Rheology and dynamics of sheared arrays of colloidal particles

Rheology and dynamics of sheared arrays of colloidal particles Jeffrey J. Graya) and Roger T. Bonnecazeb) Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712-1062 (Received 14 January 1998; final revision received 21 April 1998)

Synopsis Concentrated suspensions of colloidal particles undergo dynamical microstructural transitions under shear. During the transition from oscillating face-centered cubic twin structures to sliding layer structures, the system can exhibit hysteretic and discontinuous rheology as the shear rate is varied. We capture this behavior with a dynamic simulation of a sheared lattice of non-Brownian spherical particles with screened electrostatic interactions and hydrodynamic interactions determined using the Stokesian dynamics approximation. Rheological data are determined for a range of volume fractions, electrostatic screening lengths and shear rates or shear stresses. In controlled stress simulations, static yield stresses are observed. In controlled shear rate simulations of certain lattice orientations, plateau viscosities are observed at high and low shear rates with a high to low shear rate plateau viscosity ratio ranging from 1.4 to 2.2. Large viscosity transitions with hysteretic-like rheology are observed only in controlled shear rate simulations of face-centered cubic ~111! layers sheared parallel to the ^ 211& direction with full representation of the hydrodynamic particle interactions. Rheological curves collapse when stresses are scaled by the elastic modulus and shear rates by the elastic modulus divided by the high-shear-rate limiting viscosity. The magnitude of the hysteretic viscosity jump and the scaled critical stresses match experimental values. © 1998 The Society of Rheology. @S0148-6055~98!00205-3#

I. INTRODUCTION Colloidal suspensions, composed of particles typically ranging from 5 nm to 5 mm in size, are important in film, coatings and pulp manufacture, emulsion technology, environmental processes, and biological systems. In addition, certain new nanostructured materials for applications such as photonic devices obtain their properties from specific, highly ordered spatial arrangements on a colloidal scale. As technological interest in colloids grows, an understanding of both the mechanisms of microscopic structure formation and the relations between structure and properties becomes increasingly important. One problem at the crux of structure–property interrelationships is the rheology of dense suspensions. Viscosity is often strongly influenced by particle arrangement, especially at high volume fractions, and flow—by definition—alters the very microstructure that determines flow properties. In this study, we investigate ordered structural transitions by dynamic simulation of a sheared array of particles. We observe transitions of trajectory pathways and relate them to rheological data. a!

Electronic mail: [email protected] Author to whom all correspondence should be addressed ~Electronic mail: [email protected]!.

b!

© 1998 by The Society of Rheology, Inc. J. Rheol. 42~5!, September/October 1998

0148-6055/98/42~5!/1121/31/$20.00

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FIG. 1. Flow phases of a sheared colloidal suspension of electrostatically-repulsive particles: ~a! at low shear rates, the black particle, representative of a second hexagonally close-packed layer of particles, hops between interstices of the layer below; ~b! at higher shear rates hexagonal layers slide over each other; ~c! at still higher shear rates the particle microstructure becomes disordered.

Flow/structure relationships have been studied for some time. Concentrated suspensions of strongly repulsive colloidal particles are known to undergo dynamic phase transitions under shear. Scattering experiments performed by Ackerson and co-workers @Ackerson et al. ~1986!; Ackerson ~1990!# have elucidated the average suspension microstructural states, and a general scheme for microstructure and rheology has begun to emerge. At rest, electrostatic repulsions between particles and, at high concentrations, local entropy drive the system to a crystalline state where all particles are maximally separated @Pusey and van Megen ~1986!#. Depending on the subtleties of the particle interactions, the initial particle configuration and the system boundaries, this minimal free energy state is a body-centered cubic ~bcc!, face-centered cubic ~fcc! or hexagonally close-packed ~hcp! arrangement. Upon shearing, the system is removed from equilibrium, and the regular structure is necessarily altered. At low shear rates, a layered structure is retained and particles zigzag between interstices of the layer below such that an fcc crystal sheared parallel to a close-packed ~111! plane alternates between two mirrorimage or ‘‘twin site’’ configurations ~Fig. 1!. As the shear rate is increased, the layers separate @Tomita and Van de Ven ~1984!# and slide directly over each other, and the effective viscosity decreases. Above a second critical shear rate, the layered structure becomes unstable, and a disordered system results @Hoffman ~1974!; Hoffman ~1998!#. This latter transition is responsible for the well-known phenomenon of shear thickening. While the shear thickening transition has been heavily studied by experiment @for a review, see Barnes ~1989!# and simulation @e.g., Boersma et al. ~1995!; Dratler et al. ~1997!#, the first transition, of technological interest because it involves two ordered states, is less understood. The experimental studies of Chen et al. ~1994a!, Chen et al. ~1994b!, Chow and Zukoski ~1995!, and Imhof et al. ~1994! focused on both the structural and rheological nature of this transition in dense suspensions of latex and silica particles ranging in diameter from 50 to 250 nm. Neutron scattering patterns @e.g., Fig. 8 in Chen et al. ~1994a!# indicate that during the transition the average reciprocal lattice vectors show significantly reduced structure. Chen and co-workers label the transition regime ‘‘polycrystalline,’’ and they visually observe small ordered regions separated by grain boundaries. Rheological data @e.g., Figs. 4–6 in Chen et al., ~1994a!# highlight differences between controlled shear stress and controlled shear rate experiments. Interestingly, certain shear rates were not observed in controlled stress experiments since the shear rate jumped when the microstructure of the suspension changed. In this case, a plot of the effective viscosity versus stress is discontinuous. In controlled rate of strain tests over the same regime, a maximum in the shear stress as a function of shear rate was observed, followed by an anomalous region where stress decreased with increasing shear rate. In addition, increasing and decreasing rate-of-strain sweeps showed hysteresis of the transition point. Finally, metastable high stress states were observed in controlled shear

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rate experiments: high stress polycrystalline phases were observed for a time before the microstructure changed and the stress dropped. Other experiments on seemingly similar systems do not show explicitly anomalous rheological behavior, although there are some indications that the same phenomena are present if somewhat concealed. For example, in Fagan and Zukoski’s ~1997! study of charge-stabilized silica particles using a constant stress rheometer, there is no anomalous region where shear stress decreases with increasing shear rate, but the stress-strain rate curves show a distinct bend at a critical strain rate. Laun et al.’s ~1992! survey of several latex systems also fails to observe an anomaly, although certain samples show a sharp drop in viscosity at a critical shear stress. Laun et al.’s ~1992! data indicate that monodisperse particles are necessary to observe sharp scattering peaks, and it is likely that strict particle size control is necessary to observe the combined structural and rheological transitions observed by Chen and co-workers. The works of Kaffashi et al. ~1997! and Mackay and Kaffashi ~1995! also explore the low shear rate regime with concentrated polymethylmethacrylate suspensions. They do not observe an anomaly, but their rheological data were collected for a sparse set of shear rates, and it is unclear whether the absence of an anomaly is due to the system studied or to the lack of resolution in the rheological curves. Many microscopic physical factors can contribute to the structure of colloidal systems, including long-ranged repulsive electrostatic forces, short-ranged attractive van der Waals forces, random thermal forces, and the externally imposed shear forces. Each of these forces acts over certain time and length scales. In the experimental studies that clearly show a rheological anomaly, particle diameter ranged from 50 to 250 nm. In this regime, Brownian, viscous and colloidal forces dominate. Inertia is negligible due to the small size of the particles, and double-layer repulsive forces keep particles separated beyond the interaction length of van der Waals forces. Chow and Zukoski ~1995! and Imhof et al. ~1994! collapsed their data by scaling stresses by the elastic modulus and shear rates by the elastic modulus divided by the fluid viscosity. This scaling suggests that the physical effects that control the phase transition are derived from electrostatic forces, and we concentrate on these effects in this study. Other theoretical and numerical attempts have been made to understand the rheology of lattices and colloidal systems in general. Both Nunan and Keller ~1984! and Zuzovsky et al. ~1983! determined viscosity for regular arrays of spheres, but only for fixed cubic lattices of particles without colloidal forces. Recent Stokesian dynamics simulations, such as those of Brownian particles by Phung et al. ~1996!, have displayed ordering phenomena and shear thinning. Phung and co-workers did not include double-layer forces, however, and the hysteretic shear thinning discontinuity was not observed. Diffusivity data from these simulations showed that the Brownian shear thinning is not related to the ordering phenomena, and with their small system sizes (N 5 27), the suspensions shear thickened at flow rates slightly beyond those which induced an ordered phase. Rastogi and Wagner ~1996! and Wilemski ~1991! used nonequilibrium Brownian dynamics with large numbers of particles ~up to N 5 43 000 by Rastogi and Wagner! and observed the formation of a string phase and discontinuous shear thinning. Due to thermal forces, however, their simulations exhibited disorder at low shear rates, rather than the crystallites observed by Chen and co-workers, Chow and Zukoski, and Imhof and co-workers. Stevens and Robbins ~1993! performed nonequilibrium molecular dynamics simulations on a system of particles that exhibited fcc or bcc order at rest. Neglecting hydrodynamic interactions, they observed shear melting and a reentrant solid phase at high shear rates. The stress and viscosity curves, however, are smooth and do not show signs of the

