Rheology of non-Brownian suspensions.

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Annu. Rev. Chem. Biomol. Eng. 2014.5:203-228. Downloaded from www.annualreviews.org by Florida Atlantic University on 08/19/14. For personal use only.

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Rheology of Non-Brownian Suspensions Morton M. Denn and Jeffrey F. Morris Benjamin Levich Institute and Department of Chemical Engineering, City College of New York, New York, New York 10031; email: [email protected], [email protected]

Annu. Rev. Chem. Biomol. Eng. 2014. 5:203–28

Keywords

First published online as a Review in Advance on March 21, 2014

viscosity, normal stress, friction, granular flow, continuum model, particle mechanics, particle migration

The Annual Review of Chemical and Biomolecular Engineering is online at chembioeng.annualreviews.org This article’s doi: 10.1146/annurev-chembioeng-060713-040221 c 2014 by Annual Reviews. Copyright All rights reserved

Abstract Suspensions of non-Brownian particles are commonly encountered in applications in a large number of industries. These suspensions exhibit nonlinear flow behavior, even in Newtonian suspending fluids under conditions where inertial effects can be ignored and linearity would normally be expected. We review the observed rheological behavior, emphasizing concentrated suspensions of spheres in Newtonian fluids, and we examine both particle-level and continuum approaches to describing the nonlinear behavior. Particleparticle nonhydrodynamic interactions appear to be important in concentrated suspensions. Continuum descriptions are not yet adequate to describe the observed behavior.

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INTRODUCTION


Rheology is the study of deformation and flow. From a pragmatic perspective, developing an understanding of the deformation of a material in response to imposed stresses provides information necessary to compute flows and related engineering quantities (e.g., forces and torques). At a more basic level, rheological measurement provides information about the microstructure under flow. The former purpose can often be satisfied with experimental observation. Theory is essential for the use of rheology to deduce structural information. Coupling of visualization with rheometry is now possible as a means to access spatial heterogeneities associated with flow. Our focus here is on the rheology of suspensions of solid particles in Newtonian suspending fluids in which the particle size is roughly one micrometer or greater, so that neither interparticle potentials nor Brownian forces are generally expected to be important in determining the macroscopic behavior. The linearity of the governing equations for the fluid motion when inertial effects are negligible leads to the expectation that all stresses in a suspension of smooth non-Brownian spheres should scale linearly with the rate of deformation. In fact, shear thinning and shear thickening are observed, and the suspensions exhibit normal stress differences. These nonlinear phenomena show that a rich rheology ensues even in the absence of non-Newtonian suspending fluids. Recent monographs addressing suspension rheology include References 1 and 2. The experience of the polymer processing industries provides a useful framework for examining suspension rheology. Polymer melts and solutions are complex, nonlinear fluids in which the evolution of chain orientation determines flow properties and, ultimately, the properties of the finished product. Development of continuum constitutive equations for the stress, together with numerical methods for solution of these equations, now makes it possible to calculate flows, polymer orientation, and stress distributions under conditions that approach those of processing interest (e.g., Reference 3). This development required decades of interactions between physical and computer scientists, engineers, and mathematicians. Many issues in suspension processing mirror those already addressed in polymer processing.

RHEOLOGY Viscosity A suspension is a heterogeneous material containing distinct phases, whereas rheology is a continuum concept and thus normally requires averaging over the phases to obtain properties appropriate to a homogenized, pseudo-single phase. The first development of such an effective viscosity of a dilute suspension of spheres in a Newtonian fluid was by Einstein (4, 5; see Reference 6), who theoretically established the result ηs = 1 + 2.5φ 1. ηr = ηs m for a suspension of noninteracting spheres. Here, η (often μ) denotes a viscosity; subscripts r, s, and sm denote relative, suspension, and suspending medium, respectively; and φ is the particle volume fraction. Figure 1, from an early but typical study of the viscosities of suspensions in a Newtonian suspending fluid (7), provides an overview of the rheological issues. Linearity is restricted to small values of the solid fraction φ. The relative viscosity increases rapidly with φ, appearing to grow without bound for φ > 0.5. Clearly, at large solid fraction, small changes in concentration will lead to large changes in effective viscosity. If nonhydrodynamic interparticle forces are absent, it follows from dimensional analysis that the relative viscosity of a neutrally buoyant suspension of smooth non-Brownian spheres in a 204

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40 30

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4 3 5–10 µ 30–40 µ 45–60 µ 90–105 µ

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ϕ Figure 1 Relative viscosities of noncolloidal glass spheres of various sizes in Arochlor as a function of concentration. From Lewis & Nielsen (7), reprinted with the permission of the Society of Rheology.

Newtonian fluid should depend only on the volume fraction; additional physics must be incorporated to establish dimensionless parameters that characterize the structural state. One commonly used relative viscosity is the Krieger-Dougherty equation, φ −[η]φm , 2. ηr = 1 − φm where [η] is the intrinsic viscosity and is dependent on particle shape; [η] = 2.5 for spheres. φm is the maximum packing fraction, an empirically determined parameter ranging from 0.5 < φ m < 0.74, with a default value of 0.64 that corresponds to random close packing. The Krieger-Dougherty equation reduces to Equation 1 for φ/φm 1. Equation 2 can be derived by assuming that the differential change in ηr that results from differentially increasing φ is [η]d φ/(1 − φ/φm ), where (1−φ/φ m ) is a crowding factor (8). The linearity of the equations for noninertial fluid motion implies that the shear stress should increase linearly with the shear rate, in which case the viscosity is expected to be a constant at any given solid fraction for non-Brownian suspensions in a Newtonian fluid. In fact, a shear rate– dependent viscosity is typically observed at φ > 0.40. Zarraga and coworkers (9) and Dai and coworkers (10), for example, observed shear thinning (a decreasing viscosity) at large φ; a typical result is shown in Figure 2a. However, Fall and coworkers (11, 12) observed shear thickening (an increasing viscosity) for 0.568 < φ < 0.605 (see Figure 2b). Neither phenomenon is understood completely, nor is it obvious why shear thinning and shear thickening should be observed on systems that are largely equivalent. Differences in surface interactions may be an explanation. www.annualreviews.org • Rheology of Non-Brownian Suspensions

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Shear rate (s–1) Figure 2 (a) Relative viscosity as a function of shear rate for 43-μm glass spheres in 70 weight% corn syrup and 30% glycerine. From Zarraga et al. (9), reprinted with the permission of the Society of Rheology. (b) Shear stress as a function of shear rate for density-matched 40-μm polystyrene spheres in silicone oil, φ = 0.568 [Fall et al. (12); similar data are in Reference 11].

Particles of different shapes and aspect ratios can be accommodated by relative viscosity descriptions used for spheres by changing the value of a parameter equivalent to φ m (13). The effect of particle size distribution has not been studied extensively; a good summary is given by Metzner (13). Briefly, a bimodal distribution of particle size can cause a substantial reduction in the viscosity at a constant volume fraction when φ > 0.3 (14, 15). Simulations of bidisperse suspensions by Chang & Powell (16) suggest that the reduction of viscosity is due to a disruption of clustering that is seen in monodisperse suspensions. The viscosity of suspensions in non-Newtonian fluids was comprehensively reviewed by Metzner (13), who generally found that the relative viscosity in Newtonian and polymeric fluids 206

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followed the same relation at low shear rates. Tanner and coworkers (17) extended the differential change derivation of the Krieger-Dougherty equation to power-law matrix fluids. Gleissle and coworkers (18–20) noted that the shear stress of suspensions in polymer matrices could be shifted horizontally to obtain master curves of stress versus reduced shear rate. The applicability of Equation 2 in non-Newtonian suspending fluids may, however, be limited to concentrations well below maximum packing (21).


Rheological Measurement Rheological measurement is based on the principle that certain material properties of a continuum can be measured directly, without a priori knowledge of the material behavior. The generality of this principle greatly restricts the deformation fields that can be studied, but it permits the development of standard methodologies and provides certainty that the measured properties have a precise meaning that is independent of the particular experimental apparatus. We focus on two basic classes of nonlinear rheometric measurement that measure the components of the stress. The total stress σ in an incompressible fluid is written as σ = −pI + τ , where σ and τ are symmetric tensors except in the case of certain anisotropic liquids. τ is the extra stress. The pressure p is a dynamical variable that is defined by the flow; it is not the thermodynamic pressure, which is undefined for an incompressible fluid. In a fluid without microstructure, the pressure is defined by the condition trace (τ ) = τ 11 + τ 22 + τ 33 = 0 or, equivalently, p = −1/3 trace (σ ); in such a case, the extra stress is called deviatoric. The extra stress generally is not deviatoric for liquids with microstructure. For example, for the Maxwell model of polymeric liquids, which is a generalization of rubber elasticity to a flowing temporary network, trace (τ ) is twice the entropic free energy resulting from flow-induced chain deformation (22). A gradient in trace (τ ) can therefore cause chain migration in polymer solutions to maintain uniformity of the free energy. For a suspension, the extra stress τ is usually written as a sum of the suspending fluid contribution τ sm and the particle contribution τ p ; the particle pressure , which is a nonequilibrium osmotic pressure (23), is defined as = −1/3 trace (τ p ). A gradient in particle pressure can induce a balancing gradient in particle concentration. Rheometry is most conveniently performed in a viscometric flow, also known as a simple shear flow. Viscometric flow is steady and unidirectional in the 1-direction of an orthogonal coordinate system, with variation only in the 2-direction; the 3-direction, which is the direction of the vorticity vector, is neutral. The simplest manifestation is drag flow between two infinite parallel planes, but other flows are equivalent, including those used in rotational rheometry—torsional flow between infinite concentric cylinders, infinite parallel plates, and an infinite cone and a plane—and pressuredriven flow in an infinite circular cylinder or between infinite parallel planes. Clearly, the “infinite” designation in each case is unattainable, so approximations and end effects are a factor in every experiment. In a simple shear flow, the only nonzero components of the extra stress are τ 12 , τ 11 , τ 22 , and τ 33 , which are unique functions of the shear rate γ˙ = d v1 /d x2 , where x2 denotes the 2-direction. The viscosity is defined as η ≡ τ12 /γ˙ . Because of the incompressibility, only the normal stress differences N 1 = τ11 − τ22 and N 2 = τ22 − τ33 can be measured. η, N1 , and N2 are material functions that are the same if measured in any viscometric flow. Most measurements of normal stress functions have been carried out on flexible polymer melts and solutions, where the normal stresses are quadratic functions of the shear rate at low rates. N1 is always positive for flexible polymers, whereas N2 is negative and much smaller in magnitude than N1 . Rheometry by rotational rheometers is discussed in detail in References 24 and 25. Pressure-driven flow for rheometry may use either a cylindrical or slit geometry and requires that end effects be taken into account. The rigorous way to account for end effects is to carry out www.annualreviews.org • Rheology of Non-Brownian Suspensions

