Rheol Acta (2008) 47:621–642DOI 10.1007/s00397-008-0268-1

ORIGINAL CONTRIBUTION

High shear rate viscometry

Christopher J. Pipe · Trushant S. Majmudar ·Gareth H. McKinley

Received: 10 July 2007 / Accepted: 7 February 2008 / Published online: 29 April 2008© Springer-Verlag 2008

Abstract We investigate the use of two distinct andcomplementary approaches in measuring the visco-metric properties of low viscosity complex fluids athigh shear rates up to 80,000 s−1. Firstly, we adaptcommercial controlled-stress and controlled-rate rheo-meters to access elevated shear rates by using parallel-plate fixtures with very small gap settings (down to30 μm). The resulting apparent viscosities are gapdependent and systematically in error, but the datacan be corrected—at least for Newtonian fluids—viaa simple linear gap correction originally presented byConnelly and Greener, J. Rheol, 29(2):209–226, 1985).Secondly, we use a microfabricated rheometer-on-a-chip to measure the steady flow curve in rectangu-lar microchannels. The Weissenberg–Rabinowitsch–Mooney analysis is used to convert measurements ofthe pressure-drop/flow-rate relationship into the truewall-shear rate and the corresponding rate-dependentviscosity. Microchannel measurements are presentedfor a range of Newtonian calibration oils, a weaklyshear-thinning dilute solution of poly(ethylene oxide),a strongly shear-thinning concentrated solution ofxanthan gum, and a wormlike micelle solution that ex-hibits shear banding at a critical stress. Excellent agree-ment between the two approaches is obtained for theNewtonian calibration oils, and the relative benefits of

Paper presented at 4th Annual European RheologyConference (AERC) on April 12–14, 2007, Naples, Italy.

C. J. Pipe (B) · T. S. Majmudar · G. H. McKinleyHatsopoulos Microfluids Laboratory,Department of Mechanical Engineering,Massachusetts Institute of Technology,Cambridge, MA 02139, USAe-mail: cjpipe@mit.edu

each technique are compared and contrasted by con-sidering the physical processes and instrumental limita-tions that bound the operating spaces for each device.

Keywords Viscometry · Channel flow ·High shear rates · Microfluidics · Parallel plates

Introduction

The behavior of complex liquids at large deformationrates is relevant in many processes involving coating,spraying, lubrication, and injection molding. Althoughboth shear and extensional deformations can be im-portant in complex flows, in this study, we concentrateon the rheological behavior in steady shearing flow.From dimensional analysis, we expect γ̇ ∼ U/L, whereU is a characteristic velocity difference acting over acharacteristic distance L, indicating that, to attain highshear rates, it is necessary to either (a) increase U or (b)decrease L. Large velocities can lead to high Reynoldsnumbers Re = ρU L/η, where ρ is the density and η thedynamic viscosity of the fluid, and loss of viscometricflow. Thus, it is usual to strive to minimize L, and thelimiting factor in accessing high shear rates is the accu-racy with which flow devices or rheological test fixturescan be manufactured and aligned so that geometricperturbations remain small (δL/L < 1), even when Lis reduced.

Various experimental configurations have been pro-posed for measuring viscosity at high shear rates,including cylindrical Couette flow with narrow gaps(Merrill 1954), torsional flow between rotating parallelplates (Connelly and Greener 1985; Dontula et al. 1999;Kramer et al. 1987), flow through capillaries (Duda

622 Rheol Acta (2008) 47:621–642

et al. 1988; Talbot 1974) and slits (Erickson et al. 2002;Laun 1983; Lodge and de Vargas 1983), and impactflow between a rotating ball and plate (O’Neill andStachowiak 1996).

Measurements using narrow gap Couette flows be-tween concentric cylinders and between parallel-platefixtures have been used to measure the viscosity ofNewtonian and shear-thinning liquids up to γ̇ ∼ 106 s−1

(Connelly and Greener 1985; Davies and Stokes 2005;Dontula et al. 1999; Kulicke and Porter 1981; Merrill1954; Mriziq et al. 2004) using gaps 0.5 μm ≤ H ≤50 μm. Suggestions that Newtonian fluids might showa decrease in viscosity at shear rates γ̇ ≈ 5 × 104 s−1

presented by Ram (1961) have been reexamined byDontula et al. (1999), who argue that the fall in viscosityof a glycerin–water solution at high shear rates may wellbe due to non-viscometric flow phenomena such as vis-cous heating or hydrodynamic instabilities, rather thanarising from a true rate-dependent material property.

Flows through capillaries and slit channels have alsobeen widely used to study rheology at high shear rates(Duda et al. 1988; Erickson et al. 2002; Kang et al.2005; Laun 1983; Lodge and de Vargas 1983; Talbot1974), and capillary viscometry can be an extremelysimple and reliable technique for measuring shearviscosities. However, to minimize the importance ofnon-viscometric flow at the entrance and exit of thecapillary, it is usual to use long capillaries that can leadto long residence times for fluid elements in the shear-ing flow. Furthermore, at high shear rates, an evolvingthermal boundary layer resulting from viscous heatingmust be accounted for (Duda et al. 1988).

Beyond the realm of classical rheometry, custommicrofabricated devices have also been used to mea-sure the rheological response of complex fluids onlength scales of 1–10 μm (Clasen and McKinley 2004;Dhinojwala and Granick 1997; Mukhopadhyay andGranick 2001). With advances in micro- and nano-fabrication techniques (Marrian and Tennant 2003; Xiaand Whitesides 1998) enabling the routine and reliablemanufacture of flow devices with geometric features< 100 μm, there has been a significant increase instudies exploiting microfluidic flow geometries for rhe-ological characterization (Degré et al. 2006; Guillotet al. 2006; Hudson et al. 2004; Zimmerman et al.2006), although none of these studies have addressedthe response at high shear rates.

As the characteristic length scale of the geometryis decreased to O(1 μm), it becomes increasingly im-portant to separate the bulk rheological response ofa sample from effects due to the confining surfaces(McKenna 2006). The issue of wall slip has been shownto be significant, as very small length scales are probed,

but nearly all experimental evidence suggests that slip,if present, occurs over length scales 0–50 nm (Granicket al. 2003; Lauga et al. 2007) and is therefore usuallynegligible for most homogeneous fluids on the micronscale. However, apparent wall slip, caused by depletionor adhesion layers at the walls, can be detected whentesting heterogeneous liquids that have distinct mi-crostructural elements even in flows with length scales∼ 10 μm (Clasen et al. 2006; Degré et al. 2006).

Studies of strongly inertial flows through channelswith characteristic depths O(100 μm) are reviewedby Obot (2002) who finds that, despite some reportsthat the transition to turbulent flow in smooth-walledmicrochannels can occur at Reynolds numbers as lowas Re ≈ 200 (e.g., Peng et al. 1994), the experimentalevidence so far does not support a significant decreasebelow the usual value for macroscale flows Re ≤ 2,000.

In the present study, we compare and contrast theefficacy of two devices, a conventional rotationalparallel-plate rheometer using sample gaps in the range10–500 μm, and two microfluidic slit channels withdepths of 24.6 and 50.7 μm, to characterize shear-dependent viscosities at deformation rates up to 105 s−1.We show that the viscometric response at high ratesfor fluids ranging from constant viscosity mineral oilsto strongly shear-thinning polymer solutions and shear-banding micellar solutions can all be determined us-ing these techniques. We compare the results fromeach device to illustrate the experimental difficultiesthat can be encountered in each geometry and thelevel of reproducibility that can be obtained. We firstpresent an overview of the fluids used in this workas well as the relevant analytical framework for theparallel-plates geometry and the microfluidic channels.We then investigate the use of conventional rotationalrheometers to access high deformation rates and iden-tify errors in zeroing the gap as a primary sourceof discrepancy between measured values of apparentviscosity and the true viscosity. Using Newtonian cal-ibration oils, we then implement a calibration proce-dure originally presented by Connelly and Greener(1985) to evaluate the gap errors. We use the resultinggap corrections to obtain true shear rates and trueviscosities from the apparent values provided by therheometer software. In addition, we investigate twoadditional phenomena that can affect the viscous re-sponse at high shear rates; centrifugal stresses andviscous heating. We then proceed to investigate theviscometric response of Newtonian fluids using mi-crofluidic slit channels equipped with flush-mountedpressure sensors. Subsequently we then investigate theresponse in both types of device due to highly shear-thinning liquids. Finally, we compare and contrast the

Rheol Acta (2008) 47:621–642 623

results from each device to illustrate the principal ex-perimental difficulties associated with each geometryand the level of reproducibility that can be obtained.The material and instrumental parameters constrainingthe application of each device are then combined toproduce operating space diagrams that can guide theeffective use of each approach.

Experimental techniques

Test fluids

Liquids exhibiting a range of viscometric behaviors,from Newtonian to strongly shear-thinning, were in-vestigated experimentally. Mineral oils S60 and N1000supplied by Cannon Instrument (Pennsylvania, USA)and marketed as constant viscosity calibration fluidsfor viscometers as well as a Silicone oil (DMS T25)from Gelest (Pennsylvania, USA) were used to testthe response of Newtonian fluids with nominally con-stant viscosity. Henceforth, these fluids will be referredto by the labels N1, N3, and N2, respectively, as in-dicated in Table 1. We also study a weakly shear-thinning solution of 0.1% poly(ethylene oxide) (PEO)in a mixture of 55% glycerol–44.9% distilled water(supplied by Sigma Aldrich, MO, USA). The PEOwas polydisperse with viscosity—averaged molecularweight Mv = 2 × 106 g/mol, and the solution was pre-pared with gentle mixing and rolling to avoid degra-dation of the polymer. The aqueous PEO solution wasthen gently mixed with the glycerol for 5 min and leftto stand for a further 24 h. More complex viscometricresponses were explored using a strongly shear-thinning aqueous solution of 0.3% xanthan gum(Milas et al. 1990) supplied by CP Kelco (Georgia,USA) as well as a wormlike micellar solution of3.2 wt%/0.76 wt% cetylpyridinium chloride/sodium sal-

icylate in 0.56 wt% sodium chloride brine (Pipe et al.,in preparation), which exhibits shear-banding behaviorabove a certain critical shear rate γ̇ ≈ 1 s−1.

Parallel-plate apparatus and gap error correction

The rheological response of the Newtonian and non-Newtonian fluids were measured using parallel-platefixtures on an advanced rheometric expansion system(ARES) strain controlled rheometer (TA Instruments,New Castle DE, USA) and an AR-G2 stress-controlledrheometer (TA Instruments). In Fig. 1a, we show thetypical idealized arrangement assumed to exist with aparallel-plate geometry. The arrangement consists oftwo plates of radius R, separated by a constant anduniform gap height H. In the ARES, the top plate isfixed and attached to a torque transducer, while thebottom plate rotates with an imposed angular velocity�. In the AR-G2 the bottom plate is fixed and the topplate rotates under the action of an imposed torqueT . The radii of the plates on the ARES and AR-G2 devices were 50 and 40 mm, respectively. The gapseparations used in our experiments span the rangefrom 500 mm to 10 μm . In each case, the temperaturecontrol is provided through the Peltier plate system,which is accurate to within ±0.1◦C. In all experiments,the fluid was carefully loaded between the two platesto avoid air bubbles and the excess fluid removed toensure a smooth cylindrical interface.

