# Numerical Modeling of the Motion and Interaction of a Droplet of an Inkjet Printing Process with a Flat Surface

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Methodology for the Simulation

_{1}denotes the air density while $\mathsf{\rho}$

_{2}denotes the ink density. For the numerical simulation, the dynamic viscosity (shear viscosity) μ must be taken into account (3). It describes the relationship between the shear rate and the shear stresses in the fluid. Dynamic viscosity is calculated by:

_{1}—is the air viscosity and μ

_{2}—is the ink viscosity. The transportation of mass and momentum are described by the incompressible Navier–Stokes equations. Both ink and air can be considered as incompressible if the fluid velocity is small compared to the speed of sound.

**u**represents the fluid velocity, p denotes pressure, and

**F**st is the surface tension force. To capture the effect of the surface tension, a volume force is added at the phase interface in the computational domain where the phase interface is at present. The force depends on the surface tension coefficient and the curvature of the phase interface.

**n**is the unit vector in the normal direction, σ is the surface tension coefficient which varies according to the curve, and δ equals a Dirac delta function that is nonzero only at the fluid interface, $\mathsf{\u0138}=-\nabla \xb7n$ is the curvature. The unit vector in the normal direction is given by:

^{2}.

## 4. Obtained Results

_{d}.

_{max}determinates the droplet max diameter in the horizontal direction, while D

_{h}determinates the droplet height in vertical direction. These basic points help to define the droplet status over time t.

_{3}does not rebound and deposits least. Final point F represent the droplet in final-stable form which is reached at different times after the ejection: F

_{1}-174 µs (from D

_{1}to F

_{1}-25 µs.), F

_{2}-172 µs (from D

_{2}to F

_{2}-14 µs), F

_{3}-181 µs (from D

_{3}to F

_{3}-2 µs). Droplets from Ink

_{1}are not so viscous and after the impact, they rebound a few times up and down. That is why these inks reach stable phase later than Ink

_{2}. Ink

_{3}does not oscillate and takes final form very fast after the impact with the surface.

_{d.max}(Table 2) is the maximum droplet diameter (horizontal cross section); D

_{h}(Table 2) is droplet height (vertical cross section).

_{1}with lowest density and viscosity parameters has the smallest jet length and reveals the biggest oscillations. The droplet maximum spreading diameters is E

_{1}= 47.7 µm. The droplet sphere diameter during flight is ~22 µm. The minimum spreading diameter has Ink

_{3}which does not oscillate at all. The latter depends on droplet density and viscosity parameters—the droplet takes its final form then it touches the substrate.

_{2}reveals the best suitable result. Droplet from Ink

_{2}does not oscillate much and has the smallest time period to reach a stable form on the surface. Droplets from Ink

_{1}reach the surface faster but, in case of too low viscosity and density parameters, oscillate more strongly before they take their final form on the surface. For UV curable inks, it is very important to undergo this step in the stable phase of the droplet.

_{1}fully formed at 11 µs, Ink

_{2}-13 µs, Ink

_{3}-14 µs). When the liquid droplet is ejected, it needs a certain time and distance to reach the full spherical form and during the impact, it will lose its spherical shape and take final form on the surface. All these depend on the ejecting parameters, droplet velocity, nozzle configurations, and liquid characteristics. In order to examine the differences between ink droplets, their mass can be addressed, which are: Ink

_{1}droplet ${m}_{d}$ = 14.4·10

^{−12}kg, Ink

_{2}droplet ${m}_{d}$ = 14.8·10

^{−12}kg, Ink

_{3}droplet ${m}_{d}$ = 15.2·10

^{−12}kg. From this graphic, we can see that droplet mass stays fixed after point B when the droplet thread is pulled off.

_{1}droplet ${V}_{d}$ = 4.3491 pL, Ink

_{2}droplet ${V}_{d}$ = 4.3493 pL, and Ink

_{3}droplet ${V}_{d}$ = 4.3495 pL.

