT* ln q͒ gives still enforce the restriction on the number of bonds possible * ␳ for a single molecule when evaluating the summation ͚͗ij͘ in ͑␮*ϩT* ln q͒ϭϪ8x␳ϩT* ln Eq. ͑10͒ explicitly. Thus each molecule can participate in a 1Ϫ␳ maximum of only four bonds simultaneously. In the mean- Ϫ12␦J*͑2␳Ϫ3c*␳2͒. ͑15͒ field approximation each bond is treated independently. With this approximation, pb is the product of the individual prob- Substituting this in Eq. ͑13͒ yields abilities of each molecule in the pair pointing an arm to one P*ϭϪa*␳2ϪT* ln͑1Ϫ␳͒, ͑16͒ of its nn sites, multiplied by a factor to account for the distinguishability of D-A and A-D pairings. The probability with for one molecule to be pointing an arm to a given nn site is a*ϭ4xϩ12␦J*͑1Ϫ2c*␳͒. ͑17͒ equal to the fraction of its q orientations in which one arm is fixed in a given direction. We will denote by n the number of We see that the MF fluid equation of state has a form similar to that seen in previous, related work on network- orientations available to a molecule having the direction of 15 one arm fixed. The single molecule probability then is n/q. fluid, lattice models: a modified van der Waals equation, To derive a relationship for n in terms of q we note that with a density- and temperature-dependent attractive param- a molecule with one arm fixed can rotate the other arm͑s͒ eter a*, and a logarithmic repulsive term in the place of the through 360°, about the fixed arm. Thus n is essentially the van der Waals excluded volume term, kT/(vϪb). As we number of increments into which we discretize those 360°. show in Appendix A, this equation of state is incapable of Now consider the central atom to be at the origin of a sphere. showing a second fluid–fluid phase transition. The mean- We can point a given arm in n2/2 directions ͑i.e., rotating field solution has essentially blurred all molecular details of over 360° in one plane, and over 180° perpendicular to the bonding via the averaging of the orientational and bonding plane at each point along that circle͒. We must also include states. The loss of the details of molecular geometry and the n orientations due to rotating the other three arms about connectivity arises because the mean-field expression for pb the axis of the given arm, for each of the n2/2 directions. assumes orientations of different molecules to be indepen- This gives a total of n3/2 orientations. However, for a mol- dent, thereby neglecting the cooperative nature of bonding. ecule with two D and two A arms, this will overcount the number of distinguishable orientations by 2. Applying this to IV. EXTENDED QUASICHEMICAL SOLUTION a molecule with a waterlike geometry, we have We begin the first-order or extended quasichemical ͑QC͒ n3ϭ4q ͑11͒ solution with the canonical partition function

and Q͑M,V,T͒ϭ ⍀͑M,V,E͒eϪ␤E, ͑18͒ ͚E 2 4/3 n 2 where E is the potential energy corresponding to a conﬁgu- p ϭ2 ϭ2 , ͑12͒ b ͩ qͪ ͩ qͪ ration with a given M and V, and ⍀ is the number of distin-

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guishable ways of arranging M molecules on N(ϭV/v0) lat- ecules while preserving the energy, bonds, relative positions tice sites for a given value of E. Other notation is the same and relative orientations that define that cell configuration. as that introduced previously. In writing Eq. ͑18͒ we have Expressions for ⑀ j and ␻ j , can be written a priori in terms of omitted contributions to the partition function from kinetic the parameters of the model, and are given in Table I, where energy terms and from internal degrees of freedom of the the possible cell configurations are enumerated ͑n has the molecules; which removes a temperature-dependent, multi- same meaning in Table I as it does in our mean-field solu- plicative term. This does not affect our results, because the tion͒. The assumption in writing the ␻ j expressions in Table equation of state is derived from the derivative of ln Q with I is that each cell is independent of all others with respect to respect to volume at constant temperature ͑as shown below͒, the orientations that molecules on the vertices of the cube and because this multiplicative term contributes equally to may possess. The probability of a given cell configuration the chemical potentials of different phases at equilibrium. from among the entries in Table I is ␻ j␺ j , and thus ␺ j is the The equation of state P(␳,T) is obtained from the Helm- probability of observing a cell for which the exact number, holtz free energy A(ϭϪkT ln Q) placement, and orientations of molecules in the cell correspond to one of the degenerate arrangements of cell configu- -ln Q ration j. The values of each ␺ in the limit of infinite tem ץ Aץ PϭϪ ϭkT , ͑19͒ j .V ͪ perature are also given in Table Iץ ͩ Vͪץͩ T T The entries in Table I are to be interpreted in the follow- where AϭaN. We approximate Q by its maximum term, ing manner. Consider entry 37; this describes the cell con- reducing the problem to that of maximizing ln Q, subject to figuration shown in Fig. 3. From Table I, we see that there the constraints of ␳ and T ͑or M,N,T͒. are five occupied sites ͑A – E in Fig. 3͒, four of which are The method used here is formally identical to that de- corner sites (B – E). One bond exists ͑between molecules A 20 scribed in Debenedetti et al. The lattice is divided into a and B͒. There are two molecules (n4ϩn5ϩn6ϩn7ϭ2) number of identical, fundamental ‘‘cells.’’ The possible con- which cannot bond to A because of geometric restrictions figurations within one cell are fully enumerated with the as- imposed on the orientations of A by the A – B bond. One sumption that each cell is independent of all others. The molecule (C) weakens the A – B bond (n5ϭ1). The mol- configurations of the full lattice are then approximated by ecule D is opposite the bond and so does not weaken it assuming that cells are distributed randomly, with a distribu- (n4ϭ1). There are no molecules which simultaneously tion of cell-configuration probabilities that minimizes the weaken 2 or 3 bonds (n6ϭn7ϭ0). We have now accounted Helmholtz free energy. The geometry of the BCC lattice for four of the five molecules using the entries for n1 to n7 . does not lend itself to a tesselation procedure such as that Therefore the remaining molecule (E) occupies a site which used in Debenedetti et al.;20 such a procedure would here is not excluded from bonding with A, but is not bonded to A lead to differentiating between molecules on different sublat- because the orientations of the AE pair are not correct for tices, which is inadequate for the treatment of disordered bonding. Note that the configurations in Table I do not have different entries for differences in donor–acceptor and phases. The number of cells, Ncells , is chosen so as to preserve the number of nn site contacts existing in the actual acceptor–donor bonds with the central molecule. These differences are incorporated into the expressions for . lattice. Thus, NcellsϭN/2. The fundamental cell is chosen to ␻ j be large enough to represent all bonded and nonbonded con- The number of possible configurations of the system, figurations of the central molecule with its nearest neighbors. including molecular orientations and vacancies, is estimated Thus, the geometry of the model ͑e.g., bond orientation͒ is by assuming that the Ncells cells are independent of each preserved exactly for clusters up to nine molecules. As will other and that their m possible realizations are randomly dis- be discussed later, this approach does not account for inter- tributed on the lattice, with a frequency of occurrence to be actions of molecules on the corner sites with molecules in determined as explained below. Since the cells are independent, we write neighboring cells. The choice of NcellsϭN/2 does, however, preserve the number of nonbonding interactions; but bonds are determined only within the fundamental cell. Therefore Ncells! ⍀ϭ⍀ . ͑20͒ the lattice geometry, though used explicitly when enumerat- 0 m ␻ j ⌸ jϭ1͑͑␺jNcells͒!͒ ing the possible configurations of the fundamental cell, is not preserved when calculating the number of configurations for In Eq. ͑20͒, ⍀ is a normalization constant. We determine its the entire system, because cells are assumed to be randomly 0 value by forcing ⍀ to be accurate in the high temperature mixed. This does not pose severe problems for disordered limit, where the distribution of molecules and orientations is phases such as we consider here, but could affect the accu- random, and thus racy of calculations involving phases with long-range order.

