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Undulate phase boundaries on binary T–x diagrams
Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 89–93 www.elsevier.com/locate/calphad
Undulate phase boundaries on binary T –x diagrams Dmitri V. Malakhov ∗ , Thevika Balakumar Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4L7 Received 28 June 2007; received in revised form 7 August 2007; accepted 14 August 2007 Available online 17 September 2007
Abstract Usually, an inflection point on a phase boundary is considered as an unambiguous indication that one of phases participating in the equilibrium is internally unstable, i.e. that it is prone to separation. Subsequently, it is habitually deemed that the inflection point may appear only if a thermodynamic model of this phase contains an excess term. It is shown that in contrast to this belief, inflection points on a phase boundary may appear when a pure solid component or a stoichiometric binary phase is in equilibrium with the ideal binary solution, which is internally stable, indeed. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Binary system; Phase boundary; Inflection point; Entropy of melting
1. Introduction The Sn–Zn system was optimized by Fries and Lukas [1]. The phase diagram resulting from that assessment is shown in Fig. 1 (solid lines). Let us notice an inflection point on the liquidus. A traditional way of explaining such a shape (known as “S-shape”) is to assume that the liquid phase is prone to separation at temperatures below the liquidus. A dashed line representing a metastable miscibility gap in the liquid suggests that in this particular case, the rationalization is compelling. In this work, it is shown that such an explanation is not universal. This is done by firstly deriving general expressions for the slope and curvature of a phase boundary for the case when a binary solution is in equilibrium with a stoichiometric binary phase. Then these expressions are simplified by assuming that the solution is ideal. Finally, through a straightforward mathematical analysis, a condition resulting in the appearance of an inflection point is formulated. 2. Slope of a phase boundary
determinacy, let us assume that L is a simple (single lattice) substitutional solution. The Gibbs energies of L and α are given by (1) and (2), respectively: L 0L G L = (1 − x L )∆G 0L 1 + x ∆G 2 L L + RT ((1 − x ) ln(1 − x ) + x L ln x L ) + ∆ex G L (1) α 0α α G α = (1 − x α )∆G 0α 1 + x ∆G 2 + ∆ f G 0γ
where ∆G i is the Gibbs energy of transformation of the ith component from the structure associated with its reference state to the structure of the γ phase. The general expression 0α α α (2) reduces to G α = ∆G 0α 1 if x = 0 and to G = ∆G 2 if α x = 1. The condition of equilibrium between L and α can be written as G L + (x α − x L )G LL − G α = 0.
(3)
In (3) and below, the notation ϕ L L . . . L T T . . . T ≡ | {z } | {z } m times
Let us consider the equilibrium between a binary solution phase L and a binary stoichiometric phase α. For the sake of ∗ Corresponding author.
E-mail address: [email protected] (D.V. Malakhov). c 2007 Elsevier Ltd. All rights reserved. 0364-5916/$ - see front matter doi:10.1016/j.calphad.2007.08.002
(2)
∂ m+n ϕ ∂(x L )m ∂ T n
is
n times
used for making expressions shorter and easier to handle. A great deal of attention has been paid in the literature to calculating slopes of phase boundaries [2,3]. Despite this, it seems justified to start derivations from scratch to ensure cohesiveness of the present work as well as to enforce internal logic.
90
D.V. Malakhov, T. Balakumar / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 89–93
Fig. 1. A metastable miscibility gap in the liquid phase (dashed curve) superimposed on the equilibrium Sn–Zn phase diagram.
Let us denote the LHS of (3) as F. Since F remains equal to zero along the phase boundary, one can use implicit differentiation and write FL + FT (dT /dx L ) = 0.
(4)
The expression for the slope of the phase boundary immediately follows from (4): dT /dx L = −FL /FT
(5)
where FL = (x α − x L )G LL L FT =
G TL
α
+ (x − x
L
(6) )G LL T
−
G αT .
