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Diffusion in ordered alloys and intermetallic compounds
Acta metall, mater. Vol. 41, No. 10, pp. 2807-2813, 1993 Printed in Great Britain.All rights reserved
0956-7151/93 $6.00+ 0.00 Copyright ffS~1993PergamonPress Ltd
D I F F U S I O N IN O R D E R E D A L L O Y S A N D I N T E R M E T A L L I C COMPOUNDS C. C. WANG and S. A. AKBAR Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. (Received 12 October 1992; in revised form 5 April 1993)
Abstract--Diffusion in binary ordered alloys has been treated using the pair-approximation of the Path Probability Method (PPM) based on an atomistic model. The effect of the atomic interaction on the ordering behavior and its influenceon transport properties have been clarified. Compositional dependence of both the intrinsic diffusion and interdiffusion coefficients,correlation factor and thermodynamic factor agree very well with the Monte Carlo simulation. The calculated properties also show qualitative agreement with the experimental data on diffusivity, activity and creep in the NiA1 system.
!. INTRODUCTION Diffusion in ordered alloys recently has been the subject of considerable interest due largely to the increasing demand for the understanding of high temperature behavior of the ordered intermetallic compounds. In comparison with the disordered solid solution alloys, theoretical treatment of diffusion in ordered alloys are much more complicated because of the atomic interactions between the diffusing species. In other words, atomic jump in an ordered alloy strongly depends on the nature of ordering in the immediate environment at the microscopic level. Considering diffusion via the vacancy mechanism, when diffusing atoms exchange with vacancies, they tend to leave behind traces of disordered region and locally deviate from thermodynamic equilibrium. In order to maintain thermodynamic equilibrium, the migration of atoms must occur in a way or sequence to compensate or minimize the increase of energy during the diffusion. As a result, diffusion in ordered alloys is strongly correlated; the degree of correlation (or order) is a function of composition and temperature, and this considerably complicates the theoretical treatment of diffusion. Diffusion in ordered alloys tends to be much slower than in disordered alloys, and is often characterized with the existence of a sharp minimum in the diffusion coefficient vs composition curve and a sharp maximum in the activation energy vs composition curve around the stoichiometric composition. Slow rates of diffusion bring with them the associated advantage of improved microstructural stability at elevated temperatures and, since creep rate is proportional to the diffusion coefficient, improved creep strength. Apart from its technical importance, this problem of diffusion is also quite challenging theoretically. In the past two decades, the theoretical
treatment of diffusion in ordered alloys has mostly been developed as an extension to Manning's random alloy model in the framework of the nearest-neighbor pair-wise interactions. Bakker [1] and Stolwijk [2] have extended Manning's random alloy model to include long- and short-range order and their results show good agreement with the numerical simulations by the Monte Carlo method. However, these approaches do not provide an analytical expression for the calculation of the thermodynamic state of order as a function of composition and temperature. In the treatment of diffusion in ordered alloys, one requires firstly to construct an exact analytical expression for the change of the free energy of the system to be minimized, and secondly to derive the Onsager equation. A combined approach based on the Path Probability Method [3] and the Cluster Variation Method [4] offers a unique opportunity to treat these problems. The Cluster Variation Method (CVM), a static version of the Path Probability Method (PPM), can calculate the thermodynamic state of order as a function of temperature and composition. The PPM, on the other hand, can derive the Onsager equation for diffusion analytically based on atomistic models [5, 6]. Such analytical expressions are quite helpful in identifying measurable quantities (e.g. diffusion coefficient, ionic conductivity, etc.) in terms of fundamental microscopic parameters such as jump frequencies and interatomic interactions. If sufficient information is available on these quantities, specific predictions can be made or, conversely, diffusion experiments can be used to evaluate these atomistic parameters for which there is no straightforward calculation or measurement. It has recently been demonstrated [7] that by fitting the PPM equations to demixing experiments in oxide solid solutions, it is possible to self-consistently evaluate microscopic parameters.
2807
2808
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS
The PPM is not an alternate technique, but rather serves as an important supplement to the well established phenomenological treatment based on the Onsager formalism and other atomic theories of diffusion including computer simulations. Although some limitations have been found to appear in the original formalism of the PPM as applied to transport problems where the motion of individual particles is at stake (e.g. the calculation of the correlation factor), significant improvements have recently been made by changing some of the averaging processes. A full account of this problem can be found in Refs [8, 9]. Here, for simplicity, the original formalism will be used. The objective of this paper is to show how the PPM can be used to treat diffusion problems in ordered alloys using the well established pairapproximation. The effect of atomic interactions on the ordering behavior and its influence on transport properties will be illustrated and compared with the existing Monte Carlo simulations and experimental results in ordered intermetallic compounds such as NiA1. Compositional dependences of intrinsic diffusion and interdiffusion coefficients will be illustrated and their implication on the high temperature creep behavior of NiAI will be addressed. 2. ATOMISTIC CALCULATION The essence of an atomistic theory of diffusion within the linear approximation of the Onsager formalism is to derive the flux equations in a mutlicomponent system. For a binary alloy AB, under uniform temperature, the Onsager equations derived by the PPM can be expressed as
where, D~ is a hypothetical (or calculated) diffusion coefficient of i which is making random walk or in the absence of correlations. The term f l , sometimes referred to as the physical correlation factor, represents a collective correlation factor and corresponds to the so-called vacancy-wind effect (for random-walk motion, f~ = 1). It is important to note that D i defined in equation (3) implicitly includes the thermodynamic factor commonly recognized in the context of chemical diffusion in non-ideal multicomponent systems. Theretbre, sometimes it is convenient to express D~i in the following form Dxi = D 04"f~
(4)
where 4, accounts for deviation of the system from ideal solution behavior (4, = 1 for ideal solution). The quantity D O thus defined corresponds to the intrinsic diffusion coefficient of species i making random walk in the ideal solution, and, hence, is equivalent to the self-diffusion coefficient in the absence of correlations. This kind of expression is especially helpful in the analysis of problem of diffusion in multicomponent systems. Based on the PPM, all the above quantities can be expressed in terms of two parameters: the static and kinetic parameters, i.e. the interatomic interactions (Eij), and the jump frequencies of constituents (wi). For diffusion by the vacancy mechanism, the matrix elements Lv's are derived as: LAA = WAYAvWAfAA
(Sa)
LAB = WAYAvWAfAB
(5b)
JA = - - L A A ~ -- LABCt[
(la)
LBA = wByBv WBfaA
(5C)
JB = -- LaA~X,~-- LBB~B'
(lb)
LBa = wByBv WafBB
(5d)
wi = 0 exp(-flui)
(6a)
Here, JA and JB represents the fluxes of A and B atoms respectively, and ~((i = A, B) represents the generalized chemical potential gradient of the ith species (cti-- fl/~, where fl = 1/kT and/~ is the chemical potential). The Onsager matrix coefficient, Lij, can be related to the more familiar intrinsic diffusion coefficients as Dig = LAA/CA
-
D m = LBa/C B -
-
LAB/CB
(2a)
LBA/C A
(2b)
where, for an ordered binary alloy C~ = [Ci~l)+ C~z)]12.
