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Activation volume for self diffusion in metals
Scripta Materialia, Vol. 39, No. 7, pp. 931–936, 1998 Elsevier Science Ltd Copyright © 1998 Acta Metallurgica Inc. Printed in the USA. All rights reserved. 1359-6462/98 $19.00 1 .00
Pergamon PII S1359-6462(98)00247-4
ACTIVATION VOLUME FOR SELF DIFFUSION IN METALS G.P. Tiwari Materials Science Division, Bhabha Atomic Research Centre, Mumbai, 400 085, India (Received May 27, 1997) (Accepted June 10, 1998) Introduction It is well established that the vacancy mechanism is the dominant mode for self diffusion in metals [1]. It involves the exchange of atomic sites between a vacancy and one of its neighbouring atoms. In the course of its migration from one atomic site to the next, the diffusing atom traverses through a saddle point. At the same time there is a relaxation among the nearest neighbours of the diffusing atoms. This ensemble of the diffusing atom at the saddle point with a neighbouring vacancy and its nearest neighbour in a relaxed state is called the activated complex of the diffusion. The volume of this ensemble, called activation volume for self diffusion (DV*), is the subject of the present article. DV* has two components - the volume associated with the formation of a vacancy and the difference between the atomic volume of the diffusing atom between its equilibrium and saddle point configuration. Thermodynamically, the activation volume is the pressure derivative of the free energy of activation for diffusion. There are two possible theoretical approaches to the subject of self diffusion in metals. One of these is the theoretical ab initio calculation of the energy for vacancy formation and the additional energy required for the diffusing atom to cross the potential energy barrier at the saddle point [2]. This procedure, though difficult and time consuming has the advantage of giving a detailed picture of the micromechanism of the diffusion process. However, this procedure is not technically convenient when either a quick estimate of the diffusion rate is required or a comparison of the diffusion rates between different systems (any two metals or the compositional variation in the same metal) is necessary. In such a situation, several empirical relationships between the diffusion parameters and bulk properties of the matrix [3– 6] have proven very useful. This has led to another important route which may be called the phenomenological approach to the study of self diffusion in metals and other kinds of solids. In the phenomenological approach, a diffusion parameter - most commonly an activation energy - is related to some bulk property such as melting point [3], latent heat of fusion [4], cohesive energy [6], etc. The changes in the diffusion rate from one system to the other are then ascribed to the changes in the bulk properties. The two widely used relationship are: DH 5 K1 Tm
(1)
DH 5 K2 Lm
(2)
where DH, Tm and Lm are respectively the activation energy for self diffusion, melting point and the latent heat of fusion. K1 and K2 are constants with appropriate units. Commonly accepted numerical values for K1 and K2 are 38 and 16.5 respectively. These relationships are based on the hypothesis that 931
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the region around the vacancy is like the molten state of the matrix, and the process of diffusion is akin to the shuffling of the atoms within the region. Hence the activation energy for diffusion should be related to the melting parameter. Another approach of this kind was pioneered by Zener [7] who postulated that energy associated with the saddle point configuration originates in the straining of the surrounding lattice. Proceeding on this basis, he derived the following relation between the activation entropy for diffusion (DS) and the temperature dependence of the elastic constant of the matrix: DS 5 l(dE/Eo)/d(T/Tm)
(3)
