Field theory and polymer size distribution for branched polymers

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Field theory and polymer size distribution for branched polymers

T.C. Lubensky, J. Isaacson

To cite this version:

T.C. Lubensky, J. Isaacson. Field theory and polymer size distribution for branched polymers. Journal

de Physique, 1981, 42 (2), pp.175-188. �10.1051/jphys:01981004202017500�. �jpa-00208998�

Field theory ^and polymer size distribution for branched polymers

T. C. Lubensky and J. Isaacson

Dept. ^of Physics, University ^of Pennsylvania, Philadelphia, ^Pa. 19104, ^U.S.A.

(Reçu ^{le 21} juillet 1980, accepté le 27 octobre 1980)

Résumé.

On donne une description ^{de la} statistique des chaînes polymériques ^{basée sur} des ensembles à l’équi-

libre dont les contraintes sont le nombre de dimères, de trimères et de polymères ^{et celui des} extrémités. On montre que ^cette description correspond exactement à une théorie des champs ^avec laquelle la distribution des tailles des polyméres peut être calculée. La théorie des champs, ^dans l’approximation ^de champ ^{moyen en l’absence} d’interactions répulsives, donne le seuil de gélation ^et la distribution des tailles des polymères ^{en accord avec}

ceux calculés _par Flory ^et Stockmayer. ^On étudie les modifications de la théorie de Flory-Stockmayer ^{dues à}

l’addition d’interactions répulsives. ^{Pour les ensembles à} l’equilibre ^avec contraintes étudiés ici, ^les propriétés critiques ^de gélation ^{et de} percolation ^sont identiques.

Abstract.

The statistics of crosslinked polymer chains, produced by condensation of polyfunctional units, ^is

described by constrained equilibrium ensembles with fugacities controlling dimer, trimer, endpoint ^and polymer

number. It is shown that this description ^{can be} reproduced identically by ^a statistical field theory ^{from which} polymer ^size distribution functions can be calculated. The field theory, ^{in the mean field} approximation ^{in the}

absence of repulsive interactions, gives ^critical probabilities ^and polymer size distribution functions identical to those of Flory ^and Stockmayer. Modifications to the Flory-Stockmayer theory resulting ^from repulsive interactions

are studied. Within the context of the constrained equilibrium ensembles studied here, ^{the critical} properties ^of gelation ^and percolation ^{are identical.}

Classification

Physics ^Abstracts

05.90

36.20

61.40

82.35

1. Introduction. - A gel ^{is a} polymer ^network

formed from crosslinked polymer ^chains [1, 2, 3].

The polymer solution before sufficient crosslinking

to produce ^a gel has occurred is a sol. The sol-gel

transition has been of interest to polymer ^scientists

for many years. The first theories of this transition

were formulated by Flory [4] ^and Stockmayer [5]

over thirty years ago and remain today the standards to which other theories are compared. Percolation is the formation of an infinite network in a random medium [6, 7]. ^{It is most} precisely defined in terms of a model in which sites (bonds) ^{on a lattice are} occupied ^with probability p and absent with _pro-

bability ¹

p. Clusters are defined as groups of

adjacent occupied ^sites (or adjacent sites connected

by occupied bonds). Percolation occurs at a critical

probability, Pc, when an infinite cluster is formed.

It is now well established that percolation ^is precisely analogous ^{to a} second order thermodynamic phase

transition and is characterized by ^critical exponents and scaling ^functions [8, 9, 10]. Several authors have

pointed ^{out the} analogy ^between gelation ^and percola-

tion [6, 11, 12, 13, 14] ^{and have noted that} gelation

should be described by ^{the same set} of critical _expo- nents as percolation. ^The purpose of this _paper is to

explore ^this analogy ^{in détail} by showing ^{that both}

the classical gelation results of Flory ^and Stockmayer

and the modern critical percolation ^{results can be}

obtained from a model of reacting branched and unbranched polymer units. Our primary ^interest

will be to derive in ^some detail the field theory ^used

in reference [13], ^to reproduce ^{the classical results,}

and to explore ^how they ^{are modified} by the presence of repulsive interactions between polymer ^units.

Since the critical properties of percolation and branch- ed polymers ^have been studied in detail elsewhere [ 15, 16, 17], ^{we will} summarize, only briefly, ^the types ^of critical behaviour predicted by ^{the field} theory ^{in the}

last section. Those interested in more detail are referred to the original ^papers.

The development ^{of a} model for the sol-gel ^transi-

tion is complicated by the fact that there are many types ^of gels ^and many different paths ^to gelation.

Typically, gels ^are classified as weak (reversible)

or strong (irreversible) [2]. ^{In weak} gels, ^{bonds can}

break and reform. Thus, ^weak gels ^{will have a vanish-} ing ^zero frequency ^{shear modulus, but a} possibly significant ^finite frequency shear modulus. Weak

gels ^can often be created and destroyed by raising ^and lowering ^the temperature. ^In strong gels, ^{bonds do}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004202017500

not break ^once they have formed. These gels ^will

have a non-vanishing ^zero frequency shear modulus.

