Hydrodynamic Interactions between Two Equally Sized Spheres in

Apr 22, 2013 - Stress in a dilute suspension of spheres in a dilute polymer solution subject to simple shear flow at finite Deborah numbers. Donald L...
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Hydrodynamic Interactions between Two Equally Sized Spheres in Viscoelastic Fluids in Shear Flow Frank Snijkers,*,†,§ Rossana Pasquino,‡ and Jan Vermant† †

Department of Chemical Engineering, University of Leuven (KULeuven), W. de Croylaan 46, B-3001 Leuven, Belgium Department of Chemical Engineering, Materials and Industrial Production, University of Napels Federico II, P. le Tecchio 80, 80125 Napels, Italy

‡

ABSTRACT: The effect of using a viscoelastic suspending medium on the in-plane hydrodynamic interaction between two equally sized spheres in shear flow is studied experimentally to understand flow-induced assembly behavior (i.e., string formation). A counterrotating device equipped with a Couette geometry is used together with quantitative videomicroscopy. To evaluate the effects of differences in rheological properties of the suspending media, fluids have been selected that highlight specific constitutive features. These include a reference Newtonian fluid (N), a constant-viscosity, high-elasticity Boger fluid (BF), a wormlike micellar surfactant solution with a single dominant relaxation time (WMS), and a broad spectrum shearthinning elastic polymer solution (ST). As expected, the trajectories are symmetric in the Newtonian fluid. In the BF, the midpoints of the spheres are observed to remain in the same plane before and after the interaction, as in the Newtonian fluid, although the path lines are in this case no longer symmetric. Interactions in the ST and WMS are highly asymmetric. Two fundamentally different kinds of path lines are observed in the WMS and ST: reversing and open trajectories. The type of trajectory depends on the initial configuration of the spheres with respect to each other and on the shear rate. On the basis of the obtained results, shear-thinning of the viscosity seems to be the key rheological parameter that determines the overall nature of the interactions, rather than the relative magnitude of the normal stress differences.

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or driven by electric10 or magnetic11 fields), flow-induced selfassembly exploits the complex rheological properties of the suspending fluid (i.e., shear-thinning and elasticity) as driving forces to create self-assembled particle structures. A range of different structures have been observed and studied: strings of spherical particles aligned in the flow direction,12−16 2D sheets of colloidal crystals,17 segregation of sizes of a bidisperse distribution of particles in separate locations in the flow,18 and ordered herringbone packings of nonspherical particles.19 Interestingly, flow-induced self-assembly can be achieved at very low particle concentrations compared to most other methods, and the process can be tuned by the details of the rheological properties of the suspending medium.13,14 Despite the fact that flow-induced self-assembly has great potency for

INTRODUCTION

The self-assembly of colloidal particles is controlled by a variety of parameters (e.g., the chemistry of the particles and their surface, the shape and size of the particles, and the properties of the suspending medium1). Especially in biological systems, selfassembly can be very complex, as exemplified by the behavior of microswimmers2 and red blood cells.3 Nowadays, selfassembly is used in myriad technologies, including optical displays, designer pharmaceuticals, microelectromechanical systems, printed electronics, and solar cells.4−6 As some of the previous examples show, the self-assembly process and the resulting structures can be modulated or directed by the presence of external fields (i.e., electric, magnetic, or flow fields1). The flow-induced self-assembly of particles is probably one of the cheapest and easiest ways to create ordered macroscopic structures. Compared to other types of selfassembly (i.e., because of physicochemical and capillary interactions between particles in the bulk8,7 or at the interface9 © 2013 American Chemical Society

Received: February 20, 2013 Revised: April 17, 2013 Published: April 22, 2013 5701

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Figure 1. Schematic drawings of the different types of trajectories in a Newtonian fluid (a) with the line labeled a for a symmetric open trajectory and b for a symmetric closed trajectory. Presumed effects of nonlinearity on the trajectories (b) with c being for an asymmetric open trajectory, d being for an in-plane spiralling or kissing−tumbing−tumbling trajectory, and e being for a reversing trajectory. The trajectories all start from the left as indicated by the arrow.

