PHYSICAL REVIEW A VOLUME 35, NUMBER 8 APRIL 15, 1987

Lattice magnetic analog of branched polymers, lattice animals, and percolation.II. Correlation functions and generalizations

Agustin E. Gonzalez*

01000 Mexico 20, Distrito Federal, Mexico(Received 10 June 1986)

The correspondence, developed in an earlier communication by the author [Gonzalez, J. Phys.(Paris) 46, 1455 (1985)], between a system of branched polymers in dilute and semidilute solutions ofa good solvent on a lattice and a magnetic system consisting of n q-component spins on a relatedlattice, is extended to include correlation functions and some generalizations. We obtain explicit ex-pressions for the vertex-vertex and monomer-monomer correlations between vertices and monomersbelonging to the same or different polymers, in terms of magnetic quantities. We also obtain, interms of magnetic quantities, the correlation between vertices belonging to the same polymer (pairconnectedness). In addition, some general theorems, relating the connectedness length to the radiusof gyration of the polymers and the mean cluster size to the pair connectedness, will be proven.Furthermore, we make some generalizations considering explicitly site-bond problems on any lattice.Finally, a discussion is made on the types of gelation transitions that could be described by themodel.

I. INTRODUCTION

The gelation and vulcanization of polymers have beenstudied for a long time. ' High-polymeric substancesmay be divided according to their structure into two gen-eral classes. Linear polymers are formed when a largenumber of bifunctional monomers join to form a chainmolecule. On the other hand, if some of the reactingmonomers "alled branching units —have functionalitygreater than two, the resultant molecules are branched po-lymers. In this last case, if the reaction is allowed toproceed far enough, an infinite molecule is produced at awell-defined point called the sol-gel transition point. Thegelation of polymers occurs not only in the polymeriza-tion of reactant monomers, with some of them havingfunctionality greater than two. Formation of chemicalbonds between already formed linear polymer molecules,commonly referred to as cross linking, may also lead tothe production of branched polymers and infinite net-works. The vulcanization of rubber is the most prominentexample of a process of this sort.

The first theoretical efforts towards understanding gela-tion and vulcanization processes were done by Flory'and Stockmayer. ' Two fundamental assumptionscharacterized their work: (a) at any stage during the reac-tion, all unreacted functional groups were considered to beequally reactive, and (b) intramolecular reactions, leadingto cyclic structures, were postulated to be of negligible in-fluence. Under those assumptions, they were able to ob-tain expressions for the critical conversion factor p,(which was defined as the fraction of reacted functionali-ties at the sol-gel transition point), for the gel fraction G(fraction of monomers belonging to the infinite network),and for the weight-average molecular weight of the finitemolecules M (a quantity that diverges at p, ). Additionalcalculations in the spirit of the Flory-Stockmayer ap-

proximation were able to give the radius of gyration (Rs)of a branched polymer (a measure of its spacial extent): itwas obtained proportional to the fourth root of the num-ber of monomers ( S) it contains.

Gordon and co-workers reformulated the method interms of the theory of branching ("cascade") processes byGood' ' and were able to consider some of the effectsintroduced by intramolecular (cyclization) reactions. '

That group also discussed the critical behavior of gelationwithin the classical Flory-Stockmayer-Gordon theory.The gel fraction was seen to vanish as G-(p —p, )~,with /3=1, when p~p+; M was seen to diverge asM —

~ p —p, ~

r, with y =1, when p~p, ; whileR&-S~, when S~oo, with p= 4 as mentioned. TheFlory-Stockmayer-Gordon theory of gelation is now re-ferred to as the classical theory and the exponents P= 1,y = 1, and p= 4 are known as the classical exponents.

In 1976, theoretical predictions' ' were published ac-cording to which the critical exponents of the gelationtransition should differ drastically from the classical ex-ponents of the Flory-Stockmayer theory. The reasonsgiven were based on the analogy with other phase transi-tions and, in particular, with the percolation problem' 'and the recent advances made on it. Thus, the neglect ofcycles (something that percolation theory can take into ac-count) was proposed to be the main assumption to blamefor the differences between the classical exponents and thepercolation exponents [f3P ——0.45, y p ——1.74,

, = —'1presumed to be the real gelation exponents.

In the early experimental literature, i.e., before 1979,the problem of critical exponents was widely ignored.However, a posteriori analysis' of gel fractions andviscosities in the pre-1979 literature gave a variety of ex-ponents between the classical and the percolation ex-ponents, many times not coinciding with those of the two

35 3499 1987 The American Physical Society

3500 AGUSTIN E. GONZALEZ 35

existing theories. Later on, different laboratories tookupon themselves the job of measuring the different gela-tion exponents, as reviewed by Stauffer et al. ' The resultwas that, although sometimes the measured exponentswere closer to the percolation ones, some other times theywere nearer the classical values, not allowing us, therefore,to give a definite answer to the problem.

