Page 162 - IJB-10-5

International Journal of Bioprinting Nozzle optimization for multi-ink bioprinting

SA (I-3, molecular weight: 24 kDa) was obtained from The volume of fluid (VOF) method was employed to
KIMIKA (Tokyo, Japan). The phenol derivative of SA (SA- analyze the liquid–liquid interface, while the continuum
Ph) was synthesized according to the methods described in surface force (CSF) model was used to examine the surface
previous studies, 37,38 with the phenol groups introduced to tension between the liquids. The governing equations are
SA at a concentration of 1.8 × 10 mol/g-SA. as follows:
−4
2.2. Definition of switching efficiency
Switching efficiency (Se) is the metric used to evaluate the ∂α +∇ ⋅( ) +∇ ⋅ ( ( 1 − ) ) =α αv 0, (V)
αv
liquid behavior inside a single nozzle. Considering a liquid ∂t r
flowing at v (cm /s) into the conjunction area where two
3
inlets join with a volume of V (cm ) (Figure 1A), the ideal
3
switching time (t ) is defined as follows: v = α v + (1 − α v ) in2 , (VI)
in1
i
t = V . (I) v = v − v , (VII)
v
i
in1
in2
r
Se is defined as
where ∇ is the directional derivative; α is the ratio of
one fluid in a single nozzle; v and v are the inlet velocity
t in1 in2
Se = i , (II) from Inlets 1 and 2, respectively; and v is the relative
t m velocity. r
where t is the switching time measured through The physical property of the interface is calculated
m
numerical simulation and experimentation. Se is the ratio from α as follows:
between t and t . Ideally, efficient switching occurs at
i
m
Se = 1, while inefficient switching occurs at almost zero ρ = αρ + (1 − α ρ ) in2 , (VIII)
in1
(Figure 1B).
2.3. Numerical simulations
µ = αµ + (1 − α µ ) , (IX)
2.3.1. Governing equations for the flow inside the in1 in2
single nozzles
The fluid behavior inside a single nozzle was analyzed where α is the ratio of one fluid filled in a single nozzle.
using the numerical simulation software OpenFOAM (ver
8). This analysis involved solving the continuity equations ρ and ρ are the densities of the fluids from Inlets 1
in1
in2
and Navier–Stokes equations, which describe the motion and 2, respectively; and μ and μ are the viscosities of the
in1
in1
and interaction of incompressible fluids within the nozzle. fluid from Inlets 1 and 2, respectively. The analysis utilized
The flow was assumed to be laminar and incompressible. the viscosity data of the SA solution, which behaves as a
39,40
These equations are expressed as follows: shear-thinning fluid. The behavior is described by the
power law model as follows:
∇⋅ v f = 0, (III) μ = Kγ ˙ n–1 (X)

where µ is the viscosity of the fluid, K is the consistency
∂ ρ v +∇ ⋅(ρ v v index, γ˙ is the shear rate, and n is the flow index. K and
∂t f f f f f ) = (IV) n were determined by experimentally measuring the
−∇ +∇p 2 µ v f + ρ g + kσ n , viscosities of the inks as follows: the viscosities of SA
δ
f
f
s
solutions at concentrations of 0.5, 1.0, and 2.0 wt% were
where v , ρ , and µ are the velocity, density, and viscosity measured over a shear rate range of 0.01–10/s using a
f
f
f
of the fluid, p is the static pressure, g is acceleration gravity, rheometer (HAAKE MARS III, Thermo Fisher Scientific,
σ is the surface tension, κ is the curvature of the fluid MA, USA) equipped with a parallel plate geometry at a
interface, n is the normal vector of the interface, and δ is temperature of 25°C (Figure S1, Supporting Information).
s
the delta function.
Volume 10 Issue 5 (2024) 154 doi: 10.36922/ijb.4091