with important consequences for the behaviour of temary solutions. [3-5]. Close to a fixed point we can linearize the flow equations. Again restricting ourselves to.
J.
Phys.
(1991)
I1
211-233
1991,
PtVRIER
PAGE
211
Classification
Physics
~bstracts 61.40K
I1.10G
64.60H
calculations IIigher order multicomponent polymer Schifer,
Lothar
Ulrike
Physik
Fachbereich
(Received19
Lehr
and
accepted
25
renormalization
Christian Essen,
Universitfit
der
July 1990,
of the solutions
group
flow
for
Kappeler
D 4300
Essen,
F-R-G-
October1990)
renormalization flow of the interaction calculate three-loop order the to group polymer solutions containing several chemically different polymer species. Close to the fixed determine analysis using information the points we the flow by a Padk-Borel on asymptotic behaviour of the perturbation expansion. We furthermore derive relations which exact correction of star polymers, the connect to scaling exponents to exponents of binary solutions or and in the cross region we integrate the flow equations numerically to get a global picture. To over clarify the relation to previous work based on « direct renormalization present a detailed we » discussion of the interpenetration ratio for two chains of different chemistry or size. Abstract.
We
for
constants
1.
Introduction.
Solutions
long
of
renormalization
macromolecules
(RG)
group
are
symmetry.
among
The
the RG
most not
extensively studied only explains the
systems universal
[Ii with scaling repulsion
volume» where body limit the two functions approximate calculation of the which crossover an the noninteracting and strongly interacting limits. Using RG equations among based on a two-loop calculation, combined with theoretically well established values of critical quantitatively [2]. in fact explain many experimental data almost exponents, can we Some time ago this successful theory has been extended [3-5] to describe the physics of two chemically different polymer species in a solvent (« ternary solutions »). For that common deal with three coupling different b 2 representing 1, system we constants g~ g~~ a, of the effective interaction of type (a) and (b). The RG equations define a flow in monomers coupling which the phenomena. For ternary solutions this constant space govems crossover flow is especially rich, possessing eight different fixed points. with Furthermore, increasing concentration the system typically separates into two phases which differ in composition, and this phase transition generically is not govemed by a fixed point of the excluded volume problem but takes place in the domain. good of Thus representation the flow RG crossover a in all the accessible region is of great interest. parameter The detailed discussion of the RG flow for multicomponent present to a paper is devoted polymer solutions. Using field theoretical methods calculate the flow equations threeto we behaviour
observed
dominates, interpolate
but it also
in
the
allows
«excluded
for
=
=
JOURNAL
212
PHYSIQUE
DE
M
I
2
establish scaling relation which loop order. certain fixed points we the For a expresses in of well known important correction scaling of the exponents w~ terms to exponents binary subsystems. For the other fixed points we relate the w~~ to exponents of «star» the polymers, and we furthermore estimate resummation of our three-loop w~ by Borel of information the high orders perturbation theory. result, including on Somewhat surprisingly, part of our results differs from previous work based on direct Since in that method the coupling flow renormalization in the polymer language. is constant » appropriately normalized virial coefficients of extracted from analysis of polymer two an chains, we reconsider the nonlinear scaling properties of these so-called interpenetration « ratios » to point out the flaw in the argument of reference [3]. The organization of our paper is as follows. In section 2 we derive the RG equations. We and we discuss the flow of the calculate fixed points and correction exponents, to scaling coupling Section 3 is devoted to the analysis of the second virial coefficients and the constants. calculations of reference [3] we interpenetration ratios. With regard to the through carry a chains careful of very different size, a analysis of the limit where the two interpenetrating are reference [6] using a somewhat different limit which has been discussed before in technique. previous field Section 4 summarizes results. Our Work heavily relies theoretical our on or presented in appendices. polymer work. Some technical details are
2.
Renormalization
2.I
MAPPING
system
oF
POLYMER
containing in
0ws ~2 X
~
and
I
of («~~~)~.
insertion
~)~~~(~
interchain
according
the
to
l~i~'/~ ~
~
"
standard
Z)~~ z (I)
r(4)~~
rj
relations
(2.29i)
~' ~~
z(2)
j)
~m
~~ ~~~~~
l
,
and
using equation (2.17) together with the r((>j~(- qj q~ ; q q~) l i,
condition
~~
=
Zj~~~~ This
can
be
substituted
g
=
i~
Is
immediately
Zj~~Z)~~
=
equation (2,13)
into
Wi~
renormalization We
+
O
(2.8)
and
the
(gi~).