More recently, workers started to think that althoughthe percolation model was an improvement in that it cantake into account the cycles, it does not have to give ingeneral the real gelation exponents because it neglects animportant factor: the kinetics of the formation of the gel.In the percolation model one assumes that the monomersare fixed at the sites of a lattice, while the bonds betweennearest neighbor monomers are being formed in thecourse of the reaction. This is not what really happens inthe reaction bath. For example, in polyfunctional conden-sation one starts with the polyfunctional monomers im-mersed in a solvent, which are describing random walksdue to the Brownian motion. When two of these mono-mers touch, there is a finite probability that they react andstick together permanently, thereby forming a cluster oftwo. Afterwards, the two monomers move together at aslower pace, due to the bigger mass, and they can reactwith the monomers or clusters that they find on theirway, etc. The model of "kinetic clustering of clusters"(which simulates the above described process in a comput-er) of Meakin and Kolb et al. ' can presumably be usedto describe irreversible polycondensation. Among theresults of the model is that the formed clusters, below thesol-gel transition point, are more open when the probabili-ty of reaction at contact is one, while they are more corn-pact when this probability is less than one, i.e.,p(p, &p,, )-0.59 and 0.48, respectively. This is in strikingopposition to percolation theory, where one has a singlevalue for the exponent p(p &p, )= —,. We see, therefore,that by varying the mechanism of the reaction a little bit,we can get clusters of different compacticity or fractaldimensionality. The percolation model is too rigid todescribe such variants because it has only one degree offreedom: the probability of occupation of the bonds onthe lattice. Furthermore, it can also happen that even ifthe monomers are fixed in the course of the reaction, theresulting gelation problem falls in a different universalityclass than standard percolation, if the process of bond for-mation is not random. For example, the growth in addi-tion polymerization comes from unsaturated electrons be-ing transferred from molecule to molecule and leavingbehind a string of chemical bonds. The grown bonds aretherefore strongly correlated in space. This growth mech-anism, which produces long linear chains and cross links(when the unsaturated electrons meet four-functionalunits with two bonds already formed), is usually muchfaster than the mobility of the molecules. The model that,neglecting mobility, has been most widely used to describethe gelation via the growth of cross-linking chains is "ki-netic gelation" by Manneville and de Seze andHerrmann et al. A result of the model is that it doesnot fall in the same universality class as standard percola-tion.

Recently, there has been renewed interest in applica-

tions of critical phenomena concepts to polymers. deGennes and des Cloizeaux were the first to show thatthe statistics of linear polymers in solutions can bedescribed by the n =0 limit of an n-component P fieldtheory, in the presence of an external magnetic field.Afterwards, the analogy was established for the caseof polymer chains on a lattice, by considering the n =0limit of an n-vector model with nearest-neighbor pair in-teractions and in the presence of a magnetic field. Lateron, in the spirit of the de Gennes —des Cloizeaux analogy,Lubensky and Isaacson developed an nq-componentfield theory in the continuum to describe the statistics ofbranched polymers in dilute and semidilute solutions.

The theory presented here, which follows earlier workof the author, ' is a correspondence between a system ofbranched polymers in dilute and semidilute solutions of agood solvent on a lattice and a magnetic system consistingof nq-component spins on a related lattice. %'e will seethat, in the simplest case, we are going to have three dif-ferent fugacities: one for the number of f-functionalmonomers, another one for the number of clusters (poly-mers), and still one more for the compacticity of the clus-ters. In this sense, the model is more flexible than per-colation, allowing us to vary the compacticity of the clus-ters at our will. Due to the several fugacities involved, thepercolation critical point becomes a critical surface in thespace of the fugacities. It should be pointed out that thepresent theory is an equilibrium statistical mechanicalmodel, and when we say that it can describe certain gela-tion processes we mean that the system undergoing gela-tion can be represented by a path in the space of the fuga-cities that crosses the critical surface. One of the incen-tives for developing this lattice correspondence is the pos-sibility of performing a real-space renormalization group(RSRG) for the physically interesting cases of two andthree dimensions, which are far away from the upper crit-ical dimensionality for these problems (eight in the case oflattice animals ).

This paper is arranged as follows: in Sec. II we willbriefly describe the correspondence written in the lastcommunication ' for the case of f-functional units in atwo-dimensional space, and discussed explicitly for thecase of the square lattice (f =4). In Sec. III, expressionswill be obtained for the vertex-vertex and monomer-monomer correlations between vertices or monomers be-longing to the same or different polymers, in terms ofmagnetic quantities. In Sec. IV, the correlation betweenvertices belonging to the same polymer (pair connected-ness) will be obtained, also in terms of magnetic quanti-ties. In Sec. V, the connectedness length and its relationto the radius of gyration of the clusters will be considered,and a relation between the mean cluster size and the pairconnectedness will be obtained. In Sec. VI we will makesome generalizations, discussing explicitly site-bond prob-lems on any lattice. Finally, in Sec. VII we will conclude,discussing the types of gelation transitions that could bedescribed the model.

II. THE CORRESPONDENCE

Consider a two-dimensional regular lattice of M sitesand a coordination number f, on which one would like to

35 LATTICE MAGNETIC ANALOG OF. . . . II. 3501

describe the polymer solution. Each site of that lattice, ifoccupied, will represent an f-functional monomer. Twonearest-neighbor monomers are, by definition, bonded andbelong to the same polymer. We now construct anotherlattice with the following prescriptions: (a) put the sitesof the new lattice in the middle of the bonds of the origi-nal one, and (b) join any two of these sites with a bond,only if the bonds from which these sites come are contigu-ous and emanate from the same site in the original lattice.The new lattice of N (=fM/2) sites will have one ormore types of unit cells, some of them being special inthat they enclose a site of the original lattice. Note thateach site of the new lattice belongs to exactly two of thesecells. We assign an nq-component spin o.

