A description of the physical properties of polymer
solutions in terms of irreducible diagrams. (I)
J. Des Cloizeaux
To cite this version:
J. Des Cloizeaux.
A description of the physical properties of polymer solutions in
terms of irreducible diagrams.
(I). Journal de Physique, 1980, 41 (8), pp.749-760.
<10.1051/jphys:01980004108074900>. <jpa-00209300>
HAL Id: jpa-00209300
https://hal.archives-ouvertes.fr/jpa-00209300
Submitted on 1 Jan 1980
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Tome 41
AOÛT
No 8
LE JOURNAL DE
J.
Physique 4r (1980)
PHYSIQUE
AOÛT 1980,
749-760
749
Classification
Physics Abstracts
05.40
-
61.40K - 82.70
A description of the physical properties of
in terms of irreducible diagrams (I)
polymer solutions
J. des Cloizeaux
Service de
Physique Théorique, CEN-Saclay,
B.P. n°
2, 91190 Gif-sur-Yvette,
France
(Reçu le 10 janvier 1980, accepté le 17 avril 1980)
Les solutions de polymères peuvent être commodément étudiées dans le formalisme grand canonique;
Résumé.
dans ce cas, la pression osmotique, les fonctions de corrélations et les concentrations en polymères de différentes
longueurs peuvent être développées en fonction de potentiels chimiques et des interactions entre polymères.
On montre que ces développements peuvent être simplifiés en exprimant les quantités précédentes au moyen
de diagrammes qui sont irréductibles par rapport aux lignes d’interaction. Le procédé s’applique aux polymères
monodisperses et polydisperses. L’élimination des divergences à courte distance conduit à des lois d’échelle
2014
simples.
Abstract.
Polymer solutions can be conveniently studied in a grand canonical formalism; in this case, the
osmotic pressure, the correlations functions and the concentrations of polymers with different lengths, can be
expanded in terms of chemical potentials and polymer interactions. It is shown that these expansions can be
simplified by expressing the preceding quantities by means of diagrams which are irreducible with respect to
the interaction lines. The process applies to monodisperse and to polydisperse polymers. The elimination of
short range divergences leads to simple scaling laws.
2014
It is well known that the osmotic
1. Introduction.
in a solution can be
molecules
of
simple
pressure
easily expanded in terms of the molecular concentration [1]. The coefficients of this virial expansion
are given by connected « point-irreducible » UrsellMayer diagrams (see Fig. 1). This virial expansion
can be obtained by remarking that the usual Ursell-
Mayer diagrams obtained in the grand canonical
formalism are trees of « point-irreducible » diagrams ;
the irreducible parts are connected by articulation
points which represent molecules. Thus, the contribution of a « point-reducible » diagram is the product
of the contribution of its « point-irreducible » parts.
This approach does not apply to solutions of
monodisperse or polydisperse polymers (see Fig. 2) ;
on the corresponding diagrams, the concept of
« point-reducibility » does not exist.
However, in the case of polymer solutions, it is
possible to introduce the concept of « line-irreducibility ». A diagram is an « 1-irreducible » diagram,
if it cannot be separated into two pieces by cutting
Simple molecules in solution : a) o Point-irreducible »
Fig. 1.
diagrams ; a point represents a molecule. b) « Point-reducible »
diagram. Any reducible diagram can be considered as a tree of
irreducible diagrams. Here, the reducible diagram is made of five
irreducible parts connected by the articulation points A, B, C, D.
-
Fig. 2. Polymer solutions : the diagrams are rather
the concept of point irreducibility is meaningless.
-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004108074900
complex and
750
interaction line. A diagram is a « P-irreducible »
diagram, if it cannot be separated into two (non
trivial) pieces by cutting a polymer line (see Fig. 3).
an
solvent, repel each other with
strength which is given by the excluded volume u
monomers, in the
a
Let us call ZG(Ni,
N N) the partition function
of this set and (ZG(N,,
NN))O the same quantity
in the absence of interaction.
For investigating the asymptotic behaviour of the
solution when the numbers Nj of links are large, it
is convenient to consider the polymer chains as
continuous curves in a space of dimension d.