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FIG. 2. A lattice of colloidal particles subjected to shear.

microstructural transitions that have occurred. The origin of discontinuous shear thinning, then, is still unexplored. Since the shear thinning transition involves two ordered flow states, we have studied the shear thinning transition of a sheared suspension through dynamic simulation of particles placed on a regular lattice which evolves over time. Electrostatic interactions are included by assuming a screened Coulomb potential and neglecting multibody effects. Hydrodynamic interactions are approximated with the techniques of Stokesian dynamics, which preserve both the multibody effects and the strong lubrication forces. Brownian forces, while not negligible in experiments, are omitted for simplicity. Finally, the effect of boundaries is included by placing constraints on the allowed directions of motion of the lattice. In Sec. II we describe the lattice simulation technique including initial conditions and constraints upon the motion. In Sec. III we present rheological and structural results obtained for various initial lattices and for controlled stress and controlled rate of strain simulations. Static yield stresses are observed in controlled stress simulations, and critical stresses and shear rates are identified where phase transitions occur in controlled shear rate simulations. One particular lattice configuration produces hysteretic-like shear thinning behavior. Effects of volume fraction and ionic strength are shown to be weak when results are scaled using the elastic modulus and the high-shear-rate limiting viscosity. In Sec. IV we discuss the implications of the simulation and compare various formulations with each other and with experimental data. The observed static yield stress is shown to originate from electrostatic contributions, and the absence of the dynamic yield stress is explained by the pattern of storage and release of electrostatic energy. The occurrence of hysteretic-like rheology for only certain crystal orientations leads to speculation on the nature and dynamics of the polycrystalline phase. II. FORMULATION We study the dynamics of an array of colloidal particles subjected to a shearing motion of either constant shear rate g˙ or constant shear stress t. Shear flow is imposed in the e1 direction with the flow gradient in the e2 direction, as shown in Fig. 2. While the repulsive electrostatic forces act to restore the lattice to a state of minimum potential energy, the imposed shear continually moves the system away from equilibrium. The

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lattice trajectories that result from these competing forces are determined, and quantities such as effective viscosity, normal stresses, and hydrodynamic dissipation are calculated. Consider an infinite array of spherical particles located at the set of points

$y:y 5 n 1 x1 1n 2 x2 1n 3 x3 ;

n 1 ,n 2 ,n 3 PZ% ,

~1!

where Z is the set of integers. The three column vectors xi are the basis vectors of the lattice which is represented succinctly by the matrix X 5 @ x1 x2 x3 # .

~2!

The neutrally buoyant particles have a radius of a and are suspended in a fluid of viscosity h s and density r. The Reynolds number Re 5 r g˙ a 2 / h s or r t a 2 / h 2s is assumed to be small so that inertial forces are negligible and Stokes’ equations are valid. The volume fraction of particles f is 34 p a 3 /det X, where det X gives the volume of a unit cell of the lattice. Repulsive electrostatic interactions between the colloidal particles are approximated by a sum of two-body potentials: V~X! 5 n

(i V i ~ j i ! ,

~3!

where V is the energy of the lattice per unit volume, n 5 3 f /4p a 3 is the number density of particles, the sum is over all particles other than the reference particle, and V i is the two-particle potential function with j i representing the gap width between the reference particle and particle i. The two-particle potential is approximated with the Debye– Hu¨ckel equation 2 2

Vi~ji! 5 4p«a c

e2kji ri

,

~4!

which is valid for particles with surface zeta-potentials c , 50 mV @Verwey and Overbeek ~1948!#. « is the permittivity of the suspending medium, and r i 5 j i 12a is the distance between particle centers. The Debye–Hu¨ckel parameter or inverse screening length k is related to the ionic strength of the solution by 2

k 5

e2

ekT

( niz2i ,

~5!

where e is the charge of an electron, k is Boltzmann’s constant, T is temperature, the sum is over all ionic species i, and n i and z i are, respectively, the molar concentration and the valence of ionic species i. Since the electrostatic energy per unit volume V has the same dimensions as stress, its coefficients are used to scale the stresses and time in the problem. Hereafter, energy and stress are scaled by 3« c 2 /a 2 , time by h s a 2 /3« c 2 and length by a. These constants depend only on the nature of the particles and fluid, and leave the volume fraction and screening length to be varied as parameters. Due to the symmetry of the regular lattice, the total repulsive electrostatic force on each particle is zero. However, there is a finite stress on the lattice that acts to distort it affinely toward a lower energy configuration. This electrostatic stress is the first moment of the individual forces acting on the particle, or equivalently, the derivative of the potential of the lattice V with respect to the symmetric strain tensor E. The volumetric electrostatic stress is thus defined in terms of the lattice configuration as follows:

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FIG. 3. Base planes for three initial lattice configurations: ~a! bcc ~110! planes sheared in the ^ 111& direction, denoted bcc, ~b! fcc ~111! planes sheared in the close-packed ^ 110& direction, denoted fcc1, ~c! fcc ~111! planes sheared in the ^ 211& direction, denoted fcc2.

n^xF& 5

(i xi

S D 2

]V

] xi

52

]V ]E

5f

(i e 2 k j i

S D k1

1 xi xi

ri

r 2i

,

~6!

where the sum is again over all particles other than the origin particle and xi is the vector from the origin particle to particle i. The use of the notation n ^ xF& for the electrostatic stress tensor is consistent with standard use in suspension theory and does not imply the dyadic product. Other approximations for the electrostatic potential of interacting spheres can be used in place of Eq. ~5!. Simulations performed using the Derjaguin approximation, which is appropriate for small interparticle distances scaled by k, yield results which also collapse onto the scaled rheological curves presented later. The screened Coulomb potential of Eq. ~5! is used for results presented here. Three initial lattice configurations are used: body-centered cubic ~110! planes sheared in the ^ 111& direction, denoted bcc; face-centered cubic ~111! planes sheared in the close-packed ^ 110& direction, denoted fcc1; and face-centered cubic ~111! planes sheared in the ^ 211& direction, denoted fcc2. Base planes for these lattices are shown in Fig. 3. Layers are stacked in the shear gradient direction, and small perturbations are sometimes added to break symmetries. Note that the bcc and fcc1 lattices are sheared parallel to a close-packed direction of the lattice. The dynamics of the lattice are calculated using the equation dX dt

5 G–X,

~7!

where G is the deformation gradient tensor and t is time. This equation is integrated numerically using the Cash–Karp adaptive timestep variation of the Runge–Kutta method @Press et al. ~1992!#. After each timestep, the basis reduction algorithm of Lenstra et al. ~1982! is called to ‘‘reset’’ the lattice basis vectors, if necessary. That is, if one of the basis vectors has been excessively distorted as the flow convects it away from the reference particle, it is replaced with an equivalent lattice vector that is closer to the origin. The deformation gradient tensor G is determined using the particle suspension constitutive law, or stress balance, developed by Batchelor ~1970!: S 5 2E1n ^ S& 2n ^ xF& ,

~8!

where S is the ~externally applied! bulk stress tensor of the suspension, E 5 21 (G 1GT ) is the symmetric part of G, and S is the particle stresslet. Since 2E is the stress due to the deformation of the fluid and n ^ S& is the additional stress due to the presence of the particles, this equation balances external stress, fluid and particle hydrodynamic stress, and electrostatic stress. Brackets indicate averages taken over all particles. Since each

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particle in the lattice is equivalent, the average is obtained by simply evaluating the relevant quantities on one reference particle, and the brackets are omitted hereafter. In the Stokes approximation, the particle stresslet S is linearly related to E through a resistance tensor that encompasses the hydrodynamic interactions: S 5 RSE :E.

