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A theory of spinodal decomposition stabilized by coherent strain in binary alloys
Computational Materials Science 77 (2013) 182–188
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A theory of spinodal decomposition stabilized by coherent strain in binary alloys Wolfgang Gaudig ⇑ Schwarzwaldstrasse 102, 70569 Stuttgart, Germany
a r t i c l e
i n f o
Article history: Received 11 October 2012 Received in revised form 18 March 2013 Accepted 18 April 2013
Keywords: Binary alloys Spinodal decomposition Atomic interaction Meta-stable short-range composition modulations Coherent strain
a b s t r a c t A phenomenological theory is developed which predicts the possible existence of meta-stable chemical composition modulations in the spinodal (instable) region of the phase diagram of binary alloys. This is accomplished through a modified elastic energy term in the theory of diffuse interfaces by Cahn and Hilliard: Analogously to Cahn’s expansion of the molar chemical free energy, also the (un-relaxed) internal stress tensor due to chemical composition modulations is expanded in a Taylor series of the spatial derivatives of the local composition up to second order terms. This mathematical approach is confirmed by atomistic modeling. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
2. Theory
For some binary alloys, experimental evidence was found that the short-range order state can be described not only by the atomic pair probability parameters (Warren parameters [1]) but also in terms of ordered micro-domains [2,3]. Aubauer [4,5] developed a phenomenological theory to explain this so-called state of ‘‘disperse order’’. He assumed a dilute system of equally sized domains with a trapezium shaped composition profile. Furthermore, he assumed that the ratio of the specific interface energy to the thickness of the rim zone be a constant. This ad hoc assumption is not evident, and indeed is not supported by Cahn’s well-known theory of diffuse interfaces. Hence, Aubauer’s theory was questioned [6], and the nature of short-range decomposition and short-range order was attributed solely to statistical composition fluctuations. This discrepancy is resolved in the present paper by modifying the theory of Cahn and Hilliard [7–10]: In the same way as the local molar chemical energy, also the un-relaxed (distortion-free) internal stress tensor due to a given composition modulation is expanded in a Taylor series of the spatial derivatives of the local chemical composition up to second order terms. It is shown that this modified theory of diffuse interfaces can in principle explain the possible existence of meta-stable composition modulations in the spinodal region of the phase diagram of binary alloys, which are identified with short-range decomposition.
2.1. Problem definition
⇑ Tel.: +49 711 7803286. E-mail address: [email protected] 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.04.040
A composition modulation c(r) implies a (yet distortion-free) internal stress which together with an external pressure p0 forms a stress field r(0) (i.e. the ‘un-relaxed state’) which gives rise to a displacement field u(r) so that a point r is shifted to the point r + u(r). The displacement field u(r) induces an elastic stress field r(e), and is determined by making the total stress r = (r(0) + r(e) fulfill the elasto-mechanical equilibrium condition (stress relaxation). The thermodynamic equilibrium is characterized by a minimum of the mean molar Gibbs free energy. The mean molar Gibbs free energy G is composed of the mean molar chemical free energy Fchem, the mean molar elastic energy Eelast, and the product of the external pressure p0 and the mean molar volume V:
G ¼ F chem þ Eelast þ p0 V:
ð1Þ
The local composition c is defined as the local mole fraction of the component B of the alloy AB, and is assumed to be a periodic function of the three spatial coordinates. Thus, the volume of the sample can be thought of being built up of equivalent unit domains with the same composition distribution. The material is taken to be isotropic, and the unit domains are approximated as spheres of equal size. In the un-relaxed state under pressure, the radius of these unit spheres is R. The total volume of all of the unit spheres is set equal to the volume of the sample. The boundary condition for each unit sphere is taken to be the same as for the whole
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
sample, i.e. the unit sphere is under the hydrostatic pressure p0. In this way, the calculation of the mean molar Gibbs free energy G is approximately reduced to a spherical symmetric problem. The diameter of the unit sphere, 2R, can be interpreted as the spatial period (or wave length) of the composition modulation which is described by the function c(r) where r is the amount of the radius vector r. The spherical composition distribution function c(r) must fulfill the condition that the mean mole fraction of the component B in the unit sphere is equal to the constant alloy composition c0:
3 R
3
Z
R
cðrÞ r 2 dr ¼ c0 :
ð2Þ
0
The equilibrium value of R, and the equilibrium function c(r) are possibly obtained by minimizing G. This is a variational problem which is numerically solved by setting c(r) to be a Fourier sum, thus making G approximately a function of a finite number of variables. For this purpose, the definition of c(r), physically defined only in the range 0 6 r 6 R, is mathematically extended to R 6 r 6 R by the symmetry requirement c(r) = c(r), and to any value of r by the periodicity requirement c(r + 2R) = c(r). Taking into account the conditional Eq. (2), this Fourier sum can be written as
cðrÞ ¼ c0
n0 n0 n p r 6 X cosðn pÞ X Kn þ K n cos ; 2 2 n R p n¼1 n¼1
ð3aÞ
where the Kn(n = 1–n0) are Fourier coefficients, and n0 is a sufficiently great number. In this way, G is approximated by a function of the independent variables R and the Kn. Differentiation of c(r) with respect to r yields: n0 dc p X r ¼ K n n sin n p ; dr R R n¼1 2 n0 d c p2 X r ¼ K n n2 cos n p ; 2 2 R R n¼1 dr
ð3bÞ ð3cÞ
which is needed later. 2.2. The chemical free energy According to Cahn and Hilliard [6], the local molar chemical free energy f is a function not only of the local composition c but also of the composition distribution in the immediate environment. Thus, f is a function of c as well as of the spatial derivatives of c. Straightforwardly, Cahn and Hilliard expanded f in a Taylor series of the spatial derivatives of c up to second order terms. For cubic crystal symmetry they obtained:
f ¼ f0 ðcÞ þ j1 r2 c þ
1 j2 ðrcÞ2 ; 2
ð4Þ
@j rðj1 rcÞ ¼ j1 r c þ 1 ðrcÞ2 ; @c 1 @ j1 ðrcÞ2 ; j2 f ¼ f0 ðcÞ þ rðj1 rcÞ þ 2 @c
ð5Þ ð6Þ
where f0(c) is the molar chemical free energy of the homogeneous material of constant composition c under vacuum, and $ means the Nabla operator. The coefficients j1 and j2 are defined by:
0
1
B @f C C j1 ¼ B @ A ; @ 0
@2 c @x2k
ð7aÞ
10
2
@ f
C j2 ¼ B @ 2 A ; @
@c @xk
0
F chem ¼
1 V sample
ð7bÞ
ZZZ
f dx1 dx2 dx3 ;
ð8Þ
sample
where the integration is taken over the volume Vsample of the sample in the un-relaxed state. Substituting the expression on the righthand side of Eq. (6) for f in Eq. (8) yields:
F chem ¼
1 V sample
ZZZ
1 @ j1 ðrcÞ2 j2 2 @c ZZZ rðj1 rcÞ
f0 ðcÞ þ
sample
dx1 dx2 dx3 þ
1 V sample
sample
d x1 d x 2 d x 3 :
ð9Þ
The second volume integral on the right-hand side of Eq. (9) can be transformed to an integral of (j1 rc) over the surface of the sample, and thus must be zero because there is no atomic diffusion through the surface of the sample ð$c ¼ 0Þ. The final result for cubic crystal symmetry is then given by
F chem ¼
1 V sample
ZZZ
h
i f0 ðcÞ þ j ðrcÞ2 dx1 dx2 dx3 ;
ð10Þ
sample
where
1 2
j ¼ j2
@ j1 @c
ð11Þ
is still a function of c, but is set to be approximately constant for applications. For a spherical symmetric composition distribution in an isotropic material, a local right-handed Cartesian coordinate system X1, X2, X3 is used, with the X3-axis in the radial direction, and the X1and X2- axes tangent to the sphere r = const.. In this coordinate system, the spherical symmetry center is given by O(0, 0, r). The mole fraction c at a position P(x1, x2, x3) is only dependent on its distance to the spherical symmetry center O, and thus is defined by the function
c¼c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 þ ðx3 þ rÞ2 :
ð12aÞ
The first and second partial derivatives of c with respect to x1, x2, x3 at the position (0, 0, 0) are obtained from Eq. (12a) by using the chain rule of differentiation:
@c ¼ 0; @x1
or, by applying the mathematical identity 2
which in general are dependent on c. The index ‘‘0’’ means that the differentiations are to be carried out in the homogeneous state of the material. The coordinates xk(k = 1, 2, 3) denote a right-handed Cartesian coordinate system with axes parallel to the crystallographic h1, 0, 0i-directions. The mean molar chemical free energy is given by
@c ¼ 0; @x2
@ 2 c 1 dc ¼ ; @x21 r dr
@c dc ¼ ; @x3 dr
@ 2 c 1 dc ¼ ; @x22 r dr
@2c ¼ 0; @x1 @x2
@2c ¼ 0; @x2 @x3
ð12bÞ 2
@2c d c ¼ ; @x23 dr 2 @2c ¼ 0: @x1 @x3
ð12cÞ ð12dÞ
Substituting the first partial derivatives from Eqs. (12b) for the gradient of c in Eq. (10), and replacing the volume element dxdydz by 4pr2dr, and the volume of the sample by 4pR3/3, finally yields
F chem ¼
3 R3
Z 0
R
"
2 # dc r 2 dr; f0 ðcÞ þ j dr
ð13Þ
since the unit sphere is supposed to represent the whole sample. The function f0(c) is required to display the situation of spinodal decomposition exhibiting a miscibility gap: Two minima and one maximum in between define the A-rich phase and the B-rich phase
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
in an isostructural alloy system AB. f0(c) is here represented as a Taylor expansion about an appropriate composition c up to the mth 0 order term,
f0 ðcÞ ¼ a0 þ
m0 X am ðc cÞm ;
ð14Þ
m¼1
where the am(m = 0 to m0) are proper coefficients.