where l is designated as Zener’s constant which varies between 0.01 to 1 for metals, and E and E0 are the values of appropriate elastic constants at temperature T and 0K respectively. Following Zener, several other treatments relating self diffusion parameters to elastic constants have been published [8,9]. Being empirical in nature, all these treatments involve a constant such as l. The variation of such a constant within a narrow range for a group of metals or solids having identical physical or chemical properties is taken as a proof of their validity. In the present paper, a formal thermodynamic equation for the activation volume for self diffusion is derived on the assumption that the free energy for self diffusion in metals is a function of the specific volume of the matrix. 2.0 Derivation of the Basic Equation Mott and Gurney [10] estimated the temperature dependence of the activation energy for self diffusion in ionic crystals assuming that this parameter varies linearly with the specific volume of the matrix. However, it is more appropriate to consider Gibbs’s free energy for self diffusion as a function of the specific volume. In this context, it has been shown by Vaidya [11] that the logarithm of the self diffusion coefficients (D) for several metals varies linearly with V/V1 where V and V1 are the specific volumes of the solid at the ambient and the liquid at the melting point. In some cases, the plot shows a bend towards the volume axis but the variation of log D verses V/V1 follows a regular behaviour over the entire region. In the solid region, which is our concern here, the variation of log D with V/V1 is practically linear. More significantly, the values of D for the liquid at the melting point obtained from the extrapolation of the data points from the solid are nearly identical with the experimentally measured self diffusion coefficient for the liquid. It may, therefore, be concluded that volume is the most important single parameter affecting the self diffusion coefficient of the metallic matrices, and whatever changes in the free energy of activation (and consequently in the diffusion coefficient) occur as a result of variations in the temperature (T) and pressure (P) are solely due to their effect on the specific volume of the matrix. In solids, the equation for self diffusion coefficient is expressed [7] as D 5 f a2 n exp(2DG/RT)
(4)
or 5 f a2 n exp(DS/R) exp(2DH/RT)
(5)
where f 5 correlation factor, a 5 lattice parameter, n 5 Debye frequency, R 5 gas constant and DG is the free energy of activation for self diffusion. As discussed above, DG can be expressed as a function of the specific volume of the matrix in a Taylor series: DG 5 DG0 1
dDG d2 DG DV2 DV 1 1. . . . . . dV dV2 2
(6)
Writing V 5 V0 (11 aT) where V0 is the specific volume of the matrix at 0K and a is the volume expansion coefficient of the matrix, DV is given by
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TABLE 1 Comparison of Experimental and Calculated Values of the Activation Volume for Self Diffusion in Zinc, Cadmium and Sodium Activation volume (cm3/mole) Serial No.
Matrix
Temperature [K]
Experimental
Calculated
Difference (%)
1 2 3 4 5 6 7 8 9
Zn
573 623 673 524 549 574 592 287.8 364.3
4.28 3.97 3.72 6.90 7.14 7.65 7.49 11.46 12.40
3.20 3.22 3.44 4.84 5.08 5.49 5.87 10.23 17.25
25.2 18.9 7.5 29.8 28.8 28.2 21.6 10.7 39.1
Cd
Na
DV 5 aV0 T
(7)
By definition, activation volume, DV* can be defined as: DV* 5 (DG)/P 5 2(DG)/V.b.V
(8)
where b is the compressibility. Neglecting higher order terms and substituting from equations (7) and (8), in equation (6) one gets DG 5 DH 2
DV* DV b V
(9)
where DH has been substituted in place of DG0. Comparison of equation (9) with the standard thermodynamic relation DG 5 DH 2 T DS, yields DV* 5 DS (a/b)21(1 1 aT)
(10)
3.0 Comparison with Experimental Results Equation (10) has been applied to the data of Chhabildas and Gilder on zinc [12], of Buescher et al. on cadmium [13] and those of Mundy on sodium [14]. These are possibly the only available data of sufficient accuracy which give activation volume as a function of temperature and hence can be used to test the validity of equation (10). 3.1 Zinc and Cadmium In this case, the DS values have been calculated using equation (5) from the self diffusion data of zinc [15] and cadmium [16]. It is important to emphasise here that thermodynamically, DV* can vary with temperature irrespective of the behaviour of DS with respect to the temperature, although the temperature variations of DS and DH are themselves interrelated. Values of a and b at the desired temperatures have been taken from references [17] and [18] respectively. For the purpose of comparison, the values of DV* and DS have been taken along c-axis. The experimental and calculated values are compared in Table 1.
