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On the interrelationships between Gruneisen's parameter, shock and isothermal equation of state
Solid State Communications, Vol. 69, No. 6, pp. 667-670, 1989. Printed in Great Britain.
0038-1098/89 $3.00 + .00 Pergamon Press plc
ON THE INTERRELATIONSHIPS BETWEEN GRUNEISEN'S PARAMETER, SHOCK AND ISOTHERMAL EQUATION OF STATE H. Dandache Faculty of Sciences, Lebaneese University, Beirut, Lebanon
(Received 9 August 1988 by S. Lundqvist ) A thermodynamical method is developed which give a relation between the shock wave velocity Us and the particle velocity Up. Mie-Gruneisen equation of state is considered with coefficient 7 given by DugdaleMacDonald relation. The slope of the curve in the Us-Upplane is used to obtain an equation of state at zero temperature. The results are in good agreement with Al'tshuler's [1] experimental data for A1, Cu and Pb.
I. I N T R O D U C T I O N WE ASSUME the Mie-Gruneisen equation of state namely that the pressure P in a solid at high temperature T > 0, where 0 is the Debye temperature, is related to the pressure P,. at zero temperature by the equation [6]:
~(V)C~T
P(V, T) = P,.(V) + ~
(1)
2. EQUATION OF STATE
In this equation the electronic contributions are neglected, and also the specific heat Cv is considered as constant. Several well-known relationships, notably the Slater ?~, the Dugdale-MacDonald ?uo and the freevolume theory ?rv equations provide methods of determining ?, as a function of P,.(V). The question is now, what result can be obtained if all the ?'s are equal, i.e.: ?s = 7Mn = ?rv.
(2)
We obtain in this case only one differential equation which verifies the above equalities (2). The resulting differential equation is:
B'P,(V) = B,
(3)
where B is the bulk modulus and B' = dB/dP. The solution of equation (3) is:
Pc(V) = At/',
which is a solution of equation (3). The coefficients ai in equation (5) are generally derived from the bulkmodulus B0, the Gruneisen coefficient Y0and from the Thomas-Fermi-Dirac theory. In this work we propose an additional relation between the a~ and the slope U£ of the curve in Us-Up plane at Up equal zero.
Using the laws of conservation of mass, momentum and entropy across the shock front, the following relations are obtained [3]: 1
Vj
=
Vo Pt = c3AS
--
0vp
Up
(6)
U~
Po + OoUsUp
i> 0,
(8)
where Us is the front velocity, Up is the particle velocity in the compressed region and S is the entropy. The suffixes 1 and 0 represent the quantities in the shocked and in the unshocked regions. Combining equations (6-8) with (1) a differential equation linking the shock quantities to the pressure ones is obtained [3]:
(Us+ U" Up - -r/ dU,) { dPc~ x z(U~Up) -- ZVoP~ + V0r/-~-~-j ~> 0,
(4)
where ~/ = Vo/V is the volume ratio, V0 is the initial volume, A and 0t are constants. From equation (4) we can explain why Al'tshuler adopted the following expression for P,(V):
where Us"
=
dUs
dVp
7
P,(rl) = ~ a,q'/3+',
(5)
(7)
X
i=1
667
1+?+---
~0~"
(9)
It is easy to obtain from equation (9) the slope U~ at Up ~ 0, thus: 2U~ =
2 + 7 O + 7 o ~~1 - ~67 r / ~=' .
25
(10)
It is necessary to note here that the velocity of sound Co in bulk defined by QoC02 = (OP,/dq)I,-, is equal to Us at Up = 0. Let us now establish a connection between the parameters of the metal in its initial state and the coefficients a~ of the "cold" pressure P,. In order to obtain the necessary relations, we adopt the MacDonald's relation 7Mo and we take the pressure P~ of the form of equation (5) namely:
P,. =
Vol. 69, No. 6
GRUNEISEN'S PARAMETER
668
20 t 15 P IO-
Bo(~l 5/3 + /~1a/3 + 6rlz/3 - (o~ + t~ + 6)q).
(11) The three unknown constant parameters ~, fl and 3 can now be derived from the three equations of 70, B0 and U~. We obtain the following equations for ~, fl and 3:
I
Or5
=
3(37o + 9t + 970)
6 =
- 1 0 + 9t + 97o,
(12)
where
8o =
70
- 70 + ~
2U~
eoQ.
