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APPLIED PHYSICS LETTERS 104, 024103 (2014) The minimum flow rate scaling of Taylor cone-jets issued from a nozzle William J. Scheideler and Chuan-Hua Chena) Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708, USA (Received 13 July 2013; accepted 30 December 2013; published online 15 January 2014) A minimum flow rate is required to maintain steady Taylor cone-jets issued from an electrified nozzle. This letter reports that the well-known scaling law proportional to the charge relaxation time and independent of the nozzle diameter is only applicable to lowly viscous systems. For highly viscous systems with rapid viscous diffusion, the cone-jet transitional region spans a length scale comparable to the nozzle diameter; the minimum flow rate appears to be proportional to the capillary-viscous velocity and the cross-sectional area of the nozzle, but independent of the liquid C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862263] conductivity. V The electrohydrodynamic Taylor cone-jet1–4 is fundamental to a variety of applications including electrostatic spraying, spinning, and printing. In the most common setup, a working fluid is pumped at a flow rate Q through a round nozzle, which is electrified at a voltage V with respect to a counter electrode plate using air as the surrounding medium. Steady cone-jets are known to exist only for a limited stability window in the voltage-flow rate (V – Q) operating diagram, where the applied voltage is around the threshold giving rise to the Taylor cone2 and the imposed flow rate is above a minimum threshold necessary to sustain a steady jet.3 Although the applied voltage (field) can be rationalized by balancing the Maxwell and capillary stresses as in the case of the hydrostatic Taylor cone,2,5–7 the minimum flow rate required to sustain a steady jet is much less well-understood.4,8–13 In electrospraying systems with the common setup of a working fluid pumped through an electrified nozzle, the minimum flow rate (Qm) is known to scale as4,14–16 Qm ec ; rq (1) where e, c, r, and q are, respectively, the permittivity, surface tension, electrical conductivity, and density of the working fluid. This scaling law works well for the electrospraying of liquids in the “high-conductivity” limit ðrⲏ1 lS=cmÞ,4 and can be justified on dimensional grounds as long as the liquid viscosity (l) drops out of the picture. Note that the length scales of the overall system such as the nozzle diameter do not enter the scaling law in Eq. (1), because the jet diameter in the high-conductivity limit is typically orders of magnitude smaller than the nozzle diameter for high-conductivity liquids. The requirement of a minimum flow rate is believed to give rise to pulsating cone-jets when the supplied flow rate is smaller than this minimum value.17–20 In this sense, the minimum flow rate scaling is relevant to the transient (quasisteady) cone-jets issued from solid nozzles18,19,21–27 as well as liquid drops17,28–31 and films/pools.32–34 The absence of liquid viscosity in the minimum flow rate scaling Eq. (1) lacks a rigorous justification, which poses a severe limitation considering that highly viscous cone-jet a) Electronic mail: [email protected] 0003-6951/2014/104(2)/024103/4/$30.00 systems are critical for many practical situations. In nanoelectrospray, nozzles of micrometric or sub-micron diameter are routinely used;23,35–37 In electrospinning, viscoelastic polymeric fluids are typically employed;7,9,38,39 In electrohydrodynamic printing, fine nozzles are often used, sometimes in combination with viscous working fluids.40–44 Despite the practical relevance, systematic studies of the effects of viscosity on Newtonian cone-jets are scarce, and the few available studies have not focused on the minimum