(2c)
Here, Ci is the mole fraction of ith species, and C~l) and C~2) are the values of C~ on the two sublattices of the ordered system. The intrinsic diffusion coefficient, Dtl, can be expressed as [10] D1A = D A f k
(3a)
D m = DBf~
(3a)
where
y
wi=
L
-~i
x,
j
(6b)
The variables Yv represent the equilibrium values of the probability of finding i - j pairs (i a n d j = A, B, v), obtained by the pair-approximation of the CVM. The subscript V is used to denote vacancy, and, hence Y,v represents the probability of finding a vacancy near an ith atom. This term is conveniently called the vacancy availability factor (VAF). Here, 0 is the attempt frequency, u~ is the activation energy of motion, 2~o specifies the coordination number of the lattice (2~o = 8 for b.c,c.) and xi indicates the probability of finding the ith species on a lattice site and thus equals to C~ in the disordered state. It is obvious that while wl is the b a r e jump frequency, W, represents the effect of the surroundings on the jumping atom and is called the bond-breaking factor (BBF).
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS Therefore, it is convenient to introduce the effective jump frequency vbi as wi = wi W,
(7)
which takes into account the effect of bond breaking. Equations (5-7) show precisely how the transport of an atom depends on the availability of a vacancy in its neighborhood, and how the motion is influenced by the surrounding atoms. It should be noted that for an ordered system with two distinct sublattices, bond-breaking factors can be introduced for each sublattice. However, the product of VAF and BBF in equation (5) makes the Onsager coefficients invariant of the sublattice. The quantities f j ' s in equation (5) constitute the collective correlation factor f l in equation (3) flA = fAA -- fAB CA /CB
(8a)
fib = fBn - fBA C B / C A
(8b)
2809
sition with a negligible amount of vacancies is given by [12] 2E/kT¢ = ln[o)/(e) - 1)].
(12)
Equation (12) can be used to estimate E of a specific system at a given To. Note that equation (12) also serves to normalize the value of E with respect to k T c . For a system with known E, equation (12) gives the normalized critical temperature E/kT~. In a bcc lattice with 2~o = 8, for example, E/kTc has a value of 0.144. From equations (2), (5) and (7), the intrinsic diffusion coefficients can be expressed in the following form DIA = I~AYAv [fAA -- CAfAB/CB]/C A
(13a)
DIB = I~aYav [fBa -- CnfBA/CA]/C B.
(13b)
The vacancy availability factors YAVand Yav, which include the vacancy fraction Cv, are functions of temperature and composition, and can be evaluated Note that fan and fBA are simplified notations "JAn"erdA) by the CVM. However, in order not to introduce and f ~ , respectively, used by Allnatt and Allnatt [10] extra complications and conform to the approach and Zhang et al. [11]. taken by Zhang et aL [l l] in their Monte Carlo The PPM calculations will be divided into two computations, normalized intrinsic diffusion coparts for two different types of lattices. First, the efficients are defined as simple cubic lattice will be employed so that our results can be directly compared with those obtained DIA = DIA/[CvWB] (14a) by Monte Carlo simulations. Then, in order to compare with the experimental results of diffusion f)la = Dla/[Cv wa] (14b) coefficients in NiA1 (CsCl-type b.c.c, structure), the calculations will be extended to the b.c.c, lattice. In and are called intrinsic diffusion coefficients per unit both cases, interactions only between nearest neigh- vacancy concentration. The interdiffusion coefficient bors will be assumed and designated as EAA, EBB, [13],/), can be calculated directly from D~A and Dxa ~An(qj> 0 is taken as attractive). Interactions involved with vacancies V are assumed to be zero, i.e. EAV= ~Bv = Evv = 0.
(9)
Instead of using values of individual Eu defined above, it is more convenient to introduce an effective interaction energy parameter E of the form 4E = 2EAB -- (EAA + ~BB)"
(10)
When e is positive, it is referred to as the so called "ordering energy". In comparison with Zhang et al. [11], our definition of the ordering energy differs from theirs by a factor of 4. In the treatment of diffusion, in which the relative easiness of breaking bonds with its nearest neighbor atoms is to be considered, the difference of (AA and %B has to be taken into account. Therefore, an extra parameter U is defined as U = @AA -- E~n)/(4E)"
/) = CBD~A+ CA£)m
(15a)
and similar to equation (4), the interdiffusion coefficient can be expressed in terms of the thermodynamic and correlation factors in the following form = (Ca~)°Af I + C A £ ) ° f ~ ) ~ .
(15b)
For a given set of E, U and temperature (or a normalized temperature in terms of E/kTc), both the intrinsic diffusion and interdiffusion coefficients can be calculated as a function of composition. It should be noted the L u derived by the PPM in equation (5) implicitly includes the thermodynamic factor, 4~, which is usually defined as = flC,((?#,/SC~).
(16)
Using the relationship of CA+ CB ~ 1, equation (16) can be rewritten as [11]
(1 1)
This parameter is a measure of whether A - A or B-B bond is more easy to break and can be estimated from the energy of formation of vacancies. The effective interaction energy parameter E gives the measure of the critical temperature T¢ of the order~lisorder transformation. In the pair-approximation of the CVM, Tc at the stoichiometric compo-
= flCA CB[8(#A -- ,uB)/OCA]
(17)
which is more convenient for numerical differentiation. The chemical potential of ith species for an ordered alloy with negligible vacancies, can be derived by the CVM and is expressed as [14] /~, = fl(2~o In q i - 0.5(2o~ - 1)ln[C~mC~j)
(18)
2810
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS
where, q~ is determined by a set of simultaneous equations
1.0 0.9
Ci = qi Z qJ e-~'u
(19)
J and represents the short-range order of the system in terms of a single-site variable (it reduces to C~ in the completely disordered system). Thus, the thermodynamic factor 4~can be calculated from equation (17) by determining PA -- PB as a function of composition and performing a numerical differentiation. Furthermore, by taking the pure component as the standard state, the activity as a function of composition within the ordered phase region can be calculated by a, = exp[fl (/~, -/to/)]
(20)
where, P0~ is the chemical potential of pure i.
0.8 0.7 <<0.6 ~" 0.5
0.4 o.3 --- 0.100 0.050
0.2 0
3. RESULTS AND DISCUSSION
0 . ' 1 0 .03 .0 . 4' 20 . 5 ' 0' 6 ' 0i 7 ; 0i 8 ' Ca
0.9
.0
3.1. Simple cubic lattice--a comparison with Monte Carlo simulations
Fig. 2. Composition-dependenceofpartialcorrelation function fAA at various values of E/kT.
