Closed virial equation-of-state for the hard-disk fluid

Transcription

1 Physical Review Letter LT50 receipt of this auscript 6 Jue 00 Closed virial equatio-of-state for the hard-disk fluid Athoy Beris ad Leslie V. Woodcock Departet of Cheical Egieerig Colbur Laboratory Uiversity of Delaware, DE 976 A closed virial equatio-of-state for the low desity fluid phase of hard disks is obtaied fro the kow virial coefficiets. The equatio exhibits 6-figure accuracy for the therodyaic (MD) pressure up to the desity ρσ ~ 0.4. Iterpolatio of the discrepacy at higher desities idicates a higher-order therodyaic phase trasitio at the extesive-itesive free-volue percolatio trasitio previously located by Hoover et al. (JCP ) A closed equatio-of-state for the virial expasio i powers of desity relative to closepackig has bee recetly proposed for the hard-sphere fluid []. This virial expasio is siply writte where Z ( ) ρ Z = B () = ρ0 = pv Nk T : k B is Boltza s costat, B are the diesioless B coefficiets ρ is the uber desity ( N V ), ad ρ 0 is the crystal close packig desity. Equatio () has bee show to lead to a closed-for equatio which is extreely accurate for the hard-sphere fluid; ideed of coparable accuracy to the MD data itself all the way fro the ideal gas to the fluid freezig desity. Here we derive a siilar equatio-of-state for hard disks usig oly the kow virial coefficiets up to B 0. For D we use the sae oeclature except ow V is the area, ad the axiu packig desity, ρ0σ = 3 correspods to the hexagoal closepacked D lattice structure. The literature recoeded values for all the kow virial coefficiets for hard disks accordig to Kolafa ad Rotter [] are reproduced here i TABLE I. I referece [] the virial coefficiets are expressed i a alterative expasio i powers of the packig fractio ( y πρ 4) Z ( ) = β y () = where β is related to B i D by ( ) ( B β πρ ) =. 0 4 Also give i TABLE I are the values of the virial coefficiets expressed i powers of the desity relative to axiu close packig ρ 0 i equatio ().

2 Physical Review Letter LT50 receipt of this auscript 6 Jue 00 TABLE I Kow virial coefficiets of the D hard-disk fluid: the values of β i y- expasio, equatio, are take fro Kolafa ad Rotter []. β [] Β Β Β ( ) Eq.(3) Additioal to the coputed virial coefficiets, usig accurate MD data for disks, over the whole desity rage up to freezig, Kolafa ad Rotter also predicted the higher coefficiets B to B 6 by fittig to MD data. These fits, however, presuppose that up to the highest desity of the MD result fitted, the therodyaic equatio-of-state ad the closed virial equatio of state are the sae. Here, we do ot ake this assuptio i derivig our closed virial equatio, but rather, use oly the kow virial coefficiets to derive a closed for. The, the resultig equatio of state ca be used to ivestigate the phase trasitio whece the therodyaic pressures fro MD first begi to deviate fro the virial equatio. Ispectio of icreetal values of successive virial coefficiets, plotted i powers of desity relative to crystal close packig as i equatio ; (Fig. ) shows that beyod (B 5 -B 4 ) the icreet decrease approxiately liearly accordig to B B( ) = α0 + α ( > 5) (3) This iterpolatio of all the kow virial coefficiets suggest that sice the liitig costat A 0 is egative, the virial coefficiets will evetually becoe egative ad the correspodig virial equatio-of-state will be cotiuous i all its derivatives, evetually showig a egative pressure, with the first pole at ρ o. Equatio (3) with the paraeters obtaied fro Figure predicts the first egative coefficiet for D= is B 3. This observatio is ow cosistet with the kow fact that virial coefficiets for all hard hyper-spheres of higher diesioality tha evetually go egative. This has bee kow for alost 50 years sice Ree ad Hoover calculated B 4 for syste of various diesio (D) up to D=9 ad foud B 4 to be egative i the two cases of D=8 ad D=9. The ost recet closed-virial equatio for spheres predicts the first egative coefficiet i 3D is B 9. Nuerical values for B i iterediate D betwee D= 4-7 show that B also