To accurately measure the true viscosity in rotationalrheometers at narrow gaps and high shear rates, precisealignment of the parallel plates is crucial (Connelly andGreener 1985; Kramer et al. 1987). At very small gaps(H ≤ 10 μm), even the viscosity of air in the narrowgap between the plates while zeroing the gap has beennoted as a possible source of error (Davies and Stokes2005). In this paper, we use a calibration procedurebased on the work of Connelly and Greener (Connelly

Table 1 Overview of test fluids used in the present study and relevant physical parameters at 20◦C

Fluid η0 β k Nominal viscous behavior(Pa s) (–) (W m−1K−1)

S60 (N1) 0.137 13.5 0.1 NewtonianDMS T25 (N2) 0.485 5.5 0.16 NewtonianN1000 (N3) 2.867 5.6 0.1 NewtonianWater 0.0010 6.7 0.5 NewtonianGlycerol–water–PEO solution (T = 22.5◦C) 0.0155 0.4 Weakly shear-thinningAqueous xanthan gum solution (T = 22◦C) 11.5 Strongly shear-thinningCPyCl/NaSal micellar solution(T = 22◦C) 14.5 Shear-banding

In this study, η0 is the stated viscosity in the limit of zero shear rate, β is the thermal sensitivity (see Eq. 21), and k is the thermalconductivity of the fluid.

624 Rheol Acta (2008) 47:621–642

Upper Plate

Lower Plate

RFluid

Motor

a b δφ

ε

Zero gap

R

ε

c δφ

Fluid filled configuration

R

H

Fig. 1 Schematic diagram of a conventional rotational rheome-ter; two plates of radius R, at gap height H. The top or thebottom plate rotates with an angular velocity �. a Ideal situationin a parallel-plate rheometer, the plates are perfectly parallel. bSources of error during gap zeroing; a slight parallax (exagger-ated in the figure) causes the plates to touch when not parallel,thus introducing a gap error of size ε. c Profile of the fluid samplewhen placed between nonparallel plates

and Greener 1985) to estimate the total effective errorassociated with ‘zeroing the gap’. Figure 1b shows theprincipal source of error in the alignment of platesassociated with axial ‘run-out’ of the shaft and theresulting non-orthogonality between the plate and ro-tation axis. In a modern rotational rheometer using‘auto-gap zeroing’ based on electrical conductivity orfriction detection, the gap is considered to be ‘zeroed’when any point of the top plate touches the bottomplate. A parallax in the alignment of the plates cancause the situation shown in Fig. 1b, in which the gapis considered to be zero, but in reality, different partsof the upper fixture are at different distances from thebottom plate. The maximum distance between the topfixture and the bottom plate sets a scale for the errorincurred in zeroing the gap. This error is denoted the‘gap error’ (ε). For large gap separations (H >> ε), thiserror is expected to be negligible but it is of increas-ing importance for small gaps. The final configurationwith fluid filled between the plates is shown in Fig. 1c,where the gap is small enough (H ∼ ε) that the effectof plate misalignment is noticeable in the fluid sam-ple confined within the plates. The analogous problemhas been considered analytically for the cone-and-platerheometer by Dudgeon and Wedgewood (1994) using adomain perturbation approach. Because the measuredrheological quantities of interest such as shear rate andviscosity are dependent on the gap height, the gap errorintroduces a systematic error in measured quantities, inaddition to the intrinsic instrument accuracy.

Torque, displacement, and normal force are thefundamental quantities measured by the rheometer.

These raw measurements are then used to calculatestress, strain, shear rate, viscosity, and normal stressdifference. To eliminate the systematic discrepanciesassociated with the gap error ε and to obtain accuratevalues of the calculated quantities from the measuredvariables, it is necessary to determine the error in gapheights via calibration, and the apparent values of mea-sured quantities then have to be corrected for this gaperror.

There are several published procedures for assessingthe gap error for the parallel-plate geometry (Connellyand Greener 1985; Kramer et al. 1987). We followthis method with a slightly modified analysis, whichis presented below. The procedure consists of singlepoint tests of a Newtonian fluid of known viscosityunder steady simple shear flow at different gaps. Inthe steady shear single point test on a strain controlledrheometer, for example, a shear rate and the durationfor which the specified shear rate is to be applied isspecified by the user, and the viscosity is measuredafter the specified equilibration time. We have usedshear rates from 10 to 100 s−1 for different fluids. Thespecific shear rates used were chosen such that the mea-sured torque was well above the minimum measurabletorque. At the beginning of the experiment, the gap iszeroed to obtain a reference datum for all subsequentmeasured gap heights. The fluid is then loaded, andthe upper plate is lowered to a specified gap heightH and left undisturbed for 120 s to equilibrate at thespecified temperature. An ‘apparent shear rate’ γ̇a andthe duration of measurement are then specified in therheometer software. The duration for which the shearrate is applied was 30 s. At the end of one single pointmeasurement, an ‘apparent viscosity’ ηa is reported bythe software. The fluid is then removed, a new sampleis loaded, and the procedure is repeated at a differentgap height H. The range of specified gap heights wasdecreased steadily from 500 to 10 μm.

For a given plate of radius R, specified gap height H,and angular velocity �, the apparent shear rate γ̇a at therim of the rotating parallel-plate fixture is given by:

γ̇a = � RH

. (1)

The apparent viscosity ηa reported by the software iscomputed from the definition:

ηa = < τ >

γ̇a, (2)

Rheol Acta (2008) 47:621–642 625

where < τ > is the expected shear stress at the rimcalculated from the measured torque, T , assuming anideal torsional shear flow, and is given by:

< τ > ≡ 2 T

π R3= ηtrue γ̇true, (3)

where γ̇true and ηtrue are the true shear rate at the rimand the true viscosity, respectively. Combining Eqs. 1,2, and 3, we have,

ηa = ηtrue γ̇true

� R/H. (4)

As discussed previously, in practice, there is alwayssome error in zeroing the gap between the plates. Forerrors of the form sketched in Fig. 1, the gap is alwaysbiased toward larger values than the commandedvalue. The simplest form of correction is to postulate(Connelly and Greener 1985) that H → (H + ε) so thatthe true shear rate at the rim is

γ̇true = � R(H + ε)

. (5)

Combining Eqs. 4 and 5, we can write (Kramer et al.1987):

1

ηa= 1

ηtrue

{1 + ε

H

}. (6)

Equation 6 suggests that, for a constant gap error ε, thereciprocal of the ‘apparent’ or reported viscosity 1/ηa

should increase linearly with 1/H. A linear regressionof the apparent viscosity ηa for various gap heights Hprovides us with two relevant quantities: the interceptof the line gives 1/ηtrue and the slope gives ε/ηtrue, en-abling us to calculate the gap error ε. The analysis for astress-controlled rheometer is analogous, with the sameresulting relation between the true viscosity and thecommanded gap. We utilize this analysis in conjunctionwith gap calibration experiments on the ARES andAR-G2 rheometers to obtain typical values of the gaperrors for both instruments, when using parallel-plategeometries. We use the values of gap errors and Eqs. 1–6 to obtain true shear rates and true viscosities from theapparent values reported by the software.

For non-Newtonian fluids in a parallel-plate geome-try, we note that the correction to the viscosity resultingfrom shear thinning leads to (Bird et al. 1987):

ηa = (T /2π R3)

γ̇a

[3 + d ln(T /2π R3)

d ln γ̇a

]. (7)

Equation 7 gives the correction to the viscosity dueto inhomogeneity of the shear rate in a torsional flowbetween parallel plates. For Newtonian liquids,the logarithmic derivative term has a value of

unity, thus simplifying to Eq. 2. However, for non-Newtonian fluids, the logarithmic gradient term canbe substantially less than one, and this can changethe value of measured viscosity by up to 25%.In the section entitled “Parallel-plates geometry:Newtonian fluids,” we present calibration experimentsusing Newtonian mineral oils, implement the gaperror correction analysis outlined above, and comparethe results with independent measurements obtainedusing the microfluidic viscometer described in the nextsection. We also present a similar analysis for non-Newtonian fluids, making use of Eq. 6 to determine gaperrors and Eq. 7 to calculate rate-dependent correctionsto measured values of the apparent viscosities.

When the fluid sample confined between two parallelplates is sheared, the free surface may deviate from apurely cylindrical shape. This perturbation of the freesurface arises due to the additional viscous stresses thatact at the interface and has been considered in detail byOlagunju (1994) for Newtonian and Oldroyd-B fluids.According to this analysis, for coaxial parallel plates ofradii R and separation H, rotating with angular velocity�, the correction to the measured viscosity is of theform:

ηa = 2T

π R3γ̇a

[1 + O

(H4

R4

)]. (8)

Equation 8 shows that the effect of any free surfaceperturbations on the measured viscosity is O

( HR

)4. For

the configurations considered in the present paper, theaspect ratios for the plates of 25-mm radii range fromH/R = 0.04 for a 1-mm gap height to H/R = 0.002 fora gap height of 50 μm. Edge effects on the measuredviscosity are therefore negligibly small.

Microfluidic slit rheometer

The microchannels were fabricated by RheoSense(VROCTM, San Ramon CA, USA) and are made fromPyrex mounted on a gold-coated silicon base containingthree flush mounted microelectromechanical systems(MEMS) pressure sensors. Figure 2 indicates the fun-damental configuration of the microchannels. Values ofthe channel depth d, slit width w, aspect ratio L = d/w,hydraulic diameter dh = 4 × area/circumference, andvolume V = L × w × d, for the two devices used in thiswork, are given in Table 2. For both slits, the totalchannel length L from inlet to outlet is 12.65 mm, andthe distance between the inlet and the first pressuresensor is 2.025 mm. This latter distance is equivalentto 42dh and 20dh for channels A and B, respectively,and, for all flows considered in the present study, is

626 Rheol Acta (2008) 47:621–642

Q

ΔP

d

w

L

Pressure sensors (800x800 μm2)

Fig. 2 Sketch of RheoSense VROC slit micro-rheometer; thedimensions are given in the text

significantly larger than the entrance length needed forfully developed flow: For low Reynolds numbers Re =ρQdh/wdη, the entrance length is Le = dh(0.6/(1 +0.035Re) + 0.056Re), and for larger Reynolds num-bers before the transition to turbulence at Re ≈ 2,000,Le = dh(0.5 + 0.05Re) (Nguyen and Wereley 2002).The center-to-center distances between the first andsecond pressure sensors and between the second andthird sensors are 2.5 and 3.8 mm, respectively, resultingin pressure measurements over a total streamwise dis-tance L = 6.3 mm. The MEMS pressure sensors, eachmeasuring 800 × 800 μm2, are located along the center-line of the channel and were manufactured using simi-lar techniques to those presented in Baek and Magda(2003). The maximum absolute measurable pressurePmax of the devices used in the present work are given inTable 2; clearly, the maximum available pressure dropacross the sensor array �Pmax is less than the valuePmax in Table 2, as the latter value must also allow foradditional pressure losses due to viscous stresses for therest of the slit downstream of the final sensor before theflow exits the VROC channel at atmospheric pressure.The wet-etching process used to make the flow channelsleads to eroded interior corners that are undercut witha radius of curvature similar to the channel depth; forchannel aspect ratios L � 1, this deviation from a rec-tangular cross-section is negligible. The cross sectionalprofile of the device was measured across the chan-nel width using a mechanical surface profiler (Alpha

Step IQ, KLA-Tencor, San Jose, CA, USA) accurateto ±50 nm. Away from the undercut regions at eachsidewall, deviations in the channel height were foundto be less than 1% of the total height, which is typicalfor this fabrication technique. During experiments, thetemperature is monitored to an accuracy of ±0.25◦Cusing a temperature sensor located below the siliconmembrane at the center of the channel.