The distinction between strong ^{and weak} gels ^is

useful but is probably only ^one of time scale. A strong gel may lose its shear modulus only ^at frequencies

below inverse days ^or weeks whereas a weak gel may show no significant shear modulus even at frequencies

of inverse seconds or higher. ^{It is worth} noting ^that

weak and strong gels ^are analogous respectively

to annealed and quenched ^random systems [18].

The network topology ^remains unchanged (quenched

or frozen-in) ^{in the} presence of external strains iri strong gels but relaxes in weak gels [19]. ^The theory

we present here will be applicable ^with proper inter-

pretation ^{to both} strong ^{and weak} gels.

A common path ^{to the} gel phase is that of conden- sation of polyfunctional ^units [1]. ^{In this} path, ^bifunc-

tional and polyfunctional ^{units react in} solution, eventually producing ^a gel. ^{An alternate} path ^{is that}

of vulcanization in which crosslinking ^{units are added}

to a solution or melt of linear polymers. ^In this paper,

we will concentrate on gelation by condensation

polymerization of bifunctional units which we call dimers and denote by ^A-A (Fig. la) ^and trifunctional units which we call trimers and denote by RA3 (Fig. lb). ^{However our results can be shown to} apply

to condensation with branched units of higher ^func- tionality. ^For simplicity, ^{we will assume that all}

.

reactive end _groups, A, ^on both dimers and trimers

are identical. In this method of gel formation, average number densities, c2 > and C3)’ of dimers and trimers are introduced into solution. Before any reactions have occurred, ^{there is a} concentration

of unreacted or free ends groups. As time proceeds,

the concentration ( C1 > of free ends diminishes until

at long times, ^an equilibrium ^or quasi-equilibrium

state is reached. In strong gels, ^{the final state is} always

in the gel phase whereas in weak gels, ^it may be in the sol or gel phase depending ^{on the} temperature.

Fig. ^1.

a) ^A-A dimer; b) RA3 ^trimer.

Strictly speaking, ^the sol-gel transition (except

in the thermally driven transition in weak gels) ^{is a} dynamical transition that occurs at a critical time after reactions are allowed to begin. ^Thus many treatments of the sol-gel transition solve approximate

kinetic equations describing the chemical reactions among polymeric species [20, 21]. An alternate _ap-

proach ^{is to} attempt ^to describe the system ^{at each} instant in time in terms of a statistical ensemble

characterized by ^densities (or ^chemical potentials)

of observable quantities. ^This approach has the virtue of simplicity, but there is no proof ^{that it} correctly

describes what actually goes ^{on near} the sol-gel ^tran-

sition. In particular, ^{due to kinetic} constraints, ^the system may get trapped ^{in some} region ^of phase space and be unable to probe ^{all states} satisfying specified

constraints as required by ^an equilibrium ^ensemble.

Nevertheless, the constrained equilibrium ^ensembles yield ^a very rich behaviour and should provide a qualitative ^{if not} quantitative description of the sol-

gel transition. It is straightforward, with modern

techniques, ^to incorporate repulsive interactions among dimers and trimers using ^the equilibrium

ensemble approach, ^{but not so} straightforward using

kinetic equations. ^In this paper, ^we will use the former

approach. ^The simplest ^and physically ^{most reasonable}

ensemble is one in which all states with given dimer,

trimer and endpoint ^densities c2, c3 and ci(t) ^occur

with equal probability. ^{This is} equivalent ^{to a con-}

strained equilibrium ^ensemble (which ^we will describe in a grand ^canonical formulation) with fixed average

Cl(t) >1 ( c2 > ^and C3). ^When repulsive ^inter-

actions and loop ^{formation are} neglected, ^{the kinetic} equations ^{and the} equilibrium ^ensemble equations

can be solved exactly ^and yield ^identical polymer ^size

distribution functions at equivalent densities of un-

reacted end _groups. This result gives ^{some confidence}

that the equilibrium ^ensembles provides ^{a realistic} description ^{of the} sol-gel transition in strong ^{as well}

as weak gels ^{even when} loops ^and repulsions ^are

included. One can think of other constraints that

might ^be applied. ^{Most are} unphysical ^{or mathema-} tically intractable. We mention only ^{one in which}

the _average total polymer concentration cp > ^{as well}

as ( ci ), C2 ) and C3 ) is fixed. This ensemble, though probably unphysical, ^is important ^{in that}

it can lead to significant variation of the critical exponents associated with gelation. ^{In this} paper,

we will describe the sol-gel transition, both that which

occurs as a function of time and that in weak gels

which is thermally accessed, ^{in terms of these}

constrained equilibrium ^ensembles.

In a complete ^treatment of statistical properties

of polymers ^{in solution,} polymer-solvent interactions have to be properly treated. In this _paper, ^we will consider good ^athermal solvents in which the repulsive

interaction between polymeric ^{units and} the solvent

can be characterized by ^a single temperature ^inde- pendent ^short range potential ^{u. The formalism we} develop ^{can, however,} easily be extended to include solvent dynamical variables and to allow for different interactions between dimers and trimers. Thus, ^{it is} possible ^to generalize ^the present theory ^{to discuss} solvent-gel phase separation [22] ^{or even to include}

the possibility ^of dimer-trimer phase separation.

