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Soft Matter
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Cite this: Soft Matter, 2015,
11, 3530
Received 26th January 2015,
Accepted 16th March 2015
Customizing wormlike mesoscale structures via
self-assembly of amphiphilic star polymers
Christian Koch,a Athanassios Z. Panagiotopoulos,b Federica Lo Versoc and
Christos N. Likos*a
DOI: 10.1039/c5sm00219b
www.rsc.org/softmatter
We examine the phase behavior of end-functionalized diblock
copolymer stars by means of Grand Canonical Monte Carlo simulations.
We focus on solutions of diblock copolymer stars with a solvophobic
outer block shorter than the solvophilic inner block, which are expected
to nucleate microphase aggregates. By tuning the temperature and
rigidity of the molecules, we target specific mesoscale structures, which
can act as powerful rheology modifiers. In particular, we control the
hierarchical self-assembly of single micelles in a ‘‘pearl-necklace’’ fashion,
which eventually merge into elongated, wormlike supermicelles.
The rich phase behaviour and responsiveness to a variety of
chemical and physical stimuli, which characterize polymeric
macromolecules, make them ideal building blocks for the
engineering of new functional materials. In particular, amphiphilic
systems in solution have a strong propensity to reversibly assemble
into a variety of structures. The corresponding microphases,
which mainly depend on the architecture of the precursor
macromolecule, the ratio among solvophilic and solvophobic
units and the environmental conditions, range from conventional
micelles, through flexible cylinders (wormlike micelles), on to
bilayers and membranes.
Wormlike micelles are elongated, semiflexible aggregates
for which the spontaneous curvature of the end caps is larger
than the curvature along their cylindrical body. These aggregates
result from the self-assembly of amphiphilic molecules. In the
last two decades, wormlike micelles have received considerable
attention, from the theoretical, experimental and computational
points of view.1–6 The main reason for this interest is the rich
rheological performance of such systems.7 Their statistical and
dynamical properties resemble those of polymer solutions, which
e.g. entangle above a critical concentration. As a consequence the
a
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna,
Austria. E-mail: c.koch@univie.ac.at; Fax: +43 1 4277 73239;
Tel: +43 1 4277 73230
b
Department of Chemical and Biological Engineering and PRISM,
Princeton University, Princeton, NJ 08544, USA
c
Donostia International Physics Center DIPC, Paseo Manuel de Lardizabal 4,
´n, Spain
E-20018 San Sebastia
3530 | Soft Matter, 2015, 11, 3530--3535
solution becomes viscoelastic, and the viscoelasticity features a
single relaxation time, a property quite unusual for fluids with a
complex structure. Wormlike micelles are also related to polymers
because of their nonlinear rheological behavior.8 While applying a
steady shear, the fluid undergoes a shear-banding transition, which
is associated with a plateau in the stress versus shear rate curve, and
corresponds to a transition between an homogeneous and nonhomogeneous state of flow. The latter regime is characterized by
macroscopic regions (bands) of different share rates analogous
to the instability found in extrusion of polymer melt at high
temperature.9 In addition, there exists a large class of wormlike
micellar solutions which exhibit a dramatic increase in the
viscosity above a critical shear rate. This has been associated
with further aggregation of micelles induced by the shear
flow.10–12 Wormlike micelles have a more dynamic structure
with respect to polymers due to the undergoing of continuous
and reversible scission and recombination.5,6 This aspect of their
behavior suggests that wormlike micelles can be considered as a
model of equilibrium polymers (or living meso-polymers).4,13,14
The unique viscoelastic behaviour of these systems is exploited
to tune the rheology in numerous applications, without the use
of polymers or additives, so that they are employed as e.g.
fracturing fluids in oil fields, as drag reducing agents in hydrodynamic engineering,15,16 and in many home and personal care
products.1 A full understanding of structure and dynamics of the
wormlike micelles, namely how micellar structure is connected
to the chemical composition and architecture of the surfactants,
and how the structural features of the aggregates can be tuned by
specific control parameters, is essential to the optimization of
their practical applications.
