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Nonlinear Rheology of Colloidal Dispersions

J.M. BraderDepartment of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland

Colloidal dispersions are commonly encountered in everyday life and represent an important classof complex fluid. Of particular significance for many commercial products and industrial processesis the ability to control and manipulate the macroscopic flow response of a dispersion by tuning themicroscopic interactions between the constituents. An important step towards attaining this goalis the development of robust theoretical methods for predicting from first-principles the rheologyand nonequilibrium microstructure of well defined model systems subject to external flow. In thisreview we give an overview of some promising theoretical approaches and the phenomena they seekto describe, focusing, for simplicity, on systems for which the colloidal particles interact via stronglyrepulsive, spherically symmetric interactions. In presenting the various theories, we will consider firstlow volume fraction systems, for which a number of exact results may be derived, before moving onto consider the intermediate and high volume fraction states which present both the most interestingphysics and the most demanding technical challenges. In the high volume fraction regime particularemphasis will be given to the rheology of dynamically arrested states.

Contents

1. Introduction and Overview 1

2. Continuum Mechanics Approaches 32.1. The Lodge equation 32.2. Upper convected Maxwell equation 42.3. Material objectivity 42.4. Beyond continuum mechanics 5

3. Microscopic Dynamics 53.1. Non-interacting particles 63.2. Dimensionless parameters 73.3. Neglecting solvent hydrodynamics 73.4. Nonequilibrium states 8

4. Quiescent States 84.1. Hard-spheres 84.2. Attractive spheres 9

5. Rheological Phenomenology 105.1. Zero-shear viscosity 105.2. Shear thinning 115.3. Shear thickening 125.4. Yield stress 13

6. Theoretical approaches to fluid states 156.1. Pair Smoluchowski equation 156.2. Low volume fraction 176.3. Intermediate volume fraction 19

6.3.1. Superposition approximation 206.3.2. Potential of mean force 216.3.3. Equilibrium integral equations 216.3.4. Nonequilibrium integral equations 226.3.5. Alternative approaches 24

6.4. Temporal locality vs. memory functions 25

7. Glass rheology 267.1. MCT inspired approaches 267.2. Integration through transients 27

7.3. Translational invariance 287.4. Microscopic constitutive equation 297.5. Distorted structure factor 307.6. Applications 317.7. Schematic model 327.8. Yield stress surface 34

8. Outlook 35

Acknowledgments 36

References 36

1. INTRODUCTION AND OVERVIEW

Complex fluids exhibit a rich variety of flow behaviourwhich depends sensitively upon the thermodynamic con-trol parameters, details of the microscopic interparticleinteractions and both the rate and specific geometry ofthe flow under consideration. The highly nonlinear re-sponse characteristic of complex fluids may be readilyobserved in a number of familiar household products [1].For example, mayonnaise consists of a stabilized emul-sion of oil droplets suspended in water and behaves asa soft solid when stored on the shelf but flows like aliquid, and is thus easy to spread, when subjected toshear flow with a knife [2]. This nonlinear viscoelasticflow behaviour, known as shear-thinning, may be manip-ulated on the microscopic level by careful control of theoil droplet size distribution. In contrast, a dispersion ofcorn-starch particles in water, at sufficiently high concen-trations, exhibits a dramatic increase in shear viscositywith increasing shear rate; a phenomenon called shear-thickening [3, 4]. Even the familiar practical problem ofextracting tomato ketchup from a glass bottle presentsa highly nonlinear flow. In this case the applied shearstress, generally implemented by shaking, must exceed acritical value, the yield-stress, before the ketchup beginsto flow as desired.

2

Colloidal dispersions are a class of complex fluid whichdisplay all of the above mentioned nonlinear flow re-sponses [5]. In addition to being of exceptional rele-vance for many technological processes, the considerableresearch interest in colloidal dispersions owes much tothe existence of well characterized experimental systemsfor which the interparticle interactions may be tuned torelatively high precision (often possible by simply vary-ing the solvent conditions) [6]. The ability to control themicroscopic details of the colloidal interaction facilitatescomparison of experimental results with theoretical cal-culations and computer simulations based on idealisedmodels (see e.g. [7–9]). In particular, the size of col-loidal particles makes possible light scattering, neutronscattering and microscopy experiments which provide in-formation inaccessible to experiments on atomic systemsand which have enabled various aspects of liquid statetheory to be tested in detail [10].

The typical size of a colloidal particle lies in the range10nm to 1µm and thus enables a fairly clear separationof length- and time-scales to be made between the col-loids and the molecules of the solvent in which they aredispersed. As a result, a reasonable first approximationis to represent the solvent as a continuum fluid, gener-ally taken to be Newtonian and thus characterised bya constant solvent viscosity (see Fig.1). For suspendedparticles with a length-scale greater than approximately1µm the continuum approximation of the solvent is com-pletely appropriate. However, this becomes questionableas the average size of the particles is reduced below afew nanometers, at which point the discrete nature ofthe solvent can no longer be ignored. Colloidal particlesoccupy an intermediate range of length-scales for whicha continuum approximation for the solvent must be sup-plemented by the addition of first order Gaussian fluc-tuations (Brownian motion) about the average hydrody-namic fields describing the viscous flow of the continuumsolvent.

The Brownian motion resulting from solvent fluctua-tions not only plays an important role in determining themicroscopic dynamics; it is essential for the existence ofa unique equilibrium microstructure. With the impor-tant exception of arrested glasses and gels, the presenceof a stochastic element to the particle motion allows afull exploration of the available phase space and thusenables application of Boltzmann-Gibbs statistical me-chanics to quiescent (and ergodic) colloidal dispersions.While the specific nature of the balance between Brown-ian motion, hydrodynamic and potential interactions de-pends upon both the observable under consideration andthe range of system parameters under investigation, itis the simultaneous occurrence of these competing phys-ical mechanisms which gives rise to the rich and variedrheological behaviour of dispersions. Unfortunately, thecomplicated microscopic dynamics presented by disper-sions also serves to complicate the theoretical descriptionof these systems [15].

The present review has been written with a number of

(a) (b) (c)

FIG. 1: A schematic illustration of coarse graining as appliedto colloidal dispersions. Continuum mechanics approachestreat the dispersion as a single continuum fluid (panel (a)),whereas a fully detailed picture is obtained by treating bothcolloids and solvent explicitly (panel (c)). The theoreticalmethods considered in this work operate at an intermediatelevel (panel (b)) in which the colloids are explicitly resolvedbut the solvent may be treated as a continuum.

aims in mind. On one hand, we would like to presenta relatively concise overview of the main phenomeno-logical features of the rheology of dispersions of spher-ical colloidal particles. In order to reduce the parameterspace of the discussion, emphasis will be placed on thesimple hard sphere model for which the space of con-trol parameters is restricted to two dimensions (volumefraction and flow rate). While both attractive colloidsand the response to non-shear flows will be addressed,no attempt has been made to be comprehensive in thisrespect. Another primary aim of the present work is toprovide an overview, within the context of the aforemen-tioned phenomenology, of microscopically motivated ap-proaches to the rheology and flow induced microstructureof colloidal dispersions. Although we will discuss someless well founded ‘schematic model’ approaches, the focushere is upon ‘first-principles’ theories which prescribe aroute to go from a well defined microscopic dynamics toclosed expressions for macroscopically measurable quan-tities.

The formulation of a robust theory of dispersion rhe-ology from microscopic starting points constitutes aformidable problem in nonequilibrium statistical me-chanics. Although considerable progress has been madein this direction, a comprehensive constitutive theoryanalogous to that of Doi and Edwards for entangled linearpolymers [22–25] remains to be found. At present thereexist a number of alternative microscopic theoretical ap-proaches to dispersion rheology which, despite showingadmirable success within limited ranges of the systemparameters, have so far been unable to provide a uni-fied global picture of the microscopic mechanisms un-derlying the rheology of colloidal dispersions. Despitecommon starting points (the many-body Smoluchowskiequation) the disparate nature of the subsequent approx-imations, each tailored to capture a particular physicalaspect of the cooperative particle motion, make it diffi-cult to establish clear relations between different theoret-ical approaches. A goal of this work is thus to clarify therange of validity of the various theoretical approaches andto identify common ground. We note that the presentwork is well complemented by a number of recent reviews

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addressing dispersion rheology from both experimental[26, 27] and theoretical perspectives [20, 28, 29].The paper is organized as follows: In section 2 we will

discuss briefly some traditional continuum mechanics ap-proaches to rheology, both to give a feeling for the spiritof such work and to put into context some of the mi-croscopic results presented later (in section 7). In sec-tion 3 we will introduce and discuss in some detail theSmoluchowski equation defining the microscopic dynam-ics under consideration. In section 4 we will considerthe equilibrium and non-equilibrium phase behaviour ofhard-sphere colloids in the absence of flow, which is anecessary pre-requisite to the subsequent discussions. Insection 5 we will give a brief overview of the relevantbasic phenomenology of dispersion rheology, includingthe shear-thinning and shear-thickening of colloidal fluidsand the yielding of colloidal glasses. In section 6 we willconsider the various theoretical approaches to treatingcolloidal fluids under external flow. In particular, exactresults for the microstructure and rheology of low vol-ume fraction systems and their (approximate) extensionto finite fluid volume fractions are discussed in subsec-tions 6.2 and 6.3, respectively. In section 7, we considerthe recently developed mode-coupling based approachesto the rheology of dense colloidal suspensions which en-able glass rheology to be addressed. Finally, in section 8we will provide an outlook for future work and identifypossible new avenues for theoretical investigation.

2. CONTINUUM MECHANICS APPROACHES

Rheology is primarily an experimental discipline. In-deed, one of the simplest experiments imaginable is toexert a force on a material in order to see how it deformsin response. More precisely, in a controlled rheologicalexperiment one measures either the stress arising from agiven strain or, more commonly, the strain accumulatedfollowing application of an applied stress. In practice,both stress controlled and strain controlled experimentsare performed and provide complementary informationregarding the response of a material sample. For thepurpose of this review we will focus upon situations inwhich a homogeneous strain field is prescribed from theoutset. The description of experiments for which macro-scopic stress is employed as a control parameter posesan enormous challenge for microscopically based theo-ries and demands careful consideration of the nontrivialmechanisms by which the applied stress propagates intothe sample from the boundaries.Given the apparent complexity of any microscopic the-

ory, it is quite natural to begin first at a more coarse-grained level of description in an effort to establish thegeneral phenomenology and mathematical structure ofthe governing equations at the continuum level. Histor-ically, this methodology was pioneered by Maxwell inhis 1863 work on viscoelasticity and continued to de-velop into the following century through the efforts of

distinguished rheologists such as Rivlin and Oldroyd [30].While much of this early work aimed to acheive a morefundamental mathematical understanding of viscoelasticresponse, strong additional motivation was provided byexperiments on polymeric systems which exposed a largevariety of interesting nonlinear rheological phenomena inneed of theoretical explanation. Theoretical approachesto continuum rheology thus seek to obtain a constitutiveequation relating the stress, a tensorial quantity describ-ing the forces acting on the system [31], to the deforma-tion history encoded in the strain tensor.The typical ‘rational mechanics’ approach to this prob-

lem is to assume a sufficiently general integral or differen-tial constitutive relation between stress and strain and tothen constrain this as much as possible via the impositionof certain exact or approximate macroscopic symmetry,conservation and invariance principles [23, 30, 32]. Theclear drawback to this methodology is that the entireparticulate system is viewed as a single continuum field,thus losing any contact to the underlying colloidal inter-actions and microstructure ultimately responsible for themacroscopic response (see Fig.1). As a result, such con-stitutive theories are neither material-specific nor gen-uinely predictive in character. Despite these shortcom-ings, the continuum mechanics approach to rheology hasattained a great level of refinement and can be applied tofit experimental data from a wide range of physical sys-tems [23, 30]. Moreover, the experience gained throughcontinuum mechanics modelling may well prove usefulin guiding the construction of more sophisticated micro-scopic theories by providing constraints on the admiss-able mathematical form of the constitutive equations.

2.1. The Lodge equation

It is perhaps instructive to give an illustration of thespirit in which phenomenological constitutive relationsmay be constructed using continuummechanics concepts.The example we choose is not only of intrinsic interest,but will also prove relevant to the discussion of a recentmicroscopically based theory of glass rheology [17–19] tobe discussed in section 7. We consider a viscoelastic fluidsubject to shear deformation with flow in the x-directionand shear gradient in the y-direction (a convention wewill continue to employ throughout the present work).Suppose that we wish to determine the infinitessimalshear stress dσxy at time t arising from a small strainincrement dγ at an earlier time t′. As the material isviscoelastic, it is reasonable to assume that the influenceof the strain increment dγ(t′) = γ(t′)dt′ on the stress attime t must be weighted by a decaying function of theintervening time t−t′, in order to represent the influenceof dissipative processes. Adopting a simple exponentialform for the relaxation function it is thus intuitive towrite

dσxy(t) = G∞ exp

[

− t− t′

τ

]

γ(t′) dt′, (1)

4

where τ is a relaxation time and G∞ is an elastic constant(the infinite frequency shear modulus). Assuming linear-ity, the total stress at time t may thus be constructed bysumming up all of the infinitessimal contributions overthe entire flow history, which we take to extend into theinfinite past. We thus arrive at

σxy(t) =

∫ t

−∞

dt′ G∞ exp

[

− t− t′

τ

]

γ(t′). (2)

Partial integration leads finally to

σxy(t) =1

τ

∫ t

−∞

dt′ G(t− t′)γ(t, t′), (3)

where G(t) = G∞ exp[−t/τ ] is the shear modulus and

γ(t, t′) is the accumulated strain γ(t, t′) =∫ t

t′ds γ(s).

The simple integral relation (3) between shear stress andshear strain was first considered by Boltzmann. Indeed,the assumption that the stress increments (1) may besummed linearly to obtain the total stress is often re-ferred to as the ‘Boltzmann superposition principle’.In order to extend (3) to a tensorial relation, i.e. a true

constitutive equation, an appropriate tensorial general-ization of the accumulated strain γ(t, t′) must be iden-tified. For the spatially homogeneous deformations un-der consideration the translationally invariant deforma-tion gradient tensor E(t, t′) transforms a vector (‘mate-rial line’) at time t′ to a new vector at later time t viar(t) = E(t, t′) · r(t′), where Eαβ = ∂rα/∂rβ . An alterna-tive nonlinear choice of strain measure is the symmetricFinger tensor B(t, t′) = E(t, t′)ET (t, t′). The Fingertensor contains information about the stretching of ma-terial lines during a deformation but is invariant withrespect to solid body rotations of the material sample.For simple shear the Finger tensor is given explicitly by

B =

1 + γ2 γ 0γ 1 00 0 1

(4)

where γ ≡ γ(t, t′). The accumulated strain in the inte-grand of Eq.(3) can thus be identified as the xy element ofB(t, t′). This suggests that the Boltmann integral form(3) may be extended using the simple ansatz

σ(t) =

∫ t

−∞

dt′ B(t, t′)G∞e−(t−t′)/τ

τ, (5)

for the full stress tensor (see subsection 2.3 below formore justification of this nontrivial step). Equation (5)is known as the Lodge equation in the rheological liter-ature and is applicable in both the linear and nonlinearviscoelastic regime [23].

2.2. Upper convected Maxwell equation

The assumption of an exponentially decaying shearmodulus is generally attributed to Maxwell, who real-ized that this choice enabled an interpolation between

a purely elastic response to deformations rapid on thetime scale set by τ and a viscous, dissipative responsein the limit of slowly varying strain fields. In fact, theLodge equation derived above is simply the integral formof a nonlinear (differential) Maxwell equation. In orderto show this we first differentiate (5) to obtain

Dσ

Dt+

1

τσ =

G∞

τ1, (6)

where we have introduced the upper-convected derivative[30]

Dσ

Dt= σ(t)− κ(t)σ(t) − σ(t)κT (t), (7)

and where the velocity gradient tensor κ(t) is defined interms of the deformation gradient tensor via

∂

∂tE(t, t′) = κ(t)E(t, t′). (8)

For an incompressible material the stress is only deter-mined up to a constant isotropic term. Eq.(6) may thusbe expressed in an alternative form by first defining anew stress tensor

Σ = σ −G∞1, (9)

and substituting for σ in Eq.(6). This yields

DΣ

Dt+

1

τΣ = G∞(κ(t) + κ

T (t)). (10)

This differential form of the Lodge equation is known asthe upper-convected Maxwell equation [23] and is a non-linear generalization of Maxwell’s original scalar model tothe full deviatoric stress tensor. Historically, the upper-convected Maxwell equation was first proposed by Ol-droyd [30] directly on the basis of Maxwell’s differentialform.

2.3. Material objectivity

The assumption that one can go from (3) to (5) onthe basis of a single off-diagonal element appears at firstglance to be rather ad hoc. On one hand, this choice canbe justified retrospectively, using the fact that the Lodgeequation (5) is derivable from a number of simple molec-ular models, e.g. the dumbell model for dilute polymersolutions [23]. However, from a continuum mechanicsperspective (5) is the simplest generalization of (3) whichsatisfies the ‘principle of material objectivity’. This prin-ciple expresses the requirement that the constitutive re-lationship between stress and strain tensors should beinvariant with respect to rotation of either the materialbody or the observer, thus preventing an unphysical de-pendence of the stress on the state of rotation. That thissymmetry is an approximation becomes clear when con-sidering the material from a microscopic viewpoint: In

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a noninertial rotating frame the apparent forces clearlylead to particle trajectories which depend upon the an-gular velocity. For many systems the neglect of theseeffects on the macroscopic response of the system is anextremely good approximation. For the overdamped col-loidal dynamics considered in this work inertia plays norole and the principle of material objectivity is exact [33].Mathematically, it is straightforward to check whether

or not a proposed tensorial constitutive equation is ma-terial objective. When subject to a time-dependent ro-tation R(t) the deformation gradient tensor transformsas

E(t, t′) = R(t)E(t, t′)RT (t′), (11)

where E is the deformation gradient in the rotatingframe. The dependence of E upon the state of rota-tion arises because E contains information about boththe stretching and rotation of material lines. Insertion ofthe transformed tensor (11) into the constitutive equa-tion for the stress thus corresponds to a rotation of thematerial sample. Material objectivity is verified if theresulting stress tensor is given by

σ(t) = R(t)σ(t)RT (t). (12)

As noted, the Finger tensor B contains only informationabout the stretching of material lines and transforms un-der rotation according to

B(t, t′) = R(t)B(t, t′)RT (t). (13)

The material objectivity of the Lodge equation (5), andthus the upper convected Maxwell equation (10), followstrivially from the fact that σ is a linear functional of B.Many phenomenological rheological models thus start byassuming a general functional dependence σ(t) = F [B]in order to guarantee a rotationally invariant theory.The vast majority of microscopically motivated theo-

ries of dispersion rheology treat only a single scalar ele-ment of the stress tensor (generally the shear stress σxy).Indeed, the rarity of microscopic tensorial constitutivetheories may well be the primary reason for the apparentgap between continuum and statistical mechanical the-ories aiming to describe common phenomena. We willrevisit the concept of material objectivity in section 7when considering a recently proposed tensorial constitu-tive equation for dense dispersions.

2.4. Beyond continuum mechanics

In the last decade, significant progress has been madein understanding the response of colloidal dispersions toexernal flow on a level which goes beyond the fully coarse-grained phenomenological approaches of traditional con-tinuum rheology. Important steps towards a more re-fined picture have been provided by studies based onmesoscopic models [26, 35–37]. However, while such phe-nomenological approaches can reveal generic features of

the rheological response, they are not material specificand can therefore address neither the influence of themicroscopic interactions on the macroscopic rheology northe underlying microstructure, as encoded in the particlecorrelation functions. This deeper level of insight is pro-vided by fully microscopic approaches which start from awell defined particle dynamics and, via a sequence of ei-ther exact or clearly specified approximate steps, lead toclosed expressions for macroscopically measureable quan-tities. The symmetry, invariance and conservation princi-ples used as input in the construction of continuum theo-ries, such as the material objectivity discussed in section2.3, should then emerge directly as a consequence of themicroscopic interactions. Such an undertaking clearly re-quires the machinery of statistical mechanics.Theories founded in statistical mechanics provide in-

formation regarding the correlated motion of the con-stituent particles and are thus capable, at least in princi-ple, of capturing non-trivial and potentially unexpectedcooperative behaviour as exhibited by equilibrium andnonequilibrium phase transitions. This ability to cap-ture emergent phenomena is in clear contrast to contin-uum approaches where such physical mechanisms mustbe input by hand. An additional advantage of a sta-tistical mechanics based approach to rheology over thedirect application of continuum mechanics is that impor-tant additional information is provided regarding the mi-crostructure of the system, as encoded in the correlationfunctions. It thus becomes possible to connect the con-stitutive relations to the underlying correlations betweenthe colloidal particles and obtain microscopic insight intothe macroscopic rheological response. Additional moti-vation to theoretically ‘look inside’ the flowing system isprovided by developments in the direct visualization andtracking of particle motion in experiments on colloidaldispersions (confocal microscopy) [38–40], together withadvances in the computer simulation of model systemsunder flow [41–43].Although beyond the scope of the present work, we

note that the influence of steady shear flow on glassystates has been addressed, albeit in an abstract setting,by generalized mean-field theories of spin glasses [44, 45].Spin glass approaches have proved useful in describingthe dynamical behaviour of quiescent systems [46]. Inorder to mimic the effect of shear flow a nonconservativeforce is introduced to bias the dynamics and break thecondition of detailed balance characterizing the equilib-rium state [47]. While the abstract nature of these treat-ments certainly lends them a powerful generality, the lackof material specificity makes difficult a direct connectionto experiment.