One important result of this paper is that the gel point

depends ^{on u even in mean field} theory ⁱⁿ agreement

with Coniglio et ^al. [22].

The theory presented here, which follows earlier work of the authors [ 13], ^{is a} direct extension to branch- ed polymers of the work of de Gennes [23] ^{and des}

Cloizeaux [24] ^{on linear} polymers. ^{It is} similar in

spirt ^to earlier work by Edwards and Freed [25].

We develop, ^{in terms of an ns} component ^field tfJij,

a field theory ^whose partition function reduces to the partition function of the constrained equilibrium

ensembles describing ^the approach ^to gelation. ^We

note that these ensembles by ^{their nature are} poly- disperse. Although ^the presence of polydispersity ^is

sometimes considered a shortcoming of the field theoretic method of describing ^{to the} statistics of linear polymers [26], ^{it is} precisely ^{what is} required

for the polyfunctional condensation path ^to gelation.

For other paths, ^{such as that of} vulcanization from

a monodisperse melt of linear polymers, polydispersity

will be a problem. Presumably ^the methods of Schàfer and Witten [27] ^can be extended to treat other that

equilibrium distributions of branched polymers.

This _paper consists of six sections besides the introduction. In section 2, ^we discuss the grand ^canor

nical formulation of the equilibrium ensembles for

reacting ^chemical species. ^{In section} 3, ^{we discuss} polymer size distribution functions for the equilibrium partition function and the generating function for the

polymer size distribution. In section 5, ^we study ^mean

field theory. ^{In section} 6, ^we review results published

elsewhere on critical properties ^of gels, ^{and dilute}

branched polymers ^{obtained via} renormalization group calculations. Finally in section 7, ^we summarize

our results.

2. Grand canonical ensemble. - As discussed in the introduction, ^{we wish} principally ^to discuss the process of gelation whereby dimers and trimers are

allowed to undergo chemical reactions. As the chemi- cal reaction progresses, large ^and complex polymeric

’ species ^{are made.} Eventually ^a sample traversing polymer ^is produced. ^We envision that the system

can be described instantaneously by ^a constrained

equilibrium ensemble in which the chemical potential

for reactive endgroups ^{is fixed. In this} case, the statisti- cal properties ^{of the} system ^can be described by ^a grand canonical ensemble with chemical potentials

for the various chemical species, ^{J. For} simplicity

we will restrict our attention to systems ^with only

dimers and trimers. A molecular species J ^{is then} specified by the number of dimers, n2, trimers, n3, and unreactive end groups, nl, and by ^its topology, r :

where we introduced the notation n,,, ^{for the set} (nl, n2, n3). Altematively using the relation

valid for each polymer where n, ^{is the number of} loops

in the polymer, ^{we can write}

For a solution in which the concentration of each chemical species ^is controlled, ^{we must introduce a}

chemical potential J.l( (J) ^{for each u. Let} N p( (J) ^{be the}

number of polymers ^of species ^{J. We then have}

where Tr signifies ^{a trace over all} dynamic variables,

H is the Hamiltonian of the system, ^and

Since we are primarily interested in the distribution of polymer sizes, ^{we introduce the} concept ^of a state, G, ^{of the} system specified only by the number of

polymers ^{of each} species ^{can then be} decomposed

into a sum for all possible ^{states G :}

where

and

where Z(G) is the classical partition function for the state G and S,, ^{is a} symmetry factor for species ^J.

In this _paper we will be concerned exclusively ^with indistinguishable ^reactive endgroups. ^{Monomers can}

thus be visualized as (A-A) units and trimers as RA3

units shown in figure ^1. Symmetry factors for various

simple polymers ^{are shown in} figure ^{2. In another}

paper [28], ^{we will discuss} ARB2 trimers units and

Fig. ^2.

^This figure ^shows asymmetry factors for various simple

polymers.

A-B dimers in which only reactions between A and B

endgroups ^{are allowed.}

To facilitate contact with the field theory ^{to be} presented ^{in the next section,} it is convenient at this

point ^{to think of a ball and} spring model for the

polymers ^{that are formed as a} result of reactions between A’s. Each ball in the interior of the polymer

will consist of a reacted pair of A’s whereas each terminal ball will consist of a single ^{A as shown in} figure 3. We will assume that classical statistics ^can be applied ^to each ball and measure all lengths

relative to the thermal wavelength

of the interior A pairs, ^{which we take to have twice}

the _mass, MA, ^{of the external, A’s. We then have}

where x(G)’ represents the co-ordinates of all A engroups, A bound pairs and central _{R groups,} U(G)

the potential energy, N1(G) the total number of free ends and N3(G) the total number of trimers in state G. U can be decomposed ^{into a} part, Ub, ^measur- ing the elastic _energy of links between A’s and links between A’s and the core group R, ^a part Ue ^measur- ing the energy of ^a free A relative to a bound A and

a part, Ur measuring ^the repulsive energy between

nearby groups :

As is customary, ^{we assume short} range repulsion

and write

where p(x ; G) ^{is the} density of A’s in G. Strictly speaking, pUr should include contributions from the central R groups. In this paper, ^we will assume that these contributions ^can be included by choosing ^u correctly. ^{We note, however, that} they play ^{a central}

role in _any phase separation between dimers and trimers that might ^occur.