Finally, using D u g d a l e - M a c D o n a l d relation and equation (11), the Gruneisen parameter 7 can be written as a function of volume in the form: 37 =
0.8
0.7
0.9
V/Vo
/~ =
t =
0.6
4-9t
6~/2/3 Jr- 3flrlU 3 _ (0~ + ~ + ~) 3~r/2/3 -k- 2fltl I/3 - (o~ + fl + 3)"
(13)
3. R E S U L T S A N D D I S C U S S I O N As a first application of this theory, we consider the results of an investigation of the dynamic compression of porous substances obtained by Al'tshuler et al. [1, 2] for A1, Cu and Pb.
Fig. 1. The zero degree Kelvin isothermal Pc versus of the volume ratio V~ V0: 1 for Pb; 2 for A1 and 3 for Cu. The solid lines are from the experimental data of [1]. The circles stand for the results of the present work.
The quantities Q0, 7o and U~ for these metals are listed in Table 1. The computed parameters ~,/~, 6 and t from equation (12) are also given in Table 1. The values of the pressure P, at zero degree Kelvin computed from (11) are given in Table 2. Also the values of Gruneisen parameter and the values of the slope U / ( V ) given by equation (9) as a function of volume are listed in columns 5 & 6 respectively in Table 2. In Fig. 1, we compare our computed Pc results with the experimental data of Al'tshuler [1]. Apart from the 16% difference in the case of A1, our curves are in good agreement with the experimental data. In Figs 2-4, we compare our computed 7 from equation (13) with the data of Al'tshuler [2, 1]. Our
Table 1. Constants at the Otitial state. Qo density from [1], 70 Gruneisen parameter from [1]. U~ and t obtained from equation (12) Metal
Q0 g cm-3
70
U~
Co k m s - '
~
/~
6
t
AI Cu Pb
2.71 8.93 11.34
2.088 1.983 2.457
1.34 o 1.497 (b) 1.517 ~b)
5.2 3.92 1.91
23.692 26.383 43.339
- 55.692 - 64.302 - 110.904
- 19.9 - ! 4.536 -27.226
-2.188 - 2.487 -4.371
Vol. 69, No. 6
GRUNEISEN'S PARAMETER
669
Table 2. Values of U~, Up, II/Vo, P,., 7, Us"for A1, Cu, Pb. The shock velocity Us, the particle velocity Up and the volume ratio II/1Io are from [1]. P,. the pressure if from equation (11); 7 is from equation (13); Us' is from equation
(9) Us k m s - '
Up k m s - '
V/Vo
~l x 10 ~0 dyn cm- 2
~(V)
U~'(v )
A1 5.940 6.640 7.320 8.020 8.710 9.410 10.110 10.810 11.480 12.160 12.820 13.450
0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000 5.500 6.000
0.916 0.849 0.795 0.751 0.713 0.681 0.654 0.630 0.608 0.588 0.571 0.554
8.046 18.152 29.890 42.809 57.235 72.431 87.981 104.370 121.947 140.441 158.365 178.640
1.869 1.727 1.628 1.555 1.497 1.451 1.415 1.384 1.356 1.332 1.311 1.292
1.456 1.396 1.339 1.279 1.237 1.198 1.164 1.137 1.118 1.102 1.087 1.078
4.680 5.440 6.220 6.960 7.700 8.450 9.190 9.930 10.760
0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500
0.893 0.816 0.759 0.713 0.675 0.645 0.619 0.597 0.578
20.535 45.966 74.208 105.651 139.676 173.307 208.558 243.800 279.011
1.788 1.663 1.579 1.515 1.465 1.427 1.395 1.368 1.346
1.329 1.235 1.153 1.112 1.080 1.047 1.023 1.002 0.985
2.750 3.560 4.330 5.070 6.380 7.000 7.590 8.180
0.500 1.000 1.500 2.000 3.000 3.500 4.000 4.500
0.818 0.719 0.654 0.606 0.530 0.500 0.473 0.450
14.844 34.795 57.677 82.967 147.569 185.984 229.859 276.263
1.931 1.718 1.597 1.516 1.399 1.356 1.319 1.289
1.377 1.213 1.130 1.081 1.056 1.050 1.048 1.044
2--
f J ~' 1 . 5 -
I
0.5
I
0.6
I
0.7
I
0.8
I
0.9
I
I
V/Vo
Fig. 2. Comparison of calculated Gruneisen's parameter from (13) for A1 with the data of Al'tshuler et al [1, 2]. (The solid line from [2], the dashed line from [1] and the circles, present work).