~9!

The hydrodynamics are approximated with the techniques of Stokesian dynamics, which includes both far-field ~multibody! interactions and near-field ~lubrication! interactions @Brady and Bossis ~1988!#: ˆ 21 1R2B 2R2B` . ~10! RSE 5 M SE

SE

SE

ˆ 21 M SE

is the inverse of the leading multipole approximation of the multibody moHere, bility tensor, the resistance tensor R2B SE includes lubrication effects, and the resistance 2B` tensor RSE subtracts the leading multipole terms from the lubrication term to avoid double counting. Computation of the mobility tensor is accelerated with the summing technique of Ewald described by O’Brien ~1979!. The two-body interactions are summed over all particles within 4a using the formulas compiled by Kim and Karrila ~1991!. When the interparticle distance 0 , j i , 0.1, the resistance parameters are calculated from asymptotic lubrication formulas, and when 0.1 , j i , 4, values are interpolated from tables computed by Kim and Karrila’s publicly available boundary element code. Returning to the development of the dimensionless stress balance of Eq. ~8!, the total stress acting on the lattice T is now defined as the bulk stress plus the electrostatic stress, and the fourth-order tensor RTE is created from RSE to relate T and E: T [ S1nxF 5 2E1nS 5 2P:E1nRSE :E 5 RTE :E,

~11!

where the fourth-order idem tensor P is defined by P i jkl 5 21 ( d ik d jl 1 d il d jk 2 32 d i j d kl ). Using the fact that E is traceless and ignoring the isotropic part of the various stress tensors, an inverse of RTE can be constructed. The inverse is used to separate E into contributions arising from the electrostatic stress and from the externally applied stress: E 5 R21 TE :T 21 5 R21 TE :S1nRTE :xF

5 ES1ExF.

~12!

The dynamical model has now been described up to the specification of the externally imposed flow. The equations above relate the externally applied stress tensor S to the symmetric part of the deformation tensor E. It remains to specify at least five elements of those two symmetric traceless tensors and also to determine the projection of the symmetric E onto the total deformation tensor G. These choices are related to the boundary conditions of the simulated flow, and there are several plausible options. The experiments of Chen et al. ~1994a!, Chen et al. ~1994b!, and Chow and Zukoski ~1995! were performed in a Couette rheometer under controlled shear rate or controlled shear stress conditions. The walls of the rheometer were parallel to the flow direction. Therefore, one reasonable choice of G is:

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G5

F

E xF 11

g˙

0

0

E xF 22

0

0

E xF 23

E xF 33

G

.

~13!

Here the shear component G 12 5 g˙ is externally imposed, and the electrostatic relaxations are added in the normal and ~2,3! directions. Relaxation in the normal directions allows the crystal to ‘‘breathe’’ during deformation, that is, to expand and contract in different directions while conserving volume. This procedure is similar to the twodimensional simulations of Tomita and van de Ven ~1984! which allow layers to separate as they slide over each other. The ~2,3! component relaxation allows stacked layers to slide in the vorticity direction, effectively steering left and right as they move forward over the layer below to produce, for example, the flow behavior illustrated in Fig. 1~a!. We have investigated several configurations of G in addition to that shown in Eq. 13. By removing the normal components of G, a restricted ensemble is formed. Layers cannot separate, and the only motion allowed is the lateral sliding necessary to produce xF the zig-zag trajectory. A more liberal configuration is obtained by adding E xF 13 and E 31 in the appropriate positions of G. These components allow stacked layers to rotate relative to each other around the e2 axis. Since certain terms of G are forced to be zero in these configurations, the external contribution ES must cancel out the motions prescribed by the electrostatic stress. When the external stress is calculated with RTE :ES, the stress required to forbid these motions is then included. To simulate a controlled stress experiment, we analogously specify the same electrostatic contributions of ExF to G, but g˙ is determined to obtain the desired value of t 5 S 12 at each instant. The other components of the stress tensor are calculated subsequently. Hence, in the ‘‘controlled stress’’ simulations, only one component of the total stress tensor is specified and the other components are calculated based on the restrictions imposed on the deformation tensor. Now that the determination of the lattice trajectory is complete, the effective viscosity can be extracted using the standard definition: S12 h5 . ~14! 2E 12 Since S 12 is separable according to Eq. ~8!, the effective viscosity can also be divided into its hydrodynamic and electrostatic parts: 2E 121nS 12 hhyd 5 , ~15! 2E 12

hel 5

n ~ xF ! 12 2E 12

.

~16!

Finally, the double inner product of Eq. ~8! with E yields an energy balance: S:E 5 ~ 2E1nS! :E2nxF:E 5 T:E1 5 W 5 F1

]V ]E

dV dt

.

:E ~17!

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This balance simply states that the rate of work performed on the system by external forces W is equal to the rate of hydrodynamic dissipation F plus the rate at which electrostatic energy is stored in the lattice. III. RESULTS A. Rheology and lattice trajectories Simulations generally evolve into periodic motions with corresponding variations in the rheological properties. Results presented here represent steady-state time averages over several complete cycles. In general, low and high shear rate simulations achieve a steady cycle quickly; intermediate ~transitional! simulations require more time and

FIG. 4. Rheology of ~n! bcc, ~h! fcc1 and ~L! fcc2 lattices for controlled shear rate ~open symbols! and controlled shear stress ~closed symbols! simulations. Particle volume fraction f 5 0.33 and inverse screening length k 5 2. ~a! shear stress versus shear rate, ~b! effective viscosity vs shear stress.

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TABLE I. Normal stress plateaus for various lattices under controlled shear rate conditions with f 5 0.33 and k 5 2. Stresses are scaled by 3 ec 2 /a 2 . Low shear rates Lattice bcc fcc1 fcc2

High shear rates

S 112S 22

S 222S 33

S 112S 22

S 222S 33

0.013 0.0063 0.0009

0.0049 20.0041 20.002

0.021 0.0136 0.012

5.431024 20.0081 20.04

stricter numerical integration parameters. Convergence was checked for all simulations by reducing the maximum time step interval and error tolerance until differences between runs were no longer significant. Figure 4 shows the shear stress as a function of shear rate for the three initial lattices and for both controlled shear stress and controlled shear rate simulations with a volume fraction f 5 0.33 and an inverse screening length k 5 2. Controlled stress simulations show similar results for all three initial lattices. A static yield stress is observed, below which there is no flow. Static yield stresses are different for the three lattices: t sy 5 0.0626 for bcc lattices, t sy 5 0.0295 for fcc1 lattices, and t sy 5 0.0727 for fcc2 lattices. As the stress is increased beyond the yield stress, the suspension shear thins over half an order of magnitude of shear stress, after which a plateau in the viscosity is observed. Controlled shear rate simulations exhibit markedly different rheology than controlled shear stress simulations, and in addition, the bcc and fcc1 lattices are qualitatively different than the fcc2 lattice. For bcc and fcc1 lattices, a constant viscosity h 0 is observed ˙ ' 0.003 for fcc1 lattices! at which up to a critical shear rate ( g˙ * 1 ' 0.01 for bcc and g * 1 the suspension begins to shear thin. Above a second critical shear rate ( g˙ * ' 0.09 for 2

bcc and fcc1!, a second plateau viscosity h ` is observed, which quantitatively matches the high shear viscosity computed in the controlled shear stress simulations. Low shear rate to high shear rate plateau viscosity ratios are 1.4 and 2.2 for bcc and fcc1 lattices, respectively. The fcc2 lattice exhibits very different rheology. Shear stress increases approximately ˙ with the square root of shear rate up to a maximum stress t * 1 5 0.0477 when g * 1 5 0.0033. Stress then decreases to a minimum of t * 5 0.017 at g˙ * 5 0.0055. In this 2

2

anomalous regime, viscosity is a multivalued function of the shear stress. Above g˙ * 2 , Newtonian behavior is observed, and the viscosity matches that of the controlled stress simulations. Hydrodynamic dissipation and work follow similar behavior as the viscosity. Since the time average of dV/dt is 0 in a periodic process, Eq. ~17! requires that average rates of dissipation and work are equal, and they increase and decrease roughly as h g˙ 2 . Normal stress differences are independent of applied shear within high and low shear rate regimes, and change smoothly at the same shear rates where viscosity changes. High and low shear rate plateau normal stress differences are listed in Table I for controlled shear rate simulations. The first normal stress differences are positive for all lattices, while the second is positive for bcc lattices and negative for fcc1 and fcc2 lattices. Normal stresses in controlled stress simulations are similar. The remaining components of the stress tensor, the shear stresses S 23 and S 13 , oscillate in time but for most conditions average to zero.

