The rheology of aqueous solutions of ethyl hydroxy-ethyl cellulose (EHEC) and its hydrophobically modified analogue (hmEHEC): extensional flow response in capillary break-up, jetting (ROJER) and in a cross-slot extensional rheometer

Department of Chemical Engineering, Unive E-mail: viveks@uic.edu Hatsopoulos Microuids Laboratory, De Massachusetts Institute of Technology, Cam Okinawa Institute of Science and Technolo Onna-son, Kunigami-gun, Okinawa 904-049 Laboratory of Manufacturing and Pro Engineering, Massachusetts Institute of Tech AkzoNobel Functional Chemicals, Hamnväg AkzoNobel Research, Development and Gateshead, NE10 0JY, UK † Electronic supplementary informa 10.1039/c4sm01661k Cite this: Soft Matter, 2015, 11, 3251


Introduction
Multicomponent complex uids containing long polymer molecules are found to provide a much larger resistance to extensional ow than expected on the basis of their shear viscosity. For a Newtonian uid, the extensional viscosity is a factor of three times larger than the shear viscosity (as was rst shown by Trouton 1 ); however it can be several orders of magnitude higher for polymeric uids. 1-3 A stretching or extensional deformation is established when streamwise velocity gradients are present in a ow. Such extensional components arise in virtually all relevant processing ows, [2][3][4] including capillary contraction/expansion ows, ows through bifurcations (e.g. 'T' or 'Y' junctions), ow around obstacles and near stagnation points, [5][6][7] as well as in common industrial processes such as spraying, jet break-up, drop formation and lament stretching. [8][9][10][11] Extensional viscosity enhancement can dramatically inuence the ow characteristics of uids in these processes. 2,[4][5][6][7][8][9][10][11] Designing uids with appropriate characteristics (which are commonly expressed in heuristic terms such as sprayability, spinnability, printability and jettability) requires systematic control over the response of a complex uid to both shear and extensional deformations. Polysaccharides or carbohydrate polymers [12][13][14] are among the most common constituents of industrial and natural complex uids and thus contribute to the measurable ow resistance in many extensional deformations, inuencing spraying and coating of paints, 15 inkjet printing, lament spinning, 16 porous media ow, 17 enhanced oil recovery, 18 and turbulent drag reduction as well as in natural systems such as synovial uid, mucus and deadly viscoelastic uids found in carnivorous plants. 19,20 In this work we contrast the shear and extensional rheology of model aqueous solutions of cellulose ether, EHEC (Ethyl Hydroxy-Ethyl Cellulose), and its hydrophobically modied analogue, hmEHEC. The hmEHEC chains have hydrophobic stickers distributed along the backbone. In aqueous dispersion, the hydrophobic stickers associate together as temporary junctions that break and reform continuously under the action of thermal uctuations and in response to applied deformations. [21][22][23][24][25] Since these cellulose ethers are used as rheology modiers in multi-component uids, we probe their response over the wide range of shear and extensional rates that characterize typical process ows. In a companion study, we discuss the shear-rate-dependent response of EHEC and hmEHEC dispersions as manifested by the shear viscosity and rst normal stress difference measurements for different polymer concentrations. Interchain hydrophobic associations between the multisticker hmEHEC polymers lead to the formation of transient physical gels that provide enhancement to the low shear-rate viscosity when compared with dispersions of bare EHEC chains. At high shear rates, the gel-like microstructure breaks down and the high shear rate viscosity of the sticky, hydrophobically modied chains becomes quite similar to that of the bare polymer. We infer that the enhancement in viscosity due to stickers at low shear rates provides the requisite high viscosity for controlling sagging and slumping behavior of paints, 26 while the high degree of shear thinning associated with the break-down of the microstructure leads to a much lower shear viscosity at the high shear rates most relevant to coating applications. We also characterize the linear viscoelasticity of the physical networks formed by the sticky hmEHEC polymers using a Fractional Maxwell Model (FMM). Additionally we contrasted the measured concentration-dependent response of the linear viscoelasticity of these sticky polymer networks with the theoretical predictions of Rubinstein and Semenov. 27,28 In the present paper, we focus exclusively on a comparison between the extensional behavior of aqueous dispersions of cellulose ether, EHEC and the corresponding multisticker associative polymer, hmEHEC. We probe the inuence of transient extensional rheology at moderate extensional rates during capillary thinning break-up and at higher deformation rates using the free-surface ow during jetting. We also measure extensional contributions to the bulk pressure drop and the corresponding molecular deformation in a steady extensional ow eld generated within a cross-slot ow extensional rheometer. We specically consider the rheological properties from the perspective of design and application of water-based paints, to put into context the role of associative polymers as rheology modiers in industrial applications.
The application of paint to surfaces commonly involves the use of either roller-coating devices or spraying equipment. 29,30 Both of these application processes involve the formation of elongated uid laments that spontaneously break into droplets, due to a capillary-driven instability that seeks to minimize the total surface area of the uid sample. This surface-tensiondriven instability is manifested as sinusoidal perturbations to the initial cylindrical shape of the uid column, and this creates a uniaxial extensional ow eld within the necking lament, as the enhanced local capillary pressure squeezes uid out of the narrow neck. 8,9,31 The process of capillary thinning eventually leading to pinch-off is opposed by the presence of viscous and inertial stresses. [32][33][34] In polymeric complex uids, microstructural changes and the associated increase in the total viscous drag on the elongating polymer chains provide extra elastic stresses that can substantially delay the capillary-driven pinchoff as well as modify the drop size and the distribution of drop sizes. 8,31,[35][36][37][38][39][40][41][42][43][44] In spite of the widespread use of polysaccharides or cellulose derivatives (with and without associative stickers) as rheology modiers in applications where the response to extensional ows determines their utility, their response to extensional deformation, especially in the context of jetting and spraying, remains a less well-studied problem. 12 The lack of requisite characterization at high deformation rates for viscoelastic uids is due to the difficulties and limitations inherent to the techniques commonly used for measuring the response to extensional ow elds, as we describe next.
Extension-free simple shear ows can be readily generated in cone-and-plate or Couette geometries on a torsional rheometer 4 in order to measure the rate-dependent shear viscosity, h( _ g), as well as the rst and second normal stress differences, N 1 ( _ g) and N 2 ( _ g). Likewise, it is desirable to measure the extensional viscosity, h E (_ 3), of a complex uid as the material response to a shear-free, purely extensional deformation where the imposed extensional rate, _ 3, is controlled to be constant. However, the extensional response of complex uids in any geometry or device exhibits a strong dependence on ow parameters 9 including both the strain rate, _ 3, and the total uid strain, 3. Thus the measured extensional viscosity is a deformationhistory dependent material function, and the underlying ow kinematics in most techniques result in measurement of a transient extensional viscosity or h † E (_ 3, t). 45 Fig. 1 summarizes various techniques 2,4 that can be used for studying polymer solution extensional rheology. The typical range of zero shear viscosity and extensional rates accessed by each device are shown qualitatively. Though the comparison of different measures of extensional viscosity characterized using different techniques can show a great disparity, 45,46 understanding the dynamics of the response from a given method is extremely useful in designing complex uids with the requisite processability, or alternatively in controlling the processing parameters for a given complex uid. The use of experimental techniques with well-controlled ow elds and deformation histories is also important for developing constitutive models for complex uids. There is also considerable growth in recent interest in designing suitable, polymeric complex uids for additive manufacturing processes where relating the processability to the underlying uid rheology will be necessary for realizing economics of scale and quality in 3-D printing, ink-jet printing-based biomaterials, print electronics and print photovoltaics. 10,[47][48][49] Steady extension-dominated ows that enable control over both the local strain rate and the total accumulated strain can be realized by incorporating a stagnation point in the ow eld. [50][51][52][53] At such a singular point, the local uid velocity approaches zero but the strain rate can be large. Hence macromolecules contained in the uid elements that pass through the stagnation point become trapped in the elongational ow eld for an extended time period and can accumulate signicant strain provided the characteristic deformation rate exceeds _ 3 > 1/l, where l is the longest relaxation time of the polymeric uid. The relative magnitude of the relaxation rate (1/l) and deformation rate (_ 3) is characterized by the Weissenberg number, Wi ¼ l_ 3, and extensional ows with Wi > 1 are characterized as strong ows. Stagnation point ows can be generated using macroscopic techniques like four-roll mills and opposed jet or cross-slot devices, 54 and the measurements can be extended to higher rates by using microuidic devices 50,55 such as T-junctions, 56,57 four-roll mill analogues 58-61 and crossslots, 6,53,[62][63][64][65][66][67][68] where the small length scales and lower ow rates required to achieve high Weissenberg numbers result in reduced inertial contributions to the ow. By a suitable choice of geometry, such microuidic devices allow measurements up to extension rates of at least 10 4 s À1 and most of these techniques also allow the use of rheo-optical methods for visualizing both ow kinematics and the polymer orientation and stretch. 50 In the present study, we use a Cross-Slot Extensional Rheometer (CSER) to measure both the change in birefringence and the extra pressure drop near a stagnation point to characterize the extensional deformation of the cellulose ether solutions.
