Vol48/04/2005def 561 ANNALS OF GEOPHYSICS, VOL. 48, N. 4/5, August/October 2005 Key words silicate melts – structure – entropy – un- mixing 1. Introduction In Bernal’s words (Bernal, 1959), a liquid is an «homogeneous, coherent and essentially irreg- ular assemblage of molecules containing no crys- talline regions or holes large enough to admit an- other molecule». This acceptance of a liquid structure is particularly in line with the Random Network Model (RNM) of Zachariasen (1932, 1933) and Warren (1933) for silica melts and glasses: silicate tetrahedrals linked through cor- ners without any long-range periodicity. Already in the 1960s, Evans and King (1966) had physi- cally built up an RNM composed of Corning Glassworks tetrapods linked by bent springs, the angles of the bends having a Montecarlo-generat- ed Gaussian distribution centered at 163°. The measured radial distribution of atoms around the silicon and oxygen reference atoms of the physi- cal model appeared to be in good agreement with existing neutron and X-ray scattering data (War- ren et al., 1936; Warren, 1937; Carraro, 1964). More than half a century of experimental re- search essentially confirmed the first illuminating views of Zachariasen, Warren and Bernal, and nowadays we may basically state the following: – The structure of silicate melts and glasses has elements of long-range randomness coex- isting with short- and medium-range order (Gaskell et al., 1991). – The short-range order is dominated by the structure of the SiO4 rigid unit (Greaves et al., 1981), although the distance beyond which the radial distribution function becomes indistin- guishable from the 4π r 2ρ 0 function, represent- ing homogeneous distribution of electron den- sity, shrinks progressively with the introduction Chemical interactions and configurational disorder in silicate melts Giulio Ottonello Laboratorio di Geochimica, Dipartimento per lo Studio del Territorio e delle sue Risorse (DipTeRis), Università degli Studi di Genova, Italy Abstract The Thermodynamics of quasi-chemical and polymeric models are briefly reviewed. It is shown that the two classes are mutually consistent, and that opportune conversion of the existing quasi-chemical parameterization of binary interactions in MO-SiO2 joins to polymeric models may be afforded without substantial loss of preci- sion. It is then shown that polymeric models are extremely useful in deciphering the structural and reactive prop- erties of silicate melts and glasses. They not only allow the Lux-Flood character of the dissolved oxides to be established, but also discriminate subordinate strain energy contributions to the Gibbs free energy of mixing from the dominant chemical interaction terms. This discrimination means that important information on the short-, medium- and long-range periodicity of this class of substances can be retrieved from thermodynamic analysis. Lastly, it is suggested that an important step forward in deciphering the complex topology of the inho- mogeneity ranges observed at high SiO2 content can be performed by applying SCMF theory and, particularly, Matsen-Schick spectral analysis, hitherto applied only to rubberlike materials. Mailing address: Dr. Giulio Ottonello, Laboratorio di Geochimica, Dipartimento per lo Studio del Territorio e delle sue Risorse (DipTeRis), Università degli Studi di Ge- nova, Corso Europa 26, 16132 Genova, Italy; e-mail: giot- to@dipteris.unige.it 562 Giulio Ottonello of alkalis («network modifiers»; see later) in the system (Waseda and Suito, 1977), which is in- dicative of significant break-up of the random network. – There are no systematic structural differ- ences between silicate melts and glasses of identical composition, and the effects of ther- mal expansion are virtually negligible on both Table I. Structural data for binary silicate melts and glasses (from Brown et al., 1995). Interionic distance rij(A) approximated to ± 0.01 A; coordination number Nij approximated to ± 0.3 atoms; root mean square atomic dis- placement ∆rij1/2 approximated to ± 0.005. Join T(°C) Pair X-ray (molten state) X-ray (glassy state) Reference rij (A) Nij (atoms) ∆rij1/2 rij (A) Nij (atoms) ∆rij1/2 Pure SiO2 1750 Si-O 1.62 3.8 0.096 1.62 3.9 0.087 (1) O-O 2.65 5.6 0.124 2.65 5.5 0.102 (1) Si-Si 3.12 3.9 0.187 3.11 3.9 0.141 (1) 0.33Li2O- 1150 Si-O 1.61 3.8 0.117 1.62 3.7 0.090 (2) -0.67SiO2 Li-O 2.08 4.1 0.131 2.07 3.8 0.095 (2) O-O 2.66 5.5 0.195 2.65 5.6 0.101 (2) Si-Si 3.13 3.8 0.260 3.13 3.8 0.143 (2) 0.33Na2O- 1000 Si-O 1.62 4.1 0.095 1.62 4.0 0.086 (2) -0.67SiO2 Na-O 2.36 5.9 0.151 2.36 5.8 0.101 (2) O-O 2.66 5.6 0.202 2.65 5.2 0.112 (2) Si-Si 3.20 3.8 0.279 3.21 3.6 0.146 (2) 0.33K2O- 1100 Si-O 1.62 3.9 0.124 1.62 3.8 0.086 (2) -0.67SiO2 K-O 2.66 13.0* 0.182* 2.65 13.2* 0.120* (2) O-O 2.66 2.65 (2) Si-Si 3.23 3.7 0.257 3.23 3.5 0.154 (2) 0.50MgO- 1700 Si-O 1.62 3.9 0.109 1.63 3.7 0.096 (3) -0.50SiO2 Mg-O 2.12 4.3 0.151 2.14 4.6 0.108 (3) O-O 2.65 5.4 0.215 2.65 5.7 0.151 (3) Si-Si 3.16 3.3 0.282 3.15 3.4 0.213 (3) 0.50CaO- 1600 Si-O 1.61 3.9 0.127 1.63 3.8 0.109 (3) -0.50SiO2 Ca-O 2.35 5.9 0.171 2.43 5.9 0.125 (3) O-O 2.67 5.2 0.206 2.66 5.5 0.183 (3) Si-Si 3.20 3.1 0.264 3.23 3.4 0.199 (3) 0.66FeO- 1400 Si-O 1.62 3.9 0.147 - - - (4) -0.33SiO2 # Fe-O 2.05 3.9 0.214 - - - (4) O-O - - - - - - (4) Si-Si 3.27 3.1 0.302 - - - (4) (1) Waseda and Toguri (1990); (2) Waseda and Suito (1977); (3) Waseda and Toguri (1977); (4) Waseda and Toguri (1978); * pair correlations for K-O and O-O overlap, so that coordination numbers and peak widths could not be determined independently; # O-O pair correlation overlaps with Fe-Fe correlations, thus O-O distance could not be determined independently. 