cyclic polymers in good solvents
TRANSCRIPT
Polymer International Polym Int 49:175±183 (2000)
Cyclic polymers in good solventsAbdelhamid Bensafi,1 Ulrich Maschke2* and Mustapha Benmouna1,3
1University of Tlemcen, Institut of Science and Technology, BP 119, 13000 Tlemcen, Algeria2Laboratoire de Chimie Macromoleculaire, UPRESA-CNRS N° 8009, Universite des Sciences et Technologies de Lille, F-59665 Villeneuved’Ascq Cedex, France3Max-Planck-Institut fur Polymerforschung, Postfach 3148, D-55021 Mainz, Germany
(Rec
* CoTechE-ma
# 2
Abstract: This paper deals with the effects of excluded volume interactions on the thermodynamic and
structural properties of cyclic polymers in good solvents. Several thermodynamic properties are
discussed with particular emphasis on excluded volume interactions and their effects on the radii of
gyration and the form factors. An empirical model describing the mean square distance between two
points along a cyclic polymer is proposed and its predictions compared with those of other models.
Comparison between cyclic and linear chain properties highlights the effects of chain closure under
good solvent conditions.
# 2000 Society of Chemical Industry
Keywords: cyclic polymer; excluded volume interactions; effective exponent; swelling factor; form factor
INTRODUCTIONThe statistics of cyclic polymers are characterized by
translational invariance along the chains, in contrast to
the case for linear polymers where the two terminal
ends introduce important perturbations. Statistical
defects due to the interruption of translational
invariance near chain ends could lead to large
inaccuracies in the physical properties which are
dif®cult to evaluate precisely.
An important example of cyclic polymers is given by
the DNA macromolecule which can occur in nature as
a long ring. Knowing the key role of this molecule in
the organization of living cells,1,2 it is important to be
able to evaluate its changes of size and conformation
when it is subject to excluded volume interactions
under good solvent conditions. The effects of such
interactions together with the chain closure attribute
special features to the DNA molecule that should be
elucidated if one attempts to answer some of the key
questions currently raised in the ®eld of molecular
biology. To ascertain the speci®c features of cyclic
polymers such as DNA, a comparative analysis is made
between the swelling behaviour of cyclic and linear
chains under similar conditions.
One of the main quantities describing the chain
swelling and conformation under good solvent condi-
tions is the critical exponent n. This exponent is
accessible experimentally by exploring the scaling
behaviour of the radius of gyration Rgc as a function
of the degree of polymerization N
Rgc � N�c l �1�
eived 6 April 1999; accepted 20 October 1999)
rrespondence to: Ulrich Maschke, Laboratoire de Chimie Macromnologies de Lille, F-59665 Villeneuve d’Ascq Cedex, Franceil: [email protected]
000 Society of Chemical Industry. Polym Int 0959±8103/2000/$1
where l is the length of a unit monomer and the
subscript c is used to distinguish the properties of
cyclic chains while those of linear chains will be
indicated by the subscript l. One can introduce two
distinct exponents to allow for differences in the
scaling behaviour of cyclic and linear chain systems,
but in a real experiment, the measured exponent could
be function of N while the theoretical value of the
exponent n is valid only in the in®nite chain limit where
N → ?. However, in this limit, chain circularity is
irrelevant and nc should be identical to nl.
Very few studies have been carried out to under-
stand the scaling behaviour of single rings in good
solvents. Prentis3 performed ®eld theoretical calcula-
tions and suggested that such scaling is not affected by
the circularity condition and should be the same for
cyclic and linear chains with
�c � �l � 0:6 �2�
More recent computer simulations and ®eld theore-
tical calculations of Jagodzinski et al4 con®rmed this
result, showing that the critical exponent n is close to
0.6 for both linear and cyclic polymers in good
solvents. A similar observation was made by Pakula
et al5 using computer simulations. These results are in
contrast with those obtained in the bulk where detailed
information is available on the exponent n essentially
by computer simulation.6±15 The general observation
is that linear chains in the bulk obey Gaussian statistics
and that the radius of gyration varies with the degree of
oleculaire UPRESA-CNRS N° 8009, Universite des Sciences et
7.50 175
A Bensa®, U Maschke, M Benmouna
polymerization according to the mean ®eld power law
Rgl � N1=2l �3�Likewise, it is generally admitted that rings in the bulk
follow a much more subtle scaling behaviour repre-
sented by eqn (1) where the critical exponent nc takes
values in the interval
0:4<�c<0:45 �4�This result indicates a signi®cant deviation from
Gaussian statistics of ring polymers in the bulk.
