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Immiscibility in binary silicates: chemical and phase equilibria
. Journal of Non-Crystalline Solids 242 (1998) 1±13
Immiscibility in binary silicates: chemical and phase equilibria L.Rene Corrales
*
Environmental Molecular Sciences Laboratory, Paci®c Northwest National Laboratory, P.O. Box 999 K1-96, Richland, WA 99352, USA Received 23 March 1998; received in revised form 22 July 1998
Abstract The driving force for immiscibility in binary silicates was examined using a statistical theory which combines a Florytype theory, based on the compatibility of structural units, and a chemical equilibrium model, that describes the preferred associated species distribution of silicates. Restrictions imposed by a preferred distribution of associated species lead to a limitation of the number of ways to form a complete network structure, or equivalently to a decrease in the con®gurational entropy. An increase in con®gurational entropy is achieved by inducing a phase separation. The role of the enthalpies of species formation is examined relative to the con®gurational entropic contribution to phase coexistence. The theory leads to a rich set of phase coexistence behavior that includes three-phase equilibrium and a higher order critical point. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Immiscibility gaps in silicate melts are known to have both stable and metastable upper critical solution temperatures [1]. This is typical of a system with a positive enthalpy of mixing and a positive mixing entropy [2]. A plausible explanation for the occurence of phase separation in silicates considers the preferred coordination of free metal oxide about the non-bridging oxygen bond. The argument is that increasing the coordination about metal atoms is energetically favorable [3]. This provides a positive mixing enthalpy to the system and can lead to phase separation. However, silicate melts incorporate metal oxides into the network structure by changing a bridging oxygen bond into a pair of non-bridging oxygen bonds while maintaining the integrity of the network.
* Tel.: 1-509 375 6410; fax: 1-509 375 6631; e-mail: [email protected].
Hence, the interactions in silicate melts are not simply packing interactions. Another way to explain phase separation is to consider the incorporation of metal oxide into the network, eectively modi®ng the network structure, and describe it in terms of a heat of mixing [4]. This approach also predicts phase separation and, as a structural model, it has the advantage of determining network connectivity. However, neither of the above two approaches explain the preferred associated species distribution of silicates. An alternative approach, used in this work, is based on structural compatibility of building blocks that make up the backbone of the network [5,6]. Each building block is a tetrahedron with a central silicon atom coordinated by bridging and non-bridging oxygen bonds. In all, there are ®ve basic building blocks for a binary silicate system. A speci®ed distribution of these building blocks can be completely joined (thus saturating all the bonds) if all the building blocks are compatible with one another. When a sucient number of
0022-3093/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 7 8 2 - 0
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L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
incompatibilities exist between the building blocks, phase separation will occur in order to optimize the bond saturation. This statistical approach is based strictly on the con®gurational entropy of a solution where the distribution of the building block types is not changed. Its success in predicting phase transitions in all types of network materials, regardless of coordination, supports the idea that structural incompatibilities are the underlying force that drive phase transitions in network materials. The liquid components in silicates are present as clearly identi®able stoichiometries and free energies of formation [7]. These components are effectively the building blocks of the aformentioned statistical theory and are also refered to as the associated species. The uptake of metal oxide into the network and the formation of preferred coordination of each metal type about a silica site varies as a function of temperature, pressure and composition. The associated species distribution has been determined using species activities in a thermodynamic model [8], and by chemical equilibrium models [9] that use a linear combination of the species activities or free energies of formation. Both models lead to similar results. The latter method is used in this work because it better describes the species distribution in the dilute limits. Although these approaches are useful in determining and predicting the associated species distribution as a function of temperature and composition, they cannot by themselves predict phase coexistence in which the underlying driver is the con®gurational entropy. Alternatively, a chemical equilibrium approach lets us dierentiate between the enthalpy of forming new species (i.e. the formation of a non-bridging oxygen bond from a bridging oxygen bond reacting with a metal oxide), and the enthalpy of mixing pure silica and pure metal oxide where no reaction takes place. A better understanding of the explicit and implicit driving forces that lead to phase separation in silicate melts is achieved by partitioning the underlying interactions. A simple and elegant way to partition the interactions in a silicate melt is to consider that the network is made up of tetrahedral building blocks. In a silicate material, the molecular or associated species make up the
building blocks of the network structure as described above. The chemical equilibrium reactions serve the purpose of regulating the coordination state of the tetrahedrons. To build the network backbone the building blocks must be connected such that bridging oxygen (BO) bonds are matched with BO bonds, and non-bridging oxygen (NBO) bonds are matched with the same type of NBO bonds. The idea that the melt is made up of a network of building blocks is strictly theoretical. In fact in this work, the idea that the network backbone is built of building blocks is transposed to the idea that the network backbone is made up of BO and NBO bonds. This follows as a consequence of recognizing that the natural order parameter for phase separation in silicates is related to the formation of NBO bonds. The topology of the network dominates the con®gurational entropy because the number of ways of bringing the network together is determined by the building blocks that make up the backbone of the network structure. If the structural units are incompatible, because the building blocks cannot saturate their bonds, phase separation will occur where each phase will contain a set of compatible building blocks. There can be cases where a compatible network is only formed in which a distribution set of building blocks is not the lowest energy state in terms of the formation energies. In other words, the less probable associated species are forced to form. In the previous case, the mixing entropy drives the network into compatibility via rearranging the building blocks, whereas in the latter case the enthalpy of formation drives the system to compatibility via forming a new distribution set of building blocks. For a given metal oxide, the equilibrium constants vary as a function of temperature, pressure and composition. Thus, the equilibrium constants are eectively ®xed for a given composition, temperature and pressure. The preferred coordination of a metal oxide about a central silicon atom can lead to a distribution of incompatible building blocks. Phase separation will occur whenever unmixing is energetically favorable over redistributing the molecular species. In this work a brief description of a recently introduced statistical thermodynamic theory [10]
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
that combines a chemical equilibrium reaction model, that describes the molecular species distribution, with a statistical Flory-type treatment of the mixing interactions, that describes the connectivity of the network is presented. We show how phase equilibrium and chemical equilibrium can be theoretically partitioned, where the true equilibrium state of the system is achieved by simultaneously satisfying both chemical and phase equilibria. This clearly shows that (1) the phase equilibrium is driven by the con®gurational entropy and is identical to previous statistical theories, and (2) the chemical equilibrium portion is identical to Dorfeld's model [9] from which the associated species distribution can be obtained as a function of temperature and composition. A distinct dierence of the Corrales and Keefer theory from all other approaches is that the model is cast in terms of the bond types, which is the natural order parameter of this system. Curiously, phase separation appears to occur within the chemical equilibrium portion of the theory. However, it is shown below that the conditions of phase equilibrium are not satis®ed on that surface. In previous work, the true phase coexistence curves predicted by this theory were obtained by simultaneously satisfying both the chemical and phase equilibria conditions [10,11]. A rich set of phase coexistence behavior can be obtained that includes three-phase equlilibrium and a non-symmetric tricritical point [10] that has not been predicted by the related theories. Here we show that the true phase coexistence curve is a result of the intersection of a phase equilibrium surface and a chemical equilibrium surface such that both conditions are simultaneously satis®ed [12].
3
system that reacts with a bridging (covalent) oxygen to form two Oÿ non-bridging (ionic) oxygens [7,9,10]: BSi±O±SiB M2 O BSi±Oÿ M M Oÿ ±Si ; where M Li, Na, K, and so on. Network modi®ers are not restricted to alkali metals nor to metals with oxidation states of one, although the chemical reactions are distinct for other oxidation states. The above chemical reaction is considered to be in equilibrium. A silicon site can be four-fold coordinated by a combination of BO and NBO bonds, which correspond to a covalent oxygen and a pair of ionic oxygens, respectively. For convenience we use the Q species notation [13] to describe each of the possible coordinations of bond types about a central silicon atom as shown in Fig. 1 and described as follows. It must be made clear that a BO or a NBO are shared between the tetrahedra formed by the Q site. Hence, in the following de®nitions of the Q species a half-BO corresponds to sharing a covalent oxygen, and a half-NBO corresponds to a single ionic oxygen. The Q species de®nition are identical to the building blocks in the work by Araujo [5]. Thus, a Q4 site is made up of four half-BO bonds, a Q3 site is made up of three half-BO and one half-NBO bonds, a Q2 site is made up of two half-BO and two half-NBO bonds, a Q1 site is made up of one half-BO and two half-NBO bonds, and a Q0 site is made up of four half-NBO bonds.
2. Theoretical model Incorporation of a metal oxide into an oxide network is the result of a chemical reaction that transforms a BO bond to an NBO bond. This must be distinguished from the case in which a free metal oxide is eectively solvated by the network material, where only packing and electrostatic interactions come to play. The chemical reaction consists of a M2 O contributing an O2ÿ to the
Fig. 1. The tetrahedral building blocks in terms of Q species notations. The metal cations have been left out for clarity.
