The thermodynamics of vacancy formation in f.c.c. metals

Acta metall,mater.Vol. 43, No. 10, pp. 3721-3725, 1995 ~

Pergamon 56-7151(95)00063-1 09

ElsevierScienceLtd Copyright© 1995Acta MetallurgicaInc. Printed in Great Britain.0956.7151/9rights$9. 5All 50reserved+ 0.00

T H E T H E R M O D Y N A M I C S OF V A C A N C Y F O R M A T I O N IN f.c.c. M E T A L S R. B. MeLELLAN and Y. C. ANGEL Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, TX 77251-1892, U.S.A.

(Received 20 May 1994; in revisedform 16January 1995) Abstract--The dependence upon temperature of the monovacancy concentration in metals has been calculated using models for the perturbation of the atomic frequencies accompanying vacancy formation. The effect of the lattice thermal expansion has also been considered. The calculations, based on the quasi-harmonic model for solids, indicate that Arrhenius plots of the vacancy concentration against reciprocal temperature would exhibit a measurable curvature for A1. Much smaller degrees of curvature would be observable in such plots for Cu and Au over the temperature span of experimental vacancy concentration determinations.

1. INTRODUCTION In a recent paper [1], simple models proposed by Wautelet (W) [2], and Differt et al. (DST) [3], for the perturbation of the frequency spectrum in metals due to the creation of monovacancies were used to calculate the monovacancy concentration and the elastic properties of the lattice in the quasi-harmonic approximation. These calculations showed that, for Au and AI, there would be no significant departures from linearity in the temperature variation of the abiabatic elastic constants, nor in Arrhenius plots of the monovacancy concentration. These results were obtained in the high-temperature limit, however. The purpose of the present work is to study the implications of the W and DST models in temperature regions where the high-temperature limit is not an adequate approximation and, furthermore, to include the effects of thermal expansion in the calculation. The results found will be discussed in respect to the metals Au, Cu and AI. The metal A1 is especially interesting since the relatively large non-linearity found in the Arrhenius plot of the total vacancy concentration is traditionally interpreted in terms of divacancy formation, whereas recent work on selfdiffusion in A1 and positron annihilation indicate that the divacancy concentration in AI close to the melting point may not, in fact, be as large as earlier analyses had implied. In the present work the thermodynamics of the defect lattice are calculated on the basis of simple specific models for the form of the vibrational spectrum of the solid. The phonon spectrum is dependent on the temperature and it should also be noted that the elements of the potential energy matrix are regarded as functions of the strains. This means that the theory implies a dependence of the elastic constants

(C~i) upon temperature. In the present paper the somewhat laborious calculations of the Cij will not be made. Although the calculations clearly represent a simplification to the actual phonon spectrum of a metal lattice containing vacancies, it is reasonably clear that, once the form of the spectrum (W or DST) is chosen, the method of calculation is sufficiently accurate. The good agreement between the quasiharmonic model and the measured temperaturedependence of the adiabatic elastic constants has been discussed in a lucid manner by Weiner [4]. 2. THERMODYNAMIC CALCULATIONS The Gibbs free energy GR(P, T, n) of a crystal containing n random defects with unit orientational degeneracy is [1] Gk(P, T, n) -- Go(P, T) + ~F(p, T, n) - kT In f2 (1) where Go(P, T) is the free energy of the defect-free crystal, Gr(P, T, n) is the integral defect-free energy, and f~ the configurational degeneracy. It was shown previously [1] that, with respect to a fiducial temperature To, the atomic concentration c of defects is given by the general expression

c=exp{_~(t3IYlF(p, To,n)~

{',, r

~

}

_ 7-" x exp -~,]ro on ~F(p, U, n) du .

(2)

The only assumption made in deriving this expression is that the point defects are randomly distributed. The

3721

3722

McLELLAN and ANGEL: VACANCY FORMATION IN f.c.c. METALS

entropy and enthalpy quantities in equation (2) are simply ~F(p, T, n) = --(c~GF(P, T, n)/OT)e,, and

where fl is the thermal expansivity and the Griineisen constant ~, is considered to be independent of the volume. Since 1 >>#T, where # = fly, we may write exp(-pT)~ 1-#T and write for the Wautelet approximation

/~rF(p, T, n) = (~F(p, T, n) + TSF(p, T, n).

vw = v0(1 - #T)(1 - ~tc)

It was also shown previously [1], as a consequence of the temperature-independence of the partial defect entropy S ~ = (0~F(p, T, n)/On)e,r resulting from the W and DST approximations, that HF(P, T) = (O/'IF(p, T, n)/On)~,.r = HF(p, To) and, thus, that H F is independent of T for T > T o . This result is a consequence of the thermodynamic constraint

(9)

and for the DST case VDSx = (1 -- #T)(v')cVp[vo(1 -- ~c)](l-cvp).

