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PbO-SiO2 melts: structure and thermodynamics of mixing
Qeochimica et Cosmochimica Acta, 1075,vol. 39, pp. 671to 687. PeqamonPress. Printedin Northern Ireland
PbO-SiO, melts: structure and thermodynamics of mixing PAUL C. HESS Department
Geological Sciences, Brown University, Providence, R.I. 02912, U.S.A.
(Received
13 May 1974; accepted in revised form
7 October 1974)
Abstract-Thermodynamic
properties of PbO-SiO, melts, obtained from published data and calculated from freezing point depressions, reflect the gradual polymerization of silicate anions The free energy of mixing curve at 1OOO’C has in the melt as the SiO,/PbO ratio is increased. a minimum at 40 mole ‘A SiO, and is convex-upward between 72 and 98 mole ‘A SiO,. The latter is an indication of met&stable liquid immiscibility. The free ener,7 minimum is correlated with the maximum in the distribution of nonbridging oxygens in the melt. In SiO,-poor melts, the activities of PbO and SiO, (pure liquid standard states) show sharp negative deviations from ideality. The PbO activity reflects the paucity of free oxygen species in the melt whereas the SiO, activity reflects the depolymerized state of the silicate anions. In more SiO,-rich melts, the activity of SiO, shows s, positive deviation from ideality which is qualitatively correlated to a polymerization parameter. The heat of mixing term has a minimum of -2000 cal at 35 mole % The minimum is associated with the SiO, and a maximum of +200 cal at 90 mole% SiO,. exothermic heat effect obtained during the reaction (00) +
(02-)
= 2(0-),
whereas the maximum corresponds to the endothermic heat effect obtained when coordination The entropy of mixing curve has the same polyhedra of oxygens form around the Pb cation. form but is systematically smaller than a theoretical curve calculated on the assumption of random mixing of oxygen species. The discrepancy is due to the entropy loss obtained by the clustering of oxygen species to form complex silicate species.
INTRODUCTION IT IS NOW widely accepted
that silicate melts are ionic solut,ions containing complex silicate anions, cations and ‘free’ oxygen ions. The nature and distribution of these species are functions of melt composition, temperature and pressure. Theories attempting to explain solution non-ideality for silicate solutions, and for fluid mixtures in general, can conveniently be classified as ‘chemical’ or ‘physical’. The ‘chemical’ solution theories are based on the premise that new chemical species are formed in solution and that solution non-ideality is a consequence of these chemical The thermodynamic properties of the solution can be derived from a reactions. knowledge of the equilibrium constants corresponding to these reactions (PRAUSNITZ, 1969). According to these models, non-ideality is only apparent because it is based on an unwise choice of components. ‘Physical’ solution theories assume that the nature of the chemical species is not altered in solution and that non-ideality arises from physical intermolecular forces. Physical forces are described by simple potential energy functions. Examples of the physical theories include the regular solution model, lattice theories, and theories based on the theorem of corresponding states (PRAUSNITZ, 1969). In general, concepts from both contrasting viewpoints must be taken into account but the chemical approach is favored when intermolecular forces are strong. Various forms of the ‘chemical’ solution models have had considerable success in calculating some
672
P. C. HESS
of the thermodynamic properties of binary silicate melts (HESS, 1971, 1972, 1973; MASSON, 1965, 1968, 1972; MASSON et cd., 1970; KAPOOR and FROHBERB, 1971; BAES, 1970; PRETNAR, 1968; CHARLES, 1969; FLOOD and KNAPP, 1963; TOOP and These models adequately reproduce activitySAMIS, 1962a,b; among others). composition cnrves for the metal oxide in silica-poor melts and suggest that the
thermodynamic properties are related to and reflect the ‘internal structure’ of a silicate melt. It is the purpose of this paper to examine the interrelationship between the thermodynamic, phase and structural properties of PbO-SiO, melts. The PbO-SiO, system was chosen for study because thermodynamic data are available for the melt over 60 per cent of the compositional range. Although the system is of little direct petrological interest, it will provide the fundamental information to aid our understanding of the properties of silicate melts in general. This information is needed to better understand transport, phase and geochemical processes in naturally occurring magmas. DATA AND CALCULATIONS KOZ~JICAand SAMIS (1970) determined the activity of PbO (liquid standard state) by electromotive force measurements and calculated the activity of SiO, (solid standard state) using the Gibbs-Duhem equation. Data were obtained at 1000% and over a compositional range from 0 to 62 mole% SiO,. These data are preferred over those of RICHARDSONand WEBB (1965)) and KAPOOR and FROEBER~ ( 1971) because cell reversibility was demonstrated, oxidation of electrodes was avoided and EMF measurements were reproducible to 50.2 mV. The results at 1000% are virtually identical to those of KAPOOR and FROHBERG (1971) (Fig. 1). The
activities of SiO, were corrected to the pure liquid standard using enthalpy data from and WALDBAUM (1968). Partial molar enthalpies (liquid PbO and quartz standard states) of PbO-SiO,
-Kozuko ---Richardson .._ Kapoor
ROBIE
melts from
et al. et al. et al:
Fig. 1. Some examples of experimentally determined values of the activity of PbO (liquid standard) in PbO-SiO, melts at 1OOO’C. Activity of SiO, (solid standard state) obtained from Gibbs-Duhem calculations. NsiO, (mole ‘A SiO,).
PbO-SiO,
melts:
structure
and thermodynamics
of mixing
673
0.054 to 0.501 mole fraction SiO, were determined calorimetrically at 900°C (~STVALD and KLEPPA, 1969). The quartz enthalpies are converted to liquid standard states. In order to obtain enthalpy data for the complete range of composition it is assumed in the calculations This assumption that follow that the partial molar enthalpies are independent of temperature. is certainly a fair one for moderate temperature intervals but becomes increasingly doubtful for extreme ranges. Results by KAPOOR and FROHBERG (1971) and KOZUI~A and SAMIS (1970) suggest that the assumption is valid from at least 850 to 1050%. These partial molar enthalpies are used to calculate activities for other temperatures from d In ai d(l/T)
A& = ??-’
where ai = activity of i and Afli = partial molar enthalpy of i (pure i liquid standard state). Activities for the metastable liquid at temperatures below the SiO, (solid) liquidus are obtained as follows. Activities of SiO, (solid standard state) which are unity along the cristobalite and tridymite liquidus, are corrected to the liquid standard states using enthalpy of fusion values from ROBIE and WALDBAUM (1968). If the partial molar enthalpy of SiO, is independent of temperature then (CHBRLES, 1967, 1969)
where ysio, is the activity coefficient of SiO,. This relation cannot hold strictly since at very high temperatures the partial molar enthalpy should also approach zero. However, it is assumed without proof that the results are a good approximation in the temperature range from 900 to 1700%. Partial molar enthalpies of silica for the range of compositions from 62 to 100 mole % SiO, are compared with those determined by calorimetry in Fig. 2. The data appear to be in reasonably good agreement. If this agreement is not fortuitous, it suggests that the assumption ranges. Partial of the constancy of AHsiO, and ABp,_,, is a good one for these temperature molar enthalpies of PbO are calculated with the aid of the Gibbs-Duhem equation. The integral heat of mixing is obtained from the partial molar enthalpies (Fig. 2)
where Aif is the molar heat of mixing and X represents mole fractions. Activities of SiO, for a range of temperatures are calculated using the partial molar enthalpies and equation (2). Activities of PbO at temperatures other than 1OOO’C are obtained from Gibbs-Duhem relations. SOLUTION
MODELS
The structural evolution of a silica-poor melt to a pure SiO, melt is characterized by a continous polymerization of discrete SiO, monomers into complex silicate species until a network structure is attained (HESS, 1971, 1972; MASSON, 1965, 1968, MASSON et al., 1970; TOOP and SAMIS, 1962a,b; among others). HESS (1971), utilizing a random polymerizing model, calculated that discrete silicate species are rapidly diminished in abundance in silica-rich melts and that ‘infinitely’ large silicate species had a high probability of forming. Other authors, however, (FRAY, 1970; BOCICRIS et al., 1955, 1966) suggest that discrete silicate species persist even to high silica compositions. Figure 3 shows the results obtained by the author (HESS, 1971) where the ‘gel’ curve records the weight fraction of infinite silicate species (normalized to total silicates) in the melt. Note that in this model, the gel portion increases rapidly but smoothly from 0 to 92 wt. % in the interval from 40 to 60 mole % SiO,. If this model is correct, then the metasilicate region is a zone of transition between melts of contrasting structure. At low SiO, compositions, the melt contains small discrete silicates whereas at high SiO, compositions the melt is a partially polymerized network. These differences should be strongly reflected in some of the thermodynamic and physical properties of the melt.
P. C. HESS
674
PHASE DIAGRAM
I 0
0.2
0.4
0.6
0.8
1.0
N SiOz Pig. 2. Partial molar heats of PbO and SiO, (solid lines) determined by OSTVALD and KLEPPA (1969). Vertical bars represent uncertainty in the data. Dashed OI_UTW for A~sios calculated from freezing point depression data and dashed curve for AH,,, obtained from Gibbs-Duhem calculations. Integral heat of mixing (liquid standard states) calculated from partial molar enthalpies.
0
0.2
0.4
0.6
0.8
0
Fig. 3. Calculated weight fraction of silicate species (normalized to total silicate content) as a function of SiO, content in the melt. Gel corresponds to weight fraction of ‘infkite’ silicate species. Note that the gel curve rises abruptly and steeply at intermediate compositions.
PbO-SiO,
melts:
I
0
structure
and thermodynamics
675
of mixing
PbO -Si02
0.2
0.6
0.4
0.8
I 0
“Jsioz
Fig. 4.
Concentrations
of bridging (OO), non-bridging in PbO-SiO, melts assuming K
(O-) and free oxygen = 0.02.