Thus, in the absence of interactions, the chains
are considered as brownian. In this case, the size of
a chain of N links can be characterized by the mean
square distance RN between the extremities of the
chain
...,
...,
where 1 is an elementary length.
In the continuous limit, ZG(N1,
NN) becomes
infinite. However, this partition function can be
normalized and the quantity
...,
a) This diagram is I-irreducible and P-irreducible.
Fig. 3.
b) This diagram is 1-reducible and P-irreducible. c) This diagram
is 1-irreducible and P-reducible. d) This diagram is 1-reducible and
-
P-reducible.
remark that an « 1-reducible » diagram
can be factorized and expressed in terms if its « I-irreducible » parts. On the contrary « P-reducible »
diagrams cannot be directly factorized. Factorization
occurs only in the framework of an approximation,
which has been introduced several years ago by the
author [2] to determine the scaling properties of a
polymer solution.
For this reason, we shall start by expressing the
osmotic pressure, the correlations and the concentrations of polymers of different lengths, in terms of
« 1-irreducible » diagrams and the concept of « Pirreducibility » will be used only for analysing diver-
Now,
we
gences.
In section
2, we recall the grand canonical formalism
for polymer solutions. In sections 3 and 4, we show
that the « 1-reducible diagrams » have tree configurations and that such trees can be summed up by
means of Legendre transformations. Section 3 deals
with the osmotic pressure and section 4 with the
correlation functions. In section 5, we discuss the
elimination of short range divergences. Finally, in
section 5, simplified expressions are given for the
osmotic pressure, the polymer concentrations and the
correlation functions.
2. Osmotic pressure, polymer concentrations and
corrélations in the grand canonical formalism.
Let
, us consider, in a volume V of solution, a set of N
polymers which are chains of monomers. The numbers
of these monomers are respectively Nl,
N,,. The
-
...,
is more regular. We note that the dimensionless
factor (Vl-’)’ has been introduced to compensate
for the fact that the origins of the free chains have
arbitrary positions.
For simplicity, it will be assumed that all chains
are dissymetric which means that the origin of each
chain is marked ; however this restriction is unim-
portant.
all
numbers Nl, ..., N,, are different,
may be considered as the partition
function of the set of chains ; when all of them equal N,
the partition function is 3G(Nl,
NN)/N !.
Let us now consider a grand ensemble of polymers.
The corresponding partition function is
When
3G(Nl,
...,
NN)
...,
where
{ f } represents
the set of
The osmotic pressure lI is
fugacities
given by [1]
and the mean number CN of polymers of link number
N, per unit volume is
751
The functions 3G(N1,
NN) can be expanded in
powers of « u » and the terms of this expansion can
be represented by diagrams. As usual, the connected
diagrams play a crucial role.
The partition function can be written in the form
...,
Q{ f } is the generating function of
nected diagrams
where
the
con-
2) Two points Ml and M2 are given in the solution
and the polymer configurations are such that 1VI1
is always on a polymer with Ni links and M 2 on a
polymer with N2 links.
The restricted partition functions can be
lized as the ordinary ones and we have
The
and
Thus, the dependence of the osmotic pressure
on
1) The vector joining the origin and the extremity
of one polymer in the solution has a given value r.
the concentrations is
given by
the
corresponding grand partition
norma-
function is
H
parametric
representations
By definition,
Note also that the total number C of polymers per
unit volume and the total number c of monomers,
per unit volume, are related to CN by
we
have
These quantities can be expanded in diagrams and
be expressed in terms of connected diagrams. A
connected partition function with restrictions 5l
will be denoted by the symbol 3(Nl,
Nf,; 3t) and
we shall also introduce the connected grand partition
function (2(j f } ; 9t).
On the other hand, the general correlation functions
CG(3t) are not the most fundamental quantities.
In order to define in a precise way, correlations in a
solution, one has to determine their cumulants C(9t)
can
...,
These equations apply also when the polymer is
monodisperse. In this case, all the chains have the
same number N of links ; there is only one chemical
potential f ~ fN and Q{ f } can be written (2(j).
With these restrictions, all equations remain valid
and equations (2.10), (2.11) read
Thus, the correlation functions C(fi) can be considered as implicit functions of the
CN.
