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Volume effects in the thermodynamics of binary substitutional solid solutions at high temperatures
I Phyr Chem. Solids Vol. 43. No. 5. pp. 412420. Pnnted in Great Britain.
1982
WZZ-3697/82/050217-1~3.00/0 Pergamon Press Ltd.
VOLUME EFFECTS IN THE THERMODYNAMICS OF BINARY SUBSTITUTIONAL SOLID SOLUTIONS AT HIGH TEMPERATURES REXB. MCLELLAN Department of Mechanical Engineering, and Materials Science, Rice University, Houston TX 77001,U.S.A. (Received 27 July 1981;accepted in revised form 18 September 1981)
Ahstraet-Recent measurements of the variation with temperature and composition of the elastic properties of Fe-based solid solution containing the substitutional solutes Mn, Ni and Cr have enabled calculations to be made of the effect of the composition-dependent variation of the specific volume of the lattice upon the partial thermodynamic functions of the solute species. The calculations show that, in contrast to the case of binary interstitial solid solutions, the effects are of the same order, or less, as the normal uncertainty in the experimental determination of the partial thermodynamic functions of solid solutionsat high temperatures (1273K).
1. MTROOlJCTlON
A recent report[l] has dealt with the problem of volume effects in considering the thermodynamic functions of interstitial atoms in binary Fe-based solid solutions containing a substitutional solute (ternary austenites). Elastic measurements enabled calculations to be made which showed that, provided the concentration of the interstitial species (carbon or nitrogen) was held at essentially infinite dilution, the dilation of the binary solvent lattice concomitant to changing the composition of the substitutional solute has only a minor effect upon the partial thermodynamic functions of the interstitial species. This finding is in stark contrast to the effect of volume changes when the concentration of the interstitial species itself is increased[2-51. In this case there are large differences between the partial enthalpies and entropies of the interstitial species in the “constant pressure” and “rigid lattice” solutions. The results obtained previously in respect to ternary V-U-i systems (V = solvent, U = substitutional solute and i = interstitial solute)[l] naturally raise the question as to how the thermodynamic functions of the U-species are affected by changes in specific volume in a binary V-U system. Since the specific lattice dilations (positive or negative) for substitutional species are usually small compared to interstitial atoms (except H), it is to be expected that the comparison between “constant volume” and “rigid lattice” thermodynamic functions would not involve such large difference terms and interpretive difficulties. Since exhaustive elastic measurements have been made for Fe-O, Fe-Ni, and Fe-Mn solid solutions in a wide temperature range[l], it was decided to carry out the calculations attempting to answer the question posed above in the case of Fe-based solutions containing the substitutional solutes Cr, Ni and Mn. (J)PCS Vol. 43, No. 5-A
2. VOLUME EFFECTCALCULATIONS
The relations necessary for the present discussion may be written in the general form,
x60=
($g),, - (2),.,
(1)
where Q is an extensive thermodynamic function, x. is the mole fraction of U, 0” is the partial molar quantity, and V,,, the molar volume V,,, = V&n, t n,) and 0” is defined by
0” = (2) ”
“m.T
(2)
and nucu) denotes the number of moles of species U or
V. Generalized equations for the X& quantities have been given by both Lupis[4,5], and the present author[l-3,61. These treatments have concentrated on the treatment of interstitial solutes. Let us calculate the X& for the case of V-U binary systems and employ certain reasonable simplifications. The quantity X& is easy to derive by following the arguments pesented by Wagner[7] for the V-i-system. By expanding pU = bU(P, T, x,) using Euler’s theorem and using the identities,
and
(5) 417
REX B. MCLELLAN
418
where Vuruc., represents the partial molar volume and K is the isothermal compressibility, it is easy to show [7] that,
AV = V. - V, gives finally
(6) T aB (12) - Ti 0);iT P,X”. The entropy function Xks can be calculated in a similar way. The Euler expansion of
Note that the chemical potential pu can be written
~~=(&=G%+ S, = S,( V,Tx,) Now consider X&. The Euler expansion of J?,,= J?.(T, V,, x,) and the identity[Sl,
gives
(a, =~%)T.x” (%T)p,T +i2,“..;
(7)
(13) where a is the thermal expansivity, yields the relation, Because of the identity [5], x~~=(~(~)),rt(~),.“(v~-Y,). (14)
(8)
(15)
The factor (&,/aV,),,, can be evaluated using stardard thermodynamic identities [8] giving the result Thus,
-T(w),,,,).
