Synthesis and characterization of model cyclic block copolymers of styrene and butadiene. Comparison of the aggregation phenomena in selective solvents with linear diblock and triblock analogues

Article abstract of DOI:10.1021/ma0121565

The synthesis and characterization of model cyclic diblock copolymers of styrene (St) or perdeuterated styrene (St-d₈) and butadiene (Bd) are presented. Since conventional methods of characterization cannot separate completely the cyclic copolymer from its linear precursor, differences in the micellar behavior were used as a method for investigation of their purity. For this purpose, in addition to the cyclic and linear triblock copolymers, two linear diblocks with similar compositions, and molecular weight equal or half of the cyclic diblocks, were also synthesized. The synthetic approach of the cyclics involved the reaction of (1,3-phenylene)bis(3-methyl-1-phenylpentylidene)dilithium initiator with butadiene in the presence of sec-BuOLi, followed by polymerization of St (or St-d₈). The cyclization of the resulting α,ω-difunctional triblock copolymer was performed by using bis(dimethylchlorosilyl)-ethane, under high dilution conditions. The copolymers were characterized by size exclusion chromatography, membrane osmometry, NMR and UV spectrometry, and viscometry. The micelles formed in the selective solvents n-decane (for PBd) and dimethylformamide (for PS-d₈) were characterized by small-angle neutron scattering and dynamic light scattering. It was found that the aggregation number of the cyclic copolymers was the smallest among the different macromolecular architectures. Moreover, the SANS data for the triblocks in n-decane indicated the presence of 37% dangling chains which did not appear in the data for the corresponding cyclic copolymers. Considering that 5% of dangling chains is possible to be detected, it proves that the cyclic copolymers are at least 87% pure. A scaling model was used in order to justify the difference in the aggregation numbers between the four different copolymers.

Full text of DOI:10.1021/ma0121565

5
426
Macromolecules 2002, 35, 5426-5437
Synthesis and Characterization of Model Cyclic Block Copolymers of
Styrene and Butadiene. Comparison of the Aggregation Phenomena in
Selective Solvents with Linear Diblock and Triblock Analogues
Her m is Ia tr ou a n d Nik os Ha d jich r istid is*
Department of Chemistry, University of Athens, Panepistimiopolis, 15771 Athens, Greece
Ger d Meier , Hen r ich F r ielin gh a u s, a n d Mich a el Mon k en bu sch
Institut f u¨ r Festk o¨ rperforschung, Forscunszentrum J u¨ lich GmbH, D-52425 J u¨ lich, Germany
Received December 11, 2001; Revised Manuscript Received April 1, 2002
ABSTRACT: The synthesis and characterization of model cyclic diblock copolymers of styrene (St) or
perdeuterated styrene (St-d ) and butadiene (Bd) are presented. Since conventional methods of
8
characterization cannot separate completely the cyclic copolymer from its linear precursor, differences
in the micellar behavior were used as a method for investigation of their purity. For this purpose, in
addition to the cyclic and linear triblock copolymers, two linear diblocks with similar compositions, and
molecular weight equal or half of the cyclic diblocks, were also synthesized. The synthetic approach of
the cyclics involved the reaction of (1,3-phenylene)bis(3-methyl-1-phenylpentylidene)dilithium initiator
8
with butadiene in the presence of sec-BuOLi, followed by polymerization of St (or St-d ). The cyclization
of the resulting R,ω-difunctional triblock copolymer was performed by using bis(dimethylchlorosilyl)-
ethane, under high dilution conditions. The copolymers were characterized by size exclusion chromatog-
raphy, membrane osmometry, NMR and UV spectrometry, and viscometry. The micelles formed in the
8
selective solvents n-decane (for PBd) and dimethylformamide (for PS-d ) were characterized by small-
angle neutron scattering and dynamic light scattering. It was found that the aggregation number of the
cyclic copolymers was the smallest among the different macromolecular architectures. Moreover, the SANS
data for the triblocks in n-decane indicated the presence of 37% dangling chains which did not appear in
the data for the corresponding cyclic copolymers. Considering that 5% of dangling chains is possible to
be detected, it proves that the cyclic copolymers are at least 87% pure. A scaling model was used in order
to justify the difference in the aggregation numbers between the four different copolymers.
1
. In tr od u ction
Since the discovery in 1962 that certain DNA mol-
cal method has been presented so far. Moreover, as far
as we know, only one complete series of macrocyclic
diblock copolymers of styrene and dimethylsiloxanes has
1
ecules occur in nature in a closed loop form, many
attempts have been made to synthesize well-defined
8
been prepared by Hogen-Esch and Yin.
macromolecular cyclic polymers.2
-10
Diblock copolymers in a selective solvent exhibit
surfactant-like behavior, and their study can enhance
the understanding of the relation between molecular
structure and properties such as detergency, surfac-
tancy, colloidal dispersion, and stabilization. Although
many studies have been presented on the micellar
behavior of block copolymers in a selective solvent,1
only a few compare the micellar characteristics of
copolymers exhibiting different macromolecular archi-
tectures but similar molecular weight and composi-
Due to the absence
of free chain ends, the monomeric units of cyclic homo-
polymers are equivalent. Moreover due to the cyclic form
of the molecules, the intramolecular interactions are
increased, compared to a linear analogue. As a conse-
quence, in dilute solution, the second virial coefficient,
4-21
the dimensions, and the intrinsic viscosity2
-3,11
are
reduced. In bulk, the cyclic architecture influences the
glass transition temperature (Tg) of homopolymers, the
morphology, and the order-disorder transition of the
copolymers. The above characteristics make the cyclic
macromolecules very interesting.
Two general synthetic approaches have been reported
up to now, based either on a R,ω-homodifunctional or
R,ω-heterodifunctional polymeric chains. In the first
case the cyclization requires a difunctional coupling
agent, while in the second the cyclization requires some
type of activation for the intramolecular reaction.
Although a large variety of different functional groups
has been used for the synthesis of cyclic homo- and
copolymers, the purity of the final products are always
in question, since only limited separation methods can
separate completely cyclic homopolymer from the linear
8
2
2-26
26
1
2
tion.
micellar behavior of cyclic block copolymers of poly-
ethylene oxide) and propylene oxide in water. They
Booth et al. presented the synthesis and
(
compared the aggregation numbers and dimensions of
these copolymers with the triblock precursors and the
linear diblock analogues.
In this work the synthesis and characterization of a
series of cyclic polystyrene (PS) (or perdeuterated
polystyrene, PS-d )-b-polybutadiene (PBd) copolymers
8
with a PS fraction ranging from 15 to 70% (w/w) is
presented. Moreover, the synthesis of the linear triblock
PS (or PS-d )-b-PBd-b-PS and (PS-d )-b-PBd analogues
8
8
with either the same molecular weight and composition
(for the case of the triblocks and diblocks) or half the
molecular weight but similar composition (for the case
of diblocks) is also presented. All block copolymers have
been characterized by SEC, UV spectrometry, viscom-
1
3
precursor. In the case of cyclic copolymers, no analyti-
*
To whom correspondence should be addressed, e-mail:
hadjichristidis@chem.uoa.gr.
