b A10 39 NAVAL SURFACE WEAPONS CENTER SILVER SPRING MD F/S 7/4THEORETICAL EQUATION OF STATE FOR SIMPLE LIQUIDS AT HIGH PRESSU-ETC(U)WAR 81 M 0 J.ONES

UNCLASSIFIED MA I LOE

SLENDE7T

I

0- NSWC TR 80-419

THEORETICAL EQUATION OF STATE FOR SIMPLE

LIQUIDS AT HIGH PRESSURES

BY HERMENZO D. JONES

RESEARCH AND TECHNOLOGY DEPARTMENT DTICIfELECTE

26 MARCH 1981 JUN 1 91981.

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THEORETICAL EQUATION OF SATE FOR SIMPLE ' Sep *78 -Aug kSO.,4. PERFORMING OR4. REPORT W3bMNERI LIQUIDS AT HIGH PRESSURES,.

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Equation of State, Dense Fluids, Shock Hugoniot, Perturbation Theory

S20. A IIST RACT ('Con tn.on Ir-ir side it necssay mod Idonel l by block nmbet)A perturbation technique which is particularly well suited for high pressures

is used to calculate thermodynamic properties for a simple liquid whosemolecules interact via a Buckingham (exp-6) potential. There is gccdagreement between computations and experiments for moderate temperatureisotherms for liquid nitrogen. A calculation of the shock Hugoniot of liouidnitrogen exhibits excellent agreement with experimental data..

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NSWC TR 80-419

FOREWORD

A description of simple, dense fluids based on intermolecular forces hasbeen investigated utilizing liquid-state perturbation theory. This work is

applicable to the modeling of reaction products of explosives. Funding for thiswork was provided by both the Independent Research and Explosives 6.2 BlockPrograms at the Naval Surface Weapons Center.

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CONTENTS

Paae

INTRODUCTION ............ ................................ 4

THEORY .............. ................................... 5

RESULTS AND DISCUSSION ........ .. ........................... 7

SUMMARY ........... .................................. .10

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ILLUSTRATIONS

Figure Page

I COMPRESSIBILITY VS PRESSURE FOR LIQUID NITROGEN .... .......... 92 SHOCK PRESSURE VS RELATIVE VOLUME FOR LIQUID NITROGEN .......... 113 SHOCK VELOCITY VS PARTICLE VELOCITY FOR LIQUID NITROGEN ......... 13

TABLES

Table Page

I THERMODYNAMIC PROPERTIES FOR EXP-6 POTENTIAL .... ............ 82 SHOCK HUGONIOT FOR LIQUID NITROGEN ..... ................. .12

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INTRODUCTION

A description of even simple liquids in the high press re domain is aproblem of extreme complexity. Aside from the work of Rossy, who employs avariational technique, most efforts have been directed towards lattice theoryapproximations; but it is well known that the latter are inadequate. However,perturbation theory appears to provide a simple, but accurate characterizationof dense liquids.

The formalism of Weeks, Chandler and Anderson2 is employed to describefluids whose molecules interact via a spherically symmetric modified Buckingham(exp-6) potential. It is assumed that the repulsive forces provide the dominatecontribution to the properties of dense fluids. The intermolecular potentialis divided into a reference part which is repulsive in character and aperturbation which is attractive. Division of the potential is carried outso that the resulting free energy is a minimum or essentially constant withrespect to the break point as suggested by Ree3 . With the prescription ofVerlet and Weis (VW),4 the radial distribution function for the reference systemis akin to one for hard spheres with a diameter which is a function of thetemperature and density. First-order properties can be written analyticallyaside from several, simple numerical integrals.

Within the above framework several calculations are performed. Theisothermal compressibility and internal energy are computed for a systemgoverned by an exp-6 intermolecular potential for high temgeratures andpressures and are compared to Monte Carlo (MC) predictions . Then propertiesfor a real system, liquid nitrogen, are considered. It is assumed that theinternal modes are unchanged from the gaseous state. Retention of the freerotation is based on the observations of Smith, et a16 which indicates thatthere is a transition from free rotation in the solid-$ phase to hinderedrotation in the solid-a phase.

1 Ross, M., Phys. Rev. A8, 1466 (1973).

2Weeks, J. D., Chandler, D. and Anderson, H. C., J. Chem. Phys. 54, 5237 (1971).3 Ree, F. H., J. Chem. Phys. 64, 4601 (1976).

4Verlet, L., and Weis, J., Phys. Rev. A5, 939 (1972).

