tunable rheology of dense soft deformable colloids
TRANSCRIPT
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Tunable rheology of dense soft deformable colloids
Dimitris Vlassopoulos a,b, Michel Cloitre c
a FORTH, Institute of Electronic Structure & Laser, Heraklion 70013, Crete, Greeceb University of Crete, Department of Materials Science & Technology, Heraklion 71003, Crete, Greecec Matire Molle et Chimie (UMR 7167, ESPCI-CNRS), ESPCI ParisTech, 10 Rue Vauquelin, 75005 Paris, France
a b s t r a c ta r t i c l e i n f o
Article history:
Received 24 August 2014
Received in revised form 26 September 2014Accepted 29 September 2014
Available online 7 October 2014
Keywords:
Soft colloids
Viscoelasticity
Microgels
Star polymers
Glass/Jamming
Depletion
Particle elasticity
Viscosity
Yield stress/strain
Flow curves
In the last two decades, advances in synthetic, experimental and modeling/simulation methodologies have
considerably enhanced our understanding of colloidal suspension rheology and put theeld at the forefront of
soft matter research. Recent accomplishments include the ability to tailor the ow of colloidal materials via
controlled changes of particle microstructure and interactions. Whereas hard sphere suspensions have been
the most widely studied colloidal system, there is no richer type of particles than soft colloids in this respect.
Yet,despitethe remarkable progress in theeld,many outstandingchallenges remain in ourquest to linkparticle
microstructure to macroscopic properties and eventually design appropriate soft composites. Addressing
them will provide the route towards novel responsive systems with hierarchical structures and multiple
functionalities. Here we discuss the key structural and rheological parameters which determine the tunable
rheology of dense soft deformable colloids. We restrict our discussion to non-crystallizing suspensions of
spherical particles without electrostatic or enthalpic interactions.
2014 Elsevier Ltd. All rights reserved.
1. Introduction
1.1. Classication of soft colloids
Looking at the rich landscape of soft matter systems it is a prime
challenge to identify and classify the different types of nanoparticles
which have been investigated in reasonable depth and at the same
time are representatives of basic features in terms of phase behavior
and macroscopic properties. A reasonably comprehensive, albeit non-
exhaustive list of soft colloids includes emulsions and nanoemulsions,
surfactant onionsand liposomes, microgels,grafted coreshell particles,
block copolymer micelles, dendrimers, and star polymers[18]. Soft-
ness can be appreciated from different perspectives: particle elasticity,
diversity of soft interactions, and particle volume fraction.
1.2. Particle elasticity
Theimportance of theelasticityof Brownian objectcan be quantied
by the non-dimensional parameter = F/kTwhich represents the
ratio of theelastic free energy, F, to thethermalenergy kT[911]. Poly-
mer coils have a free energy of entropic origin withF kTindicating
that 1; they are the most deformable objects available. On the
other hand, spherical elastic particles with modulus Eand radiusRare
characterized byF=ER3 and=ER3/kT. Thus, innitely rigid hard
spheres correspond to the limit . Besides hard spheres, colloidal
star polymers and microgels are archetypical representatives of two
important classes of soft particles. Star polymers are made of a large
number of arms, f, each with the degree of polymerization Na,
which are attached to a central core. Following the DaoudCotton
model [12], each arm can be viewed as a succession of blobs whose vol-
ume V(r) r3f3/2 increaseswith distance rfromthe core. Due to theto-
pological stretching of the arms, the elastic modulus at distance rscales
as:E kT/V(r). At the periphery of the star, we haveEkTR-3f3/2 so that
f3/2. For stars with a few hundreds of arms, is of the order of a few
thousands. Microgel particles are crosslinked polymeric networks swol-
len by solvent[4,13,14]. Their modulus depends on many parameters
such as the crosslink density, the solvent quality, the presence of ions,
and the network architecture. For neutral microgels, a reasonable esti-
mate isE kT/V,whereV NxV0(Nx: number of statistical units be-
tween crosslinks; V0: volume of a statistical unit). For submicron
microgels (typically, R = 100 nm; V0 = 1 n m3; Nx = 100), we estimate
= 104. Nanoemulsion and emulsion droplets are soft colloidal objects
used in many applications. Here the elastic energyis associated with the
surface energy of the interface (typically oil/water) which resists defor-
mation [15]:F=R2 (with the interfacial tension). For R= 100 nm
and= 103 N/m, we estimate 106. For completeness, let us briey
mention multilamellar vesicles which exhibit strong analogies with
Current Opinion in Colloid & Interface Science 19 (2014) 561574
E-mail addresses: [email protected](D. Vlassopoulos), [email protected]
(M. Cloitre).
http://dx.doi.org/10.1016/j.cocis.2014.09.007
1359-0294/ 2014 Elsevier Ltd. All rights reserved.
Contents lists available atScienceDirect
Current Opinion in Colloid & Interface Science
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http://dx.doi.org/10.1016/j.cocis.2014.09.007http://dx.doi.org/10.1016/j.cocis.2014.09.007http://dx.doi.org/10.1016/j.cocis.2014.09.007mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.cocis.2014.09.007http://www.sciencedirect.com/science/journal/13590294http://www.elsevier.com/locate/cocishttp://www.elsevier.com/locate/cocishttp://www.sciencedirect.com/science/journal/13590294http://dx.doi.org/10.1016/j.cocis.2014.09.007mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.cocis.2014.09.007http://crossmark.crossref.org/dialog/?doi=10.1016/j.cocis.2014.09.007&domain=pdf -
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emulsion droplets, the effective surface elasticity resulting from a com-
bination of the bending and compressive elasticity of the smectic layers
constituting the particles [4,16]. In the context of this discussion, recent
efforts to characterize the elasticity of individual particles using various
techniques like micropipette aspiration, AFM, and microuidic ow ap-
pear very promising[1723].
Fig. 1summarizes the classication of soft colloids based on the
above approach. Clearly, colloidal particles span the softness parameter
space from ultrasoft polymer coils and star polymers to quasi-hardspheres, indicating that softness can be tuned at will during synthesis
or preparation to meet the requirements of various applications. Natu-
rally, our classication is somewhat simplied and does not account
for the variationsin parameters like star functionality, crosslinkdensity,
and interfacial tension (always in the absence of charges or enthalpicef-
fects). For example, star polymers are the simplest representatives of a
wider class of particles with heterogeneous internal microstructure. It
includes dendrimers [8,24,25], block copolymer micelles resulting
from the dissolution of diblock or multiblock copolymers in a selective
solvent[2633], and polymer-grafted (or polymer-adsorbed) particles
[11,3436]. The latter nd many applications as they provide an exqui-
site way to stabilize colloidal particles and tailor their rheology. Further
complications to our classication arise from more complex structures,
such as for example coreshell microgels consisting of a solid core cov-
ered with a network-like shell [3740]and block copolymer micelles
with a multi-compartmented coreshell microstructure [29,32]. Despite
these issues, the classication ofFig. 1offers a comprehensive descrip-
tion of the important features governing the behavior of soft colloids.
