Supercooled liquids under shear: Theory and simulation

Kunimasa Miyazaki and David R. ReichmanDepartment of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138, USA

Ryoichi YamamotoDepartment of Physics, Kyoto University, Kyoto 606-8502, Japan

and PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama, Japan(Received 26 January 2004; published 1 July 2004)

We analyze the behavior of supercooled fluids under shear both theoretically and numerically. Theoretically,we generalize the mode-coupling theory of supercooled fluids to systems under stationary shear flow. Ourstarting point is the set of generalized fluctuating hydrodynamic equations with a convection term. A nonlinearintegrodifferential equation for the intermediate scattering function is constructed. This theory is applied to atwo-dimensional colloidal suspension. The shear rate dependence of the intermediate scattering function andthe shear viscosity is analyzed. We have also performed extensive numerical simulations of a two-dimensionalbinary liquid with soft-core interactions near, but above, the glass transition temperature. Both theoretical andnumerical results show the following.(i) A drastic reduction of the structural relaxation time and the shearviscosity due to shear. Both the structural relaxation time and the viscosity decrease asg−n with an exponentnø1, whereg is the shear rate.(ii ) Almost isotropic dynamics regardless of the strength of the anisotropicshear flow.

DOI: 10.1103/PhysRevE.70.011501 PACS number(s): 64.70.Pf, 61.43.Fs, 05.70.Ln, 47.50.1d

I. INTRODUCTION

Complex fluids such as colloidal suspensions, polymersolutions, and granular fluids exhibit very diverse rheologicalbehavior [1,2]. Shear thinning is among the most well-known phenomena. Such behavior is predicted for simpleliquids as well[3,4], but the effect is too small to observe attemperatures well above the glass transition temperature. Forsupercooled liquids, however, the situation is different. Re-cently, strong shear thinning behavior was observed by ex-periments performed on soda-lime silica glasses above theglass transition temperature[5]. Yamamotoet al. have doneextensive computer simulations of a binary liquid with asoft-core interaction[6] near their glass-transition tempera-ture and found non-Newtonian behavior. The same behaviorwas found for other systems such as Lennard-Jones mixtures[7,8] and polymer melts[9], too. For all cases, the structuralrelaxation time and the shear viscosity decrease asg−n,whereg is the shear rate andn is an exponent which is lessthan but close to 1. For such systems driven far from equi-librium, the parameterg is not a small perturbation but playsa role similar to an intensive parameter which characterizesthe “thermodynamic state” of the system[10]. Such rheo-logical behavior is interesting in its own right, but under-standing the dynamics of supercooled liquids in a nonequi-librium state is more important because it has possibilities toshed light on another typical and perhaps more importantnonequilibrium problem, namely, that of nonstationary ag-ing. Aging is characterized by slow relaxation after a suddenquench of temperature below the glass transition tempera-ture. In this case, the waiting time plays a similar role to(theinverse of) the shear rate. Aging behavior has been exten-sively studied in spin glasses(see Ref.[11], and referencestherein). Aging is also observed in structural glasses[12].There have been recent attempts to study the aging of struc-

tural glasses theoretically[13] but no analysis and compari-son to experiments or simulation results[12] have been pre-sented due to the complicated nonstationary nature of theproblem.

The relationship between aging and a driven, steady-statesystem was considered using a schematic model based on theexactly solvablep-spin spin glass by Berthier, Barrat, andKurchan [14]. This theory naturally gives rise to effectivetemperatures. The validity of their idea was tested numeri-cally for supercooled liquids[7]. There have also been at-tempts to observe aging by exerting shear on the systeminstead of quenching the temperature[15]. Recently, therehave been attempts to develop the mode-coupling theory forthe sheared glasses[16,17]. Fuchs and Cates have developedthe mode-coupling theory for the sheared colloidal suspen-sions using projection operator techniques[17]. They haveanalyzed a closed equation for the correlation function for aschematic model where the shear is exerted on all directionequally (“the isotropically sheared hard sphere model”). Intheir model, shear is turned on at the initial time and there-fore the dynamics are genuinely nonstationary.

In this paper, we investigate the dynamics of supercooledfluids under shear both theoretically and numerically for arealistic system. We extend the standard mode-couplingtheory (MCT) for supercooled fluids and compare the solu-tions with the numerical simulation results. We mainly focuson the microscopic origin of the rheological behavior. Therelationship with the more generic aging problem will not bediscussed here. Since our goal is the investigation of shearthinning behavior, we have neglected violations of thefluctuation-dissipation theorem. We start with generalizedfluctuating hydrodynamic equations with a convection term.Using several approximations, we obtain a closed nonlinearequation for the intermediate scattering function for thesheared system. The theory is applicable to both normal liq-

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uids and colloidal suspensions in the absence of hydrody-namic interactions. Similar approach has been applied to theself-diffusion of the hard sphere colloidal suspension at rela-tively low densities by Indraniet al. [18]. Extensive com-puter simulations are implemented for a two-dimensional bi-nary liquid interacting with soft-core interactions. The effectsof shear on microscopic structure, the structural relaxationtime, and rheological behavior are discussed. Results bothfrom the theory and the simulation show good qualitativeagreement despite the differences between the systems con-sidered. Special attention is paid to the directional depen-dence of the structural relaxation.

The paper is organized as follows. In the next section, wedevelop the MCT for sheared suspensions. Complexitieswhich do not exist in mean-field spin glass models and howthose complexities should be treated are elucidated here. Apossible way to explore the situation without invoking thefluctuation-dissipation theorem is also discussed. In Sec. III,the model and our simulation method are explained. Theresults both from theory and simulation are discussed in Sec.IV. In Sec. V we conclude.

