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J. Non-Newtonian Fluid Mech. 160 (2009) 22–30

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

he effects of chain conformation in the microfluidic entry flow ofolymer–surfactant systems

rik Miller, Justin Cooper-White ∗

ustralian Institute of Bioengineering and Nanotechnology, University of Queensland, Brisbane 4072, Australia

r t i c l e i n f o

rticle history:eceived 22 August 2008eceived in revised form 25 February 2009ccepted 27 February 2009

eywords:icrofluidics

ontraction flow

a b s t r a c t

An associative polymer–surfactant system has been used to observe the effects of chain conformationin the entry flow through a microfabricated planar 16:1:16 contraction–expansion geometry. The well-studied system of the flexible polymer poly(ethylene oxide) (PEO) and anionic surfactant sodium dodecylsulfate (SDS) was used. Dilute polymer solutions with increasing SDS concentration were characterizedin steady and dynamic shear, as well as capillary breakup extensional rheology. Based on this character-ization, the primary quantitative difference is an increase in zero-shear viscosity as a result of the PEOchain expansion brought on by association of SDS surfactant micelles. However, these quantitatively sim-

olymer–surfactant systemlow instabilityiscoelasticityolymer conformation

ilar solutions were observed to exhibit much more qualitatively different flow patterns via fluorescentstreak imaging in the entry flow. In contrast to previous work on PEO solutions, the PEO–SDS systemswere observed to transition to a steady viscoelastic flow regime characterized by stable lip vortices atmuch lower elasticity and Weissenberg numbers. The resulting insight gained regarding the utility ofmicrofluidic flows in elucidating effects of subtle conformational changes further illustrates the potentialfor using microfabricated devices as rheometric tools for measuring the properties of dilute and weaklyviscoelastic fluids.

. Introduction

The use of microfluidic devices to access a wide range of flowegimes has become a common and popular practice. Working atery small lengthscales allows researchers to probe the response ofiscoelastic fluids at very large deformation rates in flow regimeshere the Reynolds numbers (Re) are relatively small; in otherords, where the viscous and elastic responses are not masked

y inertial effects. This method is particularly useful for investi-ating weakly viscoelastic fluids such as dilute polymer solutions.luids that fall into this category are widely used in industrial appli-ations ranging from foods, to agrochemical spraying and ink-jetrinting [1]. The specific flow and unique response of these vari-us polymeric materials in microscale channels is also becomingncreasingly important at the interface of biology and engineering,n fields such as tissue engineering where the transport of com-lex fluids containing DNA and proteins is a relevant matter [2].

urthermore, as the field of microfluidics grows, it is becomingncreasingly important to mature the technology and our under-tanding towards the goal of commercialization [3]. Thoroughnderstanding of weakly elastic flows in microchannels has great

∗ Corresponding author. Tel.: +61 7 3346 3858; fax: +61 7 3346 3973.E-mail address: j.cooperwhite@uq.edu.au (J. Cooper-White).

377-0257/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2009.02.009

© 2009 Elsevier B.V. All rights reserved.

implications for lab-on-a-chip devices and perhaps even microfab-ricated rheometric tools that can provide useful information aboutthe shear and extensional properties using a miniscule amount offluid.

There are several recent studies that have shown the enhanc-ing effects of using reduced lengthscales on the magnitude ofviscoelastic effects in dilute polymer solutions. In particular, sim-ple planar, or even axisymmetric on slightly larger lengthscales,contraction–expansions are simple but effective geometries forstudying viscoelastic response. A thorough review of flow phenom-ena in viscoelastic entry flows can be found in the collective work of[2,4,5]. One of the primary outcomes from this body of work is theability to probe a large space in terms of Reynolds and Weissenbergnumber (Wi), as well as the effects of fluid elasticity, which inthe definition of the elasticity number, El = ��/(�l2) or El = Wi/Re, isdependant only on the fluid properties (relaxation time, �, solutionviscosity, �, and fluid density, � and the characteristic lengthscaleof the device, l) [2].

It is well known that when charged surfactants are addedto dilute polymer solutions, hydrophobic association may occur.