A useful, albeit rough parameter for coreshell particle's softness is
the fraction of the shell layer [34]: s = L/(L + rc), where L is its thickness
andrcis the core radius; for hard spheres, stars (forf= 100 arms andNa= 100 statistical units per arm) and polymers, s is about 0, 0.95
and 1, respectively. In all these systems it is possible to de-swell the
outer soft layer by adjusting the physicochemical conditions (pH, tem-
perature, solvent quality, addition of small non-adsorbing polymers)
so that the system switches from soft-like to hard-like behavior.
1.3. Particle interactions
The softness of particles is linked to the softness of interactions
which can be characterized by the form of the repulsive pair potential
between two particles. For instance, star polymers or, equivalently, par-
ticles covered with long end-grafted polymer chains (such as block
copolymer micelles) are characterized by coarse-grained ultrasoft po-
tentials which exhibit long-ranged Yukawa-like repulsion at large
center-to-center distances and logarithmic repulsion at short distances
(Fig. 1)[41]. The amplitude of the potential and its range of repulsion
depend on the number of arms. The logarithmic repulsion accounts
for the microscopic interactions which develop when the arms come
into contact, interpenetrate, and deform at short distances. The number
of arms,f, is the unique control parameter for tuning the starstar inter-
actionfrom hard-sphere-like to polymer-like (Fig. 1), the potential scal-
ing withf3/2 as the normalized elastic free energy[41].This potential
has been very successful in describing the structure of stars and star-like micelles, including the crystallization and glass transition, as con-
rmed experimentally[5,27,30,4144]. On the other hand, particles
without dangling chains like microgels or emulsion droplets can be
thought of as effective hard spheres up to the point of contact, and as
elastically deformable spheres at shorter distances. When the particles
come into contact they develop facets through which they exert normal
repulsive forces which are well modeled using Hertz theory [4547].
The resulting potential is accurate up to particle deformation of about
15% but extensionshave been proposed to account for the large particle
overlaps that take place under ow[48]. These potentials successfully
apply to dense microgel suspensions or concentrated emulsions
[4550]. Hence, the key features of soft interactions that distinguish
them from their hard-sphere counterparts are their wide-ranging and
the less-steep potential at contact and beyond, which explain the
tunability of soft interactions.
1.4. Particle volume fraction
In relation to particle elasticity, a consequence of softness is the abil-
ity of the particles to deform/compress in contrast to hard spheres.
Emulsion droplets adapt their shape to steric constraints by forming
facets at contact at constant volume. Hairy particles (like star polymers,
block copolymer micelles or grafted particles) and microgels can adjust
both their shape and volume in response to external stimuli like pres-
sure, ow, pH, and temperature, which makes their properties tunable
[4]. Moreover, the hairy structure of star-like particles allows for inter-
digitation in addition to deformability. The above features profoundly
affect the denition of the volume fraction of soft particles and their
behavior at high concentrations.The volume fraction of a hard sphere colloidal suspension is usually
determined in reference to the onset of crystallization at c= 0.494 or
the maximum packing fraction atHCP = 0.74. Hard particles subjected
to a centrifugal forceeld will eventually reach the latter limit so that
desired volume fractions can be subsequently prepared by dilution
[10,51,52].This procedure does not apply to soft deformable particles
Fig. 1. Cartoons of different nanoparticles with the arrow pointing to the direction of enhanced elasticity and transition from softto hard repulsive pair interaction potential: polymeric
coil, star polymer,microgel,emulsion, andhard sphere.Dashedand dottedlines illustrate thevariability of therepulsive potentials uponchanges in molecular characteristics, which affect
particle elasticity. Particles with grafted or adsorbed polymers (not shown here) are typically positioned between stars and emulsions (see text).
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which de-swell osmotically at large volume fraction [5356]. The
volume fraction of concentrated soft particle suspensions thus remains
an ambiguous quantity[52]. Generally, the effective volume fraction
based on the single particle hydrodynamic size is agreed upon as a uni-
versal parameter, albeit different from the actual volume fraction[51,
52,57,58]. The determination of the latter is extremely challenging
and, apart from a few studies for microgels and star polymers, few re-
sults are available[53,56,59]. The effective volume fraction has been
shown to work well in terms of scaling the properties (such as zero-shear viscosity or self-diffusion) for different soft particles over a
range of sizes and concentrations[57,58,60,61]. Hereafter we shall call
itvolume fractionunless otherwise specied.
2. State diagram: crystallization, vitrication and jamming
Particle interactions are reected in the form of the potential and
eventually in the phase behavior. Since some phases are metastable,
we prefer to call the respective morphological diagrams of athermal
(i.e., possessing only entropic interactions) soft colloids as statedia-
grams. Before studying the state diagrams of soft colloids, it is useful
to revisit the reference case of hard spheres.Fig. 2a shows the behavior
of monodisperse and non-interacting colloidal spheres suspended in
Newtonianuids[62,63].With increasing volume fraction there is a
transition from the disordereduid phase to a crystal phase (at volume
fraction c) through an intermediateuidcrystal coexistence regime
where crystals are in equilibrium with the uid phase. Above volume
fractiong, the crystalline order is lost and a disordered amorphous
glass, where the positions of the particles are uncorrelated, appears in-
stead. The upper limit of the glassy region is the maximum fraction of
randomly packed spheres,J, where theparticlesare forcedinto contact
ina jammedstate [64]. AboveJ, samples must have domains of crystal-
line structure or be entirely crystallized. Polydispersity inuences the
state diagram through the values of the glass and jamming transition,
which are largerfor polydisperse samples, and more notably suppresses
crystallization[65].
The state diagram of non-interacting soft microgels is shown in
Fig. 2b for comparison. It exhibits apparent similarities with that of
hard-sphere suspensions: microgel suspensions undergo the same
sequence of transitions fromuid to crystal to amorphous solid with
increasing volume fraction. However several major quantitative differ-
ences signal the role of softness. First, particles can be compressed
well above the jamming transition where they come into contact,
because they can accommodate the increased osmotic pressure by
adjusting their volume and shape as discussed in Section 1.4[4]. Sec-ondly, crystallization and melting seem to occur at higher volume frac-
tions and exhibit a narrower phase coexistence region compared to
hard sphere suspensions[66]. Sometimes, it may happen that crystalli-
zation is suppressed, whereas the glass and jamming transition are
shifted to higher volume fractions[58].Finally, the presence of soft in-
teractions combined with the development of appropriate annealing
protocols (rapid heating and subsequent cooling after certain rest
time) [67] and variations in synthesis (for instance changing the degree
of crosslinking)[13]offers unique possibilities to manipulate the mor-
phology of soft microgel suspensions to get ordered crystalline phases
over a wide range of volume fractions. Note that the state diagram in
Fig. 2b is based on the behavior of microgel suspensions which are not
purely athermal: poly(methylmethacrylate) (PMMA) spheres swollen
in benzyl alcohol [68] and thermosensitive poly(N-isopropylacrylamide)
(PNIPAm) particles in water[39,50,66]. The state diagrams of ionic
microgels with varying softness qualitatively also conform to that
inFig. 2b[69], which supports the generality of the trends we have
identied so far.