(
(2.30)
gi~)j 2
(gad)
~
=1
standard
yield
to
+ a
condition find
+
(2.31)
O
Where
V
The
fixed
point
~
values
~
of
(gad) correlation
the
0 if 0 ~( ~ ~~~~ g
~
2 =
W~~
length
hgaa
In
z(a) $
(2.32)
Z~~~
exponent
v
are
well
known
[7]
:
(2.33)
=
/~2
~~~~~~j ~~~~~~~~~
~~
~
~~
220
JOURNAL
Differentiating
equation (2.31)
with
w
easily
respect 4
j~
g]
at
gi~
to
e
=
PHYSIQUE
DE
V(~~)
M 2
I
0
We
=
find
(2.34)
(~l)
V
equations (2.34), (2.33) give back the results of table IIIa. (2.34) estimates for w in three dimensions, be used to get reliable exact can i~ (g*) has calculated high precision nontrivial fixed point value v been since the to [9] v 1/2 exact.) results collected (Of the value v(0) is The in 0.588 for d 3. are course v 0.80 for d 3, We thus table IVa. Adding the well known [9] value of w (Eq. (2.27) : w 0. have expressions for the full matrix D at all fixed points with g[ accurate of the dimension in Hausdorff Relation (2.34) has a simple interpretation terms clearly be written Iv (g$) of the polymer coils. It l d@~ can as It is
checked
that
relation
The
=
=
=
=
=
=
=
=
w
identifying
thus
Table
wj~
Hausdorff
as
d#~
j~
=
+
dimension
of
the
of
the
two
coils.
Table
so
Uo, uj -0.70
wj2
-0.40
IVb.
g[
2.4.2
is
polymers,
0.82
0.68
0.68
0.40
fit 62
0.36
0.25
0.27
0.20
0.36
0.27
0.25
0.20
fl
1.83
1.48
1.48
fixed points with nonvanishing governing the partition function constructed by tying together endpoints
are
5~(Sj,...,54) partition function Sj, 6j(S~, 54) being composed out of monomers the following scaling law, being valid prove we
j-
~c
ii~(P) (si
+
s~)
~c
i~~(P) (s~
+
of at
of
of
four
chains
the
exponent
four-arm
star
(see Fig. 3).
the of lengths star, a arms (2), respectively. In appendix B (g(, g], g&) : point P
such
of
the
g[ appropriate
interaction
those
exponents
to
which
Consider
S
w12
For
~0.
related
U'
U
G
exp
intersection
IVa.
Go
wj~
(2.35)
d
dj~~
species fixed
=
s~)j 5~(si,
s ~)
=
,
~)~~
Here
(a),
is
the
microscopic related
to
radius
of
f
parameter
wi~(P)
potential
chemical
and Ry is the
A
of
segment
per
isolated
an
chain
is put in for
of
an
length )
dimensional
infinitely long chain of species corresponding species. The The exponent f (P) can be reasons. of the
:
~~~
I(P)
~ +
=
~
~
~~
+
7~
(gt)
+
7~
(gt)
+
8
8
+
w
j~(P)
(2.37)
M
RENORMALIZATION
2
FLOW
GROUP
221
r~
r=0
r,
r, Fig.
3.
Structure
of
four
a
polymer.
star
arm
[7, 9]
where
1~
(0)
0
ii
=
~(g*)m~
all
q
Rj equal
5*(Sj,...,
S
4)
~~~~
be used
This
relation
can
been
carried
through [10]
the
well
known
eXP
'-
relation
to at
(-
l~
5~(£,
l~~(P)(Si
l~~(P)(53
where
we
Such can
+
54)) simulations
(2.39) have
simplify (2.39) using
£ for all j.
g
y4
l~
simulations.
Carlo
Y4 -1
(-
exp
~
,
52)
+
point S
fixed =
$j
find
we
=
Monte
symmetric
(ilf~)~, §
R/f
3
f/R,
A
f (P) in
measure
the
d =
choosing
and
R
m
for
0.027
m
Taking
j~
=
(2.38)
(1+(8+°(8~))
4 ~
(2.40) ~
vi (S).
