# Scaling Behavior of Pattern Formation in the Flexographic Ink Splitting Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Experimental

#### 3.1. Elastic Printing Plates

#### 3.2. Printing Liquids

## 4. Results

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RGB | digital image color encoding format (red-green-blue) |

TIFF | Tagged image file format using lossless image data compression |

FFT | Fast Fourier transform algorithm |

EPDM | Ethylene propylene diene rubber (synthetic) |

NBR | Acrylonitrile butadiene rubber (synthetic) |

## Appendix A

#### Appendix A.1. Printing Setup

**Figure A1.**(

**Left**) Gallus RCS 330-HD printing press; and (

**Right**) Flexo printing module with printing with blade-chamber reservoir (1), anilox roller (2), printing cylinder (3), impression roller (4) and substrate (5).

**Figure A2.**Web transport and roller arrangement in the used printing unit. The paper web is held under tension with a constant force of ${F}_{W}$.

#### Appendix A.2. Sample Digitalization and Spectral Analysis

**Figure A3.**Measurement procedure for the dominant pattern wavelength ${\lambda}_{i}$ of each sample i. In the first step, a one-dimensional Fast Fourier Transform Algorithm (1D-FFT) is applied to each line of the input image in MATLAB 2018b. The 1D-FFT delivers ${n}_{\mathrm{rows}}$ = 24,568 amplitude spectra. Each amplitude spectrum shows the spectral amplitudes ${A}_{i}(k)$ versus their wave numbers k. In the second step, a sample-averaged amplitude spectrum is calculated from which a clear maximum ${k}_{i}^{\mathrm{max}}$ is obtained in the predefined region of interest. Finally, the dominant pattern wavelength ${\lambda}_{i}$ is calculated from ${k}_{i}^{\mathrm{max}}$.

#### Appendix A.3. Symbols

**Table A1.**List of symbols. In the text, physical parameters are sometimes given in derived units (e.g., mm and MPa).

Symbol | Quantity | Units (SI) |
---|---|---|

${A}_{i}(k)$ | Spectral amplitude function of the Fourier transformed digital image of sample i | - |

${\alpha}_{x}$ | Scaling exponent expressing pattern wavelength $\lambda $ as a function of a quantity x | - |

${\alpha}_{x}^{(\mathrm{grav})}$ | Scaling exponent in a gravure printing setup | - |

${\beta}_{y}$ | Scaling exponent expressing a quantity y as a function of velocity v | - |

b | Thickness of the flexo printing plate | m |

$\beta $ | Raster period of the gravure cells on a gravure printing plate | m |

$\mathrm{Ca}$ | Capillary number | - |

d | Typical lateral printing resolution | m |

${D}_{0}$ | Height of the fluid menisci of the nip | m |

$\overline{D}$ | Average liquid film thickness in the nip | m |

${d}_{\mathrm{n}}$ | Heading tool diameter in the stress-strain measurement system | m |

$\mathsf{\Delta}t$ | Typical contact time of printing plate and substrate | s |

${F}_{\mathrm{n}}$ | Compression force exerted on printing plate pieces in the stress-strain measurement system | n |

${F}_{W}$ | Web tension of the paper substrate in the printing press | $\mathrm{N}/\mathrm{m}$ |

$\dot{\gamma}$ | Shear rate of the printing liquid | ${\mathrm{s}}^{-1}$ |

h | Thickness of the liquid layer in a Hele–Shaw cell, or between printing plate and substrate | m |

$\eta $ | Dynamical viscosity of the printing fluid | $\mathrm{Ns}/{\mathrm{m}}^{2}$ |

k | Wave number defined for spectral image evaluation | ${\mathrm{m}}^{-1}$ |

${k}_{i}^{\mathrm{max}}$ | Wave number corresponding to the dominant pattern wavelength $\lambda $ | ${\mathrm{m}}^{-1}$ |

$\tilde{K}$ | Apparent compressibility modulus of the printing plate | $\mathrm{N}/{\mathrm{m}}^{2}$ |

$\kappa $ | Gap widening coefficient | ${\mathrm{m}}^{3}/\mathrm{N}$ |

ℓ | Distance between incoming and outgoing fluid menisci of the nip | m |

$\lambda $, ${\lambda}_{i}$ | Dominant pattern wavelength of the instability, on sample i | m |

$\mathrm{Ks}$ | Ratio of capillary versus elastic forces | - |

${n}_{\mathrm{pix}}$, ${n}_{\mathrm{rows}}$ | Pixel numbers of the digitalized images of the printed samples | - |