563 Chemical interactions and configurational disorder in silicate melts mean structure and single interionic distances (see, for this purpose, the review article of Brown et al., 1995, and references therein). Table I lists some important X-ray scattering data concerning the structure of binary silicate melts collected by Brown et al. (1995) and es- sentially in line with the above three points. Table II, based on the extensive compilation by Brown et al. (1995), lists concisely the mean coordination numbers experimentally observed for the various cations in melts and glasses. Al- though a generalized increase in the coordina- tion number of the central cation is observed with the increase in the Lux-Flood basicity of the corresponding oxide, the scatter in CN ob- served for basic components is a clear indica- tion that network modifiers «adapt» themselves to the geometry established by the much stronger (and essentially covalent) bond of net- work formers, and not vice versa. In this light, it is difficult to agree with the hypothesis of Gaskell et al. (1991) that compositionally simi- lar pyroxene or pyroxenoid glasses may be rep- resented as parallel random-fractal sheets of in- terconnected CaO6 octahedra interleaved by Si atoms (occupying intersitial tetrahedral sites linked in infinite chains). More probably, in the present author’s opinion, the structure of sili- cate melts or glasses locally resemble that of all-Si zeolites, in which «cages» of variable co- ordination number are determined by the rela- tive symmetry-determining arrangements of SiO4 monomeric units. Despite this topological controversy, it is quite evident that conformational disorder does exist in silicate melts and glasses. Exactly how thermodynamics can account for this confor- mational disorder may be purely phenomeno- logical (as in subregular models and polynomi- al expansions) or may be based on more sophis- ticated physical models (as in the Gaussian Random Walk method; see later). In any case, to be satisfactory, the thermodynamic model must not only reproduce the expected reactive properties of the substance, but also the inho- mogeneity ranges often observed at high SiO2 content. We discuss below the informational content arising from sound thermodynamic treatment of silicate melt energetics, and propose some conceptual guidelines to be followed in the near future, in order to decipher the complex fea- tures of the inhomogeneity ranges observed at high SiO2 content along simple MO-SiO2 bina- ries, and also in chemically complex systems. 2. Configurational disorder in quasi-chemical models Guggenheim’s quasi-chemical model (Gug- genheim, 1935; Fowler and Guggenheim, 1939) assumes that, in a binary system with compo- nents 1 and 2, «particles» 1 and 2 mix substitu- tionally in a quasi-lattice obeying short-range or- dering dictated by the equilibrium Table II. P = 1 bar mean M-O coordination number of cations in silicate melts and glasses. Listed figures are largely based on extensive compilation of Brown et al. (1995). Allocation among acidic, amphotheric and basic oxides after Ottonello and Moretti (2004) and this work (#). M Acidic Amphotheric Basic Notes Si 4 Ge# 4 Al 4 In Na-Al. .Si-O melts. Al 5.5-6 In pure Al2O3 melt. FeIII 4 Ti 4-5-6 Zr 6-8 Ni 4-5-6 Li# 4 Na 5-6 K 7-8 Rb# 8 Cs 8 Ca 4-6-7-8 Mg 4-5-6 FeII 4-5-6 Mn 4-5-6 Pb 8 564 Giulio Ottonello (2.1) The molar enthalpy change associated with equilibrium (2.1) is expressed in terms of pair bond energies ε ij (2.2) where N is the total number of atoms and Z is the coordination number. The non-configurational part of the molar entropy is also given in terms of bond contribu- tions σ ij (non-configurational entropy of the ij pair bond) (2.3) If Xij denotes the molar fraction of the ij particle with respect to the total of the particles formed in the system, since each atom i is bonded to Zj ( ) . NZ 2 2 12 11 22= - -~ h h h ( ) NZ 2 2 12 11 22= - -~ f f f [ ] [ ] [ ] .1 1 2 2 2 1 2,- + - - neighbors, it follows from mass balance that (2.4) The enthalpy of mixing then becomes (2.5) the non-configurational (excess) entropy of mixing is (2.6) and the configurational (ideal + excess) entropy of mixing is (2.7) ( )ln ln ln ln ln S R X X X X RZ X X X X X X X X X X 2 2 , conf 1 1 2 2 11 1 2 11 22 2 2 22 12 1 2 12 mixing $ $ = - + - + +c m S X 2, non conf 12 mixing = h- H X 2 12 mixing = ~ ( ) ( ) . X X X X X 2 2 12 1 11 22= - = - Fig. 1a-d. Integral enthalpy of mixing and integral entropy of mixing of quasi-chemical model for different values of interaction parameter ω along MO-Si0.5O2 join ((a) and (b), respectively). Below: if usual molar nota- tion (i.e., component SiO2 instead of Si0.5O) is adopted and parameter ω is kept unvaried, symmetricity about in- termediate composition is lost ((c) and (d) respectively). a b c d 565 Chemical interactions and configurational disorder in silicate melts because when the mixture is completely ran- dom, X11 = X12, X22 = X22 and X12 = X1X2. The equilibrium concentrations of the vari- ous pairs obey the Gibbs free energy minimiza- tion principle (2.8) This, through opportune expansions in terms of eqs. ((2.5) to (2.7)), gives (2.9) The name «quasi-chemical» assigned to the model stems from the fact that the term on the right in eq. (2.9) reduces to a constant, for con- stant values of ω and η, thus resembling the no- tation of a chemical reaction among ideal [1–1], [2–2] reactants and [1–2] product. Equation (2.9) yields (Pelton and Blander, 1986) (2.10) where (2.11) The model is resolved first in terms of equilib- rium distribution of particles (eqs. (2.10), (2.11) and (2.4)) and then in terms of enthalpy and en- tropy (eq. (2.5) and eqs. ((2.6) to (2.7)), respec- tively). The computed enthalpic and entropic terms have a typical V- and M-shaped conformation, the more pronounced as the value assigned to ω becomes more negative. The quasi-chemical en- thalpy of mixing and entropy of mixing curves computed with Z = 2, η = 0 and various ω are shown for comparison in fig. 1a,b, respectively, for a MO-Si0.5O system. When ω and η are set at zero the mixture is obviously ideal and, when η = 0 and X12 = 2X1X2, the mixture is strictly reg- ular. In all cases, mixtures are always symmet- ric in the compositional space of interest. ( ) .expX X ZRT T 1 4 2 11 2 2 1 = + - -p ~ h c m; E( 2 X X X 2 1 212 1 2= + p .exp X X X T ZRT 4 2 2211 12 2 = - -~ h^ h ; E T( ) . X G X H S 0 mixing mixing mixing 12 122 2 2 2 = = - 2.1. Modified quasi-chemical approach The symmetricity about intermediate com- position was long regarded as a severe limita- tion to the application of the quasi-chemical model to silicate melts and slags which invari- antly exhibit maximum ordering about the com- position XMO = 2/3 in the MO-SiO2 chemical space. In order to shift the maximum ordering condition depicted by the quasi-chemical