Critical nucleus composition in a multicomponent system T. Philippe, D. Blavette, and P. W. Voorhees Citation: The Journal of Chemical Physics 141, 124306 (2014); doi: 10.1063/1.4896222 View online: http://dx.doi.org/10.1063/1.4896222 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Comment on “Minimum free-energy pathway of nucleation” [J. Chem. Phys.135, 134508 (2011)] J. Chem. Phys. 136, 107101 (2012); 10.1063/1.3692688 Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems J. Chem. Phys. 125, 034705 (2006); 10.1063/1.2218845 Thermodynamically consistent description of the work to form a nucleus of any size J. Chem. Phys. 118, 1837 (2003); 10.1063/1.1531614 Thermodynamics of heterogeneous multicomponent condensation on mixed nuclei J. Chem. Phys. 113, 6822 (2000); 10.1063/1.1287615 Free energy of embryo formation for heterogeneous multicomponent nucleation J. Chem. Phys. 110, 10035 (1999); 10.1063/1.478877
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THE JOURNAL OF CHEMICAL PHYSICS 141, 124306 (2014)
Critical nucleus composition in a multicomponent system T. Philippe,1,a) D. Blavette,1 and P. W. Voorhees2 1
Groupe de Physique des Matériaux (GPM), Normandie Université, UMR CNRS 6634 BP 12, Avenue de l’Université, 76801 Saint Etienne du Rouvray, France 2 Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
(Received 7 July 2014; accepted 10 September 2014; published online 24 September 2014) The properties of a critical nucleus are derived using the capillarity theory in the framework of classical nucleation. An analytical solution for the composition of a critical nucleus is given for low supersaturation. The theory is valid for any multicomponent systems. It is found that the deviation in nucleus composition from the equilibrium tie-line is mainly due to the difference in the Hessian of the Gibbs energy of the phases and the magnitude of the deviation in composition from equilibrium is order of the supersaturation. Despite our analysis strictly holds for low supersaturation, this suggests strong deviations near the spinodal line. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4896222] I. INTRODUCTION
II. COMPOSITION OF A CRITICAL NUCLEUS
A large variety of first-order phase transformations proceeds via nucleation. In diffusive phase separations, the system evolves through the nucleation of a second phase and its subsequent growth and coarsening. The kinetic pathways involved in the early stages of the phase transformation are critical to the development of the final microstructure. On the theoretical side, phase separation implies diffusion in the matrix and transfer of atoms across the interface of the cluster of the new phase in chemical equilibrium with the matrix. In a multicomponent system, it has been shown from theoretical considerations that during coarsening the properties of the second phase, such as particle composition, result from a competition between thermodynamics and diffusion flux-couplings.1, 2 An interesting feature is that the particle composition was shown to deviate from equilibrium (predicted by the phase diagram) and this deviation was found not to lie on the equilibrium tieline, i.e., the line connecting the bulk compositions, its magnitude decreasing with time.1, 2 Various experimental and modelling works have shown similar effects in Ni-Cr-Al3–9 and Al-Zn-Ag alloys10 where significant deviations in precipitate composition from the tie-line direction have been exhibited. It has been shown that diffusion flux-couplings affect the kinetic pathway for the phase separation.8, 9 During nucleation process, if a deviation exists, it should only be due to the Gibbs-Thompson (or Ostwald-Freundlich) effect and its influence on the particle compositions,11 since there are no composition gradients in the matrix. Thus, the purpose of this paper is to determine, in the framework of classical nucleation theory12–14 and for a multicomponent system, the properties of a critical nucleus in chemical equilibrium with the matrix in an effort to determine the composition of the new critical nucleus.
