3.1. Influence of the plate radius and sample volume

Image sequences of the pinch-off dynamics are shown in figures 3(a) and 3(b) for the ${\rm PEO}_{{aq,1}}$ solution ($500\,{\rm ppm}$ PEO-4M in water) and the ${\rm PEO}_{{visc,1}}$ solution ($25\,{\rm ppm}$ PEO-4M in a $\sim 260$ more viscous solvent), respectively, illustrating the transition from a bridge shape in the Newtonian regime to a filament shape in the elastic regime. For the ${\rm PEO}_{{aq,1}}$ solution, the filament is initially cylindrical until localised pinching is observed near one of the end drops (see frame 6 in figure 3a), followed by its destabilisation into a succession of beads connected by thin filaments (which are below our spatial resolution), a phenomenon usually referred to as ‘blistering’ instability (Sattler, Wagner & Eggers Reference Sattler, Wagner and Eggers2008; Sattler et al. Reference Sattler, Gier, Eggers and Wagner2012; Eggers Reference Eggers2014; Semakov et al. Reference Semakov, Kulichikhin, Tereshin, Antonov and Malkin2015) (see frames 7 and 8). For the ${\rm PEO}_{{visc,1}}$ solution, local pinching occurs very close to breakup, and no blistering is observed. In this paper, we refer to the minimum bridge/filament radius $h$, which therefore corresponds to the pinched region if localised pinching occurs. On another note, inertio-capillary oscillations of the top and bottom end drops lead to oscillations of the filament length for the ${\rm PEO}_{{aq,1}}$ solution (see frames 3–6 in figure 3a). These oscillations are absent for the ${\rm PEO}_{{visc,1}}$ solution due to viscous damping. Note that oscillations do not lead to significant oscillations of the filament radius, implying that when the length of the filament increases, new filament is being created from the liquid in the end drops.

Figure 3. (a,b) Image sequences of the bridge/filament for the (a) ${\rm PEO}_{{aq,1}}$ and (b) ${\rm PEO}_{{visc,1}}$ solutions tested with plate diameter $2R_0 = 5\,{\rm mm}$ and sample volume $V^* \approx 2.4$. (c–f) Time evolution of the minimum bridge/filament radius $h$ in (c,e) semi-log and (d, f) lin–lin, for plate diameters $2R_0$ between $2$ and $7\,{\rm mm}$, and fixed $V^* \approx 2.4$, for the (c,d) ${\rm PEO}_{{aq,1}}$ and (e, f) ${\rm PEO}_{{visc,1}}$ solutions, and for their respective solvents (smaller data points), compared with (3.1) and (3.2), where $t_c$ is the solvent breakup time. Times with labels 1–7 and 2–5 in (c,e), respectively, for $2R_0=5\,{\rm mm}$, correspond to the snapshots in (a,b).

Time evolutions of the minimum bridge/filament radius $h$ are shown in figures 3(c–f) for four plate diameters between $2$ and $7\,{\rm mm}$, and a fixed non-dimensional sample volume $V^* \approx 2.4$ for the ${\rm PEO}_{{aq,1}}$ solution (figures 3c,d) and the ${\rm PEO}_{{visc,1}}$ solution (figures 3e, f) in semi-log (figures 3c,e) and lin–lin (figures 3d, f), the latter focusing on the transition to the elastic regime. The smaller data points correspond to the solvent alone for three of the same plate diameters, and in each case, the same $V^*$ (to within experimental reproducibility). The time reference $t_c$ corresponds to the critical time at which the bridge of solvent alone breaks up. For polymer solutions, since $t_c$ cannot be determined, curves are shifted along the time axis until overlapping their corresponding solvent curves. The good overlap between polymer solutions and their solvent at all times $t< t_1$ (before the transition to the elastic regime) confirms that polymers do not affect the pinch-off dynamics in the (hence rightfully called) Newtonian regime. For the ${\rm PEO}_{{visc,1}}$ solution, where, as is about to be discussed, capillarity is balanced by viscosity in the Newtonian regime, this solution–solvent overlap is consistent with the low polymer contribution to the total shear viscosity ($\eta _p / \eta _0 = 0.013$). The least good solution–solvent overlaps are explained by experimental differences in $V^*$.

All curves corresponding to Newtonian solvents in figures 3(d, f) overlap close to breakup, indicating a self-similar thinning regime where the initial condition, set by $R_0$ and $V^*$, is forgotten. Such overlap is also observed for droplets of different volumes for a given plate radius. For the water solvent in figure 3(d), the self-similar regime is well captured by the inertio-capillary thinning law

(3.1)\begin{equation} h = A \left( \frac{\gamma}{\rho} \right)^{1/3} (t_c-t)^{2/3}, \end{equation}

with a prefactor $A = 0.47$ that is consistent with the experimental and numerical results of Deblais et al. (Reference Deblais, Herrada, Hauner, Velikov, Van Roon, Kellay, Eggers and Bonn2018). For the $\sim 260$ times more viscous solvent in figure 3( f), the self-similar regime is well captured by the visco-capillary thinning law (Papageorgiou Reference Papageorgiou1995; McKinley & Tripathi Reference McKinley and Tripathi2000)

(3.2)\begin{equation} h = 0.0709\,\frac{\gamma}{\eta_0}\,(t_c-t). \end{equation}

This is consistent with the fact that the Ohnesorge number ${Oh} = \eta _0/\sqrt {\rho \gamma h_0}$ (see (1.5)) is up to $0.02$ for ${\rm PEO}_{{aq}}$, and up to $0.1$ for HPAM (for the smallest plate diameter where $h_0$ is lowest), i.e. ${Oh} \ll 1$, and ranges between $1.0$ and $2.0$ for the ${\rm PEO}_{{visc,1}}$ solution in the range of plate diameters considered in figure 3. In spite of the moderate Ohnesorge numbers in the latter case, we do not observe a clear transition to the inertio-visco-capillary thinning law $h = 0.0304 (\gamma /\eta _0) (t_c-t)$ (Eggers Reference Eggers1993, Reference Eggers1997; Li & Sprittles Reference Li and Sprittles2016; Verbeke et al. Reference Verbeke, Formenti, Vangosa, Mitrias, Reddy, Anderson and Clasen2020) that describes the behaviour of Newtonian fluids close to breakup; see figure 3(d).

Interestingly, the transition to the elastic regime occurs at approximately the time at which the self-similar Newtonian regime is reached in figures 3(d, f) – slightly after for the ${\rm PEO}_{{aq,1}}$ solution, and slightly before for the ${\rm PEO}_{{visc,1}}$ solution. However, in both cases, the transition radius $h_1 = h(t_1)$ increases with the plate diameter $2R_0$, which indicates that polymers already started to deform significantly before the self-similar regime. Indeed, if polymers started to deform only within the self-similar regime where the thinning dynamics no longer depends on $R_0$ or $V^*$, then the amount of polymer deformation would be independent of the initial condition, leading to a transition radius $h_1$ that would not depend on $R_0$ or $V^*$, as we discuss further in § 4.3.

After the filament formation, the thinning rate $\vert \dot {h}/h \vert$ is initially fairly constant, indicating an exponential decay, and increases close to breakup in a so-called ‘terminal regime’ where authors argue that polymer chains approach full extension and a Newtonian-like high-viscosity dynamics is recovered (Anna & McKinley Reference Anna and McKinley2001; Stelter et al. Reference Stelter, Brenn, Yarin, Singh and Durst2002; Campo-Deano & Clasen Reference Campo-Deano and Clasen2010; Dinic & Sharma Reference Dinic and Sharma2019). The (constant) filament thinning rate measured during the exponential part of the elastic regime is found to decrease with increasing plate diameter; see figures 3(c,e). This is inconsistent with the Oldroyd-B model, which predicts $\vert \dot {h}/h \vert = 1/3\tau$ (see (1.1)), where $\tau$ is the (longest) relaxation time of the polymer solution, which is a fluid property, independent of the size of the system. As we show in our previous paper (Gaillard et al. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024), this surprising dependence on the system size is also observed for the classical step-strain plate separation protocol of a commercial CaBER rheometer as well as for DoS and dripping (Rajesh et al. Reference Rajesh, Thiévenaz and Sauret2022) experiments. We show that this is not caused by artefacts such as solvent evaporation or polymer degradation, suggesting that the liquid does not change when being tested with different plate diameters. To discuss this geometry-dependent filament thinning rate, we define an apparent (or effective) relaxation time $\tau _e$ such that $\vert \dot {h}/h \vert = 1/3\tau _e$ during the exponential part of the elastic regime.

