Scaling near the point for isolated polymers in solution
A. L. Owczarek*
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
T. Prellberg
²
Institut fu ¨r Theoretische Physik, Technische Universita ¨t Clausthal, Arnold Sommerfeld Straße 6, D-38678 Clausthal-Zellerfeld, Germany
Received 19 November 2002; published 28 March 2003
Recently questions have been raised as to the conclusions that can be drawn from currently proposed scaling
theory for a single polymer in various types of solution in two and three dimensions. Here we summarize the
crossover theory predicted for low dimensions and clarify the scaling arguments that relate thermal exponents
for quantities on approaching the point from low temperatures to those associated with the asymptotics in
polymer length at the point itself.
DOI: 10.1103/PhysRevE.67.032801 PACS numbers: 61.41.+e, 05.50.+q, 05.70.Fh
Recently some interesting work has been completed on
stretched polymers in a poor solvent by Grassberger and Hsu
1GH and on collapsed polymers on a cylinder by Hsu
and Grassberger 2HG. In the course of these works vari-
ous scaling conjectures were discussed; the question arose as
to whether they can be derived from theory currently in the
literature. It is therefore timely to revisit questions such as
these: we not only address the questions raised in GH and
HG, but discuss how such questions can be answered in
general.
The basic framework of the polymer problem has been,
and still is, given by the seminal works of de Gennes and
Duplantier 3–5 which describe the long length behavior in
terms of critical phenomena. Hence the basic properties of
such polymers are argued to display scaling behavior. Much
work has been subsequently done to verify specific scaling
predictions in both two and three dimensions for examples,
from the past 10 years see Refs. 1,2,6–10. More generally,
scaling usually imposes certain relationships between critical
exponents a review of the more general scaling framework
can be found in Ref. 11—see also Ref. 12 and it is these
relationships that are addressed here.
Of particular interest here is the scaling of quantities on
approaching the point from the collapsed phase. The col-
lapsed phase itself has received attention relatively recently
1,2,13,14. Much less is known here, partially because the
long length behavior is no longer a critical phenomenon.
This is in contrast to the swollen phase and the transition
point, which are both critical. To make clear the answers to
questions such as those raised in GH and HG it is first timely
to restate, in compact fashion, the conjectured crossover scal-
ing theory for a single polymer between good and poor sol-
vents high and low temperatures respectively, and then
demonstrate how questions such as those raised can be an-
swered in general.
As temperature gets decreased, an isolated polymer in so-
lution undergoes a phase transition from a swollen coil to a
collapsed globule via a critical state at a temperature T
.
The standard description of this polymer collapse transition
is a tricritical point related to the n →0 limit of the
(
2
)
2
–(
2
)
3
O ( n ) field theory 3–5. Scaling theory can
therefore be derived, in principle, from this tricriticality 15.
The upper critical dimension for the swollen state is four
while for the state it is expected to be three. As confluent
logarithmic corrections complicate the discussion in three
dimensions, the crossover theory should be cleanest for di-
mensions strictly below the upper critical dimension. Let us
therefore concentrate our discussions on two dimensions.
Consider now, for simplicity, some quantity Q ( T , N ), as-
sociated with a property of the polymer, that is a function of
the length N and the temperature T of the polymer. More-
over, let it be a quantity that has an algebraic asymptotic
behavior for large N at any fixed value of T, such as the
radius of gyration R
g
( T , N ), for example. Such a quantity
would then be expected to possess three different behaviors:
For fixed T T
,
Q T , N a
+
T N
q
+
, 1
for T T
,
Q T , N a
-
T N
q
-
, 2
while for T =T
,
Q T , N a
N
q
, 3
each as N → . The assumption of crossover scaling 15–17
applied to this system 11 implies that there exists a cross-
over exponent such that for each fixed value of x =tN
,
where t =( T -T
)/ T
,
Q T , N N
q
G tN
as N → . 4
Note that Ref. 11 contains numerous typesetting errors in
some formulas, such as Eq. 26, that makes readability less
than optimal—see Ch. 2 in Ref. 12 for a nice summary.
Moreover, and importantly, it is assumed 17 that this
asymptotic form provides all the dominant asymptotics for
small t so that
*Electronic address: aleks@ms.unimelb.edu.au
²
Electronic address: thomas.prellberg@tu-clausthal.de
PHYSICAL REVIEW E 67, 032801 2003
1063-651X/2003/673/0328013/$20.00 ©2003 The American Physical Society 67 032801-1