1
0.1021/ma0121565 CCC: $22.00 © 2002 American Chemical Society
Published on Web 06/04/2002

Macromolecules, Vol. 35, No. 14, 2002
Model Cyclic Block Copolymers 5427
Sch em e 1
etry, and membrane osmometry. Since these analytical
methods are not sufficient to determine the purity of
cyclic copolymers, the micellar behavior of the linear
analogues was examined and compared with the be-
havior of the cyclics.
Sch em e 2
The molecular characteristics of the micelles were
obtained by using small-angle neutron scattering (SANS)
and dynamic light scattering (DLS) in n-decane and
N,N-dimethylformamide (DMF). n-Decane is a selective
solvent for PBd, while DMF is a selective solvent for
PS or PS-d8. By using mixtures of deuterated and
protonated solvents it was possible, in most cases, to
match exactly either the corona (shell) or the core of the
micelle and measure separately the structural proper-
ties of each component by SANS. The influence of the
macromolecular architecture on the micellar behavior
is also investigated, by comparing the dimensions of the
micelles formed by the linear analogues and the cyclic
block copolymer, of one composition, in two different
solvents.
Finally, the experimental observations are discussed
in terms of scaling models. The discussion is based upon
the differentiation of the free energy of the micelles
according to the macromolecular architecture; i.e., the
formation of loops in the core or the corona of the micelle
introduces an entropic penalty to the free energy of the
micelle, compared to the corresponding linear diblock
copolymer.
respectively. sec-BuLi was used as the activator of PEB, for
the preparation of (1,3-phenylene)bis(3-methyl-1-phenyl-
pentylidene)dilithium initiator (DLI).
The basic reactions used for the synthesis of the cyclic
copolymers are shown in Scheme 1.
A 5% solution (w/v) of an R,ω-difunctional PBd homopolymer
in benzene was synthesized by the sequential addition of sec-
BuOLi and butadiene to the DLI solution. The molar ratio of
sec-BuOLi/C-Li was 4/1. The polymerization of Bd took place
at 20 °C for 6 days. Then a small aliquot was removed for
2
. Exp er im en ta l Section
Syn th esis of th e Cop olym er s. Styrene (99%), styrene-d
98 atom %), butadiene (99%), n-butyllithium (1.6 M) in
hexanes, bis(dimethylchlorosilyl)ethane (BDCSE) (98%), ben-
zene (99%), n-decane (99%), DMF (99%), n-decane-d22 (99 atom
8
8
characterization and the appropriate amount of St (or St-d )
(
was added. It was found that about 1 h was required for the
crossover reaction and the polymerization was required three
more days to reach completion. Then a small amount of
butadiene was added in order to introduce three to four units
of Bd at each living end. Part of the resulted R,ω-difunctional
PS-b-PBd-b-PS triblock copolymer was neutralized by degassed
methanol (MeOH) and the rest was inserted in an ampule
equipped with a break-seal for the cyclization reaction.
The reactor used for the cyclization reaction is shown in
Scheme 2.
%
7
D) and DMF-d (99 atom % D) were purchased from Aldrich.
Purification of monomers and solvents to the standards
required for anionic polymerization has been described in
2
7
detail elsewhere. Methyllithium (Aldrich), triphenyl meth-
ylphosphonium iodide (98% Alpha) and 1,3-bis(1-phenone)
benzene (PBOP) (98%, Aldrich), were used for the preparation
of 1,3-bis(1-phenylethenyl)-benzene (PEB) according to the
2
8
method of Ignatz-Hoover. sec-Butyllithium (sec-BuLi) and sec-
butoxylithium (sec-BuOLi) were prepared in vacuo either from
sec-butyl chloride or sec-butanol and lithium dispersion,
The two ampules were attached in a way that each reactant
could be added separately. Prior the cyclization reaction, the
reactor was purged with a solution of n-BuLi in benzene. The

5
428 Iatrou et al.
Macromolecules, Vol. 35, No. 14, 2002
Ta ble 3. Molecu la r a n d Com p osition a l Ch a r a cter istics of
th e Diblock An a logu es of th e C22-28 Sa m p les
Mn(PS-d8) Mn(diblock) % PS-d8
I )
% PS-d8
3
a
3a
sample
× 10
× 10
from Mn Mw/Mn by UV
D22-28
23.2
14.0
44.0
26.0
52
54
1.03
1.03
54
56
(
D22-28)/2
a
Membrane osmometry in toluene at 37 °C.
Ta ble 4. Viscosity Mea su r em en ts of th e Tr iblock
Cop olym er s a n d th e Cor r esp on d in g Cyclics, in Tolu en e
a t 30 °C
triblock
[η] (mL/g)
diblock
[η] (mL/g)
sample
kH
kH
C45-5
71.8
74.9
64.2
51.6
48.0
0.39
0.41
0.37
0.39
0.38
53,1
56.1
47.8
38.8
35.5
0.54
0.56
0.55
0.42
0.46
C40-20
C28-22
C22-28
C15-35
F igu r e 1. SEC chromatograms during the synthesis of the
cyclic copolymer C22-28: (a) PBd homopolymer; (b) (PS-d )-
b-PBd-b-(PS-d ) triblock copolymer; (c) reaction products after
the cyclization reaction and addition of the high molecular
weight PBd chain; (d) fractionated cyclic copolymer; dotted
line: linear precursor (b).
8
The corresponding linear diblock copolymers of the sample
C22-28 were prepared by the sequential addition to the
initiator sec-BuLi, St (or St-d ) and butadiene at room tem-
perature. The polymerization of St-d was left for completion
8
8
8
for 24 h, and then a small amount of sec-BuOLi was added in
a molar ratio of [sec-BuOLi]/[CLi] 4/1 (in order to have the
same microstructure as the cyclics), followed by the addition
of the appropriate amount of butadiene. The mixture was left
for another 6 days, and the living chains were neutralized by
using degassed MeOH. The molecular characteristics of the
diblocks are given in Table 3.
Ta ble 1. Molecu la r Ch a r a cter istics of th e Cyclic
Cop olym er s a n d th e Lin ea r P r ecu r sor s
Mn(PBd) Mn(triblock)
Mw/Mn
(triblock)
Mn(cyclic) Mw/Mn
× 10
3
b
3 b
c
3 b
(cyclic)c
sample
× 10
× 10
C45-5
42.0
41.5
32.0
23.0
16.1
47.5
55.9
55.1
53.0
56.9
1.06
1.07
1.08
1.10
1.15
47.2
56.5
53.5
53.4
58.0
1.06
1.07
1.09
1.11
1.16
C40-20
C28-22
C22-28
C15-35
Ch a r a cter iza tion of th e Cop olym er s. SEC experiments
were carried out at 25 °C using a Waters 610 pump, Waters
model 410 differential refractometer, 996 diode-array UV
detector and six columns with a continuous porosity range
a
a
a
a
b
from 10 to 10 Å. THF was the carrier solvent. The number-
average molecular weights (M ) of the precursors and the final
2
6
Polystyrene block is perdeuterated. Membrane osmometry
c
in toluene at 37 °C. Size exclusion chromatography in THF at
n
2
5 °C.
products were determined by using a J upiter model 231
recording membrane osmometer (MO) at 37 °C. The solvent
Ta ble 2. Com p osition of th e Cyclic Cop olym er s a n d th e
Lin ea r P r ecu r sor s
was toluene distilled over CaH
2
. Nuclear magnetic resonance
(NMR) spectra, recorded on a Bruker 300 MHz instrument
3
with CDCl as the solvent, at 25 °C, were used for the
%
(w/w)
by NMR
PS
% PS
(w/w)
by NMR
% PS
(w/w)
by UV
% PS
(w/w)
by UV
determination of the composition and the microstructure of
PBd. In this case, due to the presence of sec-BuOLi, the 1,2-
addition is slightly higher (13%) than the conventional (8%)
one.
