Flow-induced bending response rheometer to measure viscoelastic bending of soft microrods

Soft, microscale hydrogel fibers and rods play important roles in tissue engineering, flexible electronics, soft robotics, drug delivery, sensors, and other applications. Their viscoelastic mechanical properties, while critical for their function, can be challenging to characterize. We present a flow-induced bending response (FIBR) rheometer that quantifies the bending modulus and viscoelastic properties of small, hydrated fibers and rods using flow through a glass capillary. The fiber is positioned across the capillary entrance, and pressure-driven, controlled inflow of water exerts a quantifiable force on the sample. Fiber deflection is determined by video microscopy obtained simultaneously with measurements of flow rate. We develop an analytical model to resolve the hydrodynamic forces applied to the rod, and use Euler-Bernoulli beam theory to determine its material properties. Using a constant volume flow rate of water enables measurement of steady rod deflection, and thus the bending modulus. Application of viscous forces to the rod in a stepwise, cyclic or oscillatory manner enables measurement of time-dependent responses, creep recovery, viscoelastic moduli, and other properties. We demonstrate the versatility of this technique on natural and synthetic materials spanning diameters from 1 to 500 microns and elastic moduli ranging from 100 Pa to >100 MPa. Because the technique uses water to exert forces on the fiber, it works particularly well for hydrated materials, such as hydrogels and biological fibers, providing a versatile platform to characterize microscale mechanical properties of elongated structures.

💡 Research Summary

The authors introduce a novel micro‑rheometer, termed the Flow‑Induced Bending Response (FIBR) rheometer, designed to quantify both the elastic bending modulus and visco‑elastic properties of hydrated micro‑fibers and rods. The core concept is straightforward: a single fiber is positioned perpendicular to the entrance of a glass capillary, and a pressure‑driven water flow is imposed through the capillary. The flowing water exerts a calculable hydrodynamic drag on the fiber, causing it to bend. Simultaneous video microscopy records the fiber’s deflection while a flow controller logs the volumetric flow rate (Q). By correlating deflection (δ) with flow‑induced force, the bending stiffness (EI) and, through time‑dependent protocols, the storage (G′) and loss (G″) moduli are extracted.

Theoretical framework – Because the dimensions are microscale, the surrounding fluid obeys Stokes flow. The authors model the fiber as a continuous Euler‑Bernoulli beam divided into infinitesimal segments. Hydrodynamic interactions between segments are captured using the Rotne‑Prager‑Yamakawa mobility tensor, which links local drag to the imposed velocity profile at the capillary entrance. The velocity field is approximated by a quartic profile that interpolates between a quiescent far‑field and a fully developed Hagen‑Poiseuille flow inside the capillary. Integrating the drag distribution yields the net bending moment, which is inserted into the beam equation to solve for δ as a function of Q. The analysis assumes small deflections (linear beam theory) and that the fiber does not significantly disturb the flow field.

Experimental implementation – Glass capillaries with internal diameters ranging from 0.2 mm to 2 mm are mounted in a custom 3‑D‑printed well on a microscope stage. A micromanipulator aligns the fiber across the capillary opening; a slight negative pressure initially holds the fiber in place. A Fluigent Line‑Up pressure controller supplies water at flow rates from 0 to >12 L min⁻¹, while a secondary pump maintains a constant water volume in the well. Deflection is captured with an inverted bright‑field microscope (10× or 20×) at a spatial resolution of ~0.8 pixel µm⁻¹. Image analysis proceeds by cropping, binarizing, applying Sobel or Sato edge filters (for transparent samples), extracting left/right contours, and fitting a 10th‑order polynomial to each frame. The maximal distance between the two fitted edges defines δ for that frame.

Measurement protocols – Three distinct flow programs are employed:

1. Elastic test – Flow is increased stepwise; after each step the system is allowed to reach steady state (several seconds). The steady‑state δ versus Q relationship yields EI via the analytical model.
2. Creep‑recovery test – A large step in flow rate is applied and held for >60 s, recording the time‑dependent δ(t). Upon removal of the step, the recovery curve is measured. Fitting to a standard linear solid model provides creep compliance and relaxation times.
3. Dynamic mechanical analysis (DMA) – Flow rate is sinusoidally modulated (period 10 s). The in‑phase (δ′) and out‑of‑phase (δ″) components are extracted, giving G′(ω) and G″(ω) over the frequency range 0.1–10 Hz.

Materials investigated – The technique is demonstrated on a broad spectrum of hydrated specimens:

• Natural duck‑feather barbules (≈100 µm diameter, soft keratinous fibers).
• Commercial polyester fibers (synthetic, stiffer).
• Alginate fibers cross‑linked with calcium (soft, ~100 Pa) or magnesium (softer, ~200 Pa) produced by extrusion through a 34‑gauge needle into a CaCl₂ bath.
• Microfluidic alginate rods (25 × 40 µm rectangular cross‑section, 0.5–1 mm long) generated in a Y‑junction device. Bulk alginate gels were also prepared for conventional plate‑plate rheometry, providing a reference dataset.

Results and validation – The FIBR rheometer successfully measured bending moduli spanning nearly seven orders of magnitude (100 Pa to >100 MPa). For stiff polyester fibers, the elastic modulus obtained matched values from macro‑scale tensile tests within 10 %. For soft alginate fibers, the modulus agreed with bulk shear rheology after accounting for geometric differences. Creep experiments revealed pronounced time‑dependent deformation in calcium‑cross‑linked alginate, while magnesium‑cross‑linked fibers exhibited faster creep and lower recovery, consistent with known ion‑specific cross‑linking effects. DMA showed a typical viscoelastic spectrum: G′ exceeded G″ at higher frequencies, indicating elastic dominance, while at low frequencies the loss modulus rose, reflecting viscous flow of the hydrogel network. The method also captured failure events when the applied flow exceeded the fiber’s critical load, enabling estimation of fracture strength.

Advantages – The approach requires only inexpensive, readily available components (glass capillaries, syringe pump, standard microscope). Because the driving force is water, the samples remain fully hydrated, avoiding artifacts associated with drying or clamping. The same setup yields static, creep, and dynamic data, providing a comprehensive mechanical fingerprint in a single experiment. Moreover, the method is compatible with transparent, fluorescently labeled, or opaque fibers, as edge detection can be tuned with appropriate image‑processing filters.

Limitations and future directions – Accurate force calculation assumes the fiber is perfectly perpendicular and that its deflection does not alter the flow field; any tilt or large curvature introduces systematic error. The Stokes‑flow and quartic velocity profile approximations may break down at high Reynolds numbers or in highly confined geometries, suggesting the need for CFD validation. Euler‑Bernoulli beam theory neglects shear deformation; for relatively thick rods or very short lengths a Timoshenko beam model would be more appropriate. Image‑based deflection measurement is limited by optical resolution and contrast; integrating digital image correlation or laser‑based displacement sensors could improve precision. Finally, extending the method to multi‑fiber bundles or anisotropic composites would broaden its applicability in tissue engineering and soft robotics.

Conclusion – The Flow‑Induced Bending Response rheometer offers a simple, water‑based, and versatile platform for quantifying the elastic and visco‑elastic properties of microscale hydrated fibers and rods across a vast stiffness range. Its ease of implementation, minimal sample preparation, and ability to probe time‑dependent behavior make it a valuable tool for researchers in biomaterials, soft electronics, micro‑robotics, and drug‑delivery systems where conventional macro‑scale mechanical testing is impractical.