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ISSN 1087 6596, Glass Physics and Chemistry, 2010, Vol. 36, No. 5, pp. 579­588. © Pleiades Publishing, Ltd., 2010. Original Russian Text © V.B. Polyakov, A.A. Ariskin, A.V. Shil'dt, 2010, published in Fizika i Khimiya Stekla.

Analysis of Disproportionation of Qn Structons in the Simulation of the Structure of Melts in the Na2O­SiO2 System
V. B. Polyakova, *, A. A. Ariskinb, and A. V. Shil'dt
a b, c

Institute of Experimental Mineralogy, Russian Academy of Sciences, ul. Institutskaya 4, Chernogolovka, Moscow oblast, 142432 Russia * e mail: polyakov@iem.ac.ru, vpolyakov@mail.ru b Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, ul. Kosygina 19, Moscow, 119991 Russia c Faculty of Geology, Lomonosov Moscow State University, Leninskie Gory 1, Moscow, GSP 1, 119992 Russia
Received September 16, 2009

Abstract--A new version of the STRUCTON 1.2 computer program (2009) has been presented. The pro gram combines the algorithm for calculating real distributions of Qn structons in binary silicate melts (with allowance made for their disproportionation) and the statistical simulation of molecular mass distributions of polymerized ions at different temperatures. This model has been used to perform test calculations for two melts in the Na2O­SiO2 system (Na6Si2O7, Na6Si3O9). The results of the calculations have made it possible to trace variations in the set and concentrations of chain and ring silicon­oxygen complexes with a decrease in the temperature in the order: stochastic molecular mass distribution molecular mass distribution at T = 2000 K molecular mass distribution at the liquidus temperature. The main result of these calcula tions is that the dominant species of silicon­oxygen anions at the liquidus temperatures (in contrast to the stochastic distributions) exactly correspond to the stoichiometry of the initial melts: the Si 2 O 7 chain anions and (SinO3n)3n­ ring complexes are dominant in the Na6Si2O7 and Na6Si3O9 melts, respectively. It has been established that, with a decrease in the temperature, the average size of polymer complexes varies weakly in the Na6Si2O7 melt but increases by a factor of approximately 1.5 in the metasilicate system. Key words: silicate melts, structural simulation, disproportionation of Qn particles, molecular mass distribu tion, Na2O­SiO2 system DOI: 10.1134/S108765961005007X
6­

INTRODUCTION Results of experimental investigations [1­4], theo retical analysis [5­9], statistical modeling, and molec ular dynamics simulation [10­13] have demonstrated that the structure of silicate melts is formed by discrete silicon­oxygen anions and their complexes of the general formula (SiiO3i + 1 ­ j)2(i + 1 ­ j)­, where i is the size of an anion and j is the number of self closures of terminal oxygen bonds inside a particle. Since formal constraints on the sizes of these complexes are absent, the structural state of a silicate liquid can be repre sented as a complex ion­polymer solution in which the average negative charge of an ensemble of low polymer anions (Si O 4 , Si2 O 7 , Si3 O 9 , etc.) and larger polymerized particles is compensated for by a positive charge of metal modifier cations. These con cepts underlie the polymer approach, which has been developed from the mid 1960s and for which the main problem consists in calculating molecular mass distri
4­ 6­ 6­

butions that characterize the relative proportions of silicate particles with low and high degrees of polymer ization as a function of the total composition [14­18]. The known molecular mass distributions allow one to develop thermodynamic models of polymer melts that are consistent with the experimental data on the activ ity of oxide components in molten silicates and make it possible to predict physicochemical properties of sil icate liquids at high values of the parameters P and T [6, 9]. In our previous work [19], a new statistical model was proposed for calculating molecular mass distributions of anions in binary and ternary systems MeO­Me2O­SiO2 from a known distribution of Qn structons, i.e., sili con­oxygen tetrahedra with different numbers of bridging bonds (0 n 4, Q0 is a Si O 4 ion). This model was implemented in the form of the STRUCTON com puter program, which uses the Monte Carlo method and allows one to evaluate the probability of formation
4­

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580 1.0 Mole fraction of Qn structons SiO2 0.8 Q0 0.6 0.4 Q1 0.2 Q
2

POLYAKOV et al.

p = 0.98. In tion of the ensemble of properties of
Q4

this case, we succeeded in parametriza dependence of the average size of an silicon­oxygen anions on the structural the ion­polymer solution [20].

