DOI: 10.1021/la902205x 133Langmuir 2010, 26(1), 133–142 Published on Web 09/04/2009

pubs.acs.org/Langmuir

© 2009 American Chemical Society

Rheology, Microstructure andMigration in Brownian Colloidal Suspensions

Wenxiao Pan,† Bruce Caswell,‡ and George Em Karniadakis*,†

†Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, and ‡Division ofEngineering, Brown University, Providence, Rhode Island 02912

Received June 18, 2009. Revised Manuscript Received July 27, 2009

We demonstrate that suspended spherical colloidal particles can be effectively modeled as single dissipative particledynamics (DPD) particles provided that the conservative repulsive force is appropriately chosen. The suspension modelis further improved with a new formulation, which augments standard DPD with noncentral dissipative shear forcesbetween particles while preserving angular momentum. Using the new DPD formulation we investigate the rheology,microstructure and shear-induced migration of a monodisperse suspension of colloidal particles in plane shear flows(Couette and Poiseuille). Specifically, to achieve a well-dispersed suspension we employ exponential conservative forcesfor the colloid-colloid and colloid-solvent interactions but keep the conventional linear force for the solvent-solventinteractions. Our simulations yield relative viscosity versus volume fraction predictions in good agreement with bothexperimental data and empirical correlations. We also compute the shear-dependent viscosity and the first and secondnormal-stress differences and coefficients in both Couette and Poiseuille flow. Simulations near the close packingvo-lume-fraction (64%) at low shear rates demonstrate a transition to flow-induced string-like structures of colloidalparticles simultaneously with a transition to a nonlinear Couette velocity profile in agreement with experimentalobservations. After a sufficient increase ofthe shear rate the ordered structure melts into disorder with restoration of thelinear velocity profile. Migration effects simulated in Poiseuille flow compare well with experiments and modelpredictions. The important role of angular momentum and torque in nondilute suspensions is also demonstrated whencompared with simulations by the standardDPD, which omits the angular degrees of freedom. Overall, the newmethodagrees very well with the Stokesian Dynamics method but it seems to have lower computational complexity and isapplicable to general complex fluids systems.

1. Introduction

Suspensions are a class of complex fluids and can be differ-entiated according to the physical and chemical nature of thesuspended particles and suspending fluid. In this work, colloids aresimulatedas neutrally buoyant, chemically stable (nonaggregating)hard, but not rigid, spherical particles suspended in a Newtonianfluid. We are interested in their rheological properties, microstruc-ture, and shear-induced migration.

Microstructure refers to the relative position and orientation ofparticles in suspension, and is closely correlated to the rheology ofsuspensions. When the suspension is subjected to flow, locally orfully ordered structures may be formed in dense suspensions.1-4

The anisotropic microstructure will, in turn, affect the rheologicalproperties, thereby altering the flow profile, which is usuallyreferred to as shear banding.5Hoffman6 reported the first evidencethat colloidal order can be manipulated by the application ofshear flow. Since his pioneering work, the relationship amongflow and ordered structures has received extensive attention. Inorder to investigate the microstructure of particles, microscopytechniques combined with quantitative image analysis have beendeveloped to directly probe the organization of the colloids ineither real or reciprocal space.7 Here the two-body structure canbe represented by the pair distribution function. Reciprocal-space

techniques typically rely on measuring changes in 2D scatteringpatterns, using various sources of electromagnetic radiation, i.e.,light8 and X-rays2 or neutron scattering.9 Probing the directionaldependency of the scattered radiation yields the structure factorS(q) which is related to the radial distribution function in real-space, g(r), through the Fourier transform.7 Most experimentshave been conducted in either steady-shear or oscillatory-shearflow. Here we focus on the plane steady-shear Couette andPoiseuille flow.

Shear-induced migration of colloidal particles occurs in pres-sure-driven flow of concentrated suspensions through a channel.The migration causes an initially uniform suspension to becomeless concentrated near the walls and more concentrated near thecenter of the channel, which in turn, leads to the modification ofthe velocity profile of the suspension.10Particlemigration has beenstudied extensively beginning with Leighton and Acrivos,11 butthere has been littleworkonBrownian suspensions.ForBrowniansuspensions, the gradient diffusion resulting from Brownianmotion has a significant effect of opposing migration driven byshear stresses, as observed experimentally.12 However, as the flowrate is increased, the effect of diffusion is diminishing relative toshear-driven migration because shear-driven stresses increase inmagnitude, whereas thermally driven stresses remain unchanged.This leads to a flow-rate dependent migration in Browniansuspensions, i.e., stronger migration for larger flow rates, whichis fundamentally different from non-Brownian suspensions.

*Corresponding author.(1) Ackerson, B. J. J. Rheol. 1990, 34, 553–590.(2) Versmold, H.; Musa, S.; Dux, C.; Lindner, P.; Urban, V. Langmuir 2001, 17,

6812–6815.(3) Sierou, A.; Brady, J. F. J. Rheol. 2002, 46, 1031–1056.(4) Wu,Y. L.; Brand, J. H. J.; vanGemert, J. L. A.; Verkerk, J.;Wisman,H.; van

Blaaderen, A.; Imhof, A. Rev. Sci. Instrum. 2007, 78, 103902.(5) Wu, Y. L. Utrecht University, The Netherlands, PhD thesis, 2007.(6) Hoffman, R. L. J. Rheol. 1972, 16, 155–173.(7) Vermant, J.; Solomon, M. J. J. Phys.: Condens. Matter 2005, 17, 187–216.

(8) Verduin, H.; de Gans, B. J.; Dhont, J. K. G. Langmuir 1996, 12, 2947–2955.(9) Hunt, W. J.; Zukoski, C. F. Langmuir 1996, 12, 6257–6262.(10) Lyon, M. K.; Leal, L. G. J. Fluid Mech. 1998, 363, 25–56.(11) Leighton, D. T.; Acrivos, A. J. Fluid Mech. 1987, 181, 415–439.(12) Semwogerere, D.;Morris, J. F.;Weeks, E. R. J. FluidMech. 2007, 581, 437–

451.