1 to each site ofthe new lattice, such that pi g o i =nq, withl = 1,2, . . . , q and a = 1,2, . . . , n. The Hamiltonian wechoose is

(2)

where

(~),=— Q f dn, ~ g fdn,

1+ z h g ~il 1~iI'I1, 1'

indicates an average over the spin variables, dO; is thedifferential of the solid angle in nq dimensions, and0=—g, f dQ;. By introducing (1) into (2) and expand-ing the exponential, we can rewrite the partition functionas

—PA =Kgggo, , o, , +h +go.;i, ,I 1 a i 1

+h g ~il i +1

(3)

where the first subscript in the spin variables indicates thespin of the lattice we are considering, and the other twolabel the components of that spin. Note that the magneticfield is coupled to the first q components of the spins.The first sum runs over those cells of the new latticewhich enclose a site of the original one and which have,therefore, f sites each. In Fig. 1 we show the new latticesand the chosen cells in the cases for which the originallattices were the triangular, honeycomb, square, and aBethe lattice of coordination number 3. The partitionfunction for the magnetic system will be given by

xxxxxxx,XXXXXXl~XXXXXXX,XXXXXX'XXXXXXXXXXXXXi

XXXXXXX

The well-known theorem of the moments ' states thatin the n =0 limit the only nonvanishing mean values ofthe spin variables are (oi~cri~ )0——5ii5«, and (1)0=1.Therefore, the remaining terms in the expansion of Eq. (3)do not contribute in this limit. By developing the prod-ucts in Eq. (3), we can express it, in the n =0 limit, as

+(E h q) (1+—,'qh )Q p Gt

where we have made (z/Q)o ——lim„o(z/0) and wherethe sum is over all configurations G' of occupied andempty sites of the original lattice (cf. Fig. 2). The productis over all connected clusters c in a given configuration,S' means the number of sites of the original lattice(monomers) in a particular cluster, Nf is the number ofvertices (unreacted functionalities) in a cluster, and N~ isthe number of sites of the new lattice taking part in a

(a) (b)

FIG. 1. The new lattices and the chosen cells (shaded cells) inthe cases for which the original lattices were (a) the triangular,(b) the honeycomb, (c) the square, and (d) a Bethe lattice of coor-dination number 3.

FIG. 2. Left: a section of the new lattice containing twoclusters. The biggest cluster has S'=15 (crosses), Ny ——26 (un-reacted functionalities), N~ ——43 (number of sites taking part ina cluster), NI' ——3 (number of elementary loops), and N~ ——17(solid dots). Right: the same clusters in the original lattice.

3502 AGUSTIN E. GONZALEZ 35

cluster. By using the relations Ns ——(f/2)S'+ , N—i and

Nv =fS' 2—NI'i, where NI'I is the number of reactions in acluster (cf. Fig. 2), we can transform (4) into

where

z 2VQ 0 2V —q

M

exp[(lnq)Np + (ln U)S + (ln V)NIl ]

&& Q (NI, S,Ng )

is a grand partition function for a branched polymer solu-tion in good solvents. Q(Np, S,N II ) is the'number ofways of inscribing Xp polymers, with a total of S mono-mers and Xz reactions, in the original lattice of M sites.Finally, U and V are defined by U=Khf/(1+ —,qh )fand V=(1+—,qh )/h .

The following free energy of the magnetic system in then =0 limit,

1 zgo(q, U, V) = lim —ln

x

2k~T'

will allow us to get the monomer concentrationi)}=(f/2)(Bgo/Bin U)q l, the polymer concentration

QI——(f/2)(Bgo/Blnq) U l, —(f/2) [q/(2 V —q) J, and the

reaction concentration P~ ——(f/2)(Bgo/BlnV)q U

+(f/2)[q/(2V —q)], which, in terms of the magnetiza-tion m—:(Bgo/Bh )q ~ and the internal energyu =——kIITK(Bgo/i')q h of the magnetic system, can bewritten as

colated. In the opposite limit, when U is large, the con-figurations for which many of the sites are occupied willbe weighed more and the system could be percolated. Be-sides the fugacities for the number of monomers and poly-mers, we have another for the number of reactions. Byvarying the number of reactions for a fixed number ofmonomers and polymers, we will vary the number ofloops (cycles) in the polymers, which will make the clus-ters more or less compact. This can be seen by an inspec-tion of Euler's relation cV& ——S —Xp+X~, where N~ is thetotal number of elementary loops in the system (cf. Fig.2). If we consider the space of the basic fugacities(q,K,h), it is, therefore, clear that there is going to be acritical surface separating the nonpercolated phase (whereonly finite clusters are present) from the percolated phase(where, besides the finite clusters, an infinite clusterspreads the whole lattice). In Ref. 31 several points onthis critical surface were located, which we will now men-tion.

(a) q =1, h =v'2, and K =2f p(1 —p) ' is a randomsite percolation problem, where/t is the probability of oc-cupation of the sites.

(b) The limits q~O, h~ao, and K~O, such thatqh =2 and Kh =2 'p(1 —p) ', give a tree percola-tion problem.

(c) q =0, h =1, and K = U, is a random site-animalproblem, U being the fugacity of the animal generatingfunction.

(d) q =0, h ~ oo, and K~O, such that Kh ~=X, is aloopless site-animal problem.

There is no contradiction in having an animal point onthe critical surface (one single cluster in the whole lattice),because the critical fugacity marks the apparition of theinfinite animal (one single cluster, infinite in extent).

f qh fqh u

2 2 4 k T i)q

hhm

4 III. CORRELATIONS BETWEEN VERTICESOR MONOMERS BELONGING TO ANY POLYMER

A. Correlations between vertices

f 1—f qh

4 2qh

kgT1+ hm —qh

2

(10)

The conversion factor p is obtained from (8) and (10), us-ing the relation p =2/~/fP. Finally, the osmotic pres-sure II is given by 11vo/ksT =(f/2)go —(f/2)ln(1+ —,'qh ), where vo is the "volume" of a unit cell in theoriginal lattice (an area in two dimensions).

