A description of the physical properties of polymer solutions in terms of irreducible diagrams. (I)

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1 A description of the physical properties of polymer solutions in terms of irreducible diagrams. (I) J. Des Cloizeaux To cite this version: J. Des Cloizeaux. A description of the physical properties of polymer solutions in terms of irreducible diagrams. (I). Journal de Physique, 1980, 41 (8), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1980 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 61.40K Simple Les Polymer It Polymer Tome 41 No 8 AOÛT LE JOURNAL DE PHYSIQUE J. Physique 4r (1980) AOÛT 1980, 749 Classification Physics Abstracts A description of the physical properties of polymer solutions in terms of irreducible diagrams (I) J. des Cloizeaux Service de Physique Théorique, CENSaclay, B.P. n 2, GifsurYvette, France (Reçu le 10 janvier 1980, accepté le 17 avril 1980) 2014 Résumé. solutions de polymères peuvent être commodément étudiées dans le formalisme grand canonique; dans ce cas, la pression osmotique, les fonctions de corrélations et les concentrations en polymères de différentes longueurs peuvent être développées en fonction de potentiels chimiques et des interactions entre polymères. On montre que ces développements peuvent être simplifiés en exprimant les quantités précédentes au moyen de diagrammes qui sont irréductibles par rapport aux lignes d interaction. Le procédé s applique aux polymères monodisperses et polydisperses. L élimination des divergences à courte distance conduit à des lois d échelle simples Abstract. solutions can be conveniently studied in a grand canonical formalism; in this case, the osmotic pressure, the correlations functions and the concentrations of polymers with different lengths, can be expanded in terms of chemical potentials and polymer interactions. It is shown that these expansions can be simplified by expressing the preceding quantities by means of diagrams which are irreducible with respect to the interaction lines. The process applies to monodisperse and to polydisperse polymers. The elimination of short range divergences leads to simple scaling laws. 1. Introduction. is well known that the osmotic pressure of simple molecules in a solution can be easily expanded in terms of the molecular concentration [1]. The coefficients of this virial expansion are given by connected «pointirreducible» Ursell Mayer diagrams (see Fig. 1). This virial expansion can be obtained by remarking that the usual Ursell Mayer diagrams obtained in the grand canonical formalism are trees of «pointirreducible» diagrams ; the irreducible parts are connected by articulation points which represent molecules. Thus, the contribution of a «pointreducible» diagram is the product of the contribution of its «pointirreducible» parts. This approach does not apply to solutions of monodisperse or polydisperse polymers (see Fig. 2) ; on the corresponding diagrams, the concept of «pointreducibility» does not exist. However, in the case of polymer solutions, it is possible to introduce the concept of «lineirreducibility». A diagram is an «1irreducible» diagram, if it cannot be separated into two pieces by cutting Fig. 1. molecules in solution : a) o Pointirreducible» diagrams ; a point represents a molecule. b) «Pointreducible» diagram. Any reducible diagram can be considered as a tree of irreducible diagrams. Here, the reducible diagram is made of five irreducible parts connected by the articulation points A, B, C, D. Fig. 2. solutions : the diagrams are rather complex and the concept of point irreducibility is meaningless. Article published online by EDP Sciences and available at