3. MICROSCOPIC DYNAMICS

Before addressing the phenomenology (section 5) andapproximate theories (sections 6 and 7) of colloid rheol-ogy it is rewarding to first consider in detail the micro-

6

scopic equation of motion determining the overdampedcolloidal dynamics. By a careful assessment of the funda-mental equation of motion a number of general observa-tions and comments can be made regarding the characterof nonequilibrium states, solutions in special limits, im-portant dimensionless parameters and influence of hydro-dynamics, which are independent of the specific systemor approximation scheme under consideration.We consider a system consisting of N Brownian col-

loidal particles interacting via spherically symmetricpairwise additive interactions and homogeneously dis-persed in an incompressible Newtonian fluid of givenviscosity. The probability distribution of the N -particleconfiguration is denoted by Ψ(t) and satisfies the Smolu-chowski equation [15]

∂Ψ(t)

∂t+∑

i

∂i · ji = 0 (14)

where the probability flux of particle i is given by

ji = vi(t)Ψ(t)−∑

j

Dij · (∂j − β Fj)Ψ(t), (15)

where β = 1/kBT is the inverse temperature. The hy-drodynamic velocity of particle i due to the applied flowis denoted by vi(t) and the diffusion tensor Dij de-scribes (via the mobility tensor Γij = βDij) the hydro-dynamic mobility of particle i resulting from a force onparticle j. The hydrodynamic velocity can be decom-posed into affine and particle induced fluctuation termsvi(t) = κ(t) · ri + vfl

i (t), where vfli (t) can be expressed in

terms of the third rank hydrodynamic resistance tensor[48]. The force Fj on particle j is generated from the to-tal potential energy according to Fj = −∂jUN , where, inthe absence of external fields, UN depends solely on therelative particle positions. The three terms contributingto the flux thus represent the competing effects of (fromleft to right in (15)) external flow, diffusion and interpar-ticle interactions.While the Smoluchowski equation (14) is widely ac-

cepted as an appropriate starting point for the treat-ment of colloidal dynamics, alternative approaches basedon the Fokker-Planck equation have also been investi-gated [49]. On the Fokker-Planck level of description thedistribution function retains a dependence on the parti-cle momenta. Although this makes possible the treat-ment of systems with a temperature gradient (leadingto thermophoretic effects), considerable complicationsarise when attempting to treat hydrodynamic interac-tions which make preferable the Smoluchowski equation.For the special case of monodisperse hard-spheres at

finite volume fractions under steady flow Eq.(14) can benumerically integrated over the entire fluid range usingcomputationally intensive Stokesian dynamics simulation[42, 50, 51]. This simulation technique includes the fullsolvent hydrodynamics and provides a useful benchmarkfor theoretical approaches (for an overview of the com-puter simulation of viscous dispersions we refer the reader

-1 0 12D0R

-2t

-1

0

1

2

3

4

MSD

/R2

δy2= δz

2δx

2

Pe=1

FIG. 2: The mean-squared-displacement of non-interactingcolloidal particles in flow (x), gradient (y) and vorticity (z)directions as a function of time. The MSD in flow direction ex-hibits enhanced diffusion (‘Taylor dispersion’) for values of theshear strain greater than unity. Also shown are contour plotsof the (non-normalised) probability distribution P1(r, t)N(t)(see Eq.16) in the z = 0 plane at times 2D0R

−2t = 0.15, 1 and5, demonstrating shear induced anisotropy for γ > 1 relatedto the onset of Taylor dispersion.

to [52] and references therein). While Stokesian dynam-ics simulations have focused primarily on simple shear,results have also been reported for extensional flow ge-ometries [53].

3.1. Non-interacting particles

For the special case of non-interacting particles (Fj =0) equations (14) and (15) describe the configurationalprobability distribution of an ideal gas under externallyapplied flow and may be solved analytically using themethod of characteristics [54]. For non-interacting par-ticles under steady shear the many-particle distributionfunction is given by a product of single particle functionsΨ(ri, t) = P1(r1, t)× · · · ×P1(rN , t), where P1 is givenby

P1(r, t) =1

N(t)exp

[

−x2− y2(1 + γ2

3 ) + γxy

N(t)− z2

4D0t

]

(16)

when the initial condition P1(r, 0) = δ(0) is employed.The Normalization is given by N(t) = (4πD0t)

3(1 +(γt)2/12) and the strain by γ = γt. Given a suitablylocalized initial density distribution Eq.(16) essentiallydescribes the dispersion of a colloidal droplet in a solvent(e.g. ink in water) under shear, as is apparent from Fig.2,which shows contour plots of the probability distributionat three different times for a given shear rate.Although non-interacting colloids represent a trivial

case, it is nevertheless instructive to consider the mean-squared-displacement (MSD), characterizing the diffu-sive particle motion, both parallel and orthogonal to the

7

flow direction in simple shear [55]. In both the vorticityand shear gradient directions, flow has no influence andthe equilibrium result is recovered, δz2 = δy2 = 2D0 t,with D0 the single particle diffusion coefficient. In theflow direction the MSD is enhanced by a coupling be-tween Brownian motion and affine advection, yieldingδx2 = 2D0 t(1 + γ2t2/3), where γ is the shear rate. Thephysical origin of this enhanced diffusion, termed ‘Taylordispersion’ [56], is that the random motion of a given col-loid leads to its displacement into planes of laminar flowwith a velocity different to that of the original point. Thisconstant and random ‘changing of lanes’ leads, on the av-erage, to a dramatically increased rate of diffusion in thedirection of flow. The accelerated rate of mixing achievedby stirring a dilute dispersion is thus almost entirely at-tributable to local Taylor dispersion. We note also thatanalogous effects arising from flow-diffusion coupling canalso be identified in other flow geometries, such as thepractically relevant case of Poiseuille flow along a cylin-drical tube [55].

3.2. Dimensionless parameters

The Smoluchowski equation describes the dynamicsof spherical colloidal particles dispersed in an incom-pressible Newtonian fluid and provides the fundamentalstarting point for all theoretical work to be describedin the following sections. An appropriate dimensionlessReynolds number governing the solvent flow may be de-fined as Re = ργR2/η, with γ a characteristic flow rate,ρ is the density, η the solvent viscosity and R the col-loidal length scale. Due to the small size of the colloidalparticles Re remains small for all situations of physicalrelevance and the Stokes equations, rather than the morecomplicated Navier-Stokes equations, may thus be em-ployed in treating the solvent flow.Given that Re remains small, two dimensionless pa-

rameters are of particular importance in determining theequilibrium and nonequilibrium behaviour. The first ofthese is the colloidal volume fraction φ=4πnR3/3, withnumber density n and particle radius R. The maximumvolume fraction achievable for monodisperse spheres is0.74 corresponding to an optimally packed face-centered-cubic crystal structure. For the purposes of the presentwork we will find it convenient to divide the physicalrange of volume fractions into three subregions: (i) lowpacking, φ < 0.1, (ii) intermediate packing, 0.1 < φ <0.494, and (iii) high packing, 0.494< φ. While this di-vision is somewhat arbitrary, it will later prove useful indiscussing the various theoretical approximation schemescurrently available.The second important dimensionless parameter is the

Peclet number Pe = γR2/2D0 [15]. The Peclet numberis a measure of the importance of advection relative toBrownian motion and determines the extent to whichthe microstructure is distorted away from equilibrium bythe flow field. In the limit Pe → 0 Brownian motion

dominates and the thermodynamic equilibrium state isrecovered. Conversely, in the strong flow limit, Pe →∞, solvent mediated hydrodynamic interactions may beexpected to dominate the particle dynamics, although, inpractice, surface roughness and other perturbing effectsturn out to complicate this limit [57] (see section 6.2 formore details on this point).Finally, we would like to note that there exists a fur-

ther, nontrivial dimensionless quantity implicit in themany-body Smoluchowski equation (14). An increase ineither the dispersion volume fraction or attractive cou-pling between particles is accompanied by an increasein the structural relaxation timescale of the system ταcharacterizing the temporal decay of certain two-pointautocorrelation functions. This enables the Weissenbergnumber to be defined as Wi = γτα. For intermedi-ate and high volume fractions, particularly those closeto the colloidal glass transition, it is the Weissenbergnumber, rather than the ‘bare’ Peclet number Pe, whichdominates certain aspects of the nonlinear rheological re-sponse, as has been emphasized in [58]. For the low vol-ume fraction systems to be considered in section 6.2 thestructural relaxation timescale is set by R2/2D0, leadingto Pe = Wi.

3.3. Neglecting solvent hydrodynamics

In many approximate theories aiming to describe in-termediate and high volume-fraction dispersions the in-fluence of solvent hydrodynamics beyond trivial advec-tion is neglected from the outset. For certain situations(e.g. glasses) this approximation is partially motivatedby physical intuition, however, in most cases, the omis-sion of solvent hydrodynamics is an undesirable but un-avoidable compromise made in order to achieve tractableclosed expressions. Accordingly, the expression for theprobability flux (15) is approximated in two places, whichwe will now discuss in turn.The first approximation is to set Dij = D0δij , thus

neglecting the influence of the configuration of the Ncolloidal particles on the mobility of a given particle. Forlow and intermediate volume fraction fluids this may bereasonable for Pe ≪ 1 but can be expected to break downfor Pe > 1 as hydrodynamics becomes increasingly im-portant in determining the particle trajectories. In par-ticular, the near field lubrication forces [48] which reducethe mobility when the surfaces of two particles approachcontact play an important role in strong flow and are re-sponsible for driving cluster formation and shear thicken-ing [59] (see section 5.3). For dense colloidal suspensionsclose to a glass transition the role of hydrodynamics isless clear. For certain situations of interest (e.g. glassesclose to yield) the relevant value of Pe is very small andsuggests that hydrodynamic couplings should not be ofprimary importance.The second common approximation to (15) arising

from the neglect of solvent hydrodynamics is the as-

8

sumption of a translationally invariant linear flow pro-file v(r, t) = κ(t) ·r, where κ(t) is the (traceless) time-dependent velocity gradient tensor introduced in Eq.(8).In an exact calculation the solvent flow field follows fromsolution of Stokes equations with the surfaces of the Ncolloidal particles in a given configuration providing theboundary conditions (essentially what is done in Stoke-sian dynamics simulation [42, 50, 51]). By replacing thissolvent velocity field with the affine flow, we neglect theneed for the solvent to flow around the particles and arethus able to fully specify the solvent flow profile fromthe outset, without requiring that this be determined aspart of a self consistent calculation [60]. If necessary, theassumption of purely affine flow could be corrected tofirst order. For example, under simple shear flow the sol-vent flow profile around a single spherical particle is wellknown [31] and could form the basis of a superposition-type approximation to the full fluctuating velocity field.It is important to note that the assumption of a trans-

lationally invariant velocity gradient κ(t) is potentiallyrather severe as it excludes from the outset the possibilityof inhomogeneous flow, as observed in shear banded andshear localized states. While physically reasonable forlow and intermediate density colloidal fluids, the assump-tion of homogeneity could become questionable whenconsidering the flow response of dynamically arrestedstates, for which brittle fracture may preclude plasticflow [61]. Moreover, it is implicit in the approximationv(r, t)=κ(t)·r that the imposed flow profile acts instan-taneously throughout the system. In experiments wherestrain or stress are applied at the sample boundaries afinite time is required for transverse momentum diffusionto establish the velocity field. Nevertheless, experimentsand simulations of the transition from equilibrium to ho-mogeneous steady state flow have shown that a linearvelocity profile is established long before the steady stateregime is approached, thus suggesting that the assump-tion of an instantaneous translationally invariant flow isacceptable for certain colloidal systems [65].

3.4. Nonequilibrium states

In equilibrium, the principle of detailed balance as-serts that the microscopic probability flux vanishes, ji =Ψ∂i(lnΨ + βUN ) = 0, where UN is the total interparti-cle potential energy. This balance between conservativeand Brownian forces thus yields the familiar Boltmann-Gibbs distribution Ψe = exp(−βUN )/ZN , where ZN isthe configurational part of the canonical partition func-tion. In the presence of flow (κ(t) 6= 0) there exists afinite probability current which breaks the time rever-sal symmetry of the equilibrium state and detailed bal-ance no longer applies. A nonvanishing probability cur-rent thus serves to distinguish between equilibrium andnonequilibrium solutions of (14) and rules out the possi-bility of a Boltzmann-Gibbs form for the nonequilibriumdistribution. While such a Boltzmann-Gibbs distribution

is clearly inadequate for nonpotential flows (e.g. simpleshear), for potential flows (e.g. planar elongation) it isperhaps tempting to assume such a distribution by em-ploying an effective ‘flow potential’ Uf (see e.g. [66]).The fundamental error of assuming an ‘effective equilib-rium’ description of nonequilibrium states is made veryclear by the non-normalizability of the assumed distribu-tion Ψ ∼ exp(−β(UN +Uf)). These considerations serveto emphasize the fact that the only true way to deter-mine the distribution function for systems under flow isto solve the Smoluchowski equation (14).For much of the present work we will focus on the re-

sponse of colloidal dispersions to steady flows. Whileexperiment and simulation clearly demonstrate that welldefined steady states may be achieved following a periodof transient relaxation, it is interesting to note that thereexists no mathematical proof of a Boltzmann H-theoremfor Eq.(14) which would guarantee a unique long-timesolution for the distribution function. The absence of aH-theorem for colloidal dispersions under steady flow isa consequence of the hard repulsive core of the particleswhich invalidates the standard methods of proof gener-ally applied to Fokker-Planck-type equations [20, 47].A further nontrivial aspect of Eq.(14) emerges when

considering the translational invariance properties of thetime dependent distribution function Ψ(t) ≡ Ψ(t, ri),achieved by shifting all particle coordinates by a con-stant vector r′i = ri + a (see subsection 7.3 for moredetails). For an arbitrary incompressible flow it has beenproven that a translationally invariant initial distribu-tion function leads to a translationally invariant, butanisotropic distribution function Ψ(t), despite the factthat the Smoluchowski operator [15] generating the dy-namics is itself not translationally invariant [18] . Al-though the proof outlined in [18] omitted hydrodynamicinteractions, it may be expected that the same resultholds in the presence of hydrodynamics due to the de-pendence of the diffusion tensors on relative particle co-ordinates.

4. QUIESCENT STATES

4.1. Hard-spheres

Theoretical and simulation studies based on Eq.(14)have focused largely on the hard-sphere model. In ad-dition to being mathematically convenient, the focus onthis simple model is motivated largely by the availabilityof well characterized hard-sphere-like experimental col-loidal systems [7]. In the absence of flow, a system ofmonodisperse hard-sphere colloids remain in a disorderedfluid phase up to a volume fraction of φ = 0.494, beyondwhich they undergo a first-order phase transition to asolid phase of φ = 0.545 with face-centered-cubic order(see Figure 3). This unexpected, entropically driven, or-dering transition was first observed using molecular dy-namics computer simulation in the late 1950’s [67] and

9

Fluid CrystalFluid

Crystal

0 0.494

~0.58 ~0.64

Glass

0.545 Volume Fraction

+

Phase Diagram

FIG. 3: A schematic illustration of the phase diagram ofhard-spheres as a function of volume fraction. Monodispersesystems undergo a freezing transition to an FCC crystal withcoexisting densities φ = 0.494 and 0.545. Polydispersity su-presses the freezing transition resulting in a glass transitionat φ ∼ 0.58, which lies below the random close packing valueof φ ∼ 0.64.

remains a current topic of both experimental and theo-retical research (for a recent review see [68]).

Making the system slightly polydisperse frustratescrystalline ordering and suppresses the freezing transi-tion. In sufficiently polydisperse systems [69] a disor-dered fluid remains the equilibrium state up to a volumefraction φ ≈ 0.58, at which point the dynamics becomesarrested and a colloidal glass state is formed. This dy-namical transition to a non-ergodic solid is character-ized by a non-decaying intermediate scattering functionat long times for which dynamic light scattering results[7, 8] are well described by the mode coupling theory(MCT) [11]. The standard quiescent MCT consists ofa nonlinear integro-differential equation for the transientdensity correlator which exhibits a bifurcation, identifiedas a dynamic glass transition, for certain values of thesystem parameters [11]. One of the appealing aspects ofMCT is the absence of adjustable parameters: All infor-mation regarding both the particle interaction potentialand thermodynamic state point enter via the static struc-ture factor, which is assumed to be available from eitherindependent measurements or equilibrium statisitical me-chanical calculations. For monodisperse hard-spheres,MCT predicts a dynamic glass transition at φ ≈ 0.516when the Percus-Yevick [16] approximation is used togenerate the structure factor, although other values maybe obtained using either alternative theories, simulationor experiment to determine the static equilibrium struc-ture [70]. We note that using MCT together with Percus-Yevick structure factors enables a glass transition to bestudied for monodisperse hard-spheres at volume frac-tions above freezing. Neither MCT nor PY theory iscapable of incorporating crystalline ordering effects andboth implicitly assume an amorphous microstructure.

A shortcoming of the quiescent MCT is that it predictsan idealized glass transition with a divergent structuralrelaxation time and does not incorporate the activatedprocesses which in experiment and simulation studies are

found to truncate the divergence. While extensions ofMCT aiming to incorporate additional relaxation chan-nels have been proposed [71, 72], the underlying micro-scopic mechanisms remain unclear. Despite its mean-field character, the MCT does capture some aspects ofthe heterogeneous dynamics [73–76] which have been ob-served using confocal microscopy [77].Finally, we note that a similar scenario of crystalliza-

tion and dynamical arrest may be observed also in two-dimensional systems [68, 78]. Despite the reduced dimen-sionality and new physical mechanisms associated withmelting in two-dimensions (where the hexatic phase playsan important role) the phase diagram for both monodis-perse and polydisperse hard-disc systems is qualitativelyidentical to the three-dimensional case illustrated in Fig-ure 3. The close analogies between two- and three-dimensional systems may be exploited when consideringnonequilibrium situations for which numerical calcula-tions in 3D may prove prohibitively time consuming [79].Viewing a binary mixture as the simplest form of polydis-persity, MCT has been employed to study the influenceof ‘mixing’ (variations in composition and size ratio) onthe glass transition of three dimensional hard-sphere [80]and two-dimensional hard-disc [81] systems. These stud-ies have revealed intriguing connections between glassyarrest and random close packing.

4.2. Attractive spheres

The addition of an attractive component to the hard-sphere potential can lead to an alternative form of dy-namical arrest to either a gel at low volume fraction[82, 83] or an attractive glass state at higher volumefractions [84, 85] when the interparticle attraction be-comes sufficiently strong. The origins of the attractiveinteraction are various e.g. van der Waals forces [6]or the depletion effect when nonadsorbing polymer isadded to a dispersion [86–88]. This form of dynamicalarrest has been investigated experimentally using bothdynamic light scattering (see e.g. [83–85]) and confo-cal microscopy (see e.g. [89]). There is now compellingevidence both from experiment [90] and simulation [91]that for finite densities gellation occurs via a processof arrested phase separation and that only for very di-lute, strongly attractive, suspensions does this mecha-nism cross over to one of diffusion limited aggregation.When applied to attractive colloidal systems the MCT

predicts a nonequilibrium ‘phase diagram’ which is ingood agreement with the results of experiment and qual-itatively describes the phase boundary seperating fluidfrom arrested states as a function of volume fraction andattraction strength [84, 85]. Recent studies of systemsin which the depletion attraction between particles iscomplemented by the addition of a competing long rangeelectrostatic repulsion [92] have revealed a rich and unex-pected phase behaviour, including stable inhomogeneousphases [93] and metastable arrested states [94]. In addi-

10

-3 -2 -1 0 1 2 3 4 5Shear stress [Pa]

-3

-2

-1

0

1

2S

hear

vis

cosi

ty [P

a.s

]φ = 0.50

0.470.430.340.280.180.090.00

FIG. 4: The shear viscosity of an aqueous dispersion of col-loidal latex as a function of the externally applied shear stress.Data for a range of volume fractions are shown, from diluteup to a dense colloidal liquid at φ = 0.50. Shear thinning isevident at intermediate stress values as the viscosity of thedispersion decreases due to ordering of the particles by theflow. At larger applied stresses, for sufficently high volumefraction, the dispersion shear thickens as hydrodynamic lubri-cation forces lead to cluster formation and increased disorder.(Figure adapted from [98])

tion, impressive new developments in colloid chemistryhave enabled the construction of ‘colloidal molecules’ inwhich the particle surface is decorated with a prescribednumber of attractive sites, thus rendering the total inter-action potential anisotropic [95]. For a review of thesemore recent developments we refer the reader to [96].

