Controlling extrudate volume fraction through poroelastic extrusion of entangled looped fibers

Abstract

When a suspension of spherical or near-spherical particles passes through a constriction the particle volume fraction either remains the same or decreases. In contrast to these particulate suspensions, here we observe that an entangled fiber suspension increases its volume fraction up to 14-fold after passing through a constriction. We attribute this response to the entanglements among the fibers that allows the network to move faster than the liquid. By changing the fiber geometry, we find that the entanglements originate from interlocking shapes or high fiber flexibility. A quantitative poroelastic model is used to explain the increase in velocity and extrudate volume fraction. These results provide a new strategy to use fiber volume fraction, flexibility, and shape to tune soft material properties, e.g., suspension concentration and porosity, during delivery, as occurs in healthcare, three-dimensional printing, and material repair.

Fig. 1. Concentration and velocity variations when suspensions of straight and looped fibers pass a constriction.

a, d Shapes of the straight and looped fibers, respectively. b, e Snapshots of suspensions of a straight and d 4-looped fibers flowing through a constriction when a constant flow rate of 8 ml min⁻¹ is applied. The initial fiber volume fractions ϕ_s,0 = 0.2 for both cases. c, f Velocity (x component) distributions of the straight and looped fibers from PIV measurements of the experiments in, respectively, b, e at three different times. The fiber velocity v_s is normalized by the average velocity in the nozzle χv_a. The colored triangles to the left of f are positions measured in Fig. 3b. g Distribution of magenta fiber extrudate in a tubing after extrusion from a 3 ml syringe at a flow rate of 5 ml min⁻¹. The 1- and 2-looped fibers are shown in the center along with initial volume fractions ϕ_s,0 before the extrusion.

Fig. 2. Poroelastic model for the extrusion of entangled fibers in a long tube.

a The solid phase is laterally confined and relaxed in the barrel (length L). The barrel cross section has typical width D. The ratio of the cross-sectional areas between the barrel and nozzle is χ, where χ > 1. b When a constant total flux v_a is applied from the barrel to the nozzle, the fluid phase adopts velocity v_f(x, t) and the solid phase v_s(x, t). The displacement of the free end of the suspension is denoted by x = δ(t), with δ(0) = 0. At x = L, the solid velocity v_s increases to χv_a, stretching the solids in the barrel.

Fig. 3. Poroelastic model simulations for the extrusion of looped fiber suspensions, as shown in Fig. 1e, f.

Actual experimental parameters and a single fitting parameter F ≡ k₀E_eff = 91 nN are used for the simulation, resulting in ${\overline{v}}_a=0.63$. a The computed (solid line) and experimentally measured (dots) free boundary displacement δ(t) of the suspension during extrusion. The dot size represents typical errors in the measurement. The inclined dashed line is the trajectory at constant velocity v_a. Kymograph of the center line of the setup is presented in the background. At time t_ex, all solid material has passed the constriction. The poroelastic time T_pe is defined in Equation (9). b Comparison between the computed velocity profiles and measured PIV results at locations in Fig. 1f indicated by the corresponding colored triangles. Gray: nozzle (x/L = 1); blue: near constriction (x/L = 0.75); red: farther from constriction (x/L = 0.4). c The distribution of the normalized displacement field u_s/L at times up to approximately t_ex (solid lines) and the displacement field without elasticity (dashed lines) over the same time domain. Time interval between each line is 0.15t_ex. Darker color represents later times. d The normalized elastic stress ${\sigma }_{xx}^{\prime }/E_{\mathrm{eff}}$ at early (top panel) and late stages (bottom panel) of the extrusion.

Fig. 4. Extrudate volume fraction ϕ_s,ex as a function of v¯a and χ.

a Computed ratio between the extrudate and initial volume fractions as a function of ${\overline{v}}_a$ at three different χ values. These relationships are used to calculate ${\overline{v}}_a$ using experimentally measured ϕ_s,ex/ϕ_s,0. Displayed curves use ϕ_s,0 = 0.05. b Experimentally fitted $F_{\exp }$ as a function of ϕ_s,0 at different χ values. $F_{\exp }$ is calculated from ${\overline{v}}_a$ using $F_{\exp }=v_{\mathrm{a}}\mu L/{\overline{v}}_a$. The error bars are calculated from five independent measurements. The experimental data of looped fibers from Fig. 1e is indicated by the black arrow. All experiments satisfy the condition ϕ_f > 0.7 required in Eq. (8). c Extrudate volume fractions as a function of initial volume fractions for different suspension materials. Circles are reported extrusion results from particle pastes made of ceramics ^-- and polymers ^--,. The diamonds and triangles represent individual experiments performed with looped and straight fibers shown in Figs. 1b and e, respectively. The blue shaded region represents model predictions for a range of ${\overline{v}}_a$; the dotted line represents ϕ_s,ex = ϕ_s,0 when χ = 1 or ${\overline{v}}_a=\infty$. d Computed ϕ_s,ex/ϕ_s,0 based on the poroelastic extrusion model as a function of χ and ${\overline{v}}_a$ at ϕ_s,0 = 0.1.