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Polynomial approximations to a universal equation of state
J. Phys. Chem. Solids Vol. 53, No. 4. pp. 485492. Printed in Great Britain.
1992
POLYNOMIAL
0022-3697/92 $5.00 + 0.M) 0 1992 Pergamon Press plc
APPROXIMATIONS TO A UNIVERSAL EQUATION OF STATE
F. E. PRIETO and C. RENERO Instituto de Fisica, Universidad National de Mexico, AP 20-364, 01000 Mexico, D. F. (Received
12 December
1990; accepted in revisedform
2 October
1991)
Abstract-To avoid the difficulties inherent in the use of unfamiliar and difficult-to-handle functions appearing in an otherwise seemingly useful universal equation of state at high pressures, a series expansion is made of all the volume dependent functions in the original equation. The whole problem is treated within the framework of a system-independent formulation of the thermodynamics of high pressures, but the expanded variable is essentially the relative compression. Explicit polynomials to the second and third order in this variable are obtained for the pressure as a function of volume. These polynomials are in general very good approximations to the exact PVT correlation for high pressure isotherms at any temperature. The range of validity of the approximations turns out to be related to the hardness of the material. The coefficients of the polynomial approximations are not adjustable parameters, they can be calculated as functions of the physical properties used in the formalism. Numerical values of the coefficients are obtained for 17 materials. The proposed polynomial approximations are numerically tested against the full PrietoRenero equation, also against the widely used Birch-Murnaghan equation, and finally against some of the available experimental results. In each case the validity of the approximation is discussed in detail. Keywords: Universal equation of state, high pressure.
1. INTRODUCTION
for low to moderate values of the applied pressure. Since, in order to get universality the original equation is expressed in reduced variables, the system-independent formulation of the thermodynamics of high pressures proposed by Prieto and Renero [ 11, and by Prieto [6] will be used throughout this paper. However, to simplify the use of the polynomials approximations, the final results will also be given in terms of real physical variables.
Some years ago Prieto and Renero [l] proposed a universal equation of state for materials at shock pressures. This equation was obtained from very general assumptions valid for all kinds of materials. As a consequence, the resulting equation of state (EOS) was proved to be valid for all sorts of materials: elements, compounds, mixtures, solids, liquids and gases [2,3]. The structure of the equation is in principle very simple. It contains a linear temperature term, and several volume dependent terms. Except for one of them, expressed as a combination of exponential integral functions of negative arguments [4], most of these terms are very easy to use, algebraic functions. Although the exponential integral function is tabulated, and some of its properties are well known [4], values of the function obtained by interpolation are not precise enough to be used in the EOS. Thus, computer programs to evaluate the function are required [S]. This fact may inhibit a more extended use of a seemingly useful EOS. To facilitate the use of this equation, we propose in the present paper a simplified form. This is a polynomial approximation obtained from a series expansion of all the algebraic and exponential integral functions appearing in the original equation. Since the expanded variable used is essentially the relative compression, good results are expected for small values of this variable and, as a consequence,
2. THE REDUCED VARIABLES FORMALISM The reduced variables formalism [l, 61 makes use of six physical properties of the material under study: the initial density p,, = l/I’,, the coefficients A and B of the linear relationship (U, = A + BU,) between shock and particle velocities [7], the volume coefficient of thermal expansion CL,the isothermal compressibility KT, and the specific heat at constant volume C,. The scaled variables most suitable for the thermodynamics of high pressures have been shown to be [1,61; P = PIP,,
x = Bq = B(l
-
V/V,,),
t = uBT,
(1)
in which T is the absolute temperature, t] the relative compression, and P, = poA=IB, 485
(2)
F. E.
486
hIET0
is the so-called [6] characteristic pressure or shock hardness of the material. Other thermodynamic quantities can also be reduced using these scaling factors. Of special interest are the reduced compressibility and the reduced specific heat at constant volume, respectively:
and C.
I&NERO
A Maclaurin’s series expansion of F(x) as defined by eqn (7) is then of the form F(x) =f(O) + xf(0) + +/x2~(0) + ; x3P(0) + . . . - rt,(l +mx +4m2x2+$m3x3+*..).
(12)
To obtain the derivative of the last term of the function f(x), let us first obtain the derivative of &(x). From eqn (11) It is also convenient coefficients
to define the two auxiliary d&(z)
e’ (13)
dt=z* m = (B&C”,)-‘,
(4)
and
ap
0
r= at, =CV,m.
(5)
The reduced value of the initial temperature t, = c&T,, corresponding usually to the temperature T, = 293 K at which shock experiments to determine A and B are performed, will be of ample use throughout this paper. 3. THE POLYNOMIAL
APPROXIMATIONS
m(‘-X)Ei(m, x)] &km = me-“(‘-“)Ei(m,
2-m
= ~_2;;f(mx)4)_!!(3_m)e””
(15)
y, = 1 - mrtO,
(16)
y2 = 2 - im*rt,,
(17)
y,=3-jm-im3rt
(7)
0’
(18)
Let us now denote by p,,(x, t) the polynomial obtained from the series expansion (15) truncated at the term in x”, that is PAX, t) = r(t - to) + y,x + y2x2
2(1 -x)2
+y3x3 +. . ’ + ynxn.
+ : (4m - m2 - 2)e-“” -“)Ei(m, x),
(8)
with Ei(m, x) = R(m)
(14)
with
(6)
and the auxiliary function f (x) is defined by f(x)
x) + &.
The determination of the coefficients in the series expansion (12) is not straightforward. One obtains: f(0) = 0, f(0) = 1, f”(0) = 4,f”(O) = 18 - 2m. Consequently
where F(x) =f (x) - rt0emX,
with eqn (9),
P(X,t)=r(t-t,)+Y,X+Y2X2+Y3X3+‘.‘,
When expressed in system-independent form by the use of the scaled variable defined by eqns (l)-(5), the EOS takes the form [l] p(x, t) = F(x) + rt,
Using now this result in combination one obtains
- &[m(l - x)].
(9)
(19)
For the isotherms at room temperature, i.e. t = t,,, which is the most usual case, these polynomials become functions of one single variable P”(X, to) = p,(x)
Ei(z) is the exponential integral function of negative argument, which is defined by [4]
= y,x + y2xz+ y3x3 +
. . + ynxn.
(20)
Of special interest are the first three polynomials. G(z) = or, alternatively,
m c dy, s --I Y
z > 0,
(10)
by [4]
lQz)=y
+Inz+
ZN f N=, 5-Z’
(21)
P*(x) = Ylx, P2(X) =
Ylx +
Y2X27
(22)
P,(X)
Ylx +
72x2 + Y3X3.
(23)
and (11)
=
Polynomial approximations
to a universal equation of state
We shall see later that pr(x), or eventually Pa, gives, for many materials, very good approximations to the values of pressure computed using the exact eqn (6), and to the experimental values reported by various authors as well as to the Birch-Murnaghan equation. Equations (21)-(23) can be transformed to give directly the physical pressure by inverting the transformation (1) leading to the reduced pressure. They can be written in the form p, =r,rl>
(24)
p*=r,q
+r2r12,
(25)
p,=r,q
+r2q2+r3q3,
(26)
and
where P, , P2, and P3 have now the same units of pressure as PC, and n is the relative compression. The transformed coefficients are then r, = PcBy,,
(27)
r2 =
(28)
PcB2y2,
and
r3= PcB3y3.