II. MODE-COUPLING THEORY

We shall consider a two-dimensional colloidal suspensionunder a stationary shear flow given by

v0sr d = G · r = sgy,0d, s2.1d

whereg is the shear rate andsGdab= gdaxdby is the velocitygradient matrix. Generalization to higher dimensions istrivial. We start with the generalized fluctuating hydrody-namic equations[19,20]. This is a natural generalization ofthe fluctuating hydrodynamics developed by Landau and Lif-shitz [21] to short wavelengths where the intermolecular cor-relations become important. Fluctuating fields for the num-ber densityrsr ,td and the velocityvsr ,td of the colloidalsuspension obey the following set of nonlinear Langevinequations:

] r

] t= − = · srvd,

m] srvd

] t+ m= · srv ·vd = − r ¹

dFdr

− z0rsv − v0d + fR,

s2.2d

wherem is the mass of a single colloidal particle,F is thefree energy of the system, andz0 is a bare collective frictioncoefficient for colloidal particles.z0 has a weak density anddistance dependence due to hydrodynamic interactions[22]but we shall neglect these effects. The friction term is spe-cific for the colloidal case. In the case of liquids, it should bereplaced by a stress term which is proportional to the gradi-ent of the velocity field multiplied by the shear viscosity.Both cases, however, lead to the same dynamical behavior onthe long time scales which are of interest here.fRsr ,td is therandom force which satisfies the fluctuation-dissipation theo-rem (FDT) of the second kind[23,50]

kfR,isr ,tdfR,jsr 8,t8dl0 = 2kBTrsr ,tdz0di jdsr − r 8ddst − t8ds2.3d

for tù t8, where k¯l0 is an average over the conditionalprobability for a fixed value ofrsr ,td at t= t8. Note that therandom force depends on the density and thus the noise ismultiplicative. This fact makes a mode-coupling analysismore involved as we discuss later in this section. We assumethat the second FDT holds even in nonequilibrium state sincethe correlation of the random forces are short ranged andshort lived, and thus the effect of the shear is expected to benegligible. The first term on the right-hand side of the equa-tion for the momentum is the osmotic pressure term. Here weassume that the free energyF is well approximated by thatof the equilibrium form and is given by the well-known ex-pression

bF .E dr rsr dhln rsr d/r0 − 1j

−1

2E dr 1E dr2 drsr 1dcsr12ddrsr 2d, s2.4d

where b=1/kBT, r0 is the average density, andcsrd is thedirect correlation function. We have neglected correlations ofmore than three points, such as the triplet correlation func-tion c3sr 1,r 2,r 3d, whose effect becomes important for thefluids with stronger directional interactions such as silica[24]. Under shear, it is expected thatcsrd will be distortedand should be replaced by a nonequilibrium steady stateform cNEsr d, which is an anisotropic function ofr . It is,however, natural to expect that this distortion is small on themolecular length scales which play the most important rolein the slowing down of structural relaxation near the glasstransition. The distortion of the structure under shear is givenup to linear order in the shear rate by[25]

SNEskd = SskdH1 +k · G · k

2kD0

dSskddk

J , s2.5d

where Sskd and SNEskd is the static structure factor in theabsence and in the presence of the shear, respectively.D0=kBT/z0 is the diffusion coefficient in the dilute limitk

;uk u and k ;k / uk u. The direct correlation function in theFourier representation ofcsrd is related to the structure factorby ncskd=1−1/Sskd. From Eq.(2.5), we find that the distor-tion due to the shear is characterized by the Péclet numberdefined by Pe=gs2/D0, wheres is the diameter of the par-ticle. Hereafter we shall neglect the distortion and use thedirect correlation function at equilibrium, assuming thePéclet number is very small.

We linearize Eq.(2.2) around the stationary state asr=r0+dr and v=v0+dv, where r0 is the average density.Transforming to wave vector space, we obtain the followingequations:

S ]

] t− k · G ·

]

] kDdrkstd =

ik

mJkstd,

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S ]

] t− k · G

]

] k+ k · G · kDJkstd = −

ik

bSskddrkstd

−1

mbE

qi k ·qcsqddrk−qstddrqstd −

z0

mJkstd + fRkstd,

s2.6d

where Jkstd=mr0k ·dvkstd is the longitudinal momentumfluctuation, andeq;edq / s2pd2. We have neglected the qua-dratic terms proportional toJqJk−q. Note that Eq.(2.6) doesnot contain coupling to transverse momentum fluctuationseven in the presence of shear.

The direct numerical integration of Eq.(2.6) is in prin-ciple possible but it is expensive and not theoretically en-lightening [26]. Rather we shall construct the approximatedclosure for the correlation functions, a so-called mode-coupling approximation[27–29]. There are several ap-proaches to derive mode-coupling equations, including theuse of the Mori-Zwanzig projection operator and a decou-pling approximation[27], or implementation of a loop ex-pansion developed in the context of the equilibrium criticalphenomena and generalized to the dynamic case[29,30].Both approaches lead to essentially the same equations if thesystem is at equilibrium. Under nonequilibrium conditions,however, the loop expansion approach is more straightfor-ward and flexible. We shall adopt the loop expansion ap-proach to the nonlinear Langevin equation with both multi-plicative noise and the full convection term. Equation(2.6)can be cast to a general form of the nonlinear Langevin equa-tion written as

dxi

dt= mi j xj +

1

2Vi jkxjxk + fR,i , s2.7d

wherexstd=fdrkstd ,Jkstdg is a field variable,mi j is the linearcoefficient matrix, and the nonlinear coupling coefficientVi jkis the vertex tensor which satisfies the symmetric relationVi jk =Vikj. Finally, fR,istd is the random force field which sat-isfies the second FDT,

kfR,istdfR,jst8dl0 = kBLijsxddst − t8d, s2.8d

whereLijsxd is thex-dependent Onsager coefficient which isto be expanded to the lowest order as

Lijsxd = Lijs0d + Lij ,k

s1d xk, s2.9d

where Lij ,ks1d ;]Lijsxd /]xkux=0. Comparing Eq.(2.7) with Eq.