The volume flow rate Q in the channel was con-trolled using a PHD4400 Syringe Pump supplied byHarvard Apparatus (Holliston, MA, USA) in con-junction with Hamilton Gastight glass syringes (Reno,NV, USA) with volumes from 50 μl to 2.5 ml. Twotypes of tubing of varying modulus, perfluoro alkoxyalkane (PFA; E = 0.035 GPa) and polyetheretherke-tone (PEEK; E = 3.6 GPa), were used to connect thesyringe to the VROC, and no reactions were observedbetween the tubing and the test fluids.

Before starting the experiments, the microchannelwas filled with the test fluid by syringe and then leftto relax so that all transients had decayed and steady-state baseline pressure readings were achieved with noflow. It should be noted that, for the strongly shear-thinning fluids, the time necessary for pressure tran-sients associated with the filling process to decay canbe O(1,000 s) because even small transient flow ratesgive rise to large viscous stresses as a result of the largezero shear viscosities of the fluids. At the beginningof each experiment, before starting the syringe pump,the pressure sensors are zeroed in the software. Duringexperimental runs, the pressure measured by each ofthe sensors was monitored at 67 Hz using LabViewsoftware, and each flow rate was maintained for at least60 s to ensure that a steady state was achieved. Thetotal volume of test fluid necessary for experimentsdepends greatly on the viscosity and also the durationof the transient viscoelastic response. A Newtonianliquid with viscosity O(1 Pa s) requires as little as 100 μlto fill the system (syringe, tubing, and channel) andperform measurements over the accessible range offlow rates, whereas liquids with viscosity O(0.001 Pa s)need up to 15 ml as a result of the higher volumetricflow rates needed to achieve suitable shear rates andviscous pressure drops. We note that fluids exhibit-ing a non-Newtonian viscosity can necessitate more

Table 2 Properties of VROC microchannels used in the present study (quantities are defined in the text)

Channel Depth d Width w Length L Aspect ratio L Hydraulic diameter dh Pmax Volume V(μm) (mm) (mm) – (μm) (kPa) (μl)

A 24.6 3.1 12.65 0.0079 48.2 40 1.0B 50.7 2.82 12.65 0.018 99.6 60 1.8

Rheol Acta (2008) 47:621–642 627

measurements and, thus, a greater sample volume toaccurately capture the viscous response.

To find the equilibrium flow curve connecting the im-posed flow rate and the measured pressure differenceand, hence, evaluate the shear rate-dependent viscosityof a solution, the steady flow in the microchannel isconsidered to be a fully developed two-dimensional(2-D) flow, which is a good approximation for L � 1.The pressure drop �P necessary to drive the flow astreamwise distance l0 is related to the wall shear stressτw by:

wd �P = 2l0(w + d)τw. (9)

which represents a force balance between the pressureacting across the cross-sectional area of the channeland the viscous shear stresses present at the walls. Forincompressible 2-D flow of a constant viscosity liquid ina rectangular channel, the wall shear rate γ̇w is a linearfunction of the flow rate Q:

γ̇w = 6Qwd2

. (10)

For incompressible fully developed 2-D flows of liquidswith a rate-dependent viscosity, the calculation of γ̇w ismore complex because the velocity profile is no longerparabolic. Using Eq. 10 thus results in an apparentshear rate γ̇a = 6Q/(wd2) analogous to Eq. 1 obtainedfrom the rheometer. However, because the channeldimensions are known precisely and the flow is steadyand two dimensional, the true wall shear rate can befound using the Weissenberg–Rabinowitsch–Mooneyequation (e.g., Macosko 1994):

γ̇w,true = γ̇a

3

[2 + d(ln γ̇a)

d(ln τw)

], (11)

where τw is calculated from Eq. 9. The true viscosityfunction is then computed as

η(γ̇w,true) ≡ τw

γ̇w,true= wd�P

2l0(w + d)

1

γ̇w,true. (12)

It should be noted that, although Eq. 11 is sometimescalled a ‘correction’, it is an exact solution of thesteady linear momentum equation for a generalizedNewtonian fluid with a rate-dependent viscosity. Toevaluate the derivative in Eq. 11, fitting the variationof ln γ̇a(ln τw) with a first or second order polynomial isgenerally sufficient; for the wormlike micellar solutiondiscussed below, it is necessary to fit polynomials piece-wise over specific ranges to capture the extreme shear-thinning behavior observed at a critical shear rate.

Results and discussion

Parallel-plates geometry: Newtonian fluids

Gap error calibration

We first calibrate the rheometers to quantify the gaperror. It was demonstrated above that the simplest gapcorrection model leads to a linear variation between1/ηa and 1/H (Eq. 6). In Figs. 3 and 4, we show thegap calibration for ARES and AR-G2, respectively, forthree Newtonian liquids N1, N2, and N3 (with prop-erties given in Table 1). Each fluid was subjected tosteady shear for 30 s, at gap heights varying from 10 to400 μm, and the apparent viscosity ηa was calculatedfrom the measured torque. Gap settings from 10 to100 μm were varied in steps of 10 μm and then from100 to 500 μm in steps of 100 μm. Each data point onthe plot represents the steady-state value of apparentviscosity at a specific gap height. Temperatures T andapparent shear rates γ̇a used for each fluid in the datashown are given in Table 3. The solid symbols areexperimental data points, and the dashed lines are fitsto the data using Eq. 6, using 1/ηtrue and ε as fittingparameters.

It can be seen from the figures that all three liquidsclosely follow the simple linear relationship given inEq. 6. The intercept of the straight line fit gives 1/ηtrue,and the slope gives ε/ηtrue. Combining the two, wecalculate the measured true viscosity ηtrue and the size

0.00 0.02 0.04 0.06 0.08 0.100

10

20

30

40

50

60

η-1 [

Pa s

]-1

H-1 [μm-1]

N1, T=20ºCN2, T=25ºCN3, T=20ºC

a

Fig. 3 Gap error calibration data for three Newtonian fluidsN1, N2, and N3 for the strain-controlled ARES rheometer. Theplot shows the inverse of the apparent viscosity plotted againstinverse of the gap height. The solid symbols are the experimentaldata, and the dashed lines show the fits of Eq. 6 to the data.The experimental parameters, and the results of the fit, the trueviscosities ηtrue, and the gap errors ε are given in Table 3

628 Rheol Acta (2008) 47:621–642

0.00 0.02 0.04 0.06 0.08 0.100

10

20

30

40

50 N1, T=25ºCN2, T=25ºCN3, T=25ºC

η-1 [

Pa s

]-1

H-1 [μm-1]

a

Fig. 4 Gap error calibration data for three Newtonian fluidsN1, N2, and N3 for the stress-controlled AR-G2 rheometer. Theplot shows the inverse of the apparent viscosity plotted againstinverse of the gap height. The solid symbols are the experimentaldata, and the dashed lines show the fits of Eq. 6 to the data.The experimental parameters, and the results of the fit, the trueviscosities ηtrue, and the gap error ε are given in Table 3

of the gap error ε. For example, for the fluid N3, fittingEq. 6 to the data for ARES, we find:

intercept : 1

ηtrue= 0.369 ± 0.008 Pa−1s−1

gradient : ε

ηtrue= 8.93 ± 0.22 m Pa−1s−1

which gives ηtrue = 2.704 ± 0.060 Pa s, and ε = 24 ±1 μm, where we have propagated uncertainties in boththe slope and the intercept to obtain uncertainties inthe true viscosity and the gap error. The gap errorobtained in this manner is a quantitative measure ofthe errors introduced in zeroing the gap between twoplates. The results are tabulated in Table 3, where themanufacturer’s stated viscosity, η0, is compared withthat found from the intercept of the straight line fit to

the data using Eq. 6. Also shown are the gap errorsobtained for each case.

For the two fluids N2 and N3, the discrepancy inthe stated viscosities η0 and measured true viscositiesηtrue are 8% and 6%, respectively, using the ARES,and 6% and 1% on AR-G2, respectively. In addition,the effective gap errors obtained on both rheometershave values between 25–30 μm. For the lowest vis-cosity liquid N1, the discrepancy between the statedviscosity and measured true viscosity is the largest, 21%on the ARES and 18% on the AR-G2. The gap errorsobtained are also the largest, 54 ± 1 μm on ARESand 49 ± 2μm on AR-G2. This indicates that, for lowviscosity fluids, there are additional sources of errorspossibly due to secondary flows induced between thefixtures, which cannot be captured by a simple analysisof the form given above. The discrepancy between thevalue ηtrue determined experimentally and the nominalreported value η0 of the calibration oils provides anestimate of the accuracy of the measurements afterthe gap correction has been applied to the apparentviscosity ηa.

The principal contribution to this effective gap er-ror is the intrinsic misalignment between the rotatingfixture and the stationary lower plate or base. Wemeasured this gap error directly in the absence of anyfluid using plastic shim stock of precise thicknesses. Thefixture spacing was set to a nominally zero gap using therheometer software, and strips of shim stock of increas-ing thickness (13, 19, and 25 μm) were then inserted inthe clearance between the top and the bottom platesto determine the residual gap between the plates. Forthe ARES rheometer, the physical gap clearance wasdetermined to be at least 19 μm, and for the AR-G2rheometer, the corresponding gap error was at least13 μm.

Another important source of potential error is theconcentricity of the parallel plates. We addressed thisissue by measuring the eccentricity of the rotatingbottom plate and comparing it to the eccentricity of

Table 3 Results of gap error calibration for the three Newtonian fluids

Fluid γ̇ (s−1) ARES AR-G2

T η0 ηtrue Gap error ε T η0 ηtrue Gap error ε

(◦C) (Pa s) (Pa s) (μm) (◦C) (Pa s) (Pa s) (μm)

S60 (N1) 100 20 0.137 0.107 ± 0.002 54 ± 1 25 0.101 0.120 ± 0.005 49 ± 2DMS T25 (N2) 100 25 0.485 0.522 ± 0.004 27 ± 1 25 0.485 0.512 ± 0.005 24 ± 1N1000 (N3) 10 20 2.867 2.704 ± 0.060 24 ± 1 25 2.011 2.025 ± 0.049 32 ± 1

In this paper, γ̇a is the apparent shear rate, T is the temperature at which the fluid was maintained, η0 is the manufacturer’s statedviscosity at that temperature, ηtrue and ε are the true viscosity, and the size of gap error found by fitting Eq. 6 to the calibration data,respectively.