justified provided the crystal lattice parameter does not vary too much over the chemical composition (see Appendix A, Table A.2). Plugging Eqs. (17a–g), into Eq. (15) yields:
2 X @2c @2c @c þ b þ a 2 1 2 @x @x2k @x k i i–k X @c 2 þ b2 ; @xi i–k
rð0Þ kk ¼ S0 ðcÞ þ a1
2.3. The elastic energy
rð0Þ kl ¼ c1 As well as the local chemical free energy f, also the distortionfree stress tensor r(0) at a given point r can be assumed as a function not only of the composition c at that point but also of the composition distribution in the immediate environment of r. Thus r(0) as well as f is a function not only of c but also of the spatial derivð0Þ atives of c. Analogously to f [7], the elements rkl of the tensor r(0) are expanded in a Taylor series of the spatial derivatives of c. The indices k and l (k = 1–3, l = 1–3) symbolize the three axes of a right-handed Cartesian coordinate system. The Taylor expansion ð0Þ of rkl with respect to spatial derivatives up to second order terms, about the state of uniform composition c under pressure p0, is given by
rð0Þ kl ¼ Skl ðcÞ þ
3 X
jð0Þ i;kl
i¼1
3 X 3 3 X 3 X @c X @2c ð1Þ þ jij;kl þ jð2Þ @xi i¼1 j¼i @xi @xj i¼1 j¼i ij;kl
2 dij @c @c ; @xi @xj 2
ð15Þ
where x1, x2, and x3 are the three spatial coordinates, and dij means the Kronecker symbol (dij = 1 for i = j, dij = 0 for i – j). Skl(c) denotes the stress tensor in the state of uniform composition c const. unð0Þ ð1Þ ð2Þ der pressure p0. The coefficients ji;kl ; jij;kl , and jij;kl , are defined as:
0
1 ð0Þ @ r j ¼ @ kl A ; @c @ @x i 0 0 1 ð0Þ @ r ð1Þ kl jij;kl ¼ @ 2 A ; @ @x@ @xc i j 0 0 1 2 ð0Þ @ r ð2Þ kl @ jij;kl ¼ A ; @c @c @ @x @ @x i j ð0Þ i;kl
ð16aÞ
ð16bÞ
ð16cÞ
where the index ‘‘0’’ indicates that the differentiations are to be carried out in the homogeneous state under pressure (and at temperature). For cubic crystal symmetry, choosing the Cartesian coordinate ð0Þ ð1Þ ð2Þ axes along the h1 0 0i-directions, Skl, ji;kl ; jij;kl , andjij;kl , must be invariant to the symmetry operations of a 90° and 180° rotation about any of the three fourfold coordinate axes. This, and the fact that r(0) must be a symmetric tensor, drastically reduces the number of independent coefficients – 0:
Skl ðcÞ ¼ 0 for k – l;
ð17aÞ
Skl ðcÞ ¼ S0 ðcÞ for k ¼ l;
ð17bÞ
ð0Þ ji;kl ¼ 0 for all i; kl;
ð17cÞ
¼ a1 ;
ð1Þ ij;kl
¼ b1 ;
j j
jð1Þ ij;kl ¼ c1 ;
1 ð2Þ j ¼ a2 for i ¼ j ¼ k ¼ l; 2 ij;kl 1 ð2Þ j ¼ b2 for i ¼ j – k ¼ l; 2 ij;kl ð2Þ jij;kl ¼ c2 for i ¼ k < l ¼ j or j ¼ k > l ¼ i;
jð1Þ jð2Þ for the remaining ij; kl: ij;kl ¼ 0; ij;kl ¼ 0
for k – l:
ð18bÞ
S0(c) is composed of the external hydrostatic pressure p0, and the hydrostatic pressure due to different sizes of the component atoms, and is assumed to be a linear function of c:
S0 ðcÞ ¼ p0 # ðc c0 Þ;
ð19Þ
taking into account that S0(c0) must be equal to p0. Since c is meant to be the mole fraction of component B, a positive value of # indicates that the B atoms are bigger than the A atoms. The parameter # is still dependent on the composition c, but is assumed here approximately to be a constant (see Appendix A, Table A.2). ð0Þ ð1Þ ð2Þ For an isotropic material, Skl ; ji;kl ; jij;kl , and jij;kl must be invariant to any rotation of the coordinate system which yields two additional relations,
c1 ¼ a1 b1 ; c2 ¼ a2 b2 :
ð20aÞ ð20bÞ
For a spherical composition distribution c(r) in an isotropic material, it is expedient to employ a local Cartesian coordinate system with the X3 axis in the radial direction, and the two other axes tangent to the sphere r = const., as was already done in case of the chemical free energy, Section 2.2. Using the corresponding coordinate transformation Eqs. (12b–d), and taking into account Eq. (19), Eqs. (18a–b) yield the disappearance of the shear stress comð0Þ ð0Þ ð0Þ ponents, r12 ; r13 ; r23 , and the following formulae for the normal ð0Þ ð0Þ stress components, rr ¼ r33 in the radial direction, and ð0Þ ð0Þ ð0Þ rt ¼ r11 ¼ r22 in the tangent directions: 2
0
ð1Þ ij;kl
@2c @c @c þ c2 @xk @xl @xk @xl
ð18aÞ
ð17dÞ ð17eÞ ð17fÞ ð17gÞ
The coefficients a1, a2, b1, b2, c1, c2, although in general dependent on c, are here approximately treated as material constants, what is
1 dc d c þ a1 2 r dr dr
rð0Þ r ¼ p0 # ðc c 0 Þ þ 2 b1 þ a2
2 dc ; dr
ð21aÞ 2
1 dc d c þ b1 2 r dr dr
rð0Þ t ¼ p0 # ðc c 0 Þ þ ða1 þ b1 Þ þ b2
2 dc : dr
ð21bÞ
The elastic energy is calculated by applying the linear elastic theory for an isotropic continuum. The differential equation for mechanical equilibrium in the case of spherical symmetry is
dðrr r 2 Þ ¼ 2 rt r; dr
ð22Þ
where rr and rt are the total normal stresses in the radial and the tangent directions, respectively. rr and rt are given by ð eÞ rr ¼ rð0Þ r þ rr ; ð0Þ ð eÞ rt ¼ rt þ rt ; ðeÞ
ð23aÞ ð23bÞ ðeÞ
where rr and rt are the elastic normal stresses in the radial and the tangent direction, respectively, which are induced by the displacement field (see Section 2.1). According to spherical symmetry, the displacement is zero in tangent direction, and is in radial direction given by a function u(r) to be determined. Thus, the strain in
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
radial direction is er = du(r)/dr, and in tangent direction et = u(r)/r. With these expressions, Hook’s law of elasticity for an isotropic material yields:
E 1 m duðrÞ E 2 m uðrÞ þ ; 1 2 m 1 þ m dr 12m 1þm r E 1 uðrÞ E m duðrÞ ¼ þ ; 12m 1þm r 1 2 m 1 þ m dr
rrðeÞ ¼
ð24aÞ
rtðeÞ
ð24bÞ
where E is Young’s modulus, and m is Poisson’s ratio, which are approximately treated here as constants. With Eqs. (23a–b) and (24a–b), and the boundary condition rr = p0 for r = R, the solution of the differential Eq. (22) is obtained to be:
uðrÞ 1 12m 1þm 2 12m ¼ ½2 IðrÞ þ JðrÞ þ r 3 3 E 1m E 1þm 12m 12m IðRÞ JðRÞ p0 ; 1m 1m E
ð25Þ
where I(r) and J(r) are auxiliary functions introduced for convenience which are defined as:
Z
1 ð0Þ ð0Þ dr; rr rt r 0 Z r 1 ð0Þ JðrÞ ¼ 3 dr: r 2 rð0Þ r þ 2 rt r 0
IðrÞ ¼
r
ð26aÞ ð26bÞ
The total stress can be derived from Eqs. (23a–b), (24a–b), and (25), and it was found: 2 1þm 2 12m ½IðRÞ IðrÞ ½JðRÞ JðrÞ p0 ; 3 1m 3 1m m 2 1 þ m 1 12m rt ¼ rtð0Þ rð0Þ þ ½IðRÞ IðrÞ ½2 JðRÞ þ JðrÞ p0 : 3 1m 3 1m 1m r
rr ¼
ð27aÞ ð27bÞ
The mean molar elastic energy is then given by:
1 R3 2 E r 2 dr;
Eelast ¼ V 0
3
Z
R
0
r2r þ 2 ð1 mÞ r2t 4 m rr rt
ð28Þ
where V0 is the molar volume of the homogeneous material of constant composition c0 under pressure p0. 2.4. The volume The mean molar volume of the stress-relaxed sample under the hydrostatic pressure p0 is:
V ¼ V0
uðRÞ 3 : ½R þ uðRÞ V 1 þ 3 0 R R3 1
ð29aÞ
m, a1 ¼ a1 =#, a2 ¼ a2 =#, b1 ¼ b1 =#, b2 ¼ b2 =#, p0 ¼ p0 =#. For the sake of simplicity, these parameters are treated in the following as constants. Then it is obvious that the minimization of G⁄ must yield the same result as the minimization of G.