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measurements in capillaries (slits) of different length-to-diameter (-gap) ratios at constant stress, but it is common with polymers to do measurements in a single geometry of moderate length and to account for the entry and exit losses by subtracting the pressure drop for flow through an orifice (slit) of the same diameter (gap). Normal stress differences cannot be measured in a capillary or slit. Capillary rheometry has been used to measure the viscosity of non-neutrally buoyant suspensions by maintaining continuous mixing just prior to the entry to the capillary; see, for example, Reference 26. The torque measurement from a batch mixer has been used to measure the viscosity of stiff systems by employing an analogy to torsional coaxial cylinder flow; the basic methodology is described by Bousmina et al. (27) and Estellé & Lanos (28) and applied to energetic suspensions by Guillemin et al. (29). Commercial instrumentation is available for both torsional and capillary/slit rheometry, but some of the best recent data on normal stresses in suspensions have come from noncommercial approaches different from those above; these are discussed subsequently.


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Extensional Rheometry. A uniaxial extensional flow is one in which the velocity v 1 is a function of only the 1-direction [v1 = v1 (x1 )]. For uniform extension of a cylinder, continuity requires that vz = vz (z) and vr = − 2r ddvzz in an r-θ-z coordinate system. The extension rate γ˙e = d vz /d z may be a function of time, but uniform drawdown over the entire length of the cylinder requires that the extension rate be independent of z, from which it also follows that vz is linear in z. The measured axial stress is equal to the normal stress difference τzz −τrr , with a possible correction for surface tension. This normal stress difference is a material function if a steady state is reached, but that is rarely the case. The filament stretching rheometer (30), which is not commercially available, stretches a liquid filament between two plates at a fixed rate and has been used successfully for mobile liquids; it has been used in one study of stretching of a noncolloidal suspension (31). A commercial device is available for mobile liquids that uses surface tension–driven breakup and requires a separate theoretical treatment; this geometry has been used in one study of noncolloidal suspensions (32). Uniform biaxial compression is mathematically equivalent to uniform uniaxial extension in the plane orthogonal to the compression axis and has been used to study filled systems (33, 34).

Artifacts Apparent Wall Slip. The most serious experimental artifact in shear rheometry of suspensions is apparent wall slip, wherein there appears to be a relative velocity between the rheometer surface and the adjacent fluid. Apparent slip is a well-known phenomenon in linear polymer melt rheometry and processing at high stresses (3, 35). There is an excluded-volume effect in suspensions, causing the solid volume fraction at the wall, and thus the resistance to flow, to be less than that in the bulk. Kalyon & Aktas¸ (36) discuss wall slip in suspensions in this volume. Briefly, one approach is to do measurements in rotational rheometers at different gap spacings, then to extrapolate to an infinite gap. The accessible range of gap spacings is limited because of the heuristic requirement that the minimum gap be more than an order of magnitude greater than the particle size, whereas the maximum gap must be such that the gap-to-radius ratio is small enough to ignore edge effects. Measurements using pressure-driven flow in capillary or slit rheometers require larger sample sizes and multiple dies. Equations for the slip velocity or, equivalently, the thickness of a wall layer with the properties of the suspending fluid are given, respectively, by Jana et al. (37) for torsional flow and by Kalyon (38) for capillary and slit flow. The usual approach to minimizing wall slip with suspensions is to use a rotational rheometer with roughened surfaces. This approach generally works well as long as the scale of the roughness is comparable to the particle size, but it may still require correction for slip by employing different gaps. 208

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Edge Fracture. Edge fracture, in which the free surface in a torsional cone-and-plate or parallelplate rheometer breaks up, is a well-known occurrence at high stresses for polymeric liquids. There have been several analyses of the cause, but none is definitive; one that is particularly relevant to suspensions relates the onset of edge fracture to a critical negative value of N2 (39). Edge fracture has been observed in torsional rheometry of concentrated suspensions of spheres in Newtonian suspending fluids (9, 10).


Normal Stress Measurement The first measurements of normal stress differences of spherical particles in Newtonian fluids were by Gadala-Maria (40), who observed a linear dependence of N1 − N2 on the magnitude of the shear rate. Zarraga and coworkers (9) established the general behavior, in part through a nonconventional method for measuring N1 + 2N2 , namely the rod-climbing or Weissenberg effect, in which a polymeric liquid at a free surface climbs up a rotating rod because N1 + 2N2 > 0 (effectively N1 > 0, because |N2 /N1 | is typically small for polymeric liquids). By contrast, a negative value of N1 + 2N2 requires that the surface lower as the rod is approached, and this anti-Weissenberg or rod-dipping effect was first observed by Aral & Kalyon (41) for a suspension of spheres in a polydimethylsiloxane melt; Zarraga et al. (9) showed that rod dipping occurred for a concentrated (φ > 0.5) suspension in a Boger fluid, which exhibited rod climbing in the absence of particles. Zarraga and coworkers (9) reworked the analysis for a fluid in which the normal stresses are linear in the shear rate. Combining rod-dipping measurements with cone-and-plate and torsional parallel-plate rheometry, they found negative values for N1 and N2 for volume fractions between 0.31 and 0.56, with N2 /N1 > 3 in most cases. Singh & Nott (42) found similar results using measurements in torsional flow between coaxial cylinders (with a force transducer on the outside cylinder) and parallel plates. Boyer et al. (43) and Couturier et al. (44) combined the rotating rod method with another nonstandard technique developed to measure N2 in polymeric liquids. This method uses the perturbation in the free surface in open-channel gravity-driven flow resulting from the normal stress distribution across the channel. Couturier et al. (44) recalculated the perturbation for normal stresses linear in shear rate and allowed for surface tension effects near the side walls. Their channel was rectangular, which might induce secondary flows for a fluid with a nonzero N2 . Their results spanned φ = 0.10 to φ = 0.55. No measurable normal stresses were observed for φ < 0.17, and data for N1 were not significantly different from zero. Figure 3 shows data for the normal stress coefficients α i = Ni /τ , with i = 1, 2; τ is the shear stress. The N2 data are fit over most of the range by a linear function. Two recent studies reached contradictory conclusions regarding N1 . Dai and coworkers (10) combined torsional parallel-plate rheometry with the open-channel surface deflection method. They used a semicircular open channel to avoid secondary flows possible in a rectangular channel, and they checked the free surface in the torsional instrument for fracture, which was present for φ = 0.45. Their results are generally consistent with those cited above, namely, that both normal stress differences are negative and that N2 is much larger in magnitude than N1 , but they found nonzero values of N1 . Dbouk and coworkers (45) used cone-and-plate and parallel-plate torsional methods, including distributed transducers, to measure the normal stress distribution. In contrast to Dai et al. (10) and the previous studies, they found small positive N1 . At this point, there is general agreement that N2 < 0, and this is the dominant normal stress difference, but even the algebraic sign of N1 is uncertain. At a volume fraction somewhat above 0.50, suspensions of smooth spheres may enter a region of shear jamming (46), where particles are hindered in moving past one another and frictional contacts www.annualreviews.org • Rheology of Non-Brownian Suspensions

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ϕ Figure 3 Normal stress coefficients α 2 (solid symbols) and (α2 + α1 /2) (open symbols) for neutrally buoyant suspensions. Reprinted from Couturier et al. (44) with the permission of Cambridge University Press.

appear to be important. The shear-thickening viscosity shown in Figure 2b may be a manifestation of such contact. There are very few measurements in this regime for non-Brownian particles, and issues of confinement are important: If the particle phase can expand in any direction (e.g., by protruding into a free surface) to provide additional free volume needed for relative motion, it will do so, lowering the local volume fraction and hence the local viscosity and normal stresses. The influence of surface properties was shown by Lootens and coworkers (47), who reported a transition to positive N1 for a suspension of intentionally roughened spheres at φ = 0.43, which is the point where jamming occurred for this system. The spheres were micrometer scale, so colloidal effects cannot be dismissed. Mewis & de Blyser (48) observed a decrease in N1 with increasing concentration of nonBrownian particles in a viscoelastic fluid, but they did not observe N1 < 0. The general observation for polymer melt suspending fluids is that N1 is positive and N2 is negative, but the ratio |N2 /N1 | increases with increasing φ (18, 20, 21, 41). Non-Brownian suspensions in dilute polymer solutions have more complex behavior, although data are limited (e.g., Reference 49 and references therein). Shear thickening in both the viscosity and the first normal stress difference is observed with increasing shear rate for the solutions, but the normal stress may go through a maximum and decrease at higher rates. The formation of particle strings is observed under some circumstances (50, 51, and references therein). Two-dimensional simulations of particulate flow in models of polymer solutions by Hulsen and coworkers (52–54) show viscoelastic extensional stresses that clearly play an important role in the development of structure and the continuum rheology, but the theoretical understanding is still undeveloped.