The softer star-like systems are characterized by a large uid region,
practically up to the overlap concentration (c*), a very small crystal re-
gion which may be completely masked by the glass, and a relatively
large glass-jamming region (Fig. 2c). The suppression of crystallization
in hairy suspensions is related to the fact that the dangling chains inter-
penetrate above c* and uctuate beyond the limit allowed by the
Lindeman criterion[5,35,70,71]. At sufciently long times, star glasses
eventually crystallize (in fact, this has been reported for hard spheres
as well)[43,72,73]. It appears that a large core contributes to damping
Fig. 2. Schematic one-dimensional state diagrams of nearly-monodisperse athermal soft colloidal suspensions, as function of their effective volume fraction (see Section 2). The numbers
arevaluesfor volume fractions of hardspheres;values above 0.74are relevantonly for softsystems. Frombottomto top thediagrams of a):hard spheres,b): microgels, c): star polymers
andd): polymericcoils areshown.Letters refer todifferentregions:uid(F, theblue-shaded regions), crystal(C, atc = 0.545 forhardspheres),coexistence (F + C, above volume fraction
0.494 forhard spheres), glass (G,at g = 0.58 forhard spheres), andjammed (J). RCPand HCPare therandomclosepacking andhexagonal close packing volume fractions, which forhard
spheres are determined to be 0.64 and 0.74, respectively. The solid black vertical lines represent established transitions (even if occurring at different volume fractions for different
systems) whereas red dashed lines represent transitions whose universality is still debatable. The values of the volume fraction in the horizontal scale are indicative and do not respect
the actual scale.
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theuctuations, hence promoting crystallization. This explain in part
why in general block copolymer micelles and densely grafted hard
nanoparticles can crystallize easily whereas star polymers cannot[5,
30,74,75]. When the dangling chains are short, the dynamical nature
of the micelles allows for appropriate adjustment of the number of
arms to promote crystallization[76,77].
From the above discussion we can draw some general conclusions
about the impact of softness on the state diagram of colloidal suspen-
sions. A general and important trend is that the
uid phase tends to ex-pand when the softness increases. This observation can be connected to
the behavior of polymer coil solutions which remainuid throughout
the whole range of volume fractions (Fig. 2d). For monodisperse sys-
tems, the same sequence of phases and states leading from uid phase
to glassy states through crystalline phases is qualitatively observed.
However, the actual values of the effective volume fraction marking
the transitions depend on softness. External stimuli, such as shear[78]
or thermal annealing[43,72,73], can promote local rearrangement and
even lead to crystallization.
Polydispersity systematically inhibits crystallization so that in poly-
disperse soft colloids we are left with the glassy and jammed regimes
only, which are often, but not always, the states of principal interest
for rheology. The glass transition can be described to a rst approach
by the cage model originally proposed for hard spheres where colloidal
particles are topologically constrained by their neighbors in virtual
cages[9,62,79,80]. In the jammed state, soft particles are still arranged
in a disordered amorphous way but they form polyhedral facets at con-
tact[4]. Interpenetration of star-like particles introduces an additional
complexity, which has not been investigated in detail yet [81,82].
Today there is growing evidence suggesting that solidication of poly-
disperse soft particle suspensions does not obey a universal description,
anddifferent scenarioshave been proposed as particle softness is varied.
Emulsions, which are relatively rigid particles ( 106), exhibit a glass
transition at g = 0.58 and a jamming transition around J= 0.64
[8385]. Star polymer solutions ( 103) exhibit a liquid to solid tran-
sition at a relatively well-dened concentration, but the glass or
jammed nature of the solid phase is unclear[86]. These experimental
observations qualitatively conform to the predictions of a recent
model which combines the contributions of Brownian motion and har-monic interactions[87,88]. Observations for microgels are somewhat
contradictory. The approach of the glass transition from the liquid
phase seems to becomemore gradual for softer microgels [89]. Aqueous
suspensions of submicron thermosensitive PNIPAm microgel or core
shell particles have been shown to exhibit a well-dened glass regime.
They represent an exquisite model system for studying the rheology
of entropic glasses in relation with Mode Coupling Theory (MCT) [90].
However, it is interesting to note that the rheology of very similar
PNIPAm suspensions near the liquid-solid transition has been
interpreted in terms of jamming[91]of a mixed glass-jamming state
[92]. Understanding the details of particle microstructure is important
in order to further advance the eld in this direction.
3. Mixtures and osmotic interactions
3.1. Colloidpolymer mixtures
Athermal mixtures of colloidal hard spheres with non-adsorbing
linear polymers constitute an archetypical situation where osmotic
forces induce new morphologies and original rheological behavior
(Fig. 3a)[10,93104].For example, in the colloidal gas domain (i.e.
uidphase, F),the addition of linearpolymers to hard spheres is respon-
sible for attractive depletion interactions which induce the aggregation
of particles[93104]. At low volume fraction phase separation (PS) is
observed whereas at larger volume fractions colloidal aggregates ulti-
mately ll all the available space leading to gel formation (Gel). The in-
terplay between strength and range of particle interactions (both
repulsive and attractive), and gravity remains a subject of debate
[105108].There are reports pointing to arrested phase separation as
the ultimate equilibrium state in the system (with short-range attrac-
tions) and others suggesting that percolation prevails[106112]. The
most intriguing phenomena occur in the high-volume fraction glass re-
gime. When linear polymers are added at increasing concentrations to
hard spheres at a size ratio of about 1/10, the initial repulsive glass
(RG) melts due to depletion interactions and eventually a re-entrant
attractive glass (AG) forms[99,100,103,113].
A natural question concerns the role of softness. Long-range soft po-tentials are expected to impart not only quantitative but also qualitative
differences in the behavior of soft colloidlinear polymer mixtures[71,
114]. Let us rst discuss the case of three-component mixtures made
of solvent (background), soft colloids and linear chains. Earlier work
with highly crosslinked microgel systems, which behave very much
like hard spheres, conrmed the repulsive glass to liquid to re-entrant
attractive glass or depletion occulation scenario[115118].Slightly
crosslinked microgels, which respond to external elds by both volume
and shape adjustments (Section 1.4), were found to de-swell in the
presence of small linear polymers[118121]. More systematic studies
have dealt with hairy particles and in particular star polymers
(Fig. 3b)[55,122129]. Due to the osmotic pressure of the linear poly-
mers, star polymers are expected to shrink (de-swell), causing a change
of volume fraction, and eventually experience depletion interactions.