As in the case of Monte Carlo simulations of Zhang et al. [11], the first part of our calculation was also carried out for a simple cubic (s.c.) lattice with U = 0, i.e. £AA= EBB'From equation (12), the normalized critical temperature for the s.c. lattice is calculated to be c/kTc = 0.203. Being symmetrical, only the results for the A component are shown here; those for the B component can be obtained simply by substituting B for A in the figures. Figure 1 shows the composition dependence of the thermodynamic factor • at various values of ,/kT. Note that lower temperature corresponds to higher E/kT. For E/kT >t 0.203, the ordered phase appears with the minima in ~, corresponding to the
order~tisorder phase boundaries, and with the peaks at the stoichiometric composition. Figures 2 and 3 show the dependences offAA and f m respectively at various values of E/kT. The decrease offAA to a minimum indicates the increase in ordering as CA approaching the stoichiometric composition, or, in other words, the tendency of atoms to reverse jumps becomes stronger in order to preserve local order. However, unlike fAa, )CABdoes not show a minimum, instead a drastic drop is found as CA approaches the stoichiometric composition. It is worth pointing out that, in the present calculation, the value offAA in the limit of E/kT = 0.0 and CA-~0 0.5 e/kT
e/kT
14
-- 0.333 - - 0.286 --- 0.200 --0.158 -.- 0.100
12
A 11 Jl
m 0.050
--0.333 - - 0.286 -'- 0.200 "'0.158 "" 0.100
0.4
.'..'~.Q .'>~.
10
~% • x
--
0.050
x
0.3 e
• Q •'\ • "'\
8
x
0.2
6
4 0.1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
CA Fig. 1. Composition-dependence of the thermodynamic factor • at various values of E/kT.
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CA
Fig. 3. Composition-dependenceof partial correlation function fm at various values of E/kT.
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS 1.0
2811
1.0
'..'x
E/kT
x
m 0.050 - - - O. 1 O0 - - 0.158
0.8 . . . .
0.8
b~ ~," ~,"
0.200
.~
- - 0.286
<0.6-
0,6
nr~ 0.4-
0.4
##/
.o° S
°,~lt, o'~
0.2-
V
m 0.050 -'" 0.100 "- 0.158 --- 0.200 -- 0.286 -- 0.333
0.2
0
j ~ g , '1"~ " "
S#
.i-
0
0 01, o'2 013 0'4 o'5 o'6 0'7 o'8 o'9 10
0.1
0.2
0.3
0.4
CA
""
,/
/
/
/1
.s~ I I
0.5
0.6
0.7
0.8
0.9
1.0
CA
Fig. 4. Composition-dependence of the physical correlation factor f ~ at various values of E/kT.
Fig. 5. Composition-dependence of the intrinsic diffusion coefficient /)LAat various E/kT.
is 0.714, whereas an exact treatment would give a value of 0.654, the correlation factor for self-diffusion in s.c. lattice. This discrepancy comes from an inherent time correlation problem in the original formalism of the PPM in following individual particle motion [8]. By changing some of the averaging processes in the PPM, originally introduced by Sato, the calculations can be further improved and would give a closer agreement. Figure 4 shows the composition-dependence of the physical correlation factor f~, at various values of E/kT. Note that f~, is a combination o f f a h andfAB as defined in equation (4). For E/kT > 0.203, a deep cusp manifests the so called physical correlation effect, which is reflected in the slowing down of atomic migration due to preservation of local order. Figure 5 shows the results of the intrinsic diffusion coefficient D~A at various values of e/kT as defined in equation (14). For e/kT greater than 0.203, a minimum develops in the ordered region as expected, but, due to the non-symmetrical dependence offAA andfBa on composition, a minimum is not found at the stoichiometric composition. With the results of/StA and Din, the interdiffusion coefficient/~ can be obtained, and the results are shown in Fig. 6. Again, despite the maximum in the thermodynamic factor, a minimum i n / ) develops in the ordered region. In comparison with the results of Zhang et al. based on the Monte Carlo simulation [11], our PPM calculations based on the pair-approximation agree very well with their results except for some minor differences. In the Monte Carlo calculations, a minimum is observed in fAB at the stoichiometric composition when e/kT is higher than the critical value, and also their results are systematically higher. These differences are, however, expected as we used the
original PPM, which has some inadequacies with respect to the averaging processes as indicated earlier.
AM41/IO--B
3.2. CsCl-type lattice--a comparison with the experimental data on NiAI It is known that NiAI is an ordered compound with CsCl-type structure and exits over a wide range of compositions [15]. The stoichiometric alloy retains its structure up to its melting point of 1638°C. From equation (12), the ordering energy E can be estimated to be 0,55 kcal/mol. For simplicity, the ratio of bare jump frequencies (wsi/wAl) is set to be unity. The calculations were carried out by choosing different 0.8
0.6
z~
0.4
° 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
CA
Fig. 6. Composition-dependence of the interdiffusion coefficient/5 at various values of E/kT.
2812
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS 0.6
0"5 t
- - E x p . data [17] ~ e = 0.55 kcal/mol .... e = 0.80 kcal/mol
0.4-
~
0.3-
0.2-
0.1-
/
0
.ooo-°~o...... oo°oOJ~,oW I I I I I I I I I 0 0.44 0.48 0.52 0.56
0.60
CNi
Fig. 7. Comparison of the calculated activity of Ni with the experimental data for NiAI at 1000°C in the range of Cr~i=0.44).6 [17]. The thick-solid line is calculated with c =0.55kcal/mol, while the dotted line with E = 0.80 kcal/mol.
values of ENi_Ni/EAIAI o r U. Here, the ratio of ENi-Ni/EAI Al was estimated to be 1.5 from the cohesive energies of Ni and AI [16]. It should be noted that our fitting of parameters or curves are not m e a n t to be exact, since the present model does not take into account the variation of lattice parameters and the vacancy concentration (therefore, interatomic interactions) as a function of composition. Also, the pair-approximation of the CVM does not accurately predict phase boundaries in intermetallic alloys; it is necessary to use higher-order approximations. Figure 7 shows the comparison of the calculated activity of Ni with the experimental data for NiAl at 1000°C, in the range of CNi = 0.4-0.6. The thin-solid curve represents the activity of Ni obtained by Hanneman and Seybolt [17], converted from experimental activity of A1 [18] by Gibbs-Duhem integrations. In light of the fact that this calculation is carried out using the pair-approximation with nearest neighbor interactions, the agreement is quite satisfactory. For the purpose of comparison, activity of Ni calculated with E = 0.80 kcal/mol is also plotted as the dotted curve. It can be seen that the agreement with experimental data is better for higher e, suggesting that the actual ordering energy may be higher than that estimated by the pair-approximations [equation (12)]. Figure 8 shows the composition-dependence of the intrinsic diffusion coefficient of Ni at various temperatures and U as indicated. Note that minima are observed near the stoichiometric composition for all temperatures below 1638°C, which qualitatively agree with experiments in Ref. [19]. It should also be noted that the minimum occurred is not centered at the
stoichiometric composition, a behavior sometimes observed in ordered intermetallic compounds; for example in AuCd [20]. Traditionally, the interpretation of this off-stoichiometric behavior is based on the argument of the change of defect structure across the stoichiometric composition. Our calculations suggest that both the magnitude and the sign of U play a significant role in determining the location of the minimum. In our calculations, the positive value of U means that vacancies tend to distribute in the Ni-rich region which makes the minimum offstoichiometric. In Fig. 8, composition-dependence of the intrinsic diffusivity of Ni at 1000°C for various values of U are also given for comparison. As U increases, corresponding to the increase in the ratio of eN~Ni/£AI_AI , the cusp becomes deeper. Figure 9 shows the composition-dependence of interdiffusion coefficients/) at various temperatures. Again, the off-stoichiometric behavior of the minimum is observed and is consistent with experiments [21]. Although our data show less dramatic dependence on composition in comparison with experiments, the agreement is quite satisfactory for the same reason stated previously. Another feature sometimes observed in the diffusion of ordered alloys is the temperature dependence of the composition at which the minimum occurs. Note that, in Fig. 9, the minimum i n / ) shifts toward the Ni-rich side as T increases, again in agreement with experiments [21]. An important implication of the present results is the high temperature creep behavior of ordered alloys where diffusional mechanism is operative. It has been found in NiAI that the activation energy
1.0E-9
I
I
I
~:~ 1.0E-11
1.0E-12
~ - -'---'"
1.0E-13 0.~4
i 0.5 CA
1200 1300 1500 1000 1000
(U=0.73) (U=0.73) (U=0.73) (U=0.9) (U=l.3)
016
Fig. 8. Composition-dependence of the intrinsic diffusion coefficient of Ni, DNi, in NiA1 at various temperatures, calculated with U = 0.73. At 1000°C, DN~,is also calculated with U = 0.9 and 1.3 for comparison.