${n}_{F}$ | Number of fingers across a printed sample | - |

p, ${p}_{0}$ | Fluid pressure, pressure amplitude in the nip | $\mathrm{N}/{\mathrm{m}}^{2}$ |

${r}_{1}$, ${r}_{2}$ | Radii of printing cylinder, impression roller | m |

${R}^{2}$ | Statistical correlation coefficient | - |

${r}_{\mathrm{n}}$ | Radius corresponding to the total curvature of printing cylinder and substrate | m |

$\sigma $ | Surface tension of the printing fluid | N/m |

v | Cylinder revolution and substrate velocity | $\mathrm{m}/\mathrm{s}$ |

W | Physical width of the evaluated sector of a printed sample | m |

## References

- Wegener, M.; Spiehl, D.; Sauer, H.M.; Mikschl, F.; Liu, X.; Kölpin, N.; Schmidt, M.; Jank, M.P.M.; Dörsam, E.; Roosen, A. Flexographic printing of nanoparticulate tin-doped indium oxide inks on PET foils and glass substrates. J. Mater. Sci.
**2016**, 51, 4588–4600. [Google Scholar] [CrossRef][Green Version] - Casademunt, J. Viscous fingering as a paradigm of interfacial pattern formation: Recent results and new challenges. Chaos
**2004**, 14, 809–824. [Google Scholar] [CrossRef] [PubMed] - Saffman, P.G.; Taylor, G.I. The penetration of a fluid into a porous medium or a Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A
**1958**, 245, 312–329. [Google Scholar] - Brailovsky, I.; Babchin, A.; Frankel, M.; Shivashinsky, G. Finger Instability in Water-Oil Displacement. Transp. Porous Media
**2006**, 63, 363–380. [Google Scholar] [CrossRef] - Sahimi, M. Flow phenomena in rocks: From continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Mod. Phys.
**1993**, 65, 1393. [Google Scholar] [CrossRef] - Wooding, R.A.; Morel-Seytoux, H.J. Multiphase fluid flow in porous media. Annu. Rev. Fluid Mech.
**1976**, 8, 233–274. [Google Scholar] [CrossRef] - Chen, J.-D.; Wilkinson, D. Pore-Scale Viscous Fingering in Porous Media. Phys. Rev. Lett.
**1985**, 55, 1892–1895. [Google Scholar] [CrossRef] [PubMed] - Kitsomboonloha, R.; Subramanian, V. Lubrication-Related Residue as a Fundamental Process Scaling Limit to Gravure Printed Electronics. Langmuir
**2014**, 30, 3612–3624. [Google Scholar] [CrossRef] [PubMed] - Bornemann, N.; Sauer, H.M.; Dörsam, E. Gravure Printed Ultrathin Layers of Small-Molecule Semiconductors on Glass. J. Imaging Sci. Technol.
**2011**, 55, 040201. [Google Scholar] [CrossRef] - Voß, C. Analytische Modellierung, Experimentelle Untersuchungen und Dreidimensionale Gitter-Boltzmann Simulation der Quasistatischen und Instabilen Farbspaltung. Ph.D. Thesis, Bergische Universität Gesamthochschule, Wuppertal, Germany, 2002. [Google Scholar]
- Lindner, A.; Coussot, P.; Bonn, D. Viscous Fingering in a Yield Stress Fluid. Phys. Rev. Lett.
**2000**, 85, 314–317. [Google Scholar] [CrossRef] [PubMed] - Lindner, A.; Bonn, D.; Poiré, E.C.; Amar, M.B.; Meunier, J. Viscous fingering in non-Newtonian fluids. J. Fluid Mech.
**2002**, 469, 237–256. [Google Scholar] [CrossRef] - Bico, J.; Roman, B.; Moulin, L.; Aoudaoud, A. Elastocapillary coalescence in wet hair. Nature
**2004**, 432, 690. [Google Scholar] [CrossRef] [PubMed] - Roman, B.; Bico, J. Elasto-capillarity: Deforming an elastic structure with a droplet. J. Phys. Condens. Matter
**2010**, 22, 493101. [Google Scholar] [CrossRef] [PubMed] - Kim, H.-Y.; Mahadevan, L. Capillary rise between elastic sheets. J. Fluid Mech.
**2006**, 548, 141–150. [Google Scholar] [CrossRef] - Sauer, H.M.; Daume, D.; Dörsam, E. Lubrication theory of ink hydrodynamics in the flexographic printing nip. J. Print Media Technol. Res.
**2015**, 4, 163–172. [Google Scholar] - Gaskell, P.H.; Innes, G.E.; Savage, M.D. An experimental investigation of meniscus roll coating. J. Fluid Mech.
**1998**, 355, 17–44. [Google Scholar] [CrossRef][Green Version] - Sauer, H.M.; Roisman, I.V.; Dörsam, E.; Tropea, C. Fast liquid sheet and filament dynamics in the fluid splitting process. Colloids Surf. A
**2018**, 557, 20–27. [Google Scholar] [CrossRef]

**Figure 1.**Viscous fingering instability in the printing nip creates ribbing defect on the printed sample. The cylinder revolution and substrate velocity v affects the dominant pattern wavelength $\lambda $.