(9)
If V. and o are independent of xu, the first term in eqn (8) can be approximated to VuTo(aB/axu)P.T, where B is the bulk modulus (K-l). Using this approximation and
(16) As before (eqn (4)), (aV,/ax,),., = AV, and the identity (14) can be used in a straightforward manner to calculate (aSlaV,),,, in the form, aS
(av,> -2
-
T.xu
and combining eqns (8) and (9) gives _ %V$&Td,) m
1 KV,
4 (17) aT > p,x,+m (av,
where 4 is given by eqn (II). Thus combining (16) and (17) gives, (10) p,xu
where 4 = (a(V,oB)/aP)T.x, This function may be obtained in terms of the known variables by employing well-known classical thermodynamic identities[8]. The result is,
V,(l -x,) aK K ax, (>I
P,T
(18) 4=
_a!
$i
+ ”
Vdl-L) K
aK
k>,.p] -%(%)p,,. (11)
where the approximation au/K) aP
( )
=O
These somewhat lengthy expressions for X;IE and X& may be simplified by using the approximation (l/v.) (aV,/aT), = a. This gives the final forms for X& and X& given below:
T.X,
has been employed. This approximation has been justified previously 13) Inserting do from eqn (11) into eqn (10) and writing
(19)
V&me effects
in the t~e~m~ynm~ics
AV~~~~~~)~~
X&P= X&E- TX&. Input data taken from elastic and X-ray studies wifibe used to calculate the inte~ated forms of the expressions (61,(191 andCW.
The pulse-echo method in the thin-tine limit was recently used by Yoshiharaand McLellan[l] in order to measure the elastic properties of Fe-based alloys containingNi f&-range B-&S),Mn (&range O-0.13),and Cr f&range @-&IS).The solid solutions were prepared by arc me&g MARSgrade materiak. A full descriptionof tlxe experimental detaiis has been given elsewherefl]. The lon~tudinai sound velocities were measured in the tempe~ture range 294-1700K, but, due to great attenuation, the shear waves could only be measured at low temperat~es. However, if it is assumed that Poisson’s ratio does not vary much with temperature, the buik m&uLs may be obtained from the data in the entire span of temperature and composition. The temperature of 5273K was selected for the present volume effect copulations. At this ~em~rature all the three sets of bulk modulusdata can be reduced to the linear form,
The constant B. refers of course, to pure iron at this temperature, Now p = ~~~~~T~~~~ is virtually independent of x, for the z&-rangeup to 0.1 and at 1237K. The values of y = -(~Y&,,P,T and p are given in Table 1 taken from the data of RefJl]. 3b X-Ray data The data compilationof Pearson[9] was used to caic&ate pS and VWfor fee iron-basedsolid solutions.The VU-vaiuesare obtained from the lattice constant {al data and the equation,
Tabie t. ~Oi*dyn/cm*
Ni
-QOZ1 i 0.014 f wO5
Mn Cr
p in dyu~cm*~~ 0.634x tt19 1.00 xi@ LOS x t$
6.44 7.93 7.32
419
4 ~~c~~~a The X~-~u~titjes can be integrated to give the concen~ation-v~i~~on of tire quantities QU--f& in the form
where tile superscript m denotes finite dilu~on of the ~-~om~neot. Usingthe i~n~ti~ (3, fl4f, and the fact that Bf - I? = 0 and the abbreviation
where D = AVlqeGi,, The & integralis evaluated in a similarmanuerusing standardintegraltables[to]. The result is,
+ ii,Tg fn (I+ a~)
524)
and ISSis given by,
It VW2
solid ~ludons
where a0 is the value of n when j = 0 (extrapoiated if necessary). The values of VUand AV cakuiated are included in Table 1. This tab&, together with &[I], (~)~,~” contains ali the ~nfo~at~on required to perform the CL@ calculations.
equations (61, (19)and (201show that, as expected,
U y in
of binary substitutions
can be seen from eqns (233-(25)that
AV$$
1GF= &zE- TI,.