%
PS
from Mn
(w/w)
sample
(triblocks) (cyclics) (triblocks) (cyclics)
C45-5
12
26
42
56
72
12
28
a
14
29
a
a
a
14
30
43
54
70
15
32
44
55
73
Viscometric data were analyzed using the Huggins equation:
C40-20
C28-22
C22-28
C15-35
2
a
a
η /c ) [η] + k [η] c + ...
(1)
(2)
sp
H
a
and the Kraemer equation
N/A, deuterated polystyrene.
2
lnη /c ) [η] + k [η] c + ...
r
K
procedure of the addition of the living chains and the linking
agent was as follows: 3-5 mL of the 5% solution was diluted
into 1.6 L of solvent, followed by the addition of a stoichio-
metric amount of linking agent. The reagents were left to react
for 20 min, and the procedure was repeated until all living
chains were inserted into the reactor. The addition was
performed in many steps over about 3 days. After the comple-
tion of the addition, the reaction mixture was left for another
where η
r
, ηsp, and [η] are the relative, specific, and intrinsic
and k are the Huggins and
viscosities, respectively, and k
H
K
Kraemer constants, respectively. Virtually identical intrinsic
viscosities were obtained by the two methods. All the measure-
ments were carried out at 30 °C in toluene using Cannon-
Ubbelohde dilution viscometers with a Schott-Ger a¨ te AVS 410
automatic flow timer. The viscometry results are shown in
Table 4.
3
days to react completely, and a slight excess of linking agent
(
∼10% mol/mol) was introduced. Then a large amount of high
Ch a r a cter iza tion of th e Micelles. Solu tion P r ep a r a -
tion in Selective Solven ts. Analytical grade n-decane was
molecular weight PBd living polymer (∼200 kg/mol) was
introduced in a molar ratio of living ends [PBdLi]/[LiPS-PBd-
PSLi] equal to 1:1. The reaction mixture was left for another
refluxed for 24 h over CaH
before used. DMF was dried with NaOH for 24 h and refluxed
over MgSO for another 24 h under nitrogen atmosphere. Stock
2
and fractionally distilled just
7
days, the reactor was opened, and the reaction mixture was
4
fractionated by using toluene/MeOH as the solvent/nonsolvent
pair. The synthetic procedure as well as the fractionation was
monitored by SEC. A typical example is given in Figure 1.
The total molecular weight of the triblock precursors (and
the cyclics) was approximately between 47 and 57 kg/mol. The
molecular characteristics are shown in Tables 1 and 2.
solutions were prepared by dissolving a weighted amount of
sample in the appropriate volume of the solvent with oc-
casional stirring. All samples were dissolved after 24 h at 60
°C. No polymer precipitation was observed from these solutions
after standing at room temperature for several weeks. Before
dynamic light scattering experiments, the solutions were

Macromolecules, Vol. 35, No. 14, 2002
Model Cyclic Block Copolymers 5429
Ta ble 5. Sca tter in g Len gth Den sities (Gi) of P olym er ic
avoid backscattering, under shell contrast, 2 mm quartz cells
were used (transmissions: 70-80%), whereas, under core
contrast, 1 mm quartz cells were used (transmissions: 45 and
Block s a n d Solven ts
Fi × 10-10 (cm-2
polymeric chain or solvent
)
5
5%).
When DMF was the selective solvent, it was possible to
PBd
PS-d8
0.420
6.36
match exactly the shell of the micelles, since deuterated DMF
has practically the same scattering length density with PS-
protonated n-decane
deuterated n-decane
protonated DMF
deuterated DMF
-0.489
6.58
d
8
. On the contrary, it was impossible to match exactly the
0.698
6.37
core (PBd) of the micelles, since the ∑b /v of PBd is much lower
i
i
than the one of protonated DMF (Table 5). To ensure that the
fits of the data under shell contrast are correct, one additional
contrast was measured. As a consequence, three different
contrasts were used, one in 100% (v/v) protonated DMF, one
filtered through 0.45 µm nylon filters. Deuterated solvents
were used as purchased from Aldrich, without further puri-
fication. In each case, a 1% (w/w) stock solution was prepared
and diluted to the appropriate concentration.
Sm a ll-An gle Neu tr on Sca tter in g Exp er im en ts. If the
interparticle scattering contributions are neglected, the coher-
ent macroscopic scattering cross section of a SANS experiment
in 100% (v/v) DMF-d
7
(deuterated DMF), and one with 50/50%
(v/v) protonated/deuterated DMF. One concentration was
measured for each contrast approximately 0.50% (w/v), since
it was found that in n-decane no structure factor was developed
even at 1.0% (w/v) concentration.
In the case of samples with 100% (v/v) deuterated DMF and
in the 50/50% (v/v) mixture of solvents, 2 mm quartz cells were
used, whereas solutions with 100% (v/v) protonated solvent 1
mm quartz cells were used. The transmissions were between
(
dΣ/dΩ)(Q) is given by
dΣ
dΩ
NZ
2
(Q) )
〈|A(Q)| 〉
(3)
V
7
5
5 and 85% in the first case, 55-65% in the second, and 45-
5% in the third case.
Z
where N denotes the number of scatterers, V is a reference
All SANS experiments were performed at 25 °C, on the
volume and A(Q) the intraparticle scattering amplitude. The
scattering vector Q is given by 4πsin(θ/2)/λ, where θ is the
scattering angle and λ the neutron wavelength.
For a particle consisting of a core and a shell, the scattering
amplitude AC,Sh(Q) can be written as follows:
KWSII instrument at the research reactor FRJ 2 at the
Forschungszentrum J u¨ lich GmbH. Three sample detector
distances were used: 2, 8, and 18 m. For all detector distances,
the neutron wavelength was 6.3 Å. Thus, a range of scattering
-
3
-1
-1 -1
vector of 2 × 10
Å
e Q e 1.0 × 10
Å
was covered. The
wavelength spread was ∆λ/λ ) 18%. The raw data were
corrected for background by subtracting the scattering of
empty cell. These data were then corrected for different
detector cell efficiency and calibrated to absolute units by a
AC,Sh(Q) ) V (F - F )A (Q) + V (F - F )A (Q) (4)
M
Sh
S
M
C
C
Sh
C
Here F
C
S
, FSh, and F are the scattering length densities of the
core (C), shell (Sh) and the solvent (S), respectively. The
scattering length density of a component i can be calculated
by the following equation:
29
Lupolene secondary standard according to
S
L
L
L
L
S 2
dΣ
T D (dΣ /dΩ)A (R )
S
S
S
(Q) )
(I - T B )
(9)
S
S
L
S
L 2
d(Ω)
bz
T D I A (R )
∑
Fi )
(5)
v
i
Here L and S refer to the Lupolene standard and sample, and
D and T stand for thickness and transmission, respectively.