By varying the fugacities we can weigh more certainconfigurations than others, as can be seen from Eq. (6).For example, if U is very near zero while the other fugaci-ties are different from zero or infinity, only those configu-rations for which there are just a few monomers (and po-lymers) in the whole lattice will be weighed appreciably.In this case, we should not expect the system to be per-

We will now proceed to obtain the correlation betweenvertices belonging to any polymer. The calculation will bemade, for simplicity, for the specific case of the squarelattice, but the derivation can be easily extended to anylattice. To this end, it is convenient to consider the Ham-iltonian for the inhomogeneous system, i.e., the spins ofone cell interact among them with a strength differentfrom that of any other cell, while the different spins ofthe lattice are coupled to magnetic field of different value:

P~ g KI g g ~Ii la~l2la~I3la~I&la+ g ~i g ~il 1

The partition function in the n =0 limit can be obtainedas

35 LATTICE MAGNETIC ANALOG OF. . . . II. 3503

1 ++I g g 111 I ol I oI I oI I

2h;

2 I, I p

N

g(1+ —,qh ) g(1+—,'qh )' /(1+ —,'qh )

'q

'O' S V i=1 R v

G' S(12)

where +s means a product over all occupied sites, +vis a product over all vertices, +~ is a product over all re-actions, and gl and A,; are defined by

r

[(1+—,qhl )(1+—,qhl )(1+—,qhl, )(1+—,qhl )]'1

(13)

Cv(Ij ) = & y;y, ) Gc &y; &Gc&y, &Gc (19)

where & )Gc indicates a grand canonical ensemble aver-age. Using Eq. (18), this correlation can be rewritten asfollows:

The vertex-vertex correlation function of the branchedpolymer solution, between vertices belonging to any poly-mer, can be written as

and

h;

(1+—,qh; )'1 (14)

3 ln=MC~(i,j)= lim

M oo B7 07l(20)

We will define, next, the variables

1 if site I of the original lattice is occupied0 otherwise,

r

1 if site i of the new lattice is a vertex

0 otherwise,

]n=M(q~ I gl I I&I I ) =Ngo(q~ I&1 I IhI I )

N—g ln(1+ —,qh; ) . (21)

where at the end we have to make h; =h Vi and KI ——KVI. From Eq. (17) we can get

and the chemical potentials

kl =»nl,v; =ink;,

allowing us to rewrite (z/Q)o as

(16)

We then have two coordinate systems:

fq IGI Ir 1!

with the transformation rules given by Eqs. (13)—(16).Using the chain rule, 8/B~; can be obtained as

N= g(1+ —,'qh; ):"0 (17)

where "~ is the grand partition function of the branchedpolymer solution:

2= g ( —,qh; )K[Ir] +h;(1+ ,'qh; )—

Bh;

(22)

M N=g exp (lnq)N&+ g glX1+ g rIy;G'

(18) where [iI ] denotes one of the two cells to which the site ibelongs. Therefore,

82 2

=5;1 qh; (1+ ,'qh; ) g K[Ir]—+(1+—,'qh; )(1+—,'qh; )h;a~, a~,- r=1 aK[.r]

2

+( 2qhi )( zqhl ) p K[lr']r, r'=1 BK[lr ]

2K. 8h h 1

a a[' ]g~ i + 1 X ~h [ ]~~K[ir] r=1 ~hl ~K[ir]2 a a a2

+hI( & qhl )(1+ & qhi ) g E[lr ] +h;hl(1+ ,'

qh; )(1+—,'

qhl )—r =1~h l

By using Eqs. (20), (21), and (23), the vertex-vertex correlation can be finally written as

(23)

3504 AGUSTIN E. GONZALEZ 35

Cv(i j )=5;,J[—4qh (1+—,'

qh )u +(1+—,'qh )(1+—,qh )hm —qh (2+qh )]2

2( z qh ) P K[jr'] 2( z qh )(1+ z qh ) P hj.r r'= 1 BK[j~'] r 1 Bhj

—2( —,'ql, )(1+—,'qh ) g h; +h (1+—,'qh )Bh; ' Bh. (24)

where the last four terms should be calculated at h; =h b'i

and Kl ——K VI. In this expression, m; is defined aslimz ()(Ngp)/(}h;, and m =m;( [hj ——h j, [KJ——K j )

is the magnetization per spin; ul = —lim~ „(2KI/0

f)[B(Ngp)/BK;], and u —=ul(Ihj =h j, [ KJ= Kj ) is theinternal energy per spin divided by k& T. Note that theterm k(jr ) ((}u(;r) /()K(zr l ) is essentially an eight-spincorrelation function, four of the spins belonging to the cell[iI ] and the other four to the cell [jI"]; hj. ((iu (;r}/()hj. )

is a five-spin correlation, four of the spins belonging tothe cell [i I ] and the other spin being at the site j; finally,Bm; /Bhj is a two-spin correlation between the spins at thesites i and j.