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L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
The equilibrium reactions that describe the solvation of metal oxide to produce any of the Q species from the Q4 species can be written as
The chemical equilibrium constants, or free energies of formation, are given by
units, and all the free metal oxide occupy interstitial sites o the network lattice sites and bonds. As the system is allowed to progress to chemical equilibrium, the bond types change such that metal oxide units incorporate into the bonds to form NBO bonds. Thus, the number of SiO2 units remains unchanged, this is re¯ected in the determination of the mass balance equations that follow. Each Q species contributes a total of one SiO2 unit. Hence, the total number of sites Ns or SiO2 units is given by
ÿ kT ln Ki DG0i DHi0 ÿ T DSi0 ;
Ns Q4 Q3 Q2 Q1 Q0 :
K3
Q4 12M2 O Q3 K2
Q4 M2 O Q2 ; K1
Q4 32M2 O Q1 ;
1
K0
Q4 2M2 O Q0 :
2
where we assume ideal mixing such that there are no volume changes in the formation of new species [14]. Q species can also be formed from cross reactions that are linear combinations of Eq. (1) [7]. For example, K22ÿ40
Q2 Q2 Q4 Q0 ; K22ÿ31
Q2 Q2 Q3 Q1 ;
3
K31ÿ40
Q3 Q1 Q4 Q0 ; where the last equation is a linear combination of the ®rst two reactions. Assigned to the formation of each species are the conjugate chemical equilibrium constants that act as statistical weights for each of the Qi species. The chemical equilibrium constants can be written in terms of the species concentrations, which for convenience are also de®ned by Qi . Ki
Qi
4ÿi=2
Q4 nm
;
4
where i f3; 2; 1; 0g, and nm is the number concentration of free metal oxide. In Eq. (4) the activity coecient that should appear as ck Qk [14] is approximated as ck 1 corresponding to an ideal solution. There are two mass balance equations, one for the SiO2 species and one for the total metal oxide which includes the free metal oxide and the metal oxide incorporated as NBO bonds. Initially, all sites are occupied by pure Q4 units, or in other words the system consist of strictly pure SiO2
5
In addition, each Q species, except for the Q4 species, contributes half of a M2 O unit for every half-NBO bond it contains. Thus, the total number of NBO bonds nNBO or metal oxide species incorporated into the network is given by nNBO 12Q3 Q2 32Q1 2Q0 :
6
Each unreacted species contributes one full M2 O unit such that there are nm free metal oxide units. Therefore, the total number of metal oxide species Nm in the site model is given by Nm nm nNBO :
7
The eective mass conservation equation is determined by normalizing the mass balance equations with respect to the number of sites. This allows taking the thermodynamic limit, namely where the number of sites goes to in®nity, without compromising any other aspect of the theory. Dividing Eqs. (5)±(7) by Ns de®nes the fractional amount of SiO2 units occupying Ns sites on the lattice X q 1 q4 q 3 q 2 q 1 q 0 ;
8
where Xq 1 indicates that all sites are occupied by a SiO2 units and where qi Qi =Ns . The fractional amount of metal oxide Xm occupying both NBO bonds and interstitial sites is given by Xm xm xNBO ;
9
where Xm Nm /Ns , xm nm /Ns and xNBO nNBO / Ns with nNBO given by Eq. (6). Since the free metal oxide is o the lattice it is necessary to de®ne an apparent mole fraction by
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
introducing the apparent total mass as follows. The total amount of material in the system XT is given by XT Xm Xq Xm 1:
10
Thus, the apparent mole fractions of M2 O in terms of the lattice site model is Xm 11 vsm XT and for SiO2 it is vsSiO2
Xq 1 : X T XT
12
It is now possible to obtain the distribution of the Q species as a function of the equilibrium constants and temperature. This is done by solving for the six unknowns, namely the ®ve qi s and xm , using the four equilibrium equations given by Eq. (4) and the two mass balance equations Eqs. (8) and (9). This is identical to the work by Dorfeld [9]. Solutions are obtained by specifying the equilibrium constants (or free energies) at a ®xed temperature, then sweeping through the variable xm while solving for the qi s. It is now possible to derive a partition function and, hence, write down a phenomenological free energy based on the Q species, or sites of a lattice model. In deriving the partition function, the chemical equilibrium constants of Eq. (4) are used as the statistical weights. The con®guration entropy contribution is calculated using a Flory-type theory [5,6,10,15,16]. However, the natural order parameter for this system is the degree to which the network has depolymerized. Such an order parameter is ill de®ned in a site or Q model representation. Instead the lattice site model is transposed into a lattice bond model. This allows the order parameter to be de®ned by the number concentration of broken bonds. A detailed derivation of the underlying theory is given in Ref. [10]. Here only the transposition of variables and the phenomenological free energy are given. The phase and chemical equilibrium conditions are extracted from an analysis of a standard isothermal dierential free energy and calculated from the bond model free energy. The bond model requires that the chemical equilibrium constants be de®ned in terms of the
5
site occupations. Thus, the probabilities of forming each of the speci®c Q sites in terms of the bond types (namely BO and NBO bonds) are used. The probability of forming a Q4 site from nBO BO bonds is n4BO . The probability of forming a Q3 site from nNBO NBO and nBO BO bonds is n3BO nNBO . Similarly, the probability of forming a Q2 is n2BO n2NBO , the probability of forming a Q1 is nBO n3NBO , and the probability of forming a Q0 is n4NBO . Moreover, the asymmetric species have ways of being formed that are uniquely dierent from one another in that they lead to dierent network connectivities, and so they must be counted. The Q3 and Q1 species each have four distinguishable ways of being formed on a site, and the Q2 species has six distinguishable ways of being formed on a site. The corresponding mass conservation equations for the bond model are determined as follows. The total number of bonds in the system is given by Nb nBO nNBO , or nBO Nb ÿ nNBO :
13
The total amount of SiO2 in terms of the bond model is given by substituting the probability of forming each Q species into Eq. (5). Note that the probability of forming Ns sites from the total number of bonds is given by Nb4 . 4
3
Nb4 Nb ÿ nNBO 4nNBO Nb ÿ nNBO
6n2NBO Nb ÿ nNBO 2 4n3NBO Nb ÿ nNBO n4NBO :
14
The total amount of metal oxide for the bond model Nmb is given similarly as in Eq. (7) in which the probabilities of forming each speci®c Q species are substituted into Eq. (6) 3
Nmb nm 124nNBO Nb ÿ nNBO 2
6n2NBO Nb ÿ nNBO 324n3NBO Nb ÿ nNBO 2n4NBO :
15
By expanding the polynomial and collecting terms, Eq. (15) simpli®es to Nmb nm 2Nb3 nNBO :
16
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L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
Dividing these mass balance equations by Nb4 we obtain the normalized mass balance equations for the total metal oxide content Xm Nmb =Nb4 xm 2x
17
together with the total number of silica units given by X qi 1 ÿ x4 4x 1 ÿ x3 XSiO2 i
2
2
3
4
6x 1 ÿ x 4x 1 ÿ x x 1:
18
The total mass conservation of this system is XT Xm XSiO2 xm 2x 1:
19
The apparent fraction of metal oxide in the bond representation is given by vm
2x xm ; 1 2x xm
20
with the apparent fraction of silica given by vSiO2
1 : 1 2x xm
21
To calculate the species distribution as a function of the equilibrium constants and temperature, the chemical equilibrium conditions must be determined for the bond representation of the theory. This is discussed in detail below. An approximate phenomenological free energy for the bond model derived by Corrales and Keefer [10] is
B1 ÿkT ln K3 ; K2 B2 ÿkT ln ; K32 K1 K33 ; B3 ÿkT ln K3 26 K0 K2 B4 ÿkT ln : K14 K34
The phase equilibrium and chemical equilibrium conditions are determined by analyzing a standard expression of the isothermal dierential free energy. In terms of the site model, an isothermal dierential free energy is dGT l4 dQ4 l3 dQ3 l2 dQ2 l1 dQ1 l0 dQ0 lm dnm :
4
dGT 4lBO d Nb ÿ nNBO
3lBO lNBO d4nNBO Nb ÿ nNBO 3 2
2lBO 2lNBO d6n2NBO Nb ÿ nNBO lBO 3lNBO d4n3NBO Nb ÿ nNBO
ÿ nm 2Nb3 nNBO kT ln zm 4Nb3 nNBO B1
4lNBO dn4NBO lm dnm :
6Nb2 n2NBO B2 4Nb n3NBO B3 n4NBO B4
4Nb3 Nb
24
The associated species chemical potentials can be de®ned in terms of the bond type potentials as li ilBO 4 ÿ ilNBO with i 0; 1; 2; 3; 4, and where lm is the chemical potential for the pure metal oxide. Substituting the Qi s by the formation probabilities in terms of bond types and the site chemical potentials by those in terms of the bond type chemical potentials, the isothermal dierential free energy of the bond model is
G ÿNb4 kT ln z4
4Nb3 nNBO kT ln
23
By using the conservation equations, given by Eqs. (14) and (15), expanding out the polynomial terms and rearranging, Eq. (25) simpli®es to
nNBO Nb
Nb ÿ nNBO ÿ nNBO kT ln ; Nb
25
22
where z4 and zm correspond to the activities of pure SiO2 and M2 O species, which are the initial states of the model. The Bi coecients are a linear combination of the free energies of formation of the chemical reactions of Eq. (1) given in terms of the chemical equilibrium constants de®ned by Eq. (2) [10].