A is the illuminated sample area, and R is the sample to
detector distance. I is the intensity, and B is the experimental
where b
z and v
z
i
is the coherent scattering length density of the atom
the respective volumes. V , V , A (Q), and A (Q)
M
C
M
C
denote the volumes of the overall micelle the core, and the
corresponding scattering amplitudes. The scattering length
densities of the polymeric blocks and the solvents used are
L
L
L
background. The value T D (dΣ /dΩ) for the Lupolene stan-
dard has been calibrated with vanadium to be (7.34 ( 0.03) ×
-2
1
0
.
given in Table 5. In the case FSh ) F
transformed into
S
or F
C
) F
S
, then eq 4 is
Dyn a m ic Ligh t Sca tter in g. DLS experiments were car-
ried out at 25 °C on a series 4700 Malvern system composed
of a PCS5101 goniometer with a PCS7 stepper motor control-
AC,Sh(Q) ) V (F - F )A (Q)
(6)
+
C
C
Sh
C
ler, a Cryonics variable power Ar laser, operating at 488 nm
and with (10 mW) power, a PCS8 temperature control unit,
and a RP98 pump/filtering unit. A 192-channel correlator was
used for accumulation of the data. Correlation functions were
analyzed by the cumulant method and the Contin software,
provided by the manufacturer. The correlation function was
collected at angles between 45 and 135°. There was no
indication of the presence of unimers from Contin analysis at
the concentration range studied for each sample (0.1-1.5%
w/v). In this region the equilibrium is shifted in favor of
micelles and the properties measured corresponds to those of
the micelles. The solutions for SANS measurements were
or
AC,Sh(Q) ) (F - F )(A (Q)V - A (Q)V )
(7)
C
Sh
C
C
M
M
The first situation (eq 6) will be referred as core contrast
whereas the second (eq 7) is referred to as shell contrast.
When n-decane was used as a selective solvent, two different
contrasts were used, the core and the shell contrasts, by using
solvent mixtures of deuterated and protonated n-decane. This
was possible since the coherent scattering length density of
8
PBd and PS-d is between the protonated and deuterated
within the same concentration range. The ratio µ
2
/Γ, where µ
2
n-decane (Table 5). The scattering length density of a mixture
of protonated and deuterated solvent follows, with sufficient
accuracy, a simple additivity rule:
is the second cumulant and Γ the decay rate was <0.1 for all
concentrations and angles, indicating the low polydispersity
of the micelles. Apparent diffusion coefficients at zero concen-
tration were obtained by using the following equation:
∑vb
v
bD
v
bH
M
∑
+ (1 - x)∑
)
x
(8)
Dapp ) Do,app(1 + k c)
(10)
D
where x is the volume fraction of the deuterated n-decane.
where Dapp is the diffusion coefficient measured at each
concentration and k the coefficient that gives the dependence
of the diffusion coefficient with concentration. Apparent
hydrodynamic radii, R were determined by eq 11, where k
is the Boltzmann constant, T the absolute temperature and η
Thus, core contrast (F
M
) F
s
) is obtained with x ) 0.129 and
D
shell contrast with x ) 0.97 of deuterated n-decane. Three
different concentrations, approximately 0.25, 0.50, and 1.0%
h
B
(w/v) were measured for each copolymer at each contrast. To
0

5
430 Iatrou et al.
Macromolecules, Vol. 35, No. 14, 2002
the viscosity of the solvent.
in benzene which is a very good solvent for both blocks,
PS and PBd. As a consequence, the copolymer chains
are swollen during the cyclization and the minimum
amount of permanent knots are expected to be formed.
The polydispersity index of the cyclics was equal within
experimental error to that of the corresponding linear
precursors.
k T
B
R )
(11)
h
6
πη D
o
o,app
3
. Resu lts a n d Discu ssion
Syn th esis a n d Ch a r a cter iza tion of Cyclic a n d
One indication of the formation of cyclic copolymers
is the lower molecular weight obtained by SEC com-
pared to the linear precursors. The ratio of MpC/MpT )
Lin ea r Block Cop olym er s. s-BuOLi was used as an
additive in order to reduce the propagation rate of the
polymerization of butadiene and to produce PBd with
0
.73-0.75 in all five copolymers, where MpC and MpT
3
0
low polydispersity, as was shown in a previous paper.
are the peak molecular weights of the cyclic and triblock,
respectively. This is in good agreement with the values
found by Leppoittevin et al. for ring PS homopolymers.
The polymerization started from Bd and continued with
styrene and not vice versa because the polymerization
of styrene by DLI requires the presence of a strong
promoter such as THF, which influences the micro-
structure of the PBd block. Finally the transformation
of the styrenic to the dienic anion is necessary in order
7
In that work, the purification of the cyclic copolymers
was performed by using preparative liquid chromato-
graphic separation, and information for the chemical
structure was obtained by using matrix-assisted laser
desorption/ionization time-of-flight mass spectrometry
to facilitate the linking reaction (less steric hindrance).
Ma31 has used a similar approach for the synthesis
(
MALDI-TOF). The formation of the cyclic copolymers
of cyclic PS-b-PBd copolymers. In this synthetic route,
the cyclization was performed by adding dropwise a
linking agent solution to the diluted R,ω-difunctional
living triblock copolymer. Two different linking agents
were used, either dichlorodimethylsilane or PEB. By
Ma’s way the linking agent is always added in an excess
of living chains, increasing the probability of forming
coupling products and as a consequence, lowering the
yield of cyclization.
is also supported by the presence of a peak at 0.3 ppm
in the H NMR spectrum, which is attributed to the 12
1
protons of the four -CH3 groups of the linking agent
attached to the cyclic copolymer.
In our case, the formation of the cyclic structure is
also supported by g′ ) [η]c/[η]l, where [η]c and [η]l are
the intrinsic viscosity of the cyclic and the corresponding
linear precursor before the cyclization (Table 3). g′
values between 0.73 and 0.75 were obtained. This ratio
is higher than the values of 0.66-0.68 found by Roovers
In our work, high dilution of the two reagents was
always maintained. The total volume of the solvent was
2
7
1
.8 L, keeping the concentration of the living chain ends
et al. and Leppoittevin et al. However, an expansion
to the dimensions of a cyclic copolymer is expected since
an increased number of heterocontacts is expected to
be present between incompatible polymeric blocks.
