An ordinary experiment intended to construct phase diagrams can only be carried out within some temperature range, which is as a rule very limited. Some theoretical speculations are required in order for a phase equilibrium pattern to be expanded to unstud ied temperature ranges, specifically, to low tempera tures.
This study concerns features of phase diagrams and stoichiometry and nonstoichiometry problems that follow from the third law of thermodynamics. Another point of discussion is how to gain data on hightem perature and lowtemperature states. Binary systems are in the focus; however, general conclusions will be expanded to multinary systems. Provisional consider ation is found in [1].
DECOMPOSITION OR ORDERING IN PHASES OF VARIABLE COMPOSITION
One version of the third principle of thermody namics [2] recites that, as temperature tends to abso lute zero, the system’s entropy tends to zero, too. In other words, a system that approaches the zero on the Kelvin scale in a quasiequilibrium manner enters a fully ordered state. In the context of a phase diagram, this means that all solid solutions would decompose as temperature is lowered, as the entropy of any solution is higher than the entropy of a mixture of its compo nents. On the other hand, strictly stoichiometric com pounds lacking crystal lattice defects can exist at 0 K. Such a compound represents a completely ordered system whose entropy is zero in the classical physics approximation. Thus, the decomposition of solid solutions upon cooling can follow two routes: either demixing to components or ordering, that is, decom position to compounds that have an ordered arrange ment of atoms or ions, as distinct from a random dis order in solid solutions [3]. When, for example, only continuous solid solutions are shown in a phase dia gram, this means that lowtemperature equilibria have been either unstudied or studied insufficiently care fully.
Figures 1 and 2 display representative T–x dia grams of binary systems where continuous solid solu tions experience equilibrium decomposition upon temperature depression.1 Figure 1b corresponds to, for example, phase equilibria in the NaCl–KCl sys tem. Figure 2a makes it clear that retrograde solubility, which is infrequently observed experimentally, is very likely at low temperatures given that a polymorphic transformation occurs in one component of, for example, the SrCl2–BaCl2 system.
Decomposition can be accompanied by ordering. Figure 3a shows a variation of a phase diagram where the decomposition of a continuous hightemperature solid solution is accompanied by the formation of a lowtemperature phase. This type of phase diagram is likely intrinsic to the PbF2–CdF2 system [4].
In the same way, phases of variable composition cannot exist at the absolute zero temperature; as tem perature decreases, they should either decompose or convert to strictly stoichiometric compounds. When Xray crystallography signifies a random occupation of some position by different ions or an incomplete occupation of some position, the relevant phase can not be stable at 0 K, and upon quasiequilibrium cool ing it should either decompose or experience poly morphic transition to an ordered phase.
Figure 3b shows the eutectoid decomposition scheme for a hightemperature phase of variable com position. Figure 4 shows the transformation of a disor dered hightemperature phase of variable composition to a stoichiometric compound. Figure 4a corresponds to continuous ordering, i.e., systematic reduction in the density of point defects in response to lowering temperature. Figure 4b corresponds to a sharp increase in the degree of order and the associated less ening of the homogeneity range of an intermediate phase as a result of the polymorphic transition. Figure 4a corresponds to the behavior of spinel in the MgO– Al2O3 phase [5].
1Isobaric sections of phase diagrams are considered throughout this work.
Third Law of Thermodynamics as Applied to Phase Diagrams
P. P. FedorovProkhorov Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia Abstract—Consequences from the third law of thermodynamics are analyzed from the standpoint of low temperature phase equilibria. The kinetics of attainment of lowtemperature equilibria and some ordering and decomposition features of solid solutions are considered.
The decomposition and/or ordering of phases of variable composition are two possible scenarios of the manifestations of the third law. However, ordering that occurs in an equilibrium manner is almost always (in any system but not for any composition) accompanied by decomposition. In firstorder phase transitions, only discrete compositions can exist where ordering is congruent, that is, where the compositions of two phases are identical (Fig. 3a). This situation is an ana logue of congruent melting which can occur only at discrete points of the concentration space [6].
In secondorder phase transitions, ordering can occur without compositional changes (congruently) over a range of concentration. This type of phase tran sition needs some theoretical analysis to be revealed.
A narrow homogeneity range of an ordered phase corresponds to a strong concentration dependence of the Gibbs free energy G(x) of this phase with a small curvature radius. The appearance of such a stable phase brings about a sharp narrowing of homogeneity ranges for both the parent nonstoichiometric phase and adjacent disordered phases (Fig. 5). This is the HumeRothery rule [7], which was inferred on the basis of geometric thermodynamics.
