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(1)[Science Reports of the. Yokohama Nationa! University, Sec. I, No.. 8, 1961]. Some Contribution to Lattice Model Theory of Polymer Solutions" By. Tatsuro WATARI (Received April 20, 1961) A systematic method of constructing the combinatory factor for configurational partition function of an assembly on !attice is. given in Chap. II. This method, called pseudo-assembly method, not only makes the construction of combinatory factors easier, but also clarifies their natures.. In Chap. III, it is shown that the pseudo-assembly method is very ethcacious when applied to the construction of combinatory factors of chain polymer solutions, and that it gives configurational partition functions which lead to an improved lattice model theory of chain polymer solutions, giving the second osmotic virial coefficient in terms of shape factors introduced in order to specify the shape of a polymer chain.. I. Introduction A. CIuster Variation Method In statistical thermodynamics, there are a large number of problems which can be reduced, after suitable rewritings, to those of assemblies on lattice. Although there remain, especially in the case of fluids, many problems in such. rewritings themselves, once the problems have been reduced to those on lattice, one of the central problems is to find out the configurational partition function :-. Z = 21 G, exp [- E,/le T],. s where Gs represents the number of distinguishable configurations of an assembly of lattice elements on a given lattice when the interaction energy is Es, fe is. the Boltzmann constant, and T is the absolute temperature. Lattice elements represent, for example, atoms in the case of alloys, magnetic spins in the case. of ferromagnets, molecules and holes in the case of pure liquids, and solute * Submitted to Tokyo University of Education to fulfi1 the requirements for the degree of Doctor of Science, under the title of Pseuao-assembly Methoel and its Application"to Lattice Mbael TheoTy of PolymeT Solzetions, on Nov. 18, 1959..

(2) :. l l I. i. 44 ' , T.WATARI and solvent molecules in the case of solutions.. Usually, the total interaction energy is tacitly 'assumed to be the sum of those due to the nearest neighbors. In such a case, the partition function may be written as follows:. Z-=2G({AT)},{yj,te},{hr})exp{-E({M},{yj,k},tu)lkT], (A.1). ,. {Yd,k}. where Alh represents the number of lattice elements 1', yj,ic the probability of ･s. finding a pair of cluster of lattice elements 7' and k on a pair of neighboring lattice points, G the number of distinguisl able configurations of the assembly satisfying the set {yd,k}, and E is the total interagtion energy of the assembly,. Furthermore, in the above equation, hr represents the r-th index specifying the structure of the lattice. When the lattice with Nlattice points has caN lattice bonds and the coordination･number of the lattice is 2tu, tu may be assigned. to the first specification index hi. It must be noted that, while E is determined. by giving hi=w, G is determined uniquely only when the full set {hr} is given.. In the case of a one- or two-dimensional lattice, the rigorous estimation of the partition function (A.1) has been carried out with the so-called matrix. methed by Onsager and others.i) However, the three dimensional problenLhas been left unsolved by the similar method owing to the difliculties in carrying out the calculation practically.. There has been another method more direct in its approach to this problem. Though this method has given only th,e approximate estimation of the partition function even in the case of a two-dimensional problem, it is easily applicable. to the three-dimensional problem. The original type of this method owes much to the works of Bethe,2) Fowler,3) and Takagi.`) In this method, hi=ca is used as the sole specification index of lattice structure. Hence, G in Eq. (A.1) is. constructed in terms of {M}, {yd,k} and to. In the present paper, this method is called the first approximation method.. Further developments have been made in this method by taking into account additional specification indices corresponding to the existence of higher. correlations of lattice points. In the case of a triangular lattice with tuN lattice bonds and vlV, triangular cells, for example, the existepce of triangular cells in the lattice structure is taken into agcount by assigning p tg the second specification index h2. Accordingly, the probability zj･,ic,i of finding a triangular. cluster of lattice elements 1' , le and i i's introduced to construct G. Such a. method is called the second approximation method in this paper. Similarly, by introducing other specification indices representing existences. of polygonal andlor polyhedral correlations of lattice points and by adopting.

(3) '. SomeContributiontoLatticeModelTheoryofPolymerSolutions 45 probability variables corresponding to larger clusters of lattic6 'elements in order. to construct G, it becomes possible for this method to proceed to higher approxi-. mationswithvariousdegrees. '' ･ In the present paper, the probability variables yj,k, zd,k,i, etc., are called. cluster variables, and the factor G in Eq. (A.1) is called combinatory factor. Though this method has been called by various names, it seems most suitable to call clzaster variation method, since in this method the term giving the maximum contribution to the partition function is selected from the summands of Eq. (A.1) by maximizing the contribution with respect to cluster variables,. and the free energy of the assembly is estimated from this maximum term.. ' ' B. Combinatory Factors for Higher Approximations When larger clusters of lattice elements are adopted so as to describe the. structure of a lattice in greater detail, the method becomes tremendously. diMcult in the following points: , a) Proper construction of the combinatory factor. b) Derivation of the free energy from the partition function. With regard to a), aside from the increasing diMculty in constructing the combinatory factor itself, it has been poin.ted out by several authors that the. method could not necessarily be improved by merely adopting larger clusters. ProPer construction does imply proper adoption of larger clusters as well. .. A general method for the proper construction of combinatory factors has. been given by Kikuchi.5) However, his method is somewhat diffLcult to manipulate. In the present paper, an alternative method is proposed. The method is quite systematic in constructing combinatory fabtors. Furthermore,. it not only makes the construction of combinatory factors easier, but also contributes much to clarifying the features of various degrees of approximation.. This method is tentatively called Pseudo-assembly metho4,, the details of which. may be given in the next chapter. ･ '. '. IL ,Pseudo-Assembly Method ttt .. C. Construction of Combinatory Factor for One-dimensional Lattice '. In order to facilitate the understanding of the pseudo-assembly meth,od6)･7). for constructing combinatory factors, and to explain the terminologies and abbreviated notations used in this method, an assembly of lattice elements on.

(4) 46 . T.WATARI a one-dimensional lattice is considered in the first place. ,. When an assembly of IV single-$ite lattice elements with n components on a one-dimensional lattice composed of IV lattice points is considered, and the. number of lattice elements of species 1' , 7'=1, 2,･･･, n, is written as Al]i, it follows that. n. ZM=N. . ･ (C.1). j' =1. tt. If the probability of finding a lattice element 7' on an arbitrarily selected. ' a pair cluster of lattice lattice point is denoted by xd, and that of finding elements 7' and le on a pair of lattice points neighboring each other, i.e., on a lattice bond, by yj,k, it follows that. n ik =1. xi=A7]ilN=2yj,,, (C.2). SIIxi=1,and Sl yd,k=1. (C･3). j'=1 ik==1. In Fig. 1, a fragment of an assembly of lattice elements with two components is shown, and the explanation is given of the method for this assembly,. but the formulation is made for a more general case of n components. It is --e-o--e--e-.o.-omo-.e- assumed that the assembly in (a) satisfies the {a)Assembiyon one･dimensionanattieo given sets of xe' and yik. In such a case, a. simpler expression "the assembly has a right distribution (r.d.) in y" is used. In (b), an. wwe-oalternativerepresentationofthesameassembly "-e o-mo ome. (b)Assembiyhavingrd inu iS giVeri So as to depict the assembly as a mode of configuration of pair clusters on lattice. bonds. In this figure, every fattice element. ww is represented doubly on the upper and lower {e) Q-assembiy sides of each lattice point. The problem to construct the proper combinatory factor for a one-dimensional. eoeeoooe latticeistofindoutthenumberofdistinguish-. eoeeoooe. fd}Assnmbiyhavingr,d. inx able COnfigurations G(2} of such an assembly of pair clusters as shown in (b). For this purpose the following procedure is devised.. eooeoooe (1)Thepairclustersaredistributedon. ,,)p-,e,,,.be/, e O O e O e the lattice bonds independently to construct. Fig. 1. an assembly of pair clusters having a right. ,.

(5) SomeContributiontoLatticeModelTheoryofPolymerSolutions 47 distribution in y as shown in (c) which is called a Q-assembly. In the Qassembly such a configuration as shown in (c) is necessarily included, and two different species of lattice elements occupy the same lattice point. The number. of distinguishable configurations of the Q-assembly, denoted by Q`2}, is obviously given by. Q(2)=IV!/ fi (yj,,N)!. (C.4) o',k=1 For the brevity of expression, the abbreviated notations similar to [point]L == L!/ II n(xj･L)! ,. j=1 n' [bond]L =L!1 ll (yikL)!, ik=1 and relations similar to. [point]qz=:([point]L)q,. [bond]qL=([bond]L)q, (C.5) will be used hereafter without remarks.*. Adopting the abbreviated notation given above, Eq. (C.4) may be written as. Q(2)=[bond]N. (C.6) (2) 2N lattice elements are distributed on the upper and lower sides of the lattice points to make an assembly of 21V lattice elements having right. distribution only in x as shown in (e) which is calleda P-assembly. The number of distinguishable configurations of the P-assembly is denoted by P('),. and is given by. P(i)==[point]2N=([point]N)2. (C.7) (3) In the Q-assembly, there should be configurations in which two lattice. elements located on any of the lattice points coincide with each other as shown in (b). The number of such configurations is surely G(2} in question. (4) Quite similarly, in the P-assembly, there should be configurations as shown in (d). The number of such configurations, denoted by G(i), is apparently equal to the number of distinguishable configurations having a right distribution. only in x. Hence,. G(i)= [point]N. (C.8) * In the case where L is sufficiently large, it is possible to use Stirling's formula, which gives Eq. (C.5)..

(6) 48 T. WATARI. (5) Writing the fraction of G(2) in Q{2) and that of G(i) in P(i) as T(2) and. rco, respectively, i.e･,. rC2)==G(2}IQ(2), and l"(')==G(i)!P(i), (C.9) the comparison of the role of r(2) with that of T(i) reveals an essential similarity. between them: T(2) selects G(2) configurations from Q(2) configurations of the Q-assembly while I'(i) selects G<i) configurations from P(') configurations of the. P-assembly. -- ' ' (6) Now it is assumed that the following identity holds: fromwhichitfollowsthat ' ,. r(2)=r{i), (C.10) ' ' ' G(2)=GCi)S(2), (C.11). with. S(2)==QC2)/P{i). (C.12) ' by Eqs. In the present case of a one-dimenesional lattice, G<2) is expressed '. (C.6)to(C.12)as ･. G`2)= ([point]N)-i[bond]N, (C.13) or expl･icitly by Eq. (C.5) as. G(2) == n(x,･N)!ln(yi leIV)!. (C.14) In the present paper, the P-assembly is called the Pseudo-assembly for the. Q-assembly. Since the construction of the P-assembly plays an essential role in the method outlined above, it may be called the Pseudo-assembly method. It can be shown in the following sections that the method is easily applied to two- or three-dimensional lattices.. In the present case of a one-dimensional lattice, the exact expression of G(2) for a circular lattice has been given by the present author* as. G(2)==Cn([point]N)Hi[bond]N,. nn. Cn = (il,I=,Yik h',k)1(,.I.Iil,Xd) ,. with rj,k =1/yi,k, if 1'=k,. ==1, if1'7Ek, (C.15) which holds for any N when yj,k=yk,d, i.e., the assembly is isotropic.. By inspecting the terms in G{2), the factor Cn is found to be perfectly * See Ref. 6) where C2 and C3 are given. C4 also conforms to Eq. (C.15), but the result is not published. For larger n, practical calculations are too cumbersome to carry out, and the equation is the formal generalization of the forms for C2, C3 and C4..