This has been studied in reasonable detail both theoretically and exper-
imentally[55,125,128,129]. Whereas the main physics associated with
osmotic pressure effects is the same as for polymer/hard sphere mix-
tures, the state diagrams are different (Fig. 3b). More importantly, the
re-entrant solid state is not an attractive glass (or at least not necessar-
ily) but a gel instead (Gel)[128,129]. For given attraction strength,
phase separation is observed[55]. Due to the range of the potential,
the liquid pocket (F) which exists at intermediate volume fractions of
linear chains and stars is larger than in the hard sphere case. Moreover,
polydispersity of the hard spheres may reduce or even eliminatethe er-
godic phase, but not the glass-to-glass transition[113]. These conclu-
sions are based on systematic rheological measurements supported by
Small Angle Neutron and Dynamic Light Scattering (SANS, DLS), simula-
tions and theoretical models in terms of effective interactions and Mode
Coupling Theory[55,122129]. Nonlinear rheology and in particularLarge Amplitude Oscillatory Shear (LAOS) is an exquisite tool to detect
the softening and eventual melting of the repulsive glass (RG) upon ad-
dition of linear chains through the reduction of the yield stress, yield
strain and cage modulus[128130]. It also supports the conjecture
that yielding in these metastable colloidal systems is a gradual process
involving multiple cooperative breakage events[131135]. For large
enough fractions of linear polymers, the ergodic pocket (F) disappears
and the colloidal mixture undergoes a transition from a repulsive glass
(RG) to a gel (Gel). This is unambiguously probed by linear and nonlin-
ear rheology since the LissajousBowditch stressstrain signatures of
gels and glasses measured in LAOS experiments are distinct, the former
being a nearly square curve characteristic of viscoplastic behavior and
the latter a tilted ellipse as is the case for viscoelastic materials
[132135].The star polymer paradigm has motivated or put into perspective
studies involving hairy particles, and in particular the structure and
dynamics of mixtures of polymer grafted nanoparticles and linear
polymers in good solvents [136]or block copolymers and linear
homopolymers in selective solvents[26,31,137]. In the latter case
in particular, the homopolymers not only melt the crystalline phases
formed by the block copolymer micelles but can also inuence their
aggregation number, hence providing a control parameter for self-
assembly[31,137].
Recently, the role of linear polymers in inducing liquid-like behavior
in concentrated microgel suspensions via depletion was exploited [33,
138,139]. Adding polymers to microgel glasses induced melting and
eventual re-entrance attractive glass formation, with the extension of
the ergodic region depending on the microstructure of the particles
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(e.g., density of crosslinks), the size ratio and the potentially changing
solvent quality for the polymer. Furthermore, the role of depletion in
the aggregation and phase separation of food protein systems, whichare also dominated by softness and exhibit qualitatively similar features
to those reported here, was recently reviewed[140].
Interesting phenomena also occur in the melt, i.e. in the absence of
solvent background, where grafted nanoparticles or star polymers can
be thought as solvent-free colloids[141144], with signicance for
nanocomposite systems. Although we do not review the eld of nano-
composites here, it is worth mentioning that recent work suggested
that adding linear homopolymers to polymer-grafted nanoparticles
originally in theuid state can result in the formation of anisotropic do-
mains comprising trapped particles[145147]. This brings analogies to
the asymmetric caging of binary star mixtures [148]which will be
discussed below. A further worthy example is the systematic investiga-
tions of grafted particles in polymer melts (say polystyrene-grafted
polyorganosiloxane in linear polystyrene matrices) whereby increasing
the size ratio between the radius of gyration of the polymer and the
particle radius it is possible to reduce the strength of depletion and pro-
mote dispersion of the particles, with important consequences on theirdynamic response[144,149153]. More recently, blends of crosslinked
PMMA microgels with linear PMMA of much smaller size were investi-
gated[154]. In the case of slightly crosslinked microgels the linear
chains penetrate and swellthe microgels and homogeneous dispersions
are obtained. However, highly crosslinked microgelsbehave likeimpen-
etrable hard spheres and aggregate due to the depletion effects induced
by linear chains.
3.2. Binary colloidal mixtures
The consequences of strong entropic effects discussed inSection 3.1
can be extended to more complex, yet interesting from the standpoint
of applications, systems, by replacing the linear polymer by another col-
loidal particle. For instance the morphology of binary asymmetric
Fig. 3.Schematic state diagrams of (from top to bottom): a) linear polymerhard sphere mixture atRpolymer/RHS b 0.2; b) linear polymercolloidal star mixture at Rpolymer/Rstar b 0.5;
c) binary hard sphere mixture at Rsmall/Rlarge b 0.2; d) binary star mixture atRsmall/Rlarge= 0.3; and e) hard spherestar mixture atRHS/Rstar= 0.25. Notation: F = uid, PS = phase
separation, RG = repulsive glass, AG = attractive glass, SG = single glass, DG = double glass, C = crystal, F + C = uidcrystal coexistence, APS = arrested phase separation,
AsG = asymmetric glass.
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mixtures of hard spheres was explored with emphasis given to the ex-
istence of freezing transitions and the formation of superlattice struc-
tures, driven by the maximization of entropy as the system adopts its
most efcient packing arrangement.Fig. 3c shows that, for a give size
ratio and depending on the relative volume fraction, hard sphere mix-
tures exhibit both crystalline (C) and glassy regions (SG, DG). Single
glasses (SG) and double glass (DG) form when either the small (here)
particles or both small and large particles become kinetically trapped
in amorphous structures[155,156].Mixtures of small star polymers oflow-functionality (up to 32) and large hard colloids were found to
phase separate for size ratios not exceeding 0.5 and colloid volume frac-
tions below 0.5[157]. Microgel particles were used as depletants (in-
stead of polymers) to hard spheres and it was found that gels were
formed more easily and eventually demixed[158].
A much richer behavior is observed and actually predicted in star
mixtures (with functionalities exceeding 64), placing them in the soft
colloidal domain[71,159], where the softness leads to the formation
of differenttypes of arrested states: not only those with one or bothpar-
ticle populations trapped as in hard sphere mixtures, but also a novel
asymmetric glass (AsG) where the effective large particle cages become
anisotropic due to the osmotic pressure of the small particles (Fig. 3d)
[148,160162]. Recently, mixtures involving block copolymer micelles
with frozen and livingcores and short arms were studied, and it was
found that a balance between crystallization and vitrication results
from an optimum adjustment of the number of arms[163].
Combining soft and hard interactions opens another route for ex-
ploring and tuning macroscopic properties. A mixture of small hard
spheres and large star polymers represents such a situation, where re-
pulsive pair potentials which have different ranges are coupled
[164166]. As depicted schematically in Fig. 3e, at low polymer star vol-
ume fractions the colloidal liquid phase separates upon adding hard
spheres. At higher volume fractions in the star glass regime, there is a
small domain where the glass melts due to depletion and further addi-
tion of hard spheres results in APS. Rheology was again used in order to
identify the different transitions and the strength of the moduli for dif-
ferent mixture compositions, whereas thegradual natureof the yielding
process under shear was conrmed[166].
This plethora of morphologies has important consequences on themacroscopic properties of mixtures. The various stable and metastable
states discussed above in the context of colloidal mixtures possess dif-
ferent rheological properties so that both linearand nonlinear viscoelas-
tic characterizations can very sensitively diagnose their differences. At
the same time this richness of behavior allows tailoring the ow of
soft colloidal composites at wish[4,5,167].