,,I O000 • ~°°SO0
• ••
S : : " •" I
I
I
0.5
0.7
0.9
V/Vo Fig. 3. Comparison of calculated Gruneisen's parameter from (13) for Cu with the data of Al'tshuler et al [1, 2]. (The solid line from [2], the dashed line from [1] and the circles, present work).
670
GRUNEISEN'S PARAMETER
J
2:-
2
?,
e °
/
I
Lennard-Jones potential are in good agreement with our computed values from equation (9), Table 2 and equation (10).
SSSS
~
SSSS~
1.5-
Vol. 69, No. 6
Acknowledgements - The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. The author also feels a great debt to Professor S.O. Lundqvist and to Professor F. Garcia-Moliner for their encouragement and kind help.
/., I
S
REFERENCES 0.5
I
I
I
I
0.4
0.6
0.8
I
WVo Fig. 4. Comparison of calculated Gruneisen's parameter from (13) for Pb with the data of Al'tshuler et al [1, 2]. (The solid line from [2], the dashed line from [1] and the circles, present work). curves are in good agreement with the experimental data. It is worth to note that the values of (d In ~/ d in V)lv0 obtained by Al'tshuler [1] in using the Mie-
1. 2. 3. 4. 5. 6.
L.V. Al'tshuler, S.B. Kormer, A.A. Bakanova & R.F. Trunin, Soviet Phys. J E T P 11, 573 (1960). S.B. Kormer, A.I. Funtikov, V.D. Urlin & A.N. Kolesnikova, Soviet Phys. J E T P 15, 477 (1962). H. Dandache, Internal report ICTP/86/287. B.K. Godwal, S.K. Sikka & Chidambaram, Phys. Rep. 102, 121 (1983). A. Migault, J. Phys. 32, 437 (1971). M.H. Rice, R.G. McQueen & J.M. Walsh, Solid State Phys. 6, 1 (1958).
0038-1098/89 $3.00 + .00 Pergamon Press plc
ON THE INTERRELATIONSHIPS BETWEEN GRUNEISEN'S PARAMETER, SHOCK AND ISOTHERMAL EQUATION OF STATE H. Dandache Faculty of Sciences, Lebaneese University, Beirut, Lebanon
(Received 9 August 1988 by S. Lundqvist ) A thermodynamical method is developed which give a relation between the shock wave velocity Us and the particle velocity Up. Mie-Gruneisen equation of state is considered with coefficient 7 given by DugdaleMacDonald relation. The slope of the curve in the Us-Upplane is used to obtain an equation of state at zero temperature. The results are in good agreement with Al'tshuler's [1] experimental data for A1, Cu and Pb.
I. I N T R O D U C T I O N WE ASSUME the Mie-Gruneisen equation of state namely that the pressure P in a solid at high temperature T > 0, where 0 is the Debye temperature, is related to the pressure P,. at zero temperature by the equation [6]:
~(V)C~T
P(V, T) = P,.(V) + ~
(1)
2. EQUATION OF STATE
In this equation the electronic contributions are neglected, and also the specific heat Cv is considered as constant. Several well-known relationships, notably the Slater ?~, the Dugdale-MacDonald ?uo and the freevolume theory ?rv equations provide methods of determining ?, as a function of P,.(V). The question is now, what result can be obtained if all the ?'s are equal, i.e.: ?s = 7Mn = ?rv.
(2)
We obtain in this case only one differential equation which verifies the above equalities (2). The resulting differential equation is:
B'P,(V) = B,
(3)
where B is the bulk modulus and B' = dB/dP. The solution of equation (3) is:
Pc(V) = At/',
which is a solution of equation (3). The coefficients ai in equation (5) are generally derived from the bulkmodulus B0, the Gruneisen coefficient Y0and from the Thomas-Fermi-Dirac theory. In this work we propose an additional relation between the a~ and the slope U£ of the curve in Us-Up plane at Up equal zero.