We denote the total number of cell configurations by m ␳N Ncells! N!q ͑mϭ98 in the present model͒, and denote the number of ⍀ ϭ ͑21͒ 0 m ϱ ␻ j occupied sites, the energy, and the degeneracy for the jth ⌸ jϭ1͓͑␺j Ncells͔!͒ ͓␳N͔!͓͑1Ϫ␳͒N͔! cell-configuration as n j , ⑀ j , and ␻ j , respectively. By degeneracy of a cell configuration we mean the number of equiva- in which we have denoted the infinite temperature value of ϱ lent ways of placing and orienting the given number of mol- ␺ j by ␺ j . To determine the energy we use

J. Chem. Phys., Vol. 105, No. 2, 8 July 1996

TABLE I. Conﬁgurations, energies, and degeneracies of a fundamental cell.a

ϱ jn1n2n3n4n5n6n7 ␻j ⑀j/J ␺j 10000000 1 0 ͑1Ϫ␳͒9 21000000 q 0 ␳(1Ϫ␳)8/q 3 1100000 8q 0 ␳(1Ϫ␳)8/q 4 2110000 16n2 ϪxϪ1 ␳2(1Ϫ␳)7/q2 2 2 7 2 5 2100000 8qϪ␻4 Ϫx ␳(1Ϫ␳) /q 6 2200000 28q2 0 ␳2(1Ϫ␳)7/q2 7 3220000 120n2 Ϫ2xϪ2 ␳3(1Ϫ␳)6/q3 8 3210000 48n2(qϪ3) Ϫ2xϪ1 ␳3(1Ϫ␳)6/q3 9 3211000 16n2q Ϫ2xϪ1 ␳3(1Ϫ␳)6/q3 10 3210100 48n2q Ϫ2xϪ1ϩc/3 ␳3(1Ϫ␳)6/q3 3 3 6 3 11 3200000 28qϪ⌺␻j,jϭ7...10 Ϫ2x ␳ (1Ϫ␳) /q 12 3300000 56q3 0 ␳3(1Ϫ␳)6/q3 13 4330000 48n3 Ϫ3xϪ3 ␳4(1Ϫ␳)5/q4 14 4320000 240n2(qϪn) Ϫ3xϪ2 ␳4(1Ϫ␳)5/q4 15 4320100 240n2q Ϫ3xϪ2ϩc/3 ␳4(1Ϫ␳)5/q4 16 4320010 240n2q Ϫ3xϪ2ϩ2c/3 ␳4(1Ϫ␳)5/q4 17 4310000 48n2(q2Ϫ6qϩ3n) Ϫ3xϪ1 ␳4(1Ϫ␳)5/q4 18 4311000 48n2q(qϪ3) Ϫ3xϪ1 ␳4(1Ϫ␳)5/q4 19 4310100 144n2q(qϪ3) Ϫ3xϪ1ϩc/3 ␳4(1Ϫ␳)5/q4 20 4311100 48n2q2 Ϫ3xϪ1ϩc/3 ␳4(1Ϫ␳)5/q4 21 4310200 48n2q2 Ϫ3xϪ1ϩ2c/3 ␳4(1Ϫ␳)5/q4 4 4 5 4 22 4300000 56qϪ⌺␻j,jϭ13...21 Ϫ3x ␳ (1Ϫ␳) /q 23 4400000 70q4 0 ␳4(1Ϫ␳)5/q4 24 5440000 12n4 Ϫ4xϪ4 ␳5(1Ϫ␳)4/q5 25 5430000 48n3(qϪn) Ϫ4xϪ3 ␳5(1Ϫ␳)4/q5 26 5430010 144n3q Ϫ4xϪ3ϩ2c/3 ␳5(1Ϫ␳)4/q5 27 5430001 48n3q Ϫ4xϪ3ϩc␳5(1Ϫ␳)4/q5 28 5420000 120n2(qϪn)2 Ϫ4xϪ2 ␳5(1Ϫ␳)4/q5 29 5420100 480n2q(qϪn) Ϫ4xϪ2ϩc/3 ␳5(1Ϫ␳)4/q5 30 5420010 480n2q(qϪn) Ϫ4xϪ2ϩ2c/3 ␳5(1Ϫ␳)4/q5 31 5420200 120n2q2 Ϫ4xϪ2ϩ2c/3 ␳5(1Ϫ␳)4/q5 32 5420110 480n2q2 Ϫ4xϪ2ϩc/3 ␳5(1Ϫ␳)4/q5 33 5420020 120n2q2 Ϫ4xϪ2ϩ4c/3 ␳5(1Ϫ␳)4/q5 34 541000016n2(q3Ϫ9q2ϩ9nqϪ3n2) Ϫ4xϪ1 ␳5(1Ϫ␳)4/q5 35 5411000 48n2q(q2Ϫ6qϩ3n) Ϫ4xϪ1 ␳5(1Ϫ␳)4/q5 36 5410100 144n2q(q2Ϫ6qϩ3n) Ϫ4xϪ1ϩc/3 ␳5(1Ϫ␳)4/q5 37 5411100 144n2q2(qϪn) Ϫ4xϪ1ϩc/3 ␳5(1Ϫ␳)4/q5 38 5410200 144n2q2(qϪn) Ϫ4xϪ1ϩ2c/3 ␳5(1Ϫ␳)4/q5 39 5411200 48n2q3 Ϫ4xϪ1ϩ2c/3 ␳5(1Ϫ␳)4/q5 40 5410300 16n2q3 Ϫ4xϪ1ϩc␳5(1Ϫ␳)4/q5 5 5 4 5 41 5400000 70qϪ⌺␻j,jϭ24...40 Ϫ4x ␳ (1Ϫ␳) /q 42 5500000 56q5 0 ␳5(1Ϫ␳)4/q5 43 6540001 48n4q Ϫ5xϪ4ϩc␳6(1Ϫ␳)3/q6 44 6530010 144n3q(qϪn) Ϫ5xϪ3ϩ2c/3 ␳6(1Ϫ␳)3/q6 45 6530001 48n3q(qϪn) Ϫ5xϪ3ϩc/3 ␳6(1Ϫ␳)3/q6 46 6530020 144n3q2 Ϫ5xϪ3ϩ4c/3 ␳6(1Ϫ␳)3/q6 47 6530011 144n3q2 Ϫ5xϪ3ϩ5c/3 ␳6(1Ϫ␳)3/q6 48 6520100 240n2q(qϪn)2 Ϫ5xϪ2ϩc/3 ␳6(1Ϫ␳)3/q6 49 6520010 240n2q(qϪn)2 Ϫ5xϪ2ϩ2c/3 ␳6(1Ϫ␳)3/q6 50 6520200 240n2q2(qϪn) Ϫ5xϪ2ϩ2c/3 ␳6(1Ϫ␳)3/q6 51 6520110 960n2q2(qϪn) Ϫ5xϪ2ϩc␳6(1Ϫ␳)3/q6 52 6520020 240n2q2(qϪn) Ϫ5xϪ2ϩ4c/3 ␳6(1Ϫ␳)3/q6 53 6520210 240n2q3 Ϫ5xϪ2ϩ4c/3 ␳6(1Ϫ␳)3/q6 54 6520120 240n2q3 Ϫ5xϪ2ϩ5c/3 ␳6(1Ϫ␳)3/q6 55 651100016n2q(q3Ϫ9q2ϩ9nqϪ3n2) Ϫ5xϪ1 ␳6(1Ϫ␳)3/q6 56 651010048n2q(q3Ϫ9q2ϩ9nqϪ3n2) Ϫ5xϪ1ϩc/3 ␳6(1Ϫ␳)3/q6 57 6511100 144n2q2(q2Ϫ6qϩ3n) Ϫ5xϪ1ϩc/3 ␳6(1Ϫ␳)3/q6 58 6510200 144n2q2(q2Ϫ6qϩ3n) Ϫ5xϪ1ϩ2c/3 ␳6(1Ϫ␳)3/q6 59 6511200 144n2q3(qϪ3) Ϫ5xϪ1ϩ2c/3 ␳6(1Ϫ␳)3/q6 60 6510300 48n2q3(qϪ3) Ϫ5xϪ1ϩc ␳6(1Ϫ␳)3/q6 61 6511300 16n2q4 Ϫ5xϪ1ϩc␳6(1Ϫ␳)3/q6 6 6 3 6 62 6500000 56qϪ⌺␻j,jϭ43...61 Ϫ5x ␳ (1Ϫ␳) /q 63 6600000 28q6 0 ␳6(1Ϫ␳)3/q6 64 7640002 72n4q2 Ϫ6xϪ4ϩ2c␳7(1Ϫ␳)2/q7 65 7630020 144n3q2(qϪn) Ϫ6xϪ3ϩ4c/3 ␳7(1Ϫ␳)2/q7 66 7630013 144n3q2(qϪn) Ϫ6xϪ3ϩ5c/3 ␳7(1Ϫ␳)2/q7