In the present study we examine the behavior of a specific
polymeric building block, namely a telechelic star polymer, able
to form wormlike structures via a hierarchical self-assembly
process. Telechelic star polymers are star-shaped AB-diblock
copolymers with a number of chains (functionality) f, which are
covalently anchored on a common point. In particular in our
model, solvophilic A-blocks lie at the interior of the star,
whereas the attractive, solvophobic monomers (the B-blocks)
This journal is © The Royal Society of Chemistry 2015
Communication
are the terminal reactive groups. In extremely diluted solutions
telechelic star polymers can self-organize in diﬀerent ways.17–21
Whereas at sufficiently high temperatures, the molecule presents an
open ‘‘star burst’’ configuration, lowering the temperature brings
about a collapsed conformation, namely all the solvophobic groups
physically associate into a single attractive aggregate (single watermelon19) or, as f grows, in several aggregates (partial watermelon
structures20 ). In concentrated solutions, the intra-molecular association process deeply influences the self-assembly behavior.21–25 The
main control parameters are the architecture of the building
blocks, the amphiphilicity and rigidity along the chain, and the
temperature. In our previous studies, we extensively considered
the influence of the amphiphile composition of the polymer
chains on the phase behavior of fully flexible stars while varying
the concentration of the solution.21–24 For low/intermediate
number of monomers per chain, and arm number up to
f = 10, we verified that if the terminal B block comprises at
least half of the stars’ monomers, macrophase separation
occurs. On the contrary, when the number of solvophilic
monomers exceeds the number of solvophobic ones, the system
displays a very rich microphase-separation behavior. As a
further step we considered the effect of rigidity on the macrophase separating systems.25
In this manuscript, we focus on the eﬀect of architecture,
chain rigidity and temperature as driving factors towards
the emergence of specific equilibrium mesophases, namely a
hierarchical self-assembly of wormlike micelles. Our approach
is based on Grand Canonical Monte Carlo simulations of a
lattice model. The details of the model and simulation methods
can be found in ref. 25. Each molecule is coded generically as
H(HnTm)f: the sequence in the brackets describes one of the
f arms, where the inner A-block of each chain consists of a
number n of solvophilic head beads (H), and the external
B-block of a number m of solvophobic tail beads (T). The first
letter-code, H, represents the central anchoring point. In order
to quantify the ratio among solvophilic and solvophobic units,
we introduce the parameter r = m/(n + m), stated in %. The interbead eﬀective interaction is modeled by means of the Larson
scheme for amphiphile–oil–water solutions.26 The rigidity is
modeled by introducing an energy penalty25 when the angle
between two successive bonds along a chain deviates from the
value p. The stiffness of the chain is tuned by changing the
prefactor a. A mixture of Monte Carlo steps is implemented,
namely insertion and deletion of an entire molecule, deletion
and regrowth of single chains on one molecule as well as
moving of an aggregated cluster of molecules. Every set of
parameters was run on multiple (cubic) simulation box sizes
L = 30, 40 and 50, to ensure that the mesoscale structures that
are found are no finite size effect. In what follows, monomer
concentration will be expressed in terms of the fraction of
occupied lattice sites f and temperature T* in units of the
interaction energy between neighboring solvophilic sites, also
setting kB = 1.