mod- el to the condition actually observed in MO- SiO2 melts, Pelton and Blander (1986) pro- posed substituting true molar fractions X1, X2 with «equivalent fractions» Y1, Y2 so that (2.12) (2.13) Coefficients b1 and b2 are chosen in such a way as to have b1/(b1+b2)=1/3 and b2/(b1+b2)=2/3. Clearly, when X1=2/3 and X2=1/3, we have Y1=Y2=0.5. Through this substitution it follows that, in- stead of eq. (2.4), the following equation holds (Pelton and Blander, 1986): (2.14) The molar enthalpy of mixing is now (Pelton and Blander, 1986) (2.15) the non-configurational (excess) entropy of mix- ing is (2.16) and the configurational (ideal + excess) entropy of mixing is (2.17) ( ) ( ) . ln ln ln ln ln S R X X X X RZ X Y X X Y X X Y Y X b X b X 2 2 ,mixing conf 1 1 2 2 11 1 2 11 22 2 2 22 12 1 2 12 1 1 2 2 $ $ $ $ = - + - + + + c m ( )S X b X b X 2,mixing non conf 12 1 1 2 2= + h- ( )H X b X b X 2mixing 12 1 1 2 2= + ~ ( ) ( ) . X Y YX X 2 12 1 11 2 22= - = - .Y b X b X b X 2 1 1 2 2 2 2= + Y b X b X b X 2 1 1 1 1 1 2 = + 566 Giulio Ottonello Although whatever b1, b2 couple exists, so that b1/b2 = 1/2 results in Y1/Y2 = 1/2, application of the additional condition that configurational en- tropy must attain zero when ω = − ∞ yields b1= 0.6887, b2 = 1.3774 for binary MO-SiO2 melts with maximum ordering about XMO = 2/3 (cf. Pelton and Blander, 1986, for details). Lastly, in order to assign more computation- al elasticity to the model, parameters ω and η are expanded as polynomial functions of equiv- alent fractions Y1, Y2, i.e. (2.18) (2.19) Actually, the asymmetricity of MO-SiO2 joins is due to the difference in the amounts of oxygen in the two end-member components. If we adopt Si0.5O instead of SiO2 as end-member, maxi- mum ordering will be observed at XMO = 0.5, and no readjustment in terms of equivalent fractions will be necessary. Obviously, if we wish to rep- Y .Y Y n n 0 1 2 2 2 2 2f= + + +h h h h h YY Y n n 0 1 2 2 2 2 2f= + + +~ ~ ~ ~ ~ Table IV. Non-configurational excess entropy parameters of modified-quasi-chemical model for binary silicate melts (after Pelton et al., 1995). AO BO b1 b2 η 0 η 1 η 2 η 3 η 4 η 5 η 6 η 7 AlO1.5 SiO2 2.0661 2.7548 0. 0. 0. 0. 0. 0. 0. 0. BO1.5 SiO2 2.0661 2.7548 0. −25.104 0. 0. 0. 0. 0. 0. CaO SiO2 1.3774 2.7548 −19.456 0. 0. 0. 0. 0. 0. 0. KO0.5 SiO2 0.6887 2.7548 −58.576 0. 0. 0. 0. 0. 0. 0. MgO SiO2 1.3774 2.7548 0. −37.656 0. 0. 0. 0. 0. 125.52 MnO SiO2 1.3774 2.7548 −20.92 0. 0. 0. 0. 0. 0. 62.76 NaO0.5 SiO2 0.6887 2.7548 −43.932 0. 0. 0. 0. 0. 0. −20.92 NiO SiO2 1.3774 2.7548 0. 0. 0. 0. 0. 0. 0. 125.52 PbO SiO2 1.3774 2.7548 0. 0. 0. 0. 0. 0. 0. 12.97 SiO2 TiO2 2.7548 2.7548 0. 0. 0. 0. 0. 0. 0. 0. ZnO SiO2 1.3774 2.7548 −33.472 0. 58.576 0. 0. 0. 0. 0. ZrO2 SiO2 2.7548 2.7548 0. 0. 0. 0. 0. 0. 0. 0. Table III. Chemical interaction parameters of modified-quasi-chemical model for binary silicate melts (after Pelton et al., 1995). AO BO b1 b2 ω 0 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 AlO1.5 SiO2 2.0661 2.7548 4800. 0. 0. 100784. 0. −142068. 0. 78571. BO1.5 SiO2 2.0661 2.7548 16958. −32531. 0. 0. 0. 0. 0. 0. CaO SiO2 1.3774 2.7548 −158218. −37932. 0. 0. 0. −90148. 0. 439893. KO0.5 SiO2 0.6887 2.7548 −409986. 0. 0. 0. 0. 0. −1647688. 1593677. MgO SiO2 1.3774 2.7548 −86090. −4874. 0. 0. 0. 0. 0. 328109. MnO SiO2 1.3774 2.7548 −79956. 0. 0. 0. 0. 0. 0. 228819. NaO0.5 SiO2 0.6887 2.7548 −114344. −381595. 0. 0. 0. 0. 0. 123010. NiO SiO2 1.3774 2.7548 29169. 0. 0. 0. 0. 0. 0. 509783. PbO SiO2 1.3774 2.7548 −25430. 0. 0. 0. 0. 0. −245806. 310959. SiO2 TiO2 2.7548 2.7548 28847. 52091. 0. −44484. 0. 0. 0. 0. ZnO SiO2 1.3774 2.7548 −124741. 129292. 15989. 0. 0. 0. 0. 98990. ZrO2 SiO2 2.7548 2.7548 4184. 40585. 0. 0. 0. 0. 0. −11715. 567 Chemical interactions and configurational disorder in silicate melts resent the chemical space of interest in terms of MO-SiO2 components, then it is sufficient to consider that two moles of Si0.5O correspond to one mole of SiO2, the resulting molar fractions XMO, XSiO2 are translated along the abscissa, and an apparent asymmetricity arises, as depicted in fig. 1c,d. Tables III and IV list the extensive modi- fied-quasi-chemical parameterization of binary silicate melts of Pelton et al. (1995). The listed parameters allow precise conformation of phase boundaries along the binary joins and are also consistent with observed unmixing phe- nomena. 3. Configurational disorder in polymer models In polymeric models for silicate melts, it is postulated that, at each composition, for given values of P and T, the melt is characterized by an equilibrium distribution of several ion species of oxygen, metal cations and ionic polymers of monomeric units SiO44–. The charge balance of a polymerization re- action involving SiO44– monomers may be for- mally described by a homogeneous reaction in- volving three forms of oxygen: singly bonded O-, doubly bonded (or «bridging oxygen») O0, and free oxygen O2− (Fincham and Richardson, 1954) (3.1) In fact, eq. (3.1) is similar to a reaction between monomers (3.2) Polymer chemistry shows that, the larger the various polymers become, the more their reac- tivity is independent of the length of the poly- mer chains. This fact, known as «the principle of equal reactivity of co-condensing functional groups», has been verified in fused polyphos- phate systems (which, for several properties, may be considered as analogous to silicate melts; cf. Fraser, 1977) with polymeric chains longer than 3PO44– units (Meadowcroft and .SiO Si O OSiO 24 4 2 7 6 4 4 , ++ - - -- .O O O2 melt meltmelt 0 2 , +- - Richardson, 1965; Cripps-Clark et al., 1974). Assuming this principle to be valid, equilibrium constant K18 (3.3) (in which the terms in brackets represent the number of moles per unit mole of melt) is al- ways representative of the polymerization process, independent of the effective length of the polymer chains. 