(7) Some-ContributiontoLatticeModelTheoryofPolymerSolutions 49 negligible in comparison with ([point]N)-i[bond]N. Thus, it may be said that. G(2) given in Eq. (C.13) by the pseudo-assembly method is exact from the standpoint of statistical thermodynamics,* and that the underlying assumption, given in Eq. (C.10), in the pseudo-assembly method is justified at least in the case of a one-dimensional lattice.. D. Combinatory Factor for lst Approximation Here the method developed in the previous section is applied to assemblies on a two- or three-dimensional lattice. As pointed out in Sec. A, hi =tu is used as the sole specification index of lattice structure for the combinatory factor. for the first approximation method.. Itt Consider an assembly of N single-site lattice elements on a lattice with N. lattice points and toN lattice bonds. In Fig. 2 (a), a small part of an assembly. (a) Original assembly having r.d. in y. (c) Q-assembly. (b) Assembly having r.d. in V. (d)Assernblyhavingr.d.inx (e)P-assembly. Fig. 2. * The combinatory factor giveri by Eq.･ (C.13) for a one-dimensional lattice has been proposed by variou$ authors, giving the same results as those obtained rigorously with the method of matrix. (See, for example, Re.f. 1) of the present paper.) However, the direct comparison of G(2) itself with the exact expression for any N and n does not. seem to have been made..

(8) 50 T. WATARI. with two components (n=2) is shown on a 'two-dimensional triangular lattice (tu=3), and the explanation will be given for this case, but the formulation. will be made for the general case of any n and to. In (b), an alternative representation of (a) is given, and in order to draw attention to the pair clusters, each of the lattice elements is wtitten 2to-ply. (c), (d) and (e) correspond. to those in Fig. 1, respectively. These figures themselves apparently show their constructions.. The number of distinguishable configurations of the Q-assembly, Q(2), that of the P-assembly, P(i), and that of the assembly in (d), G{'), are given respectively by. Q(2)=[bond].N=([bond]N)tu, (D.1) Pì)=[ppint]2.N==([pointliv)2W, (D.2) and. G(i}=[point]N. (D.3) By introducing these equations into Eqs. (C.11) and (C.12), the combinatory factor. G.`2)=([point]N)i-2tu([bond]N)tu, (D.4) is obtained which has already been derived by Takagi`) and others in rather complicated ways. For a one-dimensional lattice, by putting tu =1, Eq. (D.4) is reduced to Eq. (C.13).. E. Combinatory Factor for 2nd Approximation In the first approximation method, since the 6ombinatory factor is constructed by using tu as the sole specification index of lattice structure, there can be no. distinction, for example, between a two-dimensional triangular lattice and a. simple cubic lattice, both having the common coordination number, 2to=6. In order to proceed to the second approximation method, it is at least necessary to indicate the fact that the former lattice has vN triangular cells by using v as the second specification index h2 of the lattice structure. In Fig. 3 (a), there is given a duplicate of Fig. 2, but in the present case, the assembly has a right distribution in triangular variables as well. A triangular cluster variable is denoted by zj,k,i which represents the probability of finding. a triangular cluster of lattice elements 7' , le and i on a triangular cell. The. combinatory factor for the second approximation method is written as Gv(3) and v= 2 in the Present case of a two-dimensional triangular lattice, since the. lattice has 2N triangular cells. Full explanation will be given for this case, but the formulation will be made in eommon with the case of a face-centered cubic lattice, i.e., v==6. An alternative representation of (a) is given in (b),in.

(9) Some. Contribution to Lattice Model Theory of Polymer Solutions. 51. (a) Original assembly having r.d. in z. '. NMis -$MsL. tsYzsSZtwa 7xS<7xSe2xsr (b>Assemblyhavingr.dinz '. (c) Q-assembly. "a M. ."e. pt`4 pt". -e e--o o--7' /pt-ss.'by.-. .e b-o e-. 9NNNM. --o o--e o-- eM!hN.N 21NNN. us･-if" "N"N-. '-e o-ip ase--"e th"NlfN ･ (e)P-assembly･. td) Assembly having r.d. in y. Fig. 3.. '. '. which special attention is paid to triangular clusters by writing each of the pair clusters (3v/tu)-ply. (c) is the Q-assemblyfor the present case. In (d), an. assembly having a right distribution in y, but not in z, is given, but the pair clusters are written (3vlto)-play along the lattice bonds to provide convenience. forimitating (b). (e) is the P-assembly. 'It must be kept in mind that in the P-assembly two lattice elements at the same corner ofa triangular cell coincide. with each other throughout the lattice. In other words, the pair clusters are. properly connected at the corners of triangular cells.. The number of distinguishable configurations of Q-assembly is given by. -([triangle]N)", (E.1). Q(3) = [triangle] vN -. where the abbreviated notation. n [triangle]L==L!1,.,,4,,.,(z,･,k,iL)!. tt ,. which is similar to Eq. (C.5), is used.. Now, in the P-assembly, 3pN pair clusters having a right distribution in y are properly connected at the corners of the triangular cells. Hence, the number of distinguishable configurations of the P-assembly, P(2), is obtained.

(10) 52 T. WATARI. by introducing to=1 and substituting 3vN for N in Eq. (D.4).. + P(2) = ([point]3vN)-i[bond]3vN ,. =([point]N)-3'([bond]N)3'. (E.2) The.number of distinguishable configurations of the assembly in (d) is apparently G.(2) itself.. When written r(3) =Gv(3)IQ(3), and assumed that the similar assumption as expressed by Eq. (C.10) is still valid for the present case,･i.e.,. tt. I-(2) ',,,, I-(3), (E.3) it follows that. G,(3) =G.(2)SC3), (E.4) with. S(3) =Q(3)IP(2), (E.5) or, when Eq. (C.11) is introduced into Eq. (E.4),. G,(3) == G(i)S{2)S(3). (E.6) The complete expression for Gv(3} is obtained, by combining Eqs. (D.4), (E.1), (E.2) and (E.5) with Eq. (E.4), as follows:. Gv`3)=([point]N)'-2tu+3'([bond]N)to-3'([triangle]N)'. (E.7) By putting to =3 and v==2, and tu==6 and v=8, the combinatory factors of. the second approximation inethod m'ay be obtained for a two-dimensional triangular lattice and a face-centered cubic lattice, respectively.. ' ' By the exactly similar procedure, the combinatory factors of the second approximation method for various kinds of lattices may be easily constructed.. Many examples including those of the third approximation method have been given in Refs. 6) and 7).. F. Selection Rule of Cluster and Degree of Approximation The generalization of Eqs. (C.11) and (E.6) may lead easily to the following equations:. -'. G(r> = G(i)"S(s). '. {s} ==2, 3,･･･, .r, (F.1). where G(') represents the combinatory factor constructed in terms of r-lattice-. point-cluster, and ･ '. `.

(11) SomeContributiontoLatticeModelTheoryofPolymerSolutions 53. G(oo)=lim G(r), (F.2) r-oo , ' where G(oo) represents the rigorous combinatory factor with which the exact results would be obtained, in principle, by the cluster variation method.. However, there still remains a problem how to choose the proper procession of s, i.e,, the proper set {s}.. In Bethe's method, the cluster shown by (e) in Fig. 4 is used in the case. of a two-dimensional square lattice. However, it has been shown that the adoption of Bethe's cluster as well aS those shown by (b) to (d) leads to the. oo oo. as ptgb .{. L. '. gg (f). ' (a) (b) (c) (d) . .te) '. ma. {i liili. (g). (h). l]illilli. (i). Fig. 4.. same results as obtained by using pair cluster shown by (a). Hence, the procession of s=2, 3, 4, 5 along the series of figures in the first row of Fig.. 4 is trivial in improving the degree of approximation. The present author and his collaborators6)･7) have found the similar situation that the adoption of. clusters given as (g) and (h) does not improve the degree of approximation higher than the second approximation in which square cluster shown by (f) is adopted. Hence, the procession of s=4, 6, 8 along the series of figures in the. second row is of no significance. . On the other hand, it has been proved that the successive adoptions of clusters along the first column of.Fig. 4, i.e., (a), (f), (i), improye the degree. of approximation step by step. Hence the procession of s should be taken along the first column, i.e., s=2 (pair), 4 (square), 9 (quadruple square), etc.. ' Now it must be noted that the succession of clusters, (a),' (f), (i),･･･, is.

(12) 54 ' T.WATARI '. characterized by the following points: (1) Cluster (f) is closed with respect to cluster (a). (2) Cluster (i) is closed with respect to cluster (f).. The similar situation has been found in other types of lattice, and the. presentauthorhasgeneralizedthissituation . and presented a rule for the selection of a)one-dimensionaiLattice: ome. the. !r.1'iZL2eir2g]j./iC/,ei:/i.gi?:".fi20sio;Sg.,is,sb,z,.:,ig,ssg.r'th, b'[li/lii[li'i,dl'ffia:SiiaiÌi,llii,,,,,e {}i{}l{} ete. 1. CiUStTtllSearrueiegLX.egninabFoivg6s5e'emstobesome- o-o ,, cs8b ･ gEliilll ･etc what empirical, but it has been justified more c) Three-dimensional Lattice:. rigOrouslybyKurata.8) , (i)Simpieeubiciattice o-o ･ gg ･ illSilS ･ete･ Hence, Eqs. (F.1) and (F.2) may be rewritten as. (ii) Faee-centered cubie lattice. '. d-･--e,A･d{SiP ･etc･. G(tur) =G(1) II S(dis) ,. {s} , Fig.5.. {as}=ai,a2,･･･,ar, (F.3) ProperSuccessionofClusters and r->oo. G(oo)= lim G(ctr), (F.4) '. where a, represents the number of lattice elements in the s-th cluster in proper. succession of clusters as shown in Fig. 5, for example. The cluster variation method using G(ctr) as the combinatory factor should. be called the r-th approximation method. Thus the degree of approximation which has been left quite ambiguous for the lack of criterion is assigned ,easily.. The rate of improvement of the cluster variation method as r increases seems quite rapid, since the results of the second approximation method for a two-dimensional Ising lattice approach quite closely to the rigorous solution. obtained by the matrix method. The present method of constructing combinatory factors has been studied by Hijmans and De Boer9) with further examinations and applications. ' ''. Bethe Lattice and Kramers-Wannier-Kikuchi Lattice The essential features of the first approximation and the second approxi-. xxiatiQn methQds must be pointed Qut here, fQr they are very important in.