4. Linear viscoelasticity, diffusion and zero-shear viscosity
4.1. Viscoelastic relaxation spectrum and plateau modulus
The viscoelastic properties of soft particle suspensionsare exquisite-
ly reected in the dependence of the storage and loss moduli on fre-
quency. As the volume fraction increases, the rheological responsealters from viscoelastic liquid behavior where terminal regime is ob-
served over the experimental frequency window, to solid-like behavior
where the storage modulus dominates. The evolution of the viscoelastic
spectra leading to solidication hasbeen studied in detail forsoft particles
in the colloidal regime like emulsions and nanoemulsions[83,168,169],
coreshell particles[3739,90,170172], and microgel particles[40,92,
173175], which have softness parametersN104. The approach of the
glass transition is signaled by the superpositionofG and G at higher fre-
quencies beforea plateauin G anda shallowminimum in G appears. The
onset of terminal relaxation shifts to lower frequencies as the volume
fraction increases up to the point where it falls outside the experimental
frequency window. These viscoelastic properties can be qualitatively
understood in the framework of the cage model where a colloidal
particle is topologically constrained by its neighbors in a hypothetical
cage[9,79,80]. It moves locally within the cage (beta relaxation) but
will escape only if the environment of the cage can renew at long
times (alpha relaxation). The minimum ofGexpresses the transition
from in-cage motion to out-of-cage motion, and it can be also probed
in creep recovery experiments[176,177].
The shape of the viscoelastic spectra remains qualitatively the same
as the suspensions cross over the jamming transition althoughthere are
some important quantitative differences that will be discussed later on
in this section. The solidi
cation scenario is not well-known yet forsofter particles with10104 although it is denitely more gradual
[86]. Inside the solid state, the viscoelastic spectra of star-like particle
suspensions also exhibit a plateau in Gand a minimum inGbut new
features are observed at low frequencies where the relaxation of the
interdigitated arms apparently contributes to an additional relaxation
process[81]. Before alpha relaxation is activated, the arms can disen-
gage from their neighbors and deform the cage. As a result, star-like
colloidal glasses with many dangling arms often have an experimentally
accessible alpha relaxation, marking a clear departure for other systems
[178,179]. This may not necessarily be the case for colloidal block
copolymer micelles which have a larger core and smaller dangling
ends than stars and usually exhibit slightly larger particle elasticity
[180182].
In view of the huge diversity of soft particles in terms of effective
elasticity and architecture, a real challenge is to draw a one to one
comparison of their viscoelastic properties. This is a formidable task,
which in general seems inaccessible, because of the difculty to deter-
mine the glass and jamming transitions (Section 2) and the actual vol-
ume fraction of the suspensions (Section 1.4). Even for a given class of
particles such as star polymers, manifestations of softness like osmotic
shrinking and interdigitation depend on functionality, and there are
no systematic experimental studies[81,183]. An additional difculty
comes from the fact that, unlike for polymers, mean-eld theories of
colloids are in general not possible since the coordination number
does not exceed a value of 12. The rationalization of experimental re-
sults often relies on semi-phenomenological approaches or simulations.
However, recent progress in theeld allows us to identify some impor-
tant trends.
Mode Coupling Theory (MCT)[184]has proven highly successful inpredicting the linear rheology of hard colloids on approaching the glass
transition by accounting for density correlations from the static struc-
ture factor and then calculating the viscoelastic moduli[185]. It also
works well for soft glassy colloids with the same t parameters[84,
186188]. Recent extensions of MCT[189191]have been developed
to predict the frequency-dependent moduli of coreshell particles and
star polymer suspensions well-within the glassy regime, which is be-
yond the expected range of validity (up to and at the glass transition)
of the initial theory[39,192]. However, MCT is unable to capture the
low frequency relaxation which occurs in addition to alpha relaxation
as already mentioned. For the sake of comparison with predictions,
the experimental viscoelastic moduli have to be scaled with the entro-
pic elasticity of individual particles, kT/R3, while the angular frequency
is converted into the Peclet number:Pe = (6R3)/kT. These scalingforms are expected to be valid in the thermal glass regimeg b b Jdue to the entropic nature of the suspension dynamics [134]. Note
that a successful alternative, albeit similar in spirit, statistical mechani-
cal theory exists, which describes glassy dynamics based on a nonequi-
librium free energy that incorporates local cage correlations and
activated barrier hopping processes[193196].
The properties of jammed suspensions exhibit several quantitative
differences from those of entropic glasses because of the presence of
repulsive elastic interactions which exceed thermal forces. The plateau
modulus scales with the contact modulusE* of the particles (E* = E/
(1 2); Eand are the Young modulus and the Poisson ratio, respec-
tively) and exhibits a rapid linear increase with the distance to the jam-
ming transition: G0/E* ( J). This has been observed for
concentrated emulsions, microgel suspensions and even star polymer
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solutions [45,86,197]. At high angular frequency, the loss modulus of
many soft systems like emulsions and microgels increases likeG~ 0.5,
which is the sign of anomalous dissipation[83,174,178,198,199].
A recent comprehensive study combines particle scale simulations
and a microscopic mean-eld theory[200]. The particles are treated
like elastic non-Brownian spheres dispersed in a solvent (viscosity: s)
at high volume fraction, which interact through Hertz-like potential
and elastohydrodynamic lubrication forces[48]. The theory gives pre-
dictions for the pair correlation function of the suspensions, the highfrequency moduli, and the osmotic modulus [200]. Simulations also
allow computing the frequency dependence of the viscoelastic moduli
including the high frequency regime and the variations of the low-
frequency plateau modulus with volume fraction[45,135]. The predic-
tions compare well with experiments without adjustable parameters
when the moduli are scaled by the contact modulusE* of the particles
and the frequency by the characteristic time = s/E*. Other models
have been proposed to take into account the complex interfacial struc-
ture of emulsions and the non-homogeneity of coreshell particles
[201,202]. Similar investigations for star-like particles are still missing
but are highly desirable.
4.2. Zero-shear viscosity and self-diffusion
The most striking manifestation of the difference in softness (hence
interaction potential) among colloidal particles appears in the terminal
viscoelastic regime, and a relevant macroscopic quantity is the zero-
shear viscosity of the suspensions. It turns out that it is possible to
build a generic plot where the zero-shear viscosity of various systems
scaled with the solvent viscosity is represented versus the effective hy-
drodynamic volume fraction [57,203]. Thisis depicted in Fig. 4 whichfor
clarity only includes one data set from each particle type: hard spheres
[204], high functionality stars[86], block copolymer micelles[32]and
microgels[53]. Interestingly, the limited long-time self-diffusion data
available for hard spheres and stars (not shown here) are in excellent
agreement with the viscosity data when plotted accordingly, hence
conrming the validity of the StokesEinsteinSutherland relation
even at large volume fractions [81,205207]. Data for colloidal particlesof any softness (e.g., stars, grafted particles, block copolymer micelles,
microgels) could be included in such a plot, which reveals the role of
softness in reducing the relative viscosity and increasing the effective
maximum packing fraction[27,61,170,208].However, it is important
to emphasize that the transition to a weaker viscosity at higher frac-
tions, and eventually to divergence, also signals osmotic de-swelling
or shrinking. This phenomenon, which can also be triggered externally
by temperature of ionic strength variation,fullyreects particle softness
and reduces the actual volume fraction and stiffness as discussed above
[54,61,120,209,210]. Forpolyelectrolyte particlesit hasbeen shown that
the divergence of the viscosity exhibits little departure from the hard
sphere case once the volume fraction is appropriately corrected for
particle shrinkage[53].The type of representation shown inFig. 4emphasizes the fact that
softness affects only the high volume fraction regime, where particle in-
teractions come into play. At low volume fractions reduced viscosities
for all particles collapse to the EinsteinSutherland and Batchelor curves
[57]. But on increasing volume fraction the data depart according to
their softness. Recent extensive mesoscale simulation studies, accounting
for the effects of hydrodynamics as well, successfully predict the volume
fraction dependent viscosity of linear and star polymers shown inFig. 4
[211213]. Several empirical models have been proposed to describe
the viscosity departure and divergence at the glass transition. The
KriegerDougherty and Quemada models are very successful for a wide
range of systems[11,203]. The approach to glass transition is a subtle
issue because in reality there do not exist true hard sphere experimental
systems due to stabilization needs [214]. Inthis case even models used in
molecular glasses were invoked to capture the steep viscosity increase
[215]. To further characterize the variations of the viscosity before solidi-
cation, the concept of fragility was used [47,89]. It was proposed to build
Angell-type fragility plots by dening a glass transition volume fraction
and plotting the normalized viscosity of the suspension against its dis-
tance from this glass transition[89]. The outcome of this analysis is that
softer colloids make stronger glasses and indeed, a closer look at the
trend ofthe datain Fig. 4 conrms this conclusionand suggeststhat poly-
mer coils are potentially the strongest glasses.