Migration Under Shear In a revealing experiment, Gadala-Maria & Acrivos (55) observed a transient when using a concentric Couette viscometer to measure the viscosity of a suspension of neutrally buoyant polystyrene spheres (40–50-μm diameter) in a Newtonian fluid. The data for φ = 0.45 are shown in Figure 4. Following a period of rest after reaching steady state and then the start of shear flow at the previous rate, but in the reverse direction, the torque initially jumped to an intermediate value and then 210

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Time (h) Figure 4 Relative viscosity of a suspension of 40–50-μm polystyrene spheres in silicone oils, φ = 0.45 and γ˙ = 24 s−1 . Reprinted from Gadala-Maria & Acrivos (55) with the permission of the Society of Rheology.

decreased before reaching a minimum, after which it increased monotonically to a steady state at the expected magnitude. The phenomenon is shown with φ = 0.50 in Figure 5 over a decade in shear rate, where it can be seen that the reduced torque scales with strain (product of shear rate and time). Kolli et al. (56) observed a similar transient in both torque and normal stresses, in which N1 – N2 measured in a parallel-plate device went from positive to negative immediately upon flow reversal and then returned to the previous positive value. Transient behavior over roughly one strain unit in a system expected to be without memory, as seen in Figure 5, is due to microstructural rearrangement, whereas Leighton & Acrivos

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Figure 5 Reduced torque versus strain following flow reversal of 40–50-μm polystyrene spheres in silicone oils, φ = 0.5. γ˙ = 0.15 s−1 (squares), 0.38 s−1 (circles), 0.60 s−1 (inverted triangles), and 2.4 s−1 (triangles). Reprinted from Gadala-Maria & Acrivos (55) with the permission of the Society of Rheology. www.annualreviews.org • Rheology of Non-Brownian Suspensions

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R (cm) Figure 6 Concentration profiles measured across the gap between rotating concentric cylinders using magnetic resonance imaging for suspensions with mean values of φ = 0.58 (squares), 0.59 (circles), and 0.60 (triangles). Reprinted from Ovarlez et al. (61) with the permission of the Society of Rheology.

(57) showed the long-time (or large-strain) evolution of properties shown in Figure 4 to be a consequence of shear-induced particle migration. Abbott et al. (58) made definitive observations of radially outward migration in a wide-gap concentric Couette device; migration in pressuredriven flow in channels and tubes was first made by Koh et al. (59) and Hampton et al. (60), respectively. Figure 6 shows magnetic resonance imaging data of the suspensions variation across the gap of a wide-gap Couette device for suspensions of 290-μm polystyrene spheres in silicone oil by Ovarlez et al. (61). The migration in the rotational Couette device or in pressure-driven flow is from a region of high shear rate to a region of low shear rate, and the phenomenon is sometimes modeled as a diffusive-like flux, j ∝ −∇ γ˙ . This simple picture is appealing, but it gives results that conflict with several experiments in other simple-shear geometries: Chapman (62) and Chow et al. (63) found that there was little or no particle migration in torsional flow between parallel plates despite a radial gradient in the shear rate, whereas Merhi and coworkers (64) found slight outward migration. The situation in cone-and-plate devices, where the shear rate is (presumably) uniform, is less clear: Chow et al. (65) reported that particles migrate radially outward, whereas Fall et al. (12) found that there is no migration in a truncated cone-and-plate device. Flow-induced chain migration in polymeric liquids (e.g., Reference 66) can be understood to be driven by a gradient in the spatial distribution of normal stresses (67, 68). A similar approach based on the particle normal stresses has been developed for suspensions. Morris & Boulay (69), building on earlier work by Nott & Brady (70) and Jenkins & McTigue (71), showed that the particle-phase mass balance and momentum equations combine to give the following continuity equation for the particle concentration: 2a 2 ∂φ + u · ∇φ = [ f (φ)∇ · τ p ]. ∂t 9ηs m

3.

Here, u is the local velocity of the continuum and f(φ) is the hindrance function, which reflects the deviation from Stokes’s law for the concentrated suspension. Clearly, at steady state a spatial variation in the particle stress will induce a balancing gradient in the particle concentration.

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Extensional Flow It appears that there have been no studies of controlled uniform uniaxial extensional flow of non-Brownian suspensions of spheres in a Newtonian suspending fluid, but useful insight can be obtained from studies of neck formation during the breakoff of droplets. Furbank & Morris (72, 73) showed that the initial neck deformation for a suspension of spheres in a Newtonian fluid is comparable to that of a Newtonian fluid with a shear viscosity equal to that of the suspension, but that the latter stages prior to breakup are characterized by heterogeneity and particle depletion in the neck. Similar behavior has been seen in subsequent studies (e.g., 74, 75, and perhaps 76) and is to be expected when the neck diameter becomes comparable to the size of the particles. This indicates that, unlike the case of polymer solutions and melts, suspension extensional rheology can provide useful material function information only during the initial portion of the extensional deformation, where bulk behavior is maintained. Non-Brownian spherical particles seem to have little effect on the extensional flow properties of thermoplastic melts at volume fractions up to approximately 0.2, other than the expected change in the viscosity. Le Meins and coworkers (77), for example, found that the relative extensional viscosity for suspensions of polystyrene in polyisobutene at stretch rates below the onset of strain hardening is well represented by the Krieger-Dougherty equation up to φ = 0.25 (the highest value studied). There have been two interesting studies of the extensional properties of aqueous cornstarch suspensions. Bischoff White et al. (31) carried out controlled uniform uniaxial extension on a cornstarch suspension in water with φ = 0.355 (55 weight%). Photographs of the deformation indicate a substantial region of uniform uniaxial extension. The particles ranged in size from 5 to 15 μm with a “faceted shape that is approximately spherical.” This suspension showed shear thinning up to a shear rate of 10 s−1 , after which it rapidly shear thickened. However, it was highly extension thickening at a stretch rate of 0.6 s−1 , and at higher rates it experienced a solidlike brittle fracture at small strains. Clearly cornstarch suspensions respond very differently to extensional mechanics. Roché and coworkers (32) looked at more dilute corn starch suspensions of 37–39 weight% and found that the thinning dynamics become dominated by concentration heterogeneity, in which dilute regions flow and concentrated regions jam and act like solids. The more concentrated system of Bischoff White and coworkers may prevent this separation, which is perhaps why rupture from a cylindrical shape is sudden. Chopped-glass systems, which are of considerable interest for composite manufacture because of enhanced physical properties, have received the most attention among suspensions with nonspherical particles. Batchelor (78) obtained a closed-form equation for the steady-state relative extensional viscosity of a semiconcentrated fiber suspension in a Newtonian fluid, and the result was validated for glass fibers in Newtonian suspending fluids by Mewis & Metzner (79) and Weinberger & Goddard (80) using a fiber spinning apparatus and by Ooi & Sridhar (81) using a filament stretching device. Extensions to other regimes and attempts to generalize the rheology of the matrix are discussed in a comprehensive review (82). Fibers in polymer matrices raise the extensional viscosity above that of the matrix but appear to reduce the strain hardening; the field is reviewed in References 83 and 84.

Irreversibility Suspension flows in which the particles may experience Brownian or other forces that do not result from hydrodynamics or gravity are characterized in terms of the Péclet number, Pe = 6π ηs m γ˙ a 3 /kT , which is the ratio of the timescales for Brownian and shear-induced motions.