3.1. Toop-Samis model Toop and Samis (1962a,b) showed that, in a MO-SiO2 binary melt, in which MO is the ox- ide of a basic cation completely dissociated in the melt, the total number of bonds per mole of melt is given by (3.4) where NSiO 2 are the moles of SiO2 in the MO- SiO2 melt. The number of bridging oxygens is thus (3.5) Mass balance gives the number of moles of free oxygen per mole of melt (3.6) where (1 − NSiO2) are the moles of basic oxide. Equations ((3.3) to (3.6)) yield (3.7) (O–) is thus given by the quadratic equation (3.8) which may be solved for discrete values of K18 and NSiO2. ( ) ( ) ( ) ( ) ( ) K N N N O O4 1 2 2 8 1 0 SiO SiO SiO 2 18 2 2 2 - + + + + - = - - ( ) [ ( )] [ ( )] .K N N 4 O 4 O 2 2 OSiO SiO 18 2 2 2= - - - - - - ( ) ( ) .NO O 1 2SiO 2 2 = - -- - ^ h ( ) ( ) . N O O 2 4 SiO0 2= - - ( ) ( ) NO O2 4 SiO 0 2 2 =- ( ) ( ) ( ) K O O O 18 2 0 2 = - - 568 Giulio Ottonello The Gibbs free energy change involved in eq. (3.1) is (3.9) Since two moles of O− produce one mole of O0 and one of O2−, the Gibbs free energy of mixing per mole of silicate melt is given by (3.10) 3.2. Extended Polymeric Approach (EPA) Although the original Toop-Samis model is a powerful tool in deciphering the complex rela- tions existing between chemical interaction and configurational disorder, there are two minor points that must be quantified, i.e. assessment of the low polymerization limit, and extension to the multicomponent field. The usual expression of the thermodynamic activity of a component in mixture (Ottonello, 2001) yields (3.11) As the mean number of monomers in the poly- mer chain is defined as (3.12) adopting Temkin model activities of the fused salts, it is quite evident that, along any binary join, we may pose (3.13) Although the domain of eq. (3.13) spans the en- tire compositional range, it is obvious that can never be lower than one (i.e., a monomer) and, consequently, the limit of maximum de- polymerisation defines a limiting activity rep- resented by Sior ( ) ( ) ( ) . a O O SiO ,MO binary 2 2 2 Si = - o - - r ( )SiO anions2Si =or / ( ) ( ) .exp ln a RT RT K X X G 2 O 1 MO MO mixing 18 MO 2 2 = + - - > H ( ) .lnG RT K 2 O mixing 18= -∆ - .lnG RT K1818 0 = -∆ (3.14) Thus, the molar abundances of the various oxy- gen species cannot be expected to be those of the original Toop-Samis model, especially when highly basic oxides are present in the liq- uid phase. In solving thermodynamic activity on a Temkin model basis, Toop and Samis (1962a,b) observed that the mean extension of the polymer chains is univocally defined by a «polymerzsation path» depicted in terms of mean numbers of silicon atoms per polymer unit versus the stoichio- metric ratio (O–)/[(O–)+(O–)+SiIV] (cf. figs. 2 and 3 in Toop and Samis, 1962a). However, their further assumption – that a single polymeriza- tion path of general validity in the ternary sys- tem CaO-FeO-SiO2 may be proposed on the ba- sis of viscosity data – cannot be shared, because a different reaction constant, K18, pertains to each MO-SiO2 system and, as the activity of the basic dissolved oxide MO is implicitly defined by the partial derivative of the Gibbs free ener- gy of mixing at any point of the compositional space of interest, we have (Ottonello, 2001) (3.15) (3.16) (3.17) (3.18) (3.19) (3.20)( )Si SiO2IV = ( ) ( ) O X O 2MO 2 = - - - ( ) ( ) O X O 2 4 4 0 MO= - - - ( ) ( ) ( ) X X K K X X 2 4 4 2 8 2 32 4 1 MO MO P MO MO 2 18 2 2 1 - + - + - - - - -( )O = 6 @ ( ) ln X G RT K X O 2 1 MO mixing MO 182 2 2 2 = - ( ) ( ) exp RT G G anions X X O 1mixing MO MO mixing 2 2 2 = + - - ; E * 4 / ( )N anionsSiOSi 2=or / ( ) ] .SiO#[( )SiO!= ( ) ( ) ( ) a O SiO O , minMO binary 2 2 2 2 2+- - 569 Chemical interactions and configurational disorder in silicate melts where the terms in brackets denote the number of moles and XMO is the molar fraction of a generic basic oxide MO in a MO-SiO2 binary join. Distinct «polymerization paths» relating the mean number of silicon atoms in the polymeric units ( ) to the relative proportion of singly bonded oxygen in the unit (O–)/ [(O–) +(O0) +SiIV] may be calculated for the two limiting binaries CaO-SiO2 and FeO-SiO2 on the basis of the system of eqs. ((3.15) to (3.20)). These paths differ substantially from the general path pro- posed over 40 years ago by Toop and Samis (1962a,b) and, more importantly, they differ greatly from each other. The apparent complexity introduced by rec- ognizing that each oxide imposes its own im- print on the extension of polymeric units turns Sior out to be a powerful tool in deciphering the re- active properties of chemically complex melts. In fact, chemical interactions within the anion matrix of an n-component system are complete- ly fixed by the interaction properties observed along the n-1 limiting binaries. As discussed elsewhere (Moretti and Ottonello, 2003), the in- trinsic thermodynamic significance of this evi- dence is apparent in the Flood-Grjotheim ther- modynamic cycle (Flood and Grjotheim, 1952). If KP, A-F, KP, B-F are the polymerization con- stants valid along the limiting A-F, B-F bina- ries, the polymerization constant of the extend- ed Toop-Samis model turns out to be (3.21) The above equation states that, in a composi- tionally complex melt, network modifier cations A, B of charge ν+A, ν+B interact with free oxygen O2– (and hence with the polymeric units built up by network former F), with a simple propor- tionality described in terms of equivalent frac- tions (Flood and Grjotheim, 1952; Ottonello, 2001; Moretti and Ottonello, 2003) (3.22) Application of the Flood-Grjotheim thermody- namic cycle to the CaO-FeO-SiO2 system satis- factorily reproduces the bulge in the thermody- namic activity of FeO experimentally observed long ago by Elliot (1955) (fig. 2). No high-order terms are necessary to reproduce the observed complex thermodynamic behavior, except for a topologically negligible interaction in the cation sublattice. 3.3. Hybrid Polymeric Model In the Hybrid Polymeric Model (HPM) (Ot- tonello, 2001), the Gibbs free energy of mixing in binary melts is first subdivided into a (domi- nant) chemical Toop-Samis interaction term plus a (subordinate) elastic Hookian-like strain energy contribution term, as is observed in block copolymers (see later) = .N n n n A A A A A B Bf+o o o + + + +ol f+ln K+N+ln K+ln K N , ,P A P A F B P B FO= - -+ +ol l Fig. 2. FeO activity surface at 1600°C for liquid FeO-SiO2-CaO ternary system. Graphic representa- tion obtained by inverse-distance contouring of 9900 activity values generated by extended polymeric model, adopting, as limiting Kp, 0.21 for FeO-SiO2 join and 0.002 for CaO-SiO2 join, respectively. As- sumed cationic interaction WCa-Fe = − 33 kJ/mol. Strain energy contributions included (from Ottonel- lo, 2001, with modifications). 