The rules for calculating the diagrammatic expansion of 3(N,,..., NN) are the following (see Fig. 2).
concentrations
Correlations can be treated in a similar way. A
correlation function CG(3t) is the probability distribution associated with the set ail the configurations
which satisfy specific conditions or restrictions fll.
We may call ZG(N1,
N, ; 3t) the partition function
in a volume V of the set of N polymers made respecNN links and restricted by 3t. By
tively of Nl,
3t must apply to polymers
the
conditions
definition,
contained in the set ; otherwise the partition function
must vanish.
For instance, for 31, we may choose one of the following conditions :
...,
1) A
and of
diagram is made of N (solid) polymer lines
(dashed) interaction lines.
2) Each polymer line j carries a number Nj which
defines its link number.
3) A
line is
...,
where an interaction
interaction point.
point
an
joins
a
polymer
4) A polymer line j passing througli pj interaction
points is made of (pj + 1) segments containing
nj,o,
...,
nj,pj links. Thus,
we
have
752
5) Each line has a well defined origin.
6) Each solid segment and each interaction line
carry a wave vector. The free ends of each polymer
vector zero. At each interaction point,
wave vectors of the neighbouring
of
the
the sum
the interaction line vanish.
of
and
segments
7) With each solid segment carrying a wave vector
k’ is associated a factor exp(- nk,2 12/2).
carry the
wave
8) With each interaction line carrying
vector k" is associated the constant factor -
9) With the whole diagram is associated
a
...,
wave
a
factor
(2 nl-1 )d(N - 1).
10) The contribution of the diagram is obtained
by integrating over all the independent wave vectors k
and by summing over all the independent link numn.
NN) is calculated by summing
diagrams.
the
The
of
diagrams relies on the
12)
counting
each
that
polymer line has a marked end
assumption
and
that each polymer line can be
point (an origin)
from
any other one (even if both lines
distinguished
11) Finally, 3(N1,
...,
u(2 n) -’.
.
bers
therefore the contribution of an I-reducible diagram
be factorized.
The sum of the contributions of the reducible and
irreducible diagrams D with N polymer lines gives
NN)’ In the same way the sum of the contri3(N1,
butions of the diagrams which are I-irreducible and
contain N polymer lines gives 3,(Nl,
NN)
can
...,
the contributions of all the
Thus, the factorization property gives the possibility of expressing the partition functions
3(Nl, N,,) in terms of the simpler quantities
3 l( Nb, N,).
More precisely, let us consider a diagram D containing N polymer lines and let p be the number of its
irreducible parts. The corresponding tree diagram T
has p vertices connected by ( p - 1) interaction lines.
Each vertex can be labelled by an index j
1,..., p ;
mj represents the number of legs of the vertex j and Nj
the number of polymer lines contained in the subdiagram j. Thus we have
=
have the same number of links : it is more convenient
here to use labelled diagrams, without introducing
symmetry factors).
The rules for calculating 3(Nl,
N N ; 3I) are very
similar. To find them, we have only to take the restrictions 3t into account.
...,
3. Réduction of tree diagrams, expansion of the
osmotic pressure in terms of I-irreducible diagrams.
As was explained in the introduction, a diagram which
is I-irreducible cannot be separated into two pieces
by cutting an interaction line.
Consequently, an I-reducible diagram is always
made of several I-irreducible parts which are connected by interaction lines so as to form a tree structure. The tree diagram which is associated with an
I-reducible diagram is obtained by reducing each Iirreducible subdiagram to a point (see Fig. 4). The
interaction lines which connect the I-irreducible
subdiagrams carry always a wave vector zero and
-
With each polymer is associated a number N of links.
The link numbers of the polymer lines belonging to
the subdiagram j can be denoted by Nj, 1,
Nj, Nj
and the union of all sets of link numbers constitute the
set (Ni,
Nrq) corresponding to the polymer lines
of the initial diagram.
The contribution of the diagram will be denoted by
D(N1,
NN) and the contribution of the subdiagram
j will be denoted by Ij(Nj, 1,
Nj,Nj).
The interaction lines which connect the subdiagram
j to the other subdiagrams have mj end points on the
polymer lines of this subdiagram ; these points are
placed anywhere on the polymer lines belonging to the
diagram j. Thus, by applying the rules given in section 2
and by counting the number of ways the m legs can be
attached, we find that
...,
...,
...,
...,
According to equation (2.8),
Q{f} can be written
the
grand partition
function
quantity as a sum on the
by using equation (3.3).
