DoS Dos and Don'ts

TL;DR

This work establishes a quantitative framework for DoS rheometry by deriving theoretical and experimental limits on measuring the transient extensional viscosity $η_E^+(t)$ and the extensional relaxation time $τ_E$ in dilute polymer solutions. It introduces the filament capture rate as a practical figure of merit and presents both model-specific and model-agnostic operability diagrams to guide experimental design. Experiments with dilute PEO and PAM demonstrate sub-millisecond $τ_E$ measurements (theoretically down to ≈$0.04$ ms, experimentally ≈$0.23$ ms under typical imaging) and reveal gravity via Bond number effects on early thinning dynamics while leaving the elastocapillary regime largely intact. Collectively, the results provide a robust, actionable roadmap for reliably applying DoS rheometry to weakly elastic fluids and for planning measurements that resolve rapid relaxation processes.

Abstract

Dripping-onto-Substrate (DoS) rheometry is a well-established method for measuring the extensional rheology of low-viscosity liquids. However, clear guidelines on the capabilities and limitations of the technique are lacking. In the present work, we define operational limits for measuring a transient extensional viscosity directly from observation of the rate of filament thinning, as well as model-based bounds on calculating a viscosity $η$ and extensional relaxation time $τ_E$ of a liquid using DoS. Dilute solutions of polyethylene oxide (PEO) and polyacrylamide (PAM) are used to probe the lower limit of measurable $τ_E$, demonstrating that values as low as 0.1 ms can be resolved, provided (a) the intrinsic Deborah number (based on the ratio of the relaxation time and the Rayleigh breakup time scale) is $De \geq \mathcal{O}(0.1)$ and (b) an instrumental constraint related to spatial and temporal resolution is satisfied. This instrumental constraint is quantified through a new metric we define as the \textit{filament capture rate}, a ``figure of merit'' (expressed in Hz) that can be used to quantify the number of data points within the elasto-capillary regime that are available for extraction of $τ_E$. We also investigate the sensitivity to other experimental parameters including variations in nozzle radius and Bond number ($Bo$). Across the tested range ($0.2 < Bo < 0.7$), extensional relaxation times for the same fluid vary by less than $\pm16$ \%; however, experiments with low viscosity fluids at $Bo > 0.5$ exhibit damped gravitational oscillations that affect early-time dynamics. Collectively, these results provide a quantitative roadmap for reliable DoS rheometry and affirm its use for measuring sub-millisecond relaxation times in weakly elastic fluids.

Figures (7)