## 5. Discussion

## 6. Conclusions

- The droplet reaches its maximum velocity at the ejection moment when the pressure inside the nozzle is at the highest level.
- When the thread of the drop is detached from the nozzle, its speed is much lower and during flight, the speed of the drop decreases further.
- After impact, it is observed, that the drop loses its spherical shape and takes on a different form on the surface.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Roman | |

d | the diameter of the nozzle (mm); |

D_{d.max} | maximum droplet diameter (µm); |

D_{h} | droplet height (µm); |

DOD | drop-on-demand; |

F_{st} | surface tension force (N); |

g_{const} | acceleration due to gravity (m/s^{2}); |

m_{d} | droplet mass (kg); |

n | the unit vector in the normal direction; |

Oh | dimensionless coefficient; |

p | pressure (Pa); |

r | droplet radius (mm); |

g | gravitational constant (m/s^{2}); |

Re | dimensionless coefficient; |

t | time (s); |

u | fluid velocity (m/s); |

v_{d} | droplet velocity (m/s); |

V_{d} | droplet volume (pL); |

We | dimensionless coefficient; |

Z | dimensionless number. |

Greek | |

β | maximum spreading factor; |

γ | parameter which determines the repetition of initiations; |

δ | Dirac delta function; |

ε | is the representative mesh size in the area passed by the falling droplet; |

ĸ− | is the curvature; |

η | is fluid viscosity (m/s); |

μ | dynamic viscosity (N·s/m^{2}); |

ρ | density (kg/m^{3}); |

σ | surface tension coefficient (mN/m); |

ϕ | coefficient of level set interface between air and ink. |

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**Figure 1.**Droplet fall/movement. (

**a**) During the inkjet printing processes, the droplet is ejected from the inkjet nozzle and settles while changing its shape at different stages (A–F); (

**b**) Highlighted inkjet droplet phases at different stages (A–F). Phases from the left: A—ejection and formation of the droplet; B—droplet pinch off from the nozzle; C—droplet sphere forming; D—droplet before the interaction with the surface; E—droplet spreading at the surface; F—droplet in final stable form.

**Figure 5.**Representation of the droplets (Ink

_{1}, Ink

_{2}, Ink

_{3}) velocity turnover. Velocity versus time.

Substance | ρ, kg/m^{3} | μ, mN·s/m^{2} | σ, mN/m | Oh | Z |
---|---|---|---|---|---|

Air | 1.225 | 1.789·10^{−2} | - | - | - |

Ink 1st | 1050 | 7 | 72 | 0.16 | 6.09 |

Ink 2nd | 1080 | 10 | 72 | 0.23 | 4.32 |

Ink 3rd | 1110 | 14 | 72 | 0.32 | 3.13 |

Ink_{1} | Ink_{2} | Ink_{3} | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

t µs | v_{d}ms | D_{d.max}µm | D_{h}µm | t µs | v_{d}ms | D_{d.max}µm | D_{h}µm | t µs | v_{d}ms | D_{d.max}µm | D_{h}µm | |

A | 2 | 13.8 | 26.3 | 9.2 | 2 | 13.8 | 25.8 | 9.7 | 2 | 13.8 | 23.5 | 9.9 |

B | 11 | 8.51 | 20.4 | 110.9 | 13 | 7.88 | 20.2 | 115.3 | 14 | 7.28 | 20.0 | 118.7 |

C | 26 | 7.05 | 23.3 | 26.2 | 32 | 6.77 | 24.9 | 26.3 | 39 | 5.93 | 24.0 | 26.6 |

D | 143 | 4.48 | 22.2 | 21.7 | 150 | 4.28 | 22.2 | 21.8 | 169 | 3.60 | 21.9 | 22.0 |

E | 149 | 0.18 | 47.7 | 4.3 | 161 | 0.12 | 42.8 | 7.2 | 183 | 0.05 | 37.5 | 11.6 |

F | 174 | 0 | 36.0 | 11.6 | 172 | 0 | 36.1 | 12.1 | 181 | 0 | 36.3 | 11.9 |

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**MDPI and ACS Style**

Tofan, T.; Kruggel-Emden, H.; Turla, V.; Jasevičius, R.
Numerical Modeling of the Motion and Interaction of a Droplet of an Inkjet Printing Process with a Flat Surface. *Appl. Sci.* **2021**, *11*, 527.
https://doi.org/10.3390/app11020527

**AMA Style**

Tofan T, Kruggel-Emden H, Turla V, Jasevičius R.
Numerical Modeling of the Motion and Interaction of a Droplet of an Inkjet Printing Process with a Flat Surface. *Applied Sciences*. 2021; 11(2):527.
https://doi.org/10.3390/app11020527

**Chicago/Turabian Style**

Tofan, Tim, Harald Kruggel-Emden, Vytautas Turla, and Raimondas Jasevičius.
2021. "Numerical Modeling of the Motion and Interaction of a Droplet of an Inkjet Printing Process with a Flat Surface" *Applied Sciences* 11, no. 2: 527.
https://doi.org/10.3390/app11020527