(2.6), the elements of the linear coefficient matrix are givenby

mrkrk8= k · G ·

]

] kdk,k8,

mrkJk8=

ik

mdk,k8,

mJkrk8=

ik

bSskddk,k8,

mJkJk8= Sk · G ·

]

] k− k · G · kDdk,k8 −

z0

mdk,k8.

s2.10d

Nonzero elements of the vertex tensor are given by

VJkrk8rk9= −

1

bVhi k ·k8csk8d + i k ·k9csk9djdk,k8+k9,

s2.11d

whereV is the volume of the system. Finally,

LJkJk8

s0d = z0r0TVdk,−k8

LJkJk8,rk9

s1d = Tz0dk+k8,k9. s2.12d

All other components are zero. In the above expressions,

dk,k8 ;s2pd2

Vdsk − k8d

is the Dirac delta function. We construct the closure equationfor the correlation functionCijst ,t8d=kxistdxjst8dl. Since weare treating the stationary state, the time translation invari-ance holds and, thus,Cijst ,t8d=Cijst− t8d. A general schemefor the loop expansion method for the Langevin equationwith multiplicative noise has been discussed by Phythian[31]. Up to one loop, the equation for the correlation func-tion is written as[32]

dCijst − t8ddt

− miaCa jst − t8d − 2kBLias0dGa j

† st − t8d

=E−`

t

dt1oiast − t1dCa jst1 − t8d +E

−`

t8dt1 Diast − t1d

3Ga j† st1 − t8d

dGijst − t8ddt

− miaGa jst − t8d

= di jdst − t8d +Et8

t

dt1 oiast − t1dGa jst1 − t8d s2.13d

with the memory kernels defined by

oi jstd = ViabGalstdCbmstdVlm j + kBViabGalstdGbmstdLlm,j

s1d ,

Dijstd =1

2ViabCalstdCbmstdV jlm + 2kBViabGalstdCbmstdLjl,m

s1d

+ 2kBLia,bs1d Gal

† stdCbmstdV jl,m. s2.14d

In these expressions, we have introduced the propagator de-fined by

Gijst − t8d =K dxistddfR,jst8d

L . s2.15d

Gij†st− t8d;Gjist8− td is the conjugate of the propagator.

Equation(2.13) together with Eq.(2.14) are so-called mode-coupling equations.

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Without further assumption, one needs to solve thecoupled equation for the correlation function and the propa-gator. The propagator can be eliminated, however, if thefluctuation-dissipation theorem of the first kind(first FDT) isvalid. The first FDT relates the correlation function to theresponse functionxi jstd via the equation

xi jstd = −ustdkBT

dCijstddt

, s2.16d

whereustd is the heaviside function. Note that the responsefunction xi jstd is generally neither the same as, nor propor-

tional to, the propagatorGijstd. xi jstd represents the responseof the system to the perturbation via external fields or

through the boundary, whereasGijstd represents the responseto a random force field. If the Langevin equation is linear,both are the same, but this is not true in general for thenonlinear Langevin equation. To one loop, to be consistentwith the mode-coupling approximation, the response func-tion is written as[32,33]

xi jst − t8d =1

TGikst − t8dhM s0d + L s0djkj +

1

TE dt1 Gikst − t1d

3VkabGalst1 − t8dCbmst1 − t8dhM s1d + L s1djl j ,m,

s2.17d

where the tensorsMijs0d and Mij ,k

s1d are defined byMijsxd.Mij

s0d+Mij ,ks1d xk. Mijsxd is the matrix which is defined by the

reversible part of the nonlinear Langevin equation. In gen-eral, we can express the Langevin equation as

dxi

dt= visxd + Mijsxd

dS

dxj+ Lijsxd

dS

dxj+ fR,i , s2.18d

whereS is the entropy of the whole system andvisxd is aterm which originates from the nonequilibrium constraintsuch as the convection for the sheared system. From thisdefinition, the reversible term does not contribute to the en-tropy production and thus the matrixMijsxd is antisymmet-ric. For the system considered here, the nonzero elements aregiven by

MrkJk8= − MJk8rk

= − ikTsr0Vdk,−k8 + drk+k8d. s2.19d

If the system is in equilibrium, one may eliminate thepropagator in favor of the correlation function in Eq.(2.13)by using the first FDT, Eq.(2.16) [32,34,35]. If the system isout of equilibrium, one has to solve the coupled equations(2.13). For the case of thep-spin-glass model with an exter-nal drive, Berthieret al. have analyzed the equation similarEq. to(2.13) [14]. They have found that there is a systematicdeviation from the first FDT which is reminiscent of viola-tions of FDT that occur during aging. Extensive computersimulation supports these results[7]. A similar analysis forreal fluids is necessary but it is inevitably more involved dueto the complicated tensorial nature of the nonlinear coupling.In this paper, we shall not focus on the fundamental problemof the validity of the fluctuation-dissipation theorem and as-sume that the first FDT is valid. We shall show that even

with this simplification, the theory can explain qualitativeseveral aspects of the dynamical behavior very well. Using

the first FDT, as in equilibrium case, one can eliminateGijstdfrom Eq. (2.13) and the final expression is given by[32]

dCijst − t8ddt

− mikCkjst − t8d − 2kBLiks0dGkj

† st − t8d

=Et8

t

dt1 Mikst − t1dCkjst1 − t8d s2.20d

with the memory kernel given by

Mijstd = −1

2ViabCalstdCbmstdhVklm − 2LklmjCkj

−1s0d.

s2.21d

These equations are almost identical to the conventionalmode-coupling equations[34,35] except fors−2Lklmd termwhich appears in the vertex term of the memory kernel. Thethird term on the left-hand side of Eq.(2.20) enters naturallyto guarantee the time-reversal symmetry of the correlationfunction.Li jk is defined by

Li jk ; Lias0dSa jk

s3d + Lia,js1d Sak

s2d + Lia,ks1d Sa j

s2d, s2.22d

where we have definedSijs2d andSijk

s3d by

Sijs2d ; U ]2S

] xi ] xjU

x=0and Sijk

s3d ; U ]3S

] xi ] xj ] xkU

x=0.

s2.23d

The terms−2Lklmd is a new term which originates from themultiplicative noise. Since the density- and velocity-dependent part of the entropy is given by

S=E drJ2sr d

2Trsr d+

1

TF, s2.24d

Sijs2d andSijk

s3d are given by

SJkJk8

s2d = −1

mr0TVdk,−k8,

Srkrk8

s2d = −kB

r0SskdVdk,−k8,

Srkrk8rk9

s3d =kB

r02V2dk+k8+k9,0,

SJkJk8rk9

s3d =1

mr02TV2dk+k8+k9,0. s2.25d

Combining this with Eq.(2.12), we find thatLi jk =0 for allelements. Therefore, the final expression for the mode-coupling equation in the present case becomes equivalentwith the conventional one[27]. Note that the situation willchange if we consider different physical situation: For ex-ample, if we start with the diffusion equation, which is ob-tained in the overdamped limit of Eq.(2.6), Li jk plays anessential role[36].