(29)
It is to be remarked that whereas the yis are dimensionless quantities, the Tis have units of pressure. If isotherms other than those at room temperature are of interest, then a fourth coefficient r4 for the temperature term is needed. This is given by r, = urBP,,
(30)
487
and the additional term r, (T - To) must be included in eqns (24)-(26). To show the usefulness of the polynomial approximations to the second and third order in the relative compression, we will apply them to some materials. Table 1 shows, for 17 materials, values of all the quantities needed to compute the coefficients, and to perform the transformation of variables required to use the polynomial approximations. Most of these values were taken from [l], [7] and [8], where the sources of the pertinent experimental data can be found. A few comments on the values given in this table are perhaps appropriate. For most materials the slope B of the linear relationship between shock and particle velocities takes values around 1.4 [9]. Values of the parameter m are in the neighbourhood of 413, and those of the parameter r are very near to 1.0; this ‘statistical approximation’ is widely discussed in [l]. The last general remark concerning the values of these parameters is that the reduced initial temperature t,, is in general very small and even negligible compared with the integer numbers appearing in the coefficients defined by eqns (16)-( 18) or (27)-(29). But it is also to be remarked, however, that the contribution of the term in the initial temperature can be neglected only in relation to the evaluation of these coefficients; for temperatures other than t,,, the contribution of the temperature term in eqns (15) or (19) may become very important and far from negligible. The previous remarks lead immediately to a couple of approximate equation of state correlations that might be of some interest. Within the scope of validity of the statistical approximation, the coefficients of the
Table 1. Summary of numerical values of the parameters needed to use the polynomial approximations. The first column identifies the material. The second, third and fourth columns give the calculated values of m, r and r,, Columns five and six give the values of a, the coefficient of volume expansion, and E from the relation U, = A + BU, Material W MO Pt co Ni Cr V Pd Au Ti cu AS Zn Sn Cd K NaCl
m
1.390 1.309 1.731 1.624 1.274 0.780 1.072 1.382 1.964 1.488 1.327 1.535 1.300 1.501 1.403 1.143 1.200
r
t0
1.036 1.017 1.004 0.966 0.975 0.984 1.029 0.967 0.951 0.852 0.948 0.912 0.902 1.068 0.905 0.887 0.885
0.00488 0.00538 0.01206 0.01600 0.01608 0.008 11 0.00819 0.01614 0.01928 0.00570 0.02194 0.02671 0.04065 0.02744 0.0462 1 0.08536 0.04702
a
x
105(“c-‘) 1.34 1.49 2.66 4.10 3.80 1.89 2.33 3.47 4.22 2.54 5.03 5.75 8.90 6.34 9.44 24.9 11.95
B
1.237 1.233 1.544 1.330 1.445 1.465 1.201 1.588 1.560 1.089 1.489 1.586 1.559 1.476 1.671 1.170 1.343
488
F. E.
PR~ETO and C.
series development, truncated to the second or third order, take the particular values y, = 1 - ft,,
(31)
y*=2-;t,,
(32)
y, = 2.5555 - Et,, .
(33)
and
These coefficients may be taken as universal, i.e. they may have the same value for all materials, either by neglecting the terms in t, that, as previously remarked, are very small, or by performing the experiments to determine A and B at an initial temperature T, such that t, is the same for all materials [l]. In the first case, by neglecting the terms in t,, one obtains for the second and the third order approximations &(X) = x - 2x2,
(34)
p3(x) = x -2x2-2.5555x3.
(35)
and
Finally, introducing gets
the mean value 1.4 for B [9] one
p3(q)= 1.41 -4$-7713.
(36)
These approximate expressions are very good to make quick estimations of pressure. If more precision is required, one should use eqns (21)-(26), with the exact values of the coefficients for the polynomial approximations, eqns (16)-( 18) or (27)-(29). Table 2 shows, for the same materials of Table 1, values of the coefficients of the polynomial approximations up to third order. They were computed using Table 2. Values of the coefficients y, , y2, and y, as defined by eqns (16), (17) and (18), for each of the materials shown in the first column. The last column gives the values of the shock hardness PC = p. A2/B Material W MO Pt co Ni Cr V Pd Au Ti cu AS Zn Sn Cd K NaCl
YI
Y2
0.9930 0.9928 0.9790 0.9749 0.9800 0.9938 0.9910 0.9785 0.9640 0.9927 0.9725 0.9626 0.9523 0.9560 0.9413 0.9135 0.9501
1.9951 1.9953 1.9819 1.9769 1.9873 1.9976 1.9952 1.9851 1.9646 1.9946 1.9817 1.9713 1.9690 1.9670 1.9588 1.9505 1.9700
Y3 2.5344 2.5616 2.4125 2.4476 2.5699 2.6392 2.6410 2.5325 2.3215 2.5013 2.5496 2.4737 2.5532 2.4832 2.5130 2.6002 2.6000
F&NERO
the data of Table 1, and using eqns (16)-(18). It also shows, for each material, the values of the shock hardness PC. Perhaps the only pertinent comment concerning these values is that, as previously stated, yI takes values near to 1.O, y2 to 2.0, and y3 to 2.5, for all materials. The polynomial forms presented here are obtained as approximations to a complete system-independent equation of state which has been shown to give, in general, very good agreement with the experimental data for various materials [7]. A comparison between the results obtained using both, the full Prieto-Renero (PR) equation, and the polynomial approximations will be treated in the following sections. But is also seems interesting to complete the analysis by comparing these approximations with other polynomial forms currently used. Among the most recent and critical reviews on equations of state in polynomial forms we shall refer to the work by Jeanloz [lo, 111. Let us first remark that among the equations considered by Jeanloz is a recently proposed universal equation of state by Vinet et al. [12]. A detailed comparison of this equation with our full eqn (6) has been already published [ 131;thus, it will not be treated in the present paper. It is concluded in Jeanloz’s papers that the third order Birch-Murnaghan (BM) equation is the most successful among the other five polynomial forms in representing shock wave data. It is also shown that this equation is nearly identical to the linear relationship between shock and particle velocities usually obtained in shock experiments. Since our complete equation of state is based on the validity of this linear relationship, we will restrain the comparison with other polynomial forms to that with the BM. We shall use the third order Birch-Murnaghan [8] in the form P,,(f)
= 3&f(l
+ 2f)s’2[l + ?(K;, - 4Y],
(37)
where
P, (kbar) 2522.7 2171.9 1795.1 1489.9 1323.5 1319.0 1309.1 1176.9 1166.2 945.8 930.9 695.6 426.0 343.7 308.6 30.6 200.6
f= f[(V#)2”
- 11,
(38)
with & and Kh being the isothermal bulk modulus and its pressure derivative. Values of the couples (&, Kh) for the five materials here considered are: (1808.3 kbar, 5.42) for Pd [14], (1309.2 kbar, 5.65) for Cu [15], (1007.1 kbar, 5.53) for Ag [15], (31.77 kbar, 3.98) for K [16], and (238.36 kbar, 5.35) for NaCl[81. 4. APPLICATION
TO SOME
MATERIALS
We shall now analyse, for a few materials covering a wide range of shock hardness, the results obtained by the use of the polynomial approximations to the
Polynomial approximations
to a universal equation of state
489
Table 3. The first and second columns give, for copper, the experimental pressure Pa and the relative compression I, as reported by Bridgman [17]. The third column shows the corresponding values of x = Bn. The fourth and fifth columns give the calculated values of the second degreep,, and the third degree p, polynomials. The sixth column gives, for comparison, the pressure computed using the exact eqn (6). The last two columns give the physical pressure computed using the third order polynomial (P,), and the Birch-Mumaghan equation (PBM) P,(kbar) 5
10 15 20 25 30
rl
x
P2
P,
P
0.004 0.007 0.011 0.015 0.018 0.021
0.0060 0.0104 0.0164 0.0223 0.0268 0.0313
0.0059 0.0104 0.0165 0.0227 0.0275 0.0323
0.0059 0.0104 0.0165 0.0227 0.0275 0.0324
0.0059 0.0104 0.0165 0.0230 0.0275 0.0324
P,(kbar) 5.45 9.64 15.33 21.16 25.63 30.18
Pa” (kbar) 5.30 9.38 14.94 20.64 25.03 29.50
Table 4. Experimental data for copper, as reported by Keeler [18]. The column headings correspond to those of Table 3 P,(kbar) 15 20 25 30 35 40 45
rl 0.010 0.014 0.017 0.020 0.023 0.026 0.029
X 0.0149 0.0209 0.0253 0.0298 0.0343 0.0387 0.0432
P2 0.0149 0.0211 0.0259 0.0307 0.0356 0.0406 0.0457
second and third order. These results are shown in Tables 3, 4, 5 and 6 for copper, in Table 7 for silver, in Table 8 for palladium, in Table 9 for potassium, and in Table 10 for sodium chloride. With the exception of NaCl, these materials are the same ones studied in [7] where the agreement of pressure values computed using the full PR equation (6) with the experimental data was fully discussed. We include now the sodium chloride because being a pressure standard, experimental data are quite abundant. We shall now focus attention on the fast or slow conver-
P, 0.0149 0.0211 0.0259 0.0308 0.0357 0.0408 0.0459
P
0.0150 0.0210 0.0260 0.0301 0.0360 0.0408 0.0460
P,(kbar) 13.89 19.69 24.14 28.65 33.26 37.95 42.72
Par,,(kbar) 13.53 19.20 23.56 28.00 32.53 37.15 41.86
gence of the series expansion, and on the differences between pressure values computed with the exact eqn (6), those computed using the polynomials (22) and (23) to the second and third order, and those obtained through the third order Birch-Murnaghan equation. Each table lists the experimental data in pressure and in relative compression, the variable x used for the series expansion, the values of pressure computed to the second and third order, and finally, for comparison, values of pressure computed with the full eqn (6), and with the Birch-Murnaghan equation.