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TABLE I. ͑Continued.͒

ϱ jn1n2n3n4n5n6n7 ␻j ⑀j/J ␺j 67 7630030 48n3q3 Ϫ6xϪ3ϩ2c␳7(1Ϫ␳)2/q7 68 7630021 144n3q3 Ϫ6xϪ3ϩ7c/3 ␳7(1Ϫ␳)2/q7 69 7620200 120n2q2(qϪn)2 Ϫ6xϪ2ϩ2c/3 ␳7(1Ϫ␳)2/q7 70 7620110 480n2q2(qϪn)2 Ϫ6xϪ2ϩc␳7(1Ϫ␳)2/q7 71 7620020 120n2q2(qϪn)2 Ϫ6xϪ2ϩ4c/3 ␳7(1Ϫ␳)2/q7 72 7620210 480n2q3(qϪn) Ϫ6xϪ2ϩ4c/3 ␳7(1Ϫ␳)2/q7 73 7620120 480n2q3(qϪn) Ϫ6xϪ2ϩ5c/3 ␳7(1Ϫ␳)2/q7 74 7620220 120n2q4 Ϫ6xϪ2ϩ2c␳7(1Ϫ␳)2/q7 75 761110048n2q2(q3Ϫ9q2ϩ9nqϪ3n2) Ϫ6xϪ1ϩc/3 ␳7(1Ϫ␳)2/q7 76 761020048n2q2(q3Ϫ9q2ϩ9nqϪ3n2) Ϫ6xϪ1ϩ2c/3 ␳7(1Ϫ␳)2/q7 77 7611200 144n2q3(q2Ϫ6qϩ3n) Ϫ6xϪ1ϩ2c/3 ␳7(1Ϫ␳)2/q7 78 7610300 48n2q3(q2Ϫ6qϩ3n) Ϫ6xϪ1ϩc␳7(1Ϫ␳)2/q7 79 7611300 48n22q4(qϪ3) Ϫ6xϪ1ϩc ␳7(1Ϫ␳)2/q7 7 7 2 7 80 7600000 28qϪ⌺␻j,jϭ64...79 Ϫ6x ␳ (1Ϫ␳) /q 81 7700000 8q7 0 ␳7(1Ϫ␳)2/q7 82 8740003 48n4q3 Ϫ7xϪ4ϩ3c␳8(1Ϫ␳)/q8 83 8730030 48n3q3(qϪn) Ϫ7xϪ3ϩ2c␳8(1Ϫ␳)/q8 84 8730021 144n3q3(qϪn) Ϫ7xϪ3ϩ7c/3 ␳8(1Ϫ␳)/q8 85 8730031 48n3q4 Ϫ7xϪ3ϩ3c␳8(1Ϫ␳)/q8 86 8720210 240n2q3(qϪn)2 Ϫ7xϪ2ϩ4c/3 ␳8(1Ϫ␳)/q8 87 8720120 240n2q3(qϪn)2 Ϫ7xϪ2ϩ5c/3 ␳8(1Ϫ␳)/q8 88 8720220 240n2q4(qϪn) Ϫ7xϪ2ϩ2c␳8(1Ϫ␳)/q8 89 871120048n2q3(q3Ϫ9q2ϩ9nqϪ3n2) Ϫ7xϪ1ϩ2c/3 ␳8(1Ϫ␳)/q8 90 871030016n2q3(q3Ϫ9q2ϩ9nqϪ3n2) Ϫ7xϪ1ϩc ␳8(1Ϫ␳)/q8 91 8711300 48n2q4(q2Ϫ6qϩ3n) Ϫ7xϪ1ϩc␳8(1Ϫ␳)/q8 8 8 8 92 8700000 8qϪ⌺␻j,jϭ82...91 Ϫ7x ␳ (1Ϫ␳)/q 93 8800000 q8 0 ␳8(1Ϫ␳)/q8 94 9840004 12n4q4 Ϫ8xϪ4ϩ4c ␳9/q9 95 9830031 48n3q4(qϪn) Ϫ8xϪ3ϩ3c ␳9/q9 96 9820220 120n2q4(qϪn)2 Ϫ8xϪ2ϩ2c ␳9/q9 97 981130016n2q4(q3Ϫ9q2ϩ9nqϪ3n2) Ϫ8xϪ1ϩc ␳9/q9 9 9 9 98 9800000 qϪ⌺␻j,jϭ94...97 Ϫ8x ␳ /q

a n1 : number of occupied sites; n2 : number of occupied corner sites; n3 : number of bonds; n4Ϫn7 : molecules which cannot bond to the central molecule because of constraints due to existing bonds, and which simultaneously weaken existing bonds: n4 denotes the number of such molecules weakening 0 bonds; n5 , the number weakening 1 bond; n6 , the number weakening 2 bonds; n7 , the number weakening 3 bonds.

m ␺ j values are not completely independent, as the distribution EϭNcells͚ ⑀ j␻ j␺ j . ͑22͒ ͕␺ j͖ must satisfy the material balances on occupied sites and jϭ1 vacancies, Our problem is that of maximizing ln ⍀Ϫ␤E with re- 1 m spect to the ͕␺ ͖ at a given T and ␳. Note, however, that the j ␳ϭ ␺ j␻ jn j , ͑23͒ ␣ j͚ϭ1

1 m ͑1Ϫ␳͒ϭ ␺ j␻ j͑␣Ϫn j͒, ͑24͒ ␣ j͚ϭ1 with the number of sites per cell denoted by ␣͑ϭ9͒. The normalization of ͕␺ j͖ follows from the overall material balance on sites ͓which is just a linear combination of Eqs. ͑23͒ and ͑24͔͒ m

1ϭ ␺ j␻ j . ͑25͒ j͚ϭ1 Mathematically, the material balances remove two degrees of freedom from the problem of minimizing the free ener- FIG. 3. Configuration 37 from Table I. Sublattices are not distinguished in gy with respect to ͕␺ j͖. We thus have two dependent prob- this diagram. Occupied and unoccupied sites are indicated by filled and abilities, which we call ␺ and ␺ . The manner in which unfilled circles, respectively. Identity of bond arms as donor or acceptor, and c d orientations of unbonded arms are not included. Refer to text for labeling of these are included in the derivation is shown in detail in molecules A to E. Appendix C.