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Now let us substitute all matrix elements given by Eqs.(2.10) and(2.11) to Eq. (2.20), we have the set of equationsgiven by

S ]

] t− k · G ·

]

] kDCrkrk8

std = − ikCJkrk8std,

S ]

] t− k · G ·

]

] k− k · G · kDCJkrk8

std

= −ik

mbSskdCrkrk8

std −z0

mCJkrk8

std

−1

mE

0

t

dt1ok9

dzsk,k9,t − t1dCJk9rk8st1d,

s2.26d

where

ok

;V

s2pd2 E dk . s2.27d

The memory kerneldzsk ,k8 ,td is given by

dzsk,k8,td =m

2 oq1

oq2

op1

op2

ok9

VJkrq1rq2

Crq1rp1

stdCrq2rp2

std

3 VJk9rp1rp2

CJk9Jk8

−1 s0d

=b

2nVoq

op

VJkrqrk−qCrqrp

stdCrk−qr−k8−pstd

3 VJ−k8rpr−k8−p=

1

2n2NbE

qE

pVksq,k

− qdCrqr−pstdCrk−qr−k8+p

stdVk8sp,k8 − pd,

s2.28d

whereN is the total number of the particles in the system andwe have defined the vertex function

Vksq,k − qd = k ·qncsqd + k · sk − qdncsuk − qud.

s2.29d

In the derivation of Eq.(2.28), we have used the momentumequal-time correlation function given by

CJkJk8s0d =

mnV

bdk,−k8. s2.30d

If the system is in equilibrium, time and space translationalinvariance is satisfied and the correlation function becomesCasr dbsr8dstd=Casr−r8dbs0dstd or, in k space, Cakbk8

std=Cakb−k

std3dk,−k8. In the presence of shear, however, thisshould be modified as[37]

Casr dbsr8dstd = Casr−r8stddbs0dstd, s2.31d

where we defined the time-dependent position vector by

r std ; expfGtg · r = r + gtyex, s2.32d

whereex is an unit vector oriented along thex axis. In wavevector space, this is expressed as

Cakbk8std = Cakstdb−k

std 3 dkstd,−k8 = Cak8stdbk8std 3 dk,−k8s−td

s2.33d

with the time-dependent wave vector defined by

kstd = expftGtg ·k = k + gtkxey, s2.34d

wheretG denotes the transpose ofG. Figure 1(b) shows howthe shear flow advects a positional vectorr by Eq. (2.32) ina time interval of durationt. The corresponding time-dependent wave vector,(2.34), is shown in Fig. 1(c). Equa-tions(2.31) and(2.33) state that the fluctuations satisfy trans-lational invariance in a reference frame flowing with theshear contours. Using this property, Eq.(2.28) becomes

dzsk,k8,td =1

2r02Nb

EqE

pVksq,k − qdCrqr−qstd

3stdCrk−qr−kstd+qstdstdVk„td„pstd,kstd

− qstd…dp,qstddk8,kstd =1

2r02NVb

Eq

Vksq,k

− qdCrqr−qstdstdCrk−qr−kstd+qstd

stdVkstd„pstd,kstd

− qstd…dk8,kstd ; dzsk,td 3 dk8,kstd. s2.35d

Introducing the intermediate scattering function by

Fsk,td ;1

NCrks−tdr−k

std =1

Nkdrks−tdstddr−ks0dl,

s2.36d

dzsk ,td can be rewritten as

dzsk,td =1

2r0bE

qVksq,k − qdF„qstd,t…F„kstd − qstd,t…

3 Vkstd„pstd,kstd − qstd…, s2.37d

This is the memory kernel in the presence of the shear. Thishas exactly the same structure as the equilibrium one[27]except for the time dependence appearing on the wave vec-tors. The physical interpretation of Eq.(2.37) is simple: Thememory kernel has the general structure of two correlationfunctions sandwiched by two vertex functions. This means

FIG. 1. (a) Geometry of shear flow.(b) Shear advection in realspace.(c) Shear advection in Fourier space.

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that the interactions of two fluctuations with modesq andk −q scatter at a certain time and then propagate freely in a“mean field” and after a timet, they recollide and interact.Under shear, however, by the time the second interactionstake place, the fluctuations are streamed away by the flow.

We also define the cross correlation function

Hsk,td ;1

NCJks−tdr−k

std =1

NkJks−tdstddrk

* s0dl. s2.38d

Then, Eq.(2.26) can be rewritten as

dFsk,tddt

= − iks− tdHsk,td,

dHsk,tddt

= − ks− td · G · ks− tdHsk,td −iks− td

mbS„ks− td…Fsk,td

−z0

mHsk,td −

1

mE

0

t

dt8 dz„ks− td,t − t8…Hsk,t8d.

s2.39d

Note that in the above equation, the differential operatork ·G ·] /]k disappears because

dFsk,tddt

=] Fsk,td

] t− ks− td · G ·

]

] ks− tdFsk,td.

s2.40d

The memory kernel Eq.(2.37) appearing in Eq.(2.39) con-tains the nonlinear coupling with the correlation function it-self and therefore the equation should be solved self-consistently. Indrani and Ramaswamy have derived anequation similar to Eq.(2.39) for the velocity correlations ofa single tagged particle in a three-dimensional hard spherecolloidal suspension[18]. They were interested in the rela-tively low density regime, and did not solve the resultingequation self-consistently.