flow rate.34,45,46 Here, we study the role of viscosity (l) in determining the minimum flow rate (Qm) for Taylor cone-jets issued from an electrified nozzle. We show that the conventional Qm scaling Eq. (1) breaks down for cone-jets of highly viscous working fluids, and that the Qm at high viscosity can be a strong function of the nozzle diameter (D) instead of the liquid conductivity (r). Among the intrinsic properties of the working fluids (l, q, c, e, and r), only viscosity and conductivity can be varied by a few orders of magnitude without significantly affecting other fluid properties. These properties and the nozzle diameter were systematically varied and the corresponding minimum flow rates measured. (Note that permittivity can be changed by about an order of magnitude within the polarliquid limit,14,47,48 but is not systematically varied here.) As shown in Table I, the working fluids were ethanol, ethylene glycol or glycerol. These Newtonian working fluids spanned three orders of magnitude in terms of viscosity. The liquids were doped with potassium chloride to a desired range of conductivities and confirmed with a conductivity meter (Oakton 510). The conductivity was kept above the empirical threshold of 1 lS/cm for high-conductivity fluids.4 To prevent subconscious bias, the exact conductivity in most experiments was measured immediately after the measurements of the minimum flow rate (see Table S149). Note that all three working fluids are commonly used for mechanistic studies of cone-jets.50 To test the scaling laws, we focused on Taylor cone-jets issued from a nozzle electrified with respect to a conducting plate, where the working fluid was supplied at a constant flow rate by a syringe pump (Legato 180). In addition to its widespread adoption, the nozzle-to-plate configuration was the easiest for accurate Qm measurements. Cone-jets were produced on glass nozzles with outer diameters (D) ranging 104, 024103-1 C 2014 AIP Publishing LLC V 024103-2 W. J. Scheideler and C.-H. Chen Appl. Phys. Lett. 104, 024103 (2014) TABLE I. Material properties of the working fluids based on literature values at 20 C. The concentration of the glycerol-water mixtures is by weight. Working fluid Ethanol Ethylene glycol 90% glycerol 96% glycerol Glycerol l ½103 Pa s q ½103 kg=m3 c ½103 N=m e=e0 [–] 1.2 21 220 620 1400 0.79 1.11 1.24 1.25 1.26 22 48 65 64 63 25 41 47 44 41 from 125 lm to 840 lm at the nozzle exit. For smaller diameters, tapered fused-silica nozzles (New Objective PicoTips) were used with outer/inner diameters of 125/75 lm, 150/100 lm, and 210/150 lm, respectively. For larger diameters, borosilicate capillaries (Vitrocom Vitrotubes) were used with outer/inner diameters of 250/150 lm, 330/200 lm, 550/400 lm, and 840/600 lm, respectively. To assess the potential effects of nozzle conductance, 460/260 lm stainless steel capillaries (Hamilton 91026) were also used as received. All glass nozzles were cut to a uniform length of 20 mm and fixed via plastic tubing sleeves to a metallic union, through which the nozzle was grounded. The counter electrode was negatively electrified (Trek 610E) and located at a fixed distance of 1.5 mm from the nozzle tip. A textured silicon substrate was used as the counter electrode to prevent accumulation of the working fluid.51 All nozzles were oriented horizontally so that potential gravitational effects were discernable by an asymmetric cone. Our experimental procedures to extract the minimum flow rate followed the methodologies in Refs. 3, 18, and 52, and were detailed in a prior publication.19 The minimum flow rate (Qm) required to sustain a steady jet was extracted as the lowest flow rate bounding the V– Q stability window; see details in supplemental Fig. S1.49 For highly viscous working fluids containing glycerol, extra care was taken to