Using the laws of conservation of mass, momentum and entropy across the shock front, the following relations are obtained [3]: 1
Vj
=
Vo Pt = c3AS
--
0vp
Up
(6)
U~
Po + OoUsUp
i> 0,
(8)
where Us is the front velocity, Up is the particle velocity in the compressed region and S is the entropy. The suffixes 1 and 0 represent the quantities in the shocked and in the unshocked regions. Combining equations (6-8) with (1) a differential equation linking the shock quantities to the pressure ones is obtained [3]:
(Us+ U" Up - -r/ dU,) { dPc~ x z(U~Up) -- ZVoP~ + V0r/-~-~-j ~> 0,
(4)
where ~/ = Vo/V is the volume ratio, V0 is the initial volume, A and 0t are constants. From equation (4) we can explain why Al'tshuler adopted the following expression for P,(V):
where Us"
=
dUs
dVp
7
P,(rl) = ~ a,q'/3+',
(5)
(7)
X
i=1
667
1+?+---
~0~"
(9)
It is easy to obtain from equation (9) the slope U~ at Up ~ 0, thus: 2U~ =
2 + 7 O + 7 o ~~1 - ~67 r / ~=' .
25
(10)
It is necessary to note here that the velocity of sound Co in bulk defined by QoC02 = (OP,/dq)I,-, is equal to Us at Up = 0. Let us now establish a connection between the parameters of the metal in its initial state and the coefficients a~ of the "cold" pressure P,. In order to obtain the necessary relations, we adopt the MacDonald's relation 7Mo and we take the pressure P~ of the form of equation (5) namely:
P,. =
Vol. 69, No. 6
GRUNEISEN'S PARAMETER
668
20 t 15 P IO-
Bo(~l 5/3 + /~1a/3 + 6rlz/3 - (o~ + t~ + 6)q).
(11) The three unknown constant parameters ~, fl and 3 can now be derived from the three equations of 70, B0 and U~. We obtain the following equations for ~, fl and 3:
I
Or5
=
3(37o + 9t + 970)
6 =
- 1 0 + 9t + 97o,
(12)
where
8o =
70
- 70 + ~
2U~
eoQ.
Finally, using D u g d a l e - M a c D o n a l d relation and equation (11), the Gruneisen parameter 7 can be written as a function of volume in the form: 37 =
0.8
0.7
0.9
V/Vo
/~ =
t =
0.6
4-9t
6~/2/3 Jr- 3flrlU 3 _ (0~ + ~ + ~) 3~r/2/3 -k- 2fltl I/3 - (o~ + fl + 3)"
(13)
3. R E S U L T S A N D D I S C U S S I O N As a first application of this theory, we consider the results of an investigation of the dynamic compression of porous substances obtained by Al'tshuler et al. [1, 2] for A1, Cu and Pb.
Fig. 1. The zero degree Kelvin isothermal Pc versus of the volume ratio V~ V0: 1 for Pb; 2 for A1 and 3 for Cu. The solid lines are from the experimental data of [1]. The circles stand for the results of the present work.