J. Chem. Phys., Vol. 105, No. 2, 8 July 1996

Returning now to Eq. ͑21͒ and solving for ln ⍀o ͓using ␺ j ncϪn j ͑ndϪn j͒ ln ϭ ln rϪ␤ ⑀ ϩ⑀ Stirling’s approximation and Eq. ͑25͔͒ gives ͩ ␺ ͪ n Ϫn ͩ j c ͑n Ϫn ͒ c c d c d m N ͑n jϪnc͒ ϱ ϱ ϩ⑀ , ͑32͒ ln ⍀oϭ ͚ ␻ j␺ j ln ␺ j ϪN͓␳ ln͑␳/q͒ d ͑n Ϫn ͒ͪ 2 jϭ1 c d where rϵ␺ /␺ . Using this set of equations we rearrange ϩ͑1Ϫ␳͒ln͑1Ϫ␳͔͒. ͑26͒ d c Eqs. ͑23͒ and ͑24͒ to read

We now have expressions for all the terms in Eq. ͑18͒. ␳ϭ␺c␪, ͑33͒ Equating the partition function to its maximum term, we write for the Helmholtz free energy per site (aϭA/N) from 1Ϫ␳ϭ␺c␥, ͑34͒ Eqs. ͑18͒, ͑20͒, ͑22͒, and ͑26͒: which can be combined to give

m m ␪ 1 ␳͑r,T͒ϭ , ͑35͒ aϭ ͚ ⑀ j␻ j␺ jϩkT ͚ ␻j␺j ln ␺ j ␪ϩ␥ 2 ͫ jϭ1 ͩ jϭ1 with m ␳ ϱ ϱ * Ϫ ͚ ␻ j␺ j ln ␺ j ϩ2␳ ln ϩ2͑1Ϫ␳͒ln͑1Ϫ␳͒ , 1 ␺ j jϭ1 q ͪͬ ␪ϭ ␻ n ϩ␻ n ϩ␻ n r , ͑36͒ ͚ j j c c d d 9 ͩ j ␺c ͪ ͑27͒ 1 * ␺ j where the ͕␺ ͖ are to be determined by maximizing the ge- ␥ϭ ͚ ␻ j͑9Ϫn j͒ϩ␻c͑9Ϫnc͒ϩ␻d͑9Ϫnd͒r . j 9 ͩ j ␺c ͪ neric term in the partition function. We now write an expres- ͑37͒ sion for pressure using a more convenient form of Eq. ͑19͒ Here the symbol * denotes a summation over jÞc,d. The a structure of the equations is thereforeץ Pv ϭ␳ Ϫa. ͑28͒ ␳ͪ PϭP͑␳,r,T͒, ͑38͒ץ ͩ o T ␳ϭ␳͑r,T͒, ͑39͒ Using Eq. ͑28͒ gives ͑see also, Appendix C͒ ␮ϭ␮͑␳,r,T͒. ͑40͒ ␺ϱץ m ␳kT j ϱ This system is solved by ﬁxing ␳ and T; solving ͑numeri- PvoϭϪaϩ Ϫ͚ ␻j ͑1ϩln ␺ j ͒ ␳ ͪ cally͒ for r in Eq. ͑35͒; and using (r,␳,T) to calculate P andץ ͩ jϭ1 ͫ 2 ␮. 9 ␺d ␳ ϩ ␤͑⑀cϪ⑀d͒Ϫln ϩ2ln ncϪnd ͫ ͩ ␺c ͪͬ q͑1Ϫ␳͒ͬ V. RESULTS AND DISCUSSION ͑29͒ We now present phase diagrams, stability limits, and loci of density extrema for the QC and MF solutions. Calcula- ␳)T we have also tions were performed for a range of values of the modelץ/aץ)and with ␮ϭ parameters. The values for the ratio of nonbonding to bond- ␺ϱ ing strengths (x), the bond-weakening penalty as a fractionץ m kT j ϱ 9 ␮ϭ Ϫ͚ ␻j ͑1ϩln ␺ j ͒ϩ ␤͑⑀c of bond strength (c), and the number of distinguishable ori- ͫ ␳ ͪ ncϪndץ ͩ jϭ1 ͫ 2 entations of a molecule (q) were systematically varied over

␺d ␳ the ranges ͓0.1,0.53͔, ͓0.3,0.8͔, and ͓12,240͔, respectively. Ϫ⑀d͒Ϫln ϩ2ln . ͑30͒ The values for x were based on the assumption that bonding ͩ ␺c ͪͬ q͑1Ϫ␳͒ͬ interactions are approximately an order of magnitude stron- ger than nonbonding interactions e.g., hydrogen bond Provided we have the equilibrium ͕␺ ͖ at a given T and ͑ j strengths relative to van der Waals attractions3 . The values ␳, we now have P and ␮ as a function of T and ␳. In order to ͒ for c were chosen to satisfy xϽcϽ1, as discussed in Sec. II. determine the ͕␺ ͖, we use the equilibrium requirement that j The lower bound of 12 for q corresponds to the distinguish- A ͑and thus a͒ be a minimum with respect to variations in able orientations of a molecule which is able to point its arms the independent ␺ ’s at constant ␳. j at only its nn sites; the upper bound was chosen as 240 after determining that further increases in q caused only minor aץ 0ϭ jϭ1,...,m͑jÞc,d͒. ͑31͒ changes in the phase diagrams. ͪ ␺ץ ͩ j T,␳,͕␺ j͖ jÞc,d We ﬁrst consider the QC results, using the case shown in Fig. 4 ͑xϭ0.2, qϭ24, cϭ0.8͒ as a reference, and the From Eq. ͑31͒ we obtain ͑see Appendix C͒ the desired changes that result when c, x, and q are varied. We begin distribution of ͕␺ j͖ with jÞc,d. It is given by with the phase behavior; density anomalies are considered

