Capillary Breakup Extensional Rheometry (CaBER-DoS)

• Capillary Breakup Extensional Rheometry (CaBER-DoS) is an experimental suite that analyzes liquid filament thinning under capillary forces to assess transient extensional rheology.
• It employs high-speed imaging and precise experimental protocols to capture distinct thinning regimes, including inertio-capillary and elasto-capillary phases.
• The technique advances fluid mechanics by validating viscoelastic models and quantitatively characterizing low-viscosity, weakly elastic, and microstructured fluids.

Capillary Breakup Extensional Rheometry (CaBER-DoS) is a suite of experimental methodologies used to quantify the transient extensional rheological properties of liquids—particularly low-viscosity, weakly elastic, or microstructured fluids—via analysis of the thinning dynamics of a liquid filament formed between a drop and a substrate. In CaBER-DoS, the bridge formation, thinning, and eventual pinch-off occurs primarily under capillary (surface-tension) forces; extensional flow metrics such as transient extensional viscosity, Hencky strain, and relaxation time are inferred by resolving the filament radius over time and applying constitutive and similarity models. CaBER-DoS is distinct from standard step-strain CaBER by its substrate geometry and its compatibility with ultra-low viscosity and high-speed imaging regimes. Modern implementations allow measurement of polymeric fluids, suspensions, polyelectrolytes, and rate-thickening complex fluids, providing a platform for validating viscoelastic constitutive models and for mapping extensional rheological parameter spaces.

1. Device Design, Principles, and Experimental Protocols

CaBER-DoS encompasses a family of techniques characterized by initializing a fluid bridge either through direct dripping-onto-substrate, capillary jump, or acoustically actuated jetting. Key setup features include:

• Geometry and Actuation: Fluids are dispensed as ~1–10 µL pendant drops from a microcapillary (or, in microfluidic variants, an acoustically jetted droplet (McDonnell et al., 2015)) onto a hydrophobic or hydrophilic substrate. The bridge forms either passively (dripping) or actively (SAW-induced jet), and may be precisely controlled in gap height (typically 1–2 mm).
• Imaging and Resolution: High-speed cameras (up to 100,000 fps) with backlit macroscopic optics are employed to capture the evolution of the minimum bridge radius $R(t)$. Minimum observable radii down to 6–10 µm are typical, setting the measurable lower bound of relaxation times (Warwaruk et al., 21 Nov 2025, Moon et al., 2024).
• Sample Conditioning: Contact angle, substrate wettability, and environmental factors (e.g., evaporation, temperature) are controlled to maintain pinned contact lines and avoid artifacts.
• Acoustic Implementation: In the acoustically-driven variant (microfluidic CaBER-DoS), jet formation is achieved using focused surface acoustic waves (SAW) from interdigitated transducers, enabling bridge formation in ~1.5 ms and subsequent purely capillary thinning (McDonnell et al., 2015).
• Kinematic Reference: The time origin $t=0$ is commonly defined once the filament radius passes a chosen reference radius ($R_0$), excluding early transients.

2. Thinning Dynamics: Regimes, Scaling Laws, and Data Extraction

The time evolution of $R(t)$ reveals distinct physical regimes:

• Inertio-Capillary (IC) Regime: Early-time thinning is governed by a balance of inertia and capillarity. The radius follows $R(t) \sim \alpha (t_c - t)^{2/3}$, where $\alpha$ is a geometry-dependent pre-factor, $t_c$ the pinch-off time (Warwaruk et al., 21 Nov 2025, Muto et al., 2022, Moon et al., 2024).
• Viscous-Capillary (VC) or Power-Law Regime: For higher viscosity fluids (Oh > 1), viscous forces dominate, and $R(t) \sim (t_c - t)$ or a power-law $R(t) \sim (t_c-t)^n$ with $n \rightarrow 1$ for Newtonian viscosity, $t=0$0 for shear-thinning fluids (Matsumoto et al., 2024).
• Elasto-Capillary (EC) Regime: In viscoelastic solutions, for times $t=0$1 (crossover point), the filament radius decays exponentially, $t=0$2 with $t=0$3 the extensional relaxation time (Calabrese et al., 2024, Moon et al., 2024, Muto et al., 2022). The exponential regime is classically modeled by the Oldroyd-B or generalized FENE-P models.
• Terminal/Finite-Extensibility Regime: For FENE-type and entangled polymers or at high accumulated strain, thinning accelerates and departs from exponential, entering a visco-elasto-capillary or linear pinch-off regime (Zinelis et al., 2024, Calabrese et al., 2024, Du et al., 2022).