3. An arbitrary test computation In order to show that the minimization of G⁄ can in principle yield meta-stable composition modulations, an arbitrary numerical parameter set was chosen: c ¼ 0:2, m0 = 4, a1 ¼ 0:002; a2 ¼ 0:2; a3 ¼ 0, a4 ¼ 10:, m = 0.3, j⁄ = 0.005 nm2, a1 ¼ 0:1 nm2 ; b1 ¼ 0; a2 ¼ 0. b2 ¼ 0; p0 ¼ 0, and c0 was varied in the range 0.15–0.25 in steps of 0.005. The number of Fourier coefficients was taken to be n0 = 100. The sample was assumed to be under vacuum; high pressure effects shall not be treated here. f0(c) was taken to be a forth degree polynomial function displaying one maximum and two minima. j⁄ was chosen to be an arbitrary positive value. b1 ¼ b2 ¼ 0 corresponds to the assumption that the first and the second spatial derivative of the composition along any axis cause a normal stress only in the direction of this axis but not in a transverse direction. This neglect of the transverse stress effect is reasonable but not necessary for the occurrence of a meta-stable composition modulation. Furthermore, parameter study revealed that a2 , in contrast to a1 , has no effect on the stabilization of composition modulations, and therefore was set to zero. The minimization procedure of G⁄ was carried out with the aid of a personal computer equipped with a FORTRAN compiler. G⁄ was minimized by incrementally changing the variables R and the Kn(n = 1–100) in such a way that the value of G⁄ was maximally reduced (‘‘gradient procedure’’). For c0 6 0.15 and c0 P 0.25, the computer calculation yielded a uniform composition c c0. For 0.155 6 c0 6 0.195 and 0.21 6 c0 6 0.245 (in the spinodal region of the phase diagram), however, meta-stable composition modulations were obtained, Figs. 1–4. Fig. 1 displays the assumed non-dimensionalized mean molar Gibbs free energy as a function of the alloy composition c0, for the homogeneous state of the alloy as well as the state with equilibrium composition modulations. Fig. 2 shows the period of the composition modulations in terms of 2R, as a function of the alloy composition c0. Fig. 3 displays the meta-stable local composition c as a function of the non-dimensional spatial coordinate r/R, for different values of c0. As can be seen, c is monotonically decreasing with r/R for A-richer alloys, and monotonically increasing for B-richer alloys. Furthermore, adding the component B to the A-richer alloy as well as adding the component A to the B-richer
12m V ¼ V0 1 3 ðJðRÞ þ p0 Þ : E
ð29bÞ
2.5. The Gibbs free energy The mean molar Gibbs free energy G is given by Eqs. (1), (13), (14), (28), (27a–b), (26a–b), (21a–b), (29b), and (3a–c) as a function of the parameters c0, c, am(m = 0 to m0), j, m, E, V0, #, a1, a2, b1, b2, and the variables R, Kn(n = 1–n0). In order to reduce the number of parameters required, it is expedient to introduce a non-dimensionalized Gibbs free energy function
E G a0 G ¼ 2 p0 : V0 V0 #
non-dimensionalized Gibbs free energy
Using Eq. (25), this yields approximately: 0,001
homogeneous solid solution
with equilibrium composition modulation : B-enriched volume fraction < 0.5 A-enriched volume fraction < 0.5
0,0005 0 -0,0005 -0,001 -0,0015 0,05
0,1
0,15
0,2
0,25
0,3
0,35
alloy composition (mean mole fraction of component B)
ð30Þ
As can easily be verified, G⁄ can be written as a function of the parameters c0, c; am ¼ am E=ðV 0 #2 Þ (m = 1–m0), j⁄ = j E/(V0 #2),
Fig. 1. Arbitrary test calculation: Non-dimensionalized molar Gibbs free energy of the homogeneous alloy, and its decrease caused by the equilibrium composition modulation as calculated in dependence on the alloy composition c0 (given as the mean mole fraction of component B).
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
(a)
period length 2R [nm]
3
Z d/4
2,95
d
2,9 2,85
F(i,j)
2,8
j
2.5d
X
2,75 B-enriched volume fraction < 0.5 A-enriched volume fraction < 0.5
2,7 2,65 0,15
d/4 0,17
0,19
0,21
0,23
i
reference area
0,25
alloy composition (mean mole fraction of component B) Fig. 2. Arbitrary test calculation: Period length 2R of the equilibrium composition modulation as a function of the alloy composition c0.
crystal lattice segment I crystal lattice segment II
local composition c
0,4 alloy composition (mean mole fraction of component B)
0,3
(b)
0,18 0,17 0,16
0,2
Z d/4
d
0,24 0,23 0,22
0,1
2.5d Y
0 0
0,2
0,4
0,6
0,8
1
non-dimensional spatial coordinate r/R
reference area
d/4
B-enriched volume fraction
Fig. 3. Arbitrary test calculation: Local composition c (local mole fraction of component B) as a function of the non-dimensionalized spatial coordinate r/R, for different alloy compositions c0.
1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0,15
Fig. A.1. Atomistic model of the distortion-less internal stress tensor r(0) in a binary face-centered cubic 8d 8d 8d crystal (d = lattice parameter): A central 4d 4d 4d section of the crystal shows atom positions as dots in a right-handed Cartesian coordinate system. The crystal is fictively sub-divided by a reference area, i.e. the Y/Z-plane (x = 0), into two segments with atoms being denoted as i and j, respectively. Fij illustrates an example atomic interaction force between the two atoms i and j with their joining line intersecting the reference area of 2.5d 2.5d. (a) X/Z-plane (y = 0), and one adjacent crystallographic plane (y = d/4). (b) Y/Zplane (x = 0), and one adjacent crystallographic plane (x = d/4).
B-enriched volume fraction < 0.5 A-enriched volume fraction < 0.5
0,17
0,19
0,21
0,23
0,25
alloy composition (mean mole fraction of component B) Fig. 4. Arbitrary test calculation: Volume fraction enriched in component B(cB > c0), for different alloy compositions c0.
Table A.1 Morse potential parameters [12]. Metal 1
alloy gives rise to an increase of the period 2R, and an increased deviation of the profile of the composition modulation from a sinusoidal shape. In Fig. 4, the fraction of the sample’s volume which is enriched in the component B, i.e. where c > c0 holds, is plotted versus the alloy composition c0. It can be seen that there is a misfit between the two branches of the curve (B-enriched volume fraction <0.5, and A-enriched volume fraction <0.5). Actually, for c0 = 0.20 and c0 = 0.205, the local equilibrium composition c as calculated turned out to be not any more a monotonic function of r/R in the interval 0 < r/R < 1. This improper result indicates that the theoretical model in the present form (spherical unit domain) is no longer valid in this regime, and therefore the results for c0 = 0.20 and c0 = 0.205 are excluded from the graphs. A meta-stable composition modulation can not only be obtained for some special numerical example but also for extended
k (nm ) r0 (nm) D (zJ)
Cu
Ag
Al
13.588 0.2866 54.94
13.690 0.3115 53.24
11.646 0.3253 43.31
ranges of the values of the parameters. a1 seems to play a decisive role, and, according to test calculations, must necessarily be positive for the occurrence of meta-stable composition modulations. The parameters a1 and j⁄ control the value of the period 2R: Increasing values of a1 , as well as increasing j⁄ cause an increasing period 2R. 4. Discussion The modified theory of diffuse interfaces as given in the present paper predicts the possible existence of meta-stable composition
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W. Gaudig / Computational Materials Science 77 (2013) 182–188 Table A.2 Results of atomistic modeling for a-Cu–Al at various compositions, and a-Ag–Al. Input: Mole fraction Al (c0), external pressure (p0). Output: Crystal lattice parameter (d), model parameters (#, a1, a2, b1, b2, c1, c2).
a-Cu–Al
Alloy c0 (mol/mol) p0 (GPa) d (nm) # (GPa) a1/# (nm2) a2/# (nm2) b1/# (nm2) b2/# (nm2) c1/# (nm2) c2/# (nm2)
0.10 0. 0.3652 48.53 0.00974 0.00000 0.00118 0.00002 0.00470 0.00001
0.15 0. 0.3675 46.22 0.00987 0.00000 0.00117 0.00002 0.00471 0.00001
a-Ag–Al 0.20 0. 0.3698 44.03 0.01001 0.00000 0.00117 0.00001 0.00473 0.00001
0.10 0. 0.4069 2.24 0.10671 0.00000 0.03277 0.00004 0.07172 0.00002
modulations in the spinodal region of the phase diagram. The driving force of the decomposition is the decrease of the chemical free energy. The stabilization of short-range composition modulations is achieved by introducing distortion-free stress terms due to the spatial derivatives of the local composition up to the second order. It should be noted that the calculation of the elastic energy taking the unit domain of the composition distribution to be a sphere can only be an approximation. The result of this approximation seems to be reasonable. However, a more accurate calculation should be carried out by employing a more realistic unit domain, e.g. a cube, to confirm the results of this paper. This could be carried out with the aid of the finite elements method. Furthermore, crystal anisotropy should also be taken into account. The phenomenological theory as given in this paper takes explicitly into consideration composition modulations only. Order parameters could be taken implicitly into account through the function f0(c), i.e. order parameters could be assumed as fixed functions of the composition c. A rigorous theory, however, requires order parameters to be treated explicitly as independent of the composition: Accordingly, Poduri and Chen [11] extended the theory of spinodal decomposition by adding a long-range order parameter gradient term to the chemical free energy. It remains to be examined, however, to what extent order parameter gradient terms have also to be introduced in the expression of the elastic energy within the framework of the present modified theory of diffuse interfaces. To adapt the present theory to real alloy systems, it is further necessary to get an idea of the values of the newly introduced parameters a1, a2, b1, b2, c1, c2, at least the most important parameter a1. This requires atomistic modeling, i.e. the knowledge of the bilateral interaction energy of the atoms [12], as is demonstrated in the Appendix.