Surfactant molecules, or even surfactant micelles at sufficient con-centration, can form complexes on the polymer backbone, whichcauses significant conformational changes [6]. Due to repulsiveionic forces, this has the general effect of causing the polymer chainsto expand, and translates into changes in the viscous and surface

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stock solution and allowing for complete mixing on a roller devicefor a further 24 h. In addition to the PEO solution alone, four con-centrations of SDS were used to create PEO–SDS mixtures: 0.13 wt.%(∼5 mM), 0.23 wt.% (∼8 mM), 0.46 wt.% (∼16 mM), and 0.92 wt.%(∼32 mM). These surfactant concentrations cover the range of crit-

E. Miller, J. Cooper-White / J. Non-N

roperties of the solution [7]. It was the goal of the current worko explore the isolated effects of conformational change, inducedy polymer–surfactant association, on planar entry flow of diluteiscoelastic solutions.

The associative polymer–surfactant system selected for thisnvestigation was the well-known mixture of the flexible poly-

er poly(ethylene oxide) (PEO) and sodium dodecyl sulfate (SDS).he interaction and complexation of these two molecules isell-studied throughout the literature. Early conductivity studies

xplored the attractive and repulsive behaviour of the polymer andurfactant interaction [8]. Cryo-transmission electron microscopyTEM) and small angle X-ray scattering (SAXS) have also been usedo confirm the existence of spherical SDS micelles amongst PEOhains [9]. Small angle neutron scattering (SANS) has been usedo show that a PEO coil saturated with SDS micelles is swollenompared to a free coil in a good solvent, taking on a cage-liketructure rather than a fully-stretched “necklace” [10]. Further-ore, an opposing jet elongational rheometer has been used to

bserve an abrupt rise in elongational viscosity at critical valuesf strain rate, and varying with SDS concentration [11]. These stud-es have clearly shown the transition of PEO from flexible chaino semi-rigid behaviour as SDS aggregates form along the polymerackbone.

In conjunction with viscometric studies, the aforementionedorphological data has yielded a generally agreed upon mecha-

ism whereby the increased viscosity of dilute PEO solutions inhe presence of SDS occurs in three stages [10,12–14]. In the initialtage, at a concentration known as the critical aggregation concen-ration, CAC (∼4–5 mM), individual SDS molecules associate withhe PEO chain, forming aggregates on the backbone and causing alight expansion of the chain due to electrostatic repulsions. Belowhe CAC, PEO and SDS molecules are dissolved without interaction.or the PEO–SDS system, it has been consistently noted that the CACs lower than the critical micellar concentration (CMCSDS ≈ 10 mM)f SDS in aqueous solution alone, because induced dipoles on theEO chain backbone stimulate association with the anionic sur-actant [15,16]. At a concentration C2 (∼8–10 mM), the surfactant

olecules begin to form micelles, corresponding roughly with theMC. These increasingly large surfactant aggregates repel eachther and cause the polymer chain to swell further. In the finaltage, at a concentration denoted Cm (∼20 mM), the PEO chainecomes saturated with a maximum of SDS micelles and individualolecules. This corresponds with a peak in the solution viscosity

nd is attributed to the increased formation of free SDS micelleshat are not attached to the PEO. Accompanied by an increase inhe amount of free surfactant as well as the sodium ion concentra-ion, electrostatic repulsions responsible for the expansion of theEO coil are now screened. This is evident by an observed decreasen the viscosity, before once again increasing due to the transitionn the SDS micelle morphology from spherical to wormlike at muchigher SDS concentrations [10]. It should also be noted that aboveolymer molecular weights on the order of MW = 104 g/mol, therends in SDS concentration effects are independent of PEO molec-lar weight [8,17], and vary linearly with polymer concentration10].

In this study, we will build on previous work dealing with theow of dilute polymer solutions in planar contraction–expansionntry flows [2,5], and focus on observing the effects of confor-ational change in the polymer due to association with added

urfactant. The outline of this paper is as follows. In Section 2 weriefly describe our microfabricated channel geometry, introduce

he PEO–SDS solutions we have used, and present our shear andxtensional rheology characterization experiments. In Section 3 weresent our flow visualization results and pressure drop measure-ents and discuss their implications before concluding in Section

.

ian Fluid Mech. 160 (2009) 22–30 23

2. Experimental

2.1. Microchannel geometry

All flow experiments detailed within this study were carriedout in a planar 16:1:16 abrupt contraction–expansion microchan-nel geometry, similar to those used in previous studies [2,5]. Thedimensions of the geometry we used are shown in Fig. 1, where theupstream channel width, wu = 400 �m, contraction width, wc =26 �m, contraction length, Lc = 100 �m, and the uniform channeldepth, h = 45 �m. The microchannel devices were fabricated fromPDMS using standard soft-lithography and SU-8 photolithographytechniques, and bonded to PDMS-coated glass coverslips to provideuniform surface properties on all walls of the channel. Details of thespecific fabrication procedure used within our group can be foundin [2,5].

