Field theory for ARB2 branched polymers

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Field theory for ARB2 branched polymers

T.C. Lubensky, J. Isaacson, S.P. Obukhov

To cite this version:

T.C. Lubensky, J. Isaacson, S.P. Obukhov. Field theory for ARB2 branched polymers. Journal de

Physique, 1981, 42 (12), pp.1591-1601. �10.1051/jphys:0198100420120159100�. �jpa-00209356�

Field theory ^for ARB2 ^branched polymers

T. C. Lubensky, J. Isaacson

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

and S. P. Obukhov

L. D. Landau Institute of Theoretical Physics, ^USSR

(Reçu ^{le 23} juin 1981, accepté ^{le 26 août 1981)}

Résumé.

On étudie la statistique ^d’un système composé ^de dimères A-B et de trimères ARB2 où seules les réac- tions de condensation entre A et B sont permises. ^Ce système ^est décrit par ^une fonction de partition ^{de théorie}

des champs, correspondant ^{à un ensemble} grand-canonique ^de polymères ^branchés ARB2 ^{avec des} fugacités

contrôlant le nombre de dimères, trimères, extrémités et polymères. ^En l’absence d’interactions répulsives ^{entre les} monomères, la théorie du champ moyen donne des résultats identiques ^{à ceux obtenus} par Flory par ^une analyse

combinatoire. En présence d’interactions répulsives, la transition sol-gel ^est impossible. ^{Dans le cas de solutions} diluées, ^les propriétés critiques ^des grands polymères ^{sont reliées} à celles du problème ^{des animaux sur un réseau.}

Abstract.

The statistics of a system ^of condensed A-B dimers and ARB2 trimers, ^where only ^{reactions between}

A and B units are allowed is investigated. ^This system ^{is described} by ^{a field} theoretical partition ^{function corres-} ponding ^{to a} grand ^canonical ensemble of ARB2 ^branched polymers ^with fugacities controlling dimer, trimer,

end point ^and polymer number. In the absence of repulsive monomer-monomer interactions the mean field approxi-

mation gives results which are identical to the combinatorial analysis ^of Flory. ^If repulsive interactions are included,

no sol-gel transition is possible. ^{In the dilute limit,} the critical properties ^of large polymers ^{are related to those of}

the lattice animal problem.

Classification Physics ^Abstracts

05.90

36.20E - 61. 40K - 82.35

1. Introduction.

The introduction of reactive

polyfunctional units into polymeric reactions leads to the creation of polymers ^with nontrivial topologies

as shown in figure ^{1. If the} degree ^of crosslinking ^is sufficient, ^a transition from the free flowing ^sol phase

to an infinite network gel phase ^occurs [1-2] ^{and is}

manifested by ^a divergent viscocity [3]. ^{In addition}

the cluster size distribution can be obtained experimen- tally ^{for a} special ^branched polymers [4]. ^The sol-gel

transition can be observed by ^the divergence ^{of the} viscosity ^at the transition point. ^Recent experiments

have observed a sharp transition with critical expo- nents conforming ^{to the} predicted theoretical values [4].

Recently ^{a field} theory ^was developed which repro- duces generating ^{functions for the} statistics of branched

polymers ^{in solution} [5-7]. The branched polymers

studied in this work consisted of condensates of bifunctional units (dimers) ^denoted by A-A and tri- functional units (trimers) ^denoted by RA3 (Fig. 2) [5-7].

In this paper ^we will discuss a similar class of branched

polymers ^{which we} will denote by ARB2 [8]. ^These polymers ^{consist of} ARB2 trimer units and A-B dimer

units (Fig. 3). Only reactions between A and B end- Fig. ^1.

^Schematic representation of branched polymers.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420120159100

Fig. ^2.

^Dimers (A-A) and trimers (RA3) comprising RA3 ^bran-

ched polymers.

Fig. ^3.

^Dimers (A-A) and trimers (ARB2) comprising ARB2

branched polymers.

groups ^are allowed. Note that an ARB2 polymer ^can

have at most one A endgroup ^{or one internal} loop.

Both the RA3 ^and ARB2 cases were studied in the

loopless, noninteracting ^limit (Gaussian, ^mean field)

over 35 _{years ago} by Flory [8-9] ^{and were found to}

have greatly differing properties. ^The primary purpose of this paper is ^to investigate using ^{field theoretic} techniques, the effects of monomeric repulsion ^{on the}

statistics of ARB2 branched polymers ⁱⁿ good ^solvents.

