Phase Separation in Silicate Melts - American Chemical Society

Chapter 11

Phase Separation in Silicate Melts: Limits of Solubilities Downloaded by STANFORD UNIV GREEN LIBR on October 4, 2012 | http://pubs.acs.org Publication Date: September 30, 1997 | doi: 10.1021/bk-1997-0676.ch011

L . René Corrales Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, WA 99352

A statistical mechanical theory for silica melts is used to investigate the phase equilibrium behavior of binary silicates. The theory couples a Flory-type lattice model with a set of chemical equilibrium reactions that together capture the interplay between the solvation of a metal oxide into the silica network and the rearrangement of the network structure that lead to phase separation. The theory produces two-phase coexistence curves with interesting features characteristic of being in the proximity of a higher-order critical point. The theory contains the qualitative behavior and essential features of simple binary silicate melts.

1

Silicate melts are known to form multiple phases, where each phase has an inherent preference to solvate specific metal ions and, thus, form a distinct network structure. Partitioning of the metal oxide components, along with the characteristic network structure associated with the partitioning, lead to phases with different mechanical, thermodynamic and chemically reactive properties. The partitioning occurs in response to metal oxides forming specific coordinations about a central silicon, where the preferred coordination differs with each type of metal oxide. In simple binary systems, the incompatibility occurs when specific coordinated sites cannot be neighbors, due to enthalpy and entropy constraints that force a rearrangement of the network structure to minimize the free energy of the system. The interplay between the solvation of the metal oxide into the network, the formation of the preferred coordinated states, and the optimization of the network structure lead to miscibility gaps in a glass melt. Equilibrium chemical reactions govern the solvation of the metal oxide into the silica network, and determine their preferred coordination about a central silicon that define the silicate species. In principle, each type of metal oxide has a preference for the formation of specific silicate species whose concentration and distribution vary as a function of concentration and temperature. Thus, the equilibrium distribution of the silicate species is regulated by the affinity of reactions that are defined in terms of the species activities. In such a complex system, it is the net sum of reactions that regulates the species distribution that in turn drive the phase equilibrium behavior. This occurs because of structural incompatibilities between silicate species that favor 2,3

4

140

© 1997 American Chemical Society

In Supercooled Liquids; Fourkas, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

11. CORRALES

Phase Separation in Silicate Melts

141

aggregation of similar silicate species and partitioning of dissimilar silicate species, thus forming more than one phase. Effectively, the enthalpy drives the formation of silicate species, and the mixing entropy drives the phase separation. The interplay of chemical reactions and phase equilibrium is known to lead to interesting and important phase equilibria and higher-order critical phenomena. Statistical mechanics provides useful approaches to study the phase coexistence and speciation of silicate melts. The phase transition can be viewed as being driven by the entropy for a fixed species distribution. The mixing entropy is determined using Flory-Huggins theory (a combinatorial method) where the species interactions provide the favorable and disfavorable interactions that lead to phase separation. Speciation can be studied using a structural thermodynamic model where a set of reactions dictate the species formation and from which species activities can be determined. The activities define equilibrium constants that contain both enthalpy and entropy contributions of forming the species in the network. A comprehensive description of the relationship between speciation and thermodynamic properties can be provided by combining a combinatorial approach for the entropy with a chemical equilibrium theory for the speciation that is described below. For a binary silicate system, a set of chemical reactions are written that lead to the incorporation of a metal oxide into the silica network and, hence, define five possible silicate species. The net effect is to depolymerize the network by transforming a bridging oxygen (BO) bond into a nonbridging oxygen (NBO) bond by the addition of the metal oxide. The B O bond is considered to be a covalent bond and the N B O bond is an ionic bond, held together by Coulombic interactions between the anionic oxygen atoms and the metal cations. Each of the five silicate species is defined by the possible arrangements of B O and N B O bonds about a silicon atom. Excess metal oxide species, or those that have not reacted, occupy interstitial sites. In the following work, die excess metal oxide do not aggregate to form a separate phase and only tetrahedral coordinated silicate species are considered. 5