For colloidal suspensions the relaxation time of the mo-mentum fluctuations is of the order oftm=m/z0 and is muchshorter than the relaxation time for density fluctuationswhich is of the order of or longer thantd=s2/D0. In otherwords, we may invoke the overdamped limit. Thus, we ne-glect the inertial termdHsk ,td /dt. Likewise the first term ofthe right-hand side in the second equation of Eq.(2.39) isestimated to be of order ofgtm and thus should be very smallas long as the Péclet number is small. Thus, the equation forthe momentum fluctuations may be written as

0 = −iks− td

mbS„ks− td…Fsk,td −

z0

mH„k,td −

1

mE

0

t

dt8 dz„ks− t…,t

− t8…Hsk,t8d. s2.41d

Substituting this back into the first equation of Eq.(2.39), wearrive at the equation for the intermediate scattering function

dFsk,tddt

= −D0ks− td2

S„ks− t…dFsk,td −E

0

t

dt8 M„ks− td,t

− t8…dFsk,t8d

dt8, s2.42d

where the memory kernel is given by

Msk,td =D0

2r0

k

kstdEqVksq,k − qdVkstd„qstd,kstd − qstd…

3 F„kstd − qstd,t…F„qstd,t…. s2.43d

Equations(2.42) and (2.43) are the final mode-coupling ex-pressions. We have started from the equation for both thedensity and the momentum fields and proceeded via the stan-dard mode-coupling approach, taking the overdamped limitat the end. It should be possible to derive the same resultstarting from the diffusion equation with an interaction termwhich is obtained by taking the overdamped limit in Eq.(2.6). In order to arrive at the same result, however, oneneeds to use a different resummation scheme that involvesthe irreducible projection operator[38].

Equation(2.42) is a nonlinear integrodifferential equationthat can be solved numerically. An efficient numerical rou-tine to solve the mode-coupling equation is elucidated inRef. [39]. Since our equation is not isotropic ink due to thepresence of shear, the numerics are more involved than in theequilibrium case. We shall solve the equation by dividing thelower half plane into anNk3 sNk−1d /2 grid, whereNk is thegrid number which we have chosen to be an odd number. Wedo not need to consider the upper half plane because it is amirror image of the lower one.kx, for example, is discretizedaskx,0=−kc,kx,1=−kc+dk, . . . ,kx,Nk

=−kc+Nkdk=kc, wheredk

=2kc/Nk is the grid size andkc is a cutoff wave vector. Anyvalue outside the boundaryukxu , ukyu.kc is replaced by thevalue at the boundary.

III. SIMULATION METHOD

To prevent crystallization and obtain stable amorphousstates via molecular dynamics(MD) simulations, we choosea model two-dimensional system composed of two differentparticle species 1 and 2, which interact via the soft-core po-tential

vabsrd = essab/rd12 s3.1d

with sab=ssa+sbd /2, wherer is the distance between twoparticles anda,b denote particle speciessP1,2d. We take themass ratio to bem2/m1=2, the size ratio to bes2/s1=1.4,and the number of particlesN=N1+N2, N1=N2=5000. Simu-lations are performed in the presence and absence of shearflow keeping the particle density and the temperature fixed atn=n1+n2=0.8/s1

2 (n1=N1/V, n2=N2/V) and kBT=0.526e,respectively. Space and time are measured in units ofs1 andt0=sm1s1

2/ed1/2. The size of the unit cell isL=118s1. In theabsence of shear, we impose microcanonical conditions andintegrate Newton’s equations of motion

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dr ia

dt=

pia

ma,

dpia

dt= f i

a s3.2d

after very long equilibration periods so that no appreciableaging(slow equilibration) effect is detected in various quan-tities such as the pressure or in various time correlation func-tions. Here,r i

a=srxia ,ryi

a d andpia=spxi

a ,pyia d denote the position

and the momentum of theith particle of the speciesa, andf ia

is the force acting on theith particle of speciesa. In thepresence of shear, by defining the momentumpi8

a=pia

−magryia ex (the momentum deviations relative to mean Cou-

ette flow), and using the Lee-Edwards boundary condition,we integrate the so-called SLLOD equations of motion sothat the temperaturekBTf;N−1oaoispi8

ad2/mag is kept at adesired value using a Gaussian constraint thermostat toeliminate viscous heating effects[40]. The system remains atrest fort,0 for a long equilibration time and is then shearedfor tù0. Data for analysis have been taken and accumulatedin steady states which can be realized after transient waitingperiods.

We shall calculate the incoherent and the coherent parts ofthe scattering function for the binary mixture by using thedefinitions[41]

Fssk,td =1

NaKo

i=1

Na

ef−ihks−td·r iastd−k·r i

as0djgL s3.3d

and

Fabsk,td =1

NKoi=1

Na

ef−iks−td·r iastdgo

j=1

Nb

efik·r jbs0dgL , s3.4d

respectively, witha,bP1,2. Thea relaxation timeta of thepresent mixture, which is defined by

F11sk0,tad . Fssk0,tad = e−1, s3.5d

is equal tota.1800 time units in the quiescent state foruk0u=2p /s1.

IV. RESULTS

A. Microscopic structure

The partial static structure factorsSabskd are defined as

Sabskd =E dr eik·rknasr dnbs0dl, s4.1d

where

nasr d = oj

Na

dsr − r jad sa = 1,2d s4.2d

is the local number density of the speciesa. Note the dimen-sionless wave vectork is measured in units ofs1

−1. For abinary mixture there are three combinations of partial struc-ture factorsS11skd, S22skd, and S12skd. They are plotted inFig. 2(a) in the quiescent state after taking an angular aver-age overk. A density variable representing the degree ofparticle packing, corresponding to the density of an effective

one component system, can be defined for the present binarysystem by

reffsr d = s12n1sr d + s2

2n2sr d. s4.3d

The corresponding dimensionless structure factor is given by

Srrskd = s1−4E dr eik·rkdreffsr ddreffs0dl = n1

2S11skd

+ n22ss2/s1d4S22skd + 2n1n2ss2/s1d2S12skd,

s4.4d

where dreff= reff−kreffl. One can see from Fig. 2(b) thatSrrskd has a pronounced peak atk.5.8 and becomes verysmall s,0.01d at smallerk demonstrating that our system ishighly incompressible at long wavelengths. BecauseSrrskdbehaves quite similarly toSskd of one component systems,we examine space-time correlations inreffsr d rather thanthose in the partial number densitynasr d for the present bi-nary system. The usage ofreffsr d makes comparisons of oursimulation data with the mode-coupling theory developed fora one component system more meaningful.