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FIG. 5. Lattice trajectories projected into the flow-vorticity plane for sheared bcc lattices. Large circles show the initial locations and scale of particles in the x 2 5 0 plane, boxes ~h! show motions of the particle centers for the in-plane basis vectors. Lines show the trajectories of the reference particle in the sliding plane above. ~a!–~d! controlled shear rate simulations, ~e!–~g! controlled shear stress simulations. ~h! and ~i! show the location of the depicted simulations on the rheological curves selected from Fig. 4.

Lattice trajectories change with the applied shear rate or shear stress. Figures 5–7 show projections of the basis vectors for representative lattice trajectories of controlled rate of strain and controlled stress simulations for each lattice type; the run from which each trajectory is created is labeled on the accompanying rheological plots. The small motions of the in-plane basis vectors are represented with boxes; the third, out-of-plane basis vector, depicted with a line, shows the motions of the layer sliding above. In general, at low shear rates, zig-zag trajectories are observed; at high shear rates, straight trajectories are observed; trajectories at intermediate shear rates lie somewhere between. However, subtle differences in flow behavior exist between the lattice types as well as

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FIG. 6. Lattice trajectories for fcc1 lattices. ~a!–~e! constant shear rate simulations, ~f!–~h! constant shear stress simulations, ~i! and ~j! rheological curves.

between controlled shear rate and controlled shear stress runs. Controlled shear rate simulation trajectories for bcc crystals at low shear rates @Fig. 5~a!# are smooth, almost symmetrical sinusoidal curves. These oscillations decay @Figs. 5~b!–5~c!# and the peak is convected downstream as shear rate is increased until, in the limit of high flow rate, straight trajectories @Fig. 5~d!# are observed between strings of particles in the base layer. In contrast, controlled shear stress simulations at low flow rate @Fig. 5~e!# show an almost straight diagonal path until the reference particle passes between two particles below. Only at this point does the trajectory curve back toward the next interstice. Finally, there is a sharp corner as the particle begins its climb out of the interstice. As shear rate is increased @Figs. 5~f!–5~g!#, this angular path is softened toward the same limiting straight trajectory. fcc1 lattices show similar behavior for controlled rate experiments @Figs. 6~a!–

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FIG. 7. Lattice trajectories for fcc2 lattices. ~a!–~f! constant shear rate simulations, ~g!–~i! constant shear stress simulations, ~j! and ~k! rheological curves.

6~e!#: smooth sinusoidal paths decay in magnitude as shear rate is increased to a limiting straight trajectory. fcc1 lattices in controlled stress simulations at low shear stress @Fig. 6~f!# have trajectories, like those of the bcc lattice, with sharper corners. However, due to the additional symmetry of the fcc1 base plane, the two corners are mirror images of each other. As a result, an even zig-zag path is drawn rather than the asymmetric saw-tooth

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path of the bcc lattice. Again, as shear stress is increased, this path becomes smoother and straighter @Fig. 6~g!#, until the limiting straight trajectory is achieved @Fig. 6~h!#. The geometry of the fcc2 lattice produces more interesting trajectories ~Fig. 7!. A base-layer particle is located directly downstream of each interstice, so that at low shear rates, hopping between interstices creates sharp zig-zagging motions @Fig. 7~a!#. As the shear rate is increased @Fig. 7~b!#, the straight part of the trajectory extends toward the center particle before lateral perturbations grow large enough to create a sideways movement. Eventually, at increased shear rates @Figs. 7~c!–7~e!#, the straight part of the trajectory continues completely over the top of the particle in the base layer, and small swerving motions occur on the downstream side of that particle. Eventually, these swerving motions decay into a straight path that proceeds directly over the center reference particle @Fig. 7~f!#. The disappearance of lateral motions coincides with the beginning of the high shear rate plateau viscosity; the trajectory transition coincides with the multivalued viscosity regime. Trajectories for the fcc2 lattice in controlled stress simulations follow a different evolution. At low shear rates @Fig. 7~g!#, trajectories have sharp corners upon approaching a particle, but the path then curves toward the centerline of the next particle gap. As shear stress is increased @Figs. 7~h!–7~i!#, this trajectory straightens, again eventually passing directly over the center particle. B. Effect of volume fraction and screening length Rheological results are summarized in Table II for volume fractions f ranging from 5% to 42% and inverse screening lengths k ranging from 1 to 16. Note that critical shear rates and stresses, especially for the bcc and fcc1 lattices which both have smooth viscosity curves, are approximations chosen from a discrete set of runs. The maximum volume fraction f 5 0.42 was set because fcc2 lattice layers are unable to slide over each other at higher volume fractions under the restrictions of the deformation gradient tensor. Smaller k values require sums over a currently computationally prohibitive number of particles due to the long range of the interactions. On the whole, results are qualitatively similar to the runs described above. Yield stresses and critical stresses vary roughly exponentially with large to moderate volume fractions and obtain a maximum as inverse screening length is varied, following the theoretical variation of the elastic modulus according to van der Vorst et al. ~1995!: G0 ; ef. and G0 ; k2e2k, The shear or elastic modulus, defined by G0 5

]2V ]E 212

5

] n ~ xF ! 12 ] E12

,

~18!

is calculated explicitly for the bcc and fcc lattices. When stresses are scaled by the shear modulus G 0 and shear rates by G 0 / h ` , the rheological data collapse onto a master curve as shown in Figs. 8–10 and summarized in Tables III–V. In Fig. 10 it can be seen that ionic strength has a qualitative effect for fcc2 lattices at low shear rates. As k is increased ~corresponding to electrostatic forces becoming more short-ranged!, the low shear rate effective viscosity curve begins to flatten toward a

TABLE II. Summary of rheological data for various initial lattices, volume fraction, and inverse screening length. G 0 is the shear modulus, t sy is the static yield stress. t 1* , g˙ 1* , t 2* and g˙ * are critical shear stresses and shear rates. For bcc and fcc1 lattices, critical values correspond to the beginning and end of the shear thinning region. For fcc2 lattices, critical 2

Lattice

f

k

bcc bcc bcc bcc bcc

0.20 0.33 0.42 0.33 0.33

fcc1 fcc1 fcc1 fcc1 fcc1 fcc2 fcc2 fcc2 fcc2 fcc2 fcc2 fcc2 fcc2 fcc2

h0

h`

0.05 0.1 0.1 0.1 0.1

2.25 3.1 4.7 3.1 3.5

1.57 2.21 2.95 2.2 2.2

0.02 0.2 0.3 0.07 0.1

0.01 0.09 0.1 0.05 0.05

3.1 4.4 6.3 4.4 4.2

1.57 2.25 2.92 2.18 2.16

1.431024 0.0038 0.017 0.028 0.032 0.0076 0.021 0.0084 3.331024

1.531025 0.002 0.0055 0.007 0.008 0.0027 0.008 0.003 1.231024

G0

t sy

2 2 2 1 7.6

0.0916 0.326 0.571 0.160 0.183

0.0171 0.0626 0.115 0.0296 0.0332

0.006 0.03 0.04 0.014 0.02

0.003 0.01 0.01 0.005 0.005

0.08 0.2 0.3 0.2 0.2

0.20 0.33 0.42 0.33 0.33

2 2 2 1 7.6

0.107 0.366 0.630 0.163 0.255

0.00837 0.0295 0.0531 0.0126 0.0188

0.01 0.012 0.05 0.004 0.015

0.004 0.003 0.01 0.001 0.004

0.05 0.20 0.33 0.40 0.42 0.33 0.33 0.33 0.33

2 2 2 2 2 1 4 7.6 16

1.3431023 0.107 0.366 0.566 0.630 0.163 0.549 0.255 6.9031023

2.3531024 0.0199 0.0727 0.122 0.140 0.0312 0.106 0.0430 8.7631024

3.331025 0.013 0.048 0.077 0.090 0.023 0.060 0.025 9.131024

2.431025 0.0012 0.0033 0.005 0.005 0.0015 0.005 0.002 0.731024

t* 1

g˙ * 1

t* 2

g˙ * 2

— — — — — — — 60 30

1.13 1.57 2.25 2.75 2.93 2.25 2.25 2.25 2.25

SHEARED ARRAYS OF COLLOIDAL PARTICLES

values correspond the the shear stress extrema. h 0 and h ` are the limiting values of the viscosity for low and high shear rates. Critical stresses and shear rates for bcc and fcc1 lattices were difficult to determine due to the smoothness of the rheological curves, and tabulated values are approximate. Scalings are given in Sec. II.

