Theoretical Analysis of Miscibility Gaps in the Alkali-Borates

Abstract

A thermodynamic approach based on the regular solution concept is applied to the description of miscibility gap boundaries in the alkali-borate systems. It is suggested that in each system the structural units which control the entropy of mixing are the stoichiometric compounds at the apparent limit of the alkali-rich edge of the gap, and a complex boron trioxide structure. (The former is inferred by the shape of the gap, and the latter is chosen to fit the regular mixing equation.) The same boron trioxide complex is used for all the systems analyzed. The physical implications of this analysis are discussed.

1. Introduction

While miscibility gaps have been measured in a large number of systems [1]¹, there have been few attempts to analyze their shape. Concentrating on the oxide glass systems, we find that the majority of these gaps have been observed in complex systems consisting of more than two component oxides. Since it is not clear whether these systems separate into two or more phases [2], and since the exact chemical identities of the coexistent phases have largely escaped definition, a first principle analysis of the thermodynamics of their phase-separation is not possible. There are, however, several binary oxide mixtures which exhibit immiscibility at high glassformer concentrations. These are largely the alkali silicates [3, 4] and borates [5], Despite this apparent simplification, attempts to describe the shape of the miscibility gap boundaries in these binary systems have not been generally successful [6, 7].

The regular-mixing approach [8] which depends upon balancing the exothermic entropy of mixing and an endothermic enthalpy of mixing results in a coexistence or miscibility gap boundary curve which is symmetric in terms of the relative concentrations of the two immiscible liquids. The alkali-borates and silicates, which represent the major binary oxide systems with miscibility gaps, show highly asymmetric gaps. In the case of the alkali-silicates, the miscibility gap is limited at one branch by silica and at the other by the first stoichiometric crystalline compound encountered as alkali oxide is added. Assuming that the two demixing liquids are SiO₂ and this stoichiometric compound still leads to an asymmetric gap [6]. This behavior has discouraged the use of regular-mixing equations [8] to describe such gaps. The borates pose a more serious problem to this approach, since their miscibility gaps reach beyond several stoichiometric crystal compounds.

The most serious attempt to describe the thermodynamics of phase separation in binary oxide systems was recently presented by R. J. Charles [7]. Assuming that the enthalpy of mixing in the silicates resulted principally from changes of coordination and valence of oxygen ions in the melt and changes in bonding energy at oxygen sites, he was able to show that the enthalpic heat of mixing of transition metal oxide-silica mixtures was endothermic. The addition of this endothermic enthalpy contribution to the exothermic entropy contribution in the Gibbs free energy leads to phase-separation. Following the same considerations, he found that the heat of mixing for the alkali-metal and alkaline-earth oxide mixtures with silica was exothermic thus negating any possibility for phase separation. Since these systems do exhibit phase separation, it was concluded that the partial molar heat of solution of silica, which had been previously overlooked, was a prime candidate for the necessary large endothermic contribution to the heat of mixing. The implications of such an assumption are interesting, since they point to the formation upon phase-separation of a complex silica structure larger than a single SiO₄ tetrahedron.

This hypothesis, proposed by Charles, serves well to introduce the concepts of our model, although the two were developed independently. Below we propose a model to describe the thermodynamic processes which control liquid-liquid immiscibility in the binary oxide glasses. This is based upon the concept that molten oxides are made up of small structural units which control the physical properties of their melts. The configurational entropy of mixing is therefore determined by the possible rearrangements of these units. As a result, the free energy of the system is not expressed in terms of molar concentrations of the component oxides, but rather in terms of the relative concentrations of the dominant structural units. This approach allows the use of the regular mixing equation and replaces the arbitrary assumption that the glass-former, say B₂O₃, mixes as a cell containing two borons and three oxygens, with the more physically acceptable concept that the cell consists of a complex structure, (B₂O₃)_m, with m being an integer. In this paper, we will consider the alkali-metal oxide-boric oxide mixtures, since their gaps occur in similar temperature ranges, thus reducing the number of undetermined parameters. The alkali-silicates are covered in a separate paper [3].

2. Structural Units in the Melt

The existence of small stable species made up of several molecules in the melts of oxide glassformers has long been suspected. A number of authors have proposed complex species for various structures in order to describe structural, thermodynamic and kinetic measurements [10–17].

In these various studies it seems clear that the structure of the primary network formers −SiO₂, B₂O₃ and GeO₂ – extends well beyond the nearest neighbor coordination, but the definition of a molecule of glassformer is not clear and may easily depend upon the property analyzed.