Let pabe ^the probability ^{that an A} endgroup ^has

reacted. In the Gaussian limit the RA3 system ^reaches

a sol-gel transition at a critical value of pA ^{less than 1} [1].

Thus the sol-gel transition is readily reached, ^{and at}

the transition only ^an infinitesimal fraction of units

are on the infinite molecule [1]. ^{In the Gaussian limit}

the ARB2 system ^{exhibits a} transition when _PA is

equal ^to 1. This transition occurs when all possible ^A endgroups have reacted except ^{for one} per molecule.

The field theoretical approach ^{can be used to} study

the behavior of a single isolated branched polymer [6].

This is accomplished by studying the limit in which

neighboring polymers ^{are so far} apart ^that interpoly-

mer interactions ^are unimportant. ^This dilute limit for the RA3 ^{case is} isomorphic ^to the lattice animal _pro- blem which is of great importance ^{to the theories of} percolation and localization [10-13]. ^{In this} paper ^we will analyze ^{a field} theory that describes the statistics

of ARB2 polymers [5-6]. ^{We show} that in the nondilute

case when interactions are included, ^the transition to an infinite network becomes inaccessible. In the dilute limit a critical point corresponding ^to the presence of an infinite molecule ^can be reached. We will explore

this limit using general arguments ^and perturbation theory and relate the ARB2 ^{critical exponents and} scaling ^{functions to} those of the RA3 (animal) ^case.

Many points ^{which are} equally valid for this paper covered in more detail in two previous ^papers [6, 7]

on the statistics of RA3 polymers. In section 2, ^we discuss the grand canonical ensemble of reacting ARB2 polymers. ^{In section} 3, ^{we introduce a field} theory describing the statistics of ARB2 polymers.

In section 4 we analyze ^{the mean field} equations.

Finally ^{in section 5 we} analyze ^{for the} statistics of dilute ARB2 polymers.

2. Grand canonical ensemble.

In a previous

paper [6], ^a general ^{formalism was} developed ^for studying ^the grand partion function of reacting poly-

meric species. That paper dealt with molecules formed

by A-A dimers and RA3 trimers where the indistin-

guishable endgroups ^{A can react to} form chemical bonds. In this paper, ^we will treat the case of reactions among A-B dimers and ARB2 trimers where only ^A

and B endgroups ^{can react} (Fig. 4). ^{The state Q of a}

Fig. ^4.

ARB2 polymers. ^{Note that there can be at most one free A}

end in _any one polymer. The ^{polymers in figures} 4a and 4b both have three dimer units and four trimer units in different _sequences and correspond ^{to different states o.}

single polymer is defined by the number of dimers,

trimers and free A and B ends it contains, ^and by ^the

sequence and topology of these elements. We will use

the notation n2(a), n3(u), n1A( Q) ^and nIB( a) ^to specify

the number of dimers, trimers and free A and B ends in a polymer ^{in state Q. Note,} however, that these numbers _may be identical for different a’s representing

different topologies. ^For example ^{the states of the two} polymers represented ⁱⁿ figures ^{4a and 4b are different}

even though n2 = 3, n3 = 4, n1A = 1 and _nlB ⁼ 5 in both cases. In a polymeric ^{solution or} melt there will in general ^be many polymers ^{in each state u. There}

are many possible ways ^to distribute dimers and tri-

mers among the possible polymer states, and in cal-

culating equilibrium properties ^of solutions, ^we per- form a weighted average ^over the ensemble of possible

distributions of polymer ^states. We will define the

macro state, G, ^{of a member} of this ensemble by ^the

number of polymers ^{in each statuez} it contains. Thus

a macro state G is specified by ^{the set} { np(03C3, G) }

where np( a ; ^G) is the number of polymers ^{in state u}

in G. The development ^{of a} grand partition function,

for A-B, ARB2 ^molecules proceeds in very nearly ^the

same way ^as for A-A and RA3 molecules. The potential

energy, U(G, { x(G) }), depends ^{on the} coordinates

{ x(G) } of all the A, ^{B and R} groups, and ^can be

decomposed ^{into five} parts,

U2 is the elastic _energy between A and B on a dimer, U3 is the elastic _energy between endgroups ^{of a trimer}

and its central group, UA ^{is the} energy of a free A group, UB is the energy of ^a free B group and Ur ^is

the repulsive energy between nearby groups. We assume each _group A, ^{B or R can be viewed as a}

billiard ball of mass MA, MB and MR and that classical statistics can be applied ^to each ball. We measure all

lengths ^{in units of the thermal} wavelength of a reacted AB pair ^{with mass} MA ⁺ MB

