the self- avoiding-walk. Loic. Turban and. Bertrand. Berche. Laboratoire de Physique du. Solide. (*),. Universit£ de. Nancy I, BP 239,. F-54506. Vand~euvre-l~s-.
Phys.
J.
I
(1993)
3
France
925-934
1993,
APRIL
PAGE
925
Classification
Physics
Abstracts
36.20C
64.60F
05.50
geometry
Surface
and
critical
local
behaviour
self-
the
:
avoiding-walk Turban
Loic
Laboratoire
Nancy
Physique
de
Cedex,
(Received
surface
the
exponent value
(
v
analytically, agreement
the
which
using with
the
a
Nancy I, BP 239,
de
F-54506
Vand~euvre-l~s-
is
1992)
27November
polymer
Cu~
confined
chain
through
studied
considerations, behaviour
surface
a
shape
the
system
is
in
two
found
to
smaller
by
limited
which
simulations
is geometry exponent k is
surface
the
when
inside
Monte-Carlo
be
than
a
dimensions.
relevant
a
In
one.
this
vf along the parabola greater than the anisotropy ratio z adjusts itself to the direction. Mfhen k < I the transverse obtained perturbation. The the surface exponents geometry is a marginal in good approximation, either blob picture approach or a Flory the are anisotropic
becomes
in
for
of ±
=
scaling flat
the
to
system
k~'
v
with
agreement
case
Universit£
accepted
statistics
The
perturbation
(*),
Solide
du
October1992,
21
Abstracto
Berche
France
parabolic-like In
Bertrand
and
2d
simulation
with
a
radius
exponent
results.
Introduction.
lo
Isotropic critical
This
d~
systems behaviour, first result,
4
in
=
conformal
triangle comer
confined the
local
obtained
Wedges comers, magnetic exponent by Cardy ill in
in
w4-theory, has been since techniques [2-3]. The Ising comer
transfer
equations [4], matrix
[7].
Some
from exact
the
(*)
results
have
[3, 8].
URA
CNRS
n°155.
display
marginal local opening angle. e-expansion an near results in 2d using
to
with
field and through implemented by exact magnetization has also been correlation
comer-comer
~
known
are
continuously
mean
Marginal behaviour traced may be geometries. The opening angle, which is invariant identified the marginal variable. as More recently, the case of a « parabolic » surface limited by a parabola is then (SAW)
cones
varying
then
the
recursion
or
~
*
CU~
been to
in
the an
obtained
function for
the
the
deduced
from
length scale isotropic change of scale,
geometry
absence
was
of
a
considered
star-
[4-6] or using the self-avoiding walk
[9]. The
in
may
such be
system
(1)
926
JOURNAL
(Fig. I),
in 2d role
in
L~
as
perturbation which C~~ according to
=
so
k
j
)I
scaling
the
i
=
v
They satisfy
~
~ P
deduce
can
Dyll
c (I
~~
v
fifk/[I
+
function
+
k(D
=
v
+
(19)
I >1
(~~)
1)]
k(D
exponents
(i
) ~°I
v
~ l/jl
k)
k(i
defined
in
equation (7)
v
v
~~~~
)
(9) and give the asymptotic slopes given by the
dashed
lines
in
JOURNAL
932
figure 3.
chain
The
radius
exponents
vf k
When
v(
vf
I. In 2d the
=
=
Flory
=
Monte-Carlo
the
is
marginal
and
the
radius
k
0
the
chain
approximate, allow
to
us
free
trial
A
value
v
chain, dimensions,
known
to
discuss
the
upper written
critical
5
the
as
of
sum
of
energy
a
in
surface
flat ld
a
to
correlation
length
Although and
exponents
will
system.
part of entropic origin, in
a
paraboloid
the
evaluated
for
approximation.
meanfield
that
free
the
is
Ri ~~~'~~
where
we
function
ignored of
independent
Rj
quantities.
free
The
the
limit~k
l
~
kid
+
Flory
the
may
be
verified
(25) also
that
corresponds
1)]
fi~3/[3
radius
follows
for
~
D~ It
to
order
Ri ~/N.
confinement,
the
minimum
d~/[3
of
entropy
~~~~
is
energy
~R(
R~
a
not
are
to
(~~)
1)]
kid
+
Rjj
free
The
and
~
recovers
one
due
energy ~ ii
I
the
to
since,
only
~ In
fi~2
~~d-I ~j+k(d-1)
fit
contribution
transverse
a
variable
the
(23)
given by
is
energy
an
In d
V(Rjj)~C~~'R(+~~~~'~ so
to
obtained.
problem.
the
with
compare is
agreement
approach
value
geometry
in (22) to
reasonable
obtained
chain
the
the
in
introduced
be
the
elastic
an
contribution
keep
confined
can
(22)
=
exponents is
where
dimension
occupied by
volume
figure
kvf.
(
v
~)
[17] and
exact
in
done
v(1
+
altemative 10] provides an give good estimates of the
[18-19,
it is
the
=
~
=
=
3/4 is
is
energy is and a repulsive
ideal
)
I
when
This
results.
Flory theory
The
as
k(D
Otherwise
v,
N° 4
I
~
perturbation
the
I =
vf
identified
be
can
+
=
with
PHYSIQUE
DE
chain
free
a
with
fractal
the
dimension
~
(26)
=
(19)
from
when
D
D~
=
is
used
in the
blob
for
result
Rj. The
critical
upper
dimension
fractal
dimension 2.