We next examine the anisotropy in the static structurefactorSrrskd in the presence of shear flow. Figures 3–6 showSrrskd plotted on a two-dimensionalkx-ky plane(upper part)and the angular averaged curves(lower part) within the re-gions (a)–(d) obtained atg=10−4, 10−3, 10−2, and 10−1, re-spectively(in units oft0

−1). One sees that, at the lowest shearratesg=10−4d, the shear distortion is negligible but at highershear rate the distortion becomes prominent. Atg=10−3, atall regions except for the region(d), the peak heights ofSrrskd start decreasing. This is contrary to the estimate fromthe linear response theory given by Eq.(2.5) which predictsno distortion in the regions(a) and (c) but distortion to thehigher (b) and lower(d) peaks. This seems to indicate thatthe nonlinear effects due to the shear become important.Ronis has explored the higher shear regime for the structurefactor of hard sphere colloidal suspensions and concluded

FIG. 2. Partial structure factorsSabskd in (a) and Srrskd in (b)defined by Eq.(4.4) for the present binary mixture.

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that at higher shear, the peak should always be lower than theequilibrium value together with a shift that depends on thedirection [25]. Figure 6 shows that peaks in all directionshave been lowered and the peak with maximal distortion[region (b)] is shifted to the lower wave vectors while theopposite is true for the shift of the region with minimal dis-tortion [region (d)]. The qualitative agreement with Ronis’theory is good but it is not clear that our results can beexplained by a simple two-body theory such as that of Ronis.Recently Szamel has analyzedSskd for hard sphere colloidalsuspensions up to the linear order ing [42]. He took thethree-body correlations into account and found quantitativeagreement with the shear viscosity evaluated usingSskd.

It should be noted that because our system is a liquid, wecannot directly refer to the Péclet number since the bare dif-fusion coefficientD0 does not exist. Therefore, a direct andquantitative comparison with the theories discussed above isnot possible. However, estimates from the relaxation timesself-diffusion coefficient evaluated in Ref.[6] allow us toestimate a Péclet number in the range between 10−1 and 102,which corresponds to the highest shear rates explored in thetheoretical analysis of this paper.

B. Intermediate scattering function: Numerical results

Here, we examine the dynamics of the local density vari-able reffsr ,td. To this end, we defined the intermediate scat-tering function

Frrsk,td = n12F11sk,td + n2

2ss2/s1d4F22sk,td

+ 2n1n2ss2/s1d2F12sk,td s4.5d

by taking a linear combination of the partial scattering func-tions defined in Eq.(3.4). Note that Frrsk ,0d=Srrskd bydefinition. To investigate anisotropy in the scattering func-tion Frrsk ,td, the wave vectork is taken in four differentdirectionsk10, k11, k01, andk−11, where

kmn =k

Îm2 + n2smex + neyd s4.6d

and m ,nP0,1 as shown in Fig. 7. The wave vectork (inreduced units) is taken to be 2.9, 5.8, and 10[see also Fig.2(b)]. Because we use the Lee-Edwards periodic boundarycondition, the available wave vectors in our simulationsshould be given by

k =2p

Lfnex,sm− nDxdeyg, s4.7d

wheren andm are integers,Dx=Lgt is the difference in thex coordinate between the top and bottom cells as depicted inFig. 6.5 of Ref.[40]. To suppress statistical errors, we sampleabout 80 available wave vectors aroundkmn and calculateFrrsk ,td using Eqs.(4.5) and (3.4), and then we averageFrrsk ,td over sampled wave vectors. The sampled wave vec-tors are shown as the spots in Fig. 7.

FIG. 3. Srrskd for g=10−4. The solid line isSrrskd at equilib-rium. Dots represent those observed in the region indicated in theskx,kyd plane above.

FIG. 4. Srrskd for g=10−3. All symbols are as in Fig. 3.

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Figures 8–10 showFrrsk ,td /Srrskd for k=2.9, 5.8, and10, respectively. We have observed that all of the partialscattering functionsF11sk ,td, F12sk ,td, andF22sk ,td behavein a similar manner asFrrsk ,td, which demonstrates that theeffective single component scattering function typifies thedynamics of the whole system. Several features are notice-able. First, the quantitative trends as a function ofk are simi-lar for different values ofg. Secondly, shear drastically ac-celerates microscopic structural relaxation in the supercooledstate. The structural relaxation timeta decreases stronglywith increasing shear rate asta, g−n with n.1. Lastly, theacceleration in the dynamics due to shear occurs almost iso-tropically. We observed surprisingly small anisotropy in thescattering functions even under extremely strong sheargta.103. A similar isotropy in the tagged particle motions hasalready been reported in Ref.[6]. The observed isotropy ismore surprising than that observed in single particle quanti-ties. In particular, the fact that different particles labels arecorrelated in the collective quantity defined in Eq.(3.4)means that a simple transformation to a frame moving withthe shear flow cannot completely remove the directionalcharacter of the shear. Our results providepost factojustifi-cation for the isotropic approximation of Ref.[17]. This sim-plicity in the dynamics is quite different from behavior ofother complex fluids such as critical fluids or polymers,where the dynamics become noticeably anisotropic in thepresence of shear flow.

C. Intermediate scattering function: MCT results

We evaluateFsk,td for the two-dimensional colloidal sus-pension theoretically using Eq.(2.42) with Eq. (2.43). Fol-

lowing the procedure explained in Sec. II, we have solvedEq. (2.42) self-consistently. For the static correlation func-tion cskd andSskd, the analytic expressions derived by Bauset al.were used(see the Appendix) [43]. The number of gridpoints was chosen to beNk=55. We have also calculatedsolution of Eq. (2.42) for the larger grid sizes up toNk=101 but qualitative differences between results obtainedwith different grid numbers were not noticeable. We chosekcs=10p as the cutoff wave vector. The best estimate for the

FIG. 5. Srrskd for g=10−2. All symbols are as in Fig. 3. FIG. 6. Srrskd for g=10−1. All symbols are as in Fig. 3.

FIG. 7. Sampled wave vectors.