Quantitative analysis of the capillary-thinning dynamics of complex uids can also provide a measure of the transient extensional viscosity and chain unraveling timescale relevant for capillary-driven ows. 9,40,[68][69][70][71][72][73][74][75] Measurement of the capillarydriven self-thinning dynamics of a stretched liquid bridge is the underlying principle of the Capillary Break-up Extensional Rheometer (CaBER). 9,42,76,77 In this device, the practical limit to the measurable response is set by the time required to stretch the liquid bridge 77 and more specialized bespoke instrumentation must be developed for low viscosity systems. 73,74 We have recently developed a jetting-based rheometry technique, called the Rayleigh Ohnesorge Jetting Extensional Rheometer (ROJER) that provides access to the typically unchartered regime of very short relaxation time, low viscosity complex uids, 70,71 corresponding to the bottom right corner of Fig. 1. The rheometry technique is based on the understanding of the nonlinear uid dynamics underlying the jetting process, as described in the text later, and is motivated by a method originally suggested by Schümmer and Tebel. 78 In the present study, we examine the Fig. 1 Extensional rheometry of polymer solutions: typical techniques used for measuring the response of polymer solutions and their typical measurement range. The choice of measurement techniques is guided both by the magnitude of the zero shear viscosity (plotted on ordinate) and the range of extension rates accessible by the technique (on abscissa). Inset: typical rate-dependent extensional viscosity response displayed by dilute and semi-dilute solutions. Note that the ratio of extensional viscosity to zero shear viscosity plotted on the ordinate is commonly referred to as the Trouton ratio, while the abscissa corresponds to the Weissenberg number which quantifies the extensional flow strength by comparing the deformation rate with relaxation time.
capillary break-up dynamics in both CaBER and ROJER devices in order to understand the extensional behavior of cellulose ethers in free-surface ow congurations that closely mimic the deformation history and stresses encountered in jetting and spraying applications.
The typical characteristics of extensional viscosity responses for dilute and semi-dilute polymer solutions are shown in the inset to Fig. 1. The enhanced resistance to extensional deformation is most pronounced and well-studied for dilute polymer solutions where a macromolecular conformational change from a random coil to a highly stretched state underlies the observed increase in resistance to elongational ow. Extensional viscosities up to 10 2 to 10 4 times larger than the zero shear viscosity have been reported for dilute solutions 75 and the values increase with molecular weight, as indicated schematically by the inset (Fig. 1). In semi-dilute solutions, due to the overlap between different chains, the effective strain encountered by any individual coil is lower. Entangled semi-dilute solutions (blue triangles) thus typically show a regime of extensional thinning followed by extensional thickening at higher rates. [79][80][81][82][83] However, depending on the entanglement density and chain rigidity, some dilute and semi-dilute solutions show an extensional thickening regime followed by extensional thinning at high rates (see open symbols). 5,22,84,85 In this paper, the measured response of polymer chains with self-associating or 'sticky' side groups (hmEHEC dispersions) is contrasted with the corresponding response of bare polymer chains (EHEC dispersions). The understanding of the response of multisticker polymer chains is a necessary step to arrive at a rational basis for choosing or designing optimal water-soluble rheology modiers for jetting and spraying applications.

Materials
In this study, we use a hydrophobically modied ethyl hydroxyethyl cellulose ether (hmEHEC) with the degree of substitution (DS) occupied by the ethyl group given by DS Et z 0.8; the molar degree of substitution (MS) by ethylene oxide groups is MS EO z 2.5, and the degree of substitution by the hydrophobic sticker distributed along the chain is MS C14 z 0.008. 15 The hmEHEC was provided in a puried form byÅsa Söderlund, AkzoNobel Cellulosic Specialties and has a degree of polymerization DP z 800, which corresponds to a molecular weight M w z 240 000 Daltons. The molecular weight of the individual unsubstituted anhydrous glucose unit is M 0 ¼ 162 Daltons. In addition to hmEHEC, which we oen refer to as a 'sticky' polymer in the rest of this paper, we also use the corresponding unmodied or 'bare' polymer (EHEC), which has no hydrophobic stickers along the chain but has a similar molecular weight and degree of substitution. The shear rheology of both the polymers is described in detail in a companion paper.

Cross-slot extensional rheometer (CSER)
We studied the steady extensional ow of three representative EHEC and hmEHEC solutions in a microuidic cross-slot device. Cross-slot devices consist of orthogonal, mutually bisecting channels with opposing inlets and outlets, which, by virtue of their symmetry, generate a stagnation point at the exact center of the ow domain. 86 The strain rate (_ 3 CS ) is controlled by the volumetric ow rate through the device and we can perform measurements of ow-induced birefringence in the vicinity of the stagnation point together with the macroscopic pressure loss across the channel as the ow rate is progressively increased. An optical micrograph of the cross-slot geometry used in the study is shown in Fig. 2a. We dene the x and y axes as shown in the gure, with the stagnation point (marked by the blue 'x' at the centre of the intersecting channels) taken as the coordinate origin and the z-axis normal to the plane of the page. The slots are precision machined with a width of w ¼ 0.2 mm, and a depth in the z-direction of d ¼ 1050 AE 10 mm, providing an aspect ratio a ¼ d/w z 5 and hence a quasi-2D ow. The length of the inlet/outlet channels is l ¼ 1.2 mm. In all the results presented herein the inow and outow directions are as marked in Fig. 2a.
The cross-slots are fabricated from stainless steel discs by the technique of wire electrical discharge machining (wire-EDM). This method provides highly parallel, non-tapering walls and a smooth surface nish; both of which are essential features for producing a stable, symmetric ow. Annealed soda glass viewing windows are glued to the front and rear surfaces of the stainless steel ow channel (see Fig. 2b), allowing optical access to the region near the stagnation point. The rear window rst has four holes drilled through it ultrasonically, to allow ow into/out of each arm of the device. Finally, the stainless steel/glass sandwich assembly is glued onto a monolithic stainless steel back-plate, which provides connections to the external plumbing and ow loop, as shown in Fig. 2c. All bonds are made using silicone aquarium adhesive. The continuous ow through the cross-slot device is driven at a controlled rate using a precision syringe pump (Harvard PHD-Ultra). The ow from the syringe pump, labeled (1) in Fig. 2c, is split into two channels to provide the ow into the opposing inlets of cross-slot (3). The two outlets are connected together so they remain at equal pressure and do not generate an unbalanced ow. One of the inlet channels is tted with a 35 kPa gauge pressure transducer (GE Druck) (2) to measure the pressure drop across the ow cell. Effluent is ejected to a Petri dish (5) and discarded to waste.
The syringe pump delivers volume ow rates in the range 0.1 mL min À1 # Q # 10 mL min À1 . This provides nominal extension rates at the stagnation point of 40 s À1 # _ 3 CS # 4000 s À1 , that are calculated according to the expression: The Reynolds number for ow in the cross-slot device is calculated using Re ¼ rUD h /h( _ g), where D h is the hydraulic diameter, D h ¼ 2wd/(w + d), and h( _ g) is the shear rate-dependent shear viscosity. Within the cross-slot, assuming an ideal planar extensional ow given by u ¼ [_ 3 CS x, À_ 3 CS y, 0] T the corresponding value of the characteristic shear rate is _ g ¼ where II( _ g) is the second invariant of the deformation rate tensor _ g ¼ Vu + Vu T . For the range of volume ow rates investigated we obtain a range of 0.03 # Re # 40, depending on the particular uid.