(13) SomeContributiontoLatticeModelTheoryofPolymerSolutions 55 treating polymer solutions by the present method. In the first approximation method, the lattice structure is specified only. by its coordination number as pointed out frequently. Such an abstraction of the lattice structure is equivalent to replacing a real lattice with a hypothetical. lattice having a dendritic structure shown in Fig. 6. The lattice may be called Bethe lattice, since Bethe's approximation method, i.e., the first approxi-. mation method, is exact for this lattice owing to the fact that any assembly on Bethe lattice does not contain any cluster. closed with respect to pair clusters. This is the essential feature of the first approxi-. mation method. Similarly, in the second approximation. method, a real lattice is replaced with a lattice having only the smallest closed cells. with respect to lattice bonds which are connected with one another on every side. Bethe lattiee with. 2w--4 Fig. 6.. of the closed ceil as in the original lattice, nevertheless it has no larger closed. cell of lattice points. For example, in the case of a two-dimensional triangular lattice, the lattice is replaced with a hypothetical lattice having only triangular. cells connected side by side, nevertheless it lacks in hexagonal corre!ation of. lattice points. Such a lattice may be calledKramers- l7Vannier-Kikuchi lattice, since it has been found by Kikuchi5) that Kramers and Wannier's approximate. method is equivalent to the second approximation method.. G. Combinatory Factor for Intermediate Approximation It may be interesting to consider the. construction of the combinatory. factor for the cluster variation method with intermediate degree of approxima-. tion.* The combinatory factor forthe (1 +q)-th approximation method, (OEqSl),. for triangular lattices may be constructed by applying the pseudo-assembly. method only to qvN triangular cells. The combinatory factor, Ge3.), may be 'glven as. (3) G - ([point]iv)'-20+3q'([bond]iv)tu '-3qV([triangle]N)q' ,. qv -. (G.1). which may be reduced to Eqs. (D.4) and (E.7) by putting q=O and q=1, respectively.. ' When q satisfies specially the condition:. '. '. * Somewhat generalized treatments of combinatory factors for intermediate approximations are given in Ref. 7)..

(14) 56 T. WATARI '. '. q=wl(3v), (G.2). the combinatory factor takes a very simple form:. G83>,=([point]N)`hul([triangle]N)tu/3 (G.3) This is essentially equivalent to the combinatory factor given by Yang, Li and Hill.io}. '. t:/tt b.. g. '. e. ti. g fa dr. d. a. f. L bCZ. c. Wf e t/ /. 1' Z' .. (a) Original lattice (b) Yang-Li-Hill lattice. Fig.7., '. The lattice on which Eq. (G,3) holds rigorously is such a lattice as shown in Fig. 7 which may be called Ytzng-Li-Hill lattice. This lattice has no closed cell larger than a triangular cell, and triangular cells are connected with one another at their corners so as to satisfy the coordination number of the original lattice.. III. Chain Polymer Solutions Since the earlier deve}opment of the lattice model theory of polymer solu-. tions which owes much to pioneering works of Flory and Huggins, a large number of improved theories have been proposed by various authors.ii) These theories deal chiefly with the' combinatory factor or geometrical effect of chain. polymers. However, there are other two effects, energetical effect and structural. effect. The energetical effect is due to the difference in interaction energies. between two neighboring lattice elements. This kind of effect is exactly the same asin monomer solutions. On the other hand, as pointed out by Prigogine,i2) the structural effect is specific to polymer solutions and due to the difference. in external degrees of freedom of various lattice elements., However, one of the main defects of existing theories seems to rise from the incomplete estima-. tion of the geometrical effect of chain polymer molecules. It may be seen easily that the majority of existing theories correspond to the fi. rst approximation method which uses a Bethe lattice shown in Fig. 6 of. l.

(15) SorneContributiontoLatticeModelTheoryofPolymerSolutions 57 '. Sec. F, since in these theories the lattice is specified only by its coordination. number. In order to make the feature of existing theories clearer, consider,. for example, an assembly of chain polymer molecules on a two-dimensional triangular lattice. Three conse6uitive segments, labeled by 7', 7'+1 and 1'+2, of. a certain polymer chain may occupy three consecutive lattice points in various. ways. When the three segments occupy a triangular cell of the lattice, the segments 7' and 1'+2 may be found respectively on two lattice points neighboring each other. Such an intramolecular correlation of the segments 1' and 7'+2 is called an inner correlation of a polymer molecule. In existing theories, the triangular lattice is replaced implicitly by a corresponding Bethe lattice which. is lacking in triangular cells. Hence, such an inner correlation cannot be transcribed onto the Bethe lattice. Furthermore, on the Bethe lattice, once two separate chain polymer molecules approach each other, and two segments, each belonging to a separate polymer molecule, occupy two neighboring lattice points, the other parts of the two polymer molecules cap never approach each. other to occqpy another pair of neighboring lattice points as seen clearly from Fig. 6. In brief, intermolecular pair correlation of two separate chain. polymer" molecules can be taken into account once and only once. Such situations as neg' lect all of the inner correlations and as limit the intermolecular. correlations of, 'two separate polymer molecules to only once are surely drastic oversimplification of the real situations on the original triangular lattice except. one case where the polymer molecules are stretched rigidly or rod-like.. In order to take the shape of a polymer chain into account, it may be necessary to proceed to higher approximatiQn methods in which closed correla-. tionsoflatticepointsaretakenintoaccount.･ ' ･･ In the following sections of the present paper, an improved lattice model theory of chain polymer solutions is given in order to calculate the secQnd osmotic virial coefficient in terms of shape factors which define the shape of a. polymer chain. Though the method. of the calculatiQn may be applied quite generally to assemblies on various kinds of lattices, the practical calculation is carried out in the case of triangular lattices and the greater part of numerical. results for the second osmotic virial coefficient are given in a more limited case. of a two-dimensional triangular lattice for the purpose of exemplifying this g,e. nesri9niclli2thheOdo:riginai type of this improved theory has been given'in Ref･ i3),. it may be convenient for some readers to read the following sectjons in close conjunction with Parts I and II of the referred paper..

(16) 58 T. WATARI. H. Basic Formulation. Consider a lattice model polymer solution composed of Alb single-site solvent. molecules labeled by 1'=O and Allo P-site chain pplymer molecules. Each of the. polymer molecules consists of P segments which are connected by P-1 chemical bonds and are labeled by 1'--1, 2,･･･P from the head to the tail.. By writing the total number of solvent molecules and segments, which is equal to the number of lattice points, by N, and the volume fraction of polymer in the' solution by i', it follows that. IV==Aib+PATb, (H.1). and. c. v==PAilo11V. (H.2) In Fig. 8, a small part of the lattice model. polyMer solution is given on a two-dimensional. triangular lattice, and here the solvent molecules are omitted in order to simplify the. i+2. i+1. Nhs'. figure. However, the formulation is made as. j. far as possible for a triangular lattice with. wN lattice bonds and vlV triangular cells. A two-dimensional triangular lattice corresponds 'to tu =･3 and v=2, while a face-centered cubic. lattice corresponds to tu='6 and p=8.. The chemical bond connecting a segment. Fig. 8.. 1' with a segment 1'+1 is shown by a thick solid line with an arrow indicating. the sense of the chemical bond as shown in the figure. A pair of consecutive. segments 1' and 1'+1 connected by a chemical bond is written as 1'>,<7'+1 symbolically. The inner correlation of the segments 7' and 1'+2 is shown by a thin broken line with an arrow indicating the sense of this correlation as. '. shown in the figure. The broken line is called an inner correlation bond. A pair of segments 1' and 1'+2 connected with an inner correlation bond is written as 7'D,cl' +2 symbolically.. ' Restriction on intramolecular Cbrrelations In Fig. 9, is given another part of the same polymer. solution in which various kinds of intramolecular correlations are shown. However, the following restric-. tion is imposed on the occurrence of intramolecular. '. -)-. pt. '. --p >e. correlations:. "Intramolecular correlations between two segments 7' and 7'+s with sl3 are ignored, i.e., prohibited.". Fig. 9.. '.

(17) SomeContributiontoLatticeModelTheoryofPolymerSolutions 59 Hence, the existing intramolecular correlation is only the correlation between. segments 7' and 1'+2 which has been called an inner correlation as mentioned previously.. The restriction given above is partly intrinsic in the present theory and partly for the convenience of simplifying the formulation. However, the detailed consideration of this restriction will be given later in Sec. M.. ShmpePactors ･ , '. In order to specify the shape of a polymer chain on the original lattice,. shape factors Pj{S), s=1, 2,･･･, and 7'=1, 2,･･･, P, are introduced for each of. the segments of a polymer molecule in the following ways: ･ Pd('): Probability of being bent T13 at the segment 1' of three consective segments 7'-' 1, 1' and 1'+1, Hence, the probability of being not bent z/3 is 1-P,･{i).. Pj(2): Probability of being bent 2T!3 at the segment 1' of the three consecu-. tive segments not bent z13 at the segment i Hence, the probability of being bent 2z/3 at the segment 1' of the three consecutive segments is (1-Pj(i))Pd<2), and that of being kept stretched at the segment 7' i's. (1-Pj･(i))(1-P,･{2)). -. Pj(3): Probability of finding a bending in cis form of four consecutive segments 7'-2, 1'-1, 1' and 7'+1 after having been bent 2z!3 twice at. the segments 7'-1 and i Hence, the probability of being bent in cis. form of the four consecutive segments is . (1 - PS.i-) ,)PS.2-) ,(1 - Pj (i))Pj (2)pj (3) ,. and that in trans form is (1-PS･i-),)PS･22,(1-Pj(i))Pj(2)(1-Pd(3)).. For the full description of the shape of a polymer chain on the original. lattice, a large number of the other shape factors are necessary. However, tfh oremtuhlraet?.oknilndS Of Shape factors given above are sumcient for the present. ' When' &Cs) =O or 1, for all s and 7',. the polymer chain is rigid and'takes a definite shape. On the contrary, when O<Pj(s)<1, at least for some s and 7', '. the polymer chain is flexible. Some examples of the relationship between g?asPeec.fai.tOrS and the shape of a polymer chain are given in Tables s and 6. '. '.

