Abstract The large extensional viscosity of dilute polymer solutions has been shown to

Transcription

1 SAMRAT SUR, JONATHAN ROTHSTIN 1 Drop breakup dynamics of dilute polymer solutions: ffect of molecular eight, concentration and viscosity Samrat Sur, Jonathan Rothstein University of Massachusetts Amherst, MA Abstract The large extensional viscosity of dilute polymer solutions has been shon to dramatically delay the breakup of jets into drops. For lo shear viscosity solutions, the jet breakup is initially governed by a balance of inertial and capillary stresses before transitioning to a balance of viscoelastic and capillary stresses at later times. This transition occurs at a critical time hen the radius decay dynamics shift from a 2/3 poer la to an exponential decay as the increasing deformation rate imposed on the fluid filament results in large molecular deformations and rapid crossover into the elasto-capillary regime. By experimental fits of the elasto-capillary thinning diameter data, relaxation time less than one hundred microseconds have been successfully measured. In this paper, e sho that, ith a better understanding of the transition from the inertia-capillary to the elasto-capillary breakup regime, relaxation times close to ten microseconds can be measured ith the relaxation time resolution limited only by the frame rate and spatial resolution of the high speed camera. In this paper, the dynamics of drop formation and pinch-off are presented using Dripping onto Substrate Capillary Breakup xtensional Rheometry (CaBR-DoS) for a series of dilute solutions polyethylene oxide in ater and in a viscosified ater and glycerin mixture. Four different molecular eights beteen 1k and 1M g/mol ere studied ith varying solution viscosities beteen 1 mpa.s and 22 mpa.s and at concentrations beteen.4 and.5 times the overlap concentration, c*. The dependence of the relaxation time and extensional viscosity on these varying parameters ere studied and compared to the predictions of dilute solution theory hile simultaneously searching for the

2 SAMRAT SUR, JONATHAN ROTHSTIN 2 loer limit in solution elasticity that can be detected. For PO in ater, this limit as found to be a fluid ith a relaxation time of roughly 2 µs. These results confirm that CaBR-DoS can be a poerful technique characterizing the rheology of a notoriously difficult material to quantify, namely lo-viscosity inkjet printer inks. I. Introduction The addition of a small amount of moderate to high molecular eight polymer to a Netonian solvent can yield rather dramatic changes to the rheological behavior of the fluids. This is especially true in extensional flos here presence of polymers can significantly increase the resistance to stretching flos [1]. The resistance to extensional flos is characterized by the extensional viscosity of the fluid. For dilute solutions of high molecular eight polymers, the extensional viscosity can be several orders of magnitude larger than the shear viscosity. The effects of large extensional viscosity can be readily observed through the ability of these solutions to form persistent filaments and to delay the breakup into droplets hen stretched [1-7]. This polymer-induced viscoelasticity has many industrial applications, including in inkjet printing. In inkjet printing, the addition of a small amount of polymer to the ink can help minimize satellite and daughter droplet formation hich is essential for printing quality. Hoever, the addition of too much polymer to the ink can make printing impossible by delaying breakup of ink jets into drops [8, 9]. The influence of polymers in inkjet printing fluids or other lo-viscosity dilute polymer solutions is often difficult to see in standard shear rheology measurements. The shear viscosity often appears to be Netonian in steady shear measurements and, in small amplitude oscillatory tests, the relaxation time is often so small (in the range of micro to milliseconds) that it is difficult to measure using standard rheometric techniques. Hoever, for these micro-structured

3 SAMRAT SUR, JONATHAN ROTHSTIN 3 fluids, the extensional viscosity, hich is a function of both the rate of deformation and the total strain accumulated, is often clearly evident even if it is not readily measurable. Some of the most common manifestations of extensional viscosity effects in complex fluids can be observed in the dramatic increase in the lifetime of a fluid thread undergoing capillary break-up driven by interfacial tension. Depending on the composition of the fluid, viscous, elastic and inertial stresses may all be important in resisting the filament breakup resulting from capillary forces. The breakup dynamics can thus be used to obtain a number of fluid properties including the surface tension, σ, the shear viscosity, η, the extensional viscosity, η, and the relaxation time of the fluid, λ. Here, e ill be using dripping onto substrate capillary breakup extensional rheometry (CaBR-DoS) developed by Dinic et al. [1, 11] to visualize and characterize the extensional rheology of dilute, lo-viscosity polymer solutions. In the past several decades, a number of measuring techniques have been used to characterize the extensional flo rheology of complex fluid. Of those techniques, filament stretching extensional rheometry (FiSR) and capillary breakup extensional rheometry (CaBR) techniques are the most common ones [1, 4, 12-15]. In both these techniques, a small amount of liquid is placed beteen to cylindrical discs or plates. In a filament stretching device, at least one of the cylindrical discs is driven in a controlled manner so that a constant extension rate can be imposed on the fluid filament hile the stress response of the fluid to the stretching deformation is measured through a combination of a force transducer and a laser micrometer to measure the filament diameter [5, 16]. This technique is limited by the maximum strain that can be achieved ( ε max 6 ) and the maximum imposed deformation rates ( ε max 1s 1 ) that can be imposed. As a result, the use of FiSR is limited to mostly polymer melts or higher viscosity

4 SAMRAT SUR, JONATHAN ROTHSTIN 4 polymer solutions here the zero shear rate viscosity is greater than approximately η > 1 Pa.s [17]. In CaBR, a step strain is rapidly imposed on the fluid beteen the to plates by rapidly displacing the top plate by a linear motor over a short distance [18]. This extension produces a liquid filament beteen the to plates. The minimum diameter of the thinning filament is then measured as a function of time until break-up in order to calculate the apparent extensional viscosity and the relaxation time of the fluid. This is a common technique for determining the extensional rheology of viscoelastic fluids ith viscosities as lo as η = 7 mpa.s and relaxation times as small as λ = 1 ms [19]. Along ith the viscosity limit of the CaBR technique, another limitation of this method is the inertial effects resulting from the dynamics of the rapid step stretch imposed by the motor motion. At the high velocities required to measure the breakup dynamics of lo viscosity fluids, the rapid step strain can lead to oscillations in the filament that make measurement of extensional rheology difficult. Recently Campo-Deano et al. [2] used a slo retraction method (SRM) to investigate filament thinning mechanisms of fluids ith shear viscosities similar to ater and very short relaxation times. This experimental technique involves sloly separating the pistons just beyond the critical separation distance for hich a statically stable liquid bridge can exist. At this point, the filament becomes unstable and the thinning and breaking process is initiated. This SRM technique avoids inertial effects alloing the authors to extract relaxation times as short as λ = 2 μs for dilute aqueous solutions of polyethylene oxide (PO) ith a molecular eight of 6 M = 1 1 g/mol and shear viscosities beteen 1< η < 3 mpa.s [2]. Vadillo et al. [19] have further pushed the limit by measuring relaxation times as short as λ = 8 μs by using a Cambridge Trimaster rheometer