1135

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FIG. 8. Collapsed rheology for bcc lattices for a range of volume fraction f and inverse electrostatic screening length k. Closed symbols represent controlled shear stress simulations, and open symbols represent controlled shear rate simulations. ~a! shear rate vs shear stress, ~b! shear stress vs effective viscosity. Data are scaled by the shear modulus G 0 and the high-shear-rate limiting viscosity h ` .

plateau, and values for h 0 are estimated for k 5 7.6 and 16. For small values of k, the effective viscosity seems to turn upward, possibly toward an infinite viscosity limit. However, these low flow rates are difficult to probe as the ratio of flow to electrostatic time scales becomes quite large, especially for cases with small elastic moduli which require very low flow rates even in the transition region. Therefore, an infinite zeroviscosity limit or dynamic yield stress cannot be argued conclusively. C. Effect of allowed motions Results presented in the preceding sections were obtained using the deformation gradient tensor G presented in Eq. ~13!. This tensor allows motions in the normal directions

SHEARED ARRAYS OF COLLOIDAL PARTICLES

1137

FIG. 9. Collapsed rheology for fcc1 lattices: ~a! shear rate vs shear stress, ~b! shear stress vs effective viscosity.

as well as the lateral (G 32) sliding direction. Several other configurations were considered. Adding the ~1,3! and ~3,1! elements to G allows layers to rotate about the gradient (e2 ) axis. In simulations under these conditions, bcc and fcc2 layers rotate toward the fcc1 position, that is, toward a hexagonally packed layer with a close-packed ^ 111& direction parallel to the flow. The steady-state rheological results are then quantitatively comparable to that of the fcc1 lattice presented above. Additional freedom can be introduced into the system only by adding the ~1,2!, ~2,1!, or ~2,3! electrostatic contributions to G; however, this proves problematic. The ~1,2! and ~2,1! pieces allow the layers to tip forward in the flow direction about the vorticity axis. At low flow rates where electrostatics dominate, the crystal rotates rigidly without deforming. At high flow rates, the layers shear and tumble simultaneously and particle

1138

GRAY AND BONNECAZE

FIG. 10. Collapsed rheology for fcc2 lattices: ~a! shear rate vs shear stress, ~b! shear stress vs effective viscosity.

TABLE III. Reduced rheological data for bcc ~110! planes sheared along a ^ 111& direction.

f 0.20 0.33 0.42 0.33 0.33 Avg.

k

t sy /G 0

t* 1 /G 0

g˙ * 1 h ` /G 0

t* 2 /G 0

g˙ * 2 h ` /G 0

h0 /h`

2 2 2 1 7.6

0.187 0.192 0.201 0.185 0.181 0.18960.008

0.07 0.09 0.07 0.09 0.1 0.0860.01

0.05 0.07 0.05 0.07 0.06 0.0660.01

0.9 0.6 0.5 1.3 1.1 0.960.3

0.9 0.7 0.5 1.4 1.2 0.960.4

1.4 1.4 1.6 1.4 1.6 1.560.1

SHEARED ARRAYS OF COLLOIDAL PARTICLES

1139

TABLE IV. Reduced rheological data for fcc ~111! planes sheared along a ^ 110& direction ~fcc1!.

f

k

t sy /G 0

t* 1 /G 0

g˙ * 1 h ` /G 0

t* 2 /G 0

g˙ * 2 h ` /G 0

h0 /h`

0.20 0.33 0.42 0.33 0.33 Avg.

2 2 2 1 7.6

0.0782 0.0806 0.0843 0.0773 0.0737 0.078860.004

0.09 0.03 0.08 0.02 0.06 0.0660.03

0.06 0.02 0.05 0.01 0.03 0.0360.02

0.19 0.5 0.5 0.4 0.4 0.0460.13

0.15 0.6 0.5 0.7 0.4 0.560.2

2.0 2.0 2.2 2.0 1.9 2.060.1

contacts eventually occur, despite lubrication forces. Adding the ~2,3! element of G @with or without the ~1,2! and ~2,1! pieces# allows the crystal to rock left and right about the flow direction axis. In this case, layers tilt, eventually destroying the layered structure of the lattice and leading to the same catastrophic results. Meaningful rheological results are difficult to draw from these simulations. Results are obtainable, however, when the motion of the lattice is restricted further than in the base case obtained with Eq. ~13!. Simulations were performed without the normal motions, so that the only allowed motion was that of the imposed shear and a sliding of the layers in a lateral (G 32) direction. This configuration produces qualitatively similar results to those presented in Sec. III A, with some quantitative differences in the viscosities. When normal motions are forbidden, the low shear rate viscosity is somewhat elevated, while the high shear rate viscosity is reduced in the approach to the transition point g˙ * 2 . D. Effect of hydrodynamics The approximation of hydrodynamic interactions may be altered by changing the computation of the hydrodynamic resistance tensors. To study the role of the hydrodynamics in creating the anomalous rheological behavior of the fcc2 lattice, simulations were performed with two alternative levels of hydrodynamic interactions. First, the 2B` R2B SE 2RSE portion of the resistance tensor was removed, leaving only the leading ˆ 21 . Second, multibody hydrodynamic interactions were removed multipole terms in M SE completely and the resistance tensor was assumed to be the constant fourth-order idem tensor that yields a relative viscosity of 11 25f. Results for these two series of runs are shown in Fig. 11. When only leading multipole interactions are included, the difference TABLE V. Reduced rheological data for fcc ~111! planes sheared along a ^ 211& direction ~fcc2!.

f 0.05 0.20 0.33 0.40 0.42 0.33 0.33 0.33 0.33 Avg.

k

t sy /G 0

t* 1 /G 0

g˙ * 1 h ` /G 0

t* 2 /G 0

g˙ * 2 h ` /G 0

h0 /h`

2 2 2 2 2 1 4 7.6 16

0.175 0.186 0.199 0.216 0.222 0.191 0.193 0.169 0.127 0.18660.028

0.10 0.12 0.13 0.14 0.14 0.14 0.11 0.098 0.13 0.1260.02

0.013 0.018 0.020 0.024 0.023 0.021 0.021 0.018 0.023 0.02060.004

0.025 0.036 0.047 0.050 0.051 0.047 0.039 0.033 0.048 0.04260.009

0.020 0.029 0.034 0.034 0.037 0.037 0.033 0.027 0.039 0.03260.006

— — — — — — — 25 13 —

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GRAY AND BONNECAZE

FIG. 11. Effect of hydrodynamic approximations. (n) full hydrodynamics via Stokesian dynamics, ~h! leading-order multipole hydrodynamic interactions only, ~,! dilute assumption ~constant resistance tensor.! ~a! shear rate vs shear stress, ~b! shear stress vs effective viscosity. Fcc2 lattice with f 5 0.33, k 5 2 and normal motions forbidden.

between the stress maximum and minimum is not as pronounced. Without multibody hydrodynamic interactions, the anomalous rheology disappears completely, even while the low and high shear behavior is quantitatively similar. IV. DISCUSSION The imposed motion in the model is balanced by hydrodynamic and electrostatic ~colloidal or double-layer! stresses, and therefore, the behavior of the system can be interpreted in terms of these quantities. At high shear rates or stresses, electrostatic contributions are negligible regardless of configuration, and viscosities are independent