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FIG. 4. ͑a͒,͑b͒ PT and ͑c͒ T␳ projections of the binodals ͑solid lines͒, FIG. 5. ͑a͒,͑b͒ PT and ͑c͒ T␳ diagrams from the QC solution for xϭ0.2, spinodals ͑dotted lines͒, and TMD loci ͑dot–dashed lines͒ for the case cϭ0.8, qϭ12; notation is as in Fig. 4. See comments in caption to Fig. 4. xϭ0.2, cϭ0.8, qϭ24, QC solution. T*ϭkT/J; P*ϭPv0/J; v0ϭvolume per site. An enlarged view of the low P* region of ͑a͒ is shown in ͑b͒. The TMD from ͑b͒, and the spinodals for II are not resolvable on the scale of the plot in ͑a͒. Refer to the text for the explanation of a and b. qϭ12, cϭ0.8, xϭ0.2. The implications of these trends for the existence of the liquid–liquid transition will be discussed below. The salient feature is the existence of two phase tran- below. sitions ͑I,II͒ between disordered phases. For all cases in The critical temperature (Tc) and pressure (Pc) for tran- which the second ﬂuid–ﬂuid transition ͑II͒ is observed, it has sition I ͑VLE͒ are always greater than the UCT and the cor- both upper and lower consolute temperatures ͑UCT and responding pressure for II. The density of the coexisting LCT͒. We refer to the lowest density phase as the vapor, and phases is very sensitive to changes in q. This is illustrated in to the intermediate and high density phases as liquids, L1 Fig. 6 (qϭ240). Comparison with Fig. 4͑c͒ shows that in- and L2. The difference between the two liquid densities, and creasing the orientational entropy causes a density extremum between the UCT and LCT, decreases as either c or x de- to appear along the L1 branch; Tc to decrease slightly; a creases. The second phase transition disappears for low broadening of the density difference between coexisting L1 enough values of c or q. Figure 5 illustrates this for the case and L2 phases ͑transition II͒; and a decrease in the LCT. The

J. Chem. Phys., Vol. 105, No. 2, 8 July 1996

enough values of x by intersection with the spinodal branch along which L1 becomes unstable with respect to the vapor. A seemingly disconnected locus of density maxima also exists in the region where L1 is metastable with respect to the vapor. This locus always exhibits the qualitative behavior shown in Fig. 4͑a͒ and 4͑b͒: It has a negative slope in the PT plane and intersects the L1 branch of the spinodal for transition I, causing it to retrace toward higher pressures at low temperatures. The maximum temperature at which density anomalies occur increases as c is increased. These basic fea- tures of the TMD are preserved in all cases investigated. In general, the MF results are significantly different from those discussed above. The most obvious difference is that the L1–L2 transition cannot occur ͑see also Appendix A͒.In FIG. 6. T␳ diagram from the QC solution for xϭ0.2, cϭ0.8, qϭ240; addition, the TMD forms a single locus of density maxima. notation is as in Fig. 4. The MF phase behavior and density anomalies can be cat- egorized according to the behavior of the TMD locus and the liquid spinodal. In all cases studied here, the PT projection of relative strength of the nonbonding to the bonding interac- the TMD is negatively sloped over some region of the phase tions, x(ϭ⑀/J), has a pronounced effect on the vapor–liquid diagram, and has one of the following two forms: ͑1͒ The transition, but affects transition II only slightly. This is illus- TMD locus is negatively sloped at all pressures, and inter- trated in Fig. 7 ͑qϭ240, xϭ0.53͒. Comparison with Fig. 6 cepts the liquid spinodal; the intersection point is necessarily shows that weakening the bonding strength relative to the a minimum in the liquid spinodal ͑i.e., the retracing spinodal nonbonding interactions causes the density along the L1 scenario consistent with the stability-limit conjecture21͒. ͑2͒ branch of I to increase, and raises Tc . In all cases, the den- The TMD locus retraces toward low temperatures and does sity along the L1 branch of transition I approaches 5/9 in the not intersect the liquid spinodal ͑i.e., the retracing-TMD limit of zero temperature. This arises from the open ground scenario22͒. An example of the retracing TMD scenario is state ͑i.e., configuration 24 in Table I͒, in which five of the shown in Fig. 8 for qϭ24, cϭ0.8, and xϭ0.2. Upon de- nine sites are occupied. creasing c, or increasing q or x, the behavior changes to that There are pronounced density anomalies for all cases. of the retracing-spinodal scenario. Starting at high pressure and density ͓see, e.g., Figs. 4͑a͒ and As q or x is increased, Tc increases, the T – ␳ projections 4͑c͔͒ the locus of density maxima becomes a locus of density of the binodal and spinodal curves become more symmetric minima at point a. At point b the character of the locus about ␳ϭ0.5, and the TMD intersects the liquid spinodal, reverts to density maxima, and remains so as it approaches causing it to retrace. As c increases, the value of the critical the density of the L1 branch of transition I in the limit of density decreases, the maximum temperature for density zero temperature. Thus, in the PT plane the segment ab is a anomalies increases, and the TMD and liquid spinodal locus of isochore maxima, while the remaining locus repre- curves show behavior consistent with the retracing TMD sce- sents isochore minima. As can be seen in Fig. 7, the conti- nario. nuity of the locus of density extrema is interrupted at high The TMD and phase behaviors in the MF solution are similar to earlier results for solutions of simpler models with the MF approximation.15,16 This results from the MF treatment of the bonding probability as an average value for all molecules, regardless of a given molecule’s participation in other bonds. The assumption that all bonds form with probability pb significantly underestimates the chance of bonding for molecules which are already involved in one or more bonds. As an example of this, consider a molecule which is involved in one bond. The chance of it pointing one if its remaining bonding arms toward a nn site is 3/n, or 3/(4q)1/3. Assume that a completely nonbonded molecule occupies one of the nn sites with which a second bond could form. The bonding probability is then 3/n, which is greater than 7/3 4/3 pb(ϭ2 /q ) for all physically meaningful values of q ͓i.e., (3/n)/pbϭ3q/8; qу12͔. This error becomes more significant at higher densities and at lower temperatures, which FIG. 7. T␳ diagram from the QC solution for xϭ0.53, cϭ0.8, qϭ240; are the conditions of particular interest for investigating the notation is as in Fig. 4. The TMD locus is unstable ͑not shown͒ at the point denoted by a in Fig. 4. mx and mn denote respectively segments of the possibility of polyamorphic transitions. TMD for which the density extrema are maxima and minima. The errors at high density and low temperature are borne