The critical nucleus is a domain of a new phase that is in unstable equilibrium with the supersaturated matrix.12 This is equivalent to requiring that the critical nucleus reside at an extremum of an energy surface.15–20 In a multicomponent system with N chemical species, the critical nucleus (β phase) is in chemical equilibrium with the matrix α, and thus satisfies μαi C20 , C30 , . . . , CN0 , P α β β∗ β∗ β∗ = μi C2 , C3 , . . . , CN , P β for i = 1 . . . N, (1) where μi is the chemical potential of component i in the noted β∗ phase. Ci0 is the nominal composition of component i. Ci is the composition of the critical nucleus. The pressure in the nucleus (Pβ ) is not equal to that in the matrix due to the presence of a nonzero interfacial energy. Assuming that the interfacial stress and interfacial energy are identical, mechanical equilibrium requires 2σ , (2) R∗ where σ is the interfacial energy. Pα is the pressure at which the equilibrium phase diagram is measured and R∗ is the radius of the critical nucleus. We shall first consider the change in the chemical potential due to a curved interface using the Laplace equation. Assuming that the nucleus is incompressible or the partial molar volumes are pressure independent, β β∗ β∗ β∗ μi C2 , C3 , . . . , CN , P β β β∗ β∗ β∗ = μi C2 , C3 , . . . , CN , P α β 2σ ∂μi + ∗ . (3) R ∂P C β =C β∗ ,...,C β =C β∗ ,P α Pβ = Pα +
2
N
2
N
Using the result a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/141(12)/124306/5/$30.00
141, 124306-1
β ∂μi ∂P C
β
β∗ β∗ =C2 ,...,CN =CN ,P α 2
= V¯i ,
(4)
© 2014 AIP Publishing LLC
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J. Chem. Phys. 141, 124306 (2014)
β
where V¯i is the partial molar volume of the β phase, Eq. (1) becomes μαi C20 , C30 , . . . , CN0 , P α 2σ V¯iβ β β∗ β∗ β∗ = μi C2 , C3 , . . . , CN , P α + . (5) R∗ Note that μi (Pα ) is the chemical potential of a flat interface. The above equation is known as the Gibbs-Thompson equation and here gives the equilibrium condition for a critical nucleus with a curved interface in chemical equilibrium with the β∗ matrix. Multiplying Eq. (5) by Ci and summing over i from 1 to N yields N
2σ Vmβ β∗ β β∗ Ci μαi Ci0 , P α − μi Ci , P α = . R∗ i=1
(6)
Since β Vm
=
N
β∗ β Ci V¯i ,
(7)
i=1 β
where Vm is the molar volume of the β phase and is, in general, dependent on composition. In the following it is assumed that, because of small deviation from equilibrium, the molar β volume of the β phase is equal to its equilibrium value V¯m , and is therefore composition independent. The left-hand side of Eq. (6) is the driving force for nucleation, denoted in the following Gm . Equation (5) for i = 1. . . N gives N nonlinear equations in N unknowns (N − 1 independent compositions β∗ Ci and the critical radius R∗ ) and this fully sets the chemical properties of a critical nucleus in equilibrium with the matrix. In the limit of small supersaturation, it is possible to solve these equations analytically. We expand the chemical potentials about the equilibrium compositions, denoted by the overbar, to first order in the deviation in composition from equilibrium, μαi C20 , C30 , . . . , CN0 , P α = μαi C¯ 2α , C¯ 3α , . . . , C¯ Nα , P α +
N
μαi,j
Cj0
− C¯ jα
(8)
μαi,j
∂μαi = ∂Cj C
where the chemical potentials and derivatives are evaluated at the pressure of the matrix α and at the equilibrium compositions. A similar expansion in composition for the β phase yields β β∗ β∗ β∗ μi C2 , C3 , . . . , CN , P α =
+
N
β β∗ μi,j Cj
−
β C¯ j ,
β μi,j
β ∂μi = ∂Cj C
. 2
β β =C¯ 2 ,...,CN =C¯ N ,P α
N
β β C¯ i μi,j = 0
(13)
i=1
by the Gibbs-Duhem equation and at a planar interface, local equilibrium writes β β β β μαi C¯ 2α , C¯ 3α , . . . , C¯ Nα , P α = μi C¯ 2 , C¯ 3 , . . . , C¯ N , P α , (14) the second sum in Eq. (12) that involves the sum over the derivatives of the chemical potentials related to β phase can be written as N N β∗ β β β∗ β Ci − C¯ i μi,j Cj − C¯ j .
(15)
i=1 j =2
This term is second order in the difference in compositions from the equilibrium compositions, and thus the term in the β square brackets involving μi,j can be neglected in Eq. (12). Since N
C¯ iα μαi,j = 0
(16)
i=1
by the Gibbs-Duhem equation, Eq. (12) becomes Gm =
N N
β∗ Ci − C¯ iα μαi,j Cj0 − C¯ jα .
(17)
i=1 j =2
Substituting, β∗ β β∗ β Ci − C¯ iα = C¯ i − C¯ iα + Ci − C¯ i
N N β C¯ i − C¯ iα μαi,j Cj0 − C¯ jα .
(18)
(10)
(11)
(19)
i=1 j =2
The driving force for nucleation, as expected for low supersaturation, does not depend on the nucleus composition. This can be rewritten in terms of the Hessian of the molar Gibbs energy of α evaluated at C¯ iα , using the well-known relation,11, 21, 22 μαi = μα1 + Gαm,i
j =2
where
Since,
Gm =
=C¯ 2α ,...,CN =C¯ Nα ,P α
β β β β μi C¯ 2 , C¯ 3 , . . . , C¯ N , P α
j =2
(12)
(9)
, 2
j =2
i=1
in Eq. (17), to first order in the difference in compositions from their equilibrium values the second term in Eq. (18) can be neglected and thus,
j =2
with
Using Eqs. (8) and (10) leads to the following formulation of the driving force, ⎤ ⎡ N N N β∗ ⎣ β β∗ β Ci μαi,j Cj0 − C¯ jα − μi,j Cj − C¯ j ⎦. Gm =
for i > 1,
(20)
with Gαm,i the partial molar Gibbs energy of α with respect to Ciα . Differentiating Eq. (20) with respect to Cjα and since α ∂Ci /∂Cjα = δij for i > 1, j > 1 one can write μαi,j = μα1,j + Gαm,ij
for i > 1 and
j > 1,
(21)
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J. Chem. Phys. 141, 124306 (2014)
where Gαm,ij is the Hessian of the molar Gibbs energy of α. Using Eq. (21) in Eq. (19) leads to Gm =
N N β C¯ i − C¯ iα Gαm,ij Cj0 − C¯ jα .