This rule is illustrated by the BaF2–NdF3 system (Fig. 6). Two extensive regions of component’s based solid solutions, one having the fluorite structure (phase F) and the other having the tysonite structure (phase T), were revealed when hightemperature phase equilibria were studied [8]. Notably, the bound ary concentrations of the solid solutions remain prac x (a) (b) L + α L L L + α α α α1 + α2 T 0 α + β β α1 + β 0 x β
Fig. 2. Representative T–x diagrams for binary systems with continuous solid solutions between one component and the high temperature phase of the other: (a) binodal lies beneath solvus curves and (b) binodal crosses the solvus curve of the α phase. Dashed lines: metastable equilibria.
T L L + α α α1 + α2 0 L L + α L + α α1 + α2 x (a) (b) x 0 䊉
Fig. 1. Representative T–x diagrams for binary systems with continuous solid solutions between the components: (a) without minimum on melting curves and (b) with a minimum on melting curves. Dots: spinodal.
tically unchanged from the eutectic temperature to 900°С. A lowtemperature phase having trigonaldis torted cubic structure of idealized composition Ba4Nd3F17 (phase R) was also discovered in this system [9]. Phaseequilibria studies at 600°С [10] showed that the appearance of this phase strongly limits the extents of BaF2 and NdF3based solid solutions.
The manifestation of the HumeRothery rule is well defined in isothermal sections, too. By the way of example, Fig. 7 shows phase equilibria in the NaF– PbF2–BiF3 system as reported in [11]. The existence of an NaBiF4 phase (I) having a narrow homogeneity range gives rise to a characteristic dent in the concen tration region of existence of a fluorite phase of vari able composition (phase F), although these phases are not structurally related.
The same rule may be recited differently [12], as follows: if in some isothermal section a solid solution is found to exist in equilibrium with an ordered phase (α + γ at Т2 in Fig. 5c), this means that the boundary of the equilibrium homogeneity region of the disor dered phase α is strongly dependent on temperature. In light of this, it has no sense to determine the homo geneity range for a disordered phase unless the data are referred to a specific temperature; see, e.g., a study on the BaF2–NdF3 system [13].
We must note that the violation of the Hume Rothery thermodynamic rule directly signifies that the pattern of phase fields does not match thermodynamic equilibrium. Specifically, the observed ordering could have occurred upon cooling (quenching) of samples synthesized at high temperatures.
T L L + α γ α + γ γ + α 0 x L α γ β α + γ γ + β α + β (а) (b) x α L + α
Fig. 3. T–x diagrams for binary systems: (a) decomposition of a continuous solid solution with the segregation of an ordered phase γ and (b) decomposition of a phase of variable composition γ by a eutectoid scheme upon a decrease in temperature.
L α γ β α + γ γ + β T 0 x (а) (b) x γ + β β γ L α α + γ α + δ δ + β δ
Fig. 4. Transformation of a phase of variable composition γ (berthollide) in a binary system to a stoichiometric compound upon
As a example let us consider ordering in phases of variable composition having an LaF3type (tysonite) structure in CaF2–RF3 (R = Tb = Tb–Lu, Y) systems [14]. These phases are berthollides, namely, solid solu tions based on unstable trigonal RF3 polymorphs sta bilized by anionic vacancies. Xray diffraction studies of annealed and quenched samples [15, 16] in mar ginal portions of the homogeneity region revealed a new diffraction pattern of ordered phases, which dif fers from the Xray diffraction patterns of tysonite type phases (LaF3) in that the main reflections are split and superstructure reflections appear (Fig. 8). Elec tron diffraction helped to determine the monoclinic structure of these phases and an idealized composition of Ca3Y7F27 [17]. A scrupulous study of annealed and quenched (at ~200 K/s) samples revealed narrow two phase fields suspended between the tysonite (T) phase and an ordered (T') phase, the concentration bound aries of these phases being virtually temperature inde pendent (Fig. 9). Precisely in this way (with concen trationdependent ordering and a narrow twophase field) interpreted were phase equilibria in the CaF2– YF3 system [18]. However, here we are dealing with an evident violation of the HumeRothery rule. High temperature Xray diffraction showed that ordering occurs while temperature rather than concentration changes, and the tysonite phase is not quenched in some part of the concentration range creating an illu sion of concentrationdependent ordering. DTA gave order–disorder transition temperatures, which are 640°С for R = Er and 656°С for R = Y. Figures 9 and
10 make it clear that these temperatures do not match the equilibrium existence fields of disordered tysonite phases.