This operation has to be done by summing the
and
Reducible
Fig.
4.
tree
diagrams (a’)
-
diagrams (a)
(b’).
and
and
(b) and their corresponding
trees
we
want to express this
753
contribution of each
and only once.
introduced and the
remarks indicate the origin of these factors.
diagram,
Thus, symmetry factors have
following
1) By changing
may create
we
labelling of the polymer lines
diagrams (look at Fig. 5) and we
the
new
immediately
see
once
to be
that the factor which is introduced
However, any permutation of the vertices, followed
by a permutation of the content, gives a diagram which
is identical with the initial diagram. Thus, a factor
l/5’(r) has to be introduced to avoid overcounting.
The content of a vertex of a tree diagram is a vertex
function vm{ f 1 which is defined as follows
in this way is
Using this definition
we
find that
a{ f }
vertex functions
5.
Fig.
By changing
diagrams.
-
different
the
labelling
of the
polymer lines,
we
A
diagram can
and
S(T).
be constructed
content to the vertices of the
by attaching a given
corresponding tree T.
are
in terms of the
the tree diagrams T
expressed
as a sum over
m(T, j) is the number of legs of the vertex j
belonging to the tree T.
We also note that the generating function of the
vertex functions has a very simple form
where
and
obeys the relation
which shows that the vertex functions
one another by the useful equality
be
setting
create
2) The vertices of the tree diagrams are not supposed
to be a priori labelled and therefore each tree diagram
T is invariant by transformations which belong to the
group G(T) of automorphisms of the tree. The number
of elements of G(T) is the symmetry number
can
and
we
have also
related to
Now, let us calculate (21 f }. Equation (3.8) shows
that this function can be written in the form
Moreover,
new
in
a
H(y, { f }) can be expressed
in terms of
variables
simple
manner
where A(x) is an explicit function of G and of the vertex
functions vm{ I} with m &#x3E; 1.
A(x) is a generating function associated with trees
in which the 1-leg vertex function has been replaced
by a variable x. The structure of A(x) is rather complex
but the Legendre transform B(y) of A(x) is very sim-
754
ple. Thus, with the help of such a transformation, it is
possible to sum up the tree diagrams, and to express
directly (2{ f } in terms of the vm{ 1 }.
The Legendre transformation is determined by the
coupled equations
We may also
Using
now
write, in agreement with equation
equations (3.13) and (3.14),
we
find
It is well known and easy to show that B(y) is
related to the generating function of the vertex functions (see appendix). More explicitly we have
With the help of equation (3.12), we can eliminate
in this way, we find more convenient equations
Now, let
us
put x
=
y;
vl { f }. Equation (3.16) give
Incidentally,
we
note also the
following relation
which will be used in section 4.
We have now to write the concentrations as functions of the new parameters 9N. For this purpose,
the second result in
obtain the system
Bringing
we
equation (3.15),
let
us
the operator
p
p
express
deduce
we
help of these identities, and by combining
equations (2. 10) and (3.22), we obtain
and
from the
identity
successively from equation (3.23)
using this equality,
where
vo{ g } and vl { g }
Incidentally, we note
The number of
given by
a
monomers
new
variables.
Starting
With the
a
ÔfN in terms of the
we
may write
are
given by equation (3.6).
that
per unit volume is also
simple expression (see Eq. (2.9))
These results apply immediately to the monodisperse
situation in which each polymer has exactly N links.
The partition function, in this case, can be represented
,
755
by the symbol 5(N ; N) and equations (3.6) and (3 .11)
read
diagrams (see Fig. 7).
approximation
tree
which
On the other hand
equations (3.28) and (3.29) give
This amounts to the trivial
gives
This formula reminds a result which S. F. Edwards
[3] derived in 1966 by using mean field methods.