5 SAMRAT SUR, JONATHAN ROTHSTIN 5 (CTM) [21] along ith a high speed camera ith adjustable fps ith reduction of frame size for a series of solution of monodisperse polystyrene dissolved in diethyl phthalate (DP) ith concentration of polystyrene ranging from dilute to concentrated ith solution viscosity of η = 12 mpa.s. More recently Greiciunas et al. [22] have used the Rayleigh Ohnesorge jetting extensional rheometer (ROJR) technique [23] to measure relaxation times of dilute solutions of PO, ith molecular eight 5 M = 3 1 g/mol mixed in 25%/75% (by eight) glycerol/ater solution. The ROJR is one of the more technically challenging of all the extensional rheology methods for characterizing lo viscosity fluids to implement. Hoever, it does have the benefit of eliminating the need for high speed imaging hich can reduce costs substantially. In their ork, they ere able to measure relaxation time as lo as λ = 12 μs for a solution viscosity ith zero shear viscosity of η = 2.9 mpa.s. In this technique, fluid is jetted through a small diameter nozzle here a small perturbation is applied to drive a capillary instability along the liquid jet. The instability eventually gros large enough to cause the jet to break up into droplets donstream. A camera is used to capture the thinning dynamics from hich the extensional rheology of the fluids can be calculated in much the same ay as in CaBR. Amazingly, there are industrial applications like inkjet printing hich require devices that can experimentally characterize the extensional rheology of dilute solutions ith relaxation times even loer than those described above. Recently, Dinic et al. [1] have developed a dripping onto substrate CaBR technique called CaBR-DoS hich can measure relaxation times much less than λ 1 ms for lo viscosity fluids ( η 1 mpa.s ). In their experimental setup, a fluid dispensing system is used to deliver a drop of fluid at a relatively lo flo rate onto a glass substrate placed at a fixed distance belo the exit of the nozzle. As the droplet sloly drips from

6 SAMRAT SUR, JONATHAN ROTHSTIN 6 the nozzle, a filament is formed. By capturing the droplet on the substrate and not alloing it to fall further, the filament is alloed to breakup under capillary action in much the same ay that CaBR orks. The advantage is that the inertia associated ith the moving of the top plate is removed as are the acceleration and the velocity limit of the actuator. Additionally, compared to the slo retraction method, CaBR-DoS is much better suited for highly volatile fluid here evaporation can play a large role because the experiments are performed much more quickly. For visualization, a high-speed imaging system as used, ith a frame rates varying from 8 to 25, fps. Dinic et al. [1] performed a series of studies on aqueous solutions of PO having a of molecular eight of 6 M = 1 1 g/mol. In all their experiments, the concentration of PO as kept ithin the dilute region. They demonstrated a dependence of relaxation time on concentration, c, for dilute, aqueous PO solution as.65 λ c rather than the expected λ c from dilute theory [24]. This deviation as attributed to the much loer concentration required in extensional flos for a fluid to truly be ithin the dilute regime [1]. In fact, Clasen et al. [2] shoed that in extensional flos an ultra-dilute concentration belo c/c*<.1 is needed to recover the expected relaxation times and scaling s for dilute systems. Here, c/c* is the reduced concentration ith c* defined as the coil overlap concentration. Dinic et al. [1] ere successfully able to measure relaxation times as lo as λ.3 ms. In follo up papers, the authors further extended their technique by performing extensional rheometry measurements on various other complex fluids [25] such as glycerol-ater mixtures, ketchup, mayonnaise, photovoltaic ink and semi-dilute solutions of poly-acrylamide. They ere able to capture and differentiate beteen the inertio-capillary thinning, visco-capillary thinning and elasto-capillary thinning dynamics using the CaBR-DoS. Through this technique, Dinic et al. [11] captured the differences in the necking region for different fluids during pinch-off. For pure ater, it as

7 SAMRAT SUR, JONATHAN ROTHSTIN 7 observed that the necking region forms a cone close to pinch-off point as predicted by theory. For a polymer solution, a long cylindrical filament as formed and the pinch-off as found to occur in a location near the mid-plane of the filament. For a multicomponent complex fluid, such as shampoo, a non-slender liquid bridge as formed resulting in the formation of to axisymmetric cones after break-up. Through these initial studies, Dinic et al. [11] have established a quick, reliable method to perform extension rheometry on lo-viscosity fluids hile establishing the microstructured effects on the pinch-off dynamics. In this paper, e extend the ork into thinning dynamics of lo viscosity, elastic fluids using the CaBR-DoS technique by systematically probing the effects of polymer molecular eight, solution viscosity and concentration don to the point at hich measurements of extensional viscosity and relaxation time become limited by camera resolution. In this study, e focus on the transition beteen the early time inertia-capillary regime and late stage elastocapillary regime. The sharpness of this transition allos us to measure the extensional rheology of these PO solutions ith very fe data points. In fact, from the parametric studies performed here, e have developed a simple relation to determine the relaxation time of lo concentration, lo molecular eight and lo viscosity polymer fluids by capturing a single image of fluid filament before breakup. Using this technique, e have demonstrated that relaxation time measurements as lo as λ = 2 μs can be measured using the CaBR-DoS technique. II. Materials The lo-viscosity elastic fluids tested in this ork are dilute solutions of polyethylene oxide (PO) (supplied by Aldrich Chemical Co.) ith molecular eights ranging from 5 6 M = 1 1 to 1 1 g/mol in glycerol and ater mixtures. In general, commercial PO samples are knon to be polydisperse. Tirtaamadja et al. [7] measured the polydispersity index

8 SAMRAT SUR, JONATHAN ROTHSTIN 8 6 of the M = 1 1 g/mol PO sample used in their study to be PDI=1.8. Since our samples ere purchased from the same supplier, e expected them to have similar polydispersity as those used in the orks of Tirtaamadja et al. [7]. The polydispersity of the polymer can have an impact on measurements and comparisons to Zimm theory hich assumes a perfectly monodisperse polymer sample. In this ork, the polymer concentrations ere varied, hile keeping it belo the coil overlap concentration, c*, hich as calculated using the definition provided by Graessley [26] such that c*=.77/[η ]. Here [ η ] is the intrinsic viscosity of the polymer solution hich depends on the molar mass of the chain according to the Mark-Houink-Sakurda equation a [ η ] = KM, here K=.72 cm 3 /g and a =.65 for PO in ater and glycerol [7]. To arrive at the value of these Mark-Houink constants, Tirtaamadja et al. [7] used a linear regression analysis to obtain a line of best fit to all the previously published data for PO in ater and PO in ater/glycerol over a range of molecular eight spanning from 8x1 3 <M <5x1 6 of PO. They observed that all the data agreed ell ith each other and ere ithin the experimental error for variation in the solvent composition. The concentration of PO as varied from.4c* to.5c*. The glycerol and ater mixtures ere chosen such that the shear viscosity of the solution varied beteen 6 mpa.s η 22 mpa.s. The shear viscosity as measured using a cone and plate rheometer (DHR-3, TA instruments) and shoed a constant shear viscosity over the shear rate range that could be probed by the rheometer (1 s -1 < γ < 1 s 1 ). The surface tension for each glycerol and ater mixture as selected based on values available in the literature [27]. In preparing the solutions, ater as first mixed ith the required concentration of the PO and mixed in a magnetic stirrer (CIMARC) for 2 hours and then the required amount of glycerol as added, and again mixed for another 24 hours at 5 rpm at room