(7)
Having expressions (1) and (2), one can write partial derivatives of the Gibbs energies in (6) and (7) as G LL L = RT /((1 − x L )x L ) + ∆ex G LL L G TL
L
)∆S10L L
L
∆S20L L L
(8)
restrictions preventing FT in (13) from being equal to zero, infinite slopes are not prohibited by thermodynamics. Slopes tending to +∞ or −∞ are inevitable if T → 0, because all entropies of formations and transformations become infinitesimally small. It may happen than both the numerator and denominator in (5) are equal to zero. An analysis of this exotic and interesting situation is beyond the scope of the present contribution. In a particular case when x α = 0 and x L → x α , it can easily be shown (finiteness of ∆ex G LL L should be recalled) that dT /dx L = −RT1 /∆S10α→L (T1 ) < 0, where T1 is the melting point of the first component when it is in the α structure. If x α = 1 and x L → x α , then dT /dx L = RT2 /∆S20α→L (T2 ) > 0. These two expressions for limiting slopes are well known, in fact [3]. If x L → x α and if 0 < x α < 1, then FL tends to zero. Since FT becomes equal to the entropy of melting of α taken with the opposite sign, i.e. since it is always negative, an indeterminacy 0/0 is never encountered in (5). Consequently, it can be concluded that (dT /dx L )x L =x α = 0. 3. Curvature of a phase boundary In contrast to slopes, the calculation of curvatures of phase boundaries did not acquire much attention in the literature. An excellent work [4] is the only publication known to the authors in which this problem was deeply and extensively discussed. Despite the unquestionable relevance of that paper, the derivations below have a different mathematical and conceptual flavor, which is not surprising since the objective of this contribution differs quite significantly from that of [4]. Let us start with a terminological clarification. The curvature d2 ψ/dz 2 of the function ψ(z), which is [1+(dψ/dz) 2 ]3/2 , cannot be identified with its second derivative. In this work, however, for the sake of brevity, d2 T /d(x L )2 is named the curvature. A justification of such a terminological frivolity is that the curvature and the second derivative either have the same sign or are both equal to zero. Let us denote the LHS of (4) as Φ and recall that like F it remains equal to zero along the phase boundary. By using implicit differentiation again, one obtains
= −(1 − x −x + R((1 − x ) ln(1 − x ) + x ln x L ) − ∆ex S L
(9)
G LL T = ∆S10L − ∆S20L + R ln(x L /(1 − x L )) − ∆ex SLL
(10)
Φ L + ΦT (dT /dx L ) = 0
G αT = −(1 − x α )∆S10α − x α ∆S20α − ∆ f S α .
(11)
where ΦL =
Substitution of (8) in (6) gives FL = (x α − x L )(RT /((1 − x L )x L ) + ∆ex G LL L ).
(12)
Substitution of (9)–(11) in (7) yields FT = R((1 − x α ) ln(1 − x L ) + x α ln x L ) − ∆ex S L − (x α − x L )∆ex SLL + ∆ f S α .
∂ (FL + FT (dT /dx L )) ∂xL
= FL L + FL T (dT /dx L ) + FT (d2 T /d(x L )2 ) ∂ ΦT = (FL + FT (dT /dx L )) ∂T = FL T + FT T (dT /dx L ).
− (1 − x α )∆S10α→L − x α ∆S20α→L (13)
By inserting (12) and (13) in (5), the slope can be computed. It is worth mentioning that since there are no fundamental
(14)
(15)
(16)
By inserting (15) and (16) in (14), the following expression for the second derivative can be arrived at: d2 T FL L + 2FL T (dT /dx L ) + FT T (dT /dx L )2 = − . FT d(x L )2
(17)
D.V. Malakhov, T. Balakumar / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 89–93
By employing the definition of F and by making use of (1) and (2), the following formulae can be derived: RT ex L FL L = − + ∆ G LL (1 − x L )x L ! RT 1 − 2x L α L ex L + (x − x ) − (18) 2 + ∆ G LLL (1 − x L )x L R α L ex L (19) + ∆ G L LT FL T = (x − x ) (1 − x L )x L FT T = − 1 − x α − ∆ex STL
∆C 0α→L p1
∆C 0α→L p2 α
−x T T L − (x − x )∆ex SLLT + ∆ f STα . α
(20)
Substitution of (18)–(20) along with FT given by (13) and the already calculated slope in (17) finalizes the computation of the curvature of the phase boundary. 4. Simplifications Although one can write an explicit expression for the curvature, this formula will be monstrously long. Besides, it will not be very helpful unless ∆ex G L x L , T in (1) and ∆ f G α (T ) in (2) are defined. Instead of analyzing the general expression for the curvature (algebraic complexities make such an analysis virtually impossible), let us consider a trouble-free case by assuming that: (1) The solution phase is the ideal solution (∆ex G L and its partial derivatives disappear in corresponding expressions). (2) ∆ f S α = 0, i.e. ∆ f G α = ∆ f H α (the last term in the RHS of (20) is equal to zero). 0L 0α 0α (3) ∆G 0L 1 , ∆G 2 , ∆G 1 and ∆G 2 are linear functions of and ∆C 0α→L in the temperature (this entails that ∆C 0α→L p1 p2 RHS of (20) vanish). Instead of (12) and (13), one now has FL = RT (x α − x L )/((1 − x L )x L ) FT = R((1 − x α ) ln(1 − x L ) + x α ln x L ) − (1 − x α )∆S10α→L − x α ∆S20α→L .
(21)
It is worth stressing that the simplifications introduced make FT always negative! The slope is now given by Box I. The expressions (18)–(20) undergo a drastic simplification as well: FL L
RT =− ((x α − x L )2 + x α (1 − x α )) ((1 − x L )x L )2
FL T = R(x α − x L )/((1 − x L )x L )
(22) (23)
FT T = 0.