We focus first on the H(H6T4)3 model system. As mentioned
above, r = 50% corresponds to the limit below which the system
undergoes microphase separation. In particular telechelic stars
This journal is © The Royal Society of Chemistry 2015
Soft Matter
Fig. 1 Cluster size distribution of a H(H6T4)3 system at three diﬀerent
state points: (a) concentration f = 0.42, temperature T* = 4.6, (b) f = 0.5,
T* = 4.2, (c) f = 0.45, T* = 4.0. Error bars are typical for all histograms in
this work.
coded as H(H6T4)3, form, for T* 4 4.5, spherical micelles with a
well defined characteristic size.22 These micelles are not perfectly
sterically shielded by their corona of solvophilic A-blocks. In
Fig. 1 we plot the distribution probability of cluster sizes, as a
function of the number of chains per aggregate, at concentrations f D 0.45. In the cluster size histograms, the occurrence of
‘harmonics’, i.e. integer multiples of the characteristic micelle
size, marks the inter-aggregation of couples, triples, etc. of
spherical micelles into supermicelles. In panel (a) we can
observe the high temperature regime (T* = 4.6). Here the
magnitude of the macromolecular aggregates is highlighted by
few mild peaks. It is not favorable for the supermicelles to merge
further, into larger aggregates because of the entropic cost
necessary in order to push aside the solvophilic corona. Increasing
the concentration eventually leads to a percolation, namely the
coexisting micelles, supermicelles and remaining single star
molecules associate into a system spanning cluster. This cluster
has a very broad size distribution and grows continuously while
increasing the concentration.
Lowering the temperature brings about spectacular changes
in the morphology of the aggregates formed in the system. In
Fig. 1, panels (b) and (c) significant peaks of harmonics arising
in the cluster size histograms show up, while decreasing the
temperature down to T* = 4.0. These harmonics extend over the
full width of cluster sizes up to percolation. Indeed, by lowering
the system’s temperature, we enhance the propensity of the
attractive beads to maximize the number of their associations,
despite the entropic cost involved in the inter-aggregation
process. At a first stage, coexisting, stable spherical micelles
push aside their corona in order to be able to associate in a
dumbbell fashion, thus increasing the number of solvophobic
beads in contact and further minimizing the overall free energy.
Additional micelles can then associate in the same way onto
either far ends of the dumbbell, creating a super micelle
‘necklace’. The self-assembling process is illustrated in Fig. 2,
panels (a) and (b), in the case of three-micelle aggregation.
Note that this super micelle exhibits a one-dimensional
growth. A three-dimensional super micelle growth is unfavorable
by the fact that the solvophilic corona limits the size of a spherical,
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Fig. 2 Supermicelles association. Panel (a): triple aggregate. Panel (b): a
sketch of the hierarchical assembly of three micelles, first associating in a
pearl-necklace fashion aggregate and eventually merging in a quasi-1D
wormlike structure. Panel (c): a sketch of a wormlike super micelle, formed
by the hierarchical self-assembly of several micelles. The shaded regions
are guides to the eye.
purely solvophobic cluster. At the same time, a two-dimensional
equivalent in which, e.g., the micelles push aside their coronae to
expose an equatorial ring for association and form a membrane-like
super micelle are entropically highly unfavored. In fact, we did
not find any evidence for this in the range of systems investigated. A visual inspection of the system, see Fig. 3(a), confirms
that for low temperature the supermicelles start to assemble in
a pearl-necklace-like fashion and eventually maximize the
number of associated B-block beads while minimizing the
surface of their cluster: dumbbells and necklaces deform into
worm-like micelles that grow in total length with increased
concentration. For high f, these worm-like micelles also crosslink, entangle and eventually form a percolating network and at
f D 0.7, see Fig. 3(b), the network comprises all molecules
within the system. For higher temperatures, the association
mechanism is somewhat diﬀerent. Here, due to the higher
thermal fluctuations, spots of the micelle core are frequently
exposed and open a chance for small aggregates or even single,
free stars, to associate with the supermicelle. Consequently, the
cluster magnitude does not grow only in steps of the significant
micelle size, but rather in the smallest possible steps, i.e. single
chain associations. As a consequence the histogram does not
show harmonics, see Fig. 1, panel (a), but instead the distribution
grows continuously until percolation (see Fig. 4).