Fig. 3. - This figure ^{shows a} typical polymer. ^The large ^interior ball, ^labelled AA, ^{is a bound A} pair. ^The balls labelled R are core

groups and the smaller terminal balls represent single ^A endpoints.

Individual trimers are described by ^a potential

energy V 3(Yb Y2, Y3) where yi is the distance of

endgroup ^{i from the core} group. The potential ^bet-

ween A’s with separation y (either ^bound pairs ^or

free ends) ^{not on a trimer are described} by V2( y).

Finally ^we will choose the zero of _energy so that a

free A has a positive potential V 1 ^{relative to an A in a}

bound pair. Assuming that bonds remain indepen- dent, Y2 ^and V3 ^remain unchanged ^{as a result of}

interactions between A’s, ^{and we can write}

where N2(G) is the number of dimers in G, ^and

The factors of 2 - d/2 and (MR/2 M A)d/2 ⁱⁿ equations

(2.14a) ^and (2. 14c) result from the different masses

of the central R group and free A’s from the bound A

pairs.

When u is _zero, Zrep ^{is QNp(G) where Q is the volume}

of the system ^and Np(G) ^{the number of distinct poly-}

mers in G. We therefore write

where Zrep

^{1 when u}

^0. Combining equations (2.6)-(2.15), ^{we obtain}

where

C(G) is the number of configurations ^{in state G.}

The partition function z in equation (2.15) ^allows

a specification of the concentration of each chemical

species, ^{(1. In the usual} situation, only ^{the concentra-}

tions of dimers and trimers are specified, ^{and the} system ^{seeks chemical} equilibrium ^with respect ^{to all} possible ^chemical reactions. In général, molecules (1 and (1’ can interact to produce ^a third molecule (1".

Q" can have an arbitrary topology, but the reaction must conserve dimer and trimer number and reduce the number of free ends by ^{two. In} equilibrium, ^we

must, therefore, ^have

for all possible ^{Q, u’ and} a". In addition, ^{there must}

be equilibrium with respect ^{to reactions on a} single polymer ^which change the number of free ends by

two :

M(nl, n2, n3; u)

M(n, - ^2, n2, n3; 6’) (2.19)

for all a and u’. Equations (2.18) ^and (2.19) imply

that

in equilibrium. Thus, ^as expected the initial concen-

tration of dimers and trimers are all that are needed to specify ^{the final} equilibrium ^ensemble.

In order to discuss the approach ^{to the} gel point,

we need to consider a sequence of ensembles in which the concentration of endgroups and/or polymers ^is

allowed to vary. This we do by introducing ^chemical potentials ^{III and} yp ^{for end groups and} polymers,

which allows us to fix ( cl ) and _cp ). ^The grand partition function is then

In this equation, ^{we have set}

and

where Z’ ^signifies a sum over all states with

G

and

Alternatively, the number of free ends can be re-

expressed ^{in terms} of the number of loops via equa- tion (2.2) ^to yield

where NI(G) is the total number of loops ^{in G and}

From ’,-70 ^{we can calculate} averages of functions of the various densities. Let

and

Then we can express equilibrium averages ^over the states G as :

and

where the derivatives are taken keeping other fields constant. Fluctuations in densities can also be cal- culated :

where ôN,,,

N(X - N(X) ^and 8Np

Np - Np ^).

The _averages in equations (2.30bj ^and (2.31) ^involve

the total densities of dimers, trimers and endpoints, including those that might be members of the infi- nite molecule that is the gel. ^{At a} phase sepa- ration transition, ^some components ^of _X(X,p ^would diverge. ^In this paper, ^we will not consider phase separation ^{and will assume that} gaz ^{is small.}

3. Polymer ^size distribution. - The gel ^transition

involves the formation of a sample traversing ^or

infinite molecule. As the transition is approached, larger ^and larger polymers form, and it is natural to

study ^the distribution of polymer ^{sizes. Let} cp( f ^{n., } ; G)}

be the density ^of polymers ^{in the state G} containing { na } ^{units of} type ^{a. The state G occurs with} proba- bility

so that the _average value of _any function O(G) ^of

the state G is determined by

In particular, ^the generating function for the polymer

size distribution can be written as

where the sum is _{over na}

0 to oo for _every a. If

each h,,, ^{is taken to be} positive (even infinitesimal),

the term with n,,,

^{oo is excluded from the sum. Thus} f ({ ha }) generates statistics of finite polymers only.

f ({ ha }) ^{can be used to determine moments of} cp({ ^{na } ; G) on finite polymers. In particular}

The prime ^on the average brackets ( >’) emphasizes

that the _averages are only ^{over finite} polymers.

S.,, ^{is a} generalization ^to multi-component ^molecules

of the mean square cluster size of percolation theory.