Table 5. Experimental data for copper, as reported by Vaidya and Kennedy [19]. The column headings correspond to those of Table 3 P,(kbar) 5 10 15 20 25 30 35 40 45
0.0033 0.0065 0.0097 0.0128 0.0159 0.0189 0.0218 0.0247 0.0276
x
p,
p,
P
P3War)
PBMWar)
0.0049 0.0097 0.0144 0.0190 0.0237 0.028 1 0.0324 0.0368 0.0411
0.0048 0.0096 0.0144 0.0192 0.0241 0.0289 0.0337 0.0385 0.0433
0.0048 0.0096 0.0145 0.0193 0.0242 0.0290 0.0338 0.0386 0.0435
0.0048 0.0096 0.0145 0.0193 0.0242 0.0290 0.0340 0.0386 0.0440
4.47 8.94 13.47 17.94 22.49 26.99 31.41 35.90 40.48
4.36 8.69 13.11 17.49 21.95 26.36 30.71 35.14 39.65
Table 6. Experimental data for copper, as reported by Mao et al. [20]. The column headings correspond to those of Table 3 P,(kbar) 73 101 127 160 180 186 209
rl
X
P,
0.047 0.062 0.074 0.089 0.097 0.100 0.109
0.0700 0.0923 0.1102 0.1325 0.1444 0.1489 0.1623
0.0777 0.1067 0.1312 0.1637 0.1818 0.1887 0.2100
p3
0.0786 0.1087 0.1347 0.1696 0.1895 0.1972 0.2209
P
0.0787 0.1089 0.1351 0.1706 0.1909 0.1988 0.2233
P,(kbar) 73.20 101.16 125.32 157.89 176.38 183.53 205.67
PBM War) 72.19 100.38 125.03 158.73 178.12 185.66 209.21
490
F. E.
hIET0
and C. RENERO
Table 7. Experimental data for silver, as reported by Mao et ai. [20]. The column headings correspond to those of Table 3 9
x
p2
p3
P
77
0.059
1: 190 209 66 163
0.070 0.092 0.119 0,127 0.052 0.107
0.0936 0.1110 0.1459 0.1887 0.2014 0.0825 0.1697
0.1073 0.1318 0.1824 0.2519 0.2739 0.0928 0.2201
0.1094 0.1345 0.1901 0.2685 0.2941 0.0941 0.2322
0.1096 0.1350 0.1915 0.2721 0.2996 0.0943 0.2349
Pafkbar)
P,(kbar) 76.07 93.59 132.24 186.78 204.56 65.51 161.53
PBM(kbar) 72.45 89.32 126.90 181.08 199.07 62.33 155.81
Table 8. Experimental data for palladium, as reported by Mao et al. [20]. The column headings correspond to those of Table 3 PF (kbar)
?
X
A
P,
P
65 155 278 659
0.032 0.067 0.106 0.187
0.0508 0.1064 0.1683 0.2970
0.0548 0.1266 0.2209 0.4656
0.0552 0.1296 0.2330 0.5319
0.0552 0.1300 0.2358 0.5639
In Tables 6, 7, 8 and 10, the highest value of pressure shown is that for which the difference between pressure P computed using the exact eqn (6) is larger than 1% of the pressure computed using the polynomial forms. Let us now make some remarks on the results shown in these tabies. Data for copper are shown in Tables 3-6. Table 3 is based on the experimental data reported by Bridgman 1171,Table 4 by Keeler [lS], Table 5 by Vaidya and Kennedy [19]. and Table 6 by Mao et al. [20]. It can be remarked from Tables 3,4 and 5, that up to 45 kbar the two-term polynomial gives results very
P,(kbar)
PBM(kbar)
64.94 152.56 274.25 626.04
64.23 151.33 274.52 656.17
near to those computed by using the full PR equation. Thus, it is unnecessary to go to the next term. At the higher pressures shown in Table 6, the third order approximation gives agreements within 1% with values computed using the full equation. Concerning the agreement with the experimental data and with the Birch-Murnaghan equation, the following remarks are pertinent. Up to 45 kbar, both the third order approximation and the BM equation give results in good agreement between themselves. The agreement with the experimental data is in both cases quite good with Bridgman’s [17] data up to 30 kbar,
Table 9. Experimental data for potassium, as reported by Vaidya and Kennedy [21]. The column headings correspond to those of Table 3 P&bar) 5 10 15 20
9 0.1191 0.1927 0.2450 0.2868
X 0.1393 0.2255 0.2866 0.3356
PZ 0.1652 0.3051 0.4221 0.5261
s
0.1722 0.3349 0.4833 0.6244
P
0.1735 0.3453 0.5134 0.6858
P,(kbar) 5.27 10.25 14.79 19.11
PsM(kbar) 5.18 10.41 15.63 21.07
Table 10. Experimental data for sodium chloride as reported by Boebler and Kennedy [22] between 1 and 30 kbar, and that of Bridgman [23] from 10 to 100 kbar. The column headings correspond to those of Table 3
Pa@bad 1.06 5.62 11.74 19.95 25.69 29.53 9.80 19.61 29.42 39.22 49.03 58.83 68.64 78.44 88.25 98.05
v 0.0042 0.0222 0.0428 0.067 1 0.0828 0.0923 0.038 0.068 0.093 0.115 0.135 0.152 0.168 0.183 0.197 0.210
X
p2
0.0056 0.0298 0.0575 0.0901 0.1112 0.1240 0.0510 0.0913 0.1249 0.1544 0.1813 0.2041 0.2256 0.2458 0.2646 0.2820
0.0054 0.0301 0.0611 0.1016 0.1300 0.1480 0.0536 0.1032 0.1499 0.1937 0.2370 0.2760 0.3147 0.3525 0.3893 0.4247
p3
P
0.0054 0.0301 0.0616 0.1035 0.1336 0.1530 0.0540 0.1052 0.1545 0.2033 0.2525 0.2982 0.3445 0.3911 0.4374 0.4830
0.0054 0.0301 0.0616 0.1035 0.1336 0.1530 0.0540 0.1054 0.1553 0.2054 0.2566 0.3048 0.3548 0.4060 0.458 1 0.5104
P,@W
1.087 6.041 12.359 20.765 26.797 30.691 10.82 21.09 30.98 40.78 50.65 59.81 69.11 78.45 87.75 96.88
PBM@bar)
1.014 5.684 11.724 19.943 25.970 29.922 10.24 20.27 30.22 40.40 50.98 61.13 71.77 82.81 94.16 105.69
Polynomial approximations
to a universal equation of state
but computed values are too low when compared with the experimental data reported by Keeler [18], and by Vaidya and Kennedy [19] up to 45 kbar. The agreement is better with Mao’s [20] data up to 101 kbar. Beyond this pressure the BM equation gives results in better agreement with the experimental data up to 209 kbar. This indicates that our third order approximation for copper is good up to about 100 kbar, beyond this pressure it is better to use the full eqn (6), in excellent agreement with Mao’s data up to 781 kbar [7]. Table [7] shows the data for silver reported by Mao et al. [20]. It is obvious that the second order approximation is not sufficient within the pressure range of 77-209 kbar. The third order polynomial gives very good results up to 163 kbar. For this material the BM equation gives pressures systematically lower than the P, approximation, and even lower than the experimental data. The third order approximation agrees with Mao’s data up to 153 kbar. At higher pressures, up to 918 kbar, it has been shown that the full PR equation gives very good agreement with these data [7]. For palladium, shown in Table 8, also from the data by Mao et al. [20], the results are somewhat similar. The third order approximation is quite acceptable up to 155 kbar. At higher pressures it is better to use the full eqn (6), already shown to be in very good agreement with Mao’s data for this material up to 774 kbar [7]. The BM equation again gives results too low when compared with P3 up to 278 kbar, but at higher pressures the agreement with the experimental data is very good. Table 9 shows some of the scarce experimental data for potassium, as reported by Vaidya and Kennedy [21]. Only a few points are shown because it is clear that neither pz nor p3 give acceptable results beyond 10 kbar. However, the agreement with the experimental data is acceptable for both, the BM equation and the third order approximation up to the 20 kbar pressure shown in this table. For this material it is by
491
all means advisable to use the full PR equation, in quite good agreement up to 40 kbar, especially if the ‘statistical-approximation’ is used [I. Results for NaCl are shown in Table 10. The experimental data are those of Boehler and Kennedy [22] between 1 and 30 kbar, and those of Bridgman [23] from 10 to 100 kbar. Best results are obtained with the third order approximation. The agreement with the experimental data is good for both equations in the low pressure range, though values computed with the BM equation are slightly better. At higher pressures neither P:, nor P,, give good agreement with Bridgman’s data. Computed values are too high using the third order polynomial, and even higher with the BM equation. Finally, for the five materials here considered, the results obtained by the use of the polynomial approximations to the second and to the third order are summarized in Table 11. This table shows how good is the agreement of both, the third order approximation (P3) and the Birch-Mumaghan (PeM) equation with the experimental data within the pressure ranges here considered. These materials are listed in the order of decreasing values of the shock hardness PC, and for each material this table shows also the range of pressure for which one Pz or the other P, approximation give acceptable results when compared with the full PR equation (p). In the next section we will draw some conclusions from these results.