Although the formula (24) is valid regardless of whether the solution is in equilibrium with a pure component (x α = 0 or x α = 1) or a binary compound (0 < x α < 1), it is instructive to consider these cases separately. Firstly, let us consider the situation when x α = 0. If L x → x α , then instead of (22), (23) and (21) one has FL L = −RT1 , FL T = −R, FT = −∆S10α→L . Keeping in mind that dT /dx L = −RT1 /∆S10α→L , one arrives at the following expression: −RT1 − 2R(−RT1 /∆S10α→L ) d2 T = − d(x L )2 −∆S10α→L =
RT1 ∆S10α→L
(−1 + 2R/∆S10α→L ).
(25)
It is clearly seen from (25) that the sign of the curvature in the vicinity of the pure first component is determined by the magnitude of ∆S10α→L (T1 ): if the entropy of fusion is less than 2R, then the curvature is positive (the phase boundary is convex downward); if ∆S10α→L (T1 ) > 2R, then d2 T /d(x L )2 < 0 (the phase boundary is convex upward). Will the curvature of the phase boundary retain the sign it possesses in the vicinity of the pure first component if x L departs from zero? The second case when x α = 1 and x L → x α can be analyzed in the same manner. Instead of (22), (23) and (21) one has FL L = −RT2 , FL T = R, FT = −∆S20α→L . Since the limiting slope is now equal to RT2 /∆S20α→L (T2 ), the curvature is given by −RT2 + 2R(RT2 /∆S20α→L ) d2 T = − d(x L )2 −∆S20α→L RT2 = (−1 + 2R/∆S20α→L ). ∆S20α→L The conclusion is exactly the same as before: if ∆S20α→L < 2R, then d2 T /d(x L )2 > 0; if ∆S20α→L exceeds 2R, the curvature is negative. The question we would like to answer is virtually identical to that asked previously: Will the curvature of the phase boundary keep the sign it has in the vicinity of the pure second component when x L goes away from unity? Finally, if 0 < x α < 1 and x L = x α , then dT /dx L = 0, and the curvature becomes equal to d2 T RT . = α x (1 − x α )FT d(x L )2 Since FT < 0 (see (21)), d2 T /d(x L )2 < 0. Does this inequality hold if x L 6= x α ? Does, in other words, the phase boundary always remain convex upward as it is near the stoichiometric compound α? 5. Analysis
(dT /dx L )2
The last equality eliminates the term FT T in the numerator of (17), thus yielding the following much simpler expression for the curvature: d2 T FL L + 2FL T (dT /dx L ) = − . FT d(x L )2
91
(24)
For answering the question of whether the curvature can change its sign or not, the numerator of (24) has to be analyzed. If it is always negative, the curvature will hold its sign. Let us start with the case when x α = 0. The expressions (22), (23) and Box I are reduced to: FL L = −RT /(1 − x L )2 ,
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D.V. Malakhov, T. Balakumar / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 89–93
dT RT (x α − x L ) =− . L L L α L dx (1 − x )x [R((1 − x ) ln(1 − x ) + x α ln x L ) − (1 − x α )∆S10α→L − x α ∆S20α→L ] Box I.
FL T = −R/(1 − x L ),
dT dx L
RT . (1−x L )[R ln(1−x L )−∆ S10α→L ]
=
the numerator of (24) one has
For
FL L + 2FL T (dT /dx L ) R2 T RT − 2 =− (1 − x L )2 (1 − x L )2 [R ln(1 − x L ) − ∆S10α→L ] ! RT 2 =− 1+ . (26) (1 − x L )2 ln(1 − x L ) − ∆S10α→L /R It has already been demonstrated that if ∆S10α→L /2 > R, then (d2 T /d(x L )2 )x L →0+ < 0. Now let us show that if 2 is always ∆S10α→L /2 > R, then 1 + 0α→L L ln(1−x )−∆ S1
/R
positive, i.e. that the numerator remains negative. 1+
2
?
ln(1 − x L ) − ∆S10α→L /R
ln(1 − x L ) − ∆S10α→L /R + 2 ln(1 − x L ) − ∆S10α→L /R
>0
(27)
?
>0
Fig. 2. A family of phase boundaries corresponding to the equilibrium between the ideal solution and the pure solid first component constructed for various entropies of fusion.
?
ln(1 − x L ) − ∆S10α→L /R + 2 < 0 ?
ln(1 − x L ) < ∆S10α→L /R − 2.