SHEARED ARRAYS OF COLLOIDAL PARTICLES

1141

of applied shear and are similar to the results of Nunan and Keller ~1984!. Since a layered structure is required by the specification of the deformation gradient tensor G, clustering of particles and the resultant shear thickening at high g˙ is prohibited. These results are valid until the instability due to the vorticity of the flow that was examined by Hoffman ~1974, 1998! becomes important. Using the recent scaling analysis by Boersma et al. ~1995!, the critical shear rate for shear thickening is approximately one order of magnitude larger than the shear rates where our simulations show a transition. For small and intermediate values of applied shear, the effects of colloidal interactions are different depending on the details of the simulation. This is expected since we are performing nonequilibrium simulations which are sensitive to the controlled parameter and system history. In the following discussion, we first examine the controlled stress simulations at small applied shear stress and the controlled shear rate simulations for low and intermediate values of shear rate. In separate subsections we discuss the anomalous rheology of the fcc2 lattice and the normal stress differences, and finally we compare the simulation results to experiment. A. Controlled stress simulations and the static yield stress Under controlled stress conditions, flow ceases for small values of shear stress because the electrostatic stress barrier exceeds the applied shear stress. The static yield stress is the maximum electrostatic stress along the low shear trajectory of the system. For bcc and fcc1 lattices, this maximum occurs while ascending the potential hill as the third ~out-of-plane! reference particle passes between two particles in the base layer, near the sharp corner in the trajectory. For fcc2 lattices, the phenomenon is different. The maximum stress along the low shear trajectory does not occur as particle three squeezes between two base-layer particles, but when that particle is convected directly toward the ~center! base-layer particle that is directly downstream of an interstice. For 22.5 , x 31 , 21.1, the reference particle moves toward the centerline x 33 5 0 between the two upstream base layer particles. Reference particle three then approaches the center base particle directly, and here is where it encounters the greatest stress. This centering of particle three followed by a direct approach will prove crucial to all of the rheology of the fcc2 lattice. The scaled static yield stresses obtained by simulation ( t sy /G 0 ' 0.07– 0.2) are approximately an order of magnitude greater than stresses observed in experimental model systems, where t sy /G 0 ' 0.002– 0.03 @Chen and Zukoski ~1990!; Chen et al. ~1994b!; Chow and Zukoski ~1995!#. Grain boundaries and defects, present in the experimental system but not in the idealized simulated lattice, may account for the lower yield stresses, as is known to be the case for atomic solids. The relatively high static yield stress of lattices supports the hypotheses involving polycrystallinity and shear banding effects. When one crystallite can sustain the local stress, shear may occur in localized shear bands between crystal grains. B. Controlled shear rate simulations: low shear rate behavior Although simulated suspensions under controlled shear stress exhibit a static yield stress, simulated suspensions under controlled shear rate deform plastically with stresses well below t sy . Deformation below the static yield stress is possible because the shear stress is time-averaged in the controlled shear rate simulations. Thus, the stresses from the high stress configurations ~which stop motion altogether in the controlled stress simulations! are bypassed at the imposed rate of g˙ and the stress at that point is averaged

1142

GRAY AND BONNECAZE

FIG. 12. Shear stress vs time for an fcc1 lattice with f 5 0.33, k 5 2 and g˙ 5 0.001. Also shown are the hydrodynamic and electrostatic contributions to the stress, as well as the static yield stress and time-averaged shear stress ^ t & t .

equally with those of the lower stress configurations along the trajectory. Figure 12 shows the shear stress as a function of time for an fcc1 crystal with f 5 0.33, k 5 2 and g˙ 5 0.001, representative of the low shear rate regime. Note that the maximum electrostatic stress on this curve is just above the static yield stress observed in the controlled shear stress simulations. The electrostatic portion of the shear stress t el in Fig. 12 shows the electrostatic resistance to motion as the layers slide over the potential surface, storing and releasing energy. At high shear rates, this storage and release is fast and essentially symmetric: the storage of energy adds to the shear stress, and the release of energy compensates the external shear stress by an equal amount. For lower shear rates such as that depicted in Fig. 12, however, electrostatically induced motions occur on the same time scale as the externally imposed motion. The energy stored in the shear direction (G 12) is released by moving the lattice in a transverse (G 32) direction in addition to the forward (G 12) direction. In the stress versus time plot, this is exhibited as an offset in the curve: t el peaks at 10.032 and 20.027. The extra energy is dissipated into the medium, rather than assisting the shearing of the crystal, and the result is the increase in time-averaged viscosity. fcc2 lattices show an additional range of behavior at low shear rates, almost exhibiting a dynamic yield stress for low k and reaching a plateau viscosity for high k . The storage and dissipation pattern is again observed, but fcc2 lattices at low shear rates undergo rapid dissipation at one point during the cycle. As reference particle three approaches the center base layer particle, energy is stored in the lattice ~Fig. 13!. Because x 33 is near 0, the trajectory ascends the potential hill of the center base layer particle over a significant distance before the lateral instability grows large enough to move the particle. At that point, the strong electrostatic stresses move the lattice quickly, rapidly dissipating much of the stored energy without assisting the forward motion. This cycle of storage and dissipation and the accompanying k-dependent rheology can be likened to that observed in models of electrorheological fluids. In Bonnecaze and Brady’s ~1992! electrorheological fluid model, energy is stored in the straining of chains

SHEARED ARRAYS OF COLLOIDAL PARTICLES

1143

FIG. 13. Stored electrostatic energy contours for the fcc2 lattice and four low shear rate trajectories with f 5 0.33 and k 5 2. Contour interval is 0.12 « c 2 /a 2 .

of dielectric particles. Dissipation occurs when the chains break and reassemble. This snapping occurs at a critical strain, and the amount of energy that is rapidly dissipated during snapping is independent of the rate of strain and directly related to the dynamic yield stress. In our colloidal lattice simulations, the amount of energy stored depends on the shear rate. Figure 13 compares four low shear rate trajectories by plotting them over contours of stored electrostatic energy. The point of highest energy storage is at the corner before the center base particle; when the lattice trajectory moves to the side, the amount of energy dissipated is approximately the difference between this maximum and the energy level in the interstice. The point where the basis vector makes its sharp turn to the side changes depending on the shear rate since this point is convected downstream for higher shear rates. Thus, more energy is stored for higher shear rates. For k 5 2, this change in the amount of stored energy is such that the shear stress varies roughly as the square root of the shear rate. For lower values of k, electrostatic particle interactions are more longranged and the variation in the energy contours over the distance of a particle diameter are less pronounced. The amount of stored energy is less dependent on shear rate. A low shear rate limiting trajectory is approached within the range of g˙ explored, explaining the upward curve of the viscosity. For higher values of k, corresponding to shorter-ranged interactions, less energy is dissipated and more is recovered for the forward motion of the lattice, explaining the near-plateau viscosity. C. Controlled shear rate simulations: Transition behavior For bcc and fcc1 lattices, the transition between high and low shear rate behaviors is smooth. The straight trajectories observed at high shear rate are equivalent to the smallamplitude limit of the wavy trajectory observed at low shear rates. The transition, therefore, is gradual and continuous. In controlled shear rate simulations, all lattices show a transition at shear rates which produce stresses near the static yield stress. In Fig. 14, the viscosity is separated into the electrostatic and hydrodynamic components over a range of g˙ . The purely hydrodynamic contribution to viscosity is relatively constant over the range of shear rates, while the nxF contribution to stress becomes significant around the trajectory transition point. These data can be compared with those of recent experiments. Kaffashi et al. ~1997! and

1144

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FIG. 14. Total, hydrodynamic, and electrostatic contributions to suspension viscosity for ~a! fcc1 and ~b! fcc2 lattices with f 5 0.33 and k 5 2.