5. Conclusions The present study predicts the possible existence of meta-stable short-range decomposition in the spinodal (instable) region of the phase diagram, and reveals its relation to spinodal decomposition: It can be concluded that meta-stable short-range decomposition (in combination with short-range order) resembles the beginning of instable spinodal decomposition, i.e. the appearance of composition modulations. The difference is that spinodal decomposition is open to unlimited coarsening, whereas short-range decomposition is stabilized by coherent strain energy as revealed for example by former transmission electron microscopy of a-Cu–Al alloys [3]. It should be kept in mind that the occurrence of meta-stable composition modulations is conditional on appropriate values of the parameters, e.g. a1 ¼ a1 /# must be positive. As is shown in Appendix A, this condition is for example fulfilled for a-Cu–Al alloys
but is not for a-Ag–Al alloys. If this condition is not met then the theory of statistical composition fluctuations must be invoked. Since the present theory aims at an equilibrium state of the alloy system, the computation work can be restricted to any mathematical minimization routine without the need for physical relevance of the time scale. In contrast, the modeling of spinodal decomposition requires a dynamic theory, and a correspondingly much more sophisticated computation routine in order to describe the temporal development of non-equilibrium states at a real time scale. Advanced modeling was carried out at the atomic scale [13] going beyond the well-known original theory of Cahn and Hilliard [9,10], and was successfully applied to Fe–Cr-alloys. However, these models unfortunately do not include elastic strain effects, and consequently are not suited to detect eventual meta-stable states in the spinodal region of the phase diagram. A last comment refers to thin metallic films. Within the last 5– 10 years research has been performed in the field of thin films with modulated chemical composition for various technical purposes. Present modeling as described, however, is rigorously designed for application to bulk material. Films could be approximately treated like bulk material if twice its surface thickness is small (1%) compared to the bulk thickness of the film. In this context, the surface thickness means the reach of the interatomic force, which is defined as the range where the atomic pair potential function falls off to 1% of the dissociation energy (see Appendix A, Eq. (A.3a)). This assessment results in a minimum film thickness of about 150 nm, when using the data of Table A.1. Thinner films need special atomistic modeling of the surface region. Appendix A. Atomistic modeling Fig. A.1a and b shows an atomistic model to define the distortion-free internal stress tensor r(0), with the aim to get an estimate of the parameters #, a1, a2, b1, b2, c1, c2, for binary face-centered cubic crystals. The dots depict atom positions each of them occupied with probability (1-c) by an A-atom and probability c by a B-atom. The right-handed Cartesian coordinate system X, Y, Z is chosen so that each of the coordinate planes, X/Y, Y/Z; and Z/X, is a mid-plane between two adjacent parallel crystallographic {1 0 0}-planes. The total force on a reference area (2.5d square, d = lattice parameter) perpendicular to the X-axis is calculated as
X rij X rj ri F ij ¼ F ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 r ij ij ij ðx x Þ þ ðy y Þ2 þ ðz z Þ2 j
i
j
i
j
ðA:1Þ
i
where F ij means the expectation value of the atomic interaction force transmitted from an atom no. j of the crystal segment II on the right side of the reference area onto an atom no. i of the crystal segment I on the left side of the reference area, Fig. A.1a. rij is the vector pointing from atom no. i to atom no. j, and rij is its amount. ri and rj give the positions of atom i and atom j, respectively, and xi, yi, zi, and xj, yj, zj denote their components in the coordinate system X, Y, Z. The summation in Eq. (A.1) is over all pairs of atoms i and j the joining line of which intersects the reference area, with two special cases: (i) if a side of the reference square is hit then a statistical weight factor of 1/2 has to be taken into account, and (ii) if a corner is hit then a weight factor of 1/4 has to be applied. The expectation value F ij of the atomic interaction force, for a binary alloy AB, is given by
F ij ¼ ð1 cðri ÞÞ cðrj Þ F ij ðABÞ þ cðri Þ ð1 cðrj ÞÞ F ij ðBAÞ þ ð1 cðri ÞÞ ð1 cðrj ÞÞ F ij ðAAÞ þ cðri Þ cðrj Þ F ij ðBBÞ;
ðA:2Þ
where c(ri) and c(rj) denote the mole fractions of atom B at the positions ri and rj, respectively. Fij(AB), Fij(BA), Fij(AA), and Fij(BB) symbolize the interaction force of the atom pair i, j in case that the
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
positions i and j are occupied by atoms A and B, B and A, A and A, B and B, respectively. In Eq. (A.2), atomic pair probability parameters are neglected. The atomic interaction potential is assumed to be represented by the Morse function [12]:
uðrij Þ ¼ D ½expð2 k ðrij r0 ÞÞ 2 expðk ðrij r0 ÞÞ;
ðA:3aÞ
and the atomic interaction force is then given by
F ij ¼
d uðr ij Þ d rij ðA:3bÞ
where rij is the distance between two atoms, and the parameters D, k, and r0 are still dependent on the occupation of the atom pair (AB, BA, AA, BB). On the other hand, the force on the reference area (x = 0, Dy 6 y 6 Dy, Dz 6 z 6 Dz), Eq. (A.1), can be expressed as an integral of the stress tensor r(0) over that reference area (Dy = Dz = 1.25d). This yields:
r ð0Þ xy 2 Dy 2 Dz ¼ r ð0Þ xz 2 Dy 2 Dz ¼
ðA:7aÞ
ð0Þ xx ¼ p0 # Dc ð2Þ r ð0Þ xx ð3Þ r ¼ p0 þ a1 2=a2
ðA:7bÞ ðA:7cÞ
ð0Þ xx ¼ p0 þ a2 =a2 ð4Þ r
ðA:7dÞ
ð0Þ xx ð5Þ r ¼ p0 #=3 ðDyÞ2 =a2 þ b1 2=a2 þ b2 4=3 ðDyÞ2 =a4
ðA:7eÞ
ð0Þ xx ¼ p0 þ b2 =a2 ð6Þ r
ðA:7fÞ
ð0Þ xy ¼ c1 =ða: bÞ ð7Þ r
ðA:7gÞ
ð0Þ xy ð8Þ r ¼ c2 =ða bÞ:
¼ 2 k D ½expðk ðr ij r 0 ÞÞ expð2 k ðrij r0 ÞÞ;
r ð0Þ xx 2 Dy 2 Dz ¼
ð0Þ xx ð1Þ r ¼ p0
Z
Dy
Z
Dy
Z
Dy
Z
Dy
Z
ð0Þ
Dz Dz
Dz Dz
Dy Z Dy
ð0Þ
rð0Þ XX dy dz ¼
X x j xi F ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ij ðxj xi Þ2 þ ðyj yi Þ2 þ ðzj zi Þ2
ðA:4aÞ
rð0Þ XY dy dz ¼
X yj yi F ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 2 ij ðxj xi Þ þ ðyj yi Þ þ ðzj zi Þ
ðA:4bÞ
rð0Þ xz dy dz ¼
X zj zi F ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 2 ij ðxj xi Þ þ ðyj yi Þ þ ðzj zi Þ
ðA:4cÞ
Dz Dz
ð0Þ
where rxx ; rxy ; rxz correspond to the continuum-mechanical expressions, Eqs. (18a–b) and (19), of the present generalized theory of diffuse interfaces: 2
@ c rð0Þ ¼ p0 # ðc c0 Þ þ a1 @x xx 2 þ b1
þa2
@c 2 @x
þ b2
2 @c @y
þ
@c 2
h
@2 c @y2
2
þ @@z2c
i ;
ðA:5aÞ
@z
rð0Þ xy ¼ c1
@2c @c @c þ c2 ; @x @y @x @y
ðA:5bÞ
rð0Þ xz ¼ c1
@2c @c @c þ c2 ; @x @z @x @z
ðA:5cÞ
with the composition c and its spatial derivatives taken at the surð0Þ ð0Þ xy ð0Þ face of the reference area, i.e. at x = 0. The symbols r ;r xx ; r xz can be interpreted as the mean values of the stress tensor elements ð0Þ ð0Þ ð0Þ rxx ; rxy ; rxz over the reference area, respectively. In order to determine the external pressure p0, and the parameters #, a1, a2, b1, b2, c1, c2, simple test functions c = c(x, y) are chosen:
ð1Þ c ¼ c0
ðA:6aÞ
ð2Þ c ¼ c0 þ Dc
ðA:6bÞ
ð3Þ c ¼ c0 þ ðx=aÞ2
ðA:6cÞ
ð4Þ c ¼ c0 þ x=a
ðA:6dÞ
ð5Þ c ¼ c0 þ ðy=aÞ2
ðA:6eÞ
ð6Þ c ¼ c0 þ y=a
ðA:6fÞ
ð7Þ c ¼ c0 þ x=a y=b
ðA:6gÞ
ð8Þ c ¼ c0 þ x=a þ y=b;
ðA:6hÞ
where Dc is a small deviation of the alloy composition c0, and a and b are arbitrary constants which are set to 100d for the present calculation. Each of these test functions are plugged into Eqs. (A.5a–b). ð0Þ ð0Þ The resulting expressions of the tensor elements rxx and rxy are used to determine the integrals in Eqs. (A.4a–b). This yields the following equations, respectively:
ð0Þ xx ,
ðA:7hÞ ð0Þ xy
The quantities r and r are calculated for each of these cases, no. 1–8, according to the right-hand expressions of Eqs. (A.4a–b), by using a personal computer with a FORTRAN compiler. The first of these equations, Eq. (A.7a), is used to determine the crystal lattice parameter d for a given external pressure, i.e. p0=0 in the present case. The second equation, Eq. (A.7b), serves to calculate the parameter # = (Dp0/Dc)d=const., where Dc is taken to be 0.0001. The remaining six equations, Eqs. (A.7c–h), yield the six newly introduced parameters a1, a2, b1, b2, c1, c2. Values of the Morse potential parameters k, r0, and D are given in [12] for pure metals, at absolute zero temperature and zero pressure. Table A.1 shows example data for Cu, Ag, and Al. For the purpose of the present paper, the Morse potential function of atom pairs of different components (A–B) is set equal to the average of the potential functions of atom pairs of the same component (A– A and B–B):
F ij ðABÞ ¼ 1=2 ½F ij ðAAÞ þ F ij ðBBÞ:
ðA:8Þ
This corresponds to the neglect of atomic pair probability parameters, as was already assumed in Eq. (A.2). Results of the present atomistic modeling are given in Table A.2 for a-Cu–Al at various alloy compositions, and a-Ag–Al as examples. It can be seen that the empiric condition for the occurrence of meta-stable composition modulations, i.e. a1/# > 0, is fulfilled for a-Cu–Al, but is not for a-Ag–Al. Furthermore, the three parameters a2/#, b2/#, c2/# are zero or near zero so that one has only to deal with the four parameters #, a1/#, b1/#, c1/#. The most important parameters are # and a1/#. Short-range order is classified by the condition that, for next nearest neighbor atoms A and B, the value of Fij(AB), contrary to Eq. (A.8), is greater then the average value of Fij(AA) and Fij(BB), due to enhanced attraction of different sorts of atoms. This can result in considerably greater values of a1/#, as was verified by computer. The present estimation of the parameters #, a1, a2, b1, b2, c1, c2 is in the first place physical confirmation of the mathematically motivated generalized theory of diffuse interfaces, Eqs. (A.5a–c). For more accurate calculations, however, it will be necessary to take into account atomic pair probability parameters, and also electronic long-range interactions (atomic pair potential oscillations). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
B.E. Warren, B.L. Averbach, B.W. Roberts, J. Appl. Phys. 22 (1951) 1493–1502. W. Gaudig, H. Warlimont, Z. Metallk. 60 (1969) 488. W. Gaudig, H. Warlimont, Acta Metall. 26 (1978) 709–724. H.P. Aubauer, Acta Metall. 20 (1972) 165–172. H.P. Aubauer, Acta Metall. 20 (1972) 173–180. D. de Fontaine, Scripta Metall. 7 (1973) 463. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 (1958) 258–267. J.W. Cahn, J. Chem. Phys. 30 (1959) 1121–1124. J.W. Cahn, Acta Metall. 9 (1961) 795–801. J.W. Cahn, Acta Metall. 10 (1962) 179–183. R. Poduri, L.Q. Chen, Acta Mater. 44 (1996) 4253–4259. L.A. Girifalco, V.G. Weizer, Phys. Rev. 114 (1959) 687–690. J.M. Hyde, A.P. Sutton, J.R.G. Harris, A. Cerezo, A. Gardiner, Modell. Simul. Sci. Eng. 4 (1996) 33–54.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
A theory of spinodal decomposition stabilized by coherent strain in binary alloys Wolfgang Gaudig ⇑ Schwarzwaldstrasse 102, 70569 Stuttgart, Germany
a r t i c l e
i n f o
Article history: Received 11 October 2012 Received in revised form 18 March 2013 Accepted 18 April 2013
Keywords: Binary alloys Spinodal decomposition Atomic interaction Meta-stable short-range composition modulations Coherent strain
a b s t r a c t A phenomenological theory is developed which predicts the possible existence of meta-stable chemical composition modulations in the spinodal (instable) region of the phase diagram of binary alloys. This is accomplished through a modified elastic energy term in the theory of diffuse interfaces by Cahn and Hilliard: Analogously to Cahn’s expansion of the molar chemical free energy, also the (un-relaxed) internal stress tensor due to chemical composition modulations is expanded in a Taylor series of the spatial derivatives of the local composition up to second order terms. This mathematical approach is confirmed by atomistic modeling. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
2. Theory
For some binary alloys, experimental evidence was found that the short-range order state can be described not only by the atomic pair probability parameters (Warren parameters [1]) but also in terms of ordered micro-domains [2,3]. Aubauer [4,5] developed a phenomenological theory to explain this so-called state of ‘‘disperse order’’. He assumed a dilute system of equally sized domains with a trapezium shaped composition profile. Furthermore, he assumed that the ratio of the specific interface energy to the thickness of the rim zone be a constant. This ad hoc assumption is not evident, and indeed is not supported by Cahn’s well-known theory of diffuse interfaces. Hence, Aubauer’s theory was questioned [6], and the nature of short-range decomposition and short-range order was attributed solely to statistical composition fluctuations. This discrepancy is resolved in the present paper by modifying the theory of Cahn and Hilliard [7–10]: In the same way as the local molar chemical energy, also the un-relaxed (distortion-free) internal stress tensor due to a given composition modulation is expanded in a Taylor series of the spatial derivatives of the local chemical composition up to second order terms. It is shown that this modified theory of diffuse interfaces can in principle explain the possible existence of meta-stable composition modulations in the spinodal region of the phase diagram of binary alloys, which are identified with short-range decomposition.