Extensional viscosity is extracted via the local balance: $t=0$4 with $t=0$5, and $t=0$6 the surface tension (with geometric corrections for curvature as needed). The relaxation time $t=0$7 is obtained from the log-slope of the EC regime.

3. Model Selection, Data Analysis, and Measurement Limits

Accurate CaBER-DoS analysis requires careful segregation of thinning regimes, robust digital resolution, and calibration against Newtonian standards:

• Calibration and Model-Based Extraction: Measured half-times or exponential slopes are linked to dimensionless numbers (Ohnesorge, Deborah, Bond), with viscosity and relaxation extracted by inverting calibration curves or constitutive fits (McDonnell et al., 2015, Warwaruk et al., 21 Nov 2025).
• Resolution Boundaries: The minimum measurable relaxation time is set by the spatial dynamic range (filament capture rate), imaging frame rate, and minimum neck radius—practical lower limits are $t=0$8 ms for weakly elastic fluids (Warwaruk et al., 21 Nov 2025, Moon et al., 2024).
• Finite Extensibility and Size Effects: For fluids with limited molecular extensibility, pre-stretch in the viscocapillary regime causes underestimation of $t=0$9 unless FENE-P or tube models are fit across device sizes (nozzle radius or plate separation) (Hu et al., 7 Mar 2025, Gaillard et al., 2023, Calabrese et al., 2024).
• Best-Practice Guidelines: Key recommendations include maintaining $R_0$0, $R_0$1 for power-law/extensional viscosity scaling, Bond number $R_0$2 to avoid gravity-induced perturbations, and at least $R_0$3 points in the EC regime for statistical robustness (Warwaruk et al., 21 Nov 2025).

4. Constitutive Models and Applications Across Complex Fluids

CaBER-DoS is used to probe a variety of material classes, each with distinct rheological phenomenologies:

Fluid Class	Key CaBER-DoS Insights	Typical Model
Newtonian and Power-Law	Exponents $R_0$4 from $R_0$5 (IC) to $R_0$6 (viscous); Carreau model validates collapse of exponents for $R_0$7 and $R_0$8 at $R_0$9 (Matsumoto et al., 2024)	Power-law, Carreau
Weakly Elastic Polymers	Pure EC regime with $R(t)$0 matching Maxwell predictions if $R(t)$1; coil-stretch transitions visible with birefringence tracking (Muto et al., 2022, Calabrese et al., 2024)	Oldroyd-B, FENE-P
Highly Entangled Polymers	Multi-regime thinning: tube reorientation, weak exponential (apparent $R(t)$2), finite extensibility as power-law (Du et al., 2022)	Doi-Edwards, Rolie-Poly
Dense Suspensions	Master state-diagram in Pe$R(t)$3–$R(t)$4 plane separates Newtonian, yielding, ductile and brittle jammed (dilatant) regimes (Andrade et al., 2020)	Empirical/Plug-flow
Rate-Thickening Fluids	Asymptotic self-similar thinning with quadratic time-law, distinguishing geometric correction factors for extensional viscosity recovery (Du et al., 2022)	Inelastic Rate-Thickening (IRT)

For each category, the measured thinning dynamics determine, respectively, the functional form and magnitude of the transient extensional viscosity, the existence/length of the EC regime, and the presence or absence of phenomena such as dilatancy or coil-stretch transitions (Muto et al., 2022, Moon et al., 2024, Calabrese et al., 2024, Du et al., 2022, Andrade et al., 2020).