The summary phase diagram for the CaF2–YF3 system (data were borrowed from [14, 19–25]) dis played in Fig. 11 demonstrates various types of decom position and ordering. At high temperatures, grossly nonstoichiometric phases are formed in this system: a fluorite solid solution Ca1 – xYxF2 + x featuring a flat maximum on melting curves, a solid solution based on the hightemperature αYF3 phase, and a tysonite type intermediate phase of variable composition cor responding to the highpressure YF3 phase and having a maximum at 71.5 mol % YF3. As temperature decreases, all nonstoichiometric phases decompose: αY1 – yCayF3 – y and tysonite phase T decompose by a eutectoid scheme, and the fluorite solid solution decomposes to yield a series of ordered phases having various fluoriterelated structures. The dashed line corresponds to a diffuse phase transition, namely, a gradual disordering of the fluoride structure upon a rise in temperature due to the generation of a lot of antiFrenkel defects in the fluorite structure (“melt ing” of the anionic sublattice). This disordering occur ring within a temperature range is not a true phase transition; the dashed line is a conventional boundary. However, an anomalous Sshaped solvus curse is due to this transition [26, 27]. The stability boundaries of the Ca3Y7F27 ordered tysonitelike phase have not been determined. Apparently, this is a metastable phase and has no place in the phase diagram.
G γ α β T1 α + β γ + β α + γ γ x x T1 T2 T β α T2 β γ α x G (a) (b) (c) 䊉
Fig. 5. Illustration of the HumeRothery rule for a binary system where appearance of stable phase γ having a small homogeneity
range abruptly narrows the fields of phases of variable composition α and β: (a, b) Gibbs free energy curves G(x) at temperatures of T1 and Т2, respectively, and (c) T–x diagram.
METHODOLOGY OF STUDYING LOWTEMPERATURE EQUILIBRIA Lowtemperature phase equilibria are difficult to study. An ordinary way to determine the annealing time required for equilibration is by increasing the annealing time by trial and error until the phase com position and properties of samples stop changing. However, this criterion (used, e.g., in [28]) is abso lutely unsatisfactory as changes can be too slow to be noticed in laboratory studies.
Useful equilibration criteria are as follows [1]: • Convergence of results obtained in approaching equilibrium " from above" and “from below” (e.g., in
annealing of fused and unfused samples), or conver gence of DTA and annealing results;
• Sharp Xray diffraction reflections, specifically, resolved peaks from α1 and α2 radiations at far angles; here, however, quenching can have a considerable effect;
• The number of phases observed obeying the phase rule;
• Invariable unit cell parameters of phases within monovariant fields in T–x diagrams; specifically, if unit cell parameters within twophase fields in a binary system vary, either equilibrium is not attained or the system is more complex than binary.
T, °C 1500 1000 500 0 BaF2 20 40 60 80 NdF3 mol % L F F + T T F + R R + T R 䊉
Fig. 6. Summary T–x diagram for the BaF2–NdF3 system after [8, 10]. Notations: F = Ba1 – xNdxF2 + x solid solution, R = ordered fluoritelike phase of composition Ba4Nd3F17, and T = Nd1 – yBayF3 – y tysonite solid solution.
All the aforementioned criteria cannot rule out the existence of metastable equilibria.
It must be kept in mind that samples from different concentration regions of a diagram can require differ ent annealing time. In case incomplete component based solid solutions are formed, twophase assem blages are the first to sinter and solutions of intermedi ate concentrations are the last to do.
Figure 12 displays experimentally derived equili bration times versus temperature for phases of variable composition during sintering of component mixtures according to [29]. The plots show that the equilibra tion time increases nonlinearly as a function of decreasing temperature and very rapidly reaches unre alistic values. The table displays annealing times for СаF2–RF3 systems calculated according to Fig. 12. The table makes it clear that the difficulty of attaining lowtemperature equilibria very rapidly develops from technical to fundamental. The period of time equal to the existence of the Earth corresponds to equilibration at temperatures of about 200°С.