Q3

As follows from the results of investigations using Raman scattering and nuclear magnetic resonance (NMR) spectroscopy at different temperatures, the real distributions of Qn structons in silicate melts never coincide with the ideal distribution (1). This is a con sequence of the occurrence of three disproportion ation reactions [22­24]
0

0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Degree of depolymerization p
Fig. 1. Stochastic distribution of structons as a function of the degree of depolymerization p of the silicon­oxygen matrix. Calculations were performed using Eqs. (1) in the approximation of equal reactivities of nonbridging Si­O bonds. Qn

Q +Q Q +Q
2 1

0

2 KD1

2Q , 2Q , 2Q .
3 2

1

(3a) (3b) (3c)

3 KD2

Q +Q

4 KD3

(concentration) of silicate complexes from a specified distribution of Qn structons. The solution to the prob lem of the calculation of molecular mass distributions made it possible to systematically simulate polyanion ensembles involving linear­branched, ring, and vari ous spatial structures as a function of the degree of polymerization (or depolymerization) of the melt [20]. The estimates obtained are based on the ideal (stochastic) distribution of Qn structons, which follows from the combinatorial expressions [7, 21] X0 = p ,
4

X1 = 4 p ( 1 ­ p ) ,
3

3

X2 = 6 p ( 1 ­ p ) ,
4

2

2

X3 = 4 p ( 1 ­ p ) ,

X4 = ( 1 ­ p ) ,

(1)

where Xn is the concentration (fraction) of Qn struc tons and the degree of depolymerization of the melt p corresponds to the weighted mean fraction of non bridging bonds in the silicon­oxygen matrix [6, 18], which can be expressed through the relative concen trations of Q particles p = X0 + 0.75X1 + 0.5X2 + 0.25X3. (2) The character of the relation between the parameters p and Xn (expressions (1), (2)) manifests itself in a sym metric distribution of Qn structons on a polymeriza tion scale and mirror relationships for the Q0­Q4 and Q1­Q3 species (Fig. 1). The results of the calculations in the range of degrees of depolymerization 0.52 p 0.98 demonstrated that, under the assumption of the same reactivity of Si­O­ terminal bonds, size limited and stable sets of polymer complexes are formed in sil icate melts, and the average number of sets varies from 4­ 6­ 8­ 153 at p = 0.52 to 3 (Si O 4 , Si2 O 7 , Si3 O 10 ) at

The integrated result of these reactions is an increase in the concentration of tetrahedra of the Q1, Q2, and Q3 type due to the decrease in the concentration of the Q0 and Q4 species. Therefore, the changeover from the model of anionic complexes corresponding to the ideal stochastic distribution of Qn structons (1) to more realistic evaluation of molecular mass distributions of polymer particles requires a consistent inclusion of the reactions of disproportionation of Qn structons with different numbers of bridging bonds (3a)­(3c). In the case of the positive solution to this problem and calcu lations of deviations of the expected distributions of Qn structons from the stochastic distribution, the cor rected evaluations of Xn can be used as input informa tion for the performance of the statistical modeling of the molecular mass distributions with the use of the STRUCTON computer program [19]. This opens up possibilities for a systematic analysis of the evolution of the proportions of silicon­oxygen anions as a func tion of the quantity p and the temperature and, in prospects, for the reconstructions of dominant sili con­oxygen species along the liquidus curves in sili cate systems. This paper reports on the approach to the solution of the above problems for the Na2O­SiO2 binary system as an example and some preliminary results that are a development of the model proposed in our earlier works [19, 20]. ALGORITHM FOR INCLUDING THE DISPROPORTIONATION REACTIONS Let us consider reactions (3a)­(3c). The equilibrium constants for these reactions are written in the form K
Dn

=

Xn [ Xn ­ 1 ] [ X

2

n+1

]

