Tunable Rheology of Dense Soft Deformable Colloids

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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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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).

562 D. Vlassopoulos, M. Cloitre / Current Opinion in Colloid & Interface Science 19 (2014) 561574


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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



7/14

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].



8/14

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



9/14

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].



10/14

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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