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For noncolloidal suspensions in which the diameter is greater than 1 μm, the Péclet number is generally taken to be infinite and Brownian effects are neglected. Motion in a Newtonian liquid at zero Reynolds number should therefore be completely reversible because of the linearity of the governing Stokes equation. But diffusion-like behavior and irreversibility are observed in shear flow of suspensions of non-Brownian spheres. The presence of a stochastic contribution to the motion of particles under shear plays a role in the formulation of continuum constitutive descriptions. The theoretical basis for irreversibility at large strains is well established: Simulations by Drazer et al. (85) and Dasan et al. (86) have shown that the dynamics of a suspension of smooth spheres in shear flow at zero Reynolds number are chaotic, meaning that the sensitivity to small changes in position causes exponential separation of phase-space trajectories and loss of reversibility; Metzger & Butler (87) studied the relative importance of the near-contact and far-field interactions in causing irreversibility. Such changes in position could be caused by small perturbations, or alternatively by weak interparticle potentials or slight surface roughness. For purposes of simulation, to aid in avoiding overlap, the force between two spheres denoted α and β in a shear flow is typically expressed as


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F0 e −ε/rc eαβ . 4. rc 1 − e −ε/rc Here, F0 is a measure of the magnitude of the force; rc is the characteristic range of the force and ε is the distance of closest approach between the two spheres, both made dimensionless with the radius a. eαβ is the unit vector along the line of centers. Fαβ =

PARTICLE MECHANICS Pair Interaction Although irreversibility at large strains is a natural consequence of chaotic motion of the dynamics of suspensions of spheres, phenomena at the level of pairwise interactions provide insight into the physics underlying the development of constitutive equations for suspension flows. We consider a pair of freely suspended (force- and torque-free) spheres in a Newtonian fluid undergoing simple shear. In the absence of inertia, we may take the frame of reference to be centered on one sphere and examine the motion of the second sphere relative to the first. The motion exhibits fore-aft symmetry: The trajectory followed by the second particle on approach has mirror symmetry with the trajectory on separation, as illustrated in Figure 7. Also, note the presence of closed trajectories. These features were elaborated by Batchelor & Green (88). A key point is the compression of particle trajectories near contact, wherein all trajectories far upstream that have an offset from the flow (x) axis of less than one radius pass extremely close to one another in the neighborhood of the second particle, with a separation that cannot be resolved in the figure. This close approach, and the bundling of trajectories, implies that a small perturbation of scale εa near contact for particles of radius a will cause a change of order a in the trajectory as the pair separates. Hence, the reversible trajectory is extremely sensitive to near-contact perturbations, and any nonhydrodynamic perturbation will cause asymmetry or, in the context of the prior section, irreversibility of the motion. The most likely configuration when the perturbation is caused by roughness or short-range repulsion is a compressive interaction (89, 90), in agreement with experimental observations for concentrated suspensions (91, 92). Consequently, the particle pressure is positive for the hydrodynamically dominated suspension, and in general both N1 and N2 are found to be negative, as observed in most of the experiments cited above. 214

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5 4 3 Limiting closed trajectory

2 1

y

0 –1

Closed trajectory


–2 –3 –4 –5 –5

–4

–3

–2

–1

0

1

2

3

4

5

x Figure 7 Pair trajectory of two spheres of equal radius freely suspended in Stokes shear flow of a Newtonian fluid. The solid sphere represents the reference particle, and the lines are the trajectory of the second particle.

The isolated-pair problem is thus instructive in identifying the critical role of close interactions, which is the essential basis for irreversibility, but it does not contain a complete picture. The many-body interactions in a concentrated suspension introduce a random walk that leads rapidly to a chaotic trajectory, meaning that slight changes in configuration lead to large changes in the trajectory of individual particles. The understanding of these issues is not complete. However, regardless of the details of particle interactions, the shear-induced structure will result in nonlinear rheology, including a shear-rate-dependent viscosity and nonzero normal stress differences.

Simulation Stokesian dynamics is a simulation technique for suspensions in Stokes flow that operates in a gridfree environment, and it has been instrumental in establishing structure-property relations for the rheology of concentrated suspensions. Brady & Bossis (93) originally developed the method in the 1980s, and Sierou & Brady (94) developed a method known as accelerated Stokesian dynamics to speed the computation. The Stokesian dynamics technique plays a role in the study of particles suspended in a viscous liquid similar to that played by molecular dynamics for molecular gases and liquids: The motions of a many-particle system are computed (typically in shear flow), and the properties of the bulk mixture are then determined. The method is based on a Langevin equation, or stochastic differential equation, description of the motion of the particles, which allows inclusion of Brownian motion, as well as conservative forces (e.g., electrostatic) and the noted hydrodynamic interactions. The structure and rheology of suspensions have been probed by the method over a wide range of conditions, up to solid fractions exceeding φ = 0.585 for Brownian suspensions (95) and up to φ = 0.5 in non-Brownian suspensions (96). Other methodologies have been developed to carry out particle-level calculations for suspensions, some of which can incorporate inertia. The various techniques are cited by Bertevas et al. (97), who carried out a direct simulation of the rheological properties of suspensions of spheres and oblate spheroids in Newtonian suspending fluids up to φ = 0.5 using a representative elementary volume method. Their results for the viscosity and normal stresses of spheres are close www.annualreviews.org • Rheology of Non-Brownian Suspensions

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1,000

ϕ = 0.58

500

ηr

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100 50

0.54 10

0.48

5

0.42


0.01

0.02

0.05 0.10

· Γ

0.20

0.50

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Figure 8 Relative viscosity as a function of dimensionless shear rate for suspensions in which frictional contact can occur. ˙ is a shear rate made dimensionless by normalizing with the magnitude of a repulsive force. The three lines for φ = 0.54 represent differing degrees of friction. The red data sets for φ = 0.56 and φ = 0.58 indicate a 50% increase in particle stiffness [Seto et al. (46)].

to the Stokesian dynamics simulations of Sierou & Brady (96). The lattice-Boltzmann method (LBM) solves the Navier-Stokes equations. A technique using LBM for suspension simulation developed by Ladd (98–100) has been applied in several studies for spherical particles; advantages of the LBM include the relative ease of implementation of nonspherical geometries and the fact that the solution of the Navier-Stokes equations (rather than Stokes equations) allows analysis of fluid inertial effects. We also note the force-coupling-method calculations of Yeo & Maxey (101), who performed three-dimensional simulations for the flow of a suspension of monodisperse spheres in pressure-driven flow in a tube for 0.20 ≤ φ ≤ 0.40. Particle-concentration profiles are consistent with experiments. Both normal stress differences are found to be negative for φ = 0.20, but N1 takes on small positive values for φ ≥ 0.30. The particle-pressure calculations agree with the theory of Morris & Boulay (69). Simulation of highly concentrated suspensions, in which particle-particle contact is likely in combination with hydrodynamic interactions, has received only limited attention (46, 102, 103). When contact occurs, the lubrication force between particles is kept bounded, and frictional forces based on those used in simulations of dry granular systems are introduced. Shear thickening is observed when the shear stress overcomes a stabilizing repulsive interaction; this is similar to the arguments of Maranzano & Wagner (104) for Brownian suspensions, and thus it appears possible that frictional interactions are relevant for Brownian suspensions as well. Discontinuous shear thickening is argued to correspond to a transition to a shear jammed state, as shown in Figure 8 (46), and the establishment of a three-dimensional network of force chains is observed. Some simulations (e.g., Reference 105) have incorporated inertia together with friction to attempt to explain shear thickening, but the simulations show a transition at levels of inertial stresses that are orders of magnitude above those observed in experiment.

Microstructure Relating the macroscopic properties of the suspension to the particle-scale mechanics is a difficult topic in nonequilibrium statistical physics. The relationship between the microstructure, which is the spatial arrangement of particles under flow, and the rheology is of particular interest. We 216

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4

5

4 2 3

y/a 0 2 –2


1

–4 –4

–2

0

2

4

0

x/a Figure 9 Experimental distribution function for a 50 v% suspension of spherical PMMA particles, 80 ≤ a ≤ 90 μm, in a Newtonian mineral oil. Compression occurs for xy < 0. Reprinted from Blanc et al. (108) with the permission of the Society of Rheology.

focus on the pair-particle structure, which provides information about the preferred orientation of pairs under flow; anisotropy of this orientation distribution is the basic reason for non-Newtonian suspension stresses. To be precise, the pair distribution function g(r) describes the relative likelihood of finding any pair of particles at a separation vector r. Well known in the theory of liquids, g(r) for hard spheres exhibits a peak at contact and a series of minima and maxima asymptotically approaching g = 1 at large r = |r|. This structure is the result of excluded volume interactions (packing constraints). Two features of the pair microstructure are of primary interest here. The first is that at solid fractions approaching maximum packing, the likelihood of pairs near contact is very large, with g(r) known to diverge at the maximum packing fraction (106; see also the discussion in Reference 107); hence, the packing constraints are closely connected to the tendency of the suspension viscosity to diverge. The second is that, under flow, the likelihood of finding a pair is amplified in the compressional directions, where particle pairs are driven toward contact; this can be seen in Figure 9 (108), where xy < 0 is compressional. This form of g(r) reflects observations in other experiments (90) and simulations (96, 109) for noncolloidal suspensions, as well as strongly sheared Brownian suspensions (95). The essential point is that the microstructure loses isotropy, establishing a preferred direction for finding the near-contact pairs that control the observed rheology of concentrated suspensions. This anisotropy leads to normal stress differences in shear flow and the shear-induced migration phenomenon.