570 Giulio Ottonello (3.23) In eq. (3.23), is the number of monomeric units of statistical length a extended to a dis- tance x, and the remaining symbols assume the usual thermodynamic significance. The elastic strain energy depends on bending term x/a (which basically represents the effect of cova- lent bonding on the relative arrangement of monomers in the polymer chain). This term is arbitrarily expanded into a polynomial of type (3.24) Strain coefficients χ1…χn depict linear or sec- ond-order T dependence. Since the polymeriza- tion path along any join is defined in terms of versus NSiO2, the strain energy calculated along a given binary at various T conditions is intimately related to the extent of polymeriza- tion along any particular compositional path (i.e., the mean number of silicon atoms in the chain, ). The existence of a maximum de- polymerization limit and the approximate na- ture of the entropy terms depicted by the adopt- ed parameterization of bending angles means that the problem cannot be solved exactly (more precise definition of entropic intermedi- ate ordering effects on the bulk stability of the phase would require application of the self-con- sistent mean field theory – see later). To overcome these complexities, Ottonello and Moretti (2004) modified eq. (3.23) as fol- lows: (3.25) In eq. (3.25), S18 is the entropic contribution embodied in the Toop-Samis formulation of chemical interaction, and η represents the strain contributions arising from the conformational arrangements of the various polyanions in the anion matrix. This term is again arbitrarily ex- ( ) .G H T S T 2 O mixing 18 18= - - h∆ ∆ ∆ - ^ h Sior Sior f+N$|+N$|+N$|+N$|= a x SiO SiO SiO SiO1 2 2 2 3 3 4 4 2 2 2 Sior ( ) lnG G G RT K RT a x 2 O 2 3 mixing chemical strain Si 18 2 = + = + + o ∆ ∆ ∆ - r b l panded into a polynomial on NSiO2, the coeffi- cients of which are inversely dependent on T (3.26) (3.27) The new strain energy term −Tη is more akin to the modified-quasi-chemical formulation of the non-configurational entropy of mixing (Pelton and Blander, 1986) with respect to the original formulation (Ottonello, 2001) and allows better comparison of the two model energetics. Obvi- ously, eq. (3.27) implies that, here, strain energy is not purely entropic, but embodies enthalpic and entropic sublattice interactions within the anion matrix which are not explained by the original Toop-Samis formulation. This amount of energy is far less than that arising from chem- ical interaction, and important only in quantify- ing the observed unmixing phenomena at high SiO2 content in the mixture (Ottonello, 2001). Tables V and VI list the results of non-linear minimization calculations carried out to bring the modified quasi-chemical parameterization of Pelton et al. (1995) (tables III and IV) for binary MO-SiO2 melts to an HPM formulation. The polymerization constants were first calculated for a single join at various T conditions of inter- est in the specified T range, and then regressed on an Arrhenian function of absolute T. Stemming from the obtained regression coefficients, there is no doubt that the adopted functional has precise thermodynamic significance which results in ex- pansion (3.25) of the previous hybrid model. The first striking evidence that arises from HPM treatment of interactions in MO-SiO2 melts is the clear distinction between basic, am- photeric and acidic oxides. Basic oxides, which are essentially network modifiers in an MO-SiO2 liquid, exhibit purely enthalpic contributions to chemical interactions in mixture, whereas acidic oxides (network formers) give rise to athermal behavior when mixed among themselves; and, clearly, amphoteric oxides exhibit both entropic and enthalpic contributions to chemical interac- tions in the melt phase. If we now analyse the standard state en- thalpic contribution to chemical interactions in MO-SiO2 melts we note that they are simply re- .T, ,1 0 1 1 1= +| | | N$|+N$|+N$|+|= SiO SiO SiO1 2 3 2 4 3 2 2 2 h 571 Chemical interactions and configurational disorder in silicate melts Table V. Chemical interaction of HPM (from Ottonello and Moretti, 2004). Join lnKp = A/T + B R2 T range Notes A B (°C) K2O-SiO2 −31 708 0 - 1127-1527 (1) −29 540 0 - 1223 (2) −36967 0 0.9922 1500-1800 (3) Na2O-SiO2 −23336 0 0.9995 1000-1800 (3) CaO-SiO2 −15372 0 0.9958 1000-2000 (4) −14807 0 0.9943 1000-2000 (2) MgO-SiO2 −9809.5 0 0.9955 1400-2000 (3) ZnO-SiO2 −6460.1 0 - 1400-2000 (3) MnO-SiO2 −6183.8 0 0.9596 1000-2000 (3) −5649.1 0 - 1600 (1) PbO-SiO2 −5330.0 0 0.5825 1000-1800 (3) −4098.1 0 - 1273 (5) FeO-SiO2 −3600.0 0 0.9973 1000-2000 (4) Fe2O3-SiO2 7569.5 −7.2752 0.9350 1000-2000 (5) TiO2-SiO2 4667.3 −3.2092 0.9107 1500-1900 (3) ZrO2-SiOv 2685.9 −6.382 0.9658 1400-2000 (3) NiO-SiO2 1507.7 −1.7772 0.9825 1500-2000 (3) Al2O3-SiO2 0 −1.4059 - 1000-2000 (3) B2O3-SiO2 0 −1.0660 - 1000-2000 (3) (1) This work; (2) Ottonello and Moretti (2004), based on experiments of Eliezer et al. (1978); (3) Ottonello and Moretti (2004), based on modified quasi-chemical parameterization of Pelton et al. (1995); (4) Ottonello (2001); (5) Ottonello et al. (2001). Table VI. Strain energy of HPM (from Ottonello and Moretti, 2004). Join Parameterization Notes K2O-SiO2 χ1 −2.8681 −1.6701E+5 (1) χ2 −79.501 6.8787E+4 χ3 99.065 1.5980E+5 χ4 10.499 −1.9243E+3 Na2O-SiO2 χ1 −11.766 −1.4559E+5 (1) χ2 39.776 2.2133E+5 χ3 −141.53 8.0710E+4 χ4 130.52 −9.4332E+4 CaO-SiO2 χ1 −1.7859 −5.4644E+3 (1) χ2 31.535 −2.0262E+4 χ3 −171.49 2.8175E+5 χ4 260.67 4.8533E+5 MgO-SiO2 χ1 22.144 −3.4756E+4 (1) χ2 −143.77 2.2334E+5 N$|+N$|+N$|+|= =| T, , SiO SiO SiO1 2 3 2 4 3 1 0 1 1 1 2 2 2 + h | | 572 Giulio Ottonello lated to the atomistic properties of the central cation, like Pauling’s electronegativity (Paul- ing, 1932, 1960) (fig. 3) or to spectroscopically derived magnitudes, such as differences in opti- cal basicity between metal oxide and silica (Duffy and Ingram, 1974; Gaskell, 1982; Sosin- sky and Sommerville, 1986; Duffy, 1989, 1990; Ottonello et al., 2001) (fig. 4). Table VI (continued). Join Parameterization Notes MgO-SiO2 χ3 186.75 −3.4506E+5 χ4 173.75 −2.7150E+4 ZnO-SiO2 χ1 −12.840 