5 Ross, M. and Alder, B. J., J. Chem. Phys. 46, 4203 (1967).

6 Smith, A. L., Keller, W. E. and Johnston, H. L., Phys. Rev. 79, 728 (1950).

4

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Moderate temperature isotherms and a shock Hugoniot are calculated andare compared to experimental data7'8.

THEORY

The perturbation theory to be discussed here is based on the assumptionthat the repulsive intermolecular forces provide the domi ant characteristics ofthe material. This technique, originated by Weeks, et ala, should be quiteapplicable to the high pressure domain.

It is assumed that the intermolecular interaction is spherically symmetric

and is taken as

0(r) = vo(r) + w(r) , (1)

where vo(r) is the reference potential with repulsive character and w(r) isthe attractive perturbation. The reference potential is given by

o(r) - *(x) rxv0 (r) = 0 r>X (2)

and the perturbation is taken as

w(r) r>x (3)

where

0(r) = e{(6/a)exp[(1-r/rm)]- (rm/r)6}/(1-6/a). (4)

The potential parameters denote the usual quantities. Specifically, Cis the well depth, a is the steepness parameter, and rm is the position of theminimum of the potential. Historically, the potential is divided at theminimum 2 ,4 ; however, Ree 3 demonstrated that the region of applicability could beextended to the high density domain with the additional flexibility of avariable x. Hence x is here taken as a variational parameter.

7Robertson, S. L. and Bann, Jr., S. E., J. Chem. Phys. 50, 4560 (1969).

8Dick, R. D., J. Chem. Phys. 52, 6021 (1970).

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Csing standard techniques9 the free energy can be written as a perturbativeseries in inverse temperature, which is well suited for the shock wave domain.The result for the excess free energy per particle is

f = f +/2 f drw(r)g (r) + (5)

where f and g (r) are the free energy per particle and radial distributionfunctioR for tRe reference system, respectively, and the number density isdenoted by p. In this work only the two leading terms in the expansion ofEquation 5 are retained.

The steep, discontinuous behavior of v (r) is reminiscent of a hard spherepotential. Application of a functional Tay~or expansion yields the resultthat the free energy of the reference system is identical to second-orderwith that for hard spheres with a temperature and density-dependent hardsphere diameter, so that

f 0 fHs (6)

where

g (r) = e-BVo(r)Y s(r) , (7)

and

(r)(8gls (r) re)Hs YHs(r)

The hard-sphere diameter is determined from the condition

fd {e- vo(r) - e- Hs(r)}y s(r) = 0 . (9)

This is the requirement that the long wavelength structure factors for thereference system and the hard-sphere system be equal.

Combination of thiresults of VW with the Percus-Yevic description ofthe hard-sphere system yields analytic results for the thermodynamic statevariables aside from several simple, numerical integrals. The compressibilityand excess internal energy per particle are obtained from analytic derivativesof the free energy as

Z : / + pi ( La__Tpl

andE /_ -E

9Zqanzig, R. W., J. Chem. Phys. 22, 1420 (1954).

lOAnderson, H. C., Chandler, D. and Weeks, J. D., J. Chem. Phys. 56, 3812 (1972).11Wertheim, M. S., Phys. Rev. Lett. 10, E501 (1963).

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It was found that the compressibility defined in this manner rather thanthe original analysis of VW compared more favorably with MC results in higherpressure and temperature region.

RESULTS AND DISCUSSION

The isothermal properties for a typical exp-6 potential with the parameters

= 13.5

rm 3.85 x 10-10 m

c/k = 122°K

are given in Table 1. Here, the reduced temperature is defined as T* = kT/E, and

the reduced density is taken as p* = (rW/47)p. For a material such as argon, the

lower temperature isotherm ranges from 2.5 GPa to 16.6 GPa, while the highertemperature isotherm varies from 16 GPa to 100 GPa. It is seen that perturbationtheory is in excellent agreement with Monte Carlo calculations, except for thevery high density regions for the respective isotherms. In both instances theselarger deviations result from the entrance into the meta stable region for thehard-sphere reference system.

To consider a real system, moderate temperature isotherms for liquidnitrogen are computed. The values of the potential parameters are taken as

a = 13.6

rm = 4.12 x 10- 0 m

e/k = 100.OK

The compressibility as a function of the pressure is shown in Figure 1.The solid curves are the perturbative predictions, and the circles representthe experimental data It is seen that the calculation exhibits very goodagreement with the data, with a maximum error of 5%. Thus, the theorydescribes the liquid quite well in the moderate pressure and temperature range.