-
5
in a concentration range of about 3 × 10 mol/L. This
value is lower than the equilibrium concentration, Ceq
3
2
3/2
2,32
)
( /2π〈r 〉) M/NA,
which is the concentration in
3
5
which the probability of intra- and intermolecular
Iatrou et al. found a similar influence of the hetero-
contacts on the dimensions of 3-miktoarm stars of the
-
4
2
reaction is equal (∼10 mol/L). 〈r 〉 is the mean square
end-to-end distance of the chain, M is the corresponding
molecular weight and NA is the Avogadro number. For
the triblock copolymers, the sum of the end-to-end
distance of the three separate blocks was used.
The “actual” concentration of the reagents was always
kept lower than Ceq, provided that stoichiometric
amounts are always added, and the reaction is complete
within the time intervals of each step.
A B type in dilute solution. In that work, a higher
2
intrinsic viscosity was found for the miktoarm stars
compared to the intrinsic viscosity of the corresponding
3
1
star homopolymers. Ma, for a copolymer with similar
molecular characteristics, found a value of 0.79. The kH
constants of the cyclics were significantly higher which
indicates that the interactions between the polymeric
chains are increased compared to the ones of the triblock
copolymer precursor. Since there are not other results
concerning the g′ and kH values of a diblock cyclic
copolymer, it cannot be used as a reliable parameter for
the evaluation of their purity.
To ensure complete reaction within the time of each
addition, BDSCE (the two chlorine atoms on two dif-
ferent Si atoms) was used as the linking agent. It is well
established that two Si-Cl groups of each BDCSE
molecule react very quickly with C-Li anions and
present almost the same reactivity, even if one is reacted
with a living polymeric chain.33 As a consequence, the
Si-Cl group that remains after the reaction of a
difunctional living chain with one molecule of difunc-
tional linking agent exhibits reactivity similar to that
of the Si-Cl group of a free linking agent. Therefore,
the probability of cyclization compared to the reaction
of both functional groups of the polymeric chain with
two different molecules of linking agent is equal. If Me2-
SiCl2 (the two chlorine atoms on the same Si) was used
as linking agent, the Si-Cl group attached to a poly-
meric chain, for steric hindrance reasons, reacts slower
than that of the free linking agent, and therefore more
time is required between the addition of the two
reagents.34
Ch a r a cter iza tion of th e Micelles. The linear tri-
block and the corresponding cyclic copolymers C22-28,
D22-28, and (D22-28)/2 (Tables 1 and 2) constitute a
series with similar molecular characteristics (composi-
tion and total molecular weights, except (D22-28)/2,
which has half the molecular weight of the others. This
will be referred as half-diblock). To investigate the
purity of the cyclics and the influence of the macromo-
lecular architecture to the micellar behavior, the mo-
lecular characteristics of the micelles were obtained by
SANS and DLS. In n-decane, the collapsed PS-d8 chains
form the core and the dissolved PBd chains are in the
shell of the micelle that keep the aggregate in solution,
whereas in DMF the situation is reversed (Scheme 3).
SANS Mea su r em en ts in n -Deca n e. The coherent
macroscopic scattering cross sections (dΣ/dΩ) for the
cyclic copolymer C22-28, the linear precursor, and the
corresponding diblocks with the same and half the
Approximately 5 g of the triblock precursor was used
for cyclization. The overall yields of the cyclic copolymers
were 30-40%. The cyclization reaction was performed

Macromolecules, Vol. 35, No. 14, 2002
Model Cyclic Block Copolymers 5431
F igu r e 2. Scattering cross section dΣ/dΩ vs Q under core (0)
and shell (O) contrasts of the cyclic copolymer C22-28 in
n-decane at 25 °C and concentration 0.50% (w/v). The solid
line represents the calculated cross sections, and the dots
represent the experimental data.
F igu r e 3. Scattering cross section dΣ/dΩ vs Q under core (0)
and shell (O) contrasts of the triblock copolymer C22-28 in
n-decane at 25 °C and concentration 0.50% (w/v). The solid
line represents the calculated cross sections, and the dots
represent the experimental data.
Sch em e 3
F igu r e 4. Scattering cross section dΣ/dΩ vs Q under core (0)
and shell (O) contrasts of the diblock copolymer D22-28 in
n-decane at 25 °C and concentration 0.50% (w/v). The solid
line represents the calculated cross sections, and the dots
represent represent the experimental data.
Therefore, the scattering amplitude introduced in eq
for this model are given by
3
6
4
3
(sin (Qr ) - Qr cos(Qr ))
1
1
1
AM,C(Q) )
(12)
3
(Qr1)
where r1 is either the core or the outer periphery of the
micelle. To obtain a comparison on an absolute intensity
scale, the model used required the concentration of the
polymers, the molecular weight of each arm of the blocks
(in case they were split in two arms the total molecular
weight was introduced) measured by MO, the density
of the polymeric chain of the core and shell, and the
scattering length densities of the polymeric chain of the
core, shell, and solvent as a fixed input. To perform a
convolution to account for the instrumental resolution
in addition the neutron wavelength λ, the dimensions
of sample cell, the collimation length, the detector
distance, and the wavelength spreading, ∆λ/λ, were used
to model the resolution function; i.e., the calculated
scattering cross sections were convoluted by a Gaussian
type resolution function taking into account the experi-
mental parameters, according to Pedersen et al.37
molecular weights in both contrasts (core and shell) in
n-decane and 0.5% (w/v) concentration, evaluated ac-
cording to eqs 3-9, are shown in Figures 2-5.
The curve exhibits a well-pronounced secondary maxi-
mum and in some cases a weak third maximum. This
implies that the aggregates formed are relatively uni-
form in size, and present well-defined structure. More-
over, the sharpness of the minima under core and shell
contrasts indicates a high symmetry of the aggre-
gates.
To find the form of the micelles, model fitting was
performed. It was found that a spherical core/shell
structure could fit very well the experimental data.

5
432 Iatrou et al.
Macromolecules, Vol. 35, No. 14, 2002
Ta ble 6. Molecu la r Ch a r a cter istics of th e Micelles in n -Deca n e a t 25 °C of th e C22-28 Cyclic Cop olym er s, th e
Cor r esp on d in g Lin ea r P r ecu r sor a n d th e Diblock Cop olym er s
Mn(PBd)
× 10
21.0
12.0
23.0
23.0
Mn(PS-d8)
× 10
Rc
(nm)
Ro(SANS)
(nm)
σC
σSh
Rh(DLS)
(nm)
-
3
-3
sample
diblock
P
fstar
(nm)
(nm)
kd
23.2
14.0
30.0
30.0
280
220
400
210
13.2
10.0
17.2
14.5
28.3
19.6
28.0
24.0
1
1
0.37
0
2
1
3
2
2.1
1.7
2.7
1.7
28.4
20.5
31.6
26.8
5.7
3.7
(diblock)/2
triblock
cyclic
∼0
-6.2
F igu r e 5. Scattering cross section dΣ/dΩ vs Q under core (0)
and shell (O) contrasts of the diblock copolymer (D22-28)/2
in n-decane at 25 °C and concentration 0.50% (w/v). The solid
line represents the calculated cross sections, and the dots
represent the experimental data.