further away from each other (trajectory c in Figure 1b). The reversing trajectories are characterized by an initial, slow approach of the spheres toward each other in the flow direction. Meanwhile, the spheres move in the velocity gradient direction and cross the flow axis, causing them to reverse their relative motion and move away from each other in both the flow and velocity gradient direction (trajectory e in Figure 1b). Reversing trajectories can be observed only when the separation between the spheres is relatively small in the velocity gradient and large in the flow direction. In-plane spiralling trajectories are sometimes observed when the two particles begin with a very small separation from each other (nearly touching) in the vorticity plane and with their midpoints on nearly the same velocity gradient axis (trajectory d in Figure 1b). The spheres spiral around each other before departing from the close interaction and separating. This last type of interaction is observed only in a very small zone of initial separation between particles and is not apparent at all ReP. The interactions between two spheres in viscoelastic and non-Newtonian fluids have not been studied extensively. They can be expected to be influenced by the nonlinear response of the suspending fluid. A few experimental results have been published on in-plane collisions between two equally sized spheres in two different media: a strongly shear-thinning, nearly inelastic fluid and a shear-thinning elastic fluid in Couette flow31,32 and Poiseuille flow.33 In all cases, the experimental results show the existence of open trajectories with the spheres separating further away from each other in the velocity gradient direction upon interaction (trajectory c in Figure 1b), qualitatively identical to the above-described simulations concerning the effects of inertia. Because the observations in both fluids are qualitatively identical, shear-thinning seems to be the key fluid variable that determines the nature of the interactions, not elasticity. A 2D computational study showed the existence of complex kissing−tumbling−tumbling interactions for cylinders in an unconfined flow in an Oldroyd-B fluid.34 Particles initially located close to each other in the velocity gradient direction were observed to start rotating around each other while their midpoints came closer and closer to each other (trajectory d in Figure 1b). More recently, these results were extended to 3D in both confined and unconfined flows of an Oldroyd-B fluid.35 Slightly different return, pass, and tumbling dynamics were observed, with the spheres being pushed closer as the elasticity increases. The effect of elasticity was in qualitative agreement

applications, it has not been intensely studied; consequently, the phenomena are poorly understood. Particles are dispersed in fluids with a complex rheological behavior in many technological applications, as exemplified by filled polymers, nanocomposites, paints, and consumer-care products.20,21 Hence, there is a need to understand these phenomena, that depend to a large extent on the relative motions and hydrodynamic interactions of the particles. Whereas the individual motions of particles, their structure formation, and their resultant rheological behavior in Newtonian fluids under flow is relatively well understood,22−24 this is not the case for particles in viscoelastic or non-Newtonian fluids. To understand the flow-induced self-assembly phenomena in such suspensions, the effects of elasticity and the non-Newtonian rheology of the suspending medium need to be studied systematically. The hydrodynamic interaction between two equally sized, non-Brownian spheres in different viscoelastic fluids is studied here for the case of shear flow, and we report on the analysis of trajectories. A trajectory or path line is defined as the path that the midpoint of one sphere follows in time relative to the other reference sphere. Hydrodynamic interactions of this type are well-understood in Newtonian fluids in the absence of inertia.25−29 In this case, two distinct classes of trajectories are found, termed open and closed (Figure 1a). The open trajectories start far from the particles upstream and proceed past the reference sphere to infinity. For the inertialess Newtonian case, these open trajectories are symmetric about the velocity plane (trajectory a in Figure 1a). Next to open trajectories, there is a region of closed trajectories close to the particle surface. These do not extend to infinity, and spheres within this region are bound to each other (trajectory b in Figure 1a). All trajectories, whether open or closed, have fore− aft symmetry about the velocity plane. Mathematically, this comes from the linearity and reversibility of the Stokes equations. All trajectories also have a mirror image about the velocity gradient and vorticity plane. Kulkarni and Morris30 studied the effects of particle inertia on the hydrodynamic interactions in the unconfined flow of Newtonian fluids using 3D computer simulations. Inertia is found to break the fore−aft symmetry of the path lines. The closed-pair trajectories present at a particle Reynolds number ReP = 0 vanish. When considering the interactions of particles confined to the vorticity plane, they found three distinct classes of trajectories: open, reversing, and in-plane spiralling (Figure 1b). The open trajectories are similar to the inertialess case, except that in this case the spheres separate with their centers 5702

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Figure 2. Master curves for the steady-state viscosity η and first normal stress coefficient Ψ1 as function of shear rate γ̇ for the WMS (a), ST (b), and BF (c). A comparison between the fluids is made (d), where the Weissenberg number Wi is plotted as a function of shear rate γ̇ for the different fluids. Fits of the experimental data with multimode Giesekus models with a Newtonian solvent are also shown for each fluid.

fluids, selected to highlight specific constitutive features and to isolate the different rheological effects, have been selected, prepared, and characterized. These fluids include a Newtonian liquid (N), a constant-viscosity, elastic Boger fluid (BF), a shear-thinning viscoelastic polymer solution (ST), and a wormlike micellar surfactant solution with a single dominant relaxation time (WMS). Spherical particles of polystyrene with a diameter a of 430 μm have been dispersed in these fluids. Their motion is studied using a counterrotating Couette rheometer by means of quantitative videomicroscopy.

