HAL Id: jpa-00209050
https://hal.archives-ouvertes.fr/jpa-00209050
Submitted on 1 Jan 1981
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Polymers in solutions : principles and applications of a direct renormalization method
J. Des Cloizeaux
To cite this version:
J. Des Cloizeaux. Polymers in solutions : principles and applications of a direct renormalization method. Journal de Physique, 1981, 42 (5), pp.635-652. �10.1051/jphys:01981004205063500�. �jpa- 00209050�
Polymers in solutions :
principles and applications of a direct renormalization method
J. des Cloizeaux
DPh-T, CEN Saclay, B.P. 2, 91190 Gif sur Yvette, France (Reçu le 24 novembre 1980, accepti le 29 janvier 1981)
Résumé. 2014 Les propriétés des configurations de longs polymères en solution peuvent être commodément étudiées à l’aide d’une méthode de renormalisation directe récemment introduite par l’auteur. Cette méthode est décrite ici en détail et appliquée au modèle à deux paramètres. Les indices 03B3, 03BD et 03C9 sont calculés directement au deuxième ordre en 03B5 = 4 2014 d où d est la dimension de l’espace. Comme attendu, ces développements coincident avec les
développements des indices 03B3, 03BD et 03C9 de la théorie des champs de Landau-Ginzburg-Wilson à zéro composante.
Le formalisme peut décrire des ensembles monodisperses ou polydisperses et conduit directement aux lois d’échelle.
Le second coefficient du viriel, exprimé en choisissant pour échelle la taille d’un polymère isolé, est calculé exacte- ment, pour un système monodisperse, au deuxième ordre en 03B5. Le rapport du rayon de giration à la distance bout
à bout est calculé dans la limite de longues chaînes au premier ordre en 03B5 ; le résultat obtenu est en accord avec la valeur calculée par T. Witten et L. Schäfer en utilisant la théorie des champs. Les variations du gonflement et du
second coefficient du viriel dans le domaine de raccordement sont étudiées à l’ordre 03B52. On donne aussi une expres- sion pour l’entropie d’une chaîne isolée.
Abstract. 2014 The configurational properties of long polymers in solution can be conveniently studied by using a
direct renormalization method, recently introduced by the author. This method is described here in detail and appli-
ed to the two parameter model. The indices 03B3, 03BD and 03C9 are calculated directly to second order in 03B5 = 4 - d, where d
is the space dimension. As expected, these expansions coincide with the expansions of the indices 03B3, 03BD and 03C9 of the zero-component Landau-Ginzburg-Wilson field theory. The formalism may describe monodisperse or poly- disperse ensembles and leads directly to scaling equations. The second virial coefficient, expressed in terms of a scaling length which is the end to end distance of an isolated polymer, has been exactly calculated, for a mono- disperse system to second order in 03B5. The ratio of the radius of gyration to the end to end distance is calculated to first order in 03B5 ; the result obtained is in agreement with the value calculated by T. Witten and L. Schäfer using
field theory. The dependence of the expansion factor and of the virial coefficient, with respect to the interaction in the cross over domain, are studied to order 03B52. An expression is also given for the entropy of an isolated chain.
Classification
Physics Abstracts
61.40K - 64.70 - 82.70
1. introduction. - To study the properties of long polymers in solutions, it is convenient to apply directly the principles of renormalization theory, and
we have shown previously [1] how this can be done.
In the present article, we present this new method in greater detail and we show how it works on practical examples.
The efficiency of this new approach comes from its
close relation to field theory; as we use similar con-
cepts and introduce similar mathematical objects, we profit from the efforts made in field theory by many
outstanding theoreticians. For instance, we use directly,
in polymer theory, the notion of renormalization factor which has great physical significance.
Thus, the approach which is presented here is quite
different from the decimation methods which have been proposed [2] and studied [3, 4] during recent
years and which do not appear to be very practical.
It is known [5, 6] that polymer theory corresponds (to some extent) to a Lagrangian field theory of the Landau-Ginzburg-Wilson type [7, 8], in the limit
where the number n of components of the field goes to zero. However, this analogy does not allow a very
simple treatment of all the physical situations which
are encountered in the domain of polymers. By
contrast, using direct renormalization, we can study
without difficulty monodisperse or polydisperse sys- tems, linear polymers or star polymers, ordinary homogeneous polymers or polymers made of se-
quences of various monomers (copolymers).
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004205063500
Our aim is to establish a unified theory. To realize
this program it is necessary to choose a good model.