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energy of the fluid at higher pressures and lower temperatures, but this also overestimates the entropy of the fluid. The net effect on the free energy, and thus the ability of the fluid to exhibit a second fluid–fluid phase transition, cannot be reasoned from only the current results. This will be addressed in work which is presently underway to test the current predictions with computer simulations. We now consider the origin of transition II. At all temperatures and parameter values considered here, the configurational energy and entropy as a function of volume ͑at constant T͒ have the qualitative forms shown respectively in Figs. 9͑a͒ and 9͑b͒͑T*ϭ0.15; x, c, and q as in Fig. 4͒. The minimum in energy and entropy occurs near Ϫ1 v/v0ϭ␳ ϭ9/5. It corresponds to a maximally bonded configuration ͑the fully bonded ground state, configuration 24 in Table I, has v/v0ϭ9/5͒. The specific volumes of the two liquid phases at equilibrium are indicated in the figure, illus- trating the significantly higher energy and entropy of the L2 phase relative to the L1 phase. At temperatures between the LCT and UCT the entropy gain at high densities stabilizes the L2 phase; the L1 phase is always favored energetically. The essential role that the orientational entropy of the unbonded and partially bonded molecules plays in the stabilization of L2 is evident from the sensitivity of transition II to the value of q, as noted earlier when comparing Figs. 4 and 5. Because L1 is favored energetically and L2 is favored entropically, their opposing contributions to the free energy can balance over only a limited range of temperatures. Ac- FIG. 8. MF results for xϭ0.2, cϭ0.8, qϭ240, notation is as in Fig. 4. cordingly, the first-order transition between two distinct phases possesses both an LCT and an UCT. Below the LCT the entropic contributions are too small to stabilize L2; out in the markedly different behaviors that the QC solution above the UCT, the entropic contributions are too large. In predicts when compared to the MF results. Accurate treat- neither case does isothermal compression of L1 lead to a ment of just the first shell of bonding neighbors produces new phase. solutions with a second fluid–fluid transition. This provides It was noted earlier that the existence of the second an example of the cooperative nature of bond formation in phase transition is dependent upon the magnitude of the bond network fluids, and clearly shows an inherent weakness in weakening parameter c. Figure 9͑c͒ (cϭ0.3) shows the ef- the MF solution. fect on the U(v) curve of decreasing c, for the same tem- Although the QC treatment is an improvement over the perature, x, and q values as shown in Fig. 9͑a͒ (cϭ0.8). The MF solution, it is still not exact. The most serious limitations energies of the high density states are significantly reduced arise from the assumption that cells are independent. Also, a as a result of the decrease in the energy penalty on weakened notable effect of cell size in our work is that a density of 5/9 bonds. The differences between the energies of the is imposed as T approaches zero, inducing the pronounced intermediate- and high-density states are no longer large asymmetry in the TϪ␳ projection of the VLE binodal. The enough to stabilize low entropy, low energy states and phase independence of cells causes corner-site molecules in certain transition II does not occur. Based upon the preceding argu- configurations to be treated as having q possible orientations, ments the L1–L2 transition should exist for cϭ0.3 if the which denies the possibility of bonding for those molecules. orientational entropy is sufficiently low ͑i.e., small q͒. This Bonds for these molecules must form with molecules in ad- is, however, precluded in our study by the lower bound of jacent cells, and thus are neglected in the QC approximation. qϭ12 for the number of distinguishable orientations for a In Fig. 3, molecules C and D are in this category. When molecule. applied to all cells, all molecules are treated equivalently; a Additional mechanisms for the occurrence of polyamor- molecule that occupies a site such as C in Fig. 3 for one cell, phic transitions exist. In particular, we note an insightful occupies a site such as E for an adjacent cell. The overall argument by Poole et al.17 regarding the origin of a first- effect, however, is that the possibility of all molecules being order polyamorphic transition in water, which is based upon fully bonded is not realizable in the QC approximation. It is results from molecular dynamics simulations of ST2 water. not obvious what the lack of a fully bonded fluid state has on These authors observed, in addition to the usual isothermal the accuracy of the phase and TMD behavior. The under- minimum in the internal energy as a function of volume, a counting of possible bonds is expected to overestimate the second, relatively sharp minimum at somewhat larger spe-

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tion of state, which showed a second fluid–fluid transition at high pressures and low temperatures. The modification con- sisted of a bonding interaction which contributed to the energy over a relatively narrow range of fluid densities, giving it a second minimum in its internal energy. The origins of the second fluid–fluid transition predicted by our model are markedly different from those mentioned above. In our model, transition II occurs because orientational entropy stabilizes the high-density phase. In the results of Poole et al.,17 the second phase transition occurs because energy stabilizes the low-density phase. In terms of phase behavior, the most notable difference which arises from these different scenarios is that the energetic stabilization of the phase transition results in an open-loop two phase region ͑no LCT͒, while entropic stabilization of the phase transition gives rise to closed-loop phase behavior.

VI. CONCLUSIONS We have presented approximate solutions to a lattice model of network fluids that shows polyamorphic behavior at high pressure and low temperature, and exhibits density anomalies over large portions of the liquid phase diagram. Depending upon the relative strengths of the bonding and nonbonding interactions, and the number of possible molecular orientations, both the retracing-liquid-spinodal and the retracing-TMD scenarios are predicted in the mean-field approximation. This confirms earlier findings15–17 that the different scenarios used to explain the anomalous properties of supercooled water can arise from the same type of molecular interactions. The first-order solution shows two apparently disconnected loci of density extrema. One begins as density maxima in the stable liquid at high pressure, reaches a maximum temperature, and retraces to lower temperature with decreasing pressure; at still lower temperatures this locus becomes a locus of density minima, reverting finally to a locus of density maxima as T 0 K. This does not correspond to the known behaviors of→ classic network fluids such as silica or water, which show only density maxima. The other locus exists only in the region where the liquid is metastable with respect to the vapor. It is always a locus of density maxima, and it terminates at its intersection with the spinodal of the liquid relative to the vapor; which causes the spinodal to retrace to higher pressures at lower temperatures. The large FIG. 9. Reduced, configurational energy (u*ϭU/MJ) and entropy differences between the results obtained by the two methods (T*s*ϭTS/MJ) as functions of specific volume, v/v0 for T*ϭ0.15, xϭ0.2, qϭ24, cϭ0.8 ͑a͒,͑b͒ and cϭ0.3 ͑c͒. v is the volume per molecule show that for systems in which molecular orientations are and v0 is the volume per site. Specific volumes of the equilibrium L1 and important, the zeroth-order approximation is not sufficient to L2 phases in ͑a͒ and ͑b͒ are indicated by ᭹ and ᭿, respectively. represent complex fluid behavior and properties; particularly low temperature polyamorphic transitions. In general, polyamorphism refers to systems with glassy cific volumes. This corresponds to the cooperative formation states that differ in density and network topology ͑e.g., co- of a random network of hydrogen bonds over a relatively ordination number͒. The predictions of our model are for narrow of range of densities. These authors reasoned that equilibrium liquid–liquid transitions, rather than nonergodic, because A U as T 0, the additional minimum in energy glassy states. Our results remain relevant to the general prob- can drive a→ second fluid–fluid→ phase transition at low enough lem of polyamorphism, however, in light of the apparently temperatures, since this would result in a double-welled first-order LDA-HDA transition, and the questions this raises A(v) function. The plausibility of this argument was as to how such a transition would manifest itself if it oc- 17 confirmed by solutions to a modified van der Waals equa- curred at temperatures above Tg for the amorphous phases. It