134 DOI: 10.1021/la902205x Langmuir 2010, 26(1), 133–142

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Computer simulations have made significant contributions toour understanding of suspension rheology, microstructure andshear-induced migration. The major difficulty has been to ac-count correctly for the hydrodynamic interactions. The mostwidely used method for simulating suspension flow at lowReynolds number has been Stokesian Dynamics pioneered byBrady and co-workers.13-19 This method properly incorporatesthe solvent-induced multibody hydrodynamic forces, Brownianforces, and lubrication forces and enables a rigorous simulation ofthe suspensions composed of rigid spheres. Other notable simula-tion techniques are the Lattice Boltzmann method20,21 and theLagrange multiplier fictitious domain method.22,23

A relatively new method for particle simulation at the meso-scopic level is dissipative particle dynamics (DPD) proposed byHoogerbrugge and Koelman.24 Although it is a simulationtechnique similar to molecular dynamics (MD), the individualDPDparticles represent the collective dynamic behavior of a largenumber of molecules as “lumps” of fluid. These particles interactthrough effective potentials, which are much softer than the bareinteraction potentials between molecules. Therefore, in DPD,much longer time steps can be used, allowing the simulationof dynamical phenomena over much longer time scales. Also,DPD implicitly accounts for hydrodynamic interactions by em-ploying velocity-dependent dissipative forces, a feature that con-trasts with the explicit interactions of Brownian dynamics (BD).TheDPDmethod has been used with success to study a variety ofdynamical phenomena in complex fluids, including polymers andsuspensions.25-29

The DPD method was employed to investigate colloidalsuspensions first by Hoogerbrugge and Koelman, who studiedthe rheology of colloidal suspensions in simple fluids.27 However,they represented the colloidal particle as a collection of DPDparticles, an approach with two limitations. (i) Every colloidalparticle contained roughly several hundred DPD particles. Con-sequently, simulating concentrated dispersions involved a largecomputational burden. (ii) At higher volume fractions, thecolloidal particle could overlap with neighboring particles dueto the soft nature of theDPD interactions.While the latter had noimpact on properties at low volume fractions, it missed theinteresting high volume-fraction phenomena, such as shear band-ing and crystallization. Both limitations can be overcome ifcolloidal particles can be represented by single DPD particles aslong as the intrinsic length scale of eachparticle can be defined.Asshown in our previous studies,30,31 although with point centers of

repulsion, DPD particles have an intrinsic size, which is assignedby the spherical impenetrable domain occupied by each particlewhen immersed in a sea of other particles. The radius of theimpenetrable sphere is equated to the Stokes-Einstein radiusRSE

derived from the product of the coefficients of self-diffusion andviscosity of the DPD fluid.

However, in standard DPD a single DPD particle is subject tocentral pairwise forces only, which effectively ignores the non-central shear forces between dissipative particles. This deficiencyhas been recognized by Espanol and collaborators,32-34 whoproposed the fluid particlemodel (FPM).32Compared to standardDPD,24,35,36 this FPM incorporates two additional noncentralshear components into the dissipative forces, which are coupledto the random forces by means of the fluctuation-dissipationtheorem. FPM can be considered to be a generalization of DPDsince it also includes torques and angular velocities of the particles,and hence it conserves both linear and angular momenta. Dyna-mical and rheological properties of colloidal suspensions in simplefluid solvents were simulated by FPM with some success.37 Eachcolloidal particlewas representedbya singleFPMparticle, and theconservative forces for solvent-colloid and colloid-colloid inter-actions were based on Lennard-Jones potentials. Unfortunately,the drag force and torque on a solid sphere represented by a singleFPM particle in the formulation of ref 37 does not match thecontinuum hydrodynamic values.

We developed30 a new formulation of DPD in the spirit ofFPM, which deviates only slightly from standard DPD. The ideabehind the new formulation is to modify the FPM equations insuch a way that the dissipative forces acting on a particle areexplicitlydivided into two separate components: central and shear(noncentral) components. This allows us to redistribute and hencebalance the dissipative forces acting on a single particle to obtainthe correct hydrodynamics. The resulting method was shown toyield the quantitatively correct hydrodynamic forces and torqueson a single DPD particle.30 In this article we examine theapplicability of this simulation approach to reproduce the correcthydrodynamic characteristics of suspensions. Moreover, in oursimulations we employed an exponential potential for the col-loid-colloid and colloid-solvent conservative interactions, butkept the DPD quadratic potential for the solvent-solvent con-servative interactions. Itwas found that suchhybrid interactions notonly resulted in a well-dispersed colloidal phase but they alsoallowed a hard sphere type of repulsion between colloids, withoutany significant decrease of the time step, in contrast to the Lennard-Jones pontentials employed in ref 37. In particular, we haveexamined in detail the rheological behavior, structure patterns,and shear-induced migration for a range of volume fractions andshear rates, and demonstrated excellent agreement with publishedexperimental and simulation results.Wealso investigated the role ofangular momentum and torque in dense suspensions by comparingDPD simulations with and without angular variables.

The paper is organized as follows. In section 2 we describesimulation details including the formulation and its parameters.Section 3 presents our simulation results, comparisons withexperiments, and results from other simulation methods, e.g.Stokesian Dynamics. These results include suspension rheology(section 3.1), colloidal microstructure (section 3.2), shear banding(section 3.3), migration of colloids in pressure-driven channel

(13) Brady, J. F.; Phillips, R. J.; Lester, J. C.; Bossis, G. J. FluidMech. 1988, 195,257–280.(14) Nott, P. R.; Brady, J. F. J. Fluid Mech. 1994, 275, 157–199.(15) Phung, T. N.; Brady, J. F.; Bossis, G. J. Fluid Mech. 1996, 313, 181–207.(16) Morris, J. F.; Brady, J. F. Int. J. Multiphase Flow 1998, 24, 105–130.(17) Foss, D. R.; Brady, J. F. J. Fluid Mech. 2000, 407, 167–200.(18) Sierou, A.; Brady, J. F. J. Fluid Mech. 2001, 448, 115–146.(19) Brady, J. F. Chem. Eng. Sci. 2001, 56, 2921–2926.(20) Chen, S.; Doolen, G. D. Annu. Rev. Fluid Mech. 1998, 30, 329–364.(21) Hill, R. J.; Koch, D. L.; Ladd, A. J. C. J. Fluid Mech. 2001, 448, 213–241.(22) Glowinski, R.; Pan, T. W.; Hesla, T. I.; Joseph, D. D. Int. J. Multiphase