thoroughly flush the electrohydrodynamic systems to avoid trapping air bubbles, and short plastic tubing connection between the nozzle and the pump was adopted to minimize flow capacitances. The uncertainty in the Qm measurements was typically within 630%, as shown below in Fig. 3 for selected conditions. All error bars were based on 95% confidence intervals. Representative images of steady cone-jets of ethylene glycol and glycerol are shown in Fig. 1. The steady cone-jet of ethanol was similar to that of ethylene glycol. The main qualitative difference between the ethylene glycol cone-jet in Fig. 1(a) and glycerol cone-jet in Fig. 1(b) was in the FIG. 1. Electrohydrodynamic cone-jets for (a) ethylene glycol and (b) glycerol, each at 10 lS/cm with an outer diameter of 150 lm at the exit of the tapered glass nozzle. The cone-jet transition was localized around the conical apex for ethylene glycol with a relatively low viscosity, but spanned the entire cone for glycerol with a relatively high viscosity. transitional region, which bridged the Taylor cone of a dimension dictated by the nozzle diameter with a much thinner jet (that accelerated further downstream). In the ethylene glycol case with relatively low viscosity, the cone-jet transition was localized to within a few jet diameters; In the glycerol case with relatively high viscosity, the transition region extends much wider with a size comparable to the nozzle diameter. The minimum flow rates were measured for three sets of working fluids with wide-ranging viscosities. In Fig. 2(a), the nozzle diameter was fixed at 150 lm and the conductivity was varied from 1 lS/cm to 25 lS/cm for each liquid. The minimum flow rate was a strong function of conductivity for ethanol and ethylene glycol, but independent of conductivity for glycerol. In Fig. 2(b), the conductivity was fixed at 5 lS/cm, and the nozzle diameter was varied from 150 lm to 550 lm. The minimum flow rate was essentially independent of nozzle diameter at low viscosities, but became a strong function of diameter for highly viscous glycerol. The minimum flow rate of ethanol and ethylene glycol in Fig. 2 exhibited trends consistent with the conventional scaling Eq. (1). However, the parametric dependence of glycerol contradicted Eq. (1), with the minimum flow rate independent of the electrical conductivity but dependent on the nozzle diameter. The diameter-dependence for glycerol was consistent with the qualitative observation in Fig. 1(b), where the length scale of the cone-jet transitional region was comparable to the nozzle diameter. To better understand these scaling trends, the quantitative dependence on conductivity and nozzle diameter was evaluated in Figs. 3 and 4. FIG. 2. The minimum flow rate (Qm) at increasing viscosity (l). (a) With a given nozzle outer diameter of 150 lm, the Qm was dependent on conductivity for ethanol and ethylene glycol with relatively low viscosities, but not dependent on conductivity for glycerol with a much higher viscosity. (b) With a given conductivity of 5 lS/cm, the Qm did not depend on the nozzle diameter (D) at low viscosities but exhibited a strong dependence for highly viscous glycerol. 024103-3 W. J. Scheideler and C.-H. Chen Appl. Phys. Lett. 104, 024103 (2014) FIG. 3. In the low-viscosity limit, the minimum flow rate (Qm) was strongly dependent on the liquid conductivity (r) and nearly independent of the outer diameter of the nozzle (D). The dashed line was a power law fit with Qm r0:5660:07 for all ethylene glycol data. FIG. 4. In the high-viscosity limit, the minimum flow rate (Qm) was strongly dependent on the nozzle diameter (D) but nearly independent of the liquid conductivity (r). The dashed line was a power law fit with Qm D1:4560:18 for all glycerol data except the 840 lm cases. In Fig. 3, the conductivity of