(0%)
A comparatively simple yet highly instructive approach to an understanding of the properties of a silicate melt is to focus directly on the nature of the variously bonded oxygen species, FINCHAM and RICHARDSON (1954) temporarily ignoring the complexly polymerized structure. recognized that oxygen in silicate melts occurs only in three forms, singly bonded (O-) or non-bridging oxygens (nbr), doubly bonded (OO) or bridging oxygens (br) and free oxygen ions (02-) (fr). The nomenclature refers to whether the oxygen is coordinated to one, two or zero silicons. TOOP and SAMIS (1962a,b) assumed that any reaction between silicate ionic polymers is described by 2(0-) = (00) + (02-), (4) and under equilibrium
conditions,
an equilibrium K
constant
K exists, such that
= (OO)(O”)/(O-)2,
(5)
where (OO), and (02-) and (O-) are the equilibrium number of moles of the variously bonded Note that by the inclusion of mole numbers rather than activities it is assumed that oxygens. the solution is ideal with respect to mixing of the oxygen species. This simplifying step allows all oxygen ion concentrations to be calculated for a given K, composition and material balance of the system (Fig. 4). It is also possible (TOOP and SAMIS, 1962a,b) to show that the molar free energy
of mixing,
A??, is approximated
by
(6) DISCUSSION
Activities of PbO and SiO, The activity
of MO in a binary MO-SiO, %lO
=
KWr(02-)
+
melt is approximated (O-)/Z:(22
+
2)X,1,
(HESS,
1971)
by
(7)
where uMO = the activity of the component MO, x = number of silicons on a polymeric silicate ion, and X, is the mole fraction of a given x-merit ion in the melt. The model cannot be explained in detail but is based on the following assumptions : 9
P. C. HESS
676
(I) polymeric silicate ions exist only as simple chain species and their distribution is obtained by random polymerization reactions and (2) the mixing properties of the solution obey the Temkin model whereby it is assumed that the melt exists as two independent cation and anion mixtures. The activity of MO in this model is equal to the anionic mole fraction of (02-) ions and is given by the moles of oxygen ions divided by the total number of moles of anions in solution. The values of (W-), (0-j and X, are shown to be functions of K (HESS, 1971) and a suite of activity-composition curves can be calculated for different values, These correspond closely to experimentally derived curves. Figure 5 compares the experimental activity curve
l
Kozuko-Samis 1000~ c
.Nsioz Fig. 5. Activitiesof PbO (liquidstandard state)asdetermined by experiment (dots) are compared to those correspondingto a theoretical curve appropriateto K = 0.02. for PbO at 1000°C of KO~UKAand SANIS(1970) and a theoretical curve appropria~ to K = O*OZ. The theoretical curve cannot be extended beyond 39 mole% SiO, since ions of ‘infinite’ dimensions occur beyond this point (see Fig. 3). The model is clearly inappropriate for silica-rich melts and is certainly an oversimplification for even SiO%-poor melts. It is useful in a pictorial sense since the activity of PbO is correlated with the mole fraction of (02-) and to the Pb-(OS-) bonds in the melt. As (02-) ions are reduced, fewer and fewer portions of the melt resemble the reference PbO liquid and therefore the ‘escaping’ tendency of PbO as reflected in the activity is reduced. The strong negative deviation from ideal behavior is a consequence of the rapid depletion of (02-) as SiO, is added to the solution. Activity of SiO, The activity of SiOz cannot be derived (other than by Gibbs-Duhem methods) from the ionic model of HESS (1971). It is possible, however, to derive a model from a less rigorous theoretical treatment. The shape of the activity-composition curve of SiOa (liquid standard state) in the PbO-SiO, system (and in many other systems) has the following characteristics (Fig. 6). In silica-rich compositions, the activity
PbO-SiO,
melts:
structure
0.2
and thermodynamics
0.8
0.6
0.4
of mixing
677
1.0
kOZ
(a)
0.6
0.6
0.4
0.4
P
aSiO,
0.2
0.6
0.4
0.8
1.0
Go2 (b) Fig. 6. (a) Activity of SiO, curves (solid and liquid standard states) given by solid lines are compared to a theoretical P-curve corresponding to II = 0.01. (b) P-curves for a variety of K values. (Temperature = 1000°C).
curve shows a marked positive deviation from ideahty and is relative flat or sigmoidal (metastable liquid immiscibility). In intermediate compositions, the curve drops sharply from a positive to a negative deviation from ideahty, The activity in metal-rich compositions shows extreme negative deviations from ideality. The ‘central zone’ corresponds to those compositional regions marking the sudden appearance of infinite silicate species as defined by the gel curve. The correlation suggests that the activity of SiO, is directly related to the state of polymerization of the melt. A highly polymerized melt is one in which a large percentage of the silicon
P. C. HESS
678
bonds lead through a coordinated number of silicon bonds is
oxygen
2(00) + (O-)
to an adjacent =
silicon atom.
(8)
4NSi027
and the number of bonds linked to bridging oxygens is 2(O").Therefore, of all silicon bonds, P, that are bridging is
P=
2(0°) 2(0°) + (0-J
The total
the fraction
(0°)
=2N,io* 2
The value of P is then one convenient measure of the state of polymerization that is useful over the complete compositional range. Figure 6 contains P curves corresponding to various values of K, and the activity of SiO, curve at 1000°C for the PbO-SiO, system. There is a remarkable agreement between the polymerization curve of K = 0.01 and the activity of SiO, curve at 1000°C over intermediate compositions. However, the model curve cannot predict the onset of immiscibility at high SiO, compositions. Nevertheless, the similarity between the P curves with activity of SiO, curves in this, in the SnO and, to a lesser degree, in the CaO and Na,O systems (unpublished data) argues that the state of polymerization is semi-quantitatively reflected in the magnitude of asioa. Since the standard state of the activity of SiO, is the totally polymerized liquid, the hypothesis is, in fact, quite reasonable. Free energy of mixing The free energy of mixing curve calculated from the activities at 1000°Chas a minimum at 40 mole”/o SiO, and is convex-up in SiO,-rich compositions (Fig. 7). By the method of tangents, a two-liquid field is indicated between 72 and 98 mole% SiO,. The free energy changes, which are responsible for immiscibility, are less than 200 cal. A free energy of mixing curve calculated from equation (6), K = 0.06and corresponding values of (0-) adequately reproduces the experimental curve to within 299 cal (Fig. 7). The model curve, however, is concave-up rather than convex-up in the two liquid region. Moreover, the best fit of the free energy curve is given by K = 0.06,whereas the best fit to the aPbo curve is given by K = O-02. Although the simple model can reproduce, with good accuracy, the free energy curve for the PbO-SiO, system, it again has failed as a predictive tool in SiO,-rich regions. As with most strictly chemical theories, the mixing properties of the solution are assumed to be ideal once the true chemical species are correctly identified. In this model, the chemical species are (OO), (0-) and (Oz-), and are expressed in concentration units rather than in terms of activities. Due to these approximations, free energy curves calculated from equation (6)give no indication that liquid immiscibility occurs. These theories can be improved by removing the requirement that mole fractions of species are equal to activities. For example, activity coefficients may be introduced by using regular solution or van Laar models (PRAUSNITZ, 1969). Unfortunately, these modifications have the effect of introducing new adjustable parameters that may cloud rather than clarify the solution characteristics. These approaches may be useful, however, for curve fitting or extrapolation purposes.
PbO-SiO,
melts:
structure and thermodynamics
_---__ SiO,
0
-I .-__I
F C
--+J’
of mixing
679
,I
/ l’ / ,/’
PbO -2
-3
0.2
0
3
NSi02 Fig. 7. (a) Free energy of mixing of PbO-SiO, melts as determined experimentally (solid line) and calculated (light dash). Tangent to free energy curve (dotted line) indicates the compositions of the coexisting liquids (two vertical arrows). Theoretical free energy of mixing curve corresponding to K = 0.06 is given by heavy dashed line. (b) Activity coefficients of PbO and SiO, in PbO-SiO, melts aa determined experimentally (solid lines) and calculated (dashed lines).
Heat of mixing The heat of mixing is that evolved or absorbed when pure liquid PbO (free oxygen ions) is mixed with liquid SiO, (double bonded oxygen ions). The solution contains complex silicate structures and bond types that do not exist in the pure liquids. Consequently, a major portion of the heat effect arises from the change of oxygen bonding in the solution as compared to the bonding in the pure liquids (CHARLES, 1969). An analysis of the factors that contribute to the heat budget could be simplified if all activity coefficients of the oxygen species were unity as indicated by the expression for the equilibrium constant in equation (5). On this assumption, no heat effect is associated with the simple transfer of free or double bonded oxygen from the pure liquid to the solution. This analysis, however, is correct only as a first approximation. As an example, consider the bonding of bridging oxygens in crystalline silicates. Bridging Si-0 bond lengths are not constant but are, in part, dependent on the Si-0-Si bond angle with the shorter bonds being associated with wider angles. Moreover, the shorter bond lengths correlate with larger Mulliken bond overlap populations (GIBBS et al., 1972). On a qualitative basis, bond overlap
P. C. HESS
680
populations are believed to be a good criterion for relative bond strengths (COTTON and WILKINSON, 1966). Therefore, variations in both bond lengths and bond energies of bridging Si-0 bonds indicate that bridging oxygen species are not all thermodynamically equivalent and do not contribute equally to the energy of a silicate crystalline phase. These conclusions may also apply to the liquid phase. The maximum heat of mixing of the PbO-SiO, solution is -2 kcal/mole (Fig. 2). This is probably but a small fraction of the bond energies. SANDERSON (1971) calculated that the bridging Si-0 bond energy in crystalline SiOz has an energy of 1109 kcal/mole. This value multiplied by four bonds per silicon yields the atomization energy of 443.6 kcal/mole, in good agreement with the experimental value of The Pb-0 (free) bond energy, calculated by the same procedure, 445.6 kcal/mole. There is no independent check on this bond energy so that the is 72.7 kcal/mole. In Sanderson’s method, bond results must be regarded only as semiquantitative. energies are inversely related to bond lengths. Variations of 3 per cent in individual bond lengths are capable of producing heat effects comparable to those observed in the PbO-SiO, system. The heat effect, therefore, represents small energy variations between large energy terms. This makes the heat term difficult to evaluate. Nevertheless, the rough correlation between the minimum in the heat of mixing curve (Fig. 2)with the maximum in the (0-) curve (Fig. 4) suggests that the ideal model is adequate as a first approximation and that a significant portion of the heat effect is associated with the reaction 2(0-)
= (00) + (OZ-).