The apparent relaxation time $\tau _e$ and the transition radius $h_1$ are plotted against $h_0$ in figure 4 for different polymer solutions, plate diameters $2R_0$ and non-dimensional sample volumes $V^*$. The fact that data corresponding to different values of $R_0$ and $V^*$ collapse on a single curve for both the ${\rm PEO}_{{aq,1}}$ and ${\rm PEO}_{{visc,1}}$ solutions suggests that $h_0$, which is an increasing function of both $R_0$ and $V^*$ (see figure 2b), is the only relevant geometrical parameter of the problem. This is the reason why we chose $h_0$ at the relevant length scale for non-dimensional numbers such as the Ohnesorge and Deborah numbers in (1.5) and (1.6). This is in agreement with the idea that the thinning dynamics is influenced only by extensional flow in the bridge/filament, while the top and bottom end droplets act as passive liquid reservoirs.

Figure 4. (a) Effective extensional relaxation time $\tau _e$ and (b) transition radius $h_1$ against the last stable bridge radius $h_0$ for different plate radii $R_0$ and sample volumes $V^*$ for all polymer solutions. For the ${\rm PEO}_{{aq,1}}$ and ${\rm PEO}_{{visc,1}}$ solutions, three points of the same colour correspond to the same $R_0$ and three different $V^* \approx 1.3$, $2.4$ and $3.2$.

We find that $\tau _e$ seems to saturate towards a maximum value $\tau _m$ at large $h_0$; see figure 4(a). The estimated values of $\tau _m$ are reported in table 2 for the ${\rm PEO}_{{aq,2}}$, ${\rm PEO}_{{visc,2}}$ and HPAM solutions for which plate diameters up to $25\,{\rm mm}$ were used, well beyond typical CaBER plate sizes, which was needed to observe the saturation of $\tau _e$. In our previous paper (Gaillard et al. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024), we explored the possibility that $\tau _m$ could be the ‘real’ relaxation time of the solution, invoking finite extensibility effects described by the FENE-P model to explain thinning rates larger than $1/3\tau _m$ for low $h_0$. We concluded that this was a possible explanation only for the ${\rm PEO}_{{visc,2}}$ solution, not for the ${\rm PEO}_{{aq,2}}$ and HPAM solutions, suggesting that the FENE-P model misses some important features of polymer dynamics in extensional flows.

The first transition radius $h_1$ increases approximately linearly with $h_0$ for all liquids; see figure 4(b). This is in contradiction with the scaling $h_1 \propto h_0^{4/3}$ expected from the Oldroyd-B model when assuming that polymer relaxation is negligible during the time needed for the bridge to thin from $h_0$ to $h_1$; see (1.2). Since $h_0 \propto R_0$ for a fixed $V^*$ (see figure 2b), this implies that $h_1 \propto R_0$, different from the scaling $h_1 \propto R_{n}^{0.66}$ observed experimentally by Rajesh et al. (Reference Rajesh, Thiévenaz and Sauret2022) in the analogous problem of a drop falling from a nozzle of radius $R_n$. This is surprising since $R_n$ should play the same role as the plate radius $R_0$ in CaBER.

Note that the ${\rm PEO}_{{aq,2}}$ and ${\rm PEO}_{{visc,2}}$ solutions are slightly more elastic than the ${\rm PEO}_{{aq,1}}$ and ${\rm PEO}_{{visc,1}}$ solutions, since they exhibit larger apparent relaxation times; see figure 4(a). However, these differences are barely visible in figure 4(b) since the values of $h_1$ are almost the same, suggesting that the dependence of $h_1$ on $\tau _e$ is relatively weak, as we now confirm by varying the polymer concentration.

3.2. Influence of the polymer concentration

Figure 5 shows the time evolution of the minimum bridge/filament radius for the aqueous PEO solutions of table 1 of various PEO concentrations, water solvent included, that were tested for a fixed plate diameter $2R_0 = 3.5\,{\rm mm}$ and sample volume $V^* \approx 2.4$, in semi-log (figure 5a) and in lin–lin focusing on the transition to the elastic regime (figure 5b). As in figure 3, the time $t_c$ at which the solvent breaks is chosen as the time reference, and curves corresponding to polymer solutions are shifted along the time axis to maximise the overlap with the solvent for $t< t_1$. This overlap is very good up to $2500\,{\rm ppm}$ (dilute and unentangled semi-dilute solution), small deviations being attributable to slightly different sample volumes. For $10\,000\,{\rm ppm}$ (entangled semi-dilute solution), however, the bridge dynamics prior to the exponential regime is radically different from the pure solvent case, indicating that elasticity is not negligible even before the exponential regime is reached.

Figure 5. Time evolution of the minimum bridge/filament radius $h$ in (a) lin–log and (b) lin–lin focusing on the transition, for PEO-4M solutions of different concentrations in water, for a fixed plate diameter $2R_0 = 3.5\,{\rm mm}$ and sample volume $V^* \approx 2.4$. The time $t_c$ is the time at which the bridge would break up for the solvent alone, here water.

The apparent relaxation time $\tau _e$ and the transition radius $h_1$ are plotted in figure 6 as functions of the polymer concentration $c$ for the aqueous PEO solutions of table 1. Since these solutions were tested for a fixed plate diameter $2R_0 = 3.5\,{\rm mm}$ and sample volume $V^* \approx 2.4$, the initial bridge radius $h_0 \approx 460\,\mathrm {\mu }{\rm m}$, also plotted in figure 6(b), is the same for each solution. For the $500\,{\rm ppm}$ solution, labelled ${\rm PEO}_{{aq,1}}$ – which was tested for four plate diameters from $2$ to $7\,{\rm mm}$, and in each case with three different volumes – the corresponding values of $\tau _e$, $h_0$ and $h_1$ are also included in figure 6 to illustrate how data for a given concentration can shift as $h_0$ is varied. Although the plate diameters used for the ${\rm PEO}_{{aq,1}}$ solution were not large enough to estimate the high-$h_0$ limit value of $\tau _e$, $\tau _m$, the value estimated for the slightly more elastic ${\rm PEO}_{{aq,2}}$ solution (see figure 4(a) and table 2), is shown in figure 6(a) for $c=500\,{\rm ppm}$.

Figure 6. (a) Relaxation times and (b) transition radius $h_1$ and last stable bridge radius $h_0$ against polymer concentration for PEO-4M solutions of different concentrations in water. The data of (b) and the effective CaBER relaxation time $\tau _e$ in (a) correspond to a fixed plate diameter $2R_0 = 3.5\,{\rm mm}$ and sample volume $V^* \approx 2.4$, except for the 500 ppm solution, where data corresponding to different $2R_0$ (between $2$ and $7\,{\rm mm}$) and $V^*$ are shown. In (a), we also plot the relaxation time $1/\dot {\gamma }_c$ and $\varPsi _1/ 2 \eta _p$ inferred from shear rheology, as well as the Zimm relaxation time $\tau _Z$. We also show the maximum CaBER relaxation time $\tau _m$ (high-$h_0$ limit of $\tau _e$) measured for the ($500\,{\rm ppm}$) ${\rm PEO}_{{aq,2}}$ solution (which is slightly more elastic than the ${\rm PEO}_{{aq,1}}$ solution from which the other $500\,{\rm ppm}$ data points are taken; see figure 4a). Error bars are shown in (a) but are smaller than markers for $\tau _e$.

We find a weak dependence $h_1 \propto c^{0.16}$ for dilute and semi-dilute solutions (see figure 6b), consistent with Rajesh et al. (Reference Rajesh, Thiévenaz and Sauret2022), who found $h_1 \propto c^{0.15}$ in the analogous problem of a drop falling from a nozzle. We find a power law $\tau _e \propto c^{0.85}$ for the apparent relaxation time in figure 6(a), consistent with the wide range of exponents, typically between $0.6$ and $1$, reported by other authors in CaBER (Bazilevsky et al. Reference Bazilevsky, Entov, Lerner and Rozhkov1997; Clasen et al. Reference Clasen, Plog, Kulicke, Owens, Macosko, Scriven, Verani and McKinley2006b; Zell et al. Reference Zell, Gier, Rafai and Wagner2010) and pendant drop experiments (Tirtaatmadja et al. Reference Tirtaatmadja, McKinley and Cooper-White2006; Rajesh et al. Reference Rajesh, Thiévenaz and Sauret2022). Note that this exponent is expected to be a function of the solvent quality (Clasen et al. Reference Clasen, Plog, Kulicke, Owens, Macosko, Scriven, Verani and McKinley2006b) and that its potential dependence on $h_0$ is not investigated in the present work. From these scalings, we deduce a weak dependence of $h_1 \propto \tau _e^{0.19}$ on the apparent relaxation time.

