Appendix A Construction of the Phase Diagram of a Binary System AâB with Ideal Solid Solution so small that the number
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Appendices
Classic and Advanced Ceramics: From Fundamentals to Applications. Robert B. Heimann © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-32517-7
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Appendix A Construction of the Phase Diagram of a Binary System A–B with Ideal Solid Solution The variation of the Gibbs function G of phase α of a binary system A–B with the intensive variables temperature T, pressure P, and mole numbers ni can be expressed by: dG α = −s α dT + ν α dP + ∑ µiα dn iα ,
(A.1)
where s and v are the molar entropy and volume, respectively, and µ is the chemical potential. Hence, according to Gibbs’ phase rule, there are four variables in a binary (C = 2) system: the temperature T, the pressure P, and the concentrations of A and B, expressed by their mole numbers nA and nB. For a single-phase situation (α = 1), the degrees of freedom F are 3 (F = C – P + 2); that is, there are three independent variables, T, P, and either nA or nB. The molar fractions of the components A and B in phase α are defined as: X Aα = n Aα ( n Aα + nBα ) ; X Bα = nBα ( n Aα + nBα ) ; X Aα + X Bα ≡ 1
(A.2)
From this it follows g α = X Aα ⋅ µαΑ (P, T , X A ) + (1 − X Aα ) ⋅ µΒα (P, T , X A ) .
(A.3)
The lower-case letter g refers to the molar Gibbs function; that is, the Gibbs free energy per mol. For the two-phase situation (α = 2), the degrees of freedom F reduce to 2, since µ1A (P, T , X A1 ) = µ2A (P, T , X A2 ) , and µ1Β (P, T , X B1 ) = µ2Β (P, T , X B2 ) ,
(A.4)
where the superscripts (1) and (2) refer to phase 1 and phase 2, respectively. Consequently, there are only two independent variables P and T, or P and XA, or T and XA, since the molar fraction XB is fixed according to XB = 1 – XA. The molar Gibbs function of a phase of a binary mechanical mixture (A + B), gm can be derived from the thermodynamic potential equation g = h – Ts = µ·n, and can be expressed by: g m = X A ⋅ hA + X B ⋅ hB − T ( X A ⋅ s A + X B ⋅ sB ) = X A ⋅ µ*A + X B ⋅ µB* ,
(A.5)
where the enthalpy h and the entropy s are molar entities independent of the presence of the other component. This can be safely assumed, since the particles are
Classic and Advanced Ceramics: From Fundamentals to Applications. Robert B. Heimann © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-32517-7
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Appendix A Construction of the Phase Diagram of a Binary System A–B with Ideal Solid Solution µB∗
gm µA∗ gis g -T·∆sm 0
XB
1
T1
(a) T2 A
B
(b)
T3 XB gis
Figure A.1 (a) Gibbs free energy of a mechanical mixture (gm), the configurational or mixing entropy term –T·∆sm and the dependence of the Gibbs free energy of an
ideal solution (gis) on the composition for a binary system A–B; (b) Temperature dependence of the Gibbs free energy gis at temperatures T1 < T2 < T3.
so small that the number of atoms at the phase boundary can be neglected compared to those in the volume of the phases. This means that the phases are completely spatially separated, and consequently the Gibbs function is additive in terms of the pure components (see line gm in Figure A.1a). The term µ* refers to the standard chemical potential of a condensed phase. This additivity is lost in true solutions, since the particles of the components A and B interact with each other. The potential and kinetic energies of A and B in solution are different compared to their individual pure states. Both energy changes influence not only the internal energies ui of the components, but also their entropies si owing to purely kinetic effects, for example, variations of oscillation and rotation frequencies of the atoms of the two components. This is the reason why the chemical potentials µA and µB in solutions depend on the composition, and on T and P. The rules of equilibrium thermodynamics require replacement of the molar parameters h and s by the partial molar parameters ¯h and s¯. However, there is still another aspect to consider, namely the configurational or mixing entropy ∆sm. Even if the atoms of A and B were to be identical in every aspect, so that their entropy in the pure substance and the solution would be (almost) equal, their mixing would cause a state of lower order, and hence higher entropy. This increase of configurational entropy must be added to the molar Gibbs function of a solution, gs, yielding
Appendix A Construction of the Phase Diagram of a Binary System A–B with Ideal Solid Solution
g s = X A ⋅ hA + X B . hB − T ( X A ⋅ sA + X B ⋅ sB ) − T ⋅ ∆sm .
(A.6)
In a thermodynamic sense, solutions can be subdivided into ideal and real solutions, but in this context it suffices to consider only ideal solutions with a Gibbs function gis. Ideal solutions are characterized by the identities ¯hA = hA, ¯hB = hB and ¯sA = sA, ¯sB = sB. This does not mean that no interactions take place among atoms of A and B; rather, it means that the interaction A–B is (almost) identical to that of the interactions A–A and B–B in the pure components A and B. Hence, mixtures of isotopes and isomorphic components approximate ideal solutions (see below). The molar Gibbs function of an ideal solution is then g is = g m − T ⋅ ∆sm , with ∆sm = −R [ X A ⋅ ln ( X A ) + X B ⋅ ln ( X B )].
(A.7)
Using the relationship µi = µi* + RT ln(Xi), this equation can be transformed to g is = X A ⋅ µ A + X B ⋅ µB .
