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Thermodynamics of anisotropic fluids using isotropic potentials

Sorin Basteat and Francis H. Ree*

Lawrence Livermore National Laboratory,

700W East Avenue, Livermore, CA 94550

We study the effectiveness and limitations of the median potential recipe for mixtures such as

N2 + 02 and N2 + C02, that are important in detonation applications. Conversely, we treat effec-

tive spherical potentials extracted from Hugoniot experiments (e.g., N2 and 02) as median potentials

and invert them to extract atom-atom potentials. The resulting non-spherical potentials compare

remarkably well with the atom - atom potentials used in studies of solid state properties. Finally,

we propose a method to improve the median potential for stronger anisotropic fluids such as C02

and its mixtures.

[anisotropic fluids, mixtures, effective spherical potentials, median, atom-atom potentials]The equilibrium properties of anisotropic molecular

fluids can be in principle calculated in a statistical me-

chanics framework, but the theory is generally too cum-

bersome for many practical applications. Fortunately, at

high densities and temperatures the anisotropy can be

'averaged-out' by means of a density and temperature

independent spherical potential ( the 'median') that pro-

duces reliable thermodynamic data [1,2].

Lebowitz and Percus [2] pointed out some time ago

that the success of this approximation can perhaps be

understood in terms of a simple theory that treats the

asphericity as a perturbation. The idea can be summa-

rized for the more general case of a binary mixture X +Y

as follows: If Oab(r, Q1, Q2) is the anisotropic potential

between molecules of type a and b (a and b run from

1 to 2, where 1 stands for X and 2 for Y) and #5b(r)

the corresponding effective spherical potential, we write

#ab(r, Q1, Q2) = #b(r) + A[4[ab, b], with 1 > -y > 0

and A0[#ab, Oab] = 0, A1[Oab, #ab] - #ab - /#b. Then, if

we expand the Helmholtz free energy asFr = Fo + .1(8F/Oy)o + ...

(1)

the condition that the first order correction is zero,

(8F/-y)o = 0, imposes a restriction on the potentials

#8b(r). If the concentrations of the X and Y molecules

are x and y (x + y = 1), this translates into

2x2 f gxx(ri, r2)(aA x/y)lodridr2dgidQ2 +

34 f 9xy(ri,r2)(aoXY/a-y)lodridr2dlidQ2 +

1 2 f gyY(ri, r2)(OA Y/a'Y)lodridr2dQicd2 = 0

where gb are the pair correlation functions for the spher-

icalized system. The above condition can be enforced in-

dependent of density, temperature and concentrations if

we require f(8 b/ay)IodidQ2 = 0 for all a's and b's.

The form of the effective spherical potentials, for both

like-pair and unlike-pair interactions, depends of course

crucially on our choice for A07b (in fact only ,ab/Qa7)10),

which unfortunately is left undefined by the perturbation

expansion.

The proposal of Shaw and Johnson [1], which turns

out to be the so-called median potential [2], is very suc-

cessful in predicting the thermodynamics of simple flu-ids such as N2 and C02 at reasonably high pressures

and temperatures [3,4]. The median is defined by using

8aob/ay) 10 = sgn(#ab - qob), and appears to be the best

choice for hard nonspherical potentials [5]. This may ex-

plain its success for fluids at high densities, where the

hard core contribution is known to be dominant.100

a

50 k

0

5000

10000

15000

T(K)

FIG. 1. Test of median potentials for an equimolar N2+02

mixture.

Anisotropic fluids such as N2, 02, C02 appear as deto-

nation products at high densities and temperatures, gen-

erally in mixed form [6]. Therefore, we test the median

potential prescription for equimolar mixtures of N2 + 02

and N2 + C02, by comparing the results of MD simu-

lations with rigid rotor atom-atom potentials and with

sphericalized potentials in an extended range of densities

and temperatures. For the rigid rotor calculations we use

atom-atom exp - 6 potentials,

V(r) = e/(a - 6){6exp[a(1 - r/r*)] - a(r*/r)6} (2)

The N - N and O - O parameters were extracted from

Hugoniot data and for N - O we used the Lorentz-

Berthelot rule.

The N2 and 02 molecules are moderately anisotropic,

with bond lengths 1N2 = 1.098A, 102 = 1.207A, and

r* /lN2 ^ 0.29, ro /102 0.35. We show in Fig. 1 the

results of constant density simulations for the N2 + 02

mixture. The agreement between the rigid rotor and the1

- median potential plo/cm') 2.9

symbols rigid-rotor

2.6

2.3

*** * 2.3

AA 1.9

p 1.5

c0c*

0o

0

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Bastea, S & Ree, F H. Thermodynamics of anisotropic fluids using isotropic potentials, article, August 16, 1999; California. (digital.library.unt.edu/ark:/67531/metadc627772/m1/3/: accessed September 26, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.