Flow 1999, 25, 755–794.(23) Singh, P.; Hesla, T. I.; Joseph, D. D. Int. J. Multiphase Flow 2003, 29, 495–

509.(24) Hoogerbrugge, P. J.; Koelman, J. M. V. A. Europhys. Lett. 1992, 19, 155–

160.(25) Lisal, M.; Brennan, J. K. Langmuir 2007, 23, 4809–4818.(26) Malfreyt, P.; Tildesley, D. J. Langmuir 2000, 16, 4732–4740.(27) Koelman, J. M. V. A.; Hoogerbrugge, P. J. Europhys. Lett. 1993, 21, 363–

368.(28) Boek, E. S.; Coveney, P. V.; Lekkerkerker, H. N. W.; van der Schoot, P.

Phys. Rev. E 1997, 55, 3124–3133.(29) Martys, N. S. J. Rheol. 2005, 49, 401–424.(30) Pan, W.; Pivkin, I. V.; Karniadakis, G. E. Europhys. Lett. 2008, 84, 10012.(31) Pan,W.; Fedosov,D.A.; Karniadakis, G. E.; Caswell, B.Phys. Rev. E 2008,

78, 046706.

(32) Espanol, P. Phys. Rev. E 1998, 57, 2930–2948.(33) Espanol, P. Europhys. Lett. 1997, 39, 605–610.(34) Espanol, P.; Revenga, M. Phys. Rev. E 2003, 67, 026705.(35) Espanol, P.; Warren, P. Europhys. Lett. 1995, 30, 191–196.(36) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107, 4423–4435.(37) Pryamitsyn, V.; Ganesan, V. J. Chem. Phys. 2005, 122, 104906.

DOI: 10.1021/la902205x 135Langmuir 2010, 26(1), 133–142

Pan et al. Article

flow (section 3.4), andnon-Newtonian viscosity and normal stressdifferences (section 3.5).We conclude in section 4with a summaryand a brief discussion including the computational cost of themethod.

2. Formulation and Simulation Details

The simulation system consists of a collection of particles withpositions ri and angular velocitiesΩi. We define rij=ri - rj, rij=|rij|, eij=rij/rij, vij=vi -vj. The force and torque on particle i aregiven by

Fi ¼Xj

Fij

Ti ¼ -Xj

λijrij � Fij ð1Þ

Here the factor λij (introduced by Pryamitsyn and Ganesan37) isincluded as a weight to account for the different contributionsfrom the particles in different species (solvent or colloid) differ-entiated in sizes while still conserving angular momentum. It isdefined as

λij ¼ Ri

Ri þ Rj, and λij ¼ 1=2 when Ri ¼ Rj ð2Þ

where Ri and Rj denote the radii of the particles i and j,respectively. These radii are parameters as will be seen below.The force exerted by particle j on particle i is given by

Fij ¼ FCij þ FT

ij þ FRij þ ~Fij ð3Þ

The conservative (C), translational (T), rotational (R), andrandom (tilde) components are given respectively by

FCij ¼ -V 0ðrijÞeij ð4aÞ

FTij ¼ -γijTij 3 vij ð4bÞ

FRij ¼ -γijTij 3 ½rij � ðλijΩi þ λjiΩjÞ� ð4cÞ

~Fijdt ¼ ð2kBTγijÞ1=2 ~AðrijÞdWSij þ ~BðrijÞ 1

3tr½dWij�1

�

þ ~CðrijÞdWAij

i3 eij ð4dÞ

The conservative force can be that of standard DPD, whichemploys a quadratic potential V(rij), i.e.,

FCij ¼ aij 1-

rij

rc

� �eij ð5Þ

with rc being the cutoff distance. The matrix Tij in the transla-tional force and rotational force is given by

Tij ¼ AðrijÞ1 þ BðrijÞeijeij ð6Þwith

AðrÞ ¼ 1

2½ ~A

2

ðrÞ þ ~C2ðrÞ�

BðrÞ ¼ 1

2½ ~A2ðrÞ- ~C

2ðrÞ� þ 1

3½ ~B2ðrÞ- ~A

2ðrÞ� ð7Þ

where ~A(r), ~B(r), ~C(r) are scalar functions.Note that ~A(r)= ~C(r)=0 recovers standardDPDwith only central forces and no torques.Here, the scalar functions are defined as

~A ðrÞ ¼ 0,AðrÞ ¼ γS½f ðrÞ�2,BðrÞ ¼ ðγC -γSÞ½f ðrÞ�2 ð8Þ

Also, f ðrÞ ¼ 1- rrcis a weight function while the superscripts C

and S denote the “central” and “shear” components, respectively.This notation can be clarified further if we look at the particleforces. Specifically, the translational force is given by

FTij ¼ -

hγSij f

2ðrÞ1 þ ðγCij -γSij Þf 2ðrÞeijeiji3 vij

¼ -γCij f2ðrijÞðvij 3 eijÞeij -γSij f

2ðrijÞ½vij -ðvij 3 eijÞeij� ð9Þ

It accounts for the drag due to the relative translational velocity vijof particles i and j. This force can be decomposed into twocomponents: one along and the other perpendicular to the linesconnecting the centers of the particles. Correspondingly, the dragcoefficients are denoted by γij

C and γijS for the “central” and the

“shear” components, respectively. We note that the centralcomponent of the force is identical to the dissipative force ofstandard DPD.