with the trends observed in 2D.34 However, when the initial separation of two particles also had an offset in the vorticity direction, a more complex periodic motion was observed, similar to the complex orbits observed for ellipsoidal or rodlike particles.19 Compared to the previously discussed results of Kulkarni and Morris,30 the tumbling interactions seem to be the equivalents of the spiralling interactions. Note, however, that whereas for simulations including inertia the spheres start close to each other and eventually separate, for the Oldroyd-B simulations the spheres move closer and closer to each other. Yoon et al.35 suggested this to be a normal stress effect acting on streamlines clumping the particles together. Choi et al.36 performed 2D computer simulations with a Giesekus model, hence including shear-thinning. Apart from the usual pass and return behavior, they found that in a confined shear flow and in a narrow range of initial separations the particles moved to a certain fixed position relative to each other, roughly oriented in the flow direction. From the available simulation results, it is as yet not completely clear how the different rheological parameters affect the trajectories, but it is clear that the effects are pronounced. In the present work, the in-plane hydrodynamic interactions between two equally sized, non-Brownian spheres in shear flow are studied experimentally. A number of model suspending

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MATERIALS AND METHODS

Experimental Method: Counterrotating Rheometer. The interactions between two spheres in shear flow are studied by means of digital videomicroscopy and using a custom-made counterrotating rheometer.37,38 A high-precision glass Couette, custom-made by Hellma (Germany), was used as the flow geometry, and two stresscontrolled rheometers drive the cup and bob. The inner cylinder (bob) has a radius R1 of 25.00 ± 0.01 mm, and the outer cylinder (cup) has a radius R2 of 32.76 ± 0.01 mm, resulting in a gap width of 7.76 mm. The cylindrical shapes of the inner and outer geometries have been measured to be within a 0.03 mm maximal deviation. The height of the bob h, which is also the working distance of the Couette, is 60 mm. Because the rheometer is counter-rotating, there is a cylindrical surface with zero velocity somewhere in between the inner and outer cylinders. The position of this zero-velocity surface can be 5703

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window (0.01−50 s−1). Experimental data for the steady-state shear viscosity η and first normal stress coefficient Ψ1 as a function of shear rate γ̇ are shown as master curves at the indicated temperatures for the WMS, ST, and BF in Figure 2a−c, respectively. Fits of the experimental data with multimode Giesekus models with a Newtonian solvent are also shown for each fluid.37,42 The WMS is elastic and strongly shear-thinning as shown in Figure 2a. Its rheological properties can be described by a single relaxation time. This fluid displays shear-banding at shear rates above ∼3 s−1,43 and experiments are limited to rates below this shear rate. The ST, shown in Figure 2b, is a typical broad-spectrum, shear-thinning elastic polymer solution. It had to be modeled with at least four relaxation times to obtain a quantitative comparison between the fit and the experimental data. The BF is a dilute polymer solution of high-molecular-weight PIB in low-molecular-weight PIB. The macroscopic rheology (Figure 2c) shows that the viscosity is nearly constant and that the elasticity is relatively high. A fit with a two-mode Giesekus model with Newtonian solvent is shown as well. A comparison between the fluids is made in Figure 2d where the dimensionless Weissenberg number Wi, a measure of the elasticity of a fluid, is plotted as function of shear rate γ̇ for the three different elastic fluids. Because we are comparing forces acting on particles, the Weissenberg number is defined here as a ratio of elastic and viscous stresses

shifted within the gap using a potentiometer that adjusts the rotational speeds of the inner and outer cylinders while keeping the relative velocities constant. As such, it is readily possible to capture a particle and keep it in the field of view of the microscope. The pictures are taken through the bottom of the geometry using a prism and a longworking-distance microscope, and the sample is illuminated diffusely from the top. The plane of observation is the vorticity plane. Provided the gap width in the Couette cell is small compared to the radius of the inner cylinder R1, the velocity field is linear.39 In the present work, a wide-gap Couette is used, and the curvature of the flow field needs to be accounted for. In the absence of secondary flow, inertia, and wall slip and assuming infinitely long cylinders, one can obtain the following flow field for a Newtonian fluid by solving the Stokes equations40

ω(r ) =

vθ(r ) R 2R 2 ⎛ 1 1⎞ = (ω R2 − ω R1) 21 2 2 ⎜ 2 − 2 ⎟ + ω R1 r R 2 − R1 ⎝ R1 r ⎠ (1)

with ω(r) being the angular velocity field, vθ(r) being the linear velocity field, ωR1 and ωR2 being the angular velocity of the inner and outer cylinders, respectively, and r being the radial position in the gap from the midpoint of the Couette. In this case, the velocity field is not linear, and the shear rate varies over the gap, implying that, for a shearthinning fluid, the apparent viscosity will also vary over the gap. For a power-law fluid, an analytical derivation of the flow-field is possible and yields a good prediction for cases where the variation of the shear rate over the gap is relatively small and the power-law exponent is chosen carefully.39 The power-law fluid is defined as