The smallest unit which retains its identity when two liquids are mixed may be defined in several ways. Following a correlation function approach, a size corresponding to an ordered region may be obtained. A geometrical approach, emphasizing the structural units (e.g., boroxol rings for B₂O₃) may be tested by x-ray diffraction. However, it appears that the different investigators do not agree on the various interpretations of the x-ray data. A thermodynamic approach is also possible, whereby the exact nature of the structural units is not specified and only the thermodynamic behavior of the system is analyzed.

In this paper we will restrict our analysis to thermodynamic concepts, since we are only concerned with investigating the variables which determine the shape of miscibility gaps in the oxide mixtures.

3. Complex Molecule Model for Miscibility Gaps

In considering the thermodynamic properties of binary mixtures, simple mixing between the glass-formers, B₂O₃ or SiO₂, and the alkali oxides, M₂O or MO can be rejected on structural grounds, as done by other authors [6], since it is unlikely that such molecules exist in the melt. Let us analyze, instead, mixing between more complex and hopefully, more realistic structures.

As an example, in the phase diagram of the lithium-borate system it will be assumed that mixing occurs between the network former and a stoichiometric chemical compound whose composition can be obtained by extrapolating the immiscibility or coexistence curve to zero Kelvin. By inspection it appears that the coexistence curve can best be extrapolated to the compound Li₂O · 4B₂O₃ at low temperatures. The model which we propose thus considers the Li₂O · 4B₂O₃ composition as one of the mixing liquids.

The choice of this stoichiometric compound as one of the mixing liquids is not inconsistent with the existence of a crystalline compound at Li₂O · 5B₂O₃, since it is well accepted that borate melts have far different structures than the crystalline states. Thus, it is unlikely that a correspondence would develop between the structures of compounds in the melt and those of crystals, in the high B₂O₃ region of the systems.

The significant contribution presented by this paper involves choosing a complex boron molecule, [(B₂O₃)_m], as the second liquid. The complex molecule is a somewhat stable grouping of boron and oxygen atoms which can be identified as a single species in the entropy calculation. This species, however, once identified for a particular glassformer in a given temperature range, must contribute to the entropy calculations of the other miscibility gaps of that glassformer (i.e., in the alkali-borates) occurring in the same temperature range.

Without committing ourselves as to the geometrical appearance of these species, we propose to use the complex molecules, [(M₂O · nB₂O₃)] and [(B₂O₃)_m], in the calculation of the entropy of mixing. In the alkali-borates only one value of m will be used for all the different alkalis, since the miscibility gaps occur in the same temperature range. The silicates, however, present more difficulty due to the vast temperature differences between their miscibility gaps [3].

The composition of the alkali-borate melt is normally written xB₂O₃ · yM₂O, where x and y are the respective molar fractions of B₂O₃ and M₂O. In terms of the new components used here the melt composition will be expressed as $\overline{x}[{(B_2{O}_3)}_m]\cdot \overline{y}[M_{2}O\cdot n{B}_2{O}_3]$, where $\overline{x}$ and $\overline{y}$ are the respective mole fractions of [(B₂O₃)m] and [M₂O nB₂O₃]. It is easily shown that

$$\begin{gathered} \overline{y}=y/\{y+[1-(n+1)y]/m\} \\ \overline{x}=1-\overline{y} \end{gathered}$$ 1

Once the identities of the two separated liquids are found, then each original composition (molar concentrations) can be converted to relative concentrations of the two separated liquids (i.e., $\overline{x}$ and $\overline{y}$) and then the data can be compared with the ideal mixing equation for the system [(B₂O₃)_m] and [M₂O · n(B₂O₃].

The value of n in the formula [M₂O · n(B₂O₃] can be obtained in most instances by visual inspection of the data. The complex glassformer molecule must then be inferred from the data by choosing the value of m necessary to symmetrize the coexistence curve with respect to concentrations of the immiscible liquids. The crucial test then is to see whether all gaps are likewise symmetrized by the same glassformer complex molecule in the alkali-borates.