dGT 4lBO dNb4 4 D 12lm dNb3 nNBO lm dnm ;
26
D lNBO ÿ lBO ÿ 12lm :
27
where From this de®nition of D, it represents the negative anity of reaction of creating an NBO from a BO and metal oxide [14]. Hence, to satisfy chemical equilibrium, D must be equal to zero. The phase
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
equilibrium conditions are de®ned by lm , D and lBO being equal in all phases for a ®xed temperature and pressure. The use of lm is mooted as will be seen below. Therefore, only two conditions for phase equilibrium need to be satis®ed. In the original work by Corrales and Keefer [10], an error was made in the dierential free energy expression that has been corrected above. The error is in failing to make the substitution of Nm nm 2Nb3 nNBO in the free energy and the dierential free energy expressions. This error led to the incorrect expression of Eqs. (34), (36) and (37) of Ref. [10]. However, the correct expression for D was given for the wrong reason such that no errors were propagated. In this work we have corrected all the equations [17]. For ease of calculation, the dierential free energy is expressed in terms of dNb , dnNBO and dnm which leads to
equilibrium [2]. Calculating the indicated dierentials on the free energy given by Eq. (22) gives D 12 lm oG=kT 4Nb3 kT onNBO T ;Nb ;nm B1 ÿ2Nb3 ln zm 4Nb3 kT B2 B3 12Nb n2NBO 12Nb2 nNBO kT kT B4 nNBO 3 3 ; 4nNBO 4Nb ln kT Nb ÿ n
oG=kT oNb
T ;nNBO ;nm
29
oG onNBO
T ;Nb ;nm
1 4Nb3 D lm 2 4Nb3 lNBO ÿ lBO ;
oG onm
30
T ;Nb ;nNBO
lm :
Consequently, lBO is given by l nNBO 1 ÿ 3 D l 4lBO Nb 2 m 4Nb3 1 oG : B 4 G ÿ nNBO Nb onNBO
31
32
The latter equivalence is the common tangent construction that is also used to determine phase
4Nb3
l kT
ÿ4Nb3 ln z4 ÿ 12Nb2 n ln zm B1 B2 12Nn2NBO 12Nb2 nNBO kT kT B3 4n3NBO kT nNBO 4 3Nb2 nNBO ln 4Nb3 Nb Nb ÿ nNBO ; ÿ3Nb2 nNBO ln Nb
28
This equation implies that the dierentials are of the form oG 4Nb3 l oNb T ;nNBO ;nm nNBO 1 3 ; D lm 4Nb 4lBO 3 Nb 2
33
dGT 44Nb3 lBO 3Nb2 nNBO D 12lm dNb 4Nb3 D 12lm dnNBO lm dnm :
7
oG=kT onm
34
T ;Nb ;nNBO
lm ÿ ln zm : kT
Therefore, using Eqs. (32)±(34), lBO is 4lBO B2 B3 B4 ÿ ln z4 ÿ 6x2 ÿ 8x3 ÿ 3x4 kT kT kT kT 4 ln 1 ÿ x ;
35
36
where x nNBO =Nb . Also using Eqs. (33) and (35) we get x D B1 B2 B3 B4 3x 4x2 x3 ln : kT kT kT kT kT 1ÿx 37 The concentration of free metal oxide xm is determined as follows. As stated previously, the chemical equilibrium of the system is given by Eq. (27) being equal to zero which is the negative anity of the reaction in which a BO bond is transformed to an NBO bond by the uptake of a
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L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
metal oxide. Following the de®nition of the chemical equilibrium constants as in Eq. (4), the equilibrium constant for converting a BO bond to an NBO bond is x e D=kT 38 KNBO 1=2 1 ÿ xxm Hence, at equilibrium KNBO 1 and Eq. (38) leads to x x ; 39 x1=2 m D=kT 1 ÿ xe 1 ÿ x where the latter equality is true, because at chemical equilibrium D 0. In the following discussions the parameters given by Eq. (23) are rewritten as B1 kTb1 ;
40
B2 kT b2 ÿ 2b1 ; B3 kT b3 ÿ 3b2 3b1 ; B4 kT b4 ÿ 4b3 6b2 ÿ 4b1 ; where bi ÿ ln K 4ÿi are just the free energies of formation identical to Eq. (2) and corresponding to the chemical reactions of Eq. (1). This allows specifying the parameters B2 ; B3 and B4 while varying B1 (or b1 ) and eectively varying all the bi s. 3. Chemical equilibrium Knowing the chemical equilibrium condition, D 0, we can iteratively solve for x in Eq. (37) at a given temperature for a given set of equilibrium constants. In this bond representation, the free metal oxide concentration is eectively held ®xed by virtue of Eq. (39). This implies that to determine the distribution of the associated species at a given temperature for all compositions, the Bi parameters must be varied. This diers from solving for the distribution using the site representation. The dierence arises from holding density parameters versus holding the conjugate ®eld parameters ®xed. In the site representation it is the equilibrium constants that are held ®xed while varying the free metal oxide concentration. Whereas, in the bond representations it is the free metal oxide concentration that is held ®xed while the equilibrium constants are varied. The common
tie is that both methods de®ne a surface upon which chemical equilibrium is satis®ed in the temperature±composition±equilibrium constant space, where only one of the equilibrium constants is used in the plot. Experimentally, it is the composition variables that are held constant, therefore obtaining the species distribution from the site model is the preferred method. Identical results can be extracted from the bond model, but it is an arduous and unnecessary task. Chemical equilibrium given by D 0 leads to de®ning a surface in the T ; x; b1 space. A point that sits on the surface is obtained by ®xing B1 ; B2 ; B3 and B4 , speci®ng a temperature and solving for x. The temperature is then varied to obtain the entire locus. There is a range of the parameters that lead to two or more stable solutions of x for a given b1 and T , thus, a curve is obtained. (This is easily seen by graphing D vs. x and ®nding a set of the parameters that lead to solutions with D 0.) The stable solutions are determined by the condition that the second derivative of the free energy with respect to the order parameter be greater than or equal to zero. This is identical to oD=oxT ;xm P 0. The chemical equilibrium surface is obtained by sweeping through any of B2 ; B3 or B4 . Projections of a chemical equilibrium curve in the T ; x, T ; b1 and T ; D planes are shown in Figs. 2±4, respectively, as the solid lines. The ®xed parameter set of B2 ÿ1:477, B3 9:456 and B4 ÿ60:0 [10]. The value of B1 is determined at each temperature by solving Eq. (37) in terms of B1 and setting x 0:05. It is no surprise that the curve has a maximum in the T ; x plane because the condition oD=oxT ;xm 0 de®nes an extremum point. This is also one of the conditions that de®nes the critical point of this system. However, there is no reason the second critical point condition o2 D=ox2 T ;xm 0 should also be satis®ed. It is sucient that this second derivative be negative to de®ne a maximum in the curve. 4. Phase equilibrium and critical points Satisfying the phase equilibrium conditions, l0BO l00BO and D0 D00 can be done disregarding the chemical equilibrium condition. This is done
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
Fig. 2. The curve that satis®es only chemical equilibrium is the solid line. The curve satisfying only phase equilibrium is the dashed line. The phase coexistence curve satisfying both chemical and phase equilibrium is not graphically distinguishable from the pure chemical equilibrium curve given by the dotted line.
Fig. 3. These curves correspond to the curves shown in Fig. 2 where in the T ; b1 , with b1 ÿlnK3 only one curve appears because b1 is equal on both branches. The chemical equilibrium curve is the solid line, the phase equilibrium curve is the dashed lines, the critical point locus is the dotted line, and the phase coexistence curve satisfying both chemical and phase equilibrium is the dash-dot line.
9
Fig. 4. These curves correspond to the curves shown in Fig. 2 where in the T ; D only one curve appears because D is equal on both branches. The chemical equilibrium curve is the solid line, the phase equilibrium curve is the dashed lines, and the critical point locus is the dotted line. The phase coexistence curve satisfying both chemical and phase equilibrium is not shown but lies on D 0:0 just as the pure chemical equilibrium curve.
by specifying a set of B2 , B3 and B4 , such that for a ®xed b1 the two conditions of phase equilibrium can be solved for x0 and x00 at a given temperature. By sweeping through the temperature a phase equilibrium curve is determined. A shell-like surface is obtained by sweeping through both b1 and T. The curves corresponding to satisfying the phase equilibrium conditions are shown in Figs. 2± 4 for b1 0:0, B2 ÿ1:477, B3 9:456 and B4 ÿ60:0. These parameter values were chosen as a representative example where two phase equilibrium is observed with a critical point occuring at Tc 1127 K and xc 0:05 [10]. In the T ; x plane shown in Fig. 2, the phase equilibrium curve (dashed line) appears to closely overlap the chemical equilibrium curve (solid lines). However, in the T ; b1 and T ; D planes shown in Figs. 3 and 4, respectively, it is clear the two curves are displaced and nearly meet only at their maximum. In the T ; b1 plane the phase equilibrium curve (dashed line) sits on b1 0:0, while in the T ; D
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L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
plane the chemical equilibrium curve (solid lines) sits at D 0:0. The conditions for the critical point as stated above are that the second and third derivative of the free energy with respect to the order parameter be equal to zero. Equivalently, this condition is also given by 2 oD oD 0: 41 ox T ;xm ox2 T ;xm Tc ;xc
Tc ;xc
The locus of critical points is determined by varying b1 , using Eq. (40) with the same values of B2 , B3 and B4 as above, and solving Eq. (41) for Tc and xc . The critical locus is shown in Figs. 3 and 4 as the dotted line. In both diagrams the critical locus intersects the maximum of the phase equilibrium curve and approaches the chemical equilibrium curve. The maximum of the phase PE 1127:00 K with equilibrium curve is at Tmax PE xmax 0:500 identical to the critical point, and the maximum of the chemical equilibrium curve is at CE 1126:83 K with xCE Tmax max 0:499. Hence, the maximum of the chemical equilibrium curve is not a critical point.