Broadly speaking, elastic effects are expected to become signicant as the Weissenberg number (Wi ¼ _ 3l) exceeds unity, i.e. as material elements are being stretched faster than the dissolved polymers can relax. Hence, observation of elastic effects such as the onset of birefringence near the stagnation point or an enhanced pressure drop across the ow cell at a specic strain rate _ 3 c indicates the coil-stretch transition 9,54,87-90 and can be used to estimate the relaxation time of the test uid as l ¼ 1/_ 3 c .
The ow-induced birefringence arising in the uid due to macromolecular orientation in the vicinity of the stagnation point is measured by using an ABRIO birefringence microscope system (CRi, Inc.). 91,92 The cross-slot ow cell is placed on the imaging stage of an inverted microscope (Nikon Eclipse TE 2000-S) and the mid-plane of the ow cell is brought into focus using a 20 Â 0.5 NA objective. Circularly polarized monochromatic light (l ¼ 546 nm) is passed rst through the sample, then through a liquid crystal compensator optical element and nally onto a CCD array. To acquire a ow birefringence image, ve individual frames are captured with the liquid crystal compensator congured in a specic polarization state in each frame, and data from the ve individual frames are converted into a full-eld map of optical retardation and orientation angle. 91,92 The system can measure the optical retardation (R) of the polarized light to a nominal accuracy of $0.02 nm, with a spatial resolution corresponding to a pixel size $0.5 mm (with a 20Â objective lens). The relationship between the retardation and the birefringence of the sample is given by R ¼ dDn, where d is the depth of the ow cell.
By closing the two needle valves, labeled (4), the pressure drop can also be measured for steady viscous shearing ow of uid around a single corner of the cross-slot (DP shear ). Subsequently, the pressure drop is measured with the two needle valves open (DP total ) to generate a stagnation point ow and the excess pressure drop (DP excess ¼ DP total À DP shear ) arising as a result of the additional extensional component in the ow eld is computed. It has previously been shown that the extensional stress difference Ds in the stretching uid is proportional to the excess pressure drop 5,64 or Ds f DP excess . We can therefore obtain a measure of the apparent extensional viscosity of the uid thus: The Trouton ratio of the uid is the ratio of the measured extensional viscosity to shear viscosity and is calculated from the denition Tr( is the value of shear viscosity found from cone-and-plate rheometry and _ g ¼ 2_ 3 CS , as described above. The expected Trouton ratio for Newtonian uids undergoing planar elongational ow is Tr ¼ 4. 93

Extensional rheometry based on capillary break-up dynamics
In addition to the extensional viscosity measurements performed in the microuidic cross-slot rheometer, we carried out capillary thinning and jetting measurements to identify and characterize the response of cellulose ether solutions to the rapid transient extensional deformations that arise in freesurface ows realized in actual jetting, printing and spraying processes. The underlying physics and the methods are described next.
2.3.1 Relevant forces, timescales and characteristic dimensionless groups. The pressure gradient created by a local region of high curvature in the neck of a lament connecting two larger drops drives a ow out of the neck resulting in additional thinning and eventually break-up or rupture. Capillary thinning and break-up can be observed in a number of prototypical geometries 9,32 including: (I) dripping, where the pinch-off results from an interplay of gravitational drainage and capillarity, 41 (II) jetting, where convective instability develops on a uid jet as it issues out of a nozzle, 78 used in the ROJER measurements in this paper, and (III) self-thinning of a stretched liquid bridge as utilized in the Capillary Break-up Extensional Rheometer (CaBER). 77 The CaBER technique enables the assessment of the longest molecular unraveling times and extensional stresses of viscoelastic uids through monitoring of the capillary thinning and breakup dynamics of a uid thread that is formed between two circular rigid end-plates. 76,77 As the uid neck thins locally over time under the action of capillary pressure, the dynamics of the thinning process depend upon the magnitude of the viscous, inertial and elastic stresses that oppose the pinch-off. The relative magnitudes of these four contributions to the stress, and the absence of external forcing, result in similarity solutions that determine the rate of thinning. 9 The self-similar thinning dynamics can then be used to compute material parameters, such as the relaxation time for elastic uids, as discussed in the next sub-section.
We now consider the capillary thinning dynamics in the jetting geometry used in the ROJER technique. An inviscid uid jet issuing out of a nozzle is unstable to a range of different perturbations. The characteristic growth rate, a, for a perturbation of wavelength, l p (or wavenumber, k ¼ 2p/l p ), is given by an expression rst derived and discussed by Rayleigh 94-96 to be here x ¼ kR 0 is the wavenumber non-dimensionalized by the radius of the nozzle, such that 0 # x # 1 and I j (x) represent the Bessel function of order j. Here the velocity and length scales are set by the ow rate, Q, and the size of the nozzle, D 0 ¼ 2R 0 . The growth rate in eqn (3) is non-dimensionalized by a 0 , which is the inverse of the Rayleigh time t R (the timescale characterizing inviscid drop oscillations) and is dened as here r and s are the density and surface tension of the uid respectively. A free jet breaks under the fastest growing instability. The maximum growth rate can be computed from the peak of the dispersion curve (using eqn (3)). Pertubations with the maximum growth rate, a max ¼ 0.34 a 0 appear at The break-up length, L B , of the jet is predicted by linear stability analysis to be shortest for the frequency k max at which the maximum growth rate a max is achieved. The break-up length L B of inviscid or low viscosity Newtonian uid jets 97 is linearly proportional to the jet velocity, V j , and can be written as L B /D 0 ¼ CWe 0.5 . Here, the Weber number, We ¼ ReCa ¼ rV j 2 D 0 /s is the dimensionless ratio of the stagnation pressure in the jet to the capillary pressure (which acts to break the jet into drops) and typically the constant C z 12. The presence of viscous effects changes both the dispersion curve and the breakup length 97 to an expression of the form L B /D 0 ¼ CWe 0.5 (1 + 3Oh). Here the Ohnesorge number, which contrasts the relative importance of viscosity to inertio-capillary effects, is dened as Oh À2 ¼ Re/Ca ¼ rsD/h 0 2 . The Ohnesorge number scales linearly with viscosity and does not depend upon the jet velocity. In contrast to free jets, for a harmonically vibrated jet, the dominant perturbation on the jet is selected by the choice of perturbation frequency, f p , and the velocity of the jet, V j (so that wavenumber, k ¼ 2pf p /V j ). In this case, the break-up length can be measured directly as a function of imposed perturbation frequency, and for low viscosity uids, the shortest break-up length corresponds to the wavenumber The break-up of non-Newtonian or viscoelastic jets is markedly different from viscous jets because elastic effects are typically important. The relative contribution of elastic and viscous effects is characterized through an elasticity number, El ¼ h 0 l/rD 0 2 , which does not depend upon the process kinematics. Since the elasticity number depends inversely on the square of the length scale, viscoelastic effects are bound to become more signicant with any decrease in the diameter of a lament undergoing thinning. For weakly elastic jets, where El ( (1 + 3Oh) 2 , Middleman 98 used linear stability analysis to show This expression suggests that the break-up length is lower for weakly elastic uids. Furthermore linear stability analysis shows that viscoelastic jets initially have faster disturbance growth than a Newtonian uid of the same viscosity. [98][99][100][101][102][103] However, in the case of strongly strain-hardening polymer solutions, the capillarity-driven thinning is opposed at large strains by the presence of an extra stress due to polymer stretching, and this can subsequently stabilize the jet against pinch-off. 33,37,38,99,101 In some cases, the build-up of extra elastic stresses during lament thinning can lead to the development of extremely stable beads-on-a-string morphologies, and consequently lead to much longer break-up lengths. 39,43 Experimentally, the delay in pinch-off due to contributions of elastic stresses can be used to determine the apparent extensional viscosity, 9,40,104 and this underlies the basis for both CaBER and ROJER techniques. In the elasto-capillary regime, the transient extensional properties of the uid can be determined by measuring the rate of evolution in the mid-lament diameter and by using the balance between elastic and capillary forces, as described in eqn (6)-(9) below. For a cylindrical uid lament, we can dene the instantaneous strain rate (_ 3) and the accumulated Hencky strain (3 H ), as follows: 42,104 where D 0 is the initial diameter of the lament at t ¼ 0. The axial force on a cylindrical thinning uid column is given by the following stress balance: where 2s/D(t) is the capillary pressure driving the lament thinning process and Ds(t) is the total extensional stress difference in the elongating lament. Combining eqn (6) and (8), the apparent transient extensional viscosity of the stretching uid can then be evaluated using the following expression: For an aqueous polymer solution such as the cellulosic system studied in the present work, Entov and Hinch 104 argued that in the elasto-capillary thinning regime, the mid-plane diameter of the lament decays exponentially as where g is the elastic modulus and l is the longest uid relaxation time, both of which will vary with cellulose concentration. The complete solution requires a consideration of the whole relaxation spectrum, where g i and l i are respectively the modulus and relaxation time contributed by each mode i. However, for long times the self-similar thinning process means that only the longest mode is important. The relative importance of elastic effects in a jet is characterized by a dimensionless group, called the intrinsic Deborah number, De ¼ l/t R , which represents the ratio of the longest relaxation time of the uid, l, to the Rayleigh time scale, t R . Polymer molecules undergoing sustained extension can eventually approach the fully stretched state and reach the nite extensibility limit. Beyond this point eqn (10) becomes invalid and the radius eventually decays linearly with time. 9 However due to the presence of fully extended polymer chains, the pinch-off dynamics can be signicantly slower than the behavior exhibited by Newtonian uids. Finite extensibility effects and the scaling relationship between radius and time that capture the pinch-off dynamics are discussed in a subsequent section.

