When does the elastic regime begin in viscoelastic pinch-off?

J. Fluid Mech. (2025), vol.1005, A10, doi:10.1017/jfm.2024.1222 When does the elastic regime begin in viscoelastic pinch-off? A. Gaillard1,†,M.A.Herrada 2,A.Deblais 1, C. van Poelgeest1, L. Laruelle1, J. Eggers3and D. Bonn2 1Van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands 2Depto. de Mecánica de Fluidos e Ingeniería Aeroespacial, Universidad de Sevilla, Sevilla E-41092, Spain 3School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK (Received 6 June 2024; revised 30 October 2024; accepted 15 December 2024) In this experimental and numerical study, we revisit the question of the onset of the elastic regime in viscoelastic pinch-off. This is relevant to all modern filament thinning techniques, which aim to measure the extensional properties of low-viscosity polymer solutions. Examples are the slow retraction method (SRM) for capillary breakup extensional rheometry (CaBER), or the dripping method, in which a drop detaches from a nozzle. As part of these techniques, a stable liquid bridge is brought slowly to its stability threshold, where capillary-driven thinning starts. This thinning slows down dramatically at a critical radius h1, marking the onset of the elasto-capillary regime, characterised by a filament of nearly uniform radius. While a theoretical scaling exists for this transition in the case of the classical step-strain CaBER protocol, where polymer chains stretch without relaxing during the fast plate separation, we show that this theory is not necessarily valid for a slow protocol such as the SRM. In that case, polymer chains start stretching (beyond their equilibrium coiled configuration) only when the bridge thinning rate becomes comparable to the inverse of their relaxation time. We derive a universal scaling for h1, valid for both lowand high-viscosity polymer solutions. This scaling is validated by CaBER experiments with a slow plate separation protocol using different polymer solutions, plate diameters and sample volumes, as well as by numerical simulations using the FENE-P model. Key words: capillary flows, polymers †Email address for correspondence: antoine0g[email protected] © The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/ licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited. 1005 A10-1 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others 1. Introduction The elasticity of a polymer solution can be probed by stretching a drop between one’s thumb and index finger, resulting in the formation of a filament with a persistence time that is linked to the relaxation time of the solution. Such filaments are observable in many industrial free-surface flows such as spraying (Keshavarz et al. 2015,2016; Gaillard, Sijs & Bonn 2022) and inkjet printing (Christanti & Walker 2002;Senet al. 2021), where long polymer molecules can be added to a Newtonian solvent to achieve a specific flow property, as well as in ejecta produced when coughing and sneezing (Scharfman et al. 2016;Gidreta &Kim2023). The capillary-driven thinning dynamics of these filaments is the basis of numerous rheometry techniques dedicated to low-viscosity fluids, for which other techniques such as rheometric melt elongation (Meissner’s RME) and filament stretching extensional rheometry (FiSER) are not applicable. These techniques include capillary breakup extensional rheometry (CaBER), where a droplet is confined between two plates that are separated beyond the range of stable liquid bridges (Bazilevsky et al. 1997; Stelter et al. 2000; Anna & McKinley 2001), the dripping technique where a droplet detaches from a nozzle (Amarouchene et al. 2001; Tirtaatmadja, McKinley & Cooper-White 2006; Rajesh, Thiévenaz & Sauret 2022),and dripping-onto-substrate (DoS), where a solid substrate is brought into contact with a drop hanging steadily from a nozzle (Dinic, Jimenez & Sharma 2017). All these techniques aim to creae a viscoelastic filament by triggering the pinching of a liquid column via the Rayleigh–Plateau instability. Viscoelastic filaments are found to thin exponentially over time for a wide range of polymer-solvent systems and polymer concentrations (dilute and semi-dilute), consistent with the Oldroyd-B model, which predicts h=h1exp −t−t1 3τ,(1.1) where his the (minimum) filament radius,andτis the relaxation time of the polymer solution, the longest one for a multimode model (Entov & Hinch 1997; Anna & McKinley 2001). This regime corresponds to an elasto-capillary balance where the elastic stress arising from the stretching of polymer chains balances the driving capillary pressure. Experimentally, starting from an equilibrium situation where polymers are relaxed (no pre-stress), this elastic regime can be observed only once polymers have been sufficiently stretched to overcome inertia and/or viscosity, which occurs at a time t1and at a filament radius h1=h(t1)marked by a sudden deceleration of the thinning dynamics. The amount of stretching of polymer chains at times t<t1is set by the strength of the extensional flow in the pinching region. In the limit case where the thinning dynamics at times t<t1(before elasticity balances capillarity) is much faster than the solution’s relaxation time – i.e. where polymer chains deform by the same amount as the surrounding solvent itself without relaxing – Clasen et al. (2006a) showed that the Oldroyd-B model leads to h1=Gh4 0 2γ1/3 ,(1.2) where γis the surface tension, Gis the elastic modulus and h0is the radius of the ‘initial’ liquid column before the onset of thinning, i.e. when the fluid is still at rest. This formula was first derived by Bazilevsky et al. (1997) and differs by a factor 21/3from the formula proposed by Entov & Hinch (1997), who did not treat the tension in the filament properly. This ‘relaxation-free’ scenario leading to (1.2) corresponds to the step-strain CaBER protocol where the plates are separated so fast that polymer chains stretch without 1005 A10-2 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off having time to relax as the liquid bridge, connecting the two plates, stretches axially. In this step-strain protocol, the plates are separated exponentially over time to create an extensional flow with a constant extension rate ˙0that, to ensure that polymer relaxation is negligible, must be larger that the coil–stretch transition value 1/2τ(Miller, Clasen & Rothstein 2009). This corresponds to Weissenberg number Wi0=˙0τ>1/2. Once the plates have reached their final separation distance LF, the unstable liquid bridge between the two plates continues to thin, this time under the action of capillarity, until the elastic regime starts at a bridge/filament (minimum) radius h1. Miller et al. (2009) showed that, consistent with (1.2), h1does not depend on LFfor polymer solutions. However, we could not find experimental studies where h1was reported and tested against (1.2) for different plate diameters and initial gaps, which set the radius h0of the initial (unloaded) fluid sample, or for different polymer solutions. This step-strain CaBER protocol is,however, not recommended for low-viscosity polymer solutions since a fast plate separation leads to inertio-capillary oscillations of the end drops that hinder the measurement of the relaxation time (Rodd et al. 2005). Alternative protocols consist in reaching the threshold of the Rayleigh–Plateau instability slowly, e.g. by separating the plates at a constant low velocity in CaBER (slow retraction method or SRM) (Campo-Deano & Clasen 2010). In that case, the initially stable liquid bridge connecting the two end-plates becomes unstable at a critical plate separation distance, corresponding to a minimum bridge radius h0, and thins further under the action of capillarity. This is similar to the dripping method where the bridge connecting a droplet to a nozzle, from which liquid is infused at a low flow rate, becomes unstable at a critical droplet weight (Rajesh et al. 2022). In such slow protocols, (1.2) may not be valid if the time taken by the bridge to thin from its initial (minimum) radius h0to the radius h1(marking the onset of the elastic regime) is longer than the liquid’s relaxation time τ, as was already noticed by Bazilevsky et al. (1997). In that