(28)
Since the inequality (28) it undoubtedly true, the inequality (27) is also true. It can thus be concluded that if in the vicinity of the first component the phase boundary is convex upward, its curvature always remains negative. It has already been shown that if ∆S10α→L /2 < R, (d2 T /d(x L )2 )x L →0 > 0. But if ∆S10α→L /2 < R, the term 2 1+ in (26) will inevitably change its sign 0α→L L ln(1−x )−∆ S1
/R
from positive to negative, and an inflection point will appear at x L = 1 − exp(∆S10α→L /R − 2). It can thus be deduced that if in the vicinity of the first component the phase boundary is convex downward, its curvature will inevitably change its sign when x L is increasing. This is clearly exemplified by Fig. 2. It is worth repeating that whether the phase boundary is wavy or not is completely determined by the ∆S10α→L /R ratio. If x α = 1, FL L = −RT /(x L )2 , FL T = R/x L , dT RT = − L 0α→L . Consequently, the numerator of L dx L x [R ln x −∆ S2
]
(24) becomes
FL L + 2FL T (dT /dx L ) RT R2 T =− L 2 −2 (x ) (x L )2 [R ln x L − ∆S20α→L ] ! RT 2 =− L 2 1+ . (x ) ln x L − ∆S20α→L /R
if in the vicinity of the first component the phase boundary is convex upward, its curvature always remains negative, but if it is convex downward, the curvature will inevitably change its sign when x L is decreasing. The position of the inflection point is x L = exp(∆S20α→L /R − 2). Now let us write the numerator of (24) for the case when 0 < x α < 1. FL L + 2FL T
dT dx L
α L 2 α α = − ((1−xRT L )x L )2 (x − x ) + x (1 − x )
+
R(x α −x L )2 α L α R((1−x ) ln(1−x )+x ln x L )−(1−x α )∆ S10α→L −x α ∆ S20α→L
.
The curvature will remain negative if (x α − x L )2 + x α (1 − x α )
+
R(x α −x L )2 R((1−x α ) ln(1−x L )+x α ln x L )−(1−x α )∆ S10α→L −x α ∆ S20α→L
> 0. (30)
(29)
A remarkable similarity between (29) and (26) allows one to omit boring and trivial rearrangements and merely state that
The first two summands in (30) are positive, but the last one is negative. Let us notice that if x L → 0 or x L → 1, the logarithmic term tends to −∞, which translates into a negative curvature. Instead of carrying out a general algebraic analysis of the inequality (30), let us visualize its LHS for various x α , ∆S10α→L and ∆S20α→L . Fig. 3 clearly demonstrates that small entropies of fusion favor the undulation. It also is clear that since the term x α (1 − x α ) in (30) is maximized by x α = 0.5,
D.V. Malakhov, T. Balakumar / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 89–93
Fig. 3. An example of concentration dependencies of the LHS of (30) for different entropies of fusion and compositions of the stoichiometric phase.
the undulation is favored by a composition departing from the equimolar ratio. Although Fig. 3 is convincing, it might be instructive to actually construct a phase diagram with a wavy boundary. For the sake of determinacy, let us accept that: (1) The reference states are pure components having the 0L structure of the L phase, i.e. ∆G 0L 1 and ∆G 2 used in (1) are both equal to zero. 0α (2) Lattice stabilities ∆G 0α 1 and ∆G 2 employed in (2) are described by the same linear function of temperature: 0α ∆G 0α 1 = ∆G 2 = −1000 + 0.831451T J/mol. (3) The Gibbs energy of formation of the α phase is ∆ f G α = −5000 J/mol. The phase diagram resulting from these thermodynamic quantities is shown in Fig. 4. It is not surprising that the inflection points are seen at the right boundary for which (x α − x L )2 in (30) undergoes a much greater variation in comparison with the left boundary. 6. Conclusion It is not uncommon to see inflection points at phase boundaries. Usually, their existence is attributed to an internal instability of one of the phases coexisting along the boundary. It has been shown that this explanation is not universal. It has been proven that if a binary ideal solution is in equilibrium with a pure component whose lattice stability is a linear function of temperature, an inflection point at a corresponding phase boundary inevitably appears if the entropy of fusion is less than 2R. Although this criterion stems from particular simplifications and thus is not general, it tells one that small
93
Fig. 4. A phase diagram unambiguously demonstrating the presence of inflection points at the phase boundary for the case when the ideal solution is in equilibrium with a stoichiometric phase.
entropies of melting make the appearance of inflection points at phase boundaries more probable. If the solution is in equilibrium with a stoichiometric phase, inflection points are favored by small entropies of fusion as well as by a composition of the stoichiometric phase deviating from the equimolar ratio. An influence of the entropies of transformation on a shape of a phase boundary may be quite important for solid–solid transitions, for which they are normally small. Finally, it is worth mentioning that the habitual explanation is invoked when an inflection point is situated at a flat portion of a phase boundary, i.e. at the portion necessitating the S-shape of the whole boundary. As shown in the present work, this shape per se does not necessitate the flatness. Acknowledgements The authors are grateful to Professor Arthur Daniel Pelton for a stimulating discussion and valuable suggestions that stemmed from it. Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. References [1] I. Ansara, A.T. Dinsdale, M.H. Rand (Eds.), COST 507 Thermochemical Database for Light Metal Alloys, vol. 2, Office for Official Publications of the European Communities, Luxembourg, 1998, pp. 290–292. [2] H.L. Lukas, J. Weiss, E.-Th. Henig, CALPHAD 3 (1982) 229–251. [3] A.D. Pelton, Metallurgical Trans. A 19A (1988) 1819–1825. [4] D.A. Goodman, J.W. Cahn, L.H. Bennett, Bull. Alloy Phase Diagrams 2 (1981) 29–34.