The quantities Q0, 7o and U~ for these metals are listed in Table 1. The computed parameters ~,/~, 6 and t from equation (12) are also given in Table 1. The values of the pressure P, at zero degree Kelvin computed from (11) are given in Table 2. Also the values of Gruneisen parameter and the values of the slope U / ( V ) given by equation (9) as a function of volume are listed in columns 5 & 6 respectively in Table 2. In Fig. 1, we compare our computed Pc results with the experimental data of Al'tshuler [1]. Apart from the 16% difference in the case of A1, our curves are in good agreement with the experimental data. In Figs 2-4, we compare our computed 7 from equation (13) with the data of Al'tshuler [2, 1]. Our
Table 1. Constants at the Otitial state. Qo density from [1], 70 Gruneisen parameter from [1]. U~ and t obtained from equation (12) Metal
Q0 g cm-3
70
U~
Co k m s - '
~
/~
6
t
AI Cu Pb
2.71 8.93 11.34
2.088 1.983 2.457
1.34 o 1.497 (b) 1.517 ~b)
5.2 3.92 1.91
23.692 26.383 43.339
- 55.692 - 64.302 - 110.904
- 19.9 - ! 4.536 -27.226
-2.188 - 2.487 -4.371
Vol. 69, No. 6
GRUNEISEN'S PARAMETER
669
Table 2. Values of U~, Up, II/Vo, P,., 7, Us"for A1, Cu, Pb. The shock velocity Us, the particle velocity Up and the volume ratio II/1Io are from [1]. P,. the pressure if from equation (11); 7 is from equation (13); Us' is from equation
(9) Us k m s - '
Up k m s - '
V/Vo
~l x 10 ~0 dyn cm- 2
~(V)
U~'(v )
A1 5.940 6.640 7.320 8.020 8.710 9.410 10.110 10.810 11.480 12.160 12.820 13.450
0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000 5.500 6.000
0.916 0.849 0.795 0.751 0.713 0.681 0.654 0.630 0.608 0.588 0.571 0.554
8.046 18.152 29.890 42.809 57.235 72.431 87.981 104.370 121.947 140.441 158.365 178.640
1.869 1.727 1.628 1.555 1.497 1.451 1.415 1.384 1.356 1.332 1.311 1.292
1.456 1.396 1.339 1.279 1.237 1.198 1.164 1.137 1.118 1.102 1.087 1.078
4.680 5.440 6.220 6.960 7.700 8.450 9.190 9.930 10.760
0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500
0.893 0.816 0.759 0.713 0.675 0.645 0.619 0.597 0.578
20.535 45.966 74.208 105.651 139.676 173.307 208.558 243.800 279.011
1.788 1.663 1.579 1.515 1.465 1.427 1.395 1.368 1.346
1.329 1.235 1.153 1.112 1.080 1.047 1.023 1.002 0.985
2.750 3.560 4.330 5.070 6.380 7.000 7.590 8.180
0.500 1.000 1.500 2.000 3.000 3.500 4.000 4.500
0.818 0.719 0.654 0.606 0.530 0.500 0.473 0.450
14.844 34.795 57.677 82.967 147.569 185.984 229.859 276.263
1.931 1.718 1.597 1.516 1.399 1.356 1.319 1.289
1.377 1.213 1.130 1.081 1.056 1.050 1.048 1.044
2--
f J ~' 1 . 5 -
I
0.5
I
0.6
I
0.7
I
0.8
I
0.9
I
I
V/Vo
Fig. 2. Comparison of calculated Gruneisen's parameter from (13) for A1 with the data of Al'tshuler et al [1, 2]. (The solid line from [2], the dashed line from [1] and the circles, present work).
,,I O000 • ~°°SO0
• ••
S : : " •" I
I
I
0.5
0.7
0.9
V/Vo Fig. 3. Comparison of calculated Gruneisen's parameter from (13) for Cu with the data of Al'tshuler et al [1, 2]. (The solid line from [2], the dashed line from [1] and the circles, present work).
670
GRUNEISEN'S PARAMETER
J
2:-
2
?,
e °
/
I
Lennard-Jones potential are in good agreement with our computed values from equation (9), Table 2 and equation (10).
SSSS
~
SSSS~
1.5-
Vol. 69, No. 6
Acknowledgements - The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. The author also feels a great debt to Professor S.O. Lundqvist and to Professor F. Garcia-Moliner for their encouragement and kind help.
/., I
S
REFERENCES 0.5
I
I
I
I
0.4
0.6
0.8
I
WVo Fig. 4. Comparison of calculated Gruneisen's parameter from (13) for Pb with the data of Al'tshuler et al [1, 2]. (The solid line from [2], the dashed line from [1] and the circles, present work). curves are in good agreement with the experimental data. It is worth to note that the values of (d In ~/ d in V)lv0 obtained by Al'tshuler [1] in using the Mie-
1. 2. 3. 4. 5. 6.
L.V. Al'tshuler, S.B. Kormer, A.A. Bakanova & R.F. Trunin, Soviet Phys. J E T P 11, 573 (1960). S.B. Kormer, A.I. Funtikov, V.D. Urlin & A.N. Kolesnikova, Soviet Phys. J E T P 15, 477 (1962). H. Dandache, Internal report ICTP/86/287. B.K. Godwal, S.K. Sikka & Chidambaram, Phys. Rep. 102, 121 (1983). A. Migault, J. Phys. 32, 437 (1971). M.H. Rice, R.G. McQueen & J.M. Walsh, Solid State Phys. 6, 1 (1958).