However, in our notation, Edwards’ formula (corrected by M. A. Moore [4]) is
and
we see
that it contains
an
additional term. This
term can be found
as was
shown
by calculating one loop diagrams
[4].
by
M. A. Moore
4. Expansions of corrélation functions in tenns of
In section 1, we have seen
I-irreducible diagrams.
that the correlation functions were given by partition
functions with restrictions é1( {/} ; fll). The expansions
of these quantities have also a tree structure and the
concept of I-irreducibility can be applied to restricted
diagrams. However the I-irreducible diagrams have
to be precisely defined.
In general, the contribution of a diagram is calculated in momentum space and each interaction line
carries a wave vector. In the following, a diagram will
be considered as reducible, if and only if it can be
separated into two disconnected pieces by cutting
an interaction line carrying a wave vector zero. We
note that other definitions of I-reducibility are possible
and (eventually) useful but, they would lead to more
complicated discussions.
Now, we may introduce irreducible restricted
NN, iq) and we shall
partition functions 3¡(Nb
define also restricted vertex functions vm({ f } ; 9t)
-
The permutation, on a subdiagram, of the end points
Fig. 6.
A, B, C of the lines connecting this subdiagram to the other sub-
diagrams, leads
to
different
diagrams with
the
same
contribution.
Thus, the expression of the osmotic pressure
can be
of
the Ifrom
the
derived
knowledge
immediately
irreducible grand partition function QI(g).
The simplest approximation consists in assuming
that the diagrams which contribute to II are simple
...,
We have shown in the preceding section
that Q{ f } could be written in the form
Fig.
7.
-
Simple
tree
diagrams.
(Eq. (3.8))
where T({ v }, G) is a function of G and of the vertex
functions which correspond to the vertices of the tree
T.
To construct Q{ f } ; 3t) we have only to change
in all possible ways the content of one vertex of T.
Thus, if this vertex has m legs, we change in T({ v }, G)
756
vm{ f } which corresponds to this vertex,
vm({ f } ; 3t).
Analytically, this transformation can expressed
by writing
the function
to the function
To establish these formulae, we have to apply the
rules, given in section 2, which tell how the contributions of the diagrams must be calculated.
Thus, if we add to a connected diagram containing
N polymer lines, a new polymer line and an interaction
line connecting this polymer line to the other ones,
we construct a new diagram containing N + 1 polymer
A(x) is defined by (Eq. (3.15)).
Using (Eq. (3.24)), we may also write
where
where the fN and gN are related to one another by
equation (3.24).
In the preceding equation, let us now replace
vm({ f } ; 3t) by its explicit expansion given by equation (4. 1). By taking equation (3.23) into account,
we see immediately that
Thus
terms
C(3t)
can
of the
be
new
in a very
variables 9N-
expressed
simple
way in
5. Elimination of the short range divergences.
A long polymer can be considered on a small scale
as a continuous and homogeneous line. In this case,
the rules for calculating the diagrams remain unchanged but the sums over the numbers of links are replaced
-
by integrals. However, for small n the integral
dn...
does not always converge and it is necessary to introduce a cut-off no.
Thus we may write formally
The physical quantities of interest may or may not
depend explicitly on the cut-off. Consequently, it is
useful to separate the cut-off dependent contributions
from the other ones; this can be done by using a
simple renormalization technique, which is the direct
application to polymer diagrams of a standard technique of field theory.
a solution of
and
for
N &#x3E; 1, the partition
monodisperse polymers
function 3(N ; N) can be written in the form
In
fact, it will be shown that for
where c(no) is a constant which is cut-off dependent
and *3(N ; N) a regularized quantity which is of the
form
.
lines and a dimensionless factor ul -d N2 is introduced.
On the other hand, when a new interaction line is
added on a connected diagram, a factor zN is introduced (if the integrals over the numbers of links
converge !). The origin of the factors which appear
in zN is the following. The term u(21)-d/2 comes from
the interaction itself. The term N2 is related to the
fact that the number of polymer segments increases by
two when an interaction line is added and that any
number n can be considered as proportional to N.
Finally, the term (NI 2) -d/2 is related to the creation
of a new loop in the diagram. More precisely, a new
internai wave vector k appears and a new integration
over this vector has to be performed. The product of
these three factors gives zN.
The preceding remarks are valid only if the integrals
with respect to the numbers of links converge. Thus,
we must study the convergence of these integrals.