Mixtures of cyclic and linear polymers have also
been the subject of particular attention.7±9,16±18 The
scaling of linear chains is found to be weakly sensitive
to the presence of cyclic chains, while the scaling of
rings is strongly modi®ed by the presence of linear
chains. In the bulk, a ring polymer shows a compact
conformation characterized by an exponent nc of
between 0.4 and 0.45. In a matrix of linear chains, it
undergoes substantial swelling with an exponent nc
close to 0.5
THETA TEMPERATURE AND THE SECOND VIRIALCOEFFICIENTThe statistics of linear chains are relatively well
understood in dilute solutions and in the bulk. Chains
are swollen when diluted in a good solvent, while in a
dense medium they obey Gaussian statistics. These
features cannot be extended to cyclic polymers with-
out caution. Thermodynamic properties of cyclic
polymers in solution and in bulk are quite different
from those of analogous systems made of linear chains.
For example, the entropy reduction due to chain
circularity generates intramolecular excluded volume
resulting in a positive second virial coef®cient and a
substantial shift (about 6°C) in the theta temperature.
A detailed theoretical description of such differences is
still lacking and only a few attempts have been made
with simple models, among which one ®nds the work
of LeÂonard19,20 where the Flory-Huggins lattice
model21 is used to derive the mixing entropy of cyclic
chains. An extended discussion of entropies of various
mixtures is given in refs.20±22
Experimentally, Candau et al23 were the ®rst to
measure by light scattering the shift in the theta
temperature of cyclic polystyrene (PS) in solution.
Later, similar observations were made by others.24±33
The theta temperature of cyclic PS in decalin is found
to be 15°C, while for the analogous linear PS it is
19°C. In cyclohexane this temperature is 28°C for
cyclic PS and 34.5°C for the corresponding linear PS,
while in deuterated cyclohexane these temperatures
increase to 33°C and 40°C, respectively.
The relationship between the theta temperature, the
second virial coef®cient and the scaling behaviour of
cyclic polymers in solution under good solvent
conditions has not been elucidated completely. At
the ordinary theta temperature characterizing linear
176
chain solutions, one observes simultaneously unper-
turbed dimensions, zero second virial coef®cient and
mean ®eld scaling. Below this temperature, the second
virial coef®cient may decrease below zero, the chain
shrinks and eventually collapses.33 Above theta, the
second virial coef®cient increases with temperature,
and chain dimensions increase substantially. In the
following sections, the effects of excluded volume
interactions on the structural properties of cyclic
polymers are discussed, assuming that the above
observations are valid for open and closed chains.
SWELLING OF CYCLIC POLYMERS IN GOODSOLVENTSFlory’s modelOne of the ®rst models aiming to describe the swelling
of linear and cyclic polymer beyond the perturbation
limit under good solvents is given by the famous
Flory equation.21,33 According to this equation,
the static swelling factor of linear chains
�21 � R2
gl=�Nl2=6� increases with the excluded volume
parameter z as
�51 ÿ �3
1 � 1:276z �5�Likewise, the swelling factor of cyclic chains increases
with the excluded volume parameter z according to
�5c ÿ �3
c ��
2z �6�
Implicitly, the excluded volume parameter z is
assumed to be the same for linear and cyclic polymers.
If one wants to distinguish between these parameters,
it is suf®cient to attach the subscripts l and c to z.
Having this in mind, z can be related to the binary
cluster integral v and the degree of polymerization via
the de®nition33
z � 3
2�l2
� �3=2
v�����Np
�7�
The relationship between v and the second virial
coef®cient A2 is
A2 � vNav
2m20
�8�
where Nav is the Avogadro constant and m0 the
monomer molecular weight. Combining equations
(6±8) yields
A2c �8����p
NavR2g0
M2��5
c ÿ �3c� �9�
with R2g0 � Nl2=6 and M =Nm0. If one replaces the
quantity in parentheses by its approximate form for
small values of z, the perturbation result of Zimm etal34 is recovered
A2c �8����p
NavR2g0
M2��2
c ÿ 1� �10�
A similar equation holds for the properties A and a
2l lPolym Int 49:175±183 (2000)
Cyclic polymers in good solvents
characterizing linear chains with numerical differences
only. This result was used as a method of distinguish-
ing between second virial coef®cients of cyclic and
linear polymers knowing the swelling factors. Figure 1
represents the variation of �2l and �2
c as a function of zaccording to eqns (5) and (6), respectively. The initial
slopes at z close to zero are consistent with the results
of the perturbation theory. At high values of z, the
asymptotic limits are quickly reached whereby ac and
al are proportional to z1/5. The upper curve represents
cyclic chains and indicates that rings are more sensitive
to excluded volume interactions than open chains.