(10)

It is easy to calculate the following relations from equations (9) and (10)

(OH/OT)e = T(OS/dT)e.

OT\T]

In the present work, t h e vacancy concentration c will be calculated from the basic result of equation (2) by writing the non-configurational entropy in the form

--Vo(1 --~c),

-~n =

NVo(l-#T),

(111

(1V°sr-#T)'

(12a)

T25(~-~)=

S(P, T, n) = - 3 N k ln(1 - e -h~/kr) + 3Nhv/T

eh~7~_--i- (3)

rather than the high-temperature limit of this expression used in the previous calculation. In equation (3) N is the number of atoms in the crystal and v is an effective frequency which is a function of the defect concentration. We will also consider the set of v to be a function of temperature independent of the defect concentration. Noting that

On = N

v0(1---~c)

e ~/ ~-~

=

3" (12b)

Using equations (11), (12) and (5), it can be shown for the Wautelet model that

f .

T

d

__ ,.{v(p, u, n) du =

Jr0 an

3°~hvo f r ( 1 - #u) 2

0[1]

x~'~u ~

0T ~

(1-~c)

[-(1

(4)

jro

du

Uu)22" = r

-

(e ~/k~- 1) 5

it is easy to show from equation (3) that

OS(P, T,n) dn

fay\ v 0 f 3Nh~-~n) TE d - ( v ~ ~ ~

1

") ~"

(9 Now Wautelet [2] proposed that the creation of a monovacancy perturbs the de-localized phonon spectrum in a linear manner such that v = Vo(1 - ~tc)

(6)

where v0 is a constant and v is a mean (Einstein) frequency. Differt et al. [3] wrote the effective frequency in the form

v = (v')~VPlvo(1 - ~c)](1-~vp)

(7)

where v' is the average frequency of the local defect modes and Vp represents the volume fraction of the crystal in which the local perturbation occurs. Now, in addition to the effect of defect formation, the average frequencies will change with volume because of the thermal expansion. A reasonable approximation for this is [5] v (T) = v (0)exp( -- fl~T)

(8)

(1 - p u ) d u + 6#~hvo fro eh,o,-,O0-~,.)/k,__ 1

(13)

The remaining factors in equation (2) may be calculated in a similar manner, and putting To = 0 gives the result 1 d /~F(p, 0, n ) + _ _3~hvo

lncw=

kTan

kT

(1 - / t T ) 2 , ~ hvo x exp(hv0(1 _ ~c)(1 - #T)/kT) - 1 . olzct~--~

x

f ro

(1 - #u) du exp(hvo(1 - c~t)(1 - ltu)/ku) - 1

(14)

Assuming that /~t(p, 0, n) = nHF(P, 0) - nHF(0), we have

3cthvo

HE(0) In Cw =

k~

+

k----~

(1 - - g T ) 2 ×

+

exp(hvo(1 -- c~)(l -- # T ) / k T ) - 1

6#othvoI ( T) kT

(15)

McLELLAN and ANGEL: VACANCY FORMATION IN f.c.c. METALS The integral I ( T ) may be computed numerically. In the case of the DST spectrum, the calculation is somewhat more laborious and the resulting monovacancy concentration is HF(0)

In CosT = - - - kT

3hK (1 -- l z T ) 2 + -k T exp(hqo(1 - l ~ T ) / k T ) - 1

61zhK fro (1 -- #u) du -~ exp(hf0(1 - Uu)/ku) -- 1 (16) where

Vo = (v')cVp[vo( 1 -- ctc)]° -cvp)

(17)

and K=v° L (-1-~c~

E l,,l

In v o ( l V t c ~

.