η = Kγ (̇ nPL − 1)

Wi(γ )̇ =

(2)

vθ(r ) r

= (ω R2 − ω R1)

(R1R 2)2/ nPL ⎛ 1 ⎜⎜ 2/ nPL − R12/ nPL ⎝ R12/ nPL R2

−

1 ⎞ ⎟ + ω R1 r ⎠ (3) 2/ nPL ⎟

or, for the gap-dependence of the shear rate, γ(̇ r ) = r

2(ω R2 − ω R1) (R1R 2)2/ nPL 1 δω = nPL δr R 2 2/ nPL − R12/ nPL r 2/ nPL

(5)

with N1 being the first normal stress difference and σ being the shear stress. One can clearly observe that the fluids span a wide parameter space. The amount of experimental data for the Weissenberg number is, especially for the WMS, limited because of the difficulties and limitations in measuring normal forces. Because the models are fitted to the linear frequency sweep data and the nonlinear steady data at the same time,37 they can be readily used to yield predictions for the Weissenberg number at lower shear rates, as will be explained further. Experimental Procedure and Data Analysis. In the present work, the focus is solely on in-plane interactions between two spheres located in the same vorticity plane. The size of the particles enables us to position them manually in the fluid with tweezers. Using trial and error, we put two spheres in identical vorticity planes, as judged from visual observation through a long-working-distance microscope. The longitudinal positional precision is limited by the focal depth of the microscope. Because we are not able to visualize the velocity gradient plane as a result of the curvature of the cylinders, a direct measure of the distance between the particles in the vorticity direction cannot be obtained. It can be assessed qualitatively after an experiment, as will be discussed further. Particles are always positioned in the gap at about half the cylinder height h along the vertical axis to avoid end effects. The gap in the Couette is 7.76 mm, which is 18 times the particle diameter. The particles are always inserted near the center of the gap. Particles are placed at least 3 mm from the nearest walls. In one experiment, particles were inserted 2 mm from the outer cylinder to investigate possible wall effects. Comparing the resulting path lines with a similar experiment in the same fluid under the same conditions but further away from the wall showed that any possible effect of the wall is within experimental error. After setting the zero-velocity plane at a position in between the two spheres, the flow is started and sequences of images of spheres during approach, interaction, and retraction are recorded. Reversing trajectories could not be studied quantitatively, as will be discussed later. Several different pairs of particles are always used in order to avoid a bias of the results due to irregularities of the surface of a particle, slight deviations from the sphericity of the particles, and slightly different sizes of the particles. Several measures are taken to ensure an accurate prediction of the local fluid properties, such as the viscosity, first normal stress difference, and power-law index at the position of the particles in the wide-gap Couette. First, the temperature of the fluid in the gap is measured directly after each experiment using an external calibrated miniature thermocouple (hypodermic needle probe, Omega, Stamford CT) because there is no temperature control of the counterrotating

with K being a constant and nPL being the power-law exponent. The resulting equation for the flow field of a power-law fluid, using the same assumptions as before, is39,40

ω(r ) =

N1(γ )̇ σ(γ )̇

(4)

Note that setting nPL equal to 1 in eq 3 yields eq 1 as it should. Equations 3 and 4 are used in the analysis as further explained. The power-law model is used to calculate the local shear rates for the WMS and ST while the Newtonian model is used for N and BF because these fluids are not shear-thinning. Materials. The particles used are polystyrene spheres with a diameter of a = 430 ± 6 μm (Fluka). A Newtonian reference fluid (N), a viscoelastic wormlike micellar surfactant solution (WMS), a shearthinning elastic polymer solution (ST), and a dilute polymer solution with a nearly constant viscosity (so-called “Boger fluid”, BF) are used as suspending media. The fluids are chosen to separate specific constitutive properties, in particular, normal stress differences and shear-thinning. A thorough discussion of the composition of the fluids, preparation procedures, rheological properties, and constitutive modeling with a multimode Giesekus model with Newtonian solvent can be found in the literature.37,41,42 Here, only a brief summary of their nonlinear steady-shear properties and constitutive modeling will be presented. The constitutive models are used in the data analysis as further explained, but they are only of minor importance for the results presented here and are not discussed at length. The Newtonian fluid is a polyisobutylene (PIB) with a constant viscosity η of 83 Pa·s at 25 °C. No significant normal stress differences nor shear-thinning could be detected in the experimental shear rate 5704

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device. This enables one to correct all data for temperature variations using the shift factors aT of the time−temperature-shifted rheological data.41 Second, the analytical flow field for a power-law fluid in a widegap Couette (eq 3) is solved iteratively for the WMS and ST to obtain an accurate estimate of the average shear rate at the position of the particles. For N and BF, the Newtonian flow field is used (eq 1) and iterations are not necessary. To perform iterations readily, the Ellis model is fitted to the viscosity mastercurves for the two shear-thinning fluids. The Ellis model is defined as