The apparent CaBER relaxation time $\tau _e$ in figure 6(a) is compared to values estimated from shear rheology, i.e. $1/\dot {\gamma }_c$, which is estimated from shear viscosity curves featuring shear thinning, which excludes low concentrations, and $\varPsi _1 / 2 \eta _p$, which is estimated from the first normal stress difference for the only two measurable solutions exhibiting a quadratic scaling $N_1 \propto \dot {\gamma }^2$. We observe that $1/\dot {\gamma }_c$ follows a power law with an exponent close to the one found for $\tau _e$. Additionally, we find that $\tau _e$ is larger than $\varPsi _1 / 2 \eta _p$ in figure 6(a) for this specific plate radius, and would hence be even larger in the high-$h_0$ limit $\tau _m$. 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. (Reference Zell, Gier, Rafai and Wagner2010), who dedicated a paper specifically to the link between $\tau _e$ and $\varPsi _1$. In a recent perspective paper, Boyko & Stone (Reference Boyko and Stone2024) 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 ($\tau \dot {\gamma } \gg 1$), while experimentally, chains have been observed to extend only partially in strong shear flows due to their tumbling motion (Smith, Babcock & Chu Reference Smith, Babcock and Chu1999). 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 Reference Phan-Thien, Manero and Leal1984), 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 $\tau _e$ increases in the dilute regime $c< c^*$. Moreover, for the most dilute solutions, $\tau_e$ is less than the Zimm relaxation time calculated using (Clasen et al. Reference Clasen, Plog, Kulicke, Owens, Macosko, Scriven, Verani and McKinley2006b)

(3.3)\begin{equation} \tau_Z = \frac{1}{\zeta(3 \nu)}\,\frac{[\eta] M_w \eta_s}{N_a k_B T}, \end{equation}

where $N_a$ is the Avogadro number, $k_B$ is the Boltzmann constant, $T$ is the temperature, $\zeta$ is the Riemann zeta function, and $\nu$ is the solvent quality exponent. We used $\nu = 0.55$ between theta and good solvent to find $\tau _Z = 2.0$ ms. Similar results were reported by Clasen et al. (Reference Clasen, Plog, Kulicke, Owens, Macosko, Scriven, Verani and McKinley2006b), who argue that the increase of $\tau _e$ in 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. (Reference Prabhakar, Gadkari, Gopesh and Shaw2016), 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. (Reference Clasen, Plog, Kulicke, Owens, Macosko, Scriven, Verani and McKinley2006b) and Prabhakar et al. (Reference Prabhakar, Gadkari, Gopesh and Shaw2016), who considered only cases where inertia was negligible in the Newtonian regime, also show that values $\tau _e < \tau _Z$ arise 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 § 5 that values $\tau _e < \tau _Z$ at 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 Reference Campo-Deano and Clasen2010), a case where (1.1) is no longer valid, and filament thinning rates $\vert \dot {h}/h \vert > 1/3\tau$ are to be expected, as we show in our previous paper (Gaillard et al. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024).

4.1. Theory

The filament radius $h_1=h(t_1)$ marks the transition between the Newtonian regime ($t< t_1$), where the driving capillary force is balanced by inertia and/or viscosity, and the elastic regime ($t>t_1$), 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 $T$ is constant along the liquid column (bridge or filament) (Eggers Reference Eggers1997; Clasen et al. Reference Clasen, Eggers, Fontelos, Li and McKinley2006a), which, neglecting gravity and axial curvature effects, results in the zero-dimensional force balance equation (Clasen et al. Reference Clasen, Plog, Kulicke, Owens, Macosko, Scriven, Verani and McKinley2006b)

(4.1)\begin{equation} (2X-1)\,\frac{\gamma}{h} = 3 \eta_s \dot{\epsilon} + \sigma_{p,zz} - \sigma_{p,rr}, \end{equation}

for a column of radius $h$. The driving capillary pressure $\gamma /h$ is balanced by the normal stress difference $\sigma _{zz} - \sigma _{rr}$, which is the sum of the solvent viscous stress $3 \eta _s \dot {\epsilon }$ and the polymeric stress $\sigma _{p,zz} - \sigma _{p,rr}$, where $\dot {\epsilon } = -2 \dot {h}/h$ is the extension rate, the dot standing for $\mathrm {d}/\mathrm {d}t$, and $z$ is the direction of the flow. The ratio $X = T / 2 {\rm \pi}\gamma h$ may vary over time, approaching $X=0.7127$ for a Newtonian fluid (McKinley & Tripathi Reference McKinley and Tripathi2000), hence recovering (3.2) close to breakup, and approaching $X=3/2$ in the elastic regime for an Oldroyd-B fluid (Eggers, Herrada & Snoeijer Reference Eggers, Herrada and Snoeijer2020) (and not $X=1$ as originally proposed in Entov & Hinch Reference Entov and Hinch1997). When inertia is not negligible, Tirtaatmadja et al. (Reference Tirtaatmadja, McKinley and Cooper-White2006) suggested adding a term of the form $\tfrac {1}{2} \rho \dot {h}^2$ to (4.1), from which the inertio-capillary scaling of (3.1) is recovered. In the elastic regime ($t>t_1$), assuming that inertia and/or solvent viscosity has become negligible, and assuming that the axial stress dominates the radial stress, i.e. $\vert \sigma _{p,rr} \vert \ll \vert \sigma _{p,zz} \vert$, the force balance equation reduces to

(4.2)\begin{equation} (2X-1)\,\frac{\gamma}{h} = \sigma_{p,zz}. \end{equation}

The elastic regime starts when the polymeric axial stress $\sigma _{p,zz}$ – 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, when it is equal to fraction $p$ of the capillary pressure (Campo-Deano & Clasen (Reference Campo-Deano and Clasen2010) chose $p=1/2$). We hence get that $p \gamma /h_1 = \sigma _{p,zz}(t=t_1)$ where, for simplicity, the prefactor $2X-1$ of order unity has been included in the prefactor $p$. To estimate $h_1$, 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 $h_1$, 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 $\tau _e$ discussed in our previous paper (Gaillard et al. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024) (see also figures 3 and 4a), it is not yet clear whether or not it is able capture $h_1$. Finite extensibility effects on $h_1$ will be discussed in § 5 using the FENE-P model.

For an Oldroyd-B fluid with elastic modulus $G$, we have $\sigma _{p,zz} = G ({\mathsf{A}}_{zz} - 1)$, where ${\mathsf{A}}_{zz}$ is the normal part of the conformation tensor $\boldsymbol{\mathsf{A}}$ that follows (see (2.8) and Wagner, Bourouiba & McKinley Reference Wagner, Bourouiba and McKinley2015)

(4.3)\begin{equation} \skew6\dot{{\mathsf{A}}}_{zz} - 2 \dot{\epsilon} {\mathsf{A}}_{zz} ={-} \frac{{\mathsf{A}}_{zz}-1}{\tau}, \end{equation}

where $\tau$ is 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 $\dot {\epsilon } = -2 \dot {h}/h$ at the thinnest point to obtain

(4.4)\begin{equation} \skew6\dot{{\mathsf{A}}}_{zz} + \frac{4 \dot{h}}{h}\,{\mathsf{A}}_{zz} ={-} \frac{{\mathsf{A}}_{zz}-1}{\tau}. \end{equation}

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. ${\mathsf{A}}_{zz} \gg 1$, so that the right-hand side of (4.4) reduces to $-{\mathsf{A}}_{zz}/\tau$, at which point (4.4) can be integrated into

(4.5)\begin{equation} {\mathsf{A}}_{zz} h^4 \propto {\rm e}^{{-}t/\tau}, \end{equation}

with an (as yet unknown) constant prefactor. Polymer chains are expected to remain close to their equilibrium coiled size (${\mathsf{A}}_{zz}$ close to $1$) until the extension rate in the thinning bridge approaches the coil–stretch transition value $1/2\tau$ predicted by the Oldroyd-B model. Beyond this point, following Clasen et al. (Reference Clasen, Bico, Entov and McKinley2009) and Campo-Deano & Clasen (Reference Campo-Deano and Clasen2010), we assume that polymer chains unravel with negligible relaxation, i.e. that the right-hand side of (4.4) becomes negligible so that ${\mathsf{A}}_{zz} h^4$ becomes constant. More precisely,