The above data illustrate that, for each system, there is a temperature below which experimental studies involving equilibration of the system are impossible. 1 2 3 4 * PbF2 80 60 40 20 NaF + I + F NaF + F * I + F F F + T 20 40 60 80 20 40 60 80 BiF3 mol % I NaF
Fig. 7. Phase equilibria in the NaF–PbF2–BiF3 system at 400°С according to [11]: (1) singlephase samples (phase F), (2)–(4) twophase samples as determined by Xray diffraction in annealed samples, and I–NaBiF4.
60 40 20 70.5 mol % 71.0 71.5 75.0 LaF3 2θ, deg ErF3
Fig. 8. Xray diffraction patterns for samples in the CaF2– ErF3 system after annealing at 1057°С [14]. CuKα radia tion.
For example, the narrowing of an LiNbO3based homogeneity region was traced only in the range from the melting temperature to 700°С [30]; at lower tem peratures, the decomposition of a hightemperature solid solution takes too much time.
Naturally, there are great many methods for enhancing synthesis via shortening equilibration time by many orders of magnitude.
One such method consists of increasing the reac tivity of powders to be sintered. Equilibration in the BaF2–NdF3 system at 600°С was achieved by using barium fluoride that had been prepared by thermolysis
of hydrofluoride BaF2 ⋅ HF [10]. Decomposition con serves the pseudomorphose to microcrystals of the ini tial hydrofluoride, but the matter filling in the form has a very developed inner surface and is highly reac tive (Fig. 13).
A great gain in equilibration time is given by decreasing the diffusion path on account of the small sizes of reacting particles, specifically, via transition to the nanometersized level. Various lowtemperature chemical methods of preparing precursors are known to provide a considerable reduction in the process 2θ, deg 49 48 47 46 45 27 26 25 70.0 70.5 71.0 71.5 72.0 72.5 mol % ErF3 T' T' + T T
Fig. 9. Positions of Xray diffraction lines and the twophase region in the CaF2–ErF3 system at 1008°С [14]. CuKα radiation. Phase notation: T = tysonite phase and T' = ordered tysonitelike phase.
temperatures compared to solidphase syntheses of target phases (see, e.g., [3134]).
Great opportunities for attainment of lowtemper ature equilibria are opened by use of solvents, which are inert wherever possible. An additional liquid phase very strongly enhances mass transfer. Specifically, hydrothermal synthesis was used to obtain data on lowtemperature equilibria in the CaF2–YF3 (Fig. 11) and SrF2–LaF3 [20, 35] systems in good agreement with the results of hightemperature studies. The range of the temperatures was expanded by ~500°С. Hydrothermal synthesis is systematically employed to attain equilibria in silicate systems. However, water (a
very reactive substance) is a cunning solvent. A repre sentative example is the stabilization of the cubic A polymorphs of the ceriumfamily rareearth oxides by account of residual water [36, 37]. Hydrothermal syn thesis of fluoride systems inevitably brings about some substitution of hydroxide for fluoride ions in phases segregated.
Vaporphase synthesis provides a considerable enhancement of equilibration [38].
The abovedescribed methods, although allowing a considerable propagation into the range of lowtem peratures compared to mere sintering, do not deny the
mol % ErF3 T' T T, °C 1100 1000 900 800 1130 1122 1092 70 80 1117 1 2 3 L
Fig. 10. Inadequate version of a fragment of the CaF2–ErF3 phase diagram: (1) singlephase samples as probed by Xray diffrac tion in annealed and quenched samples, (2) twophase samples, and (3) DTA data. For phase notation, see Fig. 9.
earlier conclusion about the existence of temperatures below which the system cannot be equilibrated.
NONSTOICHIOMETRY AND STOICHIOMETRY
In inorganic chemistry nonstoichiometry is a gen eral case and stoichiometry is a limiting case [39].
Inasmuch the homogeneous range of a com pound/phase expands monotonically as a function of rising temperature, the following is evident: whatever ε, there is always some temperature below which the homogeneity range x of the phase is less than ε. In other words, for any image scale, there is always some temperature below which the compound is imaged in the phase diagram as stoichiometric. Accordingly, for
any accuracy of analysis, there is temperature ТC below which the homogeneity range will be less than the accuracy of analysis and the compound should be regarded as stoichiometric.
For example, careful studies did not reveal nonsto ichiometry in LiYF4 at temperatures below 700°С [40]. However, there are direct indications to nonstoichi ometry in the vicinity of its melting temperature (828°С) [41].