( _cp y, y ^and Sa,fJ ^{all depend on the} fugacities s

and f wa } ^{via their} dependence ^on P(G). ^Alternati- vely they ^{can be taken to} depend ^{on the total concen,} tration )> ^and ( cp > with the aid of _equa- tions (2.30). ^{In most} instances, ^{there is no} restriction

on the number of polymers ^{formed so that s}

¹

and ( cl >, C2 > and C3) ^{are the} only indepen-

dent variables.

The probability, ^{p, that} any pair of A units has reacted is just the total number of free ends that have reacted divided by ^{the initial number of free}

ends (eq. (1.1)) ^{[1]. This is} trivially given by

As time proceeds, ^p increases until the gel first forms when _p

Pc. p - Pc is a convenient measure of the distance from the gel point. ^In analogy with perco-

lation, ^we expect ^{the moments of} cp(j ^{na }, G) to}

display characteristic singularities ^near po. The den-

sity p,,, of units of type ^a in the infinité cluster grows

continuously ^{from zero in the} gel phase ^as [7]

The mean square cluster size in finite clusters diverges

both above and below _Pc as

and the _average number of finite clusters has the free _energy singularity

The correlation length exponent vp ^{can be intro-} duced by looking ^at spatial correlations of units within a given polymers. ^For percolation, ^the hyper-

scaling ^relation d vp

^{2 -} ap where d is the spatial dimensionality ^holds.

More generally, ^one expects f ({ ha }) ^{to scale like}

a free _energy density [10]

where Jp

fi, ⁺ yp ^{and where we} have used thé result from section 4 that the crossover exponent for all values of ^a is the same. Equation (3.9) ^{leads to}

the familiar scaling ^relation

where d - (f3p ⁺ yp) ^{and r}

² + AP p f3 P ^{and where for}

simplicity ^{we have} h2 = h3 = 0 (similar expressions

hold for other cp(n2) > ^{and (} cp(n3) > hold). ^{We will}

discuss the validity ^and limitations of this formula in section 6.

4. Perturbation theory ^{and field} theory. ^{- Lt is}

clear that a graphical perturbation theory ^{for can}

be developed along the lines discussed by ^Fixman [29]

for linear polymers ⁱⁿ solution. As for linear polymers,

the fundamental propagator ^{is that} associated with

a free linear polymer :

where P n2(X, ^{x’) is the} probability ^{that a free linear} polymer with n2 dimers has one end-point ^{at x and}

the other at x’. Note that this sum includes point polymers with n2

^0. The Fourier transform of Gf

is easily ^evaluated yielding

where

The propagator Gf ^is represented graphically by ^a

dark line. As usual repulsive interactions ^are _repre- sented by ^{a dotted line with} endpoints ^at xi and _x2 that carries a factor u l5(Xl - x2). ^Three point ^branches

are represented by ^a wavy three leg ^{vertex and carry}

a factor

where xo is the co-ordinate of the central _{R group}

and w3(x, ^x’, x") is defined so that

With these preliminaries, ^{we can} evaluate the parti-

tion function E, which includes all possible ^contri-

butions from point polymers (i.e. ^{no dimers or tri-} mers) ^{with the aid of the} following diagrammatic

rules.

1. Draw all possible graphs containing ^{an arbi-} trary number of polymers, 3-point ^{branches and} repulsive encounters. Sample graphs ^{are shown in} figure ^4.

2. With each dark line associate a factor Gf(x, x’),

Fig. ^{4. - This} figure ^{shows some} sample polymer graphs. Figure (a) represents ^a repulsive ^{encounter in a linear} polymer. Figure (b) ^is

a repulsive ^{encounter between two} legs ^{of a branched} polymer.

Figure (c) ^{is a} repulsive ^{encounter between two branched} polymers.

with each dotted line a factor u b(x - x’) ^{and with}

each 3-legged ^{vertex a factor} w3(x, ^x’, x").

3. Integrate ^over all co-ordinates.

4. Multiply ^each graph by ^{a factor SNp(G) and} l(G)

where Np(G) ^and N1(G) ^are the number of polymers

and free ends in graph ^G.

5. Divide each graph by ^the appropriate ^set of symmetry ^factors (S(1) ^{to avoid double} counting ^of

polymers.

6. Sum contributions from all possible graphs.

The above rules yield ^the following contributions

to É from the graph ^{shown in} figure ⁵

Fig. ^5.

^This figure ^{shows the} graphical representation ^of equa- tion (4.4).

Precisely ^{the same set of} diagrammatic ^rules [30] ^{can be} generated by ^the field theoretical representation for 5 introduced in reference [16]. ^{Here we will introduce a} generalized ^field theory ^{in terms of an ns} compo- nent field t/li,j ⁽ⁱ

1, ..., n ; j

1,

s) ^with anisotropic potentials that will allow us to calculate both E and f

First we introduce the notation

Then, ^{we have}

where L is decomposed ^into a sum over four parts

L2 ^{describes a} Gaussian chain with a bond weight ^factor depending ^on the direction in the _space of j ^{indices :}

where

L3 describes trimers :

where

L1 describes free ends :

Finally Lrep ^{describes the repulsive} interaction between A units :

It is straightforward ^to verify ^that

so that 8(s, s, { wa, wa }) is identical to 8(s, ^{{ wa, wa })} calculated from the perturbation ^rules just ^described.