The volumetric flow rate of the experiments was controlledusing high-precision glass syringes (SGE, 500 �L) mounted in asyringe pump (Harvard Apparatus PHD2000), and for the range ofresults within this study, was varied from 0.05 mL/h < Q < 2.00 mL/h.This range of flow rates corresponds to a characteristic deforma-tion rate of 913 s−1 < �̇c = 2Q/(w2

c h) < 3.65 × 104 s−1 in the regionof the microchannel contraction throat. Pressure taps were posi-tioned in side channels located 3 mm upstream and another 3 mmdownstream of the contraction. This configuration allowed in situpressure drop measurements to be made using a differential pres-sure transducer (Honeywell 24PCB).

2.2. Polymer–surfactant system

A weakly polydisperse, high molecular weight PEO (MW = 2 ×106 g/mol, polydispersity index = 1.13, Sigma–Aldrich) was selectedbased on previous thorough rheological characterization andcontraction flow work within our group [2]. The polymer con-centration was held constant for all solutions in this study, at[PEO] = 0.075 wt.%, known to be slightly lower than the entangle-ment or coil overlap concentration [19]. A stock solution of PEO wasprepared with distilled (MilliQ) water by stirring overnight, afterwhich PEO–SDS mixtures were made by adding SDS (Fluka) to the

Fig. 1. Photograph of the 16:1:16 planar contraction–expansion used in all exper-iments. Dimensions are the upstream channel width, wu = 400 �m, contractionwidth, wc = 26 �m, and contraction length, Lc = 100 �m; not pictured is the channeldepth, h = 45 �m.

24 E. Miller, J. Cooper-White / J. Non-Newtonian Fluid Mech. 160 (2009) 22–30

Table 1PEO–SDS solution properties and experimental parameters.

SDS concentration [mM]

0 5 8 16 32

MW , PEO [g/mol] 2 × 106

� [ms] 10�CaBER [ms] 1.75 1.63 1.85 2.35 2.21�CaBER,init [ms] 0.436 0.521 0.577 0.968 0.926�0 [mPa s] 1.52 2.18 3.44 4.52 4.23� [mN/m] 41.1 42.0 39.8�ext,dD/dt [Pa s] 5.15 5.31 5.44� 7.7 11.0 10.9R 0.04–4.54 0.03–3.46 0.03–3.69W 0.6–68 0.8–86 0.8–81E 14.9 24.9 21.9

itbctcemtt

2

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ext,fit [Pa s]e 0.10–10.3 0.07–7.16i 0.6–64 0.6–60

l 6.23 8.31

cal complexation points, and are based on previous work [7]. Forhe convenience of this study, the PEO and PEO–SDS solutions wille referred to using the following shorthand in terms of the SDSoncentration: p0sds, p5sds, p8sds, p16sds, and p32sds. As a con-rol, Newtonian analogues of glycerol–water at concentrations ofglycerol = 17 wt.% and 48 wt.% were also prepared to match the low-st and highest zero-shear rate viscosity observed in the PEO–SDSixtures, respectively. Table 1 lists the pertinent solution proper-

ies and experimental parameters of the PEO–SDS mixtures usedhroughout this study.

.3. Solution rheology

Shear rheology experiments for all solutions were performedn a stress-controlled shear rheometer (TA Instruments AR-G2).n acrylic plate geometry with a diameter of 40 mm and a rela-

ively small gap of 150 �m was used to allow access to very highhear rates, similar to those that would be experienced in the throatf the microchannel contraction (see Section 2.1). A weakly shear-hinning flow curve was observed in all cases, and the viscosityenerally increased as the SDS concentration was increased. Thexperimental data are shown in Fig. 2. A Carreau model was usedo fit the experimental data over two orders of magnitude in shearate. The resulting zero-shear viscosities, shown in Fig. 3, werebserved to increase sharply above the CAC (∼4–5 mM), begin to

evel off at C2 (∼8–10 mM), and showed a slight decrease beyondm (∼20 mM); these results are consistent with previous work usinglower molecular weight PEO (MW = 3 × 105 g/mol) [7] as well asany other studies discussed in Section 1.

Fig. 2. Steady shear rheometry experimental data for PEO–SDS systems.

Fig. 3. Zero-shear viscosity as a function of increasing SDS concentration. Resultsobtained from Carreau model fit to data in Fig. 2. Points represent experimental fitdata, while the solid line is included only as a guide.

