Critical Compressibility Factor of Two-Dimensional Lattice Gas

Transcription

1 990 Progress of Theoretical Physics, Vol. 81, No.5, May 1989 Critical Compressibility Factor of Two-Dimensional Lattice Gas Ryuzo ABE Department of Pure and Applied Sciences University of Tokyo, Komaba, Tokyo 153 (Received February 6, 1989) The critical compressibility factor Zc at the gas-liquid critical point is defined by Zc= Pc VclNkB Tc (Pc: critical pressure, Vc: critical volume, Tc: critical temperature, kb: Boltzmann's constant, N: number of molecules). By the use of the equivalence of lattice gas with the Ising model, the Zc of two-dimensional lattice gas with nearest neighbor interaction is calculated exactly. The results are Zc= (square lattice), Zc= (honeycomb lattice), Zc= (triangular lattice). 1. Introduction The critical compressibility factor Zc defined by Zc=Pc Vc/NkBTc (1 1) (Pc: critical pressure, Vc: critical volume, Tc: critical temperature, kb: Boltzmann's constant, N: number of molecules) is an important quantity*) which characterizes the property of gas-liquid critical point. If the molecules interact through the Lennard Jones potential (r)=4c[(o/r)12-(o/r)6], the Zc is shown to bel) a constant independent of E and 6. This fact is actually verified experimentally for spherical non-polar molecule such as Ar or Xe (see Table 1). In discussing the critical phenomena, the critical exponents or some critical amplitude ratios are supposed to be universal; they depend primarily on the space dimensionality, spin dimensionality and potential range.. On the contrary, as was shown by the author,z) the Zc of metallic fluid can be different from that of nonmetallic fluid. Also, the Zc of metallic fluid depends on Z (valence of ion) in general. These results are really the case l ),3) as is clear from Table 1. In this sense, the Zc is supposed to be nonuniversal. Table I. Experimental values of Zc for some From the theoretical point of view, the nonmetallic and. metallic fluids. Zc for van der Waals gas is shown to be 4 ) Substance Ar Xe Cs Rb Hg Zc *) The inverse 1/Zc is sometimes called the Kamerlingh-Onnes constant. Zc=3/8= (1 2) Also, mean field approximation for lattice gas (see 2) yields Zc=2ln 2-1= (1-3) A close agreement of (1 3) with the value

2 Critical Compressibility Factor of Two-Dimensional Lattice Gas 991 for Hg may be probably only fortuitous. Apart from (1 2) and (1 3), no theoretical estimate of Zc has been known within the author's knowledge. The purpose of this paper is to derive exact Zc in the case of two-dimensional lattice gas. In 2, the equivalence of lattice gas with the Ising model is discussed along the line of Lee and Yang. 5 ) Restricting ourselves to the nearest neighbor interaction, we will derive an expression for Zc written by the language of the Ising model We proceed in 3 to calculate Zc for square, honeycomb and triangular lattices. Finally, this paper is ended with 4 where some concluding remarks are presented. 2. Lattice gas and Ising model Let us consider a crystal composed of L lattice points and let its volume be V. In the lattice gas, each lattice point is occupied or unoccupied by a molecule. In order to describe this situation, we introduce a variable Uj which takes 1 if the jth lattice point is occupied by the molecule or takes 0 if it is unoccupied. Then, the number of molecules and the potential energy of the system are given by ~Uj and -(1/2) x ~ jkujuk, respectively, where - jk is the interaction potential between the jth and kth molecules. The grand partition function ZG of lattice gas is expressed as (2 1) where /3 = l/kb T (T: absolute temperature) and f1. is the chemical potential per molecule. Statistical mechanics states that the following relations hold: PV/kBT=lnZG, N=a(InZG)/a(/3f1.). (2 2) (2 3) We now transform the lattice gas into the Ising model along the line of Lee an<;l Yang. 5 ) For this purpose, we put with (fj the Ising spin taking values ± 1. where o is defined by Substituting (2 4) into (2 1), we find (2 4) (2 5) O=~ jk k (2 6) Here, we assume the translational invariance of the system so that o is independent of j. Introducing the exchange interaction ljk and a quantity h corresponding to magnetic field:

3 992 R. Abe (2'7) h=flj1./2+fl o/4, (2 8) we see that the sum over (Jj in (2 5) is the partition function Z(K, h) of the Ising model: Z(K, h)= <13~,<1L exp( ~ t,:kjk(jj(jk+h~(jj) (2'9) with Kjk=flJjk as usual From (2'5) and (2'9), we get In Ze= fl ol/8+ flj1.l/2+ In Z(K, h). If we substitute (2'10) into (2'3), we find (2 10) L 8 L 1 N=2-+ 28h 1nZ(K, h)=2+2~<(jj>, (2'11) where <... > indicates statistical mechanical average. Weare now in a position to discuss the critical point of lattice gas on the basis of (2'10). To do this, we notice that the above point corresponds to the transition point of the Ising model. In the following, we will denote the quantity at these points. by adding the subscript c. Since the Ising model exhibits phase transition-only at h =0, from (2'8) we obtain (2'12) Also, at the transition point of the Ising model the relation <(Jj>=O is valid. Therefore, from (2 '11) it follows that. N=L/2. (2'13) This implies that the number of molecules is just half that of lattice points at the critical point of lattice gas. Putting (2'12) into (2'10) and using (2'6) and (2'7), we see that the InZe at the critical point is expressed as where In Ze= - flclol/2+ In Z(Kc, 0), (2'14) (2'15) If we restrict ourselves to the nearest neighbor interaction, we have Jo=zJ with z the coordination number. In this case, the relation flclo = zkc holds where Kc = flel. Thus, combining (2 2) with (2 '14), we are led to Pc Vc/kBTc=ln Z(Kc, 0)-zKcL/2. (2'16) Or, using (1'1) and (2'13), we have Zc=(2/L)ln Z(Kc, O)-zKc. (2'17) The above equation enables us to calculate the critical compressibility factor of

4 Critical Compressibility Factor of Two-Dimensional Lattice Gas 993 lattice gas in terms of critical quantities of the corresponding Ising model. As a working example, let us apply mean field approximation where the relation zkc=l holds. Since this approximation is exact in the limit z->co, in calculating Z(Kc, 0) we put Kc=O. Then, from (2'9) we find that Z(Kc, 0)=2 L and therefore (2'17) leads to (1'3). If we proceed one step further taking account of l!z term, it turns out that Zc=2ln 2-1-1/2z+ O(1/z 2 ) (2'18) We are not going to enter into l/z expansion here. Instead, exact Zc will be calculated in 3 for the two-dimensional systems. 3. Z c of two-dimensional lattice gas On the basis of (2 '17), we are going to study the Zc of typical two-dimensional lattice gas in this section Square lattice The celebrated Onsager 6 ) solution for the Ising model without magnetic field is (l/l) In Zs(K, O)=ln 2 1 (n (n + 27[2)0 )0 dwdw' In (ch 2 2K -sh2k cos w-sh2k cos w'), (3 1) where the subscript S stands for square lattice. The transition point in this case is characterized by so that from (3'1) we find sh2kc=1, 1 1 (n (n LIn Zs(Kc, O)=ln 2+ 27[2)0 )0 dwdw'ln (2-cosw-cosW'). As is shown in the Appendix, it follows that (l/l)ln Zs(Kc, 0)=(1/2)ln 2+(2/7[)G, where G is Catalan's constant given by = '". By means of (3' 2), the Kc in the present case is calculated to be Kc=(1/2)ln(1 + J2). Thus, putting z=4 in (2 17) and using (3'4) and (3'5), we find Zc=ln 2+(4G/7[)-2In(1 +J2) = '". (3'2) (3 3) (3'4) (3'5). (3'6) (3'7)