(18) 60 ･ T.･WATARI '. ' Cinster Vicuriables , Consider an arbitrarily selected triangular cell abc composed of three lattice. points a, b and c on which various kinds of cluster variables are introduced. asfollows: ,. ' xj: Probability of'finding a lattice element 7' on a lattice point, say a, of. ,. the triangular cell abc. xo represents the said probability of a solvent molecule, while xi, 7'--1, 2,･･･, P, represents that of a segment 1'.. yd>,<j+t: Probability of finding a pair of segments 7' and 7'+1 connected by. a chemical bond on a lattice bond, say ab, of the triangular cell abc.. Such a pair of segments has been written 7'>,<7'+1 symbolically. yiD,cd+2: Probability of finding a pair of segments 1' and 7'+2 connected by. ,an inner ･correlation bond on a lattice bond, say ab, of the triangular. cell abc. Such a pair of segments has been written 7'),c7'+2 sym-. ･bolically. ･. zDi>,<j'+t>,<j+2a: Probability of finding a set of three consecutive segments ･ 7', 7'+1 and 7'+2 dn the triangular cell abc, 1' on a, 7･+1 on b and 7'+2. on c. Hence, the･three consecutive segments are bent T13 at the seg' ment 7'+1. These variables are connected with the volume fraction, v, of polymer by. thefollowingequation: '･･･ '''' ' xd=(v/P), ]'=1,2,･･･,P, =1--v, 1'=O, yj>,<d+i=(112to)(v!P), 1'=1,2,･･･,P-1,. L ==O 7'=O,P,''. Yj.,.j.,==(PS･i.',/2(v)(vlP), 7'--1,2,･･･,p-2,. ==O, .7'--O,P-1,P, 2D,>,<j=,i>,.j.,.=(1?R,/6v)(vlP), 1'=1,2,･･･,p-2, '. tt. ==O, 1'=O, P-1, P. (H.3). Eq. (H.3) may be obtained by the following considerations: The probability. of finding the segment 7' on the first lattice point a of the triangular cell abc. is apparently xd=(vlP), and that of finding t,he segment 1'+1 connected with. the segment 7' by a chemical bond on the second lattice point b is (112to). Hence the probability of finding 1'>,<7'+1 on the lattice bond ab is (vlP)･(112to). which is the second equation. Consider three consecutive segments 7' , 7'+1 and. 1'+2 of which y'>,<1'+1 has occupied the lattice bond ab. The probability of being bent z13 on the lattice po'int b･at the segment 7'+1 of the three consecutive. segments is PSI,),, and that of finding the segment 7'+2 on the lattic point c is. a.

(19) SomeContributiontoLatticeModelTheoryofPolymerSolutions 61 (112), where 2 is the number of triangular cells which are connected with one. another at a common lattice bond, say ab, and is given by 2=(3vlca). Hence, the total probability of finding the three consecutive segments on the triangular cell abc, ]' on a, 7'+1 on b and j'+2 on c, is yd>,<.i+i･PS･i-).i･(ca13v) which gives. the fourth equation. Since the pair of segments 7'D,c7'+2 found on ･the latti'ce. bond ac of the triangular cell abc appears 2 times on Z different triangular cells, the triangular cell abc inclusive, the probabil･ity of finding 7'D,cl'+2 on. a bond of･ an arbitrarily selected triangular cell is zDj>,<d'+i>,<j+2c;2 which leads to the third equation.. The following cluster variables are also used: yj,k: Probability of finding a pair cluster of lattice elements 7' and le on a lattice bond, say ab, of the triangular cell abc, .i on a and le on b, each. of the lattice elements belonging to a separate molecule. 2j,k,i: Probability of finding a triangular cluster of lattice elements 7', k and. ion the triangular cell abc, ]' on a, k on b and i on c, each of the. latticeelementsbelonging'toaseparatemolecule. ' zj>,<j+i,k: Probability of finding a triangular cluster of a pair'of segments 1'>,<]'+1 and a lattice element fe on the triangular cell abc, 1'>,<1'+1. on ab and le on c, each of 7'>s<1'+1 and k belonging to a separate. molecule. '･ '' '. Zjb,cj'+2,k:Probabilityoffinding Tablel. ' a triangular cluster of a pair. dHk. of segments jo,ci+2 and. ･-jj'+1 Vj>,<J'+1. -)-eij-+2 Yj-),(1'+'. 2. a lattice element le on the triangular cell abc, 7'D,c7'+2. ij'kZe',k,i. lejj'+1z">;<i+i,te. kj--j'+2Zd),(e'+2,k. e'+2 ''ie'+1Z)i>,<J'+1>,<o'+2(. on ab and k on c, each of 7'),cl'+2 and le belonging to a separate molecule.. The types of cluster shown 'by the subindices of cluster variable ate given in Table 1 which may help toward the understanding of these cluster variables.'. Second Subindexing of Cluster Vctriables t- k When special attention is paid to a segment ]' locating at a lattice point, say a; of a triangular cell, say abc,' of the original lattice, there will be found. two branches ]' , 7'-1, 1'-2,･･･, 1 and ]'. , 1'. +1, i+2,･･･, P of a polymer chain. starting the segment i The configuration ,of these .two branch ･chains on the lattice may be found in various modes with respect to the triangular cell atc.. Each of the modes of configurations is specified by an index s. When the.

(20) T. WATARI. 62. configuration of the two branch chains is found in the s-th mode, the segment ]' is written as 1' ;s symbolically, The probobility of finding a segment 1';s. on a lattice point of a triangular cell is written as xj,, by putting the index. s to xj as the second subindex. Similarly, yi,k,,,t represents the probability of finding a pair cluster of two independent segments 7' ;s and le ;t on a lattice. bond of a triangular cell. When k==O, for example, le represents solvent molecule, and the second subindex t for le is unnecessary. However, in order to keep the symmetry of the second subindexing, yd,o,,,o is used in this case, zj,k,i;r,,tee,vw represents the probability of finding a triangular cluster of three. independent segments 7' , k and i composed of i,k;s,t on ab, le,i;u,v on bc. and i,i, w,r on ca. In this case, the same segment ]', for example, is subindexed doubly by r and s for the purpose of making the decomposition of 1'. , le,i;rs, tu, i,w into i, le;s, t, le,i;u,v and i, 1';w,r easier. Similar second. subindexings are made on the other cluster variables.. Though such heavy subindexings seem to be quite cumbersome for the formulation of the present theory, they make it possible for the theory to give a rigorous formulation as will be seen later.. Since the shape of a polymer chain is fixed by shape factors, {Pj(S}}, definitely or in average, the modes of configurations of two branch chains occur in definite fractions. Thus, xj,s, for example, is a definite fraction of xj which. has been connected with the volume fraction v of polymer by Eq. (H.3). It follows, therefore, that. xd;s=￠j;sv, (H.4) where ￠j;, is a factor which will be determined by giving {Pj(S)} for a given lattice. Quite similarly,. Yi>,<e'+1;r,s=￠o'>,<j'+1;r,sV, ZljD,cd+2;r,s= ￠eb,c:j-, 2;r,sV ,. ZDe'>,<j'+i>,<3'+2c;rs,tu,vw=95Dj'>,<j'+i>,<i+2c;rs,tu,vwV, (H.5) where ip's are factors similar to ￠e･;s.. Partition 7unction ana 7ree Ertergy The following formulation is given for the (1+q)-th approximation method. When q=O, q=(to13v) and q= 1, the formulation corresponds to the first (Bethe-. Fowler-Takagi), the [1+(to13v)]-th (Yang-Li-Hill) and the second (KramersWannier-Kikuchi) approximation methods, respectively, as mentioned in Sec. D.. A term in the summands of Eq. (A.1) for the configurational partitiou function is written as.

(21) SomeContributiontoLatticeModelTheoryofPolymerSolutions 63. Z=Ge3,' exp [- Elle T], (H.6) where Ge3v) is the combinatory factor which has been formally given by Eq. (G.1) with abbreviated notations.. The abbreviated notations are given, in the present case, as follows:. fpoint]N =.ZV!ln(xd,,N)!, {bOnd]N =N!1ll(Yik;r,$N)![Yi>,<o'+i;r,sN)!'(YdD,cj'+2;r,sN)!]2, [triangle]N=.ZV!1II(zj'.k,i;rs,tu,vwlV)![(zj'>,<e'+i,k;rs,tor,vwN)!'. '(Zj'D,cj'+2,k;rs,tst,vwlV)!(ZDo'>,<d+i>,<d+2c;rs,tu,vw.ZV)!]6, (H.7). where the products are made over all of the existing cluster variables of types. written after the product signs. The powers 2 and 6 raised to the terms between square brackets come from the fact that the clusters of lattice elements corresponding to the cluster variables appearing in these bracketed terms appear. on a lattice bond and on a triangular cell in 2 and 6 ways, respectively. The interaction energy, written as E in Eq. (H.6), is given as follows:. E=toIV(￡ xjei,d+ pp 2 yj,kAej,k),. j'=O ik==O. deik=eile-(Eij'+ek,k)12, (H.8) where si,le represents the interaction energy contribution of a pair cluster of independent lattice elements 7' and le.. Instead of maximizing Z with respect to independent cluster variables, pt given below is minimized.. pt =-(leT/N)1nZ. --(leT12V)lnGe3.)+E. (H.9) The minimizations give a group of equations for the so-called quasi-chemical. equilibria between clusters of lattice elements which connect cluster variables. one another. The procedure of the minimizations is given later together with the equations for quasi-chemical equilibria. With the equations for quasi-chemical equilibria, and Stirling's formula, Eq.. (H.9) takes the following simple form after somewhat cumbersome rewritings:. pt :kT2 [(1-2tu+3qv)xj,,lnxj,, +(tu-3qv)yj',j';r,slnyid;r,s +qv2d,io`;rs,tu,vwlngiia';rs,tec,vw]. p j==o. +Ztoxjed,j, (H.10). where the first summation is made over all of the cluster variables writteh after the summation sign.. t.

(22) 64 ' T.WATARI pt given above is now regarded as the free energy per lattice point, and the chemical potential per molecule of the solvent, pto, is calculated from Eq.. (H.10) by. ' ' pto==OAIpt10M=pt-v(6pt10v). ･･ The calculation gives. ' (pto-ptoO)lkT=(1-2to+3qv)lnxo+(to-3qv)lnyo,o' ･. +qvlnzo,o,o, ' ''(H.11) where the abbreviation, ' tt. tt tt ptoO=toeo,o tt ' ,. is usedi. Equations of Quasi-Chemical Equilibria. ''. Upon minimizing pt given by Eq. (H.9) with respect to cluster variables, the choice of independent variables is made as follows:. zj,k,i;Ts,tu,.,, with the exception of Zio,o ; rs,oo,oo ,. zj'>,<d+i,k;rs,tu,v. with the exception of gj>,<p'+1,o;rs,tu.oo,. ge' ),ce'+2,k;rs,tu,vw with the exception of Zj' D,d j' +2,o ; rs,tu,oo ,. and v. ' For.the mini･mization with respect to zj,k,o,.,,t.,oo, which is a specialicase of zi･le,i,,r,,t.,..,with i---O, the following relations are used: (zj,k,o;r,,tu,'oo=z). 6zio,o ; ors,oo,oo/Oz=: - 1 ,. Ozo,k,o;oo,tu,oo!Oz='1, 02o,o,o/Oz= 1 ,. Oy,-,k,,,t/Oz==1, '. Oyio;,,olOz =-1 , Oyo,k,o,t!Oz=:-1 ,. Oyo,o lOz=1, (H.12). '. which may be obtained from the following equations connecting these variables. With ZJ',k.o;rs,tu,oo :. tt. Xj';s =Zio,o;rs,OO,OO+Zj',k,O;rs,tu,09+'''",.