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transition density isfc=0.72574 forNk=670 at equilibrium,where f=ps2r0/4 is the volume fraction. ForNk=55, ahigher value,fc=0.76645 is obtained. Figures 11–13 showthe behavior ofFsk ,td renormalized by its initial valueSskdfor f=fc3 s1−10−4d for various shear rates from Pe510−10

to 10−1, where Pe is the Péclet number defined by Pe= gs2/D0. The wave vectors were chosen to beks=3.0 (Fig.11), ks=6.7 (Fig. 12), and ks=12.0 (Fig. 13). ks=6.7 isclose to the position of the first peak ofSskd. At each wavevector, we have observedFsk ,td /Sskd for four directions de-noted by(a)–(d) in Fig. 7 and defined by Eq.(4.6). For shearrates smaller than Pe=10−10 no effect of shear is observed.For Peù10−10, we observe a large reduction of relaxationtimes due to shear. We define the structural relaxation timeta as in Eq.(3.5). We find that, although the amplitudes ofrelaxation defer depending onk, the shear dependence of therelaxation time and its shear dependence is almost indepen-

dent ofk. The shear rate dependence of the relaxation time isgiven by tasgd, g−1, consistent with the simulation resultsdiscussed in the previous subsection. The four curves for afixed shear rate but for different wave vector directions ex-hibit almost perfect isotropy, in qualitative agreement withthe behavior observed in the simulations. Thus the apparentanisotropy of the vertex function in Eq.(2.43) provides vir-tually no anisotropic scattering. Forks=3, we see perfectlyisotropic scattering. Forks=6.7 and 12.0, the curves showsmall anisotropy that is still consistent with the simulationresults. Note that the differences shown in the plateau valuefor ks=12.0 is an artifact due to our use of a square gridwhich produces an error in the radial distances depending ondirection. The differences are noticeable but small in(b) and(d) for ks=12.0. In these calculations, we have used the

FIG. 8. Fsk ,td /Sskd at ks1=2.8 for various shear rates and atthe different observing points(a) k10, (b) k11, (c) k01, and(d) k−11

as explained in Fig. 7.

FIG. 9. Fsk ,td /Sskd at ks1=5.8. All symbols are as in Fig. 8.

FIG. 10. Fsk ,td /Sskd at ks1=10. All symbols are as inFig. 8.

FIG. 11. Fsk ,td /Sskd for ks=3 for different observing pointsand for various shear rates. From the right to the left, Pe=10−10,10−7, 10−5, 10−3, and 10−1. The thick dotted lines are at(a) in Fig. 7.The thin dotted lines are at(b). The thick solid lines are at(c). Thethin solid lines are at(d). The density isf=fc3 s1−10−4d. Thetime t is scaled bys2/D0.

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static structure factor at equilibrium and the distortion of thestructure due to the shear was neglected. However, as shownin Figs. 3–6, the shape ofSskd becomes weakly anisotropicunder shear. In order to see if the anisotropy of the structuresaffects the dynamics, we have implemented the same MCTcalculation usingSskd and ncskd with an anisotropic sinu-soidal modulation mimicking those observed in the simula-tions. We found that, as long as modulations are small, noqualitative change was observed and the dynamics was stillisotropic.

It is surprising that, although the perturbation is highlyanisotropic, the dynamics of fluctuations are almost isotro-pic. The reason for the isotropic nature of fluctuations maybe understood as follows: The shear flow perturbs and ran-domizes the phase of coupling between different modes. Thisperturbation dissipates the cage that transiently immobilizesparticles. Mathematically, this is reflected through the timedependence of the vertex. This “phase randomization” occursirrespective of the direction of the wave vector, which results

in the essentially isotropic behavior of relaxation. Thismechanism is different from that of many complex fluids andof dynamics near a critical point under shear, in which an-isotropic distortion of the fluctuations at small wave vectorsby shear plays an essential role[37].

D. Viscosity

The shear-dependent viscosityhsgd is evaluated by modi-fying the mode-coupling expression for the viscosity nearequilibrium [44]. This can be done by following the sameprocedure explained in Sec. II for the total momentum bothfrom solvent molecules and colloids. The equation for thetotal momentum is given by the Navier-Stokes equation. Thenonlinear term in the equation comes from the osmotic pres-sure of the suspension particles. Neglecting the coupling ofthe density field of colloids with that of the solvent mol-ecules, one obtains the mode-coupling expression for the vis-cosity. In the presence of the shear, the same modification asin Eq. (2.37) is necessary and the wave vectors should bereplaced by their time-dependent counterparts. The final re-sult is given by

hsgd = h0 +1

2b3 E

0

`

dtE dk

s2pd2

3kxkxstd

S2skdS2„kstd…

] Sskd] ky

] Sskstdd] kystd

F2„kstd,t…, s4.8d

whereh0 is the viscosity of the solvent alone. The integralover k can be implemented for the set ofFsk,td evaluatedusing Eq.(2.42). In Fig. 14, we have plotted the shear de-pendence of the reduced viscosity defined by

hRsgd ;hsgd − h0

h0s4.9d

for various densities aroundfc. The strong non-Newtonianbehavior is observed at high shear rate and large densities,which is again in qualitative agreement with the simulationresults for liquids reported in Ref.[6]. The shear thinningexponent extracted from the data between 10−10,Pe,1 is

FIG. 12. Fsk ,td /Sskd for ks=6.7. All symbols are as inFig. 11.

FIG. 13. Fsk ,td /Sskd for ks=12. All symbols are as inFig. 11.

FIG. 14. The reduced viscosity,(4.9), versus the shear rate forvarious densities. Here Pe=gs2/D0. From top to bottom;f=0.766549, 0.76650, 0.766453, 0.76640, 0.76600, and 0.75600.The highest density is 4310−5% larger thanfc.

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hRsgd~g−n with n.0.99. This is in agreement with the ex-ponent estimated from the simulation results and the struc-tural relaxation timetasgd observed in the simulation and inthe theory discussed in the previous subsections. For largershear rates, Pe.1, the exponent becomes smaller, which isagain consistent with computer simulation[6]. However, oneshould not trust this reasoning for Pe.1. As discussed inSec. IV A, it is expected that the distortion of structurecskdand Sskd by shear becomes important and one should takethese effects into account in the theory. In this regime, how-ever, the glassy structure has already been destroyed and thesystem becomes more similar to a “liquid.” In the liquidstate, the shear-thinning exponent is always expected to besmaller than 1 and range between 0.5 and 0.8[3,4,7,45].