(22)
i=2 j =2
the deviation in composition of a critical nucleus from equilibrium in the limit of small supersaturation, ¯ T Gα C0 · Gβ−1 Cβ∗ = Gβ−1 Gα C0 − (C)
¯ V . (30) β V¯m
Using the mass conservation we introduce the tie-line vector ¯ C,
In dyadic notation, ¯ T Gα C0 , Gm = (C)
¯ ¯ C, C0 =
(23)
where Gα is the Hessian of the Gibbs energy of α evaluated ¯ = C¯ β − C¯ α for i = 2. . . N and C0 = C 0 − C¯ α at C¯ iα ,C i i j j ¯ where ¯ C ¯ is the molar for j = 2. . . N. Note that C0 = fraction at equilibrium. Using Eq. (23) in Eq. (6), the radius of the critical nucleus can therefore be written in a more general way as
(31)
¯ is the equilibrium molar fraction of β, Eq. (30) can where be written as ¯ ¯ − ( ¯ T Gα C ¯ Gβ−1 V . (32) ¯ β−1 Gα C ¯ C) Cβ∗ = G β V¯m III. DISCUSSION
β
2σ V¯m . R∗ = ¯ (C)T Gα C0
(24)
Note that Eq. (24) is similar to Eq. (10) of Ref. 11 and to Eq. (6) of Ref. 21 in other frameworks. To determine the composition of the critical nucleus, we use Eqs. (8) and (10) in Eq. (5), N j =2
μαi,j
N β 0 2V¯ σ β β∗ β α ¯ Cj − Cj = μi,j Cj − C¯ j + i ∗ . (25) R j =2
Writing Eq. (25) for i = 1, N
N β 2V¯ σ β β∗ β μα1,j Cj0 − C¯ jα = μ1,j Cj − C¯ j + 1∗ , (26) R j =2 j =2
and using Eq. (21) for i > 1 and j > 1, N α μ1,j + Gαm,ij Cj0 − C¯ jα j =2 N β β 2V¯ σ β β∗ β μ1,j + Gm,ij Cj − C¯ j + i ∗ . (27) R j =2
=
Using Eq. (26) in Eq. (27) reduces the number of equations from N to N − 1, so that for i = 2. . . N, N
Gαm,ij Cj0 − C¯ jα
j =2
=
N
β β∗ Gm,ij Cj
j =2
β β 2σ V¯i − V¯1 β ¯ − Cj + . R∗
(28)
In dyadic notation, Gα C0 = Gβ Cβ∗ +
¯ 2Vσ , R∗
(29)
β β where Gβ = Gm,ij is evaluated at C¯ i for i, j = 2. . . N, β∗ β ¯ = V¯ β − V¯ β for Cβ∗ = Cj − C¯ j for j = 2. . . N, and V i 1 i = 2. . . N. Substituting Eq. (24) in Eq. (29) and multiplying by the inverse of the Hessian of the Gibbs energy of β, Gβ − 1 , gives
The above equation is the main result of this work. One should note that Eq. (32) which determines the composition of the critical nucleus is independent of the surface tension, as found by Weinberg et al.23 in the binary case and by Wilemski.14 The composition of the critical nucleus, in general, does not lie on the equilibrium tie-line even for ¯ = 0 (same partial molar volume). In most cases the term V β−1 ¯ ¯ β G V/Vm is rather small compared to the first term. The deviation from the tie-line is therefore essentially due to the difference in the Hessian of the Gibbs energy between α and β phases, i.e., differences in local curvatures, and this deviation is order of the supersaturation. The deviation from equilibrium does lie on the tie-line only when both Hessians are ¯ = 0. Under these conditions, the magidentical and when V nitude of the deviation is exactly the supersaturation. In the two component limit, Eq. (32) gives the classical result.13 In the regions of stability, both Hessians of the molar Gibbs free energies are positive scalars. Thus, it can be easily seen from Eq. (32) that the composition of the critical β β nucleus falls outside the two-phase region when V¯2 = V¯1 , if the β phase corresponds to higher values of compositions, this β∗ β gives C2 ≥ C¯ 2 . Using a few weak assumptions and computβ∗ ing the derivative dC2 /dC20 , Weinberg et al.23 showed that the critical nucleus composition always lies beyond the equilibrium binodal composition for a binary system. The derivative of the composition of the critical nucleus with respect to the nominal composition, using Eq. (30), gives in the binary limit
β β β∗ 1 − C¯ 2α V¯1 + C¯ 2α V¯2 dC2 Gα = β , (33) β G dC20 V¯m which agrees with the result of Weinberg et al.23 in the limit of small supersaturations and for composition independent parβ β tial molar volumes. For V¯2 = V¯1 , the derivative of the critical 0 composition with respect to C2 is the ratio of the Hessians of the free energies Gα /Gβ which is positive and more generally the derivative of the critical composition with respect to the nominal composition is positive for positive partial molar volumes indicating that the composition of the critical nucleus always falls outside the binodal region. Figure 1 shows schematically, for a ternary, the deviation in composition from equilibrium. As an illustration of the
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124306-4
Philippe, Blavette, and Voorhees
J. Chem. Phys. 141, 124306 (2014)
FIG. 2. A portion of the Ni-Cr-Al phase diagram at 873 K on which is indicated the composition of the critical nucleus of the γ phase in a Ni-5.2Al14.2Cr alloy.