Following ^the steps, detained in reference [7], ^the

constrained grand partition ^{function can be written}

where n1A(G), niB(G), n2(G), n3(G) ^and np(G) ^are

the number of free A ends, ^{free B} ends, A-B dimers, ARB2 trimers and polymers ^{in state G, and} C(G) ^is

the number of configuration ^{in the state G. To be} spe-

cific, np(G) = 03A3 np(u ; ^G), n1A(G) = L ^{n1A( (J)} np( (J; ^G),

« «

etc. C(G) ^{is defined} precisely in reference [7] (eq. (2 .17))

in terms of the contribution to the grand partition

function arising ^{from the} repulsive potential U,(G).

The potentials appearing in eq. (2.3) ^{can be related}

to chemical potentials ^{and the} potential energies ^via

via

where d is the spatial ^{dimension and} /JA’ /JB’ /J2, /J3 and yp ^{are chemical} potentials for A free ends, ^{B free} ends, dimers, trimers and polymers. ^The coordinates,

y, in _eq. (2. 4c) ^and eq. (2. 4d) ^are measured in units of 03BBT ^{and the} coordinates _{yi, Y2,} y3 in ^eq. (2.4d) ^are

coordinates of A and the two B’s relative to the central R group. In situations where the number of loops,

nl, rather than the number of endpoints ^is controlled,

the following ^{relations are of use}

We can then reexpress E ^{as follows}

where Q ⁼ volume and

Densities of dimers, endpoints, ^{etc. are} easily ^obtained

from F ^--_ In Q

In this paper, ^we will be particularly interested in the dilute limit in which individual polymers ^are spatially separated. ^{In this case, we} proceed along ^the

lines detailed in reference [7] ^{and define}

where

We will be interested in moments and reduced forms of C(n2, n3, ni)’ ^In particular

and

where _n1A and _niB are given by eqs. (2.5). ^{Note that}

in _eqs. (2.9) ^to eq. (2.12), nl ^{takes on} only the values 0 and 1 since there can be at most one loop per polymer.

This can easily ^{be seen} by adding successive trimers

to any polymer. ^Each additional trimer produces ^one

additional free B end but no additional free A ends.

Thus _any polymer ^{can have at most one} free A end which could react with one free B end to produce

a polymer ^{with one} loop ^{and no} free ends. Since B’s

cannot react with themselves, ^{no further} loop produc- ing ^reactions can occur. For large n2, C(n2) ^is expected

to obey ^a power law

where is a constant and 0’ is critical exponent. ^This

implies

where W2,, = Â - ’. Other critical exponents ^{can be} introduced via the singular parts ^of Sa ^and S(lP

We use primes ^{on critical} exponents ^{for this} problem

to distinguish them from critical exponents ^of RA3 polymers. ^{One of the} principal goals of this paper will be to relate the above exponents ^for ARB2 polymers

to those from RA3 polymers.

3. Field theory. ^{- E can be} calculated from a field

theory in which the fields t/I Ai,j ^and t/lBi,j ^{are introduced}

to mark A and B free ends. As in the previous paper [7]

i ⁼ 1, ..., n and j = 1, ..., ^{s. The limit n ~} 0 is taken to eliminate Hartree loops, ^{and s is} identified with the polymer fugacities eq. (2. 4e). ^The partition ^function

is evaluated via

where

The Hamiltonian is

As we will discuss shortly É ^differs slightly ^{from z.}

In eq. (3. 3) r(x, x’) is the matrix inverse of the bare A-B propagator

where P n2(X, ^{x’) is the} probability ^{that an} ideal linear

chain consisting ^{of n2} dimers has A at point ^{x and B}

at point ^{x’. Its} Fourier transform is easily ^evaluated

where

Note that g(q ⁼ 0) ^{= 1 so that} r(q ⁼ 0) ^{== r} = 1 - w2.

W3(Xl, x2, X3) ^{describes the} ARB2 units,

where _xo is the co-ordinate of R, XI is the co-ordinate of A and _X2 and _X3 are the co-ordinates of the two B’s.