5. Phase coexistence: satisfying chemical and phase equilibria The true phase coexistence of this system is determined by simultaneously solving for both chemical equilibrium and phase equilibrium corresponding to the intersection of both surfaces. The only succinct method of obtaining a set of parameters that allows for the intersection to occur is to solve for phase and chemical equilibria simultaneously. The values of B2 ; B3 and B4 are obtained using the critical point conditions Eq. (41) and the critical values of Tc 1127 K and xc 0:05. To solve for phase equilibrium and chemical equilibrium requires determining the values of b1 ; x0 and x00 for a given T using the three conditions D 0, D0 D00 and l0BO l00BO . Note that the chemical equilibrium constraint of D 0 is solved for b1 as a function of T and x, where b1 is required to be equal in all phases. The solution to these equations is easily obtained using a Newton±
Raphson method after ®rst obtaining an approximate value at low temperature from a graphical analysis using the equations that satisfy both phase and chemical equilibrium. The temperature is raised in small increments and the previous values of b1 ; x0 and x00 are used as initial values and the sequence repeated until the critical point is reached. The phase coexistence curve that satis®es both chemical and phase equilibrium is nearly identical to the phase equilibrium curve of Fig. 2. The projection in the T ; b1 is shown in Fig. 3 as the dot±dash line that is nearly identical to the chemical equilibrium curve that does not simultaneously satisfy phase equilibrium. The projection in the T ; D plane is not interesting since it is just a line on D 0. The values of the bi s are given by the free energies of formation de®ned in Eq. (2). For the above phase coexistence curve, the free energies of formation are plotted as a function of 1=T in Fig. 5(a)±(d). From the slope and the intercept of each curve the enthalpy and entropy, respectively, of formation is obtained for each species. Eectively the enthalpies and entropies of formation can be predicted from the critical temperature and composition. (It is interesting to note that the plots are not linear, but are in fact curved due to composition dependence.) Alternatively, we can use temperature dependent enthalpy and entropy experimental data to ®t our parameters, and then predict the phase and chemical equilibrium behavior of a system along with the corresponding critical point. 6. Discussion In this work, the dierence between formation and mixing enthalpies is explained in terms of incorporating metal oxide into the network as opposed to solvating a metal oxide molecule adjacent to the network. The formation energetics come in as a consequence of chemical equilibrium reactions that dictate the incorporation of metal oxide into the network by substituting ionic bonds for covalent bonds. The idea that chemical reactions are playing an important role in phase behavior of
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
oxide materials has been previously used to study the uptake of oxygen into metallic alloy melts [18] and in the formation of slags [19]. In this theory, the reactions also serve to regulate the distribution of bond types about a central site, or regulate the distribution of associated species. Thus far, we have completely ignored the inclusion of a van der Waals-type mixing enthalpy expression. Although we can determine the enthalpy and entropy of formation given the critical temperature and composition, we have not included non-ideal mixing eects and so cannot expect our coexistence curves to be quantitative at this level of theory. We will address these points elsewhere. The question of whether the heat of formation due to the chemical reactions plays a role in the heat of mixing is raised. Mixing in solutions leads to a lowering of the free energy whenever the mixing entropy is positive. A positive mixing enthalpy between distinct species implies that packing is incompatible and so the solution will phase separate if the mixing entropy does not dominate. Upon heating, such a system will reach a temperature where the entropy of mixing again dominates and the solution will mix uniformly [20]. In solutions where van der Waals forces dominate, the roles of the mixing enthalpy and entropy are clear. Their role in silicates mixtures is less clear since chemical reactions take place between the metal oxide and the silica network. The direct eect of the chemical reactions are to form non-bridging oxygen bonds about a central silicon atom. In the network, the building blocks can only join bonds if the bond types are identical, hence the chemical reactions play an implicit role in allowing or disallowing bond formation. Such a system may appear to have a positive mixing enthalpy since the molecular species rearrange themselves to reduce the overall free energy of the system. Note that for this particular set of parameters, all the enthalpies of formation are positive, although of dierent magnitudes as indicated by the slopes of Fig. 5. A positive enthalpy of formation is consistent with a positive enthalpy of mixing, which is a criterion for observing phase separation. However, the inability to match up all the bonds leads to a phase separation where all the bridging and non-bridging oxygen bonds can be
11
saturated. This occurs with the constraint that the species distribution not change signi®cantly at a ®xed temperature, pressure and composition. It is conceivable that a redistribution of the species could occur and, therefore, completely avoid a phase separation, but at a cost of creating higher energy species. Note that if a species distribution does not change (at least not signi®cantly) the heat of reaction also does not change. Therefore, the phase separation is completely driven by the con®gurational incompatibilities. The presence of the chemical equilibrium does lead to a very rich set of phase coexistence behavior that includes three phase equilibrium. Although at ®rst thought a binary silicate system showing three phase equilibrium might appear to disobey the phase rules of thermodynamics, it in fact does not. The chemical reactions show that a pure silica molecule reacts with a pure metal oxide molecule to form a new species. In the bond representation of the theory these components correspond to the BO bond, the NBO bond and the free metal oxide, eectively a three component system. Thus, the phase rules with chemical reactions allow three phase equilibrium to occur. The theory also predicts the point at which all three phases come to simultaneous equilibrium. This is a non-symmetric tricritical point [21]. In the vicinity of a higher order critical point the phase coexistence curve shows an enhanced ¯atness not unlike that seen for the lead oxide±borate system [22]. This higher order critical phenomenon is a consequence of including the chemical equilibrium reactions. It is observed in a bond representation model by virtue of de®ning the order parameter of the silicate system as the number concentration of NBO bonds. 7. Conclusion Con®gurational entropy of a silicate network can drive phase separation whenever there is a distribution of incompatible building blocks that makeup the network backbone. The incompatibility exists as an inability to saturate the bridging and non-bridging oxygen bonds in a single phase. Inclusion of chemical equilibrium reactions that
12
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
Fig. 5. The logs of the equilibrium constant plotted as a function of Tc =T . Note that the scale on the y-axis is dierent for each plot. The enthalpy of formation corresponds to the slope and the entropy of formation corresponds to the intercept. See text.
allow for the redistribution of the BO and NBO bonds cannot prevent phase separation because the free energy to phase separate is less than that to redistribute molecular species into a set of compatible building blocks.
Acknowledgements The author would like to acknowledge K.D. Keefer for essential discussions regarding this work. This work was performed under the
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
auspices of the Division of Chemical Sciences, Oce of Basic Energy Sciences, US Department of Energy under Contract DE-AC06-76RLO 1830 with Battelle Memorial Institute that operates the Paci®c Northwest National Laboratory under Grant No. DE-FG06-89ER-75522 with the US Department of Energy. References [1] R.H. Doremus, Glass Science, 2nd Ed., Wiley, New York, 1994. [2] J.S. Rowlinson, Liquids and Liquid Mixtures, Butterworths, London, 1959. [3] R.J. Charles, Phys. Chem. Glasses 10 (1969) 169. [4] P.L. Lin, A.D. Pelton, Metall. Trans. 10B (1979) 667. [5] R.J. Araujo, J. Non-Cryst. Solids 55 (1983) 257. [6] P.J. Bray, R.V. Mulkern, E.J. Holupka, J. Non-Cryst. Solids 75 (1985) 37. [7] R. Dron, J. Non-Cryst. Solids 53 (1982) 267. [8] D.W. Bonnell, J.W. Hastie, High Temp. Sci. 26 (1990) 313. [9] W.G. Dorfeld, Phys. Chem. Glasses 29 (1988) 179. [10] L.R. Corrales, K.D. Keefer, J. Chem. Phys. 106 (1997) 6460.
13
[11] L.R. Corrales, in: J.T. Fourkas, D. Kivelson, W. Mohanty, K.A. Nelson (Eds.), Supercooled Liquids, Advances and Novel Applications, ACS Symposium Series 676, American Chemical Society, Washington, DC, 1997. [12] L.R. Corrales, J.C. Wheeler, J. Chem. Phys. 91 (1989) 7097. [13] G. Engelhardt, H. Jancke, D. Hoebbel, W. Weiker, Z. Anorg. Allg. Chem. 418 (1975) 17. [14] K. Denbigh, The Principles of Chemical Equilibrium, 4th ed., Cambridge University, Cambridge, 1981. [15] P. Flory, J. Chem. Phys. 10 (1942) 51; 12 (1944) 426. [16] C.R. Masson, I.B. Smith, S.G. Whiteway, Can. J. Chem. 48 (1970) 460; C.R. Masson, in: J. Gotz (Ed.), Glass 1977, Proc. 11th International Congress on Glass, vol. 1, Prague, 1977, pp. 3±41. [17] L.R. Corrales, K.D. Keefer, J. Chem. Phys. 109 (1998) 2044. [18] J.P. Hajra, M. Wang, M.G. Frohberg, Z. Metalkd. 81 (1990) 255. [19] H. Gaye, J. Lehmann, P.V. Riboud, J. Welfringer, Mem. Etud. Sci. Rev. Metall. 86 (1989) 237. [20] A. Findlay, A.N. Campbell, N.O. Smith, Phase Rule, 9th Ed., Dover, New York, 1959. [21] J. Bartis, J. Chem. Phys. 59 (1973) 5423. [22] See J. Zarzycki, Glasses and The Vitreous State, Cambridge Solid State Series, Cambridge University, Cambridge, 1982, p. 152, Fig. 6.4.
