1898 T. G. SLANGER AND G. BLACK

16 T. G. Slanger and G. Black, J. Chem. Phys. 53,3717 (1970). 17 G. Black, T. G. Slanger, G. A. St. John, and R. A. Young, J.

Chem. Phys. 51, 116 (1969). 18 T. G. Slanger and G. Black, J. Chem. Phys. 51, 4534 (1969). 19 E. C. Y. Inn (private communication). 20 H. Wise (private communication). 21 T. G. Slanger and G. Black, 53, 3722 (1970). 22 W. Felder, W. Morrow, and R. A. Young, J. Geophys. Res.

75,7311 (1970). 23 K. Schofield, Planetary Space Sci. 15,643 (1967). 24 H. Yamazaki and R. J. Cvetanovic, J. Chem. Phys. 40, 582

(1964); 41,3703 (1964). 25 N. G. Moll, D. R. Clutter, and W. E. Thompson, J. Chem.

Phys. 45, 4469 (1966).

THE JOURNAL OF CHEMICAL PHYSICS

26 E. Weissberger, W. H. Breckenridge, and H. Tauhe, J. Chem. Phys.47, 1764 (1967).

27 M. Arvis, J. Chim. Phys. 66, 517 (1969). 28 T. G. Slanger, J. Chem. Phys. 48, 586 (1968). 29 J. D. Simmons, A. M. Bass, and S. G. Tilford, Astrophys. J.

155,345 (1969). 30 C. A. Barth, Science 165, 1004 (1969). 31 D. A. Parkes, L. F. Keyser, and F. Kaufman, Astrophys. J.

149,217 (1967). 32 L. D. Gray Young, Icarus 11, 386 (1969). 33 D. M. Hunten, Can. J. Chem. 47, 1875 (1969). 34 S. I. Rasool, J. S. Hagan, R. W. Stewart, and L. H. Russell,

J. Atmospheric Sci. 27, 841 (1970).

VOLUME 54, NUMBER 5 1 MARCH 1971

Enskog Theory for a Rigid Disk Fluid*

DAVID M. GAsst

Lawrence Radiation Laboratory, University of California, Livermore, California 94550

(Received 3 August 1970)

The Enskog theory for a dense fluid of rigid disks is developed. The collisional contribution, which dominates in liquids, is derived and added to the kinetic term, which describes a dilute gas. Expressions for shear and bulk viscosity and for thermal conductivity are obtained. The initial correlations are evaluated via the autocorrelation function approach, and the exponentially decaying functions which result are related to the Enskog theory.

I. INTRODUCTION

Calculations in two-dimensional systems often serve as prototypes. In the investigation of long-time correla­tions by molecular dynamics computation, rigid disk systems are within the practical limitations of present day computers, whereas hard sphere systems are not.! The density expansion of the transport coefficients, based upon a generalized Boltzmann equation for a two-dimensional system, breaks down in the triple colli­sion contribution, affecting the first-order density cor­rection.' In three dimensions the density expansion breaks down beyond first order in the density, thereby complicating the numerical evaluation of the diver­gence. The Enskog theory for rigid disks can be used as a handle on these and other rigorous calculations.

The Enskog theory for a hard sphere fluid3 yields transport coefficients over the entire density range and, in the low density limit, goes over to the Boltzmann form. The Enskog theory takes into account exactly the term in the Boltzmann equation which arises from the difference in position of colliding molecules. In addition, the influence of triple and higher-order colli­sions is approximated by scaling the Boltzmann collision integral with the local equilibrium radial distribution function at contact. The same observations also apply to a rigid disk fluid. The kinetic contribution to the transport coefficients of a rigid disk fluid has been ob-

tained by Sengers.2 This contribution arises from the bodily movement of molecules between collisions. In dense fluids the collisional or potential contribution is dominant. This term, which derives from transport between molecules while they are in contact, is the one on which we focus in this paper.

In the Enskog theory, correlations in time among many molecules are only estimated, while two-body effects are calculated exactly. Correlations for short times, as reflected in the various autocorrelation func­tions, are expected to be more accurately described by this theory than those which persist for many mean collision times. This observation is important since hydrodynamic and molecular dynamics calculations4

suggest a slow decay for the autocorrelation functions, while exponentially decaying functions result from the low-density, zeroth-order Sonine polynomial solution to the Boltzmann equation. As yet, only the velocity autocorrelation function has been shown to possess a long-time tai1.4 However, since the stress and heat-flux autocorrelation functions contain a term which depends only on the velocity field at long times, it is reasonable to suppose that these functions decay slowly at such times.