(4)
The heat effect associated with the reaction above can conveniently be considered in two parts (CHARLES, 1969). The replacement of one Si-0 bond of bond strength of 111 kcal/mole by a weaker Pb-0 bond of bond strength of 73 kcal/mole allows a closer association of the remaining Si-0 pair. This is shown to be true for a large number of silicate structures where Si-0 (nbr) bonds are usually shorter than the Si-0 (br) bonds (GIBBS et al., 1972). The closer association may be the result of t,he polarization of the electron cloud of the oxygen towards and the oxygen nucleus away from the silicon (CHARLES, 1969). Si-0 bond length variations can also be rationalized in terms of a covalent bonding model wherein shorter bond lengths are correlated with large bond overlap populations (GIBBS et al., 1972). In any case, the heat effect associated only with the conversion of a bridging Si-0 bond to a non-bridging Si-0 bond should be exothermic (CHARLES, 1969). The remaining part of the heat effect in the simple two-step model is ascribed to changes in Pb-0 bonding in the solution as compared to Pb-0 bonding in the pure liquid. Some knowledge of Pb-0 bonding in the pure liquid phase can be obtained by examining the crystal structure of the polymorphs of PbO. Yellow PbO, the high temperature polymorph, is made up of layers composed of infinite chains of Pb-0 in four-fold coordination (DICKENS, 1965). The surface atoms of the layers are all of Pb, and the oxygen atoms are confined to the interior. Since the lead atoms face each other across adjacent layers, it is highly probable that the structure is not totally ionic but is characterized by covalent interactions (DICKENS, 1965). DICKENS (1965) estimated that the Pb-0 bond in yellow lead oxide had between 20 and 50 per cent ionic character, the low value corresponding to Sp3 hydridization.
PbO-SiO,
melts:
structure and thermodynamics
of mixing
681
More information about the nature of Pb-0 bonding in glasses can be obtained by several analytical methods. Nuclear Magnetic Resonance (NMR) determinations of the Pb207 chemical shift are virtually identical for PbO-Si02 glasses and compounds of identical composition (LEVENTHAL and BRAY, 1965). The results suggest that the chemical environment of the Pb ion in the glass is similar to that of the isoThe chemical shift was obtained for glasses containing chemical compound. 30-75 mole% PbO. The chemical shift of the hypothetical glass of lOOoh PbO was obtained by extrapolation and is close to the value determined for crystalline PbO (BRAY et aZ., 1963). The Pb-0 bond type in the glass therefore has a significant covalent character. However, the chemical shift approaches more ionic values as the glass becomes more SiO,-rich, although even the most SiO,-rich glasses are not characterized by a high ionic shift (LEVENTE~ALand BRAY, 1965). X-Ray diffraction data and radial distribution functions (RDF) of glasses calculated for compositions of BPbOSiO,, PbOSiO, and PbO*2SiO, (MYDLAR et uZ., 1970), indicate that Pb-Pb distances in the glasses are only slightly larger than those in corresponding crystals. Peak positions in the RDF curves are repetitive and indicate an ordering of Pb-Pb distances that are additive with respect to the shorter distances. These characteristics suggest that the Pb atoms are not randomly distributed within the silicate network but instead occur as simple and/or twisted chains of PbO, tetrahedra. The cation attempts to achieve the maximum coordinat)ion possible with the free oxygen ions in the melt. If the Pb cation does indeed prefer a coordination polyhedron of free oxygen, then the direct bonding of Pb with a free oxygen probably represents a lower energy stat,e than the bonding of Pb with a non-bridging oxygen. The conversion of a Pb-0 (fr) bond to a Pb-0 (nbr) is then an endothermic process. The heat effect associated with the transfer of Pb from the pure liquid to the solution will depend on the concentration of free oxygens in the solution. The endothermic contributionis small in SiO,-poor compositions since free oxygens are readily available (Fig. 4). In SiO,-rich melts, Pb can coordinate only non-bridging oxygens and the endothermic heat effect will become relatively more important. CHARLES (1969) has presented similar arguments and conclusions for other systems. Some of the exothermic heat gain in converting a Si-0 (br) bond to a Si-0 (nbr) bond may be lost if the remaining Si-0 (br) bonds of the given silicon increase in bond length and therefore weaken in bond strength. NOLL (1963) suggests that the polarized electron cloud of the non-bridging oxygen does an effective job of shielding the Si nucleus thereby reducing its effective positive charge. This would weaken the remaining Si-0 (br) bonds. This analysis is certainly oversimplified as residual charges on Si and 0 can be evaluated only by quantum mechanical methods (HINZE, 1970). Still, some interesting conclusions are drawn if these arguments are valid. The conversion of a weakened Si-0 (br) bond to a Si-0 (nbr) bond results in a greater release of heat than would be obtained from a ‘stronger’ Si-0 (br) bond. If the heat effect associated with the loss of the weakened Si-0 bond outweighed the loss of entropy, thereby lowering the free energy, then the formation of non-bridging Si-0 bonds would not be random but would occur in proximity to other Si-0 (nbr) bonds. Pb would not be randomly distributed but would be ordered in the structure. The proposed ordering tendency would allow Pb cations to share free oxygen ions most
682
P. C. HESS
effectively. This model may explain the general ordering tendencies observed in PbO-SiO, glasses. Within this framework, it is possible to rationalize the heat effect depicted in Pig. 2. The minimum in Ai? corresponds to a predominance of the exothermic heat effect of the conversion of Si-0 (br) to Si-0 (nbr) bonds over the small endothermic change due to Pb-0 bond changes. With increasing SiO,, the exothermic contribution to the heat effect decreases as a smaller number of Si-0 (br) bonds are converted to Si-0 (nbr) bonds. Moreover, free oxygen ions decrease to zero so that a maximum endothermic effect is obtained as Pb-0 (fr) bonds are converted to Pb-0 (nbr) bonds. At very high SiO, compositions, only a small percentage of (0-) exists making it extremely difficult for the Pb cation to achieve an oxygen coordination shell. In order to form such a coordination polyhedron of oxygens and thereby effectively shielding the cation charge, it is conceivable that considerable strain will result in the This may contribute additionally to the endothermic effect silicate structure. causing the positive heat of mixing (CHARLES, 1969). The composition dependence of the partial molar heats of mixing (Fig. 2) is also profitably discussed in the following context. When 1 mole of liquid PbO or SiO, is added to a large (in theory infinite) amount of solution, the Pb-0 or Si-0 bonds must take up the configuration and energy state of the given (average) bonds in the The partial molar heats of mixing therefore represent the heat effect solution. associated with the gain or loss of energy when the mole of added pure liquid adjusts to its new thermodynamic state. The partial molar heat of mixing of PbO is less than -200 Cal/mole from 0 to 20 mole”/o SiOZ, and then drops sharply to a minimum of -4200 cal at 68 mole”/o SiO,. The transfer of Pb and 0 (fr) to the solution is exothermic up to nearly 100 per cent SiO, and basically reflects the conversion of 0 (fr) to 0 (nbr). For Nsio, > 68 mole% the endothermic effect of the transfer Pb (pure liquid) to Pb (solution) reverses the trend to more negative heats and for iv SiO2 > 92 mole”/o it outweighs the exothermic heat effect. The partial molar heat of mixing of SiO, is strongly negative in SiO,-poor compositions but becomes positive at about 52 mole% SiO,. The exothermic effect is due to the depolymerization of SiO, pure liquid as it is added to the solution. The cross-over occurs in the compositional region where free oxygen species are minor constituents or perhaps even absent. If no free oxygen species are available, the exothermic heat contribution due to reaction (4) is not available and the heat effect must be due to the transfer of bridging oxygens from t,he pure liquid to the solution (CHARLES, 1969). The endothermic heat effect then is due to the strain of the SiO, ‘lattice’ as it attempts to coordinate and shield the Pb cation. Therefore, the positive heat of mixing is due primarily to the heat effect associated with the polymerized silicate fraction. The Pb imposes the energy demand whereas the polymerized silicate anions attempt to conform to these demands at the expense of an unfavorable energy configuration. If this hypothesis is correct, it is likely that the point where partial molar heats of mixing of SiO, change from negative to positive will be a good measure of (1) the state of polymerization of the melt and (2) the shielding demands of the cation as perhaps reflected in its field strength. HESS (1971) noted solutions containing cations of high field strength are also the most polymerized. Therefore, cross-over points of ARsio2 should prove useful in comparing the state of polymerization of different silicate melts.
PbO-SiO,
melts:
structure
and thermodynamics
of mixing
683
Entropy of mixing The entropy of mixing curve at 1000°C has two maxima at about 0.20 and 055 mole% SiO, and a minimum at 0.30 mole% SiO, (Fig. 8). In the interpretation of the entropy of mixing it is the custom to neglect changes in the vibrational contribution to the entropy on the assumption that it is small compared to the configurational contribution. The same procedure is adopted in this study although it is wise to note that the force constants of bonds linking free, single or double As a result, the vibrational contribution to bonded oxygens cannot be identical. the entropy of mixing is unlikely to be zero. The following discussion, however, will consider only configurationa,l contributions of the oxygen species to the entropy. CHARLES (1969) derived an expression for the configurational entropy of mixing on the assumption that it was mainly due to the permutations of the sites that are occupied by the oxygen ions in solution. If nonbridging oxygen ions occur as pairs in the solution, then the integral entropy of mixing one mole of solution is
COO) + (02-)
(02-) In w I ,
nw
(10)
where w = (O-)/2 + (00) + (02-). The equilibrium values of the oxygen species are obtained for different K values as before. The model entropy curves corresponding to K values near 0.01 are of remarkably similar form to those of the PbO-SiO, systems, but are systematically greater for all compositions (Fig. 8). The model entropy curves were calculated on the assumption of random mixing of (OO), (02-) and (0-) pairs of oxygen species. In the discussion on heat effects it was suggested that the formation of Si-0 (nbr) bonds could weaken the remaining Si-0 (br) bonds. This weakening would localize any additional ruptures of Si-0 (br) bonds since such reactions would be favored by the greatest release of heat. If this, in fact, were true, the distribution of oxygen species would not PbO-SiO;! K_ (0”)(0-2) (0-12
0
0.2
0.4
0.6
0.6
3
Nsros Fig. 8. Entropy and calculation
of mixing curves for PbO-SiO, melts as determined by experiment (indicated by 1000°C) and a model curve corresponding to K = 0.01.