Cases are possible where ТC is higher than the sta bility temperature of a given phase, i.e., higher than the melting temperature Тm or decomposition temper ature. Such a phase should be considered as a stoichi ometric compound over the entire temperature range of existence. T, K L 1 2 3 4 F + L αF βF F + T F + βYF3 T T + α α T + βYF3 1400 1000 600 CaF2 20 40 60 80 YF3 mol % * Y Y * * Y Y
Fig. 11. Summary T–x diagram of the CaF2–YF3 system: (1) DTA data, (2) homogeneity boundaries as probed Xray diffraction in annealed and quenched samples, (3) solubility boundary as determined in hydrothermal synthesis, and (4) conventional tem peratures of a diffuse phase transition.
Expansion of the homogeneity range is associated with pilingup of intrinsic defects. Evidently, nonsto ichiometry is less pronounced for compounds in which defect generation energies are high. These are molecular compounds or compounds containing com plex anions in their structures. Specifically, nonstoichi ometry is not intrinsic to borates. A finding of nonsto ichiometry on BaBi2O4 [42] was later disproved [43].
If a phase decomposes as temperature lowers, it is quite likely that the phase will remain nonstoichio metric over the entire temperature range of existence even as probed by lowsensitivity analytical methods.
BERTHOLLIDES AND DALTONIDES Thus, all phases are phases of variable composition in the strict meaning of this term. What is the position of Kurnakov’s daltonides [44] among these phases?
As mentioned above, there is always a certain tem perature ТC which enables, to the accuracy of an experiment, to distinguish stoichiometric and nonsto ichiometric compounds. In general, this temperature changes with the analytical tool used.
Within the homogeneity range of a nonstoichio metric phase, the composition where properties reach an extreme always deviates from the stoichiometry. Experimental studies [45, 46] and theoretical specula tions prove that a property peak is diffuse and its posi
tion deviates from the stoichiometry as temperature rises. Strictly speaking, there is no temperature at which Dalton singularities with a discontinuity of the derivative appear on a property diagram. However, we
logτ [h] 1500 1300 1100 900 700 500 T, °C 3 2 1 ZrO2–R2O3 CaF2–RF3 CdF2–RF3 MF2–RF4 PbF2–RF3 5 7 9 11 104/T, K–1
Fig. 12. Annealing time versus temperature for initial component mixtures required for equilibration according to [29].
Worksheet to calculate the sintering time for M1 – xRxF2 + x solid solutions, where M = Ca, Sr, Ba and R = Ln, as τ = 9.12 × 10–6 exp(2.07 × 104/T) [h]; Ea = 0.9 eV Time, τ, years τ, h T, K T, °C 1 8.76 × 103 1002 729 10 8.76 × 104 901 628 100 8.76 × 105 820 547 4000 3.5 × 107 715 442 10 000 8.76 × 107 693 420 40 000 3.5 × 108 662 390 2 M 1.75 × 1010 589 316 70 M 6.1 × 1011 535 261 4 Bn 3.5 × 1013 484 211 13.7 Bn 1.2 × 1014 470 197
may assume that there is some temperature, TD > TC, below which an experiment detects a homogeneity range for a phase but does not detect a significant devi ation of the property extreme from the stoichiometric composition (Fig. 14a). The (TD, TC) range is the existence range of Kurnakov’s daltonides [44]. This is a limited temperature range whose boundaries are only conventional and depend on the accuracy of analysis. Strictly speaking, all phases of variable com position are berthollides.
Situations where TD > Tm are possible. In this case, the solidus curvature radius is negligible and may be equated to zero, and it may be said that a singular point appears on the solidus curve; in other words, the soli dus curve is divided into a pair of meeting curves with a discontinuity of a derivative (Fig. 14b).
Mlodzeevskii [47] gave a phenomenological con sideration to a relationship between the liquidus and solidus curvatures and the degree of dissociation of a compound.
A matter deserving special consideration is the existence of singular points matching the stoichiome
try on property versus concentration curves in a melt (another type of singularity is possible: a temperature dependent singular point corresponding to a second order phase transition occurs in, for example, systems where liquid crystals are formed). This signifies the existence in melt of stable structural units identical to those existing in solid structures. Noteworthy, the reverse is invalid: from the existence in melt of structural units that exist in a solid phase, it cannot be deduced that this phase is stoichiometric. A counterexample in this case is fluorite solid solution M1 – x RxF2 + x where M stands for alkalineearth elements and R for rareearth elements. Figure 14d corresponds, for example, to BaB2O4 melting; the melt contains little boronoxygen rings, which are intrinsic to the BaB2O4 crystal structure [48]. This heavily hampers single crystal growth from own melt. The case in Fig. 14e illustrates the typical behavior of molecular compounds.
