602 ZHANG Zhi and CHEN Li-Rong Vol Gibbs Free Energy From the thermodynamics, the Gibbs free energy of the system can be written as G = U ; TS +

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1 Commun. Theor. Phys. (Beijing, China) 37 (2002) 601{606 c International Academic Publishers Vol. 37, No. 5, May 15, 2002 The 2D Alternative Binary L J System: Solid-Liquid Phase Diagram ZHANG Zhi and CHEN Li-Rong Deartment of Physics, Anhui Normal University, Wuhu , Anhui Province, China (Received October 15, 2001) Abstract The Lennard{Jones otential is introduced into the Collins model and is generalized to the two-dimensional alternative binary system. The Gibbs free energy of the binary system is calculated. According to the thermodynamic conditions of solid-liquid equilibrium, the \cigar-tye" hase diagram and the hase diagram with a minimum are obtained. The results are quite analogous to the behavior of three-dimensional substances. PACS numbers: D, B, 73.20, 64.60C Key words: L{J otential, Collins model, binary, solid-liquid hase diagram 1 Introduction Since R. Collins resented a simlied 2D Bernal model, [1 2] Kawamura [3 4] and Do Yi-Jing [5 6] et al. have studied the thermodynamic roerties and some tyical diagrams of the monatomic and 2D binary systems. But they only considered the case that the nearest-neighbor interaction is a constant. Recently, You-Min Yi et al. [7] introduced the Lennard{Jones (L{J) otential into Collins model of 2D monatomic system. Then Zhang et al. [8] further introduced the \molecular fraction" into this system and the whole diagram of this system is obtained by theoretical calculation. In Ref. [8], the authors gave also some discussions about the trile oint and the critical oint of this system. On the base of the above studies, we generalized this calculating method by using Collins model and studied the 2D alternative binary L{J system. The tyical solid-liquid hase diagrams of this system are obtained. The results are quite analogous to the behavior of the 3D substance. 2 Model Collins model is a close-acked system of equilateral triangles and squares in one lane. Suose there are two comonents a and b, whose distribution on the lattice site is entirely disordered. According to the acked form of the triangles and squares, we can divide the atoms of the system into four local structures, which are called the 4-atom, 5-atom, 5-atom, and 6-atom resectively, as shown in Fig. 1. Fig. 1 Four local atomic congurations. Suose N is the total number of the atoms in the system, N a and N b are the number of the comonents a and b resectively. N ar and N br are the number of r-atoms of comonents a and b resectively (r = 4 5 5, and 6). Then we have N = N a + N b = N a4 + N a5 + N a5 + N a6 + N b4 + N b5 + N b5 + N b6 : (1) Writing N ar =N = n ar and N br =N = n br, equation (1) becomes n a4 +n a5 +n a5 +n a6 +n b4 +n b5 +n b5 +n b6 = 1 : (2) For the convenience of discussing, we introduce arameters x, y, m, and z, which satisfy the following relations n 5 + n 5 = x n 5 = mx n 5 = (1 ; m)x n 4 = y n 6 = 1 ; x ; y n a = z n b = 1 ; z : (3) Then the total area of the system [5 6] is A=N = [(2 ; 3)(x + 2y)=4 + 3=2]b 2 (4) where b is the distance between the nearest neighboring lattice sites.