B. Correlations between monomers

M N

exp (lnq)NP+ g /XI+ g ~;G "(i,j ) I=1 i =1

P~(ij )= limM —+ oo M N

+exp (1nq)Np+ g /XI+ g vy;G'

(29)

where g—:gl([KJ=K j, [h =h j), r=~, (IKJ =K j., Ih.= h j ), and where the configurations G "(ij ) are that sub-set of the configurations G' for which the sites i and j arevertices of the same finite polymer. We will now work onthe subspace of the first q components of the spins (o~ ~,with I = 1,2, . . . , q). Let us introduce the orthogonaltransformation (7p ——gt &

a&(TI ~, with p = 1,2, . . . , q,satisfying

In this case, the monomer-monomer correlation func-tion, between monomers belonging to any polymer, will begiven by

I I II

I(30)

Cm(I, J)= (XIXJ )o( (Xl )o( (XJ )Q( (25)and

A look at Eq. (18) shows that this correlation can be writ-ten as

ln=MC (I,J)= lim

M~ co I J

In this case the chain rule gives

aI —= —(1,1, . . . , 1) .q

Combining (30) and (31) gives

(31)

a a' aK,(27) g a& ——0 if+&1

1=1(32)

The monomer-monomer correlation takes, therefore, thesim pie form

and

BQgC (I,J)= —2KI (28) g a~a~ =5

q I I Il 1

fi=2 q(33)

where again, it should be calculated at h; =h Vi andKI ——K VI. This is essentially an eight-spin correlationfunction, four of the spins belonging to the cell I and theother four belonging to the cell J.

IV. PAIR CONNECTEDNESS

By looking at the Hamiltonian (1), we note that(7& ——(1/Mq )g~&, o~& is just the component of the spinparallel to the magnetic field. Therefore, ofi(&=2,3, . . . , q) will be components perpendicular to thisfield. Let us now consider the spin-spin transverse corre-lations defined by

In the same way as in the field theory, the transversecorrelations between the spins will give the correlationsbetween vertices belonging to the same polymer. We willdefine the vertex-vertex pair connectedness Pz(i,j) as theprobability for the sites i and j of the new lattice to bevertices of the same finite polymer. Py(i,j ) will be givenby

E (i,j ) = (o;q(T/2) —(o;2) (. a.j2) .

where (f((T))=—limM (f(o)e ~~)p/(e ~~)p, ouraim is to show that, in the sol phase, qh Ep (i,j)=Py(i j )for i&j, where Ep(i,J')=lim„pE (i,j) First we. willshow that lim„p(o p) =0, by looking at the numerator:

35 LATTICE MAGNETIC ANALOG OF. . . . II. 3505

(lr'2e ~j0 g a 2 o 'I;lg ++ Q Q oI I rII l IJI I III I +l;=1 I lz ~s

1+—,h goj2l l +h gojl l+. . .(

J 0J J

(35)

Due to the theorem of moments, the only possibility, in the n =0 limit, for having a nonvanishing graph in the expan-sion of the products in (35) is to bring another spin variable, gl cr;l &, coming from the expansion of the magnetic field,or gl g o;l . , coming from the energy term. In the last case, the site i would be a vertex of a cluster. In both caseswe get a factor gl a2, which by Eq. (32) is identically equal to zero. Therefore, E0 (i j)=lim„0(cr;2o j2). We will nowlook at the numerator of (o;2oj2) for i+j:

( ~i 2~j 2 ~0 g g 2 2 ~il; l~jl I + ++g g IJI& llaIIII2llaIIII3llaIlII+IIal +l, =11 =1 I

k lk lk 0(36)

To search for the nonvanishing graphs in the n =0 limit, we should couple the spins o.;l 1 and ojl 1 with two more spins iC J

and j. Both could come from the magnetic field expansion, one from the magnetic field expansion and the other fromthe energy expansion, or both from the energy expansion. In the first two cases we get two factors gl a2, which areequal to zero. In the last case both sites are vertices of clusters. If the clusters to which the sites i end j belong are dif-ferent (i.e., if the sites i and j are not connected), we get again two factors gl a2, however, if i and j are vertices of thesame cluster, we get a factor gl aza2 which is, by Eq. (30), equal to one. Besides this factor of one, the nonvanishinggraph has a weight

(1/qh ) Q(K h "q) (1+—,qh )C

2(1+—,'qh ) exp[(lnq)NI +gs+rNl ]

qh

We therefore find that

2 NM N

(1+—,qh ) exp (lnq)NP+ g /XI+ g ~y;qh I =1 i =1

lim(o;2oj2e )0—— (1+ ,'qh ) g—exp (lnq)NI + g PI+ g ryiqh G (ij) I=1 i=1

(37)

The condition that the clusters to which the sites i and j belong be finite is secured by restricting ourselves to the solphase. The denominator of lim„0(a;2oI2) is none other than (z/II)0, which was obtained in (17) and (18) as

M=(1+—,qh ) g exp (lnq)NP+ g (XI+g ry;

0 6' I=1 i=1

From (37) and (38), we finally find

E0 (ij ) = lim ( o; 2oI2 ) = lim'n —+0 M~ oo

Mexp (lnq)N&+ g /XI+ g ry,

G"(t' j) I =1 i=1

qh M N

g exp (1nq)N~+ g (XI+g ry,G' I=1 i=1

(39)

which proves our assertion that, in the sol phase,qh E0 (i,j)=Pl (i,j) for i &j. If i =j, the spin i is alreadycounted twice in the numerator, which will give us a fac-tor gl(a 2) = 1. Therefore, the allowed graphs are thosefor which the site i does not take part in a cluster. This isnot what we would like for the pair connectedness withi =j. We want Pv(i, i ) to be the probability for the site ito be a vertex of a cluster, which can be obtained as

Bln-=M(y; )oc——lim

M oo 87;(40)

Pl (i,i)= —qh (1+2u )+(1+—,qh )hm . (41)

where =M is given by (18). From Eqs. (21) and (22), wereadily find, for Pl,(i,i),

3506 AGUSTIN E. GONZALEZ 35

The monomer-monomer pair connectedness will be de-fined as

exp (lnq)Xp+ g j&r+ P &3'iG"(I,J)

P (I,J)= limM~oo M N

g exp (lnqWr + g g&r + g r)';G'

(42)

where the configurations G"(I,J) are those for which thesites I and J of the original lattice are monomers of thesame finite polymer. Unfortunately, it is not obvious howto get a magnetic analog of P (I,J), except in the dilutelimit case ( q~0), when there is a single cluster or animalin the lattice. In this case P (I,J) will coincide withC (I,J), given by Eq. (28).

