Linear rheology of multiarm star polymers diluted
with short linear chainsa)
A. Miros and D. Vlassopoulosb)
FORTH, Institute of Electronic Structure and Laser and Department
of Materials Science and Technology, University of Crete,
71110 Heraklion, Crete, Greece
A. E. Likhtman
Physics Department, University of Leeds, Leeds LS2 9JT, United Kingdom
J. Roovers
NRC, Institute for Chemical Process and Environmental Technology, Ottawa,
Ontario K1A 0R6, Canada
(Received 8 July 2002; final version received 3 October 2002)
Synopsis
We present experimental results on the linear rheology of multiarm star/linear polymer mixtures, the
latter having molecular weight much smaller than the star arm molecular weight. In such a case the
linear chains act as ideal macromolecular solvents, which dilute entanglements of the arms. Using
different star polymers we show that it is possible to account for this dilution and describe the linear
rheology of the mixtures using the Milner–McLeish theory for arm relaxation, complemented by
the longitudinal modes of stress relaxation and high frequency Rouse modes. A universal
description of the isofrictional arm relaxation time as a function of the number of entanglements is
obtained for stars of any functionality and degree of dilution. The slow structural mode, related to
the diluted star’s colloidal core, also depends on the number of entanglements, but in a more
complex way. © 2003 The Society of Rheology. 关DOI: 10.1122/1.1529172兴
I. INTRODUCTION
It is now widely accepted that the tube model of entanglements can describe the
complex rheological properties of polymer melts 关Doi and Edwards 共1986兲; Marrucci
et al. 共1999兲; McLeish 共1997, 2002兲; McLeish and Milner 共1999兲; Watanabe 共1999兲兴;
these range from linear polymers, where the dominant reptation mechanism is complemented by additional modes, the most notable of which is the contour length fluctuations
关Doi 共1981兲兴, to branched polymers. The latter include star polymers, where arm relaxation takes place via an activated diffusion, because of the presence of the center branching point 关McLeish and Milner 共1999兲; Fetters et al. 共1993兲兴 and polymers that combine
linear and star behavior, such as combs 关Daniels et al. 共2001a兲; Roovers and Graessley
共1981兲兴, H polymers 关McLeish et al. 共1999兲; Roovers 共1984兲兴, and pom-pom polymers
a兲
Dedicated to Professor G. Marrucci on the occasion of his 65th birthday.
Author to whom all correspondence should be addressed; electronic mail: [email protected]
b兲
© 2003 by The Society of Rheology, Inc.
J. Rheol. 47共1兲, 163-176 January/February 共2003兲
0148-6055/2003/47共1兲/163/14/$25.00
163
164
MIROS ET AL.
关McLeish and Larson 共1998兲; Houli et al. 共2002兲兴. Branched polymers are of considerable interest because of their practical implications in the understanding and characterization of the commercial long-chain branching effects, and also because they serve as
systems for elucidating the influence of macromolecular architecture on the polymer
rheology 关Larson 共2001兲; Wood-Adams and Costeux 共2001兲; Hatzikiriakos 共2000兲兴. Inherent to the description of the dynamics of branches is the concept of dynamic dilution
at different time scales 关Ball and McLeish 共1989兲兴. In general, the effect of dilution of the
entanglement network on the rheological properties of well-defined branched polymers is
a subject of considerable interest. Linear polymers in concentrated solutions exhibit a
weight 共or volume兲 fraction, ␾, dependence of the entanglement molecular weight M e
⬃ ␾ ⫺5/4 共which corresponds to plateau modulus of G 0N ⬃ ␾ 9/4) under good solvent
conditions 关Adam and Delsanti 共1977兲; Colby and Rubinstein 共1990兲兴 and M e
⬃ ␾ ⫺4/3 共which corresponds to G 0N ⬃ ␾ 7/3) under theta conditions 关Adam and Delsanti
共1984兲; Colby and Rubinstein 共1990兲兴. On the other hand, a recent systematic study with
concentrated hydrogenated polybutadiene solutions in n-alkane solvents 关Tao et al.