Another quantity suitable for comparison between
cyclic and linear chain swelling is the ratio
� � R2gl
R2gc
� 2�2
l
�2c
�11�
Figure 2 shows that b decreases rapidly from 2 to
1.867 when z increases. This result is slightly higher
than the ®eld theoretical prediction of Prentis.3 Both
calculations predict that cyclic chains swell more than
linear counterparts. In the theta solvent limit b=2, a
result con®rmed by analytical calculations, computer
simulations and experimental data.
The radius of gyration Rg can be calculated
analytically starting from the general de®nition
R2g �
1
2N2
Xij
hr2ij i �12�
Transforming double sums into single ones and
approximating these by integrals yields
R2g �
1
N2
Z N
0
dn�N ÿ n�hr2n i �13�
where terms of the order 1/N are neglected as
compared to unity assuming that N is large.
Linear chains
For solutions of linear polymers under theta condi-
tions, the mean square distance between two points i
Figure 1. The static swelling factors of linear ��2l �and cyclic ��2
c�chains asfunctions of the excluded volume parameter z according to Flory’s eqns (5)and (6), respectively.
Polym Int 49:175±183 (2000)
and j separated by n monomers is given by
hr2ij i � ji ÿ jjl2 �14�
Under good solvent conditions, the scaling is not
described by mean ®eld exponents and it was
suggested that eqn (14) should be modi®ed according
to
hr2ij i � ji ÿ jj1�"l2 �15�
where an effective exponent increment was intro-
duced.33,35,36 In both eqns (14) and (15) end effects
are neglected.
By combining eqns (13) and (15) after some
straightforward manipulations, one obtains
R2gl �
N1�"l2
�"� 2��"� 3� �16�
In a theta solvent (ε=0), one recovers the known result
R2gl � Nl2=6. Taking the ratio of eqn (16) and the latter
expression yields
�2l �
6N"
�"� 2��"� 3� �17�
If ε=0, one ®nds �2l � 1 as expected.
Cyclic chains
For cyclic polymers, the situation is more subtle and
different models may be used to describe the mean
square distance hr2n i in the presence of excluded
volume interactions. We will mention two existing
models and suggest a new one. First, note that the
mean square distance hr2ij i for a Gaussian ring in a theta
solvent is given by
hr2ij i � l2ji ÿ jj 1ÿ ji ÿ jj
N
� ��18�
where the factor in brackets is included to meet the
closure condition. Combining eqns (12) and (18) after
some straightforward manipulations leads to the
known result
R2gc �
Nl2
12�19�
Under good solvent conditions, eqn (18) should be
modi®ed to account for the chain swelling due to
excluded volume interactions, but these modi®cations
can be introduced in various ways as we shall see in the
following sections.
The Bloomfield–Zimm modelTo investigate the effects of excluded volume interac-
tions on the radius of gyration of ring polymers,
Bloom®eld and Zimm37 suggested the following
expression for the mean square distance between two
points i and j separated by n monomers
hr2n i � l2
n1�"�N ÿ n�1�"n1�" � �N ÿ n�1�" �20�
177
Figure 2. The ratio � � R2gl=R2
gcas a function of z according to Flory’s eqns(5) and (6).
A Bensa®, U Maschke, M Benmouna
where the effective exponent increment e is assumed to
be the same for cyclic and linear chains. To simplify
notation, we write jiÿ jj=n. Calculation of the radius
of gyration from eqns (12) and (20) gives
R2gc � KBZN1�"l2 �21�
where the constant KBZ is the integral
KBZ �Z 1
0
dXX1�"�1ÿX�2�"
X1�" � �1ÿX�1�" �22�
A numerical integration with ε=0.2 yields
KBZ=0.067, which is lower than 0.083, the result
obtained in the absence of excluded volume with
ε=0.38 Taking the ratio of eqns (19) and (21) shows
that the swelling factor �2c increases according to the
power law
�2c � 12N"KBZ �23�
To illustrate the increase of �2c from unity under the
in¯uence of excluded volume interactions, we let
ε=0.2, N =103 and obtain ac�1.8. This result shows
that the swelling factor roughly doubles when the
conditions of the solution are changed from theta to
good solvent.