4. Calculation of the Regular Mixing Equation

The regular mixing equations are well known (see ref. 8, for instance) and we will only review the basic derivation here, for completeness. The free energy of mixing can be expressed in terms of an enthalpy and entropy of mixing. When the enthalpy is positive, phase separation may ultimately occur. The molar enthalpy of mixing is expressed in terms of the two concentrations $\overline{x}$ and $\overline{y}$, and an interaction energy parameter w(w > 0) as follows:

$$\mathrm{\Delta }H=\overline{x}\overline{y}w$$ 2

Contributions to the entropy of mixing of two liquids come from two sources. In general, most authors have ignored the effect due to changes in the internal degrees of freedom of each liquid, and have only been concerned with configurational states [6, 9]. Both terms will be examined here. The configurational molar entropy of mixing is written as:

$$\begin{gathered} \mathrm{\Delta }{S}_1=-R[\overline{x}\ln \overline{x}+\overline{y}\ln \overline{y}] \\ =-R[(1-\overline{y})\ln (1-\overline{y})+\overline{y}ln\overline{y}] \end{gathered}$$ 3

The molar entropy of mixing arising from changes in the internal degrees of freedom upon mixing is expressed as:

$$\mathrm{\Delta }{S}_2=\overline{x}\overline{y}\delta S$$ 4

The structural implications of the magnitude of this term will be discussed later. The molar free energy of mixing now becomes:

$$\begin{gathered} \mathrm{\Delta }G=\mathrm{\Delta }H-T(\mathrm{\Delta }{S}_1+\mathrm{\Delta }{S}_2) \\ \begin{array}{ll}\mathrm{\Delta }G\hfill & =\overline{y}(1-\overline{y})(w-T\delta S)\hfill \\ \hfill & +RT[\overline{y}ln\overline{y}+(1-\overline{y})\ln (1-\overline{y})]\hfill \end{array} \end{gathered}$$ 5

Phase separation is now obtained by minimizing the free energy of mixing with respect to composition:

$${\left(\frac{\partial \text{Δ}G}{\partial \overline{y}}\right)}_{P,T}=0=(1-2\overline{y})w\text{}+RT[ln\overline{y}+\ln (1-\overline{y})-(1-2\overline{y})\delta S/R]$$ 6

This condition leads to a temperature-composition relationship which represents the coexistence curve:

$$T_{coex}=(w/R)\cdot (1-2\overline{y})\cdot {\left\{ln\left[\frac{(1-\overline{y})}{\overline{y}}\right]+(1-2\overline{y})\delta S/R\right\}}^{-1}$$ 7

The critical temperature, T_c, corresponding to the top of the miscibility gap is found by maximizing T_coex:

$$T_c=(w/R)/(2+\delta S/R)$$ 8

The dependence of the normalized coexistence temperature on concentration can then be expressed as:

$$\begin{gathered} T_{coex}/T_c=(2+\delta S/R)\cdot (1-2\overline{y}) \\ \left\{ln\left[\frac{(1-\overline{y})}{\overline{y}}\right]+(1-2\overline{y})\frac{\delta S}{R}\right\}-1 \end{gathered}$$ 9

Once the coexistence curve has been established, it is of interest to calculate the spinodal curve, by letting the second derivative of the free energy of mixing with respect to composition go to zero:

$$\left(\frac{{\partial }^2\mathrm{\Delta }G}{\partial {\overline{y}}^2}\right)=0=-2(w-T\delta S)+RT[1/\overline{y}+1/(1-\overline{y})]$$ 10

The spinodal curve is now written as:

$$T_s=2(w/R)\left[\frac{1}{\overline{y}(1-\overline{y})}+2\delta S/R\right]-1$$ 11

$$T_s/T_c=2(2+\delta S/R)\left[\frac{1}{\overline{y}(1-\overline{y})}+2\delta S/R\right]-1$$ 12

These results may now be transformed into functions of the usual molar fractions of M₂O (e.g., y) by substituting the expression for $\overline{y}$, eq (1), into the coexistence and spinodal equations (eqs (9) and (12)). These solutions are represented in our subsequent graphs by solid and dashed lines, respectively.

5. Comparison with Data

5.1. Lithium-borate System

The data by Shaw and Uhlmann [5] were used to test the equations obtained above. The phase diagram of the lithium-borate system exhibits a liquid-liquid metastable miscibility gap between 4 and 16 mole percent Li₂O. The maximum of the coexistence curve appears at 10 mole percent Li₂O. The stoichiometric compound Li₂O · 4B₂O₃, therefore, appears to limit the gap in the high alkali region. This then will be one of the immiscible liquids at zero Kelvin.

Let us now consider the immiscible liquid on the borate-rich side of the gap. Assuming regular mixing between [B₂O₃] and [Li₂O · 4B₂O₃] yields a maximum in the coexistence curve at 50 percent (B₂O₃) and 50 percent (Li₂O · 4B₂O₃) which corresponds to the 17 mole percent Li₂O concentration. This result disagrees with the measured maximum near 10 percent Li₂O. The choice of a boroxol ring as suggested by Krogh-Moe [18] yields a maximum near 15 mole percent Li₂O, which is also unsatisfactory. The [B₄O₆] cages of Ottar and Ruigh [12] are no more successful with 14 mole percent Li₂O for the maximum. The complex (B₂O₃)_m molecule which allows symmetrization of the gap and leads to a maximum at 10 mole percent Li₂O turns out to be the [(B₂O₃)₅].