5. DISCUSSION It is evident from the previous sections that the main purpose of this paper has been accomplished. It is no longer necessary to work with unfamiliar functions, either to perform dubious interpolations, or to make computer programs to use an EOS that seems to be quite good as a PVT correlation at high pressures, in the kbar and the Mbar range. Once the coefficients are known, a quadratic or cubic
Table 11. Summary for the five materials shown in the first column of the results obtained, using the polynomial approximations to the second (pr), and third (p3) order, as well as the agreement of results obtained from the third order polynomial (Pr) and by using the BM equation (I’s,), with the experimental data
Material
PC &bar)
Pressure range (kbar)
Pd Pd cu cu Ag Ag NaCl NaCl K
1176.9 1176.9 930.9 930.9 695.6 695.6 200.6 200.6 30.6
65-155 155-659 I-105 loo-209 77-163 163-209 l-30 30-100 5-20
p2 ?
? OK
? ?
? ? ? ?
p3
OK OK OK OK OK OK OK OK ?
p3 OK ? OK ? OK ? OK ? ?
pm4 OK ? OK OK ? ? OK ? ?
492
F. E. Patnro and C. RENERO
polynomial suffices to obtain a good approximation to the pressure at any temperature. But there is a price to be paid for this simplification: the range of validity of the PVT correlation is considerably reduced. As long as the linear relationship is valid, the exact eqn (6) is also valid, no matter how high is the pressure in Mbar. The polynomial approximations, on the contrary, give good results only in the pressure range of some dozens, or at most one or two hundred kbar, depending upon the material. Furthermore, for some materials, like potassium for example, the polynomial approximations are a complete failure. Of course, if one is interested in some particular material, the best is to try the polynomial approximations and see how good are the results in the range of pressure desired. The few materials tested in the previous section suffice, however, to give some clues on what to expect from the use of these approximations for a given material. To clarify this point let us go back to Table 11. For palladium, the hardest material in our list (P, = 1177 kbar), the second order approximation is good up to 65 kbar, and the third order up to 278 kbar. For copper (P, = 931 kbar), the validity of the approximations is about the same, the second order up to 100 kbar, and the third order up to 180 kbar. The third order approximation has to be used for silver (P, = 696 kbar), between 77 and 209 kbar. This is also the case for NaCl (P, = 200 kbar), the third order approximation has to be used from 1 to 100 kbar. Finally, for potassium, a ‘very soft’ material (P, = 30.6 kbar), the polynomial approximations are useless beyond 5 or 10 kbar. These results suggest that there might be some relation between the shock hardness and the range of validity of the polynomial approximations. This can be understood if one recalls that for very hard materials a very large change in pressure is needed to induce only a very small change in volume and that, in consequence, the relative compression is very small within a very large range of pressures. For soft materials, on the contrary, a very small change in pressure suffices to induce a very large change in volume, which leads to large values of the relative compression within a very low range of pressure. Now, since the expanded variable we used is, by a factor B, the relative compression, one can understand why for hard materials the polynomial approximations are good up to pressures of several hundreds of kbar, whereas for soft materials the second or third order polynomials give very bad results even at very low pressures. In conclusion, the second or third order approximations can be safely used as follows. For hard
materials, with P, higher than 800 kbar, the second degree polynomial is good up to about 100 kbar, and the third order approximation up to 300 kbar. For medium hard materials, with P, between 300 and 800 kbar, the third degree polynomial is good up to about 200 kbar. For soft materials, with shock hardness smaller than 100 kbar, the polynomial approximations are not reliable. Concerning the comparison between our third order approximation (P,) and the Birch-Murnaghan equation, it can be remarked that in the range of pressures up to 60-100 kbar, where P, agrees well with the experimental data the results are as good as those obtained by using the BM equation. The advantage of the polynomial being, as already remarked, its simplicity. Acknowledgements-The
authors are deeply indebted to Prof. F. L. Yarger for a very careful revision of the manuscript, and to Prof. A. L. Loske for kind discussions. REFERENCES 1. Prieto F. E. and Renero C., J. Phys. Chem. SoI& 37, 151 (1976). 2. Prieto F. E. and Renero C., in High Pressure Science and Technology (Edited by K. Timmerhaus and M. S. Barker), Vol. 1, p. 50. Plenum Press, New York (1979). 3. Prieto F. E. and Renero C., in Peaceful Nuclear Explosions, Vol. V. I.A.E.A., Vienna (1978). 4. Handbook of Mathematical Functions (Edited by M. Abramowitz and I. A. Stegun), p. 230. N.B.S. (1964). 5. Prieto F. E., Renero C. and Prieto L. E., Mini calculator programs for the Prieto-Renero equations of state at high pressures. Technical report, IFUNAM (1982). 6. Prieto F. E., J. Phys. Chem. Solids 35, 279 (1974). 7. Prieto F. E. and Renero C., J. Phys. Chem. Solids 43, 147 (1982). 8. Birch F., J. geophys. Res. 83, 1257 (1978). - _ 9. Marsh S. P. (Ed.) LASL Shock Hugoniot Data, p. 658. Universitv of California Press, Berkeley (1980). 10. Jeanloz R., J. geophys. Res. 94, 5873 (1989). 11. Jeanloz R., Phys. Rev. B38, 805 (1988). 12. Vinet P., Smith J. R., Ferrante J. and Rose J. M., Phys. Rev. B35, 1945 (1987). 13. Renero C., Prieto F. E. and de Icaza M., J. Phys. Condens. Matter 2, 295 (1990). 14. Fisher E. S., Manghnani M. H. and Katahara K. in Proceedings of the IV” International Conference High Pressure, Kyoto (1974). 15. Ullmann W. and Pan’Kov V. L.. Zentralinstitut Physik a’er Era& Potsdam (1976). 16. Anderson 0. L., J. Phys. Chem. Solids 27,
W., on fur
547 (1966). 17. Bridgman P. W., Proc. Am. AC. Arts SC. 77, 1989
(1949). 18. Keeler R. N., in American Institute of Physics Handbook. McGraw Hill, New York (1970). 19. Vaidva S. N. and Kennedy G. C., J. Phys. Chem. Solids 31, 2-329 (1970). 20. Mao H. K., Bell P. M., Shaner J. W. and Steinberg D. J., J. appl. Phys. 49, 3276 (1978). 21. Vaidva S. N. and Kennedv G. C.. J. Phvs. Chem. Solids
32, 2-545(1971). r 22. Boehler R. and Kennedy G. C., J. Phys. Chem. Solids 41, 517 (1980). 23. Bridgman P. W., Proc. Am. AC. Arts SC. 76, 1, (1945).