Notice that the functional integral ⁱⁿ eq. (3.1) ^is performed ^with respect ^to the variables Ai,j ^and Bi,j rather than t/J Ai,j ^and t/JBi,j’ ^{This is to} insure that É be real for positive r(q). ^{Consider, for} example, ^the

situation when w ⁼ w1A - W2A = ^{u -} 0. Thon would be proportional ^{to the} product, fl ( - r(q)) - 1/2,

q

of imaginary ^{numbers if the measure in} eq. (3.1) ^were dt/J Aij dt/JBij’ Defining ^{the measure to be} dÎf¡ Aij dÎf¡Bij

eliminates this problem ^but changes nothing ^else.

Another, ^and perhaps ^more élégant, way ^to deal with this problem ^{is to introduce} complex ^fields t/Jij and

in place ^of t/J Aij ^and t/JBij’ ^{In this case the} Hamiltonian would be non-Hermitian with a term W3 1 t/Jt t/Jj t/Jj’

j

Averages of functions f(t/J Ai,ix), t/JBi,ix)) ^with

respect ^{to the} weighting ^{function e-pH are denoted} by angular ^brackets

Of particular ^{interest are the} propagators

In the non-interacting ^limit (u ⁼ 0) in the absence of trimers (W3 ⁼ 0), this function is diagonal ⁱⁿ ij

and i’ j’ ^and satisfies I/IBi,){X) 1/1 Ai,ix’) > ^{= G,(x, x’).}

We emphasize that correlations between 1/1 Ai,ix) ^and t/lBi,j(X) ^{and not} 1/1 Ai,j ^and I/IBi,j ^give the fundamental propagators G f(x, x’). ^{Note that a factor of} W 1 A(W 1 B)

is associated with each free A(B) end, and w1A(wla)

is conjugate ^to 1/1 Ai,iI/lBi,j)’ ^This implies ^via eq. (2.8) that 1/1 Ai,j > ⁽ I/IBi,j ^{z) is} proportional ^{to the} density

of free A-ends (B-ends). W3 is associated with ARB2

units. It appears in 03B2H ^{as a} coefficient of I/IB 1/11 ^rather

than 1/1 A 03C82 B ^{so that a} single ARB2 unit which has one

trimer, ^one free A end and two free B ends will have a

weight W3 W1A Wi 2. ^as required by ^eq. (2. 3). ^This

can easily be verified by expanding É ⁱⁿ powers of w3, w1A and _wlB.

We now consider the difference between ans Since the sum in _eq. (3.1) ^{contains a} term n2 ⁼ 0,

the partition function evaluated using eqs. (3.4)

includes contributions from point polymers ^{with no}

dimers. Each such polymer ^{carries a} weight SWIA W1B’

These unphysical polymers ^can be removed from the

partition ^function by subtracting ^out configurations

with point polymers yielding

....

were 30 ^and Fo ^{are 3 and F evaluated at} w2 ⁼ W3 ⁼ Q The external fields couple to E t/1 Al,j and E t/1Bl,j’

,

j j

It is, ^therefore convenient to introduce a rotated basis with orthonormal basis vectors a’j satisfying [14]

The order parameters ^{can be} expressed ^{in the new basis}

In equilibrium,

From _eq. (2. 8) ^and (3. 9), ^it is clear that

and

Generating ^{functions that are} natural functions of QA

and QB ^can be obtained via a Legendre transformation

QA ^and QB ^are determined by ^the equation ^{of state}

As in the case of A-A and RA3 mixtures, ^{there are}

two types ^of correlation functions one can consider : those between endpoints ^{on the same} polymer ^and

those between endpoints ^on any polymer. ^{The former}

are described by ^fields t/1 1 (x) ^{with 1 =1= 0} and the latter

by t/121 (x)

where _a, fl ^{= A or B.} The inverse of GIla.,P ^{can be}

obtained by differentiation of r

Similarly ^{we define}

The topology of polymers ^{that can} be grown from A-B dimers and ARB2 ^trimers places ^{a severe constraint}

on Gl«a, ^as discussed after _eq. (2.12). ^{No molecule}

can have more than one free A end. It is, therefore, impossible ^to propagate ^{from one A to a second A}

on the same polymer. Thus, ^{we must have}

On the other hand it is possible ^to propagate ^{from an A}

to a B or from a B to a B so that G .1 AB i= ^{0 and} G.lBB = ^0.

These conditions imply ^a constraint on G -’ :

It is possible, however, ^to _propagate ^{from an A on one} polymer ^{to an A on another} polymer. ^Thus GIIAA 1= ^0.