Since the sum in equation (4.1) includes the term with _n2

0, both Ê"s count

point polymers consisting

^{of two}

unreacted A’s at the same point ⁱⁿ space. These can be removed by dividing ---’boy S(W2

W3

0), ^{we therefore}

define

where the _w, without curly ^brackets signifies { W¡, W2

0, W3

0 }. ^From equation (4.8), ^{it is clear that both}

E and :E can be expanded in powers of (s - k) : ¹

Note Fo ^{does not} depend ^of { Wa }. Fo(k

s)

Fo(s)

is clearly ^{identical to} equation (2.28) whereas the cluster generating ^{function is}

Thus both thermodynamic and cluster functions can

be obtained from the same field theoretical partition

function.

The expectation values of fluctuating ^{fields or}

order parameters play ^{a central role in field theories}

such as thàt described by equations (4.5) ^and (4.6).

Order parameters ^become non-zero as a result of external fields or as a result of a spontaneous ^broken symmetry. ^{In our case} the linear external field is in the 1-direction in the i-indices. It treats, however,

the first k components ⁱⁿ the j ^index differently ^from

the remaining (s

k). We, therefore, ^have

This, however is not the most useful choice of varia- bles. If hl = h2 = h3

^{0, the} (1, 1, 1, ..., 1) ^direction

in the j-indices ^is special, and it is convenient to

introduce an orthonormal transformation to a co-

ordinate system ^{with one axis} along this direction via the vector di satisfying [31]

The field t/J i,j ^{can be} expressed ^{in the new} co-ordinate system. Since the 1-direction in the i-index is special,

we write

Similarly ^the coupling ^to w 1 ^can be re-expressed ^as

where

It is thus clear that when °°°1°°1 # Wl, wl ^is composed

k

of a term along ^{1= 0 and one} along £ alj. ^Introduc-

j=1

ing ^{unit vectors}

we have

where

and

We can similarly décomposez iPI > ^into parts

along e, and d, :

’

where we also derme

Unless otherwise specified, ^{we will assume} Qo ^and Ql

are evaluated at k

s. Using ^{the above} définitions,

it is _easy to see that

We are now in a position ^to relate field theoretical order parameters ^to the densities introduced in the

previous ^{two sections. We} will denote any quantity

evaluated at w2

W3

0 with a superscript ^zero.

Using ^the expression

we easily ^obtain

Similarly, ^{we fmd}

and

When w1= w1, pi = SW1(Q.i - Qf). Since there can

be no infinite cluster when w2

W3

0, ^{we conclude}

that Ql(wl

Wl)

^{0. Thus} Q1 ^is proportional ^to

the density of endpoints ^{in the infinite cluster}

Because of the complicated ^broken symmetries

in the problem,

has five independent components for k # ^{1 and four} independent components ^{for k}

^{1 :}

where ,11) ^and P(,,, ^{are the} projection operators and

Note that pIf) ^{reduces to zero when k}

^{1. It is}

now a straightforward ^{exercise to show that}

and

Since neither Go ^nor Gl ^have long range correlations,

it is clear that Gl ^{is the} divergent susceptibility ^asso-

ciated with gelation ^and Go ^with phase separation.

We close this rather formal section by defining ^the Legendre transformed generating function which , we will employ ^{in the next two sections :}

where it is understood that T also depends ^on w2,

w2, W3 and w3. ^{r can be} expanded ⁱⁿ powers of

(s

k) just ^{as F} yielding

The equation ^{of state} relating ( tJ¡’ > ^to wi ^is

From this, ^{we can determine} Qo ^and Q1 ^{evaluated at}

s = k via

Thus when k = s, Qo ^{is a function} only ^of wl, w2 and _W3 and does not couple ^to Qi. ^Inverse correlation functions can also be calculated from r

5. Mean field theory. ^{- In this} section, ^{we will} study ^the properties ^{of the} gel transition in mean field

theory. ^This theory ^reduces identically ^{to the} Flory

theory [1] ^when uAp

^{0. When} uAp > 0, ^the gel point is shifted from the Flory ^value but, ^as expected,

the mean field critical exponents ^remain unchanged.

uAp

⁰ corresponds ^{either to} having ^no repulsive

interactions between dimers or to the dilute limit.

(If uAp

^{0 and 3 body} interactions are included, Flory’s result is again ^not obtained.) Mean field

theory consists of neglecting spatial variations off

in the partition ^{sum of} equation (4.5). Using equa- tions (4.6) ^and (4.12), ^{we obtain}

where r

1 - w2 from which we find

and

where,

and

The inverse correlation function are also easily ^cal-

culated. We find

where

L2 and _i3 differ by ^{terms of order} (s

k). Thus, ⁱⁿ

mean field theory,

and

indicating ^{that the} gel point ^is determined by

The quantity (Qolwl) is finite and well defined for all values of _{wi :} for _{wi -} 0, it becomes Gl (eq. (4.27))

whereas for _riz oo it saturates because of _equa- tion (4.23). Therefore, usQ’ ^{is zero if and only if} uA p ^--_ uA p w 2 1

0. This fact will be used for future classification of critical behaviour. Note that if

us > 0, r 0 is always greater than t’ indicating ^that

fluctuations in the total density (for example leading

to phase separation) ^are unimportant ^{at the} gel

transition. Note also that t° is determined by ^{the total}

densities ( ca ) independently of the existence of

large ^{or infinite} polymers.