For this purpose, we shall use the concept of Preducibility defined in section 1. We shall consider
only diagrams containing one polymer line and
contributing to 3(l ; N). This restriction is justified
by the fact that the extension of the discussion to more
complex diagrams is trivial.
The origin of the divergences can be understood by
studying P-irreducible diagrams without P-irreducible
insertions (see Fig. 8). On the polymer line of a diagram
8.
a) A P-irreducible diagram containing a P-irreducible
insertion (a self energy diagram). b) A P-irreducible diagram
without P-irreducible insertions.
Fig.
-
of order q, belonging to this class, let ni be the abscissa
of the first interaction point and n2 the abscissa of the
last interaction point.
Let us keep n, and n2 fixed. All the integrals over
the other variables converge, as can be easily seen.
757
Dq of the diagram is given by an
insertion in the diagram. Thus, the subtracted contribution (renormalized) contribution) may ben written
in a symbolic way
The remarks made above show that, in the absence
of any divergence, the dependence of Dq on the
variables should be given by
which means that we omit all the divergent terms.
In the same way, in a general diagram, all the
divergent parts of the subdiagrams can be subtracted
step by step. Thus, with each P-irreducible diagram,
we may associate a contribution which is calculated by
subtracting the divergent parts from the contributions
of the P-irreducible divergent subdiagrams. The
contribution D of the diagram contains a term proportional to N which results from the divergence of
the diagram itself and a normal term ’D which can be
called renormalized contribution of the diagram.
On the other hand, the symbol c(no) will be used for
representing the sum of all the terms proportional
to N, which come from the contributions of the Pirreducible diagrams after regularization of the subdiagrams. Thus, all the divergent parts can be replaced
by point insertions (see Fig. 9). The total weight of
an insertion is c(no).
Thus the contribution
integral of the form
,
Consequently, homogeneity arguments show that
integral (5.5) is always of the following form
the
where A and B are constants.
Let us study more precisely the case where 4 &#x3E; d&#x3E; 2.
1 and perhaps for a few larger
We see that for q
values of q, the integral diverges when (nI - n2)-+0.
In this case, a cut-off is necessary and we may impose
the conditionn 1 - n21 &#x3E; no where no is a constant
which is not very different from one (N &#x3E; 1).
In this way, we obtain
=
Dropping the
no - 0 we may
terms which vanish in the limit
also write
9.
Separation of the normal (renormalized) contribution
and of the additional (divergent) contribution of a diagram.
Fig.
-
Now let
us
consider
a
simple diagram (without
insertions) of regularized contribution D. By making s
point insertions on the diagram, we obtain the contribution
or more
explicitly
The total contribution of the
insertions is
More
The second term is regular ; it does not depend on
the cut-off and its dependence in N is in conformity
with the scaling laws. On the contrary, the first term is
cut-off dépendent ; it is proportional to N and anomalous.
Now, we may separate the two terms, by subtracting
the first term from Dq and by considering it as a joint
generally,
we
have
diagram with all possible
by
similar arguments
where ’3(N; N) is a renormalized partition function
which is completely regular. The scaling arguments
which have been given above are applicable to
’3(N; N) and we may write
758
This formula
where N is
instance,
can
an
can
be
easily generalized
average number of links
be defined
and
using
which, for
similar arguments,
we
may also write
where
by setting
We
Similar relations are also valid for the restricted
N N ; 3t).
partition functions 3(Nl,
...,
6. Simplified expressions for the osmotic pressure
In the expressions
and for corrélation functions.
which give the osmotic pressure we may now eliminate all the cut-off dependent terms. Taking equation (5.2) into account, we replace equation (3.6) by
-
that simplifications occur, the length 1 has
from the expressions and w(t) which is
the unknown function depends only on the dimensionless parameter zN.
In particular these expressions show that II has
the form
see
disappeared
where zN is given by equation (5.4).
For instance, we see that the expression given by
3 (see
S. F. Edwards and M. A. Moore for d
is
of
this
kind.
Eq. (3.34)) precisely
On the other hand, the chemical potential y is
given by the equation
=
where
In the
are
same
way,
equations (3.27)
and
(3.28)
replaced by
func-
expressions which give the correlation
be regularized in the same way and
(4.5) can be transformed into
The
tions,
tion
can
equa-
Again, it possible to use such expressions to derive
scaling laws. For instance, it would not be difficult to
show that the mean square distance
chain in the solution is given by an
form
where
These equations which give the osmotic pressure
be written in a simple form when the polymer is
monodisperse. In this case, vm{ h } reads vm(h).