1992
POLYNOMIAL
0022-3697/92 $5.00 + 0.M) 0 1992 Pergamon Press plc
APPROXIMATIONS TO A UNIVERSAL EQUATION OF STATE
F. E. PRIETO and C. RENERO Instituto de Fisica, Universidad National de Mexico, AP 20-364, 01000 Mexico, D. F. (Received
12 December
1990; accepted in revisedform
2 October
1991)
Abstract-To avoid the difficulties inherent in the use of unfamiliar and difficult-to-handle functions appearing in an otherwise seemingly useful universal equation of state at high pressures, a series expansion is made of all the volume dependent functions in the original equation. The whole problem is treated within the framework of a system-independent formulation of the thermodynamics of high pressures, but the expanded variable is essentially the relative compression. Explicit polynomials to the second and third order in this variable are obtained for the pressure as a function of volume. These polynomials are in general very good approximations to the exact PVT correlation for high pressure isotherms at any temperature. The range of validity of the approximations turns out to be related to the hardness of the material. The coefficients of the polynomial approximations are not adjustable parameters, they can be calculated as functions of the physical properties used in the formalism. Numerical values of the coefficients are obtained for 17 materials. The proposed polynomial approximations are numerically tested against the full PrietoRenero equation, also against the widely used Birch-Murnaghan equation, and finally against some of the available experimental results. In each case the validity of the approximation is discussed in detail. Keywords: Universal equation of state, high pressure.
1. INTRODUCTION
for low to moderate values of the applied pressure. Since, in order to get universality the original equation is expressed in reduced variables, the system-independent formulation of the thermodynamics of high pressures proposed by Prieto and Renero [ 11, and by Prieto [6] will be used throughout this paper. However, to simplify the use of the polynomials approximations, the final results will also be given in terms of real physical variables.
Some years ago Prieto and Renero [l] proposed a universal equation of state for materials at shock pressures. This equation was obtained from very general assumptions valid for all kinds of materials. As a consequence, the resulting equation of state (EOS) was proved to be valid for all sorts of materials: elements, compounds, mixtures, solids, liquids and gases [2,3]. The structure of the equation is in principle very simple. It contains a linear temperature term, and several volume dependent terms. Except for one of them, expressed as a combination of exponential integral functions of negative arguments [4], most of these terms are very easy to use, algebraic functions. Although the exponential integral function is tabulated, and some of its properties are well known [4], values of the function obtained by interpolation are not precise enough to be used in the EOS. Thus, computer programs to evaluate the function are required [S]. This fact may inhibit a more extended use of a seemingly useful EOS. To facilitate the use of this equation, we propose in the present paper a simplified form. This is a polynomial approximation obtained from a series expansion of all the algebraic and exponential integral functions appearing in the original equation. Since the expanded variable used is essentially the relative compression, good results are expected for small values of this variable and, as a consequence,
2. THE REDUCED VARIABLES FORMALISM The reduced variables formalism [l, 61 makes use of six physical properties of the material under study: the initial density p,, = l/I’,, the coefficients A and B of the linear relationship (U, = A + BU,) between shock and particle velocities [7], the volume coefficient of thermal expansion CL,the isothermal compressibility KT, and the specific heat at constant volume C,. The scaled variables most suitable for the thermodynamics of high pressures have been shown to be [1,61; P = PIP,,
x = Bq = B(l
-
V/V,,),
t = uBT,
(1)
in which T is the absolute temperature, t] the relative compression, and P, = poA=IB, 485
(2)
F. E.
486
hIET0
is the so-called [6] characteristic pressure or shock hardness of the material. Other thermodynamic quantities can also be reduced using these scaling factors. Of special interest are the reduced compressibility and the reduced specific heat at constant volume, respectively:
and C.
I&NERO
A Maclaurin’s series expansion of F(x) as defined by eqn (7) is then of the form F(x) =f(O) + xf(0) + +/x2~(0) + ; x3P(0) + . . . - rt,(l +mx +4m2x2+$m3x3+*..).
(12)
To obtain the derivative of the last term of the function f(x), let us first obtain the derivative of &(x). From eqn (11) It is also convenient coefficients
to define the two auxiliary d&(z)
e’ (13)
dt=z* m = (B&C”,)-‘,
(4)
and
ap
0
r= at, =CV,m.
(5)
The reduced value of the initial temperature t, = c&T,, corresponding usually to the temperature T, = 293 K at which shock experiments to determine A and B are performed, will be of ample use throughout this paper. 3. THE POLYNOMIAL
APPROXIMATIONS
m(‘-X)Ei(m, x)] &km = me-“(‘-“)Ei(m,
2-m
= ~_2;;f(mx)4)_!!(3_m)e””
(15)
y, = 1 - mrtO,
(16)
y2 = 2 - im*rt,,
(17)
y,=3-jm-im3rt
(7)
0’
(18)
Let us now denote by p,,(x, t) the polynomial obtained from the series expansion (15) truncated at the term in x”, that is PAX, t) = r(t - to) + y,x + y2x2
2(1 -x)2
+y3x3 +. . ’ + ynxn.
+ : (4m - m2 - 2)e-“” -“)Ei(m, x),
(8)
with Ei(m, x) = R(m)
(14)
with
(6)
and the auxiliary function f (x) is defined by f(x)
x) + &.
The determination of the coefficients in the series expansion (12) is not straightforward. One obtains: f(0) = 0, f(0) = 1, f”(0) = 4,f”(O) = 18 - 2m. Consequently
where F(x) =f (x) - rt0emX,
with eqn (9),
P(X,t)=r(t-t,)+Y,X+Y2X2+Y3X3+‘.‘,
When expressed in system-independent form by the use of the scaled variable defined by eqns (l)-(5), the EOS takes the form [l] p(x, t) = F(x) + rt,
Using now this result in combination one obtains
- &[m(l - x)].
(9)
(19)
For the isotherms at room temperature, i.e. t = t,,, which is the most usual case, these polynomials become functions of one single variable P”(X, to) = p,(x)
Ei(z) is the exponential integral function of negative argument, which is defined by [4]
= y,x + y2xz+ y3x3 +
. . + ynxn.
(20)
Of special interest are the first three polynomials. G(z) = or, alternatively,
m c dy, s --I Y
z > 0,
(10)
by [4]
lQz)=y
+Inz+
ZN f N=, 5-Z’
(21)
P*(x) = Ylx, P2(X) =
Ylx +
Y2X27
(22)
P,(X)
Ylx +
72x2 + Y3X3.
(23)
and (11)
=
Polynomial approximations
to a universal equation of state
We shall see later that pr(x), or eventually Pa, gives, for many materials, very good approximations to the values of pressure computed using the exact eqn (6), and to the experimental values reported by various authors as well as to the Birch-Murnaghan equation. Equations (21)-(23) can be transformed to give directly the physical pressure by inverting the transformation (1) leading to the reduced pressure. They can be written in the form p, =r,rl>
(24)
p*=r,q
+r2r12,
(25)
p,=r,q
+r2q2+r3q3,
(26)
and
where P, , P2, and P3 have now the same units of pressure as PC, and n is the relative compression. The transformed coefficients are then r, = PcBy,,
(27)
r2 =
(28)
PcB2y2,
and
r3= PcB3y3.