,

n = 1, 2 , 3 .
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581

A combination of three Eqs. (4) with Eq. (2) and the obvious identity X0 + X1 + X2 + X3 + X4 = 1 (5)

allows us to derive the closed system of five equations with respect to five unknowns Xn (n = 0, 1, ..., 4). The solution to the nonlinear system of equations (2), (4), and (5) makes it possible to obtain the sought distribu tion of Qn structons for a specified parameter p when the constants KDn are known. The temperature dependence of the equilibrium constants of the disproportionation reactions can be described by the relationship KDn(T) = snexp (­Hn/RT), n = 1, 2, 3, (6)

Consistent information on the set of constants KDn for silicate melts can be obtained using the calcula tions based on thermodynamic databases. In order to evaluate the constants of the disproportionation reac tions in the Na2O­SiO2 system, we used the FACT thermodynamic base [31], as was previously done in [27­29, 32]. According to the data from the FACT database, the free energies of formation of Qn structons can be determined from the free energies of formation of silica and four sodium silicates that can be con structed from Qn structons in the framework of the model of associated solutions and under the assump tion of quasicrystallinity of the corresponding solu tions [27, 28, 32]: Q 4Q
3 2 1 4

SiO 2 , Na 4 Si 4 O 10 , Na 6 Si 3 O 9 , Na 6 Si 2 O 7 , Na 4 SiO 4 . (8)

where sn are the entropy factors and Hn are the enthal pies of the corresponding reactions (3a)­(3c). Our developed model based on the Monte Carlo method [19, 20] provides a maximum of the configu rational entropy. In this case, the entropy factors do not depend on the temperature, because, in the limit of infinitely high temperatures, the disproportionation constants (6) do not depend on the enthalpies of reac tions (3a)­(3c) and are determined by the ideal sto chastic distribution of Qn structons (1). Substitution of expressions (1) into relationship (6), gives the follow ing formulas for sn: s1 = s3 = KD 1 ( ) = KD3 ( ) = 8 / 3 , s2 = KD1 ( ) = 9 / 4 . (7)

3Q 2Q Q

0

The free energies of formation found for the struc tons from Eqs. (8) were used to calculate the enthalp ies of the disproportionation reactions (4) with due regard for expressions (6) and (7). For the Na2O­SiO2 system, the enthalpies of reactions (4) thus obtained are as follows: H 1 = 27.17 kJ/mol; H 2 = 32.23 kJ/mol; (9)

In terms of the model under consideration, we also assume that the enthalpies of the disproportionation reactions in relationship (6) do not depend on the temperature. This approximation has been widely used in the simulation of silicate systems (see, for example, [24­26]) in addition to other models in which the temperature dependence of the entropy fac tor and the enthalpy of disproportionation reactions is taken into account [27­29]. The constants of the disproportionation reactions are either determined from Raman and NMR spectro scopic data or calculated from thermodynamic data. Unfortunately, the direct use of NMR spectroscopy in the Me2O­SiO2 (Me = Li, K, Na) systems for melts at high temperatures faces technical problems and becomes almost impossible. Raman spectroscopy ensures the reliable determination of the dispropor tionation constant only for KD3 [28, 30], even though, in recent years, some progress in the determination of the constants KD1 and KD2 with the use of the Raman spectroscopy has been achieved using the simulation of the high frequency spectral range with allowance made for the second coordination sphere of silicon atoms [29].
GLASS PHYSICS AND CHEMISTRY Vol. 36 No. 5

H 3 = 20.75 kJ/mol.

Substitution of the determined enthalpies of the reactions (9) and the entropy factors (7) into Eqs. (6) allows us to calculate the equilibrium constants for all three disproportionation reactions as a function of the temperature. Then, by substituting the calculated con stants KDn into Eqs. (4) and solving the nonlinear sys tem of equations (2), (4), and (5) for a specified com position, we can find the distribution of Qn structons at temperatures from the liquidus to an infinite tempera ture (corresponding to the stochastic distribution). Figure 2 shows the calculated distribution of Qn structons for the Na2O­SiO2 system at a temperature of 1200°C as a function of the degree of depolymeriza tion of the melt p. On the whole, this distribution agrees with the calculations performed in [27, 33, 34], which were also carried out with the use of the FACT thermodynamic database [31] but are characterized by some differences in the formulation of the system of equations for the calculations of the structon concen trations. The positions of the peaks in the distribution of Qn structons in our calculations coincide with the corresponding positions of the peaks in [27, 33, 34]. The observed quantitative differences in the distribu
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582 1.0 Q Mole fraction of Qn structons 0.8 Q2 Q1 0.6
0