η=

η0 1 + (kγ )̇ (1 − nE)

(6)

with k being a constant, η0 being the zero-shear viscosity, and nE being the exponent. The shear-rate dependence of the power-law exponent can now be calculated using the following relation, which can be readily derived by combining eqs 2 and 6:

nPL = 1 +

nE − 1 1 + (kγ )̇ (nE − 1)

(7)

The rotation speeds of the inner and outer cylinders, the relative position of the zero-velocity plane, and the shear-rate dependence of the power-law index are now known. The shear rate of the unperturbed velocity profile in the neighborhood of the particles is calculated as an average over a fluid volume 400 μm above and below the zero-velocity plane. Initially, the power-law index nPL is set to 1 in eqs 3 and 4, and an estimate of the average shear rate of the particles is obtained. Then, eq 7 is used to get a more accurate prediction for the power-law exponent nPL for the shear rate and fluid under consideration, and the calculation of the average shear rate is repeated with this new power-law index. A new estimate of the average shear rate on the spheres is obtained, and the method can be repeated. Iterations are done until convergence is reached on nPL. Finally, this average shear rate is used to obtain the local Weissenberg number. The Weissenberg number is always calculated on the basis of the fitted constitutive models previously discussed, enabling interpolation toward the shear rates between zero shear and the region where normal force data could be obtained. Extrapolation toward higher shear rates is not performed because these rates are not reached in our interaction experiments. The recorded images are analyzed using ImageJ.44 The particle interiors are filled, and a watershed segmentation is used to erode the particles and make it easier to identify them and determine the center position, even when they touch. Subsequently, the (x, y) coordinates for the midpoints of the spheres are obtained. In some cases, when the spheres are overlapping (because of an offset in the vorticity direction), manual analysis is necessary, meaning that the coordinates of the midpoints are obtained by manual selection. The chosen frame rate to record movies is 5 frames/s except for the low shear rates, when a frame rate of 2 frames/s is sufficient. The result of the analysis is a trajectory that represents the displacement of one sphere relative to the other. Neglecting the effects of cross-stream migration, which is reasonable in all performed experiments38 given their duration and shear rates, a freely suspended single sphere in a Couette moves on a circular path around the midpoint of the Couette. The displacements one wants to observe are not displacements due to the curvature of the flow but only those due to the interaction. To be able to discern between the two displacements, a polar (r, θ)-coordinate system with the midpoint of the Couette as the origin is used to describe the positions of the spheres. The coordinate transformation is illustrated in Figure 3. After analyzing an experiment, one obtains (x, y) coordinates of the two freely moving spheres in the reference frame of the image. Fitting circles through these coordinates enables one to calculate the midpoint (xM, yM) of the Couette and the radial positions R where the particles are situated. The (x, y) coordinates of the two particles can be transformed to polar (r, θ) coordinates using the following formulas

Figure 3. Illustration of the coordinate systems. The particle trajectories are described in the (r, θ)-polar coordinate system. ⎧ 2 2 2 2 ⎪ r = x′ + y′ = (x − xM) + (y − yM ) ⎪ ⎨ ⎛ y − yM ⎞ ⎪ θ = arctan⎛⎜ y′ ⎞⎟ = arctan⎜ ⎟ ⎪ ⎠ ⎝ x′ ⎝ x − xM ⎠ ⎩

(8)

with all symbols as indicated in Figure 3. Using this transformation, one obtains the (r, θ) coordinates of the two particles. A trajectory or path line can now be calculated by a simple subtraction of the respective coordinates, obtaining Δr as a function of Δθ. For open trajectories, the radial shift will be used as a measure of the deviation from Newtonian behavior. It is defined as ΔrF − ΔrI a

(9)

with a being the particle diameter, ΔrI being an average of Δr on the approach trajectory before the interaction starts, and ΔrF being the analogue after the interaction. The radial shift is identically zero for open interactions between equally sized spheres in a Newtonian fluid in the absence of inertia.25 A positive radial shift (ΔrF > ΔrI) means that the separation between the particles in the velocity gradient direction increases as a result of the hydrodynamic interaction. Later in the text, ΔrI/a and ΔrF/a are termed the initial and final separations, respectively. One should bear in mind that the term separation in this case refers to only the separation between the spheres in the velocity gradient direction and not to the total distance between the midpoints of the spheres.