Capillary break-up extensional rheometer (CaBER).
The extensional properties of cellulose ether solutions were rst tested using a Capillary Breakup Extensional Rheometer (CaBER; Cambridge Polymer Group) to impose a predominantly uniaxial extensional deformation to the uid samples. The CaBER device uses an initially cylindrical volume of uid (V z 0.06 mL), which forms a liquid bridge between circular parallel plates of diameter D 0 ¼ 6 mm and initial separation L 0 ¼ 2 mm (initial aspect ratio To minimize gravitational sagging and obtain an approximately cylindrical liquid bridge, the initial separation was chosen to be less than the capillary length l cap ¼ ffiffiffiffiffiffiffiffiffiffi s=rg p , where s is the surface tension, r ¼ 1.0 g cm À3 is the solvent (water) density, and g ¼ 9.81 m s À2 is the acceleration due to gravity. 77 The surface tension of the aqueous cellulose ether solutions was measured to be s ¼ 60 mN m À1 using a Krüss K10ST digital tensiometer, resulting in l cap z 2.2 mm.
At time t ¼ À50 ms the top endplate was displaced upwards following an exponential prole L(t) ¼ L 0 e _ 30t to achieve a nal plate separation of L f ¼ 6 mm at time t ¼ 0 s (nal aspect ratio The subsequent evolution of the liquid lament diameter (D(t)) was monitored at the midplane between the endplates (i.e. at L ¼ L f /2) using a laser micrometer. The dynamics of the liquid bridge thinning and break-up process were also recorded at 60 frames per second using a 6 megapixel CCD camera (Casio Exilim EX-F1), with a resolution of approximately 7 mm per pixel. In uniaxial extensional ow Newtonian uids (with rate-independent shear viscosity) display an extensional viscosity that is three times the shear viscosity. Thus for Newtonian uids, the ratio of extensional viscosity to shear viscosity, dened as the Trouton ratio is Rayleigh Ohnesorge Jetting Extensional Rheometry (ROJER). The jetting rheometry technique described here was developed specically to study weakly viscoelastic uids such as cellulose ether solutions. 70,71 In the jetting rheometer, the spontaneous break-up of a uid jet issuing from a small nozzle is controlled by imposing perturbations of known frequency, f p , and amplitude. Axisymmetric sinuous perturbations are created on a uid jet by using a piezoelectric transducer that imposes radial pressure onto the nozzle. The perturbation amplitude and frequency are controlled and set by a function generator. The conguration for forced jetting experiments includes a ceramic nozzle of diameter 2R 0 ¼ 175 microns mounted on a precision stage, so that the position of the nozzle and vertical alignment can be controlled using screw micrometers. The jet is illuminated using a strobe-backlight, where the strobe frequency is either synchronized to the driving frequency, f p , of the piezoelectric transducer or maintained at a known variation, Df p , from it using a second function generator. If the perturbation frequency, f p , is out of sync with the strobe frequency, i.e. Df p > 0, the growth and evolution of the instability of the jet can be observed in an apparent slow motion. The apparent velocity of the jet, u app , as observed using a strobe frequency, f p À Df p , is related to the ow rate, Q, and the radius of the nozzle by the following equation: The rst equality relates the apparent velocity u app to the parameters from the image analysis, where M is the magnication (related to the ratio of nozzle size in microns to its size in pixels), u pix is a velocity dened in terms of pixels moved per frame, and F is the frame rate. The second equality describes the stroboscopic effect. The higher the driving frequency, the greater is the apparent slowing down of the jet. Typical operating frequencies are in the range of a few kHz, the eld of view ranges from 0.5 mm to a few mm, and the typical observable feature size is in the size range of 5-200 mm. For a driving frequency of f p ¼ 4 kHz and a frequency delay of Df p ¼ 0.1 Hz, the uid is observed at 40 000 times slower speed, allowing us to resolve the dynamics of jet thinning in a low viscosity complex uid using conventional video imaging hardware. A chargecoupled device (CCD) camera (BlueFox, Matrix Vision) with an attached zoom lens collects the images at a rate of around F ¼ 40 fps, and the movies are stored in digital, unprocessed form on an external hard-disk connected to the computer. Some of the jetting experiments are alternatively acquired using a JetXpert (ImageXpert, Nashua, NH) imaging system, where the frame rate is restricted to 6 fps. Air pressure or a precision syringe pump (Harvard PhD Ultra) is used to control the ow rate but the precision required in the uniformity of the ow rate limits us to uids with viscosity below 20 mPa s. Image analysis is carried out using ImageJ 105 (an imaging program developed by NIH, available freely), and custom programs specically written for this purpose in MATLAB (Mathworks, Natick, MA). The thinning dynamics are quantied by using a MATLAB program that follows the material deformation of a Lagrangian uid element as it moves along the jet.

Cross-slot ow extensional rheometry
In Fig. 3 we present images of the ow-induced birefringence observed near the stagnation point of the cross-slot for solutions of EHEC and hmEHEC over a representative range of strain rates. The ow-induced birefringence provides a measure of the segmental anisotropy of macromolecules as they deform and align in the ow eld and thus provides an indicator of the degree of macromolecular strain. 51,106,107 Qualitatively all of the uids display similar behavior as the strain rate is increased. At low strain rates, a narrow birefringent strand is observed that is highly localized along the central outow axis passing through the stagnation point. Macromolecules that follow uid elements along this streamline are the rst to stretch since they experience the highest uid strain. As the strain rate is increased the magnitude of the birefringence also increases and, while remaining quite localized, the birefringent strand grows in length and breadth. Similar responses are observed in PEO solutions and in worm-like micellar solutions. 5,64,66 The birefringence is greater in the more concentrated solutions, as expected. Fig. 4 provides a quantitative measure of the maximum birefringence (assessed at the stagnation point) as a function of the imposed strain rate for all six uids. We nd that the birefringence increases gradually, starting from very low extensional strain rates and approaches plateau values at high rates. This behavior is typical of that expected for rigid or semi-rigid macromolecules such as cellulose-based polymers, which readily align in extensional ow elds. One notable difference between the data from the EHEC and the hmEHEC solutions occurs at the 1% concentration, where a much higher birefringence is recorded in the EHEC at high deformation rates. We note that for _ 3 CS < 300 s À1 both uids show a similar response, however for _ 3 CS > 300 s À1 the birefringence in the hmEHEC increases more slowly than in the EHEC, due to the sticker-mediated microstructural changes in the hmEHEC solutions. The rheo-optical responses of the multisticker hmEHEC polymer dispersion and the bare EHEC dispersions both show a large absolute value of birefringence. High degrees of chain stretching at the stagnation point for _ 3 CS > 1/l have been demonstrated in cross-slot geometries by studies of ow induced birefringence 51,64,106-108 and by direct observation of uorescently labeled DNA. 7,109,110 This stretching has been shown to coincide with a signicant increase in the pressure drop measured across the cross-slot, consistent with the predicted increase in the extensional viscosity. 51,64,106-108 In Fig. 5, we show the measured extensional ow birefringence versus excess pressure drop data For EHEC solutions. The optical birefringence from macromolecular stretching is linearly proportional to the excess pressure drop measured at low extensional rates, and this provides an estimate of the stressoptical coefficient, C SOR , as the uid appears to obey the stress optical rule Dn ¼ C SOR Ds. The estimated value for the stress-optical coefficient from the data in Fig. 5 is C SOR ¼ 2.8 Â 10 À8 Pa À1 . Also shown in Fig. 5 are measurements of extensional ow birefringence versus excess pressure drop data obtained in the same geometry for two exible polymers, polyethylene oxide in water and polystyrene in DOP, an aqueous solution of wormlike micelles and an aqueous solution of hyaluronic acid (HA), which is also a polysaccharide. The measured stress-optical coefficient for the EHEC solution is comparable to values known for other polysaccharides. For example, for HA in PBS solution C SOR ¼ 1.82 Â 10 À8 Pa À1 was reported by Kulicke and coworkers 111 while Haward et al. reported C SOR ¼ 5.9 Â 10 À8 Pa À1 using cross-slot extensional rheometry measurements. 5 Kulicke et al. 112 have reported concentration-dependent values 2 Â 10 À8 Pa À1 < C SOR < 8 Â 10 À8 Pa À1 for sodium carboxymethyl cellulose. The values of ow birefringence and corresponding optical anisotropy of polymer segments in solutions of cellulose derivatives are noticeably greater than for the solutions of exible chain polymers. The data shown in Fig. 5 clearly illustrate that the EHEC segments align in the direction of ow quite easily and the behavior qualitatively illustrates the fact that the chains behave as semi-rigid polymers.