So far, it appears that a fundamental ingredient in order to
grow aggregates via a quasi-1-D assembly, are precursor
micelles with a solvophobic core not perfectly screened by the
solvophilic corona. For this reason, the system H(H8T2)3 is not a
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Fig. 3 Snapshots of a H(H6T4)3 system. (a) f = 0.45, T* = 4.0, (b) f = 0.69,
T* = 4.0. The red shapes are a geometrical encasing of clusters of
associated solvophobic tail beads. Solvophilic species are not shown.
Fig. 4 Snapshot of a H(H6T4)3 system, f = 0.42, T* = 4.6.
good candidate. Indeed for r o 20% the solvophilic corona
almost totally screens the internal core, so that the system have
a tendency to form isolated micelles. A more specific requirement is that the arm number be low enough ( f o 6) to avoid the
formation of multisite intramolecular association.19,20 Indeed, at
low T*, in the specific r and f range considered, namely r o 50%
and f o 6, the equilibrium state of the single molecule is a
collapsed ‘watermelon’ (wm) conformation, in which all the
chain solvophobic monomers form a single aggregate. If we
increase the functionality, the number of molecule association
sites (multi-wm), increases as well; e.g. at low temperature a
single star of a relative B block size of r = 40% forms one or two
wm patches for f = 6 and three or more for f = 10. Indeed the
higher number of arms, increases the repulsion at the anchoring
point, driving single stars to form more association sites and
networks instead of stable micellar aggregates.
To overcome the constraints related to the architecture and
the amphiphilic chain composition, we take advantage of another
This journal is © The Royal Society of Chemistry 2015
Communication
control parameter, namely the rigidity along the polymer
chains. This new chain property allows us to further control
the morphology of the aggregates. As previously pointed out in
the case of phase-separating systems,25 increasing the arm
stiﬀness decreases the entropic penalty due to the steric repulsion,
within a single star, while it enhances the propensity of the star
arms to collapse onto a single association site. The resulting
molecule configuration is quite narrow and eﬀectively makes the
star a colloidal particle with surfactant properties, thus drastically
stabilizing micelle formation in concentrated systems. Due to the
eﬀective stretching of the chains, those micelles have a higher
exposed core even for a low r. We fixed a = 10, namely intermediate
rigidity,25 and we started by studying the cluster size histograms
for H(H6T4)3, in order to observe the diﬀerences with respect to the
case of flexible arms discussed above. The same value of a will be
considered for all the rigid systems studied in the manuscript. As
we can see in Fig. 5, panel (a), the harmonics, related to the
hierarchical supermicelle assembly, become very pronounced
already at T* = 5.4. We stress here that a comparison of the
flexible and stiﬀ system at the same reduced temperature is not
meaningful, due to the extra energy penalty associated with
bending rigidity. Increasing the temperature, Fig. 5, panel (b)
and (c), one can see that intermediate cluster sizes are suppressed by a factor of B5. Eventually, panel (d), this suppression
evolves to a gap in the cluster size histograms. The harmonics
are always discernible. However, higher temperatures make the
intermediate aggregate sizes less probable in favor of the single
micelles (first peak on the left) or large aggregates which include
the majority of the present molecules (utmost right peak). While
for the H(H6T4)3 system, the rigidity strongly stabilizes the pearlnecklace forming of worm-like micelles, the simultaneous
increase of the temperatures promotes a faster formation of very
large, percolating aggregates. As anticipated above, stiﬀ molecule arms reduce steric repulsion eﬀects in the worm-like micelle
corona by leaving the solvophobic core more exposed.
Let us then increase the arm number up to f = 6. In contrast
to the H(H6T4)3 case, where the critical micelle concentration
lies below the percolation concentration, in the flexible, a = 0,
f = 6 system, the micellar aggregates form only after the formation
Fig. 5 Cluster size histograms of a H(H6T4)3 system with an arm rigidity of
a = 10. (a) f = 0.46, T* = 5.4, (b) f = 0.33, T* = 5.6, (c) f = 0.37, T* = 5.8,
(d) f = 0.40, T* = 6.0.