The rotational force is defined by

FRij ¼ -γSij f

2ðrijÞ½rij � ðλijΩi þ λjiΩjÞ� ð10Þ

Finally, the random force is given by

~Fijdt ¼ f ðrijÞ 1ffiffiffi3

p σCij tr½dWij�1 þ

ffiffiffi2

pσSijdW

Aij

� �3 eij ð11Þ

where σSij ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kBTγSij

qand σC

ij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kBTγCij

q. We used the gen-

eralized weight function f ðrÞ ¼ 1- rrc

� �s

with s=0.25 38 in eqs

9-11. Our numerical results in previous studies30,31 showedhigher accuracy with s=0.25 compared to the usual choice s=1. The standard DPD is recovered when γij

S � 0, i.e., when theshear components of the forces are ignored.

Initially, each colloidal particle was simulated as a single DPDparticle, of the same type as the solvent particles but of larger size.Particle size can be adjusted with the coefficient aij of theconservative force (see eq 5).However, we found that the colloidalparticles tended to aggregate into clusters in this two-phasesolvent-colloid mixture. To solve this problem, we employed anexponential conservative force for the colloid-colloid and col-loid-solvent interactions, but kept the conventional linear force(see eq 5) for the solvent-solvent interactions. It was then foundthat these hybrid conservative interactions produced a uniformcolloidal dispersionwithout significantly decreasing the time step,in contrast to the small timesteps required for the Lennard-Jonespontential (see ref 37). The exponential conservative force isdefined as

FCij ¼ aij

1-e-bijðe-bij rij=rc -e-bij Þ ð12Þ

where aij and bij are adjustable parameters. This exponential forcealong with the standard DPD linear force is sketched in Figure 1.The size of a colloidal particle can thereby be controlled byadjusting the value of acs inFcs

C (Here “c” and “s” refer to “colloid”and “solvent”, respectively, see equ. 12 and Figure 1). Fcc

C as

(38) Fan, X. J.; Phan-Thien, N.; Chen, S.; Wu, X. H.; Ng, T. Y. Phys. Fluids2006, 18, 063102.

136 DOI: 10.1021/la902205x Langmuir 2010, 26(1), 133–142

Article Pan et al.

sketched inFigure 1 helps to avoid overlap of colloidal particles insuspension.

To determine the particle size, a measurement of the coefficientof self-diffusion allows RSE to be calculated by means of theStokes-Einstein equation given by

RSE ¼ kBT

6πηsD0ð13Þ

where D0 is the diffusion coefficient of a particle subject toBrownian motion in a dilute solution and ηs is the solventviscosity. RSE is equated to the radius R of the sphere effectivelyoccupied by a single DPD particle.31 Figure 2 depicts theimpenetrable area of a colloidal particle by the number densitydistribution of its surrounding solvent particles. We note thatthere is a solvation effect indicated by a thin layer of surroundingparticleswith higher density.Wedetermined that every suspended

colloidal particle had a radius of 0.98, whereas the solventparticles had a radius of 0.27, both in reduced DPD units. Herewe did not choose larger colloidal particles for two reasons. First,it allowed us to study systems with larger number of dispersedparticles at the same concentration, which correspondingly yield amore uniform disperse phase and improved the accuracy. Second,from Stokesian Dynamics simulations by Sierou and Brady18 thesystem with the larger number of particles tends to capture moreaccurately the properties of suspensions, e.g., sedimentationvelocity and diffusion coefficient, although the viscosity is in-sensitive to the number of particles.18 With the colloidal particleradius determined, different volume fractions of the suspensionwere achieved by varying the number of colloidal particles,Nc, ina cubic domain with size of 20 (DPD units). The solvent densitywas fixed at F=3. Typically in this workNc=200-1300, which issimilar to the range of Sierou and Brady,3 corresponding tovolume fraction φ=0.1-0.64.

The radial distribution function g(r) of colloidal particles in asuspension at different concentrations (φ) is plotted in Figure 3.We see that a hard-sphere type of g(r) for the suspended particlesis achieved and that the second and third peaks become morepronounced with increasing volume fraction.

To yield correct hydrodynamics, the drag force and torque on asingle colloidal particle in the flow should satisfy Stokes laws atlow Reynolds number corresponding to the stick (no-slip) con-ditions on its surface, and hence we determined the parameters ofdissipative forces as γss

C=γssS=4.5, γsc

C=500 and γscS=1800. The

coefficients of random forces were chosen to satisfyσ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2γkBTp

. Since it is not expected that the colloidal particleswill be subject to direct interactions between each other except thehard-sphere type of repulsions, the dissipative force and randomforce between colloids had negligibly small values, i.e., γcc

C=γccS=

0.045. To simulate the Couette flow between two parallel plates,the Lees-Edwards periodic boundary condition (LEC) wasadopted to avoid the artificial wall-induced effects, e.g., densityfluctuations near the walls. The Peclet number (dimensionlessshear rate) was chosen toquantify the relative importance of shearto Brownian diffusivities, which was defined as Pe= _γR2/D0

(where R denotes the colloid radius and D0 is its diffusivity indilute solution). To keep the temperature constant as kBT=1.0,the time step Δt (inDPD units) was varied from 0.0002 to 0.0005in all the runs.

Figure 1. Conservative forces adopted in this work between sol-vent-solvent (ss), colloid-colloid (cc) and colloid-solvent (cs).Here, r= rij/rc and F= FC

ij/ aij.

Figure 2. Number density of solvent particles around a singlecolloidal particle with solvation indicated by a thin surface layerof higher density.

Figure 3. Equilibrium radial distribution function, g(r), of colloi-dal particles atdifferent volume fractions.Thecenter-to-centerpairdistance r is normalized by the colloid radius R= 0.98.

DOI: 10.1021/la902205x 137Langmuir 2010, 26(1), 133–142

Pan et al. Article

3. Results

3.1. Suspension Rheology. At low shear rates, the relativesuspension viscosity is a well studied function of solids volumeconcentration. Our simulations were restricted to the low shearrate region of Pe e 10-1 and were carried out with and withoutthe shear components of force in the DPD formulation. InFigure 4, our results are compared with some experimental datameasured at low shear rates.39-43 Also plotted are severalempirical formulas, some of which employ the maximumpackingfraction, φm, as an adjustable parameter.