2 602 ZHANG Zhi and CHEN Li-Rong Vol Gibbs Free Energy From the thermodynamics, the Gibbs free energy of the system can be written as G = U ; TS + PA (5) where U, S reresent the internal energy and the entroy of the system, resectively T is the temerature of the system, A is the area of the system, and P is the ressure of the 2D system which has the dimension of the force er unit length. 3.1 Total Internal Energy U When the system is a binary system, there are two comonents in the system and the interaction between the atoms should be divided into three tyes: interaction between a-a atom airs, interaction between b-b atom airs and interaction between a-b (or b-a) atom airs. Suose r j is the distance between the atom airs and the corresonding numbers of atom airs is N j (the relationshi between r j and N j can be seen in Ref. [8]). The three tyes of interactions can be reresented by aa (r j ), bb (r j ), and ab (r j ). We can see that the ossibility of the three tyes of atom airs is dierent, and they are z 2, (1 ; z) 2, and 2z(1 ; z) resectively. If we further consider the atom vibration around the lattice oint, the total internal energy of the 2D system U is U = X j N j [z 2 aa (r j ) + 2z(1 ; z) ab (r j ) + (1 ; z) 2 bb (r j )] + NK B T (6) where K B is the Boltzmann constant and P j corresonds to all kinds of distances between the atom airs. Now let us introduce the L{J otential (r j ) into the system, (r j ) = 4"[(=r j ) 12 ; (=r j ) 6 ] (7) where r j is the distance between the atoms, is the harddisk diameter of the close-acked system, and ;" is the minimum of the otential, as shown in Fig. 2. Suose the three tyes of interactions aa (r j ), bb (r j ), and ab (r j ) are all L{J otentials. They are aa (r j ) = 4" aa [(=r j ) 12 ; (=r j ) 6 ] bb (r j ) = 4" bb [(=r j ) 12 ; (=r j ) 6 ] ab (r j ) = 4" ab [(=r j ) 12 ; (=r j ) 6 ] (8) where ;" aa, ;" bb, and ;" ab are the minima of the otentials corresonding to the three tyes of interactions. From Fig. 2, we can see that the L{J otential (r j ) decreases raidly with the r j increasing. In Ref. [7] it is ointed out that we only need to consider seven kinds of distance of the atom airs, r j, smaller than 7 b (b 2 b 3 b b 2b 5 b, and b), and it is accurate enough to calculate the total internal energy in Eq. (6). The contribution of those atom airs with larger distance than 7 b to the total internal energy is very small and it can be neglected. So from Eqs (6) and (8) and Refs [7] and [8], we get U =NK B T = (1=K B T )[k f 1 (=b) 12 ; k f 2 (=b) 6 ] [" aa z 2 + " bb (1 ; z) 2 + " ab 2z(1 ; z)] + 1 = (1=K B T )[k f 1 (=b) 12 ; k f 2 (=b) 6 ] [z" aa + (1 ; z)" bb ; z(1 ; z)"] + 1 (9) where " is the mixed energy and " = " aa + " bb ; 2" ab k f 1 and k f 2 are two arameters related to the atom numbers of 4-atom and 5-atom. From Refs [7] and [8], they are k f 1 = 12: :0014mx ; 1:9484x ; 3:8914y k f 2 = 12: :0144mx ; 1:7926x ; 3:3789y : (10) 3.2 Entroy S The entroy of the system S is derived from the following arts: One, denoted by S cong, is the contribution from the disorder associated with the arrangement of the triangles and the squares The other, denoted by S heat, is the contribution of the thermal vibration of atoms around the lattice oint to the entroy. We write the entroy S by S = S cong + S heat : (11) Fig. 2 The Lennard{Jones interaction. The conguration entroy S cong is S cong = K B ln W 1 (12) where W 1 is the number of ways to arrange the atoms in four local structures in a lane, considering the two comonents, it can be reresented as [5]