(29)
It is to be remarked that whereas the yis are dimensionless quantities, the Tis have units of pressure. If isotherms other than those at room temperature are of interest, then a fourth coefficient r4 for the temperature term is needed. This is given by r, = urBP,,
(30)
487
and the additional term r, (T - To) must be included in eqns (24)-(26). To show the usefulness of the polynomial approximations to the second and third order in the relative compression, we will apply them to some materials. Table 1 shows, for 17 materials, values of all the quantities needed to compute the coefficients, and to perform the transformation of variables required to use the polynomial approximations. Most of these values were taken from [l], [7] and [8], where the sources of the pertinent experimental data can be found. A few comments on the values given in this table are perhaps appropriate. For most materials the slope B of the linear relationship between shock and particle velocities takes values around 1.4 [9]. Values of the parameter m are in the neighbourhood of 413, and those of the parameter r are very near to 1.0; this ‘statistical approximation’ is widely discussed in [l]. The last general remark concerning the values of these parameters is that the reduced initial temperature t,, is in general very small and even negligible compared with the integer numbers appearing in the coefficients defined by eqns (16)-( 18) or (27)-(29). But it is also to be remarked, however, that the contribution of the term in the initial temperature can be neglected only in relation to the evaluation of these coefficients; for temperatures other than t,,, the contribution of the temperature term in eqns (15) or (19) may become very important and far from negligible. The previous remarks lead immediately to a couple of approximate equation of state correlations that might be of some interest. Within the scope of validity of the statistical approximation, the coefficients of the
Table 1. Summary of numerical values of the parameters needed to use the polynomial approximations. The first column identifies the material. The second, third and fourth columns give the calculated values of m, r and r,, Columns five and six give the values of a, the coefficient of volume expansion, and E from the relation U, = A + BU, Material W MO Pt co Ni Cr V Pd Au Ti cu AS Zn Sn Cd K NaCl
m
1.390 1.309 1.731 1.624 1.274 0.780 1.072 1.382 1.964 1.488 1.327 1.535 1.300 1.501 1.403 1.143 1.200
r
t0
1.036 1.017 1.004 0.966 0.975 0.984 1.029 0.967 0.951 0.852 0.948 0.912 0.902 1.068 0.905 0.887 0.885
0.00488 0.00538 0.01206 0.01600 0.01608 0.008 11 0.00819 0.01614 0.01928 0.00570 0.02194 0.02671 0.04065 0.02744 0.0462 1 0.08536 0.04702
a
x
105(“c-‘) 1.34 1.49 2.66 4.10 3.80 1.89 2.33 3.47 4.22 2.54 5.03 5.75 8.90 6.34 9.44 24.9 11.95
B
1.237 1.233 1.544 1.330 1.445 1.465 1.201 1.588 1.560 1.089 1.489 1.586 1.559 1.476 1.671 1.170 1.343
488
F. E.
PR~ETO and C.
series development, truncated to the second or third order, take the particular values y, = 1 - ft,,
(31)
y*=2-;t,,
(32)
y, = 2.5555 - Et,, .
(33)
and
These coefficients may be taken as universal, i.e. they may have the same value for all materials, either by neglecting the terms in t, that, as previously remarked, are very small, or by performing the experiments to determine A and B at an initial temperature T, such that t, is the same for all materials [l]. In the first case, by neglecting the terms in t,, one obtains for the second and the third order approximations &(X) = x - 2x2,
(34)
p3(x) = x -2x2-2.5555x3.
(35)
and
Finally, introducing gets
the mean value 1.4 for B [9] one
p3(q)= 1.41 -4$-7713.
(36)
These approximate expressions are very good to make quick estimations of pressure. If more precision is required, one should use eqns (21)-(26), with the exact values of the coefficients for the polynomial approximations, eqns (16)-( 18) or (27)-(29). Table 2 shows, for the same materials of Table 1, values of the coefficients of the polynomial approximations up to third order. They were computed using Table 2. Values of the coefficients y, , y2, and y, as defined by eqns (16), (17) and (18), for each of the materials shown in the first column. The last column gives the values of the shock hardness PC = p. A2/B Material W MO Pt co Ni Cr V Pd Au Ti cu AS Zn Sn Cd K NaCl
YI
Y2
0.9930 0.9928 0.9790 0.9749 0.9800 0.9938 0.9910 0.9785 0.9640 0.9927 0.9725 0.9626 0.9523 0.9560 0.9413 0.9135 0.9501
1.9951 1.9953 1.9819 1.9769 1.9873 1.9976 1.9952 1.9851 1.9646 1.9946 1.9817 1.9713 1.9690 1.9670 1.9588 1.9505 1.9700
Y3 2.5344 2.5616 2.4125 2.4476 2.5699 2.6392 2.6410 2.5325 2.3215 2.5013 2.5496 2.4737 2.5532 2.4832 2.5130 2.6002 2.6000
F&NERO
the data of Table 1, and using eqns (16)-(18). It also shows, for each material, the values of the shock hardness PC. Perhaps the only pertinent comment concerning these values is that, as previously stated, yI takes values near to 1.O, y2 to 2.0, and y3 to 2.5, for all materials. The polynomial forms presented here are obtained as approximations to a complete system-independent equation of state which has been shown to give, in general, very good agreement with the experimental data for various materials [7]. A comparison between the results obtained using both, the full Prieto-Renero (PR) equation, and the polynomial approximations will be treated in the following sections. But is also seems interesting to complete the analysis by comparing these approximations with other polynomial forms currently used. Among the most recent and critical reviews on equations of state in polynomial forms we shall refer to the work by Jeanloz [lo, 111. Let us first remark that among the equations considered by Jeanloz is a recently proposed universal equation of state by Vinet et al. [12]. A detailed comparison of this equation with our full eqn (6) has been already published [ 131;thus, it will not be treated in the present paper. It is concluded in Jeanloz’s papers that the third order Birch-Murnaghan (BM) equation is the most successful among the other five polynomial forms in representing shock wave data. It is also shown that this equation is nearly identical to the linear relationship between shock and particle velocities usually obtained in shock experiments. Since our complete equation of state is based on the validity of this linear relationship, we will restrain the comparison with other polynomial forms to that with the BM. We shall use the third order Birch-Murnaghan [8] in the form P,,(f)
= 3&f(l
+ 2f)s’2[l + ?(K;, - 4Y],
(37)
where
P, (kbar) 2522.7 2171.9 1795.1 1489.9 1323.5 1319.0 1309.1 1176.9 1166.2 945.8 930.9 695.6 426.0 343.7 308.6 30.6 200.6
f= f[(V#)2”
- 11,
(38)
with & and Kh being the isothermal bulk modulus and its pressure derivative. Values of the couples (&, Kh) for the five materials here considered are: (1808.3 kbar, 5.42) for Pd [14], (1309.2 kbar, 5.65) for Cu [15], (1007.1 kbar, 5.53) for Ag [15], (31.77 kbar, 3.98) for K [16], and (238.36 kbar, 5.35) for NaCl[81. 4. APPLICATION
TO SOME
MATERIALS
We shall now analyse, for a few materials covering a wide range of shock hardness, the results obtained by the use of the polynomial approximations to the
Polynomial approximations
to a universal equation of state
489
Table 3. The first and second columns give, for copper, the experimental pressure Pa and the relative compression I, as reported by Bridgman [17]. The third column shows the corresponding values of x = Bn. The fourth and fifth columns give the calculated values of the second degreep,, and the third degree p, polynomials. The sixth column gives, for comparison, the pressure computed using the exact eqn (6). The last two columns give the physical pressure computed using the third order polynomial (P,), and the Birch-Mumaghan equation (PBM) P,(kbar) 5
10 15 20 25 30
rl
x
P2
P,
P
0.004 0.007 0.011 0.015 0.018 0.021
0.0060 0.0104 0.0164 0.0223 0.0268 0.0313
0.0059 0.0104 0.0165 0.0227 0.0275 0.0323
0.0059 0.0104 0.0165 0.0227 0.0275 0.0324
0.0059 0.0104 0.0165 0.0230 0.0275 0.0324
P,(kbar) 5.45 9.64 15.33 21.16 25.63 30.18
Pa” (kbar) 5.30 9.38 14.94 20.64 25.03 29.50
Table 4. Experimental data for copper, as reported by Keeler [18]. The column headings correspond to those of Table 3 P,(kbar) 15 20 25 30 35 40 45
rl 0.010 0.014 0.017 0.020 0.023 0.026 0.029
X 0.0149 0.0209 0.0253 0.0298 0.0343 0.0387 0.0432
P2 0.0149 0.0211 0.0259 0.0307 0.0356 0.0406 0.0457
second and third order. These results are shown in Tables 3, 4, 5 and 6 for copper, in Table 7 for silver, in Table 8 for palladium, in Table 9 for potassium, and in Table 10 for sodium chloride. With the exception of NaCl, these materials are the same ones studied in [7] where the agreement of pressure values computed using the full PR equation (6) with the experimental data was fully discussed. We include now the sodium chloride because being a pressure standard, experimental data are quite abundant. We shall now focus attention on the fast or slow conver-
P, 0.0149 0.0211 0.0259 0.0308 0.0357 0.0408 0.0459
P
0.0150 0.0210 0.0260 0.0301 0.0360 0.0408 0.0460
P,(kbar) 13.89 19.69 24.14 28.65 33.26 37.95 42.72
Par,,(kbar) 13.53 19.20 23.56 28.00 32.53 37.15 41.86
gence of the series expansion, and on the differences between pressure values computed with the exact eqn (6), those computed using the polynomials (22) and (23) to the second and third order, and those obtained through the third order Birch-Murnaghan equation. Each table lists the experimental data in pressure and in relative compression, the variable x used for the series expansion, the values of pressure computed to the second and third order, and finally, for comparison, values of pressure computed with the full eqn (6), and with the Birch-Murnaghan equation.