The ratio b is obtained from eqns (16) and (21) as
follows:
� � 1
�"� 2��"� 3�KBZ
�24�
It is interesting to note that the Bloom®eld±Zimm
model predicts that b increases in the presence of
excluded volume. Letting ε=0.2 gives b=2.112,
indicating an increase of approximately 5%. The
increase of ac and b with excluded volume interactions
is discussed later.
Figure 3. The normalized mean square distance hr2ni=�N1�� l2�as a function
of n/N. The series of dashed curves correspond to ε=0.1 and thick curvesto ε=0.2. From the bottom upwards for each series: this work, Bloomfield–Zimm model and linear chains.
The Yu–Fujita modelYu and Fujita39 suggested the following model for the
mean square distance hr2n i in terms of e and n/N to
describe the effects of excluded volume interactions in
178
cyclic polymers
hr2n i � l2n1�" 1ÿ n
N
� �1�"� ��25�
Unfortunately, this expression shows that
hr2n i 6� hr2
Nÿni and therefore the circularity condition
is not ful®lled when e is non-zero. Following the same
procedure as in the Bloom®eld±Zimm model, one can
show that this model also predicts a larger swelling of
cyclic polymers than of linear chains under the
in¯uence of excluded volume interactions, and hence
b should decrease with z. Nevertheless, eqn (25)
cannot be considered as a reliable description of cyclic
chain swelling so we shall not dwell on it any more, but
rather suggest an extension of the Yu±Fujita model to
correct for its shortcoming in satisfying the circularity
condition.
The present modelThe Yu±Fujita eqn (25) is slightly modi®ed by shifting
the exponent in the brackets to allow the circularity
condition to be ful®lled:
hr2n i � l2n1�" 1ÿ n
N
� �1�"�26�
The variations of hr2n i=�N1�"l2� versus n/N are given
in Fig 3 for ε=0.1 and 0.2 and the models of eqns (20)
and (26) together with the linear chain case. The gap
between cyclic and linear chains increases sharply with
n/N. In this representation, the present model predicts
slightly lower values than the Bloom®eld±Zimm
model. The difference is greatest at n =N/2.
We are aware that this is an empirical model which,
to be more reliable, needs justi®cation on ®rm physical
grounds. Unfortunately such a justi®cation is not
available for any model concerning the swelling of a
®nite section of cyclic chains under the in¯uence of
excluded volume effects. Given the lack of a sound
theory at the present time for the swelling of cyclic
polymers under good solvent conditions, our aim here
Polym Int 49:175±183 (2000)
Figure 5. The ratio � � R2gl=R
2gc as a function of e. From the bottom
upwards they correspond to the Bloomfield–Zimm model and the presentmodel, respectively.
Cyclic polymers in good solvents
is to explore the possibilities offered by the model
displayed in eqn (26). An attempt is made to compare
the predictions of this equation and eqns (6) and (20)
by considering data available in the literature obtained
by scattering experiments and computer simulations.
Combining eqns (12) and (26), one obtains
R2gc � K"N
1�"l2 �27�where the factor Ke is given by
K " �Z 1
0
dXX1�"�1ÿX�2�" �28�
Numerical integration for ε=0.2 yields Ke=0.0598,
which is slightly lower than the result obtained in the
Bloom®eld±Zimm model, indicating a more moderate
swelling. The quantity ac satis®es a similar scaling law
as eqn (23) with a different numerical factor
�2c � 12N"K" �29�
For ε=0.2 and N =103, one obtains �c � 1:69. The
ratio b is larger than in the Bloom®eld±Zimm model
because
� � 1
�"� 2��"� 3�K"�30�
and for ε=0.2, b=2.372, showing an increase of 20%.
This result contrasts with the ®eld theoretical calcula-
tion of Prentis and the Flory model. Figure 4
represents the variation of �2c with the effective
exponent ε. It shows that the Bloom®eld±Zimm model
predicts higher swelling due to excluded volume
interaction than does the present model. Among the
three cases represented in this ®gure, one sees that
linear chains are the most sensitive to excluded volume
repulsions and that the Bloom®eld±Zimm equation
overestimates the swelling of cyclic chains. This
behaviour is con®rmed in Fig 5 where b is shown as
a function of e. Consistent with Fig 4, the Bloom®eld±
Zimm model predicts that b increases by less than 5%
when excluded volume develops until ε=0.2 where
our model shows a strong increase.