Rheol Acta (2008) 47:621–642 629

the top plate using a swivel-head dial indicator with aresolution of 2.5 μm. The two plates were determinedto be concentric to within 17 μm, which is well withinthe manufacturer’s stated tolerances for the motoralignment.

The apparent viscosity reported by the rheometersoftware is a function of both the commanded gapheight H and the magnitude of the gap error ε intrinsicto the instrument and is thus systematically in error. Onrearranging Eq. 6, we find that the relative error in ηa

with respect to its stated value ηtrue is given by:

ηtrue − ηa

ηa= ε

H(13)

Table 4 shows the percentage error in the true viscosityand the measured or apparent viscosity at differentgap settings. It can be seen from the table that therelative percent errors increase significantly as one goesto smaller gaps; for example, assuming a representativevalue for the gap error to be 30 μm, at 2-mm gapheight, the apparent viscosity is systematically in errorby 1.5%, but at 30 μm gap height, the discrepancy is100%. Although the fractional error between measuredvalues of the apparent viscosity and the true viscositycan be very large at small gaps, the gap-corrected valuesof true viscosity using Eq. 6 fall within 10% of the statedviscosity for moderate viscosity fluids.

Ultimately, the gap error determined using this ap-proach is a composite property of the instrument andthe geometry used. Using the procedures outlinedabove, we can estimate the size of the gap error foreach instrument and the geometric fixture used. Theseerrors in the gap setting are, of course, not limitedto Newtonian fluids, as we will demonstrate in latersections, in which we apply the same procedure fornon-Newtonian fluids. The highest shear rate achiev-able in the instrument, γ̇max ∼ �max R/H, is inverselydependent to the gap height. In any experiment, thereis thus a trade-off between minimizing the errors dueto imperfect zeroing of the gap and achieving as higha shear rate as possible. Therefore, at high shear ratesand small gaps, corrections due to the systematic gap er-ror become essential in obtaining accurate viscometric

Table 4 Relative percent error in apparent viscosity and trueviscosity at different gap heights H, assuming a representativegap error ε = 30 μm

H (μm) 2,000 1,000 500 100 50 30

Error % 1.5 3.0 6.0 30.0 60.0 100.0

The errors are calculated using Eq. 13.

measurements of both Newtonian and complex fluidsas we now proceed to demonstrate.

High shear rate measurements

In this section, we describe the behavior of Newtonianand weakly shear-thinning fluids at very high shearrates. Even for Newtonian fluids, the issue of ‘apparentshear-thinning behavior’ at high shear rates is impor-tant (Dontula et al. 1999). Do some ‘Newtonian’ liquidsbecome shear thinning at high enough shear rates, oris the apparent shear-thinning behavior explained bysome other mechanism such as viscous heating? Forthe non-Newtonian case, particularly, for wormlike mi-cellar solutions, the high shear rate behavior of thesolution beyond the plateau in the flow curve remainsunclear (Radulescu et al. 2003).

We begin by describing results for Newtonian flu-ids. The three Newtonian fluids N1, N2, and N3 weresubjected to a steady shear ramp test on the ARESrheometer using a plate–plate geometry with a platediameter of 50 mm. The fluids were subjected to shearrates from 1 to 60,000 s−1, in logarithmic steps with fivepoints per decade. Each shear step was maintained for20 s. The commanded gap height was set to 50 μm, andthe temperature at the lower plate was held fixed at25◦C.

Figure 5 shows the plot of true viscosity withshear rate for the three nominally-Newtonian flu-ids. The true viscosity and the true shear rate were

10-2

10-1

100

101

100 101 102 103 104 105 106

N1N2N3

Shear rate [s-1]

η true

[Pa

s]

T = 25º C

Fig. 5 High shear rate behavior of Newtonian fluids N1, N2, andN3 in the ARES rheometer, using a 50-mm diameter parallel-plate geometry at 25◦C. The solid symbols are the gap-correctedor ‘true’ experimental data points, and the dashed lines showthe limiting curves of slope −1 given by Eq. 16, based on theminimum and the maximum measurable torque values

630 Rheol Acta (2008) 47:621–642

obtained from the apparent values reported by thesoftware, by correcting for the gap error, as de-scribed above. The gap error ε for the ARESrheometer was found as described in “Gap errorcalibration.” Once the gap error was known, Eq. 5 wasused to calculate true shear rate and the gap-adjustedtrue viscosity using Eq. 6 in the form

ηtrue = ηa

(H + ε

H

)(14)

The solid symbols in the plot show the experimen-tal data points. The two dashed lines running diago-nally across the plot represent the minimum and themaximum limiting values of the measurable torque.The minimum and maximum torque values, for theforce transducer used, are Tmin = 1.96 × 10−4 N m andTmax = 0.196 N m, respectively. These torque limitscorrespond to minimum and maximum stress valuesτmin = 7.989 Pa and τmax = 7,989 Pa, respectively, forthe 50-mm diameter plate (Eq. 3). Having determinedthe minimum and maximum shear stresses, the range ofviscosities and shear rates accessible are related by theexpressions:

ηmin = τmin

γ̇true(15)

ηmax = τmax

γ̇true(16)

where the true shear rate is given by incorporating thegap error via Eq. 5.

The viscosity of each Newtonian fluid is constantfor more than three decades in shear rate, up to γ̇ ≈104 s−1. For the fluid N2 with viscosity η0 = 0.485 Pa s,there is a visible drop in viscosity beyond shear rates of20,000 s−1. We investigate this case of apparent shearthinning in further detail below.

At very high shear rates, there are several additionalfactors such as inertial effects and viscous heating thatimpact rheometric measurements (Bird et al. 1987;Macosko 1994). At high rotation rates, centrifugalstresses may become sufficiently large to overcome thesurface tension stresses that hold the liquid between theplates resulting in liquid being thrown out of the gap;a phenomenon termed euphemistically the ‘radial mi-gration effect’ (Connelly and Greener 1985). Once theconfined fluid is partially ejected, the subsequent mea-surements are made with less fluid within the plates,which results in a drop in measured torque and, hence,in the viscosity. In particular, for a fluid with density ρ

and surface tension σ in a parallel-plate geometry withplate radius R, rotating with angular velocity �, andgap height H, the centrifugal stresses overcome the

surface tension stresses when (Connelly and Greener1985; Tanner and Keentok 1983),

3

20ρ (�2 R2) >

σ

H. (17)

The critical apparent shear rate at which the fluid be-gins to migrate outward is given by rearranging Eq. 17:

γ̇app,c =(

�RH

)

c

=√

20σ

3ρH3. (18)

In addition, incorporating the gap error correction,we get:

γ̇c =√

20σ

3ρ(H + ε)3. (19)

Equation 19 shows that the critical shear rate for radialmigration to occur decreases as gap height increases(γ̇c ∼ H−3/2). Thus, the radial migration effect can bereduced by going to very small gap heights. Further-more, it is clear that a gap error results in a criticalshear rate that is lower than the predicted critical shearrate without correcting for the gap errors. For the fluidN2 (η0 = 0.485 Pa s), the estimated critical shear ratesat which the radial migration occurs at different gapheights (assuming a gap error of 30 μm) are found tobe γ̇c = 8,096 s−1 for H = 100 μm, γ̇c = 16,770 s−1 forH = 50 μm, and γ̇c = 29,419 s−1 for H = 25 μm. It isnoteworthy that these critical rates are within the rangeof shear rates imposed in our experiments and of thesame order as the shear rates at which the viscosityshows a noticeable drop.

Many experimental and theoretical studies haveshown that viscous heating can also significantly af-fect the flow properties of Newtonian fluids (Bird andTurian 1962; Connelly and Greener 1985; Dontula et al.1999; Kramer et al. 1987; Olagunju et al. 2002; Roth-stein and McKinley 2001). In a previous study (Ram1961), a drop in viscosity of water/glycerol solutions wasreported and described as a shear-thinning transitionin an apparent Newtonian fluid at high shear rates.We reexamine this issue with fluid N2, which shows asimilar drop in viscosity at high shear rates. In additionto the radial migration effect discussed above, it ispossible that, at very high shear rates, there is sufficientviscous heating to lower the viscosity of some fluids.

The effects of viscous heating can be characterizedby the Nahme number, which is a dimensionless ratio ofthe time scales for thermal diffusion to viscous heating:

Na = η0β H2γ̇ 2true

kT, (20)

Rheol Acta (2008) 47:621–642 631

where η0 is the zero-shear viscosity, H is the true gapseparation, γ̇true is the true shear rate, k is the ther-mal conductivity, T is the temperature, and β is thelogarithmic derivative of viscosity with temperature or“thermal sensitivity” and is given by:

β = Tη0

∣∣∣∣dη

dT

∣∣∣∣T=T0

. (21)

At low shear rates, the Nahme number is very small(Na ∼ 10−9). However, as the shear rate increases, theNahme number and, hence, the magnitude of viscousheating increase quadratically with shear rate and thegap height. This means that viscous heating effectsbecome significant at lower shear rates, as the gap sizeincreases. Therefore, repeating the steady shear ratestep test with different gap separations should result innoticeable differences in the viscosity/shear-rate curveif viscous heating is important. As fluid N2 shows themost noticeable decrease in viscosity at high shearrates, we perform additional tests using this fluid atgap heights of 25, 50, and 100 μm and rotation ratescorresponding to shear rates 1–60,000 s−1. This rangein shear rate was spanned in logarithmically spacedsteps, with five points per decade, and each shear ratewas maintained for a duration of 30 s. Table 5 showsthe critical shear rates for the onset of radial migrationeffect as well as the minimum and maximum Nahmenumbers for fluid N2 at three different gap heights: Theuse of small gaps is clearly advantageous in trying toreduce the Nahme number experienced at high shearrates and thus minimize the effects of viscous heating.

Figure 6 shows the data for the apparent viscosityversus shear rate for N2; for convenience, apparentviscosities are plotted here instead of the correctedor true viscosities, as these would collapse verticallyonto each other at low rates and obscure the onset ofviscous heating. The data clearly show that the onsetof apparent shear-thinning occurs at different shearrates for different gap sizes, as indicated by the dashedvertical lines. Quantitatively, the shear rates at whichthe apparent shear-thinning begins, for 100 μm, and50-μm gap sizes are γ̇ ≈ 104 s−1 and γ̇ ≈ 2 × 104 s−1,respectively. At the smallest gap of 25 μm, viscous

Table 5 Critical shear rates for the onset of radial migrationeffect and minimum and maximum Nahme numbers at differentgap heights H for the fluid N2

H (μm) γ̇c (s−1) Namin Namax

25 29,400 8.79 × 10−10 0.12750 16,800 3.52 × 10−9 0.506100 8,100 1.41 × 10−8 2.026

100 101 102 103 104 105

0.1

0.2

0.3

0.4

0.5

25 μm gapBird-Turian Prediction50 μm gapBird-Turian Prediction100 μm gapBird-Turian Prediction

η a [

Pa s

]

Shear Rate [s-1]

T = 25º C

Fig. 6 High shear behavior of Newtonian fluid N2, at threedifferent gap sizes, H = 25 μm, H = 50 μm, and H = 100 μm.The filled symbols are the experimental data from steady shearstep tests, and the lines are the a priori predictions of the modifiedBird–Turian analysis (Eq. 22). The vertical dashed lines showthe shear rates for the onset of apparent shear-thinning. Theminimum and the maximum Nahme numbers Namin and Namaxcorresponding to the minimum and maximum shear rates aregiven in Table 5

heating effects are negligible up to the maximum shearrate γ̇ ≈ 3.5 × 104 s−1. This is consistent with the ar-gument based on the scaling Na ∼ (Hγ̇ )2: If the gapheight is halved, the shear rate necessary for the fluidto experience the same Nahme number doubles.