2.1. Problem definition
⇑ Tel.: +49 711 7803286. E-mail address: [email protected] 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.04.040
A composition modulation c(r) implies a (yet distortion-free) internal stress which together with an external pressure p0 forms a stress field r(0) (i.e. the ‘un-relaxed state’) which gives rise to a displacement field u(r) so that a point r is shifted to the point r + u(r). The displacement field u(r) induces an elastic stress field r(e), and is determined by making the total stress r = (r(0) + r(e) fulfill the elasto-mechanical equilibrium condition (stress relaxation). The thermodynamic equilibrium is characterized by a minimum of the mean molar Gibbs free energy. The mean molar Gibbs free energy G is composed of the mean molar chemical free energy Fchem, the mean molar elastic energy Eelast, and the product of the external pressure p0 and the mean molar volume V:
G ¼ F chem þ Eelast þ p0 V:
ð1Þ
The local composition c is defined as the local mole fraction of the component B of the alloy AB, and is assumed to be a periodic function of the three spatial coordinates. Thus, the volume of the sample can be thought of being built up of equivalent unit domains with the same composition distribution. The material is taken to be isotropic, and the unit domains are approximated as spheres of equal size. In the un-relaxed state under pressure, the radius of these unit spheres is R. The total volume of all of the unit spheres is set equal to the volume of the sample. The boundary condition for each unit sphere is taken to be the same as for the whole
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
sample, i.e. the unit sphere is under the hydrostatic pressure p0. In this way, the calculation of the mean molar Gibbs free energy G is approximately reduced to a spherical symmetric problem. The diameter of the unit sphere, 2R, can be interpreted as the spatial period (or wave length) of the composition modulation which is described by the function c(r) where r is the amount of the radius vector r. The spherical composition distribution function c(r) must fulfill the condition that the mean mole fraction of the component B in the unit sphere is equal to the constant alloy composition c0:
3 R
3
Z
R
cðrÞ r 2 dr ¼ c0 :
ð2Þ
0
The equilibrium value of R, and the equilibrium function c(r) are possibly obtained by minimizing G. This is a variational problem which is numerically solved by setting c(r) to be a Fourier sum, thus making G approximately a function of a finite number of variables. For this purpose, the definition of c(r), physically defined only in the range 0 6 r 6 R, is mathematically extended to R 6 r 6 R by the symmetry requirement c(r) = c(r), and to any value of r by the periodicity requirement c(r + 2R) = c(r). Taking into account the conditional Eq. (2), this Fourier sum can be written as
cðrÞ ¼ c0
n0 n0 n p r 6 X cosðn pÞ X Kn þ K n cos ; 2 2 n R p n¼1 n¼1
ð3aÞ
where the Kn(n = 1–n0) are Fourier coefficients, and n0 is a sufficiently great number. In this way, G is approximated by a function of the independent variables R and the Kn. Differentiation of c(r) with respect to r yields: n0 dc p X r ¼ K n n sin n p ; dr R R n¼1 2 n0 d c p2 X r ¼ K n n2 cos n p ; 2 2 R R n¼1 dr
ð3bÞ ð3cÞ
which is needed later. 2.2. The chemical free energy According to Cahn and Hilliard [6], the local molar chemical free energy f is a function not only of the local composition c but also of the composition distribution in the immediate environment. Thus, f is a function of c as well as of the spatial derivatives of c. Straightforwardly, Cahn and Hilliard expanded f in a Taylor series of the spatial derivatives of c up to second order terms. For cubic crystal symmetry they obtained:
f ¼ f0 ðcÞ þ j1 r2 c þ
1 j2 ðrcÞ2 ; 2
ð4Þ
@j rðj1 rcÞ ¼ j1 r c þ 1 ðrcÞ2 ; @c 1 @ j1 ðrcÞ2 ; j2 f ¼ f0 ðcÞ þ rðj1 rcÞ þ 2 @c
ð5Þ ð6Þ
where f0(c) is the molar chemical free energy of the homogeneous material of constant composition c under vacuum, and $ means the Nabla operator. The coefficients j1 and j2 are defined by:
0
1
B @f C C j1 ¼ B @ A ; @ 0
@2 c @x2k
ð7aÞ
10
2
@ f
C j2 ¼ B @ 2 A ; @
@c @xk
0
F chem ¼
1 V sample
ð7bÞ
ZZZ
f dx1 dx2 dx3 ;
ð8Þ
sample
where the integration is taken over the volume Vsample of the sample in the un-relaxed state. Substituting the expression on the righthand side of Eq. (6) for f in Eq. (8) yields:
F chem ¼
1 V sample
ZZZ
1 @ j1 ðrcÞ2 j2 2 @c ZZZ rðj1 rcÞ
f0 ðcÞ þ
sample
dx1 dx2 dx3 þ
1 V sample
sample
d x1 d x 2 d x 3 :
ð9Þ
The second volume integral on the right-hand side of Eq. (9) can be transformed to an integral of (j1 rc) over the surface of the sample, and thus must be zero because there is no atomic diffusion through the surface of the sample ð$c ¼ 0Þ. The final result for cubic crystal symmetry is then given by
F chem ¼
1 V sample
ZZZ
h
i f0 ðcÞ þ j ðrcÞ2 dx1 dx2 dx3 ;
ð10Þ
sample
where
1 2
j ¼ j2
@ j1 @c
ð11Þ
is still a function of c, but is set to be approximately constant for applications. For a spherical symmetric composition distribution in an isotropic material, a local right-handed Cartesian coordinate system X1, X2, X3 is used, with the X3-axis in the radial direction, and the X1and X2- axes tangent to the sphere r = const.. In this coordinate system, the spherical symmetry center is given by O(0, 0, r). The mole fraction c at a position P(x1, x2, x3) is only dependent on its distance to the spherical symmetry center O, and thus is defined by the function
c¼c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 þ ðx3 þ rÞ2 :
ð12aÞ
The first and second partial derivatives of c with respect to x1, x2, x3 at the position (0, 0, 0) are obtained from Eq. (12a) by using the chain rule of differentiation:
@c ¼ 0; @x1
or, by applying the mathematical identity 2
which in general are dependent on c. The index ‘‘0’’ means that the differentiations are to be carried out in the homogeneous state of the material. The coordinates xk(k = 1, 2, 3) denote a right-handed Cartesian coordinate system with axes parallel to the crystallographic h1, 0, 0i-directions. The mean molar chemical free energy is given by
@c ¼ 0; @x2
@ 2 c 1 dc ¼ ; @x21 r dr
@c dc ¼ ; @x3 dr
@ 2 c 1 dc ¼ ; @x22 r dr
@2c ¼ 0; @x1 @x2
@2c ¼ 0; @x2 @x3
ð12bÞ 2
@2c d c ¼ ; @x23 dr 2 @2c ¼ 0: @x1 @x3
ð12cÞ ð12dÞ
Substituting the first partial derivatives from Eqs. (12b) for the gradient of c in Eq. (10), and replacing the volume element dxdydz by 4pr2dr, and the volume of the sample by 4pR3/3, finally yields
F chem ¼
3 R3
Z 0
R
"
2 # dc r 2 dr; f0 ðcÞ þ j dr
ð13Þ
since the unit sphere is supposed to represent the whole sample. The function f0(c) is required to display the situation of spinodal decomposition exhibiting a miscibility gap: Two minima and one maximum in between define the A-rich phase and the B-rich phase
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
in an isostructural alloy system AB. f0(c) is here represented as a Taylor expansion about an appropriate composition c up to the mth 0 order term,
f0 ðcÞ ¼ a0 þ
m0 X am ðc cÞm ;
ð14Þ
m¼1
where the am(m = 0 to m0) are proper coefficients.