WANG and AKBAR:
DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS
1.0E-9
T°C ~1501 - - 1301 -'- 1201 1.OE-10 -
¢~
1 0E-1
1
1.0E-12
2813
is shown to have significant influence on transport properties and its derived phenomena such as the high-temperature creep in ordered alloys. Although an exact quantitative fitting or assessment of parameters is not given in the present approach, the comparisons with the Monte Carlo simulation and experimental results serve to show how the PPM can be used to deal with the complex phenomenon of diffusion in ordered alloys. The composition-dependence of the diffusivity is shown to be decided by the competition between the thermodynamic and the correlation factors. Within the pair-approximation of the PPM and ignoring vacancy-atom interactions, such a competition leads to a minimum behavior in the diffusivity in agreement with some binary alloys, but fails to predict the reverse behavior observed in others.
1.OE-13
014
015
016 REFERENCES
CNi Fig. 9. Composition-dependence of the interdiffusion coefficient D of Ni in NiA1 at various temperatures with U = 0.73 and E = 0.55 kcal/mol. for steady-state creep as a function of composition exhibits maximum at elevated temperatures [22, 23]. In addition, Shankar and Seigle [21] have found that the activation energy for interdiffusion shows similar composition-dependence, a behavior predicted by both the PPM and M o n t e Carlo simulation. The similarity between the composition-dependence of the activation energy for diffusion and steady-state creep at high temperatures implies the diffusional nature of the high-temperature creep in the ordered NiA1. It is noted that in some ordered binary alloys a m a x i m u m in the diffusion coefficient vs composition curve is exhibited. In Ti-AI system, for example, interdiffusion coefficient vs composition curves exhibit maxima in both the Ti3AI and TiA1 phases [24]. According to equation (15b), it is suspected that, unlike in the case of NiAI where f l dominates the diffusion process, • may become dominant in systems such as Ti-AI. In Ref. [25], further calculations were carried out to verify this latter conjecture. It was found that minima in composition-dependence of the diffusion coefficient persists over a wide range of j u m p frequency ratios of the constituent components. The " m a x i m u m behavior" may arise from some other factors, which affect the thermodynamic and/or correlation factors. N o t e that the present calculation does nc~ =isccifically include v a c a n c y - a t o m interactions, i.e. E ~ = 0 . It would be worthwhile to investigate the effect of non-zero v a c a n c y - a t o m interactions.
1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
4. CONCLUSION The treatment of diffusion in ordered alloys based on the Path Probability Method has shown to give insights at an atomistic level. The atomic interaction
25.
H. Bakker, Phil. Mag. A40, 525 (1979). N. A. Stolwijk, Physica status solidi (b), 105, 223 (1981). R. Kikuchi, Prog. Theoret. Phys. Suppl. No. 35, 1 (1966). R. Kikuchi, Phys. Rev. 81, 988 (1951). H. Sato, in Nontraditional Methods in Diffusion (edited by G. E. Murch, H. K. Birnbaum and J. R. Cost), p. 203. T.M.S.-A.I.M.E. Conf. Proc. Warrendale, Pa (1984). H. Sato, Mater. Trans. Japan Inst. Metals 32, 509 (1991). C. C. Wang and S. A. Akbar, J. Phys. Chem. Solids. Submitted. H. Sato and R. Kikuchi, Phys. Rev. B 28, 648 (1983). H. Sato, S. A. Akbar and G. E. Murch, in Diffusion in Solids: Recent Developments (edited by M. A. Dayananda and G. E. Murch), p. 67. T.M.S.-A.I.M.E. Warrendale, Pa (1985). A. R. Allnatt and E. L. Allnatt, Phil. Mag. A 49, 625 (1984). L. Zhang, W. A. Oates, and G. E. Murch, Phil Mag. A 58, 937 (1988). K. Gschwend, H. Sato and R. Kikuchi, J. Chem. Phys. 71, 284 (1979). J. R. Manning, Phys. Rev. B4, l l l l (1971). R. Kikuchi and H. Sato, J. Chem. Phys. 51, 161 (1969). M. Hansen, Constitution of Binary Alloys, p. 118. McGraw-Hill, New York (1958). R. Hultgren, R. L. Orr, P. D. Anderson and K. K. Kelley, Selected Values of Thermodynamic Properties of Metals and Binary Alloys. Wiley, New York (1963). R. E. Hanneman and A. U. Seybolt, Trans. T.M.S.AJ.M.E. 245, 434 (1969). A, Steiner and K. L. Komarek, Trans. T.M.S.-A I.M.E. 230, 786 (1964). G. E. Hancock and B. R. McDonnell, Physica status solidi (a) 4, 143 (1971). D. Gupta, D. Lazarus and D. S. Lieberman, Phys. Rev. 153, 863 (1967). S. Shankar and L. L. Seigle, Metall. Trans. 9A, 1467 (1978). R, R. Vandervoort, A. K. Mukherjee and J. E. Dorn, Trans. Am. Soc. Metals 59, 931 (1966). W. J. Yang and R. A. Dodd, Metal Sci. J. 7, 41 (1973). K. Ouchi, Y. Iijima and K. Hirano, Titanium '80, Science and Technology, Vol. 1, p. 559, Proc. Fourth Int. Conf. on Titanium, Kyoto, Japan (edited by H. Kimura and O. Izumi), T.M,S.-A.I.M.E., Warrendale, Pa (1980). C. C. Wang and S. A. Akbar, "Diffusion in Ordered Binary Alloys: A Microscopic Approach,' T.M.S. Fall Meeting, Chicago T.M.S.-A.I.M.E., Warrendale, Pa (1992). To be published.
