Application of Polymer Theory to Silicate Melts The System MO + MF, + SiO, BY CHARLES R. MASSON" AND WILLIAM F. CALEY Halifax, Nova Scotia, Canada B3H 321 Atlantic Regional Laboratory, National Research Council of Canada, Received 28th February, 1978 Expressions are derived for activities of MO and MF2 as functions of composition in MO + MF2 + SiOz melts. The treatment is based on the assumption that the melts consist of MZC cations and 02-, F- and an array of silicate and fluorosilicate anions of general formula Si,Ojn+ - mFL2nC2-m)- in thermodynamic equilibrium, where 1 Q n < 03 and 0 Q rn < 2n + 2. As in previous treatments, the equilibrium between the oxide and silicate ions is described in terms of an equilibrium constant k, the value of which is independent of the chain length n of the polyions.The equilibrium between the fluoride, silicate, fluorosilicate and oxide ions is described in terms of a second equilibrium constant k' the value of which is assumed to be independent of the degree of substitution of 0- by F on the silicate chains. Theoretical curves are shown for the activity of PbO as a function of com- position in PbO+PbF2+SiOz melts for the simple case in which k' = 0, i.e., for the situation in which PbFz acts solely as a diluent. Comparison with experimental data shows that the value of k' for this system is finite. Accurate determination of k' requires a knowledge of the activities of both MO and MF2. Knowledge of the value of k' allows the ion fraction of any species to be evaluated approximately in the range of composition for which the theory is applicable.In the application of polymer theory to silicate melts it is assumed that the simplest species of discrete silicate ion in the melt is the orthosilicate ion SiO$-, regarded as the monomer, and that these ions can undergo self-condensation with elimination of free oxide ions to yield an array of polysilicate ions in thermodynamic equilibrium. Application of this theory to binary melts MO + SiO, allows the average size and distribution of the silicate ions to be evaluated approximately from thermo- dynamic data in regions of composition where cyclic and network structures may be ignored. Except for the case of densities and molar volume^,^ which depend only on the mean chain length of the silicate ions and not on their size-distribution, extension of this treatment to ternary systems with more than one cation is difficult due to the competitive interactions between the anions and the various cations.This leads to significant deviations from Temkin's Law,4 assumed to hold for the binaries. In principle, however, it is possible to extend the treatment to ternary melts with a common cation. As an example of such a system we consider here melts of the ternary NO + MF2 + SO,. The treatment was developed primarily to gain some insight into the factors which govern the thermodynamic properties and constitution of fluoride-containing melts, although it is applicable, in principle, to other systems in which F- represents any univalent anion. The influence of fluoride additions on the properties of silicate melts has been studied by many investigators '-' Addition of relatively small amounts of CaF, to silicate melts markedly lowers their vis~osity.~-~ Kozakevitch showed that for acidic melts the effect is approximately twice that of CaO and suggested that monovalent F- ions, which have approximately the same and reviewed extensively.' 