justified provided the crystal lattice parameter does not vary too much over the chemical composition (see Appendix A, Table A.2). Plugging Eqs. (17a–g), into Eq. (15) yields:
2 X @2c @2c @c þ b þ a 2 1 2 @x @x2k @x k i i–k X @c 2 þ b2 ; @xi i–k
rð0Þ kk ¼ S0 ðcÞ þ a1
2.3. The elastic energy
rð0Þ kl ¼ c1 As well as the local chemical free energy f, also the distortionfree stress tensor r(0) at a given point r can be assumed as a function not only of the composition c at that point but also of the composition distribution in the immediate environment of r. Thus r(0) as well as f is a function not only of c but also of the spatial derivð0Þ atives of c. Analogously to f [7], the elements rkl of the tensor r(0) are expanded in a Taylor series of the spatial derivatives of c. The indices k and l (k = 1–3, l = 1–3) symbolize the three axes of a right-handed Cartesian coordinate system. The Taylor expansion ð0Þ of rkl with respect to spatial derivatives up to second order terms, about the state of uniform composition c under pressure p0, is given by
rð0Þ kl ¼ Skl ðcÞ þ
3 X
jð0Þ i;kl
i¼1
3 X 3 3 X 3 X @c X @2c ð1Þ þ jij;kl þ jð2Þ @xi i¼1 j¼i @xi @xj i¼1 j¼i ij;kl
2 dij @c @c ; @xi @xj 2
ð15Þ
where x1, x2, and x3 are the three spatial coordinates, and dij means the Kronecker symbol (dij = 1 for i = j, dij = 0 for i – j). Skl(c) denotes the stress tensor in the state of uniform composition c const. unð0Þ ð1Þ ð2Þ der pressure p0. The coefficients ji;kl ; jij;kl , and jij;kl , are defined as:
0
1 ð0Þ @ r j ¼ @ kl A ; @c @ @x i 0 0 1 ð0Þ @ r ð1Þ kl jij;kl ¼ @ 2 A ; @ @x@ @xc i j 0 0 1 2 ð0Þ @ r ð2Þ kl @ jij;kl ¼ A ; @c @c @ @x @ @x i j ð0Þ i;kl
ð16aÞ
ð16bÞ
ð16cÞ
where the index ‘‘0’’ indicates that the differentiations are to be carried out in the homogeneous state under pressure (and at temperature). For cubic crystal symmetry, choosing the Cartesian coordinate ð0Þ ð1Þ ð2Þ axes along the h1 0 0i-directions, Skl, ji;kl ; jij;kl , andjij;kl , must be invariant to the symmetry operations of a 90° and 180° rotation about any of the three fourfold coordinate axes. This, and the fact that r(0) must be a symmetric tensor, drastically reduces the number of independent coefficients – 0:
Skl ðcÞ ¼ 0 for k – l;
ð17aÞ
Skl ðcÞ ¼ S0 ðcÞ for k ¼ l;
ð17bÞ
ð0Þ ji;kl ¼ 0 for all i; kl;
ð17cÞ
¼ a1 ;
ð1Þ ij;kl
¼ b1 ;
j j
jð1Þ ij;kl ¼ c1 ;
1 ð2Þ j ¼ a2 for i ¼ j ¼ k ¼ l; 2 ij;kl 1 ð2Þ j ¼ b2 for i ¼ j – k ¼ l; 2 ij;kl ð2Þ jij;kl ¼ c2 for i ¼ k < l ¼ j or j ¼ k > l ¼ i;
jð1Þ jð2Þ for the remaining ij; kl: ij;kl ¼ 0; ij;kl ¼ 0
for k – l:
ð18bÞ
S0(c) is composed of the external hydrostatic pressure p0, and the hydrostatic pressure due to different sizes of the component atoms, and is assumed to be a linear function of c:
S0 ðcÞ ¼ p0 # ðc c0 Þ;
ð19Þ
taking into account that S0(c0) must be equal to p0. Since c is meant to be the mole fraction of component B, a positive value of # indicates that the B atoms are bigger than the A atoms. The parameter # is still dependent on the composition c, but is assumed here approximately to be a constant (see Appendix A, Table A.2). ð0Þ ð1Þ ð2Þ For an isotropic material, Skl ; ji;kl ; jij;kl , and jij;kl must be invariant to any rotation of the coordinate system which yields two additional relations,
c1 ¼ a1 b1 ; c2 ¼ a2 b2 :
ð20aÞ ð20bÞ
For a spherical composition distribution c(r) in an isotropic material, it is expedient to employ a local Cartesian coordinate system with the X3 axis in the radial direction, and the two other axes tangent to the sphere r = const., as was already done in case of the chemical free energy, Section 2.2. Using the corresponding coordinate transformation Eqs. (12b–d), and taking into account Eq. (19), Eqs. (18a–b) yield the disappearance of the shear stress comð0Þ ð0Þ ð0Þ ponents, r12 ; r13 ; r23 , and the following formulae for the normal ð0Þ ð0Þ stress components, rr ¼ r33 in the radial direction, and ð0Þ ð0Þ ð0Þ rt ¼ r11 ¼ r22 in the tangent directions: 2
0
ð1Þ ij;kl
@2c @c @c þ c2 @xk @xl @xk @xl
ð18aÞ
ð17dÞ ð17eÞ ð17fÞ ð17gÞ
The coefficients a1, a2, b1, b2, c1, c2, although in general dependent on c, are here approximately treated as material constants, what is
1 dc d c þ a1 2 r dr dr
rð0Þ r ¼ p0 # ðc c 0 Þ þ 2 b1 þ a2
2 dc ; dr
ð21aÞ 2
1 dc d c þ b1 2 r dr dr
rð0Þ t ¼ p0 # ðc c 0 Þ þ ða1 þ b1 Þ þ b2
2 dc : dr
ð21bÞ
The elastic energy is calculated by applying the linear elastic theory for an isotropic continuum. The differential equation for mechanical equilibrium in the case of spherical symmetry is
dðrr r 2 Þ ¼ 2 rt r; dr
ð22Þ
where rr and rt are the total normal stresses in the radial and the tangent directions, respectively. rr and rt are given by ð eÞ rr ¼ rð0Þ r þ rr ; ð0Þ ð eÞ rt ¼ rt þ rt ; ðeÞ
ð23aÞ ð23bÞ ðeÞ
where rr and rt are the elastic normal stresses in the radial and the tangent direction, respectively, which are induced by the displacement field (see Section 2.1). According to spherical symmetry, the displacement is zero in tangent direction, and is in radial direction given by a function u(r) to be determined. Thus, the strain in
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
radial direction is er = du(r)/dr, and in tangent direction et = u(r)/r. With these expressions, Hook’s law of elasticity for an isotropic material yields:
E 1 m duðrÞ E 2 m uðrÞ þ ; 1 2 m 1 þ m dr 12m 1þm r E 1 uðrÞ E m duðrÞ ¼ þ ; 12m 1þm r 1 2 m 1 þ m dr
rrðeÞ ¼
ð24aÞ
rtðeÞ
ð24bÞ
where E is Young’s modulus, and m is Poisson’s ratio, which are approximately treated here as constants. With Eqs. (23a–b) and (24a–b), and the boundary condition rr = p0 for r = R, the solution of the differential Eq. (22) is obtained to be:
uðrÞ 1 12m 1þm 2 12m ¼ ½2 IðrÞ þ JðrÞ þ r 3 3 E 1m E 1þm 12m 12m IðRÞ JðRÞ p0 ; 1m 1m E
ð25Þ
where I(r) and J(r) are auxiliary functions introduced for convenience which are defined as:
Z
1 ð0Þ ð0Þ dr; rr rt r 0 Z r 1 ð0Þ JðrÞ ¼ 3 dr: r 2 rð0Þ r þ 2 rt r 0
IðrÞ ¼
r
ð26aÞ ð26bÞ
The total stress can be derived from Eqs. (23a–b), (24a–b), and (25), and it was found: 2 1þm 2 12m ½IðRÞ IðrÞ ½JðRÞ JðrÞ p0 ; 3 1m 3 1m m 2 1 þ m 1 12m rt ¼ rtð0Þ rð0Þ þ ½IðRÞ IðrÞ ½2 JðRÞ þ JðrÞ p0 : 3 1m 3 1m 1m r
rr ¼
ð27aÞ ð27bÞ
The mean molar elastic energy is then given by:
1 R3 2 E r 2 dr;
Eelast ¼ V 0
3
Z
R
0
r2r þ 2 ð1 mÞ r2t 4 m rr rt
ð28Þ
where V0 is the molar volume of the homogeneous material of constant composition c0 under pressure p0. 2.4. The volume The mean molar volume of the stress-relaxed sample under the hydrostatic pressure p0 is:
V ¼ V0
uðRÞ 3 : ½R þ uðRÞ V 1 þ 3 0 R R3 1
ð29aÞ
m, a1 ¼ a1 =#, a2 ¼ a2 =#, b1 ¼ b1 =#, b2 ¼ b2 =#, p0 ¼ p0 =#. For the sake of simplicity, these parameters are treated in the following as constants. Then it is obvious that the minimization of G⁄ must yield the same result as the minimization of G.