POLYAKOV et al. 1 2 3 4 5 Q3 1.0 Q4 Mole fraction of Qn structons 0.8 Q0 0.6 Q1 1 2 3

SiO2

Q2 Q3 Q4

0.4

0.4

0.2

0.2

0 1.0

0.6 0.4 0.2 0.8 Degree of depolymerization p

0

0 1.0

0.6 0.4 0.2 0.8 Degree of depolymerization p

0

Fig. 2. Dependence of the distribution of Qn structons on the degree of depolymerization of the melt at a tempera ture of 1200 K. Calculations were based on the data taken from the FACT thermodynamic database [31]. Open and closed symbols indicate the results of the calculations per formed in [33, 34], respectively: (1) Q0, (2) Q1, (3) Q2, (4) Q3, and (5) Q4.

Fig. 3. Change in the distribution of Qn structons as a func tion of the temperature. T = (1) 1200, (2) 1800, and (3)3000°C.

tions of Qn structons, including the height of the peaks, can also be associated with the fact that the tempera ture dependences of the entropy factor and the enthalpy of reactions were took into account in [27, 33]. The main difference of the calculated distribu tions from the stochastic distribution (Fig. 1) is an approximately twofold increase in the maximum of the concentration of Q2 structons for the metasilicate (Na2SiO3 or Na6Si3O9, p ~ 0.5 (see below)) melt and an increase in the maxima of the Q1 and Q3 species by a factor of ~1.5. The symmetrical location of the Q1 and Q3 maxima is retained but their "mirror behavior" is violated. A larger value of the enthalpy term for reaction (3a) (H1 = 27.17 kJ/mol) as compared to reaction (3c) (H3 = 20.75 kJ/mol) results in the fact that the relative maximum of the distribution of Q1 structons appears to be systematically lower than the Q3 maximum. These relationships are retained at higher tempera tures (Fig. 3). However, with an increase in the tem perature, the peaks corresponding to different Qn structons level off, their height decreases, and they are broadened. As was noted above, in the limit of high temperatures, the distribution of Qn structons tends to the ideal stochastic distribution (Fig. 1).

SIMULATION OF THE MOLECULAR MASS DISTRIBUTIONS OF SILICON­OXYGEN ANIONS AS A FUNCTION OF THE TEMPERATURE In order to evaluate the molecular mass distribu tions of silicate melts with allowance made for the disproportionation reactions, the module for the cal culations of the constants of the disproportionation reactions KDn at a specified temperature with the use of formulas (6) and (7), the enthalpies of reactions (9), and the solution to the system of equations (2), (4), and (5) was added to the initial STRUCTON com puter program [19, 20]. The algorithm for solving this system of equations consisted in reducing the system to the fourth order equation with one unknown r = X1/X0 and its solution by the simplest half division method in the range 0­1020. The relative error in the determination of the unknown r did not exceed 10­9, and all calculations were performed with a double accuracy. The flow chart of the STRUCTON computer program (version 1.2, 2009) is shown in Fig. 4. By using this code, we performed the test calcula tions for two compositions in the Na2O­SiO2 system that correspond to two degrees of depolymerization of the melt (p = 0.75 and 0.51, Fig. 5) and represent vari ations in the molecular mass distributions as a func tion of the temperature (Tables 1, 2). The polymer the ory predicts that, for binary melts richer in SiO2 as
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ANALYSIS OF DISPROPORTIONATION OF Qn STRUCTONS
Analysis of the experimental data on the disproportionation constants

583

Determination of the enthalpies of the disproportionation reactions

Calculation of the distribution of Qn structons for a specified degree of depolymerization and temperature

Random generation of a Qn structon with the use of the random number generator

Analysis of the particle type and separation of Q0 structons If Q0 particle If Q1, Q2, Q3, or Q4 particles