Undulate phase boundaries on binary T –x diagrams Dmitri V. Malakhov ∗ , Thevika Balakumar Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4L7 Received 28 June 2007; received in revised form 7 August 2007; accepted 14 August 2007 Available online 17 September 2007
Abstract Usually, an inflection point on a phase boundary is considered as an unambiguous indication that one of phases participating in the equilibrium is internally unstable, i.e. that it is prone to separation. Subsequently, it is habitually deemed that the inflection point may appear only if a thermodynamic model of this phase contains an excess term. It is shown that in contrast to this belief, inflection points on a phase boundary may appear when a pure solid component or a stoichiometric binary phase is in equilibrium with the ideal binary solution, which is internally stable, indeed. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Binary system; Phase boundary; Inflection point; Entropy of melting
1. Introduction The Sn–Zn system was optimized by Fries and Lukas [1]. The phase diagram resulting from that assessment is shown in Fig. 1 (solid lines). Let us notice an inflection point on the liquidus. A traditional way of explaining such a shape (known as “S-shape”) is to assume that the liquid phase is prone to separation at temperatures below the liquidus. A dashed line representing a metastable miscibility gap in the liquid suggests that in this particular case, the rationalization is compelling. In this work, it is shown that such an explanation is not universal. This is done by firstly deriving general expressions for the slope and curvature of a phase boundary for the case when a binary solution is in equilibrium with a stoichiometric binary phase. Then these expressions are simplified by assuming that the solution is ideal. Finally, through a straightforward mathematical analysis, a condition resulting in the appearance of an inflection point is formulated. 2. Slope of a phase boundary
determinacy, let us assume that L is a simple (single lattice) substitutional solution. The Gibbs energies of L and α are given by (1) and (2), respectively: L 0L G L = (1 − x L )∆G 0L 1 + x ∆G 2 L L + RT ((1 − x ) ln(1 − x ) + x L ln x L ) + ∆ex G L (1) α 0α α G α = (1 − x α )∆G 0α 1 + x ∆G 2 + ∆ f G 0γ
where ∆G i is the Gibbs energy of transformation of the ith component from the structure associated with its reference state to the structure of the γ phase. The general expression 0α α α (2) reduces to G α = ∆G 0α 1 if x = 0 and to G = ∆G 2 if α x = 1. The condition of equilibrium between L and α can be written as G L + (x α − x L )G LL − G α = 0.
(3)
In (3) and below, the notation ϕ L L . . . L T T . . . T ≡ | {z } | {z } m times
Let us consider the equilibrium between a binary solution phase L and a binary stoichiometric phase α. For the sake of ∗ Corresponding author.
E-mail address: [email protected] (D.V. Malakhov). c 2007 Elsevier Ltd. All rights reserved. 0364-5916/$ - see front matter doi:10.1016/j.calphad.2007.08.002
(2)
∂ m+n ϕ ∂(x L )m ∂ T n
is
n times
used for making expressions shorter and easier to handle. A great deal of attention has been paid in the literature to calculating slopes of phase boundaries [2,3]. Despite this, it seems justified to start derivations from scratch to ensure cohesiveness of the present work as well as to enforce internal logic.
90
D.V. Malakhov, T. Balakumar / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 89–93
Fig. 1. A metastable miscibility gap in the liquid phase (dashed curve) superimposed on the equilibrium Sn–Zn phase diagram.
Let us denote the LHS of (3) as F. Since F remains equal to zero along the phase boundary, one can use implicit differentiation and write FL + FT (dT /dx L ) = 0.
(4)
The expression for the slope of the phase boundary immediately follows from (4): dT /dx L = −FL /FT
(5)
where FL = (x α − x L )G LL L FT =
G TL
α
+ (x − x
L
(6) )G LL T
−
G αT .