For dynamic rheology, a 2◦ steel cone with a diameter of 60 mmwas used to provide a uniform shear field throughout the sam-ple. In contrast to the steady shear results, very little difference

was observed in the small angle oscillatory (SAOS) data as the SDSconcentration was increased. A frequency dependant viscoelas-tic response was obtained for all solutions, but the differences indynamic moduli and crossover frequencies were negligible. Forcompleteness, Fig. 4 contains the SAOS experimental data for the

Fig. 4. Small-amplitude oscillatory shear (SAOS) rheometry results for PEO andp32sds system, showing little difference in frequency response with addition of SDS.

ewtonian Fluid Mech. 160 (2009) 22–30 25

PWsa

itu�bllwtdri

2

gwbliram(dt

3

3

qotawsPdttsb

bidltsltltaslr

accounts for some of the disappearance in inertial effects, obtainingsuccessful results in a CaBER test also relies heavily on the elonga-tional or extensional viscosity. Therefore, it can be assumed that theaddition of SDS causes the extensional viscosity of the solutions

E. Miller, J. Cooper-White / J. Non-N

EO solution and the most deviant PEO–SDS mixture (p32sds).ithin experimental deviation, the characteristic relaxation time in

hear was quantitatively indistinguishable for all solutions tested,nd therefore, a single value has been reported in Table 1.

Extensional rheology experiments were performed using a cap-llary breakup extensional rheometer (CaBER) very similar in designo the device described in [20]. A plate diameter of D = 3 mm wassed with an initial aspect ratio of �0 = 0.5, final aspect ratio off = 1.4, and a step-stretch time of t = 50 ms; these parameters are

ased on a previous study of optimal CaBER stretch parameters forow viscosity PEO solutions [20]. Subtle differences in the capil-ary thinning behaviour were observed as the SDS concentrationas increased, and will be discussed further in the following sec-

ion. The established method of fitting the decay of the filamentiameter exponentially in time [21,22] to extract the characteristicelaxation time in extension was used, and the results are includedn Table 1.

.4. Flow visualization

To observe the flow kinematics upstream of the contractioneometry, fluorescent streak imaging was used. All solutionsere seeded with epi-fluorescent (Ex./Em. = 520/580 nm), neutrallyuoyant polystyrene spheres with diameter, D = 1.0 �m (Molecu-

ar Probes, Nile Red), and at a concentration of c = 0.02 wt.%. Streakmages were acquired with a 20× objective (NA = 0.5, DOF = 0.86)esulting in a measurement depth of 12.8 �m [2,5]; images werecquired at the center plane of the microchannels. A constant illu-ination mercury lamp provided excitation through a filter cube

Nikon G-2A). Exposures were taken from the second frame of aouble-pulsed camera system (PCO Sensicam) using an exposureime of 20 ms to provide significant streaking.

. Results

.1. Rheology

The steady shear rheology data presented in this study, in auantitative sense, have served primarily as a confirmation of previ-usly observed and predicted PEO–SDS solution behaviour. In Fig. 3,he increasing viscosity is a direct result of PEO coil expansion viassociation of SDS aggregates starting at the CAC, levelling at C2ith the formation of larger micelles, and peaking at Cm with the

aturation of the polymer chain. The dynamic rheology data for theEO–SDS mixtures, shown in Fig. 4, show even less quantitativeifference. With such small elastic effects present in these systems,he dynamic measurements, while valid and significantly higherhan the baseline of the highly sensitive shear rheometer, are pre-ented in order to qualitatively highlight the lack of difference inulk measurements for the polymer–surfactant system.

What little discrepancy exists in the dynamic rheology cane explained by the change in viscosity. This observation is well

llustrated by normalizing the viscosity data; shown in Fig. 5, theimensionless viscosity, �′ = �/�0, is plotted against the dimension-

ess shear rate, �̇ = �̇ �CaBER. In the normalized data, we observehat all solutions do not fall onto a master curve, as noted for PEOolutions with changing solvent viscosity in our previous studies. Atow SDS concentration mixtures, we observe a slight deviation fromhe PEO-only solution, indicating a higher degree of shear-thinning,ikely the result of a conformational change in the PEO chains. Fur-

her deviation is observed as the SDS concentration is increased,t which point the p8sds, p16sds, and p32sds polymer–surfactantystems are found to fall onto one curve. That is, above 8 mM SDS, ateast in shear, we see self-similar shear-thinning behaviour. Theseesults are indicative of the conformational changes occurring in

Fig. 5. Steady shear rheometry experimental data from Fig. 2, normalized by zero-shear rate viscosity and extensional relaxation times for all PEO–SDS systems.

the polymer–surfactant complex with the increased addition ofSDS to the PEO backbone, resulting in the stretching and stiffen-ing of the PEO chains. This obviously has a large impact between0 < [SDS] < 8 mM, but thereafter has minimal impact on the con-stitutive response of the polymer–surfactant complex to a shearflow.