Choice of growth or cyclization of the polymer complex with the random number generator from expressions (4) and (5) Ring formation Attachment of the monomer

Choice of the unsaturated bond in the polymer for attachment of a new monomer with the use of the random number generator

Choice of a pair of unsaturated bonds in the polymer for the ring formation with the use of the random number generator

Analysis of the polymer chain: the calculation of the number of unsaturated bonds, determination of ring forming pairs of unsaturated bonds, and determination of the number of formed closures If all bonds are saturated If there are unsaturated bonds

Formation of an ensemble of polyanions: Np = Np + 1 N=N­1

Fig. 4. Schematic flow chart of the program. The dashed line separates the blocks corresponding to the STRUCTON computer program [19].

compared to the orthosilicate ( N SiO2 0.333, compo sition N2S in Fig. 5), the degree of depolymerization and the composition are related by the simple expres sion [35] p = (1 ­ N
SiO
2

number of O2­ ions in this SiO2 concentration range is negligible, the following approximation works well: 1 p 0.5 N Si
O

2

­ 1 .

(11)

­N

O

2­

)/2N

SiO

2

,

(10)

where N SiO2 is the mole fraction of silica and N O2 ­ is the number of "free oxygen" ions per mole of the melt. In the case of strongly depolymerized alkali silicate melts for which the polymerization constant accord ing to [36] does not exceed 10­3 [37] and, hence, the
GLASS PHYSICS AND CHEMISTRY Vol. 36 No. 5

Therefore, the chosen values of p are close to two sto ichiometric compositions Na6Si2O7 (40.0 mol % SiO2 at p = 0.75) and Na6Si3O9 (49.5 mol % SiO2 at p = 0.51). A small deviation to the range of melts with a higher basicity for the second composition as com pared to the sodium metasilicate is associated with the difficulties of the simulation of molecular mass distri butions under conditions of the equality of the num
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584 2000

POLYAKOV et al. N2S (p = 1) N3S2 NS N3S2

1800

1600 1400 1200

T, K

Temperature, K

1600 N2S 1400

0 0.2 0.4 0.6 Na2O SiO2 NS NS2

1200

1000 1.0

0.6 0.4 0.8 0.2 Degree of depolymerization of the melt p

Fig. 5. Phase diagram of the Na2O­SiO2 system as a function of the silica content (inset at the top) and the degree of depolymer ization of the melt. Circles indicate the melts for which the molecular mass distributions were calculated as a function of the tem perature (Tables 1, 2). Designations: N2S is Na4SiO4 (2Na2O SiO2), N3S2 is Na6SiO2O7, NS is Na2SiO3 (or N3S3 is Na6Si3O9), NS2 is Na2Si2O5, and S is SiO2.

bers of bridging and nonbridging bonds at p = 0.5 when the methods of the statistical simulation predict the beginning of an unlimited increase in silicon­oxy gen complexes and the formation of particles with a high degree of polymerization (SiiO3i + 1 ­j)2(i + 1 ­ j)­ [20].
1

For both compositions, the calculations were car ried out at two temperatures, i.e., 2000 K and the liq uidus temperature that is equal to 1397 K for the disil icate melt (p = 0.75) and 1363 K for the metasilicate melt (p = 0.51). For comparison, the molecular mass distributions that represent the stochastic distributions of Qn structons are listed in Tables 1 and 2. A compar ative analysis of these data leads to the following infer ences. As a result of the disproportionation of Qn struc tons, the model concentration of Si O 4 ions at the liquidus temperature decreases (as compared to the stochastic molecular mass distribution) by a factor of
1
4­

approximately 1.5 for the Na6Si2O7 melt and a factor of ~20 for the Na6Si3O9 melt. For both compositions, a decrease in the tempera ture leads to an increase in the number of chain spe cies; in this case, for sodium disilicate at the liquidus 6­ temperature, Si2 O 7 particles begin to dominate in the melt, which corresponds to the stoichiometry of this compound. In the case of the more acidic sodium metasilicate at the liquidus temperature, ring particles of the general formula (SinO3n)3n­ beginning with (Si3O9)6­ (n 3) are dominant in the melt. Their composition also corre sponds to the stoichiometry of the initial melt. With a decrease in the temperature, the average size of polymer complexes remains constant for the Na6Si2O7 melt but increases by a factor of approxi mately 1.5 for the metasilicate system. It should be noted that the number of species of polymer particles is several tens (11­28) in the former case and varies insignificantly from 130 to 140 in the latter case. CONCLUSIONS Thus, a new algorithm has been proposed for simu lating the temperature dependent distributions of Qn
Vol. 36 No. 5 2010

In some calculations based on the STRUCTON computer pro gram with the use of 104 initial Qn structons at p = 0.50, we obtained polymer particles containing from 500 to ~1000 silicon atoms.