3 a) 750 an interaction line. A diagram is a «Pirreducible» diagram, if it cannot be separated into two (non trivial) pieces by cutting a polymer line (see Fig. 3). monomers, in the solvent, repel each other with a strength which is given by the excluded volume u Let us call ZG(Ni,..., N N) the partition function of this set and (ZG(N,,..., NN))O the same quantity in the absence of interaction. For investigating the asymptotic behaviour of the solution when the numbers Nj of links are large, it is convenient to consider the polymer chains as continuous curves in a space of dimension d. Thus, in the absence of interactions, the chains are considered as brownian. In this case, the size of a chain of N links can be characterized by the mean square distance RN between the extremities of the chain Fig. 3. This diagram is Iirreducible and Pirreducible. b) This diagram is 1reducible and Pirreducible. c) This diagram is 1irreducible and Preducible. d) This diagram is 1reducible and Preducible. where 1 is an elementary length. In the continuous limit, ZG(N1,..., NN) becomes infinite. However, this partition function can be normalized and the quantity Now, we remark that an «1reducible» diagram can be factorized and expressed in terms if its «Iirreducible» parts. On the contrary «Preducible» diagrams cannot be directly factorized. Factorization occurs only in the framework of an approximation, which has been introduced several years ago by the author [2] to determine the scaling properties of a polymer solution. For this reason, we shall start by expressing the osmotic pressure, the correlations and the concentrations of polymers of different lengths, in terms of «1irreducible» diagrams and the concept of «P irreducibility» will be used only for analysing divergences. In section 2, we recall the grand canonical formalism for polymer solutions. In sections 3 and 4, we show that the «1reducible diagrams» have tree configurations and that such trees can be summed up by means of Legendre transformations. Section 3 deals with the osmotic pressure and section 4 with the correlation functions. In section 5, we discuss the elimination of short range divergences. Finally, in section 5, simplified expressions are given for the osmotic pressure, the polymer concentrations and the correlation functions. is more regular. We note that the dimensionless factor (Vl ) has been introduced to compensate for the fact that the origins of the free chains have arbitrary positions. For simplicity, it will be assumed that all chains are dissymetric which means that the origin of each chain is marked ; however this restriction is unimportant. When all numbers Nl,..., N,, are different, 3G(Nl,..., NN) may be considered as the partition function of the set of chains ; when all of them equal N, the partition function is 3G(Nl,..., NN)/N!. Let us now consider a grand ensemble of polymers. The corresponding partition function is where { f } represents the set of fugacities The osmotic pressure li is given by [1] 2. Osmotic pressure, polymer concentrations and corrélations in the grand canonical formalism. Let, us consider, in a volume V of solution, a set of N polymers which are chains of monomers. The numbers of these monomers are respectively Nl,..., N,,. The and the mean number CN of polymers of link number N, per unit volume is

4 751 The functions 3G(N1,..., NN) can be expanded in powers of «u» and the terms of this expansion can be represented by diagrams. As usual, the connected diagrams play a crucial role. The partition function can be written in the form where Q{ f } is the generating function of the connected diagrams 1) The vector joining the origin and the extremity of one polymer in the solution has a given value r. 2) Two points Ml and M2 are given in the solution and the polymer configurations are such that 1VI1 is always on a polymer with Ni links and M 2 on a polymer with N2 links. The restricted partition functions can be normalized as the ordinary ones and we have and The corresponding grand partition function is Thus, the dependence of the osmotic pressure H on the concentrations is given by the parametric representations By definition, we have Note also that the total number C of polymers per unit volume and the total number c of monomers, per unit volume, are related to CN by These equations apply also when the polymer is monodisperse. In this case, all the chains have the same number N of links ; there is only one chemical potential f ~ fn and Q{ f } can be written (2(j). With these restrictions, all equations remain valid and equations (2.10), (2.11) read Correlations can be treated in a similar way. A correlation function CG(3t) is the probability distribution associated with the set ail the configurations which satisfy specific conditions or restrictions fll. We may call ZG(N1,..., N, ; 3t) the partition function in a volume V of the set of N polymers made respectively of Nl,..., NN links and restricted by 3t. By definition, the conditions 3t must apply to polymers contained in the set ; otherwise the partition function must vanish. For instance, for 31, we may choose one of the following conditions : These quantities can be expanded in diagrams and can be expressed in terms of connected diagrams. A connected partition function with restrictions 5l will be denoted by the symbol 3(Nl,..., Nf,; 3t) and we shall also introduce the connected grand partition function (2(j f } ; 9t). On the other hand, the general correlation functions CG(3t) are not the most fundamental quantities. In order to define in a precise way, correlations in a solution, one has to determine their cumulants C(9t) Thus, the correlation functions C(fi) can be considered as implicit functions of the concentrations CN. The rules for calculating the diagrammatic expansion of 3(N,,..., NN) are the following (see Fig. 2). 1) A diagram is made of N (solid) polymer lines and of (dashed) interaction lines. 2) Each polymer line j carries a number Nj which defines its link number. 3) A point where an interaction joins a polymer line is an interaction point. 4) A polymer line j passing througli pj interaction points is made of (pj + 1) segments containing nj,o,..., nj,pj links. Thus, we have