0956-7151/93 $6.00+ 0.00 Copyright ffS~1993PergamonPress Ltd
D I F F U S I O N IN O R D E R E D A L L O Y S A N D I N T E R M E T A L L I C COMPOUNDS C. C. WANG and S. A. AKBAR Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. (Received 12 October 1992; in revised form 5 April 1993)
Abstract--Diffusion in binary ordered alloys has been treated using the pair-approximation of the Path Probability Method (PPM) based on an atomistic model. The effect of the atomic interaction on the ordering behavior and its influenceon transport properties have been clarified. Compositional dependence of both the intrinsic diffusion and interdiffusion coefficients,correlation factor and thermodynamic factor agree very well with the Monte Carlo simulation. The calculated properties also show qualitative agreement with the experimental data on diffusivity, activity and creep in the NiA1 system.
!. INTRODUCTION Diffusion in ordered alloys recently has been the subject of considerable interest due largely to the increasing demand for the understanding of high temperature behavior of the ordered intermetallic compounds. In comparison with the disordered solid solution alloys, theoretical treatment of diffusion in ordered alloys are much more complicated because of the atomic interactions between the diffusing species. In other words, atomic jump in an ordered alloy strongly depends on the nature of ordering in the immediate environment at the microscopic level. Considering diffusion via the vacancy mechanism, when diffusing atoms exchange with vacancies, they tend to leave behind traces of disordered region and locally deviate from thermodynamic equilibrium. In order to maintain thermodynamic equilibrium, the migration of atoms must occur in a way or sequence to compensate or minimize the increase of energy during the diffusion. As a result, diffusion in ordered alloys is strongly correlated; the degree of correlation (or order) is a function of composition and temperature, and this considerably complicates the theoretical treatment of diffusion. Diffusion in ordered alloys tends to be much slower than in disordered alloys, and is often characterized with the existence of a sharp minimum in the diffusion coefficient vs composition curve and a sharp maximum in the activation energy vs composition curve around the stoichiometric composition. Slow rates of diffusion bring with them the associated advantage of improved microstructural stability at elevated temperatures and, since creep rate is proportional to the diffusion coefficient, improved creep strength. Apart from its technical importance, this problem of diffusion is also quite challenging theoretically. In the past two decades, the theoretical
treatment of diffusion in ordered alloys has mostly been developed as an extension to Manning's random alloy model in the framework of the nearest-neighbor pair-wise interactions. Bakker [1] and Stolwijk [2] have extended Manning's random alloy model to include long- and short-range order and their results show good agreement with the numerical simulations by the Monte Carlo method. However, these approaches do not provide an analytical expression for the calculation of the thermodynamic state of order as a function of composition and temperature. In the treatment of diffusion in ordered alloys, one requires firstly to construct an exact analytical expression for the change of the free energy of the system to be minimized, and secondly to derive the Onsager equation. A combined approach based on the Path Probability Method [3] and the Cluster Variation Method [4] offers a unique opportunity to treat these problems. The Cluster Variation Method (CVM), a static version of the Path Probability Method (PPM), can calculate the thermodynamic state of order as a function of temperature and composition. The PPM, on the other hand, can derive the Onsager equation for diffusion analytically based on atomistic models [5, 6]. Such analytical expressions are quite helpful in identifying measurable quantities (e.g. diffusion coefficient, ionic conductivity, etc.) in terms of fundamental microscopic parameters such as jump frequencies and interatomic interactions. If sufficient information is available on these quantities, specific predictions can be made or, conversely, diffusion experiments can be used to evaluate these atomistic parameters for which there is no straightforward calculation or measurement. It has recently been demonstrated [7] that by fitting the PPM equations to demixing experiments in oxide solid solutions, it is possible to self-consistently evaluate microscopic parameters.
2807
2808
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS
The PPM is not an alternate technique, but rather serves as an important supplement to the well established phenomenological treatment based on the Onsager formalism and other atomic theories of diffusion including computer simulations. Although some limitations have been found to appear in the original formalism of the PPM as applied to transport problems where the motion of individual particles is at stake (e.g. the calculation of the correlation factor), significant improvements have recently been made by changing some of the averaging processes. A full account of this problem can be found in Refs [8, 9]. Here, for simplicity, the original formalism will be used. The objective of this paper is to show how the PPM can be used to treat diffusion problems in ordered alloys using the well established pairapproximation. The effect of atomic interactions on the ordering behavior and its influence on transport properties will be illustrated and compared with the existing Monte Carlo simulations and experimental results in ordered intermetallic compounds such as NiA1. Compositional dependences of intrinsic diffusion and interdiffusion coefficients will be illustrated and their implication on the high temperature creep behavior of NiAI will be addressed. 2. ATOMISTIC CALCULATION The essence of an atomistic theory of diffusion within the linear approximation of the Onsager formalism is to derive the flux equations in a mutlicomponent system. For a binary alloy AB, under uniform temperature, the Onsager equations derived by the PPM can be expressed as
where, D~ is a hypothetical (or calculated) diffusion coefficient of i which is making random walk or in the absence of correlations. The term f l , sometimes referred to as the physical correlation factor, represents a collective correlation factor and corresponds to the so-called vacancy-wind effect (for random-walk motion, f~ = 1). It is important to note that D i defined in equation (3) implicitly includes the thermodynamic factor commonly recognized in the context of chemical diffusion in non-ideal multicomponent systems. Theretbre, sometimes it is convenient to express D~i in the following form Dxi = D 04"f~
(4)
where 4, accounts for deviation of the system from ideal solution behavior (4, = 1 for ideal solution). The quantity D O thus defined corresponds to the intrinsic diffusion coefficient of species i making random walk in the ideal solution, and, hence, is equivalent to the self-diffusion coefficient in the absence of correlations. This kind of expression is especially helpful in the analysis of problem of diffusion in multicomponent systems. Based on the PPM, all the above quantities can be expressed in terms of two parameters: the static and kinetic parameters, i.e. the interatomic interactions (Eij), and the jump frequencies of constituents (wi). For diffusion by the vacancy mechanism, the matrix elements Lv's are derived as: LAA = WAYAvWAfAA
(Sa)
LAB = WAYAvWAfAB
(5b)
JA = - - L A A ~ -- LABCt[
(la)
LBA = wByBv WBfaA
(5C)
JB = -- LaA~X,~-- LBB~B'
(lb)
LBa = wByBv WafBB
(5d)
wi = 0 exp(-flui)
(6a)
Here, JA and JB represents the fluxes of A and B atoms respectively, and ~((i = A, B) represents the generalized chemical potential gradient of the ith species (cti-- fl/~, where fl = 1/kT and/~ is the chemical potential). The Onsager matrix coefficient, Lij, can be related to the more familiar intrinsic diffusion coefficients as Dig = LAA/CA
-
D m = LBa/C B -
-
LAB/CB
(2a)
LBA/C A
(2b)
where, for an ordered binary alloy C~ = [Ci~l)+ C~z)]12.