4.3. Aging
A comment on aging is in order to close this section. Disordered
crowded materials, such as glassy and jammed colloids exhibit a slow
time evolution of their dynamic and structural properties, also termedaging. There are several features of aging which are shared by many col-
loidal glasses, such as a certain time regime characterized by a logarith-
micincrease of storage modulus and slowing-downof dynamics. On the
other hand, the dependence or not of macroscopic properties on the age
and the exact form of the respective scaling, as well as the presence of
different aging regimes for different systems, represent distinct signa-
tures of softness[134,179,199,216,217].We shall not discuss aging fur-
ther, as this has been the subject of other reviews[218,219]. However
we emphasize that irrespectively of its study, aging must be properly
accounted for when analyzing properties of glassy and jammed mate-
rials. In this respect, an appropriate preparation protocol is necessary
to place the material in a reproducible state. It consists of shear-
melting the glass or jammed state in analogy to thermal annealing of
molecular or polymeric glasses, and following the dynamics over timeuntil a quasi-steady state is reached. This is not necessary a trivial task
since internal stresses may affect the dynamics, hence proper denition
of a protocol and careful implementation are important for obtaining
consistent data[179,220222]. The properties discussed above such as
the volume fraction dependence of the moduli or the frequency spec-
trum of viscoelastic moduli, all refer to aged systems.
5. Nonlinear rheological phenomena
5.1. Yielding behavior
Concentrated suspensions of soft particles exhibit solid-like proper-
ties at rest but ow at large stresses. When the applied stress is de-
creased, the material is progressively slowed-down and eventually
Fig. 4.Dependence of the zero-shear viscosity (normalized to the solvent viscosity) of
various colloidal systems on the effective volume fraction. Selected data are shown for
suspensions of hard spheres[204], microgels[53], block copolymer micelles [32], and
high functionality star polymers[86].The dashed line is the bestt of hard sphere data
to the Quemada model[53]. The low-shear viscosity data of star polymer data are well
represented by a double exponential function (dotted line)[86].
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immobilized at a nite value of the stress, called dynamic yield stress,
which for the present discussion will be simply called yield stress,y.
Note that the static yield stress represents the stress needed to apply
in order to induce macroscopicow from rest. In general, the yield
stresscan be viewed as a viscosity bifurcation: the application of a stress
is followed by ow stoppage (b y) or steady deformation ( Ny)
depending on the stress value[216,223225]. There are many other
ways of determining the yield stress[226], the most straightforward
and frequently used being the strain amplitude sweep tests[131,174,188,199,227,228]. Recently, the application and analysis of Large
Amplitude Oscillatory Shear (LAOS) has emerged as a powerful tool
for probing the yielding andow properties of glasses and jammed sus-
pensions[133,135,227,229232].
Referring to the old dilemma whether glasses ow or not, there is
consensus that yielding exists, i.e. an external eld can drastically alter
the structure of a material, hence its macroscopic properties[233,234].
However, there is stilla debate as to whether yield stress is a true mate-
rial parameter or not. A useful distinction has been made between true
yield stress materials and thixotropic materials exhibiting apparent
yield stress[11,235238].
Densely packed systems such as glassy and jammed particle suspen-
sions (sometimes termed pastes) made of soft repulsive colloids, which
are the focus in this review, are true yield stress materials. Yielding oc-
curs through a continuous break and reformation of the cages which
keep the particles motionless at rest. In this context, LAOS is a unique
tool due to the possibility to reproduce within a cycle the sequence of
events that characterize ow and yielding [133,229,239,240]. One inter-
esting outcome is the effective cage modulus, which is dened as the
slope of the stress vs. strain amplitude plot at zero stress and appears
to match the value of the bulk (plateau) modulus G0 and remains insen-
sitive to the strain or stress amplitude[130,133]. A natural question at
this point is how do yield stress and strain compare for different soft
jammed colloids and depend on volume fraction[131,134]. Due to the
prominence of elastic effects, yis often proportional toG0with the co-
efcient of proportionality being the yield strain y: y = G 0y. As
discussed inSection 4.1,the storage modulus of soft particle suspen-
sionsscales with kT/R3 in the entropic glass regime, andthe particle con-
tact modulus E* in the jammed regime. In the literature not muchattention has been paid to the yield strain [199,227,228].Simulations
and theoretical modeling have predicted that the yield strain of soft
elastic spheres in the jammed regime should increase exponentially
with the distance to the jamming transition (it varies from low values
of about 0.02 nearJto about 0.08 for = 0.9)[49]. This prediction is
relatively well obeyed by emulsions[48,49,85,168]and microgel parti-
cles [48,134]. Note that entropic hard-sphere glasses have a non-
monotonicyield strain with a maximum value of about0.15at a volume
fraction between the glassand close-packing volumefractions[176,227,
241]. Combining the scaling expected forG0andyprovides the varia-
tions of the yield stress. Due to their long-hairy conformation, star-like
colloids exhibit distinct yielding properties[86,130,134,199,228]. Since
terminal alpha relaxation is accessible in these systems even at very
high concentrations, as explained inSection 4.1, yielding exists only ina limited range of frequencies or time scales. Moreover, the yield strain
depends very weakly on concentration. These ndings call for more
detailed investigations.
Colloidal gels ofocculated dispersions with attractive interactions
mostly belong to the class of thixotropic materials[11,242,243]. The
microstructure responsible for solid-like properties at b y is
destroyed by the ow, which induces a sudden drop of viscosity and
substantial structural changes. The disruption of the structure under
ow is reversible but typically characterized by a large hysteresis as
the structural timescale is usually longer than the inverse shear rate
[243,244]. Due to their microstructure, stars in the weak glassy regime
in temperature-sensitive solvents exhibit thixotropic-like response
[245]. Although there is some confusionin the terms,the slow evolution
of thixotropic materials is distinct from the aging and rejuvenation
phenomena which are known to occur in purely repulsive systems
[246249]. On the other hand, the yield stress reects in part the
strength of bonds and/or cages. Its determination from the ow curves
poses serious problems because homogeneous ow ends-up over a
limiting range of low shear rates[250]. We will not review the eld of
thixotropy here[11,251]. However, it is worth emphasizing that some
features of thixotropic materials like non-homogeneous ow at low
shear rates are also found in repulsive star-like colloids[178,249]. This
shows that the distinction between true and apparent yield stressmaterials is not as denite as often believed, and that some materials
share properties of different classes.