case, polymer chains may indeed remain in a coiled state for a significant time, starting to stretch only when the bridge’s thinning rate becomes comparable to 1/τ. This led Campo-Deano & Clasen (2010) to derive an alternative formula for h1for their slow retraction CaBER method that, to the best of our knowledge, has never been tested experimentally. In this formula, h1is independent of h0, in sharp contrast with (1.2), which predicts h1∝h4/3 0. In a more recent experimental work from Rajesh et al. (2022), the authors proposed an empirical scaling h1∝R0.66 nin dripping experiments with low-viscosity polymer solutions, where Rnis the nozzle radius, but they did not provide a theoretical explanation for their findings. In such slow protocols, (1.2)is expected to be valid only if the time taken by the liquid bridge to thin from h0to h1is much shorter than the liquid’s relaxation time, in which case polymer chains stretch without having time to relax. This time is expected to scale as the characteristic time scale of the capillary-driven bridge thinning dynamics derived from linear stability theory, namely, the Rayleigh (inertio-capillary) time scale (Wagner et al. 2005) τR=(ρh3 0/γ )1/2,(1.3) or the visco-capillary time scale τvisc =η0h0/γ, (1.4) depending on the Ohnesorge number Oh =η0 √ργh0=τvisc τR ,(1.5) 1005 A10-3 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others where ρand η0are the liquid density and total (zero-shear) viscosity, respectively. In other words, if we define a Deborah number De =τ/τR(1.6) based on the Rayleigh time scale, then (1.2) is expected to be valid for De 1inthe inviscid case (Oh 1),andforDe/Oh =τ/τvisc 1 in the viscous case (Oh 1), which is the limit considered in most analytical studies (Clasen et al. 2006a). In this study, we aim to expand our current understanding of the transition radius h1(marking the onset of the elastic regime) to cases where polymer relaxation is not negligible during the capillary-driven thinning of the liquid bridge. This discussion follows up on our previous paper, where h1was observed to increase linearly with h0for different liquids for a slow plate separation CaBER protocol (Gaillard et al. 2024), a scaling that differs from the h1∝h4/3 0prediction of (1.2). Materials and methods are presented in § 2, and experimental results are presented in § 3. Theoretical expressions for h1are derived and tested experimentally and numerically using the Oldroyd-B model in § 4,andthe FENE-P model in § 5. 2. Materials and methods The liquids, their shear rheology and the experimental set-up and protocol are presented in §§ 2.1,2.2 and 2.3, respectively. The equations and numerical methods are presented in §2.4. 2.1. Liquids Three of the polymer solutions used in the present study are the same as in our previous paper (Gaillard et al. 2024)and have comparable ‘relaxation times’ or, more precisely, comparable filament thinning rates. Two of them are solutions of poly(ethylene oxide) (PEO) of molecular weight Mw=4×106gmol−1(PEO-4M), one in water with concentration 500 ppm, referred to as PEOaq, and one in a ∼260 times more viscous solvent with concentration 25 ppm, referred to as PEOvisc.The third solution is a 1000 ppm solution of poly(acrylamide/sodium acrylate) (HPAM) [70 : 30] of molecular weight Mw=18 ×106gmol−1in water with 1 wt% NaCl to screen electrostatic interactions and make polymer chains flexible instead of semi-rigid. Both polymers were provided by Polysciences (ref. 04030 for PEO and 18522 for HPAM). The solvent of the PEOvisc solution is a Newtonian 30wt% aqueous solution of poly(ethylene glycol) (PEG) with molecular weight 20 000 g mol−1(PEG-20K). After slowly injecting the polymer powder into a vortex generated by a magnetic stirrer, solutions were homogenised using a mechanical stirrer at low rotation speed for approximately 16 h. For the PEOvisc solution, PEG was added after mixing PEO with water. Additional solutions of PEO-4M in water were prepared from dilution of a 10 000 ppm stock solution with concentrations ranging between 5 and 10 000ppm to investigate the influence of polymer concentration. 2.2. Shear rheology The shear viscosity ηand first normal stress difference N1of polymer solutions were measured at the temperature of CaBER experiments, typically 20 ◦C, with an MRC-302 rheometer from Anton Paar equipped with a cone plate geometry (diameter 50 mm, angle 1◦, and truncation gap 53 μm) and are shown in figure 1. To measure N1, we follow 1005 A10-4 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off 101Carreau fit Power-law fit 100 10–1 10–2 10–3 100 N1 (Pa) 101 102 10–4 10–3 ∗PEOaq,1 ∗PEOaq,1 PEOaq,2 PEOvisc,2 Solvent of PEOvisc,2 HPAM PEOaq,2 PEOvisc,2 HPAM PEO in water 10 000 2500 1000 500∗ 10 000 2500 1000 500∗ 250 100 250 100 25 0 Concentration (ppm) PEO in water Concentration (ppm) 10–2 10–1 100101102103100101102103104 2 1 γ . (s–1)γ . (s–1) η (Pa s) (b)(a) Figure 1. (a) Shear viscosity ηand (b) first normal stress difference N1of the different polymer solutions against the shear rate ˙γ. a step-by-step protocol similar to Casanellas et al. (2016) in order to circumvent the instrumental drift of the normal force. This protocol consists of applying steps of constant shear rate followed by steps of zero shear, and subtracting the two raw N1plateau values. The contribution of inertia to the normal force is corrected for by the rheometer (Macosko 1994). We find that the PEOvisc solution is a Boger fluid with a constant shear viscosity, while the HPAM solution is shear-thinning, as well as the aqueous PEO solutions when concentrations are larger than 250 ppm. For shear-thinning solutions, the shear viscosity is fitted with the Carreau–Yasuda formula η( ˙γ)=η01+(˙γ/˙γc)a1(n−1)/a1,(2.1) where η0is the zero-shear viscosity, nis the shear-thinning exponent,and ˙γcis the shear rate marking the onset of shear thinning, with a1(typically 2) encoding the sharpness of the transition towards the shear-thinning regime. The polymer contribution to the shear viscosity ηp=η0−ηsincreases linearly with polymer concentration cin the dilute regime, and follows ηp=ηs[η]c, where, for the PEO solutions in water, we find an intrinsic viscosity [η]=2.87 m3kg−1. Using the expression of Graessley (1980)gives a critical overlap concentration c∗=0.77/[η]=0.268 kg m−3(268 ppm), consistent with the onset of shear thinning expected at c>c∗. For the PEOvisc solution, where only one concentration (25 ppm) was tested, assuming that the solution is dilute to calculate [η]and c∗using the same formulas leads to a larger critical overlap concentration c∗=1400 ppm, probably due to differences in polymer–solvent interactions (PEO in water versus PEO in PEG solution). The first normal stress difference is fitted by a power law N1=Ψ1˙γα1,(2.2) where we find α1=2belowc∗,andα1<2abovec∗, for aqueous PEO solutions, and α1<2 for the PEOvisc and HPAM solutions. All fitting parameters are reported in table 1 for the PEO solutions of different concentrations in water, and in table 2 for the PEOaq, PEOvisc and HPAM solutions. We also report the density ρand surface tension γmeasured with a pendant drop method and, when known, the ratio c/c∗. Note that PEO addition reduces the surface tension of water since PEO is known to adsorb at the air/water interface (Gilányi et al. 2006). Surface tensions reported in tables 1 and 2are the equilibrium ones. 1005 A10-5 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others cγc/c∗η0ηpn1/˙γcα1Ψ1 (ppm) (mNm−1)(mPa s) (mPa s) (s) (Pa sα1) 572.00.019 0.93 0.013 1 — — — 10 72.0 0.037 0.940 0.02 1 — — — 25 63.4 0.093 0.985 0.065 1 — — — 50 62.8 0.19 1.04 0.12 1 — — — 100 63.0 0.37 1.19 0.27 0.98 0.02321.2×10−6 250 63.0 0.93 1.75 0.83 0.95 0.054 2 1.2×10−5 500 62.5 1.93.00 2.08 0.95 0.12 1.15 5.9×10−3 1000 62.5 3.76.35.38 0.86 0.14 1.10 2.0×10−2 2500 62.5 9.345440.730.62 0.98 1.1×10−1 10 000 62.3 37 15 400 15 400 0.48 34 0.79 4.7×100 Table 1. Concentration c, reduced concentration c/c∗, surface tension γand shear rheological properties (from (2.1)and(2.2)) of aqueous PEO-4M solutions prepared from dilution of the same 10 000 ppm stock solution. Here, ηp=η0−ηsis the polymer contribution to the shear viscosity. The density and solvent viscosity are ρ=998 kg m−3and ηs=0.92mPas. The 