The extensional rheology data, driven by capillary thinning flow,further exaggerate differences in fundamental behaviour as a resultof increasing SDS association. Although the data for decaying fil-ament diameter, shown in Fig. 6, is quantitatively similar for allsolutions, there is an interesting qualitative difference that can beobserved. In the absence of SDS, or at very low SDS concentration,the p0sds, p5sds, and to some extent, the p8sds solutions experi-ence the effects of fluid inertia. In the data, the filament diameter isobserved to have a characteristic inertia driven oscillation in diam-eter at t > 0.025 s, common in such dilute systems [20]. The additionof SDS causes enough conformational change in the PEO chains toovercome inertia and undergo nearly unaffected capillary thinningbehaviour. While the increase in shear viscosity, as shown in Fig. 3,

Fig. 6. Capillary breakup extensional rheometry (CaBER) experimental data forPEO–SDS systems. Data for higher concentration PEO–SDS systems include fit tofunctional form in Eq. (2) for calculation of apparent extensional viscosity.

2 ewtonian Fluid Mech. 160 (2009) 22–30

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6 E. Miller, J. Cooper-White / J. Non-N

o increase. The p8sds, p16sds, and p32sds data were sufficientlyree of oscillations to attempt an analysis for apparent extensionaliscosity.

The apparent extensional viscosity, �ext, can be calculated simplyrom the change of the filament diameter, D, and knowledge of thequilibrium surface tension, �. From capillary breakup theory [22],pparent extensional viscosity is extracted from the diameter datay applying a force balance between capillary stresses and elasticensile stresses within the fluid [23]:

ext = �/Rmid(t)ε̇(t)

= −�

dDmid(t)/dt. (1)

Using Eq. (1), a direct numerical derivative of the filament diam-ter data, disregarding large fluctuations, was used to calculateext,dD/dt. In any free-surface flow containing surfactants, where newurface is generated and populated by surfactant molecules thatiffuse from the bulk, the dynamic surface tension (DST) shoulde considered. The values of dynamic surface tension of PEO–SDSystems do vary with SDS addition at very short times, but beyondpproximately 20 ms, the relative variance in DST between eachEO–SDS solution is similar to that noted for their equilibrium sur-ace tension values (refer to [7]). Given that the timescales of ouraBER experiments are on the order of 20–50 ms, it is thus sufficiento use the equilibrium surface tension values for this PEO–SDS sys-em for our analysis. Measured equilibrium surface tension values,hich are independent of PEO molecular weight, from a previous

tudy on the same polymer–surfactant system [7] were used to cal-ulate the apparent extensional viscosities of the p8sds, p16sds, and32sds systems as �ext,dD/dt = 5.15, 5.31, and 5.43 Pa s, respectively;hese data are included in Table 1.

An alternate method of determining apparent extensional vis-osity was also used. To avoid the large fluctuations that can bentroduced by taking a direct numerical gradient of experimentaliameter data, the functional form

mid(t) = A e−(1/3)Bt − Ct + E, (2)

s first used to fit the data before differentiating with respect toime. The fitting constants in Eq. (2) have physical relevance. Atntermediate times, the decay of the filament diameter can beirectly related to the extensional relaxation time, and therefore= �CaBER,fit. The final slope in the diameter data can be related to

he equilibrium value of extensional viscosity, C = �/�ext. It is thisatter aspect of the data that did not allow the p0sds and p5sdsata exhibiting the inertia driven oscillations to be fit for exten-ional viscosity; the data in the long time breakup region had largeuctuations. The fitting was performed and resulted in values ofpparent extensional viscosity for the p8sds, p16sds, and p32sdsystems of �ext,fit = 7.7, 11.0, and 10.9 Pa s; the functional form fitsre included in Fig. 6. These solutions are at the very thresholdf the operational space for the CaBER technique at which mean-ngful extensional viscosities can be extracted, and the functionalts are not perfect. However, by optimizing the fits to capture thehape of the long time breakup, a general increase in extensionaliscosity is observed. While there is a discrepancy between the val-es of �ext,dD/dt and �ext,fit, the order of magnitude is the same andhe important qualitative observation of increasing apparent exten-ional viscosity with the addition of SDS is consistent. Furthermore,oth the trend and order of magnitude for the extensional viscosi-ies have been confirmed by previous studies using opposed jetheometers, more suited to extremely dilute polymeric solutions7,11].

