8. Phase Equilibria

- 8주차 -

▪ Complete Solid Solubility

▪ For complete solid solubility to occur between two end members, the following conditions have to be satisfied:

1. Structure type: The two end members must have the same structure type. For instance, SiO2 (silica types)and TiO2 (rutile) would not be expected to form complete solid solubility.

2. Valency factor: The two end members must have the same valence. If this condition is not satisfied, compensating defects forms in the host crystal in order to maintain charge neutrality. Given that the entropy increase associated with defect formation is not likely to be compensated for by the energy required to form them over the entire composition range, complete solid solubility is unlikely.

3. Size factor: As a result of the mismatch in size of the solvent and solute ions, strain energy will develop as one is substituted for the other. For complete solid solubility to occur, that excess strain energy has to be low. Hence, in general, the size difference between the ions has to be less than 15%.

4. Chemical affinity: The two end members cannot have too high a chemical affinity for each other. Otherwise the free energy of the system will be lowered by the formation of an intermediate compound.

A typical phase diagrams for two compounds that form a complete solid solubility over their entire composition range is shown in Fig. 8.4. Both NiO and MgO crystallize in the rock salt structure, and their radii are very similar (i.e., 86 pm for Mg2+, 83 pm for Ni2+).

To illustrate the use and usefulness of phase diagrams, it is instructive to take a composition in Fig. 8.4, say, 60 mol % MgO, and examine what happens as it is cooled from the melt.

At T1, a solid solution of MgO and NiO (roughly 80 mol % MgO) will start solidifying.

At T2 ≈ 2500℃, two phases coexist: a solid solution of composition Z and a liquid solution of composition X. The relative amounts of the each phase are given by the lever rule:

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑀𝑀𝑓𝑓 𝑀𝑀𝑓𝑓𝑙𝑙𝑙𝑙𝑓𝑓𝑙𝑙 =𝑦𝑦𝑦𝑦𝑥𝑥𝑦𝑦

and mole fraction solid =𝑥𝑥𝑦𝑦𝑥𝑥𝑦𝑦

Note that as the temperature is lowered, the composition of the solid solution moves along the solidus line toward NiO, while that of the liquid moves along the liquidus line.

At T3 ≈ 2400℃, the final liquid solidifies, and the composition of the solid solution is now the same as the initial composition.

▪ Eutectic Diagrams with Partial Solid Solubility and No Intermediate Compounds▪ Given the numerous restrictions needed for the formation of complete solid

solutions, they are exception rather than the rule − most ceramic binary phase diagrams exhibit partial solubility instead.

Furthermore, the addition of one component to another will lower the freezing point of the mixture relative to the melting point of the end members. The end result is lowered liquidus curves for both end members which intersect at a point. The point of intersection defines the lowest temperature at which a liquid can exist and is known as the eutectic temperature TE.

This type of diagram is well illustrated by the MgO−CaO system shown in Fig. 8.6, where MgO dissolves some CaO and vice versa.

The limited solubility comes about mostly because the size difference between the Ca and Mg ions too large for complete solid solubility to occur. (i.e., 86 pm for Mg2+, 114 pm for Ca2+ & note that both MgO and CaO have the rock salt structure.)∴ Beyond a certain composition, the increase in strain energy associated with increasing solute content can no longer be compensated for by the increase in configuration entropy.

To illustrate the changes that occur upon cooling consider what happens when a 40 mol % CaO composition, depicted by the dotted vertical line in Fig. 8.6, is cooled from the melt.

Above 2600℃ the liquid phase is stable.

Just below 2600℃, a MgO solid solution (≈95 mol % MgO saturated with CaO) will start to precipitate out.

At 2500℃, two phases will coexist: an MgO−CaO solid solution and a liquid that is now richer in CaO (≈55 mol % CaO) than the initial composition.

Upon further cooling, the composition of the liquid follows the liquidus line toward the eutectic composition, whereas the composition of the precipitating solid follows the solidus line toward the point of maximum solubility.