5 Reducible 752 5) Each line has a well defined origin. 6) Each solid segment and each interaction line carry a wave vector. The free ends of each polymer carry the wave vector zero. At each interaction point, the sum of the wave vectors of the neighbouring segments and of the interaction line vanish. 7) With each solid segment carrying a wave vector k is associated a factor exp( nk,2 12/2). 8) With each interaction line carrying a wave vector k" is associated the constant factor u(2 n). 9) With the whole diagram is associated a factor. (2 nl1 )d(n 1) 10) The contribution of the diagram is obtained by integrating over all the independent wave vectors k and by summing over all the independent link numbers n. 11) Finally, 3(N1,..., NN) is calculated by summing the contributions of all the diagrams. 12) The counting of the diagrams relies on the assumption that each polymer line has a marked end point (an origin) and that each polymer line can be distinguished from any other one (even if both lines have the same number of links : it is more convenient here to use labelled diagrams, without introducing symmetry factors). The rules for calculating 3(Nl,..., N N ; 3I) are very similar. To find them, we have only to take the restrictions 3t into account. 3. Réduction of tree diagrams, expansion of the osmotic pressure in terms of Iirreducible diagrams. As was explained in the introduction, a diagram which is Iirreducible cannot be separated into two pieces by cutting an interaction line. Consequently, an Ireducible diagram is always made of several Iirreducible parts which are connected by interaction lines so as to form a tree structure. The tree diagram which is associated with an Ireducible diagram is obtained by reducing each I irreducible subdiagram to a point (see Fig. 4). The interaction lines which connect the Iirreducible subdiagrams carry always a wave vector zero and therefore the contribution of an Ireducible diagram can be factorized. The sum of the contributions of the reducible and irreducible diagrams D with N polymer lines gives 3(N1,..., NN) In the same way the sum of the contributions of the diagrams which are Iirreducible and contain N polymer lines gives 3,(Nl,..., NN) Thus, the factorization property gives the possibility of expressing the partition functions 3(Nl, N,,) in terms of the simpler quantities 3 l( Nb, N,). More precisely, let us consider a diagram D containing N polymer lines and let p be the number of its irreducible parts. The corresponding tree diagram T has p vertices connected by ( p 1) interaction lines. Each vertex can be labelled by an index j 1,..., p ; = mj represents the number of legs of the vertex j and Nj the number of polymer lines contained in the subdiagram j. Thus we have With each polymer is associated a number N of links. The link numbers of the polymer lines belonging to the subdiagram j can be denoted by Nj, 1,..., Nj, Nj and the union of all sets of link numbers constitute the set (Ni,..., Nrq) corresponding to the polymer lines of the initial diagram. The contribution of the diagram will be denoted by D(N1,..., NN) and the contribution of the subdiagram j will be denoted by Ij(Nj, 1,..., Nj,Nj). The interaction lines which connect the subdiagram j to the other subdiagrams have mj end points on the polymer lines of this subdiagram ; these points are placed anywhere on the polymer lines belonging to the diagram j. Thus, by applying the rules given in section 2 and by counting the number of ways the m legs can be attached, we find that According to equation (2.8), the grand partition function Q{f} can be written Fig. 4. diagrams (a) and (b) and their corresponding tree diagrams (a ) and (b ). and we want to express this quantity as a sum on the trees by using equation (3.3). This operation has to be done by summing the

6 By 753 contribution of each diagram, once and only once. Thus, symmetry factors have to be introduced and the following remarks indicate the origin of these factors. 1) By changing the labelling of the polymer lines we may create new diagrams (look at Fig. 5) and we see immediately that the factor which is introduced However, any permutation of the vertices, followed by a permutation of the content, gives a diagram which is identical with the initial diagram. Thus, a factor l/5 (r) has to be introduced to avoid overcounting. The content of a vertex of a tree diagram is a vertex function vm{ f 1 which is defined as follows in this way is Using this definition and setting Fig. 5. changing the labelling of the polymer lines, we create different diagrams. we find that a{ f } can be expressed in terms of the vertex functions as a sum over the tree diagrams T 2) The vertices of the tree diagrams are not supposed to be a priori labelled and therefore each tree diagram T is invariant by transformations which belong to the group G(T) of automorphisms of the tree. The number of elements of G(T) is the symmetry number S(T). A diagram can be constructed by attaching a given content to the vertices of the corresponding tree T. where m(t, j) is the number of legs of the vertex j belonging to the tree T. We also note that the generating function of the vertex functions has a very simple form and obeys the relation and we have also which shows that the vertex functions are related to one another by the useful equality Now, let us calculate (21 f }. Equation (3.8) shows that this function can be written in the form Moreover, H(y, { f }) can be expressed in terms of new variables in a simple manner where A(x) is an explicit function of G and of the vertex functions vm{ I} with m > 1. A(x) is a generating function associated with trees in which the 1leg vertex function has been replaced by a variable x. The structure of A(x) is rather complex but the Legendre transform B(y) of A(x) is very sim