(2c)
Here, Ci is the mole fraction of ith species, and C~l) and C~2) are the values of C~ on the two sublattices of the ordered system. The intrinsic diffusion coefficient, Dtl, can be expressed as [10] D1A = D A f k
(3a)
D m = DBf~
(3a)
where
y
wi=
L
-~i
x,
j
(6b)
The variables Yv represent the equilibrium values of the probability of finding i - j pairs (i a n d j = A, B, v), obtained by the pair-approximation of the CVM. The subscript V is used to denote vacancy, and, hence Y,v represents the probability of finding a vacancy near an ith atom. This term is conveniently called the vacancy availability factor (VAF). Here, 0 is the attempt frequency, u~ is the activation energy of motion, 2~o specifies the coordination number of the lattice (2~o = 8 for b.c,c.) and xi indicates the probability of finding the ith species on a lattice site and thus equals to C~ in the disordered state. It is obvious that while wl is the b a r e jump frequency, W, represents the effect of the surroundings on the jumping atom and is called the bond-breaking factor (BBF).
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS Therefore, it is convenient to introduce the effective jump frequency vbi as wi = wi W,
(7)
which takes into account the effect of bond breaking. Equations (5-7) show precisely how the transport of an atom depends on the availability of a vacancy in its neighborhood, and how the motion is influenced by the surrounding atoms. It should be noted that for an ordered system with two distinct sublattices, bond-breaking factors can be introduced for each sublattice. However, the product of VAF and BBF in equation (5) makes the Onsager coefficients invariant of the sublattice. The quantities f j ' s in equation (5) constitute the collective correlation factor f l in equation (3) flA = fAA -- fAB CA /CB
(8a)
fib = fBn - fBA C B / C A
(8b)
2809
sition with a negligible amount of vacancies is given by [12] 2E/kT¢ = ln[o)/(e) - 1)].
(12)
Equation (12) can be used to estimate E of a specific system at a given To. Note that equation (12) also serves to normalize the value of E with respect to k T c . For a system with known E, equation (12) gives the normalized critical temperature E/kT~. In a bcc lattice with 2~o = 8, for example, E/kTc has a value of 0.144. From equations (2), (5) and (7), the intrinsic diffusion coefficients can be expressed in the following form DIA = I~AYAv [fAA -- CAfAB/CB]/C A
(13a)
DIB = I~aYav [fBa -- CnfBA/CA]/C B.
(13b)
The vacancy availability factors YAVand Yav, which include the vacancy fraction Cv, are functions of temperature and composition, and can be evaluated Note that fan and fBA are simplified notations "JAn"erdA) by the CVM. However, in order not to introduce and f ~ , respectively, used by Allnatt and Allnatt [10] extra complications and conform to the approach and Zhang et al. [11]. taken by Zhang et aL [l l] in their Monte Carlo The PPM calculations will be divided into two computations, normalized intrinsic diffusion coparts for two different types of lattices. First, the efficients are defined as simple cubic lattice will be employed so that our results can be directly compared with those obtained DIA = DIA/[CvWB] (14a) by Monte Carlo simulations. Then, in order to compare with the experimental results of diffusion f)la = Dla/[Cv wa] (14b) coefficients in NiA1 (CsCl-type b.c.c, structure), the calculations will be extended to the b.c.c, lattice. In and are called intrinsic diffusion coefficients per unit both cases, interactions only between nearest neigh- vacancy concentration. The interdiffusion coefficient bors will be assumed and designated as EAA, EBB, [13],/), can be calculated directly from D~A and Dxa ~An(qj> 0 is taken as attractive). Interactions involved with vacancies V are assumed to be zero, i.e. EAV= ~Bv = Evv = 0.
(9)
Instead of using values of individual Eu defined above, it is more convenient to introduce an effective interaction energy parameter E of the form 4E = 2EAB -- (EAA + ~BB)"
(10)
When e is positive, it is referred to as the so called "ordering energy". In comparison with Zhang et al. [11], our definition of the ordering energy differs from theirs by a factor of 4. In the treatment of diffusion, in which the relative easiness of breaking bonds with its nearest neighbor atoms is to be considered, the difference of (AA and %B has to be taken into account. Therefore, an extra parameter U is defined as U = @AA -- E~n)/(4E)"
/) = CBD~A+ CA£)m
(15a)
and similar to equation (4), the interdiffusion coefficient can be expressed in terms of the thermodynamic and correlation factors in the following form = (Ca~)°Af I + C A £ ) ° f ~ ) ~ .
(15b)
For a given set of E, U and temperature (or a normalized temperature in terms of E/kTc), both the intrinsic diffusion and interdiffusion coefficients can be calculated as a function of composition. It should be noted the L u derived by the PPM in equation (5) implicitly includes the thermodynamic factor, 4~, which is usually defined as = flC,((?#,/SC~).
(16)
Using the relationship of CA+ CB ~ 1, equation (16) can be rewritten as [11]
(1 1)
This parameter is a measure of whether A - A or B-B bond is more easy to break and can be estimated from the energy of formation of vacancies. The effective interaction energy parameter E gives the measure of the critical temperature T¢ of the order~lisorder transformation. In the pair-approximation of the CVM, Tc at the stoichiometric compo-
= flCA CB[8(#A -- ,uB)/OCA]
(17)
which is more convenient for numerical differentiation. The chemical potential of ith species for an ordered alloy with negligible vacancies, can be derived by the CVM and is expressed as [14] /~, = fl(2~o In q i - 0.5(2o~ - 1)ln[C~mC~j)
(18)
2810
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS
where, q~ is determined by a set of simultaneous equations
1.0 0.9
Ci = qi Z qJ e-~'u
(19)
J and represents the short-range order of the system in terms of a single-site variable (it reduces to C~ in the completely disordered system). Thus, the thermodynamic factor 4~can be calculated from equation (17) by determining PA -- PB as a function of composition and performing a numerical differentiation. Furthermore, by taking the pure component as the standard state, the activity as a function of composition within the ordered phase region can be calculated by a, = exp[fl (/~, -/to/)]
(20)
where, P0~ is the chemical potential of pure i.
0.8 0.7 <<0.6 ~" 0.5
0.4 o.3 --- 0.100 0.050
0.2 0
3. RESULTS AND DISCUSSION
0 . ' 1 0 .03 .0 . 4' 20 . 5 ' 0' 6 ' 0i 7 ; 0i 8 ' Ca
0.9
.0
3.1. Simple cubic lattice--a comparison with Monte Carlo simulations
Fig. 2. Composition-dependenceofpartialcorrelation function fAA at various values of E/kT.