The mean field equation ^{of state} determining Qo

and Q1 ^are

where

H 1 = (Wl - w 1) ⁺ (W2 - W2) QO + 21W3 W3) Q2 0 (Fl ^{has a term linear in} Q1. ^{We have} incorporated

this term into H ^{-L so that} I’(’) ^{1 is not} equal ^{q to} Dr aQ 1

Equation (5. 8b) ^is easily solved for Q1 ^to yield

Using equation (5.1), ^we then obtain

When uAp

^{0, the equations for Ql and Q2} decouple, and it is more convenient to deal with these variable instead of Qo ^and Q1. ^For future refe- rence, ^we quote only the results for Q2(UAP

^{0) :}

Near the gel point, t°

0, { hx }

^{0, we can write}

the singular part of j’ ^{in a} scaling ^form

Using t Î - ( p - Pc) (which ^{we will} verify ^{in more}

detail shortly) ^and comparing equations (5.12), (5.13), (3. 6), (3. 7) ^and (3.9) ^we obtain the usual mean field critical exponents ^for percolation

The density ^of high polymers ^can be obtained by Laplace transforming f 5;ng

This implies ^{that r}

5/2 ^{in mean field} theory.

We now tum to a determination of the gel point.

Total densities can be obtained ^as discussed in section 3

by differentiation of Fo ^with respect ^{to the} appropriate fuagacities. ^{In mean field} theory, ^{we have}

where it is understood that Qo satisfies the equation

of state equation (5.8a). Thus, ^{we have}

These equations plus ^the équations ^{of state for} Qo

and Q’ 0 ^are sufficient to determine wi, w2, W3, 6o

and Q’ ^{in terms} of cl ), C2 ), C3 > ^{and u. We}

can, therefore determine the gel point to

^{0 in}

terms of initial densities and cl >. ^The general ^solu-

tion is somewhat complicated, ^{and we will concen-}

trate on the special ^{case us}

^{0. In this case,} Qo z wi and ( Cl )

SW1(QO - wl). ^The equations ^can easily

be solved for the probability ^{that an} end group is ^{on a}

three functional unit,

and the probability ^{that a reaction has taken} place (eq. (3.5)),

Similarly ^{we find}

The gel transition is usually studied in terms of the

probability, ^{a, that a branch} originating ^{at a} given

trimer terminates at another trimer. Flory calculated a

for the dimer-trimer mixture we are considering ^and

found

Using equations (5.16), (5.17) ^and (5.7) (evaluated

at u

0), ^{we obtain}

where _{Pc =} (1 ⁺ p)-1 ^{is the critical reaction} proba- bility. Equation (5.20) shows that the gel point

occurs at a

1/2 ⁱⁿ agreement ^with Flory. ^When

us > 0, ^{there is no} compact way of writing t° ⁱⁿ

terms of densities only. ^It is, however, instructive to write it in the following ^form

In the absence of the QI ^{term, this} expression agrees with that of reference [13] ^where point polymers ^were

included. Eliminating ^the point polymers ^{causes the} gel point ^{to occur at a} higher density of reacted end groups (smaller ( cp >).

We close this section with two observations. First,

it is clear that not only ( c 1 ) - ( ci 1’ ^{but also} C2 ) - C2)’ ^and C3 ) - C3)’ ^{should grow} continuously ^{from zero above the} gel point. ^{This can} easily ^{be seen} within the context 4f mean ^field theory.

We find when { ha }

⁰

showing, ^as expected ^{that both of the} quantities grow

as (p - Pcfp ^{for p} > Pc. Second, ^as already remarked,

the mean field theory presented here is identical to the Flory theory ^when uAp

^{0. To emphasize this,}

we calculate the probability P1(n2, n3) ^{that an arbi-} trary selected unreacted endgroup is attached to a

polymer with n2 dimers and _n3 trimers :

is just the coefficient of e - hl e - h3

in the expansion ^of

This is easily calculated using equation (5.12). ^We

obtain

which (using ^eq. (5.18)) is identical to equation (A. 2)

in Chap. ^{IX of} Flory’s ^book.

6. Beyond ^{mean field} theory. ^{- It is now well}

established that mean field theory breaks down below a critical dimension, dc. Critical behaviour in the vicinity of dc - e ^{can be studied} using the s-expan- sion which often suggests ^the type of scaling ^behaviour

to be expected ^{for a} general dimension. Different critical behaviours are associated with different renor-

malization _group fixed points.