9 SAMRAT SUR, JONATHAN ROTHSTIN 9 temperature. A table of the shear viscosity and surface tension of each solution tested is presented in Table S.1 (Supplementary). III. xperimental Setup The dripping onto substrate capillary breakup extensional rheometry (CaBR-DoS) setup is shon in Figure 1. CaBR-DoS requires a high speed camera (Phantom- Vision optics, V-4.2) to capture the filament break-up process, a liquid dispensing system (KD Scientific) to control the volume flo rate, a cylindrical syringe tip, a glass substrate and a high intensity light source. In CaBR-DoS, a liquid bridge is formed beteen the substrate and the nozzle by alloing a drop of liquid to drip from the nozzle onto the glass substrate. The height of the nozzle, H, from the substrate is selected such that an unstable liquid bridge is formed as soon as the drip makes contact and spreads on the substrate. In the experiments presented here, an aspect ratio of H / 3 D nozzle = as used. The high speed camera used for visualization can record at frame rates ell over 1,fps, but at those rates the number of pixels per image as quite small. For most of the measurements presented here a frame rate of 25, fps as used ith a resolution of 192x64pixel 2. The magnification attained using dmund optics long range microscope lens (O- 4.5x zoom) is 5µm/pixel. Hoever, to minimize the effect of the resolution error (± 5µm) on the diameter values reported, e do not report data belo a filament diameter length of D < 1 μm even though the edge detection algorithm e used to capture the diameter filament decay (dgehog, KU Leuven) has sub-pixel resolution. In order to calculate the extensional viscosity, the diameter decay as fit ith a spline and then differentiated as described belo. The diameter of the nozzle as D = 8 μm and the volume flo rate of Q =.2 ml/min nozzle as maintained.

10 SAMRAT SUR, JONATHAN ROTHSTIN 1 a) b) c) Figure 1. a) Schematic diagram of the drip onto substrate capillary breakup extensional rheometry (CaBR-DoS) setup ith all the major components labeled and b) a magnified image of the filament formation beteen the exit of the nozzle and the substrate along ith appropriate dimensions c) sequence of images shoing the development of the filament and subsequent thinning. IV. Methodology The dynamics of the filament thinning of lo viscosity fluids in the CaBR-DoS experiments presented here can be characterized by some of the ell-defined physical models used to characterize the dynamics of drop formation for dilute polymer solutions from dripping nozzles [7] and continuous jets [28]. In this section, e ill define some of the dimensionless numbers used to characterize the filament thinning dynamics along ith the models used to categorize the thinning dynamics into three different regimes inertio-capillary, visco-capillary and elasto-capillary. The three different regimes ill be discussed belo and highlighted in the results for different PO solutions.

11 SAMRAT SUR, JONATHAN ROTHSTIN 11 The driving force of the filament thinning in CaBR-DoS originates from the capillary pressure and depends, therefore, on the surface tension, σ, and the local curvature of the filament, κ = 1/ R1+ 1/ R2, here R 1 and R 2 are the principle radii of curvature on the filament. The capillary thinning is resisted by a combination of fluid viscosity, inertia, and elasticity depending on the fluid physical and rheological properties and the size of the filament. The important dimensionless groups characterizing this necking process are the Ohnesorge number, / ( ρσ R ) 1/2 Oh = h, hich represents a balance of the inertial and viscous forces for a slender filament; the intrinsic Deborah number, λ De =, hich represents the ratio of the ρr σ 3 / characteristic relaxation time of the fluid to the timescale of the flo; and the elasto-capillary number, c = λσ / ηr, hich represents a ratio of the characteristic relaxation time of the fluid and the viscous timescale of the flo. In this case, the intrinsic Deborah number, De = λ / t, is R a ratio of the extensional relaxation time of the fluid to the Rayleigh time, tr = ( ρr / σ). In 3 1/2 these dimensionless groups, η is the shear viscosity of the fluid, R is the radius of the filament, σ is the surface tension, ρ is the density and λ is the relaxation time in extension. Clasen et al. [29] have shon that for Ohnesorge numbers less than Oh <.2, the thinning of the fluid filament ill be dominated by a balance of inertial and capillary forces (inertia-capillary regime), hile for Oh >.2, the thinning of the fluid filament ill be dominated by a balance of viscous and capillary forces (visco-capillary regime). For a lo Ohnesorge number flo, for De > 1, the thinning process ill be dominated by a balance of elastic and capillary forces (elasto-capillary regime), hile, for De < 1, elastic forces ill play no role in the breakup dynamics [7]. Finally, for a large Ohnesorge number flo and an elasto-capillary number less

12 SAMRAT SUR, JONATHAN ROTHSTIN 12 than c < 4.7 the filament thinning ill remain visco-capillary hile for c > 4.7 the flo ill again transition to elasto-capillary thinning [29]. In the inertia-capillary regime, the radius decays ith time folloing a 2/3 poer la dependence [5], 1/3 2/3 Rt () σ t t = t t = R R t 2/3 c.64 3 ( c ).64. ρ R (1) Here, t c is the time at hich the filament breaks up, R is the initial radius and t R is the Rayleigh time hich is a characteristic timescale for the breakup of fluids in this inertia-capillary regime [1]. The prefactor in equation 1 has been reported to be beteen.64 and.8 ith some experimental measurements finding values as large as 1. [5, 1, 2, 29] For the visco-capillary regime, the radius decays linearly ith time as shon by Papageorgiou [3], Rt () σ tc t =.79 ( tc t) =.79. R ηr tv (2) Here, t = η / R σ, is the characteristic viscous timescale for breakup. v For the elasto-capillary regime, the ntov [31] shoed that, for an Oldroyd-B fluid, the radius ill decay exponentially ith time, 1/3 GR = Rt () t exp. R 2σ 3λ (3) Here, G is the elastic modulus of the fluid. Unlike the inertio-capillary regime here a conical filament is formed ith to principle radii or curvature, in the elasto-capillary regime a cylindrical filament ith a single radii of curvature is formed. In CaBR measurements, the extensional rheology of the fluid can be extracted from measurements of the diameter decay ith time. The extension rate of the filament is given by