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should also be noted that amorphous solid phases are in gen- APPENDIX A: MAXIMUM NUMBER OF FLUID–FLUID eral metastable with respect to their crystalline phase ͑s͒. PHASE TRANSITIONS FROM THE MEAN-FIELD This topic is not addressed in the present work, as ordered EQUATION OF STATE phases were not included in the solutions of the model. The We begin by considering the equation of state of the metastability of the liquid phases with respect to ordered general form P*(␳,T*). We seek solutions which represent phases will be addressed in future work. two fluid–fluid phase transitions ͑i.e., a stable vapor–liquid Previously, theoretical treatments of network-forming transition, and a liquid–liquid transition which would be fluids which predict polyamorphic transitions have been re- metastable with respect to the solid, if the solid was included stricted to models which assume bulk-density-dependent in the analysis͒ with meaningful values of ␳, T*, and the 15,17,18 bonding. Polyamorphism also arises in ‘‘two-state’’ model parameters (x,c,q). A convenient way of casting this models, in which molecules are assumed to exist in one of problem is to ask if any P*Ϫ␳ projections of the phase two distinguishable states. The first-order solution of our diagram show two van der Waals loops. Mathematically this model shows that polyamorphic transitions occur for solu- is equivalent to solving tions of the partition function which use only the micro- *Pץ scopic interactions described by the fluid’s Hamiltonian. For ϭ0 ͑A1͒ ͪ ␳ץ ͩ our model, the second fluid–fluid transition arises from a T* competition between high density states stabilized by the orientational entropy of unbonded molecules, and low density and seeking sets of (x,c,q) which permit four roots of Eq. states stabilized by molecules in low-energy, open, fully ͑A1͒. Doing this for Eq. ͑16͒ with Eq. ͑17͒ yields bonded states. 3A␳3Ϫ͑3Aϩ2B͒␳2ϩ2B␳ϪT*ϭ0, ͑A2͒ An estimation of the accuracy of our results is difficult at where we have used Aϵ(8xϩ24␦J*)c* and this stage, because the level of approximation inherent in the Bϵ4(xϩ3␦J*). This equation has at most three real, posi- first-order solution is quite large for highly bonded states in a tive roots which implies that P*(␳,T*) cannot have four network fluid. This derives from the independence of cells in turning points for a given isotherm. Two van der Waals the QC formulation, which neglects fully bonded structures loops are not possible, and we conclude that the mean-field that extend beyond the first coordination shell. The effects of equation of state is not able to predict a second fluid–fluid this approximation on the high density phase behavior are phase transition for the present model. difficult to assess. Both the energy and the entropy of the fluid are overestimated, leaving the net effect on the free APPENDIX B: MEAN-FIELD APPROXIMATION energy unknown. Improvements which we are presently ad- APPLIED TO THE POTTS VARIABLES dressing include applying a mean-field treatment to molecules on the corner sites, expanding the size of the funda- We first rewrite the exponent in Eq. ͑9͒ as mental cell, and obtaining exact solutions from Monte Carlo * simulations. J cJ ␤ ͑1Ϫc͚͒ ninj␦ijϩ ͚ ͚ ninj␦ij͑1Ϫnk͒ . Finally, we wish to emphasize the pronounced differ- ͩ 3 ͗ij͘ 3 ͗ij͘ k͑ij͒ ͪ ences between the zeroth- and first-order solutions of our ͑B1͒ model. The dramatic changes in the phase behavior and den- Using this in Eq. ͑9͒ and defining J1ϵ(J/3)(1Ϫc) and sity anomalies which occur between the two levels of ap- J ϵ(cJ/3) we then write proximation are indicative of the strong influence of states in 2 which the molecules are multiply bonded. Even within the •••ϵ Q first-order approximation, structures with extended bonding ͚ ͚ b ͕␴i͖ ͕␴i͖ are not considered. The errors which this induces are expected to be significant at the low temperatures and high Ϫ ϭ͚ e␤J1͚͗ij͘ninj␦ije␤J2͚͗ij͚͘k͑ij͒ninj␦ij͑1 nk͒. pressures in which we are interested. In light of the changes ͕␴i͖ caused only by improving the description of the cooperative ͑B2͒ nature of bond formation in the first-neighbor shell, there is an obvious need for more exact solutions, such as those men- This can be expressed equivalently as tioned above. ͚ •••ϭ͚ ⌸͗ij͑͘1ϩh1ninj␦ij͒⌸͗ij͘⌸k͑ij͒͑1 ͕␴i͖ ͕␴i͖

ϩh2ninj␦ij͑1Ϫnk͒͒, ͑B3͒ ACKNOWLEDGMENTS with

␤J1 C.J.R. would like to thank the National Science Founda- h1ϵe Ϫ1, tion for a Graduate Fellowship. P.G.D. gratefully acknowl- h ϵe␤J2Ϫ1. edges the ﬁnancial support of the U.S. Department of En- 2 ergy, Division of Chemical Sciences, Ofﬁce of Basic Energy Expanding the products of exponentials, and collecting terms Research ͑Grant No. DE-FG02-87ER13714͒. of similar order in nin j gives

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Ϫ1 ␦J2 1 ␦J2 ͚ •••ϭ͚ 1ϩh1͚ ninj␦ijϩh2͚ ͚ ninj␦ij͑1 c*ϵ ϩ . ͑B10͒ ͕␴ ͖ ͕␴ ͖ ij ij k͑ij͒ ␦J ͩ 3 ␦J ͪ i i ͩ ͗ ͘ ͗ ͘ 1 1 Substituting into Eqs. ͑B3͒ to ͑B5͒ yields Ϫn ͒ϩ••• . ͑B4͒ k ͪ ⌶͑␮,V,T͒ Now we take terms of order less than or equal to one in n n i j ͑␤␮ϩln q͚͒ n ␤⑀͚ n n ␤␦J͚* n n ͚ ͑1Ϫc*n ͒ and write Ϸ͚ e j je ͗ij͘ i je ͗ij͘ i j k͑ij͒ k , ni ͑B11͒ ͚ •••Ϸ͚ 1ϩh1͚ ninj␦ijϩh2͚ ͚ ninj␦ij͑1 ͕␴i͖ ͕␴i͖ ͩ ͗ij͘ ͗ij͘ k͑ij͒ which is Eq. ͑10͒.

Ϫnk͒ . ͑B5͒ ͪ APPENDIX C: ADDITIONAL DETAILS OF THE DERIVATION OF THE QC EQUATION OF STATE This is the first-order random-graph approximation proposed 19 by Vause and Walker; the approximation improves as tem- Writing Eqs. ͑23͒ and ͑24͒ explicitly in terms of ␺c and perature increases, i.e., as h1 and h2 become much less than ␺d we have one. The ‘‘sphericallization’’ technique treats the remaining 1 * summation over ͕␴i͖. The argument is as follows. For a ␳ϭ ͚ ␺ j␻ jn jϩ␺c␻cncϩ␺d␻dnd , ͑C1͒ given distribution of M(ϭ͚ n ) molecules on the lattice, ␣ ͩ j ͪ i i there are q ͚ini different configurations ͑q possible orienta- * tions for each molecule͒. Consider two nearest-neighbor 1 ͑1Ϫ␳͒ϭ ͚ ␺ j␻ j͑␣Ϫn j͒ϩ␺c␻c͑␣Ϫnc͒ molecules within any of these configurations. Within the ␣ ͩ j single-graph or first-order random-graph approximation, the probability of these two molecules being properly oriented ϩ␺ ␻ ͑␣Ϫn ͒ . ͑C2͒ for bonding ͑neglecting any biasing due to favorable ener- d d d ͪ gies of specific orientations͒ is p . Thus only p of the terms b b In the above expressions, the superscript * is to indicate that in the summations in Eq. ͑B5͒ have ␦ not equal to zero. We ij the summations do not include cell configurations c and d. can then write Equations ͑C1͒ and ͑C2͒ can be rearranged to allow us to write expressions for ␺ and ␺ : ͚ini c d ͚ •••Ϸq 1ϩpbh1͚ ninj ͕␴i͖ ͩ ͗ij͘ 1 * * ␺cϭ ␣␳Ϫ͚ ␻ j␺ jn jϪnd 1Ϫ͚ ␻ j␺ j , ␻c͑ncϪnd͒ ͫ j ͩ j ͪͬ ϩpbh2͚ ͚ ninj͑1Ϫnk͒ , ͑B6͒ ͗ij͘ k͑ij͒ ͪ ͑C3͒