(23) SomeContributiontoLatticeModelTheoryofPolymerSolutions 65 Xk;t =Zo,k,o;oo,tu,oo+Zj',k.o;rs,tu,oo+'''',. Xo =go,o,o+Zo,k,o;oo,tu,oo+'''', Yj',k;s,t=Zik,o;rs,tu,oo+'''', Yo' ,o ; s,o = Z.i o,o ; rs,oo,oo + ' ' ' ' ,. Yo,k ; o,t =Zo,k,q;oo,tu,oo+'･'' ,. Xo ==yo,o+yo,k;o,t+'''', (H.13) and Eqs. (H.3) and (H.4) showing x's are definite fractions of volume fraction v of polymer for a given set of shape factors.. By remembering Eq. (H.12), the minimization gives the following equation of quasi-chemical equilibrium: [(YJ',k ; s,tYo,o)!(Yio ; s,oYo,k ; o,t)]alm3qY･ '[(Xik,o ; rs,tu.oo2o,e,o)1(2io,o ; rs,oo,oozo,le,o ; oo,tu,oo)]3qV. ==[H),oHb,k)IH),le]to12. (H.14) Putting q=O, (Yik;s,tYo,o)1(Yio;s,oYo,k;o,t)=[(HIi,oHb,k)1(Elli,k]'l2, (H.15) hence, (Zik,o;rs,tu,oozo,o,o)1(Ze',o,o;rs,oo,ooZoo,k,o;oo,tu,oo) .. =[(H),oHb,k/Hli,k]i/2, '(H.16) where,. H),k=exp[(2aEi,k)1(leT)],. ' (H.17). and,byEq.(H.8), '' ･. '. '. Hh,j=1. (H.18) By exactly similar procedures, the minimizations with respect to Zo'>,<j'+1,k;rs,tu,vw. and Ze'),cj'+2,k;rs,tu,vw. give the following equations, respectively, corresponding to Eq. (H.16): (Zj'>,<i+1,k;rs,tec,v2v2o,o,o)/(2j'>,<d+i,le;rs,tu,ooZo,o,te;oe,oo,vw). =[Hb,dHli+t,oH)la,oHb,kl(H)la,j'Hli+i,k)]il2, (H.19) (2e=),co'+2,le;rs,tu,vwZo,o,o)1(Zo'),cj'+2,o;rs,tu,ooZo,o,k;oo,oo,vw). ==[Hb,j'H)+2,oH)lo,oHb,k/(th,J'H}+2,k)]`12. (H.20) Osmotic Vierial CoeMcients i. The osmotic pressure T is given, in the present case, by.

(24) 66 ' T.WATARI T = -(lcto-tteoO)/vo ,. where vo is the molecular volume of the solvent of a polYmer solution. With Eq. (H.11), the above equation becomes. (Tvo1leT)==-(1-2w+3qv)1nxo-(tu-3qv)1nyo,o. ---qvln2o,o,o, (H.21) which may be expanded in the power series of the volume fraction v of polymer as follows:. (rrvo/leT)=￡bnv", n=1. (H.22). in which bn is the n-th osmotic virial coethcient.. Writing quite formally,. Yo,o =1+ao,ov+bo,ov2+O(v3),. 2o,o,o=1+Ao,o,ov+Bo,o,ov2+O(v3), (H.23) and using Eq. (H.3) for xo, the firist and the second virial coefficients may be. obtained from Eq. (H.22) as follows:. bi=(1-2to+3qv)-(to-3qp)ao,o-qvAo,o,o, (H.24) b2=(112)(1-2to+3qv)-(tu-3qv)[bo,o-(1/2)ag,o]. -qv[B,,,,,-(y2)Ag,,,,]. (H.2s) ATbn-Athermal and Athermal Sotutions In the following formulation, the segments of polymer molecules are assumed to possess the same chemical and physical natures. It･follows, therefore, that. Aed,'k=:Ae, 1'>O and k==O, or 7'=O and le>O,. =:O, 1', le==O, or 7',k>O, (H.26) and. Hb,k==H, 1'>O and le=O, Qr 7'=O and le>O,. =1, 1', fe =O, or l', le>O. (H.27) When. As-tO,andH#1, (Non-athermal)(H.28) the polymer solution is called non-athermal.. When. Ae=O,and H=1, (Athermal) (H.29) the polymer solution is called athermal. '. '.

(25) SomeContributiontoLatticeModelTheoryofPolymerSolutions 67 ' I. First <Bethe-Fowler-Takagi) Approximation ' The first, i.e., Bethe-Fowler-Takagi, approximation method corresponds to the special case q==O of the preceding formulation. Since the first approximation method for polymer solutions has been studied extensively by many authors,. none pf new results may be obtained from the present formulation. However,. in order to render help to the understanding of the method of calculating osmotic virial coefficients by higher approximation metbods, and for the con-. venience of comparing results obtained by various approximation methods with one another, the calculation is began with the use of the first approximation. method. In the first approximation method, the original lattice is replaced by a Bethe lattice shown in Fig. 6. Since Bethe lattice has no triangular cells, the basic formulation for this approximation method is obtained by putting formally. '. Pj(i)=Oaswellasq=O. '. The equation of quasi-chemical equilibrium for the present case is given by Eq. (H.15) which may be written with Eq. (H.27) as follows:. (Yik;s,tYo,o)1(Yio;s,oYo,k;o,t)==H･ (I.1) Since,. xo =1-v. == yo,o +22 yo,j;o,s, ,. i j'>OS. xd;s =yj,o;s,o + Z 2 yj,k;s,t ,. it follows that. yo,o=1-v-2[j･O]xj,s+ 2 2 yi,k;s,t, =1+O(v) ,. o'>O ik>ts,t. yj,o;s,o==Xj;s- :IE Z Yd,k;s,t,. k>O t. =x,･,s+O(v2), (I.2). where the summation sign 2[j･O] represents that the summation must be extended over all of segments i;s which may form pair clusters 7',O;s, O. By introducing these equations into Eq. (I.1), the following equation may. beobtained: ' '' '. '. Yile;s,t=:Hlnj;,xk;t+O(v3).. With this result, Eq. (I.2) for yo,o becomes. ''. yo,o=1-v-2ij･O]xj;s+H2[ik]xj･;,xk,t+O(v3), (I.3). e'>O e',k>O ..

(26) N. 68 . T.WATARI-. ' ' ' where the summation sign 2[j･k] represents that the summation must be made over all of segments 7' ;s and le;t which may form pair clusters 7' ,. le;s, t.. In order to carry out these summations, a new cluster variable & is in-. troduced as follows: ･ 6j: Probability of finding a lattice element 7' on a lattice point, say a, at an. end of a lattice bond, say ab, with the lattice bond ab unoccupied by 'a. ,. polymer chain. '. The new cluster variable is connected with variables x and y by the fol-. lowingrelations: .. rl. &=Xj-Yi>,<d+1-Yj-1>,<j,. '. which becomes with Eq. (H.3). L. 6d =1-v, 7' =O, =[1-(112tu)](vlP), 1'=1,P, = [1-(11to)](vlp) , ti -- 2, 3,･･･, p-1, Sl] & =i-(a!to)v , j==o. '. a-=1-(11P). (I.4) Using this new variable, tl e summations appearing in Eq. (I.3) may be. expressed as ･･ ' '. ,2.>,[iOlrpi;s=,2..,8j'=,.:,E,.1,G'-eo, ,. ,.lle>,[O'ib]x7,sxk,t=(j-T,4e)2-2eo(,IPZ.=,8,･-eo),. ,. and Eq. (I.3) becornes. yo,o=1+[(altu)-2]v,+H[<ako)t1]2v2+O(v3),. from which. -. ao,o=(alto)-2, :'' bo,o=ll[(alto)-1]2. When these ao,o and bo,o are introduced into Eqs. (H.24) and (H.25), with. q:O in mind, it finally follows that ' b,=11P･, '. b2 =(112)-to[1-(1/2){(alto)-2}2+H{(alto):1}2] . (I.5) When athermal, i. e., H=1, Eq. (I.5) may be reduced to. b2=(1!2)-(112)(a21to). (Athermal) (I.6).

(27) SomeContributiontoLatticeModelTheoryofPolymerSolutions 69 Though this is the well-known result for the second osmotic virial coefficient. of athermal polymer solutions obtained by various authors, it is derived here as a special case of Eq. (I.5) for non-athermal solutions.. The calculation of varial coefficient has been carried out ,as the special. case q=O of the (1+q)-th approximation method for a triangular lattice. It must be noted, however, that the final result of virial coefificient is quite general. in any Bethe lattice, since a lattice has lost its original structure during the transcription onto a Bethe lattice.. ' J. Further Formulation for (1+q)-th Approximation The formulation given here runs parallel to Eqs. (I.1) to (I.3) of the preceding section starting equations of quasi-chemical equilibria given by Eqs. (H.16),. (H.19) and (H.20) for O<q;;ll, which may be written with Eq. (H.27) as follows:. (Zik,o;rs,tu,oo2o,o.o)/(Ze',o,o;rs,oo,ooZo,ic,o;oo,tu,oo)--H; (J.1) (Zd>,<j'+i,k;rs,tu,vwXo,o,o)1(gd>,<e'+i,o;rs,tza,oogo,o,le;oo,oo,vw)==H2, '(J.2). ' (2e'p,ci+2,k;rs,tu,vwZo,o,o)/(Zj'D,aj'+2,o;rs,tu,oo2o,o,le;oo,oo,vw)=H2. (J.3) '. Since,. xo==1-v . =zo,o,o+ 2 Z 2o,k.o;oo,tu.oo+ 2 2i zo,o,i;oo,oo,vw. k>ot i>ov '. +22Z(Zk>,<le+i,o;rs,tu,oo+ZleD,ck+2,o;rs,tu,oo), (J･4) k s,t. Xj' ,- s=gio,o;rs,oo,oo+ k￡>o:i gik,o;rs,tu,oo+ ;.lll>o4 zio,i;Ts,oo,alw .. +2ZZ(Zle>,<le+i,j';tu,vw,rs+gk=),cth+2,d;tu,vw,rs),' (J･5). ku,v .. it follows that ･ ' zo,o,o==1+O(v),. Zio,o;rs,oo,oo =:xJ';s+O(v2). (J.6) By introducing these equation into Eq. (J.1), the following equation may be obtained:. zik,o;rs,tza,oo=Hxj';$xk;t+O(v3). (J.7) Furthermore, since. ' Ye'>,<j'+1;s,t=Zd>,<j'+1,O;rs,tu,OO. +ZZZj'>,<d+i,k;rs,tu,vw, (J･8). k>O v it follows that. Zd>,<j'+i,o;rs,tpt,oo=Ys'>,<e'+i;s,t +O(v2) .. (J.9).