IV. CONCLUSION
FIG. 1. A graphical representation of the deviation in composition of the critical nucleus from equilibrium for a ternary in a particular case where the Hessian of the Gibbs energy of both phases are identical and the components 1, 2, and 3 have same partial molar volume in β (a) and in the general case (b).
capillarity effect on the composition of the critical nucleus for solid state precipitation, we have evaluated, using Eq. (32), the composition of the critical nucleus during precipitation at 873 K of the γ phase (L12 structure) in a disordered matrix (γ phase) for a concentrated Ni-5.2Al-14.2Cr alloy. Here, the ¯ V¯mβ has been neglected. The Gibbs free energy term in V/ of both phases have been computed in the framework of the CALPHAD approach using a thermodynamic database for Ni alloys.24 At this temperature, the bulk compositions for the γ and γ phases are, respectively, 0.037 and 0.161 for Al and 0.149 and 0.089 for Cr. The equilibrium molar fraction of γ is ∼0.12. The Hessians of the free energies have been evaluated numerically at the bulk compositions. Using Eq. (32) the composition of the critical nucleus is 0.169 in Al and 0.083 in Cr, as compared with the bulk phase, the critical nucleus is richer in Al (+0.8 at.%) and poorer in Cr (−0.6 at.%). As shown in Fig. 2, the composition of the critical nucleus falls outside the binodal region and does not lie on the equilibrium tie-line, with a significant deviation from equilibrium. During precipitation, it expected that L12 precipitates significantly enrich in Cr and deplete in Al. Such a result suggests that during the early stages of precipitation, significant deviations of the composition of the critical clusters from their bulk values are expected to be seen, especially near the spinodal, i.e., for high supersaturations.
In this paper, we have derived the properties of a critical nucleus in chemical equilibrium with the matrix in a multicomponent system using the capillarity theory. An analytical solution for the composition of a critical nucleus has been given for low supersaturation. In reality it is very difficult to get large supersaturations, where the difference between the composition of the matrix and the equilibrium composition is much greater than 0.1, for example. By definition, mole fractions are less than 1, hence perturbation solutions indicate that the main results of this work remain valid for many applications. It was found that, in general, the composition of a critical nucleus does not lie on the equilibrium tie-line. The deviation is mainly due to the difference in the Hessian of the Gibbs energy of the phases and the magnitude of the deviation in composition from equilibrium is order of the supersaturation. Despite our analysis strictly holds for small supersaturation only, this suggests strong deviations near the spinodal line. ACKNOWLEDGMENTS
We are grateful for the financial support of CARNOT ESP Institute and for DARPA under Contract No. W91CRB1010004. 1 T.
Philippe and P. W. Voorhees, Acta Mater. 61, 4237 (2013). J. Kuehmann and P. W. Voorhees, Metall. Mater. Trans. A 27, 937 (1996). 3 C. K. Sudbrack, K. E. Yoon, R. D. Noebe, and D. N. Seidman, Acta Mater. 54, 3199 (2006). 4 C. K. Sudbrack, R. D. Noebe, and D. N. Seidman, Acta Mater. 55, 119 (2007). 5 C. Booth-Morrison, Y. Zhou, R. D. Noebe, and D. N. Seidman, Philos. Mag. 90, 219 (2010). 6 C. Booth-Morrison, J. Weninger, C. K. Sudbrack, Z. Mao, R. D. Noebe, and D. N. Seidman, Acta Mater. 56, 3422 (2008). 7 Z. Mao, C. Booth-Morrison, C. K. Sudbrack, G. Martin, and D. N. Seidman, Acta Mater. 60, 1871 (2012). 8 L. Rougier, A. Jacot, Ch.-A. Gandin, P. D. Napoli, P. Théry, D. Ponsen, and V. Jaquet, Acta Mater. 61, 6396 (2013). 2 C.