For a Newtonian fluid containing a dilute suspension ofmonodisperse hard spheres, Batchelor44 derived that for Brow-nian suspensions in any flow

η0r ¼ 1 þ 2:5φ þ 6:2φ2 ð14Þwhere the linear term was derived by Einstein.45

Eilers’ empirical formula46 contains Einstein’s low concentra-tion limit, and is singular at the high limit as φ f φm:

η0r ¼ 1 þ 1:25φ

1-φ=φm

� �2

ð15Þ

Another formula which also captures these limits is that ofKrieger and Dougherty:47

η0r ¼ 1-φ

φm

� �-2:5φm

ð16Þ

More recently, Bicerano et al.48 examined the viscosity charater-istics of suspensions of different hard bodies, and suggested therelative viscosity of a hard-sphere suspension ηr should have thefollowing form:

η0r ¼ 1-φ

φm

� �-2

1-0:4φ

φm

þ 0:341φ

φm

� �2" #

ð17Þ

The curves defined by eqs 15-17 in Figure 4 fit the experi-mental data with φm = 0.65, the same value obtained byPowell et al.49 to fit their experimental data. Figure 4 shows avery good agreement between our simulations and all the experi-mental results and empirical predictions. Figure 4 also shows theeffect of omitting from our simulations the angular terms in thenew DPD formulation. By keeping other simulation details un-changed,we found thatwithout including the angularmomentumand torque the standard DPD model was unable to accuratelypredict viscosities of nondilute suspensions (φ g 0.2).3.2. Microstructure of Colloidal Particles. The macro-

scopic properties of suspensions, such as their rheology, are

closely related to the spatial organization of colloidal particles,usually referred to as microstructure. Since the suspension dis-plays non-Newtonian behavior, it is expected to display variousstructures at different volume fractions and shear rates. Mostexperimental techniques employ various sources of electromag-netic radiation, i.e., light andX-rays or neutron scattering to trackthe locations of particles in the flow field, with results presented asdistributions of scattered radiation intensity related to the real-space pair-distribution functions by means of Fourier trans-forms.7 Therefore, as in previous studies,17,50,51 to compare withthose experiments the pair-distribution functions of the simulatedcolloidal particles were calculated in three orthogonal planes, i.e.,the flow plane (x,y), the shear plane (x,z) and the neutral plane(y,z), to visualize the microstructure of colloids. Figure 5 showsthe pair-distribution functions in three planes over a range of Penumbers at φ=0.4. At low Pe numbers (Pe , 1), the micro-structure is isotropic and reveals a ring pattern also observed inlight-scattering experiments.1,52,53 As the Pe number increases,the structure remains isotropic in the planes perpendicular to theflow plane, but in the flow plane it becomes distorted as particlesare separated along the extensional axis (45� to the flow direction)of the flow field while being pushed together along the compres-sional axis. These findings compare well with those obtainedexperimentally50 and with those obtained from Stokesian Dy-namics simulations.17,51

At sufficiently high volume fractions (φ > 0.5), we begin tonotice flow-induced local structures of colloid particles in thesuspension, but no full long-range order is found until the volumefraction exceeds 0.6 and approaches the close packing fraction(φ = 0.64). Through the pair-distruibution functions in threeorthogonal planes at φ=0.64, Figure 6 displays the fully orderedstructure. This ordered structure consists of strings of colloidalparticles lined up along the flow direction while the pair-distribu-tion function in the neutral plane (y, z) shows the strings to bearranged in an hexagonal pattern, consistent with previous

Figure 4. Zero-shear rate relative viscosity ηr0 of the suspension at

different volume fractions (normalized) φ/φm, fitted by empiricalformulaes (lines) and compared with exprimental data (symbols),with φm = 0.65.

(39) Chong, J. S.; Christiansen, E. B.; Baer, A. D. J. Appl. Polym. Sci. 1971, 15,2007–2021.(40) Poslinski, A. J.; Ryan, M. E.; Gupta, P. K.; Seshadri, S. G.; Frechette, F. J.

J. Rheol. 1988, 32, 751–771.(41) Storms, R. F.; Ramarao, B. V.; Weiland, R. H. Powder Technol. 1990, 63,

247–259.(42) Shikata, T.; Pearson, D. S. J. Rheol. 1994, 38, 601–616.(43) Chang, C.; Powell, R. L. J. Rheol. 1994, 38, 85–98.(44) Batchelor, G. K. J. Fluid Mech. 1977, 83, 97–117.(45) Einstein, A. Investigations on the theory of the Brownian movement.

Dover: New York 1956.(46) Ferrini, F.; Ercolani, D.; Cindio, B. D. Rheol. Acta 1979, 18, 289–296.(47) Krieger, I. M.; J, D. T. Trans. Soc. Rheol. 1959, 3, 137–152.(48) Bicerano, J.; Douglas, J. F.; Brune, D. A. J.Macromol. Sci.: Part C: Polym.

Rev. 1999, 39, 561–642.(49) Stickel, J. J.; Powell, R. L. Annu. Rev. Fluid Mech. 2005, 37, 129–149.

(50) Parsi, F.; Gadala-Maria, F. J. Rheol. 1987, 31, 725–732.(51) Morris, J. F.; Katyal, B. Phys. Fluids 2002, 14, 1920–1937.(52) Ackerson, B. J.; Clark, N. A. Phys. Rev. A 1984, 30, 906–918.(53) Wagner, N. J.; Russel, W. B. Phys. Fluids A 1990, 2, 491–502.

138 DOI: 10.1021/la902205x Langmuir 2010, 26(1), 133–142

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studies.1,3,4 The full visualization of the shear-induced transitionfrom disorder to string-like structures of colloidal particles can befound in the Supporting Information.