(28) 70 ' T.WATARI When this equation and Eq. (J.6) are introduced into Eq, (J.2), the following. equation may be obtained:. Zd>,<e'+i,le;rs,tu,vw=H2Yj'>,<d+i;s,tXle;v+O(V3), (J.10) Quite similarly, from Yj'D,cj+2;s,t=Zj'D,c3'+2,o;rs,tu,oo. +Z2Zjb,c3'+2,k;rs,tu,vw (J.11). k>o v and Eq. (J.3),. 2J'D,cj'+2,le;rs,tu,vw=H2YdD,cJ'+2;s,tXk;v+O(v3), (J.12) By combining Eq. (J.5) to (J.12) with Eq. (J.4), Eq. (J.4) for zo,o,o becomes. 2o,o,o==1'(v+22[iO･O]x,･,, e'>o. +22[j'>,<j'+i･O)yi>,<j･+i;,,t ' o' >o. +22[iD,CJ'+2･O]Yo･D,cj･+2;s,t). d>o +3H( ￡ [ik･Olx,･,,xk,,. ile>o. +2H ￡ tie'>･<0'+1･k)yJ･>,<j･+1;s,tXk;v. o',k>o +2H2[eb･CJ'+2･k]y,･.,.,･+2,,,txic,.)+O(v3), (J.I3) i, k>o. where specially devised summation signs similar to those used in Eq. (I.3) are. used, meanings of which will be given in detail in the paragraph following. Eq. (J.19). '. '. Since,. YO,o=ZO,O,O+ Z Z ZO,O,k;OO,OO,vw ,. (J.14). k>o v. .. it follows, when Eqs. (J.5) to (J.13) are combined with Eq. (J.14), that yo,o =1-(v + 2 [i O･ O]x,- ,,. j'>o.. .. +22[o'>･<o'+i･Olyj･>,<j･+i,,,t. j'>o + 2 Z Lj' D･Cj+2･O] Yj･ D,c j･ +2;s,t). j' >o. ･ +H( Z [p'･k･O]x,･,,xk,, e',le>O. +4Hd,:l>o[j'>'<d+i'k]Yd>,<j'+i;s,txk;v. +4H Z [j'D･Cj'+2･k]yj･.,.d+,,,,txk,.)+O(V3) .. i k>O. From Eqs. (J.15) and (J.13), ao,o=-(v+31Ilo[iO'O]xj';s+2dZ>oEj'>'<i+i'O]ye･>,<3･+i;,,t. (J.15).

(29) SomeContributiontoLatticeModelTheoryofPolymerSolutions 71. +2￡[J'D･Cj'+2･Oly,b,.,･+2,,,t)/v, (J.16). s'>o. bo,o==H( 2 [ik･Olxe･;sxk,t. ik>o +4H Z [j'>'<J'+i' le]Ya>,<3'+i;s,tXk;v. i k>o +4H,.,;ll>o[j'P'Cj'+2'k]YJ'D,ce'+2;s,txk;v)lv2, (J.17) Ao,o,o==-(v+22[iO'O]xe-;s+22[i>'<e'+i'O]yj'>,<o'+i;s,t.. j>o j>o. +2Z[j'D･Cj'+2･O]y,b,c,･+2,,,t)!v, (J.18). e'>o. Bo,o,o=3H( 21 [ik'O]Xo';sXk;t j',k>O +2H '2 [j'>'<J'+i' k]Yo'>,<o'+i;s,tXic;v. 3',k>O. +2HZ[j'D･C3'+2･k]y,-D,c,･+2,,,txk,v)lv2. (J.19) ik>o. t. In Eqs. (J.13) to (J.19), specially devised summation signs are used in such. ways that the sign ￡[iO･O] represents a summation over all of segments 7';s which may form triangular clusters 7' ,O,O;rs,OO,OO, the sign Z[j>･<j+i･O] a. summation over all of clusters 7'>,<1'+1;s,t which may form triangular clusters 7'>,<7'+1,O;rs,tu,OO, and the sign >l[dDrCj+2･O] a summation over all of clusters 1'D, c7'+2;s,t which may form triangular clusters 7'D, c7'+2, O; rs, tu, OO. Furthermore, the sign 21[d･k･O] represents a summation over all of pairs of segments 1' ;s and k;t which may form triangular clusters 7' , k, O; rs,tu,. OO, the sign Z[j>･<j+i･k] a summation over all of pairs each of which consisting. of a cluster 1'>,<1'+1;s,t and a segment fe;v which may form a triangular cluster 1'>,<7'+1,k;rs,tu,vw, and the sign ￡[d>･<d+2･k) a summation over all of pairs each of which consisting of a cluster 1'D,c7'+2;s,t and a segment le;v which may form a triangular cluster 7'D,cl'.+2, le;rs,tu,vw. '. In order to carry out these summations, a group of new cluster variables. ' are introduced as follows: ･. ed: Probability of finding a lattice element 1' o.n a lattice point, say a, of a. triangular cell, say abc, with the lattice. bonds ab and ac unoccupied by. polymer ch.ains andlor an inner correlation bond. rpj>,<j+i: Probability of finding a cluster 1'>,<]'+1 on a lattice bond, say,. ab, of a triangular cell, say abc, with the lattice bond ac and bc unoc-. cupied by a polymer chain and an Table 2. inner correlation bond. m'D,ci+2: Probability of finding a cluster. 1'D,c7'+2 on a lattice bond, say ab, of a triangular cell, say abc, with the. Jf,. jg+iril>,<J'+l. ">-;J'i+2Pj),O'+.2.

(30) 72 T. WATARI. lattice bonds ac and bc unoccupied by a polymer chain.. These new variables are shown diagrammatically in Table 2 in order to make them easier to understand.. L. The relations corresponding to Eq. (I.4) for these variables will be given in the following sections. .. K. (1+to/3v)-th (Yang-Li-Hill),Approximation The (1+tu13v)-th, i.e., Yang-Li-Hill, approximation method corresponds to. ". the special case q =to!(3v) of the preceding formulation. In this approximation method, the original lattice is replaced by a Yang-Li-Hill lattice shown in Fig.. 7. 0nce the configuration of polymer chains has been transcribed from the original lattice onto the corresponding Yang-Li-Hill lattice, it may easily be. seen that, during this transcription, all of clusters 7'D,cl･+2,O and 1'D,cl'. +2,le have disappeared either by the disappearance of triangular cells on whieh these clusters were found or by the breakdown of inner correlation. bonds involved in these clusters. Hence the basic formulation for this approximation method is obtained by putting formally gj),cj-2,o= gjD,cj+2,k=O. asAccordingly, well Eqs. as(H.24), q==w!3v. ' (H,25), (J.18) and (J.19) must be written as follows bi=(1-tu)'(to/3)Ao,o,o, (K.1) b2=(1!2)(1-tu)-(to13)(Bo,o,oNAg,,,,/2), (K.2) Ao,o,o==-(v+22[iO'O]x,･,, e'>o. '. +2￡[j'>･<j'+i･O]yJ･>,<e･+i;,,t)/v,. .. 3>O , '. (K.3) .. Bo,o,o==3Hl(j.,iill>Io[g''k'O]xj';sXk;t. +2HiiP>o[2'>'<"+"le]Ye'>,<o'+i;;s,t Xk;v)/v2. (K.4) ,. By using the new variables introduced at the end of Sec. J, the summations in Eq. (K.3) for Ao,o,o may obviously be given as follows:. J'>O￡[j'･O･OJx,･,s= j'>O. 2] 4,･, (K.5). 2[e'>･<j'+'･O]yd>,<e･+i;,,t=2nd>,<e･+i. (K･6). d>o o'>o. For carrying out the summations in Eq. (K.4) for Bo,o,o, the following consideration on the feature of Yang-Li-Hill lattice is very useful: On this lattice, once two separate chain polymer, molecules approach each other to occupy a triangular cell of the lattice in common, the other parts of the two.

(31) 't.. '. SomeContributiontoLatticeModelTheoryofPolymerSolutions' 73 ' polymer molecules can never approach each other to occupy another triangular cell in. common.. It follows, therefore, that the summations may be factorized as follows: , ZL-ik･O]xj･,txk;s==(El[j'･O･O)xe-;,)(Z[ktO･O-ixk,t). ik>o e'>o k>o .. ==(2 Ci) (2 4k), (K.7) e'>O k>o ･ .. 2 [j'>'<j'+i'k] yi>,<j'+i;s,txic;v J',ic>O =(2]W>･<o'+1,O]yj>,<j･+1;.,t)(2[ic･O,O]xic;.). j>O k>O. =(2 rp,･>,.,･.,) (Z 4k). (K.8). j>O k>O' .. . Since,. M>,<j'+1=:Yd>,<j+1-ZDj'>,<j+1>,<j'+2c'. . -2DjLl>,<j>,<e'+lc ,. it follows from Eq. (H.3) that ･ '. ' ,l.,rp">･<j'+i= [(a12w)-(Pì'b13v)]v ,. where. a=1-(11p), b=1-(21P), and 3. P(il=(2P,･(t))1(p-2). (K.9). -,. Furthermore,since , ' '. '. G--XJ'-2(rpe'>,<j+i+rpd-i>･<d ･'. +2Ds'>,<s'+1>,<s'+2c+Z=)j-1>,<e'>,<j+lc+2DJ'-2>,<J'-1>,<j'c), '. it follows from Eqs. (H.3) and (K.9) that. ' ZCj･=[1-(2alto)+(P(Db/3v)]v. (K.10) i>o When Eqs. (K.1) to (K.10) are combined, it follows finally that b, =11P , b2=(1/2)+to[1-3(altu)+(312)(altu)2]. -toH[1-2(altu)+(113)(P(i)b!v)]･･[<1-2(a/(D) +(113) (Bì)blv)}. - +2H<(1/2)(a/tu)-(1/3)(P(i)b/v)}].. (K.11). When athermal, i.e., H=1, this equation may be reduced to. b2 == (1!2)-(112) (a21to) . -(113)(P(')blp)[a-(1!3)(P(i'btolv)], (Athermal) (K.12).