Figure 4. The static swelling factor �2c as a function of e. From the bottom
upwards thick curves correspond to the present model (eqn (29)) and theBloomfield–Zimm model (eqn (23)), respectively. The dashed linerepresents the linear chain solution and is added for comparison (N =103).
Polym Int 49:175±183 (2000)
To compare with experimental data and computer
simulations, we plot in Fig 6 the theoretical predic-
tions of log�R2gc=l
2�versus log N together with the data
available from the literature.7±9,11±13,25,28 The simula-
tion data are in the range of theoretical predictions,
whereas the light scattering data are slightly below. It is
possible to improve the ®t by choosing a negative value
of ε of the order of ÿ0.1 if one is allowed to do so
assuming that the ring experiences rather bad solvent
conditions. It is interesting to note that because the
Bloom®eld±Zimm model predicts a stronger swelling,
it could lead to lines above those shown in Fig 6. This
would mean that the discrepancy between theory and
experiment is larger, favouring the model proposed
here.
For a detailed comparison between experimental
data and the model suggested here, we have collected
in Fig 7 several series of results.7±9,11±13,25,28 One ®nds
that theoretical and experimental results for linear
chains are systematically shifted upwards, expressing a
size increase. Data are within the range of theoretical
Figure 6. Log �R2gc=l2� versus log N. Solid lines are theoretical predictions
of the present model. From below, they correspond to ε=0, 0.1 and 0.2respectively. The symbols are data taken from the literature: ^, cyclic PS/d-cyclohexane at 34.5°C, light scattering data (see refs 25 and 28); &,cyclic chain in the melt, Monte Carlo data (see refs 7–9); ~, cyclic PDMS inthe bulk at 25°C, Monte Carlo data (see refs 7–9 and 11–13); !, cyclicPDMS in the bulk at 25°C, complete enumeration data (see refs 7–9 and11–13).
179
Figure 7. Log �R2g=l2� versus log N. The series of solid lines represent
theoretical predictions for cyclic chains in the present model for ε=0, 0.1and 0.2 in ascending order. The series of dashed lines represent similarresults for linear chains. Empty and filled symbols are literature data forlinear and cyclic chain systems, respectively. ^, and ^, PS/cyclohexanelight scattering data (see refs 25 and 28); & and &, Monte Carlo data ofpolymer chain in the melt (see refs 7–9); ~ and ~, PDMS in the bulk usingRISM and Monte Carlo methods, respectively (see refs 11–13); !, PDMSin the bulk using the complete enumeration method (see refs 11–13); *and�, Monte Carlo data for unknotted rings and all knotted and/or all threeknots, respectively (see refs 28, 29 and 32); ! and *, RISM data forpolymethylene (PM) and polyoxyethylene (POE) in the bulk, respectively(see refs 11–13).
A Bensa®, U Maschke, M Benmouna
predictions for linear and cyclic chains with ε=0, 0.1
and 0.2, except for the light scattering data of PS in
cyclohexane25,28 which are slightly below. An addi-
tional ¯exibility exists in the theoretical modelling by
allowing e to be different for linear and cyclic chains.
However, one does not seek a complete agreement,
because the experimental conditions are not rigorously
similar in the theoretical modelling.
Figure 8 shows the variation of b as a function of Nobtained by computer simulations.13 If N>30 bapproaches 2, but if N<30 there is a marked deviation
from 2. Theoretical models for single chains in
solution predict constant values which are model-
dependent and function of e. In the Bloom®eld±Zimm
model, b=2.053 for ε=0.1 and b=2.112 for ε=0.2,
whereas in the present model higher values are found.
Consistent with an earlier result, we obtain b=2.176
for ε=0.1 and b=2.372 for ε=0.2, but these results
are in contrast with the value of 1.76 reported by
Prentis.3
Figure 8. The ratio � � R2gl=R2
gc as a function of N. PDMS in bulk at 25°C:~, complete enumeration method (see ref 13); *, Monte Carlo data (seeref 13). The arrows indicate the following results: b=2, theta solvent;b=2.053 and 2.112, Bloomfield–Zimm model for ε=0.1 and 0.2,respectively; b=2.176 and 2.372, this work for ε=0.1 and 0.2, respectively;b=1.76, field theoretical calculations of Prentis (see ref 3).