13 SAMRAT SUR, JONATHAN ROTHSTIN 13 2 dr ε = R t dt ( ) ( t) mid. (4) mid Hence, for an Oldroyd-B fluid, the extension rate is constant, independent of time and only dependent on the fluids relaxation time, ε = 2/3λ. The resulting filament decay has a constant Weissenberg number of Wi = λε =2/3. The Weissenberg number represents the relative importance of elastic to viscous stresses in a flo. This value is larger than the critical Weissenberg number of Wi = 1/2 needed to achieve coil-stretch transition and thus can be used to measure the extensional rheology of these polymer solutions. As seen in quation 3, the relaxation time can be calculated from the slope of the log of the radius or diameter decay ith time. In addition, the extensional viscosity can also be calculated from the measurement of diameter or radius decay ith time, η σ / R () t 2σ mid = =. ε ( t) d Rmid /dt (5) V. Results and Discussion A. CaBR-DoS of PO solutions ith constant c/c* and varying M and η Throughout the results and discussion section, e ill present CaBR-DoS results for a series of PO solutions ith varying solution viscosity, polymer molecular eight and concentration so that trends for each of these parameters can be identified and compared to theory. For most of the experiments presented here, the overarching goal as to extend the experimental characterization capabilities of CaBR-DoS to less and less elastic fluids. As e ill demonstrate, this can be done either by extending the CaBR-DoS experimental capabilities or by finding trends ith solvent viscosity, molecular eight or concentration that can be used to extrapolate from more elastic and measurable solutions to loer elasticity solutions that cannot

14 SAMRAT SUR, JONATHAN ROTHSTIN 14 be characterized even in CaBR-DoS. In all cases, the concentration levels of the PO ere maintained in the dilute region, c/c* < 1, here the results can be compared against theory. In this subsection, e ill focus on solutions ith a fixed values of reduced concentration, c/c*=constant for each polymer molecular eight. Although a ide spectrum of reduced concentrations ere tested using CaBR-DoS, only a small subset of these concentrations ill be systematically presented here. CaBR-DoS results are presented in Figures 2 and 3 for solutions of PO ith molecular eights of M = 1x1 6, 6x1 5, 2x1 5, and 1x1 5 g/mol and solution viscosity of η =6 mpa.s, 1 mpa.s and 22 mpas. Figure 2. Plot of diameter, D, as a function of time, t, for a series of PO solutions in glycerin and ater ith a) molecular eight of M =1x1 6 g/mol and a reduced concentration of

15 SAMRAT SUR, JONATHAN ROTHSTIN 15 c/c*=.2, b) M =6k g/mol and c/c*=.3, c) M =2k g/mol and c/c*=.5, and d) M =1k g/mol and c/c*=.8. In each plot the solution viscosity is varied from ηη =6 mpa.s ( ), to ηη =1 mpa.s ( ) and finally to ηη =22 mpa.s ( ). Solid lines represent the inertia-capillary and elasto-capillary fits to the experimental data from theoretical predictions. The diameter decay as a function of time is plotted in Figure 2 for each of the four different molecular eight PO solutions tested. In each subfigure, the diameter decay is shon as a function of solution viscosity at a fixed reduced concentration. Note that the value of the reduced concentration presented increases ith decreasing polymer molecular eight to insure that a transition to an elasto-capillary thinning could be observed even at the loest solvent viscosity, η = 6 mpa.s, for each molecular eight PO. In all cases, the diameter evolution in time as found to exhibit to distinct regimes: an inertial capillary regime characterized by a decay of the diameter ith a 2/3 poer la ith time folloed by an elasto-capillary regime characterized by an exponential decay of the diameter ith time. Late time deviation from the exponential decay in some data sets shos the effects of the finite extensibility of the polymer solution. A solid line as superimposed over one data set in each figure to demonstrate the quality of the fit to theoretical predictions of each of these regimes. Note that in these figures, the diameter decay begins at roughly D = 25 µm and not at the diameter of the initial filament hich as close to the diameter of the nozzle, D = 8 µm. This as a result of the high optical magnification needed to capture the late time dynamics of the extremely fine filament and to characterize the extensional rheology of the fluid. As can be observed from Figure 2, the transition from the initial poer la decay to the late stage exponential decay as extremely sharp in all cases ith no more than one or to data points spanning less than one millisecond ithin this transition regime. In the section that follos, the sharpness of this transition ill be utilized to significantly enhance the resolution

16 SAMRAT SUR, JONATHAN ROTHSTIN 16 and sensitivity of relaxation time measurements from CaBR-DoS. The transition from an inertia-capillary to an elasto-capillary response is due to the groth of elastic stresses ithin the fluid filament as it as stretched over time. Within the inertia-capillary regime, the extension rate of the fluid filament increases ith time such that ε = 4/ ( 3( t c t) ). At early times during the stretch, the extension rate is too small to deform the polymer ithin the solution because the Weissenberg number is less than one half, Wi < 1/2. As a result, in this regime, the elasticity of the fluid plays no role in the breakup dynamics. Hoever, as the time approaches the cutoff time, t c, the extension rate can gro large enough such that the Weissenberg number becomes larger than Wi > 1/2 and the polymer chain can begin to deform and build up elastic stress hich ill resist the extensional flo and dominate the breakup dynamics. In CaBR-DoS, theoretical predictions suggest that the elasto-capillary breakup should occur ith a constant Weissenberg number of Wi < 2/3. Using this Weissenberg number, one can set an extension rate of ε = 2/3λ as the theoretical criteria for the transition from the inertia-capillary to the elasto-capillary regime. Doing this, a critical radius for the transition to the elasto-capillary regime as found. This is a reasonable first approximation, but, as ill be discussed in detail in later sections, this critical radius over predicts the actual transition radius by a factor of approximately five times because it assumes that buildup of extensional deformation and stress in the polymer is instantaneous hen in reality a finite amount of strain is required to build up sufficient elastic stress to surpass the inertial forces and become the dominant resistance to the capillary forces [32]. That being said, once the flo becomes elasto-capillary, given enough data points, the diameter decay can be used to characterize both the extensional viscosity and relaxation time of the PO solutions as described in the previous section. An important observation that one can make from Figure 2 is that increasing either the molecular eight of the polymer or the solution