Mackay and Kaffashi ~1995! separate rheological data into parts of elastic and hydrodynamic origin. Although their elastic component includes both interparticle and thermal forces ~whereas our simulations neglect thermal contributions!, Fig. 2 in Kaffashi et al. ~1997! shows a remarkable qualitative resemblance to our Fig. 14~b!. Quantitative comparisons are not possible since Kaffashi et al. ~1997! do not report shear moduli. D. Anomalous rheology of fcc2 crystals In contrast to bcc and fcc1 lattices, fcc2 lattices have overlapping branches of the rheological curves and drastically different trajectories at the two extremes of g˙ . The stress versus viscosity curve @Fig. 7~k!# shows behavior that corresponds to the discontinuous and hysteretic rheology observed by Chen et al. ~1994a! and other experimental-

SHEARED ARRAYS OF COLLOIDAL PARTICLES

1145

ists. Although simulations at a given shear rate always show the same steady-state viscosity, one can imagine how the behavior of a real system may be hysteretic as the shear stress is varied. A system sheared at a low shear rate would be represented by a point on the low shear stress portion of the viscosity curve. Monotonically increasing the stress on this system moves the point along the curve until t approaches t 1* , whereupon the system must jump from the upper branch of the curve to the lower. Likewise, tracing the curve by monotonically decreasing stress from a high shear stress requires a jump from the lower branch of the curve to the upper when t approaches t * 2 . The exact jumping point may also depend on other microstructural details of the experimental system such as crystal defects. From Fig. 7 it is apparent that the viscosity transition is directly related to the microscopic lattice trajectory transition. Two factors that are necessary for the trajectory transition are the geometry of the lattice and an accurate representation of the hydrodynamic particle interactions. From the symmetry of the lattice, a straight trajectory with x 33 5 0, observed at high g˙ @Fig. 7~f!#, should be a solution for the lattice motion for all shear rates, however a portion of the straight trajectory is not stable. For x 31 , 21.15 particle three is ‘‘centered’’ as it passes between two base-layer particles, but for 21.15 , x 31 , 1.15, the center trajectory follows the peak of a potential hill and a slightly off-center particle is pushed away from x 33 5 0. The extent of these lateral motions is determined by the relative stresses due to electrostatics and the imposed shear, as shown in a scaling analysis below. Hydrodynamic interactions also play a key role in the transition. Multibody hydrodynamic forces are strongest during the unstable part of the trajectory when reference particle three is passing directly over the center base-layer particle. The hydrodynamic interactions hinder motion. Inclusion of multibody hydrodynamic interactions thus stabilizes the straight trajectory, making a crucial difference in allowing straight trajectories to occur at shear stresses low enough to undercut the upper branch of the viscosity curve. When multibody hydrodynamic interactions are neglected, the viscosity changes smoothly with t , as in Stevens and Robbins’ ~1993! nonequilibrium molecular dynamics simulations. Accurate hydrodynamic approximations are thus necessary to capture this suspension phenomenon correctly. We speculate that the behavior observed in the sheared fcc2 crystal could be a more general phenomenon applicable to imperfect lattices distorted by Brownian motion or even less-ordered systems. Consider three adjacent particles in a plane roughly perpendicular to the shear gradient, defining an interstice. A particle flowing over this gap will naturally be centered between the two most-upstream particles, which it encounters first ~Fig. 15!. If these two upstream particles are uneven in the flow direction, the passing particle will push the upstream particle with greater force, thus evening out the base-layer particles. The third base-layer particle, maximizing its distance between the first two particles, will likely be near the bisecting plane of the first two particles. The passing particle, after hopping the potential barrier into the interstice, will find itself pushing directly up toward the downstream particle. This exact situation is experienced by a sheared fcc2 crystal, and the resulting rheology may mimic the fcc2 behavior. E. Normal stress differences While normal stress differences ~Table I! have often been overlooked in suspension studies, recent controversies on issues such as shear migration have renewed interest in their importance. Early ideas on normal stress differences held that they should be zero due to the reversibility of Stokes’ flow, but experimental measurements @Gadala-Maria

1146

GRAY AND BONNECAZE

FIG. 15. A general scheme for extrapolating the fcc2 trajectory transition to nonlattice suspensions. The three white particles define an interstice in a layer perpendicular to the gradient direction, and the black particle is being convected in the layer above. ~a! As the black particle approaches the white particles, hydrodynamic interactions rotate the three white particles relative to each other. ~b! The black particle is centered between the first two white particles by electrostatic repulsions. ~c! After entering the interstice, the black particle approaches the last white particle head on, setting up a choice of a straight or zig-zagging motion, which presumably would be dependent on the shear rate, as in fcc2 crystals.

~1979!# and simulations @Brady and Bossis ~1985!# observe normal stress differences that increase linearly with the shear rate. In our formulation, the bulk stress tensor, which is a linear function of ES, includes contributions from the instantaneous applied shear rate as well as some of the electrostatically induced motions. Using Eqs. ~13! and ~12! to determine ES,

F

0

g˙ 1 21 E xF 21

S 5 RTE :ES 5 RTE : g˙ 1 21 E xF 21 E xF 13

0 1 2

E xF 23

E xF 13 1 2

G

E xF 23 . 0

~19!

The electrostatic contributions are independent of the shear rate, and when averaged over time or equivalently over the trajectory pathway, they are constant for a given trajectory. Since the trajectories are constant within the high and low shear regions, nxF contributions to normal stress differences are independent of shear rate. At high shear rates, g˙ would be expected to dominate, causing a linear growth of the normal stress differences. Upon closer inspection, normal stress differences in the lattice simulations at high shear rate oscillate in time due to the symmetries of the lattice and Stokes’ equations. While the amplitude increases linearly with the shear rate, there is no net contribution to the timeaveraged normal stress differences. The total normal stress differences are thus equal to the electrostatic contribution. For bcc and fcc1 lattices, the normal stress differences are not significant relative to shear stresses at and above the transitional shear rates. Interestingly, in fcc2 systems, normal stress differences are of the same order of magnitude as the shear stress at the critical point ( g˙ * 2 ,t* 2 ). These significant normal stress differences may play a role in the rheology and microstructural evolution of real systems at the transition. F. Scalings and comparison with experiment This numerical study was motivated by the experimental work of Chen et al. ~1994a! and subsequent similar studies @Chen et al., ~1994b!; Chow and Zukoski ~1995!; Imhof et al. ~1994!# which showed anomalous rheological behavior at low shear rates. Our study has reproduced this anomaly for certain configurations. In this subsection, we compare the simulation and experimental results. These comparisons lead to a scaling analysis to show that the critical stress is robust.

SHEARED ARRAYS OF COLLOIDAL PARTICLES

1147

Multivalued rheology is observed for simulated fcc2 crystals, and comparison with experiment indicates that the fcc2 zig-zag-to-straight trajectory transition is responsible for observed discontinuous viscosity behavior. Chow and Zukoski ~1995! determined the critical shear stress t * 1 /G 0 5 0.0460.008, averaged over a range of particle sizes and volume fractions. This value corresponds fairly well to the simulation averages of t* 1 /G 0 5 0.1260.02 and t * 2 /G 0 5 0.04260.009. Critical shear rates are more difficult to compare because of the introduction of a characteristic time. Chow and Zukoski scale their shear rates by G 0 / h s , while we scale the simulation data by G 0 / h ` . In fact, Chow and Zukoski’s reduced shear rates vary by a factor of 3 over different particle sizes and volume fractions. This is not surprising since various volume fractions have different plateau viscosities while solvent viscosity remains constant. Our scaling immediately compensates for this effect of volume fraction, but high-shear rate viscosity ( h ` ) is hard to define in experiment since real systems shear thicken at high shear rates. If we choose the minimum viscosity observed before shear thickening as h ` , experimental scaled 25 and g ˙ * h ` /G 0 critical shear rates from Chen et al. ~1994a! are g˙ * 1 h ` /G 0 5 1310 2 25 5 2.5310 . These numbers differ greatly from the simulation values of 0.020 and 0.032 obtained for the fcc2 lattice, and we will return to this discrepancy later in the discussion. Although our reduced shear rates do not match those of experiment, the simulations capture the rheological discontinuity correctly. The jump in viscosity between the two branches of the shear stress/viscosity curve is approximately one order of magnitude, depending on where the jump is measured. One consistent way to evaluate the magnitude of the transition is to compare the viscosities at the two identified critical points or extrema in the shear stress. The average simulation viscosity ratio of h * 1 /h* 2 5 4.6 compares well with the experimental ratio of 2.6 for Chen et al.’s ~1994a! data, which is representative of the other experiments. The overprediction of the viscosity jump can be attributed to the polycrystallinity of the experimental samples. Chen et al.’s ~1994a! scattering patterns indicate the predominance of crystallites with fcc1 alignment. However, the transitional behavior occurs in a polycrystalline regime with diffuse scattering peaks indicating many crystallite orientations including fcc2. The simulations show anomalous behavior only for fcc2 alignments. Measured viscosities are likely to be averages between crystals of orientation fcc1 and fcc2, as well as a range of orientations in between. The ratio of the experimental to simulation viscosity jump suggests that just over 50% of the crystallites might be oriented as fcc2. It is intriguing that in simulations that allow free rotation of layers, fcc2 lattices move towards fcc1 alignments. However, Imhof and co-workers’ ~1994! polarizing microscopy observations for pressure-driven flow between plates show that crystallites do not change their orientations during flow. Although there are differences between pressure-driven flow in a rectangular channel and flow in a Couette rheometer, it is likely that the boundaries of the rheometer or the grains themselves inhibit rotation of the crystallites, so that crystals oriented as the simulation bcc, fcc1 and fcc2 lattices hold that orientation. Returning to the critical stresses and shear rates, it at first seems unusual that the simulation results match experimental critical shear stresses and viscosity jumps but not critical shear rates. However, Mackay and Kaffashi ~1995! also found that rheological transitions occur consistently at a critical shear stress rather than at a critical shear rate, and this led us to develop a scaling analysis to show why this is true. Since the rheological behavior is linked to the microstructural evolution, we examine the trajectory pathway and use the lateral strain g 32 as an indicator of the dynamical state of the system: straight