(4.6)\begin{equation} {\mathsf{A}}_{zz} h^4 = H^4, \end{equation}

where $H$ is the (as yet unknown) bridge radius at which relaxation becomes negligible, which should correspond to the coil–stretch transition point at which ${\mathsf{A}}_{zz}$ starts to become significantly larger than $1$. In particular, at the transition to the elastic regime at time $t=t_1$,

(4.7)\begin{equation} A_1 = (H/h_1)^4, \end{equation}

where $A_1 = {\mathsf{A}}_{zz}(t_1)$ quantifies the amount of polymer stretching at the onset of the elastic regime. Since $p \gamma /h_1 = \sigma _{p,zz}(t_1) = G A_1$, we finally get that

(4.8)\begin{equation} h_1 = \left( \frac{G H^4}{p \gamma} \right)^{1/3}, \end{equation}

which is different from (1.2). Indeed, while it is assumed that $H=h_0$ in 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 ${\mathsf{A}}_{zz} h^4$ is constant and equal to $h_0^4$ since ${\mathsf{A}}_{zz} = 1$ at the onset of capillary thinning, assuming no pre-stress. In other words, in the limit where $H=h_0$, the coil–stretch transition starts at the onset of capillary thinning, which is expected to be true only if the relaxation time $\tau$ is much larger than the time taken by the liquid bridge to thin from $h_0$ to $h_1$. For completeness, in the elastic regime ($t \ge t_1$), combining (4.5) and (4.2) with $p \gamma /h_1 = G A_1$ leads to the exponential scalings $h = h_1 \exp {(-(t-t_1)/3\tau )}$ in (1.1) and ${\mathsf{A}}_{zz} = A_1 \exp {((t-t_1)/3\tau )}$.

4.2. Experiments

We can now test the generalised expression (4.8) for $h_1$ against the experimental results of § 3. In order to do so, we first need to compute $H$ from the time-evolution of ${\mathsf{A}}_{zz}$. Note that we do not experimentally measure the extension of polymer chains, unlike Ingremeau & Kellay (Reference Ingremeau and Kellay2013), 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 ${\mathsf{A}}_{zz}(t)$ using (4.4), where the extension rate $\dot {\epsilon } = -2 \dot {h}/h$ is 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 (${\mathsf{A}}_{zz} \not \gg 1$) since the point at which ${\mathsf{A}}_{zz}$ 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 (Reference Bazilevsky, Entov and Rozhkov2001), introducing a function $y(t)$ such that ${\mathsf{A}}_{zz} = y \exp {(-t/\tau )} / h^4$, which leads to $\dot {y} = h^4 \exp {(t/\tau )} / \tau$, yielding

(4.9)\begin{equation} {\mathsf{A}}_{zz} = \frac{{\rm e}^{{-}t/\tau}}{h^4} \left( h_0^4 \, {\rm e}^{t_0/\tau} + \frac{1}{\tau} \int_{t_0}^t h^4(t')\, {\rm e}^{t'/\tau} \,\mathrm{d} t' \right), \end{equation}

where the initial time $t_0$ corresponds to the onset of capillary thinning, i.e. $h(t_0) = h_0$ and ${\mathsf{A}}_{zz}(t_0)=1$ (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 $\tau$. In the following, we use either the apparent ($\tau _e$) or the maximum ($\tau _m$) relaxation time measured experimentally (see figure 4a) to calculate ${\mathsf{A}}_{zz}$ since we still do not know which is the ‘true’ one, if any.

Values of ${\mathsf{A}}_{zz}(t)$ computed from (4.9) using the experimental values of $h(t)$ with relaxation time $\tau = \tau _m$ are shown in figure 7(a) for the ${\rm PEO}_{{aq,2}}$ solution, and in figure 7(b) for the ${\rm PEO}_{{visc,2}}$ solution, for plate diameters $2R_0$ between $2$ and $20\,{\rm mm}$. The experimental values of $h(t)$ are shown on the left-hand $y$-axis, and the time reference $t_1$ corresponds to the onset of the elastic regime. We find that the amount of polymer extension $A_1 = {\mathsf{A}}_{zz}(t_1)$ at the onset of the elastic regime is fairly independent of the initial condition for the ${\rm PEO}_{{aq,2}}$, while for the ${\rm PEO}_{{visc,2}}$ solution, $A_1$ decreases as $h_0(R_0,V^*)$ increases, as mentioned in our previous paper (Gaillard et al. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024), and as we are about to discuss here in more depth. In any case, we find that ${\mathsf{A}}_{zz}$ always increases as $1/h^4$ in the Newtonian regime close enough to the transition to the elastic regime. More specifically, values of ${\mathsf{A}}_{zz}$ are well captured by $(H/h)^4$ using $H$ as a fitting parameter for each data set (close to $t_1$), from which $H$ is estimated; see figure 7.

Figure 7. Time evolution of the experimental minimum bridge/filament radius $h$ and of ${\mathsf{A}}_{zz}$, calculated from the Oldroyd-B prediction (4.9) using the experimental values of $h(t)$ with the choice of relaxation time $\tau = \tau _{m}$ (see figure 4a), for plate diameters $2R_0 = 2$, $5$, $10$ and $25\,{\rm mm}$ for the (a) ${\rm PEO}_{{aq,2}}$ and (b) ${\rm PEO}_{{visc,2}}$ solutions. Time $t_1$ marks the onset of the elastic regime, with $h_1 = h(t_1)$ and $A_1 = {\mathsf{A}}_{zz}(t_1)$. Values of ${\mathsf{A}}_{zz}$ in the Newtonian regime ($t< t_1$) are compared to $(H/h)^4$ (see (4.6)), where $H$ is used as a fitting parameter to optimise the agreement close to $t_1$.

Values of $H$ calculated using $\tau = \tau _m$, named $H(\tau _m)$, are plotted against $h_0$ in figure 8(a) for the ${\rm PEO}_{{aq,2}}$, ${\rm PEO}_{{visc,2}}$ and HPAM solutions, which are the only solutions for which sufficiently large plate diameters were used to estimate the maximum relaxation time $\tau _m$ (the high-$h_0$ limit of $\tau _e$; see figure 4a). For the ${\rm PEO}_{{aq,2}}$ and HPAM solutions, we find that $H$ is essentially equal to $h_0$ at low $h_0$, and that $H/h_0$ decreases as $h_0$ increases, down to $0.73$ for the largest plate diameter. In contrast, for the ${\rm PEO}_{{visc,2}}$ solution, the ratio $H/h_0$ takes significantly smaller values, decreasing from $0.56$ to $0.40$ as $h_0$ increases. This is why the $(H/h)^4$ fit for ${\mathsf{A}}_{zz}$ is fairly good throughout the whole Newtonian regime for the ${\rm PEO}_{{aq,2}}$ solution in figure 7(a), while it is valid only within a small time window close to the transition to the elastic regime for the ${\rm PEO}_{{visc,2}}$ solution in figure 7(b). Indeed, if $H=h_0$, then the $(H/h)^4$ fit for ${\mathsf{A}}_{zz}$ is even valid at the onset of capillary thinning where $h=h_0$ and ${\mathsf{A}}_{zz} = 1$.

Figure 8. Values of $H$ estimated from the time evolution of ${\mathsf{A}}_{zz}$ (calculated using Oldroyd-B; see figure 7) using (a) the maximum relaxation time $\tau =\tau _m$ or (b) the effective relaxation time $\tau =\tau _e$, for different polymer solutions and initial bridge radii $h_0(R_0,V^*)$, plotted against $h_0$. The line $H=h_0$ is shown in both plots.