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As seen from Table 1, the difference between the experimental and the calculated values varies from 7.5 to 25.2% for zinc and from 21.6 to 29.8% for cadmium. Considering the inherent variations in the various parameters involved, it is believed that fairly good agreement exists between calculated and the experimental values. It is also believed that the main reason for the difference between the calculated and the experimental values can be attributed to the use of compressibility data for polycrystalline instead of the single crystal materials since the activation volumes have been measured on single crystals. This observation is supported by the fact that in either case there is a tendency towards better agreement between the experimental and calculated values with the rise in temperature. The presence of grain boundaries is expected to decrease with the values of compressibility and their influence will decrease with the increase in the temperature. This could be the reason for better agreement at higher temperatures. It is also possible to evaluate the temperature dependence of the activation volume assuming a/b to be a constant. This is a valid assumption since Hanneman and Gatos [19] have earlier shown that the ratio a/b is a constant for most of the bcc and fcc metallic lattices. The same is taken to be true for zinc and cadmium here. It is further assumed that the a is also independent of temperature. Therefore, it is now possible to differentiate equation (10) to obtain the values of (DV*/T) as below: DV* 5 DS(a/b)21.a
(11)
The room temperature value of a/b has been taken for the computation since this parameter is taken to be a constant. The value of DS for self diffusion in zinc as well as cadmium is also independent of temperature and, as before, values for the c-axis have been taken. The value of ( DV*/T) for zinc and cadmium so obtained from equation (11) are 5.86 3 1024 and 7.46 3 1024 cm3/mole.K respectively. The corresponding experimental values are 6.43 3 1023 and 1.1 3 1023 cm3/mole.K [12,13]. These values are approximately 11 and 1.5 times the calculated values for zinc and cadmium respectively. This variation could be due to the assumption regarding the constancy of the ratio a/b and/or to the use of bulk compressibility in place of the values for c-direction. Again, the agreement between the calculated and experimental values can be termed as satisfactory. 3.2 Sodium The data on activation volume for self diffusion in sodium are available at 287.8 K and 364.3 K only. DS values at both temperatures have been obtained from Mundy’s work [14]. Mundy has expressed the diffusion coefficient as the sum of two straight lines representing low and high temperature regions. The low temperature diffusion data have been identified with mono vacancy diffusion, whereas the high temperature data have not been definitely identified with any mechanism, although the preference for divacancy mechanism has been indicated .The percentage contribution of low as well as high temperature processes at 287.8 K and 364.3 K has also been given and the net value of DS used here is a weighted average of the operating diffusion mechanisms. Mundy has also expressed the same diffusion data on self diffusion in sodium as a function of three straight lines. However, this has not been used here because the last procedure is neither an improvement on the statistical fit of the data nor it can be given any reasonable interpretation with respect to the operating modes of diffusion. For the thermal expansion coefficient, a value of 71 3 1026/K between 273–388 K from Smithells Metals Reference Handbook [20] has been taken. Compressibility data have been taken from the work of Bridgman [21]. Bridgman measured the compressibility at 303 K and 348 K by the piston and cylinder technique. Compressibility values at these temperatures were obtained graphically from Bridgman’s data. Actual compressibility values at the desired temperatures were again graphically evaluated through extrapolation. The results are given Table 1. At the lower temperature the difference between the calculated and
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experimental values of DV* is nearly 10%, which is considered very good. At higher temperature a difference of 39% is obtained. At this temperature sodium is very soft and Bridgman has indicated that the softness of the material has possibly affected the measured compressibility values. A part of the discrepancy between calculated and experimental values at 364.3 K may be attributed to this fact. In light of the above, the overall agreement is indeed satisfactory. 4. Discussion Several authors [22,23] have derived expressions relating activation volume to other diffusion parameters and bulk properties. In particular, Lawson [22] showed DS/DV* 5 a/b
(12)