=
(2.41)
Thus the deternfined value of y4 0.88 for d 3 with the help of equations (2.41), (2.37) values of v and Y~. This result yields wj~(S) 0.35, where we used the standard compares favourably with the higher order j~(S) 0.22 (or w 0.37) of reference [I I] estimates pm j~ wj~(S) in the section. relation (2.37) is extremely 0.40 established @Qote that next or polymers sensitive to the value of v.) Clearly simulations for appropriate would make for star of wj~(P) fixed useful also for the other points. test a =
=
=
m
m
m
PADt-BOREL
2.5 tion
of
we
the
use
the
partition
At
RESUMMATION.
asymptotic
behaviour
points expansions
fixed
of the
with to
non-vanishing interchain interacimprove our results. The expansion
function
Z(g )
£
=
Zk g~
(2.42)
k
in
the
linfit
of
high
orders
behaves
Z~
k! =
like
[12,13]
(- a)~ k~°
c
I
+
O
k
(2.43)
JOURNAL
222
This
together
behaviour
resummation.
involving
order
In
plausible
some
elucidate
to
coupling
single
a
with
PHYSIQUE
DE
the
analyticity ideas,
basic
M
I
assumptions first
we
allows
for
Padb-Borel
a
standard
the
recall
2
[13, 14]
case
constant.
Consider
z(g) where
j-
exp
j
x
(2.44)
given by equation (2.3) with
is
X
Dim
=
u~
Assuming
analyticity
Z(g)
of
in
the
g-plane
cut
Z(g)
(4
gr
m
j°
=
write
may
we
g'
"
the
down
dispersion
relation
Z(g')
~nl
dg,
(2.45)
)d/2
«
(2.46)
g
w
Expansion
in
yields
of g
powers
°
Z~
dg' g'
=
~
g'
Im Z
(2.47)
" -m
evaluate
We
easily order
Z(g')
by
[14] that
the
Im
shown
of
terms
point approximation for the «-integral (2.44). point «~~ dominating Im Z(g') also dominates expansion and leads to the form (2.43) with
saddle
a
saddle
perturbation
the
a
We
note
fluctuations
analytic
of
rotated
of
for
around
«
from
g'~
behaviour
the
of
g-expansion.
the
thus
has
It
must
show
0
Re
to
first
Wi is the
coefficient
nontrivial
now
discussed
apply and
over
fixed
nontrivial result
as
with
this
two
method
fields.
For
to
fixed
[12] from
derived «
~
R.G.
wi
~
problem our points G or
the
replaced by
a
~~'~~~
of the
the
point
fixed
at
standard
S all
results
function
flow
W
g + o
~
A
hA
g
:
(g2)
~~.~°~
involving three a priori problem essentially
S the
coupling
different
reduces
the
to
case
coupling
point G the coupling above
two
become
constants
identical
and
we
immediately
can
yielding d(S)
At
but
of I. be
above.
Specifically take
(2.43)
can
i
e ~
We
wi~
3~
or
=
w(g,
constants
factor
necessary
form
same
~
where
be
(2.48)
const.
=
yielding the e-expansion for the
can
higher
point which donfinates Im Z : the of matrix single negative eigenvalue. the In a process g'~ 0 the amplitude of this mode has to be
saddle
a
«~~
Re
imaginary axis,
the
asymptotic
behaviour
3
continuation
into
The
condition
necessary
a
Gaussian
g' X [«~~]
=
It the
(.
(2.51)
=
diagonal couplings vanish, leaving calculation gj~. Repeating the
constant
us
again sketched
with
a
above
problem of we
find
a
the
single same
:
~~~~
~'
~~'~~~
M
RENORMALIZATION
2
To
point
fixed
treat
U
0, g( # 0 # gj (or fixed
g]
:
=
g22 and
we
that
at
expansion point # takes
consider the
the
fixed
in
We
O(e)
the this
thus
corrections
with write
J~j
j
Following
fixed.
defend
To
trick
this
note
we
(e)
O
(2.54) leading
to
«~(r)
fields
l
~~~
procedure described point equations
~2 ~
A«~
(~
2
~ ~
~
gj~ r
=
above
the
saddle
keeping
+
the
2
ui~
The
write
we
are
behaviour (- d)~. replaced by complex
have
We
variables.