The initial values and slopes of the autocorrelation functions are evaluated. Then the assumption of expo­nential decay yields transport coefficients which agree with the ones calculated from the low-density, zeroth-

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ENSKOG THEORY FOR A RIGID DISK FLUID 1899

order Sonine polynomial solution to the Boltzmann equation. Initial correlations are related to terms which appear in the Enskog theory.

II. COLLISIONAL TRANSFER

In as much as the Enskog theory for dilute systems of rigid disks is already known,2.5 only the collisional con­tribution is required in order to evaluate the transport coefficients for dense fluids. This term arises from the instantaneous transfer of a molecular property across a line, I, while two disks centered on opposite sides of t are in contact.

Use of the molecular chaos approximation implies that the probable number of collisions per unit time, where c, CI, and k lie in the ranges dc, dCI, and dk, is

x(r)f[r+~ (uk) Jfl[r-~ (uk) ]u2(g·k)dkdcdcI(k·n)dl;

(1)

c and CI are the velocities of the colliding molecules, k is a unit vector in the direction of the line of centers, x(r) is the Enskog scaling factor, f is the one-particle distribution function, r+~ (uk) and r-~ (uk) are the centers of the two molecules of diameter u, g= CI- c, and n is the unit normal to t.

For each molecular property, if;, at each collision, there is a net transfer of if;' -if; across I, where if; is the value prior to collision and if;' is the value after colli­sion. The total rate of transfer across I by all collisions, per unit length, is

~[x(r)u2] 1fl (if;' -if;)f[r+~ (uk) ]fl[r-~ (uk)] g·k>O

X (g·k) (k·n)dkdcdcl. (2)

The above expression, which is of the form Vrn, gives the vector flux, V" of if;. Inasmuch as only linear trans­port coefficients are of interest here, the distribution functions in (2) can be expanded about the point of contact of the disks, r, and only terms up to the linear term in the gradien t are kept. Linearization yields the collisional contribution to the flux

V,=I+i{ilx)J f fW-if;)fORk

. (a/ar) log(j°/R) (g·k)kdkdcdcI, (3) where

fO is the local equilibrium single-particle distribution function, and all quantities are evaluated at r.

III. MOMENTUM TRANSPORT

In order to evaluate the momentum transport, we take if;=mC=m(c-co), where Co is the macroscopic velocity at the point r. Substitution of if; into Eq. (3) and integration with respect to k using the identities

(5) and (6) yieldEq. (7):

f(C'-C)k(g·k)dk= Jkk(g·k)2dk= t1l'(2gg+g2U),

(5) Jkk(k· V) (k·g)2dk= (4/15)

X[V.g(gg+g2U)/g+g(Vg+gV)], (6)

where U is the unit tensor and V denotes a vector, which will be taken as (a/ar) log(j°/R):

V m = ft (1I'mu2x) ff fh (2gg+ U g2) dcdcI

(a fO a fO)] + g g - log - + - log -0 g dcdcl. ar R ar h

(7)

Since g= CI- C and the odd moments of C and CI are zero, and since Jfdc=N / A=nand ffCCdc=n[CC]I, the first term in Eq. (7) can be written as

Vml=i{bp2x) (2[CC]I+[C2]IU), (8)

where the coarea, bp=H1I'nu2), and p=mn have been introduced. The remaining term in Eq. (7), V m2, con­tains the factor

(9)

The terms involving aT/ar in Vm2 are odd functions of C or CI and thus vanish on integration. The remaining terms are more conveniently evaluated when the vari­ables Go= (C+CI)/2 and g=CI-C are introduced. The result is

Vm2= - ~3~2; (2::TY ff exp [- k~ (G02+tg2)]

X {aco : gg(gg+g2U) ar g

+g [(:~o .g) g+g (:~o .g)]} dGodg. (10)

Performing the integration indicated in Eq. (10) yields

Vm 2= -wIHaO[co]/ar)+Hl(a/ar) • co), (11)

where w= (5n2ilx/8) (1I'mkT) 1/2 and

aO[coJii /ar= ~[( acdarj) + (acoj/ari) ]-~ (a/ar) . COOij.