For the purpose of clarification and because the values aresed in later quantitative analysis of the Weissenberg number, ithould be noted that the extensional relaxation times reported inable 1, �CaBER, were not obtained from the aforementioned func-ional form. Rather, a simple exponential form was used to fit the

Fig. 7. CaBER data from a set of experiments focusing on the early time, initial onsetof stretch behaviour. This data was used to calculate the supplemental �CaBER,init Datahave been shifted so onset of stretch coincides.

intermediate time filament diameter decay only. In this fit, theinitial thinning and long time filament breakup data are ignored,thereby obtaining a more accurate slope for the exponential decaywhich pertains to the characteristic relaxation time. This is commonpractice, particularly with such dilute systems [20].

In a CaBER experiment, the data of physical relevance are theintermediate time, typically a small plateau in diameter, and finalstretch behaviour, as evidenced by the functional fits in Fig. 6. Thevery early time data, which is not considered in fitting, is oftentoo close to a critical diameter and not cylindrical enough for thetheoretical calculations based on capillary thinning as the sole driv-ing mechanism to apply. In a qualitative sense, however, there isvalue to exploring this region. Using another set of experimentsfocused on obtaining early time stretch results, shown in Fig. 7, adifference in this initial behaviour is observed. The systems withlow surfactant concentration decrease in size much more rapidlythan the higher concentration systems, which have a more smoothtransition from the onset of capillary thinning into the intermedi-ate plateau region. A possible explanation for this discrepancy is ahigher degree of chain alignment due to the increased stiffness ofthe surfactant saturated polymer chains in the p16sds and p32sdssystems. This would result in a greater degree of extensional strainhardening present at earlier times and produce the more gradualstretching profile. For comparison, relaxation times based on theslope of the exponential decay of this initial stretch data were foundand are shown in Table 1, denoted as �CaBER,init. While these initialrelaxation times are smaller, they follow the same trend as thosebased on the more appropriate intermediate stretch data.

3.2. Pressure drop measurements

Differential pressure measurements across the abrupt con-traction for all PEO–SDS systems are shown in Fig. 8. The insetdimensional flow curves show an increase in pressure drop as afunction of volumetric flow rate that corresponds to the increasedviscosity with increasing surfactant concentration. The data areplotted with respect to Weissenberg number (defined in Section3.3) where the viscoelastic differences of the systems with increas-ing SDS concentration are accounted for by the relaxation time. In

this respect, the p8sds, p16sds, and p32sds systems deviate from thep0sds and p5sds systems more uniformly, corresponding roughlywith the abrupt change in conformation well above the CAC. Fur-thermore, the data presented in Fig. 9 compare very well with theRheometrics RFX data presented in [7] (albeit for a PEO–SDS system

E. Miller, J. Cooper-White / J. Non-Newton

Fb

ig. 8. Normalized dimensionless pressure drop as a function of Weissenberg num-er, with inset showing raw, as recorded, pressure drop vs. volumetric flow rate.

Fig. 9. Streak image visualization of the full spectrum of PEO–SDS solutions

ian Fluid Mech. 160 (2009) 22–30 27

of lower molecular weight PEO, MW = 3 × 105 g/mol), with respectto the relative order in which the pressure drop departs from thepressure trace recovered for the PEO-only solution.

In order to better isolate the effects of the varyingpolymer–surfactant behaviour, the normalized dimensionlesspressure was used, and is defined:

P ′ = Pex

Pvisc, (3)

where

Pex = P − PNewt, (4)

and

Pvisc = �0�c. (5)

In these equations, Pex is the extra pressure drop due to elas-ticity, P is simply the dimensional pressure drop upstream and

downstream of the abrupt contraction, Pvisc is the viscous pres-sure drop or critical shear stress, and PNewt is the equivalentNewtonian pressure drop which is defined as

PNewt = sQ, (6)

systems entering an abrupt 16:1 contraction at a range of flow rates.