as
l~d~(
~ ~
equivalently)
(2.53)
influence
not
which
in
Hamiltonian
the
~ ~'
should
model
toy
a
point U',
form
the
=
checked
223
#g12
=
of gi~,
powers
# where
FLOW
GROUP
(4
2
look
we
+
~12
)2
~
ar
for
~ ~
(
2)2)
(~ ~~)
2
(2.56)
)~/~
points
saddle
ml
ui~ «~
#
~2 ~2
4
of the
at,
«~
integrations.
o =
(2.57) ~"~
yield
~
2
~~~~"~~
"~
«j
~
(I)
0
«~~
=
=
l/2
~
0
«~~
«
=
j
~
=
u12
(ii)
~~
~~
c
the
denotes
~c
"
"
I
u12 g
well
explained
(I) (ii)
The
above
[12]
known
solution
With
«j
2
£
are
0
«i~
=
+
«j
=
~r( =
we
=
6«1,
~
3«j
~(-
looking
A
is
«~~ «~
(#
)
l
£
~
(iii)
~c
"
c
12
solution
A~~ As
(2.58)
#
~
"2
3X
~
solutions
the
Here
~ ~"~~
~
=
for
+
a
the
of
~/
(2.59)
0 =
point possessing
saddle
trivially stable 6«~ we find to
(1+ 2 3~j)
equation
instanton
~j)
yields
and
unstable
contribution
to
Im
mode. Z.
(6«)~
order
3«j
no
single
a
+ ~
~
+
2
£
(-
3«~ ~
~/
A
#
«~ ~
l(2.60)
~
where
we
assumed
The
second
potential
that
direction n-dimensional I in the «1~ points into space of the spin a quadratic form in (2.60) is known [12] to have one negative eigenvalue. form has a negative eigenvalue provided 1/# l, since in that case the ~ ~j attractive than the potential which by virtue of equation more =
The
components.
first
quadratic I
/#~j
is
224
JOURNAL
(2.59) possesses a # is smaller than
eigenstate at Eq. (2.54)) and
normalizable I
PHYSIQUE
DE
(compare
M
I
eigenvalue. In the saddle point (it)
zero
thus
of
case
does
2
here,
interest
not
contribute
only
one
to
Im Z.
corresponding analysis
(iii) The eigenvalue
exists.
This
therefore
a
derive
To
according
d
point (iii)
saddle
for is
relevant
the
j~ ~~
~
~
"
for
l
~
negative
yields
g/2 Wj2 (Eq. (2.15))
write
we
#)
+
It
(e2)
o
gi~ +
as
(2.62)
find
to
3 a=
w
=
equations (2.49), (2.50)
to
that
point.
(2.61)
g(2 3C(«1c, tr2c)
=
shows
saddle
Having i~(e) =
~
27
i
(2 63)
=p.
asymptotic behaviour of the coefficients e~ and the corresponding series for 3~ in the )(~
found
£
j(4-#)
w
the
)(~ of the
w
form
formal
series
power
equation (2.43)
of
we
use
k
the
standard
we
GLOBAL
2.6
solution and
are
of all
appendix
notice
we
the
(2.ll)
To
in
see
solution
temary
the
main
two-loop
a
(2.ll).
of
nontrivial
in of
symmetry
the
the
results
in
[15].
reference collected
are
of
flow
whole contains
the
in
details
Some
IVb,
table
couplings
allowed
exchange
under
binary
the
as
plot
of
the
of the flow, it may suffice to it is useful to rescale
all
~22(~ )
~12(~ ),
"
gives
4
where
from
of the
special
a
the
couplings couplings
the
Thus
case.
~ll ~
schematic
a
follows of
range
one
~
features
flow
whole
the
system
Figure
approximation.
global
The
DIAGRAM.
~ll(~ special a diagram.
discussed
The
gj/g*
equations
flow
Furthermore
is
form
C.
=
the
g~~.
in
f]
of
values
in the
method
summarized
CROSS-OVER
of
First
g~. gii
resummation
Borel
procedure also give the
of the
Furthermore
3-dimensional
evaluate
the
equations
flow
couplings
flow
with
factor
a
g*, gab(A ) to
arrive
to
be ~
dA
at
a
non
evaluated
f~~(A =
loop
order
dependent
As
the
(2.64) ,
frame
of the
flow.
The
resulting
equations
differential
are
~~
ef~[I
(1
+
32
flow of fir and f~~ perpendicular to the
is
independent flow
lines
~~
f~~
e
in
+
)
+
planes
g* fob(A
=
of the
32
ef$j
fi~(2fi~ fj~, fi~
the =
+
O(e~)
+
3
fir
(2.65i)
+
3
f~~)j
three-dimensional
0-plane.