5.2. Flow properties
An important question concerns the establishment of steady-state
ow conditions in the form of stressshear rate relationships or ow
curves. In most experiments, measurements start at high shear rates
(associated with the so-called rejuvenated state) where steady state is
reached rapidly, and the shear rate is progressively decreased down to
the yield point. Applying a stress larger but close to the yield stress is
followed by extremely long transient phenomena, after which steady
state is eventually reached[249,252254]. In practice, the shape of the
steady state ow curves constitutes a remarkable ngerprint of the
system under investigation. In many soft colloidal suspensions like
microgels, emulsions, polystyrenePNIPAm coreshell particles, and
star polymers, the onset of the glass transitionis signaled byow curves
exhibiting two branches at low and large shear rates, separated by a
slowly increasing plateau at intermediate rates (in double logarithmic
coordinates) [86,90,168]. Thelow-shear rate branchcorresponds to ter-
minal behavior, and the high shear-rate branch reects shear-thinning.
As the volume fraction increases, terminal behavior shifts to very low
shear rates due to the huge increase of the -relaxation time, up to
the point where it falls outside the experimental window; meanwhile
the intermediate plateau widens and attens. In the entropic glass re-
gime, the ow curves are conveniently represented in dimensionless
coordinates as: kT=R3 f
R2=D0
. Here, kT/R3 represents the char-
acteristic scale of the yield stress while R2=D0 is a characteristic time
scale associated with the self-diffusion of individual particles (D0isthe self-diffusion coefcient at innite dilution). In this regime, MCT
provides remarkable quantitative predictions of theow curves, sug-
gesting that the effects of softness and deformability can be effectively
lumped into the structure factor at least at low shear-rates[90,189,
190,255]. Other nonlinear features like creep behavior [256], stress
overshoots during shear ow start-up[257,258]and the built up of
residual stress[222,259]are also predicted by MCT[260].
At higher volume fractions corresponding to the regime of jammed
suspensions, the ow curves (again in double-logarithmic representa-
tion) of a wide range of viscoplasticuids with yielding properties are
characterized by a stress plateau at low-shear rates and a shear-
thinning behavior at high shear rates[217,246,261,262]. Empirically,
they arevery well represented by theHerschelBulkley phenomenolog-
ical equation: y kn
withan exponent n which for different ma-terials has beenfound to be close to 0.5. The valueof the yield stress isin
good agreement with independent measurements based on other tech-
niques. The consistency parameterkdepends on material properties. It
has been shown[262]that normalizing the stress and shear rate with
yield stressyand time scale S/G0(Sbeing the solvent viscosity), re-
spectively, collapses the data obtained for different jammed materials
onto a master curve (Fig. 5). A recent micromechanical model quantita-
tively accounts for these results and provides a physical understanding
of yielding and ow in soft colloidal particles[49]. The suspensions are
modeled as three-dimensional packings of non-Brownian elastic
spheres which are subject to repulsive central forces and
elastohydrodynamic (EHD) drag forces analogous to those at the origin
of slip near solid surfaces [263,264]. This EHD model provides quantita-
tive predictions for the ow curves[49], the microscopic dynamics and
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macroscopic rheology in LAOS[135], as well as the internal stress that
persists upon ow cessation[221].
Inan attemptto proposea unied description, computer simulations
of concentrated suspensions of soft repulsive particles involving ther-
maluctuations and viscous dissipation were performed[87,88]. They
predicted a transition in ow curves from power-law to plateau stress
vs. rate at jamming, and an empirical model of the form Glass
Jammed s
was proposed to describe the entire rheology. Scaling
relations extracted from simulations were used to model the glassy
and jamming contributions to the stress.
As discussed above, the distinction between the entropic glass re-gime and jammedregime is not alwaysclear. Recent work with PNIPAm
microgel particles at volume fractions close to the jamming transition
suggests that the stress plateau occurs only above the jamming transi-
tion whereas a non-Newtonian behavior is observed below [91,92,
265]. These results conrm that the stress plateau is a robust signature
of jamming. However, here the system jumps directly from the liquid
state to the jammed state without crossing the entropic glass regime.
The ow curves measured above and below the yield point collapse
onto two branches exhibiting power-law scalings of shear stress and
shear rates with the distance to the jamming transition. The same
type of critical scaling collapse was reported using numerical simula-
tions and simple models of soft particles interacting through repulsive
interactions, but the exact values of the exponents are largely model-
dependent[266269]. These results are qualitatively consistent withthe EHD micromechanical model, albeit again with different exponents,
leaving the issue partially unsettled. Two parameters seem to be impor-
tant: the distance to the jamming point (EHD model results are for
suspensions well inside the jammed regime) and the exact form of the
viscous drag force between particles[270].
Despite the link between the existence of a stress plateau at low
shear rates and jamming that has been mentioned in the previous
paragraph, there are situations where the plateau is not indicative of
true yielding behavior or does not even exist. A deviation of the low-
rate stressfroma true plateaumay also reect theparticle's internal mi-
crostructure, even when aging is carefully accounted for. A powerful
manifestation of these effects for systems where a yield stress is dened
from dynamic strain sweep experiments was reported for star polymers
[86]. An interpretation of this remains elusive, but with slip being
excluded, effects like the cooperative local particle rearrangement,
which for stars and long hairy particles is manifested via interpenetra-
tion[5,81,82,205]can contribute to slow relaxations visible at low
shearrates. A phenomenological descriptionof such behavior is possible
using a modied HerschelBulkley model where Am
B n
[86].
In some cases a stressplateauexists but it is associated instead to in-
homogeneousow structure where the ow splits into two bands of
different viscosities, one that is essentially uid (Ny), whereas the
other remains solid (b
y), separated along the
ow gradient directionandcoexisting at thesame shear stress. Such a situation hasbeen mainly
reported in thixotropic suspensions [225,236,237]but it also exists in
some star polymer suspensions [178,249,271] where depending on vol-
ume fraction and softness (which can be tuned via functionality as
discussedabove) a plateauor weak power-law in stresscan be observed
(Fig. 5). Interested readers are referred to different reviews on the topic
[272276]. A numberof colloidal systems have been shown experimen-
tally to exhibit gradient banding in addition to starpolymers: surfactant
solutions[277], hard sphere suspensions[278], block copolymer mi-
celles[275,279281], and carbopol microgels[282]. Presently there is
no consensus on the origin of shear-banding in soft colloidal material
and many different explanations have been proposed: coupling be-
tweenow and concentration eld[278],mechanical instability and
mobileimmobile phase coexistence in the low-shear region [178,
283], slow relaxation dueto transient bonding [44,284], competitionbe-
tween aging and rejuvenation[271,285], competition between micro-
structural time and the characteristicow time scale[286]. Returning
toFig. 5,it is remarkable that one can tune the ow properties of soft
colloids (such as stars) at a molecular level, and in particular induce or
avoid shear banding. Note that for stars with short arms, mesoscopic
simulations did not produce any shear banding for functionalitiesf 50[212,213]. The fundamental question, which is still under debate
and needs further investigations, is whether there are generic features
of banding in colloids and what is the exact role of softness. For the
star case in particular, the exact role of molecular parameters (number
and size of arms) remains elusive[212,213,271,284]. In general, the in-
terplay of shear banding, yielding, slip and aging in dense soft colloids is
subtle. Clearly, there are ample opportunities for further exploring the
nonlinear shear rheology of colloidal materials including mixtures andunderstanding the possible connection of banding to shear-induced
transitions in soft matter.