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124306-5 9 Z.
Philippe, Blavette, and Voorhees
Mao, C. K. Sudbrack, K. E. Yoon, G. Martin, and D. N. Seidman, Nat. Mater. 6, 210 (2007). 10 J. J. Hoyt, M. Sluiter, B. Clark, M. Kraitchman, and D. de Fontaine, Acta Metall. Mater. 35, 2315 (1987). 11 Q. Du, M. Perez, J. P. Warren, and M. Wells, Scr. Mater. 66, 419 (2012). 12 J. W. Gibbs, Trans. Conn. Acad. Sci. 3(108), 343 (1875–1878); see also J. W. Gibbs, The Collected Works (Longmans, Green, New York/London/Toronto, 1928), Vol. 1. 13 M. Hillert, in Lectures on the Theory of Phase Transformation, edited by H. I. Aaronson (The Minerals, Metals & Materials Society, Warrendale, PA, 1975), p. 1. 14 G. Wilemski, J. Chem. Phys. 80, 1370 (1984). 15 J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 688 (1959).
J. Chem. Phys. 141, 124306 (2014) 16 M.
Iwamatsu, J. Chem. Phys. 130, 244507 (2009). F. Lutsko, J. Chem. Phys. 128, 184711 (2008). 18 J. F. Lutsko, J. Chem. Phys. 129, 244501 (2008). 19 T. Philippe and D. Blavette, Scr. Mater. 67, 77 (2012). 20 T. Philippe and D. Blavette, J. Chem. Phys. 135, 134508 (2011). 21 J. E. Morral and G. R. Purdy, Scr. Metall. 30, 905 (1994). 22 M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformation (Cambridge University Press, Cambridge, 1998). 23 M. C. Weinberg, V. A. Shneidman, and Z. A. Osborne, Phys. Chem. Glasses 37(2), 49 (1996). 24 N. Saunders, Proceedings of the Eighth International Symposium on Superalloys (Superalloys, 1996) (The Minerals, Metals & Materials Society, Warrendale, PA, 1996), p. 101. 17 J.
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THE JOURNAL OF CHEMICAL PHYSICS 141, 124306 (2014)
Critical nucleus composition in a multicomponent system T. Philippe,1,a) D. Blavette,1 and P. W. Voorhees2 1
Groupe de Physique des Matériaux (GPM), Normandie Université, UMR CNRS 6634 BP 12, Avenue de l’Université, 76801 Saint Etienne du Rouvray, France 2 Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
(Received 7 July 2014; accepted 10 September 2014; published online 24 September 2014) The properties of a critical nucleus are derived using the capillarity theory in the framework of classical nucleation. An analytical solution for the composition of a critical nucleus is given for low supersaturation. The theory is valid for any multicomponent systems. It is found that the deviation in nucleus composition from the equilibrium tie-line is mainly due to the difference in the Hessian of the Gibbs energy of the phases and the magnitude of the deviation in composition from equilibrium is order of the supersaturation. Despite our analysis strictly holds for low supersaturation, this suggests strong deviations near the spinodal line. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4896222] I. INTRODUCTION
II. COMPOSITION OF A CRITICAL NUCLEUS
A large variety of first-order phase transformations proceeds via nucleation. In diffusive phase separations, the system evolves through the nucleation of a second phase and its subsequent growth and coarsening. The kinetic pathways involved in the early stages of the phase transformation are critical to the development of the final microstructure. On the theoretical side, phase separation implies diffusion in the matrix and transfer of atoms across the interface of the cluster of the new phase in chemical equilibrium with the matrix. In a multicomponent system, it has been shown from theoretical considerations that during coarsening the properties of the second phase, such as particle composition, result from a competition between thermodynamics and diffusion flux-couplings.1, 2 An interesting feature is that the particle composition was shown to deviate from equilibrium (predicted by the phase diagram) and this deviation was found not to lie on the equilibrium tieline, i.e., the line connecting the bulk compositions, its magnitude decreasing with time.1, 2 Various experimental and modelling works have shown similar effects in Ni-Cr-Al3–9 and Al-Zn-Ag alloys10 where significant deviations in precipitate composition from the tie-line direction have been exhibited. It has been shown that diffusion flux-couplings affect the kinetic pathway for the phase separation.8, 9 During nucleation process, if a deviation exists, it should only be due to the Gibbs-Thompson (or Ostwald-Freundlich) effect and its influence on the particle compositions,11 since there are no composition gradients in the matrix. Thus, the purpose of this paper is to determine, in the framework of classical nucleation theory12–14 and for a multicomponent system, the properties of a critical nucleus in chemical equilibrium with the matrix in an effort to determine the composition of the new critical nucleus.