This
flow
+
flow is
O
(8~) is
pictured
(2.65ii)
structured in
figure
by 5.
M
RENORMALIZATION
2
GROUP
'
exceptional
the
£q~
situation
expressed
be
can
renormalizable,
functions
~*R
~* "
(Al ) is multiplicatively
in
value
vertex
derivative
as
0,
~i
=
r)(~
of
as
m
~~m
~~,
calculated
~4
the
from
the
h
Renormalizing depends
r(((~
this ui~
on
equation with only via gi~
find
the
logarithmic
~22, ~12)
~
"
(
derivative
~
(2.29ii) for
with
respect
"
A,
«,
A
to
(~ll)
~
and
using
the
fact
+
~
that
Z~
l~
~"12
ill ~* cd,
equation expression
of
an
~
«
=
Taking
(~~~
~~ ~~
Z~~~ Z~~~
Z~
help
the we
r)()
hu12
=
(84)
u11, u22 fixed
we
find
~
(g22)
8
~~~~~
~12(gll,
g22, g12)
12
~~~~
where
Y~(g~) point
fixed
values
given in ~
P =
We
«
=
r~
where
can
model
temary
with
be
~~.
mass
that
reading
unrenormalized
~*(~ll,
fl ~j=i
"
0.) The Hamiltonian 3C' is of the r species (« ~~~, «~~~ species 1, «~~~, «
at
star
chemical
,
qj is
4)
S ,
~*(~l, Here
4 «)~(0) fl
4
~) 5~(Sj,
e~ ~J
for
checked
relation
2,
or
=
=
its
I
j=1
fields
has
easily
It is
of
j=i
fix the 0
n
arms
transform
4
o
of species Sj, 6~(S~, 54) built out of monomers 5~(Sj,..,54) partition function the star 0 spin fields : involving four n
with
star,
Laplace expectation
lm fl (d)
Each
231
polymers.
star
respectively. mapped on
with
FLOW
B.
Relafion We
GROUP
note
framework
components
(g(, g], g])
~(P)
equation =
~
(gt)
£
A
=
(2.38). At the +
~
(gt)
(86)
In Z~~~
e
fixed
w
points
i~(P)
equation (85) yields (87)
stands for fixed points G, U, U' or S. symmetric fixed point S we can calculate within the Y~~(S) also of the normal O(n) model, just interpreting the «f~, j =1,.., 4 as different °~ equal. The of a single spin field and taking all composite operator masses ~
that
at
the
JOURNAL
232
fl «)~(r) J
and
then
is in
comparison
a
the with
class
of
our
formulation
a4
allows
for
Y~
~
an
renormalized
5*(Sj,..
(2
=
from
of
in
Monte
~
reference
[8]
three-loop
to
2
(I),
order
result
equation (5) wj~(S).
for
Carlo
(Bl)
simulations
four
of
the
derive
to
(88)
~x~
[8],
reference
our
transformation
in
M
I
yields
~(s)
check
determined
be
Laplace
the
taken
be
can
independent
also
can
invert
O(e~)
to
discussed
operators
~
where
PHYSIQUE
DE
with
=
of
4, To
stars.
arm
relation
k
n
=
This
0.
show
this
we
to
its
5~(Sj,..,54)
counterpart
54)
eXp
"
(~~~~(Sj
52)
+
~~~~(53
+
54))
+
X
,
(Zj~ Z)~~)~
x
5~R (S((~, S~§~, S(§~ S((~)
Zi
(89)
z(a)
$
~
S(~~
«
=
)
(B10)
Z~~~
~
of potential infinitely long chain of species an replaced by where is of the size of a AR~li, [5] a measure S((~ can R~ length polymer coil of contour ) of the appropriate species. It, for instance, could be with the radius of gyration. I is some identified microscopic length parameter. Close to a fixed point equations (85), (86), (2.32) yield
~)~~ is
Here
(a).