5. RHEOLOGICAL PHENOMENOLOGY

As noted in the introduction, dispersions of sphericallysymmetric colloidal particles exhibit a diverse range ofresponse to externally applied flow. Much, although notall [97], of the generic rheological behaviour of colloidaldispersions is captured by the hard-sphere model intro-duced in subsection 4.1. In order to focus the discussionwe will consider the special case of hard-spheres subjectto a steady shear flow. In Fig.4 we show the results ofstress controlled experiments performed on a dispersionof spherical latex particles dispersed in water at variousvolume fractions, ranging from a dilute ‘colloidal gas’ upto φ = 0.5, corresponding to a dense colloidal liquid stateclose to the freezing transition [98]. We note that for theexperimental steady shear flow data shown in Fig.4 it isnot significant that the shear stress is employed as thecontrol parameter dictating the flow. The quiescent sys-tem is ergodic at all considered state points and qualita-tively identical results may thus be expected in an analo-gous strain controlled experiment, provided that the flowremains homogeneous.

5.1. Zero-shear viscosity

For each of the volume fractions shown in Fig.4 theshear viscosity η ≡ σxy/γ is constant for small appliedstresses (corresponding to small shear rates) and definesthe zero-shear viscosity η0. The data shown in Fig.4clearly demonstrates that the addition of colloidal parti-cles leads to a dramatic increase of η0 above that of thepure solvent (note the logarithmic scale in Fig.4).From a theoretical perspective, there are two alterna-

tive ways to understand the increase of η0 as a function ofφ. The first is to relate the viscosity to the flow distortedpair correlation functions in the limit of vanishing flowrate (see section 6.1). The leading order anisotropy ofg(r, P e → 0) captures the perturbing effect of weak flowon the microstructure and thus describes the increase ofη in terms of temporally local and physically intuitivecorrelation functions. The second method, referred to aseither the ‘time correlation’ or ‘Green-Kubo’ approach,provides an equally rigorous method in which the viscos-ity is expressed as a time integral over a transverse stressautocorrelation function (see section 7). Although thetwo approaches are formally equivalent, it is the latterwhich enables a direct connection to be made between η0and the timescale describing the collective relaxation ofthe microstructure.Within the Green-Kubo formalism the thermodynamic

colloidal contribution to the zero-shear viscosity is givenby [199]

η0 ≡ σxy

γ=

∫ ∞

0

dtGeq(t), (17)

where the equilibrium shear modulus is formally definedas a stress autocorrelation function

Geq(t) =1

kBTV〈 σxy e

Ω†eqt σxy 〉, (18)

where V is the system volume and Ω†eq is the equilibrium

adjoint Smoluchowski operator [15]. The fluctuatingstress tensor element is given by σxy ≡−∑i F

xi r

yi , and

the average is taken using the equilibrium Boltzmann-Gibbs distribution.Eq.(17) is an exact Green-Kubo relation which ex-

presses a linear transport coefficient, in this case theshear viscosity, as an integral over a microscopic auto-correlation function. For dense colloidal dispersions theshear modulus starts from a well defined initial value[99] from which it rapidly decays on a time scale set byd2/D0 to a plateau. For much later times the modulusdecays further from the plateau to zero, thus identifyingthe timescale of structural relaxation τα (see Fig.5). The‘two step’ decay of the time dependent shear modulus isa generic feature of interacting systems exhibiting botha rapid microscopic dynamics and a slower, interactioninduced, structural relaxation and is familiar from exper-iments and simulations of both colloidal and polymericsystems (where the Fourier transform G∗(ω) is typicallyconsidered, rather than G(t) directly).

11

Within the idealized mode-coupling theory (MCT) theequilibrium shear modulus (18) is approximated by [101]

Geq(t) =kBT

60π2d3

∫ ∞

0

dk k4(

S′k

Sk

)2

Φ2k(t), (19)

where T is the temperature, Sk and S′k are the static

structure factor and its derivative, respectively, and Φk(t)is the transient density correlator defined by

Φk(t) =1

Sk〈 ρ∗k(t)ρk(0) 〉, (20)

where ρk =∑

j exp(ik · rj). The collective coordinatesρk are the central quantity within mode-coupling ap-proaches and their autocorrelation (20) describes thetemporal decay of density fluctuations which slow andultimately arrest as the glass transition is approached.The mode-coupling approximation (19) arises from pro-jection of the dynamics onto density-pair modes and thusexpresses the relaxation of stress fluctuations in termsof density fluctuations. Within MCT the correlator isapproximated by the solution of a nonlinear integro-differential equation

Φq(t) + Γq

[

Φq(t) +

∫ ∞

0

dt′mq(t− t′)Φq(t′)

]

= 0, (21)

where Γq = q2/Sq and the memory function is aquadratic functional of Φk(t) which depends upon bothvolume fraction and the static structure factor (whichserves as proxy for the pair interactions). Explicit expres-sions for the quiescent memory function may be found in[103]. Equation (21) predicts that τα diverges at the glasstransition volume fraction which then leads, via Eqs.(17)and (19), to a corresponding divergence of η0. While inmany cases the quiescent MCT give a good account ofexperimental data [12] the precise nature and location ofthis apparent divergence remains a matter of debate (seee.g. [102]).

5.2. Shear thinning

Turning again to Fig.4 it is evident that for a givenvolume fraction the viscosity decreases as a function ofshear rate. This shear thinning behaviour typically setsin when the shear rate begins to exceed the inverse ofthe timescale governing structural relaxation, that is forvalues of the Weissenberg number Wi ≡ γτα > 1, whereγ is the characteristic rate of strain and τα is the struc-tural relaxation time. Within the range 0 < Wi < 1 thesystem is within the linear response regime and the flowrate is sufficently slow that the collective relaxation ofthe microstructure, characterized by, e.g. the decay ofthe transient density correlator (20), is not influenced.For Wi > 1 the rate of structural relaxation is en-

hanced by the flow field. The modulus thus becomes afunction of the shear rate and the viscosity shear thins.

-5 -4 -3 -2 -1 0 1 2 3 4log10(t D0/d

2)

0

20

40

60

80

100

120

140

G(t

) (

k BT

/d3)

-3 -2 -1 0log10(γ. t )

0

10

20

30

G(t

) (k

BT

/d3)

oG o

FIG. 5: The generalized shear modulus G(t) ≡ G(t, P e) ata volume fraction relative to the glass transition φ − φg =−1.16×10−3 calculated using the extended mode-coupling ap-proach [65]. The final relaxation of the equilibrium modulus(blue curve) serves to define the alpha relaxation timescale.As the applied shear rate is increased the relaxation timescaleis reduced. Curves are shown for Pe = 5.5 × 10−3 (green).1.1 × 10−3 (red) and 5.5 × 10−4 (black). The plateau valuewhich develops for volume fractions approaching the glasstransition is indicated by the broken line. The inset showsthe same data as a function of strain γt. (Reproduced from[65]).

To incorporate this nonlinear response Eq.(17) may begeneralized to

η(Pe) ≡ σxy(Pe)

γ=

∫ ∞

0

dtG(t, Pe), (22)

where the functional dependence on Pe has been madeexplicit. The nonlinear modulus is thus defined as

G(t, Pe) =1

kBTV〈 σxye

Ω†t σxy 〉, (23)

where Ω† is the adjoint Smoluchowski operator generat-ing the particle dynamics [15]. Despite the equilibriumaveraging employed in (18), it is important to note thatan initial stress fluctuation σxy evolves to a fluctuation atlater time t under the full dynamics, including the effectsof flow. This serves to distinguish the transient stresscorrelator (18) from that which would naturally be mea-sured in a computer simulation, where all averaging isperformed with respect to the full nonequilibrium distri-bution function. In the absence of flow Ω† = Ω†

eq and theequilibrium result (18) is recovered.Recent generalizations of the mode-coupling theory

[21] provide approximate expressions for the nonlinearmodulus G(t, Pe), see section 7. These approaches incor-porate the effects of shear flow into the memory kernelresponsible for slow structural relaxation and describethe speeding up of the relaxational dynamics. In Fig.5we show G(t, Pe) calculated using the generalized MCT

12

[65] for a volume fractions close to (but below) the glasstransition at various values of the shear rate. For Wi < 1the equilibrium result (blue curve) is not influenced bythe flow. However, for Wi > 1 the longest relaxationtime becomes dictated by the flow and τα ∼ γ−1, as isdemonstrated by the inset to Fig.5 which shows the samedata as a function of strain. From Eq.(22) it is clearthat within this generalized Green-Kubo approach thedecrease of τα with increasing γ results in shear thinningof the viscosity.An alternative, although equally valid, viewpoint is

provided by approaches focusing on the flow distortedpair correlations. Exact results for low volume fractiondispersions based on the pair Smoluchowski equation (seesection 6.1) have shown that shear thinning results from adecrease in the Brownian contribution to the shear stress[57]. In dispersions at higher volume fraction the reduc-tion in the Brownian stress is manifest in an ordering ofthe particles in the direction of flow which serves to re-duce the frequency of particle collisions. Within the shearthinning regime evidence for layered or string-like order-ing in intermediate volume fraction systems has beenprovided by Brownian dynamics simulations [104, 105]and, albeit with different characteristics, by Stokesiandynamics simulations which include hydrodynamic inter-actions [50, 106–108]. Interestingly, simulations have alsoshown that the flow induced order continues to developfollowing its initial onset. This ‘ripening’ of the orderedphase leads to a time-dependence of the viscosity knownas thixotropy [107–110] and complicates the determina-tion of flow curves in both simulation and experiment.For each shear rate the measurement time must be suffi-ciently long that the viscosity saturates to a plateau valuebefore the shear rate is updated.We note that the same ordering mechanism discussed

here for colloidal dispersions would also lead to an analo-gous shear thinning scenario for atomic liquids (e.g. liq-uid argon). In this case, however, the shear rates requiredto observe such non-Newtonian rheology are several or-ders of magnitude larger than those readily accessible inexperiment. For this reason, non-Newtonian effects inatomic systems remain largely a matter of academic in-terest.

5.3. Shear thickening

Following the regime of shear thinning, a second New-tonian plateau develops for which the viscosity attains anapproximately constant value as a function of shear rateand the flow induced ordering of the system continues todevelop. At higher shear rates the viscosity undergoes arapid increase once a critical value of the shear stress isexceeded. Such shear thickening behaviour can be eithercontinuous [59] or discontinuous [111] in character andis somewhat counterintuitive in light of the discussionpresented in 5.2 regarding flow induced microstructuralordering and its connection to shear thinning. Suspen-

sions of nonaggregating particles at intermediate volumefractions generally show reversible shear thickening, how-ever the details of the increase in viscosity depend uponthe details of the system (particle type, solvent etc.) aswell as the thermodynamic control parameters [112].

In section 5.2 we noted that the onset of shear thinningoccurs for values of the Weissenberg number Wi > 1,reflecting the essential competition between flow andstructural relaxation. In contrast, the onset of shear-thickening behaviour is determined by the value of thebare Peclet number Pe, thus serving to highlight the dif-ferent mechanisms dominating the physics of thinningand thickening states. In experiment, the difference inscaling ofWi and Pe with particle diameter d enables theextent of the second Newtonian plateau separating shearthinning and shear thickening regimes to be controlled asa function of particle size. When the microstructure un-der flow is also of interest, the contraints of instrumentalresolution (in e.g. confocal microscopy) place additionallimits on the particle size which have also to be takeninto consideration.

The earliest theoretical explanations of shear thicken-ing in colloidal dispersions proposed that the observedviscosity increase is the consequence of an order-to-disorder transition [113, 114]. Within this picture, theordered planes of particles which form within the shearthinning regime, and which persist throughout the sec-ond Newtonian viscosity plateau, begin to interact viahydrodynamic coupling at sufficiently high shear rates.This interaction pulls particles out of the layers, lead-ing to increased particle collisions, disorder, and a conse-quent increase in viscosity. The onset of shear thickeningis thus identified with the hydrodynamic instability of alayered microstructure (see [115] for a discussion of thisissue).

Despite the intuitive appeal of interpreting shear thick-ening as an order-to-disorder transition, questions wereraised by the experiments of [115, 116] in which a specificsystem (electrostatically stabilized Latex particles in gly-cols) was found to display shear thickening in the absenceof an ordered phase. These results suggested that whilean ordered phase may well precede the shear thickeningregime as the shear rate is increased, it is not a necessaryprerequisite. According to these findings, shear thicken-ing occurs via an independent physical mechanism andis not simply related to a loss of microstuctural order.Further insight into the microscopic mechanism underly-ing shear thickening was provided by Stokesian dynamicssimulations [104, 117] which identified the formation ofhydrodynamically bound particle clusters at high shearrates. Such ‘hydroclusters’ form when the shear flow issufficiently strong that the particle surfaces are drivenclosely together. At such small separations the hydrody-namic lubrication forces dramatically reduce the relativemobility of the particles such that they remain trappedtogether in a bound orbit (a point which we will later re-visit in section 6.2). Transient shear-driven hydroclusterswould appear to be the defining feature of shear thick-

13

Newtonian Shear thickeningPe

Shear thinning

FIG. 6: Schematic illustration of the microstructural orderand disorder induced in a dense colloidal dispersion by shearflow. At low values of Pe (leftmost configuration) the viscos-ity remains constant as diffusion is able to restore the equi-librium microstructure more rapidly than the shear flow candisrupt it. At intermediate shear rates (central configuration)the rate of shear exceeds the rate of structural relaxation,Wi > 1 leading to microstructural ordering and shear thin-ning. At high shear rates (right configuration) hydrodynamiclubrication forces lead to particle clustering which stronglyenhances the hydrodynamic contribution to the viscosity andresult in shear thickening.

ened states and experimental evidence for their impor-tance is accumulating [118]. Nevertheless, opinion re-mains divided regarding the fundamental mechanisms atwork [119].

The hydrodynamic mechanisms described above giverise to a continuous, albeit rapid, rise in the viscosity asa function of shear rate. An alternative scenario mayarise when, at some critical value of the shear rate, theviscosity exhibits a discontinuous jump as the system be-comes jammed [59, 113, 120–122]. While it is anticipatedthat hydrodynamics will be relevant for the descriptionof shear thickening at intermediate volume fractions (e.g.0 < φ < 0.5, as considered in Fig.4) alternative mecha-nisms may become important upon approaching the glasstransition. In [123–125] a ‘schematic’ mode coupling the-ory similar to those to be discussed in section 7 was de-veloped, in which a coupling to stress was introducedinto the nonlinear equations determining the decay of thetransient density correlator. Upon varying the model pa-rameters a range of rheological behaviour was revealed,including both continuous and discontinuous shear thick-ening, as well as a jamming transition to a non-ergodicsolid state. Within this picture, shear thickening andjamming are viewed as a type of stress induced glasstransition, for which the applied stress inhibits particlemotion, even in the absence of hydrodynamics. A num-ber of works have suggested a relationship between shearthickening and jamming [3, 4, 108, 126] although detailsof the connection between hydrodynamic cluster forma-tion and jamming transitions of the kind more familiarfrom studies of granular media [127] remains unclear.

The addition of an attractive component to thestrongly repulsive colloidal core can lead to gel formationand irreversible flocculation (see section 4.2). For suchsystems shear thickening is generally not observed, as theincrease in the hydrodynamic contribution to the viscos-ity with increasing shear rate is more than compensatedfor by the decrease in the thermodynamic contribution

arising from the attraction (see e.g. [128]). The genericbehaviour of gel and floc states is thus monotonic shearthinning as a function of shear rate [112]. It is there-fore surprising that recent experiments using attractivecarbon black particles [129] have identified a rich shearthickening behaviour for which the viscosity increases be-yond a critical value of the applied shear stress. In thiscase an additional physical mechanism has been proposedby which the forces exerted by shear flow cause flocs tobreak apart, leading to an increased surface area and thusgreater hydrodynamic dissipation [129].As pointed out in section 5.2 shear-thinning is not

unique to colloidal system and can also be observed, al-beit at high shear rates, in simple atomic liquids. Incontrast, shear thickening of the type discussed above isnot found in atomic systems (for which the ‘solvent’ is avacuum) and demonstrates clearly the breakdown of thecorrespondence between colloidal and simple liquids forstrongly nonequilibrium states. Indeed it is quite clearthat the view of colloids as ‘big atoms’ [130] will onlyhold in situations for which the influence of the solvent isnegligable and that new physics may emerge when therole of hydrodynamic interactions becomes significant.We note that an alternative type of shear thickening hasbeen observed at high shear rates in molecular dynamicssimulations of simple liquids [131–134]. In these simu-lations a profile unbiased thermostat was employed toremove artifacts which may arise when a linear flow pro-file is assumed. Shear thickening was observed in simu-lations performed at constant volume, but not in thoseperformed at constant pressure.Finally, we would like to note that the onset of shear

thickening at high flow rates has been associated withunexpected behaviour of the first normal stress differ-ence N1 = σxx− σyy. Typically, dispersions at low ormoderate shear rate exhibit a positive value of N1, indi-cating that a Weissenberg (or ‘rod climbing’) effect wouldbe observed in shear experiments performed in a Couettegeometry [2]. Experiments on dense colloidal dispersionswith repulsive interactions [121, 135, 136] have revealedthat N1 can change sign from positive to negative uponincreasing the flow rate into the regime where the viscos-ity shear thickens. Similar behaviour has been observedin Stokesian dynamics simulations [137] and in numericalsolutions of the Smoluchowski equation for dilute systems[138]. In contrast to these results for purely repulsive in-teractions, recent experiments on attractive flocculatedcolloidal dispersions display a monotonically increasingN1 throughout the shear thickening regime [129].

5.4. Yield stress

For colloidal fluid states (0 < φ < 0.494) the data pre-sented in Fig.4 represent the generic phenomenology ofdispersions of strongly repulsive colloids under shear flow.In fact, this behaviour is not limited to simple shear.Qualitatively identical behaviour is found in Stokesian

14

RH

R

FIG. 7: Many commonly studied colloidal particles (e.g.PNIPAM) consist of a polymeric core grafted with a layer ofpolymer (schematically represented on the left) which servesto stabilize against flocculation. Given a sufficiently dense andcrosslinked polymer brush the particles exhibit a strongly re-pulsive effective potential interaction approximating that ofhard-spheres with radius R. The ability of the solvent to pene-trate into the brush results in a hydrodynamic radius RH <R.These effects may be mimiced by a simple hard-sphere model(sketch on the right) in which R/RH can be used to controlthe influence of hydrodynamic interactions.

dynamics simulations for the extensional viscosity [23] ofdispersions of hard-sphere colloids under steady exten-sional flow, with the shear rate replaced by the rate ofHencky strain [53].

As already noted in section 5.1, increasing the volumefraction of a colloidal liquid leads to a strong increasein the zero-shear viscosity. Assuming that crystalliza-tion has been suppressed by polydispersity, the volumefraction can then be further increased, eventually result-ing in an apparent divergence of the zero-shear viscosity,either at the glass transition volume fraction (accord-ing to mode-coupling theory [11, 101]), or some highervolume fraction approaching random close packing [102].The variation of the viscosity as a function of shear ratefor volume fractions ranging from 0.45 to 0.57 is demon-strated in more detail by the data shown in Fig.8. Theseexperiments were performed on a system of poly(ethyleneglycol)-grafted polystyrene colloidal particles dispersedin water [139]. For the two lowest volume fractions con-sidered (φ = 0.45 and 0.48) a clear zero-shear viscositymay be be identified from the low shear rate plateau, withshear thinning evident at higher shear rates for φ=0.48.As the volume fraction is increased above 0.48 the lowshear rate plateau moves to smaller rates, out of theexperimental window of resolution, and the dispersionshows shear thinning over the entire range. Analysis ofthe intensity correlation function (related to the tran-sient density correlator (20)) measured using dynamiclight scattering leads to an estimate of the glass transi-tion for this system of 0.53 < φg < 0.55, somewhat lowerthan the typical value φg ∼ 0.58 obtained for PMMAhard-sphere-like colloids. The viscosity data for the twohighest volume fractions (φ = 0.55 and 0.57) are consis-tent with a divergence in the zero-shear viscosity at theglass transition, as predicted by the MCT (see section5.1).