As in the case of Monte Carlo simulations of Zhang et al. [11], the first part of our calculation was also carried out for a simple cubic (s.c.) lattice with U = 0, i.e. £AA= EBB'From equation (12), the normalized critical temperature for the s.c. lattice is calculated to be c/kTc = 0.203. Being symmetrical, only the results for the A component are shown here; those for the B component can be obtained simply by substituting B for A in the figures. Figure 1 shows the composition dependence of the thermodynamic factor • at various values of ,/kT. Note that lower temperature corresponds to higher E/kT. For E/kT >t 0.203, the ordered phase appears with the minima in ~, corresponding to the
order~tisorder phase boundaries, and with the peaks at the stoichiometric composition. Figures 2 and 3 show the dependences offAA and f m respectively at various values of E/kT. The decrease offAA to a minimum indicates the increase in ordering as CA approaching the stoichiometric composition, or, in other words, the tendency of atoms to reverse jumps becomes stronger in order to preserve local order. However, unlike fAa, )CABdoes not show a minimum, instead a drastic drop is found as CA approaches the stoichiometric composition. It is worth pointing out that, in the present calculation, the value offAA in the limit of E/kT = 0.0 and CA-~0 0.5 e/kT
e/kT
14
-- 0.333 - - 0.286 --- 0.200 --0.158 -.- 0.100
12
A 11 Jl
m 0.050
--0.333 - - 0.286 -'- 0.200 "'0.158 "" 0.100
0.4
.'..'~.Q .'>~.
10
~% • x
--
0.050
x
0.3 e
• Q •'\ • "'\
8
x
0.2
6
4 0.1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
CA Fig. 1. Composition-dependence of the thermodynamic factor • at various values of E/kT.
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CA
Fig. 3. Composition-dependenceof partial correlation function fm at various values of E/kT.
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS 1.0
2811
1.0
'..'x
E/kT
x
m 0.050 - - - O. 1 O0 - - 0.158
0.8 . . . .
0.8
b~ ~," ~,"
0.200
.~
- - 0.286
<0.6-
0,6
nr~ 0.4-
0.4
##/
.o° S
°,~lt, o'~
0.2-
V
m 0.050 -'" 0.100 "- 0.158 --- 0.200 -- 0.286 -- 0.333
0.2
0
j ~ g , '1"~ " "
S#
.i-
0
0 01, o'2 013 0'4 o'5 o'6 0'7 o'8 o'9 10
0.1
0.2
0.3
0.4
CA
""
,/
/
/
/1
.s~ I I
0.5
0.6
0.7
0.8
0.9
1.0
CA
Fig. 4. Composition-dependence of the physical correlation factor f ~ at various values of E/kT.
Fig. 5. Composition-dependence of the intrinsic diffusion coefficient /)LAat various E/kT.
is 0.714, whereas an exact treatment would give a value of 0.654, the correlation factor for self-diffusion in s.c. lattice. This discrepancy comes from an inherent time correlation problem in the original formalism of the PPM in following individual particle motion [8]. By changing some of the averaging processes in the PPM, originally introduced by Sato, the calculations can be further improved and would give a closer agreement. Figure 4 shows the composition-dependence of the physical correlation factor f~, at various values of E/kT. Note that f~, is a combination o f f a h andfAB as defined in equation (4). For E/kT > 0.203, a deep cusp manifests the so called physical correlation effect, which is reflected in the slowing down of atomic migration due to preservation of local order. Figure 5 shows the results of the intrinsic diffusion coefficient D~A at various values of e/kT as defined in equation (14). For e/kT greater than 0.203, a minimum develops in the ordered region as expected, but, due to the non-symmetrical dependence offAA andfBa on composition, a minimum is not found at the stoichiometric composition. With the results of/StA and Din, the interdiffusion coefficient/~ can be obtained, and the results are shown in Fig. 6. Again, despite the maximum in the thermodynamic factor, a minimum i n / ) develops in the ordered region. In comparison with the results of Zhang et al. based on the Monte Carlo simulation [11], our PPM calculations based on the pair-approximation agree very well with their results except for some minor differences. In the Monte Carlo calculations, a minimum is observed in fAB at the stoichiometric composition when e/kT is higher than the critical value, and also their results are systematically higher. These differences are, however, expected as we used the
original PPM, which has some inadequacies with respect to the averaging processes as indicated earlier.
AM41/IO--B
3.2. CsCl-type lattice--a comparison with the experimental data on NiAI It is known that NiAI is an ordered compound with CsCl-type structure and exits over a wide range of compositions [15]. The stoichiometric alloy retains its structure up to its melting point of 1638°C. From equation (12), the ordering energy E can be estimated to be 0,55 kcal/mol. For simplicity, the ratio of bare jump frequencies (wsi/wAl) is set to be unity. The calculations were carried out by choosing different 0.8
0.6
z~
0.4
° 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
CA
Fig. 6. Composition-dependence of the interdiffusion coefficient/5 at various values of E/kT.
2812
WANG and AKBAR: DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS 0.6
0"5 t
- - E x p . data [17] ~ e = 0.55 kcal/mol .... e = 0.80 kcal/mol
0.4-
~
0.3-
0.2-
0.1-
/
0
.ooo-°~o...... oo°oOJ~,oW I I I I I I I I I 0 0.44 0.48 0.52 0.56
0.60
CNi
Fig. 7. Comparison of the calculated activity of Ni with the experimental data for NiAI at 1000°C in the range of Cr~i=0.44).6 [17]. The thick-solid line is calculated with c =0.55kcal/mol, while the dotted line with E = 0.80 kcal/mol.
values of ENi_Ni/EAIAI o r U. Here, the ratio of ENi-Ni/EAI Al was estimated to be 1.5 from the cohesive energies of Ni and AI [16]. It should be noted that our fitting of parameters or curves are not m e a n t to be exact, since the present model does not take into account the variation of lattice parameters and the vacancy concentration (therefore, interatomic interactions) as a function of composition. Also, the pair-approximation of the CVM does not accurately predict phase boundaries in intermetallic alloys; it is necessary to use higher-order approximations. Figure 7 shows the comparison of the calculated activity of Ni with the experimental data for NiAl at 1000°C, in the range of CNi = 0.4-0.6. The thin-solid curve represents the activity of Ni obtained by Hanneman and Seybolt [17], converted from experimental activity of A1 [18] by Gibbs-Duhem integrations. In light of the fact that this calculation is carried out using the pair-approximation with nearest neighbor interactions, the agreement is quite satisfactory. For the purpose of comparison, activity of Ni calculated with E = 0.80 kcal/mol is also plotted as the dotted curve. It can be seen that the agreement with experimental data is better for higher e, suggesting that the actual ordering energy may be higher than that estimated by the pair-approximations [equation (12)]. Figure 8 shows the composition-dependence of the intrinsic diffusion coefficient of Ni at various temperatures and U as indicated. Note that minima are observed near the stoichiometric composition for all temperatures below 1638°C, which qualitatively agree with experiments in Ref. [19]. It should also be noted that the minimum occurred is not centered at the
stoichiometric composition, a behavior sometimes observed in ordered intermetallic compounds; for example in AuCd [20]. Traditionally, the interpretation of this off-stoichiometric behavior is based on the argument of the change of defect structure across the stoichiometric composition. Our calculations suggest that both the magnitude and the sign of U play a significant role in determining the location of the minimum. In our calculations, the positive value of U means that vacancies tend to distribute in the Ni-rich region which makes the minimum offstoichiometric. In Fig. 8, composition-dependence of the intrinsic diffusivity of Ni at 1000°C for various values of U are also given for comparison. As U increases, corresponding to the increase in the ratio of eN~Ni/£AI_AI , the cusp becomes deeper. Figure 9 shows the composition-dependence of interdiffusion coefficients/) at various temperatures. Again, the off-stoichiometric behavior of the minimum is observed and is consistent with experiments [21]. Although our data show less dramatic dependence on composition in comparison with experiments, the agreement is quite satisfactory for the same reason stated previously. Another feature sometimes observed in the diffusion of ordered alloys is the temperature dependence of the composition at which the minimum occurs. Note that, in Fig. 9, the minimum i n / ) shifts toward the Ni-rich side as T increases, again in agreement with experiments [21]. An important implication of the present results is the high temperature creep behavior of ordered alloys where diffusional mechanism is operative. It has been found in NiAI that the activation energy
1.0E-9
I
I
I
~:~ 1.0E-11
1.0E-12
~ - -'---'"
1.0E-13 0.~4
i 0.5 CA
1200 1300 1500 1000 1000
(U=0.73) (U=0.73) (U=0.73) (U=0.9) (U=l.3)
016
Fig. 8. Composition-dependence of the intrinsic diffusion coefficient of Ni, DNi, in NiA1 at various temperatures, calculated with U = 0.73. At 1000°C, DN~,is also calculated with U = 0.9 and 1.3 for comparison.