V. THE CONNECTEDNESS LENGTHAND THE MEAN CLUSTER SIZE

We will next define the vertex connectedness length gz,and the monomer connectedness length g, as the secondmoments of the pair connectedness functions:

g rr P~(rr )

gv= ' (43)Pv rr

J

and

Ps=nsS . (48)

We will now define variables gs, r), and g(rJ ) as follows:

1 if the origin is occupied

gs ——. and belongs to a cluster of size S0 otherwise,

1 if the origin is occupiedand belongs to a finite cluster

0 otherwise.

1 if the site J is connected

taining S sites, and where the prime in the summation in-dicates that the infinite cluster is not included. The sameequation can be shown to be valid here, not only in therandom percolation limit but for arbitrary values of theparameters q (&0), E, and h. To prove it we first notethat, in the general case,

N"=("-').. (47)

where Nz is the number of polymers with S monomers ina given configuration and ( )oc indicates the grandcanonical average introduced in Eq. (19). Let's be theprobability for the origin to belong to a cluster of size S;obviously,

g rrP (rr)JgP (rJ)J

(44)

to a site adjacent to the origin

or rJ ——0(r )=

0 otherwise .

1Rs ———g rrDs(rr ),S J

(45)

a quantity that varies with S as Rs-S )', as already men-tioned in the Introduction. Essam showed, in the caseof random percolation, that the square of the connected-ness length coincides with the z average of the square ofthe radius of gyration. In our notation,

where r~ is the distance from the origin to the site j in thenew lattice and rJ is the distance from the origin to thesite J in the original lattice. Both lengths g'z and g, be-ing characteristic sizes of a cluster, should diverge at thecritical surface with the same exponent v, and we willwork on this assumption. Introduce now the monomerdensity profile Ds(rj) as the probability for a site at theposition rJ to be connected to the origin, given that thislast belongs to a cluster of S monomers. The mean-

square radius of gyration, Rs, of the clusters of size S willbe defined as

and

P (rr ) = g' Ds(rJ ~psS=1

Q Ds(rj ) =S .J

To show the first, we note that P (rr ) should be given byP (rJ)=(gr)(rr))oc. On the other hand, according tothe definition of Ds(rr ) [see above Eq. (45)], this quantityshould be given by

Ds(rj)=(g(rj))„=8s

(49)

Let us define the following conditional average of a prop-erty 2:

(~q, ),As

We will first prove the following relations:

g' =(R,'), =g' nsS'RsS=1

(46)

Therefore,

S=1nsS2 g Ds(rj)/ks= g 7)s 'r)(rJ)

S=1 S=1 GC

where ns is the average number of clusters per site con- = (gg(rJ) )oc P(rr), ——(50)

3S LATTICE MAGNETIC ANALOG OF. . . . II. 3507

QDs(rj) = (S)„=S,J

(51)

which proves the second relation. We will now use (44),(45), and (50), and (51) to find

g As g rJDs(rz )S=1 J

g / s QDs(rJ)S=1 J

g nsS RsS=1

g' nsS'S=1

(52)

which proves the first relation. The second is easilyshown by summing Eq. (49) over all sites: QJDs(rJ)= ( QJ g(rj ) )„.But if the origin belongs to a cluster ofsize S, then QJ g(rJ ) =S. Therefore,

dP (I,J)(q =O, h = 1,K = U) =g' N~ (I,J)Us

Bq scan be obtained with the magnetic analogy, because itcoincides with

BC (I J)(q =O, h =1,IC =U)

Bq

obtained from Eq. (28). Now, the sum QJ Nq(I, J) isequal to S gs, where g, is the number of different an-imals of size S, whose leading asymptotic behavior isgiven by gs -S A, , 0 being a universal exponent and A. alattice constant. As will be shown in the Appendix, if wehave a set of S monomers at certain positions, the radiusof gyration can be written as

which demonstrates our assertion that g =(Rs), . Forexample, in the random percolation limit [q = 1, h =v'2,and K =2fr+(I —g) '], if p is close to /i„ it is wellknown that n, scales as'

2 1 2Rs=, griJ2S

(57)

ns=roS 'f[ri~A A. )S ]— (53)

P (I J)= limM~eo gq P sg v

G'

(55)

which shows that this is a quotient of two polynomials inq. Expanding in a Taylor series around q =0 and substi-tuting in (44), we find in the limit q =0, h = 1, and K = U

where r0 and r1 are constants and ~ and o. are universalexponents. It is also known' that the quantity nsS firstincreases up to a maximum and then decreases sharplyafter a typical cluster size Sr, which diverges itself at/i,as ~/i —p, ~

. The sums in Eq. (52) can, therefore, betruncated at this S~. For clusters smaller than S~, the to-pology and compacticity is that for the clusters at/i, ; that

2pp(g, )

is, we should use Rs-S ' . All these facts allow us toget' ' the following connection between the exponent ofthe connectedness length vz and the exponent pr(/t, ):

vp ——pp(/i, )/cr . (54)