A nal aspect of signicance in this topic is the role of particle
surface interactions in the macroscopic rheological response of jammed
materials. In a recent investigation of the ow and velocity proles of
microgel suspensions with purely repulsive interactions it was found
that surface roughness and surface chemistry have a strong effect on
ow properties[287]. Below theyield stress, wall slip is of hydrodynam-
ic origin when the surfaces are smooth and the particlesurface interac-
tions repulsive. Slip can be suppressed or reduced by attractive surfaces
[288]. When attractiveinteractionsare weak, slip canoccur as a resultof
EHD lubrication. Moreover, yielding depends on the surfaces again: it is
uniform near rough or smooth repulsive surfaces and, hence, the bulk
ow properties are recovered. It is non-uniform near smooth attractivesurfaces, suggesting that different dynamical states coexist, forming a
macroscopic boundary layer (and not an immobile band). This marks
a clear difference from bulk properties and from the behavior of non-
thixotropic materials discussed above. This important development,
which was recently extended to account for slip heterogeneities in
microchannel ows with different particlesurface interactions[289],
demonstrates another possibility to tailor the ow properties of soft
colloids[290].
5.3. Mesoscopic modeling of soft colloidal rheology
Besides MCT and micromechanical models discussed above, the rich
behavior of soft colloids at high volume fractions has stimulated a
wealth of phenomenological modeling activities. While these models
Fig. 5.Master ow curves representing normalized shear stress vs. normalized shear rate
relationship for microgels (, ) and star solutions (,;) at high effective volume
fractions eff, withf: star functionality and Ma: arm molar mass. The dotted line is at of
the data for microgels to the HerschelBulkley model[48].The dashed line is a t of the
data forstarpolymers withf= 390 andMa = 24 kg/mol tothemodied HerschelBulkley
model (see text)[86].The data for star polymers withf= 122 and Ma= 72 kg/mol col-
lapse onto the data for the other star polymers and the microgels at high shear rates;
the plateau observedat low-shear ratesis indicative of shear-banding(continuousstraightline)[178].
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involve many parameters not directly accessible in experiments, they
enjoyreasonable success in reproducingseveral features of the rheology
of dense suspensions. For example, the Soft Rheology Model (SGR) uses
trap dynamics for local mesoscopic non-ergodic events, with the effec-
tive noise temperature parameterx characterizing the jump rate out
of local traps[291,292]. Appropriate variation ofx allows to reproduce
macroscopic yield and ow dynamics observed in steady, oscillatory
and transient ows as well as viscosity bifurcation and thixotropy
[285]. Historically the SGR model has opened the route for many phe-nomenological models, which are impossible to review in depth here.
In the following we mention briey a few of them and we refer the in-
terested reader to more specialized articles [293,294]. The starting
point of all these models is a coarse-grained representation of the mate-
rial divided into mesoscopic cells, where the stress evolves according to
a prescribed law. In the uidity model, the local state of the system is
expressed by a Maxwell equation which involves the local relaxation
frequencyof thestress, calledthe uidity [172,217,295]. The timeevo-
lution of the uidity results from the competition between aging, which
decreases theuidity as time passes, andow-induced rearrangements,
which increases uidity. Further developments of the uidity model
take into account the spatial variation of the uidity, which can be
used to reproduce the occurrence of various spatial heterogeneities
like shear banding[249,271].An approach bearing similarities with
theuidity model is the non-local cooperative model which accounts
for the cooperative nature of rearrangements by relating the spatial
stress response in each cell to the global stress uctuations occurring
over the entire system[294,296298]. A characteristic cooperative
length was proposed to exist and be intrinsic material property, some-
thing that is not totally supported by existing experiments. This coopera-
tive model has been used to explain deviations observed in concentrated
emulsions from bulk rheology in conned geometries[299301].
6. Conclusions and perspectives
This partial presentation of the views of the community on the
rheology of soft colloids demonstrates the unprecedented richness
imparted by softness on the properties of colloidal suspensions. It is ev-
ident that the eld of soft colloids is picking-up. We believe that themajor advantage of these systems is their tunability which allows tailor-
ing theirow properties at molecular level. This indicates that under-
standing the ne details of internal microstructure of the particles is
often of crucial importance[4,5,128,302]. The two most important dis-
tinct features are their change of shape/size (e.g., microgels) and their
interdigitation (e.g., stars) in the dense state. It is evident from the
above discussion that several outstanding challenges remain. We list
them below.
The existence and the nature of the glass and jamming transitions in
soft dispersions pose several important questions. Does a glass to jam-
ming transition exist for all types of soft colloids (say microgels and
stars)? Is there universality in scaling the viscoelastic properties with
the actual volume fraction? In relation to this, aging and the related
role of internal stresses need further elaboration for the different softcolloidal archetypes.
Osmotic effects due to additives are shown to yield new dynamically
arrested states. New types of mixtures and other combinations of size
ratio and volume fractions must be investigated to deeply understand
the state diagrams. The rheology of these states and the possible ow-
induced transitions remain largely unexplored.
In the eld of nonlinear rheology, linking the scaling ofow curves
in the glassy and jammed regions represents a grand challenge. Along
the same lines, the high-shear and extensional rheology of soft colloids
in these states are entirely unexplored territories and their possible
combinations with rheo-physical observations of particle/structure de-
formation will advance the eld and also contribute to our understand-
ing of the relations among various types of shear-banding, yielding, and
slip with respect to particle architecture and thenature of the conning
surfaces. Moreover, the effects of softness and volume fraction on the
remarkable shear thickening phenomenon associated with colloidal
dispersions have received very little attention so far [303,304]. Recently,
discontinuous shear thickening was discussed in friction-dominated
large hard sphere suspensions (somehow equivalent to jammed parti-
cles) and the principal role of contact friction was identied[305,306].
To go beyond entropic spherical systems, variations of temperature
or pressure can change the quality of the background solvent and
hence introduce attractions. Likewise, the presence of functional end-groups can lead to clustering and hierarchical order with the possibility
of obtaining well-characterized patchy supramolecular structures[307]
whose rheology is unknown. Finally, departure from spherical shape
combined with softness is expected to change the packing of colloids
and hence their structure and rheology. These are promising new direc-
tions. The eld remains as exciting as ever!
Acknowledgments
This review was written when D.V. was visiting ESPCI ParisTech
with the support of the Michelin Chair. We thank Jan Mewis for helpful
comments on the manuscript ESPCI ParisTech is member ofPSL Re-
search University.
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