Immiscibility in binary silicates: chemical and phase equilibria L.Rene Corrales
*
Environmental Molecular Sciences Laboratory, Paci®c Northwest National Laboratory, P.O. Box 999 K1-96, Richland, WA 99352, USA Received 23 March 1998; received in revised form 22 July 1998
Abstract The driving force for immiscibility in binary silicates was examined using a statistical theory which combines a Florytype theory, based on the compatibility of structural units, and a chemical equilibrium model, that describes the preferred associated species distribution of silicates. Restrictions imposed by a preferred distribution of associated species lead to a limitation of the number of ways to form a complete network structure, or equivalently to a decrease in the con®gurational entropy. An increase in con®gurational entropy is achieved by inducing a phase separation. The role of the enthalpies of species formation is examined relative to the con®gurational entropic contribution to phase coexistence. The theory leads to a rich set of phase coexistence behavior that includes three-phase equilibrium and a higher order critical point. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Immiscibility gaps in silicate melts are known to have both stable and metastable upper critical solution temperatures [1]. This is typical of a system with a positive enthalpy of mixing and a positive mixing entropy [2]. A plausible explanation for the occurence of phase separation in silicates considers the preferred coordination of free metal oxide about the non-bridging oxygen bond. The argument is that increasing the coordination about metal atoms is energetically favorable [3]. This provides a positive mixing enthalpy to the system and can lead to phase separation. However, silicate melts incorporate metal oxides into the network structure by changing a bridging oxygen bond into a pair of non-bridging oxygen bonds while maintaining the integrity of the network.
* Tel.: 1-509 375 6410; fax: 1-509 375 6631; e-mail: [email protected].
Hence, the interactions in silicate melts are not simply packing interactions. Another way to explain phase separation is to consider the incorporation of metal oxide into the network, eectively modi®ng the network structure, and describe it in terms of a heat of mixing [4]. This approach also predicts phase separation and, as a structural model, it has the advantage of determining network connectivity. However, neither of the above two approaches explain the preferred associated species distribution of silicates. An alternative approach, used in this work, is based on structural compatibility of building blocks that make up the backbone of the network [5,6]. Each building block is a tetrahedron with a central silicon atom coordinated by bridging and non-bridging oxygen bonds. In all, there are ®ve basic building blocks for a binary silicate system. A speci®ed distribution of these building blocks can be completely joined (thus saturating all the bonds) if all the building blocks are compatible with one another. When a sucient number of
0022-3093/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 7 8 2 - 0
2
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
incompatibilities exist between the building blocks, phase separation will occur in order to optimize the bond saturation. This statistical approach is based strictly on the con®gurational entropy of a solution where the distribution of the building block types is not changed. Its success in predicting phase transitions in all types of network materials, regardless of coordination, supports the idea that structural incompatibilities are the underlying force that drive phase transitions in network materials. The liquid components in silicates are present as clearly identi®able stoichiometries and free energies of formation [7]. These components are effectively the building blocks of the aformentioned statistical theory and are also refered to as the associated species. The uptake of metal oxide into the network and the formation of preferred coordination of each metal type about a silica site varies as a function of temperature, pressure and composition. The associated species distribution has been determined using species activities in a thermodynamic model [8], and by chemical equilibrium models [9] that use a linear combination of the species activities or free energies of formation. Both models lead to similar results. The latter method is used in this work because it better describes the species distribution in the dilute limits. Although these approaches are useful in determining and predicting the associated species distribution as a function of temperature and composition, they cannot by themselves predict phase coexistence in which the underlying driver is the con®gurational entropy. Alternatively, a chemical equilibrium approach lets us dierentiate between the enthalpy of forming new species (i.e. the formation of a non-bridging oxygen bond from a bridging oxygen bond reacting with a metal oxide), and the enthalpy of mixing pure silica and pure metal oxide where no reaction takes place. A better understanding of the explicit and implicit driving forces that lead to phase separation in silicate melts is achieved by partitioning the underlying interactions. A simple and elegant way to partition the interactions in a silicate melt is to consider that the network is made up of tetrahedral building blocks. In a silicate material, the molecular or associated species make up the
building blocks of the network structure as described above. The chemical equilibrium reactions serve the purpose of regulating the coordination state of the tetrahedrons. To build the network backbone the building blocks must be connected such that bridging oxygen (BO) bonds are matched with BO bonds, and non-bridging oxygen (NBO) bonds are matched with the same type of NBO bonds. The idea that the melt is made up of a network of building blocks is strictly theoretical. In fact in this work, the idea that the network backbone is built of building blocks is transposed to the idea that the network backbone is made up of BO and NBO bonds. This follows as a consequence of recognizing that the natural order parameter for phase separation in silicates is related to the formation of NBO bonds. The topology of the network dominates the con®gurational entropy because the number of ways of bringing the network together is determined by the building blocks that make up the backbone of the network structure. If the structural units are incompatible, because the building blocks cannot saturate their bonds, phase separation will occur where each phase will contain a set of compatible building blocks. There can be cases where a compatible network is only formed in which a distribution set of building blocks is not the lowest energy state in terms of the formation energies. In other words, the less probable associated species are forced to form. In the previous case, the mixing entropy drives the network into compatibility via rearranging the building blocks, whereas in the latter case the enthalpy of formation drives the system to compatibility via forming a new distribution set of building blocks. For a given metal oxide, the equilibrium constants vary as a function of temperature, pressure and composition. Thus, the equilibrium constants are eectively ®xed for a given composition, temperature and pressure. The preferred coordination of a metal oxide about a central silicon atom can lead to a distribution of incompatible building blocks. Phase separation will occur whenever unmixing is energetically favorable over redistributing the molecular species. In this work a brief description of a recently introduced statistical thermodynamic theory [10]
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
that combines a chemical equilibrium reaction model, that describes the molecular species distribution, with a statistical Flory-type treatment of the mixing interactions, that describes the connectivity of the network is presented. We show how phase equilibrium and chemical equilibrium can be theoretically partitioned, where the true equilibrium state of the system is achieved by simultaneously satisfying both chemical and phase equilibria. This clearly shows that (1) the phase equilibrium is driven by the con®gurational entropy and is identical to previous statistical theories, and (2) the chemical equilibrium portion is identical to Dorfeld's model [9] from which the associated species distribution can be obtained as a function of temperature and composition. A distinct dierence of the Corrales and Keefer theory from all other approaches is that the model is cast in terms of the bond types, which is the natural order parameter of this system. Curiously, phase separation appears to occur within the chemical equilibrium portion of the theory. However, it is shown below that the conditions of phase equilibrium are not satis®ed on that surface. In previous work, the true phase coexistence curves predicted by this theory were obtained by simultaneously satisfying both the chemical and phase equilibria conditions [10,11]. A rich set of phase coexistence behavior can be obtained that includes three-phase equlilibrium and a non-symmetric tricritical point [10] that has not been predicted by the related theories. Here we show that the true phase coexistence curve is a result of the intersection of a phase equilibrium surface and a chemical equilibrium surface such that both conditions are simultaneously satis®ed [12].
3
system that reacts with a bridging (covalent) oxygen to form two Oÿ non-bridging (ionic) oxygens [7,9,10]: BSi±O±SiB M2 O BSi±Oÿ M M Oÿ ±Si ; where M Li, Na, K, and so on. Network modi®ers are not restricted to alkali metals nor to metals with oxidation states of one, although the chemical reactions are distinct for other oxidation states. The above chemical reaction is considered to be in equilibrium. A silicon site can be four-fold coordinated by a combination of BO and NBO bonds, which correspond to a covalent oxygen and a pair of ionic oxygens, respectively. For convenience we use the Q species notation [13] to describe each of the possible coordinations of bond types about a central silicon atom as shown in Fig. 1 and described as follows. It must be made clear that a BO or a NBO are shared between the tetrahedra formed by the Q site. Hence, in the following de®nitions of the Q species a half-BO corresponds to sharing a covalent oxygen, and a half-NBO corresponds to a single ionic oxygen. The Q species de®nition are identical to the building blocks in the work by Araujo [5]. Thus, a Q4 site is made up of four half-BO bonds, a Q3 site is made up of three half-BO and one half-NBO bonds, a Q2 site is made up of two half-BO and two half-NBO bonds, a Q1 site is made up of one half-BO and two half-NBO bonds, and a Q0 site is made up of four half-NBO bonds.
2. Theoretical model Incorporation of a metal oxide into an oxide network is the result of a chemical reaction that transforms a BO bond to an NBO bond. This must be distinguished from the case in which a free metal oxide is eectively solvated by the network material, where only packing and electrostatic interactions come to play. The chemical reaction consists of a M2 O contributing an O2ÿ to the
Fig. 1. The tetrahedral building blocks in terms of Q species notations. The metal cations have been left out for clarity.