In Fig. 6a we present the apparent extensional viscosity as a function of the strain rate for the EHEC solutions in the crossslot device. The extensional viscosity is derived from pressure drop measurements using eqn (1) and (2) and the Trouton ratio is computed using Tr . The corresponding raw pressure data are presented in the ESI. † At low ow rates, the pressure due to the extensional component of the ow eld is computed as a difference of two small numbers, and hence the error bars in the low deformation rate region are relatively high. At the lowest concentration the apparent extensional viscosity is approximately constant at h E,app z 0.01 Pa s, which equates to a Trouton ratio (Tr) very close to the expected value for Newtonian uids in planar extensional ow of Tr ¼ 4 (Fig. 6b). As the EHEC concentration is increased we observe a non-Newtonian increase in the extensional viscosity, starting from very low strain rates, and settling to plateau values at strain rates beyond 2000 s À1 . The extensional viscosity increase provides a signicant increase in the Trouton ratio to Tr z 10 and Tr > 20 for the 0.4% and 1.0% EHEC solutions, respectively. A similar plateau in extensional viscosity of hydroxypropyl-ether guar gum solutions was reported previously by Duxenneuner et al. 113 The apparent extensional viscosity and Trouton ratio data for the hydrophobically modied hmEHEC solutions are shown in Fig. 7a and b, respectively. We observe markedly different behavior from that seen with the corresponding EHEC solutions. At the lower concentrations of hmEHEC, we observe apparent extensional thinning behavior as the strain rate is increased. The Trouton ratio for these uids initially (at low extension rates) exhibits fairly high values of around Tr z 56 and Tr z 40 for the 0.2% and 0.4% solution respectively. However, the Trouton ratio of both uids drops to a low plateau value of around 10 at a strain rate of about 1000 s À1 . In contrast, the 1.0% hmEHEC solution displays an initially increasing apparent extensional viscosity, which subsequently decreases for strain rates _ 3 CS > 10 3 s À1 . This equates to a Trouton ratio that initially rises from a value Tr z 3 up to a maximum of around 10, followed by a rapid reduction.  Similar extensional thinning in associative polymers with stickers distributed along the chain was also seen in studies of a hydrophobically modied alkali soluble associative (HASE) system by Tan et al. 114 Solutions of associating HASE polymers with C-12 (12 carbon long) hydrophobic stickers showed extensional thickening at intermediate extension rates followed by a large extensional thinning at the highest rates that could be achieved using an opposed-jet rheometer. In their study, the bare hydrophilic chain exhibited a plateau value in extensional viscosity at high extension rates, qualitatively similar to our observations for EHEC solutions. These authors also studied the effect of increasing the strength of association by comparing C-12 with C-16 and C-20 modied chains and determined that chains with stronger intermolecular association show only extensional thinning as a result of the progressive disruption of the microstructure at high deformation rates.
The physics underlying extensional thinning of multisticker polymer dispersions in response to strong elongation ows can be broadly understood by using arguments from nonlinear network models developed for describing the steady and transient shear and extensional response of associative polymer solutions, including for telechelic polymers in which hydrophobic stickers are present only at the chain ends. 22,25,115 In a quiescent state, the temporary associations are formed and destroyed with a disassociation time, l do z U À1 exp(DG/k B T) which depends on the activation energy DG that in turn depends upon the number of carbon atoms (or methylene groups) present in the hydrophobic group. Under strong extensional ow conditions, the sticky chains are stretched and aligned which leads to an anisotropic enhancement in the frequency and duration of encounters between hydrophobic stickers as well as resulting in an enhanced stretching or pullout force on the stickers that acts to break down the transient junctions. As the hmEHEC solutions are exposed to an extensional ow eld, the data in Fig. 7 show that the chain alignment and stretching result in an increase in the number of associations between the bridging segments and consequently to an extensional thickening at intermediate rates. A further increase in extension rate and the onset of nite chain extensibility effects leads to a breakdown of the physical network structure and correspondingly to a large extensional thinning. In addition to associative polymers where break-up of the transient network structure leads to a decrease in extensional viscosity, extensional thinning is also observed in dilute and semi-dilute polymer solutions in experiments 84,116-118 as well as simulations. 88,118,119 In such cases, the extensional thinning can be described using  constitutive models like the Wiest model 85 and the Primitive Chain Network model 81,82 that incorporate nite extensibility along with anisotropic hydrodynamic drag. [81][82][83]85,120 The concentration-dependent birefringence measured at a given extension rate represents the effective orientation and stretching of macromolecules in EHEC and hmEHEC solutions in response to the extensional deformation, in addition to the competition between relaxation of chains and their deformation. Rescaling the birefringence with the polymer concentration, and also rescaling the extension rate by a critical extension rate, enables the data to be superimposed as shown in Fig. 8. The scaled birefringence or specic extensional ow birefringence Dn/(c 0 /r) captures the average conformational anisotropy, while the rescaled extension rate _ 3/_ 3 c can be considered to be equivalent to an effective Weissenberg number Wi, which provides a measure of the ow strength. Here the birefringence Dn is scaled with a dimensionless ratio of concentration c 0 and density, r (calculated with mass/volume units). The inset of Fig. 8a also displays extensional stress scaled by modulus, s E /G N ¼ s E /c 0 RT for EHEC solutions, as a function of the effective Weissenberg number, and the extensional response from different EHEC solutions is remarkably similar. The superposition of normalized extensional stress and effective Weissenberg number shows that the observed nonlinear response arises from macromolecules that align and stretch in a self-similar fashion in response to applied deformation of similar strength. Similar plots of specic birefringence dened as Dn/c 0 and reduced extension rate dened as r ¼ _ 3/_ 3 c À 1 have also been presented by Sasaki et al. 121 for collagen dispersions (rod-like polymers) and Fujii et al. 122 for semi-dilute solutions of hydroxypropylcellulose in water-glycerol mixtures (semi-exible polymers). The scaling carried out here for extensional ow-induced birefringence is similar in principle to the plots of specic birefringence versus reduced extension rate for dilute solutions, 5,51 where _ 3 c ¼ 1/l cs is based on the relaxation time for coil-stretch transition. Similarly the specic birefringence measured in response to shear ow of polydisperse semi-exible and rodlike polymers in solution overlaps with each other when plotted as a function of shear rate normalized by the rotational diffusion coefficient. 123,124 The superposition that follows from rescaling the birefringence by concentration suggests that the underlying fundamental contributions to the observed birefringence arise from conformational changes of individual macromolecules in EHEC dispersions. By contrast, the specic birefringence of hmEHEC dispersions in Fig. 8b exhibits a lower slope at low extension rates and the scaled data for c ¼ 1% hmEHEC dispersion show a much lower specic birefringence when compared to the corresponding EHEC solutions. This suggests the onset of interchain effects (arising from hydrophobic stickers) that limit the molecular extension and result in the decrease in the extensional viscosity observed in Fig. 7b.