As a final application of this technique, the shock compression ofliquid nitrogen is considered. The initial conditions for the material aregiven as To=770K and po=8.1 x 102 kg/m 3. The same potential parameters usedin the previous calculation of moderate temperature isotherms are used forthis computation. In this calculation a value of volume for the compressedstate was chosen, and then a value of the temperature was found by iterationto satisfy the energy conservation condition,

E - E0 (P+Po)(vo-V) , (12)

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TABLE 1

THERMODYNAMIlC PROPERTIES FOR EXP. -6 POTENTIAL

T*= 20

Monte Carlo 5 Perturbation Theory

Z BE Z B

.9 3.27 .28 3.30 .281.0 3.81 .38 3.83 .381.25 5.52 .73 5.56 .751.50 8.07 1.34 7.85 1.321.75 11.34 2.17 10.3 2.13

T* =100

1.264 3.06 .55 3.09 .571.473 3.71 .74 3.71 .741.768 4.81 1.08 4.75 1.062.431 8.17 2.20 7.57 2.032.701 9.86 2.78 9.11 2.78

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10.0

T =308.150 K

8.&0

6.0

4.0

20

0.2 0.4 0.6 0.8 1.0

P(G Pa)

FIGURE 1 COMPRESSIBILITY VS. PRESSURE FOR LIQUID NITROGEN

NSIC TR 80-419

where the zero subscript refers to material in the undisturbed state aheadof the shock front. In Equation (12), E and v are the total specific energyand specific volume, respectively.

The theoretical shock Hugoniot in the presure-volume plane for liquidnitrogen is compared with the experimental dataO in Figure 2. The datacould be interpreted as having a cusp in the 14 GPa region, signallingthe onset of a phase transformation; however, there is good agreementbetween the present calculation and experiment. Similar results areexhibited in Figure 3 in the shock velocity particle velocity plane. Thedecline in the slope is well characterized by the single-phase computation.For completeness, the theoretical Hugoniot is presented in Table 2.

SUMMARY

Thermodynamic properties for high temperatures and pressures have beencalculated using an exp-6 potential and liquid-state perturbation theory.The results were in good agreement with MC computations, except for extremelydense states which are close to the phase line of the hard sphere referencesystem. Theoretical predictions for the compressibility of liquid nitrogenfor moderate temperatures were in accord with the experimental data. Thecalculated shock Hugoniot for liquid nitrogen was also in close agreementwith experimental results. It is apparent that the theoretical techniqueutilized in the calculations provides a convenient and accurate descriptionof simple fluids in the moderate as well as high-pressure domains.

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40

0 00

30 -0 00

0 8

U0. 0

0 020 -0

10

00.3 0.4 0.5 0.6

VN,FIGUE 2SHOCK PRESSURE VS. RELATIVE VOLUME FOR LIQUID NITROGEN

FIUR

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TABLE 2

SHOCK HUGONIOT FOR LIQUID NITROGEN

Vvv P(GPa) T(OK) U(km/sec) IUp(km/sec)

.653 1.86 316.2 2.56 0.89

.628 2.47 418.3 2.84 1.06

.570 4.83 874.7 3.70 1.59

.531 7.56 1446 4.44 2.08

.501 10.7 2137 5.12 2.55

.494 11.6 2348 5.30 2.68

.468 16.0 3364 6.05 3.22

.448 20.5 4486 6.73 3.72

.432 25.3 5714 7.37 4.19

.419 30.0 6977 7.94 4.61

.409 34.2 8134 8.41 4.97

.400 39.0 9467 8.90 5.34

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06.00

0

0

4.0-

20-

2.0 4.0 6.0

UP(km/sec)

FIGURE 3 SHOCK VELOCITY VS. PARTICLE VELOCITY FOR LIQUID NITROGEN

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BIBLIOGRAPHY

1. Ross, M., Phys. Rev. A8, 1466 (1973).

2. Weeks, J. D., Chandler, D. and Anderson, H. C., J. Chem Phys. 54, 5237 (1971).

3. Ree, F. H., J. Chem. Phys. 64, 4601 (1976).

4. Verlet, L. and Weis, J., Phys. Rev. A5, 939 (1972).

5. Ross, M. and Alder, B. J., J. Chem. Phys. 46, 4203 (1967).

6. Smith, A. L., Keller, W. E. and Johnston, H. L., Phys. Rev. 79, 728 (1950).

7. Robertson, S. L. and Babb, Jr., S. E., J. Chem. Phys. 50, 4560 (1969).

8. Dick, R. D., J. Chem. Phys. 52, 6021 (1970).

9. Zwanzig, R. W., J. Chem. Phys. 22, 1420 (1954).

10. Anderson, H. C., Chandler, D. and Weeks, J. D., J. Chem. Phys. 56, 3812(1972).

11. Wertheim, M. S., Phys. Rev. Lett. 10, E501 (1963).

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