4
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
The equilibrium reactions that describe the solvation of metal oxide to produce any of the Q species from the Q4 species can be written as
The chemical equilibrium constants, or free energies of formation, are given by
units, and all the free metal oxide occupy interstitial sites o the network lattice sites and bonds. As the system is allowed to progress to chemical equilibrium, the bond types change such that metal oxide units incorporate into the bonds to form NBO bonds. Thus, the number of SiO2 units remains unchanged, this is re¯ected in the determination of the mass balance equations that follow. Each Q species contributes a total of one SiO2 unit. Hence, the total number of sites Ns or SiO2 units is given by
ÿ kT ln Ki DG0i DHi0 ÿ T DSi0 ;
Ns Q4 Q3 Q2 Q1 Q0 :
K3
Q4 12M2 O Q3 K2
Q4 M2 O Q2 ; K1
Q4 32M2 O Q1 ;
1
K0
Q4 2M2 O Q0 :
2
where we assume ideal mixing such that there are no volume changes in the formation of new species [14]. Q species can also be formed from cross reactions that are linear combinations of Eq. (1) [7]. For example, K22ÿ40
Q2 Q2 Q4 Q0 ; K22ÿ31
Q2 Q2 Q3 Q1 ;
3
K31ÿ40
Q3 Q1 Q4 Q0 ; where the last equation is a linear combination of the ®rst two reactions. Assigned to the formation of each species are the conjugate chemical equilibrium constants that act as statistical weights for each of the Qi species. The chemical equilibrium constants can be written in terms of the species concentrations, which for convenience are also de®ned by Qi . Ki
Qi
4ÿi=2
Q4 nm
;
4
where i f3; 2; 1; 0g, and nm is the number concentration of free metal oxide. In Eq. (4) the activity coecient that should appear as ck Qk [14] is approximated as ck 1 corresponding to an ideal solution. There are two mass balance equations, one for the SiO2 species and one for the total metal oxide which includes the free metal oxide and the metal oxide incorporated as NBO bonds. Initially, all sites are occupied by pure Q4 units, or in other words the system consist of strictly pure SiO2
5
In addition, each Q species, except for the Q4 species, contributes half of a M2 O unit for every half-NBO bond it contains. Thus, the total number of NBO bonds nNBO or metal oxide species incorporated into the network is given by nNBO 12Q3 Q2 32Q1 2Q0 :
6
Each unreacted species contributes one full M2 O unit such that there are nm free metal oxide units. Therefore, the total number of metal oxide species Nm in the site model is given by Nm nm nNBO :
7
The eective mass conservation equation is determined by normalizing the mass balance equations with respect to the number of sites. This allows taking the thermodynamic limit, namely where the number of sites goes to in®nity, without compromising any other aspect of the theory. Dividing Eqs. (5)±(7) by Ns de®nes the fractional amount of SiO2 units occupying Ns sites on the lattice X q 1 q4 q 3 q 2 q 1 q 0 ;
8
where Xq 1 indicates that all sites are occupied by a SiO2 units and where qi Qi =Ns . The fractional amount of metal oxide Xm occupying both NBO bonds and interstitial sites is given by Xm xm xNBO ;
9
where Xm Nm /Ns , xm nm /Ns and xNBO nNBO / Ns with nNBO given by Eq. (6). Since the free metal oxide is o the lattice it is necessary to de®ne an apparent mole fraction by
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
introducing the apparent total mass as follows. The total amount of material in the system XT is given by XT Xm Xq Xm 1:
10
Thus, the apparent mole fractions of M2 O in terms of the lattice site model is Xm 11 vsm XT and for SiO2 it is vsSiO2
Xq 1 : X T XT
12
It is now possible to obtain the distribution of the Q species as a function of the equilibrium constants and temperature. This is done by solving for the six unknowns, namely the ®ve qi s and xm , using the four equilibrium equations given by Eq. (4) and the two mass balance equations Eqs. (8) and (9). This is identical to the work by Dorfeld [9]. Solutions are obtained by specifying the equilibrium constants (or free energies) at a ®xed temperature, then sweeping through the variable xm while solving for the qi s. It is now possible to derive a partition function and, hence, write down a phenomenological free energy based on the Q species, or sites of a lattice model. In deriving the partition function, the chemical equilibrium constants of Eq. (4) are used as the statistical weights. The con®guration entropy contribution is calculated using a Flory-type theory [5,6,10,15,16]. However, the natural order parameter for this system is the degree to which the network has depolymerized. Such an order parameter is ill de®ned in a site or Q model representation. Instead the lattice site model is transposed into a lattice bond model. This allows the order parameter to be de®ned by the number concentration of broken bonds. A detailed derivation of the underlying theory is given in Ref. [10]. Here only the transposition of variables and the phenomenological free energy are given. The phase and chemical equilibrium conditions are extracted from an analysis of a standard isothermal dierential free energy and calculated from the bond model free energy. The bond model requires that the chemical equilibrium constants be de®ned in terms of the
5
site occupations. Thus, the probabilities of forming each of the speci®c Q sites in terms of the bond types (namely BO and NBO bonds) are used. The probability of forming a Q4 site from nBO BO bonds is n4BO . The probability of forming a Q3 site from nNBO NBO and nBO BO bonds is n3BO nNBO . Similarly, the probability of forming a Q2 is n2BO n2NBO , the probability of forming a Q1 is nBO n3NBO , and the probability of forming a Q0 is n4NBO . Moreover, the asymmetric species have ways of being formed that are uniquely dierent from one another in that they lead to dierent network connectivities, and so they must be counted. The Q3 and Q1 species each have four distinguishable ways of being formed on a site, and the Q2 species has six distinguishable ways of being formed on a site. The corresponding mass conservation equations for the bond model are determined as follows. The total number of bonds in the system is given by Nb nBO nNBO , or nBO Nb ÿ nNBO :
13
The total amount of SiO2 in terms of the bond model is given by substituting the probability of forming each Q species into Eq. (5). Note that the probability of forming Ns sites from the total number of bonds is given by Nb4 . 4
3
Nb4 Nb ÿ nNBO 4nNBO Nb ÿ nNBO
6n2NBO Nb ÿ nNBO 2 4n3NBO Nb ÿ nNBO n4NBO :
14
The total amount of metal oxide for the bond model Nmb is given similarly as in Eq. (7) in which the probabilities of forming each speci®c Q species are substituted into Eq. (6) 3
Nmb nm 124nNBO Nb ÿ nNBO 2
6n2NBO Nb ÿ nNBO 324n3NBO Nb ÿ nNBO 2n4NBO :
15
By expanding the polynomial and collecting terms, Eq. (15) simpli®es to Nmb nm 2Nb3 nNBO :
16
6
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
Dividing these mass balance equations by Nb4 we obtain the normalized mass balance equations for the total metal oxide content Xm Nmb =Nb4 xm 2x
17
together with the total number of silica units given by X qi 1 ÿ x4 4x 1 ÿ x3 XSiO2 i
2
2
3
4
6x 1 ÿ x 4x 1 ÿ x x 1:
18
The total mass conservation of this system is XT Xm XSiO2 xm 2x 1:
19
The apparent fraction of metal oxide in the bond representation is given by vm
2x xm ; 1 2x xm
20
with the apparent fraction of silica given by vSiO2
1 : 1 2x xm
21
To calculate the species distribution as a function of the equilibrium constants and temperature, the chemical equilibrium conditions must be determined for the bond representation of the theory. This is discussed in detail below. An approximate phenomenological free energy for the bond model derived by Corrales and Keefer [10] is
B1 ÿkT ln K3 ; K2 B2 ÿkT ln ; K32 K1 K33 ; B3 ÿkT ln K3 26 K0 K2 B4 ÿkT ln : K14 K34
The phase equilibrium and chemical equilibrium conditions are determined by analyzing a standard expression of the isothermal dierential free energy. In terms of the site model, an isothermal dierential free energy is dGT l4 dQ4 l3 dQ3 l2 dQ2 l1 dQ1 l0 dQ0 lm dnm :
4
dGT 4lBO d Nb ÿ nNBO
3lBO lNBO d4nNBO Nb ÿ nNBO 3 2
2lBO 2lNBO d6n2NBO Nb ÿ nNBO lBO 3lNBO d4n3NBO Nb ÿ nNBO
ÿ nm 2Nb3 nNBO kT ln zm 4Nb3 nNBO B1
4lNBO dn4NBO lm dnm :
6Nb2 n2NBO B2 4Nb n3NBO B3 n4NBO B4
4Nb3 Nb
24
The associated species chemical potentials can be de®ned in terms of the bond type potentials as li ilBO 4 ÿ ilNBO with i 0; 1; 2; 3; 4, and where lm is the chemical potential for the pure metal oxide. Substituting the Qi s by the formation probabilities in terms of bond types and the site chemical potentials by those in terms of the bond type chemical potentials, the isothermal dierential free energy of the bond model is
G ÿNb4 kT ln z4
4Nb3 nNBO kT ln
23
By using the conservation equations, given by Eqs. (14) and (15), expanding out the polynomial terms and rearranging, Eq. (25) simpli®es to
nNBO Nb
Nb ÿ nNBO ÿ nNBO kT ln ; Nb
25
22
where z4 and zm correspond to the activities of pure SiO2 and M2 O species, which are the initial states of the model. The Bi coecients are a linear combination of the free energies of formation of the chemical reactions of Eq. (1) given in terms of the chemical equilibrium constants de®ned by Eq. (2) [10].