which depends upon only ͕ni͖. Finally, it is convenient to 1 * * define an effective bonding interaction energy ␺dϭ Ϫ␣␳ϩ͚ ␻ j␺ jn jϩnc 1Ϫ͚ ␻ j␺ j . ␻d͑ncϪnd͒ ͫ j ͩ j ͪͬ ͑C4͒ ␦HϵϪ␦J1͚ ninjϪ␦J2͚ ͚ ninj͑1Ϫnk͒, ͑B7͒ k ij͒ ͗ij͘ ͗ij͘ ͑ This shows that the cell configurations c and d cannot be which defines ␦J and ␦J . Taking eϪ␤␦H and performing 1 2 chosen arbitrarily. They must be such that ncÞnd , otherwise the same random graph approximation as above, Equations ͑C3͒ and ͑C4͒ are no longer independent. Using Eq. ͑27͒, and since ␺c and ␺d are functions of ␳, Ϫ␤␦H e Ϸ1ϩ␦h1͚ ninjϩ␦h2͚ ͚ ninj͑1Ϫnk͒, we have ͗ij͘ ͗ij͘ k͑ij͒ ␺ץ aץ ␺ץ aץ aץ ␦J1 c d ␦h1ϭe Ϫ1, ͑B8͒ Pv0ϭ␳ ϩ ϩ Ϫa. ͬͪ ␳ץ ␺dͪͩץͩ ͪ ␳ץ ␺cͪͩץͩ ␳ͪץ ͩͫ ␦J2 T,͕␺ ͖ ␦h2ϭe Ϫ1. j ͑C5͒ Multiplying by q ͚ini, equating the resultant expression to Expressions for the partial derivatives in Equation ͑C5͒ are ͚͕␴ ͖••• , and comparing like terms gives i obtained from Eq. ͑27͒ ␤J1 ␦J1ϭkT ln͑1ϩpb͑e Ϫ1͒͒, a ␻cץ ␤J2 ϭ ͑⑀cϩkT͑1ϩln ␺c͒͒, ͑C6͒ ␺ ͪ 2ץ ͩ .J2ϭkT ln͑1ϩpb͑e Ϫ1͒͒␦ c Finally, we define a ␻dץ ϭ ͑⑀dϩkT͑1ϩln ␺d͒͒, ͑C7͒ ␺ ͪ 2ץ ͩ Jϵ␦J1/3ϩ␦J2 ͑B9͒␦ d and and

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m ϱ 4 R. G. Della Valle and H. C. Andersen, J. Chem. Phys. 97, 2682 ͑1992͒. ␺jץ a kTץ ϭϪ ␻ ͑1ϩln ␺ϱ͒ 5 C. A. Angell, Science 267, 1924 ͑1995͒. ͚ j j 6 .␳ ͪ C. A. Angell, Annu. Rev. Phys. Chem. 34, 593 ͑1983͒ץ ͩ ␳ͪ 2 jϭ1ץ ͩ T,͕␺j͖ T,͕␺j͖ 7 M. Grimsditch, Phys. Rev. Lett. 52, 2379 ͑1984͒. ␳ 8 O. Mishima, J. Chem. Phys. 100, 5910 ͑1994͒. ϩkT ln . ͑C8͒ 9 C. A. Angell, P. H. Poole, and J. Shao, Nuovo Cimento 16D, 993 ͑1994͒; q͑1Ϫ␳͒ The term polyamorphism has been suggested ͓G. H. Wolf, S. Wang, C. A. And from Table I we see that Herbst, D. J. Durben, W. J. Oliver, and Z. C. Halvorsen, in High Pressure Research: Application to Earth and Planetary Sciences, edited by Y. S. ␳ n j Manghnani and M. H. Manghnani ͑Terra Sci. Publ. Co./Am. Geophysical ϱ 9Ϫn j Union, Tokyo/Washington, 1992͒,p.503͔for the coexistence of amor- ␺ j ϭ ͑1Ϫ␳͒ , jϭ1,2,...,c,...d,...m. ͩ qͪ phous phases having the same composition but different density and structure. From Eqs. ͑C3͒ and ͑C4͒ we can write 10 O. Mishima, L. D. Calvert, and E. Whalley, Nature 314,76͑1985͒. 11 O. Mishima, L. D. Calvert, and E. Whalley, Nature 310, 393 ͑1984͒. ␺c 9ץ ϭ , ͑C9͒ 12 A notable exception to this, which provides support for the use of density- ␳ ͪ ␻ ͑n Ϫn ͒ driven polyamorphism to infer a liquid–liquid transition above T ,istheץ ͩ c c d g recent experimental work of S. Aasland and P. F. McMillan ͓Nature 369, .␺d 9 633 ͑1994͔͒ץ ϭϪ . ͑C10͒ 13 .␳ ͪ ␻ ͑n Ϫn ͒ E. G. Ponyatovsky and O. I. Barkalov, Mat. Sci. Rep. 8, 147 ͑1992͒ץ ͩ d c d 14 G. M. Bell, J. Phys. C: Solid State Phys. 5, 889 ͑1972͒; G. M. Bell and D. Equation ͑29͒ follows from Eqs. ͑C5͒ and ͑C6͒–͑C10͒ W. Salt, Trans. Faraday Soc. II 72,76͑1975͒; P. H. E. Meijer, R. Kikuchi, and P. Papon, Physica A 109, 365 ͑1981͒; P. H. E. Meijer, R. Kikuchi, and Referring now to the expression for (␺ j/␺c), Eq. ͑32͒ follows from E. van Royen, Physica A 115, 124 ͑1982͒; M. Sasai, Bull. Chem. Soc. Jpn. 66, 3362 ͑1993͒; N. A. M. Besseling and J. Lyklema, J. Phys. Chem. 98, 11610 ͑1994͒. ␺cץ aץ aץ aץ ϭ ϩ 15 S. S. Borick, P. G. Debenedetti, and S. Sastry, J. Phys. Chem. 99, 3781 .␺ ͪ ͑1995͒ץͩͪ ␺ץ ͩ ͪ ␺ץ ͩ ͪ ␺ץ ͩ j j c j T,␳,͕␺ j͖ j c,d T,␳,͕␺k j͖ Þ Þ 16 S. Sastry, F. Sciortino, and H. E. Stanley, J. Chem. Phys. 98, 9863 ͑1993͒. 17 P. H. Poole, F. Sciortino, T. Grande, H. E. Stanley, and C. A. Angell, ␺dץ aץ ϩ ϭ0 ͑C11͒ Phys. Rev. Lett. 73, 1632 ͑1994͒. ␺ ͪ 18ץͩͪ ␺ץͩ d j E. G. Ponyatovskii, V. V. Sinand, and T. A. Pozdnyakova, JETP Lett. 60, 360 ͑1994͒. in conjunction with Eqs. ͑27͒, ͑C3͒, and ͑C4͒. 19 C. A. Vause and J. S. Walker, Phys. Lett. A 90, 419 ͑1982͒. 20 P. G. Debenedetti, V. S. Raghavan, and S. S. Borick, J. Phys. Chem. 95, 1 F. Sciortino, P. H. Poole, H. E. Stanley, and S. Havlin, Phys. Rev. Lett. 4540 ͑1991͒. 64, 1686 ͑1990͒. 21 R. J. Speedy, J. Phys. Chem. 86, 982 ͑1982͒. 2 R. L. Blumberg and H. E. Stanley, J. Chem. Phys. 80, 5230 ͑1984͒. 22 P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley, Nature 360, 324 3 F. H. Stillinger, Science 209, 451 ͑1980͒. ͑1992͒.

J. Chem. Phys., Vol. 105, No. 2, 8 July 1996

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