6

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7

8

4

Description of Theory The solvation of a metal oxide into the silica network consists of transforming a bridging oxygen (BO) bond to a nonbridging oxygen bond (NBO) via insertion of a M 0 that contributes an O * to the system and forms two O" upon incorporation into the silicate lattice sSi—O—Sis + M 0 sSi—O M * NT O — S i s The chemical reaction is considered to be in equilibrium. A silicon can be four-fold coordinated by a combination of B O and NBO bonds, which correspond to a covalent oxygen and a pair of ionic oxygens, respectively. The Q species notation is used to describe each of the possible combinations of bonds. It must be made clear that a B O or a N B O are shared between silicon tetrahedra. Hence, in the following definitions of the Q species a half-BO corresponds to sharing a covalent oxygen, and a half-NBO corresponds to a single ionic oxygen. Thus, a Q site is made up of four half-BO bonds, a Q site is made up of three half-BO and one half-NBO bonds, a Q site is made up of two half-BO and two half-NBO bonds, a Q, site is made up of one halfB O and two half-NBO bonds, and a Q site is made up of four half-NBO bonds. The equilibrium reactions that describe the solvation of metal oxide to produce any of the Q species from the Q species are written as: 2

2

8,9

2

10

4

3

2

0

4


142

SUPERCOOLED LIQUIDS

Eq. 1

Q +M 0 1 2 , where the summation is over all N bonds and r[N,n ] is the number of ways of arranging the bonds on a tetrahedral lattice consisting of N bonds, given they must be arranged in such a way as to form combinations of specific Q sites. The configurational contribution, given by T, is calculated by determining the number of ways of arranging each of the sites on a lattice, keeping in mind that sites are formed by the probabilities of forming each site form the bond types. The number of ways of arranging sites on a lattice is determined following the Flory method for placing polymers on a lattice. K

A

A

nbo

Eq. 8 4

4,

6

4

jsj 14 w4 5 "L«i 4 «L«to bo

n

^ nbo

The configurational entropy of the system is calculated by taking the log of the number of ways of arranging the sites on a lattice and simplifies to Eq. 9 S = fclnQ = 4

N


144

SUPERCOOLED LIQUIDS

A n approximate free energy for the bond model can be written by considering the logarithm of Eq. 7 and including the configurational entropy given by Eq. 9 to arrive at Eq.10 3

G = -N*kTlnz

l

2

>

3

+nl*

4

3

4Ar (^-n^)*rm[^^]

+

where E q . 11 -kTlnK =AG 3

3

B =-fcrin

- A G , - 2 AG*

0

K

£, =-*rin

= A G , - 3 A G , + 3AG, A

2

J

B =-*rin|-^-

=AG -4AG +6AG -6AG

4

0

I

2

3

To understand how to extract useful thermodynamic properties, the isothermal differential of the free energy is expressed as in standard thermodynamics E q . 12 dG = fi dQ T

A

4

+ n dQ + n dQ + fi dQ + ^d% 3

3

2

2

t

+ n dn

x

m

m

where m for i = {1,2,3,4} are the chemical potentials of the Q species, \i

m

is the

chemical potential for the free metal oxide, and \i and p, are the chemical potentials corresponding to the N B O and B O bonds. In Eq. 12, the transformation from the Q site activities and species concentrations to the bond activities and site formation probabilities is made. By using Eqs. 5 and 6, expanding out the polynomial terms and rearranging Eq. 12 simplifies to tho

bo

E q . 13 4

dG = 4» dN T

bo

+4Ad[N>n ] nbo

+

i dN

t m

m

where E q . 14

*=

- A**- J* ft.

This form of the isothermal differential of the free energy reveals the conditions to satisfy both chemical equilibrium and phase equilibrium in the bond model. From the definition of A, it represents the negative affinity of reaction of creating a N B O from a B O and a metal oxide. To satisfy chemical equilibrium, A must be equal to zero in all phases that are in phase equiMbrium. The phase equilibrium conditions are then defined by p ^ and p^ being equal in all phases. In Supercooled Liquids; Fourkas, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.

11. CORRALES

145


For ease of calculation, the differential free energy is expressed in terms of dN andd n^ torn

Eq. 15 i

3

dG = 4N idN T

+ 4N Adn^

f

+fl dn

0

m

m

where E q . 16 Downloaded by STANFORD UNIV GREEN LIBR on October 4, 2012 | http://pubs.acs.org Publication Date: September 30, 1997 | doi: 10.1021/bk-1997-0676.ch011

A* = 4 ^ + 3 ^

A

From these equations, m„ is given by

Eq. 17 u. = u - 3 ^ - A = ^ — - G N N dn^ The latter equality indicates the chemical potential of the bridging oxygen bond can also be obtained from the common tangent construction. Calculating the indicated partial differentials in Eq. 15 on die free energy Eq. 10, A is

Eq. 18 _A_ =

JL +

kT

kT

3

X

+ *E± + J

* L + >BL

X

3X

kT

kT

kT

-*-)