500 ppm solution in this table is referred to as PEOaq,1in the text. For the 5 ppm solution, η0is too close to ηsto estimate ηp, and we therefore use ηp=ηs[η]cwith the intrinsic viscosity [η] extracted from the linear fit of ηp(c)for c<c∗. Name ργη scc/c∗η0ηpn1/˙γcα1Ψ1τm (kg m−3) (mN m−1) (mPa s) (ppm) (mPa s) (mPa s) (ms) (Pa sα1) (ms) PEOaq,2998 62.5 0.92 500 1.93.32.08 0.93 120 1.2 9.9×10−3240 PEOvisc,21048 56.0 245 25 0.018 248 3.31—1.65.8×10−3110 HPAM 998 72.0 0.92 1000 — 15 14 0.78 410 1.7 9.0×10−3100 Table 2. Properties of the polymer solutions used for plate diameters 2R0up to 25mm in CaBER measurements. Here, ρis the density, and γis the surface tension. See the caption of table 1 for the definition of the shear properties. Also, τmis the maximum CaBER relaxation time measured for the largest plates; see figure 4(a). The PEOvisc,1and PEOvisc,2solutions have the same shear viscosity to within less than 5%. We must mention here that two different PEOaq solutions and two different PEOvisc solutions have been used in this study, with differences in rheological properties in each case, caused by slightly different preparation protocols for a given recipe (e.g. a slightly different agitation time). The PEOaq,1solution is prepared from dilution of the same stock solution as the other aqueous PEO solutions in table 1. The PEOaq,2solution featured in table 2 exhibits a 10 % larger shear viscosity and approximately 2.5 times larger values of N1, as shown in figure 1. The PEOvisc,1and PEOvisc,2solutions have the same shear viscosity to within less than 5%, and only the latter one is presented in figure 1 and in table 2. As explained in § 2.3, the PEOaq,1and PEOvisc,1solutions were tested with (CaBER) plate diameters less than 7mm, varying the (non-dimensional) drop volume for each plate, whereas the PEOaq,2and PEOvisc,2solutions were used for plate diameters up to 25mm with a single (non-dimensional) drop volume for each plate. 2.3. Experimental set-up and slow stepwise CaBER protocol The CaBER set-up and slow stepwise plate separation protocol described here are the same as in our previous paper (Gaillard et al. 2024). A droplet of volume Vis placed on a horizontal plate of radius R0, and the motor-controlled top plate of same radius is first moved down until it is fully wetted by the liquid, i.e. until the liquid bridge 1005 A10-6 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off 0 –2.0 –1.5 –1.0 –0.5 t – t1 (s) Lp R0 (mm) 0 Capillary thinning Capillary thinningStable bridge 2R0 2R0 = 2 mm 2R0 = 20 mm Last stable bridge 0.5 101 100 10–1 100101 1.3 1 12.4 3.2 V∗ = Linear fit PEOaq,2 PEOaq,1 PEOvisc,2 PEOvisc,1 HPAM 100 h (µm) h0 (mm) 200 300 400 500 600 2hLf h1 h0 (b)(a) Figure 2. (a) Time evolution of the minimum bridge/filament radius hin our slow stepwise plate separation protocol for the PEOaq,1solution for plate diameter 2R0=3.5mm and a sample volume V∗=V/R3 0≈2.4. Inset images correspond to a stable liquid bridge (left) and a thinning filament (right) of the same liquid, with 2R0=7mmandV∗≈2.4. (b) Last stable bridge radius h0against the plate radius R0:for2R0between 2 and 7 mm, and for each plate, V∗≈1.3, 2.4and3.2forthePEO aq,1and PEOvisc,1solutions; and for 2R0between 2 and 25 mm, and a single volume (V∗≈2.4 for the smallest plates, and V∗≈0.88 for the largest plates) for the PEOaq,2,PEO visc,2and HPAM solutions. Inset images correspond to stable liquid bridges (h≥h0)for 2R0=2mm (left, PEOaq,1solution with V∗≈2.4) and 2R0=20 mm (right, HPAM solution with V∗≈1.0), the right-hand inset being taken from a phone camera because the lens of the set-up camera (used to take the other inset pictures) did not have a large enough field of view. between the plates has a quasi-cylindrical shape. The top plate is then moved up slowly (at approximately 0.5mms −1) and stopped at a plate separation distance Lpwhere the liquid bridge is still stable, as in the left-hand inset image of figure 2(a), but close to the bridge instability threshold. Then, instead of moving the top plate at a constant (lower) velocity, i.e. as in the SRM (Campo-Deano & Clasen 2010), we move it by 10 μmLp-increment steps, waiting approximately one second between each step (longer than the solution’s relaxation time), which is long enough to ensure that polymers are at equilibrium (no pre-stress) before each new step. At a certain step, the bridge becomes unstable (due to the Rayleigh–Plateau instability) and collapses under the action of surface tension, transiently leading to the formation of a nearly cylindrical filament that is the signature of viscoelastic pinch-off, as shown in the right-hand inset image of figure 2(a). We stop moving the top plate once we reach the step at which the bride collapses. The plate separation distance hence remains constant during the capillary thinning of the bridge/filament. The process is recorded by a high-magnification objective mounted on a high-speed camera (Phantom TMX 7510), and images are analysed by a Python code. A typical time evolution of the minimum bridge/filament radius is shown in figure 2(a). Throughout the paper, we use the term ‘bridge’ for times t<t1before the onset of the elastic regime, and the term ‘filament’ during the elastic regime (t≥t1). This radius, measured at the thinnest point along the bridge/filament profile (see left-hand inset image in figure 2a)and labelled ‘hmin’ by many authors, is simply referred to as hin the rest of the paper. Note that each step can trigger small inertio-capillary oscillations that increase in intensity as the Rayleigh–Plateau instability threshold is approached;seefigure 2(a), where oscillations vanish after approximately 0.2 s for the PEOaq,1solution. The purpose of this step-by-step plate separation protocol is to identify the last stable liquid bridge configuration and to 1005 A10-7 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others extract the value of its (minimum) radius h0. Since steps are small, h0can be considered to be the initial bridge radius at the onset of capillary thinning. Our image resolution is up to1pixelpermicrometre for the smallest drops, corresponding to the smallest plates, and our time resolution is 15 000 images per second to capture the fast bridge collapse from radius h0to the radius h1marking the onset of the elastic regime;seefigure 2(a). The liquid bridge becomes unstable at a critical plate separation distance Lp= L∗ pmarking the Rayleigh–Plateau instability threshold. The critical aspect ratio Λ∗= L∗ p/(2R0)depends on the liquid volume Vand on the Bond number Bo =ρgR2 0/γ , where g is the gravitational acceleration (Slobozhanin & Perales 1993; Montanero & Ponce-Torres 2020). In our experiments, we vary the plate diameter 2R0between 2 and 25 mm, as well as the non-dimensional sample volume V∗=V/R3 0.(2.3) Note that using R3 0as a reference volume is an arbitrary choice. Other authors often use the volume πR2 0L∗ pbetween the plates. As shown in figure 2(b), the last stable bridge radius increases approximately linearly with the plate radius, i.e. h0∝R0with a prefactor that increases with V∗, with no strong dependence on the liquid used since they all have comparable surface tensions. Typically, h0/R0ranges between 0.24 and 0.35 for V∗≈1.3 and 3.2, respectively. The size difference between the top and bottom end drops, visible in the inset images of figures 2(a,b), stems from Bond numbers Bo =ρgR2 0/γ increasing from 0.16 to 25 as the plate size increases (Pingulkar, Peixinho & Crumeyrolle 2021). The ‘filament’ Bond number Bof=ρgLfh1/γ , however, comparing the typical capillary pressure γ/h1in the filament to the hydrostatic pressure ρgLfover the filament length Lf,isonlyupto0.1 for the largest plate, indicating that the thinning dynamics is not driven by gravity. The filament length Lf, shown in the right-hand inset image of figure 2(a), is discussed in the Appendix. The aluminium plates are plasma-treated before each measurement to increase their hydrophilicity and hence prevent dewetting of the top plate. However, dewetting could not be avoided for plate diameters 2R0≥10 mm, as shown in the right-hand inset image of figure 2(b), featuring a stable liquid bridge (h≥h0) where the top end drop does not fully cover the top plate for 2R0=20 mm. Perhaps surprisingly, h0does not saturate at 2R0≥10 mm in spite of this lack of full coverage;seefigure 2(b). For such large plates, the top end drop is not necessarily at the centre of the top plate since the two plates are not perfectly parallel. Note that because of the plasma treatment, there is always a thin film covering the top plate. All experiments are carried out at a high relative humidity (>80 %) ensured by placing the CaBER set-up in a box with wet paper tissues. We checked that repeating an experiment several times over the course of 10 min does not lead to any monotonic increase or decrease of the filament thinning rate (defined as 1/3τe;see§3) over time, beyond small variations of less than 5%, suggesting that both evaporation and polymer degradation (which may occur during bridge/filament thinning) are negligible. 