Figure 15 shows the Gibbs free energy versus con centration curves G(x) for phases of variable composi tion. In the case of a daltonide (Fig. 15b), the curva ture is negligible and we may speak of a singular point. The image in Fig. 15c is a hypothetical case of abso lutely undissociated compound. However, such a case is unstable because a common tangent can be drawn to the two branches of the G(x) curve which corresponds to a twophase state with a greater thermodynamic sta bility. Sirota [3] interpreted this as the existence of a twophase region within each daltonide phase, this region expanding as temperature rises. Such specula tions have not been supported by an experiment. In our opinion, the instability of the situation depicted in Fig. 15c is a thermodynamic proof of the forbiddance of the existence of absolutely undissociated com pounds [49]. This speculation disavows Storonkin’s arguments in his dispute with Anosov on the definition of the term “phase” [81, 85–88].
Berthollides and daltonides are the forms of appearance of a phase in a phase diagram. Their exist ence must not be related to any type of chemical bond, though some correlations exist.
50 μm
Fig. 13. Scanning electron micrograph for barium fluoride
prepared by thermolysis of BaF2 · HF.
T (a) (b) (c) (d) (e)
TД ТС
x
ТС ТС
WHETHER A PHASE CAN HAVE AN EQUILIBRIUM HOMOGENEITY
REGION AT TEMPERATURE TENDING TO ZERO?
For an individual phase, the answer to this question is negative as follows from the foregoing. However, let us analyze the situation in a more detailed way. Cases are known where ordering is manifested as the appear ance of a plurality of ordered phases instead of a single ordered phase.
The progress in experimental methods of investiga tion of solids (Xray diffraction, electron diffraction, and highresolution electronic microscopy) helped to reveal, in many systems, socalled longperiod phases, also referred to as incommensurate or modulated phases, shear structures, intergrowth structures, or infinitely adaptive phases [50–52]. A theoretical crys tallographic description of these phases is made by means of symmetry groups for space dimensions higher than 3D. Modulated phases are exemplified by mullite [53, 54], LTa2O5 type phases [55, 56], high temperature superconductors [57], and many others.
The thermodynamic consideration of incommen surate order–disorder phase transitions describes cas cade ordering processes where an infinite set of ordered phases is formed within limited ranges of the parameters of state of the system. The term “devil’s staircase” of phase transitions [58] describes a situa tion where, specifically, the temperature spaces between phase transitions tend to zero.
In modulated structures, a modulation of site occupancies or atom displacements appears in addi tion to the set of translations corresponding to the major framework of the structure. The modulation wave length in general is not related to unit cell param eters and changes in response to changing tempera ture, pressure, and concentration. At low tempera tures, theoretically, a plurality of stoichiometric com pounds is formed at low temperatures having a rational ratio of the major translations and modulation parameters, and their density in certain concentration ranges corresponds to the density of rational numbers along the numerical axis. In reality, naturally, bound aries between these compounds are diffuse and inter growths of different compounds are observed in one sample.
A similar situation arises with quasicrystals approximants. Figure 1b represents a 900°С isother mal section of the Al–Fe–Pd system according to [59]. Four phases are revealed in the regions denoted as ε in this figure, although theoretically their number must be infinite.
A specific case of a modulated phase is a homolo gous series of compounds whose composition is described by a general formula with a variable homo logue number n. The same result is given by a family of ordered microdomain (cluster) structures.
A homologous series of compounds corresponds to the formation of compact concentration sets of sto ichiometric compounds whose composition is tem perature independent. The concentration boundary between the nearest neighbors of a homologous series tends to zero as the homologue number increases. Only some of them are observed in real samples in the form of intergrowths. Makarov [60] in 1947 noticed that such a series of stoichiometric compounds can imitate a homogeneity range.
An example is the Lu2O3–LuF3 system [61] (Fig. 17). A homologous series of LunOn – 1Fn + 2 compounds
exists in this system; they crystallize in the orthorhom bic crystal system and are structurally related to fluo rite. The homologues having n = 5–10 have been revealed and characterized.
Individual compounds in such the cases need spe cial investigative tools to be recognized, specifically electron diffraction.