THE FORM FACTORThe form factor of a cyclic chain under theta solvent
conditions was ®rst calculated by Casassa40 using the
Gaussian approximation and ε=0.41,42 The result is
the well known function
Pc�q� � 2 exp �ÿu=4����up
Z ������u=4p
0
dx exp x2 �31�
For the corresponding linear chain problem, one has
180
the classical Debye function33
P1�q� � 2
u2�eÿu � uÿ 1� �32�
This result suggests that the form factor is quite
sensitive to the chain architecture under theta condi-
tions. In the presence of excluded volume interactions,
the form factor can be easily calculated within the
model considered here letting e ≠ 0. In the Gaussian
approximation, the form factor satis®es the general
relationship in terms of the mean square distance hr2n i
Pc�q� � 2
Z 1
0
dx�1ÿ x� exp ÿ q2
6hr2
n i� �
�33�
with x =n/N and the normalization Pc (q =0)=1.
Adopting the Bloom®eld±Zimm model, one ®nds
Pc�q� � 2
Z 1
0
dx�1ÿ x� exp ÿ� �1ÿ x�1�"x1�"
�1ÿ x�1�" � x1�"
!�34�
where m depends upon the excluded volume exponent
e as
� � q2 N1�"l2
6�35�
Within the present model of chain swelling, Pc (q) is
given by the following integral:
Pc�q� � 2
Z 1
0
dx�1ÿ x� exp ÿ�x1�"�1ÿ x�1�"� �
�36�
Figure 9 shows the theoretical Kratky plots of u P (q)
against���up
=qRgc. The present model predicts a sharp
peak and a pronounced upturn at high q' values
compared to the Bloom®eld±Zimm model. It is
interesting to note that these features (ie sharp peak
and strong upturn) were also observed in the scattering
experiments, but were attributed to the rigidity and
short size effects of the polymer chains.41,42 To check
Polym Int 49:175±183 (2000)
Figure 9. Kratky plot showing u P(q) versus���up
=qRgc: solid curves, thepresent model for ε=0.1 and 0.2 from the bottom upwards; dashed curves,Bloomfield-Zimm model for ε=0.1 and 0.2 from the bottom upwards; dottedcurve, Casassa function (ε=0).
Figure 10. Kratky plot showing u P(q) versus���up
=qRgc: solid curves, thepresent model for ε=0.1 and 0.2 from the bottom upwards; dashed curvesrepresent the linear chain case for the same values of ε; empty and filledsymbols are literature data for linear and cyclic chain systems, respectively.& and &, PS/d-cyclohexane neutron scattering data at 20°C (see refs 28,29 and 32); ~ and ~, PPMS/d-benzene neutron scattering data at 20°C(see refs 28, 29 and 32); * and *, PDMS/d-benzene neutron scatteringdata at 20°C (see refs 2 and 13).
Figure 11. The form factor of half cyclic copolymer Pc1/2(q) as a function ofthe wavenumber q (Aÿ1). The insert represents the structure factor ofsymmetric HD diblock copolymers as a function of q in the case of noninteracting blocks (�HD=0) and in the optical theta condition. Dashed andthick curves correspond to ε=0 (theta solvent) and ε=0.2 (good solvent),respectively.
Polym Int 49:175±183 (2000)
Cyclic polymers in good solvents
the consistency between our model and the data
available in the literature, we show in Fig 10
theoretical and experimental Kratky plots for cyclic
and linear polymer solutions. One observes that the
curves for cyclic polymers are below those of the linear
polymers. This observation is valid for the theoretical
prediction and is also true for the experimental data.
Therefore, we can conclude that the present model
predicts results consistent with the experimental
data.2,13,28±32 The Bloom®eld±Zimm model would
lead to curves slightly above ours, as we have
mentioned earlier, which are therefore perhaps less
consistent with the data shown in this ®gure.
THE STRUCTURE FACTORWhen the polymer concentration is high enough,
interferences between different chains strongly affect
the scattering signal. A large amount of information
exists on the effects of concentration for linear chain
solutions, in contrast to the case of cyclic polymers
where the combined effects of concentration and
excluded volume on the structure factor are not
known. An attempt is made here to examine this
problem in a particular system within the model of
chain swelling presented in eqn (26). It has been
suggested29,30 that by choosing the optical conditions
properly, it is possible to express the interchain
interferences directly in terms of the single chain form
factor. This method has been termed either the optical
theta condition when dealing with light scattering, or
the zero average contrast condition when analysing
small angle neutron scattering (SANS) data. From a
theoretical point of view, these two methods are
equivalent. Because these methods are well documen-
ted in the literature,43±46 we give directly the ®nal
result of the signal corresponding to a symmetrical
mixture of deuterated and ordinary polymers in a
common solvent at the volume fraction f
�aH ÿ aD�2Sc�q� � 4
�NPc�q� ÿ 2�HD �37�
where aH and aD are the scattering lengths of the
ordinary and deuterated monomers, respectively, and
wHD is their Flory±Huggins interaction parameter.