(18)

It should be noted that when # = 0 (i.e. the crystal is completely harmonic) and we set 1 ~> c~t, equation (15) becomes lncw=--

HE(0) -4- 3°thv° (ehv°/kr- 1) -1 k---T-- k T "

(19)

3723

from the lattice dilation, and a larger positive term whose origin lies in the shear deformation [6]. The dilational contribution is proportional to the strain, and the shear term to the square of the strain and the temperature coefficient of the shear modulus. Equations (15) and (16) may be used to calculate the Arrhenius plot of In c vs 1/T provided sufficient input data are available. 3. DISCUSSION Let us consider initially the metal AI since plots of In c vs 1/T show a clear non-linearity when values of c taken from a variety of differing experimental techniques are considered [7]. These conventional methods (X-ray, dilatometry, and electrical resistivity) measure the total vacancy concentration. The curved plots of In c vs 1/T have been traditionally interpreted in terms of monovacancy (1V) and divacancy (2V) formation. In this model, with obvious notation, the total vacancy concentration c~ is given by Hw Slv c , = e x p - - ~ - exp - ~

and in the high temperature limit ( k T >>hvo) HF(0)

In Cw

- kT

(20)

so that, as shown previously, the behavior in the high-temperature limit exhibits a linear Arrhenius plot of In c vs 1/T and ~t -~ SFv/3k (using the conventional nomenclature for the monovacancy formation entropy). If the thermal expansion is ignored in the DST approach and it is noted that c~ ,~ 1, and Vp ~ 1, it is easy to show that

K ~- vo [ct - Vp ln(v "/vo)].

(21)

(Note that where x is not too large xU--*l as N ~ 0 . ) Thus, under these limiting conditions, the DST model yields a defect concentration given by HF(0 ) 3hv0[c~- Vp l n ( ~ ) l In CDST= - - - -t kT kT

and in the high-temperature limit lnCoST=

B F F [HEv - 2Hlv'~ //2S1v S~v \ + 12 exp~ k77 :exp~ -~ ) -

+ 3~

~-~ + 3 e - - V p l n

~

.

(23)

These results are interesting since they show that the form of plots of In c vs 1/T is not dependent on the actual choice of the W or DST spectrum, but in the latter case, the defect entropy in the high-temperature limit is the sum of two terms. The term 3ek is positive, and since v ' > vo, the term - 3 k V p ln(v'/Vo) is negative. This result may be compared to macroscopic classical elastic calculations of SIFVwhere is is shown that S~v corresponds to a negative term resulting

-

(24)

where superscript "B" refers to the "binding" enthalpies and entropies. Siegel [7] gives the "best fit" values, H E = 6 4 . 4 k J / m o l , SFv=0.7k, H~v= 27.03 kJ/mol, and S2av= - 1 . 2 k. The dashed upper curved line in Fig. 1 is constructed from equation (24) and the parameters given above. This line is an excellent representation of the experimental data. The lower dashed line in Fig. 1 is constructed from the first term in equation (24). In order to compare the curve of Fig. 1 to the present calculations, we require values of the parameters ~, v0 and #. HF(0) can be taken as being equal to HFv, although this is not a strict requirement. A reasonable approximation for v0 is to take v0 = kO/h, where 0 is the Debye temperature for AI (396 K) and /~ is taken from fl = 7.08 x 10 -5 °K -~ and 7 = 2.0 [8] giving/z - 1.55 x 10-4 °K -1. There is no way to give an a priori estimate of c~. Its lower bound must be SFv/3 k = 0.233. Using the input parameters delineated above and choosing ~ =0.458 (based on preliminary curve fitting) results in the solid curve shown in Fig. 1. In these calculations, the factor (1 - c o t ) was replaced by unity. The value of ~ = 0.458 was chosen by requiring the values of ln c determined from equations (24) and (15) to converge at low temperatures. The lowest line in Fig. 1 represents the first term in equation (24). It can be seen that, despite the simplicity of the frequency assumptions, the present calculation, assuming the presence of monovacancies only, is consistent with the observation of clear curvature in the