共1999兲兴 seems to support M e ⬃ ␾ ⫺1 scaling 共with G 0N ⬃ ␾ 2 ). Nevertheless, despite the
fact that the exact scaling 共between 4/3 and 1兲 is still unresolved, it is possible for the
same polymer to control its rheological properties by selectively tuning the number of
entanglements via the addition of solvent.
A recent investigation 关Daniels et al. 共2001b兲兴 examined entangled solutions of linear,
three-arm star and H-shaped polyisoprenes in oligomeric theta-like solvent squalene, and
demonstrated that by taking into account the dilution effects 共as well as some polydispersity and high frequency Rouse modes兲 the tube models can describe the entire frequency spectrum well.
In this work we consider blends of multiarm star and linear polymers within the limit
of very small linear molecular weight (M linear) compared to the star arm molecular
weight (M a ). Such a case is viewed as star polymer solutions in a macromolecular
solvent. For this case we demonstrate the validity of entanglement dilution and describe
the linear rheology of these systems over the entire frequency range 共excluding the glass兲
for a variety of blend compositions and arm molecular weights. In this respect the present
investigation represents an extension of an earlier dilution study 关Daniels et al. 共2001b兲兴
in several ways: 共i兲 different chemistry 共here 1,4-polybutadienes are employed兲, confirming the universality of the findings. 共ii兲 Very high number of arms, resembling the behavior of ultrasoft colloids 关Vlassopoulos et al. 共2001兲; Grest et al. 共1996兲; Likos
共2001兲兴, which in addition to star arm relaxation, exhibit the structural relaxation mode as
well, the latter also depending on the number of entanglements. 共iii兲 Macromolecular
solvent 共linear polymer兲 instead of a molecular or oligomeric one. 共iv兲 In a theoretical
description of the dynamics, the longitudinal modes of stress relaxation are considered
and their necessity is demonstrated. This approach adds two important additional parameters for future consideration, namely, the ratio of the linear polymer to the arm molecular
weight and the architecture of the solvent 共e.g., the star兲.
Section II describes the materials and techniques used. The main findings are presented in Sec. III, and discussed in view of the theories based on the tube model available, and the dilution of the network entanglements. Finally, a summary of the conclusions is presented in Sec. IV.
LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS
165
TABLE I. Molecular characteristics of the 1,4-polybutadiene stars.
code
PB1000
6415
6430
6460
a
Ma
共g/mol兲
f
2
60
56
61
500b
12 100
23 900
47 500
Tg
(°C)
⫺96
⫺92
⫺92
⫺92
Rga
共nm兲
1.1 共estimated兲
12.7
18.5
28
From light scattering measurements in dilute cyclohexane 共good solvent兲 solution.
The effective arm molecular weight of the linear polymer is considered to be half its total molecular weight.
b
II. EXPERIMENT
A. Materials
The 1,4-polybutadienes used in this study are listed in Table I, along with their molecular characteristics. The star polymers, with nominal functionality f ⫽ 64 and varying
arm molecular weight, were synthesized using a dendrimer scaffold and chlorosilane
chemistry 关Roovers et al. 共1993兲兴, whereas the linear one was purchased from Polymer
Source, Canada. Star-linear polymer mixtures of different compositions were prepared by
dissolution of the polymers in a good solvent cyclohexane 共about 5wt % total polymer
concentration兲 under gentle stirring for about 2 days, followed by solvent evaporation in
a vacuum oven at room temperature for another 24 h. All samples used in this work were
optically transparent. To reduce the risk of degradation, a small amount of antioxidant
2,6-di-tert-butyl-p-cresol 共0.1%兲 was added to the solution.
B. Methods
The dynamic response of the mixtures was studied with small amplitude oscillatory
shear measurements under nitrogen atmosphere over a wide range of temperatures 共from
⫺100 to 60 °C). A Rheometric Scientific ARES strain controlled rheometer was employed with a dual range force rebalance transducer 共2KFRTN1兲 and temperature control
of ⫾0.1 °C achieved via a nitrogen convection oven. The sample was placed between
two parallel plates of 8 mm diameter, reaching a gap of about 1.5 mm. Dynamic measurements consisted of strain sweeps to obtain the strain range that corresponded to the
linear response for different frequencies, time sweeps to ensure stable conditions, and
frequency sweeps in the range of 100–0.1 rad/s to obtain linear viscoelastic spectra of the
storage (G ⬘ ) and loss (G ⬙ ) moduli.