In the dilute limit, ^the density of polymers goes ^{to zero} and one would expect ^{it to become} increasingly

difficult to propagate ^{from an A to an A on different} polymers. ^{Thus, we} expect

We close this section with some expressions relating

correlation functions to moments of the cluster size

distribution in the dilute limit. From _eqs. (3.12),

(3.13) ^and (2.9), ^{we have}

Introducing the Fourier transform of the propagators in _eqs. (3.17) ^and (3.18), ^we have in the dilute limit

where S0152,p is defined in _eq. (2.12b).

4. Mean field theory.

4.1 FLORY THEORY.

Before considering ^{a mean field} analysis of the field

theory ^discussed in section 3, it is useful to review some

aspects ^of Flory’s ^{treatment of this} problem [1]. Flory

locates the gel point by setting ^the probability, ^a,

that a trimer is connected to another trimer equal ^to

extends to infinity whereas for the mole- cule is finite. a is easily calculated in terms of the

probability, ^{pB, that a B unit has} undergone ^{a reaction}

and the fraction, pA, of A’s on trimers :

The system ^is characterized by the concentrations C2 >, c3 >, cA ) ^and CB > of monomers, trimers,

A-ends and B-ends. c2 ) and c3 ) ^{remain constant} whereas cA ) and CB > change ^{as reactions} proceed.

Initially,

Thus we have

where _ps is the probability ^{that a B is on a trimer.}

Each reaction that takes place reduces the number of A-ends and B-ends each by ^{one. Thus we have}

and

where _PA is the probability ^{that an} A-unit has under- gone ^a reaction. Setting ^{a =} 1/2, ^{we find}

Thus, ^the gel point is reached only when all A-ends have reacted (apart ^{from the one} per polymer ^{that are}

not permitted ^{to react} because of the loopless approxi- mation).

4.2 MEAN FIELD EQUATIONS.

Mean field theory

is obtained by neglecting spatial fluctuations and

replacing I/J Ai,j ^and I/JBi,j ^by their average values ^as defined in _eqs. (3.12). Using eqs. (3.3) ^and (3.14),

we obtain

and

Using eq. (3.14), ^the equation ^{of state can be written}

The mean field expression ^for F121 ^and r (2) ^{can also}

be obtained in the usual _way. We find

where

The gel point ^is determined by ^{rAB = 0. Note that}

implying ^that

This equation ^{is a} consequence of the fact that each

polymer ^{has one and} only ^one free A-end in the loopless approximation ^{and can} in fact be derived from a

Ward identity [15]. Eqs. (4.9) through (4.15) ^define

for us the mean field approximation.

4.3 MEAN FIELD WITH NO INTERACTIONS.

This limit uAp ^{= 0} corresponds ^{either to} dilute solutions

or to the total neglect ^of repulsive interactions _among dimers and trimers and is the limit treated by Flory [1].

It does not correspond ^{to a mean field} 0-point ^because

at the 8-point, there would be three body ^{terms in the} potential ^{which are} important in branched polymers.

We first calculate densities and reaction probabili-

ties. When uAp ^{= 0, we have Fo = SWLA} W1B+O(uAp)

so that

From these we obtain

and

Thus the gel point ^{occurs at} PA ⁼ 1 as predicted by Flory.

Other functions of interest can be calculated ana-

lytically. ^{We find}

This implies ^that

We thus see that the critical exponents depend ^{on the} path ^to criticality. If pA ^{is varied} externally, ^the suscep-

tibility ^and correlation length exponents y and ^v

implied by eq. (4.20) ^obtain the usual mean field values of 1 and 1/2. This is the path ^{that one would}

use to describe percolation. ^{If on the other hand} A3 ^is

fixed and w2 is varied equation (4. 20) ^{leads to} y== 1/2

and v = 1/4. This is the path appropriate ^{to the des-} cription ^of polymer statistics in the dilute limit

(Ap -+ ^{0), and the exponents y and v are in agreement}

with those for the animal problem. ^{Since we} will argue

shortly ^{that the} percolation ^critical point is inacces- sible when u # 0, ^{we will} concentrate on exponents for the dilute case. We find

Equations (4.19) ^to (4.23) imply

where the subscript ^{a refers to exponents for the}

animals problems. ^{We will} argue shortly ^{that the}

relations in _eq. (4.25) between the exponents ^{for the} ARB2 problem ^{and the} RA3 problem ^{are valid even}

when mean field theory breaks down.