In the limit of a single cluster in the lattice (lattice an-imal case), we cannot use Eq. (52). However, a relationbetween the exponents v, and p, can be obtained in thiscase, starting from the defining equation (44), as follows:Eq. (42) can be rewritten as

where rlJ is the distance between monomers I and J. Thesum QJ rqqNs(I, J) in the numerator of (56) can, there-fore, be rewritten as 2+,„, ,&,~s~S Rs, where g,„, ,~,

in-

dicates a sum over all possible animals of S sites. The

quantity (g,„, „,Rs)/g~ ——Rs will be the mean-squareradius of gyration of the animals of S sites. We then find

g' U S gsRg

(animals) =2'USgg

S

(58)

U

U,

U, —U1+

U

' —S

( I+~)—5 e—se (60)

Finally, by transforming the sums into integrals, we find

2pLet us now substitute gs ——AS A, and Rs ——BS+ .

, where the ellipsis indicates the corrections to scal-ing, in the last equation. For the leading term we get

S2p. +2 —e

(animals) =2B (59)SU 2 g

U,

where U, =1/A. . If Uis below but near U„we can write

(animals) =

(56)

2 dP (I,J)grzz (q =Oh =1,K =U)

Bq

r}P (I,J)(q =O, h =1,K = U)

Bq

g rqzg' Ns(I, J)UJ S

gg N, (I,J)U'J S

+2 —~ —rdtt ' e(animals)=2B

0

—2p a (61)

&a =Pa .

But g (animals) should diverge at U, with the exponentv„which indicates that in this case

where Ns(I, J) is the number of different single clusters ofsize S containing the sites I and J of the lattice. Notethat the quantity

As in the random percolation case, it is possible to get arelation between the mean cluster size and the monomerpair connectedness. The mean cluster size is defined as

3508 AGUSTIN E. GONZALEZ 35

S=1nsS2

(62)' nsS

S=1

To evaluate the numerator we sum Eq. (50) over all J's,obtaining

rr g r

(rI ) = g As g Ds(rI ) = g'PsS = g' nsSJ S=1 J S=1 S=1

(63)

(a)

having made use of Eq. (51). Therefore,

SW

gP (rI)J

' nsSS=1

(64) 4i gi 0r 4

The denominator (64) is equal to the concentration ofmonomers belonging to the finite clusters, which coin-cides with the monomer concentration P [cf. Eq. (g)] inthe sol phase. Fortunately, this denominator remains fi-nite at the critical surface and, if we want to investigatethe divergence of S at this surface (defined by the ex-ponent y), we only need to look at the divergence ofgIP (rI), which should be equal to the divergence ofg. Pv(rI ) for which we have a magnetic analog [cf. Eq.(39)]:

P Pv(rI. ) =qh P Eo (rI ) . (65)J J

VI. SITE-BOND PROBLEMS

Suppose one would like to consider a site-bond problemon a regular lattice of M sites and coordination number f.This would represent a solution of polymers made of bi-functional (A) and f-functional (B) monomeric units, inwhich the only possible reactions are 3 with B. To thisend, we first take the new lattice constructed at the begin-ning of Sec. II and build another lattice by disjoining thechosen alternate cells from their nearest neighbors andjoining them back with an extra bond. In Fig. 3 we showthe result of that process in the cases for which the origi-

nal lattices were the triangular, honeycomb, square, and aBethe lattice of coordination number 3. In this section wewill considered explicitly site-bond problems on the squarelattice, having then to define the magnetic system on thelattice depicted in Fig. 3(c), the so-called four-eight lattice.Extensions to other lattices will be evident from the pro-cedure. Let us pick the following Hamiltonian:

M

Kl g g g ~I)la oI2laIII3la~14laI=1 1 a

2M 4m

+K, g ggoI~laoI2la+h g go l,llJ=1 I a i=1 I

(66)

where the sum over I runs over the M shaded cells in Fig.3(c), the sum over J runs over the extra bonds added whenjoining back the cells, while the sum over i runs over all1V (=4M) lattice sites. The rest of the notation is thesame as in Eq. (1). In the n =0 limit we can write, for thepartition function,

(c)

FIG. 3. The new lattices with the chosen cells and extrabonds added, to consider site-bond problems in the (a) triangularlattice, (b) honeycomb lattice, (c) square lattice, and (d) a Bethelattice of coordination number 3.

1+ —,h go;(l +h gcrlll1 0

=g II(K'K'h 'q) (1+-'qh')G' c

(67)

where the first sum in the last line is now a sum over allconfigurations 6' of occupied and empty sites and bondsin the original lattice. B' means the number of occupiedbonds in a cluster and the rest of the notation is as before.It is an easy matter to show, in this case, the following re-lations: Ns ——2S'+B'+ —,Ny and Np ——2S'+ 2 —2NI',which will allow us to transform (67) into

=(1+—,qh ) g II(U, U V 'W),

where U1, U2, V, and 8' are given by U1=E1A /(1+ ,

' qh')', U2 —Kzl(1—+,' qh'), V =(1+ , q—h ) Ih', —

and W=qh l(1+ —,'

qh ). Let us define now the follow-

3S LATTICE MAGNETIC ANALOG OF. . . . II. 3509

go ( U ~, U2, V, W) = lim —ln1 z

~~N 0and calculate

ag 4BR 2 —8'

(69)

ing free energy of the magnetic system in the n =0 limit:4 g =4+ lim gN p, (1—p, )

agM ~ M G

Xp;( I —p, )' —'=4+ (Np )~ (71)

gN,' +(U", U", V ') W 'G' c

+ limM ~ M gg(U', U', V 'W)

G' c

(70)

We will now consider several particular cases ' of thisequation.