The pair-distribution function of the ordered structures wasaveraged in all directions to yield g(r) shown in Figure 7. At lowshear rates (Pe < 1.0), this ordered structure is persistent andstable.However, shear has a dual ordering role. At low shear ratesit can induce order54, whereas at high shear rates it can melt acolloidal structure.55,56 In our simulations, at higher shear rates(e.g.,Pe=1.5) the ordered structure becamemetastable, switchingfrom one structure to another. As the shear rate was furtherincreased (e.g., at Pe ≈ 5) shear melting occurred, and thedisordered phase was recovered. This behavior is similar to thatreported in experiments.57-59 However, we note that the observa-tion differs from a recent work by Kulkarni and Morris60 whoperformed accelerated Stokesian Dynamics simulations for par-ticle volume fractions of 0.47 e φ e 0.57 and observed theordering when particle volume fraction is above 0.5 even atPe g 10. It appears that the different interparticle forces used inStokesian Dynamics and in this work play a role for thisdifference.

3.3. Shear-Banding.At sufficiently high volume fractions (φ> 0.5), the typical suspension begins to display some anisotropicorganization when subject to shear flow, i.e., the colloids becomepartially or fully crystallized. In turn, this microstructure affectsthe rheological properties, and thereby alters the flowprofile fromlinear to nonlinear, which is usually taken to be the hallmark ofshear banding.5 This was also observed in our simulations, andFigure 8 (left) gives one example of the nonlinear velocity profileatφ=0.6. The profile consists of: twohigh shear-rate regions closeto theboundarieswitha low shear-rate region inbetween.Figure8(middle) shows a more extreme case when a long-ranged orderedstructure was formed in suspension (φ=0.64) at a low shear rate,and then at sufficiently high shear rate, asmentioned above, shearmelting occurred with recovery of both the disordered phase andthe linear velocity profile (Figure 8 (right)).3.4. Migration of Colloidal Particles in Poiseuille Flow.

We also investigated the shear-induced migration of suspendedcolloidal particles relative to the solvent in pressure-drivenchannel flow. An initially uniform suspension became less con-centrated near the walls andmore concentrated near the center ofthechannel. The channel was configured as the reverse Poiseuilleflow (RPF). It consists of two parallel Poiseuille flows driven bybody forces, equal in magnitude but opposite in direction.62 RPFpermits periodic boundary conditions and thereby avoids thepitfalls of simulating real walls. To compare our results withprevious studies,12,61 we chose the same bulk volume fractionφ=0.26 and Peclet numberPe=129.The Peclet number is definedin the sameway as above, i.e.,Pe=_γR2/D0, butwith _γ=vmax(H/2),where vmax is themaximumaxial velocity of the suspension andHis the width of the channel. In Figure 9, the resulting volumefraction profile across the channel is compared with the measuredprofile of Semwogerere et al.,61 and the predicted profile of Franket al.12 who used a continuum mixture flow model. In theirexperiment, Semwogerere et al.12 investigated the influence ofBrownian motion on shear-induced particle migration of mono-disperse suspensions of micrometre colloidal particles by pump-ing the suspension through a glass channel of rectangular cross-section. In the flow model, a constitutive law was proposed to

Figure 5. The pair-distribution function in different planes, g(x,y), g(x, z) and g(y, z), atPe=0.1, 1.0, 20 and φ=0.4. The displayregion is 6 � 6 colloid diameters.

Figure 6. The pair-distribution function of the string-like orderedstructure of colloids on different planes at Pe=0.1 and φ=0.64.The display region is 7 � 7 colloid diameters.

Figure 7. Radial distribution function g(r) of colloidal particles intheir fully ordered state. Pair distance normalized by the colloidradius R. Here, φ= 0.64 and Pe= 0.1.

(54) Ackerson, B. J.; Pusey, P. N. Phys. Rev. Lett. 1988, 61, 1033–1036.(55) Hoffman, R. L. J. Colloid Interface Sci. 1974, 46, 491–506.(56) Tomita, M.; van de Ven, T. G. M. J. Colloid Interface Sci. 1984, 99, 374–

386.(57) Ackerson, B. J.; Clark, N. A. Phys. Rev. Lett. 1981, 46, 123–126.(58) Ashdown, S.; Markovic, I.; Ottewil, R. H.; Lindner, P.; Oberthur, R. C.;

Rennie, A. R. Langmuir 1990, 6, 303–307.(59) Yan, Y. D.; Dhont, J. K. G. Physica A 1993, 198, 78–107.(60) Kulkarni, S. D.; Morris, J. F. J. Rheol. 2009, 53, 417–439.

(61) Frank, M.; Anderson, D.; Weeks, E. R.; Morris, J. F. J. Fluid Mech. 2003,493, 363–378.

(62) Backer, J. A.; Lowe, C. P.; Hoefsloot, H. C. J.; Iedema, P. D. J. Chem. Phys.2005, 122, 154503.

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Pan et al. Article

describe the suspension normal stresses as a function of both φ

and the localPe number, and the particle migration is then drivenby the spatially varying normal stresses.

Figure 9 shows that at the same volume fraction and Pecletnumber, compared with the model predictions by Frank et al.61

our simulations yield a slightly better agreement with the experi-ments. However,the observed migration is still noticeably stron-ger than that in the experiment. The discrepancy may result fromthe fact that in the experiment, the charge on the particles causesan electrostatic repulsion between the particles that may besufficiently strong to resist the migration of particles toward thechannel center.12

The particle migration can lead to a modification of the velo-city profile of the suspension. In particular, the profile is no longerparabolic, but becomes more flattened on the center as shownin Figure 10. Furthermore, the migrationis flow-rate depen-dent as expected for Brownian suspensions. Figure 11 plots thevolume fraction profiles along the channel width over a rangeof Pe numbers. The migration causes larger concentrations inthe low-shear regions near the center of the channel forlarger flow rates, which is a typical characteristic of Browniansuspensions.