684
P. C. HESS
be random, and the entropies would be reduced. In general, any deviation from the condition of a random distribution would result in lower entropy values than those predicted by equation (10). The entropy of mixing curve for the PbO-SiO, solution can be viewed as a composite curve of two distinctly different mixing patterns. The curve in the interval from 0 to 20 mole% SiO, describes a mixing process between polyhedra of PbO, and SiO, ionic species. According to HESS (1971), Si04 monomers make up more than 90 per cent of the silicate ionic species in this compositional range. The entropy of mixing from 30 to 100 mole”/o SiO, describes a smooth curve reaching a maximum near 55 mole”/o SiO,. The curve lies below an extrapolation of the entropy curve corresponding to the hypothesized mixing of monomers. The entropy curve at compositions from 30 to 100mole”/o SiO, therefore is believed to represent mixing properties of complexly polymerized silicate and plumbate ionic species. The lower entropy values below those of the extrapolated depolymerized curve and also below the model entropy curve reflects the clustering of oxygen species forming the complex ionic configurations. The fact that the entropy curve varies smoothly in this range supports, but does not prove, the contention of HESS (1971) that the polymerization The local minumum of the entropy process is smoothly varying and continuous. = 0.30 corresponds to those compositions where the extent of curve at Ns,, polymerizatiodin the PbO-SiO, at 1000°C increases rapidly. TWO liqtd jield The inverted S-shaped cristobalite liquids at 1670°C (CALVERT and SHAW, 1970) points to the existence of a metastable two-liquid field at lower temperatures or a stable two-liquid field at higher temperatures. Free energy calculations indicate that the two-liquid field occurs metastably with an upper consulate point. The existence of an upper critical solution point could have been predicted from and is consistent with the endothermic heat of mixing in the critical region (PRIGOQINE etal.,1962). Unfortunately, the simplifying approximations used to calculate the free energy curves introduce errors that are large in comparison to the small energy differences needed to differentiate between one and two phase fields. The uncertainTherefore, the author does ties grow even larger as the critical point is approached. not give the T-x coordinate of the solvus since the numbers would be of little value. Liquid immiscibility occurs if the free energy of the system is minimized on splitting the solution into two phases. The free energy of mixing contains two opposing terms, the enthalpy and entropy (-T AB) contribution. These terms cannot individually be minimized at the same time. The enthalpy of mixing curve of the metastable solution is within a few hundred calories of zero in the two-liquid composition range, whereas it falls to -2000 cal at 40 mole% SiO,. The endothermic heat effects in high silica melts are at least two-fold: (1)Pb cations are coordinated only by nonbridging oxygens and cannot obtain a complete coordination polyhedron of oxygens if the latter are not ordered and (2) the silicate structure is deformed in order to accommodate and provide a shield of oxygens for the cation (CHARLES, 1969). The influence of these endothermic heat effects is reduced by forming two liquid phases, one relatively cation-rich, one cation-poor. The cation-rich melt is less polymerized and contains greater numbers of free as well as nonbridging oxygens,
PbO-SiO,
melts:
structure
and thermodynamics
of mixing
685
thereby allowing Pb easy access to a coordination polyhedron of oxygens. The cation-poor melt is characterized by a framework-like structure of silicate tetrahedra that provide the Pb cation with very few non-bridging oxygens. However, since the Pb are few they will have only a small effect on the enthalpy. This explains why the cation-poor liquids are nearly of pure SiO, at all temperatures except near the critical point.
SUMMARY AND CONCLUSION PbO-SiO, melts (and probably melts in other binary silicate systems) are characterized by phase and thermodynamic properties that reflect the gradual polymerization of silicate species as the SiO, content is increased. Some of the more important conclusions are summarized below. (1)The activity of PbO is adequately described by a chemical solution model that equates anionic mole fractions of (02-) to the activity. The model fails in SiO,-rich solutions because discrete silicate ions are supplanted by The marked negative deviation of the PbO ‘infinitely’ large silicate species. activity from ideality is due to the strong depletion of (02-) species as SiO, is added to PbO melts. (2) The activity of SiO, is semiquantitatively related to the state of polymerization of the silicate fraction in the melt. Polymerization curves are remarkably similar to the activity curves over intermediate compositions but diverge in SiO,-rich regions. The activity curve shows a marked positive deviation from ideality in SiO,-rich melts and a marked negative deviation in SiO,-poor melts. This fact clearly reflects the contrasting structural states of SiO,-poor vs SiO,-rich melts. (3) The free energy of mixing curve at 1000°C has a minimum at 40 mole% SiO, and is convex-up, indicating liquid immiscibility in SiO,-rich compositions. The free energy curve is adequately fitted by a model curve that correlates t’he minimum with the maximum in the distribution of (0-) oxygen species. The model curve, however, cannot confirm the occurrence of a two-liquid field in SiO,-rich compositions. (4) The free energy of unmixing is less than 200 cal. The two-liquid field is determined by the method of tangents but the results are uncertain since small energy changes can radically change the temperature-composition coordinates of the solvus. (5) The heat of mixing curve has a minimum of -2000 cal at 35 mole% SiO, and a maximum of 200 cal at 90 mole”/o SiO,. The minimum corresponds t,o the exothermic heat effect associated with the reaction 00 + 02- = 20-. The maximum corresponds to the endothermic heat effect associated with the difficulty of coordinating a Pb cation with a polyhedron of oxygens. (6) The entropy of mixing curve at 1000°C has a small concave-up portion (a local minimum) near 30 mole% SiO, and a maximum at 55 mole% SiO,. A model entropy curve derived on the assumption of random mixing of oxygen species is similar in form but systematically greater than the experimental
686
P. C. HESS
(and calculated) curve. The discrepancy is explained by assuming that the oxygen species are not randomly distributed. This explanation agrees with evidence obtained from the X-ray analysis of PbO-SiO, glasses. The concaveup portion of the entropy curve corresponds to those melts where polymerization is expected to increase rapidly with SiO, content. In this paper, some theoretical concepts were introduced that have met with some success in describing and interpreting the properties of silicate solutions. One of the essential requirements for a successful theory is the ability to simplify a problem without losing sight of physical reality. This or any other theory cannot be expected to hold rigorously, since real solutions are much too complex to understand completely at the present state of theoretical knowledge. However, the model presented in this paper provides a useful framework in which the properties of silicate melts can be correlated, compared, and, hopefully, better understood. Acknowledgements-The paper has benefited from the review work was supported by NSF Grant GA-31785.
of ALEXANDRA
NAVROTSEY.
This
REFERENCES BAES C. F., JR. (1970) A polymer model for BeF, and SiO, melts. J. SoZidStateChem. 1,159-169. BOCKRIS J. O’M., MACKENZIE J. D. snd KITCHENER J. A. (1955) Viscous flow in silica and binary liquid silicates. Trans. Faraday SOC. 51, 1734-1748. BOCKRIS J. O’M., TOMLINSON J. W. and WHITE J. L. (1956), Structure of liquid silicates partial molar volumes and expansivities. Trans. Faraday Sot. 52, 299-310. BRAY P. J., LEVENTHAL M. and HOOPER H. 0. (1963) Nuclear magnetic resonance investigations of the structure of lead borate glasses. Phys. Chem. Glasses 4, 47-66. CALVERT P. D. and SHAW R. R. (1970) Liquidus behavior in the silica-rich region of tho system PbO-SiO,. J. Amer. Ceram. Sot. 53, 350-352. CHARLES R. J. (1967). Activities in L&O-Na,Oand K,O-SiO, solutions. J. Amer. Ceram. Sot. 50, 631-664. CHARLES R. J. (1969) The origin of immiscibility in silicate solutions. Phys. Chena. Glasses 10, 169-178. COTTON A. F. and WILKINSON G. (1966) Advanced Inorganic Chemistry, 1971 pp. John Wiley. DICKENS B. (1965). The bonding in the yellow form of lead monoxide. J. Inorg. ~Vucl. Chem. 27, 1495-1501. FINCHAM C. J. B. and RICHARDSON F. D. (1954) Behavior of sulfur in silicate and aluminate melts. Proc. Roy. Sot. London &23, 40-62. FLOOD H. and KNAPP W. J. (1963), Acid-base equilibria in the system PbO-SiO,. J. Amer. Ceram. Sot. 46, 61-65. FRAY D. J. (1970) The structure of alkali silicate melts. Phys. Chem. Glasses 11, 219-222. GIBBS G. V., HAMIL M. M., BARTELL L. S. and YOW H., 1972, Correlations between Si-0 bond length, Si-0-Si angle and bond overlap populations calculated using extended Hiickel molecular orbital theory. Amer. Mineral. 57, 1578-1613. HESS P. C. (1971) Polymer model of silicate melts. Geochim. Cosmochim. Acta 35, 289-306. HESS P. C. (1972) Structural models of silica-rich melts. (A&S’) EOS 53, 540. HESS P. C. (1973) Thermodynamic of a binary silicate melt. (A&S’) EOS 54, 483. HINZE J. (1970) Heteropolar bonds. In Physical Chemistry, An Aduanced Treatise, (editor H. Eyring), Chapter 4. Academic Press. KAPOOR M. IL., and FROHBERG M. G. (1971) Die bestimmung der thermodynamischen Eigenschafter des systems PbO-SiO, mit Hilfe von E. M. K.-Messungen. Arch. Eisenhiittenw. 42, 5-8. KOZT~A Z. and SAMIS C. S. (1970) Thermodynamic properties of molten PbO-SiO, systems. Met. Trans. 1,871-876.