While this choice of a complex [(B₂O₃)₅] molecule appears arbitrary in this case, it must be remembered that since the other alkali-borates have miscibility gaps in the same temperature range, we are restricted to the same complex for these other gaps. The data are plotted in terms of concentrations of the two demixing liquids [(B₂O₃)₅ and (Li₂O · 4B₂O₃)] in figure 1. The regular-mixing solution is also plotted, and appears to fit the data well. In this case the entropy contribution from changes in vibrational states of the molecules appears to be negligible (ΔS₂ = 0). A plot of the data and the coexistence and spinodal curves in terms of Li₂O molar concentration is shown in figure 2 and also demonstrates good agreement between the data and the equations.

Figure 1.

Miscibility gap in the lithium-borate system, plotted in terms of concentrations of the two demixing liquids: [(B₂O₃)₅] and [(Li₂O · 4B₂O₃)].

Figure 2. Miscibility gap in the lithium-borate system, plotted in terms of Li₂O concentration.

The solid and dashed lines are the calculated coexistence and spinodal boundary curves, respectively.

5.2. Other Alkali-Metal Oxide-Borate Systems

While we have four adjustable parameters in this analysis, m, n, w and δS, we may only freely adjust use, in fitting the shape of the coexistence curve in the other alkali-borate systems. The value of maximum coexistence temperature, T_c, in general cannot be adjusted by more than 1 percent and determines w directly. The identities of the two demising liquids at zero Kelvin (n and m) are fixed by the data at the high-alkali branch, and by the choice of [(B₂O₃)₅] on the boric oxide side.

Shaw and Ullmann’s measurements in the potassium-borate, rubidium-borate and cesium-borate systems show phase separation occurring between 4 and 20 mole percent K₂O, 4 and 14 mole percent Rb₂O, and 4 and 18 mole percent CS₂O, respectively. The data for the potassium and cesium borate systems indicates that the compound limiting one of the branches of the coexistence curves is at 25 mole percent alkali oxide. A lack of data in the rubidium-borate system prevents a similar analysis, so we assumed that it would follow the same behavior as the other two. Therefore, the demising liquid at the high-alkali side of the gap is [M₂O · 3B₂O₃], where M₂O stands for the three alkali oxides in question.

A regular mixing between [M₂O · 3B₂O₃] and [(B₂O₃)₅] leads to a maximum in the coexistence curve near 11 percent M₂O. Inspection of the data shows good agreement for the K₂O and Cs₂O mixtures. Figures 3 and 5 show that the data appear symmetric when plotted as a function of concentration of these immiscible liquids. The regular-mixing equation gives a good fit of the data with the vibrational contribution to the entropy, use, chosen at +0.5R and −0.8R for the K₂O and Cs₂O systems, respectively. The data for the rubidium system show only fair agreement with the model and require a choice of use = −0.5R for the fit shown (fig. 7). Plots of the data and the coexistence and spin odal curves in terms of alkali-oxide molar concentrations are shown in figures 4, 6, and 8. In general, the model satisfactorily describes the phase-separation data.

Figure 3.

Miscibility gap in the potassium-borate system, plotted in terms of concentrations of the two demixing liquids: [(B₂O₃)₅] and [(K₂O · 3B₂O₃)].

Figure 5.

Miscibility gap in the cesium-borate system, plotted in terms of concentrations of the two demixing liquids: [(B₂O₃)₅] and [(Cs₂O · 3B₂O₃)].

Figure 7.

Miscibility gap in the rubidium-borate system, plotted in terms of concentrations of the two demixing liquids: [(B₂O₃)₅]. and [(Rb₂O · 3B₂O₃)].

Figure 4. Miscibility gap in the potassium-borate system, plotted in terms of K₂O concentration.

The solid and dashed lines are the calculated coexistence and spinodal boundary curves, respectively.

Figure 6. Miscibility gap in the cesium-borate system, plotted in terms of Cs₂O concentration.

The solid and dashed lines are the calculated coexistence and spinodal boundary curves, respectively.

Figure 8. Miscibility gap in the rubidium-borate system, plotted in terms of Rb₂O concentration.

The solid and dashed lines are the calculated coexistence and spinodal boundary curves, respectively.

The above results confirm our choice of the species: [(B₂O₃)₅] as the molecular complex controlling the configurational entropy of the system over the observable temperature range, and therefore, support the merit of this approach. We see that in addition to describing the shape of the coexistence curve, a calculation of the location of the spin odal region is also possible. This has potential usefulness in the interpretation of measurements on glasses with compositions far from the critical composition.

