---
title: Capillary Breakup Extensional Rheometry (CaBER-DoS)
url: https://www.emergentmind.com/topics/capillary-breakup-extensional-rheometry-caber-dos
type: topic
---

# Capillary Breakup Extensional Rheometry (CaBER-DoS)

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 [1502.06305]) 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 [2511.17360, 2410.15132].
- **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 [1502.06305].
- **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 [2511.17360, 2204.13450, 2410.15132].
- **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, $n < 1$ for shear-thinning fluids [2407.15378].
- **Elasto-Capillary (EC) Regime:** In viscoelastic solutions, for times $t > t^*$ (crossover point), the filament radius decays exponentially, $R(t) = R_e \exp\left[-(t-t_e)/(3\lambda_E)\right]$ with $\lambda_E$ the extensional relaxation time [2403.04103, 2410.15132, 2204.13450]. 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 [2404.06947, 2403.04103, 2206.06539].

Extensional viscosity is extracted via the local balance:
\[
\eta_E^+(t) = \frac{\sigma}{R(t) \dot{\varepsilon}(t)},
\]
with $\dot{\varepsilon}(t) = -2\,d\ln R/dt$, and $\sigma$ the surface tension (with geometric corrections for curvature as needed). The relaxation time $\lambda_E$ 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 [1502.06305, 2511.17360].
- **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 $O(0.1)$ ms for weakly elastic fluids [2511.17360, 2410.15132].
- **Finite Extensibility and Size Effects:** For fluids with limited molecular extensibility, pre-stretch in the viscocapillary regime causes underestimation of $\lambda_E$ unless FENE-P or tube models are fit across device sizes (nozzle radius or plate separation) [2503.05897, 2309.08440, 2403.04103].
- **Best-Practice Guidelines:** Key recommendations include maintaining $De \gtrsim 0.1$, $Oh > 1$ for power-law/extensional viscosity scaling, Bond number $Bo \lesssim 0.5$ to avoid gravity-induced perturbations, and at least $n\geq10$ points in the EC regime for statistical robustness [2511.17360].

## 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 $n$ from $2/3$ (IC) to $1$ (viscous); Carreau model validates collapse of exponents for $\eta_S$ and $\eta_E$ at $Oh>1$ [2407.15378]   | Power-law, Carreau     |
| Weakly Elastic Polymers    | Pure EC regime with $\lambda_E$ matching Maxwell predictions if $L\gg1$; coil-stretch transitions visible with birefringence tracking [2204.13450, 2403.04103]       | Oldroyd-B, FENE-P      |
| Highly Entangled Polymers  | Multi-regime thinning: tube reorientation, weak exponential (apparent $\lambda_e \ll \tau_d$), finite extensibility as power-law [2206.06539]                  | Doi-Edwards, Rolie-Poly |
| Dense Suspensions          | Master state-diagram in Pe$_{\rm ext}$–$t_{\rm br}/t_{\rm B}$ plane separates Newtonian, yielding, ductile and brittle jammed (dilatant) regimes [2007.04141]  | Empirical/Plug-flow    |
| Rate-Thickening Fluids     | Asymptotic self-similar thinning with quadratic time-law, distinguishing geometric correction factors for extensional viscosity recovery [2206.06314]          | 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 [2204.13450, 2410.15132, 2403.04103, 2206.06539, 2007.04141].

## 5. Size Effects, Finite Extensibility, and Correction Protocols

A critical advance is the recognition that the apparent relaxation time $\lambda_e$ obtained from EC thinning depends on the initial filament size due to pre-stretch in the viscocapillary regime and finite polymer extensibility [2309.08440, 2503.05897]. For accurate $\lambda$ extraction:

- Apparent relaxation times scale as $\lambda_e \sim D_0^\alpha$ for small filaments, transitioning to a device-independent plateau at larger sizes, but still typically underestimating the true $\lambda$ when finite extensibility is substantial [2503.05897].
- Correction methods employ multi-size datasets and phase diagrams ($\lambda_e/\lambda$ vs $Ec_e$ and $b$) to back out material relaxation times and extensibility parameters [2503.05897].
- In certain regimes (particularly for semi-flexible polyelectrolytes or low $L$), standard exponential data fitting may underestimate characteristic times by orders of magnitude unless corrected (see FENE-P and tube model fits) [2403.04103, 2410.15132].

## 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 [2511.17360, 1502.06305]. Its utility encompasses:

- Quantifying coil–stretch transitions and chain orientation via simultaneous optical measurements (e.g., birefringence) [2204.13450].
- Mapping operational limits across parameters—frame rate, nozzle size, dynamic range—and defining figures of merit such as the filament capture rate [2511.17360].
- Providing practical tools for identifying onset concentrations for viscoelastic overlap (e.g., $c_E^*$) in polyelectrolyte solutions and connecting EC regime emergence with “stringiness” in applications [2410.15132].
- Enabling model selection via statistical criteria (Bayesian Information Criterion) for automated discrimination among physically plausible constitutive models [2206.06314].

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 [2404.06947].

## 7. Contemporary and Emerging Directions

Current research leverages CaBER-DoS for elucidating:
- Active-matter rheology, where swimming microbes modulate macroscopic extensional viscosities [1502.06305].
- 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 [2407.15378].
- Quantitative state-diagram construction for dense suspensions under uniaxial extension, providing new diagnostics for soft-matter jamming, dilatancy, and yielding [2007.04141].
- Direct experimental access to the physics of highly nonequilibrium chain stretch and coil–stretch hysteresis (beyond Oldroyd-B) [2309.08440, 2206.06539].

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.

Source: https://www.emergentmind.com/topics/capillary-breakup-extensional-rheometry-caber-dos