4.4 MEAN FIELD THEORY WITH INTERACTIONS.

When uAp ^{is non-zero,} neighboring polymers ^interact

repulsively. This leads to a shift in the gel point ^{and a}

différence between correlations of endpoints ^{on a} single polymer ^{and on different} polymers ^{which show}

up mathematically ^{as the} splitting sT between F(2)

and rf¡2). ^{In the case of A-A polymers,} the shift leads to

percolation critical behavior. In the present ^case, the shift in fact makes the gel point inaccessible.

Using equations (4.12) ^and (4.15), ^{we have}

Solving ^for rAB, ^we find

the quantities r, W3, u, s and QA ^{are related to} physical potentials and all densities are real. One can easily

see from eq. (4.27) ^{that for u} > 0, rAB becomes zero

only ^for complex ^{values of r} for fixed values of the other parameters. r is, however, ^a linear function

(eqs. (3.4) ^to (3.6)) of the dimer fugacity ^{and must be}

real. We therefore, ^conclude that, ^at least within mean

field theory, ^the gel point (determined by rAB ⁼ 0)

cannot be reached in ARB2 polymers ^when repulsive

solvent effects are included. Stated differently, ^the equilibrium ^{state of} reacting A-B dimers and ARB2

trimers (or ^more generally Arbi - 1 ^{f-mers) in a good}

solvent will never contain an infinite ^{network. We have}

verified that this result does not change when lowest order fluctuations about mean field theory ^{are inclu-}

ded. We expect ^{it to be true to all} orders, ^{in renorma-} lized perturbation theory though ^{we have not verified}

this explicitly.

5. &yond ^{mean field} theory. Dilute limit.

5.1 GENERAL CONSIDERATIONS.

In the last section,

we saw that mean field critical exponents ^{for dilute} ARB2 polymers ^{are related to} those for the animal

problem. ^In this section we will _argue that these relations summarized in table 1 are in fact generally

valid.

Each polymer ^{can have at most one} loop ^{or one}

free A-end. In the dilute limit, ^this implies

where f i ^{counts those} single polymer configurations

with no loops and f2 those with ^one loop. ^{We know}

from our study of RA3 ^animals [6] ^that loop fugacity ^is

an irrelevant variable, i.e. that the leading ^critical exponents ^for loopless ^{animals are identical to those}

for animals with ^an arbitrary ^{number of} loops. ^We, therefore, expect f2 ^{to have weaker} singularities ^than fl. ^{Now consider the} diagrams determining fl. ^A single ^root is chosen to be the unreacted A-site and

Table 1.

Critical exponents for RA3 ^and ARB2

branched polymers. ^{Column one} gives the values to first

order in e ⁼ 8 - d. The second column gives ^{the values} obtained from ^the Flory approximation for}’ IX ^discussed

in reference [16] and the relations

and Pa ⁼ Ba - ^{1 discussed in} references [6] ^and [13].

The Flory approximation gives ^{the exact answer} [12, 13]

in two and three dimensions and - remarkably good

results [13] ⁱⁿ all dimensions between 2 and 8. Columns three and four ^{give the exact answers} for exponents obtained from ^the Flory formula for ^{d = 2 and d = 3.}

all possible ^branched polymers ^with arbitrary repulsive

interactions terminating ^{in unreacted B ends are drawn}

as shown in figure ^{5. It} is clear that these are the

exactly ^{the same} diagrams ^{as one would use to eva-}

luate the order parameter Qa ^for loopless ^animals

Fig. ^5.

Diagrams contributing ^to fl. ^{Solid lines} represent ^AB propagators. ^These diagrams ^{are identical to} those for the order parameter Qa ^{for the case of} RA3 polymers.

of RA2 polymers. The rules for evaluating ^the graphs

in the two cases are the same with GAB ^the propagator for ARB3 polymers ^and GAA ^for RA3 polymers. ^We

therefore _argue that fl (W 1 B) ^m Qa(w1B) Using ^this

relation, ^{we have} (ignoring f2)

These equations immediately imply ^{relations between} ARB2’s ^and RA3’s ^critical exponents ^{as was obtained} in mean field theory eq. (4. 25).