PbO-SiO,
melts: structure and thermodynamics of mixing
687
LEVENTHAL M. and BRAY P. J. (1965) Nuclear magnetic resonance investigations of compounds and glasses in the systems PbO-B,Os and PbO-SiO,. Phys. Chem. Glasses 6, 113-125. MASSONC. R. (1965) An approach to the problem of ionic distribution in liquid silicates. hoc. Roy. Sot. London AZ%‘, 201-221. MASSONC. R. (1968) Ionic equilibria in liquid silicates. J. Amer. Cerum. SOC. 51, 134-143. MASSON C. R. (1972) Thermodynamics and constitution of silicate slags. J. Iron Steel Inst.
210,89-96. C. R., SMITH I. B. and WHITEWAY S. G. (1970) Activities and ionic distribution in
MASSON
liquid silicates: application of polymer theory. Can. J. Chem. 48, 1456-1464. MYDLAR M. F., KREIDL N. J., HENDREN J. K. and CLAYTON G. T. (1970) X-Ray diffraction study of lead silicate glasses. Phys. Chem. Glasses 11,196-203. NOLL W. (1963) Die silicatische Bindung vom Standpunkte der Electronentheorie. Angew. Chem. 75, 123-130. OSTWALD T. and KLEPPA 0. J. (1969) Thermochemistry of the liquid system lead oxide-silica at 900%. Inorg. Chem. 8, 78-82. PRAUSNITZ J. M. (1969) Molecular Thermodynamics of Fluid Phase Equuilibria,523 pp. PrenticeHall. PRETNAR V. CT.,(1968) Beitrag zur Ionentheorie der Silikatschmelzen. Ber. Bunseng. r&773-778. PRIGOGINE I., DEFAY R. and EVERETT D. H. (1962) Chemical Thermodynamics, 543 pp. John Wiley. RICHARDSONF. D. and WEBB L. E. (1955) Oxygen in molten lead and thermodynamics of lead oxide-silica melts. Bull. Inst. illining Met., 584, 529-564. ROBIE R. A. and WALDBAUM D. R. (1968) Thermodynamic properties of minerals and related substances at higher temperatures. U.S. Geol. Surv. Bull. 1259, 256 pp. SANDERSONR. T. (1971) Chemical Bonds and Bond Energy, 222 pp. Academic Press. TOOP G. W. and SAMIS C. S. (1962a) Activities of ions in silicate melts. Trans. Met. Sot. AIME 224,
878-887.
TOOP G. W. and SAMIS C. S. (1962b) Some new ionic concepts of silicate slags. Can. Met. Quart.
1,129%152.
PbO-SiO, melts: structure and thermodynamics of mixing PAUL C. HESS Department
Geological Sciences, Brown University, Providence, R.I. 02912, U.S.A.
(Received
13 May 1974; accepted in revised form
7 October 1974)
Abstract-Thermodynamic
properties of PbO-SiO, melts, obtained from published data and calculated from freezing point depressions, reflect the gradual polymerization of silicate anions The free energy of mixing curve at 1OOO’C has in the melt as the SiO,/PbO ratio is increased. a minimum at 40 mole ‘A SiO, and is convex-upward between 72 and 98 mole ‘A SiO,. The latter is an indication of met&stable liquid immiscibility. The free ener,7 minimum is correlated with the maximum in the distribution of nonbridging oxygens in the melt. In SiO,-poor melts, the activities of PbO and SiO, (pure liquid standard states) show sharp negative deviations from ideality. The PbO activity reflects the paucity of free oxygen species in the melt whereas the SiO, activity reflects the depolymerized state of the silicate anions. In more SiO,-rich melts, the activity of SiO, shows s, positive deviation from ideality which is qualitatively correlated to a polymerization parameter. The heat of mixing term has a minimum of -2000 cal at 35 mole % The minimum is associated with the SiO, and a maximum of +200 cal at 90 mole% SiO,. exothermic heat effect obtained during the reaction (00) +
(02-)
= 2(0-),
whereas the maximum corresponds to the endothermic heat effect obtained when coordination The entropy of mixing curve has the same polyhedra of oxygens form around the Pb cation. form but is systematically smaller than a theoretical curve calculated on the assumption of random mixing of oxygen species. The discrepancy is due to the entropy loss obtained by the clustering of oxygen species to form complex silicate species.
INTRODUCTION IT IS NOW widely accepted
that silicate melts are ionic solut,ions containing complex silicate anions, cations and ‘free’ oxygen ions. The nature and distribution of these species are functions of melt composition, temperature and pressure. Theories attempting to explain solution non-ideality for silicate solutions, and for fluid mixtures in general, can conveniently be classified as ‘chemical’ or ‘physical’. The ‘chemical’ solution theories are based on the premise that new chemical species are formed in solution and that solution non-ideality is a consequence of these chemical The thermodynamic properties of the solution can be derived from a reactions. knowledge of the equilibrium constants corresponding to these reactions (PRAUSNITZ, 1969). According to these models, non-ideality is only apparent because it is based on an unwise choice of components. ‘Physical’ solution theories assume that the nature of the chemical species is not altered in solution and that non-ideality arises from physical intermolecular forces. Physical forces are described by simple potential energy functions. Examples of the physical theories include the regular solution model, lattice theories, and theories based on the theorem of corresponding states (PRAUSNITZ, 1969). In general, concepts from both contrasting viewpoints must be taken into account but the chemical approach is favored when intermolecular forces are strong. Various forms of the ‘chemical’ solution models have had considerable success in calculating some
672
P. C. HESS
of the thermodynamic properties of binary silicate melts (HESS, 1971, 1972, 1973; MASSON, 1965, 1968, 1972; MASSON et cd., 1970; KAPOOR and FROHBERB, 1971; BAES, 1970; PRETNAR, 1968; CHARLES, 1969; FLOOD and KNAPP, 1963; TOOP and These models adequately reproduce activitySAMIS, 1962a,b; among others). composition cnrves for the metal oxide in silica-poor melts and suggest that the
thermodynamic properties are related to and reflect the ‘internal structure’ of a silicate melt. It is the purpose of this paper to examine the interrelationship between the thermodynamic, phase and structural properties of PbO-SiO, melts. The PbO-SiO, system was chosen for study because thermodynamic data are available for the melt over 60 per cent of the compositional range. Although the system is of little direct petrological interest, it will provide the fundamental information to aid our understanding of the properties of silicate melts in general. This information is needed to better understand transport, phase and geochemical processes in naturally occurring magmas. DATA AND CALCULATIONS KOZ~JICAand SAMIS (1970) determined the activity of PbO (liquid standard state) by electromotive force measurements and calculated the activity of SiO, (solid standard state) using the Gibbs-Duhem equation. Data were obtained at 1000% and over a compositional range from 0 to 62 mole% SiO,. These data are preferred over those of RICHARDSONand WEBB (1965)) and KAPOOR and FROEBER~ ( 1971) because cell reversibility was demonstrated, oxidation of electrodes was avoided and EMF measurements were reproducible to 50.2 mV. The results at 1000% are virtually identical to those of KAPOOR and FROHBERG (1971) (Fig. 1). The
activities of SiO, were corrected to the pure liquid standard using enthalpy data from and WALDBAUM (1968). Partial molar enthalpies (liquid PbO and quartz standard states) of PbO-SiO,
-Kozuko ---Richardson .._ Kapoor
ROBIE
melts from
et al. et al. et al:
Fig. 1. Some examples of experimentally determined values of the activity of PbO (liquid standard) in PbO-SiO, melts at 1OOO’C. Activity of SiO, (solid standard state) obtained from Gibbs-Duhem calculations. NsiO, (mole ‘A SiO,).
PbO-SiO,
melts:
structure
and thermodynamics
of mixing
673
0.054 to 0.501 mole fraction SiO, were determined calorimetrically at 900°C (~STVALD and KLEPPA, 1969). The quartz enthalpies are converted to liquid standard states. In order to obtain enthalpy data for the complete range of composition it is assumed in the calculations This assumption that follow that the partial molar enthalpies are independent of temperature. is certainly a fair one for moderate temperature intervals but becomes increasingly doubtful for extreme ranges. Results by KAPOOR and FROHBERG (1971) and KOZUI~A and SAMIS (1970) suggest that the assumption is valid from at least 850 to 1050%. These partial molar enthalpies are used to calculate activities for other temperatures from d In ai d(l/T)
A& = ??-’
where ai = activity of i and Afli = partial molar enthalpy of i (pure i liquid standard state). Activities for the metastable liquid at temperatures below the SiO, (solid) liquidus are obtained as follows. Activities of SiO, (solid standard state) which are unity along the cristobalite and tridymite liquidus, are corrected to the liquid standard states using enthalpy of fusion values from ROBIE and WALDBAUM (1968). If the partial molar enthalpy of SiO, is independent of temperature then (CHBRLES, 1967, 1969)
where ysio, is the activity coefficient of SiO,. This relation cannot hold strictly since at very high temperatures the partial molar enthalpy should also approach zero. However, it is assumed without proof that the results are a good approximation in the temperature range from 900 to 1700%. Partial molar enthalpies of silica for the range of compositions from 62 to 100 mole % SiO, are compared with those determined by calorimetry in Fig. 2. The data appear to be in reasonably good agreement. If this agreement is not fortuitous, it suggests that the assumption ranges. Partial of the constancy of AHsiO, and ABp,_,, is a good one for these temperature molar enthalpies of PbO are calculated with the aid of the Gibbs-Duhem equation. The integral heat of mixing is obtained from the partial molar enthalpies (Fig. 2)
where Aif is the molar heat of mixing and X represents mole fractions. Activities of SiO, for a range of temperatures are calculated using the partial molar enthalpies and equation (2). Activities of PbO at temperatures other than 1OOO’C are obtained from Gibbs-Duhem relations. SOLUTION
MODELS
The structural evolution of a silica-poor melt to a pure SiO, melt is characterized by a continous polymerization of discrete SiO, monomers into complex silicate species until a network structure is attained (HESS, 1971, 1972; MASSON, 1965, 1968, MASSON et al., 1970; TOOP and SAMIS, 1962a,b; among others). HESS (1971), utilizing a random polymerizing model, calculated that discrete silicate species are rapidly diminished in abundance in silica-rich melts and that ‘infinitely’ large silicate species had a high probability of forming. Other authors, however, (FRAY, 1970; BOCICRIS et al., 1955, 1966) suggest that discrete silicate species persist even to high silica compositions. Figure 3 shows the results obtained by the author (HESS, 1971) where the ‘gel’ curve records the weight fraction of infinite silicate species (normalized to total silicates) in the melt. Note that in this model, the gel portion increases rapidly but smoothly from 0 to 92 wt. % in the interval from 40 to 60 mole % SiO,. If this model is correct, then the metasilicate region is a zone of transition between melts of contrasting structure. At low SiO, compositions, the melt contains small discrete silicates whereas at high SiO, compositions the melt is a partially polymerized network. These differences should be strongly reflected in some of the thermodynamic and physical properties of the melt.