According to the extended MCT [21, 58, 140], forglassy states the slowest relaxation time is τα ∼ γ−1,which leads, via Eq.(22), to η ∼ G(t → ∞)γ−1, whereG(t → ∞) is the plateau modulus (see Fig.5), thus re-producing the power law decay of the viscosity demon-strated by the data in Fig.8. For the idealized glassystates considered by MCT, where τα is infinite in the ab-sence of flow, this power law dependence extends to thelimit γ → 0, resulting in a true divergence. In real col-loidal experiments, higher order relaxation processes willalways endow the quiescent system with a finite value ofτα and the viscosity divergence will be truncated. Thelow shear divergence of the viscosity and power law shearthinning η ∼ γ−1 suggested by Fig.8 are supported byindependent experiments performed on thermosensitivecore-shell particles [63, 64]. However, some recent exper-iments on sterically stabilized PMMA particles providecontradictory evidence and have suggested a nontrivialdependence of the relaxation time on shear rate, namelyτα ∼ γ−0.8, which remains to be understood [141].

The inset to Fig.8 shows the shear stress as a functionof shear rate and provides an alternative representation ofthe viscosity data shown in the main panel. For the twohighest volume fraction samples (φ = 0.55 and 0.57) theshear stress becomes constant for the lowest shear ratesconsidered, thus identifying a dynamic yield stress forglassy states. Both the shear thinning as a function of Peand appearance of a dynamic yield stress as a function ofφ evidenced by Fig.8 are well described by the extendedMCT [63, 64].

The relationship between the dynamic yield stress andthe more familiar static yield stress mirrors that be-tween stick and slip friction in engineering applications(σstat

y > σdyny is thus to be expected). Indeed, it may be

argued that the dynamic yield stress is, in fact, a morewell defined quantity than the static yield stress. The lat-ter is typically defined as the step stress amplitude whichmust be exceeded such that the system will flow at longtimes [142]. The point of static yield may therefore be de-pendent upon details of the system preparation, with theconsequence that nonstationary properties, such as sam-ple age in colloidal glasses, could influence the outcomeof a given experiment [143]. Moreover, the existence ofcreep motion, for which the strain increases sublinearlywith time, makes difficult an unambiguous identificationof the static yield stress. In contrast, the dynamic yieldstress is defined as the limiting stress within a sequence ofergodic, fluidized steady states and is thus independentof prior sample history.

It is apparent from Eq.(22) that a dynamic yieldstress can only exist in the event that τα ∼ γ−ν , withν = 1. Values of ν less than unity result in a shear stressσxy(γ → 0) = 0, despite the fact that the viscosity di-verges. We thus note that a low shear rate divergence ofthe viscosity is a necessary but not sufficient condition forthe existence of a yield stress. While the results presentedin [141] apparently cast doubts on the existence of a dy-namic yield stress for certain colloidal glasses, complica-

15

10-3

10-2

10-1

100

101

102

γ (s-1)

10-2

10-1

100

101

102

103

104

105

106

η (P

as)

0.4089 0.570.3949 0.550.3595 0.520.3570 0.510.3486 0.500.3307 0.480.2994 0.45

10-3

10-2

10-1

100

101

102

103

γ (s-1)

10-1

100

101

102

σ (P

a)

.

.

c (g/mL) φHS

FIG. 8: Shear thinning and the dynamic yield stress of a con-centrated aqueous dispersion of poly(ethylene glycol)-graftedpolystyrene colloidal particles which, to a good approxima-tion, behave as hard spheres. The main figure and inset showthe viscosity and shear stress, respectively, as a function ofshear rate. (Reprinted with permission from [139]. Copy-right: American Physical Society)

tions due to inhomogeneous, shear localized flow makethis a subject of ongoing debate [144].The interplay between static and dynamic yield has

been investigated in simulation studies of a glass form-ing binary Lennard-Jones mixture (the Kob-Andersonmodel) using molecular dynamics simulations [145, 146].In these simulations the mixture was confined betweentwo atomistic walls, one of which was then subjected toeither a constant stress or constant strain in order toinduce shear flow. It is important to note that due tothe application of shear through the boundaries, the flowprofile within the confined fluid/glass is an output of thenumerical calculation and is not constrained to be linear.In Fig.9 we show the simulated flow curve for this sys-

tem (analogous to that shown in the inset to Fig.8) for aglassy statepoint, calculated by applying a fixed rate ofstrain to one of the bounding walls [145, 146]. When theshear stress is plotted as a function of the total strain rate(which may differ from the local rate of strain) a dynamicyield stress can clearly be identified. It was observed inthe simulations that at sufficiently low (total) shear rates,an inhomogeneous flow profile develops in which a staticlayer coexists with a fluidized region exhibiting a linearflow profile. In a complementary set of simulations alower bound for the static yield stress was identified byslowly (stepwise) increasing the shear stress until viscousflow could be detected at long times. The static yieldstress thus obtained was found to provide a criterion fordetermining the onset of inhomogeneous flow. These ob-servations may be consistent with experiments on PMMAcolloids exhibiting inhomogeneous flow [141] but are ap-

parently at odds with experiments on core-shell particleswhich do not give indications of banding or shear local-ization effects [63, 64, 139]. While these discrepanciesremain to be understood, it seems possible that the soft-ness of the potential interaction in the core-shell systemsstudied in [63, 64, 139] may play a role in maintaininghomogeneous flow.

6. THEORETICAL APPROACHES TO FLUID

STATES

There currently exist several alternative theoretical ap-proaches to first-principles calculation of the microstruc-ture and macroscopic rheology of colloidal dispersionssubject to externally applied flow. Each of the availableapproximation schemes is tailored to capture the phys-ically relevant aspects of the correlated particle motionwithin a restricted range of volume fractions. Theoriesaiming to treat low and intermediate volume fraction dis-persions take as their common starting point the pairSmoluchowski equation, which is an exact coarse grainedreduction of the many-body Smoluchowski equation (14).At high volume fractions close to the glass transition thepair Smoluchowski equation no longer provides a conve-nient starting point and an alternative approach capableof capturing slow structural relaxation is required. Thisis provided by the recently developed integration throughtransients mode-coupling theory [17, 18, 21, 140]. In thefollowing, we will first introduce the pair Smoluchowskiequation before proceeding to follow the ‘volume fractionaxis’ to give an overview of the current state of researchon the theory of flowing states.

6.1. Pair Smoluchowski equation

While Eq.(14) provides a well defined microscopic dy-namics, it has been found useful to start from an equiva-lent coarse grained level of description by integrating outuneccessary degrees of freedom from the outset. Assum-ing spatial translational invariance, integration of Eq.(14)over the center-of-mass coordinate of a pair of particlesand the remaining N−2 particles leads to an equation forthe flow distorted pair correlation function as a functionof r = r2 − r1 (see e.g. [57, 147–149])

∂g(r)

∂t+∇r ·

[

v(r) g(r) −D(r) · ∇rg(r)]

(24)

= −∇r ·[

D(r) · β F(r) g(r)]

where we have suppressed explicit time-dependence inthe function arguments for notational convenience andwhere we have introduced the gradient operator ∇r =∇2−∇1. The conditional probability to find particles atcoordinates r3 · · · rN , given that the first two are knownto be at locations r1 and r2, respectively, is given by

16

P (r3, · · · , rN |r1, r2) = P (r1, · · · , rN )/P (r1, r2) and is re-quired to calculate the functions F(r),D(r) and v(r) en-tering Eq.(24). The first of these functions, F(r), de-scribes the force acting between our chosen pair of parti-cles due to both direct potential interaction v(r), takenhere to be pairwise additive, and indirect interactionstransmitted via the surrounding N−2 particles

F(r) = −∇r v(r) −n

2

∫

dr3g(3)(r1, r2, r3)

g(r1, r2)×

×(

∇2v(|r2 − r3|)−∇1v(|r1 − r3|))

, (25)

where g(3)(r1, r2, r3) is the nonequilibrium triplet distri-bution function. The diffusion tensor is similarly ob-tained by conditional averaging and contains details ofthe hydrodynamic interactions

D(r) = 2D0

( rr

r2G(r) +

(

δ − rr

r2

)

H(r))

, (26)

where rr denotes a dyadic product and the scalar hydro-dynamic functions G(r) and H(r) remain to be specified.Finally, the relative velocity of a pair of particles is givenby

v(r) = κ · r+C(r) : κ (27)

where κ is the symmetric rate-of-strain tensor κ = (κ+κT )/2 and C is the (third rank) hydrodynamic resistancetensor describing the disturbance of the affine flow dueto the presence of the particles

C(r) : κ =−r

(

rr · κ · rr3

A(r) +(

δ− rr

r2

)

· κ · rr

B(r)

)

(28)

The tensor C arises from purely geometrical considera-tions and is not material specific. It is interesting to notethat the addition of hydrodynamic interactions preventsadvection leading to unphysical hard core overlap. A pairof approaching particles thus ‘flow around’ each other inthe solvent flow, an effect taken care of by the secondterm in Eq.(27).In the dilute limit, much is known about the hydro-

dynamic functions A,B,G and H , as only an isolatedpair of spheres must be considered. For both large andsmall separations analytical expressions for these func-tions exist [48] and are supplemented by tabulated nu-merical data for intermediate ranges [150]. At higher vol-ume fractions approximations are required to obtain thehydrodynamic functions and a number of schemes havebeen developed which aim to incorporate the effects ofmany-body hydrodynamics [28, 50, 100, 151, 152]. Inthe absence of hydrodynamic interactions A = B = 0and G = H = 1 leading to considerable simplification. Itshould be noted that C = 0 in this limit, with the con-sequence that affine motion alone can lead to hard-coreoverlaps. While an exact treatment of the thermody-namic part of the problem would lend such unphysical

10-5

10-4

10-3

10-2

dγtot

/dt

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

shea

r st

ress

shear "banding"

homogeneous

yield stress, σy

flow regime

FIG. 9: Molecular dynamics simulation results for a glassforming binary Lennard-Jones mixture. The full line showsthe shear stress as a function of the total shear rate underconditions for which a constant rate of strain is applied toone of the bounding walls. The square (labelled σy) indi-cates the static yield stress obtained by sequentially increasingthe applied stress until the system begins to exhibit viscousflow. For values of the stress between dynamic and static yieldpoints, the system was found to exhibit inhomogeneous flow.(Reprinted with permission from [145]. Copyright: AmericanInstitute of Physics.)

configurations zero statistical weight, care must be exer-cised in approximate treatments which may satisfy onlypartially this important geometrical constraint.

Although the coarse grained pair Smoluchowski equa-tion (24) is still exact (under the assumption of homo-geneity), it does not provide a closed expression for themicrostructure, as encoded in g(r). Evaluation of theintegral term required to determine the force (25) de-mands knowledge of the nonequilibrium triplet distribu-tion function, which remains unknown and contains theresidual influence of the surrounding particles which havebeen integrated out. This situation is familiar from theBBGKY heirarchy [16] for which the triplet correlationsmust be approximated in terms of the pair correlations(using e.g. the Kirkwood superposition approximation)in order to arrive at a closed equation. In recent years,accurate approximations for the equilibrium triplet cor-relations of certain model systems have been developed[153, 154]. Less is known regarding the nature of thetriplet correlations in nonequilibrium situations. Recentsimulations [156] using accelerated Stokesian dynamics[42, 51] have revealed the existence of aligned particletriplets under shear and it may be hoped that such mi-crostructural insights will eventially lead to improvedtheories by guiding the development of approximate clo-sures for the triplet correlations. Some of the approachesto be reported in section 6.3 have attempted to makeprogress in this direction by approximating explicitly theintegral term on the right hand side of (24).

17

We note that, although the assumption of spatial ho-mogeneity underlying (24) is mathematically convenient(for a translationally invariant system the only physicallyrelevant coordinate is the separation vector r = r2 − r1)it may not be appropriate under all conditions. The pres-ence of spatial inhomogeneity induced by either externalpotential fields, shear banded or shear localized statescomplicates the coarse graining procedure, resulting inan inhomogeneous version of (24). While these issuesshould pose no difficulty at low volume fractions, forwhich the right hand side of (24) may be disregarded,caution should be exercised when treating systems athigher volume fraction.Once g(r) is known, calculation of the stress tensor

describing the macroscopic rheological response becomespossible. Although exact expressions relating the stresstensor to to the flow distorted microstructure are knownformally, the situtation is complicated by the appear-ance of unknown conditionally averaged hydodynamicfunctions in the expressions. However, reliable approxi-mations for these functions are available and enable thestress to be evaluated directly from g(r) [28]. For the sim-pler case of a system interacting via a pair potential andin the absence of hydrodynamic interactions, the stresstensor may be completely determined by a simple inte-gral over the pair correlation function [155]

σ = −nkBT1 + 5ηsφ κ − n2

2

∫

drrr

rv′(r)g(r), (29)

where v′(r) is the derivative of the pair potential, ηs thesolvent viscosity, 1 the identity matrix and rr denotes adyadic product. Note that the second term on the righthand side of Eq.(29) assumes that the particles possess awell defined hard-core from which the solvent is excluded.In the absence of flow the stress tensor is diagonal with

the osmotic pressure given by Π = −Trσ/3. For a sys-tem of pure hard-spheres the familiar equation of stateβΠ/n = 1+4φg(d) is thus recovered. When under shearflow Eq.(29) yields a shear viscosity due to the colloidsη = 5φ ηs/2 + O(φ2), where the first term correspondsto Einsteins classic dilute limit result [157] and the cor-rections to higher order in φ come from the anisotropy ofg(r) inside the integral term. It is clear from Eq.(29) thatflow induced microstructural anisotropy can give rise tothe finite normal stress differences N1 = σxx− σyy andN2 = σyy−σzz characteristic of non-Newtonian rheology.The dyadic weight factor entering the integral term hasthe consequence that if g(r) possesses a mirror symmetryabout the x = 0 plane then the integral term will be equalto zero and the rheology will thus be Newtonian. Whilethis ‘fore-aft’ symmetry of the pair distribution functionis an exact mathematical consequence of the ‘pure hy-drodynamic limit’, in which the motion of the particlesis determined by Stokes flow alone [158, 159], chaoticmany-body particle motion and experimental perturba-tions, such as particle surface roughness, present in realcolloidal systems break the symmetry and result in a non-Newtonian rheology [57].

6.2. Low volume fraction

Efforts to obtain a microscopic understanding of colloidrheology began with the seminal 1906 work of Einstein inwhich it was shown how the shear viscosity of a dilute dis-persion of hard spherical colloids increases with colloidalvolume fraction, assuming that both the volume frac-tion and the shear rate remain small (η = ηs(1 + 5φ/2))[157]. Einstein’s study addressed the one-body problemof a single colloid suspended in a Newtonian fluid. Thenext step is naturally to consider the interaction betweenpairs of colloidal particles, thus making possible a discus-sion of the pair correlation functions and their relationto rheological functions at low volume fraction. Study ofthe two-particle dilute limit was initiated by Batchelor[31, 158–160] whose fundamental work formed the basisfor the more recent investigations by Brady and cowork-ers [57, 138, 161].At low volume fraction, Eq.(24) admits analytical so-

lution in the limits Pe → 0 and Pe → ∞ [57, 161] andprecise numerical results have been obtained for interme-diate values of Pe, both with and without hydrodynamicinteractions [138, 164]. Exact results are made possibleby the fact that, in the dilute limit, triplet correlations inthe pair Smoluchowski equation may be neglected leadingto a closed expression for g(r). Neglecting the difficultintegral term in Eq.(24) incurs an O(φ) error which be-comes irrelevant as φ →0 and yields a closed equation forg(r) which is exact to lowest order in the volume fractionand valid for all Pe values. For the simple special caseof hard-spheres Eq.(24) thus reduces to the equation ofmotion

∂g(r)

∂t+∇r ·

[

v(r)g(r) − Pe−1D(r) · ∇g(r)

]

= 0, (30)

where we have scaled distance and time with particle ra-dius and flow rate, respectively, such that Pe appearsexplicitly. In order to fully specify the problem Eq.(30)must be supplemented with appropriate boundary con-ditions enforcing both the requirement that the particlesdo not penetrate, via a no-flux condition at r = d, andthat g(r) → 1 as r → ∞ [6]. The first of these bound-ary conditions is clearly an exact physical requirementand is valid also at higher volume fractions. The secondcondition assumes the decay of ‘wake’ structures whichdevelop in g(r) downstream from the reference particleat higher flow rates. Detailed analysis of (30) has shownthat the range of the wake scales linearly with Pe, thusjustifying the choice of boundary conditions.For systems interacting via a spherically symmetric

pair potential it can be shown that, regardless of vol-ume fraction, in the weak flow limit Pe→0 a steady flowfield acts on the spherically symmetric equilibrium distri-bution geq(r) to produce an O(Pe) perturbation [6, 160]

g(r) = geq(r)

[

1 − Per · κ · r

r2f(r)

]

, (31)

where κ = κ/√2κ : κ, with κ is defined below Eq.(29).

18

-4 -2 0 2 4x/R

-2

0

2y

/R

FIG. 10: For mathematically perfect hard-spheres with ahydrodynamic radius equal to the radius of potential interac-tion, RH =R, particle pairs exhibit well defined trajectories.Taking a reference frame in which one particle is fixed at theorigin (yellow) the second particle (red) follows the trajecto-ries shown in the ‘pure hydrodynamic limit’ of large Pe [31].The figure shows some sample trajectories in the z = 0 flow-gradient plane (see e.g. [6]). Closed orbits are indicated inred and open in black. The apparent fore-aft mirror symme-try gives rise to Newtonian rheology.

In the dilute limit geq(r) = Θ(−r−1) and substitution of(31) into (30) yields a differential equation for the dimen-sionless function f(r) which has been solved for severalinteraction potentials of interest [6]. Although analyticalexpressions exist for certain systems, a numerical inte-gration is still required to obtain f(r) in the special caseof hard-spheres with hydrodynamic interactions [6, 160].The extension of Eq.(31) to higher order in the Peclet

number has been analyzed in considerable detail [161].It can be shown that g(r) has a regular perturbationexpansion to O(Pe2) but that calculation of the next or-der term requires singular perturbation theory, yieldingan O(Pe5/2) correction. The calculation of higher or-der terms in the Pe expansion requires use of matchedasymptotic expansions which rapidly become intractableand make preferable a numerical solution of (30). Theexpansion of the distorted structure is given by [161]

g(r) = 1 + f1Pe+ f2Pe2 + f5/2Pe5/2 + · · · , (32)

where comparison with (31), and noting that geq(r >2R) = 1 for hard-spheres at low volume fraction, enablesidentification of the coefficient f1. To O(Pe) the rhe-ology is predicted to be Newtonian with normal stressdifferences identically equal to zero [31]. Non-Newtonianrheology first occurs at O(Pe2), which is sufficient tocapture both non-zero normal stresses and the first flowinduced correction to the osmotic pressure [161].Analytic solutions to Eq.(30) exist also in the ‘pure

hydrodynamic limit’ of strong flows (Pe→∞) and havehighlighted the subtle balance between hydrodynamicand potential forces in determining the rheological re-sponse [31, 159]. For large Peclet number steady flows thesolution of Eq.(30) is well approximated by the solution

of the the simplified equation ∇ · [v(r) g(r)] = 0 (subjectto the boundary condition g(r) = 1 at r = ∞), despitethe fact that the approximation neglects the boundarylayer and thus violates the no-flux condition at contact.Subject to certain conditions on the trajectories of par-ticle pairs, Batchelor and Green proved the surprisingresult that the simplified equation predicts a sphericallysymmetric radial distribution function for hard spheres,leading to Newtonian rheology [159]. This clear predic-tion is a direct consequence of the fore-aft symmetry ofg(r) (see subsection 6.1) inherent in the assumed Stoke-sian solvent flow. In Figure.10 we show sample trajec-tories of a hard-sphere in shear flow as it moves arounda second sphere held fixed at the coordinate origin. Ofparticular interest is the existence of closed trajectoriesalong which the particles become trapped in bound orbitsand which are connected to the lubrication force requiredto displace solvent from the region between the particles[48]. In fact, the lubrication force acting between a pairof perfect spheres at separation r shows a divergence,F lub ∼ (r/R − 2)−1, corresponding to surface contact.Crucially, the time-reversibility of the Stokes equationsdictating the solvent flow implies that the force requiredto push particles togther is identical to that required topull them apart, with the consequence that particle tra-jectories exhibit the fore-aft mirror symmetry apparentin Fig.10.