The critical nucleus is a domain of a new phase that is in unstable equilibrium with the supersaturated matrix.12 This is equivalent to requiring that the critical nucleus reside at an extremum of an energy surface.15–20 In a multicomponent system with N chemical species, the critical nucleus (β phase) is in chemical equilibrium with the matrix α, and thus satisfies μαi C20 , C30 , . . . , CN0 , P α β β∗ β∗ β∗ = μi C2 , C3 , . . . , CN , P β for i = 1 . . . N, (1) where μi is the chemical potential of component i in the noted β∗ phase. Ci0 is the nominal composition of component i. Ci is the composition of the critical nucleus. The pressure in the nucleus (Pβ ) is not equal to that in the matrix due to the presence of a nonzero interfacial energy. Assuming that the interfacial stress and interfacial energy are identical, mechanical equilibrium requires 2σ , (2) R∗ where σ is the interfacial energy. Pα is the pressure at which the equilibrium phase diagram is measured and R∗ is the radius of the critical nucleus. We shall first consider the change in the chemical potential due to a curved interface using the Laplace equation. Assuming that the nucleus is incompressible or the partial molar volumes are pressure independent, β β∗ β∗ β∗ μi C2 , C3 , . . . , CN , P β β β∗ β∗ β∗ = μi C2 , C3 , . . . , CN , P α β 2σ ∂μi + ∗ . (3) R ∂P C β =C β∗ ,...,C β =C β∗ ,P α Pβ = Pα +
2
N
2
N
Using the result a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/141(12)/124306/5/$30.00
141, 124306-1
β ∂μi ∂P C
β
β∗ β∗ =C2 ,...,CN =CN ,P α 2
= V¯i ,
(4)
© 2014 AIP Publishing LLC
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124306-2
Philippe, Blavette, and Voorhees
J. Chem. Phys. 141, 124306 (2014)
β
where V¯i is the partial molar volume of the β phase, Eq. (1) becomes μαi C20 , C30 , . . . , CN0 , P α 2σ V¯iβ β β∗ β∗ β∗ = μi C2 , C3 , . . . , CN , P α + . (5) R∗ Note that μi (Pα ) is the chemical potential of a flat interface. The above equation is known as the Gibbs-Thompson equation and here gives the equilibrium condition for a critical nucleus with a curved interface in chemical equilibrium with the β∗ matrix. Multiplying Eq. (5) by Ci and summing over i from 1 to N yields N
2σ Vmβ β∗ β β∗ Ci μαi Ci0 , P α − μi Ci , P α = . R∗ i=1
(6)
Since β Vm
=
N
β∗ β Ci V¯i ,
(7)
i=1 β
where Vm is the molar volume of the β phase and is, in general, dependent on composition. In the following it is assumed that, because of small deviation from equilibrium, the molar β volume of the β phase is equal to its equilibrium value V¯m , and is therefore composition independent. The left-hand side of Eq. (6) is the driving force for nucleation, denoted in the following Gm . Equation (5) for i = 1. . . N gives N nonlinear equations in N unknowns (N − 1 independent compositions β∗ Ci and the critical radius R∗ ) and this fully sets the chemical properties of a critical nucleus in equilibrium with the matrix. In the limit of small supersaturation, it is possible to solve these equations analytically. We expand the chemical potentials about the equilibrium compositions, denoted by the overbar, to first order in the deviation in composition from equilibrium, μαi C20 , C30 , . . . , CN0 , P α = μαi C¯ 2α , C¯ 3α , . . . , C¯ Nα , P α +
N
μαi,j
Cj0
− C¯ jα
(8)
μαi,j
∂μαi = ∂Cj C
where the chemical potentials and derivatives are evaluated at the pressure of the matrix α and at the equilibrium compositions. A similar expansion in composition for the β phase yields β β∗ β∗ β∗ μi C2 , C3 , . . . , CN , P α =
+
N
β β∗ μi,j Cj
−
β C¯ j ,
β μi,j
β ∂μi = ∂Cj C
. 2
β β =C¯ 2 ,...,CN =C¯ N ,P α
N
β β C¯ i μi,j = 0
(13)
i=1
by the Gibbs-Duhem equation and at a planar interface, local equilibrium writes β β β β μαi C¯ 2α , C¯ 3α , . . . , C¯ Nα , P α = μi C¯ 2 , C¯ 3 , . . . , C¯ N , P α , (14) the second sum in Eq. (12) that involves the sum over the derivatives of the chemical potentials related to β phase can be written as N N β∗ β β β∗ β Ci − C¯ i μi,j Cj − C¯ j .
(15)
i=1 j =2
This term is second order in the difference in compositions from the equilibrium compositions, and thus the term in the β square brackets involving μi,j can be neglected in Eq. (12). Since N
C¯ iα μαi,j = 0
(16)
i=1
by the Gibbs-Duhem equation, Eq. (12) becomes Gm =
N N
β∗ Ci − C¯ iα μαi,j Cj0 − C¯ jα .