III. RESULTS AND DISCUSSION
Figure 1 depicts master curves of G ⬘ and G ⬙ for the 6460/PB1000 mixture at different
compositions, including the pure components. They were obtained by horizontal shifting
of the individual frequency sweeps at different temperatures along the frequency axis,
according to the time–temperature superposition principle 关Ferry 共1980兲兴. Except for the
PB1000 linear component, all other samples exhibited entangled polymer behavior. As
can be noted in Fig. 1共a兲, however, the plateau modulus G 0N drops significantly upon
addition of the small linear chains, which effectively dilute the entanglements created by
the star arms; the corresponding entanglement molecular weight increases. In this respect
the short linear chains act as ideal macromolecular solvents. In fact, as long as the
relation 关deGennes 共1979兲兴 M linear ⬍ M a holds 共see Table I兲, the linear chains should
penetrate the stars, although this is quite rough since the role of the functionality needs
166
MIROS ET AL.
FIG. 1. Typical master curves of storage, G ⬘ 共a兲, and loss, G ⬙ 共b兲, moduli for a 6460/PB1000 mixture at
different compositions 共from the top: 100/0, 䊉; 80/20, 䉭; 50/50, 䉲; 30/70, 䊊; 0/100, 䊏兲, with a reference
temperature of 190 K.
further consideration. The data in Fig. 1 conform to the picture of short linear chains
nearly uniformly penetrating the star molecules; this was experimentally confirmed using
dynamic light scattering measurements 关Vlassopoulos et al. 共1999兲兴. Furthermore, from
Fig. 1共b兲 one can appreciate the effects of the decreasing number of entanglements 共by
adding solvent兲 in reducing the G ⬙ minimum.
To quantify the dilution of the linear chains, we determined the plateau modulus and
checked it against the Colby–Rubinstein 共1990兲 prediction. Note that the G 0N determination from the G ⬘ ( ␻ ) curve in Fig. 1共a兲 is somewhat ambiguous, because G ⬘ ( ␻ ) exhibits
a weak power law rather than a true plateau, a feature expected for branched polymers
关Graessley and Roovers 共1979兲; Roovers 共1985兲; Vlassopoulos et al. 共2001兲; McLeish
and Milner 共1999兲; Islam et al. 共2001兲兴. Consequently, G 0N was also estimated from
LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS
167
FIG. 2. Dependence of the plateau modulus G 0N 共a兲 and entanglement molecular weight M e 共b兲 on the volume
fraction of the star 共␾兲 in star-linear polymer 共PB1000兲 blends: 6460: 䊐; 6430: 䉭; 6415: 䊊. Dotted and solid
lines represent the scaling predictions for G 0N 共7/3 and 2, respectively兲 and M e (⫺4/3 and ⫺1, respectively兲,
discussed in the text.
integration of the G ⬙ ( ␻ ) curve around the terminal region, according to G 0N
⫹⬁
⫽ 2/␲ 兰 ⫺⬁
关 G ⬙ ⫺G s⬙ 兴 d ln ␻ 关Ferry 共1980兲; Roovers 共1985, 1986兲兴; the subscript s refers
to the contribution from the high frequency 共Rouse-like transition兲 region, which is
practically identical to the linear case. However, this procedure involves some uncertainty, since additional mechanisms of relaxation such as contour length fluctuations and
longitudinal modes, which also relate to the cut-off value of G s⬙ , were not considered.
Nevertheless, the plateau values from this integration were comparable to those estimated
from G ⬘ ( ␻ ) directly.
Figure 2 depicts the dependence of G 0N and M e on the star volume fraction ␾ in the
three star/linear polymer mixtures investigated. The entanglement molecular weight was
determined from M e ⫽ ( ␳ RT)/G 0N with ␳ being the density; the prefactor 共4/5兲 was
consistently omitted throughout this work, since the theoretical star relaxation model does
168
MIROS ET AL.