The first effects of viscous heating in the cone-and-plate geometry have been considered asymptoticallyby Bird and Turian (Bird et al. 1987; Bird and Turian1962), and the analysis has been extended more re-cently by Olagunju (2003) for parallel-plate fixtures offinite radius R with walls held at a constant tempera-ture. The measured torque on the rotating plate isgiven by:

T = π R3 γ̇a η0

2

[1 + A Na + O(Na2)

], (22)

where γ̇a = �R/H and A = −1/18 + 0.104 (H/R) +O[(H/R)2]. Detailed computational analysis showsthat this asymptotic result is accurate for a wide rangeof aspect ratios up to Na ≈ 1 (Zhang and Olagunju2005). By combining Eqs. 20–22, we can calculate theshear stress and apparent viscosity at each shear rate.In Fig. 6, we show the predictions of the asymptotictheory (solid lines) and the data. The general trendsare similar; however, the decrease in viscosity observedexperimentally is larger than the asymptotic theorypredicts at very high shear rates. The reasons for thismay include several factors, such as edge effects, thethermal variation of the thermal sensitivity β(T), or

632 Rheol Acta (2008) 47:621–642

most importantly the radial migration effect discussedabove. For gap heights of 50 and 100 μm, beyond ashear rate of γ̇ ∼ 104 s−1, both viscous heating andradial migration contribute synergistically to the dropin viscosity.

To confirm the presence of viscous heating and iso-late its effects from those of radial migration effect,we perform thixotropic loop tests on fluid N2, at agap height of 50 μm following the protocol of Con-nelly and Greener (1985). In this test, fluid N2 wassubjected to a continuous shear ramp from 1,000 to20,000 s−1 and then brought back down to 1,000 s−1

in the same manner. The duration of the thixotropicloop, tL, was varied from 4 to 40 s. The maximum shearrate was chosen using Eq. 18 and Table 5 such thatno significant radial migration occurs. In a thixotropicloop test, viscous heating is manifested in the form ofhysteresis in the stress–strain rate curve. If the areabetween the ‘up’ and ‘down’ sweeps increases with looptime, it signifies greater viscous heating due to a longerduration of shearing. In Fig. 7, stress—shear rate datafor four different loop times, tL = 4, 10, 20, and 40 s, areshown. For clarity, the stress values for tL = 10 s, tL =20 s, and tL = 40 s are shifted vertically by 500, 1,000,and 2,000 Pa, respectively. In agreement with Connellyand Greener (1985), we observe that, as the loop timeis reduced, the hysteresis decreases. For the smallestloop time of 4 s, viscous heating is negligible, even for amaximum imposed shear-rate γ̇ = 20,000 s−1.

The relevant timescales in these thixotropic loopexperiments are the thermal timescale, the viscous

0 5000 10000 15000 20000

0

2

4

tL = 40 s

tL = 20 s

tL = 10 s

tL = 4 s

Stre

ss [

k Pa

] (a

rb. s

cale

)

Shear rate [s-1]

ARES: 50 mm Plate, 50 μm gap

T = 25º C

Fig. 7 Thixotropic loops showing variation of stress with shearrate at four different loop times, tL = 4 s, tL = 10 s, tL = 20 s,and tL = 40 s. The latter three curves are shifted vertically by500, 1,000, and 2,000 Pa for clarity of presentation only

40

30

20

10

0

P(x)

[kP

a]

1086420

Streamwise distance, x [mm]

Fluid N1, Channel B:Q = 50 μl/min: ,Q = 22 μl/min: ,Q = 11 μl/min: ,

Fig. 8 Fluid S60 in channel B, steady state pressure P(x) mea-sured at each pressure sensor for 10 s at three flow rates Q. Thelines are a least-squares fit of a first order polynomial

heating timescale, and the loop time. The thermaldiffusion timescale tdiff = H2/DT (based on the gapheight H and the thermal diffusivity of the fluid DT =k/ρCp) is a measure of the time over which thermalenergy diffuses to the boundaries. The viscous heatingtimescale is a measure of the rate at which the fluidrheology changes due to viscous heating and is givenby tvisc = ρCpT0/(βη0γ̇ ) = tdiff/Na. This timescale de-creases rapidly with increasing shear rate; hence, theneed to perform rapid thixotropic loops in which thetotal time that the sample is held at very high shearrates is minimized. The question then arises as towhether the test sample is in local thermal equilibrium.A thixotropic loop enables us to probe the steadyshear viscosity of a fluid provided tdiff � tL. In ourcase, for fluid N2 with thermal diffusivity DT = 1.1 ×10−7 m2 s−1, at a gap of 50 μm, between plates of radiiR = 25 mm, the thermal diffusion timescale is tdiff =23 ms. For a continuous shear rate ramp, the ratioof the thermal diffusion time and the smallest looptime (tL = 4 s) is tdiff/tL = ρCp H2/ktL = 0.0058. Thusthe flow is quasi-steady at every intermediate shearrate with a fully developed temperature profile acrossthe gap.

To conclude, we have shown that modern rotationalrheometers are capable of measuring accurate visco-metric properties at high shear rates up to γ̇ ∼ 5 ×104 s−1 provided the gap heights are kept small. Usingthe gap correction procedure outlined in “Parallel-plateapparatus and gap error correction,” accurate measure-ments of viscosity can be obtained, at least for constantviscosity Newtonian fluids. We have also shown that theeffects of viscous heating and centrifugal stresses canboth be appreciable at high shear rates and can mani-fest themselves as apparent shear-thinning behavior for

Rheol Acta (2008) 47:621–642 633

nominally Newtonian fluids. In view of this, we nowturn to the use of microchannel devices, which are notsusceptible to the effects of either viscous heating orcentrifugal stresses.

Rectilinear flow of Newtonian fluids in microchannels

Fundamentally, the VROC microchannel device allowsus to measure the pressure P(x, t) at various stream-wise locations along the center of a straight channelfor an imposed flow rate Q. The steady-state pressureas a function of streamwise location x for fluid N1is shown in Fig. 8 at three different flow rates 10 <

Q ≤ 50 μl min−1. The pressure readings are sampledat 67 Hz and averaged for more than 10 s to show thesteady-state pressure for each flow rate. Fitting a firstorder polynomial shows that, as one expects, for a givenflow rate, the streamwise pressure gradient dP/dx isconstant for a straight channel of uniform cross-section.

The transient pressure response to a step changein flow rate is illustrated clearly in Fig. 9: In thesetests, the fluid is initially at rest, and then, the syringepump is started impulsively at time t = 0 s; thereafter,the commanded flow rate is reduced every 60 s. Thepressure is sampled at 67 Hz, and the data output fromthe sensor software is treated with a moving averagefilter applied over 25 consecutive samples. The pres-sure drop �P over the array of pressure sensors asa function of time is shown for two different typesof tubing, PFA (elastic modulus E ≈ 0.035 GPa) andPEEK (E ≈ 3.6 GPa) tubing. The residual noise of thepressure sensor is approximately ± 0.25% full scale ofthe sensor, corresponding to ±150 Pa for channel B,and limiting the lowest practical working pressure dropto approximately 300 Pa (by averaging over a large

25

20

15

10

5

0

ΔP [

kPa]

300250200150100500

Time [s]

Q = 50 μl/min Q = 22 μl/min Q = 11 μl/min Q = 5 μl/min Q = 2.2 μl/min

PFA tubingPEEK tubingPEEK tubing with air bubble

Fig. 9 Fluid S60 in channel B, pressure drop �P as a function oftime at several flow rates Q

40

30

20

10

0

ΔP [

kPa]

806040200

Flow rate, Q [μl/min]

Channel B:N1: ,N2: ,N3: ,

Fig. 10 Pressure drop �P as a function of flow rate Q in VROCchannel B for Newtonian fluids N1, N2, and N3; experimentaldata (symbols) and linear fit (solid line)

number of samples). However, much greater noise inthe measured pressure can be caused by periodic fluc-tuations in the flow rate from the syringe pump: At50 μl min−1, the error is ±5% for the two sets of datawith no bubble in the syringe. Fluctuations in flow ratecan be damped significantly by introducing a compliantair bubble into the syringe, and, in this case, the erroris ±1%. Hence, it is important to average over a largenumber of samples to account for periodicity in the flowrate from the syringe pump.

The transient response in the pressure differenceto a step change of Q is clear, and it is important towait for the signal to attain a steady state to calculatethe equilibrium values of the pressure difference �P.This transient response is highly dependent on anyair bubbles in the system as well as the viscosity ofthe fluid. For fluid N1 shown in Fig. 9, the transientpressure drop with an air bubble present is well fittedby a decaying exponential with a time constant of 3 s,while with no bubbles in the system, the time constantis < 1 s. Therefore, to avoid long transient flows, it isimportant to ensure that the syringe and tubing arefree of bubbles. This is critical for the shear-thinningviscoelastic liquids discussed in the following sectionwhere the time needed to reach a steady state may beO(1,000 s) depending on the flow rate.

In Fig. 10, we show that the pressure difference�P for the three constant viscosity liquids N1, N2,and N3 is a linear function of flow rate Q passingthrough the origin as expected. At the highest flowrate shown in Fig. 10, Q = 80 μl/min, the Reynoldsnumber based on the hydraulic diameter of the channelis Re = ρQdh/(ηwd) = 8 × 10−3, indicating that theflow is dominated by viscous stresses and far from theonset of any inertial effects or turbulence. The Nahme

634 Rheol Acta (2008) 47:621–642

number (Eq. 20) describes the importance of viscousheating, and using the properties listed in Fig. 1, we findfor all flow rates Na < 10−4 for fluids N1–3 indicatingthat viscous heating is insignificant even at the largestflow rates used in this work. Thus, we consider Eqs. 9and 10 for the wall shear rate and wall shear stress toprovide an accurate description of the flow curve.

Viscosity data from VROC channel B for four con-stant viscosity fluids are shown in Fig. 11. The scatterin the measured value of η(γ̇ ) is less than 5%, andthe measured viscosity is independent of shear rateas expected. As we show in Table 1, the data arein good agreement with the gap-corrected measure-ments from the parallel-plate fixture using a standardrheometer. The upper and lower sensing limits of theVROC channel B are indicated by the dashed linesin Fig. 11: In the parameter space of viscosity andshear rate, a specified pressure drop �Pmin correspondsto a fixed minimum wall shear stress (ηγ̇ )min = τmin =wd�Pmin/(2L(w + d)) from Eq. 9. This corresponds toa slope of −1 on a log–log plot of viscosity versus shearrate. A similar analysis of course also applies for themaximum pressure drop, and thus, for fluids with alower viscosity, higher shear rates can be attained.