(7)
Having expressions (1) and (2), one can write partial derivatives of the Gibbs energies in (6) and (7) as G LL L = RT /((1 − x L )x L ) + ∆ex G LL L G TL
L
)∆S10L L
L
∆S20L L L
(8)
restrictions preventing FT in (13) from being equal to zero, infinite slopes are not prohibited by thermodynamics. Slopes tending to +∞ or −∞ are inevitable if T → 0, because all entropies of formations and transformations become infinitesimally small. It may happen than both the numerator and denominator in (5) are equal to zero. An analysis of this exotic and interesting situation is beyond the scope of the present contribution. In a particular case when x α = 0 and x L → x α , it can easily be shown (finiteness of ∆ex G LL L should be recalled) that dT /dx L = −RT1 /∆S10α→L (T1 ) < 0, where T1 is the melting point of the first component when it is in the α structure. If x α = 1 and x L → x α , then dT /dx L = RT2 /∆S20α→L (T2 ) > 0. These two expressions for limiting slopes are well known, in fact [3]. If x L → x α and if 0 < x α < 1, then FL tends to zero. Since FT becomes equal to the entropy of melting of α taken with the opposite sign, i.e. since it is always negative, an indeterminacy 0/0 is never encountered in (5). Consequently, it can be concluded that (dT /dx L )x L =x α = 0. 3. Curvature of a phase boundary In contrast to slopes, the calculation of curvatures of phase boundaries did not acquire much attention in the literature. An excellent work [4] is the only publication known to the authors in which this problem was deeply and extensively discussed. Despite the unquestionable relevance of that paper, the derivations below have a different mathematical and conceptual flavor, which is not surprising since the objective of this contribution differs quite significantly from that of [4]. Let us start with a terminological clarification. The curvature d2 ψ/dz 2 of the function ψ(z), which is [1+(dψ/dz) 2 ]3/2 , cannot be identified with its second derivative. In this work, however, for the sake of brevity, d2 T /d(x L )2 is named the curvature. A justification of such a terminological frivolity is that the curvature and the second derivative either have the same sign or are both equal to zero. Let us denote the LHS of (4) as Φ and recall that like F it remains equal to zero along the phase boundary. By using implicit differentiation again, one obtains
= −(1 − x −x + R((1 − x ) ln(1 − x ) + x ln x L ) − ∆ex S L
(9)
G LL T = ∆S10L − ∆S20L + R ln(x L /(1 − x L )) − ∆ex SLL
(10)
Φ L + ΦT (dT /dx L ) = 0
G αT = −(1 − x α )∆S10α − x α ∆S20α − ∆ f S α .
(11)
where ΦL =
Substitution of (8) in (6) gives FL = (x α − x L )(RT /((1 − x L )x L ) + ∆ex G LL L ).
(12)
Substitution of (9)–(11) in (7) yields FT = R((1 − x α ) ln(1 − x L ) + x α ln x L ) − ∆ex S L − (x α − x L )∆ex SLL + ∆ f S α .
∂ (FL + FT (dT /dx L )) ∂xL
= FL L + FL T (dT /dx L ) + FT (d2 T /d(x L )2 ) ∂ ΦT = (FL + FT (dT /dx L )) ∂T = FL T + FT T (dT /dx L ).
− (1 − x α )∆S10α→L − x α ∆S20α→L (13)
By inserting (12) and (13) in (5), the slope can be computed. It is worth mentioning that since there are no fundamental
(14)
(15)
(16)
By inserting (15) and (16) in (14), the following expression for the second derivative can be arrived at: d2 T FL L + 2FL T (dT /dx L ) + FT T (dT /dx L )2 = − . FT d(x L )2
(17)
D.V. Malakhov, T. Balakumar / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 89–93
By employing the definition of F and by making use of (1) and (2), the following formulae can be derived: RT ex L FL L = − + ∆ G LL (1 − x L )x L ! RT 1 − 2x L α L ex L + (x − x ) − (18) 2 + ∆ G LLL (1 − x L )x L R α L ex L (19) + ∆ G L LT FL T = (x − x ) (1 − x L )x L FT T = − 1 − x α − ∆ex STL
∆C 0α→L p1
∆C 0α→L p2 α
−x T T L − (x − x )∆ex SLLT + ∆ f STα . α
(20)
Substitution of (18)–(20) along with FT given by (13) and the already calculated slope in (17) finalizes the computation of the curvature of the phase boundary. 4. Simplifications Although one can write an explicit expression for the curvature, this formula will be monstrously long. Besides, it will not be very helpful unless ∆ex G L x L , T in (1) and ∆ f G α (T ) in (2) are defined. Instead of analyzing the general expression for the curvature (algebraic complexities make such an analysis virtually impossible), let us consider a trouble-free case by assuming that: (1) The solution phase is the ideal solution (∆ex G L and its partial derivatives disappear in corresponding expressions). (2) ∆ f S α = 0, i.e. ∆ f G α = ∆ f H α (the last term in the RHS of (20) is equal to zero). 0L 0α 0α (3) ∆G 0L 1 , ∆G 2 , ∆G 1 and ∆G 2 are linear functions of and ∆C 0α→L in the temperature (this entails that ∆C 0α→L p1 p2 RHS of (20) vanish). Instead of (12) and (13), one now has FL = RT (x α − x L )/((1 − x L )x L ) FT = R((1 − x α ) ln(1 − x L ) + x α ln x L ) − (1 − x α )∆S10α→L − x α ∆S20α→L .
(21)
It is worth stressing that the simplifications introduced make FT always negative! The slope is now given by Box I. The expressions (18)–(20) undergo a drastic simplification as well: FL L
RT =− ((x α − x L )2 + x α (1 − x α )) ((1 − x L )x L )2
FL T = R(x α − x L )/((1 − x L )x L )
(22) (23)
FT T = 0.