Slightly abovefc, plastic behavior is observed, which im-plies the presence of the yield stress. Note that this is anartifact of the theory becausefc predicted from the mode-coupling theory is much lower than the real glass transitiondensity. Below the real glass transition densityfg, it is ex-pected that shear thinning behavior similar to that predictedby MCT for f,fc will result.

V. CONCLUSIONS

In this paper, we presented the derivation of the mode-coupling equation for therealistic supercooled fluids undershear and compared the results with the molecular dynamicsimulation. Our starting point is fluctuating hydrodynamicsextended to the molecular length scales. A simple closedequation for the intermediate scattering function was de-rived. We applied the theory to a two-dimensional colloidalsuspension with hard-core interactions. The numerical analy-sis of the equation revealed very good agreement with simu-lation results for a related system: a binary liquid interactingwith a soft-core potential. Theory and simulation showedcommon features such as(i) drastic reduction of relaxationtimes and the viscosity and(ii ) nearly isotropic relaxationirrespective of the direction of the shear flow. The fact thatthe dynamics is almost isotropic supports the validity of theschematic models proposed so far, in which the anisotropicnature of the nonequilibrium states was not explicitly consid-ered[14,17].

The mode-coupling theory developed in this paper is farfrom complete. The most crucial approximation is the use ofthe first FDT, which was employed when we close the equa-tion for dynamical correlators. It is already known that thefirst FDT is violated for supercooled systems under shear aswell as aging systems[7]. Without the first FDT, one has tosolve simultaneously the set of mode-coupling equations forthe propagator and correlation function, which couple eachother through the memory kernels. Research in this directionis under way. A simple argument, however, should suffice toexplain the success of the present theory with regard to shearthinning behavior. Consider a system that evolves out ofequilibrium with one effective temperatureTeff. Crudely, wecan use the effective steady state version of FDT violation inthe form

xi jstd = −ustd

kBTeff

dCijstddt

s5.1d

to eliminate the response functionxi jstd in favor of the cor-relation functionCijstd from the set of mode-coupling equa-

tions. We thus expect for the behavior of the structural relax-ation timetasgd will be unaffected as long as Eq.(5.1) holds.Another important approximation was to neglect the smalldistortion of the structure[cskd and Sskd] due to shear. Thesimulation supports that the distortion is negligibly small atthe low shear rates(small Péclet number). The constructionof the equation for equal-time correlation functions such asthe structure factor might be more subtle and should be con-sidered in future. Simulation results show that the structure isdistorted in a noticeable and anisotropic manner at high shearrates; the peak height of the first peak ofSskd was lowered asmuch as 10% atg=10−1. It is interesting that the dynamics isstill isotropic and qualitative behavior is not affected evenfor such high shear rates.

In the absence of shear, MCT is known to break down atdensities well below the real glass-transition density, whereMCT incorrectly predicts a fictitious nonergodic transition.Beyond this MCT crossover density, activated hopping be-tween the local minima of the free energy surface is expectedto dominate the dynamics of the system. Sollichet al.[46,47] have analyzed a schematic model for hopping pro-cesses and explained similar shear thinning behavior. Lacks[48] has also rationalized shear thinning behavior in terms ofchanges of the free energy barriers due to shear. It is inter-esting that the totally different picture given by MCT leads tosome qualitatively similar conclusions. The analysis of theviolation of the first FDT and effective temperatures has thepossibility of clarifying the difference between these distinctpictures. Barratet al. have shown by simulation that the firstFDT is violated in a sheared liquid and observed nontrivialeffective temperatures[7]. This is consistent with the conclu-sion of Berthieret al. [14] for the nonequilibriump-spinmodel. On the other hand, the trap model predicts multipleeffective temperatures[47]. Future effort will be directed to-wards extracting effective temperatures from our fluctuatinghydrodynamic approach.

In Sec. II we have mentioned that there are subtle prob-lems in constructing an equation for the intermediate scatter-ing function alone: The “correct” MCT equation can be de-rived if you start from the nonlinear Langevin equations forthe density and momentum fields, taking the overdampedlimit at the end. But difficulties arise if the overdamped limitis taken for the Langevin equation at the beginning. Theseproblems exist even for systems at equilibrium and are re-lated to generic problems that exist in loop expansion meth-ods. These subtleties were already recognized in the mid1970’s[34] and recently pointed out in the context of glassysystems by Schmitzet al. [49]. A detailed study of these andissues related to the field-theoretic approach to out-of-equilibrium glassy liquids will be discussed in a future pub-lication [32].

ACKNOWLEDGMENTS

The authors acknowledge support from NSF Grant No.0134969. The authors would like to express their gratitude toProfessor Matthias Fuchs for for useful discussions. K.M.and R.Y. would like to thank A. Onuki for helpful discus-sions.

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APPENDIX: STATIC STRUCTURE FOR THETWO-DIMENSIONAL FLUIDS WITH THE

HARD-CORE INTERACTION

Bauset al.have derived an approximated analytic expres-sion of the direct correlation functioncskd for thed-dimensional hard sphere fluids[43]. For the two-dimensional system, it is given by

ncskd = − f] hfZsfdj

] f F4s1 − a2fdfsksd

+ a2fHa2SJ1saks/2daks/2

D2

+16

pE

1/a

1

dx s1

− x2d1/2SJ1sksdks

− saxd2J1saksxdaks

DJG , sA1d

where Jnsxd is the Bessel function of the first kind,

f=ps2r0/4 is the packing fraction,r0 is the number density,and s is the diameter of spheres.a is a parameter which isdetermined by solving

2

pFa2sa2 − 4darcsinS1

aD − sa2 + 2dÎa2 − 1G =

1

f2F1 − 4f

− H ] hfZsfdj] f

J−1G . sA2d

Zsfd;p/r0kBT is the compressibility factor which is ex-panded by a rescaled virial series as

Zsfd = s1 − fd−2S1 + on=1

`

cnfnD , sA3d

wherecn is a coefficient which is related to the virial coeffi-cients via a recurrence relation. We truncated the series atn=6. The coefficientscn are given byc1=0, c2=0.1280,c3=0.0018,c4=−0.0507,c5=−0.0533, andc6=−0.0410.

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