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BROWNIAN MOTION, INERTIA, AND GRAVITY Our objective is to study the effects of medium viscoelasticity, excluding possible effects of Brownian motion, inertia, and sedimentation. The Péclet number Pe, particle Reynolds number ReP, and sedimentation speed u0 have been calculated for the worst-case scenarios. The smallest Péclet number Pe = (ηγ̇a3)/(kT), with k being the Boltzmann constant and T being the absolute temperature, is obtained for BF at γ̇ = 0.02 s−1 and is O(108), and it can be concluded that the particles are nonBrownian. The largest particle Reynolds number ReP = (ρSγ̇a2)/ 5705

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Figure 4. Open trajectories for two spheres in N (a) and radial shift together with ((ΔrMAX)/a) − 1 vs shear rate γ̇ (b).

Figure 5. Sequence of microscopy pictures of an open path line between two spheres in the WMS. One of the spheres is labeled for clarity. Experimental parameters: γ̇ = 0.05 s−1, Wi = 0.09, nPL = 0.99, ΔrI/a = 0.079, and ΔrF/a = 0.53. The specific values for Δr/a and Δθ are shown for each snapshot.

(ηγ̇) is encountered in the WMS at γ̇ = 0.9 s−1 when the fluid viscosity η(γ̇) = 20 Pa·s. With the fluid density ρS = 1000 kg/ m3, the generalized Reynolds number is on the order of O(10−6), which is much smaller than 1. As such, it can be concluded that inertial effects play no role.30 Finally, the sedimentation speed u0 is calculated using u0 = ((ρP − ρS)ga2)/ (18η), with g being the gravitational acceleration, for the sedimentation of a single sphere in a Newtonian fluid. The most extreme situation is encountered in the ST. The speed u0 is found to be 1.4 mm/h with a particle density of ρP = 1050 kg/m3, a fluid density of ρS = 800 kg/m3, and a fluid viscosity at the relevant shear rate of η = 67 Pa·s. As such, given the time scale of an experiment (10 times longer). One can expect that the unsteadiness of the macroscopic flow field in the beginning will have no effect on the trajectory when the gap between the particles becomes small, in which the viscous lubrication forces and pressure field in between the particles can be assumed to have an overwhelming effect on the interaction. Only open trajectories are observed in the BF. Reversing or closed trajectories are not observed under the reported experimental conditions (i.e., shear rate and initial configuration of the spheres). Practically, it is far more difficult to obtain clean measurements of the trajectories in the BF than in the other fluids for multiple reasons. First, the BF has a very high extensional viscosity,37,45 rendering accurate positioning of the particles difficult. Second, because interactions propagate over long distances in the flow direction, it is impossible to record trajectories based on only a single magnification of the microscope alone, and we are obligated to swap magnification during the course of an experiment to be able to follow the particles over long enough interparticle distances. Because the curvature has to be assessed for the different magnifications separately, the experiments and analysis become significantly more elaborate. 5708

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Figure 8. Selected open trajectories for two spheres in the BF (a) and radial shift as a function of the initial separation for all obtained trajectories in the BF (b).

Figure 9. Comparison among the open interactions in the four different fluids. Radial shift as a function of the initial separation (a) and final separation as a function of the Weissenberg number (b).

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experimental error. The final separation shows a slight upward trend with Weissenberg number and is nearly independent of the initial separation. The data for the BF and N are not included in this plot because in these fluids the radial shift of the interactions is identically zero within experimental error, and as such the final separation is strongly dependent on the initial separation, not on Wi. The earlier separation of the particles in the shear-thinning fluids points to a weaker hydrodynamic coupling. Note in this respect that we do not observe a tendency for the radial shift to converge toward zero, even at Wi values as low as 0.1 (Figure 6b). In principle, one could reasonably expect that the Newtonian result (i.e., symmetric trajectories) would be recovered in the zero-shearrate limit. However, this limit may be hard to reach experimentally because during the approach and release the local deformation rates can remain relatively high, even when the macroscopic shear rate is low. The most surprising difference between the fluids is the totally different nature of the interactions in the BF as compared to that in the WMS and ST, as shown for example in Figure 9a. The three main differences in flow behavior between the BF and the other two viscoelastic fluids are the constant viscosity and hence high-viscosity contribution of the solvent,

COMPARISON AND DISCUSSION It is not feasible to map the entire trajectory space with experiments because the type of interaction depends on both the initial relative configuration of the spheres and the shear rate. Here we have restricted the study to particles in the same plane for four different fluids and a range of initial separations and Wi. Some general trends can be deduced from the experimental observations. Reversing interactions are observed in the WMS and ST on some limited occasions not in the BF or N. The conditions necessary to observe reversing path lines are a small initial separation in the velocity gradient direction and a large initial distance between the particles in the flow direction (large Δθ). Open path lines are more easy to study experimentally and are observed in all fluids because they fill the largest part of the trajectory space. A comparison between the open trajectories in the different fluids is made in Figure 9. Figure 9a shows the radial shift as a function of the initial separation. The radial shift is zero within experimental error for the interactions in the N and BF and positive for the interactions in the WMS and ST. Figure 9b shows the final separation as function of the Weissenberg number for the open interactions in the WMS and ST. The final separation in the ST is slightly lower than in the WMS, but the difference is within 5709