P. C. HESS
674
PHASE DIAGRAM
I 0
0.2
0.4
0.6
0.8
1.0
N SiOz Pig. 2. Partial molar heats of PbO and SiO, (solid lines) determined by OSTVALD and KLEPPA (1969). Vertical bars represent uncertainty in the data. Dashed OI_UTW for A~sios calculated from freezing point depression data and dashed curve for AH,,, obtained from Gibbs-Duhem calculations. Integral heat of mixing (liquid standard states) calculated from partial molar enthalpies.
0
0.2
0.4
0.6
0.8
0
Fig. 3. Calculated weight fraction of silicate species (normalized to total silicate content) as a function of SiO, content in the melt. Gel corresponds to weight fraction of ‘infkite’ silicate species. Note that the gel curve rises abruptly and steeply at intermediate compositions.
PbO-SiO,
melts:
I
0
structure
and thermodynamics
675
of mixing
PbO -Si02
0.2
0.6
0.4
0.8
I 0
“Jsioz
Fig. 4.
Concentrations
of bridging (OO), non-bridging in PbO-SiO, melts assuming K
(O-) and free oxygen = 0.02.
(0%)
A comparatively simple yet highly instructive approach to an understanding of the properties of a silicate melt is to focus directly on the nature of the variously bonded oxygen species, FINCHAM and RICHARDSON (1954) temporarily ignoring the complexly polymerized structure. recognized that oxygen in silicate melts occurs only in three forms, singly bonded (O-) or non-bridging oxygens (nbr), doubly bonded (OO) or bridging oxygens (br) and free oxygen ions (02-) (fr). The nomenclature refers to whether the oxygen is coordinated to one, two or zero silicons. TOOP and SAMIS (1962a,b) assumed that any reaction between silicate ionic polymers is described by 2(0-) = (00) + (02-), (4) and under equilibrium
conditions,
an equilibrium K
constant
K exists, such that
= (OO)(O”)/(O-)2,
(5)
where (OO), and (02-) and (O-) are the equilibrium number of moles of the variously bonded Note that by the inclusion of mole numbers rather than activities it is assumed that oxygens. the solution is ideal with respect to mixing of the oxygen species. This simplifying step allows all oxygen ion concentrations to be calculated for a given K, composition and material balance of the system (Fig. 4). It is also possible (TOOP and SAMIS, 1962a,b) to show that the molar free energy
of mixing,
A??, is approximated
by
(6) DISCUSSION
Activities of PbO and SiO, The activity
of MO in a binary MO-SiO, %lO
=
KWr(02-)
+
melt is approximated (O-)/Z:(22
+
2)X,1,
(HESS,
1971)
by
(7)
where uMO = the activity of the component MO, x = number of silicons on a polymeric silicate ion, and X, is the mole fraction of a given x-merit ion in the melt. The model cannot be explained in detail but is based on the following assumptions : 9
P. C. HESS
676
(I) polymeric silicate ions exist only as simple chain species and their distribution is obtained by random polymerization reactions and (2) the mixing properties of the solution obey the Temkin model whereby it is assumed that the melt exists as two independent cation and anion mixtures. The activity of MO in this model is equal to the anionic mole fraction of (02-) ions and is given by the moles of oxygen ions divided by the total number of moles of anions in solution. The values of (W-), (0-j and X, are shown to be functions of K (HESS, 1971) and a suite of activity-composition curves can be calculated for different values, These correspond closely to experimentally derived curves. Figure 5 compares the experimental activity curve
l
Kozuko-Samis 1000~ c
.Nsioz Fig. 5. Activitiesof PbO (liquidstandard state)asdetermined by experiment (dots) are compared to those correspondingto a theoretical curve appropriateto K = 0.02. for PbO at 1000°C of KO~UKAand SANIS(1970) and a theoretical curve appropria~ to K = O*OZ. The theoretical curve cannot be extended beyond 39 mole% SiO, since ions of ‘infinite’ dimensions occur beyond this point (see Fig. 3). The model is clearly inappropriate for silica-rich melts and is certainly an oversimplification for even SiO%-poor melts. It is useful in a pictorial sense since the activity of PbO is correlated with the mole fraction of (02-) and to the Pb-(OS-) bonds in the melt. As (02-) ions are reduced, fewer and fewer portions of the melt resemble the reference PbO liquid and therefore the ‘escaping’ tendency of PbO as reflected in the activity is reduced. The strong negative deviation from ideal behavior is a consequence of the rapid depletion of (02-) as SiO, is added to the solution. Activity of SiO, The activity of SiOz cannot be derived (other than by Gibbs-Duhem methods) from the ionic model of HESS (1971). It is possible, however, to derive a model from a less rigorous theoretical treatment. The shape of the activity-composition curve of SiOa (liquid standard state) in the PbO-SiO, system (and in many other systems) has the following characteristics (Fig. 6). In silica-rich compositions, the activity
PbO-SiO,
melts:
structure
0.2
and thermodynamics
0.8
0.6
0.4
of mixing
677
1.0
kOZ
(a)
0.6
0.6
0.4
0.4
P
aSiO,
0.2
0.6
0.4
0.8
1.0
Go2 (b) Fig. 6. (a) Activity of SiO, curves (solid and liquid standard states) given by solid lines are compared to a theoretical P-curve corresponding to II = 0.01. (b) P-curves for a variety of K values. (Temperature = 1000°C).
curve shows a marked positive deviation from ideahty and is relative flat or sigmoidal (metastable liquid immiscibility). In intermediate compositions, the curve drops sharply from a positive to a negative deviation from ideahty, The activity in metal-rich compositions shows extreme negative deviations from ideality. The ‘central zone’ corresponds to those compositional regions marking the sudden appearance of infinite silicate species as defined by the gel curve. The correlation suggests that the activity of SiO, is directly related to the state of polymerization of the melt. A highly polymerized melt is one in which a large percentage of the silicon
P. C. HESS
678
bonds lead through a coordinated number of silicon bonds is
oxygen
2(00) + (O-)
to an adjacent =
silicon atom.
(8)
4NSi027
and the number of bonds linked to bridging oxygens is 2(O").Therefore, of all silicon bonds, P, that are bridging is
P=
2(0°) 2(0°) + (0-J
The total
the fraction
(0°)
=2N,io* 2
The value of P is then one convenient measure of the state of polymerization that is useful over the complete compositional range. Figure 6 contains P curves corresponding to various values of K, and the activity of SiO, curve at 1000°C for the PbO-SiO, system. There is a remarkable agreement between the polymerization curve of K = 0.01 and the activity of SiO, curve at 1000°C over intermediate compositions. However, the model curve cannot predict the onset of immiscibility at high SiO, compositions. Nevertheless, the similarity between the P curves with activity of SiO, curves in this, in the SnO and, to a lesser degree, in the CaO and Na,O systems (unpublished data) argues that the state of polymerization is semi-quantitatively reflected in the magnitude of asioa. Since the standard state of the activity of SiO, is the totally polymerized liquid, the hypothesis is, in fact, quite reasonable. Free energy of mixing The free energy of mixing curve calculated from the activities at 1000°Chas a minimum at 40 mole”/o SiO, and is convex-up in SiO,-rich compositions (Fig. 7). By the method of tangents, a two-liquid field is indicated between 72 and 98 mole% SiO,. The free energy changes, which are responsible for immiscibility, are less than 200 cal. A free energy of mixing curve calculated from equation (6), K = 0.06and corresponding values of (0-) adequately reproduces the experimental curve to within 299 cal (Fig. 7). The model curve, however, is concave-up rather than convex-up in the two liquid region. Moreover, the best fit of the free energy curve is given by K = 0.06,whereas the best fit to the aPbo curve is given by K = O-02. Although the simple model can reproduce, with good accuracy, the free energy curve for the PbO-SiO, system, it again has failed as a predictive tool in SiO,-rich regions. As with most strictly chemical theories, the mixing properties of the solution are assumed to be ideal once the true chemical species are correctly identified. In this model, the chemical species are (OO), (0-) and (Oz-), and are expressed in concentration units rather than in terms of activities. Due to these approximations, free energy curves calculated from equation (6)give no indication that liquid immiscibility occurs. These theories can be improved by removing the requirement that mole fractions of species are equal to activities. For example, activity coefficients may be introduced by using regular solution or van Laar models (PRAUSNITZ, 1969). Unfortunately, these modifications have the effect of introducing new adjustable parameters that may cloud rather than clarify the solution characteristics. These approaches may be useful, however, for curve fitting or extrapolation purposes.