We must mention that the existence of a homolo gous series of compounds described by a general for mula does not necessarily mean the existence of an infinite number of compounds. Specifically, the series of ordered compounds existing in MF2–RF3 (M = Ca, Sr, Ba; R = Ln) are described by general formulas that reflect the packing pattern of R6F36,37 clusters in the fluorite matrix [24], but the overall number of these compounds is likely to be finite (Fig. 11).
DISEQUILIBRIUM
As a rule, samples studied at room temperature are in disequilibrium, regardless of whether they have been cooled from high temperatures or via lowtem perature syntheses, which is intrinsic to the vast majority of materials and is thereby of paramount practical importance. In the context of phase equilib ria, this creates problems of how to gain reliable data on lowtemperature and hightemperature states.
(a) (b) (c)
x G
Fig. 15. Gibbs free energy versus concentration for phases
of variable composition in binary systems: (a) berthollide, (b) daltonide, and (c) unstable case of absolutely nondis sociated compound.
10 20 30 40 M Al5Fe2 Al2Fe 90 80 70 60 50 50Al50Fe 10 20 30 40 50Al50Pd Pd, at. % A l, at . % Fe, at. % Al C C1 δ O ε C2 L β
Fig. 16. Isothermal section (900°С) of the Al–Pd–Fe system according to [59]. Dots indicate the compositions of the samples
studied. T, K 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 1.5 1.6 1.8 1.9 2.0 2.1 2.2 2.3 2.8 2.9 3.0 1800 1600 1400 1200 1000 ?
LuO1.5 + Lu(O,F)1.91 Lu(O,F) 1.91 + Lu(O,F)2.10 LuO1.5 + LuOF LuOF + Lu(O,F)2.10 Lu(O,F)2.20 + LuF3(lq) Lu(O,F)2.20 + αLuF3 Lu(O,F)2.20 + βLuF3 x in Lu(O,F)x Vernier region
Fig. 17. Lu2O3–LuF3 system [61]. phase
Phases obtained in lowtemperature syntheses usu ally obey the Ostwald step rule in evolution from less equilibrated to more equilibrated states. An example is the synthesis of aluminum yttrium garnet via precipi tation from aqueous solution. The initially formed precursor is amorphous. Crystallization occurs upon heating, as a rule, to yield first a nonstoichiometric phase having the YАlO3 structure that then transforms to a virtually stoichiometric compound Y3Аl5O12 [32, 62, 63]. Lowtemperature synthesis frequently pro duces a disordered phase instead of an ordered one. An example is the synthesis of Ba4R3F17 compounds from aqueous solution where cubic phases were obtained instead of trigonaldistorted ones [64].
Let us consider the limitations of the annealand quench method in the context of studies of hightem perature equilibria. Presumably, samples are equili brated at a high temperature and then rapidly cooled in order to conserve their state.
First, it is doubtful whether the annealand quench method is useful without DTA. For example, when Poulain et al. [65] studied systems of ZrF4 and HfF4 with di and tervalent metal fluorides, annealing brought samples to melt [66, 67] and quenching gave extensive solid solutions which are absent in the rele vant equilibrium phase diagrams.
As a rule, phases obtained by the quench method are not equilibrated because of leaving the thermody namic stability region. In some cases, decomposition is fairly noticeable. Rapid cooling allows some extent of control over the decomposition. However, there are cases where decomposition or ordering is unavoidable. The retention of a hightemperature state seems an unsolvable problem if the system passes through the spinodal and enters a labile state. First, these are near secondorder phase transitions. Systematically they take place in, for example, compounds having the elpasolite structure [68]. A second case is spinodal decomposition [69]. This type of decomposition is inevitable (a phase in this region has no potential bar rier, and the process must be selfaccelerating). Some researchers neglect this process (see, e.g., [70]).
In practice, a spinodal solution is not always noticeable. A system drops into a potential well, but the dropping rate at low temperatures can be low so as to be undetectable.
In some cases, a system leaves the labile region via nonequilibrium ordering. One example of this type of ordering of tysonite phases in CaF2–RF3 systems has been considered above. Various types of ordered phases (γ1 and γ2) appear when attempts are made to quench samples from the hightemperature region of Cd1 – хMnхF2 solid solutions [71] (Fig. 18). Interest ingly, solid solution samples while being quenched retain their stability against the segregation of a second phase (MnF2), as verified by the match of DTA and powder Xray diffraction data. Similar phenomena take place in CaF2–MnF2 [72] and Li2SiO3–Na2SiO3 [73] systems.