Just above TE, that is, at TE + δ, a solid solution of CaO in MgO and a liquid with the eutectic composition XE ≈ 65% CaO coexist.

Just below TE, however, the following reaction

𝐿𝐿 ⇒ 𝑆𝑆1 + 𝑆𝑆2

known as a eutectic reaction, occurs, and the liquid disproportionates into two phases of very different compositions − a calcia-rich and a magnesia-rich solid solution.

At the eutectic temperature, three phases coexist and there are no degrees of freedom left.In other words, the coexistence of three phases in a two-component system can occur only at a unique temperature (TE), pressure, and composition (XE).

▪ Partial Solid Solubility with Formation of Intermediate Compounds

▪ One of the conditions for the existence of a wide solid solution domain is the absence of a strong affinity of the end members for one another. That is not always the case − in many instances, the two end members react to form intermediate compounds.

For instance, the compound 𝐴𝐴𝑥𝑥𝐵𝐵𝑦𝑦𝑂𝑂2 can be formed by the reaction

𝑥𝑥𝐴𝐴𝑂𝑂1/𝑥𝑥 + 𝑦𝑦𝐵𝐵𝑂𝑂1/𝑦𝑦 ⇒ 𝐴𝐴𝑥𝑥𝐵𝐵𝑦𝑦𝑂𝑂2

where the free energy change for the reaction exceeds that for the simple mixing of the two end members to form a solid solution.

Under these conditions, intermediate compounds appear in the phase diagram, which in analogy to the end members either can be line compound (i.e., solubility of end members in the intermediate compound is small) or can have a wide range of stoichiometry.

Furthermore, these intermediate phases can melt either congruently or incongruently.

Congruently melting intermediate phases

▪ Na2O⋅SiO2 and Na2O⋅2SiO2, shown in Fig. 8.7a, are examples of line compounds that melt congruently, i.e., without a change in composition.Note that in this case the resulting phase diagram is simply split into a series of smaller simple eutectic systems (e.g., Fig. 8.7a, for compositions greater than 50 mol % SiO2).

▪ Spinel, MgO⋅Al2O3, however, which also melts congruently and splits the phase diagrams into two simple eutectic systems (Fig. 8.7b), is not a line compound but readily dissolves significant amounts of MgO, and Al2O3.

Incongruently melting intermediate phases

▪ If the intermediate compound melts incongruently, i.e., the compound dissociates before melting into a liquid and another solid, then the phase diagram becomes slightly more complicated.

A typical example of such a system is the SiO2 − Al2O3 system (Fig. 8.8) where mullite, 2SiO2⋅3Al2O3, melts at ≈1828℃, by the formation of a liquid containing ≈40 mol % Al2O3 and “pure” alumina according to the reaction

𝑆𝑆1 ⇒ 𝐿𝐿 + 𝑆𝑆2

This reaction is known as a peritectic reactionand quite common in ceramic systems.

Other examples of incongruently melting ternary compounds are and 2Na2O⋅SiO2 (Fig. 8.7a) and 3Li2O⋅B2O3 (Fig. 8.9).

This brings up the topic to be considered in this section which is nothing but a variation of the aforementioned case, and in which a ternary compound will dissociate into two other solid phases upon either cooling or heating.

For example, according to Fig. 8.9, at about 700℃, 2Li2O⋅5B2O3 will dissociate into the 1:2 and 1:3 compounds.

8.5 Ternary Systems▪ Ternary phase diagrams relate the phases to temperature in a three-

component system, and the four variables to be considered are temperature, pressure, and the concentration of two components (the composition of the third is fixed by the other two).

A graphical representation is possible if the three components are represented by an equilateral triangle, where the apexes of the triangle represent the pure components, and temperature on a vertical axis as shown in Fig. 8.10a.

The two-dimensional representation of the same diagram is shown in Fig. 8.10c, where the intersection of two surfaces is a line, the intersection of three surfaces is a point, and the temperature is represented by isotherms.The temperature of the liquidus surface is depicted by isothermal contours, as shown in Fig. 8.10c.