D
for
For the
free
the
=
expression,
Flory
the
SAW
~
usual
chain
anisotropic
for
from
deduced
N~~~
value is
energy
«
least
at
equation (27). equation (23),
ideal
=
I +
chain
value.
higher repulsive repulsive part is
I =
due
It
way
which
follows to
reach
of
the
is
that
the
the
at
equal
to
the
ideal
so
=
chain
that, using
confinement,
4 at the
between
the
d$ blob
Nl~
order
chain
conclusion like
This
case.
result
may
also
(24) [10]. Replacing RI by the
confined
scales =
of
most
same
d$
isotropic in
term
energy
system
D~ given by (26) to
the
1/2
(27)
in the
than
of the
volume
the
k(d Finally, keeping D Flory theory as if,
d~~~(d)
Rjj, the O(I).
Another
to
km
+
=
d~ is
systems,
behaviour
the
corresponding
4 =
gets
one
d~
As
d$
is
chain, it will be given by v((d~)
confined
is
R(~~,
is to
with
critical
upper
~~l~~
~~Y~
while
ideal
above
consider
that,
an
the
blob
result
fractal
dimension
were
still
elastic
given according
in
d~
effective
dimension
and d~,
the
be
ideal
to
dimension
given by (27). agrees
increased
with
the
above
the
SURFACE
N° 4
5.
GEOMETRY
AND
CRITICAL
LOCAL
BEHAVIOUR
933
Discussion.
An anisotropic critical system is left invariant by b~ in the parallel direction and b in the given by the ratio vii /v~. When this scaling in (I), equation (2) becomes
Thus
the
verified
surface
the
invariant
is
geometry
for
walk,
directed
one,
is the
z
applied
is
to
of the length scales anisotropy exponent surface
the
geometry
=b~~~~C~~ the
and with
where
transformation
C'~~
been
anisotropic change
the
under transverse
z
marginal
system
2,
=
(28)
confined
for
inside
k'~.
z
a
This
=
parabola
with
already
has
[20-
1/2
k =
2l]. For
the
influence
SAW
studied
of the
relevant
original isotropic fixed point is perturbation, the system is driven adjusting ratio self-consistently z
here,
the
surface
and,
unstable towards
a
under
the
anisotropic new marginal value
point with an the exponent to k~ invariant, a necessary condition for the existence of a fixed and leaving the geometry mere point. It would be interesting to study the number of walks in order to obtain the « magnetic » Simulations using the grand canonical ensemble would also exponents at the new fixed point. fixed
determine critical fugacity which has the mapping of the half-space on the parabola critical fugacity [9].
useful
be
bulk As
extension
an
approximation. Polymer
of
Let
the
us
chain
in
work
present
just a
some
B-solvent
[22]
~~ 2
polymer
Branched
in
good
a
polymer
Branched
in
~~~ The
may fractal
blob be
method
extended
already
has to
the
dimension
D
for
this
approach
Finally behaviour
from
radius
transverse
parabolic
the
r~
start a
~,
A
geometry.
JOURNAL
DE
PHYSIQUE
left
are
original
value
i
T
systems
one
since
length in
the
the
the
at
Flory
:
~l'
(30)
~~'
~~~~
~
:
~~
study
to
~~~~
k'
growth
in
fractal growth process on a strip [26]. This parabolic The knowledge of the geometry. gives vf when inserted into equation (22).
growth immediately generalized to study systems displaying anisotropic can Then, one has to consider anisotropic blobs with vii /v~ z. and piling up length r~ j, containing n~ r(~ (~ r(I' monomers gives simple calculation be
critical with
=
a
in the
~
+
exponents
used
~
~~
free
=
Both
[25]
fractal
vf
keeps its [20, 21].
[23-24]
+4/(d-1)
been
of
case
8
bulk
correlation
polymer
other
~~
6+2k(d-1) >solvent
a
a
the
from
finite
:
1)
solvent
~~
to
:
k(d
+
leads
consider
may results
one
mention
different
be
to
to
conformal
unchanged for
3, N' 4,
any
APRIL
iW3
v(
zk(I/vj
k if
in
marginal
the vi
I, =
kvf
k
=
1)
a
case
situation
k =
«
z~ ~.
(32)
z~
The
encountered
parallel with
the
exponent
also
directed
walk
3~
JOURNAL
934
DE
PHYSIQUE
N°
I
4
Acknowledgments. Discussions
with
Ingo
Peschel
and
Ferenc
Ig16i
at
an
early stage
of
this
work
are
gratefully
acknowledged.
References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
iii
[12] [13]
[14] [15] [16] [17]
[18]
CARDY
J. L.,
CARDY
J. L.,
CARDY
J. L.,
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BARBER
M.
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PESCHEL
I., Phys.
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I.,
PEARCE
P. A.,
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l10A
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j19j FISHER M. E., J. Phys. Soc. Japan Suppl. 26 (1969) 44. j20] TURBAN L., J. Phys. A 25 (1992) L127. j21] IGLOI F., Phys. Rev. A 45 (1992) 7024. j22] RAPHAEL E., PINCUS P., J. Phys. II France 2 (1992) 1341. L469. 41 (1980) France j23] ISAACSON J., LUBENSKY T. C., J. Phys. Lett. 1359. France 42 (1981) j24] DAOUD M., JOANNY J. F., J. Phys. Macromolecules j25] DAOUD M., PlNcus P., STOCKMAYER W. H., WITTEN Jr T., j26] TURBAN L., DEBIERRE J. M., J. Phys. A17 (1984) L289.
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