17 SAMRAT SUR, JONATHAN ROTHSTIN 17 viscosity leads to an increase in breakup time of the filament due to an increase in the relaxation time of the polymer solution. Note also that the inertia-capillary dynamics, hich manifest before the transition point, appears to be nearly independent of concentration. This variation in the relaxation time is plotted as a function of solutions viscosity and molecular eight in Figure 3. Figure 3. Plot of extensional relaxation time, λλ, as a function of solution shear viscosity, ηη, and molecular eight, M, for a series of PO solutions in glycerin and ater ith a) molecular eight of M =1x1 6 g/mol and a reduced concentration of c/c*=.2 ( ), M =6k g/mol and c/c*=.3 ( ), M =2k g/mol and c/c*=.5 ( ) and M =1k g/mol and c/c*=.8 ( ) and b) solution viscosity of ηη =6 mpa.s and c/c*=.5 ( ), ηη =6 mpa.s and c/c*=.1 ( ) and ηη =1 mpa.s and c/c*=.5 ( ). Solid lines represent the fits to the experimental data. For the case of PO ith a molecular eight of 6 M = 1 1 g/mol, the relaxation time for a solution viscosity ith shear viscosity of η = 6 mpa.s as found to be λ =.4 ms. The relaxation time increased to λ =.72 ms as the shear viscosity of the solution as increased to η = 22 mpa.s. Similarly, for PO ith molecular eight of M = 6k g/mol, the relaxation time for a solution ith a shear viscosity of η = 6 mpa.s as found to be λ =.42 ms and λ =.8 ms for a solution ith a shear viscosity of η = 22 mpa.s. A reduction of close to a

18 SAMRAT SUR, JONATHAN ROTHSTIN 18 factor of four as observed for the relaxation times measured for the loer molecular eights polymer solutions as compared to the higher molecular eights polymer solutions. Similar trends ere observed for the loer molecular eights polymer solutions here for the case of PO ith molecular eight of M = 1k g/mol, the relaxation time for a solution ith shear viscosity of η = 6 mpa.s as found to be λ =.45 ms. The relaxation time increased to λ =.13 ms as the shear viscosity of the solution as increased to η = 22 mpa.s. From theory it has been shon that the relaxation time of a dilute polymer solution should increase linearly ith the viscosity as shon in equation 6, a linear increase ith solvent viscosity as observed for all of our experimental measurements. According to kinetic theory, the longest relaxation time of an isolated polymer coil in dilute solution is proportional to the solvent viscosity as [2, 6] Where U ητ o 1 [ ηη ] sm λ z =. (6) U RT ητ = λ / λ is the universal ratio of the characteristic relaxation time of a dilute polymer η solution system λ η and the longest relaxation time λ. In addition, M is the molecular eight, η s, is the solvent viscosity, [ηη] is the intrinsic viscosity, R is the universal gas constant and T is the absolute temperature. The numerical value of the universal ratio depends on the relaxation spectrum of the specific constitutive model [2]. Given the molecular eight dependence of the intrinsic viscosity for PO in ater and glycerol as shon earlier, the Zimm model can be used to compare the dependence of the relaxation time on the molecular eight of the PO. For a constant solvent viscosity, the relaxation time for the PO should scale ith molecular eight as λ 1.65 z M. In our experiments, a poer la dependence of λ M as observed. This value is close, but does not precisely match the predictions of the Zimm theory as seen in Figure 3b. It 1.4

19 SAMRAT SUR, JONATHAN ROTHSTIN 19 is important to note that a similar discrepancies beteen the Zimm theory and experimental measurements have been observed in the past by Tirtaamadja et al. [7] and others. In their experiments, the relaxation time of a series of different molecular eights of PO in glycerol/ater mixtures as measured by monitoring a droplet formation from a nozzle due to gravity. Although they do not quantify it, a scaling of λ M can be fit to their data. As ff 1.2 ith the measurements here, the measured value of the poer-la coefficient as less than the value predicted by the Zimm model. As e ill discuss in detail later, this discrepancy is likely due to the fact that even though these solutions all have polymer concentrations less than coil overlap concentration, c *, they are not truly dilute in extensional flos until the reduced concentration is belo c/c*<1-4 [2]. Thus the Zimm scaling in quation 6 may not hold in extensional flos until the concentration becomes extremely small, c/c*<1-4. The effect of solution viscosity on the evolution of the apparent extensional viscosity, η, as a function of strain, ε, can be found in Figure 4. An important point to note here is that the strain reported in figure 4, hich is defined as ε = 2ln( R / Rt ( )), depends heavily on the value of the initial radius, R, used to define it. The strain therefore requires a consistent and repeatable choice for R. One possibility is to choose the radius of the syringe tip hich can be made a priori ithout knoledge of the fluid rheology. Here, hoever, e chose to use the radius at hich the dynamics of the filament decay begins to transition from an inertia-capillary to an elasto-capillary flo. This transition point is defined as the radius at hich the Weissenberg number becomes greater than Wi = ½. This is a more physically correct choice for R because, at larger radii, the Weissenberg number is less than Wi < ½ and no appreciable strain is accumulated in the polymer chains. Hoever, for radii smaller than R < R, strain is

20 SAMRAT SUR, JONATHAN ROTHSTIN 2 continuously accumulated in the polymer chains up until the point of filament failure. Using equations 1, 3 and 4, a value for R becomes ( ) 1/3 R = 1.23 λσ / ρ [6, 19, 2]. The only donside to this choice is that extensional relaxation time of the fluid must be measured from the data before the strain can be determined, hoever, because all the data presented here had a measurable relaxation time, this as not an issue. In Figure 4, an increase in the extensional viscosity as observed ith increasing strain for all solutions tested. At large strains, the extensional viscosity in each case approached a steady-state value signifying that the polymer chains had reached their finite extensibility limit. At this point, the resistance to filament thinning is not viscoelastic but rather returns to a viscous response albeit ith a viscosity several orders of magnitude larger than the zero shear viscosity [19]. For the case of PO ith molecular eight of 6 M = 1 1 g/mol, a steady state extensional viscosity of η = 8 Pa.s and η = 3 Pa.s ere observed for a solution ith shear viscosities of η = 22 mpa.s and η = 6 mpa.s respectively. The resulting Trouton ratio, Tr = η / η, as beteen Tr 4 and Tr 5 for all the high molecular eight PO solutions tested. This value is much larger than the Netonian limit of Tr =3 shoing the degree of strain hardening by these high M PO solutions. Additionally, the roughly linear increase the value of the steady state extensional viscosity ith increasing solution viscosity as observed for each of the different molecular eight POs tested. The result as a collapse of the data hen the Trouton ratio as plotted as a function of Hencky strain for a given molecular eight obtained at fixed concentration but variable shear viscosity. This data is presented in supplementary material Figure S.2. These trends ith solvent viscosity conform to the predictions of the FN-P model as long as the value of the steady state Trouton ratios are significantly larger than Tr =3.