The sum of Eqs. (8) and (11) yields the collisional contribution to the momentum flux and is the potential part of the pressure tensor. The rate of transport due to molecular motion between collisions is p[CCJ. There-

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1900 DAVID M. GASS

fore, the total pressure tensor is

F= p[l +~ (bpx) J[CCJ+i (bp2x) [C2JU

-wi t(aO[coJ/ar) +tU(a/ar) • col. (12)

To first order in the gradient,

p[CCJ = nkTU -x-l [l +~ (bpx) J21)0(aO[coJ/ar), (13)

where 1)0= (1/20') (mkT/Tr) I/2bo(N) is the coefficient of shear viscosity of the dilute gas. Further, the hydro­static pressure, PO, is given by

P=knT(1+bpx) - (4w/5) (a/ar) ·co. (14)

The deviation of the pressure tensor from the hydro­static pressure, from Eqs. (12)-(14) and the identity [C2J=2kT/m is found to be

tlF= -I (21)0/x)[1+Hbpx)J2+twl (aO[coJ/ar). (15)

The definitions of wand 1)0 are used, and the identifica­tion of the coefficient of -2(aO[coJ/ar) with the co­efficient of shear viscosity yields

1)= 1)obp«(l/bpx) + 1 + bpx Ii+ [2/Trbo(N) Jl). (16)

The third Sonine polynomial approximationS gives bo(3) = 1.022 and therefore

1) = 1)obp[ (l/bpx) + 1 +0.8729bpx]. (17)

Vm 2 depends on only the local equilibrium distribution function and therefore does not depend on the Sonine polynomial correction factor, bo(N).

The coefficient of bulk viscosity, c/J, is identified with the coefficient of - (a/ar) ·coO in Eq. (12) and there­fore

c/J= 4w/5 = [4x/ll'bo(iV) J (bp )21)0, (18)

which becomes c/J = 1)obp (1.246bpx) ,

when bo(3) = 1.022 is used.s

IV. ENERGY TRANSPORT

(19)

In order to evaluate the energy transport, we take 1/I=mC2/2. Integration with respect to k in Eq. (3) using Eqs. (20)-(22) yields Eq. (23), which is the collisional contribution to the heat flux:

J (C'LC2)k(g·k)dk= Jk· (g+2C)k(g·k)2dk

=2J(k·Go)k(g.k)2dk, (20)

2J (k· Go)k(g·k) 2dk= ill'[2g(g· Go) + g2GoJ, (21)

J (C'2-C2) (k· V) (g·k)kdk

= !s{ Go· g[g(g. V) +g2VJ/ g

+g[g(Go· V) +Go(g· V) Jl, (22)

Ve= /6 (ll'mO'2X) J Jffl[2g(g· Go) + g2GoJdcdcl

+ /5 (mO'3X)J JjOR[ (Go· g) (gg+g2U) /g

+g(gGo+Gog)]- VdcdcI, (23)

where V= (ajar) log(f°/R). As 2g(g·GO)+g2GO= ![CI2CI+C2CJ, the first term in Eq. (23) is

(24)

Use of (9), where now the terms involving aco/ar are the integrals of odd functions of C and CI, and the introduction of the variables Go and g yield

+g(g.Go) (gGo+Gog). aT] dGodg. (25) ar

Integration over Go and then g yields

(26)

where Cv= kim and w= ~ (uSn2x) (ll'mkT)I/2. The collisional contribution of the heat flux, given

by Eqs. (23), (24), and (26), is augmented by the energy transfer due to molecular motion, (p/2) [C2CJ, and the total energy flux, Q, is thus found to be

Q= ~p[l + (3bpx/4) J[C2CJ-c vw(aT /ar). (27)

To first order in the temperature gradient,

~[pC2CJI = - [1 + (3bpx/4) J(Ao/x)(aT /ar) , (28)

where Ao= (2/0') (k 3T/mll')I/2al (N) is the low-density formulation of the coefficient of thermal conductivity. The insertion of (28) into (27) yields

Q= - ['\o/X[l+ (3bpx/4) J2+ cvwJ(aT /ar) , (29)

and thereby the coefficient of thermal conductivity, A, can be expressed as

A = (AD/X) [1 + (3bpx/4) J2+ cvW

= Aobp«(l/bpx) +!+bpx{ 196 + [l/ll'al (N) Jl). (30)

The third Sonine polynomial approximationS gives al(3) = 1.029 and therefore

A= Aobp[ (l/bpx) +!+0.8718bpx]. (31)