3.5. Non-NewtonianViscosity andNormal Stress Differ-

ences. At higher shear rates colloidal suspensions usually exhibitnon-Newtonian viscosity and normal stress differences, whichwere also calculated from our simulations in both Couette flowand Poiseuille flow. As above, we simulated Couette flow withLEC and Poiseuille flowwithRPF. The non-Newtonian viscosityand the first and second normal stress differences and coefficientsare calculated from the following expressions

η ¼ Sxy

_γ,N1 ¼ Sxx-Syy,N2 ¼ Syy -Szz,

Ψ1 ¼ Sxx -Syy

_γ2,Ψ2 ¼ Syy -Szz

_γ2ð18Þ

For each simulation of Couette flow, the shear stress (Sxy) andnormal stresses (Sxx, Syy, Szz) are calculated corresponding to asingle shear rate ( _γ). Therefore, it appears to be expensive whenwe need to study the rheological properties over a wide range ofshear rates.However, in Poiseuille flow, a range of shear rates canbe extracted from a single velocity profile, which improves theefficiency significantly. Particularly, in Poiseuille flow we extractthe shear rates from the measured velocity profile by numerical

Figure 8. Velocity profiles in Couette flow for concentrated colloidal suspensions (left) at φ = 0.6; (middle) at φ = 0.64 after the orderedstructure of colloids was formed in suspension and (right) after the ordered structure of colloids melted into the disordered phase. Thecoordinate y is normalized by the colloid radius R.

Figure 9. Volume fraction profiles across the channel in Poiseuilleflow at φ = 0.26 and Pe = 129, compared with experimentaldata of Semwogerere et al.12 and model predictions of Franket al.61 Here, all the channel widths have been normalized to unity.

Figure 10. Velocity profile across the channel in Poiseuille flow atφ= 0.26 and Pe= 129.

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differentiation of fitted even-order polynomial for the centralregion, and cubic splines for the wall region.

The regularity assumptions of continuum theory allow for theexpansion ofV(y) in powers of y2, and for the low shear rates nearthe centerline the leading terms are

VðyÞ ¼ Vc -nf

2η0y2 þOðy4Þ ð19Þ

where f is the imposed body force per unit mass in RPF and n isthe number density. This suggests that the central region of thevelocity profile can be fitted well with even-order polynomials in ymeasured from the centerline, with the coefficient of y2 furnishingthe zero-shear rate viscosity ηr

0. We employ a fourth-orderpolynomial fitted near the centerline by careful limitation of theregion so that the term y4 is not dominant. For the near-wallregion where polynomial interpolation is known to perform

rather poorly, the profile velocity profile is then interpolatedwith cubic splines and projected onto grid points yi, defined as yi=iΔy, i=0,... ,M, where M=º0.5H/Δyß with uniform spacing ay.The required shear rates at points yi + 0.5 are then calculated fromthe second-order central difference (V(yi+ 1)-V(yi))/Δy. For thehigher flow rate profile, the zero-shear rate plateau cannot beobtained accurately because the central region of the velocityprofile is very narrow, and not resolvable by fittingwith low-orderpolynomials. Hence, full curves of viscosity and normal-stresscoefficients for a particular system were obtained through twosimulations: first with a low flow rate profile to resolve zero-shearviscosity plateau and subsequently with a high flow rate profile toresolve the high-shear rate region.

The imposed shear stress is Sxy=fn (H/2- y) and hence is freeof noise. Dividing the imposed shear stress by shear rate obtainedby the differentiation of velocity profile is a less noisy operationthan using the calculated shear stress. Thus, normal-stress coeffi-cients are noisier since normal stresses have to be calculated.

Figure 12. Relative viscosity ηr of the suspension vs Pe number.

Figure 11. Volume fraction profiles across the channel in Poi-seuille flow with different flow rates at φ= 0.26.

Figure 13. The shear-dependent first and second normal-stresscoefficients, Ψ1 and Ψ2, scaled into dimentionless units as Ψ1/Ψ1

0,Ψ2/Ψ20, and λ0 _γ.

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The relative viscosities of different concentrated suspensions asa function of Peclet number are plotted in Figure 12, where resultsfrom both LEC and RPF are presented. The symbols denoted by“direct” are extracted from the RPF simulations without regardfor the cross channel concentration distribution. We find notice-able disagreement compared to results from theLEC simulations.Also, the “direct” curves are not continuous for the two flowrates. These discrepancies result from the nonuniform concentra-tion of colloidal particles due to their stress-gradient drivenmigration across the channel (refer to section 3.4). Since viscosityis a function of both concentration and shear rate, we need toincorporate the local volume fraction information of colloidalparticles in order to correct for its effect on viscosity. This can bedone through a superposition procedure. We form our super-position shift factor as

aφ ¼ η0r ðφbulkÞη0r ðφlocalÞ

ð20Þ

where φlocal is the local volume fraction, φbulk is the bulk volumefraction and ηr

0 is the zero-shear rate relative viscosity determinedby eq 17. The shift for ηr at different shear rates can be preformedas ηraφ and for the shear rate as _γ/aφ. The curves denoted by“superposition” are thereby obtained based on the local volumefraction profile and the superposition shift factor aφ. We find thatafter superposition, theRPF curves become continuous and agreewell with the LEC results. These results demonstrate that densesuspensions (e.g., at φ≈ 0.5) are significantly shear-thinning withNewtonian limiting behavior at both low and high shear rates.However, for relatively dilute suspensions (e.g., at φ ≈ 0.3), thenon-Newtonian behavior is not significant and only slight shear-thinningwas observed. These observations are consistentwith theexperiments of Krieger et al.63

The superposition shift was also performed for the normal-stress coefficients obtained inRPF simulations asΨ1aφ

2 andΨ2aφ2.

After the superposition,we further normalized the data asΨ1/Ψ10,

Ψ2/Ψ20, and λ0 _γ, where Ψ1

0 and Ψ20 are the plateau values of

normal-stress coefficients at low shear rates, λ0=Ψ10/(2ηr

0ηs) is themean relaxation time and ηs is the solvent viscosity. Figure 13shows the normalized first and second normal-stress coefficientsof suspensions at different concentrations plotted as functions ofλ0 _γ.We note that the RPF results agree well with the LEC results,but discrepancies exist between different concentrations of sus-pensions, especially for the second normal-stress coefficient. Thedeviations can be explained by large errors in the plateau values ofnormal-stress coefficients. These errors result fromthe noise in thecalculated normal stresses, especially in the second normal stress,and the numerical differentiation errors of local shear rates ( _γ) inthe neighborhood of the Poiseuille flow centerline. And they arefurther magnified after dividing the normal stresses by _γ2 toobtain the normal-stress coefficients.