The boundary curves represent equilibrium between two solids and the liquid (three phases), and the intersection of the boundary curves represents four phases in equilibrium (three solid phases and a liquid). This point is the lowest temperature at which a liquid can exist and, in complete analogy to the binary case, is called the ternary eutectic.

The composition at any point is found by drawing lines parallel to the three sides of the triangle. For example, the composition of point X (Fig. 8.10b) is 40 mol % AO, 20 mol % BO, and 40 mol % MO, while that at point Y is 80 mol % MO, 10 mol % AO, with a balance of BO.

Any ternary phase diagram that does not include a solubility region will consist of a number of compatibility triangles, shown as solid lines in Fig. 8.10d, with the apexes of the triangles representing the solid phases that would be present at equilibrium.

For example, if the starting mixture is point X in Fig. 8.10d, then at equilibrium the phases present will be MO, 2AO⋅MO, and AO⋅BO. Similarly, for an initial composition at point Z, MO, BO, and the quaternary phase AO⋅2MO⋅2BO are the equilibrium phases, and so forth.

Once the compatibility triangles are known, both the phases present at equilibrium and their relative amounts can be determined. At equilibrium, composition X (Fig 8.10d) would comprise the phases MO, 2AO⋅MO, and AO⋅BO in the following proportions:

𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑀𝑀𝑓𝑓 𝑀𝑀𝑂𝑂 =𝑋𝑋𝑓𝑓

𝑓𝑓 − 𝑀𝑀𝑂𝑂

𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑀𝑀𝑓𝑓 2𝐴𝐴𝑂𝑂 � 𝑀𝑀𝑂𝑂 =𝑋𝑋𝑋𝑋

𝑋𝑋 − 2𝐴𝐴𝑂𝑂 � 𝑀𝑀𝑀𝑀

𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑀𝑀𝑓𝑓 𝑀𝑀𝑓𝑓 𝐴𝐴𝑂𝑂 � 𝐵𝐵𝑂𝑂 = 𝑋𝑋𝑋𝑋−𝐴𝐴𝐴𝐴�𝐵𝐵𝐴𝐴

8.6 Free-Energy Composition and Temperature Diagrams▪ The previous sections dealt with various types of phase diagrams and their

interpretations. What has been glossed over, however, is what determines their shape.

In principle, the answer is simple: the phase or combination of phases for which the free energy of the system is lowest is by definition the equilibrium state.

If the free-energy function for all phases in a given system were known as a function of temperature and composition, how could one construct the corresponding phase diagram?

In other words, what is the relationship between free energies and phase diagram? Two examples are considered next.

Note that in going from a ternary to a binary representation, a dimension is lose: planes become lines and lines become points. Thus a quaternary phase is a point, and the edges of the triangles represent the corresponding binary phase diagrams.

▪ Polymorphic Transformations in Unary Systems▪ Congruent melting of a compound, or any of the polymorphic transformations

discussed earlier, is a good example of this type of transformation.

To illustrate, consider the melting of a compound. The temperature dependence of the free-energy function for liquid is

𝐺𝐺𝑇𝑇,𝑙𝑙𝑙𝑙𝑙𝑙 = 𝐻𝐻𝑇𝑇,𝑙𝑙𝑙𝑙𝑙𝑙 − 𝑇𝑇𝑆𝑆𝑙𝑙𝑙𝑙𝑙𝑙

while that for the solid phase is

𝐺𝐺𝑇𝑇,𝑠𝑠 = 𝐻𝐻𝑇𝑇,𝑠𝑠 − 𝑇𝑇𝑆𝑆𝑠𝑠

where 𝐻𝐻 and 𝑆𝑆 are the enthalpy and entropy of the solid and liquid phases, respectively.

The two functions are plotted in Fig. 9.1, assuming they are linear functions of temperature, which is only valid as long as (1) the heat capacities are not strong functions of temperature and (2) the temperature range considered is not too large.

𝐺𝐺𝑇𝑇,𝑙𝑙𝑙𝑙𝑙𝑙 is steeper than 𝐺𝐺𝑇𝑇,𝑠𝑠 because the entropy content of the liquid is larger (more disorder) than that of the solid.