WANG and AKBAR:
DIFFUSION IN ORDERED ALLOYS AND INTERMETALLICS
1.0E-9
T°C ~1501 - - 1301 -'- 1201 1.OE-10 -
¢~
1 0E-1
1
1.0E-12
2813
is shown to have significant influence on transport properties and its derived phenomena such as the high-temperature creep in ordered alloys. Although an exact quantitative fitting or assessment of parameters is not given in the present approach, the comparisons with the Monte Carlo simulation and experimental results serve to show how the PPM can be used to deal with the complex phenomenon of diffusion in ordered alloys. The composition-dependence of the diffusivity is shown to be decided by the competition between the thermodynamic and the correlation factors. Within the pair-approximation of the PPM and ignoring vacancy-atom interactions, such a competition leads to a minimum behavior in the diffusivity in agreement with some binary alloys, but fails to predict the reverse behavior observed in others.
1.OE-13
014
015
016 REFERENCES
CNi Fig. 9. Composition-dependence of the interdiffusion coefficient D of Ni in NiA1 at various temperatures with U = 0.73 and E = 0.55 kcal/mol. for steady-state creep as a function of composition exhibits maximum at elevated temperatures [22, 23]. In addition, Shankar and Seigle [21] have found that the activation energy for interdiffusion shows similar composition-dependence, a behavior predicted by both the PPM and M o n t e Carlo simulation. The similarity between the composition-dependence of the activation energy for diffusion and steady-state creep at high temperatures implies the diffusional nature of the high-temperature creep in the ordered NiA1. It is noted that in some ordered binary alloys a m a x i m u m in the diffusion coefficient vs composition curve is exhibited. In Ti-AI system, for example, interdiffusion coefficient vs composition curves exhibit maxima in both the Ti3AI and TiA1 phases [24]. According to equation (15b), it is suspected that, unlike in the case of NiAI where f l dominates the diffusion process, • may become dominant in systems such as Ti-AI. In Ref. [25], further calculations were carried out to verify this latter conjecture. It was found that minima in composition-dependence of the diffusion coefficient persists over a wide range of j u m p frequency ratios of the constituent components. The " m a x i m u m behavior" may arise from some other factors, which affect the thermodynamic and/or correlation factors. N o t e that the present calculation does nc~ =isccifically include v a c a n c y - a t o m interactions, i.e. E ~ = 0 . It would be worthwhile to investigate the effect of non-zero v a c a n c y - a t o m interactions.
1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
4. CONCLUSION The treatment of diffusion in ordered alloys based on the Path Probability Method has shown to give insights at an atomistic level. The atomic interaction
25.
H. Bakker, Phil. Mag. A40, 525 (1979). N. A. Stolwijk, Physica status solidi (b), 105, 223 (1981). R. Kikuchi, Prog. Theoret. Phys. Suppl. No. 35, 1 (1966). R. Kikuchi, Phys. Rev. 81, 988 (1951). H. Sato, in Nontraditional Methods in Diffusion (edited by G. E. Murch, H. K. Birnbaum and J. R. Cost), p. 203. T.M.S.-A.I.M.E. Conf. Proc. Warrendale, Pa (1984). H. Sato, Mater. Trans. Japan Inst. Metals 32, 509 (1991). C. C. Wang and S. A. Akbar, J. Phys. Chem. Solids. Submitted. H. Sato and R. Kikuchi, Phys. Rev. B 28, 648 (1983). H. Sato, S. A. Akbar and G. E. Murch, in Diffusion in Solids: Recent Developments (edited by M. A. Dayananda and G. E. Murch), p. 67. T.M.S.-A.I.M.E. Warrendale, Pa (1985). A. R. Allnatt and E. L. Allnatt, Phil. Mag. A 49, 625 (1984). L. Zhang, W. A. Oates, and G. E. Murch, Phil Mag. A 58, 937 (1988). K. Gschwend, H. Sato and R. Kikuchi, J. Chem. Phys. 71, 284 (1979). J. R. Manning, Phys. Rev. B4, l l l l (1971). R. Kikuchi and H. Sato, J. Chem. Phys. 51, 161 (1969). M. Hansen, Constitution of Binary Alloys, p. 118. McGraw-Hill, New York (1958). R. Hultgren, R. L. Orr, P. D. Anderson and K. K. Kelley, Selected Values of Thermodynamic Properties of Metals and Binary Alloys. Wiley, New York (1963). R. E. Hanneman and A. U. Seybolt, Trans. T.M.S.AJ.M.E. 245, 434 (1969). A, Steiner and K. L. Komarek, Trans. T.M.S.-A I.M.E. 230, 786 (1964). G. E. Hancock and B. R. McDonnell, Physica status solidi (a) 4, 143 (1971). D. Gupta, D. Lazarus and D. S. Lieberman, Phys. Rev. 153, 863 (1967). S. Shankar and L. L. Seigle, Metall. Trans. 9A, 1467 (1978). R, R. Vandervoort, A. K. Mukherjee and J. E. Dorn, Trans. Am. Soc. Metals 59, 931 (1966). W. J. Yang and R. A. Dodd, Metal Sci. J. 7, 41 (1973). K. Ouchi, Y. Iijima and K. Hirano, Titanium '80, Science and Technology, Vol. 1, p. 559, Proc. Fourth Int. Conf. on Titanium, Kyoto, Japan (edited by H. Kimura and O. Izumi), T.M,S.-A.I.M.E., Warrendale, Pa (1980). C. C. Wang and S. A. Akbar, "Diffusion in Ordered Binary Alloys: A Microscopic Approach,' T.M.S. Fall Meeting, Chicago T.M.S.-A.I.M.E., Warrendale, Pa (1992). To be published.