Despite the sound mathematical evidence provided in[159], serious doubts were cast by subsequent experi-ments on intermediate volume fraction hard-sphere-likecolloidal dispersions, which seemed to contradict the the-oretical predictions by identifying a non-Newtonian rhe-ology at large flow rates [162]. It should be noted thatthe reversibility of Stokes flow implies that fore-aft sym-metry in the pure hydrodynamic limit holds also for finitevolume fractions and so the value φ = 0.4 employed inthe experiments of [162] cannot be held responsible forthe apparent discrepancy. The situation was eventuallyresolved by Brady and Morris, who analytically identifieda boundary-layer in the region close to particle contactin which Brownian motion balances advection [57]. Theanalysis of [57] indeed recovers the findings of [159] inthe case that the hydrodynamic radius is equal to theexcluded volume radius, as would be the case for mathe-matically perfect hard-spheres with no surface roughness.However, when the excluded volume radius exceeds thehydrodynamic radius, even by a very small amount, theresidual Brownian motion within the anisotropic bound-ary layer of g(r) leads to a non-Newtonian rheology inthe strong flow limit. Fig.7 shows a sketch of the modelemployed in [57]. The physical origin of these symmetrybreaking surface effects remains an open problem and islikely to be a function of various system specific parame-ters (e.g. surface roughness). We note that the existenceof a boundary-layer structure was originally identified instudies of the distorted structure factor of colloids undershear [163].

The analytical results for Pe→0 and Pe→∞ for hard-

19

-4 -2 0 2Pe

0

0.2

0.4

0.6

0.8

AB

2

4

6

AH

3

4

5

6

A

~Pe-1

R/RH =1.0011.011.1

Total

Hydrodynamic

Brownian

FIG. 11: The pair contributions to the relative shear vis-cosity of a dilute colloidal dispersion under steady shear flowfor three values of the ratio R/RH (see Fig.7). To secondorder in φ the relative viscosity is given by ηr ≡ η/ηs =1 + 5φH/2 + A(Pe,R/RH)φ2, where φH = (RH/R)3 φ, andA(Pe,R/RH) can be divided into hydrodynamic and Brown-ian contributions A = AH+AB. As R/RH → 1 the dispersionshows shear thickening at large Pe values due to the increasein the hydrodynamic contribution to the shear stress. As thevalue of R/RH is increased the increase in AH becomes bal-anced by the decrease in AB and only shear thinning remains.(Figure adapted from [138])

spheres under steady shear have been both confirmed andsupplemented by full numerical solutions at all values ofPe [138, 164]. These accurate numerical studies revealedthat the dilute dispersions described by Eq.(30) demon-strate not only shear thinning and finite normal stressesat intermediate flow rates (accessible from the expansion(32) to O(Pe2)), but also shear thickening at high flowrates [138]. Shear thinning in dilute systems is correlatedwith a nonvanishing distortion of the structure factor inthe plane perpendicular to the flow direction [165].In Figure 11 we show some of the numerical results

obtained in Ref.[138] for the shear viscosity of a dilutedispersion as a function of Pe. In these calculations theeffective sphere model sketched in Fig.7 was employedand results are shown for three values of the ratio of po-tential to hydrodynamic radius, R/RH . To second orderin φ the relative viscosity may be expressed as

ηr ≡ η/ηs = 1 + 5φH/2 +A(Pe,R/RH)φ2, (33)

where φH = (RH/R)3 φ is the volume fraction withrespect to the hydrodynamic radius and the functionA(Pe,R/RH) contains the effects of microstructural dis-tortion (note that calculation of g(r) to O(φ) yields theviscosity to O(φ2)). The numerically determined func-tion A(Pe,R/RH) may be further split into hydrody-namic and Brownian contributions, A = AH+AB, whichare independently accessible from the numerical calcula-

tions of [138].The top panel of Fig.11 shows A as a function of Pe.

For shear rates up to Pe ∼ 1 the qualitative variationof the viscosity is independent of the value of the sizeratio R/RH , displaying a low-shear Newtonian plateaufollowed by shear thinning. The R/RH independence ofthe form of the curves for Pe < 1 reflects the fact thathydrodynamic interactions are not central to the mech-anisms underlying shear thinning and only influence theabsolute value of the viscosity. For Pe> 1 the viscositybegins to increase as a function of Pe and the disper-sion shear thickens. In contrast to the shear thinningbehaviour, the viscosity increase is strongly sensitive tothe value of R/RH . In the Limit R/RH → 1 the trajecto-ries sketched in Fig.10 are recovered and Batchelor’s purehydrodynamic limit is realized with a viscosity indepen-dent of Pe. In [138] it is also shown that for R/RH → 1the normal stresses also vanish as Pe → ∞, indicatinga Newtonian response. As R/RH is increased the mag-nitude of the shear thickening reduces strongly and byR/RH=1.1 is lost entirely. This trend strongly indicatesthe important influence of short range lubrication forceson shear thickening (see section 5.3), which can effectivelybe turned-off by slightly reducing the hydrodynamic ra-dius below that of the repulsive potential interaction.The central and lower panels of Fig.11 show the in-

dividual hydrodynamic and Brownian pair contributionsto the total stress as a function of Pe. For Pe < 1 it isapparent that the shear thinning is due to a reduction inthe Brownian contribution with increasing Pe. For val-ues of R/RH close to unity the reduction in AB is morethan compensated by an increase in the hydrodynamicstress afor Pe > 1, leading to shear thickening. However,increasing R/RH above unity rapidly supresses the influ-ence of lubrication and the hydrodynamic contribution isoverwhelmed by the strong drop in AB .Comparing the first panel of Fig.11 with Fig.5, it is

remarkable the extent to which the qualitative rheologi-cal response observed in systems at finite volume fractionis reproduced by calculations based on the dilute limit.However, the fact that Wi = Pe at low volume fractionsdoes not permit investigation of potentially interestinginteraction effects between shear thinning and thicken-ing. Despite the extensive understanding of the responseof dilute dispersions to steady flows, analogous solutionsfor time-dependent flows remain to be investigated. Thisleaves open many interesting questions regarding tran-sient response and non-steady states.

6.3. Intermediate volume fraction

The simplest way to extend the dilute-limit results tofinite volume fraction is via the introduction of empiri-cal volume fraction dependent scale factors [57, 152, 161,167]. Analysis of Eq.(30) has provided two importantinsights. Firstly, at finite volume fractions the value ofg(r) outside the boundary layer close to the surface of

20

a reference particle should asymptote to the solution of∇ · [v(r) g(r)] = 0, for which the no-flux boundary con-dition is ignored. This amounts to assuming that, out-side the thin boundary layer, it is sufficient to solve apurely advective problem. Secondly, that the appropri-ate Peclet number in the presence of hydrodynamics isPeH = γR2/2Ds(φ), where Ds(φ) is the short time dif-fusion coefficient (which differs from D0 due to hydro-dynamic interactions with neighbouring particles). Forweak flows (Pe ≪ 1) these ideas are manifest in a modi-fied perturbation to the quiescent pair correlations

g(r) = geq(r)

[

1 − PeHr · κ · r

r2f(r)

]

. (34)

The finite volume fraction equilibrium radial distributionfunction geq(r) is an external input to the theory andcan be calculated using either simulation or equilibriumintegral equation theory [16]. The function f(r) is deter-mined by substitution of (34) into the dilute limit equa-tion (30), or its generalization for non-hard-sphere poten-tials. Many-body effects are thus included via the equi-librium radial distribution function and the short-timediffusion coefficient entering PeH. It should be notedthat Brady’s approach assumes an input geq(r) which di-verges at random close packing (φ ≈ 0.64).Despite considerable success, the scaling approach suf-

fers from two significant drawbacks: (i) the mathematicalstructure of the nonequilibrium part of the theory is thatof the dilute system. Thus, regardless of rescaling, thisapproach does not admit the occurrence of possible ad-ditional physical mechanisms which may only occur asa consequence of cooperative behaviour at finite volumefraction, (ii) the equilibrium microstructure geq(r) is re-quired as an external input and does not emerge from anapproximate treatment of many-body correlation effectswithin the theory. These issues may be addressed byapproaches which aim to approximate the triplet distri-bution function entering the pair Smoluchowski equation(24) via the effective force between a pair of particles(25). In order to arrive at a closed theory it is neces-sary to relate g(3)(r1, r2, r3), either explicitly or implic-itly (by approximating weighted integrals over the tripletdistribution), to g(r) and the pair potential v(r) using anappropriate ‘closure’ hypothesis.

6.3.1. Superposition approximation

Guidance in developing an appropriate closure rela-tion to treat non-equilibrium states is provided by ex-perience from equilibrium liquid-state integral equationtheory. One of the earliest approximation schemes aim-ing to arrive at a closed equation for the equilibriumpair correlations was developed by Born and Green [168]who employed the Kirkwood superposition approxima-tion g(3)(r1, r2, r3) ≈ g(r12)g(r23)g(r13) in combinationwith the second member of the exact Yvon-Born-Greenheirarchy (Eq.37) [16, 169]. Numerical solution of the

resulting Born-Green equation yields acceptable resultsonly for weak coupling. The superposition approxima-tion is asymptotically correct for large particle separa-tions but is poor when particles come close to contact,leading to a failure of the Born-Green equation at inter-mediate volume fractions (a breakdown which was erro-neously taken as an indicator for the first order freezingtransition of hard-spheres). Subsequent attempts haveaimed to systematically improve upon the superpositionapproximation by including additional Mayer cluster di-agrams (see e.g. [170]).One of the earliest attempts to close the non-

equilibrium Eq.(24) using the superposition approxima-tion, albeit in the absence of hydrodynamics, was madeby Ohtsuki [171] (self diffusion was addressed using ananologous approach in [172]). Numerical solution of theclosed integro-differential equation resulting from this ap-proximation was performed for charged hard-spheres atintermediate volume fraction. Although the theoreticalresults for the zero-shear viscosity were found to be inreasonable agreement with those of experiment, the paircorrelation g(r) was found to be in considerable error.These findings are supported by the work of Wagner andRussel who investigated a similar superposition based ap-proach [147].The observed discrepancies in g(r) arising from super-

position are not surprising: In equilibrium, the pair fluxin Eq.(24) may be set equal to zero, resulting in an exactequation for the pair correlation function

kBT ∇r ln geq(r) = F(r), (35)

where F(r) is given by Eq.(25) and contains the unknowntriplet distribution function. Use of superposition to ap-proximate g(3)(r1, r2, r3) in Eq.(25) leads directly to theBorn-Green equation for the equilibrium correlations, theshortcomings of which have been noted above. It is there-fore nontrivial that, despite a relatively poor descriptionof the microstructure, the results for the zero-shear vis-cosity presented in [171] turn out to be rather good agree-ment with experimental data. A similar situation is en-countered in a number of the theories to be outlined inthis section and serves to highlight the fact that goodvalues for integrated quantities (e.g. the viscosity) doesnot neccessarily imply that the underlying correlationsare treated adequately.When applying superposition to tackle non-

equilibrium problems it should be borne in mindthat the approximation represents an uncontrolledansatz and only possesses a firm statistical mechanicalbasis in the limit of vanishing flow rate. In equilibrium,the superposition approximation represents both theexact low volume fraction limit of the triplet correlationfunction and recovers correctly the long range asymp-totic behaviour. Analogous limiting results for thenonequilibrium triplet correlations which could motivatea more appropriate superposition-type approximationare currently lacking and would require a detailed anal-ysis of the triplet Smoluchowski equation in the dilute

21

limit. Despite these shortcomings, superposition maynevertheless turn out to be useful for some applications.As noted at the end of section 5.2, very recent simulationresults investigating the microstructure of hard-spherefluids under shear flow revealed the importance of lineartriplet configurations [109]. From equilibrium studiesit is known that the error of the superposition approx-imation is reduced for such linear configurations [173],thus raising the interesting possibility that superpositionmay be appropriate for treating certain non-equilibriumstates.

6.3.2. Potential of mean force

Inspection of the exact expression for the force (25)shows that only a certain weighted integral of the tripletdistribution is required to determine the pair correla-tions. It is thus not strictly neccessary to know thefull details of the triplet distribution and schemes canbe developed which aim to approximate directly the in-tegrated quantity. The simplest approximation possibleis to neglect entirely the integral contibution to the ef-fective force (25) and set

F(r) = −∇r v(r). (36)

Combining this crude approximation with Eq.(24) recov-ers the dilute limit equation of motion for g(r) for anarbitrary spherically symmetric pair potential v(r). Forthe special case of hard-spheres this leads to Eq.(30), butwith an additional delta function term on the right handside. As demonstrated by Cichocki [174] the resultingequation is completely equivalent to (30), with zero righthand side, supplemented by a no-flux boundary conditionat contact.As already noted, in equilibrium the pair Smolu-

chowski equation (24) reduces to the Yvon-Born-Greenequation for the pair correlations

kBT ∇r ln geq(r) = −∇r v(r) −n

2

∫

dr3g(3)eq (r1, r2, r3)

geq(r1, r2)×

×(

∇2v(|r2 − r3|)−∇1v(|r1 − r3|))

≡ −∇vmf(r), (37)

where vmf(r) is the equilibrium potential of mean force[16], defined by geq = exp(−βvmf) in analogy with thelow density limit of the pair correlations. A first steptowards improving the zeroth order approximation (36)is thus to approximate the nonequilibrium force (25) bythe equilibrium potential of mean force, leading to

F(r) = −kBT ∇r ln geq(r). (38)

This approximation, developed by Russel and Gast[148], incorporates equilibrium thermodynamic many-body couplings but, as is clear from Eq.(37), neglects theinfluence of flow on the triplet correlations. It should be

noted that the approximation (38) does not provide anyinformation regarding the nonequilibrium triplet correla-tion function. This is in contrast to superposition basedapproaches from which the triplet function can be recon-structed using a product of the self-consistently deter-mined pair correlation functions. It should also be notedthat within the Russel-Gast approach the function geq(r)is an input, which can be calculated using either simula-tion or equilibrium statistical mechanical approximations(see subsection 6.3.3).The linear equation resulting from combining equa-

tions (24) and (38) has been solved for hard-spheresin weak shear flow [148]. In these calculations, sim-ple approximations were employed to determine the hy-drodynamic functions A,B,G and H entering equations(26) and (28) defining the conditionally averaged hydro-dynamic tensors which represent the effective medium(solvent+ (N − 2) colloids) in which the chosen pair ofparticles are immersed. Results were obtained for thezero-shear viscosity, linear response moduli G′(ω), G′′(ω)under small amplitude oscillatory shear and the leadingorder flow induced distortion of g(r) (via determinationof the function f(r), see Eq.(31)). For φ < 0.3 goodagreement with experiment was obtained for the inte-grated quantities η0, G

′ and G′′, despite providing onlya poor description of the microstructure (see Fig.6 in[175]).An interesting feature of the Russel-Gast theory is that

the predicted zero-shear viscosity becomes very large inthe vicinity of random-close-packing (φ ≈ 0.64), althoughthe precise nature of this rapid increase as a function ofvolume fraction remains to be studied in detail. Theapparent divergence of η0 is a non-trivial output of thetheory, given that the approximate Verlet-Weiss expres-sion for geq(r) used as input diverges only at φ = 1.0[176]. When viewed within the context of the time-correlation/Green-Kubo formalism (see subsection 5.1)it is tempting to infer that the observed growth in η0 isrelated to the development of an underlying slow struc-tural relaxation time. The solution of Eq.(38) for smallamplitude oscillatory shear at finite frequencies wouldenable this issue to be addressed.

6.3.3. Equilibrium integral equations

The Russel-Gast theory outlined above neglects the in-fluence of external flow on the triplet correlation function,which leads to a force F(r) generated from the equilib-rium potential of mean force (Eq.(38)). In order to gobeyond this close-to-equilibrium ansatz it is neccessaryto express the force F(r) as a functional of g(r), suchthat both functions can be determined self-consistently.A promising approach in this direction is to generalizeequilibrium liquid-state integral equation theory [16] totreat the nonequilibrium problem. It will thus be use-ful to review briefly some concepts from the equilibriumtheory before moving on to more unfamiliar territory in

22

the next subsection.Integral equation theories generally aim to calcu-

late the equilibrium pair correlation function geq(r)from knowledge of the interaction potential in a non-perturbative fashion. Fundamental to the integral equa-tion approach is the Ornstein-Zernicke (OZ) equationwhich, for a translationally invariant system, is given bythe convolution form [16]

heq(r12) = ceq(r12) + n

∫

dr3 ceq(r13)heq(r32), (39)

where heq(r) = geq(r)−1. The direct correlation functionceq(r) defined by (39) is a function of simpler structurethan heq(r) and is thus easier to approximate. When sup-plemented by an independent closure relation betweenceq(r) and heq(r), containing details of the interaction po-tential under consideration, (39) provides a closed equa-tion for the pair correlations ( note that for pairwise addi-tive potentials the triplet distribution does not enter ex-plicitly). The task of the integral equation practitioner isthus to find numerically tractable closures which capturethe essential physics of the problem under consideration.Using diagrammatic techniques [177] it can be shown

that a formally exact closure relation is given by

geq(r) = exp[−βv(r) + heq(r) − ceq(r) + beq(r)], (40)

where beq(r) is the unknown ‘bridge function’ containingthe difficult to evaluate ‘irreducible’ Mayer cluster di-agrams [16]. Two closures of particular merit are theHyper-Netted-Chain (HNC) and Percus-Yevick (PY),given by

beq = 0 (HNC) (41)

beq = − ln(1+heq−ceq)− (heq−ceq) (PY) (42)

respectively. An additional practical advantage of the-ories based on (39) over superposition-type approachesis that the convolution form of the integral term enablesself consistent solutions to be obtained using efficient it-erative numerical algorithms [178].Although the majority of integral equation theories fo-

cus on the pair correlations, triplet correlations can alsobe handled within the same framework [154]. In addi-tion to providing a higher level of resolution, the devel-opment of triplet-level integral equations is motivated bythe desire for an improved description of the pair corre-lations. Exact relations, such as the YBG equation (37),connect the triplet to the pair correlations and the ex-

pectation is that errors in an approximate g(3)eq may be

averaged out by integration to the pair level. Duringthe mid-1960s many of the leading liquid-state theoristsproposed integral equations for the triplet correlations(e.g. Verlet [179–183], Wertheim [184], Baxter [185],Stell [186]), all of which showed considerable promise.However, the complexity of solving the equations hashindered progress along this route and somewhat sim-pler, numerically tractable, theories now seem preferable[154, 187, 188].

An integral equation which will be of particular rel-evance for the following section was derived by Scher-winski [189]. Within the Scherwinski approximationthe triplet correlation function is obtained from self-consistent solution of the following linear equation

geq(r1, r2, r3) = geq(r12)geq(r13)geq(r23) + ngeq(r12)×

×∫

dr4

(

g(3)eq (r1, r3, r4)

geq(r14)− geq(r13)

)

heq(r14)heq(r24).

(43)

Iteration of Eq.(43) yields an infinite series expressing thetriplet correlation function as a functional of the pair cor-relation function geq(r). Substitution of this series intothe exact YBG equation (37) yields a diagrammatic ex-pansion for geq(r) in perfect agreement with that arisingfrom solution of Eqs.(39), (40) and (41). In this sense,Eq.(43) represents a triplet generalization of the morefamiliar pair-level HNC theory. The more powerful ap-proximations proposed by earlier workers [179–186] prob-ably provide a superior description of the triplet correla-tions than Eq.(43) and would, upon substitution into theYBG equation, lead to improved (i.e. better than stan-dard HNC) estimate of the pair correlations. However,the Scherwinski approximation has a number of purelytechnical advantages which make it particularly suitablefor application to non-equilibrium situations and whichare convenient for numerical implementation (see also[147], which predates [189], but contains several of thekey ideas).

6.3.4. Nonequilibrium integral equations

In order to go beyond the Russel-Gast approximation[148] outlined in subsection 6.3.2, Lionberger and Russelemployed the Scherwinski equation for the triplet corre-lation function (43) in order to estimate the force (25) en-tering the pair Smoluchowski equation [190]. Nonequilib-rium pair and triplet correlations are thus determined selfconsistently and are both influenced by the externally im-posed flow. It should be noted that the direct applicationof an equilibrium relation, such as Eq.(43), to nonequi-librium ignores the fact that nonequilibrium states areintrinsically different from equilibrium and thus repre-sents a major approximation.The original version of the Lionberger-Russel theory

presented in [190] neglected hydrodynamic interactionsand has been implemented numerically for weak flowsonly. No results beyond leading order in Pe have beenpresented, although in principle the theory remains validalso in the nonlinear regime. For φ < 0.4 the LR the-ory makes predictions for the zero-shear viscosity, self-diffusion cooefficient and distorted microstructure in rea-sonable agreement with available computer simulationresults. For φ ≥ 0.45 significant quantitative deviationsappear and the theory becomes unreliable. We note that

23

the study [190] considered a suspension interacting viaa continuous repulsive potential which was then mappedonto a hard-sphere system. The input equilibrium mi-crostructure was generated using the Rogers-Young in-tegral equation [191] (an interpolation between PY andHNC), despite the fact that the Pe → 0 limit of thetheory reduces to the HNC approximation for geq(r).