3. An arbitrary test computation In order to show that the minimization of G⁄ can in principle yield meta-stable composition modulations, an arbitrary numerical parameter set was chosen: c ¼ 0:2, m0 = 4, a1 ¼ 0:002; a2 ¼ 0:2; a3 ¼ 0, a4 ¼ 10:, m = 0.3, j⁄ = 0.005 nm2, a1 ¼ 0:1 nm2 ; b1 ¼ 0; a2 ¼ 0. b2 ¼ 0; p0 ¼ 0, and c0 was varied in the range 0.15–0.25 in steps of 0.005. The number of Fourier coefficients was taken to be n0 = 100. The sample was assumed to be under vacuum; high pressure effects shall not be treated here. f0(c) was taken to be a forth degree polynomial function displaying one maximum and two minima. j⁄ was chosen to be an arbitrary positive value. b1 ¼ b2 ¼ 0 corresponds to the assumption that the first and the second spatial derivative of the composition along any axis cause a normal stress only in the direction of this axis but not in a transverse direction. This neglect of the transverse stress effect is reasonable but not necessary for the occurrence of a meta-stable composition modulation. Furthermore, parameter study revealed that a2 , in contrast to a1 , has no effect on the stabilization of composition modulations, and therefore was set to zero. The minimization procedure of G⁄ was carried out with the aid of a personal computer equipped with a FORTRAN compiler. G⁄ was minimized by incrementally changing the variables R and the Kn(n = 1–100) in such a way that the value of G⁄ was maximally reduced (‘‘gradient procedure’’). For c0 6 0.15 and c0 P 0.25, the computer calculation yielded a uniform composition c c0. For 0.155 6 c0 6 0.195 and 0.21 6 c0 6 0.245 (in the spinodal region of the phase diagram), however, meta-stable composition modulations were obtained, Figs. 1–4. Fig. 1 displays the assumed non-dimensionalized mean molar Gibbs free energy as a function of the alloy composition c0, for the homogeneous state of the alloy as well as the state with equilibrium composition modulations. Fig. 2 shows the period of the composition modulations in terms of 2R, as a function of the alloy composition c0. Fig. 3 displays the meta-stable local composition c as a function of the non-dimensional spatial coordinate r/R, for different values of c0. As can be seen, c is monotonically decreasing with r/R for A-richer alloys, and monotonically increasing for B-richer alloys. Furthermore, adding the component B to the A-richer alloy as well as adding the component A to the B-richer
12m V ¼ V0 1 3 ðJðRÞ þ p0 Þ : E
ð29bÞ
2.5. The Gibbs free energy The mean molar Gibbs free energy G is given by Eqs. (1), (13), (14), (28), (27a–b), (26a–b), (21a–b), (29b), and (3a–c) as a function of the parameters c0, c, am(m = 0 to m0), j, m, E, V0, #, a1, a2, b1, b2, and the variables R, Kn(n = 1–n0). In order to reduce the number of parameters required, it is expedient to introduce a non-dimensionalized Gibbs free energy function
E G a0 G ¼ 2 p0 : V0 V0 #
non-dimensionalized Gibbs free energy
Using Eq. (25), this yields approximately: 0,001
homogeneous solid solution
with equilibrium composition modulation : B-enriched volume fraction < 0.5 A-enriched volume fraction < 0.5
0,0005 0 -0,0005 -0,001 -0,0015 0,05
0,1
0,15
0,2
0,25
0,3
0,35
alloy composition (mean mole fraction of component B)
ð30Þ
As can easily be verified, G⁄ can be written as a function of the parameters c0, c; am ¼ am E=ðV 0 #2 Þ (m = 1–m0), j⁄ = j E/(V0 #2),
Fig. 1. Arbitrary test calculation: Non-dimensionalized molar Gibbs free energy of the homogeneous alloy, and its decrease caused by the equilibrium composition modulation as calculated in dependence on the alloy composition c0 (given as the mean mole fraction of component B).
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
(a)
period length 2R [nm]
3
Z d/4
2,95
d
2,9 2,85
F(i,j)
2,8
j
2.5d
X
2,75 B-enriched volume fraction < 0.5 A-enriched volume fraction < 0.5
2,7 2,65 0,15
d/4 0,17
0,19
0,21
0,23
i
reference area
0,25
alloy composition (mean mole fraction of component B) Fig. 2. Arbitrary test calculation: Period length 2R of the equilibrium composition modulation as a function of the alloy composition c0.
crystal lattice segment I crystal lattice segment II
local composition c
0,4 alloy composition (mean mole fraction of component B)
0,3
(b)
0,18 0,17 0,16
0,2
Z d/4
d
0,24 0,23 0,22
0,1
2.5d Y
0 0
0,2
0,4
0,6
0,8
1
non-dimensional spatial coordinate r/R
reference area
d/4
B-enriched volume fraction
Fig. 3. Arbitrary test calculation: Local composition c (local mole fraction of component B) as a function of the non-dimensionalized spatial coordinate r/R, for different alloy compositions c0.
1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0,15
Fig. A.1. Atomistic model of the distortion-less internal stress tensor r(0) in a binary face-centered cubic 8d 8d 8d crystal (d = lattice parameter): A central 4d 4d 4d section of the crystal shows atom positions as dots in a right-handed Cartesian coordinate system. The crystal is fictively sub-divided by a reference area, i.e. the Y/Z-plane (x = 0), into two segments with atoms being denoted as i and j, respectively. Fij illustrates an example atomic interaction force between the two atoms i and j with their joining line intersecting the reference area of 2.5d 2.5d. (a) X/Z-plane (y = 0), and one adjacent crystallographic plane (y = d/4). (b) Y/Zplane (x = 0), and one adjacent crystallographic plane (x = d/4).
B-enriched volume fraction < 0.5 A-enriched volume fraction < 0.5
0,17
0,19
0,21
0,23
0,25
alloy composition (mean mole fraction of component B) Fig. 4. Arbitrary test calculation: Volume fraction enriched in component B(cB > c0), for different alloy compositions c0.
Table A.1 Morse potential parameters [12]. Metal 1
alloy gives rise to an increase of the period 2R, and an increased deviation of the profile of the composition modulation from a sinusoidal shape. In Fig. 4, the fraction of the sample’s volume which is enriched in the component B, i.e. where c > c0 holds, is plotted versus the alloy composition c0. It can be seen that there is a misfit between the two branches of the curve (B-enriched volume fraction <0.5, and A-enriched volume fraction <0.5). Actually, for c0 = 0.20 and c0 = 0.205, the local equilibrium composition c as calculated turned out to be not any more a monotonic function of r/R in the interval 0 < r/R < 1. This improper result indicates that the theoretical model in the present form (spherical unit domain) is no longer valid in this regime, and therefore the results for c0 = 0.20 and c0 = 0.205 are excluded from the graphs. A meta-stable composition modulation can not only be obtained for some special numerical example but also for extended
k (nm ) r0 (nm) D (zJ)
Cu
Ag
Al
13.588 0.2866 54.94
13.690 0.3115 53.24
11.646 0.3253 43.31
ranges of the values of the parameters. a1 seems to play a decisive role, and, according to test calculations, must necessarily be positive for the occurrence of meta-stable composition modulations. The parameters a1 and j⁄ control the value of the period 2R: Increasing values of a1 , as well as increasing j⁄ cause an increasing period 2R. 4. Discussion The modified theory of diffuse interfaces as given in the present paper predicts the possible existence of meta-stable composition
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W. Gaudig / Computational Materials Science 77 (2013) 182–188 Table A.2 Results of atomistic modeling for a-Cu–Al at various compositions, and a-Ag–Al. Input: Mole fraction Al (c0), external pressure (p0). Output: Crystal lattice parameter (d), model parameters (#, a1, a2, b1, b2, c1, c2).
a-Cu–Al
Alloy c0 (mol/mol) p0 (GPa) d (nm) # (GPa) a1/# (nm2) a2/# (nm2) b1/# (nm2) b2/# (nm2) c1/# (nm2) c2/# (nm2)
0.10 0. 0.3652 48.53 0.00974 0.00000 0.00118 0.00002 0.00470 0.00001
0.15 0. 0.3675 46.22 0.00987 0.00000 0.00117 0.00002 0.00471 0.00001
a-Ag–Al 0.20 0. 0.3698 44.03 0.01001 0.00000 0.00117 0.00001 0.00473 0.00001
0.10 0. 0.4069 2.24 0.10671 0.00000 0.03277 0.00004 0.07172 0.00002
modulations in the spinodal region of the phase diagram. The driving force of the decomposition is the decrease of the chemical free energy. The stabilization of short-range composition modulations is achieved by introducing distortion-free stress terms due to the spatial derivatives of the local composition up to the second order. It should be noted that the calculation of the elastic energy taking the unit domain of the composition distribution to be a sphere can only be an approximation. The result of this approximation seems to be reasonable. However, a more accurate calculation should be carried out by employing a more realistic unit domain, e.g. a cube, to confirm the results of this paper. This could be carried out with the aid of the finite elements method. Furthermore, crystal anisotropy should also be taken into account. The phenomenological theory as given in this paper takes explicitly into consideration composition modulations only. Order parameters could be taken implicitly into account through the function f0(c), i.e. order parameters could be assumed as fixed functions of the composition c. A rigorous theory, however, requires order parameters to be treated explicitly as independent of the composition: Accordingly, Poduri and Chen [11] extended the theory of spinodal decomposition by adding a long-range order parameter gradient term to the chemical free energy. It remains to be examined, however, to what extent order parameter gradient terms have also to be introduced in the expression of the elastic energy within the framework of the present modified theory of diffuse interfaces. To adapt the present theory to real alloy systems, it is further necessary to get an idea of the values of the newly introduced parameters a1, a2, b1, b2, c1, c2, at least the most important parameter a1. This requires atomistic modeling, i.e. the knowledge of the bilateral interaction energy of the atoms [12], as is demonstrated in the Appendix.