2.4. Equations and numerical methods The numerical simulations discussed in §§4and 5are performed using the FENE-P model, which aims to describe the stretching and finite extensibility of polymer chains. We consider a cylindrical axisymmetric (r,z)coordinate system aligned with the vertical axis 1005 A10-8 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off of the liquid bridge. In the simulations, we integrate the mass and momentum conservation equations of general form ∇·v=0,(2.4) ρDv Dt=−∇p+∇·σ,(2.5) where ρ,v=vr(r,z,t)er+vz(r,z,t)ezand p(r,z,t)are the density, velocity and (reduced) pressure fields (accounting for gravity), respectively, and D/Dtis the material derivative. These equations are completed with the constitutive relationships for the stress tensor σ=σs+σp, where σs=ηs∇v+(∇v)T(2.6) is the contribution of the solvent of viscosity ηs,and σpis the polymer contribution. In the FENE-P model (Snoeijer et al. 2020), this contribution is calculated as σp=ηp τ(fA−I), f=1 1−tr(A)/L2,(2.7a,b) where ηpis the polymer contribution to the zero-shear viscosity η0=ηs+ηp,τrepresents the relaxation time,andL2denotes the finite extensibility limit, with Ithe identity matrix. The conformation tensor Ais calculated from the nonlinear relaxation law DA Dt−(∇v)T·A+A·∇v=−1 τ(fA−I). (2.8) The free-surface location is defined by the equation r=h(z,t). The boundary conditions at that surface are ∂h ∂t+hzw−u=0,(2.9) −p+gz −hhzz −1−h2 z h(1+h2 z)3/2+n·σ·n=0,(2.10) t·σ·n=0,(2.11) where hz≡∂h/∂z,hzz ≡∂2h/∂z2,gis the gravitational acceleration, nis the unit outward normal vector,andtis the unit vector tangential to the free-surface meridians. Equation (2.9) is the kinematic compatibility condition, while (2.10)and(2.11) express the balance of normal and tangential stresses, respectively. The anchorage condition h=R0is set at z=0andz=Lp, where Lpis the plate separation distance. The no-slip boundary condition is imposed at the solid surfaces in contact with the liquid. The liquid volume Vof the initial configuration is prescribed (and conserved), namely, πLp 0 h2dz=V.(2.12) We start the simulation from a liquid bridge at equilibrium with a plate separation distance Lpjust below (very close to) the critical one. The breakup process is triggered by applying a very small gravitational force perturbation. We refer to the minimum radius of the (stable) liquid bridge just before the perturbation is applied as h0due to the similarities with the experimental stepwise plate separation protocol described in §2.3. 1005 A10-9 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others The apparent CaBER relaxation time τein figure 6(a) is compared to values estimated from shear rheology, i.e. 1/˙γc, which is estimated from shear viscosity curves featuring shear thinning, which excludes low concentrations, and Ψ1/2ηp, which is estimated from the first normal stress difference for the only two measurable solutions exhibiting a quadratic scaling N1∝˙γ2. We observe that 1/˙γcfollows a power law with an exponent close to the one found for τe. Additionally, we find that τeis larger than Ψ1/2ηpin figure 6(a) for this specific plate radius, and would hence be even larger in the high-h0 limit τm.Hence even for dilute solutions exhibiting weak shear thinning and quadratic normal stresses, for which the Oldroyd-B model could describe the shear rheology, there is no quantitative agreement between the relaxation time measured from normal stresses and that from filament thinning rheometry. This impossibility to quantitatively describe both shear and elongation properties with the Oldroyd-B model has also been reported by Zell et al. (2010), who dedicated a paper specifically to the link between τeand Ψ1. In a recent perspective paper, Boyko & Stone (2024) suggest that the reason why elastic dumbbell models such as Oldroyd-B and FENE-P cannot quantitatively predict both shear and elongation properties of polymer solutions is because these models assume that, as in elongation flows, polymer chains approach full extension in strong shear flows (τ˙γ1), while experimentally, chains have been observed to extend only partially in strong shear flows due to their tumbling motion (Smith, Babcock & Chu 1999). The authors hence suggest that flows with both shear and elongational components should be described using more complex models such as the FENE-PTML model (Phan-Thien, Manero & Leal 1984), which captures this partial extension under shear. In this paper, we consider only filament thinning elongational flows for which Oldroyd-B and FENE-P are suitable model candidates. As shown in figure 6(a), the apparent CaBER relaxation time τeincreases in the dilute regime c<c∗. Moreover, for the most dilute solutions, τeis less than the Zimm relaxation time calculated using (Clasen et al. 2006b) τZ=1 ζ(3ν) [η]Mwηs NakBT,(3.3) where Nais the Avogadro number, kBis the Boltzmann constant, Tis the temperature, ζ is the Riemann zeta function,andνis the solvent quality exponent. We used ν=0.55 between theta and good solvent to find τZ=2.0 ms. Similar results were reported by Clasen et al. (2006b), who argue that the increase of τein the dilute regime is caused by a self-concentration effect where chains start to interact while unravelling well beyond their equilibrium size in strong extensional flows. This was later rationalised by Prabhakar et al. (2016), who argue that the Zimm relaxation time is relevant only close to equilibrium, even for dilute solutions, and is not expected to accurately describe the relaxation behaviour of polymer chains in strong extensional flows such as in filament thinning. Clasen et al. (2006b) and Prabhakar et al. (2016), who considered only cases where inertia was negligible in the Newtonian regime, also show that values τe<τ Zarise at low polymer concentration where elasticity is too weak to fully overcome the solvent viscosity in the elastic regime. This effect should,however, be negligible for the aqueous PEO solutions of figure 6(a) since it is inertia (and not the solvent viscosity) that dominates in the Newtonian regime (see figure 5b). We show in § 5that values τe<τ Zat low concentrations are consistent with polymer chains approaching their finite extensibility limit at the onset of the elastic regime (as anticipated by Campo-Deano & Clasen 2010), a case where (1.1)is no longer valid, and filament thinning rates |˙ h/h|>1/3τare to be expected,asweshow in our previous paper (Gaillard et al. 2024). 1005 A10-16 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off 4. Oldroyd-B prediction for h1 In order to rationalise the experimental findings of § 3, we need to expand the polymer-relaxation-free theory leading to (1.2)forh1to cases where polymer relaxation is not negligible in the Newtonian regime. A generalised theoretical expression for h1based on the Oldroyd-B model is first derived in § 4.1. This expression is then tested against experimental results in § 4.2,before we use numerical simulations in § 4.3 to validate the theory and further explore the role of non-negligible polymer relaxation on h1. 