CONTINUUM MODELING Polymers There is useful context for suspension rheology in the experience of formulating constitutive equations for polymer solutions and melts. The understanding that stress constitutive equations must be frame invariant (i.e., the physical response must not depend on the frame of reference of the observer) was recognized more than 100 years ago, but it had little impact on practice until after the Second World War, with the growth of the synthetic rubber and polymer www.annualreviews.org • Rheology of Non-Brownian Suspensions

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industries, as well as the publication of unexpected nonlinear phenomena. Two distinct approaches developed: A molecular approach, based on the statistical mechanics of a chain, led to the development of Fokker-Planck equations for the probability distribution of the chain conformation, from which equations for the full stress tensor can sometimes be obtained without approximation for simple models of chain behavior. Approximations in taking moments of the Fokker-Planck equation have resulted in constitutive equations that describe real polymeric fluid behavior. Computing power has reached the point that the Fokker-Planck equation and the corresponding moments to obtain the stress can sometimes be solved on the fly for local velocity gradients in a complex flow, permitting integration of the molecular model directly into the continuum calculation; however, this development is still in its infancy. A second approach is completely rooted in continuum mechanics and contains no molecular information. Here, one starts with a statement of the variables on which the stress depends, most generally the history of the velocity gradient. Representation theorems that ensure frame invariance then provide the most general relation for the stress. The generality rigorously establishes the existence of material functions, such as the viscosity and the normal stress differences, but it provides little that is useful without approximations. Both approaches share one feature: There are explicit predictions of the viscosity and normal stress differences in shear flow, and most of the constitutive equations in use have no more parameters than can be measured by combining a linear viscoelastic experiment with measurements in simple shear and uniform uniaxial extension. The current state of the art permits the computation of velocity and stress distributions in complex flows of polymer melts under conditions that approach processing interest; see, for example, Reference 3. The development of large extensional stresses in elementary continuum constitutive equations for polymers led to serious numerical difficulties (known as the high Weissenberg number problem). Schunk & Scriven (110) and Ryssel & Brunn (111, 112) suggested one remedy to this problem, but it never gained widespread acceptance. This approach was to use purely viscous (memoryless) formulations that accurately reflected the shear and extensional data independently, then to establish a frame-invariant way to mix these functions to compute the stresses in general flow fields. This approach would hardly merit mention except that it has found use in the formulation of stress constitutive equations for suspensions.


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Dilute Non-Brownian Suspensions Continuum theory for dilute non-Brownian suspensions of ellipsoidal particles in a Newtonian suspending fluid is well developed and is instructive to consider. The particle contribution to the total stress is computed from Jeffery’s (113) exact solution for the creeping flow of an ellipsoid in a shear flow. n is a unit vector that describes the rotational axis of the ellipsoid, and the total stress is then obtained in terms of the orientation distribution tensor A = , where represents an ensemble average over all particle orientations, as 1 σ = − pI + ηs m D + φ μ0 D + μ2 D :< nnnn > + μ3 (D · A + A · D) , 2

5a.

where the evolution equation for the structural tensor A is DA r2 − 1 =A·−·A+ 2 [D · A + A · D − 2D :< nnnn >] . Dt r +1 218

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Here, D = ∇v + ∇vT and = ∇v − ∇vT . r is the aspect ratio of the ellipsoid, which equals unity for a sphere and approaches infinity for a long slender fiber. Equation 5b is simply a statement that n rotates with the local fluid but retains a unit length. These equations require the fourth-order moment . This is a classical closure problem, which was first addressed for Brownian suspensions by Hinch & Leal (114) and subsequently by many authors in the contexts of both fiber suspensions and liquid crystals (see References 3, 115); the simplest form, used by Lipscomb et al. (116), is


D :< nnnn >≈ (D : A)A. In that case, Equations 5a and 5b become, respectively, 1 σ = − pI + ηs m D + φ μ0 D + μ2 (D : A)A + μ3 (D · A + A · D) 2

6.

7a.

and r2 − 1 DA =A·−·A+ 2 [D · A + A · D − 2(D : A)A]. 7b. Dt r +1 The coefficients are known exactly for any aspect ratio; the two most interesting sets are for the limits r → 1 (sphere) and r → ∞ (long slender fiber): r → 1 : μ0 = 2.5ηsm ,

μ2 = μ3 = 0,

and r → ∞ : μ0 = 2ηs m ,

μ2 = ηs m r 2 / ln r, μ3 = 0.

The former simply gives Einstein’s result for the relative viscosity, ηr = 1 + 2.5φ. The parameter values for r → ∞ are a good approximation for all r ≥ 10. The same continuum equations are obtained from the representation theorem approach, but with undefined coefficients. A principal result that follows from considering long fibers is the extreme sensitivity of the flow field to microstructure. Figure 10 compares results for creeping flow through a 4:1 contraction for a Newtonian fluid and a suspension of long rods with φμ2 /ηsm = 5, corresponding to a volume fraction of 0.2% for r = 100. The large change in the corner vortex is striking and is observed experimentally (116).

a

0·9

0·0 0·1

b

0·9

0·0 0·1

Figure 10 Computed streamlines for creeping flow through a 4:1 contraction. (a) Newtonian fluid. (b) Suspension of long rods with φμ2 /ηsm = 5 [Lipscomb et al. (116)]. www.annualreviews.org • Rheology of Non-Brownian Suspensions

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The dilute regime assumes that the particles are noninteracting, which requires φr2 1. The same set of equations, but with a value of μ2 /ηsm that depends on the actual dimensions of the particles and the mean spacing, is valid in the semidilute regime, φr 1; calculations have been done in this regime, assuming that the spacing is that of a random orientation of rods. The most challenging regime is that of highly concentrated fiber suspensions, which is also the case most relevant to this review. Here, there have been two approaches for rodlike particles. One has used direct simulation of particle motion and interactions, with results that are instructive but do not lead to continuum constitutive equations. The other has been to extend the dilute suspension theory into the concentrated regime by introducing irreversibility. Folgar & Tucker (117) pioneered this approach by adding an ad hoc diffusion term to the equation for the orientation distribution, based on the notion that fiber interactions cause diffusion-like dispersion; Equation 7b is thus written as


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r2 − 1 DA =A·−·A+ 2 [D · A + A · D − 2(D : A)A] + 2Ci γ˙ (I − 3A). Dt r +1

8.

γ˙ is the square root of one-half of the second invariant of the deformation rate, which reduces to the shear rate for a viscometric flow. The apparent diffusion coefficient Ci is a phenomenological coefficient that depends on rφ (118). This model is employed for mold-filling simulations of fiber-filled polymer composites; see, for example, Reference 119.

Concentrated Non-Brownian Suspensions of Spheres in Newtonian Fluids The formulation of constitutive equations for polymers through structural theories is built around the elementary unit of the chain segment, whereas for dilute suspensions of ellipsoids the structure tensor is based on the unit vector defining the axis of rotation. In these cases and others where a structural tensor can be identified (liquid crystals, for example), the particle contribution to the stress can be formulated in terms of the instantaneous rate of deformation and the structure tensor through representation theorems for tensor functions, following a procedure usually attributed to Hand (120). The fundamental unit for constructing a structure tensor (often called a fabric tensor) for concentrated suspensions of spheres is elusive, and several formulations have been proposed, all of which are connected in some way to the pair-distribution function. Phan-Thien (121) adopted the unit vector along the line of centers of near-contact spherical pairs, denoted p, as the basic structural unit, with the structure tensor defined as A = . He proposed that the relative motion of two spheres is affine with the surrounding continuum, except for a non-Brownian isotropic and stationary random fluctuation, which he took to have a correlation that decays exponentially in time. The primary dissipative interaction between spheres was assumed to be the lubrication forces associated with squeezing flow. Phan-Thien’s constitutive formulation for the particle contribution to the stress is then 1 9a. σ p = η p (φ) D :< pppp > + K γ˙ A . 2 ηp (φ) is chosen empirically to match the shear viscosity. The evolution equation for the structural tensor A is 1 3 DA = ∇v · A + A · ∇vT − 2D :< pppp > − K γ˙ A − I . 9b. Dt 2 3 The K γ˙ terms result from the hypothesized random forces; this term in the evolution equation incorporates irreversibility, whereas Goddard (122) has noted that the term in the stress equation is a viscous analog of hypoelasticity in elastic solids. Phan-Thien chose a closure approximation 220

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developed by Hinch & Leal (114) for the term D: rather than Equation 6. Equation 9b with closure is essentially equivalent to Folgar & Tucker’s (117) modification of the equation for ellipsoids in the limit r → ∞. This formulation predicts a transient on flow reversal, as observed by Gadala-Maria & Acrivos (55), Kolli et al. (56), and Narumi et al. (123), but it gives the incorrect signs and relative magnitudes of the normal stresses in simple shear and is therefore largely of historical interest for opening the field to study. Phan-Thien and coworkers (124, 125) generalized this work to be more flexible by incorporating a second-rank tensor K associated with the stochastic contribution. They did not introduce a closure approximation but rather carried out simulations of the underlying stochastic equations to obtain the relevant ensemble averages ( and ). The predicted viscosity as a function of volume fraction does not agree well with experimental data, and the particular parameter set employed results in positive normal stress differences. [Phan-Thien et al. (125) stated that different parameters for K could result in negative normal stress differences.] They performed a simulation of sedimentation of a sphere in a cylinder with the same set of parameters for φ = 0.25 and φ = 0.55 and obtained effective viscosities in reasonable agreement with the Krieger-Dougherty equation (125). The theory underpredicts the time required to reach steady state in shear (123). Goddard (122, 126) used representation theorems to develop a general fabric tensor. His stress equation differs from that of Phan-Thien (121) in that it retains terms up to third order in A and hence has several additional parameters. The structure evolution equation is taken to be a history integral involving a particular strain measure Ht (t). Goddard used a memory function with two dynamical modes and exponential relaxation to fit the transient shear data of Kolli et al. (56) and Narumi et al. (123). His two time constants differed by four orders of magnitude, and he suggested association of the shorter time constant with a nonhydrodynamic force of the type used in discrete particle simulations to introduce irreversibility (cf. Equation 4). In the work of Stickel and coworkers (127), a length scale interpreted as a mean-free path, defined as l m f = a(φ −1 − φm−1 )/3, was introduced, and the directional dependence of the reciprocal of this length was used to construct the structure tensor. Both the stress and structure tensor equations were truncated to be linear in A, D, and a/lmf , with some coefficients in each equation taken to be proportional to γ˙ and to a function h(6π ηs m a 2 γ˙ )/F0 ; F0 is the strength of the particleparticle nonhydrodynamic force, and h is taken empirically to have a −1/8 power-law dependence on its argument. Stickel and coworkers (127, 128) evaluated the coefficients to match Stokesian dynamics simulations of steady and unsteady shear flow and then applied the model to the transient shear data of Kolli et al. (56) and Narumi et al. (123). The model predictions exhibit many of the qualitative features of the shear experiments, but the predicted normal stress difference N1 − N2 is an order of magnitude smaller than the measurement, whereas the predicted shear stress is half that measured, and the predicted recovery of the steady state upon reversal of the shear direction is slightly more rapid than in experiment. Yapici and coworkers (129), with a revised set of parameters, included an equation for particle migration and used the model to simulate steady and oscillatory shear flow. The steady simulations agree well with published concentration profile data for φ = 0.2 and φ = 0.3, but the agreement with data for φ = 0.4 and φ = 0.5 is less satisfactory. Morris & Boulay (69) and Miller et al. (130) adopted a fundamentally distinct methodology that most closely approximates the approach of Schunk & Scriven (110), in which there is an explicit separation of the shear and normal stress contributions that arise from the presence of the particle anisotropy. They wrote the particle contribution to the stress as σ p = ηs m η p (φ)D − ηs m γ˙ Q(φ),