FIG. 3. Temperature dependence of the frequency shift factors for various stars, linear polymers, and star-linear
blends, at a reference temperature of T ref ⫽ 190 K.
not include it either 关Milner and McLeish 共1997, 1998兲兴. In addition, we note that the M e
value of the pure stars was 1815 g/mol, conforming well to values reported in the
literature 关Ferry 共1980兲; Milner and McLeish 共1998兲兴. In this consistent manner, the
comparison between model and experimental data is satisfactory for star polymers, as
will be further discussed below. It is evident that, within experimental error, the data lie
between the two slopes 关Colby and Rubinstein 共1990兲; Tao et al. 共1999兲兴, namely, 7/3 and
2 and 4/3 and 1. Therefore, determination of the exact scaling of G 0N and M e with ␾
remains an unresolved problem that requires carefully designed experiments that involve
a large variety of polymer systems and concentrations 关Fetters et al. 共1994, 1999兲; Colby
共1997兲兴, and this is beyond the scope of the present work. For the purposes of this study
it is sufficient that the dilution works, and we choose the Colby–Rubinstein 共1990兲
scaling exponents 7/3 and 4/3. Daniels et al. 共2001b兲 have reported a similar observation
of dilution for three-arm star and H-polyisoprenes with the oligomeric solvent squalene;
note however, that squalene is probably a better solvent for polyisoprene than PB1000 for
polybutadiene 关Brandrup and Immergut 共1989兲兴. Depsite the exact value of the exponent,
in the present case it is interesting that given the topology of the multiarm stars the
macromolecular dilution still holds; moreover, the macromolecular solvent here, PB1000,
having 22 repeat units, barely qualifies as an oligomeric solvent. There is a small part of
the arm near the center in close contact to its neighbors that forms an effective core and
may be partially stretched, and it is not known how much the short linear chains can
penetrate. However, the measured G 0N of the stars is found to be very similar to that of
linear chains, within experimental error 关Kapnistos et al. 共1999兲; Pakula et al. 共1998兲;
Roovers 共1986兲兴, neglecting any possible small temperature dependence 关Graessley
共1982兲兴; this suggests that a small amount of stretching of the arms does not contribute
appreciably to the plateau modulus. In addition, the deviations of the data in Fig. 2 from
the theoretical slopes 共here taken for theta solvent conditions, the difference from good
solvent being very small兲 are really small. It is thus safe to conclude that the dilution
effect is universal and that the multiarm star topology essentially does not affect it
quantitatively.
All mixtures exhibited the same thermorheologically simple behavior, as seen in Fig.
3, which depicts the temperature dependence of the frequency shift factor ␣ T . The
well-known WLF expression 关Ferry 共1980兲兴, log ␣T ⫽ ⫺c1(T⫺Tref)/(c 2 ⫹T⫺T ref) represented all data for various stars, linear polybutadiene and their mixtures well, with
T ref ⫽ 190 K and values of c 1 and c 2 being about 10 and 60 K, respectively.
LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS
169
The terminal region is characterized by two-step relaxation. The faster of the two
relaxation processes is well established as corresponding to star arm relaxation. A theoretical account of this process was recently presented by Milner and McLeish, who used
the concept of dynamic dilution with appropriate scaling of the entanglement length
关Milner and McLeish 共1997兲; Vega et al. 共2002兲兴. This theory was developed within the
framework of the tube model and was proven successful in describing the arm relaxation
of stars of any functionality and of different chemistry, without adjustable parameters
关Kapnistos et al. 共1999兲; Milner and McLeish 共1998兲兴. The only ones used, namely, the
entanglement molecular weight M e , respective plateau modulus G 0N , and the 共Rouse兲
relaxation time of an entanglement segment ␶ e ⫽ N 2e ␨ b 2 /3␲ 2 k B T, with ␨ the monomeric friction coefficient and b the Kuhn segment 共entanglements set at ␻ ⬍ 1/␶ e ) can
be obtained from the data. The relaxation modulus G MM(t) is obtained from work by
Milner and McLeish 共1997兲
GMM共 t 兲 ⫽ 共 x⫹1 兲 G 0N
冕
1
0
ds 共 1⫺s 兲 x exp关⫺t/␶共s兲兴,
共1兲
where s is the relaxed fraction of the arm and x ⫽ 4/3 is the dilution exponent ( MM
stands for Milner and McLeish兲. The total arm relaxation time, ␶ (s)
⫽ 关 (e ⫺U eff(s))/␶early(s) ⫹ 1/␶ activated(s) 兴 ⫺1 , where U eff is the effective potential, incorporates early fast diffusion of the free end of the arm and activated arm retraction, and
depends on ␶ e and the number of entanglements per arm; the latter is reduced by the
dynamic dilution effect. The analysis of the multiarm star data using this theory considers
that a small fraction of the arm near the center is included in the core and does not
contribute to this process.