Equation (12) was obtained on the assumption that vibration frequency of the matrix atoms is a function of volume only. Equation (12) was also derived by Lawson et al. [23] from the dynamical theory of diffusion formulated by Rice [24]. Since equation (10) reduces to equation (12) by putting T 5 0, it can be concluded that the equation derived here is more general. Equation (12) represents a special case of equation (10) when DV* is independent of temperature. Another relation for DV* was obtained by Keys [25] from empirical considerations: DV* 5 4bDH
(13)
Equations (12) and (13) are useful for displaying the functional relationships between the diffusion and bulk parameters. In the field of diffusion, the predictive ability of equation (10) is comparable to equations (1) and (2). Zener’s equation (3) relating DS to temperature dependence of the elastic constants aims to predict only its sign and the order of magnitude. More significantly, equations (12) and (13) are based on hypothetical premises. On the other hand, equation (10) is based on experimental observation and yields better quantitative agreement. It also represents a formal thermodynamic relationship between the diffusion parameters and the bulk properties of the matrix. The use of equation (6) implies that the activated complex is a system with a well defined energy and characteristic volume, and is amenable to the application of equilibrium thermodynamics. It has been argued that owing to its short lifetime, which is less than any vibrational period in the lattice, it is not justifiable to accord the status of a thermodynamic state to the activated complex in the process of diffusion [26]. However, Flynn [27] has pointed out that the saddle point configuration of the diffusing atom forms a part of the ensemble of the matrix in full contact with other similar systems. The individual points pass the saddle point depending upon the probability of achieving correct jump frequency in the ensemble; hence the reaction rate theory approach to diffusion is perfectly valid. The existence of an activated complex as a prerequisite for diffusion is the basis of the reaction rate theory of diffusion. Employing rate theory methods, Zener [7] has interpreted DG as the isothermal isobaric work done in the lattice by moving the diffusing atom from the trough of its equilibrium position to the height of the potential energy barrier. Therefore, the diffusion parameters should bear a relationship to the matrix properties as depicted in equation [10]. It is worth emphasising that DG is the sum total of energies associated with the diffusing atom, its associated vacancy and the nearest neighbouring atoms, which together constitute the activated complex. Satisfactory agreement between the experimental results and equation (10) provides support for the reaction rate approach to the process of diffusion. In the dynamical theory of diffusion in a crystal via the vacancy mechanism, the diffusion coefficient is defined in terms of normal vibrational modes of the crystals [27]. Flynn and his coworkers [28] have succeeded in establishing a link between the total spectrum of the normal modes of the lattice and the
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actual motion of an atom during an elementary diffusion jump. If the diffusion is dependent upon the normal vibrational modes of the lattice, the diffusion parameters such as activation volume should exhibit anisotropy in the case of anisotropic matrices. Experimentally measured activation volumes, in zinc as well as cadmium, are identical within the precision of measurements in both cases. Hence the measurement of activation volume in these metals does not support the dynamical theory of diffusion. Conclusion The expression for activation volume for self diffusion derived here is considered to be an improvement over the heretofore obtained relationships between the diffusion parameters and the bulk properties because it is based on a transparent assumption and does not involve any arbitrary constants. Further, it satisfactorily accounts for the experimental measurements in zinc, cadmium and sodium. Owing to its thermodynamic basis, the present equation should be applicable for other systems having different types of chemical bonding as well. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