(32) 74 T. WATARI '. and further to Eq. (I.6), by putting P(i)= Pd(i)=O.. ' L. Second (Kramers-Wannier-Kikuchi) Approximation '. The second, i.e., Kramers-Wannier-Kikuchi, approximation method corre-. sponds to the special case q==1 of the preceding formulation. In this. '. approximatioh method, the calculation of osmotic virial coefficients is extremely long. Hence, the calculation of the first virial coefficient is given in the first. place. Furthermore, since the calculation of the second virial coefficient is. g. very much complicated, it is given only for the case of a two-dimensional triangular lattice in order to illustrate the method of calculation. Full explanation of the present method will be given in a separate paper including some applications to three-dimensional lattices.. First Osmotie V]lrial CoeMc. p'ent .. The first virial coefficient may be obtained by combining Eqs. (J.16) and. (J.18) with Eq. (H.24). The summations appearing in Eq. (J.16) for ao,o and Eq. (J.18) for Ao,o,o may be written by using new variables introduced a' t the end of Sec. J as follows:. ,2.>,F"'O' O]xd;s= ,2. >,4J', (L,1) IZ[o'>･<d+i･O]y,･>,<j-+i,,,t==Zrpj->,<,･+i, (L.2). o>o j>o. jl>o[jb'Cj'-"2'OIYjb,cj'+2,s,t=j2.>orpj'D,cj'+2･ (L･3) Since, in the present case, .. rp3'>,<o'+1=Yd>,<j'+1ptZ)j'>,<j'+1>,<j'+2c-Z)d-1>,<i>,<j'+lc,. M]],cj'+2== Zld ),cj+2'ZDj>,<d+1>,<j+2c ,. it follows from Eq. (H. 3) that. ' ' Zrp3'>･<e'+i==[(a12tu)-(P(i'b13v)]v, j'>o. Z rpjD,[j+2== [(P(i)b12(v)-(P(i)b16,)]v ,. . j>o. where, as in Eq. (K.9), '. a==1-(11p), b-1-(2/p),. and ' Furthermore, since. d. P(i)=(ZPj(i))!(p-2). (L.4).

(33) 1. SomeContributiontoLatticeModelTheoryofPolymerSolutions 75 4j･--xj･-2(rpi>,<j･+i+va-1>,<J-+rpj-D,cj'+2+rpjL2D,cs'. +ZDe'>,<j'+1>,<J'+2c+g=j-1>,<o'>,<j'+lc+ZDpL2>,<J'-1>,<p'c), '. it follows from Eqs. (H.3) and (L.4) that j' >o. 24j---[1-(2a/w)-(2P{')bltu)+(Pì)blv)]v. ･ (L.5). When Eqs. (L.1) to (L.5) are combined with Eqs. (J.16) and (J.18), it follows that. '. '. '. ao,o==-[2-(altu)-(P(i)b!to)], (L.6). ' Ao,o,o==-[3-(3alw)-(3Pì)bfto)+(P")b!v)]v. (L.7) By introducing these equations into Eq. (H.24), the first virial coefficient. may be found to be. bi=11P, (L.8) which may legitimate the foregoing calculation.. Second Osmotic J7iriat CoeMcient(Two-aimensional Lattice) As mentioned in the earlier part of this section, the practical calculation of the second virial coefficient is made for a two-dimensional triangular lattice,. w==3 and v=2, for which Eq. (H.25) is simplified, after introducing Eqs. (J.17) and (J.19) into Eq. (H.25), as. b2=(112)-3H 2 ["'ic'Oix,･,,xk,tlv2-(112)(3ag,o-2A3,o,o) , (L.9) ik>o '. leaving only the summation over clusters 1'. ,le,O.. In order to carry out this summation, the following consideration on the feature of Kramers-Wannier-Kikuchi lattice is very useful: On this lattice,. there may be found sequences of triangular cells connected side by side on their lattice bonds. It may be seen, therefore, that if two separate chain polymer molecules approach each other to occupy a triangular cell of the lattice. in common, there remains still another possibility that the other parts of the. two molecules might approach each other to occupy another triangular cell in common. Hence, if this summation is written as (Z 4j-)(Z 4k) as in Eq. (K.7). j>O k>O. of the previous section, there may be included impossible pairs of segments i, s and le;t of two separate polymer molecules of which two branch chains, starting 7' ;s and le;t, respectively, would overlap at a certain lattice point outside,the triangular cell on which the pair 1';s and le;t have been found.. It follows, therefore, that the summation.must be written as follows;. ￡ [ik･O]x,-,,xle,t==(2 4i)2-rcv2, (L.10). ik>O j'>O.

(34) T. WATARI. 76. where rc is a correction factor for the aforementioned overlapping. rc is called. the overlapping correction factor, which may be expressed quite formally as follows:. p s=2. rc==￡rc,, forP-mer,. rc,==Zrc,(r). (L.11). r. ,". rc, is called the s-mer correction factor, since correction factors up to rc, should. be taken into account so as to correct the overlapping of s-mers. The mode. .. of overlapping of two branch chains corresponding to rc,(') is illustrated schematically in Table 3.. Table 3. -. rc2. etc,. S4. rc3. rc3(1). rc4(1). S2. rc3(2). rc. C3). 4. s4. C6). etc. t. N4U). e-"-o Indicates the segments are. conneetecl with either chemical or' inner. correlat･ion bond. .. (a) Cluster Variables for Estimations of OverlaPPing Correction I7txctors. For the estimations of rc2 and rc3, new variables given in Table 4 are introduced.. With these new variables, rc2 and rc3 may be given as follows: rc2==(2 4e'(i))2!v2 ,. .5i)==2(￡cSP･i)){z(rpS-O.･1).,.,+rpS･is9).j.i ･ +rpS･Ofi,).,･.,+rpS･i60,).,･.,)}/v2, rc52)=2{z(rpSPs9)ES?.,+rpS･ig(,08ny).,+rpS･tt9).'}).,+rpS･ig(,O.･O,).,)}･. '{Z(rpS'iS9'20e?+i+rpS'O>'10(ie"+i+rpS"S9'EOd'+2+rp;'OD'10ei,?+2)}!v2･ (L･12). The estimation of rc,, s>3, may be performed in exactly similar manner by introducing cluster variables representing probabilities of finding larger. N.

(35) Some Contribution to Lattice Model Theory of Polymer Solutions. portion of polymer chain.. 77. Table 4.. However, in the present paper, the final results will be given. (. for some simpler cases.. 4￡P. For the evaluation of the. N"->g. l t. nSP,;i&･.,. Nu(['. l. l. y.y. oS･O,:i<l･.,. l. cgp,i). Y.,.¥. t t. vS5･,02･,,. t I. Y..Y. vSl:?},5. t t. '. first two summations in Eq.. 1. (L.12), consider, on the original. triangular lattice, a triangular. rps'Lo&{?, rpsg)s.o:o.),. rpS･;:2'f･O+'i oSg'f.Ot?i. cell, say abc, of which a lattice. Y.-Y, #..Y. point, say a, has been occupied t. byasegmenti Furthermore,. nsp,:o,}(i,), p;･l)s?J･o.),. Y.Y y.y rpS :O&{O.',. ns･o))so(x),. consider a group of six trian-. gular cells surrounding- the Indicates lattice point a, of which one is. the lastice bond. is oecupied by either chemical or inner eorrelation bond.. group of six triangular cells, there will be found the triangular cell abc. On this. with respect to the triangular cell abc, of two various modes of configurations,. and7',]'+1,7'+2,･･･,startingthesegmenti Each of the modes of configurations isspecified by index a. When the configuration. branch chains 7' , 7'-1, 1'-2,･''. of the two branch chains on thisgroup of six triangular cells is found in the a-th mode, the segment 7' iswritten as 7';a. The subindexing a corresponds to. a coarse-grained grouping of the subindexing s used hitherto. The probability of point of a triangular cell is written as xj,at. finding a segment 7':a onlattice a on the occurrence of intramolecular correlations By the restriction imposed. at the beginning of the present formulation, the subindexing a may be made by observing whether each the of six lattice bonds, ab, ac, etc., diverging from the lattice point a, is occupiedor not by either chemical or inner correlation. bond.. The probability of finding acertain mode of occupancy of the six lattice. bonds by chemical and inner correlation bonds is determined by shape factors PS･'-'i, PS･", PS･'+'i and PS･2'. Hence,xj,di. for the present formulation may be given. in terms of v and these four shape factors after elementary but laborious calculation.. By summing xj,tu over all of a which are compatible with 6d(i) and &{O･i),. summing respectively,andalsoby ' them over 7'>O, the first two summations in the equations for rc2 and rc3{i). of Eq. (L.12) may be obtained. The final. results will be given shortly.. Quite similarly, by considering the configuration of two branch chains, one. starting the segment 7' and the other starting the segment 7'+1, (7'+2), of cluster 7'>,<7'+1, (1'D,c7'+2),on a group of ten triangular cells surrounding.

(36) T. WATARI. 78. the cluster 1･>, <7･+1, (7'D, cl'+2), the summatlons over rp in Eq. (L.12) may be obtained.. (b) Second Coqfiicient for Tetramer. For tetramer, P==4, the summations appearing in Eq, (L.12) may be given as follows:. .. 24,･(i)=('1!24){P,(i)+B,{i)+(1-P,(i})+(1-P,(i))+2P2(i)P3(i). +P2(i)(1-P3{i))+P3(i)(1-P,(i))+P,(2)P,(i)(1-P,(i)). i. +P3(2)P2(i)(1-P,(i))+(1-P,{2))P,(i)(1-P,(i}). +(1-P,{2))P2(i)(1-P3(i))+P,(2)(1-P,{i))(1-P,(i)). +P,{2)(1-P,(i))(1-P,Ci)). +2(1-B,(2))(1-P,(i})(1-P,(i)) ･ + 2(1 - P,(2))(1 - P2(i))(1 - P,(i})}v ,. 24d{O･i)=(1!24){P,(i)+P,(i)+(1-P,(i})+(1-P,(i))+2P,{i)P,Ci) + P2(i)(1 - P3(i)) + P,{i)(1 - P,(i)). +P,{2)(1-P,(i))(1-P,(i))+P,(2)(1-P,{i})(1-P3(i))}v, 2?S･O.･l).,-.,=(1/48){2P3(2)P,(i)(1-P,(i))+P,(2)(1-P,(i))(1-I3,Ci)). +(1-P,(2))P2(i)(1-P,(i)) +P2(2)P,(2)(1-P2(i))(1-P,(i})(1-P,(3)). +P,(2)(1-P,(2))(1-P,(i))(1-P,(i))}v, IZrpS･isO,).i.,==(1/48){2P,{2)P,(i)(1-P,(i))+P,(2)(1-P,(i))(1-P,(i)). +(1-P,(2))P,(i)(1-P,(i)) +P,C2)P,C2)(1-P,(i))(1-P3{i))(1-P,(3)). +P,(2}(1-P2(2))(1-P,(i))(1-P,Ci))}v, ZrpS-is9).i.,=(P2(i)/48){(1-P,(2))(1-P,(i))}v, ￡rpS･O.'','.,･+,==(P3<i'148){(1-P2(2')(1-I32ì))}v,. st. ' 21vS･OsO,)2i,･).,==(1!48){P2<2)P,{i)(1-P,(i))+(1-P,(2))P,(i)(1-P,(i)) , +(1-P,(2))P2(i)(1-P,(i)) ,. + 2(1 - P,(2))(1 - P2(i))(1 - P,(i)). P2(2)(1-P,(2})(1-P,(i))(1-P,<i)) + 2(1 - P,C2))(1 - P,(2))(1 - P,{i))(1 - P3(Z.>)}v , ,2rpS･i.)!OkO,)･.,=(1148)<P3(2)P,Ci)(1-P,Ci})+(1-P,(2))P,(i)(1-P,(i)). +(1-P,(2>)P3(i)(1-P,(i)) + 2(1 --- P,(2))(1 - P,Ci))(1 - P,Ci)). +P3(2)(1-P2<2))(1-P,(i))(1-P,(i･) +2(1-P,(2))(1-f3,(2))(1-l3,<i))(1-P,(i))}v, ZrpS･O.･O,)Ei,-).,=:(P2(i)148)<P3(2)(1-p,ci))}v,. 2vS･'BIOeO,･).,:(P3(i)48){i32(2)(1-P2(i))}v, ,.