VI-*/

where x = n ^ / N . By determining |X and using Eq. 17, j i ^ is

Eq. 19 2

3

4

^ = [ - I n z - 6x 5 L - 8* ^ - 3x 2*- + 4 ln(l - x)] v kT L *r *r fer 'J One of the conditions for phase equilibrium is given by Eq. 19. The second equilibrium condition, determined by being equal in each phase, does not lead to an explicit expression in terms of the field parameters, the BjS, and the order parameter, n . To this end, A can be used, recognizing that it contains the chemical potential of the free metal oxide, , as well as the chemical potential of the metal oxide incorporated into the network, p, . Thus, A + = j i - 1/2 ( i is used for the second phase equilibrium condition. This phase equilibrium condition must be coupled with the chemical equilibrium condition that requires A = 0 also be satisfied in each of the phases in equilibrium. Alternatively, A is used for Maxwell's construction or to satisfy the equal area rule. Thus, A can be used for the second phase equilibrium condition while simultaneously ensuring that it is equal to zero in all phases that are in equilibrium. Both approaches lead to equivalent results, and require that one of the field parameters, namely B , , be searched and solved in a self-consistent manner. Thus, two-phase equilibrium of the system is determined by searching for the solution of m^x, T) = H j x " , T) and A(x, B , , T) = A(x", B„ T) = 0 at fixed values of B , B , B , and T. The entire coexistence curve is then determined by varying T and solving for x andx . 4

nb0

nbo

2

3

n b o

4


m

146

SUPERCOOLED LIQUIDS

The critical point condition is given by the second and third partial derivative of the free energy with respect to the order parameter be equal to zero and is identical to Eq.20 2

(dA/kT\ V dx

Jd A/kT^ )

2

{

TcXc

dx

,

=0

These conditions are easily calculated from Eq. 18. The parameters B and B can be determined using the critical temperature and composition of a system while keep B fixed. The role of B is to introduce an asymmetric stretch in the coexistence curve, and so in principle can also be fit to available data. Using the definitions of the equilibrium constants in terms of the activities of each species given by Eq. 3, the average number density of each Q species is calculated from Eq. 10 using 2

3

4


4

E q . 21 4

9G/N kT

,

x

The total number of silica units is given by Eq.22 4

^

2

3

4

= 5 > , = 0 - * ) + 4x(l - xf + 6* (1 - xf + 4* (1 - x) + x = 1 i

Using Eq. 6 along with Eq.23

V

m l

d(-]nz ) N* where x = nJN* the total numter of metal oxide units in the system is given by m

m

y

Eq.24 N X =—7- = x m

+2x

m

Thus the total mass conservation of this system is given by Eq.25 The apparent fraction of metal oxide in the system is Eq. 26 x

m

+2x

m

\ + x +2x and the apparent mole fraction of silica in the system is m

Eq.27 1 X

s

i

C

h

~ \ +

x+2x


11. CORRALES


147

Phase Diagrams Two-phase coexistence curves are determined for a fixed value of the parameter B = -4.99 and of the critical temperature T =1000 K, corresponding to B /kT = -0.43. The critical composition is varied from the dilute side to the high concentration side of the metal oxide fraction. The parameters B and B are determined from the critical composition and critical temperature using the equations for the critical point obtained from Eq. 19. The resulting phase diagrams are shown in Figures 1-5. In Figure l a , the critical point sits on the dilute side at % = 0.44 and corresponds to x = 0.3 for the number concentration of NBO bonds, as defined above. This value of B shows only a slight asymmetric stretch of the coexistence curve on the right branch. The shape of the coexistence curve is similar to those observed in binary silicates containing N^O. In Figure lb, the critical point is at X M 2 O 0-55, corresponding to x = 0.4. Note the drastic change in the shape of the right-hand branch of the coexistence curve, that emphasizes the asymmetric stretch with a shoulder to the right of the critical point. A similar coexistence curve is observed in binary silicates containing CaO or MgO. The phase coexistence curve shown in Figure l c is significantly flatter in appearance than all the other coexistence curves for the same values of B . The critical composition occurs at % = 0.66, corresponding to x = 0.5, which is identical to the critical composition of the nonsymmetric tricritical point that is known to exist for this theory. The parameter values lie very close to this higher order critical point and is the reason why the coexistence curve has such a pronounced flatness. In Figure Id, the critical point occurs at % = 0.77, corresponding to x = 0.6. This coexistence curve nearly mirrors that in Figure 2, with its shoulder to the left of the critical point. In Figure le, the critical point occurs at % = 0.87, corresponding to x = 0.7. This coexistence curve has a pronounced asymmetry on the left branch compared to that in Figure la. 4