Although the Lionberger-Russel theory [190] providesa sophisticated treatment of the microstructural distor-tion by incorporating the triplet correlations into theself-consistency loop, it is interesting that the results forη0 are inferior to those from the much simpler Russel-Gast theory [148] at high volume fractions. In partic-ular, the former predicts only a relatively weak growthof η0 with volume fraction, whereas the latter suggests adivergence. Calculations of the linear moduli G′, G′′ asa function of frequency have been performed using theLR theory and, perplexingly, reveal a structural relax-ation time which decreases with increasing volume frac-tion. This unphysical prediction would appear to be atodds with the growth of η0 output from the theory andrepresents a weak point of the approach, in need of clar-ification [192]. We note that the only approximation in-voked by the Lionberger-Russel theory is the Scherwinskiclosure (43) for the triplet correlation function. It wouldtherefore be interesting, albeit numerically demanding,to see whether any of the more sophisticated triplet clo-sures available [179–188] can improve the performance athigher volume fractions.

The theory developed in [190] omitted hydrodynamicinteractions in order to make clearer the approximationsto the many-body thermodynamic couplings and to fa-cilitate comparision with Brownian dynamics simulationresults. In [192] hydrodynamic interactions were includedinto the theory of [190] in order to make closer contactwith experiment. The hydrodynamic approximations de-veloped were found to be of the correct magnitude buterrors in the underlying thermodynamic approximation,namely the Scherwinski equation (43), led to an under-estimation of the magnitude of the nonequilibrium struc-ture. It was identified that the magnitude of the flowinduced structural distortion is determined by the slowstructural relaxation in the system (a finding supportedby subsequent theoretical studies [193]), which, taken to-gether with the results for the volume fraction depen-dence of the linear response moduli, suggests that theintegral equation approach does not capture the slow dy-namics characteristic of dense systems.

Wagner and Russel [147] developed an integral equa-tion approach closely related to that of Lionberger andRussel in which the nonequilibrium triplet correlationsare approximated using a closure motivated by the PYequilibrium theory (42). Although use of a PY-type clo-sure ensures that excluded-volume packing constraintsare treated realistically, at least close to equilibrium, thetheory of [147] included hydrodynamic interactions byemploying only the low density limit of the pair Hydro-dynamic functions A,B,G and H entering Eqs.(26) and

(28). The simultaneous introduction of hydrodynamicand thermodynamic approximations in [147] served toobscure the validity of the proposed nonequilibrium PYapproximation.When using integral equation methods to tackle the

triplet corelations in nonequilibrium it is important tobear in mind that the physical situation is intrinsicallydifferent from that in equilibrium. Consequently, cau-tion must be exercised when attempting to apply trustedand familiar results from equilibrium statistical mechan-ics to system under flow. In order to appreciate moreclearly the approximations involved in applying equilib-rium triplet closures to the pair Smoluchowski equationboth Lionberger and Russel [190] and Szamel [175] haveinvoked the concept of a ‘fictitious’ flow dependent two-body potential u(r, γ). In a study of the kinetic theoryof hard-spheres Resibois and Lebowitz [194] assumed theexistence of a two-body potential u(r, γ) which, if em-ployed in an equilibrium calculation, reproduces exactlythe nonequilibrium pair correlation function

g(r) = geq(r; [u ]), (44)

where the square brackets indicate a functional depen-dence and where geq is anisotropic as a result of the aniso-topy of u. The fictitious potential thus serves as proxyfor the flow field acting on the real system. It shouldbe made clear that the assumption that an effective two-body potential can yield the correct g(r) is quite distinctfrom the (erroneous) assumption that an effective one-body external potential field can represent the one-bodyflow induced force acting on the particles (see the end ofsection 3.4). Eq.(44) implicitly assumes that a homoge-neous one-body density distribution n is not a functionof the flow rate, thus neglecting possible dilation effects.Given equation (44), it is natural to go one step furtherand assume that the relation can be uniquely inverted,such that

u(r, γ) = u(r; [ g ]). (45)

By construction, u(r, γ) reproduces the nonequilibriumpair correlations. However, if the same fictitious two-body potential is used in an equilibrium statistical me-chanical calculation of the triplet correlation function,the exact nonequilibrium g(3) will not be reproduced.The missing part of the triplet correlation is referred toas the ‘irreducible’ term

g(3)(r1, r2, r3, γ)=g(3)eq (r1, r2, r3; [ g ]) + g(3)irr (r1, r2, r3; γ),

(46)

where we have assumed that (45) is valid. A key approx-

imation in the work of [190] and [175] is to set g(3)irr = 0,

which essentially amounts to assuming that equilibriumrelations such as (43) may be used to connect triplet andpair functions in nonequilibrium.The nonequilibrium integral equation method consid-

ered in [147, 190, 192] represents a synthesis of the exact

24

dilute-limit results with statistical mechanical descrip-tions of dense systems at equilibrium. While this ap-proach is promising for weak flows (Pe ≪ 1), data islacking for stronger flows which would enable the close-to-equilibrium character of the closure approximations tobe better tested. In particular, it has been emphasizedby Szamel [175] that a major challenge for theories ofthe distorted microstructure is to account for the nonzerodistortion of g(r) in the vorticity-gradient plane perpen-dicular to an applied shear flow. This distortion cannotbe detected to leading order in Pe and makes desireablenumerical studies of the existing closure approximationsunder strong flow. Applications of the nonequilibriumintegral equation method have been restricted to shearflow and, with the exception of small amplitude oscil-latory shear [190, 192], have neglected time-dependentflows entirely.

6.3.5. Alternative approaches

The nonequilibrium pair correlations and rheology ofcolloidal dispersions under weak shear flow was investi-gated by Szamel [175], who employed functional methodsto approximate the unknown integral term in Eq.(25).Hydrodynamic interactions were neglected entirely. As-suming that the irreducible term in (46) can be neglected,the triplet correlation function can be developed in afunctional Taylor expansion about the equilibrium state.To first order this is given by

g(3)(r1, r2, r3; [ g ]) = g(3)eq (r1, r2, r3) (47)

+

∫

dr4

∫

dr5δg(3)(r1, r2, r3)

δg(r4, r5)(g(r4, r5)− geq(r45)) ,

where the functional derivative is evaluated for a con-stant one-body density. Equation (47) is clearly a close-to-equilibrium approximation. Insertion of (47) into theexpression for the force (25) and rearrangement of termsleads to a closed equation for g(r) which requires both

g(3)eq (r12, r23, r13) and c

(4)eq (r1, r2, r3, r4), a higher order

equilibrium direct correlation function, as input. Fol-lowing appropriate decoupling approximations for theunknown higher order equilibrium correlations and lin-earizing with respect to the shear flow, Szamel obtaineda closed equation for g(r) which only requires geq(r) asinput (for which the Verlet-Weiss approximation was em-ployed [176]).The Szamel theory [175] is considerably simpler to im-

plement than the integral equation approach of Lion-berger and Russel [190] and provides comparable, per-haps even slightly better, results for the zero-shear vis-cosity as a function of volume fraction, as is evidentfrom Fig.12. Both g(r) and its Fourier transform S(k)were found to be qualitatively similar to those from theLionberger-Russel theory, substantially underestimatingthe magnitude of the distortion from equilibrium whencompared to Brownian dynamics simulation.

0 0.1 0.2 0.3 0.4 0.5φ

0

1

2

3

4

5

6

7

∆η/η

s

0.2 0.3 0.4 0.50

5

10

FIG. 12: The reduced zero-shear viscosity as a function ofvolume fraction as predicted by various theories based on thepair Smoluchoski equation: Szamel (black) [175], Brady (red),Lionberger-Russel (blue) [190], Ronis (green) [195]. The in-set shows the results of the Szamel theory compared withBrownian dynamics simulation data (circles). For φ > 0.43the theory strongly underestimates the zero-shear viscosity.(Adapted from Ref.[175])

In an early study, Ronis took an alternative ap-proach based on fluctuating hydrodynamics, in whichphenomenological fluctuating terms are added to themacroscopic equations of hydrodynamics [195]. The the-ory of stochastic processes may then be employed to cal-culate nonequilibrium time-correlation functions and thedistorted microstucture. Of all the theories described inthis and the previous section, Ronis theory provides themost accurate results for the distortion of the structurefactor at low Pe values. Nevertheless, it can be shownthat the Ronis approximation leads to a vanishing of themicrostructural distortion in the vorticity-gradient planeat all Pe, in contradiction to experiment and simula-tion results [166]. This deficit is related to the fact thatthe hard-sphere ‘core condition’ g(|r| < 1) = 0 is vio-lated within this approach. The Ronis theory reduces tothe closely related theory of Dhont [163] in the low Pe,linearized limit. In another early work, Schwarzl andHess [196] postulated a phenomenological equation forg(r) involving a number of empirical parameters repre-senting the relaxation times in the system. However, dueto the phenomenological nature of both the fluctuatinghydrodynamics approach and the equation-of-motion forg(r) proposed by Schwarzl and Hess, the foundation ofthe theoretical treatments presented in [195] and [196] innonequilibrium statistical mechanics is unclear.

Finally, we note that Wagner has assessed approachesbased on the pair Smoluchowski equation using theGENERIC framework of beyond equilibrium thermody-namics [197]. This formalism enables any proposed clo-sure of the pair Smoluchowski equation to be checkedfor thermodynamic consistency. The study presentedin [197] identified the thermodynamically admissable ex-pression for the stress tensor and clarified the nature ofthe inconsistencies which can occur when separate deriva-tions are performed for the equation-of-motion for g(r)

25

and the stress tensor.

6.4. Temporal locality vs. memory functions

In sections 5.1 and 5.2 we introduced briefly the Green-Kubo expressions for calculating the shear stress in boththe linear (17) and nonlinear regimes (22). Within theGreen-Kubo framework, transport coefficients are relatedto integrals over time correlation functions [199]. Thegrowth of the shear viscosity as a function of volumefraction is thus related to an increasingly slow decayof the stress autocorrelation function. What is lack-ing thus far in the discussion of the pair Smoluchowskiequation and its approximate solutions, is the connec-tion between the temporally local Eqs.(24) and (29) andthe nonlocal expressions (17) and (22) fundamental tothe time-correlation function formalism. Moreover, thememory kernels characteristic of the time-correlation ap-proach (and widely employed in continuum mechanicsapproaches, see section 2) make no explicit appearancewithin the pair Smoluchowski framework.The key to understanding the connection between

the nonlocal time-correlation formalism and approachesbased on the local pair Smoluchowski equation lies inthe study of time-dependent external flow fields. In-deed, it is the absence of time-dependent data from thepair Smoluchowski approach which has served to ob-scure the relationship between these two methods, de-spite the fact that they are formally equivalent. Theonly available Smoluchowski-based time-dependent cal-culations were performed using the Lionberger-Russeltheory [28, 190, 192] for small amplitude oscillatory shearflow. As pointed out in subsection 6.3.4, the available re-sults for the volume fraction dependence of G′(ω) andG′′(ω) within the LR theory reveal underlying problemsresulting from the approximations employed (i.e. a re-duction of τα with increasing φ) which would otherwisehave gone unnoticed. This serves to emphasize the im-portance of going beyond steady flow calculations in ap-plications of pair Smoluchowski theories.The generalization of Eq.(22) to general time-

dependent shear is given by integration over the entireflow history [17]

σxy(t) =

∫ t

−∞

dt′ γ(t′)G(t, t′), (48)

where G(t, t′) ≡ G(t, t′; [γ ]) is the nonlinear shear mod-ulus. The lack of time-translational invariance in themodulus arises from a functional dependence on the shearrate. The microscopically derived Eq.(48) should be con-trasted with the more familiar phenomenological result(2).Using the generalized Green-Kubo result (48) it is in-

structive to consider a simple special case: The stress re-sponse of hard-spheres to the onset of steady shear flowin the absence of hydrodynamic interactions. Specifically,

we consider a shear field which is switched from zero toa constant value, γ(t) = γΘ(−t). For this choice of shearfield Eq.(48) reduces to

σxy(t) = γ

∫ t

0

dt′ Gss(t′), (49)

where Gss(t) is the time translationally invariant shearmodulus under steady shear flow. On the other hand,within approaches based on the distorted microstructurethe interaction contribution to the stress is given by (seeEq.(29))

σxy(t) = − n2

2

∫

drrr

rv′(r) g(r, t), (50)

where g(r, t) is calculated from the pair Smoluchowskiequation (24) subject to the switch-on shear flow underconsideration. The expressions (49) and (50) are formallyequivalent. Equating the time derivatives thus leads tothe exact relation

Gss(t) = − n2

2γ

∫

drrr

rv′(r)

∂g(r, t)

∂t. (51)

The quiescent shear modulus is thus recovered in the slowflow limit

Geq(t) = − n2

2

∫

drrr

rv′(r) lim

γ→0

(

1

γ

∂g(r, t)

∂t

)

. (52)

The right hand side of Eq.(52) can be further reduced tothe determination of the function f(r, t) by substitutionof the first order expansion (31) for g(r, t).As the volume fraction is increased, the time deriva-

tive on the r.h.s. of Eq.(52) must give rise to a growth inthe timescale determining the relaxation of stress fluctu-ations, as described by Geq(t). The fact that this is notcaptured by the Lionberger-Russel theory [28, 190, 192]simply reflects the failings of the approximate closure(43) relating the nonequilibrium triplet and pair corre-lation functions. However, it is not clear that even anexact equilibrium expression for the triplet correlationswould be sufficient to resolve these difficulties. If the keysource of error lies in the neglect of the irreducible termapprearing in (46) then considerable new insight into thenature of nonequilibrium states will be required in orderto make further progress using integral equation meth-ods. Nevertheless, these considerations should serve tomotivate time-dependent studies of the Russel-Gast [148]and Szamel [175] theories.While the particular example chosen (switch-on shear

flow) provides access to the steady shear rate dependentmodulus, other choices of time-dependence may enableconnection to be made between non-time translationallyinvariant correlation functions (e.g. G(t, t′)) and time-dependent solutions of the pair Smoluchowski equation.The issue of whether temporally local constitutive equa-tions are preferable to nonlocal functionals for describingcomplex fluids has been addressed within the framework

26

of nonequilibrium thermodynamics [198]. The nonequi-librium thermodynamics approach requires identificationof an appropriate set of structural variables which, in ad-dition to the standard hydrodynamic variables of mass,momentum and internal energy density, contain all in-formation about the state of the system at a given timethat is necessary to determine the macroscopic quantitiesof interest. The pair Smoluchowski approaches discussedin this section essentially introduce g(r) as an additionalstructural variable [197]. While this appears to be a validapproach for low and intermediate volume fractions, acorrect identification of the structural variables appropri-ate for describing glass formation and dynamical arrestremain to be found. For this reason the most promis-ing approaches to treating high volume fraction statesare based on the generalized Green-Kubo relations andmode-coupling theory.

7. GLASS RHEOLOGY

Assuming that crystallization effects can be suppressed(see section 4) the volume fraction can be increased tothe point at which the individual particles are unableto diffuse beyond the cage of nearest neighbours anda dynamically arrested glassy state is formed. In or-der to visualize the amorphous cage structure in sucha glassy state Figure 13 shows a configuration snapshottaken from a Brownian dynamics simulation of a binaryhard-sphere mixture in two dimensions (hard-discs) [79].The two-dimensional volume fraction φ2D = 0.81 of thesimulation is above the estimated glass transition pointof φ2D ≈ 0.79 and the size ratio of large to small discradii is 1.4, a value empirically found to frustrate crys-tallization in two dimensional systems (which occurs atφ2D = 0.69 for monodisperse discs). In both two- andthree-dimensional systems the physics of the glass tran-sition becomes important for determination of both therheology and flow distorted microstructure of high vol-ume fraction systems.The response to externally applied flow of states close

to, or beyond, the glass transition is only beginning to beunderstood and establishing the basic principles of glassrheology remains a challenging task. At present, the onlytruly microscopic theories available are provided by re-cent extensions of the quiescent MCT which enable theeffects of external flow to be incorporated into the formal-ism and thus make possible a theoretical investigation ofthe complex interaction between arrest and flow.

7.1. MCT inspired approaches

Extending earlier work on the low volume fraction selfdiffusion of colloidal dispersions [200], Miyazaki and Re-ichman constructed a self-consistent mode-coupling-typeapproach to describe collective density fluctuations fordense colloidal fluids under shear below the glass transi-

FIG. 13: A snapshot from a Brownian dynamics simulation ofa quiescent binary hard-disc mixture (using a size ratio of 1 :1.4 to supress crystallization). The simulation was performedat a two-dimensional volume fraction of φ2D = 0.81, which

lies above the estimated glass transition packing φ(g)2D ∼ 0.79,

with 50% large discs and 50% small discs. (Figure courtesyof F. Weyßer.)

tion [201–203]. The Miyazaki-Reichman theory consid-ers time-dependent fluctuations about the steady stateand thus requires the (unknown) flow distorted structurefactor S(k) as an input quantity. Approximating S(k)by the quiescent correlator, results have been presentedfor colloidal dispersions in two-dimensions under steadyshear [201, 202] and in three dimensions (subject to addi-tional isotropic approximations) under oscillatory shear[203]. Applications to glassy states have been avoided asthe theory relies upon an ergodic fluctuation-dissipationtheorem. An alternative extended-MCT approach hasbeen proposed by Koblev and Schweizer [204] and Saltz-man et al [205] which is built on the idea that entropicbarrier hopping is the key physical process driving themicroscopic dynamics and rheology of glassy colloidalsuspensions. Due to the activated nature of the barrierhopping process the ideal glass transition described byquiescent MCT (see section 4) plays no role. A nonlinearrheological response results from a stress induced modi-fication of the barrier heights.

A currently promising method of extending quiescentMCT to treat dense systems under flow involves integra-tion through the transient dynamics, starting from anequilibrium Boltzmann distribution in the infinite past.In contrast to [201–203] the distorted microstructure isan output of this approach. The initial form of the the-ory was outlined by Fuchs and Cates for steady shear flow[140] and presented two essential developments: Firstly,that integration through the transient dynamics leads di-rectly to exact generalized Green-Kubo formulas, relat-ing average quantities to integrals over microscopic time-correlation functions. Secondly, that MCT-type projec-tion operator approximations reduce the formal Green-

27

Kubo expressions to closed equations involving transientcorrelators (which can be calculated self-consistently). Astrong prediction of the ITT-MCT theory resulting fromcombining these two steps is that the macroscopic flowcurves exhibit a dynamic yield stress (see section 5.4) inthe limit Pe→0, for states which would be glasses or gelsin the absence of flow. Moreover, the yield stress appearsdiscontinuously as a function of volume fraction, in con-trast to mesoscopic approaches [35–37]. The ITT-MCTthus provides a scenario for a nonequilibrium transitionbetween a shear-thinning fluid and a yielding amorphoussolid which is supported by considerable evidence fromboth colloidal experiments [63, 64, 139, 206] and Brown-ian dynamics simulation [41, 79].

Due to the numerical intractability of the microscopictheory of [140], subsequent work focused on the construc-tion of both isotropically averaged approximations to thefull anisotropic equations and simplified schematic mod-els inspired by these [58]. Comparison of the theoret-ical predictions with experimental data for thermosen-sitive core-shell particles (see Fig.7) has proved highlysuccessful [63, 64, 207–209]. The original formulation ofthe ITT-MCT (more details of which can be found in[210]) has subsequently been superseded by a more ele-gant version [21]. It is interesting to note that the signif-icant technical changes to the ITT-MCT formalism in-troduced in [21] lead to expressions which resemble moreclosely those of Miyazaki and Reichman [201–203]. Giventhe very different nature of the formal derivation (fluctu-ating hydrodynamics vs. projection operator methods)and approximations employed, the similarity of the fi-nal expressions is reassuring and serves to highlight therobustness of MCT-based approaches. For a comprehen-sive overview of the status of the steady shear theory werefer the reader to the recent review [20].

Going beyond steady shear, the original formulation ofITT-MCT [58, 140, 210] has been generalized to treat ar-bitrary time-dependent shear [17]. These developmentsnot only enable shear fields of particular experimentalrelevance to be investigated (e.g. large amplitude oscil-latory shear flow), but have also revealed an underlyingmathematical structure which is not apparent from con-sideration of steady flows alone. The theory has beenapplied (albeit subject to various simplifying approxi-mations) to investigate the build-up of stress and corre-sponding microscopic particle motion, as encoded in themean-squared-displacement, following the onset of shear[65]. More recently, the modern version of ITT-MCT [21]has been extended to describe time-dependent flow of ar-bitrary geometry [18], thus making possible the study ofnon-shear flow and enabling the full tensorial structure ofthe theory to be identified. The developments presentedin [18] elevate the ITT-MCT approach to the level of afull constitutive theory for dispersion rheology and maybe regarded as the most up-to-date formulation of thetheory. While the development of numerical algorithmsto efficiently solve the fully microscopic theory [18] iscurrently in progress, this task is made computationally

demanding by the combination of spatial anisotropy andlack of time-translational invariance presented by manyflows of interest. In [19] a simplified theory was pre-sented which contains the essential physics of the fullmicroscopic equations, including the tensorial structure,but which is much more convenient for numerical solution(see subsection 7.7).

