Modeling thermal inkjet and cell printing process using modified pseudopotential and thermal lattice Boltzmann methods

Pseudopotential lattice Boltzmann methods (LBMs) can simulate a phase transition in high-density ratio multiphase flow systems. If coupled with thermal LBMs through equation of state, they can be used to study instantaneous phase transition phenomena with a high-temperature gradient where only one set of formulations in an LBM system can handle liquid, vapor, phase transition, and heat transport. However, at lower temperatures an unrealistic spurious current at the interface introduces instability and limits its application in real flow system. In this study, we proposed new modifications to the LBM system to minimize a spurious current which enables us to study nucleation dynamic at room temperature. To demonstrate the capabilities of this approach, the thermal ejection process is modeled as one example of a complex flow system. In an inkjet printer, a thermal pulse instantly heats up the liquid in a microfluidic chamber and nucleates bubble vapor providing the pressure pulse necessary to eject droplets at high speed. Our modified method can present a more realistic model of the explosive vaporization process since it can also capture a high-temperature/density gradient at nucleation region. Thermal inkjet technology has been successfully applied for printing cells, but cells are susceptible to mechanical damage or death as they squeeze out of the nozzle head. To study cell deformation, a spring network model, representing cells, is connected to the LBM through the immersed boundary method. Looking into strain and stress distribution of a cell membrane at its most deformed state, it is found that a high stretching rate effectively increases the rupture tension. In other words, membrane deformation energy is released through creation of multiple smaller nanopores rather than big pores. Overall, concurrently simulating multiphase flow, phase transition, heat transfer, and cell deformation in one unified LB platform, we are able to provide a better insight into the bubble dynamic and cell mechanical damage during the printing process.

Figure 1

Calculation of interparticle force on boundary nodes using ghost points (shown in red).

Figure 2

Maxwell construction of the Peng-Robinson (P-R) EOS at various subcritical temperatures. Highest pressure at $T=T_c$ is shown with a red solid line (dark gray), while lowest pressure at $T=0.5T_c$ is shown in a solid cyan line (light gray).

Figure 3

Maxwell construction perdition for P-R EOS versus simulation results of our coupled pseudopotential-thermal model using reduced EOS. The relative density, $\rho /{\rho }_c$, is demonstrated in two linear and logarithmic scales at right and left, respectively.

Figure 4

Mesh dependency analysis of phase separation at $T_{\mathrm{init}}=0.5T_c$. Relative density data along dotted line shown in Fig. 3.

Figure 5

Liquid-vapor separation for different reduced parameter. Mesh size is $150\times 150\times 150$. From highest density $\left[\mathrm{at}T=0.5T_c,k^{\prime }=0.1,\mathrm{cyan}\left(\mathrm{light}\mathrm{gray}\right)\right]$ to lowest density $\left[\mathrm{at}T=0.5T_c,k^{\prime }=0.05,\mathrm{blue}\mathrm{line}\left(\mathrm{dark}\mathrm{gray}\right)\right]$.

Figure 6

Spurious current in lattice units for various temperatures and reduced parameters. The red stars represent the spurious current at $k^{\prime }=1$ using the original pseudopotential formulation by Gong and Cheng [8].$0.5T_c$ and $0.8T_c$ are colored violet (dark gray) and blue (light blue), respectively. The velocity distribution contours demonstrate an unrealistic spurious current at the phase interface.

Figure 7

Droplet on a surface for different liquid-solid interaction strengths: (a) $g_s=0$, (b) $g_s=-10$, (c) $g_s=10$. These simulations are carried out with an initial temperature of $0.5T_c$ with $120\times 240\times 600$ lattice nodes. (d) Relative density across the dashed line for various liquid-solid interaction strengths.

Figure 8

(a) Cross section of the HP60 nozzle plate. (b) Schematics of the microinkjet nozzle in a computational model. Heating resistor is shown in red.

Figure 9

Relative density distribution during initialization. $t_{\mathrm{lattice}}=7000$ marks $t=0\mathrm{s}$ in reference units.

Figure 10

Linear variation of reduced parameter versus relative temperature. $k^{\prime }$ equals $0.03$ at $0.45T_c$.

Figure 11

Solution process flow chart.

Figure 12

Time sequence of droplet injection process at nozzle cross section. Color bar represents relative density.

Figure 13

Time sequence of droplet injection process at nozzle cross section. Color bar represents relative temperature.

Figure 14

Relative density distribution during nucleation, bubble growth, and bubble collapse (a) before and (b) after bubble coalescence. The density data shown are related to dashed lines shown in Fig. 12 at (a) $t=4\mu \mathrm{s}$ and (b) $t=13\mu \mathrm{s}$. (a) Black and gray solid lines represent density distribution at $t=4\mu \mathrm{s}$ and $t=11.5\mu \mathrm{s}$, respectively. (b) Black and gray dashed lines represent density distribution at $t=13.5\mu \mathrm{s}$ and $t=35.5\mu \mathrm{s}$, respectively.

Figure 15

Comparison of drop head velocity between experiment [44], CFD simulation of Tan et al. [44], and pseudopotential LB method in this study.

Figure 16

The history of (a) minimum vapor density and (b) maximum local temperature from the pseudopotential LB method.

Figure 17

Maximum local pressure derived from the pseudopotential LB method in this study compared to vapor pressure used in Ref. [13].

Figure 18

Time sequence of cell deformation as it gets squeezed out of the nozzle printer.

Figure 19

(a) Areal strain, (b) stretching speed, (c) maximum axial stress, and (d) maximum shear stress distribution on cell membrane at $6.25\mu \mathrm{s}$. (e) The probability of pore formation $R\left({\varepsilon }_{\mathrm{A}}\right)$ at various stretching speeds [54].