MARs. 1991, PAGE. 313. Classification. PfiysibsAbsmwts. 05.50. 75.40. Shom. Communication. Flory approximant for self-avoiding walks near the theta-point on.
J
Phys.
(1991)
I 1
313-316
1991,
MARs
PAGE
313
Classification
PfiysibsAbsmwts 05.50
75.40
Shom
Communication
Flory
approximant
fractal
on
Chang (I)
Iksso
for
self-avoiding
walks
near
theta-point
the
structures AJnnon
and
Aharony
(~>
~)
(1)
School of Physics and Astronomy, Raymond lbl Aviv University, lbl Aviv 69978, Israel
(~)
Department
of
Physics, University
(Received 7Janua~1991, accepted
of
and
Beverly
Faculty
Sackler
of
Sciences,
Exact
Oslo, Norway
Janua~1991)
lo
We present a Flory approximant for the size exponent and the exponent of a crossover This approximant involves the three fractal walk at the theta-point on ~actal structures. dimensionalities resistance of the fractal for the backbone, the minimal path, and the structures.
Abstract.
self-avoiding
There
has
randomness action
been
[3-16].
between
two
action, usually this randomness.
In the
much At
interest
non-bonded
statistics
is
presence
nearest
described of
statistics
in the
high temperature
of
a
single
linear
polymer [1, 2]
with
quenched
good solvents, where the excluded volume neighbor prevails over their attractive monomers by that of self-avoiding walks (SAWS) [1, 2] with the
dilution
T
or
in
it has
been
shown
that
interintersame
for p > pc ~p is the probability SAWS on a dilute lattice belong
being occupied and pc is the percolation threshold) universality class of SAWS on a pure lattice [3, 5 7, 10, 14]. For SAWS on the incipito same infinite cluster at the percolation threshold Aharony and Harris (AH) [13] presented a Flory ent approximant for the exponent v describing the scaling of the mean square distance end-to-end N2v of SAWS of N-steps on fractal [12,15]. For a long time there had been a (R~) structures debated question whether the randomness changes the universality class of SAWS at p pc from Recently Meir and Harris [14] answered this question and showed by the that on a pure lattice. renormalization group theory that it is described by a fixed point different from the one on a pure lattice. Also their results, obtained using series, are very close to the prediction from numerical Euclidean AH and exclude the pure lattice values for dimensions higher than 2. solvent becomes is special 8 at which As T decreases, the there temperature T poor, or a volume and the interaction the (two body) excluded attractive compensate each other, therefore the three body interaction chain behaves closer starts to play a role [1, 2, 17, 18]. As a result, the Also a single linear polymer at its e-point is usually modeled by self-attracting SAWS on to ideal. lattice. It is interesting to see how the quenched affects the asymptotic statistical randomness a behavior of SAWS at the e-point (8-SAWS). In thin paper we present a new Flory approximant of each
bond
the
~-
=
=
for the size
exponent
we
and
the
crossover
exponent # of e-SAWS
on
fractal
structures.
JOURNAL
314
starting point is based on the Flory formulae [1, 2]. In general
work
Our for
have
to
monomers
~d~,B F
first
The
random
entropic
is the
term
distance
end-to-end
whose
walks
N2
walks, has been argued by
(R
and
R
B
the
single
a
procedure polymer of N
standard linear
R
II)
+
~~ B
due to the probability of where dw,B = DB +
is
finding a regular random walk (R is the fractal diJnension of The value of a, for typical random
exponent.
resistance
a
Aharony [Ill
and
follows of
~g3
jw
+
N-steps,
after
Harris
(j)
jv
contribution
backbone
the
on
as
+
h R
ill, 13, lti~ and the probability exp(-F), with write
°
~
=
can
R
N°3
of AH
one
dimension
linear
a
PHYSIQUE I
DE
to be
,~~~ "
"
(~)
~
~
m~
minimal (or chemical) path. The second term in equacontrAution, which may be repulsive or attractive depending interaction the parameter v, and the third term is the three body interaction conon tribution, with the interaction parameter w. The last two terms use the mean-field approximaof concentration concentration tion, in the sense that one replaces a by the average monomers c
dma
where
tion
?