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velocity gradient direction upon interaction. The separation effect is far less apparent in the mentioned simulation studies, although the Oldroyd-B model does not incoporate shearthinning. In fact, the separation effect in the open trajectories is more apparent in the simulation study on the effects of inertia on the two-particle interactions in Newtonian fluids.30 As outlined in the Introduction, shear-induced alignment of particles in viscoelastic fluids has been observed in several experimental studies12−17,50 as well as in computer simulations.51−53 The phenomenon is poorly understood, but especially because string formation can be observed at low particle concentrations, it seems obvious that it should start with two particles and then evolve. The observations of tumbling interactions34,35 using the Oldroyd-B model and the observations of a fixed position between particles in the confined shear flow of a Giesekus fluid36 also point in that direction, as regardless of the differences between the observations the two particles stay together upon interaction. In our experiments, the particles always separate further away from each other upon interaction. Even when tumbling interactions would be observed, the alignment of two particles may occur at even higher Wi or must be a higher-order hydrodynamic effect related to either multiple interactions or interactions with the wall. The latter refers to both the role of confinement effects or indirect wall effects after cross-stream lateral migration of the particles toward the wall. The latter ideas are in agreement with recent experimental works15,16 and simulations.53 Here, we have been rather limited in terms of accessible rates (Wi of order 1). This is not different from the mentioned simulations on two-particle interactions, but it is on the low side for the typical observations on alignment.13 In a recent simulation study,54 an attempt was made to explain the origin of alignment. The study showed that shear flow creates a local concentration gradient between two colloidal, nonrotating particles. The concentration gradient induces depletion attractions. For our case, this explanation is most likely not applicable because first we observe separation and hence the depletion forces are not strong enough for the cases reported here. Furthermore, the simulations are for very small nonrotating particles (size of the same order of magnitude as the polymers), and experimental observations on alignment are in the noncolloidal, non-Brownian regime and on freely rotating particles. Finally, we point out that strong attractions between particles, due to either depletion as in ref 54 or hydrodynamic interactions as for the in-plane spiralling interactions in ref 35, should eventually lead to an overall motion of the two spheres like a single nonspherical, ellipsoidallike particle, and hence it should exhibit the typical orbital motion (so-called “Jeffery orbits”).19,22 Hence, the observations of open interactions with particles separating further away from each other in the velocity gradient direction in the shearthinning fluids may be in qualitative agreement with the observations of alignment because the interactions are “weaker” and the orbital motion of the two particles is surpressed, opening up the possibility for particles to line up. The observations of shear-thickening in filled Boger fluids at relatively low particle concentrations49 may be rationalized on the basis of the observed path lines. Shear-thickening is well known for suspensions based on Newtonian fluids and occurs at high volume concentrations. In Boger fluids, the phenomenon was found to occur at lower concentrations of around 7 vol %.49 Particles similar to those studied by Scirocco et al.49 in different Boger fluids had also been studied in a Newtonian

long relaxation time, and higher extensional viscosity of the BF. Because irreversible (i.e., asymmetric) open interactions have also been observed in shear-thinning nearly inelastic fluids,31 the most plausible parameter responsible for the differing behavior of the trajectories is whether the suspending fluid has a constant viscosity or is shear-thinning. A hypothetical and qualitative explanation can be found when considering the effect of shear-thinning on the lubrication forces in the small gap between the particles. A shear-thinning viscosity is known to weaken the lubrication forces acting between particles near contact significantly.46 Moreover, the effects of shear-thinning in squeeze flows are generally strong and dominant compared to the effects of normal forces,47 which are generally dominated by the viscous lubrication pressure. These effects of normal forces become important only when the particles are close and the fluid in the gap is at a relatively high shear rate.47 A further, more detailed comparison of the radial shift data for the open interactions in the WMS and ST as shown in Figure 9a reveals that the radial shift in the ST goes to zero at slightly lower initial separations (∼ 0.5) compared to the WMS (∼ 0.6) and that the radial shift is slightly lower in the ST in most cases. Although the differences are within the limits and the data set is limited, the differences point to an effect of the higher degree of shear-thinning in the WMS compared to that in the ST. The strong differences in the rheological behavior of particles in Boger fluids as compared to the behavior in shear-thinning elastic fluids have been observed on many occasions: single particles in Couette flow were found to migrate in shearthinning elastic fluids, not in Boger fluids;38 the alignment of particles was observed in shear-thinning fluids, not in Boger fluids;13,14 the value of the drag coefficient was found to be very high in some specific Boger fluids as compared to the drag coefficient in Newtonian and shear-thinning elastic fluids under comparable flow conditions;48 and filled Boger fluids display shear-thickening at relatively low particle concentrations as opposed to filled shear-thinning or Newtonian fluids.49 Comparing our experimental findings to the results of computer simulations,34−36 we can conclude that there appears to be less tendency to tumble in the experiments than in the simulations,34,35 although this might be in appearance only as the tumbling dynamics are observed only in a very narrow range of initial separations (and furthermore in Oldroyd-B fluids). Although we did, in principle, reach these conditions (i.e., they are in the range of our data in terms of initial separations and shear rates), the exact, narrow regime of initial separations was perhaps missed. Choi et al.36 did not observe tumbling. They observed a motion of the particles toward a fixed relative position in a mildly confined flow of a Giesekus fluid (hence including shear-thinning) again in a very narrow range of initial separations. We also did not observe this phenomenon in the experiments, perhaps for similar reasons. The simulation studies do, however, show a clear qualitative difference between the constant viscosity Oldroyd-B fluid,34,35 where particles are shown to tumble, and the shear-thinning Giesekus model,36 where particles reach a certain steady relative position, pointing toward the importance of the details of the rheological behavior of the fluid (i.e., the level of shearthinning) on the nature of the two-particle interactions, which is also clearly important in the experiments. In the reported experiments, the open interactions are by far the most abundant. They appear to be more asymmetric than those reported in the simulation studies. In the shear-thinning fluids, the particles clearly separate strongly from each other in the 5710