F. Seitz, Acta Crystall., 3, 355 (1950). H. B. Huntington and F. Seitz, Phys. Rev., 76, 1728 (1949). A. D. Leclarie, in Diffusion in Body Centred Cubic Metal, Editors, F. R. Winslow and T. S. Lundy, ASM (1965) p. 11. N. H. Nachtriebs and G. H. Handler, Acta Metall., 2, 797 (1954). G. P. Tiwari and R. V. Patil, Trans. Jpn. Inst. Metals., 17, 476 (1976). R. V. Patil and G. P. Tiwari, Ind. J. Pure Appl. Phys., 42, 206 (1977). C. Zener, W. Shockley, et al., (Editors), Imperfections in Nearly Perfect Crystals, John Wiley, New York (1950) p. 210. A. D. Leclaire, Acta Metall., 1, 438 (1953). F. S. Buffington and M. Cohen, Acta Metall., 2, 660 (1954). N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals, p. 30, The Clarendon Press, Oxford (1946). S. N. Vaidya, J. Phy. Chem. Solids., 42, 621 (1981). L. C. Chhablidas and H. M. Gilder, Phys. Rev. B. 5, 2135 (1972). B. J. Buescher, H. M. Gilder, and N. Snea, Phys. Rev. B. 7, 2261(1513). J. N. Mundy, Phys. Rev., 3, 2431 (1971). N. L Peterson and S. J. Rothamnn, Phys. Rev. 163, 645 (1967). E. S. Wajda, G. A. Shrin and H. B. Huntington, Acta Metall., 3, 39, (1955). B. G. Childs, Rev. Mod. Phys., 25, 665 (1953). A. M. Magomedev and A. T. Ragakov, Teplofizika Vysofikh Temp. 17, 728, (1979). R. E. Hanneman and H. C. Gatos, J. Appl. Phys., 36, 1794 (1965). E. A. Brandes, ed., Smithhells Metals Reference Handbook, 6th edn., Butterworths, London (1983). P. W. Bridgman, Collected Experimental Papers of Bridgman, vol. 3, p. 43–1583, Harward University Press (1964). W. Lawson, J. Phys. Chem. Solids., 3, 250 (1957). A. W. Lawson, S. A. Rice, R. D. Corneliussen and N. H. Nachtrieb, J. Chem. Phys., 32, 447 (1960). S. A. Rice, Phys. Rev. 112, 804 (1958). R. W. Keys, J. Chem. Phys., 29,467 (1958). P. G. Shewmon, Diffusion in Solids, p. 60, McGraw-Hill, New York (1963). C. P. Flynn, Point Defects and Diffusion, p. 323, The Clarendon Press, Oxford (1972). G. Jaucoci, M-Toller, G. De Lorenzi, and C. P. Flynn, Mater. Sci. Forum., 1, 187 (1984).
Pergamon PII S1359-6462(98)00247-4
ACTIVATION VOLUME FOR SELF DIFFUSION IN METALS G.P. Tiwari Materials Science Division, Bhabha Atomic Research Centre, Mumbai, 400 085, India (Received May 27, 1997) (Accepted June 10, 1998) Introduction It is well established that the vacancy mechanism is the dominant mode for self diffusion in metals [1]. It involves the exchange of atomic sites between a vacancy and one of its neighbouring atoms. In the course of its migration from one atomic site to the next, the diffusing atom traverses through a saddle point. At the same time there is a relaxation among the nearest neighbours of the diffusing atoms. This ensemble of the diffusing atom at the saddle point with a neighbouring vacancy and its nearest neighbour in a relaxed state is called the activated complex of the diffusion. The volume of this ensemble, called activation volume for self diffusion (DV*), is the subject of the present article. DV* has two components - the volume associated with the formation of a vacancy and the difference between the atomic volume of the diffusing atom between its equilibrium and saddle point configuration. Thermodynamically, the activation volume is the pressure derivative of the free energy of activation for diffusion. There are two possible theoretical approaches to the subject of self diffusion in metals. One of these is the theoretical ab initio calculation of the energy for vacancy formation and the additional energy required for the diffusing atom to cross the potential energy barrier at the saddle point [2]. This procedure, though difficult and time consuming has the advantage of giving a detailed picture of the micromechanism of the diffusion process. However, this procedure is not technically convenient when either a quick estimate of the diffusion rate is required or a comparison of the diffusion rates between different systems (any two metals or the compositional variation in the same metal) is necessary. In such a situation, several empirical relationships between the diffusion parameters and bulk properties of the matrix [3– 6] have proven very useful. This has led to another important route which may be called the phenomenological approach to the study of self diffusion in metals and other kinds of solids. In the phenomenological approach, a diffusion parameter - most commonly an activation energy - is related to some bulk property such as melting point [3], latent heat of fusion [4], cohesive energy [6], etc. The changes in the diffusion rate from one system to the other are then ascribed to the changes in the bulk properties. The two widely used relationship are: DH 5 K1 Tm
(1)
DH 5 K2 Lm
(2)
where DH, Tm and Lm are respectively the activation energy for self diffusion, melting point and the latent heat of fusion. K1 and K2 are constants with appropriate units. Commonly accepted numerical values for K1 and K2 are 38 and 16.5 respectively. These relationships are based on the hypothesis that 931
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the region around the vacancy is like the molten state of the matrix, and the process of diffusion is akin to the shuffling of the atoms within the region. Hence the activation energy for diffusion should be related to the melting parameter. Another approach of this kind was pioneered by Zener [7] who postulated that energy associated with the saddle point configuration originates in the straining of the surrounding lattice. Proceeding on this basis, he derived the following relation between the activation entropy for diffusion (DS) and the temperature dependence of the elastic constant of the matrix: DS 5 l(dE/Eo)/d(T/Tm)