7 754 ple. Thus, with the help of such a transformation, it is possible to sum up the tree diagrams, and to express directly (2{ f } in terms of the vm{ 1 }. The Legendre transformation is determined by the coupled equations We may also write, in agreement with equation Using now equations (3.13) and (3.14), we find It is well known and easy to show that B(y) is related to the generating function of the vertex functions (see appendix). More explicitly we have With the help of equation (3.12), we can eliminate y ; in this way, we find more convenient equations Now, let us put x = vl { f }. Equation (3.16) give Incidentally, we note also the following relation which will be used in section 4. We have now to write the concentrations as functions of the new parameters 9N. For this purpose, Bringing the second result in equation (3.15), we obtain the system let us express the operator a in terms of the new p p ÔfN, variables. Starting from the identity we deduce successively from equation (3.23) With the help of these identities, and by combining equations (2. 10) and (3.22), we obtain and using this equality, we may write where vo{ g } and vl { g } are given by equation (3.6). Incidentally, we note that The number of monomers per unit volume is also given by a simple expression (see Eq. (2.9)) These results apply immediately to the monodisperse situation in which each polymer has exactly N links. The partition function, in this case, can be represented

8 The Simple In 755 by the symbol 5(N ; N) and equations (3.6) and (3.11) read tree diagrams (see Fig. 7). This amounts to the trivial approximation which gives On the other hand equations (3.28) and (3.29) give This formula reminds a result which S. F. Edwards [3] derived in 1966 by using mean field methods. However, in our notation, Edwards formula (corrected by M. A. Moore [4]) is and we see that it contains an additional term. This term can be found by calculating one loop diagrams as was shown by M. A. Moore [4]. Fig. 6. permutation, on a subdiagram, of the end points A, B, C of the lines connecting this subdiagram to the other subdiagrams, leads to different diagrams with the same contribution. Thus, the expression of the osmotic pressure can be immediately derived from the knowledge of the I irreducible grand partition function QI(g). The simplest approximation consists in assuming that the diagrams which contribute to II are simple 4. Expansions of corrélation functions in tenns of Iirreducible diagrams. section 1, we have seen that the correlation functions were given by partition functions with restrictions é1( {/} ; fll). The expansions of these quantities have also a tree structure and the concept of Iirreducibility can be applied to restricted diagrams. However the Iirreducible diagrams have to be precisely defined. In general, the contribution of a diagram is calculated in momentum space and each interaction line carries a wave vector. In the following, a diagram will be considered as reducible, if and only if it can be separated into two disconnected pieces by cutting an interaction line carrying a wave vector zero. We note that other definitions of Ireducibility are possible and (eventually) useful but, they would lead to more complicated discussions. Now, we may introduce irreducible restricted partition functions 3 (Nb..., NN, iq) and we shall define also restricted vertex functions vm({ f } ; 9t) We have shown in the preceding section (Eq. (3.8)) that Q{ f } could be written in the form Fig. 7. tree diagrams. where T({ v }, G) is a function of G and of the vertex functions which correspond to the vertices of the tree T. To construct Q{ f } ; 3t) we have only to change in all possible ways the content of one vertex of T. Thus, if this vertex has m legs, we change in T({ v }, G)