V. AUTOCORRELATION FUNCTIONS

At low densities exponentially decaying autocorrela­tion functions lead to coefficients of shear viscosity and thermal conductivity equal to the ones calculated from the zeroth-order Sonine polynomial solution to the Boltzmann equation. This result was first observed for the shear viscosity in rigid sphere systems.6

We next consider the shear and bulk viscosity in some detail. The presentation is based on that of Ref. 6. The coefficient of shear viscosity, 1), for long-wave-

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ENSKOG THEORY FOR A RIGID DISK FLUID 1901

length, zero-frequency processes is given by

71= (AkT)-ll'" (jXy(t)jXy(t+r) )dr o

It

• J'" = (AkTts)-l jXy(t)dt jXY(s)ds, o t

(32)

where ergodic theory considerations and a variable change from t+r to s allow the passage to the second formulation of Eq. (32), and where

N N JXY(t) = m L Ci""(t)Ciy(t) + L Rix(t)Fiy(t)

i=l i=l

N

= m L Cix(t) Ciy(t) i=l

m +"2?;j uki/kil(kij"gij)O(t-te), (33)

Cix(t) and Ciy(t) are the x and y velocity components of particle i at time t and Fiy(t) is the y component of the force on i at time t. The direction of the line of centers from j to i is given by k ij and the relative velocity gij=Cj-Ci' The second formulation of Eq. (33) is a specialization to rigid disk systems where te is the time of collision c. The zeroth-order Sonine polynomial solu­tion of the Boltzmann equation for the coefficient of

W(O), 5(0), and «(RcxY)2> can be evaluated exactly:

shear viscosity yields

7Joo=7Jo/bo(N) = (1/2u) (mkT/rr) 1/2,

and the Enskog expression for the collision rate of rigid disks is r= (2Nbpx/u) (kT/rrm) 1/2. With the help of these expressions Eq. (32) can be converted into

71 [srl'" - = X-I - W(t)dt 710° N 0

(34)

The effects of considering rigid disks, as compared to hard spheres, are included in the coefficients of the integrals of Eq. (34), which can be compared to Eq. (S) of Ref. 6. The definitions of W, 5, and R/Y are given in Eq. (7) of this reference.

'" «(ReXY )2) and L (RexYRe+kXY) k-l

arise from correlations of molecular forces. The former term is a zero-time correlation effect and represents the contribution of instantaneous correlations to the transport coefficient.

m2 N

W(O)= 4N(kT)2E (c/c/)=t, (35)

( m )3/2 J J J dcdc1dk exp[ - (m/2kT) (C2+CI2) ] (g·k)2kXkY(cXCY+ClxCIY)

5(0) = - 2kT J J J dcdc1dk exp[ - (m/2kT) (C2+CI2) ](g·k) . (36)

First, the coordinate transformations, c_= C-Cl and C+= C+Cl, are made and the integrations over C+ are carried out. Next k is projected onto L and a vector orthogonal to it, CJ.:

k=(L/i Li) coso+(cJ./i cJ.i) sinO, (37)

where 0 measures the angle between c_ and k. The expression of kx and ky in terms of 0 using (37) and the use of the identity (g·k)2= (cOS20)C2 allow the integrations in (36) to be carried out with the result

5(0) = - (1r)1/2/SV2,

«(RexY )2)= (~) J J J dcdc1dk exp[ - (m/2kT) (C2+CI2) ](g·k)3(kXkY)2 2kT If J dcdc1dk exp[ - (m/2kT) (C2+CI2) ](g·k)

_lc -4'

The contribution of the «(ReXY )2) term to 71 is given by Eqs. (39) and (34), and is

71(0) = 7Jobp[2bpx/1rbo(N)],

(3S)

(39)

(40)

where the zero in the parentheses indicates it is the zero-time contribution. 71(0) is identical to the unperturbed single-particle distribution function's contribution in the Enskog theory.

The initial slope of Wet) can be calculated exactly and that of Set) can be calculated using the molecular chaos approximation. It is then found that both decay as e-', where s is the mean number of collisions per particle.

