Block Copolymers, Melt Rheology of
The ordered phases of block copolymer melts have
long-range order that is reminiscent of lyotropic
surfactant systems (Seddon 1990). Owing to energetic
repulsion, the different monomers in block copolymers
prefer to phase separate, but cannot because of their
covalent connection. Depending on the architecture of
the copolymer, different microphase-separated, ordered phases are formed. In order of increasing
monomer-content asymmetry, the equilibrium phases
are lamellar, bicontinuous gyroid, hexagonal cylinders, and b.c.c. spheres (see Block Copolymer Phase
BehaŠior). Each of these phases has its own characteristic linear viscoelastic response (detailed below).
However, there are three common rheological features
of all ordered phases of nearly monodisperse block
copolymer melts that are in stark contrast to ordinary
flexible polymer melts, and appear to be direct
consequences of the long-range order (Colby 1996,
Fredrickson and Bates 1996).
(i) The low-frequency linear viscoelastic response
does not exhibit a terminal relaxation time beyond
which the material flows like a simple liquid. The
viscoelasticity persists on time scales that are many
orders of magnitude longer than the time required for
individual block copolymers to diffuse distances of the
order of molecular dimensions. Thus, despite extensive
molecular rearrangement, the long-range order is
preserved on very long time scales and makes block
copolymers highly viscoelastic.
(ii) The critical yield strain, below which linear
response is observed, becomes smaller at lower frequencies. It is not uncommon for this yield strain to be
considerably less than 0n01.
(iii) The rheological properties are extremely sensitiŠe to deformation history. The long-range order is
perturbed by strong shear or extensional flows, with
direct consequences on viscoelasticity. This observation also allows nearly monodisperse block copolymer melts to be aligned by strong flows, as
discussed in Block Copolymers, Flow Alignment of.
To understand why the long-range order is important, one may compare these observations to
systems that microphase separate without having
long-range order. An example is graft copolymers—
multiple chains of one type of monomer grafted onto
a backbone chain of another type of monomer. Since
the grafting reaction occurs randomly along the
backbone, graft copolymers are polydisperse with no
long-range order. These materials microphase separate, but the order is very local, persisting only
slightly beyond molecular dimensions. None of the
three characteristic features are observed for graft
copolymers (Ge et al. 1999). Polydisperse graft copolymers are simple viscoelastic liquids at low frequencies, with a wide range of strain corresponding to
linear response, and are not sensitive to shear history.
Hence, the graft copolymers are quite similar to
ordinary flexible linear polymers because they do not
have any long-range order in their microphaseseparated state.
The rest of this chapter summarizes the present
understanding regarding these three observations. The
distinct rheological characteristics of each ordered
phase will be discussed. These unique signatures make
viscoelastic measurements useful for phase identification and for determining transition temperatures.
1. Low-frequency Linear Viscoelasticity
The linear viscoelastic response of the ordered phases
of nearly monodisperse block copolymer melts is
shown schematically in Fig. 1 (Kossuth et al. 1999),
through the frequency dependence of the component
of the complex modulus (G*) that is in phase with the
applied sinusoidal strain (Gd). All phases, including
the disordered phase, have essentially the same response at the highest frequencies shown in Fig. 1. This
includes the entangled rubbery plateau (with plateau
modulus G!N) that is a characteristic feature of all
polymer melts that have sufficiently long chains for
entanglement effects (Ferry 1980). Over a wide range
of frequency, entangled linear polymers (including all
phases of block copolymers) exhibit a frequencyindependent plateau modulus that indicates chains
must escape from their entanglements before additional stress can relax at longer time scales (or lower
frequencies). The plateau modulus is related to the
length scale of the entanglement spacing a (Doi and
Edwards 1986):
GN! %
kT
a#b
(1)
where T is absolute temperature, k is Boltzmann ’s
constant, and b is the (average) monomer size. Flexible
polymers typically have plateau moduli in the range
10& GN! 2i10' Pa (Fetters et al. 1996), and block
copolymers are no exception. The relaxation time of
single molecules, defined as the time scale for a
molecule to diffuse a distance equal to its coil size,
corresponds to the low-frequency end of the rubbery
plateau, where Gh falls as frequency is lowered (indicating relaxation). For disordered phases far above
the ordering transition temperature, this drop becomes
the classical response of all viscoelastic liquids:
Gh " ω#. Disordered phases that are closer to the
ordering transition temperature have fluctuations that
impose additional barriers to motion, delaying the
true terminal response somewhat (Jin and Lodge
1997), but at low enough frequency all disordered
liquids show Gh " ω#.
