Contrib. Plasma Phys. 33 (1993) 516, 578

Interpolated Mean Spherical Approach to the Energy Equation of State of the Classical One-Component Plasma Fluid

TORSTEN KAHLBAUM (a) and HUGH E. DEWITT (b)

(a) WIP-AG Niedertemperatur-Plasmaphysik, D-10117 Berlin, Germany; (b) Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.A.

Abstract We present a fit formula for the excess internal energy of the one-component plasma in the fluid phase that was obtained from the analytic solution of the mean spherical approximation for charged hard spheres in a uniform neutralizing background by utilizing a new relation for the effective packing fraction. Our results are in much better agreement to recent Monte Carlo data for strong coupling, I? 2 1, than previous results from similar calculations, and extend down to the Debye limit, l? -+ 0.

1. Introduction In the past twenty years several attempts have been made to derive the equilibrium properties of the one-component plasma (OCP) from that of the charged hard-sphere fluid at the same Coulomb coupling strength within the mean spherical approxi- mation (MSA) by determining the hard-sphere packing fraction 7 from various con- straints on the MSA thermodynamic functions. To be specific, the maximum condition d ~ ~ s A / d q = 0 leads to GILLAN’S [l, 21 continuous (or soft) mean spherical approach (CMSA), while the minimum condition f ~ ~ r s ~ ) / a q = 0 gives MACGOWAN’S [3] discontinuous mean spherical approach (DMSA), where UMSA, ~ M S A , and f ~ s are the MSA Coulomb contributions to the internal energy and to the Helmholtz free energy, and the excess Helmholtz free energy of uncharged hard spheres (HS), respectively. The self-consistent mean spherical approach (SMSA) was introduced by MACGOWAN [4], too, who fixed 7 by imposing thermodynamic consistency between the virial and com- pressibility forms of the resulting OCP equation of state. Finally, ROSENFELD [5] considered a special case of an interpolation between CMSA and DMSA that follows from a modification of the Percus-Yevick (PY) virial form of the HS free energy.

As can be seen from Fig. 1, all approaches listed above reproduce the data from precision Monte Carlo (MC) simulation experiments by SLATTERY et al. [6] and STRINGFELLOW et al. [7] not sufficiently good enough, compared to phenomenological fits [7, 81, and DMSA and SMSA do not yield any results for I‘ < 53.6 and I‘ < 8.5, respectively. In order to remove these drawbacks, we propose another variant of an MSA-based model for the OCP that appears as the generalization of ROSENFELD’s [5] interpolation and, hence, is termed interpolated mean spherical approach (IMSA).

579

Uth

3

2.5

2

1.5

1 0 50 100 150 200

r Fig. 1: OCP fluid thermal energy 'Ilth vs. coupling strength I? from MC simu- lations and the MSA approaches described in the text. The DMSA and IMSA curves were both computed for the virial case, (Y = 1, being the best choice for the former and corresponding to ROSENFELD'S [5] original guess for the latter.

2. A New Formula for the OCP Internal Energy The common starting-point for all approaches is the analytic solution of the MSA for charged hard spheres in a penetrating background given by PALMER and WEEKS [9],

from which fMSA(r, q ) can be calculated analytically [3] by the usual r-integration. The uncharged hard-sphere system is represented by f ~ s ( ~ ) , or by the related HS

excess compressibility factor ZHS = q d f ~ s l d q , that are approximated in the literature [3,5] by any one of the corresponding PY compressibility (C) form, PY virial (V) form, or Carnahan-Starling (CS) arithmetic-mean combination form, e.g.,

Instead of performing the following calculations for each one of these alternatives separately, we proceed with the geometric-mean (GM) combination form [lo] for ZHS,

that is a more effective interpolation between ZC for a = 0 and ZV for (Y = 1 than the analogous arithmetic-mean combination form which includes the CS form for LY = 1/3.

580

4

3.5

3 CMSA . . * DMSA * . . -

2.5

Q , $ 2

1.5

1

0.5

0

. . . . . . . . I . . . . . . .

0 0.2 0.4 0.6 0.8 I 77

Fig. 2: Functions Q and 1(, vs. packing fraction 7. Q, Eq. (2), is plotted for three values of r by which the curves are labeled. The DMSA and IMSA curves for 1(,, Eqs. (7) and (8), were both computed for the compressibility case, a = 0.

The solutions of the extremum conditions for U M S A and ~ H S + ~ M S A can be both expressed as Q ( r , q ) = +(q) [3] or, using Eq. (2), as an inverse relation for q(r) [5],

where the defining functions $J for CMSA and DMSA, shown in Fig.

Note that Eq. (7) generalizes MACGOWAN’S + D M ~ A for the virial [3], to an arbitrary ZHS and corresponds to ROSENFELD’S Eq. (84) in

case, Eq. (48) in [5]. Taking from

now on ZHS M Z c ~ ( c r ) , Eq. (4), the curve for $DMSA is terminated at qrnjn(cr), 0.25 5 v,;, 5 0.27, so that equality between Q and $ in the DMSA can only be achieved for I? >_ rrnin(a), 53.6 5 Fmin 5 72.0. Two attempts made by MACGOWAN [Ill to extend the DMSA to lower values of were only moderately successful. On the other hand, the curve for $CMSA, Eq. (6 ) , is defined over the whole range 0 5 71 5 1 and intersects the &-curves for all r 2 0, thus providing OCP results for the entire fluid region. However, it is evident from the comparison of the thermal energy in Fig. 1 that the DMSA where it exists yields the best fit to the MC data of all the MSA models for the OCP so far. Therefore, it would be much desirable to maintain the DMSA results for Uth at strong coupling and to change the behavior only at weaker coupling under the condition that the function T(r) preserves its smoothness.