Capillary break-up extensional rheometry of cellulose ether dispersions
In the Capillary Break-up Extensional Rheometer (CaBER) experiment, 2% by weight aqueous dispersions of EHEC and hmEHEC both exhibit a linear decrease in lament radius with time as shown in Fig. 9. A linear decrease in radius with time suggests that a visco-capillary response, visible when Oh $ O(1), is observed here. This viscocapillary response can be described by the following expression: 9,125 where Oh ¼ h 0 = ffiffiffiffiffiffiffiffiffiffi ffi rsD 0 p is the Ohnesorge number and the numerical pre-factor is found from the solution for the selfsimilar shape of capillary-driven thinning of a viscous uid lament. 125 Using the zero shear viscosity of EHEC h 0 ¼ 0.26 Pa s and hmEHEC h 0 ¼ 1.85 Pa s, the estimates for the Ohnesorge number (Oh) are Oh ¼ 0.5 and Oh ¼ 3.6 respectively. The longer pinch-off time, t p , for the hmEHEC dispersion (t p ¼ 0.33 s) compared with EHEC solution (t p ¼ 0.08 s) is due to the higher shear viscosity of the hydrophobically modied polymer dispersion. The slope of the radius versus time plots shown in Fig. 9 can be used to estimate the effective shear viscosity, and for viscous, Newtonian uids, the estimated viscosity is within 5% of the values obtained in shear measurements. 125 However, in the present experiments, the slopes correspond to h eff ¼ 0.021 Pa s (i.e. Oh ¼ 0.04) for EHEC and h eff ¼ 0.08 Pa s (i.e. Oh ¼ 0.16) for hmEHEC dispersions, implying that the effective solution viscosity during capillary thinning is an order of magnitude lower than the zero shear viscosity. Both solutions are highly shear thinning and the lower effective viscosity is a result of this deformation rate-dependent viscosity. The presence of associating hydrophobic stickers in hmEHEC dispersions as well as the possibility of association through hydrogen bonding and hydrophobic side-groups in EHEC solutions provides these dispersions with a complex microstructure. The initial deformation of the sample required to generate the liquid bridge in the CaBER device leads to microstructural changes that result in lower values of the effective solution viscosity. A similar effect of initial step-strain is also observed for wormlike micelles and immiscible polymer blends; 126,127 however the measurements for dilute polymer solutions are found to be insensitive to the amplitude and rate of step strain.
The typical concentration of the cellulose ethers used in sprayable paints (<1%) is usually lower than the 2% concentration examined here. However as the lower concentration EHEC and hmEHEC solutions have viscosity h < 100 mPa s, the time required to initially establish the stretched liquid bridge in the CaBER device ($50 ms) is longer than that required for the capillary-driven pinch-off. Rodd et al. 77 examined the practical limits of the capillary break-up rheometer (with a CaBER device) and note that the device cannot be used for weakly elastic (De ¼ 1 or l < 1 ms) or low viscosity uids (Oh < 0.1). For the range of cellulose concentrations studied (0.2 < c < 2.4%), only solutions with c > 2% are sufficiently viscous to sustain a viscocapillary balance. Shear thinning further reduces the effective viscosity of the solutions implying that capillary break-up of typical water-borne cellulose solutions occurs in a low Oh regime, where elastocapillary effects are hard to observe on experimentally accessible length-scales.
However, it is well-established that the response of highly elastic uids can be measured for lower viscosity uids (Ohnesorge number Oh < 0.1), 77 if the Deborah number, De $ O(1). For a dilute polymer solution, the relaxation time can be estimated using Zimm's theory, using the following expression: where M w is the polymer molecular weight, N A is Avogadro's number, k B is the Boltzmann constant, T is the absolute temperature and [h] is the intrinsic viscosity. The pre-factor can be approximated by the Riemann zeta function ð1=i 3y Þ in which y represents the solvent quality exponent, and it varies from n ¼ 0.5 for an ideal chain in a theta solvent to n ¼ 0.588 for a coil in a good solvent. 128 Using [h] ¼ 3.33 dL g À1 and M w ¼ 240 000 Daltons (from our shear characterization), we get l Z ¼ 0.014 ms. The characteristic time for relaxation as determined in extensional experiments using techniques like CaBER is typically found to be concentrationdependent even for solutions considered to be dilute i.e. c < c*. 18,69,129 The overlap concentration for the EHEC dispersions is estimated using c* z 1/[h] to be c ¼ 0.3% by weight (where c in weight percent is equivalent to the absolute value in units of g dL À1 ). Above the overlap concentration, the relaxation time for exible coils in an unentangled semi-dilute solution in a good solvent increases with concentration, c or volume fraction f (ref. 128) as here b K is the Kuhn length and N K is the number of Kuhn segments per chain, and as mentioned previously the volume fraction f is related to the mass concentration c 0 by the expression f ¼ c 0 N K b K 3 N A /M w . Using Zimm's formula for good solvents, we can rewrite the above expression as l SD z l Z N K 2À3y f (2À3y)/(3yÀ1) , and using y ¼ 0.588, the estimate simplies to be l SD z l Z N K 0.33 f 0.31 . At the overlap concentration, f ¼ f* z 1/N K 3yÀ1 , and therefore l SD /l Z z (f/f*) 0.31 or equivalently, l SD /l Z z (c/c*) 0.31 . Thus even for a 2 wt% solution of EHEC (with c/c* z 7), the estimated relaxation time l SD z 0.025 ms of the EHEC solution is much less than a millisecond for these solutions, and it is difficult to establish an elastocapillary balance in such a weakly elastic uid.
Only a few studies have probed the extensional response of dilute and semi-dilute polysaccharide solutions using CaBER. In contrast to the aqueous cellulose ether dispersions examined here, either higher molecular weights or more viscous solvents were used to make measurements possible. In capillary breakup studies of semi-dilute dispersions made with methylhydroxyethyl cellulose (MHEC), Plog et al. 130 used aqueous NaOH solutions to suppress associations citing experimental difficulties with pure aqueous solutions. In their study, the MHEC solutions in aqueous NaOH contained a cellulose backbone with a higher degree of polymerization and higher intrinsic viscosity (3.56-17.38 dL g À1 compared to 3.33 dL g À1 for hmEHEC), and therefore 2% by weight solutions provided a measurable extensional viscosity and relaxation time. We have also probed the capillary break-up of cellulose (DP ¼ 800) in an ionic liquid -1-ethyl-3-methylimidazolium acetate (EMIAc)for which the solvent viscosity was h s ¼ 0.1 Pa s (i.e. 100 times higher than the viscosity of water), and found relaxation times in the range 30-1700 ms for concentrations in the range 0.5-8% by weight. 16 Since the relaxation time scales with the solvent viscosity, a simple scaling estimate for an aqueous solution of any cellulose ether with similar DP would give a relaxation time of 0.6 ms for a 2% solution. Duxenneuner et al. 113 reported CaBER measurements for another polysaccharide, hydroxypropyl ether guar gum (HPEGG) in aqueous solutions with concentrations as low as c ¼ 0.17c*. Again the much higher molecular weight (M w ¼ 2.3 Â 10 6 Daltons) and a higher intrinsic viscosity of 13.2 dL g À1 for the HPEGG dispersions facilitated CaBER measurements of the extensional relaxational time, with relaxation times l > 1 ms and Tr > 10 for both dilute and semi-dilute dispersions.
Like the cellulose ethers studied here, many of the complex uids that are used in spraying applications or in inkjet printing are only very weakly elastic, i.e. the relaxation times of such uids are small (typically signicantly less than 1 ms), and most sprayable or jettable uids possess viscosities less than 100 mPa s, which makes extensional rheology measurements with the CaBER instrument very difficult. Furthermore, the extension rates encountered in a real processing application can be as large as 10 6 s À1 and are much higher than the values that can be established in a capillary thinning experiment, where the maximum extension rate resolved in elasto-capillary thinning is limited 77 to around 1000 s À1 . Therefore, we turn to detailed studies of capillary-driven thinning and break-up during jetting as a means of observing pinch-off dynamics for low viscosity and weakly elastic uids, as described in the next section.