1148

GRAY AND BONNECAZE

trajectory states have g 32 5 0, while zig-zag trajectory states have a nonzero g 32 . The lateral strain that occurs over half a cycle can be found by integrating G 32 between some t 0 and t 1 which correspond to the beginning and end of the unstable part of the straight fcc2 trajectory:

g32 5

E

t1

t0

~20!

G 32 dt.

The integration variable can then be changed using the fact that g˙ 5 d g /dt, and G 32 can be related to the electrostatic stress with Eq. 12 so that:

g32 5

E

21 g 1 ~ RTE :nxF! 32

g˙

g0

dg,

~21!

where g 0 and g 1 are the shear strains that correspond to the beginning and the end of the unstable part of the straight lattice trajectory. Inserting g˙ 5 t /h,

g32 5

E

21 g 1 ~ RTE :nxF! 32

h 21 t

g0

dg.

~22!

Since RTE is the tensor representation of the viscosity, the magnitude of R21 TE scales with the reciprocal of h. These two terms cancel, and we obtain the scaling

g32 ;

K

n ~ xF ! 32

t

L

;

G0

^t&

,

~23!

where the brackets indicate a mean value over the trajectory segment. n(xF) 32 is determined by the physical parameters relating to the colloidal interactions between particles, and its magnitude is proportional to the elastic modulus G 0 . Therefore, for a given suspension the applied shear stress t corresponds directly to certain trajectories including the critical trajectory at the transition point. The trajectories of the system and thus the rheological behavior can be correlated by t /G 0 . The specific range of g 32 that corresponds to the transition region has t /G 0 ranging from 0.04 to 0.12. Thus for electrostatically stabilized colloidal suspensions, the variation of the critical stresses with parameters such as particle size, ionic strength, or surface zeta potential can be determined using values of the elastic modulus from our calculations or from models such as that of van der Vorst et al. ~1995!. One can physically interpret the development of this scaling by considering the competition between stresses and rates in the lateral ~32! and shear ~12! directions. The displacement g 32 at the end of the cycle depends on the relative rates G 32 and G 12 5 g˙ . In Eq. ~22!, the viscosity terms cancel because the resistance to motion in the lateral and forward directions scale equivalently, even as various complications change the viscosity, and the applied shear stress is then the sole variable which controls the trajectory. Equation ~22! can be contrasted with Eq. ~21!, in which the viscosity term does not cancel and a knowledge of the hydrodynamic resistance tensor is required to predict the critical shear rate that corresponds to the transition point. Concisely, contributions to the viscosity will affect the critical shear rates but not the critical shear stresses. In the case of our simulations, the neglect of thermal effects is the most likely cause for the discrepancy in our critical shear rates. In experiments that observe the rheological anomaly, Peclet numbers (Pe 5 6 p h s a 3 g˙ /kT, where k is Boltzmann’s constant and T is

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temperature! range from 1027 to 10, and thermal shear thinning is known to occur up through Pe ' 1. Viscosities near the transition point are higher in experiment than in our simulations, accounting for the lower critical shear rates. From the analysis above, though, the simulations have still reproduced the correct trajectory transition, critical stresses, and viscosity jump. Finally, we note and try to explain some experimental cases where the rheological anomaly is not observed. First, Chen and Zukoski ~1995! and Imhof et al. ~1994! both see anomalous behavior disappear at certain parameter extremes. As particle volume fraction is lowered, viscosity discontinuities diminish and disappear near the volume fraction corresponding to the equilibrium solid/liquid boundary. Obviously, the lattice assumption is not valid for systems that are not ordered, but we also note that the extremely small elastic modulus for these systems indicates that the simulation-predicted transition point would occur at shear rates and stresses below those which were experimentally measured. Chow and Zukoski’s large particles (2a 5 463 and 514 nm! also have small elastic moduli and have similar experimental difficulties. The distinct inflections in the rheological curves presented by Fagan and Zukoski ~1997! occur at scaled shear stresses t /G 0 near 0.05 where the zig-zag to straight trajectory transition is expected, and it likely that the behavior is of the same origin. The qualitative difference in the rheology may simply be a result of the transition occurring over a wider range of applied shear rate due to the polydispersity of the system or differences in the details of the surface chemistry. The aqueous silica systems may have a somewhat different interparticle potential which could affect the balance of fcc and bcc crystallites. Laun et al. ~1992! unfortunately do not report the shear modulus of their systems, so it is difficult to determine whether the sharp shear thinning that they observe also corresponds to the transition. Assuming that the thinning does indeed arise from the same phenomenon, the broad transition region that they observe can be attributed to the polydispersity of their particles. Several questions remain open in the explanation of discontinuous shear thinning and structure/property relationships of electrostatically stabilized colloidal suspensions. Scattering experiments and simulations have shown the types of crystallites present at various shear rates, but explanations for the creation and destruction of crystal grains are lacking. Particularly intriguing is the creation of fcc2 crystallites from high or low shear states that contain predominately fcc1 crystallites, especially when lattice simulations that allow lattice plane rotation show fcc2 lattices moving towards fcc1 alignments. Explanations may require a multiple-crystallite mesostructure description which includes grain boundaries and shear bands. Simulation approaches to this problem will require overcoming the challenge of accurate hydrodynamic simulation of large numbers of particles. V. CONCLUSION Simulations have been performed on the shearing of a lattice of strongly repulsive colloidal particles for three lattice orientations and a range of particle volume fractions and inverse electrostatic screening lengths. In agreement with experimental studies, low shear rate or shear stress simulations show a dynamical transition from zig-zag to straight trajectory pathways, directly corresponding to the rheological transition. Constant shear stress and constant rate of strain simulations produce qualitatively different results. Constant stress simulations exhibit a static yield stress below which there is no flow, while controlled rate of strain simulations at low shear rates produce stresses orders of magnitude lower than the static yield stress. bcc and fcc lattices sheared parallel to close-packed directions exhibit two plateau viscosities; however, fcc lattices sheared parallel to the ~111! planes and along the ^ 211& direction exhibit an anomalous stress region where

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stress decreases with increasing shear rate, creating an hysteretic-like viscosity curve when plotted against shear stress. This anomalous behavior arises only when including multibody hydrodynamic effects which contain increased resistance for particles passing directly over each other, thus stabilizing the high-shear-rate straight trajectories. These results suggest that the zig-zag to straight trajectory transition of fcc2 crystallites embedded in a polycrystalline suspension is responsible for the hysteretic and discontinuous shear thinning observed experimentally. Also, the rheology of flowing dense suspensions can be explained by the bulk shearing of crystal grains rather than motion only at grain boundaries, although boundaries and rheometer geometry maintain grain orientation. Experimental studies with larger particles that experience lower Brownian forces would also help corroborate the findings of the simulations, and more work on the relation of crystals to shear bands, grain boundaries, and defects will help complete the descriptions of the shearing of strongly repulsive colloidal systems.

ACKNOWLEDGMENTS This research was supported by NSF Grant No. CTS-9358409. Graduate fellowships for one of the authors ~J.J.G.! were provided by NSF and the W. M. Keck Foundation.

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