4.1. Theory The filament radius h1=h(t1)marks the transition between the Newtonian regime (t< t1), where the driving capillary force is balanced by inertia and/or viscosity, and the elastic regime (t>t1), where capillarity is balanced by elastic stresses arising from the stretching of polymer chains. If inertia is negligible, then slender filament theory predicts that the total (unknown) tensile force Tis constant along the liquid column (bridge or filament) (Eggers 1997; Clasen et al. 2006a), which, neglecting gravity and axial curvature effects, results in the zero-dimensional force balance equation (Clasen et al. 2006b) (2X−1)γ h=3ηs˙+σp,zz −σp,rr,(4.1) foracolumnofradiush. The driving capillary pressure γ/his balanced by the normal stress difference σzz −σrr, which is the sum of the solvent viscous stress 3ηs˙and the polymeric stress σp,zz −σp,rr, where ˙=−2˙ h/his the extension rate, the dot standing for d/dt,and zis the direction of the flow. The ratio X=T/2πγhmay vary over time, approaching X=0.7127 for a Newtonian fluid (McKinley & Tripathi 2000), hence recovering (3.2) close to breakup, and approaching X=3/2 in the elastic regime for an Oldroyd-B fluid (Eggers, Herrada & Snoeijer 2020) (and not X=1asoriginally proposed in Entov & Hinch 1997). When inertia is not negligible, Tirtaatmadja et al. (2006) suggested adding a term of the form 1 2ρ˙ h2to (4.1), from which the inertio-capillary scaling of (3.1) is recovered. In the elastic regime (t>t1), assuming that inertia and/or solvent viscosity has become negligible, and assuming that the axial stress dominates the radial stress, i.e. |σp,rr||σp,zz|, the force balance equation reduces to (2X−1)γ h=σp,zz.(4.2) The elastic regime starts when the polymeric axial stress σp,zz – which increases over time in the Newtonian regime as polymer chains are progressively stretched by the extensional flow in the thinning bridge – becomes of the order of the capillary pressure, say,whenitisequaltofractionpof the capillary pressure (Campo-Deano & Clasen (2010) chose p=1/2). We hence get that pγ/h1=σp,zz(t=t1)where, for simplicity, the prefactor 2X−1 of order unity has been included in the prefactor p. To estimate h1, we hence need to choose a constitutive equation to express the polymeric stress. Since our main goal is to understand the effect of polymer relaxation during the Newtonian regime on h1, which, when negligible, leads to (1.2) for a single-mode Oldroyd-B fluid, we choose to use this model for simplicity. Indeed, although we know that the Oldroyd-B model is unable to capture the system-size dependence of the apparent relaxation time τediscussed in our previous paper (Gaillard et al. 2024) (see also figures 3 and 4a), it is not yet clear whether or notitisablecaptureh1. Finite extensibility effects on h1will be discussed in §5using the FENE-P model. 1005 A10-17 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others For an Oldroyd-B fluid with elastic modulus G,wehaveσp,zz =G(Azz −1), where Azz is the normal part of the conformation tensor Athat follows (see (2.8) and Wagner, Bourouiba & McKinley 2015) ˙ Azz −2˙Azz =− Azz −1 τ,(4.3) where τis the relaxation time. Since we are interested in the location of highest polymer extensions along the bridge, we use the expression of the extension rate ˙=−2˙ h/hat the thinnest point to obtain ˙ Azz +4˙ h h Azz =− Azz −1 τ.(4.4) Some (as yet unknown) time after the onset of capillary thinning of the liquid bridge, polymer chains will have stretched well beyond their equilibrium size, i.e. Azz 1, so that theright-handsideof(4.4) reduces to −Azz/τ, at which point (4.4) can be integrated into Azzh4∝e−t/τ ,(4.5) with an (as yet unknown) constant prefactor. Polymer chains are expected to remain close to their equilibrium coiled size (Azz close to 1) until the extension rate in the thinning bridge approaches the coil–stretch transition value 1/2τpredicted by the Oldroyd-B model. Beyond this point, following Clasen et al. (2009) and Campo-Deano & Clasen (2010), we assume that polymer chains unravel with negligible relaxation, i.e. that theright-handsideof(4.4) becomes negligible so that Azzh4becomes constant. More precisely, Azzh4=H4,(4.6) where His the (as yet unknown) bridge radius at which relaxation becomes negligible, which should correspond to the coil–stretch transition point at which Azz starts to become significantly larger than 1. In particular, at the transition to the elastic regime at time t=t1, A1=(H/h1)4,(4.7) where A1=Azz(t1)quantifies the amount of polymer stretching at the onset of the elastic regime. Since pγ/h1=σp,zz(t1)=GA1,wefinallygetthat h1=GH4 pγ1/3 ,(4.8) which is different from (1.2). Indeed, while it is assumed that H=h0in the polymer-relaxation-free theory leading to (1.2), this is actually true only in the limit where polymer relaxation is negligible throughout the whole Newtonian regime so that Azzh4 is constant and equal to h4 0since Azz =1 at the onset of capillary thinning, assuming no pre-stress. In other words, in the limit where H=h0,thecoil–stretch transition starts at the onset of capillary thinning, which is expected to be true only if the relaxation time τis much larger than the time taken by the liquid bridge to thin from h0to h1. For completeness, in the elastic regime (t≥t1), combining (4.5)and(4.2)with pγ/h1=GA1leads to the exponential scalings h=h1exp (−(t−t1)/3τ) in (1.1)and Azz =A1exp ((t−t1)/3τ). 1005 A10-18 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off 4.2. Experiments We can now test the generalised expression (4.8)forh1against the experimental results of § 3. In order to do so, we first need to compute Hfrom the time-evolution of Azz. Note that we do not experimentally measure the extension of polymer chains, unlike Ingremeau & Kellay (2013), who confirmed the transition from a coiled to a stretched state in viscoelastic pinch-off using fluorescently labelled DNA. Rather, since our goal is to test a specific constitutive equation, here Oldroyd-B, we calculate its prediction for Azz(t) using (4.4), where the extension rate ˙=−2˙ h/his taken from experimental values of h(t). In other words, we calculate the prediction of the model for the experimental history of extension rates in the bridge/filament. In particular, we do not assume large polymer extension (Azz  1) since the point at which Azz starts to become significantly larger than 1 is precisely what sets H.Equation (4.4) can in fact be integrated, as shown by Bazilevsky, Entov & Rozhkov (2001), introducing a function y(t)such that Azz =yexp (−t/τ)/h4, which leads to ˙y=h4exp (t/τ)/τ, yielding Azz =e−t/τ h4h4 0et0/τ +1 τt t0 h4(t)et/τ dt,(4.9) where the initial time t0corresponds to the onset of capillary thinning, i.e. h(t0)=h0 and Azz(t0)=1 (no pre-stress). Since the h(t)history is set by the experimental data, the only adjustable parameter of (4.9) is the relaxation time τ. In the following, we use either the apparent (τe) or the maximum (τm) relaxation time measured experimentally (see figure 4a) to calculate Azz since we still do not know which is the ‘true’ one, if any. Values of Azz(t)computed from (4.9) using the experimental values of h(t)with relaxation time τ=τmare shown in figure 7(a)forthePEO aq,2solution,andinfigure 7(b) for the PEOvisc,2solution, for plate diameters 2R0between 2 and 20 mm. The experimental values of h(t)are shown on the left-hand y-axis, and the time reference t1corresponds to the onset of the elastic regime. We find that the amount of polymer extension A1= Azz(t1)at the onset of the elastic regime is fairly independent of the initial condition for the PEOaq,2,while for the PEOvisc,2solution, A1decreases as h0(R0,V∗)increases, as mentioned in our previous paper (Gaillard et al. 2024), and as we are about to discuss here in more depth. In any case, we find that Azz always increases as 1/h4in the Newtonian regime close enough to the transition to the elastic regime. More specifically, values of Azz are well captured by (H/h)4using Has a fitting parameter for each data set (close to t1), from which His estimated;seefigure 7. Values of Hcalculated using τ=τm, named H(τm), are plotted against h0in figure 8(a) for the PEOaq,2,PEO visc,2and HPAM solutions, which are the only solutions for which sufficiently large plate diameters were used to estimate the maximum relaxation time τm (the high-h0limit of τe;seefigure 4a). For the PEOaq,2and HPAM solutions, we find that His essentially equal to h0at low h0,andthatH/h0decreases as h0increases, down to 0.73 for the largest plate diameter. In contrast, for the PEOvisc,2solution, the ratio