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 chains stretch almost without relaxing in the Newtonian regime for the low solvent viscosity solutions (${\rm PEO}_{{aq,2}}$ and HPAM), while relaxation is not negligible for the high solvent viscosity (${\rm PEO}_{{visc,2}}$) solution. This is because the Newtonian thinning dynamics is slower for the most viscous solution; see e.g. figure 7, where, for $2R_0 = 2\,{\rm mm}$, the bridge takes only approximately $6$ ms to thin from $h_0$ to $h_1$ for the ${\rm PEO}_{{aq,2}}$ solution, much less than $\tau _m$ (chains do not have enough time to relax), while it takes approximately $900$ ms for the ${\rm PEO}_{{visc,2}}$, much more than $\tau _m$ (not visible in figure 7(b), where we focus on times close to $t_1$). The fact that $H/h_0$ increases as $h_0$ increases can therefore be interpreted by a longer time to thin from $h_0$ to $h_1$ as $h_0$ increases, consistent with the fact that the Rayleigh and viscous time scales $\tau _R = (\rho h_0^3 / \gamma )^{1/2}$ and $\tau _{{visc}} = \eta _0 h_0 / \gamma$ both increase with $h_0$.

The dependence of $H$ on $h_0$ and on the solvent viscosity explains why, in figure 7 and in our previous paper (Gaillard et al. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024), $A_1$ is independent of $h_0$ for low solvent viscosity solutions (${\rm PEO}_{{aq,2}}$ and HPAM), while $A_1$ decreases as $h_0$ increases for the high solvent viscosity solution (${\rm PEO}_{{visc,2}}$). For the former, where $H$ is close to $h_0$ (negligible polymer relaxation in the Newtonian regime), $A_1 = (H/h_1)^4$ (see (4.7)) is close to $(h_0/h_1)^4$, which is fairly constant since $h_1 \propto h_0$ according to figure 4(b). For the latter, however, where $H \propto h_0^{0.74}$ in our range of $h_0$ according to figure 8(a), we get $A_1 = (H/h_1)^4 \propto h_0^{-1.04}$.

In figure 8(b), we plot the values of $H$, named $H(\tau _e)$, calculated using the apparent relaxation time $\tau = \tau _e$ when computing ${\mathsf{A}}_{zz}$ from (4.9) (instead of its large-$h_0$ limit $\tau _m$ in figure 8a). All polymer solutions are now featured since $\tau _e$ is 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 ${\rm PEO}_{{aq,2}}$, ${\rm PEO}_{{visc,2}}$ and HPAM solutions, as in figure 8(a), but also the ${\rm PEO}_{{aq,1}}$ and ${\rm PEO}_{{visc,1}}$ solutions, 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 $(R_0,V^*)$ set corresponding to $h_0 \approx 460\,\mathrm {\mu }{\rm m}$. The data corresponding to the PEO solutions with different polymer concentrations $c$ show how, for a given flow history in the Newtonian regime (same $h(t)$ curves for $t< t_1$ for all concentrations; see figure 5), $H$ increases with $c$ via the increase in the relaxation time (here $\tau = \tau _e$), reaching the upper limit value $h_0$ at large $\tau$. This is because for large $\tau$ values, polymer relaxation is negligible throughout the whole Newtonian regime, while for low $\tau$ values, ${\mathsf{A}}_{zz}$ remains equal to $1$ for most of the Newtonian regime, increasing only when $\dot {\epsilon }$ is finally of the order of $1/2\tau$ close 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 $h_1$ at the onset of the elastic regime. The value of the elastic modulus, $G = \eta _p/\tau$ in the Oldroyd-B model, is, however, not uniquely defined, since while $\eta _p = \eta _0 - \eta _s$ can be calculated unambiguously from the shear rheology, the relaxation time $\tau$ could be either the apparent one $\tau _e$ or the maximum one $\tau _m$, 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

(4.10)\begin{equation} G_H = \gamma h_1^3 / H^4, \end{equation}

where $h_1$ is the value measured experimentally, which should be $G_H = G/p$ according to (4.8).

In figure 9(a), $G_H$ is plotted against $G$ for the choice of relaxation time $\tau = \tau _m$ for the ${\rm PEO}_{{aq,2}}$, ${\rm PEO}_{{visc,2}}$ and HPAM solutions, which are the only solutions for which sufficiently large plate diameters were used to estimate $\tau _m$. More precisely, values of $G_H(\tau _m)$ calculated from $H(\tau _m)$ are plotted against $G(\tau _m) = \eta _p / \tau _m$, which takes a unique value for each solution since the relaxation time is unique. We find that values of $G_H(\tau _m)$ are, however, not unique for the ${\rm PEO}_{{aq,2}}$ and HPAM solutions, and monotonically decrease as $h_0$ increases. For the ${\rm PEO}_{{visc,2}}$ solution, however, values of $G_H$ vary between $0.11$ and $0.23$ without clear monotonic trend as $h_0$ increases. This is because, since $h_1 \propto h_0$ and $H \propto h_0^{k_H}$ in the range of $h_0$ values investigated, with $k_H \le 1$ (see figures 4b and 8a), $G_H \propto h_1^3 / H^4 \propto h_0^{3-4k_H}$, which means that $G_H$ is expected to be independent of $h_0$ only for $k_H = 0.75$, very close to the value $0.74$ found for the ${\rm PEO}_{{visc,2}}$ solution in figure 8(a). This suggests that the prediction of (4.8) for the choice $\tau = \tau _m$ is potentially valid for only one of our three solutions, the most dilute and viscous one.

Figure 9. Values of $G_H$ defined in (4.10) against the elastic modulus $G = \eta _p/\tau$, where $H$ and $G$ are calculated from (a) the maximum relaxation time $\tau = \tau _m$ or (b) the effective relaxation time $\tau = \tau _e$, for various polymer solutions and initial bridge radii $h_0(R_0,V^*)$. Values of $h_1$ are those measured experimentally. In (b), for the aqueous PEO solutions of different concentrations (see table 1), we show the effect replacing $\tau _e$ by the Zimm relaxation time $\tau _Z$ when calculating $G$ and $H$ for the lowest concentrations $c = 5$ and $10\,{\rm ppm}$, which exhibit values of $\tau _e < \tau _Z$; see figure 6(a). The line $G_H = 3.7 G$ is shown in both plots.

By contrast, as shown in figure 9(b), when choosing $\tau _e$ instead of $\tau _m$ for the relaxation time to calculate $G_H(\tau _e)$ from $H(\tau _e)$ and $G(\tau _e) = \eta _p/\tau _e$, data points fall on a single curve for all three solutions (${\rm PEO}_{{aq,2}}$, ${\rm PEO}_{{visc,2}}$ and HPAM) as well as for the ${\rm PEO}_{{aq,1}}$ and ${\rm PEO}_{{visc,1}}$ solutions (for which $V^*$ is varied for the four smallest plate diameters), and we find the linear relationship $G_H = G/p$ predicted by (4.8) with $p \approx 0.27$. It is quite 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 $G_H = 3.7 G$ line can be observed in figure 9(b) at low 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 $c < 100\,{\rm ppm}$, $G_H$ decreases sharply with concentration, while is it almost constant for $100\,{\rm ppm} \le c < 2500\,{\rm ppm}$, and increases sharply with concentration for $c \ge 2500\,{\rm ppm}$. These trends can be explained by the fact that $G_H \propto h_1^3/H^4$, where $h_1$ increases very slowly with concentration as $h_1 \propto c^{0.16}$ for $c<2500\,{\rm ppm}$, and increases sharply for higher concentrations (see figure 6b), while $H$ increases sharply with concentration for typically $c < 100\,{\rm ppm}$, becoming almost constant and equal to $h_0$ for larger concentrations (see figure 8b). These strong deviations from the $G_H = 3.7 G$ line 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 $\tau _Z = 2$ ms for $c = 5$ and $10\,{\rm ppm}$ (see figure 6a). In figure 9(b), we show the effect of choosing $\tau = \tau _Z$ instead of $\tau _e$ as the relaxation time for $c = 5$ and $10\,{\rm ppm}$, which changes the value of both $G_H$ via $H(\tau )$, and $G = \eta _p/\tau$. We find that this correction leads to data points significantly closer to the $G_H = 3.7 G$ line, mainly stemming from values of $H$ larger than in figure 8(b), although one order of magnitude deviation from the $G_H = 3.7 G$ line remains. We show in § 5 that finite extensibility effects can explain this deviation (i.e. values of $h_1$ higher than the Oldroyd-B prediction) as well as the values of $\tau _e < \tau _Z$ for low polymer concentrations.