(17)
i=1 j =2
Substituting, β∗ β β∗ β Ci − C¯ iα = C¯ i − C¯ iα + Ci − C¯ i
N N β C¯ i − C¯ iα μαi,j Cj0 − C¯ jα .
(18)
(10)
(11)
(19)
i=1 j =2
The driving force for nucleation, as expected for low supersaturation, does not depend on the nucleus composition. This can be rewritten in terms of the Hessian of the molar Gibbs energy of α evaluated at C¯ iα , using the well-known relation,11, 21, 22 μαi = μα1 + Gαm,i
j =2
where
Since,
Gm =
=C¯ 2α ,...,CN =C¯ Nα ,P α
β β β β μi C¯ 2 , C¯ 3 , . . . , C¯ N , P α
j =2
(12)
(9)
, 2
j =2
i=1
in Eq. (17), to first order in the difference in compositions from their equilibrium values the second term in Eq. (18) can be neglected and thus,
j =2
with
Using Eqs. (8) and (10) leads to the following formulation of the driving force, ⎤ ⎡ N N N β∗ ⎣ β β∗ β Ci μαi,j Cj0 − C¯ jα − μi,j Cj − C¯ j ⎦. Gm =
for i > 1,
(20)
with Gαm,i the partial molar Gibbs energy of α with respect to Ciα . Differentiating Eq. (20) with respect to Cjα and since α ∂Ci /∂Cjα = δij for i > 1, j > 1 one can write μαi,j = μα1,j + Gαm,ij
for i > 1 and
j > 1,
(21)
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124306-3
Philippe, Blavette, and Voorhees
J. Chem. Phys. 141, 124306 (2014)
where Gαm,ij is the Hessian of the molar Gibbs energy of α. Using Eq. (21) in Eq. (19) leads to Gm =
N N β C¯ i − C¯ iα Gαm,ij Cj0 − C¯ jα .
(22)
i=2 j =2
the deviation in composition of a critical nucleus from equilibrium in the limit of small supersaturation, ¯ T Gα C0 · Gβ−1 Cβ∗ = Gβ−1 Gα C0 − (C)
¯ V . (30) β V¯m
Using the mass conservation we introduce the tie-line vector ¯ C,
In dyadic notation, ¯ T Gα C0 , Gm = (C)
¯ ¯ C, C0 =
(23)
where Gα is the Hessian of the Gibbs energy of α evaluated ¯ = C¯ β − C¯ α for i = 2. . . N and C0 = C 0 − C¯ α at C¯ iα ,C i i j j ¯ where ¯ C ¯ is the molar for j = 2. . . N. Note that C0 = fraction at equilibrium. Using Eq. (23) in Eq. (6), the radius of the critical nucleus can therefore be written in a more general way as
(31)
¯ is the equilibrium molar fraction of β, Eq. (30) can where be written as ¯ ¯ − ( ¯ T Gα C ¯ Gβ−1 V . (32) ¯ β−1 Gα C ¯ C) Cβ∗ = G β V¯m III. DISCUSSION
β
2σ V¯m . R∗ = ¯ (C)T Gα C0
(24)
Note that Eq. (24) is similar to Eq. (10) of Ref. 11 and to Eq. (6) of Ref. 21 in other frameworks. To determine the composition of the critical nucleus, we use Eqs. (8) and (10) in Eq. (5), N j =2
μαi,j
N β 0 2V¯ σ β β∗ β α ¯ Cj − Cj = μi,j Cj − C¯ j + i ∗ . (25) R j =2
Writing Eq. (25) for i = 1, N
N β 2V¯ σ β β∗ β μα1,j Cj0 − C¯ jα = μ1,j Cj − C¯ j + 1∗ , (26) R j =2 j =2
and using Eq. (21) for i > 1 and j > 1, N α μ1,j + Gαm,ij Cj0 − C¯ jα j =2 N β β 2V¯ σ β β∗ β μ1,j + Gm,ij Cj − C¯ j + i ∗ . (27) R j =2
=
Using Eq. (26) in Eq. (27) reduces the number of equations from N to N − 1, so that for i = 2. . . N, N
Gαm,ij Cj0 − C¯ jα
j =2
=
N
β β∗ Gm,ij Cj
j =2
β β 2σ V¯i − V¯1 β ¯ − Cj + . R∗
(28)
In dyadic notation, Gα C0 = Gβ Cβ∗ +
¯ 2Vσ , R∗
(29)
β β where Gβ = Gm,ij is evaluated at C¯ i for i, j = 2. . . N, β∗ β ¯ = V¯ β − V¯ β for Cβ∗ = Cj − C¯ j for j = 2. . . N, and V i 1 i = 2. . . N. Substituting Eq. (24) in Eq. (29) and multiplying by the inverse of the Hessian of the Gibbs energy of β, Gβ − 1 , gives
The above equation is the main result of this work. One should note that Eq. (32) which determines the composition of the critical nucleus is independent of the surface tension, as found by Weinberg et al.23 in the binary case and by Wilemski.14 The composition of the critical nucleus, in general, does not lie on the equilibrium tie-line even for ¯ = 0 (same partial molar volume). In most cases the term V β−1 ¯ ¯ β G V/Vm is rather small compared to the first term. The deviation from the tie-line is therefore essentially due to the difference in the Hessian of the Gibbs energy between α and β phases, i.e., differences in local curvatures, and this deviation is order of the supersaturation. The deviation from equilibrium does lie on the tie-line only when both Hessians are ¯ = 0. Under these conditions, the magidentical and when V nitude of the deviation is exactly the supersaturation. In the two component limit, Eq. (32) gives the classical result.13 In the regions of stability, both Hessians of the molar Gibbs free energies are positive scalars. Thus, it can be easily seen from Eq. (32) that the composition of the critical β β nucleus falls outside the two-phase region when V¯2 = V¯1 , if the β phase corresponds to higher values of compositions, this β∗ β gives C2 ≥ C¯ 2 . Using a few weak assumptions and computβ∗ ing the derivative dC2 /dC20 , Weinberg et al.23 showed that the critical nucleus composition always lies beyond the equilibrium binodal composition for a binary system. The derivative of the composition of the critical nucleus with respect to the nominal composition, using Eq. (30), gives in the binary limit