To calculate critical properties ^{in the} E-expansion,

one begins ^{with the} partition ^{function defined in} equations (4.5) ^and (4.6). Since tfJi,j) ^will always

have a non-vanishing ^{value, we shift} tfJ i,j ^via

The components ^of _({Ji,j ^{with i} > 1 are non-critical in the vicinity ^{of the} gel point (since their inverse propagators ^{have no} contributions from

We, therefore, eliminate them by integration and obtain a new action L’ that is only ^a

function of qJ lj ^or

In principal ^{L’ should have} potentials renormalized

by ^the non-critical degrees of freedom. We will ignore

this problem ^{and write} using the summation conven-

tion on repeated ^indices

where Lo

QFMF ^and ri,l, ^{apart from a constant} rescaling factor is defined in equation (5.3) ^to (5.5)

FMF ^{is the mean field} free energy (eq. (5.1)) ^evaluated

at the actual values of Qo ^and Ql. Strictly speaking,

there should be additional third and fourth order terms, ^but they ^are irrelevant to our present discussion.

Even in cases where only ^{units of} functionality higher than three are present ^the reduced action

(eq. (6.3)) ^{has the same} symmetry ^as equation (6.3)

with an explicit third order term. Thus all critical behaviour as described below apply equally ^{well to} systems ^with higher functional units. All of the fixed

points and associated critical behaviour arising ^from

a s-expansion analysis of the above field theory ^have

been studied in references [15, ^{16 and} 17]. ^{Here we}

will summarize the types of behaviour that can be

expected.

The symmetries of the fixed points ^{are determined} by the number of components ^of qi ^{which are} cri-

tical. These in tum are determined by ^{which of thç} potentials’ (appropriately ^{shifted to} include effects of interactions) 1"0’ 1" , i2 and _1"3 go ^{to zero.} We now list the various symmetry ^critical points ^{that occur.}

6.1 DILUTE BRANCHED POLYMERS (ANIMALS).

^All components ^of ({J’ ^are simultaneously critical. This

occurs only ^when uAp

^{0. This critical} point ^has dc

^{8, and its} properties within the E-expansion

were analysed in detail in reference [16]. ^{There have}

been also extensive series and Monte Carlo studies of statistics of animals [32-35]. ^The polymer ^{size dis-}

tribution behaves as

where is a constant and, 9 ^{is a critical} exponent that varies from 5/2 ^{for d} > de ^to unity ^{for d}

² [32].

1

6.2 PERCOLATION.

Here all components ^of

qi except (po ^are simultaneously ^critical [13, 18]. ^The (s

1) critical field have the same symmetry ^{as the}

s-state Potts model [8, 9, 15]. Thus the critical point corresponding ^to gelation ^{is in the same} universality

class as the s-state Potts model. The one-state Potts model is usually associated with percolation though

one can definé s-state percolation (s

⁰ corresponds

to loopless percolation) [36], dc

^{6 for this} problem.

The critical properties ^of percolation ^{have been} extensively documented elsewhere [8, 9, 10, 18, 37].

In the ^case of gelation s ^{can be} interpreted ^{as the} product ^of fugacites rip A,.

Ô . 3 ANIMALS.

When 61 ^is negative, ^{as occurs}

when _any or all of the fields, ha’ ^are negative, ’t" 1 andr3

go ^{to zero} before To andr2 ^{and one is lead to a} theory

in which the s - k fields in the subspace ^defined by di di ^{and p(3) are critical. This critical} point ^was analysed in detail in reference [17] for k = 1, s --. 1

and is in the same universality ^{class as animals} (fixed point 1). The presence of this fixed point ^for negative h,,, ^{leads to} a crossover in the function

cp(na) > ^from percolation type behaviour. In parti- cular, ^{for d 6} the function Y(x) equation (3.10)

which is of order e-x for _p Pc dies off ^as x-a+t e-x for large ^x leading ^to

For 0 d 6 cp(n(X) > ^{has a slightly more} complicated ^{form discussed} in reference [17].

7. Summary. - ^{In this} paper, ^we have developed ^a

formalism to describe the statistics of branched poly-

mers and the sol-gel transition in terms of constrained equilibrium ensembles. This formalism is field theo- retical and is a natural extension to branched polymers

of that of de Gennes [23] and des Cloizeaux [24] ^for

linear polymers. Perturbation series in the repulsive potential, ^{u, between monomer units} produced by ^the

field theory agree ^term by ^term with those produced by ^an extension of the Fixman [29] theory ^{to branched} polymers. ^{In mean field} theory, this formalism _repro- duces exactly ^{the theories of} Flory [1] ^{and Stock-}

mayer [5] ^when uAp

^{0 where} Ap ^{is the fugacity for}

polymers. ^When uAp ^{is greater than zero, in mean}

field theory, ^{there is a} shift of the gel point ^{from the} Flory value. The formalism is set up ^so that contact with the previously ^{studied critical} problems ^of percolation and the statistics of lattice animals can

easily be made. The various critical behaviours that

can result are reviewed in section 6.

Acknowledgments. ^{- We} acknowledge ^financial

support from the National Science foundation under

grant ^{No. DMR} 79-10153 and from the office of

Naval Research under grant No. N00014-0106.