Although the formula (24) is valid regardless of whether the solution is in equilibrium with a pure component (x α = 0 or x α = 1) or a binary compound (0 < x α < 1), it is instructive to consider these cases separately. Firstly, let us consider the situation when x α = 0. If L x → x α , then instead of (22), (23) and (21) one has FL L = −RT1 , FL T = −R, FT = −∆S10α→L . Keeping in mind that dT /dx L = −RT1 /∆S10α→L , one arrives at the following expression: −RT1 − 2R(−RT1 /∆S10α→L ) d2 T = − d(x L )2 −∆S10α→L =
RT1 ∆S10α→L
(−1 + 2R/∆S10α→L ).
(25)
It is clearly seen from (25) that the sign of the curvature in the vicinity of the pure first component is determined by the magnitude of ∆S10α→L (T1 ): if the entropy of fusion is less than 2R, then the curvature is positive (the phase boundary is convex downward); if ∆S10α→L (T1 ) > 2R, then d2 T /d(x L )2 < 0 (the phase boundary is convex upward). Will the curvature of the phase boundary retain the sign it possesses in the vicinity of the pure first component if x L departs from zero? The second case when x α = 1 and x L → x α can be analyzed in the same manner. Instead of (22), (23) and (21) one has FL L = −RT2 , FL T = R, FT = −∆S20α→L . Since the limiting slope is now equal to RT2 /∆S20α→L (T2 ), the curvature is given by −RT2 + 2R(RT2 /∆S20α→L ) d2 T = − d(x L )2 −∆S20α→L RT2 = (−1 + 2R/∆S20α→L ). ∆S20α→L The conclusion is exactly the same as before: if ∆S20α→L < 2R, then d2 T /d(x L )2 > 0; if ∆S20α→L exceeds 2R, the curvature is negative. The question we would like to answer is virtually identical to that asked previously: Will the curvature of the phase boundary keep the sign it has in the vicinity of the pure second component when x L goes away from unity? Finally, if 0 < x α < 1 and x L = x α , then dT /dx L = 0, and the curvature becomes equal to d2 T RT . = α x (1 − x α )FT d(x L )2 Since FT < 0 (see (21)), d2 T /d(x L )2 < 0. Does this inequality hold if x L 6= x α ? Does, in other words, the phase boundary always remain convex upward as it is near the stoichiometric compound α? 5. Analysis
(dT /dx L )2
The last equality eliminates the term FT T in the numerator of (17), thus yielding the following much simpler expression for the curvature: d2 T FL L + 2FL T (dT /dx L ) = − . FT d(x L )2
91
(24)
For answering the question of whether the curvature can change its sign or not, the numerator of (24) has to be analyzed. If it is always negative, the curvature will hold its sign. Let us start with the case when x α = 0. The expressions (22), (23) and Box I are reduced to: FL L = −RT /(1 − x L )2 ,
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dT RT (x α − x L ) =− . L L L α L dx (1 − x )x [R((1 − x ) ln(1 − x ) + x α ln x L ) − (1 − x α )∆S10α→L − x α ∆S20α→L ] Box I.
FL T = −R/(1 − x L ),
dT dx L
RT . (1−x L )[R ln(1−x L )−∆ S10α→L ]
=
the numerator of (24) one has
For
FL L + 2FL T (dT /dx L ) R2 T RT − 2 =− (1 − x L )2 (1 − x L )2 [R ln(1 − x L ) − ∆S10α→L ] ! RT 2 =− 1+ . (26) (1 − x L )2 ln(1 − x L ) − ∆S10α→L /R It has already been demonstrated that if ∆S10α→L /2 > R, then (d2 T /d(x L )2 )x L →0+ < 0. Now let us show that if 2 is always ∆S10α→L /2 > R, then 1 + 0α→L L ln(1−x )−∆ S1
/R
positive, i.e. that the numerator remains negative. 1+
2
?
ln(1 − x L ) − ∆S10α→L /R
ln(1 − x L ) − ∆S10α→L /R + 2 ln(1 − x L ) − ∆S10α→L /R
>0
(27)
?
>0
Fig. 2. A family of phase boundaries corresponding to the equilibrium between the ideal solution and the pure solid first component constructed for various entropies of fusion.
?
ln(1 − x L ) − ∆S10α→L /R + 2 < 0 ?
ln(1 − x L ) < ∆S10α→L /R − 2.