On the same figure, we show data for water; here, wecan clearly see the advantage the VROC offers for lowviscosity fluids, allowing shear rates 103 ≤ γ̇ ≤ 105 s−1

to be obtained for shear viscosities η ∼ 1 mPa s. Forthese measurements, the Reynolds number is in therange 1 < Re < 100, which is still significantly below

10-4

10-3

10-2

10-1

100

101

η [P

a s]

100 101 102 103 104 105

Shear rate [s-1]

N3

N2

N1

Fig. 11 Measured viscosity as a function of shear rate for thethree viscous Newtonian calibration fluids N1, N2, N3, and wa-ter. Solid symbols are for data from the VROC microchannelobtained using channel A, and hollow symbols are for dataobtained using plate–plate geometry on ARES. The dashed linesshow the minimum and maximum operating pressures for themicrochannel

the onset of turbulent flow in a channel Re = 2,000. Atthe maximum shear rate of γ̇ = 8 × 104 s−1, the Nahmenumber Na < 10−3 and viscous heating are negligibleeven at these high shear rates.

We now proceed to investigate the viscosity of aweakly shear-thinning PEO–water–glycerol solution.In Fig. 12, we show data for the viscosity as a function ofshear rate obtained using VROC channel A. Comple-mentary measurements with a 60-mm diameter cone-and-plate fixture on an AR-G2 shear rheometer showthat the PEO solution has a zero shear viscosity η0 =0.015 Pa s at 22.5◦C. With the cone-and-plate fixture,shear rates of up to 1,000 s−1 can be attained, andthe fluid exhibits slight shear thinning for γ̇ � 100 s−1.Agreement between the viscosity measured using thecone-and-plate fixture and the microchannel is excel-lent, and using the VROC, we are able to extend theshear rate range over which we can probe the viscousresponse to γ̇ = 10,000 s−1. We also indicate by theerror bars the magnitude of the fluctuations in thepressure signal. At low flow rates, the viscous stressesfall below the stated resolution of the VROC, andthere is considerable noise in the data; for example, at�Pmin = 55 Pa, we measure τmin = 0.12 ± 0.11 Pa.

The PEO solution is only weakly shear thinning, andin Eq. 11, the derivative term d(ln γ̇a)/d(ln τw) = 1.06,resulting in only 2% difference between γ̇a and γ̇w. Thiserror is of the same order as other experimental errorsthat we would expect due to fluctuations in tempera-ture, imposed flow rate and the precision and accuracyof the pressure transducers, and it is sufficiently accu-rate to process the measurements for such a weaklyshear-thinning fluid in the same way for a Newtonianliquid.

10-3

10-2

10-1

η [P

a s]

100 101 102 103 104

Shear rate [s-1]

PEO solution (22.5ºC) Channel A AR-G2 60mm Cone & Plate 55% glycerol-45% water (CRC Handbook)

Lower and upper pressure limits of channel A

Fig. 12 Viscosity as a function of shear rate for a shear-thinningPEO solution (55% glycerol, 44.9% water, and 0.1% PEO). Datafrom measurements performed using VROC channel A

Rheol Acta (2008) 47:621–642 635

In summary, we have seen how streamwise pres-sure measurements along a straight microfluidic chan-nel allow us to calculate the viscosity for severalNewtonian liquids and weakly shear-thinning liquidsover a wide range of shear rates. The measurementsagree extremely well with data from a narrow gapparallel-plate geometry and have the additional advan-tage that no fluid is ejected from the device due to largerotation rates, and the flow remains in a low Reynoldsnumber and low Nahme number regime.

Measurements of fluids with a rate-dependent viscosityusing parallel plates and microchannels

In the two sections above, we have demonstrated twocomplementary techniques for accurately measuringthe shear viscosities of Newtonian liquids at large shearrates by reducing the characteristic length scale of thedevice to l ∼ 30–50 μm. For weakly shear-thinningfluids, the small rate-dependent change in viscosityallows the same analysis as for a constant viscosityliquid, to within experimental error. We now extendthe analysis to complex liquids with a strongly rate-dependent viscosity. To investigate the possibilities andlimitations of the two techniques, we present results foran aqueous xanthan gum and a CPyCl/NaSal micellarsolution, which are both known a priori to have shearviscosities that change by several orders of magnitudeover certain shear rate ranges. The xanthan gum so-lution is a strongly shear-thinning liquid (Milas et al.1990), while the micellar solution shows a yield-likebehavior at a critical shear stress that is associated withthe onset of shear-banding flow (Rehage and Hoffmann1991; Pipe et al., in preparation).

We first consider gap correction of measurements ina parallel-plate fixture with narrow gaps from 500 to10 μm. The method for determining the gap error andthe resulting analysis to obtain true shear rates and vis-cosities remain the same, as described in “Parallel-plateapparatus and gap error correction.” In Fig. 13, weshow gap error calibration using aqueous xanthan gumand for the CPyCl/NaSal micellar solution using theAR-G2. Both fluids were subjected to steady shearfor a duration of 60 s, at constant apparent shearrates of γ̇a = 1 s−1 and γ̇a = 0.006 s−1 for CPyCl/NaSalmicellar solution and xanthan gum, respectively. Theshear rates were chosen such that the viscometric re-sponse is expected to remain in the zero shear rateplateau, even though the actual shear rates in thesample for small gaps may be substantially largerthan the nominal values (see Table 4). The solidsymbols in the plot are the experimental data, andthe solid lines are linear regression fits to the data

0.00 0.02 0.04 0.06 0.08 0.10 0.120.0

0.1

0.2

0.3

0.40.3% aqueous xanthan gum

ηtrue

= 17.7 Pa s, ε = 31 μm

CPyCl/NaSalη

true= 19.1 Pa s, ε = 53 μm

T = 22 ºC

H-1 [μm-1]Fig. 13 Gap error calibration data for two non-Newtonian fluids(CpyCl/NASal and xanthan gum) on the AR-G2 rheometer. Thegap separations range from 10 to 500 μm. For CPyCl/NaSal theapparent shear rate was held constant at γ̇a = 1 s−1, and forxanthan gum, γ̇a = 0.006 s−1. The lines are fits of Eq. 6 to thedata, and the errors in the least-squares linear fit are reported inthe text

using Eq. 14, with true viscosity 1/ηtrue and size ofthe gap error ε as fitting parameters. The results ofthe fitting show that, for xanthan gum, ηtrue = 17.7 ±0.6 Pa s, and ε = 31 ± 1 μm. For the micellar solu-tion, ηtrue = 19.1 ± 0.4 Pa s, and ε = 53 ± 1 μm. Thesecalculated values of the viscosity at low shear ratesare slightly higher but in good agreement with thezero shear rate viscosities given in Table 1. Althoughthe analysis presented in “Parallel-plate apparatus andgap error correction” is only valid for Newtonian fluids,it also appears to apply, at least empirically, to the datain Fig. 13. However, interpretation of the gap errorcorrection is complicated by the fact that it is now fluiddependent. The value of the gap error obtained withCPyCl/NaSal is much larger than the value obtainedwith xanthan gum solution, which is similar to that ob-tained for Newtonian fluids N2 and N3. Furthermore,the ‘gap-correction’ is found to be dependent on theimposed shear rate γ̇a when higher deformation ratesbeyond values corresponding to the zero shear rateplateau are employed.

In particular, for the aqueous xanthan gum solution,the apparent shear rate chosen was selected to beat the edge of the constant viscosity plateau; therefore,the ‘true viscosity’ obtained may not correspond tothe zero shear rate viscosity. The range of applicableshear rates in the case of xanthan gum is limited bythe minimum measurable torque on AR-G2, which, inpractice, is Tmin = 1.0 μNm, and the true shear rateat which shear-thinning behavior begins, which is at

636 Rheol Acta (2008) 47:621–642

0.1

1

10

ΔP [

Pa]

0.01 0.1 1 10 100 1000

Q [μl/min]

1

CPyCl/NaSal Aqueous xanthan gum solution

Fig. 14 Pressure drop �P versus flow rate Q for the CPyCl/NaSal micellar fluid and the xanthan gum solution measuredusing VROC channel A

γ̇true = 0.1 s−1. If the gap-corrected true shear rate isgreater than the shear rate at which shear-thinningoccurs, the analysis for gap correction needs furthermodification. Instead of the simple linear model forthe gap error correction, corrections to the calculatedviscosity due to inhomogeneity of shear rate must alsobe included. Thus, Eq. 6 should be modified to includevariations in the shear rate given by Eq. 7. Detailsof the resulting nonlinear modification to the simplelinear model of gap error correction are given in theAppendix. For the shear rates used here (close to thezero shear rate plateau), the additional correction is lessthan 5%, which is within the precision of the measureddata. We, therefore, retain the simpler linear analysis ofEq. 6.

In Fig. 14, the measured pressure drop �P in themicrofluidic channel shown as a function of flow rateQ for the CPyCl/NaSal and xanthan gum solutions. Forboth fluids, d(ln �P)/d(ln Q) = 1, indicating that theviscometric response is non-Newtonian over the rangeof flow rates studied. As with a constant viscosity liq-uid, we can still use conservation of linear momentum(Eq. 9) to relate the measured pressure drop to the wallshear stress τw, but to find the true shear rate at the wall(γ̇w), we use Eqs. 10 and 11. To determine the gradientterm in Eq. 11, the values of ln γ̇a = g(Q) evaluatedfrom Eq. 10 are plotted as a function of ln τw as shownin Fig. 15. Polynomial functions can be fitted to thisdata, which can then easily be differentiated. For thestrongly shear-thinning xanthan gum solution, a single,second-order polynomial is sufficient, but the highlynonlinear response of the micellar solution necessitatesfitting polynomials piecewise to the data. At flow rateswhere shear banding occurs in the CPyCl/NaSal so-

lutions, the gradient becomes very large; that is, thesystem “spurts” (Méndez-Sánchez et al. 2003). Fittinga polynomial in this region leads to a large uncertainty,which is reflected in subsequent calculations. However,the Newtonian-like regime at low shear rates, the on-set of shear-thinning, as well as the high shear rateregime observed after the shear-banding regime is overare well described by the respective polynomial fits.It should be noted that the value of d(ln γ̇a)/d(ln τw)

is significantly different to unity for the CPyCl/NaSaland xanthan gum solutions and is therefore essentialin calculating the true shear rate at the wall (γ̇w) usingEq. 11.