F igu r e 6. Volume fraction of the coronal bulk polymer as a
function of the distance of the surface of the core: (A) constant
density profile (cyclics in both solvents); (B) starlike density
profile (diblocks and triblock in DMF); (C) mixed density profile
Instead of an idealized sphere form factor (eq 12), a
smeared out density profile (æc(r)) for the micelle core
was used
(triblock in n-decane).
0
1
a parameter fstar was introduced, which is the fraction
of the bulk volume VSh,tot of the shell 2. In this case fstar
describes the relative polymer amounts in the shells
æ (r) ) æc
(13)
c
r - r1
σC
1
+ exp
(case c). A detailed description of the fitting functions
is given in the Appendix. The aggregation number and
therefore the bulk volume of the shell polymer were
calculated from the fit of the data under core contrast.
The scattering length density of the mixture of polymer
and solvent was the average, calculated from the
volumes of the bulk polymer and solvent.
The density of PBd used was 0.91 g/mL. The fit
parameters under shell contrast were fstar, σi and di. A
simultaneous fit under core and shell contrast was
always performed. P, fstar, σi, RC, and Ro of the micelles
obtained are given in Table 6. Here we set Ro ) r3
identical.
o
where æc is the density of the PS-d8 in the core and σC
is a fit parameter for the width of the core boundary,
due to its roughness or/and deformation from the
spherical shape. The Fourier transform of the density
profile æc(r) replaces the scattering amplitude of eq 12.
Only in the limit of an infinite thin smearing (σcf0) is
eq 12 retained.
First, the scattering curve under core contrast was
fitted independently using eq 4. Thus, the core radius
was calculated by using
1
/3
3
PMC
From Table 6, the fstar value of 1, obtained in the case
of the two diblock copolymers, indicates that a starlike
density profile can fit very successfully the experimental
R )
(14)
C
(
)
4
πF NA
C
-
4/3
where MC is the total molecular weight of the PS-d8
blocks, FC ) 1.12 g/mL, the density of PS-d8, and NA is
the Avogadro number. The only adjustable parameters
were the aggregation number P and σC (note that Rc
and r1 are identical here). The obtained values are given
in Table 6. The very small smearing indicates a low
polydispersity or/and low interface roughness of the
core.
The model used for the fit (Figure 6) under shell
contrast had the option of splitting the bulk volume of
the shell polymer in two parts, shell 1 close to the core
with constant density profile and radius r2 (case a), and
shell 2, in the outer periphery with a density profile of
a star polymer and radius r3 (case b). For that purpose,
data of the shell. This density profile follows the r
expansion of the PBd polymeric chains presented by
3
8
Daoud and Cotton for the density profile of a starlike
molecule. The low values of σ verify the already indi-
cated low polydispersity of the micelles and their
spherical form.
In the case of triblock copolymer, interchain micelli-
zation is possible, since n-decane is a poor solvent for
PS-d8 end blocks. However, no indication of the forma-
tion of such aggregates was found, since no indication
of a modulation at low Q due to a structure factor was
observed in the experimental data under core or shell
contrast. The data were fitted by a simple form factor
of a sphere. An aggregation number of 400 was found

Macromolecules, Vol. 35, No. 14, 2002
Model Cyclic Block Copolymers 5433
for the triblock copolymer, which is much higher than
the one of the corresponding diblock. A mixed shell
density profile composed of a 63% constant and 37%
starlike density profile (fstar ) 0.37) was necessary to
be used in order to fit the data. This indicates that the
micelles contain some triblock copolymers with the form
of dangling chains (Scheme 3). The PS-d8 blocks which
are in the outer periphery are collapsed, and therefore
they are very small in volume and randomly distributed.
As a consequence, they do not contribute significantly
in the overall scattering under core contrast. In the case
of 37% of the PS-d8 dangling chains, only half of the
amount of PS-d8 contributes to the formation of the core
and therefore the actual aggregation number is 520. The
increased σ value of 27 Å found in the case of the
triblock copolymers is higher that the corresponding
values of the diblock and cyclic copolymers. This indi-
cates an increased roughness of the outer periphery or/
and slight deviations from the spherical form. The
density profile of the micelles is shown in Figure 6. The
formation of spherical noninterconnected micelles has
Ta ble 7. Geom etr ica l Ch a r a cter istics of th e Micelles of
th e C22-28 Cyclic Cop olym er s, th e Cor r esp on d in g
Lin ea r P r ecu r sor , a n d th e Diblock Cop olym er s in
n -Deca n e a n d DMF a t 25 °C
n-decane
DMF
Rc
(nm)
H
corona
Ac
(nm )
Rc
(nm)
H
corona
Ac
(nm )
2
(nm)a
2
sample
(nm)a
diblock
diblock/2
triblock
cyclic
13.2
10.0
17.2
14.5
15.1
9.6
10.8
9.5
7.8
12.6
8.8
21.3
14.3
19.0
16.5
7.5
5.0
5.9
5.1
5.4
4.3
3.2
3.8
12.6
a
Hcorona ) Ro - Rc.
3
9
been observed in similar systems, by Tanaka et al. and
2
2
Balsara et al. for copolymers with composition close
to 50%. However, in these works, a lower aggregation
number was found, compared to the one of the linear
diblock analogues.
Surprisingly, in the case of cyclic copolymer the data
were better fitted when a 100% (fstar ) 0) constant
density profile was used. The constant density profile
can be explained by the formation of loops of the PBd
chains. Since there is no free chain end, all monomeric
units are almost equivalent, having the same excluded
F igu r e 7. Scattering cross section dΣ/dΩ vs Q under core (0),
intermediate (g), and shell (O) contrasts of the cyclic copolymer
C22-28 in DMF at 25 °C and concentration 0.50% (w/v). The
solid line represents the calculated cross sections, and the dots
represent the experimental data.
-
4/3
volume, and cannot expand with an r
dependence of
the density profile of a starlike molecule38 (Figure 6).
The corresponding σ value for the cyclic is much lower,
supporting the conclusion that the micelles are nearly
monodisperse.
In the case of cyclic and triblock copolymers, the
calculated area has been divided by two, since there are
two junction points per PBd chain. It can be seen that
the coronal thickness of the cyclic and half-diblock are
equal, while the one of the triblock is slightly larger and
the one of the diblock the largest. This order is expected
since the mean square radius of gyration of a ring
polymer of length N is identical to the mean square
The aggregation numbers obtained from the fits are
on the order of
P > P > P >PC
(15)
T
D
D/2
where PT, PD, PD/2, and PC are the aggregation numbers
of the triblock, diblock, half-diblock, and cyclic copoly-
mers, respectively.
40,41
radius of gyration of a linear chain with length N/2.
These numbers are comparable since the areas per
junction point, and therefore the stretchings of the
polymeric chains due to crowding effects, are identical.