21 SAMRAT SUR, JONATHAN ROTHSTIN 21 Note that in Figure 4, both the reduced concentration and the molecular eight ere varied beteen subfigures. Here, the reduced concentration of the loest molecular eight samples as purposefully increased in order to obtain coherent filaments from hich clean extensional viscosity measurements could be extracted. In fact, unlike the PO solutions at c/c*=.2, measurements of extensional rheology for the M = g/mol M = 1k g/mol at c/c*=.2 did not result in the formation of a viscoelastic filament, but broke up ithout transitioning from an inertia-capillary decay. Thus a natural loer limit in the measurable extensional viscosity of about η =.1 Pa.s as obtained, although e ill sho in the sections that follo that this is not truly a loer limit on extensional viscosity, but a loer limit on extensional relaxation time as the transition radius described above gets smaller and smaller ith

22 SAMRAT SUR, JONATHAN ROTHSTIN 22 decreasing extensional relaxation time and eventual becomes so small that it cannot be resolved optically. Figure 4. Plot of extensional viscosity, ηη, as a function of strain, ε, for a series of PO solutions in glycerin and ater ith a) molecular eight of M =1x1 6 g/mol and a reduced concentration of c/c*=.2, b) M =6k g/mol and c/c*=.3, c) M =2k g/mol and c/c*=.5 and d) M =1k g/mol and c/c*=.8 at solution viscosity of ηη =6 mpa.s ( ), ηη =1 mpa.s ( ) and ηη =22 mpa.s ( )

23 SAMRAT SUR, JONATHAN ROTHSTIN 23 B. CaBR-DoS of PO solutions ith a fixed solution viscosity η = 6 mpa.s and varying c/c* and M Similar to the discussions in the last section, CaBR-DoS results for a series of PO solutions ith varying polymer concentration and molecular eight are presented in this section so that trends for each of these parameters can be identified and compared to theory. Once a relation has been established, the relationship can be used to extrapolate the data to loer concentration or loer M solution hich are not measurable using CaBR-DoS technique. In all cases, the concentration levels of the PO ere maintained in the dilute region, c/c* < 1, so that the results could be compared against dilute theory. In this subsection, results are presented for values of reduced concentration varying from, c/c*=.4 to.5, but ith the shear viscosity fixed at η = 6 mpa.s. A small subset of the CaBR-DoS results are presented in Figures 5 and 6 for solutions of PO ith each sub figure corresponding to a fixed molecular eights of M = 1x1 6, 6x1 5, 2x1 5, and 1x1 5 g/mol.

24 SAMRAT SUR, JONATHAN ROTHSTIN 24 Figure 5. Plot of diameter, D, as a function of time, t, for a series of PO solutions in glycerin and ater at a fixed solution viscosity of ηη =6 mpa.s and molecular eight varying from a) M =1x1 6 g/mol, b) M =6k g/mol, c) M =2k g/mol and d) M =1k g/mol. In each subfigure, reduced concentration is varied from c/c*=.4 ( ), c/c*=.5 ( ), c/c*=.2 ( ), c/c*=.3 ( ), c/c*=.5 ( ), c/c*=.8 ( ), c/c*=.1 ( ) and c/c*=.5 ( ). Solid lines represent the inertia-capillary and elasto-capillary fits to the experimental data from theoretical predictions. The diameter decay as a function of time is plotted in Figure 5 for each of the four different molecular eight PO solutions tested. In each subfigure, the diameter decay is shon as a function of reduced concentration at a fixed solution viscosity. For a given molecular eight, an increase in the reduced concentration as found to result in an increase in both the relaxation time and, subsequently, the time required for the filament to breakup. The increased

25 SAMRAT SUR, JONATHAN ROTHSTIN 25 relaxation time can be seen qualitatively as a decrease in the slope of the linear region of the data in this semi-log plot. The breakup times of the filaments ere greatly affected by both the concentration and molecular eight. For the highest molecular eight PO tested, the breakup times ere an order of magnitude larger than the loest molecular eights PO tested at the same reduced concentration. The sensitivity of the break up time to change in concentration as not as strong as to molecular eight. For instance, a tenfold increase in the breakup time could be induced by increasing the molecular eight by a factor of roughly six at a given reduced concentration. Hoever, to achieve the same tenfold increase in the break up time at a fixed molecular eight required an increase in the reduced concentration by a factor of one hundred.

26 SAMRAT SUR, JONATHAN ROTHSTIN 26 Figure 6. Plot of extensional relaxation time, λλ, as a function of reduced concentration, c/c*, for a series of PO solutions in glycerin and ater at fixed shear viscosity of ηη =6 mpa.s ith a molecular eight of M =1x1 6 g/mol ( ), M =6k g/mol ( ), M =2k g/mol ( ) and d) M =1k g/mol ( ). Solid lines represent a poer la fit to the data having the form λλ ~(c/c*).7. The hollo symbols shos the Zimm time at ηη =6 mpa.s for the molecular eight of ( ) M =1x1 6 g/mol, ( ) M =6k g/mol, ( )M =2k g/mol and ( ) M =1k g/mol. In Figure 6, the relaxation time, λ, is presented as a function of reduced polymer concentration, c/c*, for a series of PO solutions in glycerin and ater at a fixed shear viscosity of η =6 mpa.s. One can observe that, ith an increase in the reduced concentration, an increase in the relaxation time as observed. For the case of PO ith molecular eight of 6 M = 1 1 g/mol, the relaxation time increased from λ =.16 ms to.4 ms as the reduced concentration as increased from c/ c * =.4 to.5. These relaxation times are similar to those measured by Dinic et al. [1, 11] using the CaBR-DoS technique for a similar 6 M = 1 1 g/mol aqueous PO solution. Note that belo a concentration of c/c*=.4, a

27 SAMRAT SUR, JONATHAN ROTHSTIN 27 viscoelastic filament could not be observed given the limitations in the temporal and spatial resolution of our high speed camera. Similar trends observed for each of the different molecular eight PO solutions tested. With increasing molecular eight, the relaxation time measured for a given concentration as found to decrease. Additionally, the minimum concentration that could be characterized using CaBR-DoS as found to increase quite significantly ith decreasing molecular eight as the cut-off appears to be limited by a minimum relaxation time of just belo λmin 1μs that can be confidently characterized using CaBR-DoS. For the case of PO ith molecular eight of M = 1k g/mol, the relaxation time as found to increase from λ =.6 ms to.22 ms as the reduced concentration as increased from c/ c * =.8 to.5. For this molecular eight, the minimum concentration that could be characterized in CaBR-DoS as tenty times larger than that of the highest molecular eight sample. The increase in the relaxation time for all the different molecular eights PO solutions tested as found to have poer la dependence on the reduced concentration such that ( / *).7 λ c c. Similar observation ere made by Dinic et al. [1] here they observed a poer la dependence of relaxation time on reduced concentration as.65 λ ( c/ c*) for their set of measurements on a series of aqueous PO solution of a single molecular eight of, 6 M = 1 1 g/mol [1]. Tirtaamadja et al. [7] also observed a poer la dependence of the relaxation time on reduced concentration as ( c/ c*).65 λ for their set of measurements on a series of PO in ater/glycerol solution. This observation is counter-intuitive because according to equations 6 and 7, the relaxation time of isolated coils ithin dilute solution should be independent of the concentration. Clasen et al. [2] also noted a dependence of the longest relaxation time in extension on the reduced concentration, c/c*. They rationalized this by noting