(28)
Since the inequality (28) it undoubtedly true, the inequality (27) is also true. It can thus be concluded that if in the vicinity of the first component the phase boundary is convex upward, its curvature always remains negative. It has already been shown that if ∆S10α→L /2 < R, (d2 T /d(x L )2 )x L →0 > 0. But if ∆S10α→L /2 < R, the term 2 1+ in (26) will inevitably change its sign 0α→L L ln(1−x )−∆ S1
/R
from positive to negative, and an inflection point will appear at x L = 1 − exp(∆S10α→L /R − 2). It can thus be deduced that if in the vicinity of the first component the phase boundary is convex downward, its curvature will inevitably change its sign when x L is increasing. This is clearly exemplified by Fig. 2. It is worth repeating that whether the phase boundary is wavy or not is completely determined by the ∆S10α→L /R ratio. If x α = 1, FL L = −RT /(x L )2 , FL T = R/x L , dT RT = − L 0α→L . Consequently, the numerator of L dx L x [R ln x −∆ S2
]
(24) becomes
FL L + 2FL T (dT /dx L ) RT R2 T =− L 2 −2 (x ) (x L )2 [R ln x L − ∆S20α→L ] ! RT 2 =− L 2 1+ . (x ) ln x L − ∆S20α→L /R
if in the vicinity of the first component the phase boundary is convex upward, its curvature always remains negative, but if it is convex downward, the curvature will inevitably change its sign when x L is decreasing. The position of the inflection point is x L = exp(∆S20α→L /R − 2). Now let us write the numerator of (24) for the case when 0 < x α < 1. FL L + 2FL T
dT dx L
α L 2 α α = − ((1−xRT L )x L )2 (x − x ) + x (1 − x )
+
R(x α −x L )2 α L α R((1−x ) ln(1−x )+x ln x L )−(1−x α )∆ S10α→L −x α ∆ S20α→L
.
The curvature will remain negative if (x α − x L )2 + x α (1 − x α )
+
R(x α −x L )2 R((1−x α ) ln(1−x L )+x α ln x L )−(1−x α )∆ S10α→L −x α ∆ S20α→L
> 0. (30)
(29)
A remarkable similarity between (29) and (26) allows one to omit boring and trivial rearrangements and merely state that
The first two summands in (30) are positive, but the last one is negative. Let us notice that if x L → 0 or x L → 1, the logarithmic term tends to −∞, which translates into a negative curvature. Instead of carrying out a general algebraic analysis of the inequality (30), let us visualize its LHS for various x α , ∆S10α→L and ∆S20α→L . Fig. 3 clearly demonstrates that small entropies of fusion favor the undulation. It also is clear that since the term x α (1 − x α ) in (30) is maximized by x α = 0.5,
D.V. Malakhov, T. Balakumar / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 89–93
Fig. 3. An example of concentration dependencies of the LHS of (30) for different entropies of fusion and compositions of the stoichiometric phase.
the undulation is favored by a composition departing from the equimolar ratio. Although Fig. 3 is convincing, it might be instructive to actually construct a phase diagram with a wavy boundary. For the sake of determinacy, let us accept that: (1) The reference states are pure components having the 0L structure of the L phase, i.e. ∆G 0L 1 and ∆G 2 used in (1) are both equal to zero. 0α (2) Lattice stabilities ∆G 0α 1 and ∆G 2 employed in (2) are described by the same linear function of temperature: 0α ∆G 0α 1 = ∆G 2 = −1000 + 0.831451T J/mol. (3) The Gibbs energy of formation of the α phase is ∆ f G α = −5000 J/mol. The phase diagram resulting from these thermodynamic quantities is shown in Fig. 4. It is not surprising that the inflection points are seen at the right boundary for which (x α − x L )2 in (30) undergoes a much greater variation in comparison with the left boundary. 6. Conclusion It is not uncommon to see inflection points at phase boundaries. Usually, their existence is attributed to an internal instability of one of the phases coexisting along the boundary. It has been shown that this explanation is not universal. It has been proven that if a binary ideal solution is in equilibrium with a pure component whose lattice stability is a linear function of temperature, an inflection point at a corresponding phase boundary inevitably appears if the entropy of fusion is less than 2R. Although this criterion stems from particular simplifications and thus is not general, it tells one that small
93
Fig. 4. A phase diagram unambiguously demonstrating the presence of inflection points at the phase boundary for the case when the ideal solution is in equilibrium with a stoichiometric phase.
entropies of melting make the appearance of inflection points at phase boundaries more probable. If the solution is in equilibrium with a stoichiometric phase, inflection points are favored by small entropies of fusion as well as by a composition of the stoichiometric phase deviating from the equimolar ratio. An influence of the entropies of transformation on a shape of a phase boundary may be quite important for solid–solid transitions, for which they are normally small. Finally, it is worth mentioning that the habitual explanation is invoked when an inflection point is situated at a flat portion of a phase boundary, i.e. at the portion necessitating the S-shape of the whole boundary. As shown in the present work, this shape per se does not necessitate the flatness. Acknowledgements The authors are grateful to Professor Arthur Daniel Pelton for a stimulating discussion and valuable suggestions that stemmed from it. Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. References [1] I. Ansara, A.T. Dinsdale, M.H. Rand (Eds.), COST 507 Thermochemical Database for Light Metal Alloys, vol. 2, Office for Official Publications of the European Communities, Luxembourg, 1998, pp. 290–292. [2] H.L. Lukas, J. Weiss, E.-Th. Henig, CALPHAD 3 (1982) 229–251. [3] A.D. Pelton, Metallurgical Trans. A 19A (1988) 1819–1825. [4] D.A. Goodman, J.W. Cahn, L.H. Bennett, Bull. Alloy Phase Diagrams 2 (1981) 29–34.