3.5427E+4 (1) χ2 −6.3910 −1.4814E+5 χ3 164.050 −6.1594E+4 χ4 −117.200 9.1512E+4 MnO-SiO2 χ1 −7.3391 1.3047E+4 (1) χ2 −18.670 3.5311E+4 χ3 92.477 −1.8607E+5 χ4 −23.422 3.1079E+4 PbO-SiO2 χ1 21.546 −3.5593E+4 (1) χ2 −59.309 2.6842E+4 χ3 132.23 −3.4732E+4 χ4 −58.665 −1.2444E+4 FeO-SiO2 χ1 10.214 −2.6396E+4 (2) χ2 −19.638 6.5843E+4 χ3 67.340 −2.1477E+5 χ4 −37.220 1.2462E+5 TiO2-SiO2 χ1 −11.099 1.7398E+4 (1) χ2 −20.417 −8.0525E+2 χ3 7.2065 2.5437E+4 χ4 30.269 −4.3948E+4 ZrO2-SiO2 χ1 −15.898 2.2410E+4 (1) χ2 −18.918 −1.7096E+4 χ3 16.482 −1.2937E+3 χ4 14.203 −1.3222E+4 NiO-SiO2 χ1 −3.2377 2.7449E+3 (1) χ2 21.656 4.9178E+3 χ3 −49.883 −4.5618E+4 χ4 18.565 5.9709E+4 Al2O3-SiO2 χ1 −3.7071 −6.1142E+2 (1) χ2 26.748 −1.5242E+4 χ3 −64.828 4.2276E+4 χ4 77.543 −7.0683E+4 B2O3-SiO2 χ1 0.0928 −6.1472E+3 (1) χ2 11.819 7.9109E+3 χ3 −41.175 −2.0305E+4 χ4 36.358 3.8358E+5 (1) Based on modified quasi-chemical parameterization of Pelton et al. (1995); (2) based on modified quasi- chemical parameterization of Pelton and Blander (1986) and Pelton (pers. comm.). N$|+N$|+N$|+|= =| T, , SiO SiO SiO1 2 3 2 4 3 1 0 1 1 1 2 2 2 + h | | 573 Chemical interactions and configurational disorder in silicate melts The discrimination capability of optical ba- sicity in terms of Lux-Flood basicity is even clearer when looking at polymerization con- stant KP. As shown in fig. 5, the relationship be- tween the natural logarithm of KP and the opti- cal basicity difference is so evident as to allow it to be used as a predictive tool for the chemi- cal interaction of basic oxides with silica. Each Fig. 3. Reaction enthalpy of equilibrium 2 plotted against Pauling’s electronegativity of MO metal cation. Fig. 4. Reaction enthalpy of equilibrium 2 plotted against difference in optical basicity of MO and SiO2. Adopt- ed basicity values are those of column 6 in table A1 of Ottonello et al. (2001). Selection of other basicity values within uncertainty ranges (columns 1 to 6 in same table) do not greatly alter observed relations. Fig. 5. Polymerization constant in MO-SiO2 melts at various T (with MO = basic oxide) plotted against differ- ence in optical basicity of MO and SiO2 (adopted basicity values are those of column 6 in table A1 of Ottonel- lo et al., 2001). Selection of other basicity values within uncertainty ranges (columns 1 to 6 in same table) do not greatly alter observed relations. Dashed lines: extrapolations at T beyond T limits investigated here for most basic oxides. Solid heavy line: functional form adopted by Ottonello et al. (2001) and Moretti and Ottonello (2003) when role of T was still unclear. 3 4 574 Giulio Ottonello of the depicted straight-line dependencies in fig. 5 has a correlation coefficient R 2= 0.9985, and their slopes and intercepts also exhibit a second-order dependence on absolute T. 4. Configurational disorder in block copolymers Although there is no experimental evidence about the possibile formation of local block copolymeric structures in silicate melts or glass- es, recent findings in this branch of materials sci- ence are also important for proper understanding of the unmixing phenomena which take place in SiO2-rich melts. The physics of microphase sep- aration in block copolymers is understood in terms of an expanded Flory-Huggins model. The Flory-Huggins model belongs to the class of quasi-chemical approaches. Flory him- self defined his method as «a restricted variation of the quasi-chemical method used by Guggen- heim» (cf. Flory, 1953, p. 507). Basically, the main merit of the Flory-Huggins approach is to acknowledge that molar fractions are unsuitable chemical parameters whenever molecules differ- ing greatly in size interact among themselves and with much smaller stoichiometric units in the melt phase. Volume fractions are thus substituted for molar fractions, based on the identity (4.1) where φi is the volume fraction of the ith compo- nent in mixture, ni its molar amount, υi its partial molar volume, and the summation at the denom- inator is extended to all k components in mixture. Block copolymers are composed of two or more distinct blocks interconnected by linear or branched sequencing. The relative arrangements of the various blocks gives rise to an astonishing number of distinct configurations, which is re- sponsible for the many useful and desirable prop- erties exploited by the industry of rubberlike ma- terials. The simplest architecture is the linear AB diblock: a long sequence of type A monomers co- valently bonded to a chain of type B monomers. The relative arrangements of the two blocks is n n i i i i k i i 1 =z y y = / characterized by «a fluidlike disorder on the mo- lecular scale and a high degree of order at longer length scales» (Bates and Fredrickson, 1999). In interpreting the structural and reactive properties of diblock copolymers, the following working assumptions are commonly made: – Each molecule in the melt is composed of ν segments, f A-type and (1−f) B-type monomers. – A and B segments represent a sufficient length of polymers such that they can be treat- ed as Gaussian, where the internal conforma- tional states of the segments produce a Hookian entropic penalty of stretching. – The statistical length of a segment is its average RMS end-to-end length when no ten- sion is applied, and is related to the effective spring constant. – The average segment concentration is forced to be uniform (incompressibility con- straint). The interaction between A and B is de- scribed in terms of a χAB Flory-Huggins interac- tion parameter which, in units of thermal ener- gy kT, is expressed as follows: (4.2) where Z is the number of nearest-neighbor monomers to a copolymer configuration cell (note the analogy with eq. (2.2)). If χAB is large, chemical interaction leads to a macroscopic segregation of A and B (binodal or spinodal de- composition). If instead χAB is sufficiently low, the thermodynamic forces driving separation are counterbalanced by entropic restoring con- tributions (i.e., «chain elasticity»; intimately connected to the covalent character of the bond but Hookian in nature) reflecting the require- ment that, to keep the dissimilar A and B por- tions of each molecule apart, copolymers must adopt extended configurations (Bates and Fredrickson, 1999). 