This journal is © The Royal Society of Chemistry 2015
Soft Matter
Fig. 6 Cluster size histograms of a H(H6T4)6 system with an arm rigidity of
a = 10. (a) f = 0.44, T* = 5.2, (b) f = 0.375, T* = 5.4.
of the percolating network.25 Applying a bending stiﬀness to
the H(H6T4)6 system we stabilize the formation of worm-like
micelles in a pearl-necklace-like fashion, similarly to the H(H6T4)3
case. The cluster size histograms show multiple harmonics that,
once again, are most pronounced at lower temperatures, see Fig. 6.
As already pointed out above, rigidity promotes the collapse of the
single star to a narrow watermelon conformation. As a consequence,
for very diluted systems the single wm configuration is stable up to
much higher temperature than in the case of flexible chains, while
for concentrated solutions, micellization and worm-like aggregates
are the expected scenarios.
As a further step, we tuned the amphiphilicity along the
chain by reducing the number of terminal groups. In accordance
with previous investigations,22 stars with r = 20% form stable,
spherical micelles of well-defined size. Compared to the H(H6T4)f
system ( f = 3, 6), however, the external solvophilic corona is
much thicker. The possibility for the corona to push aside the
solvophilic part and release the solvophobic core, necessary for
an aggregation of two micelles, is more diﬃcult, the compression of the corona being penalized due to steric repulsion. Next
to that, the number of additional associations that can be gained
by the inter-aggregating micelles is much lower compared to the
case r = 40%. We argue that the formation of supermicelles is
unfavored and the histograms, relative to the probability size
distribution, feature significant cluster sizes only up to the first
harmonic, or few more lowering the temperature. The spherical
micelles are stable up to very high temperature and concentration, eventually leading to percolation and to the formation of
a bicontinuous fluid phase.
In analogy to the r = 40% case, applying a bending rigidity to
the arms of the H(H8T2)3 system reduces the entropic cost of an
inter-micelle aggregation. Indeed we observed pearl-necklacelike formation of supermicelles highlighted by the harmonics
in P(Nch) of up to fourth order at T* = 3.2, see Fig. 7. In contrast
to the fully flexible H(H8T2)3 system, rigidity promotes the
formation of wormlike micelles due to a strong reduction of
the sterically repulsive eﬀects in the micelle coronae. As a
consequence the coronae is globally more flexible with respect
to the possibility of pushing aside the solvophilic part as well as
to expose the solvophobic core. For high concentrations, the
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Fig. 7 Cluster size histogram of the of H(H8T2)3 system with an arm
rigidity of a = 10 at f = 0.5, T* = 3.2.
harmonics become less pronounced. In this case free stars can
not form another micelle. They then, instead, bring one or
more arms into association with an existing micelle or super
micelle, thus changing its cluster size. We did not find evidence
of a system-spanning cluster. The application of rigidity promotes
the formation of worm-like micelles that show – in contrast to the
r = 40% systems – little tendency to form cross links. The majority
of supermicelles comprises well defined worms that quite heavily
bend and even entangle. The histogram shows a clear trend of
decreasing probability for long necklaces, which indicates that a
one-dimensional percolation is a finite size artifact, which indeed
disappears above a certain lattice size.
This scenario is extremely interesting while considering
rheological properties. For the fully flexible H(H8T2)3 system,
we have found that the formation of percolating clusters is
possible at suﬃciently high temperature and concentration. In
fact increasing the temperature, the entropic eﬀects in the micelle
corona become so strong that the micelles melt and the formation
of a fractal percolating network is favored. Rigidity decreases the
entropic eﬀects in the micelle corona, thus stabilizing micellar
aggregates. A much higher temperature is needed to melt these
micelles and to allow for a percolation transition. At given high
concentration, the sol–gel transition and thus, the rheological
properties can therefore not only be tuned by changing the system
temperature, but also at constant temperature by tuning the
rigidity via e.g. salinity of the solution, so that micelle formation
is stabilized sufficiently that the percolation transition is suppressed. Indeed, from an experimental point of view, a tunable
rigidity can be realized for instance by periodic ionization of the
polymer chains and control of the screening strength by changing
the salinity.