The higher shell thickness of the triblock copolymer
maybe due to the presence of the dangling chains. It is
well-known that the ratio of mean square radius of
gyration between a cyclic and linear chain is equal to
A lower aggregation number of the half-diblock com-
pared to the diblock copolymer was expected since the
aggregation number strongly depends on the molecular
weight of the collapsed chain. Moreover, a lower ag-
gregation number of the cyclic copolymer was expected
compared to the one of the corresponding linear diblock
copolymer, due to the necessity of the formation of loops
of the dissolved chains, which decreases the possible
conformations and therefore introduces an entropic
penalty. More quantitative analysis on the aggregation
numbers of several architectures will be discussed below
on the basis of scaling models.
The geometrical characteristics of the micelles are
shown in Table 7. In this table, the coronal thickness
H ) Ro - RC along with the core area per junction point
AC is calculated. The coronal thickness can be compared
with the end-to-end distance of a polymeric chain in a
Θ solvent. The core area per junction point AC was
calculated by using the following equation:
0
.5. On the basis of this, the ratio of the shell thickness
between the linear diblock and the cyclic copolymer
should be equal to the square root of 0.5, i.e., 0.7. The
ratio was found equal to 0.63, and the difference may
be due to the smaller area per junction point of the
linear diblock copolymer which shows that the PBd
chains are more stretched and therefore more elongated.
SANS Mea su r em en ts in DMF . The coherent mac-
roscopic scattering cross sections (dΣ/dΩ) for the copoly-
mers in DMF evaluated according to eqs 3-9 are shown
in Figures 7-10.
The relatively sharp secondary maximum and in some
cases a weak third maximum implies that the ag-
gregates formed are relatively uniform in size and
symmetric. The form of the scattering cross sections
obtained under core contrast in this solvent is similar
2
4
πRC
A )
(16)
C
P

5
434 Iatrou et al.
Macromolecules, Vol. 35, No. 14, 2002
Ta ble 8. Molecu la r Ch a r a cter istics of th e Micelles in DMF a t 25 °C of th e C22-28 Cyclic Cop olym er s, th e Cor r esp on d in g
Lin ea r P r ecu r sor , a n d th e Diblock Cop olym er s
Mn(PBd)
× 10
Mn(PS-d8)
× 10
Rc
(nm)
Ro(SANS)
(nm)
σC
(nm)
σSh
(nm)
Rh(DLS)
(nm)
-
3
-3
sample
P
fstar
kd
diblock
diblock/2
triblock
cyclic
21.0
12.0
23.0
23.0
23.0
14.0
30.0
30.0
1050
585
680
450
21.3
14.3
19.0
16.5
28.8
19.3
24.9
21.4
1
1
1
0
2
2
3
2
2.1
2.0
2.5
2.3
30.1
19.9
26.4
23.2
6.6
4.8
6.1
5.2
F igu r e 10. Scattering cross section dΣ/dΩ vs Q under core
(0), intermediate (g) and shell (O) contrasts of the diblock
copolymer (D22-28)/2 in DMF at 25 °C and concentration
0.50% (w/v). The solid line represents the calculated cross
sections, and the dots represent the experimental data.
F igu r e 8. Scattering cross section dΣ/dΩ vs Q under core (0),
intermediate (g), and shell (O) contrasts of the triblock
copolymer C22-28 in DMF at 25 °C and concentration 0.50%
(
w/v). The solid line represents the calculated cross sections,
and the dots represent the experimental data.
The aggregation number of the half-diblock was ex-
pected to be lower than that of the diblock like in
n-decane. The aggregation number of the triblock is
expected to be lower than the one of the corresponding
diblock due to the entropic penalty of the looped PBd
chains in the core and the crowding of the PBd chains
in the shell. Finally the aggregation number of the cyclic
should be lower than the corresponding triblock due the
additional entropic penalty from the PBd loops in the
corona. More quantitative analysis concerning the ag-
gregation numbers will be made below.
The geometrical characteristics of the micelles are
shown in Table 7. The same order in coronal thickness,
H, was found like in the case of the micelles in n-decane.
The lower ratio (0.6) of the coronal thickness between
the diblock and cyclic copolymer maybe due to the much
lower area per junction point of the diblock copolymer
which would oblige the PS-d8 chains to become stretched
and, as a consequence, would increase the dimensions
of the PS-d8 coronal chains.
Dyn a m ic Ligh t Sca tter in g. The Rh values of the
micelles formed in both solvents along with the kD
values obtained by DLS are shown in Tables 6 and 8.
The obtained values are 5-10% larger than the one
obtained from SANS. For hard spheres, the static and
dynamic radii should be identical. The difference may
be due to experimental uncertainty or to the fuzziness
of the outer periphery of the micelle.
F igu r e 9. Scattering cross section dΣ/dΩ vs Q under core (0),
intermediate (g), and shell (O) contrasts of the diblock
copolymer D22-28 in DMF at 25 °C and concentration 0.50%
(
w/v). The solid line represents the calculated cross sections,
and the dots represent the experimental data.
to the one obtained in n-decane. As a consequence, a
model of a spherical core/shell structure was used in
order to fit the data.
The same fit model described above was used. The
molecular characteristics of the micelles formed in this
case are shown in Table 8.
Surprisingly, the value of kD of the triblock in n-
decane was almost zero, while the corresponding value
of the cyclic was negative. kD is composed of thermo-
dynamic as well as frictional components. For a dilute
solution of homopolymers, kD can be expressed as
In this case, a constant density profile was used to
fit the data only of the cyclic copolymer, whereas a
starlike density profile could fit the experimental data
of the triblock, and two diblock copolymers.
The aggregation numbers obtained in DMF are of the
order
k ) 2A M - k
f
(18)
D
2
w
where A2 is the second virial coefficient that depends
on intermolecular interactions, Mw is the weight-aver-
P > P > PD/2>P
C
(17)
D
T

Macromolecules, Vol. 35, No. 14, 2002
Model Cyclic Block Copolymers 5435
age molecular weight of the polymer, and kf is related
Ta ble 9. Ca lcu la ted Aggr ega tion Nu m ber s in n -Deca n e
a n d DMF
to the frictional drag that opposes the motion of the
4
2
molecules. Since the friction coefficient for similar
polymer solutions is related to their concentration, their
values are expected to be similar for the four different
copolymers with three different architectures, because
they were measured in similar concentrations. As a
consequence, the lower kD values may be due to the
lower A2 values in the case of cyclics, compared to the
one of the triblock and the diblocks. Similar results were
obtained in DMF (Table 7).
n-decane
DMF
Pexpa
Pcalcb
Pexpa
Pcalcb
sample
cyclic
210
360
280
220
450
680
1050
585
triblock
diblock
diblock/2
506
1070
686
324
216
a
Experimentally found aggregation numbers. b Calculated ag-
gregation numbers by the scaling models.
numerical results from the theory, the following scaling
relation was established:
4
. In ter p r eta tion in Ter m s of Sca lin g Mod els
The free energy F of a single chain incorporated in a
micelle far away from the critical micelle concentration
is approximated as a sum of three contributions:
Peq
)
2
1/2
1/2
4
πm (γl /k T) + (4π/3)m
+ (4π/3)mSh (R /H)
C
S
B
C
C
F ) Fint + FSh + FC
(19)
-1/3
2
1
+ mC
+ (m /mSh)(H/RC)
C
(
22)
The first term Fint denotes the interfacial contribution
to the free energy associated with the core-corona
interface given by
In this equation lS denotes the characteristic length of
a solvent molecule, which is calculated from its volume
1
/3
2
lS ) vS , mC,Sh are the ratios of the volumes of the core
or shell blocks and a solvent molecule: mC,Sh ) vC,Sh/vS.