3 No. 5 The 2D Alternative Binary L{J System: Solid-Liquid Phase Diagram 603 W 1 = N! N 4!N 5!N 5!N 4!N a!n b! (n 4 + n 5 ) (4N 4+N 5 )=2 (n 5 + n 5 ) (2N 5+4N 5 )=2 (n 4 + n 5 + n 5 ) (2N 5+2N 5 +4N 4 )=2 (n 5 + n 5 + n 6 ) (6N 6+2N 5 +N 5 )=2 (13) where N!=(N 4!N 5!N 5!N 4!N a!n b!) is the number of ways of arranging the a or b atoms in four local structures quite randomly, and the remaining factor is a correction due to the geometrical constraints in the neighboring sites. Using Eqs (3), (12), (13), and Stiring formula, we get where S cong =NK B = ;f(m x y) ; z ln z ; (1 ; z)ln(1 ; z) (14) f(m x y) = y ln y + mx ln(mx) + (1 ; m)x ln[(1 ; m)x] + [1 ; (x + y)] ln[1 ; (x + y)] ; 1 2 (4y + mx)ln(y + mx) ; (2 ; m)x ln x ; (x + 2y)ln(x + y) ; 1 [6 + (m ; 5)x ; 6y]ln(1 ; y) : (15) 2 On the other hand, S heat is due to the thermal vibration, and related to the free area of the atom vibration. By making use of the free volume theory, [9;11] we get S heat =NK B = z ln(2m a K B T=h 2 ) + (1 ; z)ln(2m b K B T=h 2 ) + ln( ) + ln[1 + C(x + 2y)] + 1 (16) where = b= ; 1, M a is the mass of the atom a, M b is the mass of the atom b, h is the Planck constant and C = (2 3 ; 3)= PA Term Suose that the total area of the system at the close acking A 0 is denoted by A 0 = 3 2 N=2, then using Eq. (4), we have the total area A, A=A 0 = [1 + C(x + 2y)](b=) 2 : (17) If let t = K B T=" s and = PA 0 =N" s, t and are the normalized temerature and the normalized ressure resectively, " s is the mixed energy of the solid hase. So the PA term can be written as PA=NK B T = (PA 0 =NK B T )[1 + C(x + 2y)](b=) 2 = (=t)[1 + C(x + 2y)](b=) 2 : (18) So using Eqs (5), (9), (11), (14), (16), and (18), the Gibbs free energy of the system can be written as G=NK B T = U=NK B T ; TS=NK B T + PA=NK B T = (1=t)[k f 1 (=b) 12 ; k f 2 (=b) 6 ][z" aa =" s + (1 ; z)" bb =" s ; z(1 ; z)"=" s ] + f(m x y) + z ln z + (1 ; z)ln(1 ; z) ; z ln(2m a K B T=h 2 ) ; (1 ; z)ln(2m b K B T=h 2 ) ; ln( ) ; ln[1 + C(x + 2y)] + (=t)(b=) 2 [1 + C(x + 2y)] : (19) According to our model, when x 6= 0, because of the existence of 5-atom, the system shows the short-range order, corresonding to the liquid. It means that equation (19) just corresonds to the Gibbs free energy of the liquid hase. When x = 0 and y = 0, there is only 6-atom conguration existing in the system which corresonds to the solid. So from Eq. (19), the Gibbs free energy of solid hase can be obtained easily. For the convenience of discussion, the suerscrits l and s reresent the liquid and solid hases resectively in all kinds of hysical arameters (such as the Gibbs free energies G l and G s, lattice oint distances b l and b s, minimum otentials ;" l aa, ;" l, bb ;"s aa, ;" s bb, the mixed energies " l and " s, the ercentagies of the a atoms z l and z s, and the arameters l, s, k f 1, k f 2 and k s1, k s2, etc.). Then from Eq. (19) we can get the Gibbs free energies of solid and liquid hases in the L{J system resectively. They are and G l =NK B T = (1=t)[k f 1 (=b l ) 12 ; k f 2 (=b l ) 6 ][z l " l aa ="s + (1 ; z l )" l bb ="s ; z l (1 ; z l )" l =" s ] + f(m x y) + z l ln z l + (1 ; z l )ln(1 ; z l ) ; ln(2 3 2 l 2 ) ; ln[1 + C(x + 2y)] ; z l ln(2m a K B T=h 2 ) ; (1 ; z l )ln(2m b K B T=h 2 ) + (=t)(b l =) 2 [1 + C(x + 2y)] (20) G s =NK B T = (1=t)[k s1 (=b s ) 12 ; k s2 (=b s ) 6 ][z s " s aa ="s + (1 ; z s )" s bb ="s ; z s (1 ; z s )] + z s ln z s + (1 ; z s )ln(1 ; z s ) ; ln(2 3 2 s 2 ) ; z s ln(2m a K B T=h 2 ) ; (1 ; z s )ln(2m b K B T=h 2 ) + (=t)(b s =) 2 (21)