The
the
chemical
segment
variables
be
z(a) z~
(2.36), (2.37) result,
Equations
Appendix
Given
the
power
asymptotic
behaviour
(x)
being
of A~
where
b is
it
l
used and
adjustable as inverting the
(x) =
This
has
been
pointed
out
z
in
the
main
text.
of the
A~ x~
form
parameter to be discussed transformation find we A
(1)
discussion
our
to
us
by
(ci)
(2.43)
we
consider
the
Borel
transform
"~(~r(k+b+1)~
~b~~~
~
of
series
=
la,
basis
[13].
formal
a
the
(g~)
(Bl I)
form
A
with
~
+
~ ~ ~.(gii,~ g22, g12)
~
which
2
C.
resummafion
Borel
ljv(gz~)
~
~
)~ B.
dt t~ e~
Duplantier.
~~~~
below.
B~ (xi)
The
series
(C2)
converges
for
(C3) ,
RENORMALIZATION
M 2
Assuming
singularities
all
that
A(x)
of
('
~ ~
map the transform.
domain
integration
of
(I
the U~
possible
be
can
calculated
from
singularity
type
power
introducing
minimizing the series of B~,~ in restricted
to
exponents
it
the
powers
of at
turns
U-expansion
original
negative
real
axis
the
use
we
l
+
+ l
into
~c~~
region
the
of
convergence
of the
Borel
z Uk(U(xt))~
order. out
is
that
our
restricted
Following the
in
(xi)
(i
=
~
The
a.
[9]
furthermore
we
extract
the the
few
x
e-expansion
m
final
result
in
fit
first other fact
a
(c6)
for
e
I is =
close
to
a
determined by by restricting the the results fairly are the U-expansion are e-expansion of to
b parameters a, constructed
It
terms.
applications is
B~(xi)
approximations
successive
with
and
reference
form
u(xi))a
parameter
U to hand
As
I
(C5)
=
between
Having
third
fit
second
the
difference
b.
to
axt)~/~ axt)~/~
the A~. U
at
B~,
insensitive
+
equation (C3)
in
"
thus
the
on
233
yields
This
Bb(Xt) where
located
are
FLOW
mapping
conformal
to
GROUP
turns
of
that
out
O(e~)
to
that
we
in
method
are
fairly insensitive to the order to which simple (2, 2)-Padb approximant to the
series.
References
[1] DES
CLOISEAUX
J.,
JANNINK
G., Les Polymdres
en
Solution
(l~es
Editions
de
Physique,
France)
1988.
L., Macromolecules SCHAFER 17 (1984) 1357. [2] See, for instance, [3] JOANNY J.-F., LEIBLER L., BALL R., J. Chem. Phys. 81 (1984) 4640. Ch., J. Phys. France 46 (1985) 1853. [4] SCHAFER L., KAPPELER Ch., Coil. Polymer Sci., in press. KAPPELER [5] SCHXPER L., [6] STEPANOW S., ~cta Po/ym. 32 (1981) 98. Critical Transitions and Phenomena, Eds. C. [7] BREzIN E., LE GUILLOU J. C., ZINN-JUSTIN J., Phase (Academic Press, New York) Vol. 6 (1976) 125. Domb and M. S. Green [8] WALLACE D. J., ZIA R. K. P., J. Phys. C.' Solid State Phys. 8 (1975) 839. Phys. Rev. B 21 (1980) 3976. [9] LE GUILLOU J. C., ZINN-JUSTIN J., J. Phys. France 50 (1989) 1365 [10] BATOULIS J., KREMER K., Camp. Phys. R 7 (1988) 259. [I I] BROSETA D., LEIBLER L., JOANNY J.-F., Macromolecules 20 (1987) 1935.
[12] [13] [14] [15] [16] [17j
[18]
E., LE GUILLOU J. C., ZINN-JUSTIN J., Phys. Rev. D IS (1977) 1544. J., Phys. Lett. C 70 (1981) 109. Mechanics, D. J., Applications of Statistical WALLACE Some Instantons in Linear Structure Dynamics in Condensed Oxford (1978). Matter, on LE GUILLOU J. C., ZINN-JUSTIN J., J. Phys. Lett. France 46 (1985) L137. Math. WITTEN T. A., SCHXPER L., J. Phys. A Gen. 11 (1978) 1843. Yeshiva meeting on GENNES P. G., Talk at the statistical mechanics, Fall DE also WITTEN T. A., PRENTIS J., J. Chem. Phys. 77 (1982) 4247. see Ch., SCHXPER L., FUKUDA T., Macromolecules, in press. KAPPELER BREzIN
ZINN-JUSTIN
Proc.
1980
Symp.
Non