It seems that, at the present time, there are only two really good models. One model represents a polymer
in solution as a random walk on a lattice with contact interactions. In this case, all interesting physical quantities can be defined in a simple way and are
finite. Moreover, this model is convenient for com-
puter simulations. However the choice of the lattice remains arbitrary and it is difficult to make analytical
calculations with this type of representation. From
the analytical point of view, it is better to consider
a polymer as a continuous curve in a continuous space of dimension d. This is the basis of the so-
called « two parameter model ». A priori, using a
continuous representation seems difficult. As the number of configurations of the polymer is conti- nuously infinite, various divergences occur. For this model, it is not possible to associate a probability
with each configuration. The probabilities have to be replaced by weights. The averages are not given by
sums but by functional integrals. Renormalization factors have to be introduced to define physical quantities. However, all divergences can be eliminated without real difficulty by renormalizing the partition
functions.
To study the asymptotic properties of long chains,
three successive renormalizations have to be per- formed. In the model, the unperturbed chains are
Brownian chains which are the continuous limits of chains with independent links; the partition function
of a Brownian chain is infinite because the chain hag
a continuously infinite number of continuous degrees
of freedom. This is why the first renormalization is needed. The introduction of interactions produces
new short range divergences, and they are eliminated by the second renormalization (« ultra-violet » renorT
malization). In this way, we define the « two para- meter model » which is good because it is rather
simple and can be studied extensively by perturbation theory. In this model, an isolated polymer depends
on only one dimensionless parameter (usually denoted by z) and a polymer solution on only two dimension-
less parameters.
When the size of the primitive Brownian chain
becomes large, the size of the chain with interactions also becomes large and this limit corresponds to the
case of long polymers. To study their properties, we
need a third renormalization of the partition function (infrared renormalization). This third renormalization has a deep meaning; it is related to the existence of critical indices and universal functions.
Of course, the simple « two parameter theory »
cannot describe faithfully the behaviour of all non-
ionic polymers in solution. However, in the asymp- totic domain corresponding to very long chains and good solvents, we expect a universal behaviour.
Moreover, in the intermediate range, for rather long
chains and moderately good solvents, the model
may give a fairly good description of the properties
of flexible polymers.
In section 2, we introduce continuous models. In subsection 2.1, we define a continuous model with
two-body interactions and a « short ranges cut-off.
This model will be used as a starting point but it is
shown in subsection 2.2 that generalizations are possible.
In section 3, we study the diagram expansions of
the partition functions associated with the model.
In section 4, we show how the cut-off can be elimi- nated by a simple « short range » renormalization of the partition functions. This renormalization is de- scribed in subsection 4.1 and the two parameter model is defined in subsection 4.2.
In section 5, we study the asymptotic behaviour
of long polymers. The basic partition functions are presented in subsection 5.1. A new basic scale is introduced in subsection 5.2. Renormalization fac- tors are defined in subsection 5.3 and the corres-
ponding critical indices in subsection 5.4. The renor-
malization strategy is explained in subsection 5.5. It is applied in subsection 5.6 to derive expansions of
critical quantities to second order in 8 = 4 - d,
where d is the dimensionality of space. Subsection 5. 7 deals with the radius of gyration.
In section 6, we study the variation of physical quantities in the cross over domain. The second virial coefficient is considered in subsection 6.1.
A general formula concerning the variation of any renormalization factor is given in subsection 6.2. In subsection 6.3, we look more precisely at the varia-
tion of the swelling factor and subsection 6.4 deals with the entropy of an isolated chain.
2. The continuous model. Partition functions and first renormalization. - 2.1 THE SIMPLE MODEL WITH CUT-OFF. - Many years ago, S. F. Edwards [9]
showed that a polymer in a good solvent can be represented by a Brownian chain with contact inter- actions and this is the model which will be adopted
here. The weight associated with a Brownian chain,
in a space of dimension d, can be written :
Thus, the properties of a Brownian chain depend only on one parameter S. Moreover, as the argument of the exponential must be a pure number, this parameter S has the dimension of an area (S oc L2).
These facts have a fundamental origin : the Brownian chain depends only on one parameter, because it is
a critical object; this parameter is an area because the Hausdorff dimensionality of a Brownian chain is two whereas the Hausdorff dimensionality of an ordinary curve is one (see Appendix A). So, a Brownian
chain is a curve which looks like a surface.
Using this weight, we may calculate
which is given by the functional integral
In the numerator, we may write
As the integrals are Gaussian, the integration contour in the numerator can be shifted in the complex planes of the variables r(s). Thus, the Gaussian integrals cancel and we find
By comparing (2.2) and (2. 3), we obtain also
In order to represent a polymer in a good solvent, we now introduce two-body interactions. The weight
associated with one polymer becomes I
A cut-off so at short distances is also introduced in order to ensure the convergence of the physical quantities which will be calculated (see sections 3 and 4) from this weight. It will be assumed that, in
the double integral, we always have I s" - s’ I > so where so S.