Jetting and break-up of associative cellulose ethers
A representative set of images showing instability growth and pinch-off dynamics for jets of a 0.23% hmEHEC solution as a function of perturbation frequency (proportional to dimensionless wavenumber, kR 0 ) are shown in Fig. 10 and 11. The images incorporate detailed information about the nature of viscoelastic jet break-up (symmetric at high wavenumbers, k becoming progressively more asymmetric at lower wavenumbers) and the formation of satellite drops (Movies M1-M3 added as ESI † showing kR 0 ¼ 0.53, 0.64 and 0.78).
The dotted line shown in Fig. 10 superimposes the dispersion curve for an inviscid uid (computed using eqn (3)). A particularly simple and symmetric mode of break-up is observed for frequencies above the critical frequency, f*, at which the shortest breakup length is measured. In this regime, jet break-up can be used to extract extensional viscosity information. 70,71 The critical frequency corresponds to the wavenumber, k max R 0 ¼ 0.69 for an inviscid jet, and the peak shis to a slightly lower value for low viscosity, weakly viscoelastic jets. 71,102 At lower perturbation frequencies (longer wavelength modes), additional higher frequency modes with higher growth rates compete with the primary excitation mode that corresponds to the applied frequency. Computational analysis shows that these higher frequency modes lead to the formation of ner scale features such as satellite drops. However, if the excitation frequency is higher than the critical frequency, so that f > f* the shorter wavelength modes that correspond to the development of asymmetries now grow more slowly than the primary varicose mode and are thus suppressed. In Fig. 11 we show a sequence of images of the temporal evolution of the capillary thinning process observed during jetting. In contrast to low viscosity, Newtonian uid jets (at the same We and Oh), the jets of the cellulose ether solutions examined in this study, show no satellite drop formation. Thus it is clear that nonlinear effects due to tensile stresses generated by the deformation of polymer chains in response to the elongational ow within the neck delay pinch-off sufficiently to suppress satellite formation.
The spatiotemporal evolution of a viscoelastic jet can be described in terms of the jet radius R(z, t) where z represents the distance from the nozzle and t is the time. A Lagrangian observer "P" located on the jet moving with velocity V j reports the change in radius as R ¼ R P (z ¼ V j t). From the sequence of images in Fig. 11, the progressive decrease in the radius of the neck R ¼ R P (z ¼ V j t) is extracted for an excitation with kR 0 ¼ 0.78 and the data are shown, plotted as a function of dimensionless time in Fig. 12. The corresponding extensional rates computed using eqn (6) are also shown. We note that the Rayleigh time used for non-dimensionalizing the abscissa is t R ¼ 0.11 ms, and thus the whole capillary pinch-off event is effectively complete in a time of 1.8 ms. This is much faster than the typical time required in our CaBER experiment for establishing a stretched liquid bridge. Though Schümmer and Tebel 78 were probably the rst to motivate the use of a jetting-based elongational rheometer, their own data show large variability with perturbation amplitude and imposed frequency (which was varied using a speaker) and jet velocity. By identifying the signicant role of perturbation wavenumbers in dictating the kinematics of capillary-driven thinning and break-up during jetting, and by limiting ourselves to forcing frequencies in the range f > f* for analysis, we were able to create repeatable measurements of the transient extensional viscosity and relaxation time, and we obtained similar values with different perturbation frequencies, jet velocities and perturbation amplitudes. By using high driving frequencies and jet velocity, we can apply extensional deformations at rates that are higher than the estimated Zimm relaxation rate of the chain, and also faster than the disassociation rate of stickers present on the hmEHEC chains. We note that in Fig. 12 the neck radius decreases linearly with time close to break-up. In the previous section, we noted that a linear decrease of radius with time is characteristic of the viscocapillary response of Newtonian uids. 125 If we use eqn (12) and the measured slope (0.09) of the curve shown in Fig. 12, we obtain an effective viscosity h eff ¼ 22h 0 that is substantially larger than the value of the zero shear viscosity, h 0 value measured for 0.23% hmEHEC solution. The apparent Ohnesorge number Oh ¼ 0.55 is computed to be 22 times higher than the value based on zero shear viscosity, Oh ¼ 0.025. Unlike the CaBER observations with the 2% cellulose ether solutions that yielded a lower apparent Oh than computed on the basis of zero-shear viscosity, the value of Oh measured during jetting is substantially larger, and clearly, the presence of the hmEHEC chains has modied the nal stage of the pinch-off dynamics.
In contrast to the Oldroyd-B model in which only an elastocapillary regime with exponentially slow thinning is predicted, nonlinear models that include anisotropy in the drag acting on polymer (e.g. the Giesekus model) or nitely extensible dumbbell models (such as the FENE-P model) allow a nite time break-up accompanied by a linear decrease in radius close to the pinch-off time i.e. the minimum radius in the neck should scale as R(t)/R 0 z B(t c À t). 9,131,132 In the FENE-P model, the polymer chains reach a fully stretched state during elastocapillary thinning and behave as a dilute suspension of rigid rods that increase the total effective viscosity of the uid. 132, 133 Entov and Hinch 104 used the FENE model to predict a linear decrease in radius with time of the form given above with B ¼ s/6h eff R 0 where h eff is the effective viscosity of the fully stretched dumbbells. Renardy, and Fontelos and Li argued that the prefactor B is dependent on the constitutive model used for describing nite extensibility effects and the conformation-induced change in the magnitude of drag coefficient. 9,131,134,135 The evolution and break-up of jets of viscoelastic uids of Giesekus and FENE-P type both predict a self-similar and linear decrease in neck diameter close to pinch-off, quite similar to the dynamics observed in Fig. 11c and d and 12. The fore-a symmetry of the thin thread or ligament connecting the large primary drops indicates that uid inertia is not important during the break-up process.
The extensional ow induced stretching of polymer chains results in a linear decrease in radius of the thinning ligament with time which can be written in the following dimensionless form: Fitting the linear decrease in radius with time observed in Fig. 12 with eqn (15) gives a slope of 0.05(Tr N Oh) À1 ¼ À0.09,  and consequently the limiting value of Trouton ratio obtained for fully stretched chains is Tr N ¼ 22. Irrespective of the details of the model used, the observation of such a linear scaling implies that the limiting value of Trouton ratio, Tr N , associated with a high strain, and high extension rate (i.e. large Wi) ow is being measured in this experiment. Due to the enhanced resistance to pinch-off, the cylindrical ligament remains slender for a longer time than the value expected from the viscocapillary scaling based on the zero shear viscosity. The enhanced viscoelastic resistance to pinch-off also leads to the absence of satellite drops at different perturbation frequencies, as shown in Fig. 11 and 12.
In recent work using drop on demand inkjet printing with weakly elastic uids, 72,73,136 it has been observed that the macromolecules approach their maximum elongation in the thinning neck between drops. Macromolecules reach a high degree of extension in a transient ow when two conditions: 5,90 1) a critical rate condition and 2) the accumulated strain condition are satised. The rst condition requires the velocity gradient (or deformation rate), _ 3, to exceed the relaxation rate of the macromolecules, 1/l. For dilute solutions, the simplest estimate of this rate is based on _ 3 > l Z or Zimm relaxation time, 137 which is the longest relaxation time of the molecule in its coiled state. The accumulated strain condition requires a large strain rate to be maintained for long enough for large molecular strains to develop (exible high molecular weight polymers can require stretching by 100-fold to become fully stretched). According to the relaxation times given above and the magnitudes of the measured extension rates shown in Fig. 12 and 13, condition (1) is satised in the jetting experiments. The experimental results indicate that the observed pinch-off dynamics are characteristic of fully stretched and oriented chains, and the maximum observable uid strain in the current set-up is estimated using eqn (7) to be 3 H z 7. To determine whether the cellulose ethers are likely to become fully stretched in these jetting experiments, we need to estimate the extensibility of the polymer chains.