ethylene glycol was varied by two orders of magnitude, and the resulting minimum flow rate followed the scaling trend in Eq. (1). However, the power-law dependence for ethylene glycol was notably weaker with Qm r0:5660:07 , consistent with the observation in Ref. 19. Interestingly, ethanol followed a similar trend as ethylene glycol, with an arguably larger power law coefficient. In contrast to the strong dependence on liquid conductivity, the minimum flow rate did not show any obvious dependence on the nozzle diameter (Fig. 3 inset). The measurement was also insensitive to the choice of conducting versus insulating nozzles. The qualitative trend of decreasing minimum flow rate at increasing conductivity is consistent with prior reports.4,14–16 In sharp contrast to the low-viscosity cases in Fig. 3, the minimum flow rate of glycerol in Fig. 4 was essentially independent of the liquid conductivity, but depended strongly on the nozzle diameter. For glycerol with the outer diameter ranging from 125 lm to 550 lm, the power law was extracted as Qm D1:4560:18 . Because of experimental limits, the nozzle diameter could be neither much smaller nor much larger. For smaller nozzles with a diameter well below 100 lm, it was extremely difficult to consistently pump a highly viscous liquid like glycerol through the nozzle. For larger nozzles with a diameter well over 500 lm, gravitational effects could no longer be neglected; In addition, a larger nozzle diameter could interfere with the constant nozzle-to-plate separation of 1.5 mm. With the limited range of nozzle diameters, it was difficult to test if the Qm measurements correlated more strongly with the inner or outer diameter, and the wetted diameter (outer diameter here) was adopted in all data analysis to be consistent with prior studies.20,24,25 Similar to pure glycerol, the minimum flow rate of 96% glycerol exhibited a strong dependence on nozzle diameter (Fig. 4) but nearly no dependence on liquid conductivity (Fig. 4 inset). The drastically different scaling regimes of the minimum flow rate in Figs. 3 and 4 are directly related to differing liquid viscosities, consistent with the effects of viscosity observed in Fig. 2. The different scaling regimes can be tentatively rationalized with the following analysis on the governing time scales. The flow rate scaling in Eq. (1) is proportional to the charge relaxation time, te ¼ e=r, indicating that te governs the electrohydrodynamic processes at low viscosities. With increasing viscosity, the viscous diffusion time which scales as tv ¼ qD2 =l may become comparable or smaller than the charge relaxation time. When the rapid viscous diffusion process dominates over the charge relaxation process, the minimum flow rate scaling will adopt an alternative form Qm cD2 ; l (2) which can be obtained by replacing te with tv in Eq. (1). Note that with the parametric space explored here for pure glycerol as well as glycerol mixtures (see supplemental Table S149), pffiffiffiffiffiffiffiffi ffi the Ohnesorge number defined as Oh ¼ l= qcD is consistently greater than 1, which indicates that viscous effects will dominate over inertial effects and the liquid density (q) drops out of the scaling. The scaling law in Eq. (2) can also be viewed as the capillary-viscous velocity ðc=lÞ multiplied by a characteristic cross-sectional area, which scales as D2 but is likely smaller in magnitude as in Fig. 1(b). With the caveat of the somewhat limited parametric range, particularly the narrow range of nozzle diameters, our existing experimental data on highly viscous working fluids were consistent with the capillary-viscous scaling Eq. (2). The power law dependence on the nozzle diameter extracted from Fig. 4 appeared to be close to the Qm D2 scaling. The independence on conductivity was observed for pure glycerol