6-1 2942C.R. MASSON AND W . F . CALEY 2943 radius as bivalent 02- ions, can disrupt the -Si-0-Si- linkages in a silicate net- work, with formation of 02-, which can further disrupt the network : I I I I I I I I I I I I I I I I I I I I (1) (2) -Si-0-Si- + 2F- + -Si-F F-Si- + 02- -Si-O--Si- + 02- + -Si-O- + -0-Si- . Reaction (1) yields fluorosilicate structures wtih Si-F linkages. Bockris and Lowe l 9 proposed that F can replace 0- groups in silicate structures by depolymer- ization reactions of the type (3) with formation of discrete fluorosilicate anions.Direct evidence for Si-F bonding in quenched fluorosilicate melts was obtained by Kumar et aLIO from infrared absorption studies. The loss of fluorine as SiF4 from fluorosilicate melts has also been reported.l0. l1, Si3O$- + 2F- -+ Si03F3- + Si206F5- THEORETICAL It is assumed that the anionic constitution of a ternary MO+MF2+Si02 melt can be described in terms of 02-, F- and an array of silicate and fluorosilicate ions in thermodynamic equilibrium. For a complete description of the system it is necessary to consider all kinds of polyions which can arise by self-condensation of the monomeric units and their interaction with F- ions. This includes cyclic and net- work structures as well as linear and branched chains.Such a complete description is beyond our capability at present due to our inability to formulate exact expressions for the total number of moles or ions in the system when cyclic and network struc- tures are present, a limitation so far inherent in polymer theory generally. In the present treatment we confine attention to the case in which only linear chains are allowed. Although an oversimplification, this facilitates enormously the evaluation of theoretical relationships. As shown previously 20* 21 this approach also provides a reasonable description of the thermodynamic properties of many binary systems at low silica contents (Xsioz < 0.5).For this simple case the silicate ions have the general formula Si,O:2,",+12)- (1 B n < 00) and the equilibrium between the oxide and silicate ions may be written for which the equilibrium constant k , expressed in terms of the ion fractions of the participating species, is given by* and is independent of the chain length n. Activities of MO calculated on this basis for binary MO + SiO, melts are given by the expression SiOt- + S i , O $ ~ ~ ~ ) - + Sin+ 10(2n+4)- 3n+4 + 02- (4) k = Nsi,+,o,,+,No/Nsio,Nsi,o,,+ 1 (5) 1 1 -- - 2+- - 1 Xsioz 1 - UMO 1 + U M O ( l / b - 1) where Xsioz is the mole fraction of SO2. * For simplicity, values of ionic charge are omitted in the expressions for ion fractions.2944 POLYMER THEORY OF SILICATE MELTS For ternary melts MO+MF,+SiO,, the silicate ions again have the general formula SinO'32,",fi2'-.Fluorosilicate ions are considered to arise by substitution of 0- groups by F groups on the silicate chains. The silicate and fluorosilicate ions thus have the general formula Sin03n+l-mF~n+2-m)- where 1 < n < 00 and 0 < m < 2n-1-2. The equilibrium between the silicate and oxide ions is again given by eqn (4). For the equilibrium between the fluoride, silicate, fluorosilicate and oxide ions we may write where the equilibrium constant k' is given by and, for linear chains, is again independent of n. As emphasized previously, the assumption that equilibrium constants for reactions of this nature are independent of the chain lengths of the reacting species must be regarded as an approximation which cannot be expected to hold rigorously for the smallest ions but which should become increasingly reliable as n increases.It is further assumed for simplicity in the present treatment that k' is independent of the value of m in eqn (8). This implies that the reactivity of the 0- groups is independent of their degree of substitu- tion by F groups on the silicate chains. Without these simplifications the treatment appears hopelessly complicated at present. F- +Sin03,+ I-mF~2nf2-m)- 3n-m m + l +02- (7) (8) = Si,O F ( 2 n + l - m ) - k' = NSinO~n-mFm+ iNOINFNSinOsn+i -mFm From eqn ( 5 ) it follows that N s i 2 0 7 = ( k N ~ i 0 s / N o W S i 0 4 ~ S i 3 0 1 0 = ( k N s i o 4 / ~ o ) ~ s i z o , = ( k N S i 0 4 / N 0 ) 2 N S i 0 4 Nsi&13 = ( k N S i 0 4 / ~ o ) ~ s i 3 0 1 0 = ( k N s i o 4 / N o > 3 N s i o 4 or, in general where From eqn (8) we obtain NSin03n + I = An- Nsio4 A = kNsio4/No.or, in general where From eqn (9) and (11) : which yields the ion fraction of any species in terms of Nsio4 if k, k', No and NF are known. For the sum of the ion fractions of anions of chain length n we have, from eqn (13) : Nsin03n + 1 - mFm - - BmNsin03, + 1 (1 1) B = k'NF/No. (12) (1 3) ~ S i , , 0 3 , , + 1 -mFm = An- lBmNSi04 2n+2 2n+2 1 - p + 3 (14) 1-B C NSin03n+l-,nFm = A"-1NSi04 C Bm = An-1NSi04 m=O m u 0C . R. MASSON AND W. F. CALEY 2945 Hence, for the sum of the ion fractions of all silicate and fluorosilicate anions : AS the sum of the ion fractions of all anions is unity so that, from eqn (15) and (16) : 1-No-NF = - which yields NSio4 in terms of k and k' if No and NF for the melt under consideration are known or can be determined experimentally. When k' = 0 (ie., when MF2 acts merely as a diluent) B = 0 in eqn (17) and the expression for Nsio4 becomes Resubstituting for A from eqn (9) yields (NSiO.Jk'=O = (1 -No-NF)(l - A ) which is identical with the expression derived previously [eqn (6), ref.(l)] for the binary MO + SiOz except for replacement of the term (1 -No) by (1 -No - NF). In the general case for which k' is finite, substitution for A from eqn (10) in eqn (17) yields a quadratic expression in Nsio4 whose solution is No(Nal - B5)2 i- K YZ{2N:( 1 + B5) + K YZ])* - (19) NJNO(1 -B5)+KY(1-B)(1 +B2)] 2KB2[N0(B3 - 1) - K Y( 1 - B)] Nsio, = where Y = I-NO-NF (20) and Z = (1 -B)(l- B2).(21) To complete the treatment it is necessary to express the composition of the melt in terms of the ion fractions of the individual constituents. We have : 8 From eqn (14) and (22) : 1 -B5 1-B7 1 - ~ 9 moles Si02 = Nsio4 - +2A&o, - +3A2Nsio4 - 4- ... 1-B 1-B 1-B Similarly it may be shown (see Appendix) that moles MO= N o + L [ Nsio 1 {-+1--}+-{ 1 B B5 -+l+L}] 1 (24) 1-B 1-A 1-A 1-B 1-AB2 1-AB2 1-B2946 POLYMER THEORY OF SILICATE MELTS and B 2 Eqn (23)-(25), in combination with eqn (19), are the expressions required to yield No and NF in terms of k , k' and the composition of the melt. The resulting equations, though cumbersome, may be handled with the aid of a computer. To compare the predictions of theory and experiment it is necessary to know the activities of both MO and MF2 in the system. It has been shown previously that for binary MO+Si02 melts the value of aMO is given approximately by the Temkin equation : For the binary PbO+PbF, it has been shown l5, 22* 2 3 that the activity of PbO, and hence the activity of PbF2, is also given approximately by the Temkin equation and this is substantiated by the results in the accompanying publi~ation.~~ Hence we may write, with some justification : Substitution for No and NF in eqn (19) and (23)-(25) and combination of the resulting expressions with the relationships aMO = No.(26) a,,, = N : . (27) moles MF2 X M F 2 moles SiO, X s i o z - - - and moles MF, X M F 2 moles MO XMo -- - yields aMO and aMF2 in terms of k, k' and the composition of the melt.The complexity of the resulting relationships precludes the derivation of theoretical activity against composition curves for all except the simplest cases. One of these is the situation in which k' = 0 and it is instructive to compare the predictions of theory with experiment for this simple case. MODEL WITH k' = 0 For this case B = 0 in eqn (23)-(25) inclusive and we have Substituting for Nsio4 from eqn (18) and A from eqn (10) in eqn (30) : NFNO ~ ( ~ - N ~ - N F ) ( N ~ + ~ ( ~ - N O - N F ) ) ' Similarly it may be shown thatC. R. MASSON AND W. F. CALBY 2947 For comparison with experiment, use has been made of recent experimental data 23 for activities of PbO in the system PbO+PbF,+SiO,. The value of k for this system has been determined previously as 0.196 from experimental data on the binary PbO+SiO,.The following procedure was used to derive the theoretical activity against composition relationships for this system for k’ = 0. 1.0 0.8 0.6 0.A 0.2 0 0 0.2 0.4 0.6 08 1.0 XPbO FIG. 1 .