(3)
where l is designated as Zener’s constant which varies between 0.01 to 1 for metals, and E and E0 are the values of appropriate elastic constants at temperature T and 0K respectively. Following Zener, several other treatments relating self diffusion parameters to elastic constants have been published [8,9]. Being empirical in nature, all these treatments involve a constant such as l. The variation of such a constant within a narrow range for a group of metals or solids having identical physical or chemical properties is taken as a proof of their validity. In the present paper, a formal thermodynamic equation for the activation volume for self diffusion is derived on the assumption that the free energy for self diffusion in metals is a function of the specific volume of the matrix. 2.0 Derivation of the Basic Equation Mott and Gurney [10] estimated the temperature dependence of the activation energy for self diffusion in ionic crystals assuming that this parameter varies linearly with the specific volume of the matrix. However, it is more appropriate to consider Gibbs’s free energy for self diffusion as a function of the specific volume. In this context, it has been shown by Vaidya [11] that the logarithm of the self diffusion coefficients (D) for several metals varies linearly with V/V1 where V and V1 are the specific volumes of the solid at the ambient and the liquid at the melting point. In some cases, the plot shows a bend towards the volume axis but the variation of log D verses V/V1 follows a regular behaviour over the entire region. In the solid region, which is our concern here, the variation of log D with V/V1 is practically linear. More significantly, the values of D for the liquid at the melting point obtained from the extrapolation of the data points from the solid are nearly identical with the experimentally measured self diffusion coefficient for the liquid. It may, therefore, be concluded that volume is the most important single parameter affecting the self diffusion coefficient of the metallic matrices, and whatever changes in the free energy of activation (and consequently in the diffusion coefficient) occur as a result of variations in the temperature (T) and pressure (P) are solely due to their effect on the specific volume of the matrix. In solids, the equation for self diffusion coefficient is expressed [7] as D 5 f a2 n exp(2DG/RT)
(4)
or 5 f a2 n exp(DS/R) exp(2DH/RT)
(5)
where f 5 correlation factor, a 5 lattice parameter, n 5 Debye frequency, R 5 gas constant and DG is the free energy of activation for self diffusion. As discussed above, DG can be expressed as a function of the specific volume of the matrix in a Taylor series: DG 5 DG0 1
dDG d2 DG DV2 DV 1 1. . . . . . dV dV2 2
(6)
Writing V 5 V0 (11 aT) where V0 is the specific volume of the matrix at 0K and a is the volume expansion coefficient of the matrix, DV is given by
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TABLE 1 Comparison of Experimental and Calculated Values of the Activation Volume for Self Diffusion in Zinc, Cadmium and Sodium Activation volume (cm3/mole) Serial No.
Matrix
Temperature [K]
Experimental
Calculated
Difference (%)
1 2 3 4 5 6 7 8 9
Zn
573 623 673 524 549 574 592 287.8 364.3
4.28 3.97 3.72 6.90 7.14 7.65 7.49 11.46 12.40
3.20 3.22 3.44 4.84 5.08 5.49 5.87 10.23 17.25
25.2 18.9 7.5 29.8 28.8 28.2 21.6 10.7 39.1
Cd
Na
DV 5 aV0 T
(7)
By definition, activation volume, DV* can be defined as: DV* 5 (DG)/P 5 2(DG)/V.b.V
(8)
where b is the compressibility. Neglecting higher order terms and substituting from equations (7) and (8), in equation (6) one gets DG 5 DH 2
DV* DV b V
(9)
where DH has been substituted in place of DG0. Comparison of equation (9) with the standard thermodynamic relation DG 5 DH 2 T DS, yields DV* 5 DS (a/b)21(1 1 aT)
(10)
3.0 Comparison with Experimental Results Equation (10) has been applied to the data of Chhabildas and Gilder on zinc [12], of Buescher et al. on cadmium [13] and those of Mundy on sodium [14]. These are possibly the only available data of sufficient accuracy which give activation volume as a function of temperature and hence can be used to test the validity of equation (10). 3.1 Zinc and Cadmium In this case, the DS values have been calculated using equation (5) from the self diffusion data of zinc [15] and cadmium [16]. It is important to emphasise here that thermodynamically, DV* can vary with temperature irrespective of the behaviour of DS with respect to the temperature, although the temperature variations of DS and DH are themselves interrelated. Values of a and b at the desired temperatures have been taken from references [17] and [18] respectively. For the purpose of comparison, the values of DV* and DS have been taken along c-axis. The experimental and calculated values are compared in Table 1.