5. Conclusions The present study predicts the possible existence of meta-stable short-range decomposition in the spinodal (instable) region of the phase diagram, and reveals its relation to spinodal decomposition: It can be concluded that meta-stable short-range decomposition (in combination with short-range order) resembles the beginning of instable spinodal decomposition, i.e. the appearance of composition modulations. The difference is that spinodal decomposition is open to unlimited coarsening, whereas short-range decomposition is stabilized by coherent strain energy as revealed for example by former transmission electron microscopy of a-Cu–Al alloys [3]. It should be kept in mind that the occurrence of meta-stable composition modulations is conditional on appropriate values of the parameters, e.g. a1 ¼ a1 /# must be positive. As is shown in Appendix A, this condition is for example fulfilled for a-Cu–Al alloys
but is not for a-Ag–Al alloys. If this condition is not met then the theory of statistical composition fluctuations must be invoked. Since the present theory aims at an equilibrium state of the alloy system, the computation work can be restricted to any mathematical minimization routine without the need for physical relevance of the time scale. In contrast, the modeling of spinodal decomposition requires a dynamic theory, and a correspondingly much more sophisticated computation routine in order to describe the temporal development of non-equilibrium states at a real time scale. Advanced modeling was carried out at the atomic scale [13] going beyond the well-known original theory of Cahn and Hilliard [9,10], and was successfully applied to Fe–Cr-alloys. However, these models unfortunately do not include elastic strain effects, and consequently are not suited to detect eventual meta-stable states in the spinodal region of the phase diagram. A last comment refers to thin metallic films. Within the last 5– 10 years research has been performed in the field of thin films with modulated chemical composition for various technical purposes. Present modeling as described, however, is rigorously designed for application to bulk material. Films could be approximately treated like bulk material if twice its surface thickness is small (1%) compared to the bulk thickness of the film. In this context, the surface thickness means the reach of the interatomic force, which is defined as the range where the atomic pair potential function falls off to 1% of the dissociation energy (see Appendix A, Eq. (A.3a)). This assessment results in a minimum film thickness of about 150 nm, when using the data of Table A.1. Thinner films need special atomistic modeling of the surface region. Appendix A. Atomistic modeling Fig. A.1a and b shows an atomistic model to define the distortion-free internal stress tensor r(0), with the aim to get an estimate of the parameters #, a1, a2, b1, b2, c1, c2, for binary face-centered cubic crystals. The dots depict atom positions each of them occupied with probability (1-c) by an A-atom and probability c by a B-atom. The right-handed Cartesian coordinate system X, Y, Z is chosen so that each of the coordinate planes, X/Y, Y/Z; and Z/X, is a mid-plane between two adjacent parallel crystallographic {1 0 0}-planes. The total force on a reference area (2.5d square, d = lattice parameter) perpendicular to the X-axis is calculated as
X rij X rj ri F ij ¼ F ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 r ij ij ij ðx x Þ þ ðy y Þ2 þ ðz z Þ2 j
i
j
i
j
ðA:1Þ
i
where F ij means the expectation value of the atomic interaction force transmitted from an atom no. j of the crystal segment II on the right side of the reference area onto an atom no. i of the crystal segment I on the left side of the reference area, Fig. A.1a. rij is the vector pointing from atom no. i to atom no. j, and rij is its amount. ri and rj give the positions of atom i and atom j, respectively, and xi, yi, zi, and xj, yj, zj denote their components in the coordinate system X, Y, Z. The summation in Eq. (A.1) is over all pairs of atoms i and j the joining line of which intersects the reference area, with two special cases: (i) if a side of the reference square is hit then a statistical weight factor of 1/2 has to be taken into account, and (ii) if a corner is hit then a weight factor of 1/4 has to be applied. The expectation value F ij of the atomic interaction force, for a binary alloy AB, is given by
F ij ¼ ð1 cðri ÞÞ cðrj Þ F ij ðABÞ þ cðri Þ ð1 cðrj ÞÞ F ij ðBAÞ þ ð1 cðri ÞÞ ð1 cðrj ÞÞ F ij ðAAÞ þ cðri Þ cðrj Þ F ij ðBBÞ;
ðA:2Þ
where c(ri) and c(rj) denote the mole fractions of atom B at the positions ri and rj, respectively. Fij(AB), Fij(BA), Fij(AA), and Fij(BB) symbolize the interaction force of the atom pair i, j in case that the
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W. Gaudig / Computational Materials Science 77 (2013) 182–188
positions i and j are occupied by atoms A and B, B and A, A and A, B and B, respectively. In Eq. (A.2), atomic pair probability parameters are neglected. The atomic interaction potential is assumed to be represented by the Morse function [12]:
uðrij Þ ¼ D ½expð2 k ðrij r0 ÞÞ 2 expðk ðrij r0 ÞÞ;
ðA:3aÞ
and the atomic interaction force is then given by
F ij ¼
d uðr ij Þ d rij ðA:3bÞ
where rij is the distance between two atoms, and the parameters D, k, and r0 are still dependent on the occupation of the atom pair (AB, BA, AA, BB). On the other hand, the force on the reference area (x = 0, Dy 6 y 6 Dy, Dz 6 z 6 Dz), Eq. (A.1), can be expressed as an integral of the stress tensor r(0) over that reference area (Dy = Dz = 1.25d). This yields:
r ð0Þ xy 2 Dy 2 Dz ¼ r ð0Þ xz 2 Dy 2 Dz ¼
ðA:7aÞ
ð0Þ xx ¼ p0 # Dc ð2Þ r ð0Þ xx ð3Þ r ¼ p0 þ a1 2=a2
ðA:7bÞ ðA:7cÞ
ð0Þ xx ¼ p0 þ a2 =a2 ð4Þ r
ðA:7dÞ
ð0Þ xx ð5Þ r ¼ p0 #=3 ðDyÞ2 =a2 þ b1 2=a2 þ b2 4=3 ðDyÞ2 =a4
ðA:7eÞ
ð0Þ xx ¼ p0 þ b2 =a2 ð6Þ r
ðA:7fÞ
ð0Þ xy ¼ c1 =ða: bÞ ð7Þ r
ðA:7gÞ
ð0Þ xy ð8Þ r ¼ c2 =ða bÞ:
¼ 2 k D ½expðk ðr ij r 0 ÞÞ expð2 k ðrij r0 ÞÞ;
r ð0Þ xx 2 Dy 2 Dz ¼
ð0Þ xx ð1Þ r ¼ p0
Z
Dy
Z
Dy
Z
Dy
Z
Dy
Z
ð0Þ
Dz Dz
Dz Dz
Dy Z Dy
ð0Þ
rð0Þ XX dy dz ¼
X x j xi F ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ij ðxj xi Þ2 þ ðyj yi Þ2 þ ðzj zi Þ2
ðA:4aÞ
rð0Þ XY dy dz ¼
X yj yi F ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 2 ij ðxj xi Þ þ ðyj yi Þ þ ðzj zi Þ
ðA:4bÞ
rð0Þ xz dy dz ¼
X zj zi F ij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 2 ij ðxj xi Þ þ ðyj yi Þ þ ðzj zi Þ
ðA:4cÞ
Dz Dz
ð0Þ
where rxx ; rxy ; rxz correspond to the continuum-mechanical expressions, Eqs. (18a–b) and (19), of the present generalized theory of diffuse interfaces: 2
@ c rð0Þ ¼ p0 # ðc c0 Þ þ a1 @x xx 2 þ b1
þa2
@c 2 @x
þ b2
2 @c @y
þ
@c 2
h
@2 c @y2
2
þ @@z2c
i ;
ðA:5aÞ
@z
rð0Þ xy ¼ c1
@2c @c @c þ c2 ; @x @y @x @y
ðA:5bÞ
rð0Þ xz ¼ c1
@2c @c @c þ c2 ; @x @z @x @z
ðA:5cÞ
with the composition c and its spatial derivatives taken at the surð0Þ ð0Þ xy ð0Þ face of the reference area, i.e. at x = 0. The symbols r ;r xx ; r xz can be interpreted as the mean values of the stress tensor elements ð0Þ ð0Þ ð0Þ rxx ; rxy ; rxz over the reference area, respectively. In order to determine the external pressure p0, and the parameters #, a1, a2, b1, b2, c1, c2, simple test functions c = c(x, y) are chosen:
ð1Þ c ¼ c0
ðA:6aÞ
ð2Þ c ¼ c0 þ Dc
ðA:6bÞ
ð3Þ c ¼ c0 þ ðx=aÞ2
ðA:6cÞ
ð4Þ c ¼ c0 þ x=a
ðA:6dÞ
ð5Þ c ¼ c0 þ ðy=aÞ2
ðA:6eÞ
ð6Þ c ¼ c0 þ y=a
ðA:6fÞ
ð7Þ c ¼ c0 þ x=a y=b
ðA:6gÞ
ð8Þ c ¼ c0 þ x=a þ y=b;
ðA:6hÞ
where Dc is a small deviation of the alloy composition c0, and a and b are arbitrary constants which are set to 100d for the present calculation. Each of these test functions are plugged into Eqs. (A.5a–b). ð0Þ ð0Þ The resulting expressions of the tensor elements rxx and rxy are used to determine the integrals in Eqs. (A.4a–b). This yields the following equations, respectively:
ð0Þ xx ,
ðA:7hÞ ð0Þ xy
The quantities r and r are calculated for each of these cases, no. 1–8, according to the right-hand expressions of Eqs. (A.4a–b), by using a personal computer with a FORTRAN compiler. The first of these equations, Eq. (A.7a), is used to determine the crystal lattice parameter d for a given external pressure, i.e. p0=0 in the present case. The second equation, Eq. (A.7b), serves to calculate the parameter # = (Dp0/Dc)d=const., where Dc is taken to be 0.0001. The remaining six equations, Eqs. (A.7c–h), yield the six newly introduced parameters a1, a2, b1, b2, c1, c2. Values of the Morse potential parameters k, r0, and D are given in [12] for pure metals, at absolute zero temperature and zero pressure. Table A.1 shows example data for Cu, Ag, and Al. For the purpose of the present paper, the Morse potential function of atom pairs of different components (A–B) is set equal to the average of the potential functions of atom pairs of the same component (A– A and B–B):
F ij ðABÞ ¼ 1=2 ½F ij ðAAÞ þ F ij ðBBÞ:
ðA:8Þ
This corresponds to the neglect of atomic pair probability parameters, as was already assumed in Eq. (A.2). Results of the present atomistic modeling are given in Table A.2 for a-Cu–Al at various alloy compositions, and a-Ag–Al as examples. It can be seen that the empiric condition for the occurrence of meta-stable composition modulations, i.e. a1/# > 0, is fulfilled for a-Cu–Al, but is not for a-Ag–Al. Furthermore, the three parameters a2/#, b2/#, c2/# are zero or near zero so that one has only to deal with the four parameters #, a1/#, b1/#, c1/#. The most important parameters are # and a1/#. Short-range order is classified by the condition that, for next nearest neighbor atoms A and B, the value of Fij(AB), contrary to Eq. (A.8), is greater then the average value of Fij(AA) and Fij(BB), due to enhanced attraction of different sorts of atoms. This can result in considerably greater values of a1/#, as was verified by computer. The present estimation of the parameters #, a1, a2, b1, b2, c1, c2 is in the first place physical confirmation of the mathematically motivated generalized theory of diffuse interfaces, Eqs. (A.5a–c). For more accurate calculations, however, it will be necessary to take into account atomic pair probability parameters, and also electronic long-range interactions (atomic pair potential oscillations). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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