β β β∗ 1 − C¯ 2α V¯1 + C¯ 2α V¯2 dC2 Gα = β , (33) β G dC20 V¯m which agrees with the result of Weinberg et al.23 in the limit of small supersaturations and for composition independent parβ β tial molar volumes. For V¯2 = V¯1 , the derivative of the critical 0 composition with respect to C2 is the ratio of the Hessians of the free energies Gα /Gβ which is positive and more generally the derivative of the critical composition with respect to the nominal composition is positive for positive partial molar volumes indicating that the composition of the critical nucleus always falls outside the binodal region. Figure 1 shows schematically, for a ternary, the deviation in composition from equilibrium. As an illustration of the
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124306-4
Philippe, Blavette, and Voorhees
J. Chem. Phys. 141, 124306 (2014)
FIG. 2. A portion of the Ni-Cr-Al phase diagram at 873 K on which is indicated the composition of the critical nucleus of the γ phase in a Ni-5.2Al14.2Cr alloy.
IV. CONCLUSION
FIG. 1. A graphical representation of the deviation in composition of the critical nucleus from equilibrium for a ternary in a particular case where the Hessian of the Gibbs energy of both phases are identical and the components 1, 2, and 3 have same partial molar volume in β (a) and in the general case (b).
capillarity effect on the composition of the critical nucleus for solid state precipitation, we have evaluated, using Eq. (32), the composition of the critical nucleus during precipitation at 873 K of the γ phase (L12 structure) in a disordered matrix (γ phase) for a concentrated Ni-5.2Al-14.2Cr alloy. Here, the ¯ V¯mβ has been neglected. The Gibbs free energy term in V/ of both phases have been computed in the framework of the CALPHAD approach using a thermodynamic database for Ni alloys.24 At this temperature, the bulk compositions for the γ and γ phases are, respectively, 0.037 and 0.161 for Al and 0.149 and 0.089 for Cr. The equilibrium molar fraction of γ is ∼0.12. The Hessians of the free energies have been evaluated numerically at the bulk compositions. Using Eq. (32) the composition of the critical nucleus is 0.169 in Al and 0.083 in Cr, as compared with the bulk phase, the critical nucleus is richer in Al (+0.8 at.%) and poorer in Cr (−0.6 at.%). As shown in Fig. 2, the composition of the critical nucleus falls outside the binodal region and does not lie on the equilibrium tie-line, with a significant deviation from equilibrium. During precipitation, it expected that L12 precipitates significantly enrich in Cr and deplete in Al. Such a result suggests that during the early stages of precipitation, significant deviations of the composition of the critical clusters from their bulk values are expected to be seen, especially near the spinodal, i.e., for high supersaturations.
In this paper, we have derived the properties of a critical nucleus in chemical equilibrium with the matrix in a multicomponent system using the capillarity theory. An analytical solution for the composition of a critical nucleus has been given for low supersaturation. In reality it is very difficult to get large supersaturations, where the difference between the composition of the matrix and the equilibrium composition is much greater than 0.1, for example. By definition, mole fractions are less than 1, hence perturbation solutions indicate that the main results of this work remain valid for many applications. It was found that, in general, the composition of a critical nucleus does not lie on the equilibrium tie-line. The deviation is mainly due to the difference in the Hessian of the Gibbs energy of the phases and the magnitude of the deviation in composition from equilibrium is order of the supersaturation. Despite our analysis strictly holds for small supersaturation only, this suggests strong deviations near the spinodal line. ACKNOWLEDGMENTS
We are grateful for the financial support of CARNOT ESP Institute and for DARPA under Contract No. W91CRB1010004. 1 T.
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