is the
(I) is the the sign of
N/ROB,
=
backbone, and v
since
0 and
>
w
dimension
body
interaction
where
contain
w
fractal
two
DB is the
fractal
of I
/kBT,
0, AH [13]
dimension
and
thus
minimized
The v
same
vanbhes
imization
that
recover
Dmw
"
back
"
D~
+
(the polymer is restricted to the Both u out of dangling bonds). high T. For regular SAWS, with that
~~~
adw,~
expression was also found by others [12, @, using different arguments. At the 8 -point, since the (two body) repulsive and attractive interactions compensate each other. Minof equation (I) with respect to R immediately yields our new Rory approximant, "~
Note
backbone
it from
become
F in
~~
of the
coming unimportant at equation (I) and found
prevents
self-avoidance
its
factors =
of the
for the
the
usual
non-dilute
Floty
case
~p
=
I), DB
d, dw,B
=
3/(d + 2) and vmw predictions kom equation(4) results
~~~
2DB+ adw,B
De
=
=
Me
=
2 and dn~~n 2/(d + I).
=
I, and equations (2)
(4)
percolation clus> 6 respectively. ter pc First, we notice that vmw > we for every d, which makes sense physically. Second, we is always ) and ) for d 2 and > 3), which tells us that the dilularger than the pure lattice values (we tion of sites swells the polymer. Thin h to be expected: the dilution of sites increases effectively the excluded interaction (or introduces a long range excluded volume interaction) between volume since it eliminates spatial region where a chain can be embedded. Therefore it monomers, some makes a chain swollen than a chain on a pure lattice for a given Euclidean dimension d. We more also observe that the inequalities, d~n;n < DsAw < De < DB hold for every d. These inequalities exact for our approximants, and follow kom the fact that dm~ < DB. are quoted above work only for d < du, with the upper on regular lattices, the Flory formulae In table I at p
=
we
lit
the
together
with
those
for SAWS
kom
AH
dimensions
du
=
4 for
SAWS
and du
ford
on
=
the
infinite
2, 3, 4, and
=
=
critical
e-SAWS
for
~Eq.(3))
=
3 for
e-SAWS.
For d > du,
one
recovers
the
315
SAWNEARIHEePOINTONFRACLU£
N°3
lbble
1.
e~ponent # cited
are
for the Floly approdmant for the incipient in finite pelcolation cluster
Esdhlates
she
the
at p
on
d
dm;n
DB
dw,B
DSAW
2
1,13
1.62
2.61
1.31ci
3
1.36
1.83
3.14
1.53
4
1.62
1.94
3.53
1.73
4
2
2
2
from
for d > 6, and
identifies
crossover
N, keeping both
Thh
expression
two
body
=
6
and the
crossover
of dn~a, DB~
and
dw,B
l/0.72
0.17
ci
1/0.65
1.62
ci
1/0.62
0,14
ci
1/0.58
1.79
ci
1/0.56
0.09
dimension.
2
behavior and
body
R
N"~
"
the
as
R
probably
with
small
a
interaction
N
results
in the
of
absence
~
+
«
e-point
a
v
«
(T
8).
One
describe
can
the
by minimizing equation (I) with respect terms (v # 0, w # 0). This yields
(j p~j W
scaling
~DB
~
to
(5)
form
N"e f
~w
behavior
dimension.
that of SAWS
three
analogous
the
recover
0
=
Thin
e-point, to
never
1/0.5
=
De = DB for d > 6. This is a direct for d > 6, and the polymer simply follows the
critical
upper
an
of the
#
~
1/0.5
=
I/Ue
"
1.39
irrelevant
are
one
as
written
be
can
in
as
8-SAWS
the
from
d
loops
that
vicinity
the
to
turn
now
fact
the
backbone,
connected
We
values
1/0.76
behavior N « R~. In our case we However, equations (3) and (4) yield DsAw
consequence
De
I/VSAW
"
non-interacting
singly
and we,
vmw
The
p~
=
Jtom r~fierence [13].
/ 6
R~»>B.
e~ponent
e) N~)
((T
(6) ,
and
the
exponent # b
crossover
identified
as
(7)
#=DBwe-1.
resulting
The
in z
table
I. The
+cx>.
-
scaling
The
For T < 8, I.e. dimension of the
(I).
of # for the
values
function
0,