Table 5. Experimental data for copper, as reported by Vaidya and Kennedy [19]. The column headings correspond to those of Table 3 P,(kbar) 5 10 15 20 25 30 35 40 45
0.0033 0.0065 0.0097 0.0128 0.0159 0.0189 0.0218 0.0247 0.0276
x
p,
p,
P
P3War)
PBMWar)
0.0049 0.0097 0.0144 0.0190 0.0237 0.028 1 0.0324 0.0368 0.0411
0.0048 0.0096 0.0144 0.0192 0.0241 0.0289 0.0337 0.0385 0.0433
0.0048 0.0096 0.0145 0.0193 0.0242 0.0290 0.0338 0.0386 0.0435
0.0048 0.0096 0.0145 0.0193 0.0242 0.0290 0.0340 0.0386 0.0440
4.47 8.94 13.47 17.94 22.49 26.99 31.41 35.90 40.48
4.36 8.69 13.11 17.49 21.95 26.36 30.71 35.14 39.65
Table 6. Experimental data for copper, as reported by Mao et al. [20]. The column headings correspond to those of Table 3 P,(kbar) 73 101 127 160 180 186 209
rl
X
P,
0.047 0.062 0.074 0.089 0.097 0.100 0.109
0.0700 0.0923 0.1102 0.1325 0.1444 0.1489 0.1623
0.0777 0.1067 0.1312 0.1637 0.1818 0.1887 0.2100
p3
0.0786 0.1087 0.1347 0.1696 0.1895 0.1972 0.2209
P
0.0787 0.1089 0.1351 0.1706 0.1909 0.1988 0.2233
P,(kbar) 73.20 101.16 125.32 157.89 176.38 183.53 205.67
PBM War) 72.19 100.38 125.03 158.73 178.12 185.66 209.21
490
F. E.
hIET0
and C. RENERO
Table 7. Experimental data for silver, as reported by Mao et ai. [20]. The column headings correspond to those of Table 3 9
x
p2
p3
P
77
0.059
1: 190 209 66 163
0.070 0.092 0.119 0,127 0.052 0.107
0.0936 0.1110 0.1459 0.1887 0.2014 0.0825 0.1697
0.1073 0.1318 0.1824 0.2519 0.2739 0.0928 0.2201
0.1094 0.1345 0.1901 0.2685 0.2941 0.0941 0.2322
0.1096 0.1350 0.1915 0.2721 0.2996 0.0943 0.2349
Pafkbar)
P,(kbar) 76.07 93.59 132.24 186.78 204.56 65.51 161.53
PBM(kbar) 72.45 89.32 126.90 181.08 199.07 62.33 155.81
Table 8. Experimental data for palladium, as reported by Mao et al. [20]. The column headings correspond to those of Table 3 PF (kbar)
?
X
A
P,
P
65 155 278 659
0.032 0.067 0.106 0.187
0.0508 0.1064 0.1683 0.2970
0.0548 0.1266 0.2209 0.4656
0.0552 0.1296 0.2330 0.5319
0.0552 0.1300 0.2358 0.5639
In Tables 6, 7, 8 and 10, the highest value of pressure shown is that for which the difference between pressure P computed using the exact eqn (6) is larger than 1% of the pressure computed using the polynomial forms. Let us now make some remarks on the results shown in these tabies. Data for copper are shown in Tables 3-6. Table 3 is based on the experimental data reported by Bridgman 1171,Table 4 by Keeler [lS], Table 5 by Vaidya and Kennedy [19]. and Table 6 by Mao et al. [20]. It can be remarked from Tables 3,4 and 5, that up to 45 kbar the two-term polynomial gives results very
P,(kbar)
PBM(kbar)
64.94 152.56 274.25 626.04
64.23 151.33 274.52 656.17
near to those computed by using the full PR equation. Thus, it is unnecessary to go to the next term. At the higher pressures shown in Table 6, the third order approximation gives agreements within 1% with values computed using the full equation. Concerning the agreement with the experimental data and with the Birch-Murnaghan equation, the following remarks are pertinent. Up to 45 kbar, both the third order approximation and the BM equation give results in good agreement between themselves. The agreement with the experimental data is in both cases quite good with Bridgman’s [17] data up to 30 kbar,
Table 9. Experimental data for potassium, as reported by Vaidya and Kennedy [21]. The column headings correspond to those of Table 3 P&bar) 5 10 15 20
9 0.1191 0.1927 0.2450 0.2868
X 0.1393 0.2255 0.2866 0.3356
PZ 0.1652 0.3051 0.4221 0.5261
s
0.1722 0.3349 0.4833 0.6244
P
0.1735 0.3453 0.5134 0.6858
P,(kbar) 5.27 10.25 14.79 19.11
PsM(kbar) 5.18 10.41 15.63 21.07
Table 10. Experimental data for sodium chloride as reported by Boebler and Kennedy [22] between 1 and 30 kbar, and that of Bridgman [23] from 10 to 100 kbar. The column headings correspond to those of Table 3
Pa@bad 1.06 5.62 11.74 19.95 25.69 29.53 9.80 19.61 29.42 39.22 49.03 58.83 68.64 78.44 88.25 98.05
v 0.0042 0.0222 0.0428 0.067 1 0.0828 0.0923 0.038 0.068 0.093 0.115 0.135 0.152 0.168 0.183 0.197 0.210
X
p2
0.0056 0.0298 0.0575 0.0901 0.1112 0.1240 0.0510 0.0913 0.1249 0.1544 0.1813 0.2041 0.2256 0.2458 0.2646 0.2820
0.0054 0.0301 0.0611 0.1016 0.1300 0.1480 0.0536 0.1032 0.1499 0.1937 0.2370 0.2760 0.3147 0.3525 0.3893 0.4247
p3
P
0.0054 0.0301 0.0616 0.1035 0.1336 0.1530 0.0540 0.1052 0.1545 0.2033 0.2525 0.2982 0.3445 0.3911 0.4374 0.4830
0.0054 0.0301 0.0616 0.1035 0.1336 0.1530 0.0540 0.1054 0.1553 0.2054 0.2566 0.3048 0.3548 0.4060 0.458 1 0.5104
P,@W
1.087 6.041 12.359 20.765 26.797 30.691 10.82 21.09 30.98 40.78 50.65 59.81 69.11 78.45 87.75 96.88
PBM@bar)
1.014 5.684 11.724 19.943 25.970 29.922 10.24 20.27 30.22 40.40 50.98 61.13 71.77 82.81 94.16 105.69
Polynomial approximations
to a universal equation of state
but computed values are too low when compared with the experimental data reported by Keeler [18], and by Vaidya and Kennedy [19] up to 45 kbar. The agreement is better with Mao’s [20] data up to 101 kbar. Beyond this pressure the BM equation gives results in better agreement with the experimental data up to 209 kbar. This indicates that our third order approximation for copper is good up to about 100 kbar, beyond this pressure it is better to use the full eqn (6), in excellent agreement with Mao’s data up to 781 kbar [7]. Table [7] shows the data for silver reported by Mao et al. [20]. It is obvious that the second order approximation is not sufficient within the pressure range of 77-209 kbar. The third order polynomial gives very good results up to 163 kbar. For this material the BM equation gives pressures systematically lower than the P, approximation, and even lower than the experimental data. The third order approximation agrees with Mao’s data up to 153 kbar. At higher pressures, up to 918 kbar, it has been shown that the full PR equation gives very good agreement with these data [7]. For palladium, shown in Table 8, also from the data by Mao et al. [20], the results are somewhat similar. The third order approximation is quite acceptable up to 155 kbar. At higher pressures it is better to use the full eqn (6), already shown to be in very good agreement with Mao’s data for this material up to 774 kbar [7]. The BM equation again gives results too low when compared with P3 up to 278 kbar, but at higher pressures the agreement with the experimental data is very good. Table 9 shows some of the scarce experimental data for potassium, as reported by Vaidya and Kennedy [21]. Only a few points are shown because it is clear that neither pz nor p3 give acceptable results beyond 10 kbar. However, the agreement with the experimental data is acceptable for both, the BM equation and the third order approximation up to the 20 kbar pressure shown in this table. For this material it is by
491
all means advisable to use the full PR equation, in quite good agreement up to 40 kbar, especially if the ‘statistical-approximation’ is used [I. Results for NaCl are shown in Table 10. The experimental data are those of Boehler and Kennedy [22] between 1 and 30 kbar, and those of Bridgman [23] from 10 to 100 kbar. Best results are obtained with the third order approximation. The agreement with the experimental data is good for both equations in the low pressure range, though values computed with the BM equation are slightly better. At higher pressures neither P:, nor P,, give good agreement with Bridgman’s data. Computed values are too high using the third order polynomial, and even higher with the BM equation. Finally, for the five materials here considered, the results obtained by the use of the polynomial approximations to the second and to the third order are summarized in Table 11. This table shows how good is the agreement of both, the third order approximation (P3) and the Birch-Mumaghan (PeM) equation with the experimental data within the pressure ranges here considered. These materials are listed in the order of decreasing values of the shock hardness PC, and for each material this table shows also the range of pressure for which one Pz or the other P, approximation give acceptable results when compared with the full PR equation (p). In the next section we will draw some conclusions from these results.