(a) W=1, V=1, U&——p&(1 —g&) ', and U2 ——p2(l

—p2) ', with 0(p&,&2( l. In terms of E„K2, h, and q,this implies q =1, h =~2, I

~ =4+&(1—p&)K2 ——F2(1—g2) '. We then find

where ( ) denotes a random site-bond percolationaverage per site, (Np)& giving the average number ofclusters per site. In the limit&&~1, one gets a randombond percolation problem, while when+2~1, the randomsite percolation problem is recovered. It is well knownthat, in these two limits, (Np)& is singular with an ex-ponent 2 —ap at the percolation thresholdg, . Therefore,Bg/BS'is also singular at the corresponding K, .

(b) W=1, V~O, U& ——p&(1 —p, ) ', and U2 ——+2(1—p2) ', which implies that q~O, h~oo, E&~0, andK2 ——2+2(1 —p2) ' such that qh =2 and IC~h=8g, (1—g, ) '. In this case one gets

y NG'~S( 1 ~ )M —S~B(1 ~ )2M —8

4 =4+ limBg . 1 G"

M M y~S( 1 ~ )M —S~B( 1 ~ )2M —8G"

=4+(N, )„"", (72)

where the allowed configurations G" contain clusters withno loops and ( )& is a loopless site-bond percolationaverage per site. Again, the limitsp~~ 1 andp2~1 will

give the corresponding bond and site percolation prob-lems, respectively.

(c) W=0, V= 1, which implies q =0, h =1, K~ ——U&,and K2 ——Uz. In this other case we get

4 =2+ lim g U)U2 ——2+ggsBU)U2Bg . & S B S BM MG„, S,B

(73)

where the configurations 6'" contain one single cluster oranimal and gsB is the total number of animals per sitewith S sites and B bonds. The second term of the rhs of(73) is the site-bond animal generating function.

(d) Finally, let us take the case W =0 and V~O, whichimplies q =0, h~oo, K&~0, and K2 ——U2 such thatK~h = U~. We then obtain

4 =2+ lim g UfU2 ——2+ggs8 U~U2,aw

(74)

where the configurations 6"" contain one single clusteror animal without loops, and gsB is the total number ofloopless animals per site with S sites and B bonds.

It is also easy to see that the longitudinal correlationsbetween the spins will give the correlations between ver-tices or monomers belonging to any polymer, while the

transverse correlations will proportion the correlations be-tween vertices belonging to the same polymer.

VII. DISCUSSION

We have already seen that, in the simplest case, we havethree different fugacities for describing the branched poly-mer solution: one for the number of monomers, anotherfor the number of polymers, and still one more for thecompacticity of the polymers. This gives us more flexibil-ity than random percolation to model the sol-gel transi-tion. In an actual polymerization process, the formingpolymers do not have to be as close as they appear in ran-dom percolation. This can be achieved in the presentmodel by a small monomer concentration for a given po-lymer density. Besides this, we can vary the compactnessof the clusters to fix it at the value of the actual growingpolymers. There are, of course, shortcomings in themodel. For example, as in the de Gennes —des Cloizeauxanalogy for the linear polymer solution, we cannot varythe polydispersity at our will, but is is fixed for givenvalues of the fugacities. Nevertheless, if we overlook thisdeficiency, we may ask for the types of gelation processesthat could be described by the present theory, following apath in the space of the fugacities that crosses the criticalsurface. It seems that the formed polymers have to behomogeneous and isotropic. In this respect, we could notmodel highly inhomogeneous clusters, like DI A clus-ters. Anisotropic clusters, such as those that appear indirected percolation or directed animals, also could notbe described by the model.

Another important question, is the different kinds of

3510 AGUSTIN E. GONZALEZ 35

critical behavior that could be obtained when we cross thecritical surface following different paths. This could onlybe answered once a partial solution like a RSRG solutionhas been obtained for the magnetic system.

APPENDIX

In this appendix we want to show that if we have a setof S monomers at certain positions, then the radius ofgyration may be written as [Eq. (57)]

But

SRs = g (ro+hr )'(ro+hr )

S1 S

=ro+ —ro. $ hr+ —gbr .I=1 I=1

Sro —— g hr. hJS I J=1

2 1 2Rs=z g "u2S2 IJ

where rIJ is the distance between monomers I and J.Proof. The square of the radius of gyration of a set of

S monomers, all of the same mass, is defined as

aIl,d

S—ro. g h, =—S

Therefore,

Shr. hr .

S I J=1

SRs= —g rr,2 1 2

S

where rI is the distance from the center of mass to themonomer I. Let rI be the vector from the center of massto the monomer I, ro the vector from the center of massto some origin, and hI the vector from that origin to themonomer I. Therefore,

&I =rO+hI .

1 2 1Rs= g hr g—hr hJS I=1 I,J=1

On the other hand, rIJ can be written as

rrr ——(hr —hr) =br+br —2hr. hr,2 2 2 2

allowing us to rewrite Rs asS S

Rs= g hr —2 g —(br+br ru)—2 2 2

SI=1 S I J=12S

By the definition of the center of mass, we have, rr ——0 or ro ———(1/S)gr, hr. Then which coincides with Eq. (57).

*Also at Departamento de Fisica, Universidad AutonornaMetropolitana, Iztapalapa, Apartado Postal 55-534, 09340Mexico 13, Distrito Federal, Mexico.

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