-Theoretical curves of activity against mole fraction of PbO in PbO+ PbF2 + Si021melts for various ratiosXpbFz/XSio2 = (a) 00, (6) 10, (c) 4, (d)2, (e) 1, (f) 0.5 and (g) 0. The curves correspond to the case k’ = 0, i.e., to the situation in which fluorosilicate ions are absent, and PbF2 acts solely as a diluent. Broken lines indicate compositions for which Xsi02 > 0.3 ; the theory is not expected to hold accurately in this region. k = 0.196. An arbitrary value of aPbO (0 < apbo < 1) was first chosen for a melt of selected ratio XPbF2/XSiO2.This value was substituted for No in eqn (31) and the value of NF calculated, using k = 0.196. The value of NF thus obtained was substituted, along with the value of No, in eqn (33) to yield xpbF2/xpbO. This fixed the composition of the melt. The procedure was repeated for other values of apbo to yield apbo as a function of x p b o for the selected ratio XpbF2/&02. The entire procedure was then repeated for other selected ratios of &bF2/&02 to yield the theoretical curves shown in fig. 1. The curve for &bF2/&02 = 03, which corresponds to the Temkin equation for the binary MO + MF2, is given by ~~Mo~xMF2,xs*o~ = 00 = X M 0 / ( 2 -&lo) (34) and is readily obtained by substituting (1 -No) for NF and (1 - X&) for XMF2 in eqn (33).The curves in fig. 1 predict that, if PbFz behaves solely as a diluent, the effect of replacing SO2 by PbF2 in the binary PbO + Si02 is to cause a slight lowering of the activity of PbO for melts with x p b o > 0.775. For melts with x p b o < 0.775 the2948 POLYMER THEORY OF SILICATE MELTS theory predicts that, at constant XPbO, apbo will be raised by the substitution of PbF2 for Si02 until it eventually attains its value in the binary PbO +PbF2. At Xp,, = 0.775, no change is predicted in the value of amO at all levels of PbF2. The com- position at this point is readily obtained by substituting for apbo from eqn (34) in 0' 0.4 0.6 0.8 I .O XPM) xpbo FIG. 2.-Comparison of theoretical curves for k' = 0 with experimental data for PbO + PbFz + SiOz melts.The points are interpolated from the data of ref. (23). XmFz/XSiOz = (a) 0.5, (b) 1, (c) 4 and (d) 10. As shown previously,1* the theory outlined above (with k = 0.196) describes activities in binary PbO + Si02 melts when Xsioz < 0.3 and it appears reasonable to expect that the theoretical curves in fig. 1 will also be applicable in this region. The range of applicability of the theory expected on this basis is shown by the bold lines in fig. 1. Broken lines indicate compositions for which XsiOz > 0.3.C. R. MASSON AND W. P. CALEY 2949 In form, the curves in fig. 1 closely resemble the experimental data 23 for the system PbO +PbF2 + SO2. In particular, the theory correctly predicts the point of intersection at XpbO = 0.775.A detailed comparison of theory and experiment for melts with various levels of XpbFz/XSiO2 is shown in fig. 2. The theory provides a good representation of the data for melts with XpbO 2 0.8 at all levels of XpbFz/&02. For these melts XsIoz does not exceed 0.14. Similarly, good agreement between theory and experiment is observed for melts with Xpb0 > 0.5 when XpbF2/&02 = 10. For these melts, Xsio2 is always < 0.05. For melts with higher silica contents, however, the theoretical curves for aPbo are lower than the experimental values and the discrepancy becomes greater as Xsio2 increases. These differences are well beyond experimental error or the uncertainties involved in interpolation of the data. Thus the simple theory with k’ = 0 does not adequately describe the effect of PbF2 on the activity of PbO in this system.For a more quantitative representation of the data a finite value of k’ is required. The effect of introducing a finite value of k’ will be to raise the theoretical values of aPbo due to the displacement of 0- groups by F groups on the silicate chains, with liberation of free 02- ions according to reaction (7). The value of k’ required to provide a more detailed