9 a) 756 the function vm{ f } which corresponds to this vertex, to the function vm({ f } ; 3t). Analytically, this transformation can expressed by writing where A(x) is defined by (Eq. (3.15)). Using (Eq. (3.24)), we may also write where the fn and gn are related to one another by equation (3.24). In the preceding equation, let us now replace vm({ f } ; 3t) by its explicit expansion given by equation (4. 1). By taking equation (3.23) into account, we see immediately that Thus C(3t) can be expressed in a very simple way in terms of the new variables 9N 5. Elimination of the short range divergences. A long polymer can be considered on a small scale as a continuous and homogeneous line. In this case, the rules for calculating the diagrams remain unchanged but the sums over the numbers of links are replaced by integrals. However, for small n the integral dn... does not always converge and it is necessary to introduce a cutoff no. Thus we may write formally To establish these formulae, we have to apply the rules, given in section 2, which tell how the contributions of the diagrams must be calculated. Thus, if we add to a connected diagram containing N polymer lines, a new polymer line and an interaction line connecting this polymer line to the other ones, we construct a new diagram containing N + 1 polymer lines and a dimensionless factor ul d N2 is introduced. On the other hand, when a new interaction line is added on a connected diagram, a factor zn is introduced (if the integrals over the numbers of links converge!). The origin of the factors which appear in zn is the following. The term u(21)d/2 comes from the interaction itself. The term N2 is related to the fact that the number of polymer segments increases by two when an interaction line is added and that any number n can be considered as proportional to N. Finally, the term (NI 2) d/2 is related to the creation of a new loop in the diagram. More precisely, a new internai wave vector k appears and a new integration over this vector has to be performed. The product of these three factors gives zn. The preceding remarks are valid only if the integrals with respect to the numbers of links converge. Thus, we must study the convergence of these integrals. For this purpose, we shall use the concept of P reducibility defined in section 1. We shall consider only diagrams containing one polymer line and contributing to 3(l ; N). This restriction is justified by the fact that the extension of the discussion to more trivial. complex diagrams is The origin of the divergences can be understood by studying Pirreducible diagrams without Pirreducible insertions (see Fig. 8). On the polymer line of a diagram The physical quantities of interest may or may not depend explicitly on the cutoff. Consequently, it is useful to separate the cutoff dependent contributions from the other ones; this can be done by using a simple renormalization technique, which is the direct application to polymer diagrams of a standard technique of field theory. In fact, it will be shown that for a solution of monodisperse polymers and for N > 1, the partition function 3(N ; N) can be written in the form where c(no) is a constant which is cutoff dependent and *3(N ; N) a regularized quantity which is of the form. Fig. 8. A Pirreducible diagram containing a Pirreducible insertion (a self energy diagram). b) A Pirreducible diagram without Pirreducible insertions. of order q, belonging to this class, let ni be the abscissa of the first interaction point and n2 the abscissa of the last interaction point. Let us keep n, and n2 fixed. All the integrals over the other variables converge, as can be easily seen.

10 Separation 757 Thus the contribution Dq of the diagram is given by an integral of the form insertion in the diagram. Thus, the subtracted contribution (renormalized) contribution) may ben written in a symbolic way The remarks made above show that, in the absence on the of any divergence, the dependence of Dq variables should be, given by Consequently, homogeneity arguments show that the integral (5.5) is always of the following form where A and B are constants. Let us study more precisely the case where 4 > d> 2. We see that for q 1 and perhaps for = a few larger values of q, the integral diverges when (ni n2)+0. In this case, a cutoff is necessary and we may impose the condition n 1 n2 1 > no where no is a constant which is not very different from one (N > 1). In this way, we obtain which means that we omit all the divergent terms. In the same way, in a general diagram, all the divergent parts of the subdiagrams can be subtracted step by step. Thus, with each Pirreducible diagram, we may associate a contribution which is calculated by subtracting the divergent parts from the contributions of the Pirreducible divergent subdiagrams. The contribution D of the diagram contains a term proportional to N which results from the divergence of the diagram itself and a normal term D which can be called renormalized contribution of the diagram. On the other hand, the symbol c(no) will be used for representing the sum of all the terms proportional to N, which come from the contributions of the P irreducible diagrams after regularization of the subdiagrams. Thus, all the divergent parts can be replaced by point insertions (see Fig. 9). The total weight of an insertion is c(no). Dropping the terms which vanish in the limit no 0 we may also write Fig. 9. of the normal (renormalized) contribution and of the additional (divergent) contribution of a diagram. Now let us consider a simple diagram (without insertions) of regularized contribution D. By making s point insertions on the diagram, we obtain the contribution or more explicitly The total contribution of the diagram with all possible insertions is More generally, we have by similar arguments The second term is regular ; it does not depend on the cutoff and its dependence in N is in conformity with the scaling laws. On the contrary, the first term is cutoff dépendent ; it is proportional to N and anomalous. Now, we may separate the two terms, by subtracting the first term from Dq and by considering it as a joint where 3(N; N) is a renormalized partition function which is completely regular. The scaling arguments which have been given above are applicable to 3(N; N) and we may write