1.1 Bicontinuous Gyroid and b.c.c. Spheres
The bicontinuous gyroid and b.c.c. spheres phases
1
Block Copolymers, Melt Rheology of
data do not exist. ABA triblocks for which A is the
cylinder and B is the matrix are solids at stress levels
below a critical yield stress (Spaans and Williams
1995), when the cylinders are approximately aligned.
This result is hardly surprising, as many chains will
simultaneously have A segments in two different
cylinders, making a network that can only relax if the
A blocks are forced into the B phase.
log Gh
GNo
1.3 Lamellar
log x
Figure 1
Schematic representation of the linear viscoelastic response
of various phases of diblock copolymers, using the
frequency dependence of the part of the complex modulus
that is in-phase with the applied sinusoidal strain (used
with permission from Kossuth et al. 1999).
both have three-dimensional cubic structure, which
makes them viscoelastic solids for both diblock and
triblock copolymers (Kossuth et al. 1999). This is seen
in Fig. 1, since Gh is independent of frequency at the
lowest frequencies, and is much larger than the out-ofphase modulus (Gd not shown). The low-frequency
modulus is related to the domain spacing d for each of
these cubic phases (Kossuth et al. 1999):
! %
Gcubic
40kT
d$
(2)
The cubic phases (particularly the b.c.c. spheres
phase) are analogous to ordinary crystalline solids,
but with much larger domain spacings that make their
terminal modulus many orders of magnitude smaller
!
(typically 10# Gcubic
10' Pa).
1.2 Hexagonal Cylinders
The hexagonal-packed cylinder phase has the complex
modulus at low frequency approximately obeying a
power law in frequency (Ryu et al. 1997):
G*hexagonal(ω) " (iω)"/$
(3)
The 1\3 slope for the in-phase part of the complex
modulus [Gh(ω)] is shown in Fig. 1. Equation (3)
indicates that Gd(ω) will also be proportional to the
1\3 power of frequency, and be a factor of 1.7 smaller
than Gh. The only available theory for the hexagonal
cylinder phase predicts that the exponent is 1\4
(Rubinstein and Obukhov 1993). Presumably ABA
triblocks for which A is the matrix and B is the cylinder
would also obey Eqn. (3) at low frequencies, although
2
The lamellar phase of diblock copolymers is by far the
most extensively studied in terms of rheology. The
complex modulus obeys a power law in frequency at
low frequencies (Rosedale and Bates 1990, Koppi et
al. 1992):
G*lamellar(ω) " (iω)"/#
(4)
Such a power law is also anticipated by several
theories (Kawasaki and Onuki 1990, Rubinstein and
Obukhov 1993). It is not yet clear whether the power
laws of Eqns. (3) and (4) can be the true low-frequency
limiting response of either class of materials. Although
some theories do predict that these anomalous power
laws can be the actual low-frequency limit (Kawasaki
and Onuki 1990, Rubinstein and Obukhov 1993,
Sollich et al. 1997), the most likely scenario is that
these materials will behave as either solids or liquids at
sufficiently low frequency. Lamellar phases are analogous to smectic liquid crystals, which are known to
be solids (Colby et al. 1997). Specifically, there should
be a small but finite modulus associated with the
network of defects in the material, described by a
defect spacing ξ:
G!defect %
kT
ξ$
(5)
Typical defect spacings are 0.1–10 µm, so we would
expect the modulus to be smaller than 1 Pa. The small
value of the modulus, coupled with the increased
sensitivity to strain as frequency is lowered (discussed
below), have so far precluded observation of the defect
modulus of Eqn. (5). Lamellar triblock copolymers are
viscoelastic solids, with Gh(ω) independent of frequency at low frequency (Riise et al. 1995), since many
chains of ABA triblocks will simultaneously have A
segments in two different A layers.