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NS2


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585

Table 1. Characteristics of polyanion ensembles representing stochastic and temperature dependent distributions of Qn structons for 40 mol % SiO2 in the melt (p = 0.75) T, K Fraction of structon Q0 Q1 Q2 Q3 Q4 Monomer Linear and branched chains (j = 0)
4­ 4 6­ 7 8­ 10 10 ­ 13

Stochastic distribution 2000 0.3164 0.4219 0.2109 0.0469 0.0039 Anion concentration, mol % 0.2149 0.5750 0.2054 0.0047 0 (~10­5) 1397 0.1859 0.6297 0.1829 0.0015 0 (~10­6) 36.76 ± 0.43 37.89 ± 0.54 17.15 ± 0.44 5.72 ± 0.27 62.85 ± 0.49 0.30 ± 0.07 0.06 ± 0.02 0.02 ± 0.02 0.39 ± 0.08 ­ ­ ­ ­ 0.00 100.00 11 ± 1 1.99 ± 0.01

SiO

61.21 ± 0.51 15.03 ± 0.49 9.10 ± 0.30 5.42 ± 0.21 36.18 ± 0.58 0.57 ± 0.13 0.65 ± 0.15 0.46 ± 0.09 2.50 ± 0.14 0.02 ± 0.02 0.02 ± 0.03 0.02 ± 0.01 0.09 ± 0.02 0.02 ± 0.05 100.00 28 ± 1 1.95 ± 0.01

42.87 ± 0.42 30.91 ± 0.52 15.93 ± 0.38 6.56 ± 0.40 56.49 ± 0.36 0.43 ± 0.06 0.15 ± 0.06 0.05 ± 0.03 0.64 ± 0.07 ­ ­ ­ ­ 0.00 100.00 13 ± 1 1.98 ± 0.01

Si 2 O Si 3 O Si 4 O Total

Planar and branched ring complexes (j = 1)

Si 3 O Si 4 O Si 5 O Total

6­ 9 8­ 12 10 ­ 15

Simple three dimensional complexes (j = 2)

Si 4 O Si 5 O Si 6 O

6­ 11 8­ 14 10 ­ 17

Total The other high polymer anions (j 3) Total of anions Anion species Average anion size

Note: The temperature T = 1397 K corresponds to the liquidus temperature of the sodium disilicate Na6Si2O7. Average values were obtained from the results of ten calculations for the initial systems containing 104 Qn structons, the standard deviations correspond to ±1, and j is the number of bridging oxygen atoms in the particle.

structons. This algorithm has been employed to modify the STRUCTON computer program (version 1.3, 2009), which allows one to calculate the molecular mass distributions of polymer anions in silicate melts at different temperatures. This model has been used to perform test calculations for two compositions in the Na2O­SiO2 system (Na6Si2O7, Na6Si3O9). The results of the calculations have made it possible to trace variations in the set and concentrations of chain and ring silicon­ oxygen complexes with a decrease in the temperature in the order: stochastic molecular mass distribution
GLASS PHYSICS AND CHEMISTRY Vol. 36 No. 5

molecular mass distribution at T = 2000 K molecu lar mass distribution at the liquidus temperature. The main result of these calculations is that the dominant species of silicon anions at the liquidus temperatures (in contrast to the stochastic distributions) exactly corre spond to the stoichiometry of the initial melts: the 6­ Si2 O 7 chains anions and (SinO3n)3n­ ring complexes are dominant in the Na6Si2O7 and Na6Si3O9 melts, respectively. This is not a trivial or predicted result, because the STRUCTON program does not include any special
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POLYAKOV et al.