4 604 ZHANG Zhi and CHEN Li-Rong Vol. 37 where s = b s =;1 and l = b l =;1. And from Eq. (10) we get k s1 = 12:0194 k s2 = 12:6319 : 4 Solid-Liquid Phase Diagram 4.1 Equations of Solid-Liquid Phase Equilibrium Curve According to the thermodynamic conditions describing the coexistence of two hases in a binary system, we have F s ; z = F l ; z l (22) s F s ; (1 ; z s ; z s ) = F l ; (1 ; z l ; z l ) (23) where F s and F l reresent the free energies of the system at solid and liquid hases resectively. Equations (22) and (23) can also be written as (G s ; PA s ) ; z ; PA s s = (G l ; PA l ) ; z ; PA l l (24) (G s ; PA s ) ; (1 ; z s ; PA s ; z s ) = (G l ; PA l ) ; (1 ; z l ; PA l ; z l ) (25) where A s and A l are the total areas of the system at solid and liquid hases resectively. From equation (17), we can get A s = (b s =) 2 A 0 and A l = [1 + C(x + 2y)](b l =) 2 A 0. Also the Gibbs free energies should satisfy the stability conditions of hase equilibrium, they = = 2 > 0 2 > 0 = l ) = s ) = 2 > 0 l ) 2 > 0 2 G s ) 2 > 0 : (30) If we substitute Eqs (20) and (21) into Eqs (24) (30), we can get a grou of seven equations of solid-liquid hase equilibrium curve in 2D binary system. 4.2 Phase Diagram We should notice that there are two factors ln(2m a K B T=h 2 ) ln(2m b K B T=h 2 ) in Eqs (20) and (21), but after we substitute Eqs (20) and (21) into Eqs (24) and (25), the two factors are cancelled and they do not aear in the grou of equations. So if the values of the attractive interactions, " s aa ="s, " s bb ="s, " l aa ="s, " l bb ="s, and " l =" s, are given roerly and the normalized ressure is xed to a constant, we can get the values of the seven arameters, m x y b s =, b l =, z s, and t, according to dierent z l (e.g., z l = 0 0:1 0:2 ::: 1). Then we can get the solid curve (z s t) and liquid curve (z l t) in the hase diagram for 2D binary system in theory. Now, let us assign two grous of the values to the attractive interactions " s aa ="s, " s bb ="s, " l aa ="s, " l bb ="s, " l =" s, and the normalized ressure. Substituting these values into Eqs (24) (30), we can get the hase diagram of the `cigar-tye' and the hase diagram with a minimum of the 2D binary system between the solid and liquid, as shown in Figs 3 and 4. The calculated results are listed in Tables 1 and 2 resectively. Of course, if we change the values of the attractive interactions " s aa ="s, " s bb ="s, " l aa ="s, " l bb ="s, " l =" s and the normalized ressure, we can also get other tyes of hase diagrams of the 2D system. Because the length of the aer is limited, we do not give further discussions. Table 1 The calculated results of the cigar-tye hase diagram. The values taken for the attractive interactions and normalized ressure are " s aa = ;0:5, ="s " s bb = ;0:8, ="s " l aa = ;1:0, ="s " l bb = ;2:0, ="s " l =" s = 1:0, and = 30. z l z s t m x y b l = b s = 0 0 2:3206 0:5766 0:8867 0:0063 1: : :1 0:2782 2:5650 0:5738 0:8997 0:0076 1: : :2 0:5412 2:9696 0:5750 0:9104 0:0095 1: : :3 0:6708 3:2268 0:5752 0:9146 0:0105 1: : :4 0:7454 3:3923 0:5760 0:9160 0:0110 1: : :5 0:7962 3:5211 0:5762 0:9167 0:0112 1: : :6 0:8373 3:6456 0:5732 0:9166 0:0113 1: : :7 0:8755 3:7700 0:5776 0:9166 0:0115 1: : :8 0:9137 3:9144 0:5766 0:9169 0:0116 1: : :9 0:9546 4:0926 0:5773 0:9172 0:0117 1: : :1978 0:5776 0:9151 0:0120 1: :126048