The high frequency region ( ␻ ⭓ 1/␶ e ), in which tube constraints do not significantly
affect the relaxation modes of the stars, proceed via Rouse dynamics as follows 关Doi and
Edwards 共1986兲兴:
GR共t兲 ⫽
G0N
Na
兺
Na /Ne n ⫽ Na /Ne
exp共⫺2n2t/␶R兲,
共2兲
with the Rouse time being ␶ R ⫽ (N a /N e ) 2 ␶ e 关Likhtman and McLeish 共2002兲兴. Calculation of the contribution to the dynamic response is obtained through the appropriate
Fourier transform.
It is now straightforward to calculate the dynamic response of the mixtures in frequency space that encompasses the frequency range from arm relaxation to the Rouse
modes using Eqs. 共1兲 and 共2兲. Typical results for melt 6460 and the diluted star 6460/
PB1000 mixture are presented in Figs. 4共a兲 and 4共b兲, respectively 共dotted lines兲. The
necessary parameters of the theory used in the fitting procedure were obtained selfconsistently from the experimental data, discussed in detail by Kapnistos et al. 共1999兲.
Both the plateau modulus and entanglement molecular weight were determined from the
frequency spectra 共Fig. 4兲 described above and shown in Fig. 2 as well. We did not obtain
the friction coefficient 共␨兲 by fitting; instead, we determined the Rouse time of an entanglement segment ␶ e from the high frequency limit of the rubber plateau 关Kapnistos
et al. 共1999兲; Ferry 共1980兲兴, and then as a check we determined the friction coefficient at
300 K using b ⫽ 0.7 nm 关Fetters et al. 共1994兲兴 and the shift factors in Fig. 3; we found
that for the star systems considered here, ␨ varies from ⫺1⫻10⫺10 to ⫺3.1
⫻10⫺10 N m/s, which is in good agreement with values reported in the literature 关Ferry
共1980兲; Milner and McLeish 共1998兲; Vega et al. 共2002兲兴. The agreement between theory
170
MIROS ET AL.
FIG. 4. Comparison of experimental data and theoretical predictions for G ⬘ and G ⬙ for the 6460 star melt 共a兲
and two star-linear 6460/PB1000 blends: 70/30 and 80/20 at T ref ⫽ 190 K; the latter data were shifted vertically by a factor of 100 to facilitate a comparison. Solid lines represent Eq. 共4兲 where the longitudinal relaxation
was accounted for; dotted curves represent theory without the longitudinal modes 关Eqs. 共1兲 and 共2兲 only兴. The
values of ␶ e used were 2 s 共6460 star兲, 4 s 共70/30兲, and 2.8 s 共80/20兲.
and experimental data is very good over the whole frequency range 共except for onset to
the Rouse-like transition zone and the slow terminal relaxation mode, which will be
discussed later兲 and confirms that the Milner–McLeish approach captures the basic physics of star arm relaxation for any functionality when the dilution effects of macromolecular solvents are properly accounted for. Furthermore, these results seem to indicate that
dynamic dilution does not discriminate between intra- and intermolecular entanglements
关Vlassopoulos et al. 共2001兲; Grest et al. 共1996兲兴. The former would be expected to be
more prevalent in 64-arm stars than in 3-arm stars and linear polymers.
The greatest disagreement between theory and experimental data observed in Fig. 4
that lies in the area of minimum of G ⬙ needs some further consideration; to this end, we
have added the longitudinal stress relaxation to the original Milner–McLeish expressions,
that was recently calculated for the case of linear chains 关Likhtman and McLeish 共2002兲兴.
LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS
171
To describe the physical origin of this mode, let us consider small step deformation of the
isotropic entangled melt. Because different tube segments are oriented differently before
deformation, some of them will stretch and some of them will contract. The stress after
the Rouse time of one entanglement segment ␶ e will be G 0N ⫽ ␳ RT/M e . However, after
time ␶ e , chain segments can become redistributed along the tube as a result of new
segment lengths, i.e., some chain segments will move from compressed segments to
stretched segments. After this relaxation, Doi–Edwards theory predicts the stress to be
equal to 54 G 0N , i.e., 1/5 of the stress stored in the tube can relax after time t ⬎ ␶ e as a
result of the longitudinal mechanism described. This longitudinal relaxation must not be
confused with contour length fluctuations 共CLFs兲. Whereas both mechanisms are derived
from the bead-and-spring model of the chain inside a tube, the CLF mechanism describes
escape from the original tube by fluctuations; on the other hand, longitudinal relaxation is
due to motion inside the original tube. As mentioned earlier, Likhtman and McLeish
共2002兲 calculated the dynamics of this process for linear chains. Repeating the same
derivation for the case of a star we get
Glong共 t 兲 ⫽
1
G 0N
5 N a /N e
(N a /N e )⫺1
兺
p⫽0
exp
冉
⫺共 p⫹1/2兲 2 t
␶R
冊
.
共3兲
The complete equation now reads
G共t兲 ⫽ GMM共 t 兲 ⫹G R 共 t 兲 ⫹G long共 t 兲 .
共4兲
It should be noted that although Eq. 共4兲 is adequate for the present discussion, it is still
not the final quantitative prediction. The calculation of both the early and the late times in
the Milner–McLeish approach is somewhat approximate, and an exact calculation is
beyond the scope of the present work
Figure 4共a兲 shows the experimental linear viscoelastic data for the 6460 star melt,
along with the predictions of Eq. 共4兲, by the solid curve. The effect of the longitudinal
modes is remarkable indeed. Based on this comparison, it can be stated that Eq. 共4兲
describes the full spectrum of star relaxation 共except for the segmental dynamics and the
ultraslow dynamics of colloidal nature兲 well and that longitudinal relaxation should be
accounted for. In similar manner, Fig. 4共b兲 demonstrates the success of this approach for
the two diluted 6460 stars at different linear chain concentrations 共and thus a different
number of arm entanglements兲. The longitudinal mechanism again captures the data
around the G ⬙ minimum well. Note that the 70/30 data exhibit more noise, but, on the
other hand, no vertical shifting was utilized.
Despite the satisfactory description of the experimental data using Eq. 共4兲, it should be
kept in mind that the Milner–McLeish 共1997兲 model has limitations, which have been
recently discussed in the literature; in particular, dynamic dilution apparently breaks
down a few entanglement segments near the branch point 共which probably follow constraint release dynamics兲 关Watanabe et al. 共2002兲; Shanbhag et al. 共2002兲兴. However, it
still remains the most complete and accurate model for star arm relaxation at the moment
and as such it was employed in this comparison.
Figure 5 presents the dependence of the arm relaxation time under isofrictional conditions, normalized to the segmental time and scaled with the number of arm entanglements ( ␶ a / ␶ s )(M a /M e ) ⫺5/2, on the number of entanglements per arm M a /M e for a
variety of star polymer melts with functionality ranging from 4 to 128, all being 1,4polybutadienes 关Vlassopoulos et al. 共2001兲兴, as well as the present star polymer blends;
in the latter case solvent-mediated dilution of entanglements 共Fig. 2兲 is taken into account
in the horizontal axis (M a /M e ) in Fig. 5. The arm relaxation time was determined
172
MIROS ET AL.
