Page 131 - IJB-10-6

International Journal of Bioprinting Fluid mechanics of extrusion bioprinting

mathematical equations that relate the stress and Generalizing Equation V, the apparent viscosity (η )
app
deformation tensors of the fluid. The normal components can be defined as 70
of stress are defined as :
70
η app = τ (XV)
σ = p + τ xx γ i
xx
σ = p + τ (XIII) By replacing the dynamic viscosity (η) with apparent
yy yy
σ = p + τ zz viscosity (η ), Equation V can describe the flow behavior
app
zz
of time-independent non-Newtonian fluids. For an
isothermal flow, wherein Newtonian fluids maintain
where p is the isotropic pressure. The components τ , a constant η , non-Newtonian fluids exhibit η as
xx
τ , and τ are known as the deviatoric normal stresses a function of strain rate tensor magnitude and other
app
app
yy
zz
for Newtonian fluids, representing the contributions of parameters that characterize the rheological behavior of
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the flow to the normal stresses on fluid elements. For an the fluid. Considering the process forces during extrusion,
incompressible Newtonian fluid, the isotropic pressure is the shear stress level is low at the syringe barrel, increased
the average of normal stresses, and the deviatoric normal to a maximum at the dispensing nozzle, and then decreases
stresses are identically equal to zero. 70,78 to very low levels as the filament is deposited on the
Newtonian fluids display a linear relationship between stage. A shear-thinning bioink, whose apparent viscosity
the stress and strain rate tensors, expressed by Newton’s decreases with shear stress, provides favorable behavior
law of viscosity : for extrusion bioprinting, as it requires a lower driving
70
force for extrusion and exerts lower stress on cells. After
= = . fiber deposition on the stage, the low shear stress increases
τ = hγ (XIV) the viscosity of the bioink, improving shape fidelity and
printability. In contrast, a shear-thickening fluid, with
where h is the dynamic viscosity of the fluid. In contrast,
non-Newtonian fluids cover a wide range of fluids with a viscosity that increases with shear stress, is not suitable for
extrusion as its viscosity increases inside the nozzle, which
non-linear relationship between their stress and strain rate.
compromises cell viability, and decreases after deposition,
For non-Newtonian fluid flow, the normal stress reducing shape fidelity.
components τ , τ , and τ are known as extra stresses, and There are various mathematical models to represent
xx
zz
yy
they may take non-zero values (contrary to Newtonian the flow behavior of a time-independent non-Newtonian
fluid). Non-Newtonian fluid can be divided into three fluid. Among these models, the power-law, 80,81
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sub-classes, i.e., time-independent, time-dependent, and Carreau, Carreau-Yasuda, Cross, and Herschel–
77
83
84
82
viscoelastic fluids. Figure 6 illustrates the characteristic Bulkley models are more compatible with the
85
behaviors of these sub-classes.
flow behavior of biomaterials. Table 3 illustrates the
Generally, bioinks exhibit a combination of features mathematical equations and parameters governing these
from all three non-Newtonian sub-classes. There are models.
36
many studies on the rheology of bioinks that demonstrate The Herschel–Bulkley model describes the flow
the viscoelastic behavior of bioinks. 36,79 To understand the behavior of yield-pseudoplastic fluids characterized by the
complex behavior of viscoelastic fluids, various rheological presence of a yield stress (τ ), which must be exceeded for
tests are employed to gain information on flow behavior, the fluid to deform or flow. If the externally applied stress
0
which can be used to predict printability and cell viability is lower than the yield stress, the material will exhibit
in the bioprinting process.
elastic deformation.
3.1. Time-independent flow behavior The determination of the yield stress typically involves
For time-independent fluids, the instantaneous strain at each conducting steady shear tests at low shear rates. The data
point is determined by the stress value at that moment and obtained from these experiments are then utilized to
location. The time-independent flow behavior of a fluid can be extrapolate and predict the yield stress when the shear rate
evaluated using a steady shear test with a rotational rheometer, approaches zero. 36,77 An alternative approach to measuring
employing cone plates, parallel plates, or concentric cylinders the yield stress is to use vane geometries with rotational
(bob-cup). In this test, the shear rate is gradually increased/ rheometers. The vane geometry enables the measurement
decreased while recording the corresponding shear stress, of shear stress corresponding to the onset of flow under
resulting in a flow curve that displays the variation of shear specified test conditions, typically within the time scale of
stress with strain rate (Figure 6A). the measurement. 77

Volume 10 Issue 6 (2024) 123 doi: 10.36922/ijb.3973