10.

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1

1 0.95

0.8 0.9 0.85

0.6

x/H

0.8 0.4

0.75 0.7


0.2 0.65 0

8

9

10

11

12

0.6

y/H Figure 11 Particle fraction at steady state for flow through a 4:1 contraction with a uniform volume fraction φ/φ m = 0.735 at 10 channel widths upstream. The large channel width is 18 particle diameters [Miller et al. (130)].

where Q is responsible for the normal stresses and ηp is the relative particle contribution to the total viscosity. Morris & Boulay (69) were concerned only with shear flows, and they took Q to be a constant diagonal tensor multiplied by a concentration-dependent normal stress viscosity function. The stress equation was combined with the overall momentum equation and the particle-phase continuity equation to calculate the stresses. Setting Q33 = 0.5 gave results on particle migration that are consistent with the torsional parallel-plate data of Chapman (62) and Chow et al. (63) and the cone-and-plate data of Chow et al. (65), as well as with data in concentric cylinder Couette flow. The results of Merhi et al. (64) suggest that Q33 < 0.5. Miller and coworkers (130) generalized this approach to arbitrary flows in two spatial dimensions by incorporating interpolation functions that depend on a relative rotation vector that is the difference between the local angular velocity of a fluid element and the local rotation of the principal axes of the rate of strain tensor (111, 112), which in two dimensions are unambiguously tension and compression. They carried out finite-volume calculations in two planar complex flow fields, a 4:1 contraction followed by a 1:4 expansion and rectilinear flow past a square cavity with a depth equal to the channel width. There is particle migration away from the walls, causing a reduction in pressure drop. A striking feature of both flows is the occurrence of particle migration across streamlines into vorticity-dominated regions of slow recirculation. Figure 11 shows the computed particle contour plot in the contraction. Few data sets are available to validate computations in complex geometries like these. The results are in qualitative agreement with magnetic resonance imaging studies (131, 132), but quantitative comparisons are premature.

LOOKING AHEAD Understanding the mechanics that leads to irreversibility over a scale of less than one strain unit, as seen in Figure 5, appears to be the primary need to develop continuum equations for the rheology of concentrated suspensions that have predictive ability in complex flows. This phenomenon is incorporated in an ad hoc way in the various particle-level simulation techniques 222

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through the use of a close-contact force, like Equation 4, but the actual physics underlying the force and the way in which it might depend on phenomena such as surface roughness, frictional interactions, or slight deformation are undetermined. As a consequence, the force, which is clearly important in suspensions that are close to maximum packing, is not explicitly contained in any of the current approaches to continuum modeling. Goddard’s continuum formulation, in which the structure tensor is assumed to depend on the history of the strain, captures this short time constant empirically, but it contains no mechanism. Indeed, because the short time irreversibility is undoubtedly related to nonhydrodynamic particle-particle interactions, it is not evident that dynamics associated with irreversibility at small strains should arise from a strain history integral. The published continuum approaches have promise for moderately concentrated suspensions of spheres, but they cannot be expected to apply close to maximum packing. There has also been some progress on continuum representation of dilute and semidilute suspensions of high-aspect-ratio particles, especially in elongational flows, but nonspherical particles with small aspect ratios will be sphere-like but more complex in behavior. The continuum analysis of the rheology of nondilute particle systems in non-Newtonian matrices is inherently more difficult than the Newtonian case and has received little attention. This area of study is likely to advance through the recent development of simulations, such as the work by Hulsen and coworkers (52–54). Suspensions near maximum packing have similarities to dry granular systems, where particleparticle contacts are an essential component in the mechanics and jamming can occur (see, for example, Reference 133). Force chains, which are contact networks that provide the primary mechanism for supporting the stress, play an essential role in granular mechanics (see, for example, References 134 and 135). Incorporation of frictional contacts in particle-scale modeling of suspensions of spheres leads to discontinuous shear thickening and the development of force chain networks (46); the study of force chains and their dynamics is undoubtedly relevant to understanding the role of nonhydrodynamic interparticle forces in dense suspensions, but this area is largely unexplored at present.

DISCLOSURE STATEMENT The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS This article is based in part on a report prepared for the International Fine Particle Research Institute. We are grateful to Daniel Bonn, Elizabeth Guazzelli, and Jan Mewis for many useful conversations during the preparation. LITERATURE CITED 1. Guazzelli E, Morris JF. 2012. A Physical Introduction to Suspension Dynamics. New York: Cambridge Univ. Press 2. Mewis J, Wagner NJ. 2012. Colloidal Suspension Rheology. New York: Cambridge Univ. Press 3. Denn MM. 2008. Polymer Melt Processing: Foundations in Fluid Mechanics and Heat Transfer. New York: Cambridge Univ. Press 4. Einstein A. 1906. Eine neue Bestimmung der Molekuldimensionen. Ann. Phys. 19:289–305 ¨ 5. Einstein A. 1911. Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Molekuldimensionen. Ann. ¨ Phys. 34:591–92 6. Happel J, Brenner H. 1965. Low Reynolds Number Hydrodynamics. Englewood Cliffs, NJ: Prentice-Hall www.annualreviews.org • Rheology of Non-Brownian Suspensions

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7. Lewis TB, Nielsen LE. 1968. Viscosity of dispersed and aggregated suspensions of spheres. Trans. Soc. Rheol. 12:421–33 8. Roscoe R. 1952. The viscosity of suspensions of rigid spheres. Br. J. Appl. Phys. 3:267–69 9. Zarraga IE, Hill DA, Leighton DT Jr. 2000. The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J. Rheol. 44:185–220 10. Dai S-C, Bertevas E, Qi F, Tanner RI. 2013. Viscometric functions for non-colloidal sphere suspensions with Newtonian matrices. J. Rheol. 57:493–510 11. Fall A, Bertrand F, Lemaˆıtre A, Bonn D, Overlez G. 2010. Shear thickening and migration in granular suspensions. Phys. Rev. Lett. 105:268303 12. Fall A, Denn MM, Bonn D. 2013. Why Is (Wet) Granular Rheology So Complicated? Amsterdam: Inst. Phys., Univ. Amsterdam (unpublished manuscript) 13. Metzner AB. 1985. Rheology of suspensions in polymeric liquids. J. Rheol. 29:739–75 14. Chong JS, Christiansen EB, Baer AD. 1971. Rheology of concentrated suspensions. J. Appl. Polym. Sci. 15:2007–21 15. Farris RJ. 1968. Prediction of the viscosity of multimodal suspensions from unimodal data. Trans. Soc. Rheol. 12:281–301 16. Chang C, Powell RL. 1993. Dynamic simulation of bimodal suspensions of hydrodynamically interacting spherical particles. J. Fluid Mech. 253:1–25 17. Tanner RI, Qi F, Housiadad KD. 2010. A differential approach to suspensions with power-law matrices. J. Non-Newton. Fluid Mech. 165:1677–81 18. Ohl N, Gleissle W. 1993. The characterization of the steady-state shear and normal stress functions of highly concentrated suspensions formulated with viscoelastic liquids. J. Rheol. 37:381–406 19. Gleissle W, Baloch MK. 1984. Reduced flow functions of suspensions based on Newtonian and nonNewtonian liquids. Proc. IX Int. Congr. Rheol. 2:549–56 20. Mall-Gleissle SE, Gleissle W, McKinley GH, Buggisch H. 2002. The normal stress behavior of suspensions with viscoelastic matrix fluids. Rheol. Acta 41:61–76 21. Le Meins J-F, Moldenaers P, Mewis J. 2002. Suspensions in polymer melts. 1. Effect of particle size on the shear flow behavior. Ind. Eng. Chem. Res. 41:6297–304 22. Marrucci G. 1972. The free energy constitutive equation for polymer solutions from the dumbbell model. Trans. Soc. Rheol. 16:321–30 23. Yurkovetsky Y, Morris JF. 2008. Particle pressure in a sheared Brownian suspension. J. Rheol. 52:141–64 24. Macosko CW. 1995. Rheology: Principles, Measurements, and Applications. New York: VCH 25. Morrison FA. 2001. Understanding Rheology. New York: Oxford Univ. Press 26. Leong YK, Boger DV, Christie GB, Mainwairing DE. 1993. Rheology of low-viscosity, highconcentration brown-coal suspensions. Rheol. Acta 32:277–85 27. Bousmina M, Ait-Kadi A, Faisant JB. 1999. Determination of shear rate and viscosity from batch mixer data. J. Rheol. 43:415–33 28. Estellé P, Lanos C. 2008. Shear flow curve in mixing systems—a simplified approach. Chem. Eng. Sci. 63:5887–90 29. Guillemin JP, Menard Y, Brunet L, Bonefoy O, Thomas G. 2008. Development of a new mixing rheometer for studying rheological behaviour of concentrated energetic suspensions. J. Non-Newton. Fluid Mech. 151:136–44 30. Anna SL, McKinley GH, Nguyen DA, Sridhar T, Muller SJ, et al. 2001. An interlaboratory comparison of measurements from filament-stretching rheometers using common test fluids. J. Rheol. 45:83–114 31. Bischoff White EE, Chellamuthu M, Rothstein JP. 2010. Extensional rheology of a shear-thickening cornstarch and water suspension. Rheol. Acta 49:119–29 32. Roché M, Kellay H, Stone HA. 2011. Heterogeneity and the role of normal stresses during the extension thinning of non-Brownian shear-thickening fluids. Phys. Rev. Lett. 107:134503 33. Walberger JA, McHugh AJ. 2000. A comparison of the rheology of reactive filled systems using lubricated squeezing flow. J. Rheol. 44:743–58 34. Kalyon DM, Tang H, Karuv B. 2006. Squeeze flow rheometry for rheological characterization of energetic formulations. J. Energy Mat. 24:195–212