The large deviation from the $G_H = 3.7 G$ line for the entangled $10\,000\,{\rm ppm}$ solution in figure 9(b) (the only solution for which polymers affect the thinning dynamics even before the exponential regime; see figure 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 ($t< t_1$), (1.2) should be replaced by (4.8), which gives $h_1 \propto H^{4/3}$ where $H \le h_0$. In our experiments, the power-law dependence of $H$ on $h_0$ (see figure 8) leads to the fairly proportional relationship between $h_1$ and $h_0$ observed in figure 4(b), differences in slopes among different liquids stemming from differences in elastic moduli $G$. Now that we have established that it is $H$ (and not $h_0$) 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 $h_1$ marking the onset of the elastic regime, we now consider numerical simulations using the Oldroyd-B model ($L^2 = + \infty$) with a single relaxation time $\tau$ (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. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024).

In order to capture $h_1$, we first need to capture the critical bridge radius $H$ at which polymer relaxation (the right-hand side of (4.3) or (2.8)) becomes negligible in the Newtonian regime ($t< t_1$). The bridge radius H marks the onset of the coil–stretch transition at which polymer chains start to extend significantly beyond their equilibrium shape, i.e. at which ${\mathsf{A}}_{zz}$ starts becoming significantly larger than $1$; see § 4.1. We already know that $H \to h_0$ in the limit where the relaxation time $\tau$ is 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 $h/h_0$ for a fixed Ohnesorge number ${Oh} = 2.07$, viscosity ratio $S = 0.988$ and $h_0/R_0 = 0.23$, 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 ${\mathsf{A}}_{zz}$ at this minimum-radius position along the bridge/filament since this is where polymer chains are the most stretched. This maximum value, simply denoted ${\mathsf{A}}_{zz}$ from now on, is plotted in figure 10 on the right-hand $y$-axis. Note that all $h/h_0$ curves are identical in the Newtonian regime, diverging only at the transition to the elastic regime at different radii $h_1$. The time reference $t_c$ corresponds 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 $h/h_0 = 0.0709(t_c-t)/({Oh}\, \tau _R)$ 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. Numerical time evolution of the non-dimensional (minimum) bridge/filament radius $h/h_0$ and of the (maximum) polymer extension ${\mathsf{A}}_{zz}$ for ${Oh} = 2.07$, $S=0.988$ and $h_0/R_0 = 0.23$, and three different Deborah numbers. Values of ${\mathsf{A}}_{zz}$ are compared with $(H/h)^4$ close to the onset of the elastic regime, where $H$ is used as a fitting parameter. The self-similar viscous regime of (3.2), or equivalently $h/h_0 = 0.0709 (t_c-t)/({Oh}\,\tau _R)$, is also plotted, where $t_c$ is the time at which the filament would break if the transition to an elastic regime, at $h=h_1$, did not occur.

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 ${De} \gg 1$, relaxation is always negligible in the Newtonian regime, and ${\mathsf{A}}_{zz}$ therefore increases as $(H/h)^4$ where $H=h_0$; see the discussion in § 4.1. For ${De} \ll 1$, however, the flow 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 ${Oh} \gg 1$, follows (3.2). In that case, ${\mathsf{A}}_{zz} = 1$ for most of the Newtonian regime, increasing only close to the transition to the elastic regime, following ${\mathsf{A}}_{zz} = (H/h)^4$ where $H \ll h_0$ is the characteristic bridge radius at which ${\mathsf{A}}_{zz}$ starts increasing. Values of $H$ are estimated by fitting ${\mathsf{A}}_{zz}$ with $(H/h)^4$, using $H$ as a fitting parameter (see figure 10), as was done in figure 7 for experimental results.

Values of $H/h_0$ are 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 $h = B (t_c-t)^{\beta }$, with $B \sim (\gamma / \rho )^{1/3}$ and $\beta = 2/3$ in the inviscid limit (${Oh} \ll 1$; see (3.1)), and $B \sim \gamma / \eta _0$ and $\beta = 1$ in the viscous limit (${Oh} \gg 1$; see (3.2)). The coil–stretch transition occurs when the extension rate $\dot {\epsilon } = -2 \dot {h}/h = 2 \beta /(t_c-t)$ becomes of the order of $1/\tau$, i.e. at a time $t_H = t_c - 2 \beta \tau$. Therefore, the bridge radius $H = h(t_H)$ marking the onset of the coil–stretch transition scales as

(4.11)\begin{equation} H \sim (\gamma \tau^2 / \rho)^{1/3} \quad \Leftrightarrow \quad H/h_0 \sim {\textit{De}}^{2/3} \end{equation}

in the inviscid limit (${Oh} \ll 1$), as first derived by Campo-Deano & Clasen (Reference Campo-Deano and Clasen2010), or as

(4.12)\begin{equation} H \sim \gamma \tau / \eta_0 \quad \Leftrightarrow \quad H/h_0 \sim {\textit{De}}/{\textit{Oh}} = \tau/\tau_{{visc}} \end{equation}

in the viscous limit (${Oh} \gg 1)$. The scaling of (4.12) is shown in figure 11(a) with a prefactor $0.2$, and shows good agreement with the values of $H$ for the two largest Ohnesorge numbers. Unfortunately, no simulations could be performed for ${Oh} \ll 1$ to test (4.11). Note that in this peculiar limit where polymer chains start stretching only within the self-similar thinning regime, $H$ – and therefore $h_1$ given by (4.8) – does not depend on $h_0$ and is therefore independent of the size of the system, in sharp contrast with the high-${De}$ limit, where $H=h_0$ and therefore $h_1 \propto h_0^{4/3}$; see (1.2). In fact, inserting (4.11) or (4.12) into (4.8) gives

(4.13)\begin{equation} h_1 \sim (G (\gamma \tau^8 / \rho^4)^{1/3} )^{1/3} \end{equation}

in the inviscid limit (${Oh} \ll 1$), or

(4.14)\begin{equation} h_1 \sim (G \gamma^3 \tau^4 / \eta_0^4)^{1/3} \end{equation}

in the viscous limit (${Oh} \gg 1)$.

Figure 11. Numerical values of $H/h_0$ against (a) the Deborah number ${De} = \tau /\tau _R$ based on the inertio-capillary time scale $\tau _R$, and (b) the general Deborah number ${De}_N = \tau /\tau _N$ based on the general time scale $\tau _N = \tau _R (1 + \alpha {Oh})$ with $\alpha = 4.3$, for various parameters (the same in (a) and (b)). Dots ($\bullet$) correspond to $h_0/R_0 = 0.23$ and $S = 0.988$, with ${Oh} = 0.207$ (blue) ${Oh} = 2.07$ (purple) and ${Oh} = 20.7$ (red), and ${De}$ ranging between $8.92 \times 10^{-2}$ and $8.92 \times 10^{3}$ (last ${De}$ excluded for the largest ${Oh}$). Triangles ($\blacktriangle$) correspond to $h_0/R_0 = 0.23$ with ${De} = 8.92 \times 10^{-2}$ (yellow) and ${De} = 8.92 \times 10^{1}$ (green), varying both $S$ (between $0.1$ and $0.988$) and ${Oh}$ while keeping $S\,{Oh}$ constant and equal to $9.88$. Stars ($\star$) correspond to $h_0/R_0 = 0.362$ with ${Oh} = 16.6$ and $S = 0.988$, and ${De}$ ranging between $4.59 \times 10^{-2}$ and $8.92 \times 10^{3}$.

In the high-${De}$ limit, $H/h_0 \to 1$ since polymer relaxation becomes negligible even at the onset of capillary thinning where $h=h_0$. However, while all curves in figure 11(a) have the same shape, the Deborah number at which $H/h_0$ reaches $1$ depends on the Ohnesorge number. This is because we chose to express the Deborah number as ${De} = \tau /\tau _R$, where $\tau _R = (\rho h_0^3 / \gamma )^{1/2}$ is 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 $\tau _{{visc}} = {Oh}\,\tau _R = \eta _0 h_0 / \gamma$, and we would hence expect that $H/h_0 = O(1)$ not when $\tau /\tau _R = O(1)$ but when $\tau /\tau _{{visc}} ( = {De} / {Oh} ) = O(1)$. In figure 11(b), we show that values of $H/h_0$ indeed rescale on a single curve when plotted against a generalised Deborah number ${De}_N = \tau /\tau _N$, where $\tau _N$, defined as

(4.15)\begin{equation} \tau_N = \tau_R (1 + \alpha\,{\textit{Oh}}), \end{equation}

is an empirical attempt at expressing the general time scale of the thinning dynamics in the Newtonian regime for any ${Oh}$, connecting the low- and high-${Oh}$ scalings $\tau _R$ and $\tau _{{visc}}$, where $\alpha = 4.3$ is a fitting parameter. This scaling ensures that $H/h_0 = O(1)$ when ${De}_N = O(1)$ for any Ohnesorge number. However, according to (4.11) and (4.12), we expect different scalings for ${De}_N \ll 1$, namely, $H/h_0 \sim {De}_N^{2/3}$ for ${Oh} \ll 1$ and $H/h_0 \sim {De}_N$ for ${Oh} \gg 1$.