5 No. 5 The 2D Alternative Binary L{J System: Solid-Liquid Phase Diagram 605 Table 2 The calculated results of the hase diagram with a maximum oint. The values taken for the attractive interactions and normalized ressure are " s aa = ;0:03, ="s " s bb = ;0:02, ="s " l aa = ;1:0, ="s " l bb = ;0:5, ="s " l =" s = 0:0005, and = 35. z l z s t m x y b l = b s = 0 0 4:0983 0:5763 0:9039 0:0090 1: : :1 0:0352 3:9725 0:5769 0:9030 0:0091 1: : :2 0:0756 3:7016 0:5768 0:8984 0:0085 1: : :3 0:1246 3:4323 0:5767 0:8932 0:0078 1: : :4 0:1891 3:1618 0:5775 0:8861 0:0070 1: : :5 0:2902 3:0241 0:5744 0:8814 0:0066 1: : :6 0:4622 2:9304 0:5724 0:8759 0:0062 1: : :7 0:6954 2:8150 0:5755 0:8733 0:0058 1: : :711 0:711 2: :5756 0:8733 0:0060 1: : :8 0:8432 2:9322 0:5750 0:8803 0:0064 1: : :9 0:9339 3:0047 0:5867 0:8827 0:0070 1: : :0893 0:5769 0:8882 0:0071 1: : Fig. 3 The binary solid and liquid hase diagrams of the `cigar-tye', where the normalized ressure = 30. Fig. 4 The binary solid and liquid hase diagrams with a minimum, where the normalized ressure = 35.

6 606 ZHANG Zhi and CHEN Li-Rong Vol Discussions (i) By considering the nearest neighbor interaction that is a constant, H. Kawamura [3 4] and Do Yi-Jing et al. [5 6] have studied the 2D Collins model in the monatomic system and in the binary system. Their results have quite large dierence with the real system, so the results are rough and unreasonable. You-min Yi et al. [7] introduced the L{J otential into Collins model, but they only studied the monatomic system. Here we introduced L{J otential into the 2D Collins binary system and this is an imrovement on the base of their studies. The obtained two tyical hase diagrams (`cigar-tye' hase diagram and the hase diagram with a minimum oint) of the 2D binary system are quite similar to the 3D substance. It is roved that our imrovement is more reliable and reasonable. (ii) The grou of seven equations of solid-liquid hase equilibrium curve has not an analytical exression. So we can only get the results by numerical method. The tyes of the hase diagrams relate closely to the chosen values of attractive interactions. How to choose the attractive interactions roerly will be discussed in another aer. (iii) We can foresee that other otentials (such as the Morse otential) can also be introduced into the Collins model or we can generalize this model into threecomonent system (ternary system) and many-comonent system. These are also very interesting studies in the future. References [1] J.D. Bernal, Proc. Roy. Soc. A280 (1964) 229. [2] R. Collins, Proc. Phys. Soc. 83 (1964) 553. [3] Hikaru Kawamura, Prog. Theor. Phys. 61 (1979) [4] Hikaru Kawamura, Prog. Theor. Phys. 63 (1980) 24. [5] DO Yi-Jing, CHEN Li-Rong, and YAN Tzu-Tong, J. Phys. C: Solid State Phys. 15 (1982) [6] DO Yi-Jing, CHEN Li-Rong, and YAN Tzu-Tong, J. Phys. C: Solid State Phys. 15 (1982) [7] You-Min YI and Zhi-Chun GUO, Commun. Theor. Phys. (Beijing, China) 11 (1989) 7. [8] Zhi ZHANG and Li-Rong CHEN, J. Phys.: Conden. Matter 13 (2001) [9] J.E. Lennard-Jones and A.F. Devonshire, Proc. Roy. Soc. A163 (1937) 63 ibid. A165 (1938) 1. [10] L.G. Caron, J. Chem. Phys. 55 (1971) [11] T. Ichimura, N. Ogita, and A. Ueda, J. Phys. Soc. Jaan 45 (1978) 252.