PbO-SiO,
melts:
structure and thermodynamics
_---__ SiO,
0
-I .-__I
F C
--+J’
of mixing
679
,I
/ l’ / ,/’
PbO -2
-3
0.2
0
3
NSi02 Fig. 7. (a) Free energy of mixing of PbO-SiO, melts as determined experimentally (solid line) and calculated (light dash). Tangent to free energy curve (dotted line) indicates the compositions of the coexisting liquids (two vertical arrows). Theoretical free energy of mixing curve corresponding to K = 0.06 is given by heavy dashed line. (b) Activity coefficients of PbO and SiO, in PbO-SiO, melts aa determined experimentally (solid lines) and calculated (dashed lines).
Heat of mixing The heat of mixing is that evolved or absorbed when pure liquid PbO (free oxygen ions) is mixed with liquid SiO, (double bonded oxygen ions). The solution contains complex silicate structures and bond types that do not exist in the pure liquids. Consequently, a major portion of the heat effect arises from the change of oxygen bonding in the solution as compared to the bonding in the pure liquids (CHARLES, 1969). An analysis of the factors that contribute to the heat budget could be simplified if all activity coefficients of the oxygen species were unity as indicated by the expression for the equilibrium constant in equation (5). On this assumption, no heat effect is associated with the simple transfer of free or double bonded oxygen from the pure liquid to the solution. This analysis, however, is correct only as a first approximation. As an example, consider the bonding of bridging oxygens in crystalline silicates. Bridging Si-0 bond lengths are not constant but are, in part, dependent on the Si-0-Si bond angle with the shorter bonds being associated with wider angles. Moreover, the shorter bond lengths correlate with larger Mulliken bond overlap populations (GIBBS et al., 1972). On a qualitative basis, bond overlap
P. C. HESS
680
populations are believed to be a good criterion for relative bond strengths (COTTON and WILKINSON, 1966). Therefore, variations in both bond lengths and bond energies of bridging Si-0 bonds indicate that bridging oxygen species are not all thermodynamically equivalent and do not contribute equally to the energy of a silicate crystalline phase. These conclusions may also apply to the liquid phase. The maximum heat of mixing of the PbO-SiO, solution is -2 kcal/mole (Fig. 2). This is probably but a small fraction of the bond energies. SANDERSON (1971) calculated that the bridging Si-0 bond energy in crystalline SiOz has an energy of 1109 kcal/mole. This value multiplied by four bonds per silicon yields the atomization energy of 443.6 kcal/mole, in good agreement with the experimental value of The Pb-0 (free) bond energy, calculated by the same procedure, 445.6 kcal/mole. There is no independent check on this bond energy so that the is 72.7 kcal/mole. In Sanderson’s method, bond results must be regarded only as semiquantitative. energies are inversely related to bond lengths. Variations of 3 per cent in individual bond lengths are capable of producing heat effects comparable to those observed in the PbO-SiO, system. The heat effect, therefore, represents small energy variations between large energy terms. This makes the heat term difficult to evaluate. Nevertheless, the rough correlation between the minimum in the heat of mixing curve (Fig. 2)with the maximum in the (0-) curve (Fig. 4) suggests that the ideal model is adequate as a first approximation and that a significant portion of the heat effect is associated with the reaction 2(0-)
= (00) + (OZ-).
(4)
The heat effect associated with the reaction above can conveniently be considered in two parts (CHARLES, 1969). The replacement of one Si-0 bond of bond strength of 111 kcal/mole by a weaker Pb-0 bond of bond strength of 73 kcal/mole allows a closer association of the remaining Si-0 pair. This is shown to be true for a large number of silicate structures where Si-0 (nbr) bonds are usually shorter than the Si-0 (br) bonds (GIBBS et al., 1972). The closer association may be the result of t,he polarization of the electron cloud of the oxygen towards and the oxygen nucleus away from the silicon (CHARLES, 1969). Si-0 bond length variations can also be rationalized in terms of a covalent bonding model wherein shorter bond lengths are correlated with large bond overlap populations (GIBBS et al., 1972). In any case, the heat effect associated only with the conversion of a bridging Si-0 bond to a non-bridging Si-0 bond should be exothermic (CHARLES, 1969). The remaining part of the heat effect in the simple two-step model is ascribed to changes in Pb-0 bonding in the solution as compared to Pb-0 bonding in the pure liquid. Some knowledge of Pb-0 bonding in the pure liquid phase can be obtained by examining the crystal structure of the polymorphs of PbO. Yellow PbO, the high temperature polymorph, is made up of layers composed of infinite chains of Pb-0 in four-fold coordination (DICKENS, 1965). The surface atoms of the layers are all of Pb, and the oxygen atoms are confined to the interior. Since the lead atoms face each other across adjacent layers, it is highly probable that the structure is not totally ionic but is characterized by covalent interactions (DICKENS, 1965). DICKENS (1965) estimated that the Pb-0 bond in yellow lead oxide had between 20 and 50 per cent ionic character, the low value corresponding to Sp3 hydridization.
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More information about the nature of Pb-0 bonding in glasses can be obtained by several analytical methods. Nuclear Magnetic Resonance (NMR) determinations of the Pb207 chemical shift are virtually identical for PbO-Si02 glasses and compounds of identical composition (LEVENTHAL and BRAY, 1965). The results suggest that the chemical environment of the Pb ion in the glass is similar to that of the isoThe chemical shift was obtained for glasses containing chemical compound. 30-75 mole% PbO. The chemical shift of the hypothetical glass of lOOoh PbO was obtained by extrapolation and is close to the value determined for crystalline PbO (BRAY et aZ., 1963). The Pb-0 bond type in the glass therefore has a significant covalent character. However, the chemical shift approaches more ionic values as the glass becomes more SiO,-rich, although even the most SiO,-rich glasses are not characterized by a high ionic shift (LEVENTE~ALand BRAY, 1965). X-Ray diffraction data and radial distribution functions (RDF) of glasses calculated for compositions of BPbOSiO,, PbOSiO, and PbO*2SiO, (MYDLAR et uZ., 1970), indicate that Pb-Pb distances in the glasses are only slightly larger than those in corresponding crystals. Peak positions in the RDF curves are repetitive and indicate an ordering of Pb-Pb distances that are additive with respect to the shorter distances. These characteristics suggest that the Pb atoms are not randomly distributed within the silicate network but instead occur as simple and/or twisted chains of PbO, tetrahedra. The cation attempts to achieve the maximum coordinat)ion possible with the free oxygen ions in the melt. If the Pb cation does indeed prefer a coordination polyhedron of free oxygen, then the direct bonding of Pb with a free oxygen probably represents a lower energy stat,e than the bonding of Pb with a non-bridging oxygen. The conversion of a Pb-0 (fr) bond to a Pb-0 (nbr) is then an endothermic process. The heat effect associated with the transfer of Pb from the pure liquid to the solution will depend on the concentration of free oxygens in the solution. The endothermic contributionis small in SiO,-poor compositions since free oxygens are readily available (Fig. 4). In SiO,-rich melts, Pb can coordinate only non-bridging oxygens and the endothermic heat effect will become relatively more important. CHARLES (1969) has presented similar arguments and conclusions for other systems. Some of the exothermic heat gain in converting a Si-0 (br) bond to a Si-0 (nbr) bond may be lost if the remaining Si-0 (br) bonds of the given silicon increase in bond length and therefore weaken in bond strength. NOLL (1963) suggests that the polarized electron cloud of the non-bridging oxygen does an effective job of shielding the Si nucleus thereby reducing its effective positive charge. This would weaken the remaining Si-0 (br) bonds. This analysis is certainly oversimplified as residual charges on Si and 0 can be evaluated only by quantum mechanical methods (HINZE, 1970). Still, some interesting conclusions are drawn if these arguments are valid. The conversion of a weakened Si-0 (br) bond to a Si-0 (nbr) bond results in a greater release of heat than would be obtained from a ‘stronger’ Si-0 (br) bond. If the heat effect associated with the loss of the weakened Si-0 bond outweighed the loss of entropy, thereby lowering the free energy, then the formation of non-bridging Si-0 bonds would not be random but would occur in proximity to other Si-0 (nbr) bonds. Pb would not be randomly distributed but would be ordered in the structure. The proposed ordering tendency would allow Pb cations to share free oxygen ions most
682
P. C. HESS
effectively. This model may explain the general ordering tendencies observed in PbO-SiO, glasses. Within this framework, it is possible to rationalize the heat effect depicted in Pig. 2. The minimum in Ai? corresponds to a predominance of the exothermic heat effect of the conversion of Si-0 (br) to Si-0 (nbr) bonds over the small endothermic change due to Pb-0 bond changes. With increasing SiO,, the exothermic contribution to the heat effect decreases as a smaller number of Si-0 (br) bonds are converted to Si-0 (nbr) bonds. Moreover, free oxygen ions decrease to zero so that a maximum endothermic effect is obtained as Pb-0 (fr) bonds are converted to Pb-0 (nbr) bonds. At very high SiO, compositions, only a small percentage of (0-) exists making it extremely difficult for the Pb cation to achieve an oxygen coordination shell. In order to form such a coordination polyhedron of oxygens and thereby effectively shielding the cation charge, it is conceivable that considerable strain will result in the This may contribute additionally to the endothermic effect silicate structure. causing the positive heat of mixing (CHARLES, 1969). The composition dependence of the partial molar heats of mixing (Fig. 2) is also profitably discussed in the following context. When 1 mole of liquid PbO or SiO, is added to a large (in theory infinite) amount of solution, the Pb-0 or Si-0 bonds must take up the configuration and energy state of the given (average) bonds in the The partial molar heats of mixing therefore represent the heat effect solution. associated with the gain or loss of energy when the mole of added pure liquid adjusts to its new thermodynamic state. The partial molar heat of mixing of PbO is less than -200 Cal/mole from 0 to 20 mole”/o SiOZ, and then drops sharply to a minimum of -4200 cal at 68 mole”/o SiO,. The transfer of Pb and 0 (fr) to the solution is exothermic up to nearly 100 per cent SiO, and basically reflects the conversion of 0 (fr) to 0 (nbr). For Nsio, > 68 mole% the endothermic effect of the transfer Pb (pure liquid) to Pb (solution) reverses the trend to more negative heats and for iv SiO2 > 92 mole”/o it outweighs the exothermic heat effect. The partial molar heat of mixing of SiO, is strongly negative in SiO,-poor compositions but becomes positive at about 52 mole% SiO,. The exothermic effect is due to the depolymerization of SiO, pure liquid as it is added to the solution. The cross-over occurs in the compositional region where free oxygen species are minor constituents or perhaps even absent. If no free oxygen species are available, the exothermic heat contribution due to reaction (4) is not available and the heat effect must be due to the transfer of bridging oxygens from t,he pure liquid to the solution (CHARLES, 1969). The endothermic heat effect then is due to the strain of the SiO, ‘lattice’ as it attempts to coordinate and shield the Pb cation. Therefore, the positive heat of mixing is due primarily to the heat effect associated with the polymerized silicate fraction. The Pb imposes the energy demand whereas the polymerized silicate anions attempt to conform to these demands at the expense of an unfavorable energy configuration. If this hypothesis is correct, it is likely that the point where partial molar heats of mixing of SiO, change from negative to positive will be a good measure of (1) the state of polymerization of the melt and (2) the shielding demands of the cation as perhaps reflected in its field strength. HESS (1971) noted solutions containing cations of high field strength are also the most polymerized. Therefore, cross-over points of ARsio2 should prove useful in comparing the state of polymerization of different silicate melts.