7.2. Integration through transients

The integration through transients (ITT) approachoriginally developed by Fuchs and Cates [140] and subse-quently developed in [17, 18] provides a formal expres-sion for the nonequilibrium distribution function Ψ(t)required to calculate average quantities under flow. Inessence, ITT provides a very elegant method of derivinggeneralized (i.e. nonlinear in κ(t)) Green-Kubo relationswhich invite mode-coupling-type closure approximations.The current formulation of the theory neglects hydrody-namic interactions (HI) entirely. On one hand this omis-sion is made for purely technical reasons, but it is alsohoped that HI will prove unimportant for the microscopicdynamics of the dense states to which the final theorywill be applied. In the following we will briefly outlinethe key points of ITT, employing throughout the modernformulation of [18].The starting point for ITT is to re-express the Smolu-

chowski equation (14) in the form

∂Ψ(t)

∂t= Ω(t)Ψ(t), (53)

where, in the absence of HI, the Smoluchowski operatorcontrolling the dynamical evolution of the system is givenby [15]

Ω(t) =∑

i

∂i · [D0(∂i − βFi)− κ(t) · ri ]. (54)

Equation (53) may be formally solved using a time-ordered exponential function (which arises because Ω(t)does not commute with itself for different times [211])

Ψ(t) =

[

exp+

∫ t

−∞

dsΩ(s)

]

Ψeq, (55)

where Ψeq is the initial distribution function in theinfinite past, which is taken to be the equilibriumBoltzmann-Gibbs distribution corresponding to the ther-modynamic state-point under consideration. The as-sumption of an equilibrium distribution is clearly accept-able in situations for which the quiescent state is one ofergodic equilibrium. However, the role of the initial stateis less clear for statepoints in the glass and the depen-dence, if any, on the initial condition may depend uponthe details of the flow between t = −∞ and the presenttime t. The absence of a general proof that Ψ(t) is inde-pendent of Ψ(−∞) leaves open the possibility that cer-tain flow histories do not restore ergodicity and that thesystem thus retains a dependence on the initial state.

28

The solution (55) is formally correct, but not partic-ularly useful in its present form. A partial integrationyields an alternative solution of (53) which is exactlyequivalent to (55), but more suited to approximation

Ψ(t) = Ψe +

∫ t

−∞

dt1 Ψe κ(t1) : σ e

∫t

t1dsن(s)

− , (56)

where σαβ = −∑

i Fαi r

βi and the ‘double dot’ notation

familiar from continuum mechanics, A :B = TrA · B[212], has been employed. As a result of the partial in-tegration the dynamical evolution in Eq.(56) is dictatedby the adjoint Smoluchowski operator

Ω†(t) =∑

i

[D0(∂i + βFi) + ri · κT (t) ] · ∂i. (57)

Equation (56) is the fundamental formula of the ITTapproach and expresses the nonequilibrium distributionfunction as an integral over the entire transient flow his-tory. Both solutions (55) and (56) are valid for arbitraryflow geometries and time-dependence. The relation be-tween the two formal solutions is analogous to the Heisen-berg and Schrodinger pictures of quantum mechanics inwhich the time evolution of the system is attributed toeither the wavefunction (equation (55)) or the opera-tors representing physical observables (equation (56)). Itshould be understood that the ITT form (56) is an oper-ator expression to be used with the understanding thata function to be averaged is placed on the right of theoperators and integrated over the particle coordinates.A technical point to note is that in cases for which phasespace decomposes into disoint pockets (‘nonmixing’ dy-namics) the distribution (56) averages over all compart-ments. A general function of the phase-space coordinatesf(t, ri) thus has the average

〈f〉ne = 〈f〉+∫ t

−∞

dt1 〈κ(t1) : σ e

∫t

t1dsن(s)

− f 〉, (58)

where 〈f〉ne denotes an average over the non-equilibriumdistribution (56). Equation (56) generalizes the orig-inal formulation of ITT [140] to treat arbitrary time-dependence [18].

7.3. Translational invariance

Before applying mode-coupling-type approximationsto the exact result (58) we first address an important con-sequence of assuming homogeneous flow (reflected in thespatial constancy κ(t) appearing in Eq.(53)). On purelyphysical grounds, it seems reasonable that for an infinitesystem the assumed translational invarance of the equi-librium state (crystallization is neglected) will be pre-served by the Smoluchowski dynamics. However, prov-ing this for a general time-dependent flow is mathemat-ically nontrivial, due to the fact that the Smoluchowskioperator (54) is itself not translationally invariant. By

considering a constant vectorial shift of all particle co-ordinates, r′i = ri + a, Brader et al have shown thatthe nonequilibrium distribution function Ψ(t) is transla-tionally invariant (but anisotropic) for any homogeneousvelocity gradient κ(t) [18].Given the translational invariance of Ψ(t) it becomes

possible to investigate the more interesting invarianceproperties of the two-time correlation functions. Thecorrelation between two arbitrary wavevector-dependentfluctuations δfq = fq − 〈fq〉ne and δgk = gk − 〈gk〉neoccuring at times t and t′ is given by

Cfqgk(t, t′) = 〈δf∗

q(t)δgk(t′)〉ne. (59)

It is clear that in a homogeneous system the correlationfunction (59) must be translationally invariant. How-ever, in this case, shifting the particle coordinates by aconstant vector a yields

Cfqgk(t, t′) = e−i( q(t,t′)−k )·aCfqgk

(t, t′), (60)

where

q(t, t′) = q · e−∫

t

t′dsκ(s)

− . (61)

The only way in which the required translational invari-ance of the correlation function can be preserved is if theexponential factor in (60) is equal to unity. This require-ment has the consequence that a fluctuation at wavevec-tor k = q(t, t′) at time t′ is correlated with a fluctuationwith wavevector q at time t as a result of the affine sol-vent flow. Eq.(61) thus defines the advected wavevectorwhich is central to the ITT-MCT approach and whichcaptures the affine evolution of the system in approachesfocused on Fourier components of fluctuating quantities(e.g. the density ρk) rather than particle coordinates di-rectly. The wavevector q(t, t′) at time t′ evolves due toflow-induced advection to become q at later time t. Itshould be noted that various definitions and notations forthe advected wavevector have been employed in the lit-erature documenting the development of ITT-MCT andwhich could provide a source of confusion. In the presentwork we exclusively employ the modern definition usedin [18, 19, 21].Although Eq.(61) arises from microscopic considera-

tions it is nevertheless fully consistent with the contin-uum mechanics approaches outlined in section 2, de-spite the very different mindset underlying the two meth-ods. Eq.(61) simply describes the affine deformation ofmaterial lines in Fourier space and can thus be usedto define the inverse deformation gradient tensor viaq(t, t′) = q · E−1(t, t′) in complete accord with contin-uum approaches. Doing so leads to the identification

E−1(t, t′) = e−

∫t

t′dsκ(s)

− . (62)

As the deformation gradient tensor simply describes theaffine distortion of a material line under flow, it is nat-ural to define also a reverse-advected wavevector result-ing from the inverse transformation q(t, t′) = q ·E(t, t′),

29

where

E(t, t′) = e∫

t

t′dsκ(s)

+ . (63)

The choice of using either advected or reverse-advectedwavevectors in treating the effects of affine motion withina microscopic theory has parallels with the choice be-tween Lagrangian and Eulerian specifications of theflow field in continuum fluid dynamics approaches [31].Within a continuum mechanics framework the deforma-tion gradient would simply be defined as the solution ofthe equation

∂

∂tE(t, t′) = κ(t)E(t, t′), (64)

for a given flow κ(t). According to the rules of time-ordered exponential algebra [211], Eq.(63) is the formalsolution of (64), thus demonstrating the consistency be-tween the Fourier-space microscopic approach of [18] andtraditional real-space continuum mechanics.The advected wavevector introduced above provides a

convenient way to keep track of the affine deformationin a particulate system. Mode-coupling-type approxima-tions (to be discussed below) seek to factorize the av-erage entering Eq.(58) by projecting the dynamics ontothe subspace of density fluctuations ρq [103]. For a flow-ing system a fluctuation at wavevector q(t, t′) at time t′

evolves (in the absence of interactions and Brownian mo-tion) to one at q at time t. It thus becomes essential toproject onto density fluctuations at the correct advectedwavevectors in order to avoid spurious decorrelation ef-fects in the resulting approximations.

7.4. Microscopic constitutive equation

In order to address dispersion rheology the specialchoice f = σ/V is made in (58), leading to an exactgeneralized Green-Kubo relation for the time-dependentshear stress tensor [18]

σ(t) =1

V

∫ t

−∞

dt1 〈κ(t1) : σ e

∫t

t1dsن(s)

− σ〉, (65)

noting that there are no ‘frozen in’ stresses in the equilib-rium state, 〈σ〉 = 0. The adjoint Smoluchowski operatorΩ†(t) has a linear dependence on κ(t) and equation (65)is thus nonlinear in the velocity gradient tensor. Equa-tion (65) is a formal constitutive equation expressing thestress tensor at the present time as a nonlinear func-tional of the flow history. Although the result (65) doesnot provide an exact description of the physical systemunder consideration (particle momenta are assumed tohave relaxed and hydrodynamic interactions are absent),it has a formal status equivalent to that of Eq.(56). Forthe special case of steady shear flow (65) is consistentwith (22) with a shear modulus given by (23).

Application of MCT-type projection operator factor-izations [18] to the average in (65) leads to a compli-cated, but closed, constitutive equation expressing thedeviatoric stress in terms of the strain history [18, 19]

σ(t) = −∫ t

−∞

dt′∫

dk

32π3

[

∂

∂t′(k·B(t, t′)·k)kk

]

×

×[(

S′kS

′k(t,t′)

kk(t, t′)S2k

)

Φ2k(t,t′)(t, t

′)

]

, (66)

where Sk and S′k are the equilibrium static structure

factor and its derivative, respectively. The influence ofexternal flow enters both explicitly, via the Finger ten-sor B(t, t′) (see subsection 2.1), and implicitly throughthe reverse-advected wavevector. As noted above, thereverse-advected wavevector, which provides an impor-tant source of nonlinearity in (66), enters as a result ofjudicious projection of the dynamics onto appropriatelyadvected density fluctuations ρk(t,t′). The normalizedtransient density correlator describes the decay underflow of thermal density fluctuations and is defined by

Φk(t, t′) =

1

NSk〈 ρ∗ke

∫t

t′dsΩ†(s)

− ρk(t,t′)〉. (67)

The occurance of the advected wavevector in (67) en-sures that trivial decorrelation effects are removed (i.e.that in the absence of Brownian motion and potentialinteractions Φk = 1 for all times).In order to close the constitutive equation (66) we re-

quire an explicit expression for the transient correlator(67). Time-dependent projection operator manipulationscombined with the theory of Volterra integral equationsyield an exact equation of motion for the time evolu-tion of the transient correlator containing a generalizedfriction kernel - a memory function formed from the au-tocorrelation of fluctuating stresses. Mode-coupling typeapproximations to this kernel yield the nonlinear integro-differential equation [17–19]

Φq(t, t0) + Γq(t, t0)

(

Φq(t, t0) (68)

+

∫ t

t0

dt′mq(t, t′, t0)Φq(t

′, t0)

)

= 0

where the overdots denote partial differentiation with re-spect to the first time argument. Here the ‘initial de-cay rate’ obeys Γq(t, t0) = D0q

2(t, t0)/Sq(t,t0) with D0

the diffusion constant of an isolated particle. The for-mal manipulations presented in [17, 18] have revealedthat imposing a time-dependent flow results in a mem-ory kernel which depends upon three time arguments.The presence of a third time argument, which would havebeen difficult to guess on the basis of quiescent MCT in-tuitition, turns out to have important consequences forcertain rapidly varying flows (e.g. step strain [17]) andis essential to obtain physically sensible results in such

30

cases. The memory kernel mq(t, t′, t0) entering (68) is

given by the factorized expression [18, 19]

mq(t, t′, t0)=

ρ

16π3

∫

dkSq(t,t0)Sk(t′,t0)Sp(t′,t0)

q2(t′, t0)q2(t, t0)(69)

× Vqkp(t′, t0)Vqkp(t, t0)Φk(t′,t0)(t, t

′)Φp(t′,t0)(t, t′),

where p = q− k, and the vertex function obeys

Vqkp(t, t0)= q(t, t0) · (k(t, t0)ck(t,t0)+ p(t, t0)cp(t,t0))

(70)

with Ornstein-Zernicke direct correlation function ck =1−1/Sk (see Eq.(39)). In the linear regime Eqs.(66) and(68) reduce to the standard quiescent MCT forms (19)and (21), respectively.An important feature of Eqs.(66)–(70) is that they of-

fer a closed constitutive equation requiring only the staticstructure factor and velocity gradient tensor κ(t) as in-put to calculating the stress tensor. The equilibrium Sq

is determined by the interaction potential and thermo-dynamic statepoint and, as in quiescent MCT, serves asproxy for the pair potential (an interpretation arisingfrom field-theoretical approaches to MCT [213]). Therole of Sq within the ITT-MCT should be contrastedwith that within the Miyazaki-Reichman theory [201–203], discussed in subsection 7.1, where it enters as an ap-proximation to the flow-distorted structure factor S(k).In subsection 2.3 we introduced the principle of mate-

rial objectivity; an approximate symmetry requiring thata valid constitutive relation be rotationally invariant.While verification of rotational invariance is straightfor-ward for the phenomenlogical Lodge equation introducedin subsection 2.1, proof becomes more demanding for themicroscopic constitutive theory given by Eqs.(66)–(70).Nevertheless, substitution of Eqs.(11) and (13) into (66)–(70) yields the result (12), thus verifying that the ITT-MCT constitutive equation is indeed material objectiveas desired [18]. Material objectivity is an important con-sistency check for constitutive theories based on over-damped Smoluchowski dynamics, for which it representsan exact symmetry constraint.Possibly the most exciting feature of the ITT-MCT

constitutive equation (66)–(70) is that it incorporates amechanism for describing the slow structural relaxationleading to dynamical arrest. The predicted rheologicalresponse thus goes from that of a viscous fluid to thatof an amorphous solid, characterised by an elastic con-stant, upon variation of the thermodynamic control pa-rameters. The ability of the theory to unify the descrip-tion of fluid and glassy states stems from the underlyingmode-coupling-type approximations which are tailoredto capture the cooperative particle motion in dense col-loidal dispersions, ultimately leading to particle cagingand arrest. Mathematically, this scenario arises froma bifurcation in the solution of the nonlinear integro-differential equation (68) at suffiently high volume frac-tion/attraction strength associated with a diverging re-laxation time of the transient density correlator. Glass

formation within the MCT-ITT approach is a purely dy-namical phenomenon, as the equilibrium Sq used as in-put varies smoothly accross the transition. The fluid-solid transition contained within the ITT-MCT equationscan be better appreciated by considering the small strainlimit of Eq.(66) which yields the linear response result[18]

σl(t)=

∫ t

−∞

dt′∫

dk

16π3(k·κ(t′)·k)kk

(

S′kΦk(t−t′)

kSk

)2

(71)

where Φk(t) is the correlator from quiescent MCT [103].In the glass the correlator does not relax to zero for longtimes and a partial integration of (71) followed by takingthe limit of small strain leads to the result

σ(t) = 2G(t→∞)ǫ(t), (72)

where ǫ(t) is the infinitessimal strain tensor and G(t→∞) is an elastic modulus obtained from Eq.(19) (alsoknown as Lame’s second coefficient, the first being zerohere due to incompressibility). Eq.(72) is essentiallyHookes law, describing the small strain response of aglassy solid. Going beyond linear response, Eq.(66) incor-porates the fluidizing effect of flow and thus makes pos-sible investigation of a large number of time-dependentrheological situations in which externally applied flowfields compete with glass formation and slow structuralrelaxation (see subsection 7.6).

7.5. Distorted structure factor

The microscopic ITT-MCT constitutive equation dis-cussed above enables comparisons to be made with tra-ditional continuum rheological modelling (section 2) forwhich the macroscopic stress tensor is the fundamentalquantity of interest. However, the formal ITT result (58)also enables calculation of the distorted structure fac-tor, S(k, t) = 1 + n

∫

dr (g(r, t)− 1) eik·r, which makespossible a comparison with the microstructure obtainedfrom approaches based on the pair Smoluchowski equa-tion (section 6.1). In particular, setting f = ∆ρ∗kρk ≡ρ∗kρk − 〈 ρ∗kρk〉 in Eq.(58) yields the formal result

S(k, t)=〈 ρ∗kρk〉+∫ t

−∞

dt′〈κ(t′) : σ e∫

t

t′dsΩ†(s)

− ∆ρ∗kρk 〉.

(73)

Mode-coupling projection operator steps analagous tothose leading to (66) yield the ITT-MCT expression forthe distorted structure factor

S(k, t) = Sk −∫ t

−∞

dt′∂Sk(t,t′)

∂t′Φ2

k(t,t′)(t, t′) (74)

where the transient density correlator is given by solu-tion of Eqs.(68)–(70) for given Sk and κ(t) and where an

31

isotropic term has been suppressed. Eq.(74) has the ap-pealing interpretation that flow-induced microstructuralchanges are built-up by integration of the affinely shiftedequilibrium structure factor over the entire flow history,weighted by the transient density correlator describingthe fading memory of the system. The temporally non-local character of Eq.(74) is in striking contrast to thelocal approximations based on the pair Smoluchowskiequation. The former consists of a history integral overa memory function which is itself determined by solutionof a nonlocal integro-differential equation (68) whereasthe latter are purely Markovian approximations.Within the pair Smoluchowski approach, in the ab-

sence of HI the stress tensor is exactly related to the dis-torted pair correlation function by Eq.(29). It is thereforeof interest to inquire whether a similar connection holdswithin the approximate ITT-MCT approach. It is a rel-atively simple exercise to show that, subject to a certaincontraint to be discussed below, Eqs.(66) and (74) areconnected by the relation [18]

σ(t) = −Π1− nkBT

∫

dk

16π3

kk

kc′k δSk(t), (75)

where δSk(t;κ) = Sk(t;κ) − Sk and Π is the equilib-rium osmotic pressure. For shear flow σxy(t) from (75)coincides with a result of Fredrickson and Larson [214]for sheared copolymers, reflecting the Gaussian statis-tics underlying both the field-theory approach of [214]and the ITT-MCT factorization approximations. Equa-tion (75) thus connects stresses to microstructural distor-tions, which build up over time via the affine stretchingof density fluctuations competing with structural rear-rangements encoded in Φk(t, t

′).For the off-diagonal stress tensor elements Eq.(75) con-

nects Eqs.(66) and (74) directly. For the diagonal ele-ments contibuting to the osmotic pressure Eq.(75) is alsovalid, providing that the following approximate ‘sum-rule’ is obeyed

∂Π

∂φ=

1

6π

∫

dk

(

∂ lnSk

∂ ln k

)(

∂ lnSk

∂ lnφ

)

. (76)

Although it is not at all obvious that the above relationshould hold, numerical calculations for hard spheres (us-ing e.g. the Percus-Yevick approximation for Sk [16])show that it represents a rather good approximation.It should be emphasized that the application of projec-tion operator approximations to (65) and (73) yield ap-proximate expressions for the stress and structure factor,respectively, which are not neccessarily self-consistent,in the sense that integration of the approximate S(k, t)leads to the approximate σ(t). The fact that the ITT-MCT S(k, t) is almost consistent with the direct ITT-MCT approximation to the stress is a testament to theunderlying robustness of the method.Although the caging mechanism is expected to be most

important for statepoints close to the glass transition,ITT-MCT calculations of S(k) for hard-spheres under

weak shear flow [193] suggest that the (truncated) diver-gence of the structural relaxation timescale at the pointof dynamical arrest (be it at the idealized glass transi-tion point, as predicted by MCT, or some higher vol-ume fraction [102]) remains relevant for volume fractionswell removed from the singularity and has a range ofinfluence which extends back to dense equilibrium fluidstates. More recently the MCT-ITT S(k) has been eval-uated numerically for two-dimensional hard-discs undershear at finite values of Pe [79], without invoking anyadditional isotropic approximations (see subsection 7.6).These calculations show only qualitative agreement withBrownian dynamics simulation results and overestimatethe magnitude of the distortion by around an order ofmagnitude, a failing which is attributed to the fact thatITT-MCT apparently underestimates the speeding up ofstructural relaxations induced by the shear flow. Thisis to be contrasted with pair Smoluchowski-based ap-proaches (e.g. [175, 190]) which underestimate the mag-nitude of the low-shear distortion for dense fluid states(φ ∼ 0.5) by around an order of magnitude. It there-fore appears that neither the pair Smoluchowski nor theITT-MCT approach can account adequately for the shearinduced distortion of the microstructure.