In Fig. 16, we show the composite steady-state flowcurve for the CPyCl/NaSal solution. The measuredshear stress as a function of shear rate from bothmicrochannels is in excellent agreement with the datameasured using an ARES-controlled rate rheometer.The plate–plate data from the ARES, for gap heightsof 500 and 50 μm, were first corrected for the gaperror using Eq. 5 and subsequently corrected for theinhomogeneous shear rate using Eq. 7. At low shearrates, the Newtonian-like (constant viscosity) responseand the onset of shear thinning are captured accurately.In the steady-state stress plateau regime, the errorin determining the imposed shear rate, indicated bythe error bars, is large as explained above becaused(ln Q)/d(ln �P) diverges. The stress increases slowlyonce more after the stress plateau, and data are shownup to γ̇ = 3,000 s−1 for channel A and γ̇ = 10,000 s−1

for channel B. Cone-and-plate geometries that are fre-quently used to characterize the behavior of micellarsolutions can cause the sample to foam at high shear

10

8

6

4

2

0

-2

g(Q

)

43210

ln τw

g(Q) = 0.72 ln τw2 - 2.21 ln τw + 1.02

g(Q) = 4.89 ln τw - 11.6

g(Q) = -0.16 ln τw2 + 2.72ln τw + 1.42

CPyCl/NaSalAqueous xanthan gum solution

Fig. 15 g(Q) = ln γ̇a versus ln τw for CPyCl/NaSal and xanthangum solution (measured using VROC channel A). The polyno-mial functions fitted are given next to the data

Rheol Acta (2008) 47:621–642 637

10-1

100

101

102

103Sh

ear

stre

ss [

Pa]

10-1 100 101 102 103 104

Shear rate [s-1]

VROC: Channel A; Channel B ARG2 plate-plate: h=50 m; h=500 mμμARES 50mm C&PLower and upper pressure limits:

Channel A ; channel B

Fig. 16 Flow curve for CPyCl/NaSal. The error bars for theVROC data in the stress plateau are discussed in the text

rates above the stress plateau due to the large visco-elastic stresses acting at the free surface. However,the microchannels do not have a free surface in themeasurement section of the channel, and this allowssignificantly higher deformation rates to be imposedwithout causing foaming or incorporation of air. In thishigh rate region of the flow curve, the measured stressas a function of the shear rate is in good agreement be-tween the two microchannels; however, the results aresystematically lower than the data from the plate–plategeometry. The difference between the two techniquescould be due to foaming or different wetting conditions;the flow after the shear stress plateau is highly nonlin-ear (Pipe et al., in preparation), and the interaction of

10-3

10-2

10-1

100

101

102

η [P

a s]

10-3

10-2

10-1

100

101

102

103

104

105

Shear rate [s-1]

Xanthan gum solution, T = 22 ºC VROC Channel A ARG2, 40 mm parallel plates ARG2, 60 mm cone and plate Fit of Carreau-Yasuda equation Operating envelope of channel A

Fig. 17 Viscosity as a function of shear rate for aqueous xanthangum solution from 10−3 < γ̇ < 3 × 104 s−1

velocity fluctuations and interfacial tension may play arole in determining the flow established in this regime.

The strong monotonic shear-thinning behavior of thexanthan gum solution is clearly illustrated in Fig. 17.Using a 60-mm cone-and-plate geometry with thecontrolled stress rheometer (AR-G2), the zero shearviscosity and the onset of shear-thinning behaviorare readily documented. Measurements with a 40-mmplate–plate geometry also show very similar behavior.The plate–plate data were corrected for the gap aswell as for the shear rate inhomogeneity using Eqs. 6and 7. Measurements performed with VROC chan-nel A are also able to quantify the viscosity in theshear-thinning region and capture the viscosity out toshear rates γ̇ = 25,000 s−1. In this study, the shear-thinning behavior begins to decrease in severity, andthe viscosity tends toward an infinite shear rate limitη∞; however, as noted by Kulicke and Porter (1981),shear rates in excess of 106 s−1 would be requiredto accurately determine η∞. The microfluidic channelprovides nearly an extra two decades of informationon the shear viscosity of the xanthan gum solutioncompared to using a classical cone-and-plate geometryand nearly an extra decade compared to the narrow gapparallel-plate technique. This is of considerable interestfor industrial processes such as coating and sprayingfor which deformation rates may often approach γ̇ ∼O(104 s−1). A Carreau–Yasuda model (Bird et al. 1987)can be fitted to the complete set of cone-and-plate andmicrochannel data and describes the rate-dependentviscosity well over seven decades of shear rate.

It should be noted that the good agreement betweenthe cone-and-plate and VROC measurements at highrates strongly suggests that there is no apparent wallslip present in the VROC for the aqueous xanthan gumsolutions. As discussed by Degré et al. (2006), apparentwall slip due to depletion layers is dependent on thesurface chemistry of the system as well as the ratiobetween the solvent and solution viscosity, and here, weconclude that the hydrophilic xanthan gum polymer isnot significantly repelled by the glass or gold surfaces ofthe microfluidic channel.

Conclusions

In this paper, we have demonstrated methodologiesfor exploring the steady-state viscous response of New-tonian and non-Newtonian fluids at high shear ratesusing conventional controlled stress and controlled raterheometers as well as a new microfluidic channel-baseddevice. For both types of device, the key to accessing

638 Rheol Acta (2008) 47:621–642

high shear rates involves minimizing the length scaleover which shearing occurs.

Using the conventional rotational rheometers, it isimportant to accurately calibrate the errors incurredin zeroing the gap when narrow gaps (H � 250 μm)are employed. We have demonstrated that the simplelinear approach proposed by Connelly and Greenercan be used to evaluate gap errors and correct theapparent viscosities obtained at different gaps to obtaintrue viscosities. We have implemented this approachfor both Newtonian and non-Newtonian fluids anddemonstrated that the method works surprisingly wellfor both. We observe that, for moderate viscosity New-tonian fluids N2 and N3, the true viscosities calculatedvia gap calibration (Eq. 6) are within 7% of the statedzero-shear viscosities. The systematic gap offset errorsobtained with these two fluids are ε ≈ 30 μm for bothrheometers used. The lowest viscosity Newtonian fluidN1 shows significant deviation from these values. Thismay be due, at least partially, to larger levels of noise inthe data at the lower torque limits of the instrument,but it serves to remind us that this approximate gapcorrection approach involves both the geometry andthe test fluid. In the case of non-Newtonian fluids, wehave demonstrated that the same linear approximatecorrection works, although not uniformly well. Thegap error calibration approach for the viscous aqueousxanthan gum yields values for the gap error consis-tent with the two Newtonian fluids N2 and N3, butthe gap error obtained with a strongly shear-thinningmicellar solution yields higher values similar to thosemeasured for the low viscosity Newtonian fluid N1.Thus, the approach is valid for both Newtonian andnon-Newtonian fluids, but with differing (and a priori)unknown accuracy.

We also studied the high shear rate behavior ofNewtonian and non-Newtonian fluids. To achieve highshear rates using conventional rheometers, very smallgap heights (H < 100 μm) must be used, leading toincreased differences between the apparent viscositiesand true viscosities (see Table 4), and it is essentialto correct for gap errors. Using such protocols maylead to apparent shear-thinning behavior even in New-tonian calibration oils. We identified two key systematicsources of error that could account for these observa-tions, radial migration and viscous heating. Relevantdimensionless scalings show that both effects becomeimportant in the vicinity of shear rates 104–105 s−1.

We have also investigated the viscous response ofNewtonian and non-Newtonian fluids in straight mi-crofluidic channels. For constant viscosity liquids, weare able to measure η0 over two and a half decadesof shear rate, with the attainable range of shear rates

depending on the viscosity of the fluid. For any givenfluid, there are resolution limits associated with theresolution of the pressure sensor, at the highest shearrates, the percentage error is O(±0.25%); whereas theerror at the very lowest shear rates is O(±50%), al-though this can be reduced by sampling over longertimes to determine an appropriate mean value. Ad-ditional errors in the measured steady-state viscositycan be introduced by the syringe/syringe pump setup.To ensure that these errors are small (< 5%), it isimportant to select a combination of syringe/syringepump that can provide a steady and constant flow atthe desired flow rate. In our experimental setup, thisconstraint limits measurements of high zero shear rateviscosity liquids at low shear rates due to the difficultyin imposing very low flow rates with sufficient accuracy.Step changes in applied flow rate lead to transient pres-sure drops in the microchannel, and we associate thiswith the stiffness of the tubing between the channel andthe syringe and also at the channel exit. It is especiallyimportant to reduce flow transients when using highlyshear-thinning viscoelastic liquids, and we show that atransient pressure response with a time constant < 1 scan be obtained by using stiff PEEK tubing.

At moderate shear rates O(100 s−1), the measuredviscosities of Newtonian calibration oils N1–3 are inexcellent agreement with those obtained from con-ventional rheometers. Shear rates O(104 s−1) can beobtained using lower zero shear-viscosity liquids, andthe viscosity of water was measured up to a shear rate of8 × 104 s−1. The viscosities of two highly shear-thinningfluids calculated using the Weissenburg–Rabinowitsch–Mooney equation (Eq. 11) were also found to be ingood agreement with results from cone-and-plate mea-surements, and the effective shear rate range that canbe accessed is significantly increased due to the de-crease in viscosity associated with increasing shear rate.Thus, we were able to characterize the viscosity of anaqueous xanthan gum solution up to γ̇ = 30,000 s−1 andcapture the approach to the infinite shear rate viscosity.Furthermore, for a shear-banding worm-like micellarliquid, we have been able to explore the steady-stateflow curve up to γ̇ = 10,000 s−1, charting the viscousresponse at shear rates significantly beyond the end ofthe plateau in the steady shear stress.

The results presented here show that both conven-tional rheometry and microchannel rheometry havetheir specific domains of applicability. To access highshear rates, both viscometric approaches can be used,but each has particular strengths and weaknesses. Inrotational rheometers, the effects of centrifugal stressesand viscous heating can be significant. Viscous heatingeffects can be substantially reduced by applying shear

Rheol Acta (2008) 47:621–642 639

ramps in a very short interval of time. Radial migrationeffects can be mitigated by moving to very small gaps,but the resulting systematic gap error grows. In addi-tion, the magnitude of the gap correction that must beapplied to the measured data appears to be dependenton the viscosity of the fluids under investigation.

Microfluidic-based rheometry offers several distinctadvantages over conventional rheometry. We havedemonstrated that high shear rates can be achievedusing the microchannels while still minimizing inertialand viscous heating effects, avoiding the need for an adhoc correction. On the other hand, the dynamic rangeof the pressure transducers mounted on the microchan-nels constrains the range of fluid viscosities that can beeffectively studied. For constant viscosity fluids with azero shear rate viscosity beyond 1 Pa s, the applicabil-ity of the technique is also limited by the range of flowrates that can be reached by the syringe/syringe pumpsetup.