From (2.5), we deduce immediately that b has dimensions b - Ld - 4 (L is a length). Thus, from pure dimensional analysis, we see that the value d = 4
plays a crucial role in the problem.
It will be useful to express the results in terms of the dimensionless parameter
In order to relate S and b to more conventional quantities (1) we may also set
(1) The fact that natural parameters S and b have been introduced only very recently [1] is rather strange and this critical remark
applies to many articles including those written [10] recently by the present author.
Here, N can be interpreted as the numbers of links, as the characteristic length of one link and v
as the excluded volume. However, it must be empha-
sized that this interpretation is rather fictitious. The quantities S and b must be considered as purely phenomenological parameters. One may try for instance to compare a lattice model to the two para- meter model by applying directly equations (2.7), However, it appears that the coefficient b which represents the interaction of two continuous pieces
of chain depends on the microstructure of the chain;
on the contrary v depends only on the two-body
interaction of points of the discrete chain. Thus one
expects that, if a comparison is made between the models by using equations (2.7), slight discrepancies
will occur even for large values of z.
Consider now a set of N polymers made of the
same monomers but containing different numbers
of them. Each polymer will be labelled by an index j.
The chain j has an area Sj; it is represented by a
function rj(s) (0 s Sj). The weight associated
with a configuration of this set of polymers can be
written
Again, the existence of a cut-off implies the inequa- lity [ s" - s’ I > so in the integrals corresponding to
the terms i = j in the double sum.
All physical quantities can be deduced from « res-
tricted partition functions ». These partition functions
are obtained by counting the number of configura-
tions of a system of N polymers subjected to definite
constraints.
For instance, we may fix the position of p given points belonging to given polymers. Such a point,
called a correlation point, will be labelled by an index q (1 q p). It is assumed that the point of index q belongs to a given chain jq of area Sq, that
its coordinate on the chain is sq (0 sq Sjq) and
that it occupies in space a definite position, defined by a given vector rq, Thus, our requirement is that, for all q, 1 q p :
In this way, we can define a restricted partition
function +3G(r1, ..., rp ;j1’ ...Ijp; sil ... 9 sp; 81, ..., 8Nj)
where the symbol G means that we deal with a general
non-connected partition function and the sign +
means that a cut-off so has been added.
By definition, we set
In particular, the total partition function is defined by
(For polymers in a large volume.)
We note that such partition functions are given by
functional integrals and that the denominator is a
normalization factor which prevents these quantities
from being infinite. (First renormalization.) We note
also the presence of 6 functions in the denominator;
they prevent from renormalizing out the perfect gas terms.
2.2 POSSIBLE GENERALIZATIONS OF THE MODEL. -
In the present article, we shall consider only the pro- blem of long polymers in good solvents (b > 0) and
our model applies to this case. However, generaliza-
tions can be found for describing other situations.
Thus, a more complicated model is needed for
describing the behaviour of long polymers in poor solvents, an important problem which has been
studied recently by B. Duplantier [11]. For negative
values of b (and z), equations (2.5) and (2.8) cannot
be properly used for representing physical situations.
In order to avoid a complete collapse, an other para- meter has to be introduced. For instance, for an iso-
lated chain, we may write
The third term represents a three-body interaction.
We see from the equation, that c has dimensions
Thus we find immediately the well known fact that
the second term is « marginal » for d = 3.
We may note that actually the interaction between
polymers contains only a small three-body contribu-
tion. The potentials consists essentially of two-body
forces but the interactions are non-local; they consist
of a hard core and of a smooth attractive potential.
Again the coefficient c must be considered as a pheno- menological parameter which depends in a compli-
cated way on the chemical structure.
3. Perturbation expansions. - In the expressions defining the partition functions, the weight P { r }
can be expanded in powers of b and each interaction term can be written as an integral
Thus, the expansion of the Fourier transform of a
partition function consists of various terms and each term is the average of a product of exponentials of
the form exp[ik.r(s)]. This average is over a set of Brownian chains and is performed with the help of equations (2.2) and (2.3),
The diagrams representing the terms of order n
are made of Fl polymer lines and n interaction lines which connect some of the polymer lines (see Fig. 1).
Fig. 1. - A diagram contributing to +3G(51, ..., Ss). The polymers
are represented by solid lines, the interactions by dashed lines.
The diagram consists of two separated « connected parts ».
The contributions of the connected diagrams are especially interesting because all physical quantities
can be expressed in terms of the sums [ + 3G(* * *)]connected
of these contributions. We define the connected par- tition functions +3(...) by setting
When the volume V of the system increases these
partition functions become independent of V. How-
ever, they are not pure numbers; for instance, equa- tions (2.11) and (3.2) show immediately that
(L is a length) and this important remark will be used in the following sections.