For a comparison with our results, we used the radii
from Tables 5 and 7, as obtained from the model fitting
of SANS data. Concerning the results in n-decane, the
models were applied only for the diblock copolymers,
because the triblock copolymer is more complicated, the
aggregation number cannot be calculated. The models
will be applied in the results in DMF. With lS of
n-decane equal to 6.87 Å, mC of the cyclic and triblock
copolymers are equal to 132.9 for both, mC for the
diblock copolymer is equal to 116.1, and mC for the half-
diblock is equal to 62. The corresponding values in the
same solvent for the shell are 123.5 for the cyclic,
triblock, and diblock copolymers and 61.8 for the half-
diblock copolymer. The lS value for DMF is 5.04 Å, while
mC for the diblock, triblock, and cyclic copolymer is
4
pRC
γ
Fint
)
(20)
Peq
where γ denotes the interfacial tension between the
solvent and the collapsed chains, and Peq is the equi-
librium aggregation number. The contributions FC and
FSh are related to chain stretching in the core and the
shell, respectively. Iatrou et al.24 found that a scaling
model of an intermediate regime, i.e., micelles with H
e RC, where H ) Ro - RC, could describe the experi-
mental results of the linear diblock PS-PI in n-heptane
or n-decane along with the (PI)3PS-d8(PI)3 super-H block
copolymers. In that work the aggregation number of the
super-H block copolymers stars were found to be 1 order
of magnitude lower than the corresponding linear
diblocks. It was found that the intermediate regime
occurs when
3
1
13.8. The corresponding value for the half-diblock is
57. The mSh values are 337.5 for the triblock and cyclic
copolymers, 294.7 for the diblock, and 157.4 for the half-
1
5/11 2/5
18/ 11 7/6
NPI
f
, NPS , NPI
f
(21)
diblock copolymer. An interfacial tension of γ ) 5.1 dyn/
4
3,23
where f is the number of PI chains per PS chain of the
super-H copolymers, and NPS,PI the number of mono-
meric units of PS or PI. In the case of diblocks, f equals
cm between PS and n-decane was used,
whereas a
value of γ ) 12.5 dyn/cm was used for the PBd and DMF
pair. The interfacial surface tension between DMF and
PBd was measured from the contact angle between the
two liquids. The aggregation numbers obtained are
shown in Table 9.
1
. Although this model is valid for comparison of the
aggregation numbers between copolymers with different
architectures, the values calculated are much lower
than the experimental ones.
To calculate the aggregation number of a triblock
copolymer in DMF, where the middle part is collapsed,
one has to take into account the entropic penalty that
comes from the stretching of the PBd chain in the core
in order to bring the junction points on the interface
and the increased repulsive monomer-monomer inter-
actions within the starlike corona, since the number of
arms are doubled. Since the core is considered to be in
the melt state, the PBd chains are already deformed in
order to fill the space of the core. As a consequence, the
entropic penalty of the deformation of PBd chains to
bring the two junction points to the interface is much
smaller than the repulsive interactions of the crowded
monomers within the corona. This penalty has been
calculated to reduce the aggregation number by a factor
Nagarejan and Ganesh23 developed a more compre-
hensive model which relates the core radius and coronal
thickness H of micelles of AB diblock copolymers in
selective solvents. Six different contributions to the
micellar free energy are considered in this model. The
reference state for the calculation of the energy is a
single block copolymer in the selective solvent. Each free
energy of the core and shell consists of two terms. The
first term includes entropy contributions when passing
from the reference to micellar state. The second term
includes contributions due to chain deformation. The
interfacial free energy is equal to the one given in eq
2
0. In this model, the equilibrium micelle is obtained
from the minimization of the free energy per molecule
with respect to the independent variables of core and
outer radius. By comparison of the experimental data
from three different copolymer/solvent systems and
-
7/6 24
of f
,
where f is the number of chains in the corona
per collapsed chain in the core (for triblocks f ) 2). This
reduces the calculated aggregation number to 506. This

5
436 Iatrou et al.
Macromolecules, Vol. 35, No. 14, 2002
is much lower than the experimentally found, because
PBd arm to expand like an arm of a star. On the basis
of this difference, the purity of the cyclics is higher than
87%.
-
7/6
the term f
was applied for the super-H miktoarm
star copolymers, where the functionality was 6, and as
a consequence is overestimated in the case of the
triblocks.
Ack n ow led gm en t. The authors would like to ac-
knowledge the BMBF for financial support, Dr. J .
Roovers and Dr. L. Willner for fruitful discussions, and
Prof. Spyros Anastasiadis for the surface tension mea-
surements.
To calculate the aggregation number of the cyclic
copolymers, one has to take into account all the entropic
and enthalpic penalties already mentioned for the
triblock copolymers with a collapsed middle chain and,
additionally, the reduction in entropy due to loop
formation of the connected end blocks in the corona.
Ap p en d ix
4
4
Hadjiioannou and ten Brinke calculated the free
energy of the micelle, treating the looped block as two
separate linear chains with half molecular weight. The
contribution to the total free energy from the reduction
in entropy, according to that work, is estimated to be
The equation for the density profile of the shell 1 (case
a) is given by
0
1
1
φsh1 ) φsh1 1 -
(A1)
r - r1
r - r2
σsh1
1
+ exp
1 + exp
(
)(
)
σc
3
PkT
2
Floop ) â
ln(NA)
(23)
[
]
The equation for the density profile of the shell 2 (case
b) is given by
where â is a correction factor close to unity, NA is the
number of monomeric units of the dissolved chain, P is
the aggregation number, and k is the Boltzmann
constant. However, this equation does not relate the
aggregation number of a cyclic polymer with the corre-
sponding value of a linear diblock copolymer analogue.
Booth et al.26 in a system of ethylene oxide and
propylene oxide (PPO) block copolymers, found that
triblock copolymer exhibited the lowest aggregation
number, by comparing the micellar behavior of the cyclic
copolymer and the linear diblock and triblock analogues.
In their triblocks, the collapsed chain was the middle
PPO block.
0
1
1
1
φsh2 ) φsh2 1 -
4/3
r - r2
r - r3
r
1 + exp σsh1 1 + exp
(
)(
)
σsh2
(A2)
The case c combines the two shells. Here the parameter
fstar describes the relative amounts of shells 1 and 2 by
the following definition:
∫
φ
sh2 dr
fstar
)
∫
φ
sh1 dr + φsh2 dr
∫
5
. Con clu sion s
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(
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