- 0.775 +o&ls 0
If we restrict our considerationsto xu = 0.t (IO At. % of U) then the ~prox~ma~onsI> d2 and in (1+ ox,J * ax” can be used. Introducingthe numericaldata given in
REX B.
420
Section 3 and using the approximations given above yields the expressions given below for 0” - 0. when combined with the values of @- 0: taken from eqns (22). For the Fe-Ni system: G, - 1’, = 56.14 x,, kJ/mol. fi,, = 8, = 45.91- 5.967 x,
kJ/mol.
s, - 5, = 36.068- 39.41 X, J/deg/mol For the Fe-Mn system: G,, - pU = -59.26 x,
kJ/mol.
fi,, - i?-, = 56.57 + 24.64 xU kJ/mol. S, - 3. = 44.4 t 65.9 kJ/deg/mol. Since AV = 0 for Fe-Cr, volume effects in the sense of the QU- 6. quantities are essentially zero. 5. CONCLUSIONS The three binary systems chosen represent “typical”
systems in the sense that in one case (Fe-Mn) the U-element causes considerable positive lattice dilation (AV = 0.615) and in the other case (Fe-Ni) there is roughly the same degree of negative dilation (AV = -0.776). The Fe-Cr system represents the case where changes in the partial thermodynamic fuctions of the U-species directly attributable to change in the specific volume of the lattice may be ignored. Let us consider the Fe-Ni system as being typical for most systems, i.e. \AV]-0.5-1.0. Since it is normal[ll] to evaluate the strength of the V-U-interactions, e.g. through the Wagner coefficients[ I I], at the V-rich end of the system, let us restrict x, to a maximum value of xu = 0.05 (5 At. % U). The difference between G. and FU at x,, = 0.05 is 2.81 kJ/mol (i.e. 0.67 kcol/mol or 0.03eV.). The difference between I?, and I?,, not counting the constant (fiz- 8:) is 0.298 kJ/mol (i.e. 0.071 kcal/mol or 0.003 eV). The corresponding entropy difference is -0.237 k. Now the quantities (a:- 6:) are to be regarded as “corrections” to &values calculated from constantvolume models in the limit of infinite dilution of the
MCLELLAN
U-component and thus they may be simply taken as a constant addition to I?, and s,. The numbers show that up to 8. = 0.05, the composition-dependent part of 0” arising from the lattice dilation is of the same order or less than that involved in the experimental measurement of 0” in high-temperature systems Thus in many such binary systems (especially those approaching the state of affairs for Fe-Cr, i.e. IAVl = O0.3, the composition-dependent “correction terms” may be safely ignored. Even for systems like Fe-Mn or Fe-Ni the terms are small and are essentially negligible. In the true high-temperature limit of course Iss makes no contribution to the configurational entropy and is merely a term contributing to the partial excess entropy. This simple state of affairs does not hold however for binary interstitial system (V-i-systems), where theconfigurational nature of the constant-pressure and rigid lattice solutions may by greatly different so that considerable interpretive difficulties arise in considering the looquantitiestC61. These difficulties do nor arise due to inadequacies in the classical thermodynamic treatment, but due to the lack of well-founded statistical mechanical models for solid solutions with non-rigid lattices.
Acknowledgement-The author is grateful for the financial support provided by the National Science Foundation under the Metallurgy Program (Grant No. D.M.R. 78-01306)and the Robert A. Welch Foundation. REFERENCFS I. Yoshihara M. and McLellan R. B., J. Phys. Chem. Solids 42(9), 767 (1981). 2. Farraro R. J. and McLellan R. B.. J. Phys. Chem. Solids 39,
781 (1978). 3. McLellan R. B., In Treatiseon MaterialsScience(Edited by H. Herman). Academic Press, New York (1975). 4. Lupis C. H. P., Acta Metall 25, 751 (1977). 5. Lupis C. H. P., Acta Metal/ 26, 211 (1978). 6. McLellan R. B., Acta Metall 27, 793 (1979). 7. Wagner C., Acta Metall 19, 843 (1971). 8. Guggenheim E. A., Thermodynamics. North-Holland, Amsterdam (1959). 9. Pearson W. B., A Handbook of Lattice Spacings and Structures of Metals and Alloys. Pergamon Press, New York (1958). IO. Gradshteyn I. S. and Ryzhik I. M., Tablesof Integrals, Series and Products.Academic Press, New York (1966). Il. Parris D. C. and McLellan R. B., Mats. Sci. Engng 9, 181 (1971).