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8 E. Miller, J. Cooper-White / J. Non-N

here

= limQ→0

dP

dQ. (7)

Defining s as the slope of the dimensional pressure drop curvet low flow rates to obtain the Newtonian pressure was foundo be more accurate than measuring pressure drop for Newto-ian analogues. The polymer and surfactant in solution have andditional viscous contribution which is not captured correctly byn analogue, and therefore, determining s for each system bet-er normalized the results. This approach was used to remove theewtonian component of pressure drop and allows a more directomparison of the elastic and viscous contributions from each sys-em [24]. The normalized data, shown in Fig. 8, show a deviationn the behaviour of the p8sds, p16sds, and p32sds systems. Theseata also give some indication of ranges of critical Wi at whichome changes in flow behaviour may occur; this is discussed inhe following section.

.3. Flow visualization

Streak images for all solutions at a range of flow rates werebtained and are shown in Fig. 9. In the first two columns of Fig. 9,orresponding to p0sds and p5sds, we observe steady Newtonian-ike flow for the entire range of flow rates. Because the p5sds isery close to the CAC, this provides a good confirmation of the rel-tive lack of association of the polymer with the surfactant. As theDS concentration is increased, beyond the CAC, the appearance ofighly stable lip vortices at the entry of the contraction is observed

n the p8sds. As the concentration is again increased to p16sds, theip vortices are observed to appear at lower flow rates, and grow

ith increasing flow rate. Based on the viscosity data in Fig. 3, weould expect the p32sds sample to behave in a very similar fashion

o the p16sds. The PEO chain should be nearly saturated with SDSn both solutions and therefore, the only difference should be theresence of free surfactant micelles in the solution, which in a flowf this nature should have little effect. In the p32sds, however, weee the onset of an unsteady diverging flow pattern at Q = 1.00 mL/h,hich is not observed in the p16sds until higher flow rates. As a con-

rol experiment, glycerol–water solutions (detailed in Section 2.2)ere visualized at the same flow conditions in order to rule out

ny purely viscous effects. No qualitative difference was observedetween these viscous analogues and the PEO–SDS solutions belowhe CAC, ruling out viscosity effects alone.

The solution and flow parameters can be more uniformly

escribed using the dimensionless Reynolds, Weissenberg, andlasticity numbers, defined:

i = ��̇c = �V̄c

wc/2= �Q

hw2c /2

, (8)

ig. 10. Streak image visualization at the observed onset of steady viscoelastic flow exhibi

ian Fluid Mech. 160 (2009) 22–30

Re = �V̄cDh

�0= 2�Q

(wc + h)�0, (9)

and

El = 2��

�wcDh= ��(wc + h)

�w2c h

= Wi

Re. (10)

These formulations are dependant only on the microchan-nel geometry, where wc = 26 �m is the contraction width andh = 45 �m is the channel depth, the flow rate, Q, and the solutionproperties of characteristic relaxation time, �, and fluid density, �.Throughout this study, we have used the relaxation times foundusing CaBER, as shown in Table 1, and for such low concentrations,the density was assumed to be that of water.

Using Wi instead of flow rate, we can take another look at theflow patterns shown in Fig. 9. The photographs shown in Fig. 10with corresponding flow rate and Wi indicate that the appear-ance of the highly stable lip vortices occurs at approximately thesame Weissenberg number, 4 > Wi > 7, for all the PEO–SDS systemsthat exhibited this flow pattern. The lip vortices are observed togrow outward from the abrupt contraction entry in a very per-sistent, highly symmetric, and very stable manner, suggesting theflow pattern is steady but due to the complex streamlines is clearlyviscoelastic. In the case of the p32sds, p16sds, and p8sds, anotherchange in the flow pattern is observed as the lip vortices becomelarge enough to fill the corners of the contraction geometry. At thispoint, the flow becomes unstable with the streamlines varying intime from one side to the other. This “bouncing” of streamlinesacross the entry is characteristic of a diverging flow. The Weis-senberg numbers at which the flow regimes are observed to changecorrespond well to the increased normalized pressure drop devia-tions shown in Fig. 8, especially in the case of the transition fromsteady viscoelastic with stable lip vortices to divergent flow. Weshould note that the error associated with the pressure drop mea-surements, at these low flow rates are estimated to be between 5and 10% of the absolute value of the measured pressure. For thisreason we have avoided attempting to explicitly correlate pressuredrop with transitions from one regime to another as a function ofWi. Further experiments will focus on improving the accuracy ofour pressure measurements.