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As seen from Table 1, the difference between the experimental and the calculated values varies from 7.5 to 25.2% for zinc and from 21.6 to 29.8% for cadmium. Considering the inherent variations in the various parameters involved, it is believed that fairly good agreement exists between calculated and the experimental values. It is also believed that the main reason for the difference between the calculated and the experimental values can be attributed to the use of compressibility data for polycrystalline instead of the single crystal materials since the activation volumes have been measured on single crystals. This observation is supported by the fact that in either case there is a tendency towards better agreement between the experimental and calculated values with the rise in temperature. The presence of grain boundaries is expected to decrease with the values of compressibility and their influence will decrease with the increase in the temperature. This could be the reason for better agreement at higher temperatures. It is also possible to evaluate the temperature dependence of the activation volume assuming a/b to be a constant. This is a valid assumption since Hanneman and Gatos [19] have earlier shown that the ratio a/b is a constant for most of the bcc and fcc metallic lattices. The same is taken to be true for zinc and cadmium here. It is further assumed that the a is also independent of temperature. Therefore, it is now possible to differentiate equation (10) to obtain the values of (DV*/T) as below: DV* 5 DS(a/b)21.a
(11)
The room temperature value of a/b has been taken for the computation since this parameter is taken to be a constant. The value of DS for self diffusion in zinc as well as cadmium is also independent of temperature and, as before, values for the c-axis have been taken. The value of ( DV*/T) for zinc and cadmium so obtained from equation (11) are 5.86 3 1024 and 7.46 3 1024 cm3/mole.K respectively. The corresponding experimental values are 6.43 3 1023 and 1.1 3 1023 cm3/mole.K [12,13]. These values are approximately 11 and 1.5 times the calculated values for zinc and cadmium respectively. This variation could be due to the assumption regarding the constancy of the ratio a/b and/or to the use of bulk compressibility in place of the values for c-direction. Again, the agreement between the calculated and experimental values can be termed as satisfactory. 3.2 Sodium The data on activation volume for self diffusion in sodium are available at 287.8 K and 364.3 K only. DS values at both temperatures have been obtained from Mundy’s work [14]. Mundy has expressed the diffusion coefficient as the sum of two straight lines representing low and high temperature regions. The low temperature diffusion data have been identified with mono vacancy diffusion, whereas the high temperature data have not been definitely identified with any mechanism, although the preference for divacancy mechanism has been indicated .The percentage contribution of low as well as high temperature processes at 287.8 K and 364.3 K has also been given and the net value of DS used here is a weighted average of the operating diffusion mechanisms. Mundy has also expressed the same diffusion data on self diffusion in sodium as a function of three straight lines. However, this has not been used here because the last procedure is neither an improvement on the statistical fit of the data nor it can be given any reasonable interpretation with respect to the operating modes of diffusion. For the thermal expansion coefficient, a value of 71 3 1026/K between 273–388 K from Smithells Metals Reference Handbook [20] has been taken. Compressibility data have been taken from the work of Bridgman [21]. Bridgman measured the compressibility at 303 K and 348 K by the piston and cylinder technique. Compressibility values at these temperatures were obtained graphically from Bridgman’s data. Actual compressibility values at the desired temperatures were again graphically evaluated through extrapolation. The results are given Table 1. At the lower temperature the difference between the calculated and
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experimental values of DV* is nearly 10%, which is considered very good. At higher temperature a difference of 39% is obtained. At this temperature sodium is very soft and Bridgman has indicated that the softness of the material has possibly affected the measured compressibility values. A part of the discrepancy between calculated and experimental values at 364.3 K may be attributed to this fact. In light of the above, the overall agreement is indeed satisfactory. 4. Discussion Several authors [22,23] have derived expressions relating activation volume to other diffusion parameters and bulk properties. In particular, Lawson [22] showed DS/DV* 5 a/b