Table 2. Characteristics of polyanion ensembles representing stochastic and temperature dependent distributions of Qn structons for 49.5 mol % SiO2 in the melt (p = 0.51) T, K Fraction of structon Q0 Q1 Q2 Q3 Q4 Monomer Linear and branched chains (j = 0)
4­ 4 6­ 7 8­ 10 10 ­ 13

Stochastic distribution 2000 0.0677 0.0069 0.2600 0.1836 0.3747 0.6559 0.2400 0.1499 0.0576 0.0037 Anion concentration, mol % 55.85 ± 1.46 7.76 ± 0.52 12.10 ± 0.70 5.20 ± 0.70 3.30 ± 0.12 26.32 ± 1.13 0.31 ± 0.10 0.82 ± 0.25 0.70 ± 0.29 6.87 ± 0.51 ­ 0.21 ± 0.14 ­ 2.89 ± 0.12 8.07 ± 0.90 100.00 137 ± 8 8.10 ± 0.31 8.68 ± 1.24 6.91 ± 1.15 5.71 ± 0.53 44.16 ± 1.50 5.29 ± 0.84 3.85 ± 0.87 2.91 ± 0.38 33.21 ± 1.75 0.11 ± 0.12 0.14 ± 0.14 0.22 ± 0.18 9.40 ± 0.15 5.47 ± 0.75 100.00 140 ± 4 11.97 ± 0.56 1363 0.0022 0.1411 0.7522 0.1037 0.0009 2.74 ± 0.33 5.16 ± 0.68 5.02 ± 0.74 4.34 ± 0.47 39.56 ± 1.50 8.69 ± 0.75 6.18 ± 0.60 4.00 ± 0.60 46.13 ± 1.59 0.09 ± 0.07 0.07 ± 0.07 0.12 ± 0.09 8.50 ± 0.10 3.07 ± 0.08 100.00 130 ± 4 11.51 ± 0.49

SiO

Si 2 O Si 3 O Si 4 O Total

Planar and branched ring complexes (j = 1)

Si 3 O Si 4 O Si 5 O Total

6­ 9 8­ 12 10 ­ 15

Simple three dimensional complexes (j = 2)

Si 4 O Si 5 O Si 6 O

6­ 11 8­ 14 10 ­ 17

Total The other high polymer anions (j 3) Total of anions Anion species Average anion size

Note: The temperature T = 1363 K corresponds to the liquidus temperature of the sodium metasilicate Na2SiO3. Average values were obtained from the results of ten calculations for the initial systems containing 104 Qn structons, the standard deviations correspond to ±1, and j is the number of bridging oxygen atoms in the particle.

constraints on the formation of silicon­oxygen com plexes, and the molecular mass distributions calcu lated at high temperatures in the given composition range are characterized by the dominance of Si O 4 monomers over other particles. Here, it is appropriate to recall that, in the study of polymer properties of sil
2
4­

2

icate melts, Esin [6] noted that there should appear relative maxima of concentrations of small sized par ticles (SiiO3i + 1 ­ j)2(i + 1 ­ j)­ that, on the composition scale, should coincide with points of stoichiometric compounds, which usually coincide with maxima in the liquidus curves. However, by using the pure sto chastic models of the distribution of silicon­oxygen anions, he failed to reveal reliable maxima for Si2 O 7 dimers, (SinO3n)3n­ ring species, and other complexes with respect to Si O
4­ 4 6­

Except that two arbitrarily chosen Qn structons can be linked together only by one bridging bond. This constraint is equivalent to the fact that the silicate tetrahedra cannot be shared by edges or faces and the silicon­oxygen bonds cannot be formed through the vertices of the tetrahedra.

ions. It is evident that the results
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of our calculations with due regard for the reactions of disproportionation of Qn structons in the melt confirm this prediction (Tables 1, 2). A more detailed analysis of the molecular mass distributions for silicon­oxy gen anions along the liquidus in the Na2O­SiO2 sys tem is given in our earlier work [35]. ACKNOWLEDGMENTS This study was supported by the Russian Founda tion for Basic Research (project no. 08 05 00194). REFERENCES
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