By using the dimensionless parameters, we can plot all experi-ments on a Wi–Re space, and represent all solutions by a uniqueEl; this is shown in Fig. 11. In this space, the shift in flow pat-tern, from Newtonian to steady viscoelastic with stable lip-vortexflow appears to be most affected by the elasticity. However, basedon the dynamic rheology discussed in Section 2.3, there is little

quantitative difference in elasticity of the solutions and, there-fore, another factor must be present. This becomes more evidentif we compare the current results with previous work using dilutePEO–glycerol solutions, where El was varied simply by increasingthe glycerol concentration and thereby only the Newtonian solvent

ting stable lip vortices at given flow rates and corresponding Weissenberg numbers.

E. Miller, J. Cooper-White / J. Non-Newton

Fig. 11. Summary of flow patterns in terms of Wi–Re space, with dependence on El.The data include all solutions, and flow patterns are denoted: Newtonian-like (�),steady viscoelastic with stable lip-vortices (©), and diverging (�).

Fig. 12. Summary of flow patterns in Wi–Re space, with dependence on El. Flowpatterns include: Newtonian-like (�), steady viscoelastic with stable lip-vortices(©), steady viscoelastic data of [2] (), and diverging (�). Data from Rodd et al. [2]fdPr

vhoiWTlarpotdt

4

i

or dilute PEO–glycerol solutions (denoted with dashed lines) has been included withata from current study (denoted with dotted lines) for comparison. Inset highlightsEO–SDS solutions at 15 < El < 25 that do not seem to fit proposed Newtonian flowegime in given Wi–Re space.

iscosity component [2]. In Fig. 12, the data of Rodd et al. [2], whichave been reformulated in terms of a Weissenberg number basedn CaBER relaxation time instead of Maxwell relaxation time, arencluded with the data from the current study. In this expanded

i–Re space, the effect of El on the flow pattern is less distinct.he data for PEO–glycerol at El = 43 clearly exhibit a Newtonian-ike flow pattern in a range of Wi and Re where the PEO–SDS,t much lower El = 25, are well within a steady viscoelastic flowegime, exhibiting the development of stable lip vortices. In the32sds and p16sds systems at El = 22, 25, respectively, the onsetf a diverging flow is seen at much lower Wi in the PEO–SDS sys-ems. These results are highlighted in the inset of Fig. 12 and clearlyemonstrate the importance of subtle conformational changes inhe polymer–surfactant system on the resultant flow behaviour.

. Conclusion

In light of the pre-existing knowledge about the role of viscos-ty and elasticity in the entry flow of dilute polymer solutions, the

ian Fluid Mech. 160 (2009) 22–30 29

primary goal of this study was to make isolated observations aboutthe effects of polymer conformation on flow behaviour. By usingthe well-studied associative polymer–surfactant system, PEO–SDS,we have shown that chain expansion and the inherent chain stiff-ening this involves, accelerates the onset of steady viscoelastic flowpatterns with vortex growth at the entry lip, as well as the transi-tion to unsteady diverging flow. In contrast to previous results, thisoccurs with less dependence on Reynolds and Weissenberg num-ber, and more so as a function of elasticity. All the dilute polymersolutions tested in this study, although affected by the surfactantcomplexation somewhat, were well below the polymer overlapconcentration, in a regime where flow behaviour is not affectedby chain entanglements.

It is expected that, although the system investigated here isa polymer–surfactant complex, similar changes in the conforma-tion of a polymer (only) in solution, from flexible through torigid rod would have similar outcomes in flows through micro-fabricated contraction–expansion geometries. We have previouslyhighlighted the effects of polymer concentration, polymer–solventinteractions [2], and more recently, downstream geometry (that is,the length of the contraction), on the flow regimes observed andthe transitions between them as a function of Reynolds and Weis-senberg numbers. This work has provided another insight into howthe dynamics of a polymer chain under an extensional flow, whichare constitutively related to the intrinsic conformation of a polymer,must be considered as another relevant (and potentially dominant)parameter with regards to onset of elastic instabilities and the out-comes of complex flow events of complex fluids in general.

As more parameters that affect the flow through a microfab-ricated contraction flow are investigated in such a manner, wemove closer to being able to extract meaningful data from flows ofunknown fluids through microscale rheometers. At present, unlessthe flow of polymer- and surfactant-based solutions (i.e. polymer,polymer–surfactant, and wormlike surfactant systems) throughavailable microfluidic rheometers have simultaneous measure-ment of both the pressure drop and the flow field, their applicationto measuring extensional properties of complex fluids is limited.

Acknowledgement

The authors would like to thank the Australian Research Council(ARC) Discovery Grants Scheme for funding this work.

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