H/h0 takes significantly smaller values, decreasing from 0.56 to 0.40 as h0increases. This is why the (H/h)4fit for Azz is fairly good throughout the whole Newtonian regime for the PEOaq,2solution in figure 7(a), while it is valid only within a small time window close to the transition to the elastic regime for the PEOvisc,2solution in figure 7(b). Indeed, if H=h0, then the (H/h)4fit for Azz is even valid at the onset of capillary thinning where h=h0and Azz =1. Figure 8(a) hence suggests that although all three solutions have comparable relaxation times (see figure 4a), the thinning dynamics in the Newtonian regime is such that polymer 1005 A10-19 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others 0 0.5 1.0 h (mm) 1.5 2.0 (a)(b) 100 101 102 103 104 105 0 0.2 0.4 0.6 0.8 1.0 1.2 100 101 102 103 104 105 106 –50 –40 –30 –20 t – t1 (ms) –10 0 10 –120 –100 –80 –60 t – t1 (ms) –40 –20 0 20 h Azz 2R0 (mm) 25 PEOaq,2 PEOvisc,2 A1 h1 A1 h1 10 5 2 (H/h)4 Azz Figure 7. Time evolution of the experimental minimum bridge/filament radius hand of Azz, calculated from the Oldroyd-B prediction (4.9) using the experimental values of h(t)with the choice of relaxation time τ=τm(see figure 4a), for plate diameters 2R0=2, 5, 10 and 25 mm for the (a)PEO aq,2and (b)PEO visc,2solutions. Time t1marks the onset of the elastic regime, with h1=h(t1)and A1=Azz(t1).ValuesofAzz in the Newtonian regime (t<t1) are compared to (H/h)4(see (4.6)), where His used as a fitting parameter to optimise the agreement close to t1. 102 102103 h0 (µm) 102103 10 000 ppm 5 ppm 10 ppm h0 (µm) H(τm) (µm) 103 (a)(b) 1 1 1 1 1 0.74 1 0.82 102 H(τe) (µm) 103 Power-law fit H = h0 PEOaq,2 PEOvisc,2 HPAM Power-law fit H = h0 PEOaq,2 PEOvisc,2 HPAM PEOaq,1 PEOvisc,1 2R0 (mm) 75 3.5 2 PEO in water Various concentrations h0 = 460 µm Figure 8. Values of Hestimated from the time evolution of Azz (calculated using Oldroyd-B; see figure 7) using (a) the maximum relaxation time τ=τmor (b) the effective relaxation time τ=τe, for different polymer solutions and initial bridge radii h0(R0,V∗), plotted against h0. The line H=h0is shown in both plots. chains stretch almost without relaxing in the Newtonian regime for the low solvent viscosity solutions (PEOaq,2and HPAM), while relaxation is not negligible for the high solvent viscosity (PEOvisc,2) solution. This is because the Newtonian thinning dynamics is slower for the most viscous solution;see e.g. figure 7,where, for 2R0=2 mm, the bridge takes only approximately 6 ms to thin from h0to h1for the PEOaq,2solution, much less than τm(chains do not have enough time to relax), while it takes approximately 900 ms for the PEOvisc,2, much more than τm(not visible in figure 7(b), where we focus on times close to t1). The fact that H/h0increases as h0increases can therefore be interpreted by a longer time to thin from h0to h1as h0increases, consistent with the fact that the Rayleigh and viscous time scales τR=(ρh3 0/γ )1/2and τvisc =η0h0/γ both increase with h0. 1005 A10-20 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off The dependence of Hon h0and on the solvent viscosity explains why, in figure 7 and in our previous paper (Gaillard et al. 2024), A1is independent of h0for low solvent viscosity solutions (PEOaq,2and HPAM), while A1decreases as h0increases for the high solvent viscosity solution (PEOvisc,2). For the former, where His close to h0(negligible polymer relaxation in the Newtonian regime), A1=(H/h1)4(see (4.7)) is close to (h0/h1)4, which is fairly constant since h1∝h0according to figure 4(b). For the latter, however, where H∝h0.74 0in our range of h0according to figure 8(a), we get A1=(H/h1)4∝h−1.04 0. In figure 8(b), we plot the values of H, named H(τe), calculated using the apparent relaxation time τ=τewhen computing Azz from (4.9) (instead of its large-h0limit τmin figure 8a). All polymer solutions are now featured since τeis measured for any experiment from an exponential fit of h(t)in the exponential part of the elastic regime (see figure 3), i.e. the PEOaq,2,PEO visc,2and HPAM solutions, as in figure 8(a), but also the PEOaq,1 and PEOvisc,1solutions, where three different non-dimensional sample volumes V∗were tested for each of the four smallest plate diameters, as well as the PEO solutions in water with various polymer concentrations (table 1), which were tested for a single (R0,V∗) set corresponding to h0≈460 μm. The data corresponding to the PEO solutions with different polymer concentrations cshow how, for a given flow history in the Newtonian regime (same h(t)curves for t<t1for all concentrations;seefigure 5), Hincreases with cvia the increase in the relaxation time (here τ=τe), reaching the upper limit value h0 at large τ. This is because for large τvalues, polymer relaxation is negligible throughout the whole Newtonian regime,while for low τvalues, Azz remains equal to 1 for most of the Newtonian regime, increasing only when ˙is finally of the order of 1/2τclose to the transition to the elastic regime. Now that we know the value of H, we can test the validity of (4.8) for the filament radius h1at the onset of the elastic regime. The value of the elastic modulus, G=ηp/τ in the Oldroyd-B model, is,however, not uniquely defined,since while ηp=η0−ηscan be calculated unambiguously from the shear rheology, the relaxation time τcould be either the apparent one τeor the maximum one τm, since we do not know yet which one is the ‘true’ one, if any. We hence need to test for both. To this end, we define GH=γh3 1/H4,(4.10) where h1is the value measured experimentally, which should be GH=G/paccording to (4.8). In figure 9(a), GHis plotted against Gfor the choice of relaxation time τ=τmfor the PEOaq,2,PEO visc,2and HPAM solutions, which are the only solutions for which sufficiently large plate diameters were used to estimate τm. More precisely, values of GH(τm)calculated from H(τm)are plotted against G(τm)=ηp/τm, which takes a unique value for each solution since the relaxation time is unique. We find that values of GH(τm) are, however, not unique for the PEOaq,2and HPAM solutions,and monotonically decrease as h0increases. For the PEOvisc,2solution, however, values of GHvary between 0.11 and 0.23 without clear monotonic trend as h0increases. This is because, since h1∝h0and H∝hkH 0in the range of h0values investigated,withkH≤1(seefigures 4band 8a), GH∝h3 1/H4∝h3−4kH 0, which means that GHis expected to be independent of h0only for kH=0.75, very close to the value 0.74 found for the PEOvisc,2solution in figure 8(a). This suggests that the prediction of (4.8) for the choice τ=τmis potentially valid for only one of our three solutions, the most dilute and viscous one. By contrast, as shown in figure 9(b), when choosing τeinstead of τmfor the relaxation time to calculate GH(τe)from H(τe)and G(τe)=ηp/τe, data points fall on a single curve 1005 A10-21 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others 101 (a)(b) PEOaq,2 PEOvisc,2 PEOaq,1 PEOvisc,1 HPAM PEOaq,2 PEOvisc,2 HPAM 100GH = 3.7G 10–1 GH (τm) = γh1 3/[H(τm)]4 (Pa) 10–2 101 100 10–1 5 ppm 10 ppm 10 ppm 10 000 ppm 5 ppm 1 1 1 1 25 ppm GH (τe) = γh1 3/[H(τe)]4 (Pa) 10–2 10–2 10–1 G(τm) = ηp/τm (Pa) 100 h0 increases h0 increases 10110–2 10–1 G(τe) = ηp/τe (Pa) 100101 PEO in water Various concentrations h0 = 460 µm τ = max(τe, τZ) τ = τe 2R0 (mm) 75 3.5 2 Figure 9. Values of GHdefined in (4.10) against the elastic modulus G=ηp/τ,whereHand Gare calculated from (a) the maximum relaxation time τ=τmor (b) the effective relaxation time τ=τe, for various polymer solutions and initial bridge radii h0(R0,V∗).Valuesofh1are those measured experimentally. In (b), for the aqueous PEO solutions of different concentrations (see table 1), we show the effect replacing τeby the Zimm relaxation time τZwhen calculating Gand Hfor the lowest concentrations c=5 and 10 ppm, which exhibit values of τe<τ Z;seefigure 6(a). The line GH=3.7Gisshowninbothplots. for all three solutions (PEOaq,2,PEO visc,2and HPAM) as well as for the PEOaq,1and PEOvisc,1solutions (for which V∗is varied for the four smallest plate diameters),and we find the linear relationship GH=G/ppredicted by (4.8)withp≈0.27.Itisquite remarkable that the Oldroyd-B model, derived for ideal dilute chains, is able to capture the transition to the