5 994 R. Abe 3.2. Honeycomb lattice In this case, the partition function of the Ising model without magnetic field is expressed as 7 ) The transition point is given by ch 2Kc=2, whence we obtain -sh 2 2K(cos (O+cos (Or +cos«(o+ (Or))]. (3'8) (3'9) (I/L)ln ZH(Kc, O)=ln 2+(1/4)ln(3/2)+ I, (3 10) where I is defined by 1 [21C (21C 1= 1671"2)0 )0 d(od(or In [3-(cos (O+cos (Or +cos«(o + (Or))]. (3'11) To calculate I, we write cos (Or +cos«(o+ (Or) =2 cos«(o/2)cos«(or + a) with a given by tan a=sin (0/(1 +cos (0) and integrate over (Or by the use of (A '2). Then, introducing the variable x= (0/2, we are led to (3'12) Unfortunately, analytical calculation of I seems to 'be impossible. However, since the integrand is rather simple, numerical calculation of I can be performed by a pocket computer (e.g., Casio FX-860PVC) in about 30 seconds. The result is 1= '". (3 13) The above I is related to C2 discussed in Domb's review: 8 ) 1= (In 3) / 4 + Cd 4. The numerical value C2= '" obtained by Houtappel 9 ) is consistent with (3 13). From (3'9) we find Kc=(1/2)ln(2+J3). (3 14) As a result, by putting z=3 in (2'17), the Zc is calculated to be Zc=2ln 2+(1/2)ln(3/2)+21-(3/2)ln(2+J3) = '". (3 '15)

6 Critical Compressibility Factor of Two-Dimensional Lattice Gas Triangular lattice The equation corresponding to (3'8) is 7 ) lin ZT(K, O)=ln 2+ 8;21 2 "1 2 " dwdu/ In [ch 3 2K +sh 3 2K - sh2k(cos w + cos w' +cos(w + w'»]. (3 16) At the transition point, the following relations are valid whence Kc is calculated as K c=(1/4)ln 3. Also, it is easily seen that (I/L)ln ZT(Kc, O)=ln 2-(1/4)ln In this way, putting z=6 we have Zc=2ln 2-21n 3+41 = '". 4. Concluding remarks (3'17) (3'18) (3'19) (3'20) In this paper we calculated the critical compressibility factor Zc of the typical two-dimensional lattice gas. From our results it may be concluded that Zc increases as z increases. This tendency is also seen from (2 '18). Although the exact solution is possible for the two-dimensional Ising model without magnetic field, the same thing is impossible for the three-dimensional system. One way of dealing with such a system is to apply high temperature expansion for In Z(K, 0). We hope that the exact results obtained here are useful to examine the accuracies of the approximations employed. These problems wait for a further investigation. Appendix The details of calculation of (I/L) In Zs(Kc, 0) have been discussed in the original Onsager's paper. 6 ) However, for the convenience of the reader, we present its calculation in this appendix. Using the relation, w + w' w _ w' cos w + cos w = 2cos 2 cos 2 ' introducing variables x=(w+ w')/2, y=(w- w')/2 and noting the symmetries of integrand in (3'3), we have

7 996 R. Abe lin Zs(Kc, O)=ln 2+ 2~21"1"dXdy In(2-2cos x cos y). (A-I) By applying a formula (a>lbl): (A-2) we see that LIn 1 Zs(Kc, O)=ln 1 1" 2+ 0 In(I +sin x)dx. 27r The integral on the right-hand side is expressed aslo) 1" In (1 +sin x)dx= -J[ In 2+4G, where G is Catalan's constant given by (3-5). References (A-3) (A-4) 1). J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley, New York, 1954), p ) R. Abe, J. Phys. Soc. Jpn. 58 (1989), ) F. Hensel, High Pressure Chemistry and Biochemistry, ed. R. van Eldik and J. Jonas (Reidel, Dordrecht, 1987), p ) S. Koide, S. Hyodo and R. Abe, General Physics, Part I (Shokabo, Tokyo, 1983), p. 192 [in Japanese]. 5) T. D. Lee and C. N. Yang, Phys. Rev. 87 (1952), ) L. Onsager, Phys. Rev. 65 (1944), ) See, for example, G. F. Newell and E. W. Montroll, Rev. Mod. Phys. 25 (1953), ) C. Domb, Adv. Phys. 9 (1960), ) R. M. F. Houtappel, Physica 16 (1950), ) I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York and London, 1980), p. 527.