FIG. 5. Semilogarithmic representation of the normalized isofrictional arm relaxation time ( ␶ a / ␶ s )
(M a /M e ) ⫺5/2 vs the number of entanglements per arm (M a /M e ) for various star polymers ( f ⫽ 128, 〫; 64,
䊊; 32, 䉮; 18, 䉰; 4, 䉭兲 and star/linear 共1000兲 mixtures 共6415, ⫻; 6460, *; 6430, ⫹).
consistently for all samples from the inverse crossover frequency to the terminal region,
whereas the segmental time from the inverse crossover frequency to the Rouse-like
transition 关Pakula et al. 共1998兲兴. This type of representation stems from development of
the tube theory for arm relaxation 关see, e.g., Milner and McLeish 共1997兲兴, that predicts
␶ a ⬃ (M a /M e ) 5/2 exp (␥⬘/2 M a /M e ) with ␥ ⬘ being the spring constant of the quadratic
potential. This plot suggests rather universal behavior and differs from that reported by
Watanabe and Kotaka 共1983兲 and Watanabe et al. 共1996a兲, who studied the viscoelastic
relaxation of a mixture of a styrene 共core兲–butadiene 共arms兲 diblock copolymer with low
molecular weight polybutadiene. They found that the micelle arm relaxation times were
much longer 共as much as two orders of magnitude兲 compared to the corresponding star
arm relaxation, and concluded that this fast relaxation mechanism is similar but not
completely the same in the two systems. Apart from the fact that the ratio of linear to star
共or micelle兲 arm molecular weights, M linear /M a , is not the same in the two systems
共although it conforms to the above mentioned penetration criteria in both cases兲, an
important difference relates to the larger core/shell ratio of the micelle 关Watanabe et al.
共1996a兲兴 compared to that of the star 关Vlassopoulos et al. 共2001兲兴. Therefore, this discrepancy provides further evidence of the difference between multiarm stars and block
copolymer micelles 关Halperin 共1987兲; Vlassopoulos et al. 共1999兲兴. Another important
difference that possibly affects the above results relates to the fact that whereas the
polystyrene core of the micelles 关Watanabe et al. 共1996b兲兴 is glassy and thus rigid, that of
the stars is rather fuzzy, i.e., soft and can deform 关Vlassopoulos et al. 共2001兲兴.
The present well-defined star systems provide a clear physical picture of the diluted
arm relaxation mechanisms. However, a few additional remarks are in order. Within the
uncertainty due to scattering of the data, we find a value of the effective spring constant
of the quadratic potential ␥ ⬘ that is about 0.7 for M a /M e ⬍ 20, which is smaller than
the extracted value of 0.96 from lower functionality polyisoprenes 关Fetters et al. 共1993兲;
Rubinstein and Colby 共2002兲兴. For more than 20–25 entanglements per arm, the arm
relaxation times in Fig. 5 apparently level off; this is probably not physical, but due rather
to the procedure of extracting the relaxation times as well as to the 5/2 power which
LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS
173
FIG. 6. Double logarithmic representation of the isofrictional structural relaxation time ( ␶ slow / ␶ s ) vs
f 2.5(M a /M e ) 5 for stars ( f ⫽ 128, 䊏; 64, 䉱兲 and star/linear mixtures 共6460, 䉮; 6415, 䊊兲. The line is to guide
the eye.
maybe too large. One can also observe small deviation of the blend data from the single
star data, which however does not alter the conclusions drawn here. This could also relate
partly to the extraction of the relaxation times and the number of diluted entanglements.
The slow relaxation process, detected at the lowest frequencies in Fig. 1, is established
as related to the cooperative structural rearrangements of the weakly ordered 共liquid-like兲
stars 关Kapnistos et al. 共1999兲兴. This type of ordering has been documented in the literature for stars with functionality f ⫽ 128 or 64 arms, based on small angle x-ray scattering measurements 关Pakula et al. 共1998兲兴. However, in the case of the blends, the star–star
distances increased due to the presence of linear chains that shift the ordering peak to
outside of the detectable wave vector range. Despite the dilution effect, the slow mode
can still be detected in a high-star content blend; as the linear chain concentration increases, this mode becomes weaker and eventually disappears 共at intermediate concentrations it fuses with arm relaxation兲. The mean-field scaling approach developed to
account for star melt structural relaxation is applicable to the present diluted case as well.