Interestingly, this result is similar to that of symme-
trical blends. Symmetric diblock copolymers have
received comparatively little attention. The scattering
signal Sc1/2 (q) corresponding to the optical theta
conditions is
�aH ÿ aD�2Sc1=2�q� �
4
�N �Pc1=2�q� ÿ Pc�q�� ÿ 2�HD �38�
where Pc (q) is the total chain form factor and Pc1/2 (q)
is the form factor of a single block (half chain). Figure
11 shows the variation of the latter quantity with and
without excluded volume interactions. In the presence
of excluded volume interactions, interferences of
scattered waves are stronger and the signal weaker.
181
A Bensa®, U Maschke, M Benmouna
The insert shows the total structure factor assuming
compatibility of the two blocks and wHD=0. The peak
of the dashed curve shifts to the right, indicating that
the wavelength of the fundamental mode of ¯uctua-
tions increases with excluded volume interactions.
The two curves cross each other showing that in the
small q range, the scattering signal is higher in good
solvents. Excluded volume generates a long range
interaction which is better identi®ed at small q values.
CONCLUSIONSThermodynamic and structural properties of cyclic
polymers under good solvent conditions are discussed.
Two models describing chain swelling under the
in¯uence of excluded volume interactions are used to
calculate properties of chains related to their size,
conformation and structure. The ®rst model was
suggested long ago by Bloom®eld and Zimm while
the second is a modi®ed Yu±Fujita model to ful®ll the
circularity condition. This modi®cation is introduced
in the present work and describes the stretching of
rings under good solvents assuming that the mean
square distance between two points along the chain
scales with the quantity n[1ÿ(n/N)] following a power
law de®ned by the exponent 1�e, where e is allowed to
have values ranging from zero to 0.2.
Predictions of these two models are compared, and
differences between linear and cyclic chain polymers
are highlighted via thermodynamic and structural
properties. The results are also compared with
computer simulations, neutron and light scattering
data.
A complete understanding of the physical properties
of cyclic polymers remains a challenging task. The
dif®culty is inherent essentially in the chemistry of
polymer rings and the problem of synthesizing long
cyclic chains with well de®ned molecular weight and
high yield. When dealing with excluded volume
effects, an additional dif®culty arises because a
compromise should be reached between the weak
excluded volume interactions at low molecular weight
and the decreasing effects of chain-ends as the mol-
ecular weight increases. One should ®nd the appro-
priate range of molecular weights suitable for the
capabilities of ring polymer synthesis and the strength
of excluded volume interactions leading to distinct
physical properties as compared to the theta state and
the behaviour of analogous linear polymer systems.
When this compromise is reached, the present
theoretical calculations could give the appropriate
framework for the analysis of thermodynamics, con-
formation and structural properties of ring polymers in
solution.
It is clear that the Yu±Fujita model cannot be
retained as a possible description of chain swelling
under good solvent conditions. The Bloom®eld±
Zimm model tends to overestimate the effects of
excluded volume, predicting a chain swelling close to
that of linear chains. Comparison between experi-
182
mental data and computer simulations seems to favour
the present model which gives results consistent with
these data.
ACKNOWLEDGEMENTSMB acknowledges interesting discussions and useful
comments from Professor TA Vilgis of the Max-
Planck Institut fuÈ r Polymerforschung (MPI-P, Mainz)
and from Professor JF Joanny from the Centre de
Recherche sur les MacromoleÂcules (CRM, Stras-
bourg). This paper was written while MB was staying
at the MPI-P. He expresses his gratitude to Professor
Kurt Kremer for his kind invitation.
REFERENCES1 Weil R and Vinograd J, Proc Natl Acad Sci USA 50:730 (1963).
2 Semlyen JA, in Cyclic Polymers, Ed by Semlyen JA, Elsevier,
London. Chapter 1. (1986).
3 Prentis JJ, J Chem Phys 76:1574 (1982).
4 Jagodzinski O, Eisenriegler E and Kremer K, J Phys I, France
2:2243 (1992).