2. Limit of Linear Response: The Yield Strain
With amplitudes γ of the applied sinusoidal strain
! the yield strain γ , the complex
that are larger than
y
modulus decreases as strain amplitude is increased.
For smectic liquid crystals (Colby et al. 1997) the
Block Copolymers, Melt Rheology of
following form has been found to describe the nonlinear response of Gh.
Gh(ω,0)
Gh(ω,γ )l
! 1jγ \γ
! y
(6)
At large strain amplitudes (γ γc) Eqn. (6)
!
corresponds to a constant stress amplitude,
which is
the yield stress σy. The yield strain has been most
extensively studied in the cubic phases (Kossuth et al.
1999). Unentangled diblock copolymers in the gyroid
phase obey Eqn. (6), and have a yield strain that varies
roughly as the square root of applied frequency
(γy " ω"/#). More entangled copolymers in either the
bicontinuous gyroid or b.c.c. spheres cubic phases do
not obey Eqn. (6), and have generally larger yield
strains that have much weaker frequency dependences.
A yield strain increasing with frequency is opposite to
observations for disordered materials (such as flexible
linear homopolymers). Disordered materials have
larger yield strains as frequency is lowered, which is
very useful for their viscoelastic characterization
because the strain amplitude can be raised to keep the
torque measurable. This cannot be done for ordered
phases while maintaining linear response, making
linear response of the hexagonal cylinder phase and
particularly the lamellar phase very difficult to study at
low frequency.
Surprisingly, a stronger dependence of modulus on
strain amplitude than predicted by Eqn. (6) has been
reported, for both the gyroid phase (Zhao 1995) and
the b.c.c. spheres phase (Kossuth et al. 1999). This
stronger dependence means that the stress amplitude is
actually smaller at larger strain amplitudes. This
interesting observation may indicate that the strong
flow has induced a phase transition in the material.
Strong shear is known to increase the order–disorder
transition temperature (Colby 1996), so perhaps it
affects transition temperatures between ordered
phases as well.
3. Steady Shear and Shear History Effects
Strong shear and\or elongational flows affect the
structure and hence the rheology of all ordered phases
of block copolymers (see Block Copolymers, Flow
Alignment of). The lamellar phase has been most
extensively studied for the interrelation between shear
and morphology. Large-amplitude oscillatory shear
(LAOS) provides an efficient means of aligning the
lamellar layers (Koppi et al. 1992). Improved alignment lowers the complex modulus at low frequency,
while maintaining Eqn. (4) (Rosedale and Bates 1990,
Riise et al. 1995). Coupled with diffusion data that
show the power law of Eqn. (4) is only seen on time
scales longer than molecular diffusion times (Colby
1996), the reduction in G*lamellar(ω) as alignment improves is strong evidence that defects are responsible for
the low-frequency power law.
The LAOS alignment procedures for lamellar diblocks have been optimized to create samples with
superb alignment parallel to the external plates (Polis
and Winey 1998). When this well-aligned state is
steadily sheared, a reduction of the lamellar spacing is
both predicted (Williams and MacKintosh 1994) and
observed (Pinheiro et al. 1996). At a strain of γ $ 0n3,
kink bands start to form (Polis and Winey 1998). Kink
bands are defects that are known to form in a variety
of layered systems, such as sedimentary rocks (Weiss
1980).
Both LAOS and steady shear have similarly strong
influences on structure and rheology of b.c.c. spheres
(Watanabe 1997) and hexagonal cylinders (Winter et
al. 1993, Morrison et al. 1993, Ryu et al. 1997). The
general trend of QG*(ω)Q decreasing as order is improved
appears to also hold for these ordered phases.
Bibliography
Colby R H 1996 Block copolymer dynamics. Curr. Opin. Coll.
Interface Sci. 1, 454–65
Colby R H, Ober C K, Gillmor J R, Connelly R W, Duong T,
Galli G, Laus M 1997 Smectic rheology. Rheol. Acta 36,
498–504
Doi M, Edwards S F 1986 The Theory of Polymer Dynamics.
Clarendon, Oxford
Ferry J D 1980 Viscoelastic Properties of Polymers, 3rd edn.