Important questions thus arise with regard to theoptimal choice of rheometer for investigating the highshear rate viscometry of a given fluid. To assess whichapproach is more suitable, it is helpful to have a pictureof the operating space in terms of the fluid viscositiesand the range of shear rates achievable for each class ofdevice and compare and contrast them. In Fig. 18, we

show the relevant operating spaces in terms of viscosityand shear rate for a microfluidic channel (Fig. 18a)and a controlled strain device (ARES) with typicalfixture settings (Fig. 18b). The lines 1 and 2 in Fig. 18arepresent the lower and upper bounds for the VROCchannel B based on the minimum and maximum mea-surable pressure differences, �P. According to Eq. 9,the measured pressure difference is directly related tothe wall stress, and the apparent shear rate is directlyrelated to the volume flow rate by Eq. 10. Thus, for thelimiting values of minimum and maximum measurablepressure differences, we get:

τw,min = wd�Pmin

2l0(w + d), τw,max = wd�Pmax

2l0(w + d). (23)

Combining Eq. 23 with the relation τ = ηγ̇ gives twolimiting curves of the form:

ηmin = τw,min

γ̇w(24)

ηmax = τw,max

γ̇w. (25)

These lines are power laws of slope −1 on a log–logplot. For the VROC channel B with the minimum

a b

Fig. 18 Operating region in the viscosity/shear rate space for amicrofluidic rheometer (VROC) and a strain-controlled rheome-ter (ARES). a The operating region for VROC channel B. Thelines labeled 1 and 2 of slope −1 are the limiting curves based onthe minimum and the maximum measurable pressure differences�P (Eqs. 9 and 10). Line 3 is based on the shear rate viscositycutoff at which the flow approaches a Reynolds number Re =2,000 and can become turbulent. The vertical line 4 is a low shearrate cutoff (γ̇ = 0.0132 s−1) based on the minimum flow rate

achievable. b Equivalent operating space for a strain-controlledrheometer (ARES). The lines 1 and 2 are the limiting curvesbased on the minimum and the maximum measurable torques(Eq. 16). The vertical line 3 is the high shear rate cutoff (γ̇ =2 × 104 s−1) for the onset of radial migration and viscous heatingat a typical gap setting (H = 50 μm). The vertical line 4 is thelow shear rate cutoff (γ̇ = 0.002 s−1) based on the minimumcontrollable rotation rate of the fixture

640 Rheol Acta (2008) 47:621–642

and maximum pressure differences given in Table 2,we find τmin = 0.8 Pa and τmax = 160.0 Pa. Anotherimportant limiting case arises when the flow rate inthe slit is so large that the Reynolds number for thefully developed viscous flow in the channel exceedsRe ≡ ρQdh/(ηwd) = 2,000. In this limit, the onset ofturbulence results in non-viscometric flow, althoughit should be noted that some studies in microfluidicchannels (Peng et al. 1994) suggest that the turbulenttransition may happen considerably earlier at Re ≈200. Using Eq. 10, we can cast the expression for thecritical Reynolds number in terms of the shear rate andviscosity relation at which Re = 2,000:

2,000 = ρ dh d γ̇w

6η(26)

which gives:

(γ̇w)crit = 12,000 η

ρ dh d. (27)

At the other extreme, a minimum attainable flow ratelimited by the choice of syringe and the syringe pumpsets a lower cutoff for the shear rate based on Eq. 10,which, in this case, is γ̇min = 0.014 s−1, for the minimumflow rate of Qmin = 1 × 10−3 μl/min.

Analogously, we obtain the limiting curves and,hence, operating space for a conventional rheometersuch as the ARES as shown in Fig. 18b. Lines 1 and2 represent the limit curves based on the minimum andmaximum measurable torques of the transducer. Theselimiting curves are the same as those given by Eq. 16and are of the form ηmin = τmin/γ̇ and ηmax = τmax/γ̇ .Based on the minimum and maximum torques Tmin =1.96 × 10−4 N m and Tmax = 0.196 N m, the values ofthe minimum and maximum shear stresses are τmin =7.989 Pa and τmax = 7989 Pa for a parallel-plate fixtureof radius R = 25 mm. Again, these are power laws witha slope of −1. The vertical line 3 is the high shearrate cutoff at γ̇ = 2 × 104 s−1, based on the onset ofradial migration (Eq. 18) and viscous heating (Eq. 20)for a typical gap height of 50 μm. As the gap spacingis increased, this limit shifts to lower shear rates (seeEq. 19). The vertical line 4 is a low shear rate cutoff con-strained by the minimum achievable rotation rate of theplate. The minimum rotation rate of the ARES motoris � = 2 × 10−6 rad s−1, and the minimum achievableshear rate is given by γ̇min = R�min/H, which for aplate of radius R = 25 mm, and a typical gap heightH = 50 μm, gives γ̇min = 0.001 s−1.

These operating space diagrams show the range ofshear rates that can be accessed for Newtonian fluidsand shear-thinning fluids of differing viscosities. For

the VROC, the viscosity scale spans 7 orders of mag-nitude from η = 10−5 Pa s (providing the low Knudsennumber limit Kn = λ/d < 0.001, is respected, where λ

is the mean free path of molecules in a gas) to η =102 Pa s; whereas for the ARES, the viscosity scalespans 9 orders of magnitude, but from η = 10−3 Pa s toη = 106 Pa s. This points to one important differencebetween the two approaches: For high viscosity liquids,a conventional modern rheometer typically offers abroader dynamic range compared to the solid statepressure sensors in the microfluidic channel. For exam-ple, if the fluid has a zero shear viscosity of 10 Pa s,for the VROC, the minimum and maximum shear ratesapplicable are γ̇min = 0.1 s−1 and γ̇max = 20 s−1, respec-tively, whereas using the ARES with a plate–platefixture of radius R = 25 mm and gap height H = 50 μm,the minimum and the maximum shear rates are γ̇min =1 s−1 and γ̇min = 1,000 s−1, respectively. On the otherhand, inertial effects are significant for low viscosityfluids using the ARES rheometer, severely constrainingthe useful range at high shear rates, while the lowestmeasurable torque limit curtails the accessible shearrates at the low shear rate end. For a fluid with zeroshear viscosity of 0.001 Pa s (1 cPs), the dynamic rangeof shear rates for the microfluidic device VROC is fromγ̇min = 1,000 s−1 to γ̇max = 200,000 s−1, whereas for theARES, the effective range with a gap of 50 μm is fromγ̇min = 8,000 s−1 to γ̇max = 20, 000 s−1.

Furthermore, it should be noted that theWeissenberg–Rabinowitsch–Mooney equation (Eq. 11)can be applied robustly to any pressure–flow raterelationship obtained in the microfluidic device. Bycontrast, the simple linear gap correction applied tothe narrow gap results for the parallel-plate deviceis strictly applicable only to Newtonian fluid data.Our measurements suggest that a simple gap errorcalibration can be applied to measurements with morecomplex shear-thinning fluids; however, the gap offsetcorrection ε is both gap and fluid dependent. Ideally,to confirm the applicability of the simple linear gapcorrection algorithm for a given fluid, it is necessary tocompare the gap-corrected data with a direct measureof the true shear rate-dependent material functionsobtained, for example, using the microfluidic channel.

The flexibility of microfluidic fabrication technolo-gies also enables a wide range of different flow channelconfigurations to be considered. Having demonstratedthe efficacy of a simple rectilinear slit device to deter-mine flow curves for a range of complex fluids up toshear rates of 105 s−1, we hope in the future to usethis robust pressure-sensing technology to also measureentrance pressure drops and planar elongational vis-cosities using other rheometer-on-a-chip designs.

Rheol Acta (2008) 47:621–642 641

Acknowledgments The authors would like to thank Dr. S.-G.Baek and Dr. M. Yi of Rheosense for providing us with prototypemicrochannel designs. This research was supported in part byNASA (grant NNC04GA41G), NSF (DMS-0406590) and a giftfrom the Procter and Gamble Company.

Appendix

In this appendix, we outline the difficulties in adaptingthe expression for the rate-dependent viscosity mea-sured using a parallel plate device (Eq. 7):

ηtrue(γ̇true) = T /2π R3

γ̇true

[3 + d(ln T /2π R3)

d(ln γ̇true)

], (28)

to include the linear gap error analysis of Connelly andGreener. We start by noting that the analysis can followtwo distinct approaches: one, to find the gap error ε

where the apparent shear rate γ̇a is held constant andthe gap height H is altered and, the other, to find thetrue rate-dependent viscosity where the gap height His held constant and the shear rate or rotation rate isaltered.

Where possible the analysis to find the gap heighterror ε should be performed in the zero shear rateregion so that the fluid viscosity is not a function ofapplied shear rate. However, for fluids that start shearthinning at very low shear rates, this can result inshear stresses below the resolution of the rheometertorque transducer, and it is not possible to access thisregion. In this case, we expect the apparent viscosityto be a function of both the true shear rate and thegap setting, and we want to find the gap error ε as afunction of the apparent shear rate γ̇a = �R/H, whichcan be commanded in the software. We start from theexpression for the true shear rate γ̇true = �R/(H + ε):

γ̇true = γ̇a

(H

H + ε

), (29)

and, differentiating using the chain rule, we find:

dγ̇true

dγ̇a

∣∣∣∣�

=(

∂γ̇true

∂ H

)

�

dHdγ̇a

=(

HH + ε

)2

. (30)

Thus, we can rewrite Eq. 28 in terms of the apparentshear rate:

ηtrue = T /2π R3

γ̇a

(H + ε

H

)×

×[

3+(

HH + ε

)d(ln T /2π R3)

d(ln γ̇a)

∣∣∣∣�

]. (31)

To solve for the gap error ε, we cast Eq. 31 in terms ofthe apparent viscosity ηa (c.f. Eq. 6):

1

ηa= 1

ηtrue

(1 + ε

H

) 1

4×

×[

3+(

1

1 + ε/H

)d(ln T /2π R3)

d(ln γ̇a)

∣∣∣∣�

], (32)

and we note that, for a complex liquid outside the zeroshear-viscosity regime, 1/ηa is a nonlinear function of1/H. For a Newtonian viscous response in which τ ∼γ̇a, we recover Eq. 6, and 1/ηa varies linearly with 1/H.

We now turn to the case where we wish to deter-mine the rate-dependent viscosity as a function of theapparent shear rate for a fixed gap height and knowngap error. In this instance:

dγ̇true

dγ̇a

∣∣∣∣H

=(

∂γ̇true

∂�

)

H

d�

dγ̇a=

(H

H + ε

), (33)

and, for measurements performed over a range ofrotation rates at a fixed gap height, Eq. 28 becomes (cf.Eq. 7):

ηtrue(γ̇true) = ηa

(1 + ε

H

) 1

4

[3 + d(ln T /2π R3)

d(ln γ̇a)

∣∣∣∣H

].

(34)

It is therefore clear that, for shear-thinning fluids,slightly different results can be obtained dependingon whether measurements are performed at constantgap (varying rotation rate) or constant rotation rate(and varying the shear rate by changing the gap). Itis instructive to compare the magnitude of the leadingorder correction due to (a) the error in gap height ε

and (b) the derivative of the logarithmic term due toa rate-dependent viscosity. At small gaps, H ∼ 50 μm,ε ∼ 30 μm, and the correction provided by the gapcorrection term in Eq. 34 can be O(100%). On theother hand, the corrections associated with the shear-thinning term in square brackets are much smaller. Atlow apparent shear rates, a ratio of Eq. 32 or 34 with thelinear form of Eq. 6 in the text shows that the results inEqs. 32 and 34 differ by at most [3 + (1 + ε/H)−1]/4. Inthe case of an extremely shear-thinning fluid in which

d(ln T /2π R3)

d(ln γ̇a)→ 0,

both Eqs. 32 and 34 approach the same limit andthe additional correction to the viscosity is only 25%.Given that the majority of fluids do not shear thinthis strongly, the additional error due to a shear-rate-

642 Rheol Acta (2008) 47:621–642

dependent viscosity in many cases will be less than 10%of that due to gap error. Thus, we conclude that, in themajority of experiments, applying the gap error analysisalone will provide a good estimate of the shear-rate-dependent viscosity.

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