5. Size Effects, Finite Extensibility, and Correction Protocols

A critical advance is the recognition that the apparent relaxation time $R(t)$5 obtained from EC thinning depends on the initial filament size due to pre-stretch in the viscocapillary regime and finite polymer extensibility (Gaillard et al., 2023, Hu et al., 7 Mar 2025). For accurate $R(t)$6 extraction:

• Apparent relaxation times scale as $R(t)$7 for small filaments, transitioning to a device-independent plateau at larger sizes, but still typically underestimating the true $R(t)$8 when finite extensibility is substantial (Hu et al., 7 Mar 2025).
• Correction methods employ multi-size datasets and phase diagrams ($R(t)$9 vs $R(t) \sim \alpha (t_c - t)^{2/3}$0 and $R(t) \sim \alpha (t_c - t)^{2/3}$1) to back out material relaxation times and extensibility parameters (Hu et al., 7 Mar 2025).
• In certain regimes (particularly for semi-flexible polyelectrolytes or low $R(t) \sim \alpha (t_c - t)^{2/3}$2), standard exponential data fitting may underestimate characteristic times by orders of magnitude unless corrected (see FENE-P and tube model fits) (Calabrese et al., 2024, Moon et al., 2024).

6. Broader Implications, Limitations, and Best Practices

CaBER-DoS has emerged as a robust platform for extensional rheology at low viscosities and sub-millisecond temporal resolution, extending measurable domains far below commercial plate-driven CaBER instruments (Warwaruk et al., 21 Nov 2025, McDonnell et al., 2015). Its utility encompasses:

• Quantifying coil–stretch transitions and chain orientation via simultaneous optical measurements (e.g., birefringence) (Muto et al., 2022).
• Mapping operational limits across parameters—frame rate, nozzle size, dynamic range—and defining figures of merit such as the filament capture rate (Warwaruk et al., 21 Nov 2025).
• Providing practical tools for identifying onset concentrations for viscoelastic overlap (e.g., $R(t) \sim \alpha (t_c - t)^{2/3}$3) in polyelectrolyte solutions and connecting EC regime emergence with “stringiness” in applications (Moon et al., 2024).
• Enabling model selection via statistical criteria (Bayesian Information Criterion) for automated discrimination among physically plausible constitutive models (Du et al., 2022).

Limitations include sensitivity to flow history, size and extensibility artifacts, nonuniform contact-line pinning, and the requirement for precisely measured capillary parameters. Advanced computational rheology (e.g., Basilisk simulations with log-conformation FENE-P) now allows direct numerical reproduction of experimental CaBER-DoS dynamics, aiding parameter extraction—especially in complex or multi-physics regimes (Zinelis et al., 2024).

7. Contemporary and Emerging Directions

Current research leverages CaBER-DoS for elucidating:

• Active-matter rheology, where swimming microbes modulate macroscopic extensional viscosities (McDonnell et al., 2015).
• The non-Newtonian behavior of industrial and biological ‘living’ materials across capillarity-dominated processing flows [multiple refs].
• Systematic connection between extensional and shear rheology, exploiting experimentally validated Carreau model scaling (Matsumoto et al., 2024).
• Quantitative state-diagram construction for dense suspensions under uniaxial extension, providing new diagnostics for soft-matter jamming, dilatancy, and yielding (Andrade et al., 2020).
• Direct experimental access to the physics of highly nonequilibrium chain stretch and coil–stretch hysteresis (beyond Oldroyd-B) (Gaillard et al., 2023, Du et al., 2022).

The adoption of robust measurement protocols, systematic model-based correction for pre-stretch and finite extensibility, and multi-scale imaging and simulation integration will continue to refine the precision and interpretive power of CaBER-DoS for extensional flow characterization.