4.1. Mean field approximation theories Helmholtz free energy F (the incompress- ibility constraint) (Fredrickson et al., 1994, Fredrickson and Liu, 1995) is kT Z 2 1 AB AB AA BB= - +| f f f] g: D 575 Chemical interactions and configurational disorder in silicate melts (4.3) where q is the reciprocal of the distance in the search space and SA, SB are form factors of the A and B block copolymers (Fredrickson et al., 1994; Fredrickson and Liu, 1995; Chen et al., 2000, 2002). These «form factors» assume pre- cise values depending on the extent and geom- etry of the block. For a linear chain L of monomers of statistical Kuhn length b, the form factor is simply SL = νLg(x) with νL = degree of polymerization and (4.4) (4.5) For branched polymers the form factors are more complex and are obtained by a combina- ( ) ( ) .g x x x2 e 1x 2= + -- x q b 6L 2 2= o dqlnq= ( ) ( )F kT S q S q 4 A A B B2 2 0 + r z z 3 7 A# tion of arm and backbone contributions (see Fredrickson et al., 1994, 1995, for appropriate treatment). 4.2. Matsen-Schick self-consistent mean field method The spectral analysis developed by Matsen and Schick in a series of articles (Matsen and Schick, 1994a,b; Matsen and Bates, 1996; Mat- sen, 1998) is a powerful predictive tool in deci- phering the complex microphase separation processes taking place in rubberlike materials. The theory stems from the Self-Consistent Mean Field (SCMF) treatment of Hong and Noolandi (1981). The form factor functionals are substituted here by space-occupation func- tions rα(s) describing the space curve occupied by the αth copolymer, and s is the parameteri- zation variable. In a system of n starblock copolymers com- posed of M AB diblock arms joined together at a central core, s = 0 at the core, s = f at the AB junction, and s = 1 at the end of the first arm. In the second arm, s is 1, 1+ f, 2, and so on, until the end of the last arm, where s = M is reached. The space occupation function is thus piece- wise continuous with discontinuities at integer values (fig. 6). The partition function for a system of p polymers of polymerization ν and density ρ0 is (4.6) P is the probability density functional for a giv- en curve (see eq. 2 in Matsen and Schick, 1994a, for the approximate form adopted to de- scribe P); , are (dimensionless) A and B monomer density operators (4.7) (4.8)( ) ( ) ( )ds s sr r r1B M f p 0 0 $= - -z t o c d + t ^ h 6 @# ( ) ( ) ( )ds s sr r rA M f p 0 0 $= -z t o c d + t 6 @# ( )rBz t( )rAz t pD ; ( ) ( ) .exp Q P M f d r r r r r 1p A B A BAB 0 # # = + - - - d z z o t | oz z t t t t 6 8@ B ' 1 # # Fig. 6. Positional parametrization of starblock co- polymers. Copolymer melt is composed of starblock copolymers made up of alternating blocks consist- ing of f A-type and (1−f) B-type monomeric frac- tions. Positional vector defining relative position of various branches of the ith copolymer is defined in terms of polymer number and branch number. Space function is piecewise continuous with discontinu- ities at integer values. 576 Giulio Ottonello γ(s) is a discontinuity function which attains a value of 1 when s corresponds to an A-monomer region and a value of 0 for a B-monomer region. The product of density operators in eq. (4.6) represents the various space-dependent interac- tions in the copolymer melt. To render the problem mathematically tractable, the following steps are performed: 1) A Delta functional integral (4.9) allows the chain density operator to be replaced by segment volume fractions φΑ. 2) Segment volume fractions are expanded as linear combinations of basis functions (4.10) 3) Basis functions (Fourier transforms) pos- sessing the symmetry of the phase in question are selected, i.e., for the gyroid phase with sym- metry (Matsen and Schick, 1994b) (4.11) with X= 2π x/d and d = size of the cubic (gyroid) unit cell. Several passages yield SCMF equations, re- ducing the problem to that of average densities of the A, B monomers at r in the ensemble of non-interacting polymers subject to self-consis- tent external fields (cf. eqs. 11-15 in Matsen and Schick, 1994a). In various configurational states, Helm- holtz free energy F = − ktlnQ is computed, and the minimum value (saddle point), attained through steepest descent techniques, is selected as representative of the stable configuration. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) / ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) / cos sin sin cos sin sin cos sin sin cos cos cos coscos cos f r f r X Y Z Y Z X Z X Y f r X Y Z Z X f r f r Y 1 2 2 2 3 2 2 2 2 2 2 8 3 4 1 4 2 3 5 $ $ $ $ f f = = + + = + + = = 6 6 @ @ Q 3Ia d- ( ) ( ) ( ) ( )r r f r f rf, 2 ,A A A1 2 331 f= + + +z z z z D1 A A A= -d zΦ Φ t 8 B# 5. Discussion For the sake of clarity, the discussion con- cerning information arising from the thermody- namic treatment of binary silicate melt interac- tions is divided into three distinct sections: 1) in- sights into short- and medium-range structural arrangement; 2) chemical interaction and long- range disorder; 3) and inhomogeneity ranges. 5.1. Short- and medium-range disorder: insights from the EPA In fig. 7, the polymerization paths of the EPA for various binary joins overlap the dis- crete values for structural units thought to be present in sili- cate melts and glasses (Toop and Samis, 1962a,b). For comparative purposes, in the same plot, the consolute point in the high-SiO2 portion of each binary join also overlaps. This sort of plot is very informative, since the fol- lowing evidence may be retrieved: – In most joins, the melt is fully depoly- merized over a large part of the compositional range. ( ) ( ) / [( ) ( ) ]O O O SiSi IV0- + +o - -r Fig. 7. Polymerization paths of EPA model com- pared with compositions of discrete polymeric units which may form in melt at various SiO2 contents. Adopted polymerization constants in constructing various curves are those of table V. Location of con- solute points overlap, for comparative purposes. 