(37) SomeContributiontoLatticeModelTheoryofPolymerSolutions 79 2rpS･i.･O,)EO,･).,==(1148){P,(2)P,(i}(1-P,(i))-l-P,(2)P,(i)(1-P,(i)). +(1-P,(2))P,(i)(1-P,(i))+P,(2)(1-P,(i))(1-P,(i)) + P,(2)P,(2)(1 - P,Ci))(1 - P3(i))(1 - P,(3))}v , ￡?7S･O./10(i,.).,=(1148){P,(2)P,(i)(1-/Ei,{i))+P,(2)IEI,{i)(1-P,(i)). +(1-P,(2))P,(i)(1-P,Ci))+P,(2)(1-P,(i))(1-P,(i)) +P,{2)P2(2)(1-P,(i))(1-P,(i))(1-P,{3))}v, ZvY.･O?EOi).,=(P3(i)148){(1-P,(2))(1-P,(i))}v,. ￡)7SP.)10d,L?.,=(P2(i)148){(1-i(3,(2))(1-P,(i))}v. (L.13) With these equations, the overlapping correction factors rc2 and rc3 may be calculated explicitly by Eq. (L.12). rc4 is evaluated in a similar manner in the. case ofa rigid polymer chain and is given in Table 5. In the case of a flexible polymer chain, rc4 is ignore" With these correction factors,the second. virial coefficient for tetramer may be calculated by combining Eq. (L.9) with Eqs. (L.6), (L.7), (L.10), (L.5) and (L.11). The numerical results for athermal. solution, H=1, are summarized in Table 5. The rigorous values of the second virial coefficient are given in the last. row of the table which are obtained by computing geometrically the excluded ' ' volume of polymer chain. The details of the method have been given in Sec.. Hof Ref. 13). ' '. For rigid tetramer chain, there may be found perfect accordance between. the calculated and rigorous values of the second virial coefficient. However, in the case of flexible tetramer chain, since the tetramer correction factor has. been ignored, the calculation gives an under-estimated value. Nevertheless, the calculated value seems to be correct within an error of 2%... (c) Second Coofcient for Extremely Long Chain POIymer In the case of extremely long chain polymer, P-->oo, rigid shape chains are of little interest. Hence, shape factors are chosen in advance as. Pj'a)=Pa), Pe'C2)=Pc2), and. Pd(3)=112, foralll'. ,(L.14) The summations appearing in Eq. (L.12) may be given, after making p.oo, as follows: ￡9(i)=(116){P7,,+2P?,,(1-P{b)+3P,i,(1-P,,,)2. +(2-P,2,)(1-P,,,)3}v, 24d{O･i)=(116){2P?,)(1-P,i,)+P(i,(1-P(i,)2 + P,2,(1 - P,,))3}v ,.

(38) T. WATARI. 80. Table 5 Rigid* B2(1) =B3(1)==O,. l92(1)= B3(1)=O,. B2(2)=B3{2)=O,.. I92(2) == O, B3(2) = 1,. B2(!} = B3(1) = o, S2(2) = B3(2) = 1,. B2(2) = S3{2) == 1,. B3(3) = O. B3(3) = O. B3{3)=O. B3(3) = 1. (a). (Z<j)2/v2 rc2. (b). (cl). A. e-･-e-.-or･-e. e--V'. "VNe. (12/24)2. (12/24)2. (12/24)2. (12124)2. (6124)2. (5124)2. (4/24)2. (4/24)2. rc3(i>1!96. rc3(i)1136. rc3(i)1/72. o. rc3. (c). B2(1) == t?3(1) = O,. rc3(2)1/384 rc4(t)11144. rc4{2)1/1152. o. rc4. o. rc4(4)3/1152. rc4(7)1/288 rc4(8)1/288. [i k,O]. ZX･X /v2. 3116. 73/384. 7/36. 7/36. - ao,o. 7/4. 714. 7/4. 714. -Ao,o,o. 914. 9/4. 914. 9/4. b2 calc.. 156f384. 148!384. 153f384. 148/384. b2. . rlgoToecs. * ** ***. U (b2 rigorous 116/384) is omitted, since it (a)t･v(g) occur in the following mole fractions:. . violates the restriction on lntra(a) 2/18, (b) 4118, (c) 1118,. (ol) ll18,. Weighted mean of b2 for (a)N(g). .. ZvS･O.･1).,.,==2rpSl.･9).,.i. =(1/24)({2IIIII,,(1-P<,,)+(3+2P,,,)(1-Ill,,,)2P2,,, + 2(1 + P(,,)P,,,(1 - P,,,)3 + 2Pc2,(1 - P,,,)(1 - P,,,)4}v , ZrpS･O.･l).,.,==2vS･i.･9).,.2. ==(P,i}/12){P?,,(1-P<i,)+(2-P,,,)P?,,(1-P,,))2 + Pci,(1 - P(i>)3 + P,,,(1 - P,,,)(1 ----- P,,,)4}v ,. 2rps･osg)s)･.,=zrpsigso<t?.,. :(1/24){(3-2P(2,)P{,,(1-Pci))2. +(6-5Pc,,)P<,,(1-P{,,)3. +2(2-3P,2)+P?,,)(1-Pci))4}v,.

(39) Some Contribution to Lattice Model Theory of Polymer Solutions. 81. Tetramer (p=4) .Elcxible B2'(a" =･ B,(i) l= 1,. B.2tt) =1, B3(1) == O,. B2･<･1;) = 1, B3(1>= O,. .P2(2) .F B,3(2) = O,. B2(2)=O, B3(-2)=1, t. B2i(2') = B3'('2') =- O,. B3(3,) -=i(E). B3(3) = O. B3<3} = O. .･. (e). ". (f). (g). N. ･B2(i} = B,(t) = 1/3, B2t(･20 = B3(2) = 112, B3(3} == 112. N. Perfectly flexN le**. ,(IJ,/24)2. <1 1124)2. (10124)2. (17X36)2. <4124)2. <4124)2. (4124)2. (bl27)2. K3(i)1,l96. ]. .ng(i･>1./96. ua"2･).1, el288. ･rc3, -<i)420!(36.26). o. rc3(i}1431(36･26). lg3(('e-,),1st288. '. o. s. 97/576. 9ny576. -7148. 5/3. 513. 19112. 61,f36. Z18. ･2. '7'ew36. IZ/8. 1'. ignQrea. o. 27471,(･35･26) .... fe137.4!3.84. 1321384. X> 1161384. 1321384. '. rv139.3/384***. molecular correlations. (e) 4118, (f) 4118, (g) 2f18.. 2?7SO.･,9'.'i,･'+, :2v}S･tt'1O.'0i'+2. =:(P(i,112){Pt,,+P?,,(1-P(i,)+2P,,,P?,,(1-P,,,)2 + P(2) P(i)(1 - P(i,)3 + P?,)(1 - P(,))4}v , Ei7S･g."Q,)21(//),., =- Xo7/(:OilltE02Elil9･.,. =(1124):C2P?,,(1-19,,,)+2(1+ptl),2,)P?,,(1-P(t,)2 +3P(2,Pci,<'1-P,i,.)3+ft,)(1-P,,i)4F}-v, Z?7St'i9,'eS/'･+2=:}Il.:i7S-O,,',,!O.e,},?.i-,g. '. =:gP(iJl12)K.P/.,t"s>(1-,311(iJ)2+Kl-,33c2p)Psi.pt(1-,(3sii)3])Tv .. With these equations, rc2 and rc3 may be calculated by are all ignored. The numerical results for athermal. Eq. (L.12).. (L.15) rc,, s>3,. solution, H =:1, are. ,.

(40) T.. 82. WATARI･ i･. ,. ,'/i:r'･'X1"'fll//i'''. 1;" "}･Vl1. Table 6 Extremely long chain polymer (p･co) Flexible. Rigid B{1) = O, B{2) == O,. B(3) = O. Ba)= 1, B(2)= O,. l9(1)=O, B{,)=1,. Bc3)==O. B{3)=lf2. Ba)=113, B(2)=1/2, S(3)==1f2 peirvflectly .17exible. ww ww (2<j)2/v2. (6118)2,. (3118)2. ,(6118)2. (5118)2. rc2. (6/18)2. (3118)2. (3/18)2. (29/162)2. rc3. o. o. rc3(i)232f(3･94). 'rc,(1}1/36. rc3(2)9801(3･9s) rcs(s > 3). [ik,o]. Zx･x lv2 - ao,o. 'Ao,o,o. o. o. ignoTe(l. ignoTecl. o. o. 1118. 49241(3･95). 5f3. 413. 5!3. 14f9. 2. 312. 2. 1116. L. b2. calc.. b2 Tlgorous. 349791(22･95). 2/!2 4/12. =vO.333. 1/12. ftrO.083. f=O.148. fivO.167 b2. nv rw. 114. = O.250 3116 A O.188. 119 `=vO.111. summarized in Table 6.. '. As seen from Eq. (L.14), two cases, Pa)=P(2)=O and Pa) =1, Pc2)=O, eorrespond. to a rigid stretched cha,jn and a zigzag chain, respectively. The results for these two special cases are ihcluded in the table. The case ,P(i)=113, P(2)==112 and P(3)==1/2 corresponds to a perfectly flexible chain. Since rcs for s>3 has been ignored, the value of the second virial coethcient for a flexible chain is. under-estimated. However, the calculated value seems to be quite reasonable.. :.