This mode is considered an activated process that involves partial disentanglement of the
interpenetrating stars followed by displacement of the star into a neighboring cell, separated by a distance of its size, a process controlled by the free energy of corona elastic
deformation 共arm stretching兲. The net result of this analysis suggests the following scaling relation 关Kapnistos et al. 共1999兲兴:
␶slow
␶s
⬃
⫺1
␣ ⫺1/3 f 11/9N 26/9
a Ne
冋
exp
X1 f 5/3
␣ N 1/3
a
⫹X 2 ␣
2
N 11/3
a
N 3e f 4/3
册
,
共5兲
where ␣ ⫽ (a 2 ␷ ⫺2/3) and a is the monomer size, and X 1 and X 2 are unspecified numerical constants. The main outcome is the strong dependence of structural relaxation on
both the functionality and arm molecular weight, supporting the experimental findings.
Based on the data from star melts with 64 and 128 arms, a plot of the isofrictional slow
time ␶ slow / ␶ s vs f 2.5(M a /M e ) 5 has been proposed as representing the structural mode of
all stars 关Vlassopoulos et al. 共2001兲兴. For the case of mixtures, since arm disentanglement
participates in structural relaxation, dilution by the macromolecular solvent should be
considered. The is shown in Fig. 6 which is an attempt to describe the structural mode of
174
MIROS ET AL.
all diluted stars. Given the complex nature of the scaling in Eq. 共5兲, as well as the
difficulty in accurate experimental determination of ␶ slow 关from the intersection of the
terminal G ⬘ ⬃ ␻ 2 and G ⬙ ⬃ ␻ lines; see also Pakula et al. 共1998兲兴, especially when
low enough frequencies were not reached, the clear message in Fig. 6 is a universal trend
of the experimental data, in qualitative agreement with the prediction. For completeness,
we note again the difference of the present results from block copolymer micelles in a
nonentangled matrix, where the slow mode was assigned to the Stokes–Einstein diffusion
of micelles 关Watanabe et al. 共1996a, 1998兲; Gohr and Schärtl 共2000兲兴.
A issue that remains is the role of the size ratio M linear /M a , as already mentioned.
Whereas the present results support the penetration of small chains into the stars, at
higher linear chain molecular weights the entropic cost of penetration is too high
关Raphaël et al. 共1993兲; Halperin and Alexander 共1988兲; Leibler and Pincus 共1984兲兴 and
the conformation of the mixture as well as its properties is different; in such a case it
should be treated as a star-linear mixture in which both arm relaxation of the star and
reptation of the linear chain participate and should be accounted for 关Milner et al. 共1998兲;
Roovers 共1987兲; Struglinski et al. 共1988兲兴. At the same time, the importance of star
functionality should not be underestimated, since it can lead to a non-negligible core size
and eventually the mixture can exhibit many similarities to micelle/linear polymer
systems—the slow relaxation mechanism being the most notable one 关Watanabe et al.
共1996b兲; Watanabe and Kotaka 共1984兲; Gohr et al. 共1999兲; Gohr and Schärtl 共2000兲兴.
Naturally, the crossover of M linear from macromolecular solvent behavior to a star-linear
blend is of particular interest, and it will be addressed in the future.
IV. CONCLUDING REMARKS
The linear rheology of mixtures of multiarm star and linear polymers having molecular weight much smaller than the star arm molecular weight as investigated. These systems were considered solutions of stars in macromolecular solvents which dilute entanglements of the arms. Using a variety of mixtures 共different star arm molecular
weights and compositions兲 we were able to describe the response of diluted stars over the
entire frequency spectrum 共excluding the glass兲, based on the the Milner–McLeish theory
for arm relaxation, the longitudinal stress relaxation 共which was introduced and calculated for the first time for star polymers兲, and the high frequency Rouse modes. A virtually universal description of isofrictional arm relaxation time as a function of the number
of entanglements was obtained for stars of any functionality and degree of dilution. The
slow structural mode, related to the diluted star’s colloidal core, also depends on the
number of entanglements, as was indicated by a recent mean field scaling approach, but
in a rather complex way; nevertheless, good qualitative agreement with the data was
attained.
ACKNOWLEDGMENTS
This work was carried out at the Institute for Theoretical Physics, University of California, Santa Barbara; two of the authors 共A.E.L. and D.V.兲 would like to acknowledge
support by the National Science Foundation under Grant No. PHY99-07949. Additional
support was received from the European Union 共Grant No. HPRN-CT-2000-00017兲.
LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS
175
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