We have omitted a description of the sodium-borate system, because the data appeared very different from that for the other systems. Attempts to symmetrize the coexistence curve were relatively unsuccessful. An attempt to measure the gap as a check on the data was equally unsuccessful due to the quick hydration of the samples.

5.3. Lead-Borate System

Data for the lead-borate system was reported by two authors [19, 20]. One of us, J. H. Simmons, has also recently reported the results of careful measurements of phase separation in this system [21]. Figure 9, reprinted from his paper, shows his data and the fit of the regular-mixing equations. The two immiscible liquids at zero kelvin are [(B₂O₃)₅] and [(Bo · 4B₂O₃)]. Because of the pronounced flatness of the coexistence curve, the contribution to the entropy of mixing from changes in the vibrational states was large, use = + 10R. The implications of this result are discussed in the next section.

Figure 9. Miscibility gap in the lead-borate system, reprinted from ref [21].

The solid and dashed lines are the calculated coexistence and spinodal boundary curves, respectively.

6. Changes in the Internal Degrees of Freedom

The contribution to the entropy of mixing arising from changes in the internal degrees of freedom or vibrational states of the immiscible liquids, ΔS₂, is the parameter which most affects the shape of the miscibility gap boundaries in the fits presented above. Its effect on the regular mixing equation serves to either broaden (use > 0), or narrow (use < 0), the coexistence curve while maintaining symmetry. This is achieved through its effect on the critical temperature, T_c: eq (8). A positive value for the additional entropy corresponds to an increase in the internal degrees of freedom. The additional states developed by mixing thus will tend to suppress phase separation. The evidence is mounting in support of the two-dimensional structural description of pure boron trioxide glass. The addition of alkali-metal oxides tends to convert three-coordinated planar boron’s to four-coordinated boron’s. This helps form a three dimensional network. The change is small, however, since only the boron ions are affected. Because of their low valence the alkali ions cannot form cross linking. In addition, however, one must consider the effect of changes in the molecular vibrational states. As a simple example, we may look at the phonon density of states of a linear chain with alternating different masses (mixed state) and compare it with that of two chains with equal masses (phase-separated state). The result is a decrease in the available states or internal degrees of freedom upon mixing. The additional entropy, use, would be negative thus raising T_c. The effect is more pronounced for the heavier alkali-borate compounds.

The table below lists the various alkali-oxide compounds postulated with their formula weights and the additional entropy terms necessary for fitting the regular mixing equation. The potassium oxide compound is the lightest and it appears that the entropy contribution from the formation of a three dimensional structure dominates. As the mass of the compound increases to the cesium oxide system, the change in vibrational states dominates and the entropy term becomes increasingly negative.

The lead oxide-boric oxide system is also included. Despite its heavy compound, the entropy term is positive. This is due to the large contribution from the change in dimensionality of the system. The lead ions with their high valence become tetrahedral coordinated in solution and add a large number of internal degrees of freedom to the system. This apparently dominates over the effect of the mass change.

7. Conclusion

We have shown that the thermodynamic equations of regular mixing can be applied to phase separation in the alkali-borate glasses. The concept, which must be introduced to accomplish this, is the assumption that the glass former phase is structurally represented by a complex molecule and the glass former-modifier phase by a stoichiometric compound. This transformation symmetrizes the coexistence curve.

The regular mixing equation appears to represent the data well when an additional entropy of mixing arising from changes in the internal degrees of freedom of the system is included in the calculation. Following this approach, five systems were treated successfully. The complex boron trioxide molecule [(B₂O₃)₅] was used in the regular mixing analyses of all the borate systems considered, since the miscibility gaps occurred approximately within the same temperature range.

The implications of such an approach are interesting but not surprising. As mentioned before, by other authors and in some of our other work, the existence of complex glass-former structures in the melt appears to offer a good explanation for the behavior of an increasing number of physical properties of molten oxide glasses.

An actual geometrical description of the complex boron trioxide structure is not accessible to us by this analysis since thermodynamic treatments do not afford structural descriptions.

Table 1.

Immiscible compound	Molecular weight	δS/R	[2+δS/R]−1
[K2O·3B2O3]	303	+0.5	0.4
[Li2O·4B2O3]	308	0	0.5
[Rb2O·3B2O3]	396	−0.5	0.67
[Cs2O·3B2O3]	491	−0.8	0.83
[PbO·4B2O3]	501	+10.0	0.083
[(B2O3)5]	348