577 Chemical interactions and configurational disorder in silicate melts – Ring connectivity at the consolute point depends on the Lux-Flood basicity of the dis- solved oxide (i.e., 4-rings-4-membered for FeO, 10-rings-6-membered for Na2O). Figure 8 compares experimental evidence concerning the speciation of the various forms of oxygen present in a CaO-SiO2 melt at T= = 1600°C, P = 1 bar, obtained by Park and Rhee (2001) by XPS (X-ray Photoelectron Spec- troscopy) with EPA predictions. This plot, be- sides experimentally confirming for the first time (as far as we know) the actual presence of oxide ions in the silicate melt, leaves few doubts about the soundness of the polymeric approach in depicting oxygen speciation. The fit between calculated and observed specia- tion is particularly satisfactory at low SiO2 amounts. At high silica content, the polymeric model appears to underestimate the degree of condensation of the polymeric units to some extent. This may be interpreted (in line with the preceding observations) as due to strain energy contribution effects (not accounted for in the model), which may stabilize high-mem- bered rings in the vicinity of the two-phase re- gion. Fig. 9. HPM themodynamic parameterization of CaO-SiO2 system (solid lines; Ottonello and Moretti, 2004), compared with estimates of modified quasi-chemical model (dashed lines; Pelton et al., 1995). Fig. 8. XPS measurements of oxygen speciation in quenched CaO-SiO2 melts (after Park and Rhee, 2001), compared with EPA predictions (Ottonello, 2001). – Incipient decomposition (intrinsic insta- bility) tendentially takes place when ring con- formations change (e.g., 3-to-4 members, 4-to- 5 members, and so on). 578 Giulio Ottonello 5.2. Chemical interaction and long-range disorder As already mentioned, the HPM allows us to discriminate chemical (enthalpic + entropic) from elastic strain energy contributions to the bulk Gibbs free energy of mixing in MO-SiO2 joins, and this discrimination is typical of the basic-amphotheric and acidic Lux-Flood be- havior of MO oxides. The quasi-chemical to HPM conversion does not affect the bulk value of the Gibbs free energy of mixing of the join in question, although it does modify to some ex- tent the relative magnitudes of enthalpic and entropic terms in the two models. As we see in fig. 9, for the CaO-SiO2 join, entropic interac- tions observed for a Lux-Flood basic oxide arise entirely from elastic strain in the anion sublattice. Moreover, enthalpic chemical inter- actions are dominant, and the bulk strain ener- gy contribution to the Gibbs free energy of mix- ing is far less and practically confined to the high-SiO2 portion of the system. Instead, when a Lux-Flood acidic oxide is dissolved in the sil- icate melt (Al2O3-SiO2 join, fig. 10), although strain energy contributions are again confined to high SiO2 content, both enthalpic and entrop- ic chemical interaction terms are present over the entire compositional range, leading to an overall «regular» appearance of the bulk Gibbs free energy of mixing. Undoubtedly, CaO per- turbs the long-range periodicity of the silicate network with local depolymerization effects whereas, at least apparently, «clusters» of over- all SiO2 (tectosilicate) stoichiometry coexist stably with clusters of overall Al2O3 stoichiom- etry throughout the compositional range. Obvi- ously, the picture is more complex for oxides with less definite Lux-Flood reactivity. 5.3. Inhomogeneity ranges Although we can envisage the role of chem- ical interaction in determining (or modifying) short-, medium- and long-range periodicity in silicate melts and glasses, our appraisal is still in- sufficient to appreciate the complex phenomena which take place within the instability region. In the present author’s opinion, better appraisal may be achieved by tentatively applying SCMF procedures to this topologically complex region. Application of the Matsen-Schick theory to silicate polymer melts seems particularly prom- Fig. 10. HPM thermodynamic parameterization of Al2O3-SiO2 system (solid lines; Ottonello and Moretti, 2004), compared with estimates of modified quasi-chemical model (dashed lines; Pelton et al., 1995). 579 Chemical interactions and configurational disorder in silicate melts ising. As anticipated in the introduction, the short- and medium-range periodicity of silicate melts and glasses locally resemble that of all-Si zeolites, with cationic «cages» of variable di- mensions, depending on silicate network con- nectivity (fig. 11). In this sort of arrangement, it is still possible to isolate polymeric subunits (in- set, fig. 11) of quasi-crystalline short- and medi- um-range order, but their relative positions per- turb long-range periodicity. The following working assumptions are thus proposed, to ren- der the problem feasible for SCMF treatment: 1) A simple, irreducible, A-A starblock-like representation of the network is selected as re- sponsible for the short- and medium-range or- der observed in silicate melts and glasses (figs. 6 and 11). 2) It is assumed that, in the two-phase re- gion, coexisting liquids locally mimic the struc- tures of crystalline phases observed at the mono- tectic point. 3) Basic functions representative of the geometry of fictive intermixing A-A starblocks (or A-B starblocks, in the case of aluminosili- cate melts) are assumed to be represented by the Fourier expansions of the spatial groups of the two solid phases at the monotectic point. 4) Calculations are conducted in terms of Gibbs free energy on the basis of HPM parame- terization, after subtracting chemical interac- tion terms and adding PV contributions. 6. Conclusions Polymeric models provide an extremely useful tool in deciphering the structural and re- active properties of silicate melts and glasses. They not only establish the Lux-Flood charac- ter of the dissolved oxides through opportune conversion of existing quasi-chemical parame- terization (Pelton and Blander, 1986; Pelton et al., 1995; Pelton, pers. comm.), but also dis- criminate the subordinate strain energy contri- butions to the Gibbs free energy of mixing from the dominant chemical interaction terms. This discrimination allows us to retrieve important information about the short-, medium- and long-range periodicity of this class of sub- stances from thermodynamic analysis. Howev- er, the conceptual models developed for silicate melts and glasses are still insufficient to allow thorough appraisal of the complex phenomena which take place within the inhomogeneity range. 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