dGT 4lBO dNb4 4 D 12lm dNb3 nNBO lm dnm ;
26
D lNBO ÿ lBO ÿ 12lm :
27
where From this de®nition of D, it represents the negative anity of reaction of creating an NBO from a BO and metal oxide [14]. Hence, to satisfy chemical equilibrium, D must be equal to zero. The phase
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
equilibrium conditions are de®ned by lm , D and lBO being equal in all phases for a ®xed temperature and pressure. The use of lm is mooted as will be seen below. Therefore, only two conditions for phase equilibrium need to be satis®ed. In the original work by Corrales and Keefer [10], an error was made in the dierential free energy expression that has been corrected above. The error is in failing to make the substitution of Nm nm 2Nb3 nNBO in the free energy and the dierential free energy expressions. This error led to the incorrect expression of Eqs. (34), (36) and (37) of Ref. [10]. However, the correct expression for D was given for the wrong reason such that no errors were propagated. In this work we have corrected all the equations [17]. For ease of calculation, the dierential free energy is expressed in terms of dNb , dnNBO and dnm which leads to
equilibrium [2]. Calculating the indicated dierentials on the free energy given by Eq. (22) gives D 12 lm oG=kT 4Nb3 kT onNBO T ;Nb ;nm B1 ÿ2Nb3 ln zm 4Nb3 kT B2 B3 12Nb n2NBO 12Nb2 nNBO kT kT B4 nNBO 3 3 ; 4nNBO 4Nb ln kT Nb ÿ n
oG=kT oNb
T ;nNBO ;nm
29
oG onNBO
T ;Nb ;nm
1 4Nb3 D lm 2 4Nb3 lNBO ÿ lBO ;
oG onm
30
T ;Nb ;nNBO
lm :
Consequently, lBO is given by l nNBO 1 ÿ 3 D l 4lBO Nb 2 m 4Nb3 1 oG : B 4 G ÿ nNBO Nb onNBO
31
32
The latter equivalence is the common tangent construction that is also used to determine phase
4Nb3
l kT
ÿ4Nb3 ln z4 ÿ 12Nb2 n ln zm B1 B2 12Nn2NBO 12Nb2 nNBO kT kT B3 4n3NBO kT nNBO 4 3Nb2 nNBO ln 4Nb3 Nb Nb ÿ nNBO ; ÿ3Nb2 nNBO ln Nb
28
This equation implies that the dierentials are of the form oG 4Nb3 l oNb T ;nNBO ;nm nNBO 1 3 ; D lm 4Nb 4lBO 3 Nb 2
33
dGT 44Nb3 lBO 3Nb2 nNBO D 12lm dNb 4Nb3 D 12lm dnNBO lm dnm :
7
oG=kT onm
34
T ;Nb ;nNBO
lm ÿ ln zm : kT
Therefore, using Eqs. (32)±(34), lBO is 4lBO B2 B3 B4 ÿ ln z4 ÿ 6x2 ÿ 8x3 ÿ 3x4 kT kT kT kT 4 ln 1 ÿ x ;
35
36
where x nNBO =Nb . Also using Eqs. (33) and (35) we get x D B1 B2 B3 B4 3x 4x2 x3 ln : kT kT kT kT kT 1ÿx 37 The concentration of free metal oxide xm is determined as follows. As stated previously, the chemical equilibrium of the system is given by Eq. (27) being equal to zero which is the negative anity of the reaction in which a BO bond is transformed to an NBO bond by the uptake of a
8
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
metal oxide. Following the de®nition of the chemical equilibrium constants as in Eq. (4), the equilibrium constant for converting a BO bond to an NBO bond is x e D=kT 38 KNBO 1=2 1 ÿ xxm Hence, at equilibrium KNBO 1 and Eq. (38) leads to x x ; 39 x1=2 m D=kT 1 ÿ xe 1 ÿ x where the latter equality is true, because at chemical equilibrium D 0. In the following discussions the parameters given by Eq. (23) are rewritten as B1 kTb1 ;
40
B2 kT b2 ÿ 2b1 ; B3 kT b3 ÿ 3b2 3b1 ; B4 kT b4 ÿ 4b3 6b2 ÿ 4b1 ; where bi ÿ ln K 4ÿi are just the free energies of formation identical to Eq. (2) and corresponding to the chemical reactions of Eq. (1). This allows specifying the parameters B2 ; B3 and B4 while varying B1 (or b1 ) and eectively varying all the bi s. 3. Chemical equilibrium Knowing the chemical equilibrium condition, D 0, we can iteratively solve for x in Eq. (37) at a given temperature for a given set of equilibrium constants. In this bond representation, the free metal oxide concentration is eectively held ®xed by virtue of Eq. (39). This implies that to determine the distribution of the associated species at a given temperature for all compositions, the Bi parameters must be varied. This diers from solving for the distribution using the site representation. The dierence arises from holding density parameters versus holding the conjugate ®eld parameters ®xed. In the site representation it is the equilibrium constants that are held ®xed while varying the free metal oxide concentration. Whereas, in the bond representations it is the free metal oxide concentration that is held ®xed while the equilibrium constants are varied. The common
tie is that both methods de®ne a surface upon which chemical equilibrium is satis®ed in the temperature±composition±equilibrium constant space, where only one of the equilibrium constants is used in the plot. Experimentally, it is the composition variables that are held constant, therefore obtaining the species distribution from the site model is the preferred method. Identical results can be extracted from the bond model, but it is an arduous and unnecessary task. Chemical equilibrium given by D 0 leads to de®ning a surface in the T ; x; b1 space. A point that sits on the surface is obtained by ®xing B1 ; B2 ; B3 and B4 , speci®ng a temperature and solving for x. The temperature is then varied to obtain the entire locus. There is a range of the parameters that lead to two or more stable solutions of x for a given b1 and T , thus, a curve is obtained. (This is easily seen by graphing D vs. x and ®nding a set of the parameters that lead to solutions with D 0.) The stable solutions are determined by the condition that the second derivative of the free energy with respect to the order parameter be greater than or equal to zero. This is identical to oD=oxT ;xm P 0. The chemical equilibrium surface is obtained by sweeping through any of B2 ; B3 or B4 . Projections of a chemical equilibrium curve in the T ; x, T ; b1 and T ; D planes are shown in Figs. 2±4, respectively, as the solid lines. The ®xed parameter set of B2 ÿ1:477, B3 9:456 and B4 ÿ60:0 [10]. The value of B1 is determined at each temperature by solving Eq. (37) in terms of B1 and setting x 0:05. It is no surprise that the curve has a maximum in the T ; x plane because the condition oD=oxT ;xm 0 de®nes an extremum point. This is also one of the conditions that de®nes the critical point of this system. However, there is no reason the second critical point condition o2 D=ox2 T ;xm 0 should also be satis®ed. It is sucient that this second derivative be negative to de®ne a maximum in the curve. 4. Phase equilibrium and critical points Satisfying the phase equilibrium conditions, l0BO l00BO and D0 D00 can be done disregarding the chemical equilibrium condition. This is done
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
Fig. 2. The curve that satis®es only chemical equilibrium is the solid line. The curve satisfying only phase equilibrium is the dashed line. The phase coexistence curve satisfying both chemical and phase equilibrium is not graphically distinguishable from the pure chemical equilibrium curve given by the dotted line.
Fig. 3. These curves correspond to the curves shown in Fig. 2 where in the T ; b1 , with b1 ÿlnK3 only one curve appears because b1 is equal on both branches. The chemical equilibrium curve is the solid line, the phase equilibrium curve is the dashed lines, the critical point locus is the dotted line, and the phase coexistence curve satisfying both chemical and phase equilibrium is the dash-dot line.
9
Fig. 4. These curves correspond to the curves shown in Fig. 2 where in the T ; D only one curve appears because D is equal on both branches. The chemical equilibrium curve is the solid line, the phase equilibrium curve is the dashed lines, and the critical point locus is the dotted line. The phase coexistence curve satisfying both chemical and phase equilibrium is not shown but lies on D 0:0 just as the pure chemical equilibrium curve.
by specifying a set of B2 , B3 and B4 , such that for a ®xed b1 the two conditions of phase equilibrium can be solved for x0 and x00 at a given temperature. By sweeping through the temperature a phase equilibrium curve is determined. A shell-like surface is obtained by sweeping through both b1 and T. The curves corresponding to satisfying the phase equilibrium conditions are shown in Figs. 2± 4 for b1 0:0, B2 ÿ1:477, B3 9:456 and B4 ÿ60:0. These parameter values were chosen as a representative example where two phase equilibrium is observed with a critical point occuring at Tc 1127 K and xc 0:05 [10]. In the T ; x plane shown in Fig. 2, the phase equilibrium curve (dashed line) appears to closely overlap the chemical equilibrium curve (solid lines). However, in the T ; b1 and T ; D planes shown in Figs. 3 and 4, respectively, it is clear the two curves are displaced and nearly meet only at their maximum. In the T ; b1 plane the phase equilibrium curve (dashed line) sits on b1 0:0, while in the T ; D
10
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
plane the chemical equilibrium curve (solid lines) sits at D 0:0. The conditions for the critical point as stated above are that the second and third derivative of the free energy with respect to the order parameter be equal to zero. Equivalently, this condition is also given by 2 oD oD 0: 41 ox T ;xm ox2 T ;xm Tc ;xc
Tc ;xc
The locus of critical points is determined by varying b1 , using Eq. (40) with the same values of B2 , B3 and B4 as above, and solving Eq. (41) for Tc and xc . The critical locus is shown in Figs. 3 and 4 as the dotted line. In both diagrams the critical locus intersects the maximum of the phase equilibrium curve and approaches the chemical equilibrium curve. The maximum of the phase PE 1127:00 K with equilibrium curve is at Tmax PE xmax 0:500 identical to the critical point, and the maximum of the chemical equilibrium curve is at CE 1126:83 K with xCE Tmax max 0:499. Hence, the maximum of the chemical equilibrium curve is not a critical point.