The Fourier transforms of the connected partition
functions can be defined by
When V becomes infinite in all directions
remains finite
but we define it only for k1 -t-"’+kp==0.
These partition functions are defined here in a
very general way. However, very often, the points
which are fixed on the chains are the end points
and in this case, we use the simplified notation
which corresponds to the situation where p = 2 N and
Thus, for instance :
The terms in the expansion of ’3(Sl, ..., SN) or in
the expansion of
with respect to b are represented by connected dia-
grams.
The contributions of the diagrams follow directly
from the preceding considerations. These diagrams
are constructed in all possible ways and calculated
by applying the following rules :
1) A diagram is made of N polymer lines; each polymer line is labelled by an index j and has a given
area Sj (Figs. 1 and 2).
2) The polymer lines are connected by interaction lines; the extremities of the interaction lines on the
polymer lines are the interaction points (actually the
end points of an interaction line coincide in space but not on the diagram, see figures 1 and 2).
Fig. 2. - A diagram with correlation points where external vectors
are injected (here k1 + k2 + k3 = 0).
3) Sometimes, on the polymer lines, there are also
p correlation points.
4) The interaction points and the correlation points separate the polymer lines into polymer segments.
Each segment has an area. The sum of the areas of the polymer segments contained in a polymer line j equals Si.
5) Each polymer segment and each interaction line
carries a wave vector. At the correlation points,
external wave vectors (kl, ..., kp) are injected (see Fig. 22). Each segment at a free end of a polymer line
carries a wave vector 0. At the interaction points and
at the correlation points, the flow of wave vectors is conserved.
6) With each interaction line is associated a fac- tor - b.
7) With each polymer segment of area s, carrying
a wave vector q, is associated a factor e -sq2/2 8) We take the product of all factors and we perform
an integration of the form 1 (2 1t fddq... with respect
to each independent internal wave vector q (one per internal loop).
9) We perform an integration of the form ds...
with respect to each independent area s arid the
domains of integration are determined by the cons-
traints (see prescription (4)).
10) When performing this summation, we keep the
area of each polymer segment larger than the cut-
off so, in order to avoid short range divergences.
The cut-off is necessary to ensure convergence but inconvenient for two reasons. It is an additional parameter and it makes the calculation of the dia- grams complicated. However, as we shall see in the
next section the contribution of the cut-off can be eliminated by a trivial renormalization and the renormalized quantities can be calculated by modi- fying slightly prescription (10).
4. Elimination of the short range cut-off. - 4. 1 SECOND RENORMALIZATION. - The cut-off can be eliminated by a second renormalization of the parti-
tion functions. This renormalization is of the form
where
When so - 0, 3(S 1, ..., S N ) and 3(...; S1’ ..., SJ remain
well defined (whereas the exponential diverges). This
renormalization has been studied elsewhere [10] and corresponds in field theory to a simple (so-called)
mass subtraction.
Using our new notation, we may repeat the argu- ments as follows. Consider an N-polymer connected diagram of order n (n >, N - 1) contributing to
Before integration with respect to the independent
wave vectors and the independent areas the contri-
bution of a diagram is the product of b" by a dimension-
less finite factor; this factor is a function of the areas
and wave vectors. The number of independent areas
is (2 n). The number of loops (i.e. the number of inde-
pendent wave vectors) is t = n - N + 1. After inte-
gration, with respect to the C wave vectors, we obtain
an homogeneous function b" D { s I of the internal areas,
Integration with respect to the internal independent
areas { s } may produce divergences when so -+ 0. To
study them, we need the concept of P-reducibility.
A diagram is P-reducible if it can be separated into
two disconnected non-trivial pieces by cutting one polymer line ; otherwise it is P-irreducible (see Fig. 3).
Fig. 3. - Connected diagrams : (a) A P-reducible diagram. (b) A P- irreducible diagram.
If a diagram is P-reducible into two pieces 1 and 2,
we may write :
because, on the diagram, the wave vector carried by
the polymer segment joining 1 and 2 depends only
on the external wave vectors; consequently, the integrals on the wave vectors are factorized.
Thus, the short range divergences occurring in pieces 1 and 2 are also factorized. If the diagram is P-irreducible, the number of independent « internal »
areas which appear in D { s I is (2 n - N). Thus
such a diagram is primitively divergent at short
distances if (see (4.3))
which can be written
For d 4, the product. on the left hand side is
positive. Therefore, for N > 1, there are no primiti- vely divergent diagrams.
For N = 1, the condition is
For d 2, all diagrams converge at short distances;
no cut-off and no renormalization are needed. For