1982
WZZ-3697/82/050217-1~3.00/0 Pergamon Press Ltd.
VOLUME EFFECTS IN THE THERMODYNAMICS OF BINARY SUBSTITUTIONAL SOLID SOLUTIONS AT HIGH TEMPERATURES REXB. MCLELLAN Department of Mechanical Engineering, and Materials Science, Rice University, Houston TX 77001,U.S.A. (Received 27 July 1981;accepted in revised form 18 September 1981)
Ahstraet-Recent measurements of the variation with temperature and composition of the elastic properties of Fe-based solid solution containing the substitutional solutes Mn, Ni and Cr have enabled calculations to be made of the effect of the composition-dependent variation of the specific volume of the lattice upon the partial thermodynamic functions of the solute species. The calculations show that, in contrast to the case of binary interstitial solid solutions, the effects are of the same order, or less, as the normal uncertainty in the experimental determination of the partial thermodynamic functions of solid solutionsat high temperatures (1273K).
1. MTROOlJCTlON
A recent report[l] has dealt with the problem of volume effects in considering the thermodynamic functions of interstitial atoms in binary Fe-based solid solutions containing a substitutional solute (ternary austenites). Elastic measurements enabled calculations to be made which showed that, provided the concentration of the interstitial species (carbon or nitrogen) was held at essentially infinite dilution, the dilation of the binary solvent lattice concomitant to changing the composition of the substitutional solute has only a minor effect upon the partial thermodynamic functions of the interstitial species. This finding is in stark contrast to the effect of volume changes when the concentration of the interstitial species itself is increased[2-51. In this case there are large differences between the partial enthalpies and entropies of the interstitial species in the “constant pressure” and “rigid lattice” solutions. The results obtained previously in respect to ternary V-U-i systems (V = solvent, U = substitutional solute and i = interstitial solute)[l] naturally raise the question as to how the thermodynamic functions of the U-species are affected by changes in specific volume in a binary V-U system. Since the specific lattice dilations (positive or negative) for substitutional species are usually small compared to interstitial atoms (except H), it is to be expected that the comparison between “constant volume” and “rigid lattice” thermodynamic functions would not involve such large difference terms and interpretive difficulties. Since exhaustive elastic measurements have been made for Fe-O, Fe-Ni, and Fe-Mn solid solutions in a wide temperature range[l], it was decided to carry out the calculations attempting to answer the question posed above in the case of Fe-based solutions containing the substitutional solutes Cr, Ni and Mn. (J)PCS Vol. 43, No. 5-A
2. VOLUME EFFECTCALCULATIONS
The relations necessary for the present discussion may be written in the general form,
x60=
($g),, - (2),.,
(1)
where Q is an extensive thermodynamic function, x. is the mole fraction of U, 0” is the partial molar quantity, and V,,, the molar volume V,,, = V&n, t n,) and 0” is defined by
0” = (2) ”
“m.T
(2)
and nucu) denotes the number of moles of species U or
V. Generalized equations for the X& quantities have been given by both Lupis[4,5], and the present author[l-3,61. These treatments have concentrated on the treatment of interstitial solutes. Let us calculate the X& for the case of V-U binary systems and employ certain reasonable simplifications. The quantity X& is easy to derive by following the arguments pesented by Wagner[7] for the V-i-system. By expanding pU = bU(P, T, x,) using Euler’s theorem and using the identities,
and
(5) 417
REX B. MCLELLAN
418
where Vuruc., represents the partial molar volume and K is the isothermal compressibility, it is easy to show [7] that,
AV = V. - V, gives finally
(6) T aB (12) - Ti 0);iT P,X”. The entropy function Xks can be calculated in a similar way. The Euler expansion of
Note that the chemical potential pu can be written
~~=(&=G%+ S, = S,( V,Tx,) Now consider X&. The Euler expansion of J?,,= J?.(T, V,, x,) and the identity[Sl,
gives
(a, =~%)T.x” (%T)p,T +i2,“..;
(7)
(13) where a is the thermal expansivity, yields the relation, Because of the identity [5], x~~=(~(~)),rt(~),.“(v~-Y,). (14)
(8)
(15)
The factor (&,/aV,),,, can be evaluated using stardard thermodynamic identities [8] giving the result Thus,
-T(w),,,,).
