Extensional Relaxation Times of Dilute, Aqueous Polymer Solutions

We show that visualization and analysis of capillary-driven thinning and pinch-off dynamics of the columnar neck in an asymmetric liquid bridge created by dripping-onto-substrate can be used for characterizing the extensional rheology of complex fluids. Using a particular example of dilute, aqueous PEO solutions, we show the measurement of both the extensional relaxation time and extensional viscosity of weakly elastic, polymeric complex fluids with low shear viscosity η < 20 mPa·s and relatively short relaxation time, λ < 1 ms. Characterization of elastic effects and extensional relaxation times in these dilute solutions is beyond the range measurable in the standard geometries used in commercially available shear and extensional rheometers (including CaBER, capillary breakup extensional rheometer). As the radius of the neck that connects a sessile drop to a nozzle is detected optically, and the extensional response for viscoelastic fluids is characterized by analyzing their elastocapillary self-thinning, we refer to this technique as optically-detected elastocapillary selfthinning dripping-onto-substrate (ODES-DOS) extensional rheometry. A of a dilute amount, even 1−400 ppm (parts per million), of a high molecular weight polymer like poly(ethylene oxide) (PEO, Mw > 10 6 g/mol) to a solvent like water is observed to significantly change the fluid response to extensional or stretching flows. Examples include enhanced pressure drop in porous media flows, suppression of rebound in drop impact studies, a discernible birefringence around a stagnation point in cross-slot flows, delayed breakup in dripping, spraying or jetting, and possibly turbulent drag reduction. The influence of polymers is even more remarkable for dilute, aqueous solutions as the measured shear viscosity η(γ)̇ appears to be Newtonian, and elastic modulus, relaxation time, and the first normal stress difference are not measured, or manifested, in steady shear or oscillatory shear tests carried out on the state-of-the-art torsional rheometers. Macromolecular solutions typically exhibit a large and measurable resistance called extensional viscosity, ηE, to streamwise velocity gradients characteristic of extensional flows and undergo stress relaxation with a characteristic extensional relaxation time λE. However, for dilute, aqueous solutions, quantitative measurements of both ηE and λE remain beyond the capability of commercially available devices like CaBER (capillary breakup extensional rheometer). A countable few measurements of extensional relaxation time in dilute aqueous solutions presented in the recent literature require bespoke instrumentation not available or easily replicable in most laboratories. The aim of the present study is 3-fold: to describe an extensional rheometry protocol that can be recreated virtually in any laboratory (quite inexpensively for high viscosity fluids), to characterize the extensional viscosity and extensional relaxation time for dilute, aqueous polymer solutions, and to provide a scaling argument that captures the concentration-dependent variation of extensional relaxation time. Free surface flows of polymer solutionsunderlying drop formation and liquid transfer in jetting and printing, dripping, and microfluidic drop/particle production9 involve the formation of columnar necks that spontaneously undergo capillary-driven instability, thinning, and pinch-off. The progressive self-thinning of the neck is often characterized by self-similar profiles and scaling laws that depend on the relative magnitude of capillary, inertial, and viscous stresses for simple (Newtonian and inelastic) fluids. Macromolecular stretching and orientation in response to extensional flow field within the thinning columnar necks (recently visualized using DNA solutions) leads to extra viscoelastic stresses that change the thinning and pinch-off dynamics. Pioneering studies by Schümmer and Tebel as well as Entov, Yarin, and collaborators developed the idea of characterizing capillary-driven thinning for evaluating the role of added polymers in terms of an extensional viscosity and an extensional relaxation time. The extensional relaxation time, λE, is distinct and often larger in magnitude than the value of relaxation time obtained in oscillatory shear or stress relaxation experiments. Such extensional rheometry measurements are realized in several prototypical geometries: (I) Dripping, where the pinch-off results from an interplay of gravitational drainage and capillarity. (II) Jetting, where convective Received: June 13, 2015 Accepted: July 10, 2015 Letter pubs.acs.org/macroletters © XXXX American Chemical Society 804 DOI: 10.1021/acsmacrolett.5b00393 ACS Macro Lett. 2015, 4, 804−808 instability develops on a fluid jet and the Rayleigh Ohnesorge jetting extensional rheometer (ROJER) measurements and analysis are based on understanding of the nonlinear fluid dynamics underlying the jetting process. (III) Selfthinning of a stretched liquid bridge formed by applying a discrete step-strain to a drop between two parallel plates, and utilized in CaBER. In this letter, we show that visualizing and analyzing thinning of a stretched liquid bridge formed by dripping-onto-substrate (see Figure 1) can be used for extensional rheometry characterization. As an extreme application of the drippingonto-substrate protocol and to outline its efficacy, we carry out measurements of extensional relaxation time and extensional viscosity for low viscosity (η < 20 mPa·s), low elasticity fluids (λ < 1 ms). Such measurements are inaccessible in a standard CaBER as the pinch-off is completed even before the typical commercial instruments can stretch the liquid bridge. Campo-Deano and Clasen recently modified the CaBER protocol to create the slow retraction