as well as 96% glycerol. The scaling dependence on viscosity was harder to test, given the difficulty in obtaining Newtonian liquids with a viscosity higher than that of glycerol. Although binary mixtures were not ideal working fluids with potential complications such as demixing at high electric fields53,54 and preferential solvent evaporation, preliminary tests with 100%, 96%, and 90% glycerol did indicate increasing minimum flow rate with decreasing viscosity, albeit much weaker than the l1 power law; see for example, the three cases with 10 lS/cm and 150 lm in 024103-4 W. J. Scheideler and C.-H. Chen Table S1.49 It should be noted that the minimum flow rate of 90% glycerol was no longer completely independent of conductivity (see Table S149), so Eq. (2) may not be fully applicable at this viscosity. We should stress that the capillary-viscous scaling in Eq. (2) is one of the many possible scaling relations yielding a dependence on the nozzle diameter. Given the dominance of viscosity with Oh > 1 and the experimentally established independence on liquid conductivity, Eq. (2) appears to be a reasonable choice on dimensional grounds. Our work may be related to a recent study of pulsating cone-jets emitted from dielectric films.34 In that study, the capillary-viscous velocity ðc=lÞ appeared in the scaling laws validated for highly viscous working fluids, which satisfied tv te with their extremely low conductivity.34 On the other hand, we shall note that there are different views on the role of charge relaxation time (te) in cone-jet dynamics,4,10 and that the actual viscous time scale may be much shorter than the simple scaling of tv qD2 =l. (For example, the viscous decay time is tv/20 during drop oscillation,55 which is sometimes used as a first approximation to cone oscillation.24,56) In future, the capillary-viscous scaling Eq. (2) should be further assessed with experiments over a wider parametric range and with a full electrohydrodynamic model accounting for all the competing processes. In summary, we experimentally studied the scaling laws governing the minimum flow rate of steady Taylor cone-jets issued from an electrified nozzle. Contrary to the wellknown scaling for low-viscosity liquids which is proportional to the charge relaxation time as in Eq. (1), an alternative scaling regime was identified for high-viscosity liquids which seemed to be proportional to the viscous diffusion time as in Eq. (2). The “charge-relaxation” scaling is strongly dependent on the liquid conductivity, whereas the “capillary-viscous” scaling is strongly dependent on the nozzle diameter. Our findings suggest precautions when applying existing scaling models of lowly viscous cone-jets to highly viscous electrohydrodynamic systems, such as those in miniaturized electrospraying and electroprinting. This work was supported by the National Science Foundation (Grant No. CBET-08-46705). We thank D. B. Bober, O. M. Knio, and X. Qu for useful discussions. 1 J. Zeleny, Phys. Rev. 10, 1 (1917). G. I. Taylor, Proc. R. Soc. London, Ser. A 280, 383 (1964). M. Cloupeau and B. Prunet-Foch, J. Electrostat. 22, 135 (1989). 4 J. Fernandez de la Mora, Annu. Rev. Fluid Mech. 39, 217 (2007). 5 J. Zeleny, J. Franklin Inst. 219, 659 (1935). 6 O. A. Basaran and L. E. Scriven, J. Colloid Interface Sci. 140, 10 (1990). 7 A. L. Yarin, S. Koombhongse, and D. H. Reneker, J. Appl. Phys. 90, 4836 (2001). 8 R. P. A. Hartman, D. J. Brunner, D. M. A. Camelot, J. C. M. Marijnissen, and B. Scarlett, J. Aerosol Sci. 30, 823 (1999). 9 Y. M. Shin, M. M. Hohman, M. P. Brenner, and G. C. Rutledge, Appl. Phys. Lett. 78, 1149 (2001). 10 A. M. Ganan-Calvo and J. M. Montanero, Phys. Rev. E 79, 066305 (2009). 11 F. J. Higuera, Phys. Fluids 21, 032104 (2009). 12 F. J. Higuera, J. Fluid Mech. 648, 35 (2010). 