7.6. Applications

Explicit numerical solution of the ITT-MCT constitu-tive equation (Eqs.(68)–(70)) has been performed for aone-component system of hard spheres under steady flowsof various geometry [18]. However, the computational re-sources required to solve the anisotropic Eqs.(68)–(70) inthree-dimensions over many decades of time are dauntingand the efficient numerical algorithms required to reducethe computational load are still under development. Nev-ertheless, it is hoped that much of the essential physicsmay be captured by solving a simplified set of equationsin which the advected wavevectors are approximated byan isotropic average k(t, t′) → kis(t, t

′), where

k2is(t, t′) =

1

4π

∫

dΩ k2(t, t′). (77)

This technical approximation has been successfully ap-plied to the case of simple shear [58, 203] and enablesthe angular integrals entering Eqs.(68)–(70) to be per-formed analytically. For two-dimensional systems algo-rithms have been developed which enable accurate nu-merical solution of the ITT-MCT equations without ad-ditional isotropic approximation [79].In Fig.14 flow curves for hard-spheres resulting from

solution of Eqs.(68)–(70) subject to the isotropic approx-imation (77) are shown for both steady shear and steadyplanar extensional flow. For these two choices of flow thedefining velocity gradient tensors are given by

κs =

0 γ 00 0 00 0 0

κe =

γ 0 00 −γ 00 0 0

. (78)

32

-8 -6 -4 -2 0 2Log

10(Pe

0)

-4

-3

-2

-1

0

1L

og10

(σ)

(a)

(b)

(c)

-8 -6 -4 -2 0Log

10(Pe

0)

1.5

2

2.5

3

3.5

4

Tro

uton

rat

io

(a)

(b)

(c)

Planar

-8 -6 -4 -2 0Log

10(Pe

0)

1.5

2

2.5

3

Tro

uton

rat

io

(a)(b)

(c)

Uniaxial

FIG. 14: Flow curves from the microscopic ITT-MCT ap-proach of [18] for hard-spheres at three different volume frac-tions close to the glass transition φc. Full lines show thesteady shear stress, σ = σxy in units of kBT/(2R)3 undershear flow as a function of Pe. Broken lines show the stressdifference σ = σxx−σyy (related to the extensional viscos-ity) under planar extensional flow. Each curve is labelledaccording to the distance in volume fraction from the glasstransition, ∆φ = φ − φc. (a) and (b) are fluid states with∆φ = −10−4 and−10−3, respectively. State (c) is in the glass,∆φ =10−4, and exhibits a dynamic yield stress for both flowgeometries. The inset shows a possible realization of planarextensional flow. The lower two panels show the Trouton ra-tio σxx−σyy/σxy as a function of Pe for both uniaxial andplanar extensional flow.

Specifically, Fig.14 shows σxy under shear flow and ∆σ ≡σxx − σyy under planar extension for hard-spheres asa function of Pe, for various volume fractions aroundthe glass transition. The equilibrium structure factorsused as input for these calculations were provided by themonodisperse Percus-Yevick theory [16]. For extensionalflows it is natural to plot the stress difference ∆σ as afunction of Pe, as this is simply related to the exten-sional viscosity ηe = ∆σ/γ. Flow curves below the glasstransition show a regime of linear response, character-ized by a constant viscosity, followed by shear thinningas Pe is increased. On approaching φg from below thelinear regime moves to lower values of Pe and disappearsentirely on crossing the glass transition. The resultingplateau in the flow curves identifies a dynamical yieldstress (see subsection 5.4) for both of the considered flowgeometries.

In the lower panels of Fig.14 the Trouton ratio (σxx −σyy)/σxy [30] is shown as a function of Pe for both pla-nar and uniaxial extensional flow [18]. For fluid statesin the linear regime Trouton’s rules assert that the ratio

of extensional to shear viscosity ηe/ηs takes the values4 and 3 for planar and uniaxial extension, respectively.These characteristic ratios arise from purely geometricalconsiderations and emerge naturally from the ITT-MCTapproach as the Pe → 0 limiting values of the Troutonratio. For glassy states (curves labelled (c) in Fig.14)the structural relaxation time diverges and the linear re-sponse regime vanishes. As a consequence, the classicalTrouton ratios are not recovered for glassy states in thelimit Pe → 0 and non-trivial values dictated by the dy-namical yield stress may be identified. The results for ex-tensional flow presented in [18] alongside those for simpleshear [140] thus provide the first steps towards the predic-tion from first-principles theory of a dynamic yield stresssurface for glasses [218]. Calculation of the yield surfacefrom a simplified schematic version of Eqs.(68)–(70) willbe discussed in subsection 7.8 below.Going beyond steady flow, in [65] experiments on

PMMA colloidal dispersions, molecular dynamics simu-lation and the ITT-MCT approach of [17] were combinedto study the evolution of stresses during start-up shearflow for high volume fraction fluids close to glassy ar-rest. The sudden onset of a steady shear flow leads tothe build-up of stresses in the systems as a function ofthe accumulated strain γ ≡ γt. For small values of γthe response is elastic (as described by Eq.(72)), whereasfor large strains system enters steady state viscous flowwith a stress independent of γ. In between these two lim-its, typically at strains around 10%, the stress exhibitsa maximum as the local microstructure is broken up bythe external flow [145]. Although a stress ‘overshoot’ inresponse to start-up shear flow is a rather generic featureof the rheology of complex fluids, its microscopic originsremain poorly understood. A central novel aspect of [65]was thus to connect the stress overshoot to anomolousbehaviour in the mean-squared-displacement (‘superdif-fusion’) identified in both simulation and confocal mi-croscopy experiment. From the exact Eq.(49) it is clearthat the only way in which the shear stress can exhibita maximum is if the modulus under steady shear, Gss(t),becomes negative at long times. Fig.5 shows that Gss(t)from the ITT-MCT approach (employing approximation(77)) indeed predicts negative values at long times. Theinset to Fig.5 shows the same data as a function of strainand demonstrates that the negative region of Gss(t) oc-curs at around 10% strain, consistent with the positionof the stress overshoot.

7.7. Schematic model

The microscopic ITT-MCT constitutive equation out-lined in subsection 7.4 provides a route to first-principlesprediction of the rheological behaviour of arrested col-loidal states. However, the anisotropic, wavevector de-pendent expressions are rather intractable for three di-mensional flows, hindering both their practical use andinterpretation. In order to facilitate numerical calcula-

33

tions for flows of interest a simplified ‘schematic’ versionof the tensorial microscopic theory has very recently beenproposed [19]. Such schematic models have proved in-valuable in the analysis of mode-coupling theories andprovide a simpler set of equations which retain the es-sential mathematical structure of the microscopic theory[58, 103].Applying the isotropic approximation (77) to the mi-

croscopic ITT-MCT expression for the stress (66) enablesthe angular integrals to be performed explicitly, leadingto the simplified form

σ(t) =

∫ t

−∞

dt′[

− ∂

∂t′B(t, t′)

]

G(t, t′), (79)

where B is the Finger tensor and an explicit expressionfor G(t, t′) may be found in [19]. By disregarding allwavevector dependence the modulus can be expressed interms of a single-mode transient density correlator

G(t, t′) = vσΦ2(t, t′) (80)

where vσ = G(t, t) is a parameter measuring the strengthof stress fluctuations (typically taking values of the order100kBT/R

3 for hard-sphere-like colloids). A schematicequation-of-motion for Φ(t, t′) may be obtained by ne-glecting the wavevector dependence of the microscopicexpression (68), leading to

Φ(t, t′) + Γ

(

Φ(t, t′) +

∫ t

t′dsm(t, s, t′)Φ(s, t′)

)

= 0,

(81)

where the single decay rate Γ simply sets the time scaleand may thus be set to unity. Experience with the con-struction of schematic MCT models both in the quiescent[103] and steady shear cases [58] combined with analysisof the way in which strain enters the microscopic mem-ory (69) lead to the following schematic ansatz for thememory function

m(t, t′, t0) = h(t, t0)h(t, t′) [ ν1Φ(t, t

′) + ν2Φ2(t, t′) ].(82)

The parameters ν1 and ν2 represent in an unspecifiedway the role of Sq in the microscopic theory and, fol-lowing standard MCT practice, are given by v2 = 2 andv1 = 2(

√2 − 1) + ǫ/(

√2 − 1). The separation parame-

ter ǫ encodes the distance from the glass transition, withnegative values corresponding to fluid states and posi-tive values corresponding to glass states. Finally, theh−function is given by

h(t, t0) =γ2c

γ2c + ν(I1(t, t0)− 3) + (1− ν)(I2(t, t0)− 3)

,

(83)

where γc sets the strain scale (typically γc ≈ 10%) andν is a mixing parameter (0 < ν < 1). The invariants ofthe Finger tensor, I1 = TrB and I2 = TrB−1 incor-porate the fluidizing influence of flow into the memory

FIG. 15: The state of stress of a material under appliedforce can be represented by a point in the three dimensionalspace of principal stresses. The cylinder shown here is thesurface defined by the von Mises criterion (86). Stress stateslying within the cylinder are deformed by the applied stress,but do not yield, whereas states outside the cylinder exhibitplastic flow. Due to invariance along the ‘hydrostatic’ axiss1 = s2 = s3 (arising from incompressibility) the yield surfaceis usually viewed in the ‘deviatoric’ plane perpendicular tothis (the lower left representation in this figure).

function. Requiring that flow enter via the Finger tensoralone guarantees that the resulting schematic theory ismaterial objective (see subsection 2.3), consistent withthe fully microscopic theory.The schematic expression for the stress (79) is closely

related to the Lodge equation of continuum mechanics(see subsection 2.1). Integrating Eq.(79) by parts yields

σ(t) =

∫ t

−∞

dt′ B(t, t′)

(

∂

∂t′Φ2(t, t′)

)

. (84)

Comparison of this expression with Eq.(5) shows thatthe present schematic theory goes considerably beyondthe standard Lodge equation by incorporating memorywhich is both nonexponential and a function of two timearguments, reflecting the loss of time-translational invari-ance under time-dependent flow. Replacing the correla-tor with a simple exponential trivially recovers the Lodgeequation (5).The tensorial schematic theory given by Eqs.(79)–(83)

has been applied to predict flow curves under shear andextensional flow, normal stresses under shear and thetransient stress response to step strain. In all cases testedso far the predictions of the schematic model are in goodqualitative agreement with those available from the mi-croscopic theory using the isotropic approximation (77)in three-dimensions [17, 18, 58, 140] and exact numerical

34

-0.2 0 0.2

-0.2

0

0.2

S3

S2

S1

-0.25 -0.2 -0.150.05

0.1

0.15

0.2

Planar Extension

Uniaxial Extension

von Mises

ITT-MCT

FIG. 16: Left Panel: The dynamical yield surface from ITT-MCT [19] for a glass with ǫ ≈ 3(φ − φgt) = 10−3, where φgt

is the volume fraction at the hard-sphere glass transition, inthe space of principal stresses (s1, s2, s3) as viewed along thehydrostatic axis s1 = s2 = s3 (stress in units of kBT/(2R)3).The red points correspond to planar extensional flow and theblue points to uniaxial extensional flow. Right Panel: A closerview reveals that the surface is not a perfectly circular cylinder(as predicted by the von Mises criterion for static yielding)and that maximal deviation from circularity occurs at pointsof pure uniaxial extension. These deviations are connectedwith the existence of finite normal stress differences.

solution in two-dimensions [79]. The schematic theorypredicts a positive value for the first normal stress dif-ference N1 = σxx−σyy under shear flow in accord withmicroscopic ITT-MCT calculations. On the other hand,the second normal stress difference N2=σyy−σzz is fromEq.(79) identically zero, in disagreement with both an-alytical low Pe analysis of the pair Smoluchowski equa-tion [161] and colloidal experiments [215] (finite negativevalues are found). The disappearance of N2 from theschematic theory can be traced back to the isotropic ap-proximation leading to Eq.(79) which effectively kills offthis feature of the fully microscopic theory.

7.8. Yield stress surface

A striking feature of the theory developed in [19] is thatit permits direct calculation of a dynamic yield stress sur-face for glasses. Related static yield surfaces have beenempirically postulated and employed for over a centuryin the engineering community to study the yielding ofamorphous solids (see also subsection 5.4). The two clas-sical criteria for determining the onset of plastic yieldare due to Tresca [216] and von Mises [217]. The Trescacriterion asserts that a material will yield when the max-imum shear stress due to the deformation exceeds a crit-ical value. Recalling that an external force imposed ona material can be represented as a stress tensor whichcan be diagonalized to obtain values for the three prin-cipal stresses s1, s2 and s3, the Tresca criterion can becompactly stated in the form [218]

Max ( | s1 − s2|, |s2 − s3|, | s1 − s3| ) =√3 σy

ss , (85)

where σyss is the shear stress at yield under a simple shear

deformation. According to Eq.(85) knowledge of σyss is

thus sufficient to determine the mechanical stability of amaterial under an arbitrary applied force. Alternatively,the vonMises critierion requires that the distortion strainenergy exceeds a critical value at yield [218]

1

6

(

(s1 − s2)2 + (s2 − s3)

2 + (s1 − s3)2)

= (σyss)

2 . (86)

Both the Tresca and von Mises criteria have proven tobe in reasonable qualitative agreement with yield experi-ments on crystalline metals [219]. Two main assumptionsunderly Eqs.(85) and (86): (i) the microscopic rearrange-ments leading to plastic deformation do not lead to sig-nificant dilation of the material, (ii) that residual stressesarising from the deformation history of the sample do notinfluence the yielding (i.e. there is no Bauschinger effect).It is useful to interpret equation (86) geometrically in

the space of principal stresses, where it describes a sur-face separating elastically deformed states from states ofplastic flow. Equation (86) defines a circular cylinder

with axis along the line s1 = s2 = s3 and radius√2 σy

ss.The symmetry about this ‘hydrostatic’ axis is a geomet-rical reflection of the fact that the yield condition (86)is independent of hydrostatic pressure. The plane whichpasses through the origin and which lies perpendicularto the cylinder axis is the so-called deviatoric plane. Allyield stress surfaces which are independent of hydrostaticpressure may be projected without loss of informationonto the deviatoric plane. In the case of the von Misesand Tresca criteria this generates a circle (see Fig.15) anda hexagon, respectively.The results presented in [19] reveal intriguing connec-

tions between the static yielding discussed above and dy-namic yield, as determined from the Pe → 0 values of

FIG. 17: The yield stress surface in the τrθ (rotational shearstress) τrz (squeeze shear stress) plane, where the stresses havebeen scaled by the yield stress τc in simple shear. The blueand red squares show results for two different emulsions withdiffering values of τc (28Pa and 52Pa). The green squaresare data taken using a carbopol gel (τc = 70Pa). In this rep-resentation the von Mises criterion (86) becomes a circle andis indicated by the solid line. Reprinted by permission fromMacMillan Publishers Ltd: Nature Materials 9 115, copyright(2010).

35

the flow curves. Within the ITT-MCT approach, for anygiven steady flow field (e.g. shear, uniaxial extension)there exists for glassy states a finite stress tensor in thelimit of vanishing flow rate. This stress tensor at yieldmay be diagonalized to obtain three eigenvalues and plot-ted as a point in the cartesian space of principal stresses.By considering all possible nondegenerate flows (madepossible by a suitable parameterization of the velocitygradient tensor [19]) a closed locus of points may be con-structed in the space: The dynamic yield stress surface.In [19] this procedure was realized using the schematicmodel Eqs.(79)–(83).In Fig.16 we show the deviatoric projection of the dy-

namical yield stress surface of a colloidal glass, as pre-dicted by the schematic model. To a first approximationthe numerically calculated dynamical yield surface fromthe theory agrees well with the empirical von Mises crite-rion for static yield (86). However, closer inspection (seeFig.16, right panel) reveals that deviations at around thepercent level occur, with the maximal deviation locatedat points of pure uniaxial extension. A careful analysisof the schematic equations reveals that this fine struc-ture of the surface can be attributed to the existenceof a finite first normal stress difference [19]. Expandingthe schematic model result to first order in Ny

1 , the firstnormal stress difference at yield, provides an explicit ex-pression for the schematic model yield surface [19]

1

6

(

(s1−s2)2 + (s2−s3)

2 + (s1−s3)2)

= (σyss)

2

+1

12(Ny

1 )2 +

3(1−A2)

(3 +A2)3/2Ny

1 σyss, (87)

where 0<A< 1 parameterizes the geometry of the im-posed flow. Eq.(87) thus describes a noncircular cylinderwith a radius which varies according to the value of theparameter A, each value of which corresponds to a givenazimuthal angle about the hydrostatic axis. Higher orderterms in the expansion (87) exist (and can be explicitlycalculated) but remain numerically negligable due to thesmallness of N1/σ

yss.

Very recent experiments on yielding soft materials haveattempted to determine the shape of the yield surface[220]. The novel rheometer employed in [220] applies acombined squeeze and rotational shear flow to a materialsample loaded between two parallel plates. The claim isthat by independently varying the rotation and squeezerate it may become possible to explore the entire familyof flows in a way analogous to the mathematical param-eterization of the velocity gradient tensor employed in[19] to calculate the schematic model yield surface. How-ever, it remains uncertain whether the superposition oftwo shear flows (radial Poiseuille and tangential homoge-neous shear) is really sufficient to map the yield surface.Fig.17 shows experimental data taken using three differ-ent yielding materials: a Carbopol gel and two differentemulsions [220].The yield data are shown in the rotational shear stress

(τrθ), squeeze shear stress (τrz) plane. While the data

shown in Fig.17 are not inconsistent with the von Misescriteria, it may well be that the chosen superposition flowactually constrains the surface to be spherical, regardlessof the true form of the yield surface. It would certainlybe remarkable if the soft materials considered in [220]obey yield criteria developed for crystalline solids, de-spite the very different underlying microscopic plasticitymechanisms [221].Whether the microscopic theory (Eqs.(68)–(70)) in-

deed predicts a similar dynamic yield surface, as ex-pected, and its relationship to static yielding in glassymaterials remain important open problems. Neverthe-less, the results from the schematic model are promisingand represent a considerable step towards a microscopicderivation of material specific yield surfaces from firstprinciples.

8. OUTLOOK

In this review we have attempted to provide anoverview of the rheological phenomenology presented bycolloidal dispersions and to outline some of the leadingtheoretical approaches aiming to rationalize this. It isclear that much remains to be done and that exisitingtheories have met with only partial success in solving thecomplex many-body problem of driven, strongly inter-acting colloidal systems. We hope that both the presen-tation and choice of topics contained within the presentwork serve to emphasize the common ground between dif-ferent theories (continuum mechanics, pair Smoluchowskitreatments and Green-Kubo based approaches), as wellas to highlight where progress still needs to be made.One of the clear deficiencies of approximate closures

of the pair Smoluchowski equation is their apparent in-ability to describe, even qualitatively, the slow structuralrelaxation time present in colloidal dispersions at finitevolume fraction. This failing is inherent in the Marko-vian character of the approximate closures, which followsas a natural consequence of applying equilibrium statisit-ical mechanical relations to connect pair and triplet cor-relation functions. A clear challenge to future theo-ries which attempt to improve this situation is thus totackle directly the intinsic difference between nonequlib-rium and equlibrium by confronting the irreducible term(see Eq.46) containing the missing physics. Moreover,the existing numerical data in the literature is restricted,largely for technical reasons, to the low Pe limit. Whilethis enables interesting analysis of the zero-shear viscos-ity and leading order microstructural distortion, there isan absence of data for the nonlinear rheology as a func-tion of Pe.Despite the success of the ITT-MCT approach in de-

scribing the nonlinear response of states close to dynamicarrest, there remain gaps in the theoretical formulationwhich should be filled and many fundamental questionsare yet to be addressed. A notable omission is that thepresent formulation of the theory does not enable incor-

36

poration of hydrodynamic interactions. The influence ofhydrodynamics on the rheology of densely packed glassystates is largely unexplored and their incorporation intothe theory, even at a crude approximate level, wouldtherefore be of considerable interest. Moreover, due tothe recent nature of the ITT-MCT theory and the com-plexity of numerically solving the equations, many flowsof rheological interest remain to be explored in detail.Of particular importance is to assess the predictions ofthe theory for nonsteady flows where interesting inter-action effects between yielding and the time-dependentstrain field may be envisaged (e.g. large amplitude oscil-

latory shear, where higher harmonics will contribute tothe stress response).

Acknowledgments

I would particularly like to thank M. Fuchs, M.E.Cates, Th. Voigtmann and M. Kruger for many stim-ulating discussions. Support was provided by the SFBTR6 and the Swiss National Science Foundation.

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