3724

McLELLAN and ANGEL: VACANCY FORMATION IN f.c.c. METALS

plots of In c vs 1/T. It should also be noted that additional computer fitting calculations show that the variation with composition of the defect entropy, i.e. the factor ( 1 - c o t ) , is completely negligible. This result does not mean that all of the observed curvature is due to the effects considered here and that the divacancy concentration in A1 is negligible, but it does indicate that the large values of C2v close to the melting point concomitant with equation (24) are doubtful. The positron lifetime spectroscopy studies of Schaefer et al. [9] indicate that the divacancy concentration in A1 near the melting point may, indeed, be much smaller than had previously been supposed. This conclusion, however, may be open to some question because the observation may be due to a small difference between the positron lifetimes in monovacancies and divacancies [9]. However, recent nuclear magnetic resonance studies of self-diffusion in Al in the range 500-800 K [10] also indicate that the contribution of divaeancies to the self-diffusivity is small. Let us now turn our attention to the metals Au and Cu. In the case of Cu, Mehrer [11] has combined five sets of self-diffusion data spanning the temperature range 623-1350 K. His analysis, and that of Mehrer and Seeger [12], conclude that the monovacancy mechanism is dominant in the entire temperature range. However, later work [13] on the diffusion of Au in Cu and Cu self-diffusion indicates that there is a slight curvature in both these diffusivities in the T range 623-1350 K. Older compilations of data sets for the total vacancy concentration in Cu [7] and the recent positron lifetime spectroscopy studies of

A(T):

3cthvo

(1 - #T) 2

k T exp{hv0(1 - # T ) / k T } - 1

q-

-6 \ \

•

Inc

Schaefer et al. [14] fail to detect contributions ascribable to divacancies. Thus, in the case of Cu the diffusivity data are consistent with a slight curvature in Arrhenius plots of the diffusivity with reciprocal temperature, but no curvature is observed in the existing vacancy concentration data. The vacancy concentration data for Au have been compared and analyzed by Siegel [7] and by Sahu et al. [15]. Data spanning the temperature range 675-1320K indicate a slight curvature, and Siegel et al. [7, 15] have compiled values of the parameters of equation (24) consistent with the data. The bulk of the self-diffusion measurements in Au refer to temperatures less than 1220 K [16]. These data do not exhibit any significant deviation from linearity. The self-diffusion data of Herzig et al. [17], extending up to 1333 K, do exhibit a positive curvature consistent with the analyses of the vacancy data and a small, but significant, divacaney concentration close to the melting point. The model presented in this work has been used to calculate c as a function of T using v0-values taken from the Debye temperatures, HF(0) = HiFv, ),-values from the literature [8, 16], and s-values taken from preliminary curve fitting in the low-temperature region. For both Cu and Au the curves of In c vs 1/T do not exhibit an "optical" non-linearity from 500 K to the melting temperature. However, they do indicate a positive curvature as T increases. This curvature is optimally exhibited by calculating the quantity

""', \ N

-9

.

-12

10

. . . 104/T(K "~)

15

Fig. 1. Dependence of the calculated vacancy concentration c in aluminum in the form of Arrhenius plots of In c vs reciprocal temperature. The uppermost dashed line represents the "best" fit of the experiment data using the monovacancy/divacancy model. The lowest dashed line represents the curve of In c vs 1/T for monovacancies only (using the monovacancy fitting parameters for the uppermost curve). The solid line depicts the present model.

6#cthVoI( T) kT

(25)

This function, as can be seen from equation (15), represents the amount by which In c deviates from the extrapolation to high temperatures of the low temperature vacancy concentration values. In A(T), we have set (1 - cot) = 1. The A(T)-values are shown in Fig. 2. The solid lines cover the lowest experimental measurement temperature up to the melting temperature. It is clear that for the three metals compared, the model indicates that for Al these would be the greatest departure from linearity over the observation temperature span of the In c vs 1/T plots and the least for Cu. As the temperature increases A(T) increases from zero, but eventually the rate of increase of A(T) begins to decrease with increasing temperature. This is because the second term in equation (15) is approaching its high-temperature limit of 3ct(1 - #T) ~- 3~t. The term 61~thvoI(T)/kT increases with temperature, but is small compared with the second term. Values of 61~hvoI(T)/kT = J ( T ) vs T for A1 are shown in Fig. 3. The A(T) values for all three metals considered are not negligible when a

McLELLAN and ANGEL:

VACANCY FORMATION IN f.c.c. METALS

2.1

0.3

'

'

+

'

I

'

3725

'

J(T)

'

,

Cu

A(T)

0.2

AI

1.5 Au

0.1 0.9

. . . .