5 Pakula T and Jeszka K, Macromolecules, 32:6821 (1999).
6 Flory PJ and Jernigan RL, J Chem. Phys 42:3509 (1965).
7 Pakula T, Macromolecules 20:679, 2909 (1987).
8 Pakula T and Geyler S, Macromolecules 20:2909 (1987).
9 Geyler S and Pakula T, Makromol Chem Rapid Commun 9:617
(1988).
10 Croxton CA, Macromolecules 25:4352 (1992).
11 Edwards CJC, Stepto RFT and Semlyen JA, Polymer 23:869
(1982).
12 Edwards CJC, Rigby D, Stepto RFT, Dodgson K and Semlyen
JA, Polymer 24:391, 395 (1983).
13 Edwards CJC and Stepto RFT, in Cyclic Polymers, Ed by
Semlyen JA, Elsevier, London. Chapter 4 (1986).
14 Cates ME and Deutsch JM, J Phys (Paris) 47:2121 (1986).
15 MuÈller M, Wittmer JP and Cates ME, Phys Rev E 53:5063
(1996).
16 Santore MM, Han CC and McKenna GB, Macromolecules
25:3416 (1992).
17 Kholkhov AR and Nechaev SK, J Phys II, France 6:1547 (1996).
18 Benmouna M, Khaldi S, Bensa® A and Maschke U, Macromol-
ecules 30:1168 (1997).
19 LeÂonard J, J Phys Chem 93:4346 (1989).
20 LeÂonard J, J Polym Sci Polym Phys Ed 31:1496 (1993).
21 Flory PJ, Introduction to Polymer Chemistry, Cornell University
Press, Ithaca (1956).
22 Benmouna M, Bensa® A, Vilgis TA, Maschke U and Ewen B,
Recent Res Dev Polym Sci 1:175 (1997).
23 Candau F, Rempp P and BenoõÃt H, Macromolecules 5:627 (1972).
24 Hild G, Strazielle C and Rempp P, Eur Polym J, 16:843 (1983).
25 Lutz P, McKenna GB, Rempp P and Strazielle C, Makromol
Chem Rapid Commun 7:599 (1986).
26 McKenna GB, Hadziioannou G, Lutz P, Hild G, Strazielle C,
Straupe C, Rempp P and Kovacs AJ, Macromolecules 20:498
(1987).
27 Higgins JS, Ma K, Nicholson LK, Hayter JB, Dodgson K and
Semlyen JA, Polymer 24:793 (1983).
28 Hadziioannou G, Cotts PM, ten Brinke G, Han CC, Lutz P,
Strazielle C, Rempp P and Kovacs AJ, Macromolecules 20:493
(1987).
29 ten Brinke G and Hadziioannou G, Macromolecules 20:3 (1987).
30 Roovers J and Toporowski PM, Macromolecules 16:843 (1983).
31 Roovers J, J Polym Sci Polym Phys Ed 23:1117 (1983).
32 ten Brinke G and Hadziioannou G, Macromolecules 20:480
(1987).
33 Yamakawa H, Modern Theory of Polymer Solutions, Harper and
Row, New York (1971).
Polym Int 49:175±183 (2000)
Cyclic polymers in good solvents
34 Zimm BH, Stockmayer WH and Fixmann M, J Chem Phys
21:1716 (1953).
35 Weill G, Loucheux C and BenoõÃt H, J Chim Phys 55:540 (1958).
36 Peterlin A, J Chem Phys 23:2462 (1955).
37 Bloom®eld VA and Zimm BH, J Chem Phys 41:315 (1966).
38 Weill G and Des Cloizeaux J, J Phys (Paris) 40:99 (1979).
39 Yu H and Fujita H, J Chem Phys 52:1115 (1970).
40 Casassa EF, J Polym Sci Part A 3:605 (1965).
41 Burchard W and Schmidt M, Polymer 21:745 (1980).
Polym Int 49:175±183 (2000)
42 Burchard W, in Cyclic Polymers, Ed by Semlyen JA, Elsevier,
London. (Chapter 2) (1986).
43 Higgins J and BenoõÃt H, Neutron Scattering and Polymers,Oxford
University Press, Oxford (1994).
44 Benmouna M and Hammouda B, Prog Polym Sci 22:49 (1997).
45 Benmouna M, Borsali R and BenoõÃt H, J Phys II France 3:1041
(1993).
46 Borsali R, BenoõÃt H, Legrand JF, Duval M, Picot C, Benmouna
M and Farago B, Macromolecules 22:4119 (1989).
183