THE POWER OF CAD/CAM LASER BIOPRINTING AT THE SINGLE-CELL LEVEL: EVOLUTION OF PRINTING - 3D Bioprinting and Nanotechnology in Tissue Engineering and Regenerative Medicine - page 85

As shown in Figure 4.6 , the computational domain used to simulate the thermoelastic stress gen-
eration is treated as 2D. This domain is due to the axisymmetric characteristic of the laser bioprinting
process under a typical round laser spot. For the stress wave governing equation, the second order central
difference scheme is used to approximate spatial derivatives, and the backward difference scheme is
used for the time derivative computation. The Crank-Nicolson method, which has second-order time
accuracy, is used to solve the stress wave governing equation. For a finite thickness coating, the pres-
sure wave reflection at the coating-air and the coating-glass interfaces has to be considered. Pressure
reflection occurs at the coating-air and the coating-glass interfaces due to their acoustic impedances.
The interface reflectivity is equal to − 1 for a free surface (interface) and 1 for a rigid surface (interface).
Thus the reflected stress at the rigid transparent support does not change the sign of stress due to the
very high acoustic impedance in the rigid support, while it does change its sign due to the reflection at
the free surface. The stress wave may be canceled by the reflected stress wave in the near vicinity of the
coating-air free surface since the reflected stress wave has an opposite sign.
Figure 4.7 shows the pressure profile at a fixed location, 50 m m below the laser spot center, for
the first 140 ns ( Wang et al., 2011 ). It is found that a bipolar pressure pulse is developed. A bipolar
pulse such as this was also observed in the study of the acoustic wave field generated in front of a
submerged fiber tip by Paltauf et al. (1998) . At about 33 ns after laser radiation, a positive compres-
sive pressure arrives at this fixed location, immediately followed by a negative tensile pressure. The
first pressure peak (13.9 MPa magnitude) originates from the compressive pressure of a plane wave,
and the subsequent tensile pressure ( − 14.4 MPa magnitude) emits from the edge of the laser spot.
Both compressive and tensile components physically coexist for the sake of the momentum conser-
vation ( Vogel and Venugopalan, 2003 ). They are experienced 4.6 ns apart on the order of 10 MPa at
this fixed location, 50 m m below the center of the laser spot. At approximately 66 ns, the compres-
sive pressure wave reaches the free surface and is reflected back into the coating medium as a tensile
stress wave. Then, at approximately 100 ns, the first reflected wave reaches the fixed location with a
peak magnitude of − 6.4 MPa, and another compressive wave is observed with an even higher peak
magnitude of 10.3 MPa; that is, a negative tensile pressure is followed by a larger positive compres-
sive pressure. The second pressure pair is formed due to pressure reflection at the coating-air free
surface, which changes the sign of pressure upon reflection. Since the wave energy is transmitted into
the surrounding coating during propagation, both components of the second pressure pair decrease in
magnitudes, as seen in Figure 4.7 .
FIGURE 4.6
Schematic of the computational domain ( Wang et al . , 2009 ).