The salient point here is that at the temperature above which the lines intersect, the liquid has the lower energy and thus is the more stable phase, whereas below that temperature the solid is.

Not surprisingly, the intersection temperature is the melting point of the solid.

𝑓𝑓𝑝𝑝 = (𝑙𝑙𝑙𝑙𝑙𝑙𝑇𝑇)𝑝𝑝= (

𝑙𝑙𝐻𝐻𝑙𝑙𝑇𝑇)𝑝𝑝

where 𝑓𝑓𝑝𝑝 is the heat capacity measured at constant pressure.

▪ Complete Solid Solutions

▪ The free-energy versus composition diagram for a system that exhibits complete solid solubility is shown in Fig. 8.11.

The components of the diagram are the two vertical axes that represent pure AO (left) and pure BO (right). The point labeled 𝜇𝜇𝐴𝐴𝐴𝐴0 represents the molar free energy of formation ∆𝐺𝐺𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 of AO from its elements, and similarly, 𝜇𝜇𝐵𝐵𝐴𝐴0

for BO.

If, for simplicity’s sake, the solution is assumed to be ideal, that is, ∆𝐻𝐻𝑓𝑓𝑙𝑙𝑥𝑥 = 0(no heat of mixing), then the free energy of mixing of AO and BO is given by

∆𝑮𝑮𝒎𝒎𝒎𝒎𝒎𝒎 < 𝟎𝟎

𝐺𝐺 = 𝑋𝑋𝐴𝐴𝐴𝐴𝜇𝜇𝐴𝐴𝐴𝐴0 + 𝑋𝑋𝐵𝐵𝐴𝐴𝜇𝜇𝐵𝐵𝐴𝐴0 + ∆𝐻𝐻𝑓𝑓𝑙𝑙𝑥𝑥 − 𝑇𝑇∆𝑆𝑆𝑓𝑓𝑙𝑙𝑥𝑥

where 𝑋𝑋𝑙𝑙 represents the mole fraction of 𝑓𝑓.

The entropy change of mixing ∆𝑆𝑆𝑓𝑓𝑙𝑙𝑥𝑥 [see Eq. (5.11)] is given by

∆𝑆𝑆𝑓𝑓𝑙𝑙𝑥𝑥 = −𝑅𝑅(𝑋𝑋𝐴𝐴𝐴𝐴𝑀𝑀𝑓𝑓𝑋𝑋𝐴𝐴𝐴𝐴 + 𝑋𝑋𝐵𝐵𝐴𝐴𝑀𝑀𝑓𝑓𝑋𝑋𝐵𝐵𝐴𝐴)

Combining these two equations and plotting 𝐺𝐺 versus composition yields the curve 𝜇𝜇𝐴𝐴𝐴𝐴0 − 𝑀𝑀 − 𝜇𝜇𝐵𝐵𝐴𝐴0 shown in Fig. 8.11.

Using the same arguments, a free-energy versus composition function for the liquid solution can be determined. Superimposing the two functions as a function of temperature results in the curves depicted in Fig. 8.12b to d.

It is, in principle, from these types of curves that the corresponding phase diagram shown in Fig. 8.12a can be plotted.

At T1 the free energy of the liquid solution is lowest at all compositions (Fig. 8.12b) and is the only phase that exists at that temperature.

∆𝐺𝐺𝑓𝑓𝑙𝑙𝑥𝑥before mixing

Conversely, at T3, the solid solution is the most stable phase (Fig. 8.12d).

At some intermediate temperature T2, the free-energy versus composition curves have to intersect (Fig. 8.12c), from which it is obvious that, as depicted in Fig. 8.12a,

(1) Between pure AO and point M, the lowest energy of the system is that of the liquid solution.

(2) Between BO and N, the solid solution has the lowest energy.

(3) Between compositions M and N, the system’s lowest energy state is given by the common tangent construction. In other words, the system’s lowest free energy occurs when two phases (a solid phase and a liquid phase) coexist.