581

3

2.5

2

1.5

1

0.5

0

u t h

0 50 100 150 200 r Fig. 3: OCP fluid thermal energy '1Lth vs. coupling strength l". The full curve represents our best result for Uth derived from the IMSA, Eqs. (l), (5), and (8), in the compressibility case, (Y = 0; the crosses mark the MC simulation data [8].

A simple "procedure" that fulfills these requirements and can be directly applied to Eq. (7) is now being presented: Since the termination of $ D M ~ A is caused by the square root, we replace d m with 1 - f d ( q ) . . . , thereby extending + down to q = 0. This is reasonable because d ( q ) c( (1 - q)'+" tends to zero for q -+ 1, and no problems arise in the opposite limit q -+ 0 when b(q) approaches a constant. Keeping the first two terms in the expansion with respect to 4, we arrive at the defining function $ I M ~ A for the new interpolated mean spherical approach,

Eq. (8) interpolates between $iMsA(q -+ 0) = 247 and $ k M s A ( q -+ 1) = 2a(1-q)1fa, see Fig. 2, and includes ROSENFELD'S Eq. (88) in [5] as a special case for a = 1 though it was obtained in a completely different manner.

With the new $J, the packing fraction q(r) is computed from the numerical in- version of Eq. ( 5 ) and the OCP fluid excess internal energy U O C P then follows from Eq. (1) according to uocp(r) = u M S A ( r , q ( r ) ) . Of course, one can expect that the interpolation between CMSA for 7 -+ 0 and DMSA for 77 -+ 1 will be mapped onto the corresponding r scale and also work for uOcp(r). This is true at weak coupling, I? < 1, and, in fact, the uocp values from IMSA and CMSA closely agree to each other even up to r - 1. More precisely, IMSA and CMSA both exhibit the exact Debye limiting behavior for r' -+ 0, as was first demonstrated by PALMER [12] for the latter. At strong coupling, I? >_ 1, on the contrary, it turns out that the numerical IMSA and DMSA results considerably differ in the physical fluid region below the melting-value rm = 178 [8], which is illustrated in Fig. 1 for the virial form of Uth, and only merge for I' -+ 00 due to the asymptotic character of the theory.

582 Through the introduction of the “mixing”-parameter cy in Eq. (4), the OCP internal

energy has become a one-parameter fit, uocp(r, a), where cy can be adjusted to match the MC fluid data. We used the standard deviation u between uocp and uMC at 22 I? values in the strong-coupling range 1 5 I? I 200 [8] as a measure of the quality of the fit and found the best agreement in the IMSA for cy = 0, the compressibility case, with u = 0.0546 being much smaller than for any other previous fit. Fig. 3 shows the corresponding OCP thermal energy uth(r) where Uth 5 uocp + 0.9r. This result clearly contradicts previous findings in the variational hard-sphere approach (VHSA) [13] and the DMSA [3] that the best fits are generally obtained from the PY virial equation of state for cy = 1. Furthermore, it does not confirm expectations [8, 101 that choosing cy = 112 or any other intermediate cy value between 0 and 1 could give an improvement on the best VHSA, DMSA, and IMSA fits for cy = 1 and 0, respectively. Finally, we mention that the variation of a has practically no influence on the IMSA results for uocp up to I’ - 1 and that these results are fairly close to the data from hypernetted-chain and MC calculations in the weak-coupling range 0.1 5 < 1 [14].

3. Summary The IMSA based on a new recipe for the effective packing fraction provides an excellent quasi-analytic description of the OCP fluid thermodynamics in the range from r = 0 to the melting-value rm for the case that the PY compressibility equation of state is used for the €IS reference system. Analytic solutions of this model in the asymptotic limits --+ 0 and r -+ 00 will be the subject of a forthcoming paper.

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[7] STRINGFELLOW, G. S., DEWITT, H. E., and SLATTERY, W. L., Phys. Rev. A 41

[8] DEWITT, H. E., SLATTERY, W. L., and STRINGFELLOW, G. S., in: Strongly Coupled

[9] PALMER, R. G. and WEEKS, J. D., J. Chem. Phys. 58 (1973) 4171.

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[lo] KAHLBAUM, T. and DEWITT, H. E., in: Strongly Coupled Plasma Physics, ed. H. M.

[ll] MACGOWAN, D., J . Phys. C 16 (1983) L7. [12] PALMER, R. G., J. Chem. Phys. 73 (1980) 2009. [13] DEWITT, H. E. and ROSENFELD, Y. , Phys. Lett. 75A (1979) 79. [14] SLATTERY, W. L., DOOLEN, G. D., and DEWITT, H. E., Phys. Rev. A 21 (1980)

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