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 $h_0/R_0 = 0.23$. In order to further investigate the generality of the $H/h_0$ dependence on ${De}_N$ identified in figure 11(b), we therefore performed additional simulations. Two sets of simulations were performed for ${De} = 0.089$ and $89.2$, respectively, keeping $h_0/R_0 = 0.23$, where both $S$ and ${Oh}$ were varied while keeping $S\,{Oh} = \eta _s/\sqrt {\rho \gamma h_0}$ constant and equal to $9.88$. In each case, $S$ is varied between $0.1$ and $0.988$, where the total (constant shear) viscosity $\eta _0 = \eta _s + \eta _p$ is respectively dominated by the polymer and by the solvent contribution. All these additional data points in figure 11(a) rescale on the same curve identified in figure 11(b). This is because all these cases correspond to ${Oh} \gg 1$, where $\tau _{{visc}} = \eta _0 h_0 / \gamma$ is the relevant time scale of the thinning dynamics in the Newtonian regime (and not $\eta _s h_0 / \gamma$, for example), regardless of the value of $S$.

Additionally, a set of simulations was performed for a larger (non-dimensional) sample volume corresponding to $h_0/R_0 = 0.36$, varying ${De}$ for fixed ${Oh} = 16.6$ and $S = 0.988$. These additional data points in figure 11(a) also rescale on the same curve identified in figure 11(b). This is because $h_0$ is the relevant length scale of the problem and sets $\tau _N$ (see (4.15)), in agreement with our experimental results, which show that $h_1$ increases when increasing the sample volume for a given plate diameter due to the increase in $h_0$ (see figure 4b).

We can finally test the prediction of (4.8) for the transition radius $h_1$ between the Newtonian and elastic regimes, which in non-dimensional terms reads

(4.16)\begin{equation} \frac{h_1}{h_0} = \left( \frac{H}{h_0} \right)^{4/3} \left[ \frac{(1-S)\,{\textit{Oh}}}{p\,{\textit{De}}} \right]^{1/3}. \end{equation}

Figure 12(a) shows that $h_1/h_0$, which is estimated from the numerical $h/h_0$ curves for the same sets of parameters as in figure 11, does not monotonically increase or decrease with ${De}$. This is because at low ${De}$ (more specifically at low ${De}_N = \tau /\tau _N$; see figure 11), $H/h_0 \propto {De}$ for ${Oh} \gg 1$ according to (4.12), implying that $h_1/h_0 \propto {De}$ according to (4.16), while at high ${De}$$_N$, $H/h_0 = 1$, implying that $h_1/h_0 \propto {De}^{-1/3}$ according to (4.16). We find that all values of $h_1/h_0$ indeed rescale on a single master curve when potted against $(H/h_0)^{4/3} ((1-S)\,{Oh} / {De})^{1/3}$ in figure 12(b), where we find that the value of $p$ in (4.16) should be $p \approx 1.6$. Note that the value of $p$ depends on the exact definition of $h_1$, since the transition between the Newtonian and elastic regimes is not necessarily sharp.

Figure 12. Numerical (non-dimensional) transition radius $h_1/h_0$ against (a) the Deborah number ${De} = \tau /\tau _R$, and (b) $(H/h_0)^{4/3} ((1-S)\,{Oh} / {De})^{1/3}$. The dash-dotted line in (b) is the line of equation $y = 0.854 x$. The legend and range of parameters (${Oh}$, ${De}$, $S$ and $h_0/R_0$) are the same as in figure 11.

The validation of (4.16) proves that while (1.2) is valid for a fast plate separation protocol, relaxation of polymer chains must be taken into account to allow for values of $H< h_0$ for a slow plate separation protocol. It also proves that when we are interested in the minimum bridge/filament radius and maximum polymer extension at that point, the full two-dimensional problem can be reduced to a simple force balance equation such as (4.1) (which strictly applies only to a cylindrical thread) without losing predictive power, since (4.16) (or equivalently, (4.8)) is based on (4.1).

5.1. Simulations and theory

Figure 13(a) shows how, for fixed ${Oh}$, ${De}$, $S$ and $h_0/R_0$, decreasing $L^2$ leads to an increase in $h_1$. This can be seen as counterintuitive since a decrease in $L^2$ implies that chains are shorter and therefore less elastic, which should imply a delayed transition to the elastic regime (smaller $h_1$). As we discuss in § 6, this apparent contradiction is resolved by considering that shorter chains have shorter relaxation times, leading to smaller values of $H$ and therefore of $h_1$, which is not taken into account in the simulations of figure 13(a), where the Deborah number is kept constant.

Figure 13. (a) Numerical time evolution of $h/h_0$ using the FENE-P model with $L^2$ ranging between $10^2$ and $10^7$, as well as $L^2 = + \infty$ (Oldroyd-B limit), for fixed ${Oh} = 2.07$, ${De} = 44.6$, $S = 0.988$ and $h_0/R_0 = 0.23$. The inset is a zoomed version for a better visualisation of the transition to the elastic regime at $h=h_1$. Simulations start at $t=0$ used as the time reference. (b) Values of $Y - 1$ against $\varphi$ (see (5.2a–c)) (where $Y = h_1 / h_{1,{OB}}$, for $h_{1,{OB}}$ the Oldroyd-B limit of $h_1$) for $L^2$ ranging between $10^2$ and $10^8$, with ${Oh} = 0.207$, $2.07$ and $20.7$, and fixed values ${De} = 44.6$, $S = 0.988$ and $h_0/R_0 = 0.23$. Values are compared with the analytical solution of (5.2a–c) and with the limit scalings of (5.4).

Note that in figure 13(a), the thinning rate $\vert \dot {h}/h \vert$ in the elastic regime ($t>t_1$) is larger than $1/3\tau$ for the lowest $L^2$ values, while for the largest $L^2$ values, the elastic regime initially follows (1.1). This is because polymer chains are already close to being fully extended at the onset of the elastic regime for the lowest $L^2$ values, as discussed in our previous paper (Gaillard et al. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024), where we explored the possibility of invoking this effect to explain variations of the apparent relaxation time $\tau _e$ (see e.g. figures 3c,e).

The reason why $h_1$ increases as $L^2$ decreases for a fixed ${De}$ is because the elastic stress $\sigma _{p,zz} \approx G f {\mathsf{A}}_{zz}$ (assuming ${\mathsf{A}}_{zz} \gg 1$ at the transition; see (2.7a,b)) increases faster during the Newtonian regime ($t< t_1$) as $L^2$ decreases. This is because $f \approx 1/(1-{\mathsf{A}}_{zz}/L^2)$ (assuming ${\mathsf{A}}_{zz} \gg 1 > {\mathsf{A}}_{rr}$) diverges as ${\mathsf{A}}_{zz}$ approaches $L^2$ to model the stiffening of polymer chains as they approach full extension. Formally, assuming that the transition occurs when the elastic stress reaches a fraction $p$ of the capillary pressure, i.e. when $\sigma _{p,zz} = p \gamma / h$ (where the ($2X-1$) prefactor is integrated into $p$; see § 4.1), we get

(5.1)\begin{equation} p\,\frac{\gamma}{h_1} = G\,\frac{A_1}{1 - A_1/L^2}, \end{equation}

where $h_1 = h(t_1)$ and $A_1 = {\mathsf{A}}_{zz}(t_1)$ are the values at the transition. We assume that the bridge radius $H$ marking the onset of the coil–stretch transition is unaffected by finite extensibility effects since ${\mathsf{A}}_{zz}$ is still close to 1 at the onset of this coil-stretch transition, i.e. $f \approx 1$. Assuming that relaxation becomes negligible between the onset of the coil–stretch transition and the onset of the elastic regime, i.e. for $h_1 < h < H$, we use ${\mathsf{A}}_{zz} = (H/h)^4$ (see (4.6)), from which we get $A_1 = (H/h_1)^4$. Injecting this scaling into (5.1) leads to a polynomial equation for $h_1$ in the form