28 SAMRAT SUR, JONATHAN ROTHSTIN 28 that the polymer viscosity can have a concentration dependence depending on the model used. For instance, using the Martin equation, Clasen et al. [2] ere able to fit the relaxation time data to an exponential dependence on the reduced concentration predicted by the Martin equation. This model also shoed that the concentration dependence should disappear as one moves farther and farther from the critical overlap concentration into the ultra-dilute regime here the relaxation time measurement as found to approach the Zimm relaxation time. Compared to the results of Clasen et al. [2], the range of concentrations for our orking fluids in Figure 6 may be too narro to observe a true dilute response as the data in Figure 6 do not appear to reach an asymptotic limit. From Clasen et al. [2], it is clear that only for reduced concentrations less than c/ c* 1 4 < can a truly dilute value of the extensional relaxation be recovered. Thus, even though under quiescent conditions all of the PO solutions studied in the present ork ere ell ithin the dilute regime, c/ c*<1, hen deformed by an extensional flo the fluids all appear to exhibit semi-dilute solution behavior. This semi-dilute behavior arises from chain-chain interactions resulting from the increase in pervaded volume of the stretched chain. Thus, even though c/c*<1, excluded volume interaction beteen neighboring chains can become important. Using semi-dilute theory [33], Dinic et al. [1] shoed through an alternate theoretical approach to Clasen et al. [2] that a poer la dependence similar to that observed experimentally here, λ ( c/ c*).7, could be rationalized. An important final observation from Figure 6 should be made. From Figure 6 it is clear that the measured relaxation time do not asymptote to the Zimm relaxation time as the concentration as reduced. For the lo molecular eight cases, M =1k g/mol and 2k g/mol, all the values of the measured relaxation time ere larger than the Zimm relaxation time. Hoever, for both the higher molecular eight POs cases, M =1x1 6 g/mol and 6k g/mol,

29 SAMRAT SUR, JONATHAN ROTHSTIN 29 the measured relaxation times ere found to decrease ith decreasing concentration and, at the loest concentrations tested, ere measured to be smaller than the Zimm relaxation time. For the case of M =1x1 6 g/mol and 6k g/mol, the Zimm relaxation time is λ z =.6 ms and λ z =.24 ms respectively for reduced concentrations less than c/c*<.1. The value of the Zimm relaxation time for all the fluids tested here are presented in Table 1. Note that there are a number of differences beteen these CaBR experiments and the conditions under hich the Zimm model predictions of relaxation time are made. First, the relaxation time predicted by the Zimm model is the shear relaxation time and these measurements are of the extensional relaxation time. Second, and perhaps more importantly, the Zimm model predictions assume a small deformation of the polymer chains, hile the extensional flo studied here can result in a nearly fully extended polymer chain. Thus, it is not unreasonable to expect that differences ould be observed beteen the Zimm relaxation time and the extensional relaxation time measured through CaBR. It is important to note here that these relaxation time measurements do not appear to be in error. ach experiment ere repeated multiple times and the exponential fits to the data used to determine the relaxation times ere fit to beteen 1-2 pts of data over nearly a decade of time and diameter variation. Clasen et al. [2] observed similar trend and argued that for the cases here λ < λzthat the data as interpreted incorrectly. They fit their data ith the assumption that the finite extensibility limit had been reached during the inertia-capillary decay and thus the relaxation time measurement ere in error. Unfortunately, this argument does not hold for our data for several reasons. As seen in Figure 7, the filaments in our experiments, even as the relaxation time as found to drop belo that predicted by the Zimm model, bore the expected cylindrical shape of an elasto-capillary thinning.

30 SAMRAT SUR, JONATHAN ROTHSTIN 3 Figure 7. A sequence of images shoing the formation of a slender filament and subsequent thinning for a PO of M =1x1 6 g/mol at c/c*=.2 and shear viscosity of ηη =6 mpa.s. In addition, folloing the experimental decay in the diameter as in Figures 2 and 5, a deviation as observed at late time hich is clearly the onset of the finite extensibility limit ell after the relaxation time has been fit to the data. Finally, it should be noted that all of the concentrations used in our experiments ere an order of magnitude higher than the theoretical minimum concentration, c lo, above hich theory predicts that a true elasto-capillary balance should be observable [19]. We are thus confident in the measurements of relaxation time and note that similar values of relaxation times in extensional flos have been observed in the past both in Clasen et al. [2] and Bazilevskii et al. [34]. The reason for these observations is still not fully understood, but requires further theoretical development along ith and further experimental testing and development of this and other techniques to probe smaller and smaller concentrations and extensional relaxation times. Molecular Weight M =1x1 6 g/mol Solution Zimm Viscosity time, [mpa.s] λ [ms] z η Molecular Weight M =6k g/mol Solution Zimm Viscosity time, [mpa-s] λ [ms] z η Molecular Weight M =2k g/mol Solution Zimm, Viscosity time, [mpa-s] λ [ms] z η Molecular Weight M =1k g/mol Solution Zimm Viscosity time, [mpa-s] λ [ms] z η Table 1. Zimm relaxation times for the PO solutions.

31 SAMRAT SUR, JONATHAN ROTHSTIN 31 Figure 8. Plot of extensional viscosity, ηη, as a function of Hencky strain, ε, for a series of PO solutions in glycerin and ater at fixed shear viscosity of ηη =6 mpa.s ith a molecular eight of a) M =1x1 6 g/mol, b) M =6k g/mol, c) M =2k g/mol and d) M =1k g/mol. The reduced concentration is varied from c/c*=.4 ( ), c/c*=.5 ( ), c/c*=.2 ( ), c/c*=.3 ( ), c/c*=.5 ( ), c/c *=.8 ( ), c/c*=.1 ( ) and finally to c/c*=.5 ( ). In Figure 8, the extensional viscosity, η, is plotted as a function of Hencky strain, ε, for a series of PO solutions in glycerin and ater at fixed shear viscosity of η =6 mpa.s and varying reduced concentration, c/c* and molecular eights, M. In all cases, a steady-state extensional viscosity as reached at large values of Hencky strain. For the case of PO ith molecular eight of M =1x1 6 g/mol, the steady-state extensional viscosity observed varied

32 SAMRAT SUR, JONATHAN ROTHSTIN 32 from η =1 Pa.s to 4 Pa.s as the reduced concentration as varied from c/c*=.4 to.5. Trouton ratios in the range of 2 4 Tr = 1 1 are observed for the highest molecular eight PO. Similar variations ere observed for all the molecular eight tested. For the case of loest molecular eight PO tested, M = 1k g/mol, the steady state extensional viscosity observed varied from η =.7 Pa.s to 3 Pa.s as the reduced concentration as varied from c/c*=.8 to.5. This corresponds to Trouton ratios of the order of roughly 2 Tr 1. As observed from the diameter decay in Figure 5, at later times a deviation from the viscoelastic exponential fit as observed. As seen in Figure 8, this deviation as the result of the polymer chain approaching finite extensibility limit here the resistance to thinning returns to a viscous response albeit at a much higher viscosity. C. A method for extending CaBR-DoS to make micro-seconds relaxation time measurements In this section, a ne approach to extend CaBR-DoS beyond the inherent limitations of the technique in order to make relaxation time measurements on the order of micro-seconds by taking advantage of the sharp transition beteen the inertia-capillary dominated thinning at early time and the elasto-capillary dominated thinning at later times is presented. As shon in the previous sections, CaBR-DoS can measure relaxation times greater than λ =.1 ms ithout running into any temporal or spatial resolution limits of the chosen high speed camera. This is true independent of the solutions shear viscosity. Hoever, the relaxation times ere driven don even further by decreasing the shear viscosity of the solution or decreasing molecular eight of the polymers. The resulting rheological measurements ere limited by the temporal resolution hich limited the number of data points that could be captured in the elasto-capillary