5. DISCUSSION It is evident from the previous sections that the main purpose of this paper has been accomplished. It is no longer necessary to work with unfamiliar functions, either to perform dubious interpolations, or to make computer programs to use an EOS that seems to be quite good as a PVT correlation at high pressures, in the kbar and the Mbar range. Once the coefficients are known, a quadratic or cubic
Table 11. Summary for the five materials shown in the first column of the results obtained, using the polynomial approximations to the second (pr), and third (p3) order, as well as the agreement of results obtained from the third order polynomial (Pr) and by using the BM equation (I’s,), with the experimental data
Material
PC &bar)
Pressure range (kbar)
Pd Pd cu cu Ag Ag NaCl NaCl K
1176.9 1176.9 930.9 930.9 695.6 695.6 200.6 200.6 30.6
65-155 155-659 I-105 loo-209 77-163 163-209 l-30 30-100 5-20
p2 ?
? OK
? ?
? ? ? ?
p3
OK OK OK OK OK OK OK OK ?
p3 OK ? OK ? OK ? OK ? ?
pm4 OK ? OK OK ? ? OK ? ?
492
F. E. Patnro and C. RENERO
polynomial suffices to obtain a good approximation to the pressure at any temperature. But there is a price to be paid for this simplification: the range of validity of the PVT correlation is considerably reduced. As long as the linear relationship is valid, the exact eqn (6) is also valid, no matter how high is the pressure in Mbar. The polynomial approximations, on the contrary, give good results only in the pressure range of some dozens, or at most one or two hundred kbar, depending upon the material. Furthermore, for some materials, like potassium for example, the polynomial approximations are a complete failure. Of course, if one is interested in some particular material, the best is to try the polynomial approximations and see how good are the results in the range of pressure desired. The few materials tested in the previous section suffice, however, to give some clues on what to expect from the use of these approximations for a given material. To clarify this point let us go back to Table 11. For palladium, the hardest material in our list (P, = 1177 kbar), the second order approximation is good up to 65 kbar, and the third order up to 278 kbar. For copper (P, = 931 kbar), the validity of the approximations is about the same, the second order up to 100 kbar, and the third order up to 180 kbar. The third order approximation has to be used for silver (P, = 696 kbar), between 77 and 209 kbar. This is also the case for NaCl (P, = 200 kbar), the third order approximation has to be used from 1 to 100 kbar. Finally, for potassium, a ‘very soft’ material (P, = 30.6 kbar), the polynomial approximations are useless beyond 5 or 10 kbar. These results suggest that there might be some relation between the shock hardness and the range of validity of the polynomial approximations. This can be understood if one recalls that for very hard materials a very large change in pressure is needed to induce only a very small change in volume and that, in consequence, the relative compression is very small within a very large range of pressures. For soft materials, on the contrary, a very small change in pressure suffices to induce a very large change in volume, which leads to large values of the relative compression within a very low range of pressure. Now, since the expanded variable we used is, by a factor B, the relative compression, one can understand why for hard materials the polynomial approximations are good up to pressures of several hundreds of kbar, whereas for soft materials the second or third order polynomials give very bad results even at very low pressures. In conclusion, the second or third order approximations can be safely used as follows. For hard
materials, with P, higher than 800 kbar, the second degree polynomial is good up to about 100 kbar, and the third order approximation up to 300 kbar. For medium hard materials, with P, between 300 and 800 kbar, the third degree polynomial is good up to about 200 kbar. For soft materials, with shock hardness smaller than 100 kbar, the polynomial approximations are not reliable. Concerning the comparison between our third order approximation (P,) and the Birch-Murnaghan equation, it can be remarked that in the range of pressures up to 60-100 kbar, where P, agrees well with the experimental data the results are as good as those obtained by using the BM equation. The advantage of the polynomial being, as already remarked, its simplicity. Acknowledgements-The
authors are deeply indebted to Prof. F. L. Yarger for a very careful revision of the manuscript, and to Prof. A. L. Loske for kind discussions. REFERENCES 1. Prieto F. E. and Renero C., J. Phys. Chem. SoI& 37, 151 (1976). 2. Prieto F. E. and Renero C., in High Pressure Science and Technology (Edited by K. Timmerhaus and M. S. Barker), Vol. 1, p. 50. Plenum Press, New York (1979). 3. Prieto F. E. and Renero C., in Peaceful Nuclear Explosions, Vol. V. I.A.E.A., Vienna (1978). 4. Handbook of Mathematical Functions (Edited by M. Abramowitz and I. A. Stegun), p. 230. N.B.S. (1964). 5. Prieto F. E., Renero C. and Prieto L. E., Mini calculator programs for the Prieto-Renero equations of state at high pressures. Technical report, IFUNAM (1982). 6. Prieto F. E., J. Phys. Chem. Solids 35, 279 (1974). 7. Prieto F. E. and Renero C., J. Phys. Chem. Solids 43, 147 (1982). 8. Birch F., J. geophys. Res. 83, 1257 (1978). - _ 9. Marsh S. P. (Ed.) LASL Shock Hugoniot Data, p. 658. Universitv of California Press, Berkeley (1980). 10. Jeanloz R., J. geophys. Res. 94, 5873 (1989). 11. Jeanloz R., Phys. Rev. B38, 805 (1988). 12. Vinet P., Smith J. R., Ferrante J. and Rose J. M., Phys. Rev. B35, 1945 (1987). 13. Renero C., Prieto F. E. and de Icaza M., J. Phys. Condens. Matter 2, 295 (1990). 14. Fisher E. S., Manghnani M. H. and Katahara K. in Proceedings of the IV” International Conference High Pressure, Kyoto (1974). 15. Ullmann W. and Pan’Kov V. L.. Zentralinstitut Physik a’er Era& Potsdam (1976). 16. Anderson 0. L., J. Phys. Chem. Solids 27,
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32, 2-545(1971). r 22. Boehler R. and Kennedy G. C., J. Phys. Chem. Solids 41, 517 (1980). 23. Bridgman P. W., Proc. Am. AC. Arts SC. 76, 1, (1945).