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fluid, and a volume concentration of roughly 55 vol % was found to be necessary for the onset of shear-thickening for these particles.55 The trajectories in the Newtonian fluid are compared to those in the Boger fluid in Figure 10. The

Article

CONCLUSIONS The effect of medium viscoelasticity on the interaction between two equally sized spherical particles in shear flow was studied experimentally. Symmetric open interactions with a zero radial shift are measured first in the reference case of a Newtonian fluid, and good quantitative agreement was obtained. In the BF, the open path lines have a zero radial shift, but the trajectories are not completely symmetric. In the WMS and ST, two distinct types of trajectories have been observed: reversing and open trajectories. For the open trajectories, the spheres are shown to separate from each other in the velocity gradient direction upon interaction. The final separation between the particles in the velocity gradient direction is found to be nearly constant with only a slight dependence on the Weissenberg number. Furthermore, the open interactions in the WMS and ST are nearly identical within experimental error for the range of conditions studied here. The reversing path lines are observed only when the initial separation between the spheres is relatively small in the velocity gradient and large in the flow direction. On the basis of the presented results, shear-thinning, rather than elasticity, seems to be the key rheological parameter of the fluid that dominates the overall nature of the hydrodynamic interactions.

Figure 10. Comparison between the open trajectories in the Newtonian fluid and the Boger fluid.

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trajectories are comparable, in the sense that the radial shift is zero in both fluids and different because the interactions in the Boger fluid are not completely symmetric and act over significantly longer distances. In the Boger fluid, the particles “feel” each other earlier, before the interaction, and much longer after the interaction, possibly because of effects related to the extensional viscosity. Comparing more quantitatively, one can say that in the Newtonian fluid the particles feel each other up to a distance of roughly Δθ ≈ 2°, whereas in the BF this distance is roughly doubled to about Δθ ≈ 4°. On the basis of this difference, one could argue that the spheres in the Boger fluid have an apparent size that is twice as large as in the Newtonian fluid. This apparent size gives rise to an apparent volume fraction that is the real volume fraction multiplied by 8. This factor of 8 rationalizes the difference in volume fraction for the onset of shear-thickening: 7 vol % in the Boger fluid and 55 vol % in the Newtonian fluid. Although the Boger fluid used in this study and the Boger fluids studied by Scirocco et al.49 are not completely identical and multiparticle interactions are important in shear-thickening, the scaling with relative particle size works amazingly well. Shear-thickening in Newtonian fluids has, from a microstructural point of view, been explained by the formation of hydrodynamically induced aggregates, so-called hydroclusters.56−59 One could speculate that in the Boger fluid hydroclusters based on the apparent size of the spheres form, with particles separated further away from each other in lessdense clusters. However, in Newtonian fluids strong lubrication forces between particles very close to each other have been found to be responsible for the strong shear-thickening, making the driving force behind shear-thickening most likely from a different nature in the BF. For the BF, one could speculate that the additional energy dissipation relates to the high extensional viscosity of Boger fluids45 and the possible large effect this might have on the interaction between the particles upon separation, as exemplified by the large tail in the trajectories in Figure 10a.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address §

Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS F.S. acknowledges a graduate fellowship from the Fund for Scientific Research-Flanders (FWO-Vlaanderen). J.V. acknowledges financial support by ESA, the Belgian Prodex Office, and the Fund for Scientific Research-Flanders (FWO-Vlaanderen) through grant G.0554.10.

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