For the Kuhn chain, the maximum extension or nite extensibility limit of a chain can be estimated by here the mean square end-to-end distance hR 0 2 i gives an estimate for coil size and the contour length of the chain. R max gives the maximum size to which a chain can be stretched. We can use our measured value of [h] ¼ 3.88 dL g À1 for the EHEC solution to obtain an estimate of the root-mean-squared end-toend length at equilibrium, hR 0 2 i 1/2 using the formula where M w is the molecular weight of the cellulose ether (M w z 240 kDa for EHEC of DP ¼ 800) and F N ¼ 2.1 Â 10 23 mol À1 is the Fox-Flory constant for a good solvent. 138,139 We obtain a value of hR 0 2 i 1/2 z 75 nm. We can estimate R max z 412 nm for the contour length of the DP ¼ 800 cellulose ether molecule, by taking the length of a single AGU repeat unit to be l AGU ¼ 0.515 nm (ref. 12 and 140) (for detailed computation, see the work by Tsvetkov 141 ). From eqn (16), this gives N K 1Àn ¼ 5.5 and since water is a good solvent for EHEC, 142 we use the exponent y ¼ 0.588, to get N K ¼ 62 and b K z 6.6 nm, or K z 12l AGU . Most importantly, the ratio of contour length to root-mean-squared end-to-end length is very modest, L ¼ R max /hR 0 2 i 1/2 z 5.5, suggesting that only moderate strains are required to completely extend this semi-exible macromolecule.
To further test the validity of these arguments, we carried out additional jetting-based rheometry studies on a dilute polyethylene oxide (PEO) solution (M w ¼ 300 000 Daltons) dissolved in a glycerol-water (46 : 54) mixture (viscosity matched with  (6)) as a function of scaled time. cellulose ether solution). The molecular weight of the PEO is chosen to be comparable to the cellulose ethers, and the concentration chosen, c ¼ 0.01%, corresponds to an extremely dilute solution (c[h] z 0.03). Because of the exibility of PEO chains, we obtain N K ¼ 3750, in contrast to N K ¼ 62 for the cellulose ethers. The corresponding extensibility for PEO chains is L z 27, which is nearly ve times larger than the extensibility of the cellulose ethers. The capillary-driven thinning behavior exhibited by this dilute viscoelastic solution is shown in Fig. 13. The more exible, extensible PEO solution displays exponential thinning dynamics, a signature feature of the elastocapillary scaling discussed previously (see eqn (10)). For PEO solutions, a relaxation time of l ¼ 0.16 ms ¼ 3l Z , and a Trouton ratio as high as Tr ¼ 300 can be computed. In the context of spraying applications, it is clearly an advantage to have cellulose-based thickeners or rheology modiers as their limited extensibility ensures that extensional viscosity effects are bounded.
Analysis of capillary break-up during jetting of 0.4% EHEC and hmEHEC solutions shown in Fig. 14 also shows a qualitatively similar response to the 0.23% hmEHEC solutions. The radius versus time data are shown along with the corresponding extension rate for both the solutions. Using eqn (15), we can again estimate limiting Trouton ratios of Tr N ¼ 10 for 0.4% EHEC solution and Tr N ¼ 8 for hmEHEC solutions. The 0.4% EHEC and hmEHEC solutions can be considered semi-dilute, but since the intrinsic viscosity of these solutions is 3.88 dL g À1 and 3.33 dL g À1 respectively, the effect of stickers is hardly visible in the extensional response of the hmEHEC solutions.
An estimate for the upper bound of the extensional viscosity or Trouton ratio for fully extended EHEC chains can be made by assuming that the fully extended chains behave like rods of length R max . The estimated value of the Trouton ratio Tr B N computed using Batchelor's theory for rods 88,143 gives: where all variables are the same as before and the volume fraction f is related to the concentration c 0 in mass/volume by the expression f ¼ c 0 N K b K 3 N A /M w . The estimated value of Tr B N z 111 using eqn (18) is much higher than the measured value, and suggests that the semi-exible EHEC chains are not fully aligned. For nitely extensible polymer molecules with extensibility L, the rate-dependent FENE-P model gives an alternative (and better) estimate for the steady extensional viscosity value at close to full extension: For the cellulose ether chains at c*, using the appropriate values of extensibility, L ¼ 5.5 and the measured polymer contribution to shear viscosity, h P ¼ 0.7 mPa s we nd the expected maximum value of the Trouton ratio to be Tr FP N z 64. The experimental value of Tr N z 22 for 0.23% HMEHEC measured using jet rheometry is thus reasonably close to the estimate given that the polymer samples used in our study are polydisperse.

Conclusions
The rheological response of EHEC and hmEHEC solutions to a range of extensional deformations was studied by using excess pressure drop and ow induced birefringence measurements during steady planar extensional ow in a microuidic crossslot device and by a recently developed technique referred to as Rayleigh Ohnesorge Jetting Extensional Rheometry or ROJER that follows the capillary thinning and break-up of thin uid ligaments that form between drops during jetting of weakly viscoelastic uids. The reduction of the characteristic geometric dimensions down to the microscale in both the ROJER and cross-slot instruments, together with the use of electro-optic imaging techniques, enables accurate measurements of the extensional viscosities of these polymer solutions using relatively small scales, and allows access to the high extension rates that are of most relevance to applications such as paint spraying and coating operations. The ROJER technique can be used with opaque uids and access comparatively higher extensional rates than cross-slot extensional rheometry, which requires optically transmissive uids. Both techniques are ideal for use with other polysaccharide-based biological samples and industrial uids, as the evaporation of volatile solvents is negligible and temperature control can be implemented relatively simply. We hope that the techniques and methodology described here will nd widespread use in formulating and studying specialized inks for electronic, photovoltaic and biomedical print-based manufacturing technologies. 47,48,144 Quantitative comparison between the extensional rheology of EHEC solutions and the hydrophobically modied analogue, hmEHEC, shows that the presence of additional C-14 hydrophobic groups as stickers provides the hmEHEC dispersions with a much larger extensional viscosity at low extension rates. The extensional viscosity and ow-induced birefringence of the semiexible EHEC solutions increase monotonically with extensional rate and with concentration. By contrast, the hmEHEC solutions exhibit extensional thinning behavior and a reduction in the specic birefringence at higher concentrations, indicative of inter-chain effects. The dramatic extensional thinning behavior at high extension rates shown by hmEHEC dispersions is quite reminiscent of their response to high shear rates. At the highest shear and extensional rates, the physical associations between the chains are no longer effective and the hydrophobically modied cellulose ethers display low viscosities, which make hmEHEC dispersions ideal for use in jetting and spraying applications. In the jetting experiments, analysis of the pinch-off dynamics exhibited by the cellulose ether solutions reveals a linear decay of neck radius with time, which is a scaling characteristic of nite extensibility. The EHEC and hmEHEC solutions display quite moderate values of the limiting Trouton ratio (typically no more than Tr N z 20), and the experimental values are in agreement with estimates based on nite extensible nonlinear elastic (FENE-P) models. Again this bounded increase in the extensional viscosity is favorable for spraying and coating operations as it eliminates the stringiness and beads-on-strings structures that plague many other systems such as dilute solutions of exible long chain macromolecules. 136,145,146 Of course, the jetting or spraying behavior of a complex multicomponent uid such as paint is inuenced not only by the polymer viscoelasticity and extensibility, but also by the combined and coupled effects of colloidal particles, surfactants and other additives that are present in the dispersion. 32,33,136 However the present study illustrates that the ow-induced destruction of the transient network that is initially established in solutions of associative polymer thickeners can help produce the requisite low viscosities required for drop formation in jetting or spraying. The creation of a physical network at low deformation rates provides the dispersions with the high viscosity and large gel strength needed to stick to surfaces and confers such paints and coatings with desirable leveling and sagging properties. In spite of the extensional thinning effects, the cellulose ether dispersions still present a greater extensional resistance to pinch-off than corresponding Newtonian uids with similar shear viscosities, and this delays the nal pinch-off event substantially, therefore reducing the size of satellite drops (if present at all) below experimental resolution. Tailoring the number of stickers and the strength of the association in the transient junctions and controlling the distance between hydrophobic stickers along the hydrophilic backbone each provide possible mechanisms for the design of a stable and sprayable multicomponent complex uid. In each case, an understanding of the rate-dependent response of the uid to the very large imposed shear and extensional deformation rates characteristic of coating operations is essential. The different extensional rheometer congurations presented in this paper show that both the transient and steady extensional rheological response of weakly viscoelastic uids can now accurately be measured at deformation rates up to _ 3 z 10 4 s À1 .