(12)
Equation (12) was obtained on the assumption that vibration frequency of the matrix atoms is a function of volume only. Equation (12) was also derived by Lawson et al. [23] from the dynamical theory of diffusion formulated by Rice [24]. Since equation (10) reduces to equation (12) by putting T 5 0, it can be concluded that the equation derived here is more general. Equation (12) represents a special case of equation (10) when DV* is independent of temperature. Another relation for DV* was obtained by Keys [25] from empirical considerations: DV* 5 4bDH
(13)
Equations (12) and (13) are useful for displaying the functional relationships between the diffusion and bulk parameters. In the field of diffusion, the predictive ability of equation (10) is comparable to equations (1) and (2). Zener’s equation (3) relating DS to temperature dependence of the elastic constants aims to predict only its sign and the order of magnitude. More significantly, equations (12) and (13) are based on hypothetical premises. On the other hand, equation (10) is based on experimental observation and yields better quantitative agreement. It also represents a formal thermodynamic relationship between the diffusion parameters and the bulk properties of the matrix. The use of equation (6) implies that the activated complex is a system with a well defined energy and characteristic volume, and is amenable to the application of equilibrium thermodynamics. It has been argued that owing to its short lifetime, which is less than any vibrational period in the lattice, it is not justifiable to accord the status of a thermodynamic state to the activated complex in the process of diffusion [26]. However, Flynn [27] has pointed out that the saddle point configuration of the diffusing atom forms a part of the ensemble of the matrix in full contact with other similar systems. The individual points pass the saddle point depending upon the probability of achieving correct jump frequency in the ensemble; hence the reaction rate theory approach to diffusion is perfectly valid. The existence of an activated complex as a prerequisite for diffusion is the basis of the reaction rate theory of diffusion. Employing rate theory methods, Zener [7] has interpreted DG as the isothermal isobaric work done in the lattice by moving the diffusing atom from the trough of its equilibrium position to the height of the potential energy barrier. Therefore, the diffusion parameters should bear a relationship to the matrix properties as depicted in equation [10]. It is worth emphasising that DG is the sum total of energies associated with the diffusing atom, its associated vacancy and the nearest neighbouring atoms, which together constitute the activated complex. Satisfactory agreement between the experimental results and equation (10) provides support for the reaction rate approach to the process of diffusion. In the dynamical theory of diffusion in a crystal via the vacancy mechanism, the diffusion coefficient is defined in terms of normal vibrational modes of the crystals [27]. Flynn and his coworkers [28] have succeeded in establishing a link between the total spectrum of the normal modes of the lattice and the
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ACTIVATION VOLUME
Vol. 39, No. 7
actual motion of an atom during an elementary diffusion jump. If the diffusion is dependent upon the normal vibrational modes of the lattice, the diffusion parameters such as activation volume should exhibit anisotropy in the case of anisotropic matrices. Experimentally measured activation volumes, in zinc as well as cadmium, are identical within the precision of measurements in both cases. Hence the measurement of activation volume in these metals does not support the dynamical theory of diffusion. Conclusion The expression for activation volume for self diffusion derived here is considered to be an improvement over the heretofore obtained relationships between the diffusion parameters and the bulk properties because it is based on a transparent assumption and does not involve any arbitrary constants. Further, it satisfactorily accounts for the experimental measurements in zinc, cadmium and sodium. Owing to its thermodynamic basis, the present equation should be applicable for other systems having different types of chemical bonding as well. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
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