(9)
If V. and o are independent of xu, the first term in eqn (8) can be approximated to VuTo(aB/axu)P.T, where B is the bulk modulus (K-l). Using this approximation and
(16) As before (eqn (4)), (aV,/ax,),., = AV, and the identity (14) can be used in a straightforward manner to calculate (aSlaV,),,, in the form, aS
(av,> -2
-
T.xu
and combining eqns (8) and (9) gives _ %V$&Td,) m
1 KV,
4 (17) aT > p,x,+m (av,
where 4 is given by eqn (II). Thus combining (16) and (17) gives, (10) p,xu
where 4 = (a(V,oB)/aP)T.x, This function may be obtained in terms of the known variables by employing well-known classical thermodynamic identities[8]. The result is,
V,(l -x,) aK K ax, (>I
P,T
(18) 4=
_a!
$i
+ ”
Vdl-L) K
aK
k>,.p] -%(%)p,,. (11)
where the approximation au/K) aP
( )
=O
These somewhat lengthy expressions for X;IE and X& may be simplified by using the approximation (l/v.) (aV,/aT), = a. This gives the final forms for X& and X& given below:
T.X,
has been employed. This approximation has been justified previously 13) Inserting do from eqn (11) into eqn (10) and writing
(19)
V&me effects
in the t~e~m~ynm~ics
AV~~~~~~)~~
X&P= X&E- TX&. Input data taken from elastic and X-ray studies wifibe used to calculate the inte~ated forms of the expressions (61,(191 andCW.
The pulse-echo method in the thin-tine limit was recently used by Yoshiharaand McLellan[l] in order to measure the elastic properties of Fe-based alloys containingNi f&-range B-&S),Mn (&range O-0.13),and Cr f&range @-&IS).The solid solutions were prepared by arc me&g MARSgrade materiak. A full descriptionof tlxe experimental detaiis has been given elsewherefl]. The lon~tudinai sound velocities were measured in the tempe~ture range 294-1700K, but, due to great attenuation, the shear waves could only be measured at low temperat~es. However, if it is assumed that Poisson’s ratio does not vary much with temperature, the buik m&uLs may be obtained from the data in the entire span of temperature and composition. The temperature of 5273K was selected for the present volume effect copulations. At this ~em~rature all the three sets of bulk modulusdata can be reduced to the linear form,
The constant B. refers of course, to pure iron at this temperature, Now p = ~~~~~T~~~~ is virtually independent of x, for the z&-rangeup to 0.1 and at 1237K. The values of y = -(~Y&,,P,T and p are given in Table 1 taken from the data of RefJl]. 3b X-Ray data The data compilationof Pearson[9] was used to caic&ate pS and VWfor fee iron-basedsolid solutions.The VU-vaiuesare obtained from the lattice constant {al data and the equation,
Tabie t. ~Oi*dyn/cm*
Ni
-QOZ1 i 0.014 f wO5
Mn Cr
p in dyu~cm*~~ 0.634x tt19 1.00 xi@ LOS x t$
6.44 7.93 7.32
419
4 ~~c~~~a The X~-~u~titjes can be integrated to give the concen~ation-v~i~~on of tire quantities QU--f& in the form
where tile superscript m denotes finite dilu~on of the ~-~om~neot. Usingthe i~n~ti~ (3, fl4f, and the fact that Bf - I? = 0 and the abbreviation
where D = AVlqeGi,, The & integralis evaluated in a similarmanuerusing standardintegraltables[to]. The result is,
+ ii,Tg fn (I+ a~)
524)
and ISSis given by,
It VW2
solid ~ludons
where a0 is the value of n when j = 0 (extrapoiated if necessary). The values of VUand AV cakuiated are included in Table 1. This tab&, together with &[I], (~)~,~” contains ali the ~nfo~at~on required to perform the CL@ calculations.
equations (61, (19)and (201show that, as expected,
U y in
of binary substitutions
can be seen from eqns (233-(25)that
AV$$
1GF= &zE- TI,.
- 0.775 +o&ls 0
If we restrict our considerationsto xu = 0.t (IO At. % of U) then the ~prox~ma~onsI> d2 and in (1+ ox,J * ax” can be used. Introducingthe numericaldata given in
REX B.