Wiley, New York
Fetters L J, Lohse D J, Colby R H 1996 Chain dimensions and
entanglement spacings. In: Mark J E (ed.) Physical Properties
of Polymers Handbook. AIP Press, pp. 335–40
Fredrickson G H, Bates F S 1996 Dynamics of block copolymers: theory and experiment. Annu. ReŠ. Mater. Sci. 26,
501–50
Ge S, Guo L, Rafailovich M H, Sokolov J, Peiffer D G, Schwarz
S A, Colby R H, Dozier W D 1999 Surface-induced ordering
in graft copolymer thin films. Langmuir 15, 2911–15
Jin X, Lodge T P 1997 Fluctuation regime in the viscoelastic
properties of block copolymer solutions. Rheol. Acta 36,
229–38
Kawasaki K, Onuki A 1990 Dynamics and rheology of diblock
copolymers quenched into microphase-separated states. Phys.
ReŠ. A 42, 3664–6
Koppi K A, Tirrell M, Bates F S, Almdal K, Colby R H 1992
Lamellae orientation in dynamically sheared diblock copolymer melts. J. Phys. II 2, 1941–59
Kossuth M B, Morse D C, Bates F S 1999 Viscoelastic behavior
of cubic phases in block copolymer melts. J. Rheol. 43, 167–96
Morrison F A, Mays J W, Muthukumar M, Nakatani A I, Han
C C 1993 Shear-induced morphological structures in triblock
copolymers. Macromolecules 26, 5271–3
Pinheiro B S, Hajduk D A, Gruner S M, Winey K I 1996 Shearstabilized bi-axial texture and lamellar contraction in both
diblock copolymer and diblock copolymer\homopolymer
blends. Macromolecules 29, 1482–9
Polis D L, Winey K I 1998 Controlling kink band morphology in
block copolymers: threshold criteria and stability. Macromolecules 31, 3617–25
3
Block Copolymers, Melt Rheology of
Riise B L, Fredrickson G H, Larson R G, Pearson D S 1995
Rheology and shear-induced alignment of lamellar diblock
and triblock copolymers. Macromolecules 28, 7653–9
Rosedale J H, Bates F S 1990 Rheology of ordered and
disordered symmetric poly(ethylenepropylene)-poly(ethylethylene) diblock copolymers. Macromolecules 23, 2329–38
Rubinstein M, Obukhov S P 1993 Power-law-like stress relaxation of block copolymers: disentanglement regimes. Macromolecules 26, 1740–50
Ryu C Y, Lee M S, Hajduk D A, Lodge T P 1997 Structure and
viscoelasticity of matched asymmetric diblock and triblock
copolymers in the cylinder and sphere microstructures. J.
Polym. Sci., Polym. Phys. 35, 2811–23
Seddon J M 1990 Structure of the inverted hexagonal (H )
""
phase, and non-lamellar phase transitions of lipids. Biochim.
Biophys. Acta 1031, 1–69
Sollich P, Lequeux F, Hebraud P, Cates M E 1997 Rheology of
soft glassy materials. Phys. ReŠ. Lett. 78, 2020–3
Spaans R D, Williams M C 1995 Non-linear viscoelasticity of
ABA block copolymer melts: Stress relaxation and recovery.
Ind. Eng. Chem. Res. 34, 3496–507
Watanabe H 1997 Rheology of diblock copolymer micellar
systems. Acta Polym. 48, 215–33
Weiss L E 1980 Nucleation and growth of kink bands. Tectonophysics 65, 1–38
Williams D R M, MacKintosh F C 1994 Shear of diblock
copolymer lamellae: width changes and undulation instabilities. Macromolecules 27, 7677–80
Winter H H, Scott D B, Gronski W, Okamoto S, Hashimoto T
1993 Ordering by flow near the disorder-order transition of a
triblock copolymer styrene-isoprene-styrene. Macromolecules
26, 7236–44
Zhao J 1995 Phase behavior of diblock copolymer mixtures:
Binary diblock blends and diblock solutions. PhD thesis,
University of Minnesota
R. H. Colby
Copyright ' 2001 Elsevier Science Ltd.
All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted
in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the publishers.
Encyclopedia of Materials : Science and Technology
ISBN: 0-08-0431526
pp. 727–730
4