(A.7a)
The configurational entropy ∆sm is always positive for all compositions and finite temperatures. Consequently, it holds that gis < gm; that is, under ideal conditions a solution is always more stable than a mechanical mixture. Likewise, a homogeneous solution is more stable than a mechanical mixture of solutions. To construct the phase diagram of an ideal solid solution (A,B) of the two components A and B, the concentration dependence of the configurational entropy will be considered, as shown in Figure A.1a. The convex curvature of the configurational entropy term, and hence g, increase with increasing temperature (Figure A.1b). This is also shown in Figure A.2. At high temperature, T6, the free energy gL of the melt is lower than that of the solid gs over the entire range of compositions (gL < gs); hence, the only phase stable is that of the liquid melt. At decreasing temperature, the free energy of the liquid increases faster than that of the solid phase, so that the former will eventually overtake the latter. Consequently, at T5 the two curves intersect at the composition of pure A. This is the melting point, TA. At this point liquid A and solid A are in equilibrium, that is, their chemical potentials and hence their free energies are equal. A further decrease in temperature to T4 leads to an intersection of the free energy curves gL and gs at X1. At this point, the state with the lowest possible free energy is a mixture of a solid with composition a on the gs curve and a liquid of compositions b on the gL curve, given by the common tangent. Between the limits Xa and Xb the liquid has always the composition Xa and the solid always the composition Xb, even though zonation is possible. The relative amounts of liquid and solid can be determined by the lever rule (see below). At T3, the point of intersection of the two free energy curves has shifted to a composition richer in B, and at T2 the melting point of the pure component B, TB, has been reached. At this point the last trace of melt disappears, and at temperatures lower than T2 the system contains only a single solid solution (gs < gL). Based on the dependence of free energy on temperature and composition shown in Figure A.2, the phase diagram can now be constructed as shown in Figure A.3. The temperatures T1–T6 define unequivocally the compositions of the liquid and solid phases in equilibrium with each other. The curve connecting all freezing
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Appendix A Construction of the Phase Diagram of a Binary System A–B with Ideal Solid Solution
T6
T5
T4
TA S
S
L
L
A
B
Gibbs free energy gis
504
A
T3
a
S
B
A
d X2
TB
T6 > T5 > T4 > T3 > T2 > T1
L S
S B
B
X1
T1
L
c
L
b
T2
L
A
S
A
B
A
B
Composition XB
Figure A.2 Gibbs free energy curves of the liquid (L) and solid (s) phases of an ideal solid solution system between T1 (low temperature) and T6 (high temperature).
points of the liquid is the liquidus, while the curve connecting all melting points of the solid solutions is the solidus. Lines of constant temperature between those two curves are called conodes. Hence, the liquidus and the solidus divide the phase diagram into three regions: (i) the single-phase field of the liquid; (ii) the singlephase field of the solid solutions; and (iii) the two-phase liquid-solid field between the curves. The amounts of coexisting melt and solid can be determined by the “lever rule,” using the simple geometric relation that the amounts of the coexisting phases are inversely proportional to their distance from the point of the global composition measured along a conode. For example, at T4 the relationship holds
( n As + nBs ) ( n LA + nBL ) = m s m L = Ob Oa ,
(A.8)
where m s = ( n As + nBs ) is the total mass of solid and m L = n LA + nBL is the total mass of liquid. There are several real solid solution systems that approximate this ideal system, for example, the binary systems Ge–Si and Cu–Ni owing to the energetic similarities of the two limiting components that crystallize in identical space groups (Si and Ge: S.G. Fd3m (No. 227); Cu and Ni: S.G. Fm 3m (No. 225)). It should be briefly mentioned that the construction of binary phase diagrams with invariant phase transformations requires three Gibbs free energy curves – two for the solid phases α and β, and one for the liquid phase, L. At equilibrium temperature a common tangent exists for gL, gα and gß. The position of the
Appendix A Construction of the Phase Diagram of a Binary System A–B with Ideal Solid Solution
T6 T5
TA liquid b
T4
a
O
Liq
uid
us
cu
rv
So
lid
us
cu
e
rv
T3
e
d
c
solid TB
T2
T1
X1
A
X2
B
Figure A.3 Equilibrium phase diagram of an ideal solid solution, constructed from the Gibbs free energy curves shown in Figure A.2.
(a)
(b)
β
Gibbs free energy g
Gibbs free energy g
L α
A
Composition XB
B
α
β
L
A
Composition XB
Figure A.4 Gibbs free energy curves for invariant phase transitions, showing an eutectic (a) and a peritectic (b) reaction between a liquid (L) and two solid phases (α and β).
B
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Appendix A Construction of the Phase Diagram of a Binary System A–B with Ideal Solid Solution Table A1 Types of three-phase invariant reactions in binary phase diagrams. (Smith, W.F.
(1996) Principles of Materials Science and Engineering, 3rd edition, McGraw-Hill, Inc., New York, p. 464). Type of reaction
Reaction equation
Eutectic Eutectoid Peritectic Peritectoid Monotectic
L→α+β α→β+γ α+L→β α+β→γ L1 → α + L2
minimum of gL between those of the solid phases gα and gß characterizes a eutectic reaction (Figure A.4a). If the minimum gL is outside the minima of gα and gß, a peritectic reaction takes place during which the liquid reacts with one of the solid phases to form a second solid phase of different composition (Figure A.4b). If the decomposing phase and product phases are all solid, then the reaction type is called eutectoid. Likewise, if two solid phases react to form a new solid phase, one speaks of a peritectoid reaction. Hence, peritectic and peritectoid reactions are the inverse of the corresponding eutectic and eutectoid reactions. The relations are depicted schematically in Table A1.