5. Phase coexistence: satisfying chemical and phase equilibria The true phase coexistence of this system is determined by simultaneously solving for both chemical equilibrium and phase equilibrium corresponding to the intersection of both surfaces. The only succinct method of obtaining a set of parameters that allows for the intersection to occur is to solve for phase and chemical equilibria simultaneously. The values of B2 ; B3 and B4 are obtained using the critical point conditions Eq. (41) and the critical values of Tc 1127 K and xc 0:05. To solve for phase equilibrium and chemical equilibrium requires determining the values of b1 ; x0 and x00 for a given T using the three conditions D 0, D0 D00 and l0BO l00BO . Note that the chemical equilibrium constraint of D 0 is solved for b1 as a function of T and x, where b1 is required to be equal in all phases. The solution to these equations is easily obtained using a Newton±
Raphson method after ®rst obtaining an approximate value at low temperature from a graphical analysis using the equations that satisfy both phase and chemical equilibrium. The temperature is raised in small increments and the previous values of b1 ; x0 and x00 are used as initial values and the sequence repeated until the critical point is reached. The phase coexistence curve that satis®es both chemical and phase equilibrium is nearly identical to the phase equilibrium curve of Fig. 2. The projection in the T ; b1 is shown in Fig. 3 as the dot±dash line that is nearly identical to the chemical equilibrium curve that does not simultaneously satisfy phase equilibrium. The projection in the T ; D plane is not interesting since it is just a line on D 0. The values of the bi s are given by the free energies of formation de®ned in Eq. (2). For the above phase coexistence curve, the free energies of formation are plotted as a function of 1=T in Fig. 5(a)±(d). From the slope and the intercept of each curve the enthalpy and entropy, respectively, of formation is obtained for each species. Eectively the enthalpies and entropies of formation can be predicted from the critical temperature and composition. (It is interesting to note that the plots are not linear, but are in fact curved due to composition dependence.) Alternatively, we can use temperature dependent enthalpy and entropy experimental data to ®t our parameters, and then predict the phase and chemical equilibrium behavior of a system along with the corresponding critical point. 6. Discussion In this work, the dierence between formation and mixing enthalpies is explained in terms of incorporating metal oxide into the network as opposed to solvating a metal oxide molecule adjacent to the network. The formation energetics come in as a consequence of chemical equilibrium reactions that dictate the incorporation of metal oxide into the network by substituting ionic bonds for covalent bonds. The idea that chemical reactions are playing an important role in phase behavior of
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
oxide materials has been previously used to study the uptake of oxygen into metallic alloy melts [18] and in the formation of slags [19]. In this theory, the reactions also serve to regulate the distribution of bond types about a central site, or regulate the distribution of associated species. Thus far, we have completely ignored the inclusion of a van der Waals-type mixing enthalpy expression. Although we can determine the enthalpy and entropy of formation given the critical temperature and composition, we have not included non-ideal mixing eects and so cannot expect our coexistence curves to be quantitative at this level of theory. We will address these points elsewhere. The question of whether the heat of formation due to the chemical reactions plays a role in the heat of mixing is raised. Mixing in solutions leads to a lowering of the free energy whenever the mixing entropy is positive. A positive mixing enthalpy between distinct species implies that packing is incompatible and so the solution will phase separate if the mixing entropy does not dominate. Upon heating, such a system will reach a temperature where the entropy of mixing again dominates and the solution will mix uniformly [20]. In solutions where van der Waals forces dominate, the roles of the mixing enthalpy and entropy are clear. Their role in silicates mixtures is less clear since chemical reactions take place between the metal oxide and the silica network. The direct eect of the chemical reactions are to form non-bridging oxygen bonds about a central silicon atom. In the network, the building blocks can only join bonds if the bond types are identical, hence the chemical reactions play an implicit role in allowing or disallowing bond formation. Such a system may appear to have a positive mixing enthalpy since the molecular species rearrange themselves to reduce the overall free energy of the system. Note that for this particular set of parameters, all the enthalpies of formation are positive, although of dierent magnitudes as indicated by the slopes of Fig. 5. A positive enthalpy of formation is consistent with a positive enthalpy of mixing, which is a criterion for observing phase separation. However, the inability to match up all the bonds leads to a phase separation where all the bridging and non-bridging oxygen bonds can be
11
saturated. This occurs with the constraint that the species distribution not change signi®cantly at a ®xed temperature, pressure and composition. It is conceivable that a redistribution of the species could occur and, therefore, completely avoid a phase separation, but at a cost of creating higher energy species. Note that if a species distribution does not change (at least not signi®cantly) the heat of reaction also does not change. Therefore, the phase separation is completely driven by the con®gurational incompatibilities. The presence of the chemical equilibrium does lead to a very rich set of phase coexistence behavior that includes three phase equilibrium. Although at ®rst thought a binary silicate system showing three phase equilibrium might appear to disobey the phase rules of thermodynamics, it in fact does not. The chemical reactions show that a pure silica molecule reacts with a pure metal oxide molecule to form a new species. In the bond representation of the theory these components correspond to the BO bond, the NBO bond and the free metal oxide, eectively a three component system. Thus, the phase rules with chemical reactions allow three phase equilibrium to occur. The theory also predicts the point at which all three phases come to simultaneous equilibrium. This is a non-symmetric tricritical point [21]. In the vicinity of a higher order critical point the phase coexistence curve shows an enhanced ¯atness not unlike that seen for the lead oxide±borate system [22]. This higher order critical phenomenon is a consequence of including the chemical equilibrium reactions. It is observed in a bond representation model by virtue of de®ning the order parameter of the silicate system as the number concentration of NBO bonds. 7. Conclusion Con®gurational entropy of a silicate network can drive phase separation whenever there is a distribution of incompatible building blocks that makeup the network backbone. The incompatibility exists as an inability to saturate the bridging and non-bridging oxygen bonds in a single phase. Inclusion of chemical equilibrium reactions that
12
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
Fig. 5. The logs of the equilibrium constant plotted as a function of Tc =T . Note that the scale on the y-axis is dierent for each plot. The enthalpy of formation corresponds to the slope and the entropy of formation corresponds to the intercept. See text.
allow for the redistribution of the BO and NBO bonds cannot prevent phase separation because the free energy to phase separate is less than that to redistribute molecular species into a set of compatible building blocks.
Acknowledgements The author would like to acknowledge K.D. Keefer for essential discussions regarding this work. This work was performed under the
L.R. Corrales / Journal of Non-Crystalline Solids 242 (1998) 1±13
auspices of the Division of Chemical Sciences, Oce of Basic Energy Sciences, US Department of Energy under Contract DE-AC06-76RLO 1830 with Battelle Memorial Institute that operates the Paci®c Northwest National Laboratory under Grant No. DE-FG06-89ER-75522 with the US Department of Energy. References [1] R.H. Doremus, Glass Science, 2nd Ed., Wiley, New York, 1994. [2] J.S. Rowlinson, Liquids and Liquid Mixtures, Butterworths, London, 1959. [3] R.J. Charles, Phys. Chem. Glasses 10 (1969) 169. [4] P.L. Lin, A.D. Pelton, Metall. Trans. 10B (1979) 667. [5] R.J. Araujo, J. Non-Cryst. Solids 55 (1983) 257. [6] P.J. Bray, R.V. Mulkern, E.J. Holupka, J. Non-Cryst. Solids 75 (1985) 37. [7] R. Dron, J. Non-Cryst. Solids 53 (1982) 267. [8] D.W. Bonnell, J.W. Hastie, High Temp. Sci. 26 (1990) 313. [9] W.G. Dorfeld, Phys. Chem. Glasses 29 (1988) 179. [10] L.R. Corrales, K.D. Keefer, J. Chem. Phys. 106 (1997) 6460.
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[11] L.R. Corrales, in: J.T. Fourkas, D. Kivelson, W. Mohanty, K.A. Nelson (Eds.), Supercooled Liquids, Advances and Novel Applications, ACS Symposium Series 676, American Chemical Society, Washington, DC, 1997. [12] L.R. Corrales, J.C. Wheeler, J. Chem. Phys. 91 (1989) 7097. [13] G. Engelhardt, H. Jancke, D. Hoebbel, W. Weiker, Z. Anorg. Allg. Chem. 418 (1975) 17. [14] K. Denbigh, The Principles of Chemical Equilibrium, 4th ed., Cambridge University, Cambridge, 1981. [15] P. Flory, J. Chem. Phys. 10 (1942) 51; 12 (1944) 426. [16] C.R. Masson, I.B. Smith, S.G. Whiteway, Can. J. Chem. 48 (1970) 460; C.R. Masson, in: J. Gotz (Ed.), Glass 1977, Proc. 11th International Congress on Glass, vol. 1, Prague, 1977, pp. 3±41. [17] L.R. Corrales, K.D. Keefer, J. Chem. Phys. 109 (1998) 2044. [18] J.P. Hajra, M. Wang, M.G. Frohberg, Z. Metalkd. 81 (1990) 255. [19] H. Gaye, J. Lehmann, P.V. Riboud, J. Welfringer, Mem. Etud. Sci. Rev. Metall. 86 (1989) 237. [20] A. Findlay, A.N. Campbell, N.O. Smith, Phase Rule, 9th Ed., Dover, New York, 1959. [21] J. Bartis, J. Chem. Phys. 59 (1973) 5423. [22] See J. Zarzycki, Glasses and The Vitreous State, Cambridge Solid State Series, Cambridge University, Cambridge, 1982, p. 152, Fig. 6.4.