33 SAMRAT SUR, JONATHAN ROTHSTIN 33 region and the spatial resolution hich limited the minimum size filament that could be accurately resolved. To directly measure the relaxation time, an exponential decay must be fit to the diameter decay data in the elasto-capillary thinning regime. In order for this fit to be accurate, a minimum of 5-1 data points is required. Here e sho that a full exponential fit is not necessary to measure the relaxation time. In fact, in order to make relaxation time measurements for extremely lo viscosity-lo elasticity solutions only a single data point in the elastocapillary region needs to be captured, thus increasing the resolution of CaBR-DoS by a full order of magnitude. This extension of CaBR-DoS takes advantage of the sharp transition beteen the inertia-capillary and the elasto-capillary thinning to define an experimentally observed transition radius, R *. By calibrating the experimentally observed transition radius against the transition radius predicted from theory, R * theory, e demonstrated that an empirical correlation beteen the experimentally observed transition radius and the relaxation time can be formed. This finding significantly enhanced the resolution and sensitivity of the relaxation time measurements obtainable through CaBR-DoS. Thinning dynamics of lo viscosity dilute polymer solutions ith Ohnesorge number, Oh<.2 are knon to be dominated by inertial decay [29]. As the diameter decays, the extension rates can eventually become large enough that the polymer chains to under-go a coil-stretch transition. This is knon to occur for Weissenberg numbers greater than Wi >1/2. At these Weissenberg numbers, elastic stresses ill gro ith increasing accumulated strain. As they gro, the elastic stresses ill become increasingly important to the flo and ill eventually dominate the breakup dynamics of the liquid bridge [29]. In a CaBR experiments of an Oldroyd-B fluid, the thinning dynamics of the liquid bridge are knon to occur at a constant Weissenberg number of Wi = 2/3. Thus, a reasonable approach taken by a number of previous

34 SAMRAT SUR, JONATHAN ROTHSTIN 34 groups [6, 19, 2] to determine the transition point is to assume that the transition from inertiocapillary to elasto-capillary flo occurs hen the extension rate induced by the intertio-capillary flo gros to ε = 2/3λ. Using this value of the extension rate in combination ith quations 1 and 4, a relation for the theoretical transition radius can be obtained, R * theory 1/3 2 λ σ = 1.1. ρ (7) If equation 7 is valid, the extension relaxation time can be estimated by measuring the initial radius of the cylindrical filament that is characteristic of the elasto-capillary flo. An example of a radial decay shoing the transition beteen inertia-capillary and elasto-capillary thinning is shon in Figure 9. From the data in Figure 9, one can independently determine the experimental transition radius, * R, from the intersection of the inertia-capillary and elasto-capillary region of the radial decay and the relaxation time from the late stage exponential decay of the data. Figure 9. Radial decay, R, as a function of time, t, for the PO of molecular eight M =1x1 6 g/mol in ater at a reduced concentration of c/c*=.5 and shear viscosity of ηη =1 mpa.s ( ) shoing the transition from an inertia dominated thinning to an elasticity dominated thinning. Solid lines represents the inertial ( ) and exponential fits (- -) to the experimental radius. An inset figure is provided ith the magnified image to demonstrate the sharpness of the transition point.

35 SAMRAT SUR, JONATHAN ROTHSTIN 35 By comparing the theoretical transition radius obtained from equation 7 and the experimentally determined transition radius, it can be seen from Figure 9 that the theoretical transition radius predictions as significantly larger than the experimentally measured value. From the data presented in Figure 9, for a PO solution ith 6 M = 1 1 g/mol, 1 mpa.s η = and c/c*=.5, the transition radii ere found to be R * theory =.12mm and * R =.22mm. The magnitude of the theoretical over prediction in the transition radius as found to be consistent across molecular eight, shear viscosity and reduced concentration. If one takes the ratio of the experimental and theoretical transition radius, a correction factor can be found such that β = R * * / Rtheory =.18. Similar values of the correction factor, β, ere observed for all the PO solutions tested. The average correction factor for all the PO solution data as found to be β =.18 ±.1. The small size of the uncertainty in the correction factor data is remarkable given the data spans several orders of magnitude in M, and / * η c c. Additionally, it gives us confidence that an experimentally observed transition radius can be used to predict the relaxation time of lo viscosity and lo concentration PO solution by capturing just a single data point. Reriting equation 7 in terms of the experimentally observed transition radius, R *, and theoretically and the average correction factor β =.18 as *3 1/2 R ρ λ = 13.1 σ (8)

36 SAMRAT SUR, JONATHAN ROTHSTIN 36 t=.1ms t=.14ms t=.18ms t=.22ms Figure 1. Filament thinning dynamics of an aqueous solution of PO ith a molecular eight of M =2k g/mol at a reduced concentration of c/c*=.5 and a shear viscosity of ηη =1 mpa.s. Images ere captured at 5, frames per seconds. An example shoing the poer of this technique is shon in Figure 1 for CaBR-DoS breakup of a PO ith molecular eight of M =2k g/mol, and reduced concentration of c/c*=.5 in ater ith ηη =1 mpa.s. Note that in the images in Figure 1, the breakup of the fluid is in the inertia-capillary regime up until the last to frames. At time t =.18 ms, hints of a cylindrical filament can be seen and at t =.22 ms a clear cylindrical filament is produced. From those to images and the last t = 4 μs of the data, the relaxation time of the fluid can be calculated from equation 8. For this fluid, the extensional relaxation time as found to be λ = 22 μs. A similarly late stage cylindrical filament as observed for M =1k g/mol PO in ater at c/c*=.8. For that solution, the extensional relaxation time calculated using equation 8 as found to be λ = 2 μs. To demonstrate the viability of our technique for that the relaxation time data for the 1k g/mol PO solution as superimposed over the relaxation time data in Figure 3a obtained through the standard exponential fit to the time diameter decay data. As can be seen in Figure 3a, the relaxation time obtained ith our experimental technique fits quite nicely in the trend line extrapolated from the higher relaxation time data. Given the maximum magnification of our high speed camera setup, the minimum resolvable radius can be used to