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35. Denn MM. 2001. Extrusion instabilities and wall slip. Annu. Rev. Fluid Mech. 33:265–87 36. Kalyon DM, Aktas¸ A. 2014. Factors affecting the rheology and processability of highly filled suspensions. Annu. Rev. Chem. Biomol. Eng. 5:229–54 37. Jana S, Kapoor B, Acrivos A. 1995. Apparent wall-slip velocity coefficients in concentrated suspensions of noncolloidal particles. J. Rheol. 39:1123–32 38. Kalyon DM. 2005. Apparent slip and viscoplasticity of concentrated suspensions. J. Rheol. 49:621–40 39. Tanner RI, Keentok M. 1983. Shear fracture in cone-plate rheometry. J. Rheol. 27:47–57 40. Gadala-Maria F. 1979. The rheology of concentrated suspensions. PhD Diss., Stanford Univ., Stanford, CA 41. Aral BK, Kalyon DM. 1997. Viscoelastic material functions of noncolloidal suspensions with spherical particles. J. Rheol. 41:599–620 42. Singh A, Nott PR. 2003. Experimental measurements of the normal stresses in sheared Stokesian suspensions. J. Fluid Mech. 490:293–320 ´ 2011. Dense suspensions in rotating-rod flows: normal stresses and 43. Boyer F, Pouliquen O, Guazzelli E. particle migration. J. Fluid Mech. 686:5–25 ´ Boyer F, Pouliquen O, Guazzelli E. ´ 2011. Suspensions in a tilted trough: second normal 44. Couturier E, stress difference. J. Fluid Mech. 686:26–39 45. Dbouk T, Lobry L, Lemaire E. 2013. Normal stresses in concentrated non-Brownian suspensions. J. Fluid Mech. 715:239–72 46. Seto R, Mauri R, Morris JF, Denn MM. 2013. Discontinuous shear thickening of frictional hard-sphere suspensions. Phys. Rev. Lett. 111:218301 47. Lootens D, van Damme H, Hémar Y, Hébraud P. 2005. Dilatant flow of concentrated suspensions of rough particles. Phys. Rev. Lett. 95:268302 48. Mewis J, de Blyser R. 1975. Concentration effects in viscoelastic dispersions. Rheol. Acta 14:721–28 49. Sirocco R, Vermant J, Mewis J. 2005. Shear thickening in filled Boger fluids. J. Rheol. 49:551–67 50. Ponche A, Dupuis D. 2005. On instabilities and migration phenomena in cone and plate geometry. J. Non-Newton. Fluid Mech. 127:123–29 51. Sirocco R, Vermant J, Mewis J. 2004. Effect of the viscoelasticity of the suspending fluid on structure formation in suspensions. J. Non-Newton. Fluid Mech. 117:183–92 52. Hwang WR, Hulsen MA, Meier HEH. 2004. Direct simulation of particle suspensions in a viscoelastic fluid in sliding bi-periodic frames. J. Non-Newton. Fluid Mech. 121:15–33 53. Choi YJ, Hulsen MA, Meier HEH. 2010. An extended finite element method for the simulation of particulate viscoelastic flows. J. Non-Newton. Fluid Mech. 165:607–24 54. Hwang WR, Hulsen MA. 2011. Structure formation of non-colloidal particles in viscoelastic fluids subjected to simple shear flow. Macromol. Mater. Eng. 296:321–30 55. Gadala-Maria F, Acrivos A. 1980. Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol. 24:799–814 56. Kolli VG, Pollauf EJ, Gadala-Maria F. 2002. Transient normal stress response in a concentrated suspension of spherical particles. J. Rheol. 46:321–34 57. Leighton DT, Acrivos A. 1987. The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181:415–39 58. Abbott JR, Tetlow N, Graham AL, Altobelli SA, Fukushima E, et al. 1991. Experimental observations of particle migration in concentrated suspensions: Couette flow. J. Rheol. 35:773–95 59. Koh CJ, Hookham P, Leal LG. 1994. An experimental investigation of concentrated suspension flows in a rectangular channel. J. Fluid Mech. 266:1–32 60. Hampton RE, Mammoli AA, Graham AL, Tetlow N, Altobelli SA. 1997. Migration of particles undergoing pressure-driven flow in a circular conduit. J. Rheol. 41:621–40 61. Overlez G, Bertrand F, Rodts S. 2006. Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging. J. Rheol. 50:259–92 62. Chapman BK. 1990. Shear-induced migration phenomena in concentrated suspensions. PhD Diss., Univ. Notre Dame, IN 63. Chow AW, Sinton SW, Iwamiya JH, Stephens TS. 1994. Shear-induced particle migration in Couette and parallel-plate viscometers: NMR imaging and stress measurements. Phys. Fluids 6:2561–76 www.annualreviews.org • Rheology of Non-Brownian Suspensions

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119. Zheng R, Kennedy P, Phan-Thien N, Fan X-J. 1999. Thermoviscoelastic simulation of thermally and pressure-induced stresses in injection moulding for the prediction of shrinkage and warpage for fibrereinforced thermoplastics. J. Non-Newton. Fluid Mech. 84:159–90 120. Hand GL. 1962. A theory of anisotropic fluids. J. Fluid Mech. 13:33–46 121. Phan-Thien N. 1995. Constitutive equation for concentrated suspensions in Newtonian liquids. J. Rheol. 39:679–95 122. Goddard JD. 2006. A dissipative anisotropic fluid model for non-colloidal particle dispersions. J. Fluid Mech. 568:1–17 123. Narumi T, See H, Honma Y, Hasegawa T, Takahashi T, Phan-Thien N. 2002. Transient response of concentrated suspensions after shear reversal. J. Rheol. 46:295–305 124. Phan-Thien N, Fan X-J, Khoo BC. 1999. A new constitutive model for monodisperse suspensions of spheres at high concentrations. Rheol. Acta 38:297–304 125. Phan-Thien N, Fan X-J, Zhong R. 2000. A numerical simulation of suspension flow using a constitutive model based on anisotropic particle interactions. Rheol. Acta 39:122–30 126. Goddard JD. 2008. A weakly nonlocal anisotropic fluid model for inhomogeneous Stokesian suspensions. Phys. Fluids 20:040601 127. Stickel JJ, Phillips RJ, Powell RL. 2006. A constitutive equation for microstructure and total stress in particulate suspensions. J. Rheol. 50:379–413 128. Stickel JJ, Phillips RJ, Powell RL. 2007. Application of a constitutive model for particulate suspensions: time-dependent viscometric flow. J. Rheol. 51:1271–302 129. Yapici K, Powell RL, Phillips RJ. 2009. Particle migration and suspension structure in steady and oscillatory plane Poiseuille flow. Phys. Fluids 21:05302 130. Miller RM, Singh JPB, Morris JF. 2009. Suspension flow modeling for general geometries. Chem. Eng. Sci. 64:4597–610 131. Altobelli SA, Fukushima E, Mondy LA. 1997. Nuclear magnetic resonance imaging of particle migration in suspensions undergoing extrusion. J. Rheol. 41:1105–15 132. Moraczewski T, Tang H, Shapley NC. 2005. Flow of a concentrated suspension through an abrupt axisymmetric expansion measured by nuclear magnetic resonance imaging. J. Rheol. 49:1409–28 133. Boyer F, Guazzelli E, Pouliquen O. 2011. Unifying suspension and granular rheology. Phys. Rev. Lett. 107:188301 134. Walker DW, Tordesillas A, Thornton C, Behringer RP, Zhang J, Peters JF. 2011. Percolating contact subnetworks on the edge of isostaticity. Granul. Matter 13:233–40 135. Tordesillas A, Lin Q, Zhang J, Behringer RP, Shi J. 2011. Structural stability and jamming of selforganized cluster conformations in dense granular materials. J. Mech. Phys. Solids 59:265–96