c

4

2

c

3


M20

4

=

4

mo

M20

M20

Discussion This theory shows a rich assortment of phase behavior that includes three-phase equilibrium, critical endpoints and a nonsymmetric tricritical point, that have not been shown here. The nonsymmetric tricritical point is where three phases that are in equilibrium simultaneously coalesce to form a single phase. The critical endpoints is where two of the three phases have coalesced to a critical point that remains in equilibrium with the third phase. Experimentally determined phase coexistence curves for binary silicates have focused on two-phase coexistence curves. This work shows the theory is capable of capturing the wide behavior, in terms of the shapes of the coexistence curves, seen in binary silicate melts. The shapes of the curves are strongly coupled to the proximity of the parameters to the nonsymmetric tricritical point, as well as to the position of the critical composition of the system. Although a number of approximations have been used, that includes the use of Flory theory for computing the mixing entropy, a more severe approximation has been to keep the free metal oxide off the lattice. Incorporating the metal oxide into the lattice will result in a mixing entropy contribution to the system, that will be significant in the high concentration regime of the phase diagram. Additionally, it would be desirable to turn on the interaction energy of the metal oxide with the other components of the system, that can lead to aggregation of a pure metal oxide phase and to favoring further uptake of the metal oxide into the network.


148


SUPERCOOLED LIQUIDS

920.0 900.0 0.00

0.20

0.40

0.60

0.80

1.00

1020.0 1000.0 980.0 T O O 960.0 940.0 920.0 900.0 0.00

0.20

0.40

0.60

0.80

1.00

Figure 1. Phase coexistence curves at T = 1000 K at critical compositions of c

) XM2O = > b) % = 0.55, c) x = 0.66, d) % = 0.77, and e) x = 0.87. The series of coexistence curves pass through the vicinity of a nonsymmetric tricritical point that leads to the asymmetric broadening of the coexistence curves. A

0

M

m

o

M 2 0

M 2 0


M 2 0


CORRALES

149


900.0 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.% 1.00 *M20

900.0 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 ^M20

Figure 1. Continued.

Continued on next page



SUPERCOOLED LIQUIDS



11. CORRALES


151

The proximity of the coexistence curves, presented here, to the nonsymmetric tricritical point leads to the broadening asymmetric behavior. In general, the coexistence curves of binary silicate systems are observed to occur in the dilute regime. In the high concentration regime other phase transformations may occur, such as metal oxide aggregation. Due to temperature constraints, most phase separation of silicates is observed in the solid phases. Thus, phase domains may be glass or crystalline depending upon quench history and the propensity for a phase to crystallize from the melt. Phases rich in metal oxide may be expected to crystallize thereby driving a phase separated system into a single phase. Such phenomenon must be further explored. Expansion of this theoretical approach to include multiple network forming and network modifying components can be carried out, although the analytic solutions may become intractable. However, solutions can in principle be obtained using Monte Carlo simulation techniques. Acknowledgements This work was performed under the auspices of the Division of Chemical Sciences, Office of Basic Energy Sciences, U.S. Department of Energy under Contract DEAC06-76RLO 1830 with Batelle Memorial Institute that operates the Pacific Northwest National Laboratory under Grant No. DE-FG06-89ER-75522 with the U.S. Department of Energy. References 1

R. H. Doremus, Glass Sience, (John Wiley & Sons Inc., New York), 2nd Edition, 1994. D. R. Uhlmann and A.G. Kolbeck, Phys. Chem. Glasses, 1976, 17, 146 R.J. Charles, Phys. Chem. Glasses, 1969, 10, 169. L.R. Corrales and K.D. Keefer, J. Chem. Phys., 1997, 106, 6460. L.R. Corrales and J.C. Wheeler, J. Chem. Phys., 1989, 91, 7097. P. Flory, J. Chem. Phys., 1942, 10, 51; 1944, 12, 426. R.J. Araujo, J. Non-Cryt. Solids, 1983, 55, 257; P.J. Bray, R.V. Mulkern and E.J. Holupka, J. Non-Cryst. Solids, 1985, 75, 37. W.G. Dorfeld, Phys. Chem. Glasses, 1988, 29, 179. R. Dron, J. Non-Cryst. Solids, 1982 53, 267. G. Englehardt, H. Jancke, D. Hoebbel and W. Weiker, Z. Anorg. Alg. Chem., 1975, 418, 17. 2

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Phase Separation in Silicate Melts - American Chemical Society

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