Thus, if some phase appears in quenching prod ucts, it is not necessarily that this phase takes place in the equilibrium phase diagram.
Figure 19 illustrates nonequilibrium ordering in the marginal parts of Ca1 – хRхF2 + х fluorite solid solu tions where R stands for an yttriumfamily rare earth [14, 74, 75]. Powder Xray diffraction patterns feature a set of superstructure reflections associated with ordered phases whose ideal composition is Ca8R5F31. As distinct from ordered tysonite phases in the systems under consideration, these compounds are shown in phase diagrams, but their compositions lie outside the stability field of solid solutions and the upper temper ature bounds as determined by DTA are lower than annealing temperatures. Processing the data shown in Fig. 19b allows estimating the domain size of the ordered phase in quenched samples. Calculations from the Selyakov–Scherrer relationship [76]
D = Kλ/βcosθ, (1)
where D is coherence length, λ is Xray wavelength, θ is diffraction angle of the peak position, β is Xray dif fraction peak halfwidth, and К is particle form factor equal to 0.9, shows that D increases from 3.4 nm for 35 mol % LuF3 to 22 nm for 37.5 mol % LuF3. Char acteristically, proceeding from Fig. 19, these D values are independent of both temperature and annealing time.
Thus, in studying quenched samples, one deals with decomposition products of a phase rather than the phase itself.
Of paramount importance is that, in studying sam ples of decomposing phases (for example, by electron microscopy), one must be extremely careful in expanding inferences about the structure (twining, ordering, and so on) to equilibrium phases (see, e.g., [77]). Let us quote from Tret’yakov [78]: “Great cau tion must be taken in inferring the structure of … phases at high temperatures from crystallographic studies carried out on quenched samples. Even upon very rapid cooling from temperatures at which equili bration was attained, … atoms (ions) can shift by 20 to 50 Å before diffusion process are completely frozen. In view of the tendency of systems to ordering upon cool ing, expected is considerable local rearrangement of atoms to form structures not intrinsic to a hightem perature equilibrium state of crystals.”
CONCLUSIONS
The existence of temperature bounds below which equilibrium is unattainable should be taken into account in studying phase diagrams [79]. It is neces sary to analyze the thermal behavior of an equilibrium homogeneity region. As a rule it is inadequate to image a phase with a temperatureindependent homogeneity range (as a lamppost). Accordingly, the initial data set for thermodynamic modeling needs to be carefully studied to reject an experiment where equilibrium has
not been attained. In Mamontov [80], for example, the data set for the thermodynamic modeling of the Pb–Pd system included the homogeneity boundary of the palladiumbased solid solution that likely did not correspond to an equilibrium state, as this boundary was virtually temperature independent and could not be extrapolated to zero concentration at Т 0 K.
Whether the term “berthollide” or “daltonide” is applied to an individual intermediate phase is dictated by epistemological reasons, namely, the degree of scrutiny of the phase, accuracy of investigative tools, and reasonable degree of idealization. The abstract concept of an absolutely nondissociated compound with solid solutions on its basis is selfcontradictory.
A great caution must be taken in transferring the results of studies performed at low temperatures, espe cially on samples of unknown thermal history, to high temperature equilibria. There is an urgent need for hightemperature in situ studies (hightemperature Xray diffraction, vapor pressure measurements, and others).
It is absolutely inadequate to reject the phenome non of nonstoichiometry and the existence of phases of variable composition reducing them to, for exam ple, a set of stoichiometric compounds.
On the other hand, it must be kept in mind that “phase” is a macroscopic concept. The concept of homogeneity, which underlies the definition of a T, °C 1100 900 700 CdF2 20 40 60 80 MnF2 1 2 3 L F γ1 γ2 F + MnF2 L + F mol %
Fig. 18. CdF2–MnF2 system [71]: (1) DTA data, (2) singlephase samples, and (3) twophase samples as determined by Xray diffraction in annealed and quenched samples.
phase, is a conventional concept and depends on the methods of investigation used [81]. The atomic struc ture of the matter apart, classical thermodynamics suggests a continuous variation of concentration and a possibility of differentiating thermodynamic functions over concentration. Defect clusters transfer the homo geneity problem to the nanometer scale [82–84]. Likely, different solutions can exist in each particular case to decide whether a sample is singlephase, two phase, or manyphase.
ACKNOWLEDGMENTS
The author thanks S.V. Lavrishchev for carrying out scanning electron microscopy of barium fluoride.
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