To compare with the results from other work, we also show thefirst and second normal-stress differences scaled by the solventviscosity and shear rate in Figure 14. We find that our results arecomparable to those of Stokesian Dynamics simulations,17 buthere we also include predictions in the low shear-rate region(Pe < 10�) which we obtained using the RPF approach.

4. Conclusion

The aim of this work has been to demonstrate that a newformulation of DPD model allows accurate and economicalsimulations of colloidal suspensions by representation of thecolloidal particles as single DPD particles. We found thatemploying exponential conservative interactions for colloid-solvent and colloid-colloid particles yield a well-dispersedphase of colloidal particles differentiated from solvent particlesby size, and by essentially hard sphere repulsions betweencolloidal particles as demonstrated by their radial distributionfunction, g(r).

In particular, we examined in detail the rheological behavior,the microstructure, and shear-induced migration of colloidalsuspensions in plane shear flows (Couette and Poiseuille) over arange of volume fractions and flow rates. The relationship ofrelative viscosity versus volume fraction at the low shear-rate limit(Pe e 10-1) was computed in good agreement with availableexperimental data and empirical correlations. The shear-depen-dent viscosity and the first and second normal-stress differencesand coefficients were also investigated at different concentrationsin both Couette and Poiseuille flows. When the suspensionunderwent higher shear-rate flow, it displayed shear-thinning.

Figure 14. The shear-dependent first and second normal stressdifferences scaled by ηs _γ.

(63) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 111–136.

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In this work, the limit of attainable Peclet numbers is in the orderof 100. However, this is not a fundamental limitation and can bereadily increased by different interaction envelopes and/or in-creasing the cutoff radius for particle interactions at the cost ofsmaller time step.

Since the rheological behavior of suspensions is closely corre-lated with the microstructure of colloidal particles, we alsoexamined the microstructure of colloidal particles at differentshear rates. The structure pattern was measured by the pairdistribution function on different planes. We observed that atlow Pe numbers (Pe , 1) the microstructure was isotropic andrevealed a ring-like pattern, and as the Pe number increased thestructure remained isotropic on the planes perpendicular tothe flow plane. However, in the flow plane the structure becamedistortedwith a depletion of particles along the extensional axis ofthe flow field and concentration of particles along the compres-sional axis. These findings agree well with experimental observa-tions1,50,52,53 and with predictions by Stokesian Dynamicssimulations.17,51

For dense suspensions (φ > 0.5), a flow-induced locallycrystallized structure of colloids was found in the suspension atlow shear rates, but no long-range fully ordered structure wasfound until the volume fraction was near the close packingfraction(φ=0.64). Our simulations demonstrated flow-inducedstring-like structures of colloids formed at the close packingvolume fraction at relatively low shear rates (Pe < 1.0), thestrings being arranged in an hexgonal pattern. This observation isconsistentwith previous studies.1,4 Such flow-induced anisotropicorganization occurred in dense suspensions, in turn, affected therheological properties, thereby altering the Couette flow profilefrom linear to nonlinear; this led to shear banding phenomenon.Upon sufficient increase of the shear rate, the crystallized struc-turemelted into the disorderedphasewith restoration of the linearvelocity profile.

The shear-induced migration of colloidal particles was inves-tigated with the suspension in Poiseuille flow at φ=0.26. Thecomputed migration effect in terms of the volume fraction profileof colloidal particles is comparable with experimental dataand model predictions.12,61 Furthermore,we found that the mi-gration effect was enhanced by increasing the flow rate, whichis a typical difference between Brownian and non-Browniansuspensions.

Finally, we quantified the role of the angular momentum andtorque in the hydrodynamics of colloidal suspensions by simulat-ing the rheological behavior of colloidal suspensions through thestandard DPD approach with central dissipative interactionsand thereby linear momentum only, preserving other simulation

details unchanged. We noted that without angular variables thestandard DPD model was unable to simulate the correct rheolo-gical properties of nondilute suspensions (φg 0.2). Furthermore,we did not observe the shear banding by the standard DPDmodel, even after the ordered structure of colloids was formedwhen the volume fraction of suspension approached or evenexceeded the close-packing fraction (φ=0.64).

With regards to computational complexity, in the presentDPDmodel the computational cost is proportional to the total numberof particles in the system (o(N)), which is about 10 times thenumber of suspended particles, Nc, for an average volumefraction. Hence, the speed-up factor compared to the conven-tional method typically using 200 DPD particles for each sus-pended particle is a factor of about 20. It is also interesting;although not straightforward;to compare with the cost ofStokesian Dynamics (SD) that achieves very accurate results forBrownian suspensions. Since we do not have access to any of theSD codes currently, we can attempt to provide a rough estimate.To this end, we note that the cost of the conventional StokesianDynamics scales as o(C1Nc

3) whereas the cost of the acceleratedStokesian Dynamics is o(Nc

1.25lnNc) .64 However, the heavy over-

head associated with the accelerated version is expressed by thevalue of the constant C2, which is about 4 orders of magnitudegreater than C1 in the scaling of the conventional SD (This resultis based on the break-even point reported in ref 64, which is aboutNc=300.) Hence, indirectly we can deduce that the current DPDmodel is possibly up to 3 orders of magnitude faster than theaccelerated version of Stokesian Dynamics, however a directcomparison of costs is currently missing. Also, other indirectcosts associated with both methods may change these estimates.

In conclusion, the above results in terms of accuracy andefficiency support the use in large-scale simulations of the newDPD formulation that represent a colloidal particle with a singleDPD particle. The advantage of this approach is that theessential physics of colloidal suspensions is captured correctlyand economically. In addition, it is a very general method forunbounded or confined domains and will allow, for example,economical studies of polydispersed colloidal suspensions anddispersion in viscoelastic media.

Supporting InformationAvailable: The full visualization ofthe shear-induced transition from disorder to 3D string-likestructures of colloidal particles projected on different planesat Pe=0.1 and φ=0.64. This material is available free ofcharge via the Internet at http://pubs.acs.org.

(64) Banchio, A. J.; Brady, J. F. J. Chem. Phys. 2003, 118, 10323.