elastic regime for solutions that are as diverse in terms of solvent viscosity, concentration and, potentially, solvent quality exponents. Strong deviations from the GH=3.7Gline can be observed in figure 9(b)atlow polymer concentrations for the data corresponding to the PEO solutions with various polymer concentrations. This data set can be broken down into three subsets: for typically c<100 ppm, GHdecreases sharply with concentration, while is it almost constant for 100 ppm ≤c<2500 ppm, and increases sharply with concentration for c≥2500ppm. These trends can be explained by the fact that GH∝h3 1/H4, where h1increases very slowly with concentration as h1∝c0.16 for c<2500 ppm, and increases sharply for higher concentrations (see figure 6b), while Hincreases sharply with concentration for typically c<100 ppm, becoming almost constant and equal to h0for larger concentrations (see figure 8b). These strong deviations from the GH=3.7Gline observed at low polymer concentrations in figure 9(b) could be partially explained by the fact that apparent relaxation times (measured from exponential fitting of h(t)) are less than the Zimm relaxation time τZ=2msforc=5 and 10 ppm (see figure 6a). In figure 9(b), we show the effect of choosing τ=τZinstead of τeas the relaxation time for c=5 and 10 ppm, which changes the value of both GHvia H(τ ),and G=ηp/τ . We find that this correction leads to data points significantly closer to the GH=3.7Gline, mainly stemming from values of Hlarger than in figure 8(b), although one order of magnitude deviation from the GH=3.7Gline remains. We show in § 5that finite extensibility effects can explain this deviation (i.e. values of h1higher than the Oldroyd-B prediction) as well as the values of τe<τ Zfor low polymer concentrations. The large deviation from the GH=3.7Gline for the entangled 10 000 ppm solution in figure 9(b) (the only solution for which polymers affect the thinning dynamics even before 1005 A10-22 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off the exponential regime;seefigure 5)is probably due to the fact that such solutions cannot be described by non-interacting polymer theories such as Oldroyd-B. In conclusion, when polymer relaxation is not negligible in the Newtonian regime (t<t1), (1.2) should be replaced by (4.8), which gives h1∝H4/3where H≤h0.In our experiments, the power-law dependence of Hon h0(see figure 8) leads to the fairly proportional relationship between h1and h0observed in figure 4(b), differences in slopes among different liquids stemming from differences in elastic moduli G.Nowthatwehave established that it is H(and not h0)that sets the transition to the elastic regime, we discuss in § 4.3 how it scales with the parameters of the problem, using numerical simulations to cover a wide range of parameters. 4.3. Simulations To further investigate the effect of polymer relaxation on the transition radius h1marking the onset of the elastic regime, we now consider numerical simulations using the Oldroyd-B model (L2=+∞) with a single relaxation time τ(finite extensibility effects will be discussed in § 5). In this paper, numerical simulations are not directly compared to experiments but are rather used to validate theoretical expressions that are then compared with experiments (see our previous paper for direct experiment–simulation comparisons; Gaillard et al. 2024). In order to capture h1, we first need to capture the critical bridge radius Hat which polymer relaxation (the right-hand side of (4.3)or(2.8)) becomes negligible in the Newtonian regime (t<t1). The bridge radius Hmarks the onset of the coil–stretch transition at which polymer chains start to extend significantly beyond their equilibrium shape, i.e. at which Azz starts becoming significantly larger than 1;see§4.1. We already know that H→h0in the limit where the relaxation time τis so large that polymer relaxation is always negligible in the Newtonian regime, a limit where (4.8) reduces to the classical formula (1.2). The goal of this subsection is therefore to expand our knowledge to cases where relaxation is not negligible in the Newtonian regime using the Oldroyd-B model. Figure 10 shows the numerical time evolution of the non-dimensional minimum bridge/filament radius h/h0for a fixed Ohnesorge number Oh =2.07, viscosity ratio S=0.988 and h0/R0=0.23, with three Deborah numbers De spanning four orders of magnitude (see (1.5), (1.6)and(2.13) for definitions). We consider only the value of Azz at this minimum-radius position along the bridge/filament since this is where polymer chains are the most stretched. This maximum value, simply denoted Azz from now on, is plotted in figure 10 on the right-hand y-axis. Note that all h/h0curves are identical in the Newtonian regime, diverging only at the transition to the elastic regime at different radii h1. The time reference tccorresponds to the time at which the bridge would break if this transition did not occur. This is highlighted by the fact that the self-similar viscous thinning law (3.2), which becomes h/h0=0.0709(tc−t)/(Oh τR)with our choice of non-dimensionalisation, and which is plotted in figure 10, fits numerical results close to the transition. Note that simulations could often not be continued long after the transition. Figure 10 shows how, for a given flow history in the Newtonian regime, polymer chains start stretching at different times depending on the Deborah number. For De 1, relaxation is always negligible in the Newtonian regime,andAzz therefore increases as (H/h)4where H=h0;see the discussion in § 4.1.ForDe 1,however,theflow becomes strong enough to start stretching polymers (beyond their equilibrium shape) only at small bridge radii where the thinning dynamics has become self-similar and, for Oh 1, follows 1005 A10-23 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others 0 0.2 –40 –30 –20 (t – tc)/τR –10 0 10 0.4 (H/h)4 De = 892 De = 8.92 De = 0.089 h/h0 0.6 0.8 Viscous regime Oh = 2.07, S = 0.988 h/h0Azz 10–1 100 101 102 Azz 103 104 105 106 107 A1 h1/h0 Figure 10. Numerical time evolution of the non-dimensional (minimum) bridge/filament radius h/h0and of the (maximum) polymer extension Azz for Oh =2.07, S=0.988 and h0/R0=0.23, and three different Deborah numbers. Values of Azz are compared with (H/h)4close to the onset of the elastic regime, where His used as a fitting parameter. The self-similar viscous regime of (3.2), or equivalently h/h0=0.0709(tc− t)/(Oh τR), is also plotted, where tcis the time at which the filament would break if the transition to an elastic regime, at h=h1, did not occur. (3.2). In that case, Azz =1 for most of the Newtonian regime, increasing only close to the transition to the elastic regime, following Azz =(H/h)4where Hh0is the characteristic bridge radius at which Azz starts increasing. Values of Hare estimated by fitting Azz with (H/h)4, using Has a fitting parameter (see figure 10), as was done in figure 7 for experimental results. Values of H/h0are plotted against the Deborah number in figure 11(a) for Ohnesorge numbers between 0.2 and 20. The low-De behaviour corresponds to cases where polymers start stretching only within the self-similar regime where the thinning dynamics follows a scaling of the form h=B(tc−t)β,withB∼(γ /ρ)1/3and β=2/3 in the inviscid limit (Oh 1;see(3.1)),andB∼γ/η 0and β=1 in the viscous limit (Oh 1;see(3.2)). The coil–stretch transition occurs when the extension rate ˙=−2˙ h/h=2β/(tc−t)becomes of the order of 1/τ, i.e. at a time tH=tc−2βτ. Therefore, the bridge radius H=h(tH) marking the onset of the coil–stretch transition scales as H∼(γ τ2/ρ)1/3⇔H/h0∼De2/3(4.11) in the inviscid limit (Oh 1), as first derived by Campo-Deano & Clasen (2010), or as H∼γτ/η 0⇔H/h0∼De/Oh =τ/τvisc (4.12) in the viscous limit (Oh 1). The scaling of (4.12) is shown in figure 11(a)with a prefactor 0.2,and shows good agreement with the values of Hfor the two largest Ohnesorge numbers. Unfortunately, no simulations could be performed for Oh 1totest (4.11). Note that in this peculiar limit where polymer chains start stretching only within the self-similar thinning regime, H– and therefore h1given by (4.8) – does not depend on h0 and is therefore independent of the size of the system, in sharp contrast with the high-De limit, where