3 Physical Review Letter LT50 receipt of this auscript 6 Jue 00 go egative betwee B 4 ad B, ad, for soe values of D ca oscillate i sig. The 0 th virial coefficiet is egative for all kow diesios greater tha D=4 [4]. Equatio (3) for D= therefore, is cosistet with what is geerally kow about hyper-sphere virial coefficiets i all diesios. 0.5 B-B(-) y = 4.45x R = / Figure : Differece betwee successive virial coefficiets (B ) i the expasio i powers of desity relative to close packig: for all kow > 4 the differece B - B (-) decreases as /, ad approaches the costat α 0 = whe. ρ ρ ρ 0 the closed-virial equatio-of-state for disks the takes the for (APPENDIX ) α ( ) ρ * α Z = B ρ * ( B ( ρ *) α0 ) l ( ρ *) ( ρ *) + + (4) = ( ρ * ) ρ *( ρ *) * If we use ( ) where B is the highest kow virial coefficiet, presetly B 0. Usig oly the kow coefficiets B 5 to B 0 of Kolafa ad Rotter we obtai the liitig value of B - B (-), i.e. α 0 = , ad the slope α = It is also iterestig to ote that the costat α0 is very close to half of the close-packed scaled volue ( ) V0 σ = 3 4 Curiously, the correspodig liitig costat for 3D spheres 3 is very close to V σ []. 0 3

4 Physical Review Letter LT50 receipt of this auscript 6 Jue 00 Whe the pressure is predicted fro the closed virial equatio (4), ca be copared with the therodyaic pressure fro MD coputatios i the viciity of eltig. I figure we copare with the data of Kolafa ad Rotter [] up to freezig, ad our ow ew MD data, obtaied fro very log rus of systes o N=0000 i the two-phase regio ad crystal at eltig...5 fluid phase area (A). crystal phase pressure (p) FIG. : Equatio-of-state for the hard-disk fluid i the viciity of the first-order freezig trasitio; virial equatio-of-state equatio 4 (solid lie) crystal equatio-of-state [5] (dashed lie), MD results Kolafa ad Rotter (p < 9) ad preset MD data (N=0000: 5000 illio collisios each data poit) for p > 9 ad crystal brach. The copariso betwee the virial pressure ad the fluid therodyaic pressure i Figure shows that the discrepacy is ot siply a pre-eltig pheoea as has bee suggested previously, or eve a artifact of MD sall systes [7]. A previous closed equatio [6] used fitted coefficiets B -B 5 which prejudiced the coparisos. The preset copariso shows that the deviatio is real, ad is begiig at a uch lower desity. This begs the questio: where is the first deviatio ad what is the uderlyig 4

5 Physical Review Letter LT50 receipt of this auscript 6 Jue 00 sciece? Sice the virial expasio becoes exact at very low desity; with covergece up to ρσ = 0.05 havig bee rigorously prove by Lebowitz ad Perose [6], ad sice the virial equatio is cotiuous i all its derivatives, if the therodyaic pressure deviates, it ust be sigaled by a therodyaic phase trasitio of secod or higher-order. To ivestigate this possibility, we have plotted the deviatio of equatio (4) fro MD pressures as a fuctio of desity for all the MD data poits above ρσ = 0.4 i Figure 3. This plot suggests that the deviatio is origiatig i the viciity of that foud by Hoover et al. to be the oset of the free volue percolatio trasitio for the hard-disk fluid [8]. Z(MD)-Z(virial) percolatio trasitio desity (ρ p ) desity (ρ ρ σ ) Figure 3: Desity depedece of pressure differece betwee closed-virial equatio-ofstate (equatio (4): =0) ad therodyaic pressures obtaied fro MD siulatios by Kolafa ad Rotter (Ref.).The percolatio trasitio desity deteried by Hoover et al is idicated by the vertical arrow. This percolatio trasitio ca be defied as the desity at which the ea accessible cofiguratioal itegral of a sigle disk, withi the equilibriu eseble, chages fro beig extesive at low desity to itesive at high desity. This free volue (v f ) is related to the therodyaic pressure accordig to σ s f Z = + (5) D v f 5