420
Section 3 and using the approximations given above yields the expressions given below for 0” - 0. when combined with the values of @- 0: taken from eqns (22). For the Fe-Ni system: G, - 1’, = 56.14 x,, kJ/mol. fi,, = 8, = 45.91- 5.967 x,
kJ/mol.
s, - 5, = 36.068- 39.41 X, J/deg/mol For the Fe-Mn system: G,, - pU = -59.26 x,
kJ/mol.
fi,, - i?-, = 56.57 + 24.64 xU kJ/mol. S, - 3. = 44.4 t 65.9 kJ/deg/mol. Since AV = 0 for Fe-Cr, volume effects in the sense of the QU- 6. quantities are essentially zero. 5. CONCLUSIONS The three binary systems chosen represent “typical”
systems in the sense that in one case (Fe-Mn) the U-element causes considerable positive lattice dilation (AV = 0.615) and in the other case (Fe-Ni) there is roughly the same degree of negative dilation (AV = -0.776). The Fe-Cr system represents the case where changes in the partial thermodynamic fuctions of the U-species directly attributable to change in the specific volume of the lattice may be ignored. Let us consider the Fe-Ni system as being typical for most systems, i.e. \AV]-0.5-1.0. Since it is normal[ll] to evaluate the strength of the V-U-interactions, e.g. through the Wagner coefficients[ I I], at the V-rich end of the system, let us restrict x, to a maximum value of xu = 0.05 (5 At. % U). The difference between G. and FU at x,, = 0.05 is 2.81 kJ/mol (i.e. 0.67 kcol/mol or 0.03eV.). The difference between I?, and I?,, not counting the constant (fiz- 8:) is 0.298 kJ/mol (i.e. 0.071 kcal/mol or 0.003 eV). The corresponding entropy difference is -0.237 k. Now the quantities (a:- 6:) are to be regarded as “corrections” to &values calculated from constantvolume models in the limit of infinite dilution of the
MCLELLAN
U-component and thus they may be simply taken as a constant addition to I?, and s,. The numbers show that up to 8. = 0.05, the composition-dependent part of 0” arising from the lattice dilation is of the same order or less than that involved in the experimental measurement of 0” in high-temperature systems Thus in many such binary systems (especially those approaching the state of affairs for Fe-Cr, i.e. IAVl = O0.3, the composition-dependent “correction terms” may be safely ignored. Even for systems like Fe-Mn or Fe-Ni the terms are small and are essentially negligible. In the true high-temperature limit of course Iss makes no contribution to the configurational entropy and is merely a term contributing to the partial excess entropy. This simple state of affairs does not hold however for binary interstitial system (V-i-systems), where theconfigurational nature of the constant-pressure and rigid lattice solutions may by greatly different so that considerable interpretive difficulties arise in considering the looquantitiestC61. These difficulties do nor arise due to inadequacies in the classical thermodynamic treatment, but due to the lack of well-founded statistical mechanical models for solid solutions with non-rigid lattices.
Acknowledgement-The author is grateful for the financial support provided by the National Science Foundation under the Metallurgy Program (Grant No. D.M.R. 78-01306)and the Robert A. Welch Foundation. REFERENCFS I. Yoshihara M. and McLellan R. B., J. Phys. Chem. Solids 42(9), 767 (1981). 2. Farraro R. J. and McLellan R. B.. J. Phys. Chem. Solids 39,
781 (1978). 3. McLellan R. B., In Treatiseon MaterialsScience(Edited by H. Herman). Academic Press, New York (1975). 4. Lupis C. H. P., Acta Metall 25, 751 (1977). 5. Lupis C. H. P., Acta Metal/ 26, 211 (1978). 6. McLellan R. B., Acta Metall 27, 793 (1979). 7. Wagner C., Acta Metall 19, 843 (1971). 8. Guggenheim E. A., Thermodynamics. North-Holland, Amsterdam (1959). 9. Pearson W. B., A Handbook of Lattice Spacings and Structures of Metals and Alloys. Pergamon Press, New York (1958). IO. Gradshteyn I. S. and Ryzhik I. M., Tablesof Integrals, Series and Products.Academic Press, New York (1966). Il. Parris D. C. and McLellan R. B., Mats. Sci. Engng 9, 181 (1971).