H=h0and therefore h1∝h4/3 0;see(1.2). In fact, inserting (4.11)or(4.12) into (4.8)gives h1∼(G(γ τ8/ρ4)1/3)1/3(4.13) 1005 A10-24 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off 100 (a)(b) 10–1 10–2 10–3 10–4 100 10–1 10–2 10–3 10–4 H/h0 10–2 10–1 100101 De = τ/τRDeN = De/(1 + α Oh) 10210310410510–4 10–2 100104 102 h0/R0 = 0.23 Vary S and Oh S Oh = 20.5 De = 0.089 De = 89.2 h0/R0 = 0.36 Vary De S = 0.988 Oh = 16.6 Vary De S = 0.988 Oh = 0.207 Oh = 2.07 Oh = 20.7 0.2De/Oh Figure 11. Numerical values of H/h0against (a) the Deborah number De =τ/τRbased on the inertio-capillary time scale τR,and(b) the general Deborah number DeN=τ/τNbased on the general time scale τN=τR(1+αOh)with α=4.3, for various parameters (the same in (a)and(b)). Dots (•) correspond to h0/R0=0.23 and S=0.988, with Oh =0.207 (blue) Oh =2.07 (purple) and Oh =20.7 (red), and De ranging between 8.92 ×10−2and 8.92 ×103(last De excluded for the largest Oh). Triangles () correspond to h0/R0=0.23 with De =8.92 ×10−2(yellow) and De =8.92 ×101(green), varying both S(between 0.1 and 0.988) and Oh while keeping SOhconstant and equal to 9.88. Stars () correspond to h0/R0=0.362 with Oh =16.6andS=0.988, and De ranging between 4.59 ×10−2and 8.92 ×103. in the inviscid limit (Oh 1), or h1∼(Gγ3τ4/η4 0)1/3(4.14) in the viscous limit (Oh 1). In the high-De limit, H/h0→1 since polymer relaxation becomes negligible even at the onset of capillary thinning where h=h0. However, while all curves in figure 11(a)have the same shape, the Deborah number at which H/h0reaches 1 depends on the Ohnesorge number. This is because we chose to express the Deborah number as De =τ/τR, where τR=(ρh3 0/γ )1/2is the inertio-capillary time scale, which is not relevant for the moderate to large Ohnesorge number featured in figure 11(a). The relevant time scale for the thinning dynamics at large Oh is τvisc =Oh τR=η0h0/γ , and we would hence expect that H/h0=O(1)not when τ/τR=O(1)but when τ/τvisc(=De/Oh)=O(1).Infigure 11(b), we show that values of H/h0indeed rescale on a single curve when plotted against a generalised Deborah number DeN=τ/τN, where τN,definedas τN=τR(1+αOh), (4.15) is an empirical attempt at expressing the general time scale of the thinning dynamics in the Newtonian regime for any Oh, connecting the lowand high-Oh scalings τRand τvisc, where α=4.3 is a fitting parameter. This scaling ensures that H/h0=O(1)when DeN= O(1)for any Ohnesorge number. However, according to (4.11)and(4.12), we expect different scalings for DeN1, namely, H/h0∼De2/3 Nfor Oh 1andH/h0∼DeNfor Oh 1. So far, we have varied De and Oh for a fixed viscosity ratio S=0.988 and a fixed sample volume characterised by a fixed value of h0/R0=0.23. In order to further investigate the generality of the H/h0dependence on DeNidentified in figure 11(b), we therefore performed additional simulations. Two sets of simulations were performed for De =0.089 and 89.2, respectively, keeping h0/R0=0.23, where both Sand Oh were varied while 1005 A10-25 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press A. Gaillard and others and h1∝M(7ν−1)/2 wfor Oh 1, according to (4.11)and(4.12)forH, using L∝M1−ν w(see (5.6)). Our work suggests that a FENE-P description is sufficient to predict the transition radius h1to the elastic regime provided that the apparent relaxation time τe, measured from an exponential fit of h(t)in the elastic regime, is chosen as ‘the’ relaxation time of the polymer solution or, more precisely, max(τe,τ Z)(see §5.2). However, the surprising increase of τewith h0(see figure 4a)cannot be rationalised using a FENE-P description, as detailed in our previous paper (Gaillard et al. 2024), where we chose the high-h0limit of τe, named τm, as ‘the’ relaxation time. The results of this study allow us to further comment on whether it is τeor τmthat should be considered as the ‘true’ relaxation time, if any. Indeed, we saw in §4.2 that using τmas ‘the’ relaxation time instead of τeworks only for the PEOvisc solution, i.e. the most dilute one in the most viscous solvent. This would suggest that it is τe(and not τm)that is the ‘true’ relaxation time since it can be used to predict h1for any solution. However, if τereally measures the ‘true’ relaxation, then it implies that some rheological property of a polymer solution somehow ‘changes’ when being tested with a different system size (plate diameter and sample volume) via a mechanism that we could not identify and which is unlikely to be evaporation or polymer degradation (Gaillard et al. 2024). Another possibility is that the solution in fact does not change, meaning that the system-size dependence of τeis not an artefact, in which case it would be only by coincidence that we could successfully capture experimental values of h1using τe. This would imply that the Oldroyd-B and FENE-P models miss some important features of polymer dynamics in extensional flows, strengthening the already established need for better constitutive equations. Future works will determine if more sophisticated models such as conformation-dependent drag models, accounting for the action of both chain stretching and intermolecular hydrodynamic interactions on the friction coefficient (Prabhakar et al. 2016,2017), are able to rationalise our experimental results on the system-size dependence of both τeand h1. Funding. M.A.H. acknowledges funding from the Spanish Ministry of Economy, Industry and Competitiveness under grant PID2022-140951O. Declaration of interests. The authors report no conflict of interest. Author ORCIDs. A. Gaillard https://orcid.org/0000-0003-1775-2682; M.A. Herrada https://orcid.org/0000-0003-0388-8001; A. Deblais https://orcid.org/0000-0002-3574-2480; J. Eggers https://orcid.org/0000-0002-0011-5575; D. Bonn https://orcid.org/0000-0001-8925-1997. Appendix. Filament length Lf The filament length Lf, introduced in the right-hand inset of figure 2(a), is plotted in figure 15 against the initial bridge radius h0for all polymer solutions, plate diameters 2R0, and (non-dimensional) sample volumes V∗considered in this study. All data points collapse on a single curve, indicating that Lfdoes not depend on rheological properties. This is particularly true for data points corresponding to different PEO concentrations in water where, since h0is kept constant by keeping the same plate diameter and sample volume, the filament length also remains constant. This is easily understood by considering that the filament length actually corresponds to the distance between the top and bottom end drops after pinch-off (i.e. after rupture of the filament), which should be the same 1005 A10-32 https://doi.org/10.1017/jfm.2024.1222 Published online by Cambridge University Press Onset of the elastic regime in viscoelastic pinch-off 1 2R0 (mm) Lf = 2.0h0 7 5 3.5 2 2 30 1 2 3 PEO in water Various concentrations h0 = 460 µm PEOaq,2 PEOvisc,2 PEOaq,1 PEOvisc,1 HPAM Lf(mm) h0 (mm) Lb Lf Lt Lp ∗ Figure 15. Filament length Lfagainst the initial bridge radius h0for all polymer solutions, plate diameters and sample volumes. The inset shows a sketch of the top and bottom end drops after pinch-off. regardless of the bulk mechanical properties of the liquid (as long as the plates are separated slowly). This distance can be expressed as Lf=L∗ p−Lt−Lb, where L∗ pis the final plate separation distance, Ltis the distance between the top plate and the bottom of the top end drop,andLb(≥Ltdue to gravity) is the distance between the bottom plate and the top of the bottom end drop;see the inset of figure 15. Note that our protocol is such that L∗ pis the critical plate separation distance at which the bridge becomes unstable to the Rayleigh–Plateau instability. All these quantities should indeed be functions only of the Bond number Bo =ρgR2 0/γ ,ofV∗, and of the contact angle of the liquid with the plates. We find that Lf≈2.1h0for plates of diameters typically 2R0≤7mm, while lower Lfvalues are observed for larger plates. This might be caused by the dewetting of the top plate which, as discussed in § 2.3, could not be avoided for such large plates in spite of the plasma treatment, hence resulting in a smaller ‘effective’ top plate, which might result in values of L∗ plower than expected. 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