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Annual Review of Chemical and Biomolecular Engineering

Contents


Volume 5, 2014

Plans and Detours James Wei p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1 Simulating the Flow of Entangled Polymers Yuichi Masubuchi p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p11 Modeling Chemoresponsive Polymer Gels Olga Kuksenok, Debabrata Deb, Pratyush Dayal, and Anna C. Balazs p p p p p p p p p p p p p p p p p p p35 Atmospheric Emissions and Air Quality Impacts from Natural Gas Production and Use David T. Allen p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p55 Manipulating Crystallization with Molecular Additives Alexander G. Shtukenberg, Stephanie S. Lee, Bart Kahr, and Michael D. Ward p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p77 Advances in Mixed-Integer Programming Methods for Chemical Production Scheduling Sara Velez and Christos T. Maravelias p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p97 Population Balance Modeling: Current Status and Future Prospects Doraiswami Ramkrishna and Meenesh R. Singh p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 123 Energy Supply Chain Optimization of Hybrid Feedstock Processes: A Review Josephine A. Elia and Christodoulos A. Floudas p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 147 Dynamics of Colloidal Glasses and Gels Yogesh M. Joshi p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 181 Rheology of Non-Brownian Suspensions Morton M. Denn and Jeffrey F. Morris p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 203 Factors Affecting the Rheology and Processability of Highly Filled Suspensions Dilhan M. Kalyon and Seda Akta¸s p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 229

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Continuous-Flow Differential Mobility Analysis of Nanoparticles and Biomolecules Richard C. Flagan p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 255


From Stealthy Polymersomes and Filomicelles to “Self ” Peptide-Nanoparticles for Cancer Therapy Nuria ´ Sancho Oltra, Praful Nair, and Dennis E. Discher p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 281 Carbon Capture Simulation Initiative: A Case Study in Multiscale Modeling and New Challenges David C. Miller, Madhava Syamlal, David S. Mebane, Curt Storlie, Debangsu Bhattacharyya, Nikolaos V. Sahinidis, Deb Agarwal, Charles Tong, Stephen E. Zitney, Avik Sarkar, Xin Sun, Sankaran Sundaresan, Emily Ryan, Dave Engel, and Crystal Dale p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 301 Downhole Fluid Analysis and Asphaltene Science for Petroleum Reservoir Evaluation Oliver C. Mullins, Andrew E. Pomerantz, Julian Y. Zuo, and Chengli Dong p p p p p p p p p p 325 Biocatalysts for Natural Product Biosynthesis Nidhi Tibrewal and Yi Tang p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 347 Entangled Polymer Dynamics in Equilibrium and Flow Modeled Through Slip Links Jay D. Schieber and Marat Andreev p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 367 Progress and Challenges in Control of Chemical Processes Jay H. Lee and Jong Min Lee p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 383 Force-Field Parameters from the SAFT-γ Equation of State for Use in Coarse-Grained Molecular Simulations Erich A. Muller ¨ and George Jackson p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 405 Electrochemical Energy Engineering: A New Frontier of Chemical Engineering Innovation Shuang Gu, Bingjun Xu, and Yushan Yan p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 429 A New Toolbox for Assessing Single Cells Konstantinos Tsioris, Alexis J. Torres, Thomas B. Douce, and J. Christopher Love p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 455 Advancing Adsorption and Membrane Separation Processes for the Gigaton Carbon Capture Challenge Jennifer Wilcox, Reza Haghpanah, Erik C. Rupp, Jiajun He, and Kyoungjin Lee p p p p p 479 Toward the Directed Self-Assembly of Engineered Tissues Victor D. Varner and Celeste M. Nelson p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 507 Ionic Liquids in Pharmaceutical Applications I.M. Marrucho, L.C. Branco, and L.P.N. Rebelo p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 527

Contents

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Perspectives on Sustainable Waste Management Marco J. Castaldi p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 547 Experimental and Theoretical Methods in Kinetic Studies of Heterogeneously Catalyzed Reactions Marie-Fran¸coise Reyniers and Guy B. Marin p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 563 Indexes Cumulative Index of Contributing Authors, Volumes 1–5 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 595 Annu. Rev. Chem. Biomol. Eng. 2014.5:203-228. Downloaded from www.annualreviews.org by Florida Atlantic University on 08/19/14. For personal use only.

Cumulative Index of Article Titles, Volumes 1–5 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 598 Errata An online log of corrections to Annual Review of Chemical and Biomolecular Engineering articles may be found at http://www.annualreviews.org/errata/chembioeng

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Annual Reviews It’s about time. Your time. It’s time well spent.

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Annual Review of Statistics and Its Application Volume 1 • Online January 2014 • http://statistics.annualreviews.org


Editor: Stephen E. Fienberg, Carnegie Mellon University

Associate Editors: Nancy Reid, University of Toronto Stephen M. Stigler, University of Chicago The Annual Review of Statistics and Its Application aims to inform statisticians and quantitative methodologists, as well as all scientists and users of statistics about major methodological advances and the computational tools that allow for their implementation. It will include developments in the field of statistics, including theoretical statistical underpinnings of new methodology, as well as developments in specific application domains such as biostatistics and bioinformatics, economics, machine learning, psychology, sociology, and aspects of the physical sciences.

Complimentary online access to the first volume will be available until January 2015. table of contents:

• What Is Statistics? Stephen E. Fienberg • A Systematic Statistical Approach to Evaluating Evidence from Observational Studies, David Madigan, Paul E. Stang, Jesse A. Berlin, Martijn Schuemie, J. Marc Overhage, Marc A. Suchard, Bill Dumouchel, Abraham G. Hartzema, Patrick B. Ryan

• High-Dimensional Statistics with a View Toward Applications in Biology, Peter Bühlmann, Markus Kalisch, Lukas Meier • Next-Generation Statistical Genetics: Modeling, Penalization, and Optimization in High-Dimensional Data, Kenneth Lange, Jeanette C. Papp, Janet S. Sinsheimer, Eric M. Sobel

• The Role of Statistics in the Discovery of a Higgs Boson, David A. van Dyk

• Breaking Bad: Two Decades of Life-Course Data Analysis in Criminology, Developmental Psychology, and Beyond, Elena A. Erosheva, Ross L. Matsueda, Donatello Telesca

• Brain Imaging Analysis, F. DuBois Bowman

• Event History Analysis, Niels Keiding

• Statistics and Climate, Peter Guttorp

• Statistical Evaluation of Forensic DNA Profile Evidence, Christopher D. Steele, David J. Balding

• Climate Simulators and Climate Projections, Jonathan Rougier, Michael Goldstein • Probabilistic Forecasting, Tilmann Gneiting, Matthias Katzfuss • Bayesian Computational Tools, Christian P. Robert • Bayesian Computation Via Markov Chain Monte Carlo, Radu V. Craiu, Jeffrey S. Rosenthal • Build, Compute, Critique, Repeat: Data Analysis with Latent Variable Models, David M. Blei • Structured Regularizers for High-Dimensional Problems: Statistical and Computational Issues, Martin J. Wainwright

• Using League Table Rankings in Public Policy Formation: Statistical Issues, Harvey Goldstein • Statistical Ecology, Ruth King • Estimating the Number of Species in Microbial Diversity Studies, John Bunge, Amy Willis, Fiona Walsh • Dynamic Treatment Regimes, Bibhas Chakraborty, Susan A. Murphy • Statistics and Related Topics in Single-Molecule Biophysics, Hong Qian, S.C. Kou • Statistics and Quantitative Risk Management for Banking and Insurance, Paul Embrechts, Marius Hofert

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Rheology dynamics of aggregating colloidal suspensions.

Rheology of Confined Non-Brownian Suspensions.

Rheology of Lignocellulose Suspensions and Impact of Hydrolysis: A Review.

Rheology of semi-dilute suspensions of carboxylated cellulose nanofibrils.

Relaxation times and rheology in dense athermal suspensions.

Microfluidic rheology of active particle suspensions: Kinetic theory.

The rheology of three-phase suspensions at low bubble capillary number.

Structure and rheology of mixed suspensions of montmorillonite and silica nanoparticles.

Flow regime transitions in dense non-Brownian suspensions: rheology, microstructural characterization, and constitutive modeling.

Rheology of fluoride gels.

Blood rheology and ischaemia.

Rheology of clustering protein solutions.

Erythrocyte rheology.

Blood rheology and aging.

Basics of dermal filler rheology.

Rheology of concentrated perfluorocarbon emulsions.

Technological advances in blood rheology.

Dynamics and rheology of nonpolar bijels.

Structural Rheology of the Smectic Phase.

The cellular and molecular rheology of malaria.

Rheology of composite filling material pastes.

Membrane filtration of food suspensions.

Dynamic fracture of nonglassy suspensions.

Blood rheology in lupus erythematosus.