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1902 DAVID M. GASS

The diagonal elements of the stress autocorrelation function characterize the coefficient of bulk viscosity, ¢:

¢+1)= (AkT)-! [" (Jq,XX (O)Jq,xx (t) )dt, o

(41)

where Jq,xx(t) =Jxx(t) - (px(t», with pI given by (33) when x replaces y, leads directly to

¢P+1)P= 1)00 [16 (bpX

)2 o «(Rcxx)2)+ f (R/xRc+kXX»]. X 7r k=!

(42)

Only the term involving intermolecular forces in px has been written in Eq. (42), as it is the only nonzero con-tribution to ¢ in the Enskog theory. Further,

«( )2 ( m )fffdCdCldkexp[-(m/2kT)(C2+C!2)](gok)kX4

Rxx )- -c - 2kT f f fdcdc!dk exp[ - (m/2kT) (C2+CI2) ](gok)

=1· (43)

A comparison of Eq. (18) with Eqs. (42) and (43) indicates that the only contribution to the bulk viscosity in the Enskog theory arises from zero-time correlations of the diagonal elements of the stress tensor. The only non­zero contribution to ¢ from the initial autocorrelation of the microscopic stress tensor is due to the correlation of the impulsive force term with itself at the time of collision of two molecules. Evidently the deviations from local equilibrium (in the distribution approach) give rise to correlations of more than two particles and to cor­relations separated in time.

Similar calculations can be performed for the coefficient of thermal conductivity. The results again show an agreement between an exponentially decaying autocorrelation function and the zeroth-order Sonine polynomial solution to the Boltzmann equation. For the heat flux autocorrelation function the decay is found to be ~e-·/2.

VI. DISCUSSION

This paper presents the Enskog theory for a dense fluid composed of rigid disks. The collisional contribu­tion, which dominates in dense fluids, is derived and added to the low-density kinetic term. By determining the initial values of the autocorrelation functions, we conclude that, at low density, an exponential decay of the microscopic stress tensor and microscopic heat flux autocorrelation functions gives results identical to the zeroth-order Sonine polynomial solution to the Boltz­mann equation. The exponent of the decay agrees with that found for the cross term (which is first order in the density) under the molecular chaos approximation.

The collisional contribution to 1) and to A contains one term dependen t upon the single-particle equilib­rium distribution function and one term dependent upon deviations from this distribution. The former term, which is more than twice the latter for 1) and more than half the latter for A, is equal to the contribu­tion of the instantaneous correlations of the stress tensor and heat flux to the transport coefficients. Thus, the integration of the exact autocorrelation functions from time zero to a few mean collisions times, when e-S

and e-s/2 are essentially zero, should give transport co­efficients not unlike those predicted by the Enskog theory. The early time behavior of Wet) and S(t) is not expected to be greatly dependent on the density, whereas

(Rc i.iRc+! i.i)

might vary by a factor of 2 over the fluid density range, as it does in systems of hard spheres.!

At long times in rigid disk systems, correlations are not believed to decay rapidly. This discrepancy from the Enskog theory reflects the approximate treatment of multiple correlations by the scaling of two-body interactions. It is hoped, however, that the Enskog theory, which represents the "best" scaled, binary colli­sion theory, might be grafted onto a realistic long-time description, involving a hydrodynamic model. The availability of such a theory will provide the means for the accurate determination of transport coefficients for rigid disk systems.

ACKNOWLEDGMENTS

The author would like to express his appreciation to T. E. Wainwright, W. G. Hoover, and B. J. Alder for helpful discussions.

* Work performed under the auspices of the U.S. Atomic Energy Commission.

t Present address: Physics Dept., University of Toronto, Toronto 5, Ontario, Canada.

1 B. J. Alder, D. M. Gass, and T. E. Wainwright, ]. Chem. Phys. 53, 3813 (1970).

2]. V. Sengers, in Lectures in Theoretical Physics, Kinetic Theory, edited by W. Brittin (Gordon and Breach, New York, 1967), Vol. IXC, p. 335.

3 S. Chapman and T. G. Cowling, The klatlzematical Theory of Non-Uniform Gases (Cambridge U. P., London, 1939). The notation of this reference is used throughout this paper.

4 B. J. Alder and T. E. \Vainwright, Phys. Rev. 1, A18 (1970). 5 ]. V. Sengers, "Triple Collision Effects in the Thermal Con­

ductivity and Viscosity of Moderately Dense Gases," Arnold Engineering Development Center, Tenn., 1969, Tech. Rept. AEDC-TR-69-68.

6 T. Wainwright, J. Chem. Phys. 40,2932 (1964).

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