5.2 PERTURBATION THEORY.

In this section,

we will outline how the general results of the previous

section can be obtained using diagrammatic pertur- bation theory. This exercise is of interest since it is not obvious at the outset which potentials ^{are the most} important ^and how the animals fixed point ^{is reached}

under renormalization _group transformations. We

begin by slightly simplifying ^{the field} theory. ^We ignore ^the non-critical fields t/J ij ^{for i} > 2 and shift the fields t/J IX Ij

and obtain the Hamiltonian

where rrxp ^and Trxp ^{are given by eqs. (4.12) and (4.13)}

and where h a ^is w - 1 la T 1MF where rMF ^{is the mean}

-

s ^aMF aMF

field form of La(1) given in eqs. (4.9). ^In eq. (5.4), ^we

have omitted fourth order terms and two third order terms with different symmetry, because, ^{as shown}

in detail in reference [6], they ^are irrelevant to the critical behavior under study. ^{The omitted} third order terms though irrelevant for branched polymers ⁱⁿ good

solvents ^are relevant for branched polymers ^{in 0-}

solvents.

We now develop rules for diagrammatic pertur- bation theory along the lines used by Parisi and Sourlas [12] ^{in their treatment} of the animal problem.

First, ^{we note} that the inverse bare and renormalized propagators satisfy

LE JOURNAL DE PHYSIQUE

T. 42, ? 12, DÉCEMBRE 1981 1

where El’) ^and 1 (2) ^{are self} energies ^{to be determined} perturbatively. Eqs. (5.5) imply that bare and renor-

malized propagators have the form

We will do perturbation theory ^{in terms of the unre-}

normalized propagators Sflo ^and Rexop rather than in terms of the renormalized propagators SexP ^and R,,p

because some relations _among critical exponents emerge ^more naturally ^{in terms of the former. In the}

limit s ^~ 0, ^{we have}

Diagrammatic ^{rules for} evaluating El(’) ^{and other}

quantities ^{in terms of} S(X0) ^and R(X°p ^{are similar to those}

developed for the animal problem by Parisi and Sourlas in analogy ^{with the} problem ^{of an} Ising ferromagnet ^{in a} random external field [17]. ^{In order}

to survive the limit s ^~ 0, ^each diagram ^{can have at} most 1, RflOp propagators ^where 1 is the number of loops

in the graph. ^Since R ^{is more divergent than} SI

for _any oefl, ^{the most} divergence graphs will have 1,

R(X°p propagators. ^{In the} present problem, ^{one should}

also use the most divergent R(X, ^{namely R3B or RAB}

rather than AA ^whenever possible. ^{However, the fact}

that SAA ^{= 0} severly ^{restricts this choice.}

In figure ^{6a we show the most} divergent diagrams contributing ^to EXB ^{to order w4. Note that} only ^the propagators SXB ^and RXA appear and that the dia- grams ^are identical to those obtained by ^{Parisi and}

Sourlas [17] ^{for the animals} problem. ^{One can see}

what happens ^to arbitrary ^{order in W3 as follows.}

In order to survive the s ^~ 0 limit, ^{there must exist for}

each _W3 vertex in _any graph ^for EAB ^a path along Sgo

propagators ^to the extemal legs ^{of the} graph. Thus,

the first step ^{in the} construction of arbitrary complex graphs ^for E AB ^{would be to draw} arbitrary ^trees emanating ^{from a} continuous line connecting ^{the end} points A and B of the graph ^{as shown in} figure ^7a.

Each branch of the tree represents ^an S,,o ,, propagator.

The next step ⁱⁿ constructing ^the graph ^{is to close the}

104

loop ⁱⁿ figure ^7a by drawing ^a continuous line connec-

ting ^{A to B} passing through all free ends of the trees in figure ^{7a with} R’, propagators represented by

dotted lines as shown in figure 7b. It is clear that since

s 0 ^{= 0 the} Solo, R Op propagators ^can only ^be S 1

and AA propagators. Finally ^{the most} general graph

can be constructed from figure ^7b by connecting any

pairs ^of SAB ^lines by RÂA ^{lines as shown in} figure ^7c.

Thus we see that the most divergent contributions to each diagram ^for ’AB ^contains only SXB ^and R 0 pro- pagators and that each loop ^{contains one and} only ^one RÂA propagator ^as required ^to make the identification with the Yang-Lee problem [17] ^{in two} dimensions.

Fig. ^6.