500

t

. . . .

1000 T(K) 1500

Fig. 2. Dependence of the quantity A(T) on the temperature for Cu, Au and A1. The segments cover temperature ranges of vacancy measurements. Cu: 0 = 315 K, ct =0.78, p = 1.02 × 10 -4 ° K - I ; Au: 0 = 180 K, ct = 0.45, ~ = 1.04 x 10-4 OK-t.

large temperature range is considered. The reason why curvature in the In c vs l I T plot is well defined for A1, but not for Cu, is because of the much larger values of HF(0) resulting in smaller values of c at a given temperature and restricting the temperature range over which accurate vacancy concentrations are possible. The rank of the HF(0) = HiFv values is as follows: 64.4kJ/mol (A1), 90.TkJ/mol (Au), and 122.5 kJ/mol (Cu). The W and D S T models are physically distinct and the latter frequency approximation is clearly more reasonable. However, when c < 1 (i.e. Vp ~ 1) the resulting expression for the vacancy concentration does not differ in form from that calculated from the simpler W-spectrum. In conclusion, it may be said that despite the simple model used to represent the frequency spectrum of the solid, observable curvature would be expected in experimental data for the Arrhenius plot of In c vs 1/T for AI even if divacancies are not considered. Some degree of curvature would be expected for Au and less for Cu, but the extent would be considerably less than would be seen for A1. The significance of the results obtained lies in the fact that some degree of curvature is to be expected in the plots of In c vs 1/T in metals where large volumes of experimental data on the total vacancy concentration have been interpreted in terms of the monovacancy/divacancy approach. If it is indeed true that a substantial part of this curvature is not due to the presence of divacancies, then it is clearly not possible to evaluate such data in terms of equations like (24) and derive accurate values of the monovacancy and divacancy formation enthalpies and entropies.

0

1 . . . .

;00

I

,

,

,

,

1000 T(K) 1500

Fig. 3. The variation with temperature of the quantity 6#cthvol(T)/kT for AI, Cu and Au.

It is not intended to imply that multivacancy formation is not a primary cause of curvature in the Arrhenius plots of diffusion or vacancy concentration data, but it is clear that the interpretation of such curvature is not unambiguous.

Acknowledgement--R. B. McLellan is grateful for support

provided by the Robert A. Welch Foundation.

REFERENCES

1. R. B. McLellan and Y. C. Angel, Acta metall, mater. 40, 3497 (1992). 2. M. Wautelet, Phys. Lett. A 108, 99 (1985). 3. K. Differt, A. Seeger and W. Trost, Mater. Sci. Forum 15, 99 (1987). 4. J. H. Weiner, Statistical Mechanics of Elasticity. Wiley, New York (1983). 5. S. M. Collard and R. B. McLellan, Acta metall, mater. 39, 3143 (1991). 6. R. B. McLellan, Trans. Am. Inst. Min. Engrs 245, 379 (1969). 7. R. W. Siegel, J. nucl. Mater. 69, 117 (1978). 8. D. B. Fraser and A. C. H. Hallett, in Proc. 7th Int. Conf. Low Temp. Phys. University of Toronto Press, Toronto, Ontario (1960). 9. H.-E. Schaefer, R. Gugelmeier, M. Schmolz and A. Seeger, Mater. Sci. Forum 15, 111 (1987). 10. S. Dais, R. Messer and A. Seeger, Mater. Sci. Forum 15, 419 (1987). 11. H. Mehrer, J. nucl. Mater. 69, 38 (1978). 12. H. Mehrer and A. Seeger, Phys. status solidi 35, 313 (1969). 13. S. Fujikawa, M. Werner, H. Mehrer and A. Seeger, Mater. Sci. Forum 15, 431 (1987). 14. H.-E. Schaefer, W. Stuck, F. Banhart and W. Bauer, Mater. Sci. Forum 15, 117 (1987). 15. R. P. Sahu, K. C. Jain and R. W. Siegel, J. nucl. Mater. 69, 264 (1978). 16. N. L. Peterson, J. nucl. Mater. 69, 3 (1978). 17. C. Herzig, H. Eckseler, W. Bussman and D. Cardis, J. nucl. Mater. 69, 61 (1978).