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Entropy of mixing The entropy of mixing curve at 1000°C has two maxima at about 0.20 and 055 mole% SiO, and a minimum at 0.30 mole% SiO, (Fig. 8). In the interpretation of the entropy of mixing it is the custom to neglect changes in the vibrational contribution to the entropy on the assumption that it is small compared to the configurational contribution. The same procedure is adopted in this study although it is wise to note that the force constants of bonds linking free, single or double As a result, the vibrational contribution to bonded oxygens cannot be identical. the entropy of mixing is unlikely to be zero. The following discussion, however, will consider only configurationa,l contributions of the oxygen species to the entropy. CHARLES (1969) derived an expression for the configurational entropy of mixing on the assumption that it was mainly due to the permutations of the sites that are occupied by the oxygen ions in solution. If nonbridging oxygen ions occur as pairs in the solution, then the integral entropy of mixing one mole of solution is
COO) + (02-)
(02-) In w I ,
nw
(10)
where w = (O-)/2 + (00) + (02-). The equilibrium values of the oxygen species are obtained for different K values as before. The model entropy curves corresponding to K values near 0.01 are of remarkably similar form to those of the PbO-SiO, systems, but are systematically greater for all compositions (Fig. 8). The model entropy curves were calculated on the assumption of random mixing of (OO), (02-) and (0-) pairs of oxygen species. In the discussion on heat effects it was suggested that the formation of Si-0 (nbr) bonds could weaken the remaining Si-0 (br) bonds. This weakening would localize any additional ruptures of Si-0 (br) bonds since such reactions would be favored by the greatest release of heat. If this, in fact, were true, the distribution of oxygen species would not PbO-SiO;! K_ (0”)(0-2) (0-12
0
0.2
0.4
0.6
0.6
3
Nsros Fig. 8. Entropy and calculation
of mixing curves for PbO-SiO, melts as determined by experiment (indicated by 1000°C) and a model curve corresponding to K = 0.01.
684
P. C. HESS
be random, and the entropies would be reduced. In general, any deviation from the condition of a random distribution would result in lower entropy values than those predicted by equation (10). The entropy of mixing curve for the PbO-SiO, solution can be viewed as a composite curve of two distinctly different mixing patterns. The curve in the interval from 0 to 20 mole% SiO, describes a mixing process between polyhedra of PbO, and SiO, ionic species. According to HESS (1971), Si04 monomers make up more than 90 per cent of the silicate ionic species in this compositional range. The entropy of mixing from 30 to 100 mole”/o SiO, describes a smooth curve reaching a maximum near 55 mole”/o SiO,. The curve lies below an extrapolation of the entropy curve corresponding to the hypothesized mixing of monomers. The entropy curve at compositions from 30 to 100mole”/o SiO, therefore is believed to represent mixing properties of complexly polymerized silicate and plumbate ionic species. The lower entropy values below those of the extrapolated depolymerized curve and also below the model entropy curve reflects the clustering of oxygen species forming the complex ionic configurations. The fact that the entropy curve varies smoothly in this range supports, but does not prove, the contention of HESS (1971) that the polymerization The local minumum of the entropy process is smoothly varying and continuous. = 0.30 corresponds to those compositions where the extent of curve at Ns,, polymerizatiodin the PbO-SiO, at 1000°C increases rapidly. TWO liqtd jield The inverted S-shaped cristobalite liquids at 1670°C (CALVERT and SHAW, 1970) points to the existence of a metastable two-liquid field at lower temperatures or a stable two-liquid field at higher temperatures. Free energy calculations indicate that the two-liquid field occurs metastably with an upper consulate point. The existence of an upper critical solution point could have been predicted from and is consistent with the endothermic heat of mixing in the critical region (PRIGOQINE etal.,1962). Unfortunately, the simplifying approximations used to calculate the free energy curves introduce errors that are large in comparison to the small energy differences needed to differentiate between one and two phase fields. The uncertainTherefore, the author does ties grow even larger as the critical point is approached. not give the T-x coordinate of the solvus since the numbers would be of little value. Liquid immiscibility occurs if the free energy of the system is minimized on splitting the solution into two phases. The free energy of mixing contains two opposing terms, the enthalpy and entropy (-T AB) contribution. These terms cannot individually be minimized at the same time. The enthalpy of mixing curve of the metastable solution is within a few hundred calories of zero in the two-liquid composition range, whereas it falls to -2000 cal at 40 mole% SiO,. The endothermic heat effects in high silica melts are at least two-fold: (1)Pb cations are coordinated only by nonbridging oxygens and cannot obtain a complete coordination polyhedron of oxygens if the latter are not ordered and (2) the silicate structure is deformed in order to accommodate and provide a shield of oxygens for the cation (CHARLES, 1969). The influence of these endothermic heat effects is reduced by forming two liquid phases, one relatively cation-rich, one cation-poor. The cation-rich melt is less polymerized and contains greater numbers of free as well as nonbridging oxygens,
PbO-SiO,
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thereby allowing Pb easy access to a coordination polyhedron of oxygens. The cation-poor melt is characterized by a framework-like structure of silicate tetrahedra that provide the Pb cation with very few non-bridging oxygens. However, since the Pb are few they will have only a small effect on the enthalpy. This explains why the cation-poor liquids are nearly of pure SiO, at all temperatures except near the critical point.
SUMMARY AND CONCLUSION PbO-SiO, melts (and probably melts in other binary silicate systems) are characterized by phase and thermodynamic properties that reflect the gradual polymerization of silicate species as the SiO, content is increased. Some of the more important conclusions are summarized below. (1)The activity of PbO is adequately described by a chemical solution model that equates anionic mole fractions of (02-) to the activity. The model fails in SiO,-rich solutions because discrete silicate ions are supplanted by The marked negative deviation of the PbO ‘infinitely’ large silicate species. activity from ideality is due to the strong depletion of (02-) species as SiO, is added to PbO melts. (2) The activity of SiO, is semiquantitatively related to the state of polymerization of the silicate fraction in the melt. Polymerization curves are remarkably similar to the activity curves over intermediate compositions but diverge in SiO,-rich regions. The activity curve shows a marked positive deviation from ideality in SiO,-rich melts and a marked negative deviation in SiO,-poor melts. This fact clearly reflects the contrasting structural states of SiO,-poor vs SiO,-rich melts. (3) The free energy of mixing curve at 1000°C has a minimum at 40 mole% SiO, and is convex-up, indicating liquid immiscibility in SiO,-rich compositions. The free energy curve is adequately fitted by a model curve that correlates t’he minimum with the maximum in the distribution of (0-) oxygen species. The model curve, however, cannot confirm the occurrence of a two-liquid field in SiO,-rich compositions. (4) The free energy of unmixing is less than 200 cal. The two-liquid field is determined by the method of tangents but the results are uncertain since small energy changes can radically change the temperature-composition coordinates of the solvus. (5) The heat of mixing curve has a minimum of -2000 cal at 35 mole% SiO, and a maximum of 200 cal at 90 mole”/o SiO,. The minimum corresponds t,o the exothermic heat effect associated with the reaction 00 + 02- = 20-. The maximum corresponds to the endothermic heat effect associated with the difficulty of coordinating a Pb cation with a polyhedron of oxygens. (6) The entropy of mixing curve at 1000°C has a small concave-up portion (a local minimum) near 30 mole% SiO, and a maximum at 55 mole% SiO,. A model entropy curve derived on the assumption of random mixing of oxygen species is similar in form but systematically greater than the experimental
686
P. C. HESS
(and calculated) curve. The discrepancy is explained by assuming that the oxygen species are not randomly distributed. This explanation agrees with evidence obtained from the X-ray analysis of PbO-SiO, glasses. The concaveup portion of the entropy curve corresponds to those melts where polymerization is expected to increase rapidly with SiO, content. In this paper, some theoretical concepts were introduced that have met with some success in describing and interpreting the properties of silicate solutions. One of the essential requirements for a successful theory is the ability to simplify a problem without losing sight of physical reality. This or any other theory cannot be expected to hold rigorously, since real solutions are much too complex to understand completely at the present state of theoretical knowledge. However, the model presented in this paper provides a useful framework in which the properties of silicate melts can be correlated, compared, and, hopefully, better understood. Acknowledgements-The paper has benefited from the review work was supported by NSF Grant GA-31785.
of ALEXANDRA
NAVROTSEY.
This
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