• Figure 1: A sample measurement of the minimum filament radius $R_{\mathrm{min}}$ versus time $t$ (represented by the black symbols) for an $M_v = 8.1$ MDa aqueous PEO solution with $c/c^* = 0.06$, for a pendant droplet expelled from a nozzle with $R_0 = 0.359$ mm. The gray shaded region represents the lowest measurable filament radius or $R_{\mathrm{min}}<R_{\mathrm{res}}$. The blue and red lines denote the time-dependent evolution in the minimum filament radius predicted in the inertio-capillary (IC) thinning regime $R_{\mathrm{ic}}$ from Eq. (\ref{['eqn:1']}) and in the elasto-capillary (EC) thinning $R_{\mathrm{ec}}$ domain from Eq. (\ref{['eqn:5']}), respectively.
• Figure 2: (a) Annotated three-dimensional model of the experimental DoS setup, and (b) a sample image taken from the high-speed camera for an 8 MDa PEO solution, with $c = 0.14c^*$, $R_0 = 0.635$ mm, and $\varLambda = 1.5$. The green contour line in (b) represents the radial profile $\hat{R}(\hat{z})$ extracted from the edge detection algorithm. The white dashed line in (b) represents the location of the glass substrate.
• Figure 3: Measurements of the reduced and inherent viscosity for (a) PEO $M_v = 8.1$ MDa, and (b) PAM with a reported molecular weight in the range of $M_v = 17-23$ MDa. Solid lines represent fits of the reduced viscosity using the Huggins Eq. (\ref{['eqn:26a']}), while dashed lines are fits of the inherent viscosity using the Kraemer Eq. (\ref{['eqn:26b']}).
• Figure 4: Plots of (a) the minimum filament radius $R_{\mathrm{min}}$ versus time $t$ and (b) the time-varying Weissenberg number $Wi$ (Eq. \ref{['eqn:9']}) versus time $(t-t^*)/\tau_E$ for the $M_v = 2.5$ MDa PEO solution at different concentrations $c/c^*$ and in the nozzle radius $R_0 = 0.359$ mm. Similarly (c) shows the evolution in the minimum radius $R_{\mathrm{min}}$ versus $t$ and (d) the evolution in the Weissenberg number $Wi$ versus $(t-t^*)/\tau_E$ for the $M_v = 17-23$ MDa PAM solution at different $c/c^*$ and $R_0 = 0.359$ mm. Solid coloured lines in (a) and (c) correspond to fits of the elasto-capillary (EC) regime using Eq. (\ref{['eqn:5']}), while the black dashed line is a fit of the inertio-capillary (IC) regime for the fluid with the lowest $c/c^*$. More information about the flows is listed in Table \ref{['tab:3']}. The grey shaded region in (a) and (c) denote $R_{\mathrm{res}} = 0.002$ mm. The black solid line in (b, d) represents $Wi = 2/3$ for the EC regime, where the grey region represents a 50 % confidence region. The dashed black line (b, d) represents the analytical expression for Weissenberg number in the IC regime $Wi(t) = \dot{\varepsilon}_{\mathrm{ic}}\tau_E$. Insets in (b, d) shows the shape of the filament, coloured according to the different $c/c^*$, while the dark shaded region represents the nozzle. All profiles in the inset of (b) correspond to the same time instance of $(t-t^*)/\tau_E = 3.5$, while the inset in (d) corresponds to $(t-t^*)/\tau_E = 2.0$. The bottom panels show images of the droplet thinning for the $c/c^* = 0.25$ PEO solution at different time instances $t$ are shown in the bottom panels. The time $t = 0$ corresponds to the time instance when the minimum radius equals the nozzle radius, $R_{\mathrm{min}} = R_0$.
• Figure 5: Filament thinning DoS measurements all for a $M_v = 8.1$ MDa, $c/c^* = 0.14$ PEO solution, showing (a) the minimum filament radius $R_{\mathrm{min}}$ as a function of time $t$ and (b) the time-varying Weissenberg number $Wi(t)$ (Eq. \ref{['eqn:9']}) as a function of $(t-t^*)/\tau_E$ in the $R_0 = 0.635$ mm nozzle at different aspect ratios $\varLambda$ or Bond numbers $Bo$. Similarly, (c) shows $R_{\mathrm{min}}(t)$ and (d) $Wi$ versus $(t-t^*)\tau_E$ for the same PEO solution as (a,b), but in a $R_0 = 1.055$ mm nozzle, at different $\varLambda$ and $Bo$. More information about each experimental configuration is listed in Table \ref{['tab:4']}. The grey shaded region in (a) is bounded by $R_{\mathrm{res}} = 0.003$ mm, and in (c) is $R_{\mathrm{res}} = 0.007$ mm. Coloured solid lines in (a) and (c) correspond to fits of the elasto-capillary (EC) regime using Eq. (\ref{['eqn:5']}). The dashed black line in represents a fit of the inertio-capillary (IC) regime using Eq. (\ref{['eqn:1']}) for $Bo = 0.25$ for (a) and $Bo = 0.35$ in (c). The inset axis in (a, c) shows the minimum radius rescaled in such a way that the IC prediction given by Eq. (\ref{['eqn:1']}) is linear with a slope of $-\alpha^{3/2}$. The black solid line in (b, d) represents thew value $Wi = 2/3$ for the EC regime, where the grey shaded bar represents a 50 % confidence region. The dashed black line in (b, d) represents the analytical expression for the Weissenberg number in the IC regime $Wi(t) = \dot{\varepsilon}_{\mathrm{ic}}\tau_E$. Insets in (b, d) shows the different shapes of the filament, coloured according to the different $Bo$, while the dark shaded region denotes the nozzle. All profiles in the inset of (b) and (d) correspond to the same time instance of $(t-t^*)/\tau_E = 3$.