Page 137 - IJB-10-6

International Journal of Bioprinting Fluid mechanics of extrusion bioprinting

secondary flow can significantly affect the outcomes of extrudate swelling at the dispensing nozzle exit due to the
extrusion bioprinting. Barus effect. 105–107 The values of Wi for the onset of vortices
cr
Despite the crucial role of normal stress differences in in the contraction area or extrudate swell phenomenon may
100
the behavior of viscoelastic fluids, there is no published differ depending on various flow parameters. Figure 8A
research reporting the measured values of these differences illustrates the extrudate swell effect for polymeric fluid
for biomaterials. Existing data come from tests on industrial extruding from a circular nozzle. This swelling can
polymeric fluids that differ from the biomaterials used significantly change the filament diameter and affect
97
in bioprinting. The first normal stress difference can be printability. To compensate for filament swelling, the
measured using a rotational rheometer with axial force filaments should be printed with a smaller diameter than
measurement capability during a steady shear test with a the original design. This can be achieved by reducing
cone-plate geometry. The second normal stress difference extrusion pressure, increasing print speed, or using
77
97
N is typically much smaller than N and is often negligible, smaller nozzles.
1
2
especially for dilute solutions. Nevertheless, for polymer Empirical and semi-empirical correlations often relate
melts and concentrated or entangled polymer solutions, the swell ratio (d /d) to the first normal stress difference
ex
N may have a value of −0.1N to 0.3 N , affecting the N . Tanner suggested a relationship based on the Phan-
108
2
1
1
1
flow pattern by generating secondary flows. Due to Thien-Tanner (PTT) viscoelastic model that aligns with
103
geometrical symmetry, N does not affect flow in circular experimental observations:
2
or coaxial pipes, so its effect can be ignored for the flow
of biomaterial melts or concentrated bioinks within
bioprinting nozzles. 16/
d  1  N  2 
Normal stress differences are characteristics of a ex = 1 +  1   (XXIV)

viscoelastic fluid. Therefore, a comparison of the first d   22τ w   
normal stress difference with the shear stress can be
a measure of viscoelastic behavior. To quantify this where d represents the diameter of filament emerging
104
ex
behavior, the relaxation time of a fluid (λ), also known as from the needle; the subscript w indicates that N and τ are
1
Maxwell relaxation time, is defined as : the first normal stress difference and shear stress values at
77
the wall of the needle.
N 3.3.3. Extensional viscosity in a viscoelastic fluid
λ = 1 (XXII)
i
2 τγ yx The response of a viscoelastic fluid to extensional flow (as
yx
discussed in Section 2.3) can differ drastically from that of
a Newtonian fluid. Despite the crucial role of extensional
To compare the elastic and viscous forces in the flow,
the Weissenberg number is defined as : viscosity in cell viability assessments, there are no published
104
studies on the extensional viscosity of bioinks. Extensional
rheology can provide crucial data for analyzing the stresses
τ − τ developed in the viscoelastic bioink as it is extruded
2
Wi = xx yy = λγ i (XXIII) through the dispensing nozzle. The extensional viscosity
τ yx yx can be measured using a filament stretching rheometer
This number compares the first normal stress difference (FiSER). 109,110 Figure 9 illustrates a schematic of the fluid
and shear stress as characteristic values for elastic and filament stretched inside a FiSER. In this method, a
.
viscous forces. In a simple shear flow, τ and γ denote the cylindrical bridge of liquid is created between two parallel
yx
yx
shear stress and strain rate, respectively. plates. The upper plate is then moved up, stretching the
liquid bridge and creating a tensile stress from the tensile
The Weissenberg number quantifies the significance force (F ) exerted by plates. The upper plate velocity profile
E
of viscoelastic behavior in a flow. For example, depending U is adjusted to create an exponential decrease in midpoint
u
on the contraction ratio, bioink flow inside a chamfered radius (R) with a constant stretching rate γ . 111 :
nozzle with Wi > Wi may develop symmetrical (or e
cr
asymmetrical) vortices in the contraction area, even at low 2 dR
i
Reynolds numbers. The critical Weissenberg number Wi γ =− (XXV)
e
cr
determines when elastic forces within the viscoelastic fluid R dt
are strong enough to affect the regular viscous behavior The Hencky strain (γ ) at the midpoint of the liquid
e
of fluid. Conversely, for Wi > Wi , the bioink may exhibit bridge is influenced by the stretching history and can be
cr
Volume 10 Issue 6 (2024) 129 doi: 10.36922/ijb.3973