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

TL;DR

The paper introduces a flow-induced bending rheometer (FIBR) that characterizes bending modulus and viscoelastic properties of small, hydrated fibers by driving flow through a capillary and measuring deflection with video microscopy. Hydrodynamic drag on discretized beads is calculated with a Rotne–Prager–Yamakawa mobility framework and coupled to Euler–Bernoulli beam theory to extract the Young's modulus from deflection data, with a derived practical relation $E = B(R,d)rac{ abla ext{η} Q}{ ext{δ}}$ (where $Q$ is flow rate and $δ$ is deflection). The method is demonstrated on alginate-based Ca- and Mg-crosslinked fibers and 25×40 μm alginate rods, covering elastic moduli from 10^2 to 10^8 Pa and flexural stiffnesses from 10^{-18} to 10^{-12} Pa m^4, and extends to time-dependent and failure behaviors via creep and dynamic testing. The FIBR technique operates in hydrated environments, is simple and inexpensive, and provides a versatile platform for microscale mechanical characterization of bio-inspired and hydrogel-based elongated structures, with potential applications in tissue engineering, soft robotics, and bio-materials research.

Abstract

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.

Figures (9)

• Figure 1: (a) Schematic of the experimental setup. (b) Image analysis pipeline: (1) Raw microscope image with a dotted box to represent the crop used in the analysis, (2) The cropped image with an applied Sato filter, (3) Polynomial fit of the contour found in the Sato filter and comparison of this polynomial, in red, to the baseline, zero-deflection frame, in blue.
• Figure 2: (a) Schematic illustration of a fiber (grey) placed inside a capillary tube, showing the flow velocity profile, Eq. \ref{['eq:quatric_flow']}, and the force profile acting on the fiber, Eq. \ref{['eq:force_quartic']}. (b) A continuous "thin" fiber is discretized as a collection of mutually connected beads, each pair of neighboring beads being in point contact. This approach allows the computation of hydrodynamic forces acting on the fiber using Eq. \ref{['eq:RPY_ij_thin']}. (c) A continuous "thick" fiber is discretized as a set of overlapping, connected beads. This approach allows the computation of hydrodynamic forces acting on the fiber using Eq. \ref{['eq:RPY_ij_thick']}. (d) Schematic illustration of a deformed fiber: the capillary tube has radius $R$, the fiber has thickness $d$, and the deflection is denoted by $\delta$. The distribution of hydrodynamic forces acting on the fiber is also sketched. (e) Plot of the dimensionless Young's modulus $E \delta/U_{max}\eta$, as a function of the ratio of the tube radius to the fiber thickness, $k=R/d$, for ‘thin’ fibers (Eq. \ref{['eq:E_thin_cylinder']}) in orange and ‘thick’ fibers (Eq. \ref{['eq:E_thick_cylinder']}) in blue. The dashed lines indicate the continuation of the relationship, extrapolated into regimes where it is no longer valid (e.g., the ‘thick’ regime extended into the ‘thin’ regime). The modulus plotted in green represents the exact expression (Eq. \ref{['eq:E_exact']}) for thin fibers, relaxing the assumption of large aspect ratio $k$.
• Figure 3: (a) Elastic behavior of a polyester fiber. The solid line shows the flow rate of water pulled into the capillary, while the dotted line shows the deflection of the fiber. (b) Elastic behavior in a range of materials, where the polyester fiber from (a) is in blue.
• Figure 4: Cyclic loading reveals fiber creep. (a) The applied flow rate (black) and deflection (colored by time) of a Ca-alginate fiber during 3 square-wave loading cycles. (b) Creep data for the 3 loading cycles from (a) fitted with the Burgers model (dashed lines). The average fit parameters are listed in the lower right. (c) The applied flow rate (black) and deflection (colored by time) of a Mg-alginate fiber during a 3 square-wave loading cycle. (d) Creep data for the 3 loading cycles from (c) fitted with the Burgers model (dashed lines). The average fit parameters are listed in the lower right.
• Figure 5: Cyclic loading reveals fiber properties. (a) An example of a sinusoidal cyclic loading test in a Ca-alginate fiber with 1% NaCl. Flow rate is plotted in black, deflection is plotted as a dotted line in black. (b) Comparison of the loading/unloading from the first and last cycles. The deflection increases slightly over the course of the experiment. (c) A similar test on a Mg-alginate Fiber shows a much larger creep in deflection.