Similarly to the fully flexible H(H8T2)3 systems, for H(H8T2)6
spherical micelles are stable and super micelle formation is
only found at concentrations above f E 0.5. Also like H(H8T2)3,
the f = 6 case allows for a percolation through the lattice, for
temperature high enough to contrast the screening of the
micellar aggregate. Below such a temperature, despite the high
arm number, even at very high concentrations, the micellar
aggregates are so well sterically shielded that no system spanning
cluster is found. As in all investigated systems, the application of
rigidity to the stars’ arms promotes the formation of worm-like
micelles. In addition, for this system we observed the occurrence
of ordered phases. For a temperature range 3.6 o T* o 3.8, the
3534 | Soft Matter, 2015, 11, 3530--3535
Fig. 8 Snapshot of a H(H8T2)6 system with an arm rigidity of a = 10 at
concentration f = 0.64 and temperature T* = 3.8.
worm-like micelles that form, start to arrange in a columnar
hexagonal ordering at concentrations f E 0.6, as can be seen in
Fig. 8. For higher concentrations, additional micelles form, the
columns start to associate with one another and the ordering
breaks down into a fractal network. In our study we have used a
finite size simulation with periodic boundary conditions.
Whereas all disordered phases were found to be independent
of the simulation box size, the hexagonal ordering in this case
most certainly is. However, the mere tendency of the system to
arrange in an ordered phase is witnessed in simulations of
diﬀerent box size, where, instead, we found the micelles to
arrange on a cubic lattice.
There have been no prior studies on micelle nucleation of
telechelic stars. Indeed, the propensity to aggregate into wormlike micelles via a pearl-necklace fashion outlined in this work,
is novel. However, there are several studies of THT-triblock
copolymers, their eﬀects on oil-in-water microemulsions and
pre-assembled micelles and the formation and viscoelasticity of
transient networks formed by the telechelic polymer chains
interconnecting microemulsion droplets and vesicles.27–31
More recently, Ramos et al.32 investigated the effect of the
addition of telechelic PEO to a surfactant system of worm-like
micelles, and Nakaya-Yaegashi et al.33 extended these studies
by focusing on the linear viscoelasticity of these bridged worm-like
micelles. Telechelic star polymers represent a one-component, very
simple, system that features both the micellization process typical
of surfactant molecules and the networking due to the telechelic
linkers which are expected to show viscoelastic properties similar to
multi-component mixtures of spherical and/or wormlike micelles
and telechelic polymers.34
To summarize, a very rich microphase behavior can be
obtained by a proper design of telechelic stars. A necessary
condition for wormlike micelles to occur, are building blocks
(individual telechelic stars) that collapse onto a single physical
association site, such as the H(H6T4)3-system. However, this is not a
suﬃcient condition, since systems with a thicker solvophilic corona,
such as the H(H8T2)f-ones ( f = 3, 6) will not form wormlike micelles
despite the fact that they have single association cites. To overcome
the stabilizing eﬀect of the thick solvophobic corona, rigidity should
This journal is © The Royal Society of Chemistry 2015
Communication
be introduced to the arms, and in this way wormlike micelles
are stabilized. Rigidity also brings about wormlike micelle
formation for systems with more than one physical association
sites, such as the H(H6T4)6-system. The consequences of mesostructure formation on the rheology of such systems is the
subject of current work.
Acknowledgements
The authors acknowledge most helpful discussions with Emanuela
Del Gado and Dimitris Vlassopoulos. This work has been supported by the Marie-Curie ITN-COMPLOIDS (Grant Agreement
no. 234810). Work at Princeton was supported by the Princeton
Center for Complex Materials (U.S. NSF Grants DMR-0819860
and DMR-1420541).
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