11 In 758 This formula can be easily generalized and using similar arguments, we may also write where N is an average number of links which, for instance, can be defined by setting where Similar relations are also valid for the restricted partition functions 3(Nl,..., N N ; 3t). 6. Simplified expressions for the osmotic pressure and for corrélation functions. the expressions which give the osmotic pressure we may now eliminate all the cutoff dependent terms. Taking equation (5.2) into account, we replace equation (3.6) by where We see that simplifications occur, the length 1 has disappeared from the expressions and w(t) which is the unknown function depends only on the dimensionless parameter zn. In particular these expressions show that II has the form where zn is given by equation (5.4). For instance, we see that the expression given by S. F. Edwards and M. A. Moore for d = 3 (see Eq. (3.34)) is precisely of this kind. On the other hand, the chemical potential y is given by the equation In the same way, equations (3.27) and (3.28) are replaced by The expressions which give the correlation functions, can be regularized in the same way and equation (4.5) can be transformed into where Again, it possible to use such expressions to derive scaling laws. For instance, it would not be difficult to show that the mean square distance RN of an isolated chain in the solution is given by an expression of the form These equations which give the osmotic pressure can be written in a simple form when the polymer is monodisperse. In this case, vm{ h } reads vm(h). We may set : and we see immediately from equations (5.13) and (6.1 ) that We might also express the results in terms of the surface SN = Nl2 which defines the size of a brownian chain in the absence of interaction and b = ul 4 which defines the strength of the interaction. In fact, we have (see Eq. (5.4)) Thus equations (5.2) give Incidentally, we note that this coefficient b is just the parameter which defines the interaction in Lagrangian theories [2]. All the preceding expressions describe situations in which N is large but zn might be small. Thus, they apply in the cross over domain between the brownian and the excluded regime.

12 In Tree Rooted 759 In the excluded regime, zn > occur 1 and simplifications In this limit, there remains only one significant surface in the system namely esn and therefore, as was shown previously [2], the osmotic pressure takes the form where 0(n) is a (d dependent) universal function which has the following properties [2] Thus in the semidilute regime, we have 7. Conclusion. this article, we have shown that polymer solutions can be studied by diagrammatic methods and that, in this case, it is sufficient to calculate the Iirreducible diagrams to determine all the properties. Moreover, we have shown that the short range divergences could be directly eliminated. These remarks are very simple but basic because they are general and apply both to monodisperse and polydisperse systems. Thus, the principles given here can be used as a starting point for more precise studies and, in forthcoming articles, applications will be given. Fig. 10. diagrams and their symmetry numbers. Each tree diagram T will be labelled by the number m of its extemal legs (m > 2) and by an additional index a. Thus, Sm,a will be the symmetry number of the diagram (m, a). With each diagram (m, a), we associate a contribution Am,a which is a product of factors, determined as follows. With each internai or extemal interaction line, we associate the factor G ; with each nleg vertex (n > 1), we associate the factor vn. The function A(x) is defined as the generating function In the same way, A (x) can be defined as the generating function of rooted tree diagrams. A rooted diagram is obtained from a simple tree diagram by marking the extremity of one external leg (see Fig.11 ). APPENDIX Legendre transformation and tree diagrams. The Legendre transforme B( y) of a function A(x) is by definition related to A(x) by the coupled equations The function A(x) is arbitrary but we want to show that, if A(x) is the generating function of tree diagrams, the function B( y) is the generating function of the vertices of this diagrams. The tree diagrams which are considered here are connected diagrams which are made out of nleg vertices (n > 1) and out of interaction lines. Two vertices in a diagram are connected by zero or one interaction line and this property defines the tree structure (see Fig. 10). The diagram made of only one interaction line is considered as a tree diagram. Fig. 11. tree diagrams and their symmetry numbers. Each root is characterized by a small circle. Each rooted tree diagram T will be labelled by the number m of its (marked or unmarked) external lines and has a symmetry number Sm,«. The contributions A,p of these rooted tree diagrams are calculated exactly as the contributions Am,rJ. Thus, we see immediately that

13 760 We see now that the series y = A (x) can be constructed easily by iteration and the iteration process can be described by the equation But according to (A. 2), A (0) = 0 and therefore (A. 3) shows that 0(0) = 0. Thus, the preceding equation can be integrated and gives From this equation, we deduce immediately the result which is the announced result. References [1] HILL, T. L., Statistical Mechanics (Mc Graw Hill) [2] DES CLOIZEAUX, J., J. Physique 36 (1975) 281. [3] EDWARDS, S. F., Proc. Phys. Soc. 68 (1966) 265. [4] MOORE, M. A., J. Physique 38 (1977) 265 ; see also DES CLOI ZEAUX, J., J. Physique 41 (1980) 761.