6 Physical Review Letter LT50 receipt of this auscript 6 Jue 00 ad s f is the surface area of the hole i which the disk is trapped at high desity, or the etworked uio of holes at desities below the percolatio trasitio. The desity at which <v f > chages fro beig a itesive state fuctio to a extesive state fuctio defies the percolatio trasitio. It follows fro equatio (5) that <s f > ust chage likewise at the sae trasitio poit. At low desity, the cell-free volue (<v f >) becoes equivalet to the so-called spare volue (<v s >), which is the relative probability of addig a extra sphere aywhere i a equilibriu cofiguratio, ad hece is exactly related to the excess Gibbs cheical potetial (G), for large N dg vs µ = kbt l (6) dn V Now, we deduce that if equatios (5) ad (6) are rigorous, ad for desities below tha ρ p (the percolatio trasitio) v = v ; ρ < ρ (7a) f s p whereas for desities above the percolatio trasitio we have vs v f = O ; ρ > ρ N p (7b) It sees that there is a discotiuity i a higher derivative of <v s > i equatio (6) ad a deviatio betwee therodyaic pressure ad the cluster expasio at the poit whe a cluster of size N spas the whole syste. Therodyaically, fluctuatios i uber desity are exactly related to the secod-derivative of the cheical potetial dµ/dρ. At the percolatio trasitio desity whe oe cluster spas the whole syste, a class of desity fluctuatios becoe froze out. Oe therefore expects ot a secod, but a thirdorder therodyaic trasitio, i.e. a discotiuity i the derivative of the isotheral copressibility, which is the equivalet of heat capacity Cp for disks as a cosequece of siple scalig. This observatio ay explai the apparet high accuracy of the closedvirial equatio i 3D alost up to freezig, eve though it ay belog to a differet phase at liquid desities. The percolatio trasitio is weaker i three diesios. We should ow have a closer look at the hard-sphere percolatio trasitio i 3D as this result has further iplicatios for the geeral developet of theories of the liquid state ad the origi of critical poit pheoea. Traditioal approaches based upo itegral equatio closures for hard spheres all assue the essetial correctess of the Mayer cluster expasio with cotiuity up to liquid desities. It ow appears that this ay ot be so. 6

7 Physical Review Letter LT50 receipt of this auscript 6 Jue 00 Refereces [] M Baera, L Lue ad L V Woodcock, J. Che. Phys (00) [] J. Kolafa ad M.. Rotter, Molecular Physics, 04, (006) [3] F. Ree ad W.G. Hoover, J. Che. Phys., (964) [4] N. Clisby ad B.M. McCoy, J Statistical Physics 5-57 (006) [5] B.J. Alder, W.G. Hoover ad D.A.Youg, J. Che. Phys., (968) [6] L.V. Woodcock, Ar. Xiv cod-at (008) [7] J.L. Lebowitz ad ). Perose J Math. Phys (964) [8] W.G. Hoover, N.E. Hoover ad K. Haso, J. Che. Phys (979) 7

8 Physical Review Letter LT50 receipt of this auscript 6 Jue 00 APPENDIX: Derivatio of closed virial equatio for hard-disk fluid: If x = ρ/ρ 0, fro equatios () ad (3) we have α Z = B x B x = = + l= + l ( ) ( ) α0 (A) However, ote that we ca use the followig expressios for the suatio of the geoetric series appearig i the secod ter i the above equatio ( α ) x x ( ) ([ ] ) ( = α ) 0 0 = + l= + = + 0x x = + = α ([ ]) ( ) ( k ) d k = α0x kx = α0x x k = 0 dx k = 0 d α x = α x = dx x x 0 0 ( ) (A) ( ) ( ) x α x α = + l= + l= + = l l= + ( l ) α x l = l = l ( x) α l ( l ) = x x ( x ) x l l l l = = x α ( l ) ( l ) = z dz x ( x) x 0 l= l= l x α = dz ( x) x z 0 l= x l ( x) ( l ) α l = x ( ) ( l ) x x l= l (A3) ( ) ( ) ( B ) x = B x = B (A4) = + = + x x Whe Eqs. (A-A4) are used withi Eq. (A) it is easy to see that we get Eq. (4). 8