2 3 Appl. Phys. Lett. 104, 024103 (2014) 13 A. M. Ganan-Calvo, N. Rebollo-Munoz, and J. M. Montanero, New J. Phys. 15, 033035 (2013). J. Fernandez de la Mora and I. G. Loscertales, J. Fluid Mech. 260, 155 (1994). 15 A. M. Ganan-Calvo, J. Davila, and A. Barrero, J. Aerosol Sci. 28, 249 (1997). 16 A. Barrero and I. G. Loscertales, Annu. Rev. Fluid Mech. 39, 89 (2007). 17 J. Fernandez de la Mora, J. Colloid Interface Sci. 178, 209 (1996). 18 R. Juraschek and F. Rollgen, Int. J. Mass. Spectrom. 177, 1 (1998). 19 D. B. Bober and C. H. Chen, J. Fluid Mech. 689, 552 (2011). 20 C. H. Chen, “Electrohydrodynamic stability,” in Electrokinetics and Electrohydrodynamics in Microsystems, edited by A. Ramos (Springer, New York, 2011), pp. 177–220. 21 M. Cloupeau and B. Prunet-Foch, J. Aerosol Sci. 25, 1021 (1994). 22 I. Marginean, L. Parvin, L. Heffernan, and A. Vertes, Anal. Chem. 76, 4202 (2004). 23 M. Alexander, M. Paine, and J. Stark, Anal. Chem. 78, 2658 (2006). 24 I. Marginean, P. Nemes, L. Parvin, and A. Vertes, Appl. Phys. Lett. 89, 064104 (2006). 25 C. H. Chen, D. A. Saville, and I. A. Aksay, Appl. Phys. Lett. 89, 124103 (2006). 26 H. Choi, J. Park, O. Park, P. Ferreira, J. Georgiadis, and J. Rogers, Appl. Phys. Lett. 92, 123109 (2008). 27 U. Stachewicz, C. Yurteri, J. Marijnissen, and J. Dijksman, Appl. Phys. Lett. 95, 224105 (2009). 28 A. Gomez and K. Tang, Phys. Fluids 6, 404 (1994). 29 D. Duft, T. Achtzehn, R. Muller, B. A. Huber, and T. Leisner, Nature 421, 128 (2003). 30 W. D. Ristenpart, J. C. Bird, A. Belmonte, F. Dollar, and H. A. Stone, Nature 461, 377 (2009). 31 R. T. Collins, K. Sambath, M. T. Harris, and O. A. Basaran, Proc. Natl. Acad. Sci. U.S.A. 110, 4905 (2013). 32 L. Oddershede and S. R. Nagel, Phys. Rev. Lett. 85, 1234 (2000). 33 R. T. Collins, J. J. Jones, M. T. Harris, and O. A. Basaran, Nat. Phys. 4, 149 (2008). 34 P. Kim, C. Duprat, S. S. H. Tsai, and H. A. Stone, Phys. Rev. Lett. 107, 034502 (2011). 35 M. S. Wilm and M. Mann, Int. J. Mass Spectrom. 136, 167 (1994). 36 S. Castro and J. Fernandez de la Mora, J. Appl. Phys. 105, 034903 (2009). 37 G. T. T. Gibson, S. M. Mugo, and R. D. Oleschuk, Mass Spectrom. Rev. 28, 918 (2009). 38 J. Doshi and D. H. Reneker, J. Electrost. 35, 151 (1995). 39 D. H. Reneker, A. L. Yarin, E. Zussman, and H. Xu, Adv. Appl. Mech. 41, 43 (2007). 40 O. Yogi, T. Kawakami, M. Yamauchi, J. Y. Ye, and M. Ishikawa, Anal. Chem. 73, 1896 (2001). 41 C. H. Chen, D. A. Saville, and I. A. Aksay, Appl. Phys. Lett. 88, 154104 (2006). 42 J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, Nature Mater. 6, 782 (2007). 43 D. Y. Lee, Y. S. Shin, S. E. Park, T. U. Yu, and J. Hwang, Appl. Phys. Lett. 90, 081905 (2007). 44 J. Choi, Y. J. Kim, S. Lee, S. U. Son, H. S. Ko, V. D. Nguyen, and D. Byun, Appl. Phys. Lett. 93, 193508 (2008). 45 S. N. Jayasinghe and M. J. Edirisinghe, J. Aerosol Sci. 33, 1379 (2002). 46 H. B. Zhang, M. J. Edirisinghe, and S. N. Jayasinghe, J. Fluid Mech. 558, 103 (2006). 47 D. R. Chen and D. Y. H. Pui, Aerosol Sci. Technol. 27, 367 (1997). 48 M. Gamero-Castano, J. Fluid Mech. 662, 493 (2010). 49 See supplementary material at http://dx.doi.org/10.1063/1.4862263 for Figure S1—example V–Q operating windows for steady cone-jets; Table S1—minimum flow rate measurements used in Figs. 2–4. 50 J. M. Grace and J. C. M. Marijnissen, J. Aerosol Sci. 25, 1005 (1994). 51 Y. Zhao, D. B. Bober, and C. H. Chen, Phys. Rev. X 1, 021007 (2011). 52 L. Parvin, M. C. Galicia, J. M. Gauntt, L. M. Carney, A. B. Nguyen, E. Park, L. Heffernan, and A. Vertes, Anal. Chem. 77, 3908 (2005). 53 Y. Tsori, F. Tournilhac, and L. Leibler, Nature 430, 544 (2004). 54 Y. Tsori, Rev. Mod. Phys. 81, 1471 (2009). 55 S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford Univ. Press, Oxford, 1961; Dover, New York, 1981). 56 W. Deng and A. Gomez, Microfluid. Nanofluid. 12, 383 (2012). 14