We may set :
can
and
we see
RN of an isolated
expression
of the
might also express the results in terms of the
surface SN Nl2 which defines the size of a brownian
We
=
chain in the absence of interaction and b
which defines the strength of the interaction.
In fact, we have (see Eq. (5.4))
=
ul - 4
immediately from equations (5.13) and
(6 .1 ) that
Thus
equations (5.2) give
Incidentally, we note that this coefficient b is just
the parameter which defines the interaction in Lagrangian theories [2].
All the preceding expressions describe situations
in which N is large but zN might be small. Thus, they
apply in the cross over domain between the brownian
and the excluded regime.
759
In the excluded
regime, zN &#x3E;
1 and
simplifications
occur
In this limit, there remains only one significant
in the system namely eSN and therefore, as
was shown previously [2], the osmotic pressure takes
the form
surface
where 0(n) is
which has the
(d dependent) universal function
following properties [2]
a
Fig.
10.
-
Tree
diagrams and their symmetry numbers.
Each tree diagram T will be labelled by the number
of its extemal legs (m &#x3E; 2) and by an additional
index a. Thus, Sm,a will be the symmetry number
of the diagram (m, a).
With each diagram (m, a), we associate a contribution Am,a which is a product of factors, determined
as follows. With each internai or extemal interaction
line, we associate the factor G ; with each n-leg vertex
(n &#x3E; 1), we associate the factor vn.
The function A(x) is defined as the generating
function
m
’
Thus in the semi-dilute
regime,
we
have
In this article, we have shown that
7. Conclusion.
can
solutions
be studied by diagrammatic
polymer
methods and that, in this case, it is sufficient to calculate the I-irreducible diagrams to determine all the
properties. Moreover, we have shown that the short
range divergences could be directly eliminated.
These remarks are very simple but basic because
they are general and apply both to monodisperse and
-
In the same way, A’(x) can be defined as the genepolydisperse systems.
rating function of rooted tree diagrams. A rooted
Thus, the principles given here can be used as a diagram is obtained from a simple tree diagram by
starting point for more precise studies and, in forth- marking the extremity of one external leg (see Fig.11 ).
coming articles, applications will be given.
APPENDIX
Legendre transformation and tree diagrams.
The Legendre transforme B( y) of a function A(x) is by
definition related to A(x) by the coupled equations
-
11.
Rooted tree diagrams and their symmetry numbers.
Each root is characterized by a small circle.
Fig.
The function A(x) is arbitrary but we want to show
that, if A(x) is the generating function of tree diagrams,
the function B( y) is the generating function of the
vertices of this diagrams.
The tree diagrams which are considered here are
connected diagrams which are made out of n-leg
vertices (n &#x3E; 1) and out of interaction lines. Two
vertices in a diagram are connected by zero or one
interaction line and this property defines the tree
structure (see Fig. 10). The diagram made of only
one interaction line is considered as a tree diagram.
-
Each rooted tree diagram T’ will be labelled by the
number m of its (marked or unmarked) external lines
and has a symmetry number Sm,«. The contributions
A’,p of these rooted tree diagrams are calculated
exactly as the contributions Am,rJ.’
Thus, we see immediately that
760
We see now that the series y
A’(x) can be constructed easily by iteration and the iteration process
can be described by the equation
=
0 and therefore (A. 3)
But according to (A. 2), A (0)
0. Thus, the preceding equation
shows that 0(0)
can be integrated and gives
=
=
From this
result
equation,
we
deduce
immediately
the
which is the announced result.
References
[1] HILL, T. L., Statistical Mechanics (Mc Graw Hill) 1956.
[2] DES CLOIZEAUX, J., J. Physique 36 (1975) 281.
[3] EDWARDS, S. F., Proc. Phys. Soc. 68 (1966) 265.
[4] MOORE, M. A., J. Physique 38 (1977) 265 ; see also DES CLOIZEAUX, J., J. Physique 41 (1980) 761.