method (SRM) to access shorter relaxation times. But the initial step-strain required in both CaBER and SRM measurements can disrupt the fluid microstructure, influencing the observed extensional rheology response for highly structured fluids. Jetting always requires higher flow rates than dripping, so the effect of preshear induced within the nozzle is less important in dripping-ontosubstrate. The presence of substrate also averts issues associated with dripping: the released drop no longer hangs from the thinning neck, and changing drop volume/weight has no effect on dynamics. Furthermore, the fixed Eulerian location of the thinning neck facilitates visualization in contrast to dripping (higher viscosity and more viscoelastic fluids form longer necks). Figure 1. Introducing optically-detected elastocapillary self-thinning dripping-onto-substrate [ODES-DOS] extensional rheometry. Self-thinning dynamics of the necked region in a stretched liquid bridge formed by dripping-onto-substrate are captured using an imaging system (a light source, a diffuser, and a camera). Extensional viscosity and extensional relaxation time can be obtained from the analysis of neck-thinning dynamics. Figure 2. Image sequences and radius evolution plots obtained using the ODES-DOS method. Images, 3 ms apart, show capillary-driven thinning and breakup of a stretched liquid bridge for aqueous PEO solutions, Mw = 10 6 g/mol: (a) c = 0 wt. % (pure water), (b) c = 0.02 wt. %, (c) c = 0.1 wt. %, and (d) c = 0.2 wt. % (c/c* = 0, 0.01, 0.45, and 1.1), respectively. (e) Radius evolution plots obtained using the ODES-DOS method are shown with the time axis shifted such that the transition point tc overlaps. Radius evolution (blue, squares) for an aqueous c = 0.1 wt % PEO solution shows two distinct regimes: inertio-capillary regime, fit by eq 1 (blue dashed line) before tc and elasto-capillary regime described by eq 2 (blue dotted line) after tc. ACS Macro Letters Letter DOI: 10.1021/acsmacrolett.5b00393 ACS Macro Lett. 2015, 4, 804−808 805 For dripping-onto-substrate experiments, a discrete fluid volume delivered at a relatively low flow rate, Q, is deposited onto a glass substrate placed at a fixed distance H below the nozzle. An unstable, stretched liquid bridge, bounded by the nozzle and a sessile drop on the substrate (see Figure 1), is formed, and its necked region undergoes capillary-driven thinning. Unlike CaBER that relies on a laser-based diameter measurement, neck shape and diameter are both extracted from movies captured at a rate of 8000−25 000 frames per second (fps). Analysis is carried out by using specially written codes in ImageJ and MATLAB. The imaging system consists of a light source, a diffuser, and a Photron Fastcam SA3 high-speed camera equipped with a Nikkor 3.1X zoom lens (18−55 mm) and an additional super macrolens. Each measurement is repeated at least five times for the chosen nozzle (diameter: inner, Di = 0.838 mm and outer, Do = 1.270 mm), aspect ratio (H/Di = 3), and dispensing rate, and a good reproducibility is observed (see Supporting Information). As the neck radius is optically detected, and the elongational viscosity as well as relaxation time are deduced by an analysis of the elasto-capillary self-thinning regime (described later), we christened this method as optically-detected elastocapillary self-thinning dripping-onto-substrate (ODES-DOS) extensional rheometry. Aqueous solutions of PEO (Sigma-Aldrich, average molecular weight Mw = 1.0 × 10 6 g/mol) were prepared by successively diluting a stock solution (0.4% PEO in water), prepared by slowly adding polymer powder to deionized water. The solutions are placed on a roller for at least 5 days to achieve homogeneous mixing. High deformation rate mixers and flows were avoided as these are known to cause chain scission. In shear rheology, solutions are considered dilute if c/c* < 1, and for such solutions, the concentration-dependent solution shear viscosity η = ηs(1 + c/c*) is comparable to solvent viscosity, ηs. Here the critical overlap concentration (c* = 0.17 wt %), i.e., the concentration value at which unperturbed polymer coils start to overlap, was computed using the formula c*[η] ≈ 1, together with Mark−Houwink−Sakurada equation: [η] = KMw a . Intrinsic viscosity, [η], depends on Mw, and for PEO, the values of coefficient K = 1.25 × 10−2 mL/g and the exponent a = 0.78 are listed in the polymer handbook data. ODES-DOS extensional rheometry characterization was carried out for PEO solutions, with concentration c = 0.005−0.3 wt % spanning a range above and below c*. Representative snapshots of the stretched liquid bridge formed by dripping-onto-substrate and the necked region undergoing thinning are shown in Figure 2. For pure water, the progressive thinning of neck results in the formation of a coneshaped morphology (Figure 2a), displaying a characteristic feature of the potential flow solution obtained for inviscid fluids. A distinct slender liquid filament appears for polymer solutions (see Figure 2c−d), and often beads-on-astring structures can be observed in the last stages of thinning (e.g., see Movie, included as Supporting Information). Clasen et al. and Bhat et al. showed that the region where the viscoelastic filament thread connects with the drop develops a sharp corner that evolves self-similarly. Qualitatively similar corner profiles are observed for polymer solutions (see Figure 2c−d). The image sequences obtained by the DOS setup were analyzed to track thinning of the neck radius over time. Two distinct regimes can be observed for the PEO solutions: an initial power law regime, followed by a slower exponential decay. The initial neck-thinning dynamics, dominated by inertial and capillary stresses only, can be described quite well by the following expression for inertio-capillary (IC) scaling