Diagrams contributing ^to ’AB (6a), ZAA (6b) ^and FB(’) (6c).

represents ^the SXB propagator, ^z. S3B, - - - - RÂA ^and

R 0 AB-

In figure 6b, ^we show low order diagrams ^contri- buting ^to ZAA. ^{Note that} they ^{are identical to those}

for E AB ^{with one} SÂB propagator replaced by ^an SBs

propagator ^{in all} possible places. ^An analysis ^similar

to that just discussed for EAB ^{shows this to be true to}

all orders in w3. We, therefore, conclude that

where the derivative is taken at constant QA ^and QB.

We have

Fig. ^7.

a) ^{Trees of} S,,O ,, propagators connecting ^{to a continuous}

line of S.0 ,, propagators running ^{from the} endpoint ^{A to} endpoint ^B

of a graph for ZAB. b) Graph for EAB ^obtained by connecting ^{A to B}

with a continuous line of R0153op propagators passing through ^{all free}

ends of the trees of figure 7a. At this points it becomes clear that S0153,P

must be SXB(-) ) and R0153p ^{must be} RXA- c) ^A higher ^order graph

obtained from figure ^7b by connecting pairs ^of SXB lines with R 0

lines.

where rO(Q0152) is ^evaluated at constant QA ^and QB ^and

where y(Q ) ^{is the constant order} parameter suscepti- bility exponents for the animals problem which is in turn equal ^{to the constant order} parameter suscepti- bility exponent ^{for the} Yang-Lee edge singularity ⁱⁿ

two lèss dimensions [12,17].

To obtain SBB ^{as a} function of the fugacities wl A and _{W 1 B,} we use the equation ^{of state to} express QA

and QB in ^{terms of} rAB, WlA and WlB. From references

[6, 13] ^{we know that the constant order} parameter exponent is related to the constant fugacity exponent

via

(In ^ref. [6] ^{this was} expressed ^as

where 1À. ^{is a critical} exponent controlling the behavior of the first order vertex. In reference [13] ^{it was shown}

that _M3 ⁼ 1 + y(Q ) implying eq. (5 . 9).) ^{Thus, we have}

where rc(Wla) ^{is the critical value of r evaluated as a}

function of fugacities. Finally, ^{we need to evaluate} QB- Figure ^{6c shows} diagrams contributing ^to ril).

They clearly correspond ^to QB -Y(AlB) implying ^{that the}

leading singularity ⁱⁿ QB ^satisfies eq. (4.14) ^{to all}

orders in perturbation theory. ^{Thus we have}

in agreement ^with eq. (5.2).

6. Summary. - ^In this paper ^we have demonstrated how to express the statistics of polymers having ^both

AB dimers and ARB3 ^trimers using ^{a field} theoretical

partition function. This formalism is capable ^{of trea-} ting the effects of monomer-monomer repulsion. ^We

have studied these statistics within the ^mean field

approximation ^and by renormalization group tech-

niques. ^{The mean field} approximation in the absence of monomer-monomer repulsion (u ⁼ 0) ^{is shown to} produce exactly the results of Flory. ^{If the monomer-}

monomer repulsion ^{is non-zero} (u = 0) ^{then the}

transition is inaccessible. The dilute limit has a transi- tion and we relate it’s exponents ^and correlation functions to the exponents ^and correlation functions of the lattice animal problem. Note that correlation functions for this problem ^{with no direct} analog ⁱⁿ

the lattice animals case e.g. GBB, diverge ^more strongly

than GAA of the lattice animal case. Nevertheless exponents for these functions can also be expressed

in terms of those for lattice animals.

Acknowledgments.

^{Two of us} (T.C.L. ^and J.I.)

are grateful for financial support from N.S.F. under contract # DMR79-10153 and O.N.R. under contract

# N0014-0106. This work was initiated at the US- USSR Research Group ^on Condensed Matter Theory

held at Lake Sevan in Armenia from Sept. 1 ^to Sept. ^24,

1979.

References

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Press, Ithaca) ^1969.

[2] ^DE GENNES, ^P. G., Scaling Concepts ⁱⁿ Polymer Physics (Cor-

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[3] ^{See for} example ADAM, ^{M. and} DELSANTI, M., ^J. Physique-

Lett. 40 (1979) ^L-523.

[4] ^VON SCHULTHESS, ^G. K., BENEDEK, ^{G. B. and DE} BLOIS, ^R. W., Macromolecules 13 (1980) ^939.

[5] LUBENSKY, ^{T. C. and} ISAACSON, J., Phys. Rev. Lett. 41 (1978) 829; 42 (1979) ⁴¹⁰ (E).

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