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International Journal of Bioprinting Fluid mechanics of extrusion bioprinting

on a rotational rheometer. 78,93 The controlled oscillatory frequency. This process allows for the determination of the
rotation of the shaft can generate a sinusoidal shear strain storage and loss moduli of the bioink at various frequencies
γ on the fluid stored between parallel plates or a cone and (Figure 6G).
a plate.
In addition to loss and storage moduli, researchers use
the loss tangent to assess the printability of bioinks. The
γ = γ sin (ωt) (XVI) loss tangent indicates the balance between the elastic and
0
viscous behaviors of the material. Biomaterials with a loss
tangent of less than 0.1 act as strong gels, providing excellent
where γ denotes shear-strain amplitude; ω and t are structural integrity in the printed construct but potentially
0
angular velocity and time, respectively. The rheometer causing inconsistent extrusion due to gel fracture in the
measures the transient shear stress (based on transient printing nozzle. Conversely, weak gels have a loss tangent
resistive torque), greater than 0.1 and can be extruded uniformly, but
may collapse after deposition unless crosslinked during
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τ = τ sin (ωt + δ) (XVII) deposition to achieve good shape fidelity. Therefore,
0
the loss tangent plays a crucial role in optimizing bioink
where τ is shear-stress amplitude and δ represents the printability. Petta et al. found that tyramine hyaluronan
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0
phase shift between stress and strain, known as loss angle derivative hydrogels with a loss tangent between 0.5 and
(Figure 6E). The storage and the loss moduli are defined 0.6 exhibited optimal printability. In a similar study, Cheng
as et al. identified that biomaterials with a loss tangent of
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0.2–0.7 are suitable for 3D printing with good printability.
τ τ 43
δ
δ
G’ = 0 cos (); G" = 0 sin () (XVIII) Gao et al. identified an optimal region for bioprinting
γ 0 γ 0 using gelatin-alginate biomaterials. They summarized their
results in a map that illustrates the optimal extrudability
The loss tangent of material is calculated by and cell viability based on loss and storage moduli, as well
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as the loss tangent (Figure 7). Their experiments with
vvarious gelatin–alginate composite bioinks suggested
G "
tanδ = (XIX) that a loss tangent in the range of approximately 0.25–0.45
’ G results in smooth fiber extrusion without compromising
structural integrity. However, increasing the moduli raises
For a viscoelastic material, the loss angle can take the extrusion pressure, negatively affecting cell viability
values between 0 (elastic material) and π/2 (pure fluid) due to increased bioink viscosity.
(Figure 6E). The storage modulus (Gʹ) indicates solid-like
behavior and the material’s ability to store energy, reflecting 3.3.2. Normal stress differences in a viscoelastic fluid
the elastic shape recovery or cell suspension capabilities of As aforementioned, additional stress terms in normal
bioinks. Conversely, the loss modulus (G˝) represents the stresses may take non-zero values and can lead to complex
liquid-like behavior of the bioink and relates to the energy behaviors in viscoelastic fluids. In these fluids, there is no
dissipated during extrusion and mixing. 78 simple expression for the isotropic pressure; instead, the
first and second normal stress differences play a crucial
The storage and loss moduli can vary under different role. Assuming the flow of a viscoelastic fluid in the
conditions, including shear rate, stress, and temperature. x-direction (Figure 5), the first and second normal stress
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Therefore, to obtain accurate Gʹ and G˝ values as fluid differences are defined as :
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properties, oscillatory tests should be performed within
the linear viscoelastic region (LVR) of the bioink, where N = τ – τ (XX)
both moduli are independent of stress/shear. The LVR can 1 xx yy
be determined through an oscillatory amplitude (stress or N = τ – τ zz (XXI)
2
yy
shear) sweep, where the shear stress or shear rate gradually
increases while maintaining a constant frequency (typically Normal stress differences are more readily measurable
1 or 2 Hz). The LVR is identified as the region where Gʹ than the individual normal stresses, and they account for
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and G˝ remain unaffected by the stress or shear amplitude most of the unusual behaviors in viscoelastic fluids, such as
(Figure 6F). Once the LVR is established, a frequency extrudate swell (Barus effect), rod climbing (Weissenberg
sweep test (SAOS) can be conducted by applying a fixed effect), and the development of secondary flows at low
stress or shear amplitude within the LVR and varying the Reynolds numbers (Figure 8). Both extrudate swell and
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Volume 10 Issue 6 (2024) 128 doi: 10.36922/ijb.3973