test of the theory cannot be estimated from measurements of aPbO alone, due to lack of knowledge of the value of NF required in the computations. For this purpose, values of apbFz must be deter- mined. The experimental determination of apbF2 and a more detailed discussion of the constitution of PbO +PbF2 + SOz melts form the subject of a separate corn- munication.We thank Miss S. Taylor and Mr. J. Uher for assistance with the computations. C. R. Masson, Proc. Roy. SOC. A , 1965,287, 201. C. R. Masson, J. Amer. Ceram. SOC., 1968,51, 134. A. E. Grau and C. R. Masson, Canad. Metal. Quart., 1976, 15, 367. M. Temkin, Zhur. fiz. Khim., 1946, 20, 105. G. H. Herty, Jr., F. A. Hartgen, G. L. Frear and M. B. Royer, US. Bur. Mines Rep. Invest., 1934, 3232. F. Hartmann, Stahl u. Eisen, 1934,54,564 ; 1938, 58,1029 ; Arch. Eisenhiittenw., 1938,10,45. ’ L. Schwerin, Metals and Alloys, 1934,5,118. P. Kozakevitch, Rev. Metal., 1954, 51, 569. P. M. Bills, J. Iron and Steel Inst., 1963, 201, 133. lo D. Kumar, R. G. Ward and D. J. Williams, Disc. Faraday Soc., 1961, 32, 147. l 1 A. Mitchell, Trans.Faraday Soc., 1967, 63, 1408. l2 J. R. Michel and A. Mitchell, Canad. Metal. Quart., 1975, 14, 153. l3 I. D. Sommerville and D. A. R. Kay, Metal. Trans., 1971, 2, 1727. l4 G. J. W. Kor, Metal. Trans., 1977, SB, 107. l 6 M. W. Davies, Chemical Metallurgy of Iron and Steel, ed. B. B. Argent and M. W. Davies l7 B. J. Keene and K. C. Mills, N.P.L. Report Chem., no. 60, 1976. l9 J. O’M. Bockris and D. C. Lowe, Proc. Roy. Soc., 1954,226,423. 2o C. R. Masson, Chemical Metallurgy of Iron and Steel, ed. B. B. Argent and M. W. Davies (Iron and Steel Inst., London, 1973), p. 3 ; J. Iron and Steel Inst., 1972, 210, 89, 369, C. R. Masson, Glass ’77, Proc. XIth Int. Congr. Glass. Prague, ed. J. Gotz (CVTS-Durn Techniky Praha, 1977), p. 1 ; J. Non-Cryst. Solids, 1977, 25, 1.22 M. G. Frohberg, M. L. Kapoor and G. M. Merohtra, Special Electrochemistry (Proc. Inter- national Symposium, Kieve, U.S.S.R., 1972), vol. 2, p. 175. 23 A. E. Grau, W. F. Caley and C. R. Masson, Canad. Metal. Quart., 1976, 15, 267. 24 W. F. Caley and C. R. Masson, J.C.S. Faraday I, 1978,74,2952. P. Perrot and M. F. El Ghandour, Silic. ind., 1976,41, 407. (Iron and Steel Inst., London, 1973), p. 43. K. C. Mills, N.P.L. Report Chem., no. 65, 1977. (PAPER 8/374)2950 POLYMER THEORY OF SILICATE MELTS APPENDIX The following summations were used in the derivations : co 1 -- (0 < x < 1) I-x c xn-I - 11'1 co 1 n m=O For the derivation of eqn (24) and (25) we require to evaluate the summations 2n+ 2 2 n + 2 f c mfvsin03n+l-mFm and n = l m=O n = l m = O 2 c nNsin03,,+1-nlFm* Introducing the value of NSj,103n+l+n,Fm from eqn (13), the first of these is given by From eqn (A4) and (A5) Expanding the summation and using eqn (Al) and (A2) we obtain Similarly, the second summation is given by which, withIthe aid of eqn (A3) yields Again, expanding the summation and using eqn (Al) and (A2) we obtain CA 10) which is the required result. From eqn (15) and (A10) : B5 1 Nsio4 - - -[ { +l}--{-+I}]. 1-AB2 1-AB2 ( A l l ) 1-B 1-A 1-A We now use these relationships to derive the expressions for the number of moles of MO and MF2 in eqn (24) and (25). We first note that all the Si02 in the melt isC. R . MASSON AND W. F . CALEY 295 1 associated with the silicate and fluorosilicate anions. This yields eqn (22) and (23) directly. For polyions with O/Si > 2, the excess oxygen must be ascribed to MO. For polyions with O/Si < 2 the oxygen deficit required for SiOz must be derived from MO and this is reflected as negative terms in the summations. Thus we have A 6 From eqn (A7), (All) and (A13) we obtain moles MO= B which is the result in the text. write As the MF2 is associated solely with the fluoride and fluorosilicate anions we may Substituting for from eqn (A7) in eqn (A15) we obtain -- - B5 { +2+---!-}] (25) B (1-A)(l-B) 1-AB2 1-AB2 1-B which completes the derivations.