(5.2a–c)\begin{equation} h_1 = h_{1,{OB}} \times Y(\varphi) , \quad Y (Y^3 - 1) = \varphi, \quad \varphi = \frac{A_{1,{OB}}}{L^2}, \end{equation}

where

(5.3a,b)\begin{equation} h_{1,{OB}} = \left( \frac{G H^4}{p \gamma} \right)^{1/3} \quad \mathrm{and} \quad A_{1,{OB}} = \left( \frac{H}{h_{1,{OB}}} \right)^4 \end{equation}

are the values of $h_1$ and $A_1$ predicted in the Oldroyd-B limit $L^2 = + \infty$ ($\varphi = 0$); see (4.8) for $h_{1,{OB}}$. The two limit scalings of (5.2a–c) corresponding to weak ($\varphi \ll 1$) and strong ($\varphi \gg 1$) finite extensibility effects are

(5.4)\begin{equation} Y = \begin{cases} 1 + \varphi/3, & \text{for } \varphi \ll 1, \\ \varphi^{1/4}, & \text{for } \varphi \gg 1 . \end{cases} \end{equation}

In the weak limit ($\varphi \ll 1$), polymer chains are still far from full extension at the onset of the elastic regime, while in the strong limit ($\varphi \gg 1$), the transition occurs significantly sooner than the Oldroyd-B prediction due to the fact that polymer chains have almost already reached full extension at the transition. Indeed, in the strong limit, the transition occurs when ${\mathsf{A}}_{zz} = (H/h)^4$ becomes of the order of $L^2$, leading to

(5.5)\begin{equation} h_1 \sim \frac{H}{L^{1/2}}, \end{equation}

which is equivalent to (5.4) for $\varphi \gg 1$.

Values of $Y$ estimated from numerical simulations using the FENE-P model are plotted against $\varphi$ in figure 13(b) for three different Ohnesorge numbers, varying $L^2$ between $10^2$ and $10^8$ in each case, while keeping ${De}$, $S$ and $h_0/R_0$ constant, using an extra Oldroyd-B simulation as a reference to get $h_{1,{OB}}$. We find that all data points collapse on a single curve corresponding to the solution of (5.2a–c). In particular, no prefactor is required in the $\varphi \gg 1$ limit. Note that we choose to plot $Y - 1$ instead of $Y$ in order to better visualise the $\varphi \ll 1$ regime. Note that the values of $H/h_0$, estimated for each simulation as in figure 10, are found not to depend on $L^2$, and take values between $1$ and $0.36$ for the data of figure 13(b). Equations (5.2a–c) therefore generalise (1.2) to cases where both finite extensibility effects and polymer relaxation effects are not negligible. In § 5.2, we apply this updated theory to rationalise some of the discrepancies discussed in § 4.2 between the Oldroyd-B theory and experiments.

5.2. Experiments

Experimentally, cases where both polymer relaxation and finite extensibility effects are expected to play a role correspond to low polymer concentrations $c$. Indeed, as $c$ decreases, the transition to the elastic regime is delayed (i.e. $h_1$ decreases; see figures 5b and 6b) and polymer chains are therefore expected to be increasingly stretched at the onset of the elastic regime, assuming that their relaxation time becomes equal to the Zimm relaxation time $\tau _Z$, which is independent of polymer concentration $c$ (see (3.3)). Polymer chains should therefore ultimately approach full extension (at the onset of the elastic regime) below a critical concentration $c_{{low}}$ introduced by Campo-Deano & Clasen (Reference Campo-Deano and Clasen2010). For $c< c_{{low}}$, the elasto-capillary balance leading to the exponential decay of (1.1) is therefore no longer valid since $A_1 \sim L^2$, leading to filament thinning rates $\vert \dot {h}/h \vert > 1/3\tau$, as shown numerically in figure 13(a) and discussed in our previous paper (Gaillard et al. Reference Gaillard, Herrada, Deblais, Eggers and Bonn2024). This would explain why fitting the elastic regime ($t>t_1$) with a exponential leads to apparent relaxation times $\tau _e$ that are smaller than $\tau _Z$, as reported in figure 6(a) for aqueous PEO-4M solutions of concentrations $c \le 10\,{\rm ppm}$. This would also explain why, as discussed in § 4.2 (see triangle symbols in figure 9b), transition radii $h_1$ measured for low polymer concentrations cannot be captured by the Oldroyd-B prediction even when replacing values of $\tau _e<\tau _Z$ by $\tau _Z$.

This idea is tested in figure 14, where experimentally measured values of $h_1$ ($h_{1,{exp}}$) are plotted against the FENE-P theoretical prediction $h_{1,{th}} = h_{1,{OB}} \times Y(\varphi )$ (see (5.2a–c) and (5.3a,b)) for all polymer solutions and initial bridge radii. We choose $L^2 = \infty$ ($Y = 1$) and $\tau = \tau _e$ as model parameters for data points corresponding to the ${\rm PEO}_{{aq}}$ (1 and 2, $500\,{\rm ppm}$ PEO-4M solution in water), ${\rm PEO}_{{visc}}$ (1 and 2, $25\,{\rm ppm}$ PEO-4M solution in a more viscous solvent) and HPAM solutions, since we already know from figure 9(b) that these $h_1$ values are consistent with the Oldroyd-B prediction of (4.8) (we choose $p = 0.27$ to match the prefactor found in figure 9b). The experimentally measured values of $h_1$ corresponding to aqueous PEO-4M solutions of various concentrations in figure 14 (orange triangle symbols) are higher than the Oldroyd-B prediction at low concentrations, as we saw in figure 9(b), where replacing values of $\tau _e < \tau _Z$ by $\tau _Z$ was found to be insufficient to explain the discrepancy. The blue triangle symbols in figure 14 show that this discrepancy can be rationalised using the FENE-P model, where we chose a value $L^2 = 1 \times 10^4$ that is sufficiently small to allow for values of $Y$ sufficiently larger than $1$ (i.e. polymer chains close to being fully extended at the onset of the elastic regime) to ‘fill the remaining gap’, while using $\tau = \max (\tau _e,\tau _Z)$ at the model relaxation time.

Figure 14. Experimentally measured $h_1$ values ($h_{1,{exp}}$) against the FENE-P theoretical prediction $h_{1,{th}} = h_{1,{OB}} \times Y(\varphi )$ (see (5.2a–c) and (5.3a,b)) for various polymer solutions and initial bridge radii. Values of the FENE-P model parameters $\tau$ and $L^2$ are indicated in the legends ($\eta _p$ values are the ones from shear rheology). The discrepancy between experiments and the Oldroyd-B prediction for $h_1$ at low polymer concentration can be rationalised by finite extensibility effects; see main text.

This value $L^2 = 1 \times 10^4$ has the same order of magnitude as the value expected from the microscopic formula (Clasen et al. Reference Clasen, Plog, Kulicke, Owens, Macosko, Scriven, Verani and McKinley2006b)

(5.6)\begin{equation} L^2 = 3 \left[ \frac{j \sin^2(\theta/2)\,M_w}{C_{\infty} \, M_u} \right]^{2(1-\nu)} , \end{equation}

which gives $L^2$ between $1.6 \times 10^{4}$ and $1.3 \times 10^{5}$ for PEO of molecular weight $M_w = 4 \times 10^{6}\,{\rm g}\,{\rm mol}^{-1}$, for solvent quality exponents $\nu$ between $3/5$ (good solvent) and $1/2$ (theta solvent), where $M_u$ is the monomer molecular weight, $\theta = 109^{\circ }$ is the C-C bond angle, $j=3$ is the number of bonds of a monomer, and $C_{\infty } = 4.8$ is the characteristic ratio (Brandrup & Immergut Reference Brandrup and Immergut1999). Assuming that $\nu$ is between $1/2$ and $3/5$, the discrepancy can be explained by a value $M_w$ less than $4 \times 10^{6}\,{\rm g}\,{\rm mol}^{-1}$ stemming from polymer degradation during mixing when preparing the stock solution. Note that choosing $L^2 = 1 \times 10^4$ leads to values of $Y$ close to $1$ (up to $1.18$) for the ${\rm PEO}_{{aq}}$ (1 and 2), ${\rm PEO}_{{visc}}$ (1 and 2) and HPAM solutions, consistent with the agreement between experiments and the Oldroyd-B theory for these solutions in figure 14, as shown by the small difference between the $L^2 =\infty$ (orange) and $L^2 = 1 \times 10^4$ (blue) triangle symbols for $c = 500\,{\rm ppm}$, which corresponds to the ${\rm PEO}_{{aq,1}}$ solution.