**Figure 2.**Principal design of a flexographic printing unit. A, B, incoming and outgoing ink meniscus; N, printing nip; ${D}_{0}$, meniscus height; ℓ, width of the wetted zone.

**Figure 3.**Experimental setup to determine the gap widening coefficient $\kappa $. A cylindric heading tool (1) with diameter ${d}_{\mathrm{n}}$ is used to deform the printing plate (2) by $-\mathsf{\Delta}b$. The applied force is ${F}_{\mathrm{n}}$.

**Figure 5.**The finger wavelength $\lambda $ as a function of printing velocity v for different printing forms and fluids in double logarithmic plots: (

**Left**) compressible printing plates with printing fluid at standard formulation; (

**Middle**) compressible printing plates with diluted fluid; and (

**Right**) incompressible elastomer printing plate with diluted fluid.

**Table 1.**Measured apparent compressibilities and gap widening coefficients of the used compressible (${P}^{\mathrm{comp}}$) and incompressible (${P}^{\mathrm{inc}}$) printing plates.

Printing Plate | $\tilde{\mathit{K}}$ [MPa] | $\mathit{\kappa}$ [$\mathbf{mm}/\mathbf{MPa}$] |
---|---|---|

${P}_{soft}^{\mathrm{comp}}$ | $9.57\pm 0.18$ | $0.159\pm 0.003$ |

${P}_{hard}^{\mathrm{comp}}$ | $11.19\pm 0.33$ | $0.136\pm 0.004$ |

${P}_{1}^{\mathrm{inc}}$ | $3.57\pm 0.05$ | $1.961\pm 0.028$ |

${P}_{2}^{\mathrm{inc}}$ | $4.86\pm 0.05$ | $1.440\pm 0.015$ |

${P}_{3}^{\mathrm{inc}}$ | $7.97\pm 0.12$ | $0.878\pm 0.013$ |

${P}_{4}^{\mathrm{inc}}$ | $8.81\pm 0.06$ | $0.795\pm 0.005$ |

Ink ID | Printing Liquid | $\mathit{\eta}$ [mPas] at $26\pm 1$ ${}^{\circ}$C | $\dot{\mathit{\gamma}}$ [${\mathit{s}}^{-1}$] | $\mathit{\sigma}$ [$\mathbf{mN}/\mathbf{m}$] at $24\pm 1$ ${}^{\circ}$C |
---|---|---|---|---|

LS1 | Siegwerk magenta std. formulation | $305\pm 21$ $100\pm 5$ | 1–7 2000 | $38.1\pm 0.4$ |

LS2 | 4:1 diluted with water | $19.3\pm 1.7$ $19\pm 1$ | 50–100 2000 | $37.1\pm 0.4$ |

LK1 | Kappaflex red std. formulation | $46.6\pm 0.9$ | 6–50 | $39.0\pm 0.4$ |

LK2 | + tenside | $51.3\pm 5.1$ | 6–50 | $28.8\pm 0.6$ |

**Table 3.**The ratios of the observed dominant pattern wavelengths in samples printed with the inks LS1 and LS2 of viscosities ${\eta}^{(LS1)}=200\pm 100$ mPas and ${\eta}^{(LS2)}=19$ mPas, respectively, but identical surface tensions $\sigma =38\pm 1\phantom{\rule{0.277778em}{0ex}}\mathrm{mN}/\mathrm{m}$, using printing plates with the same velocities v and gap widening coefficients $\kappa $.

v [m/min] | $\mathit{\kappa}$ [mm/MPa] | Ratio ${\mathit{\lambda}}^{\mathit{LS}1}/{\mathit{\lambda}}^{\mathit{LS}2}$ |
---|---|---|

10 | 0.136 | 0.6110 |

20 | (plates ${P}_{hard}^{\mathrm{comp}}$) | 0.6306 |

60 | 0.6059 | |

100 | 0.5880 | |

160 | 0.6953 | |

10 | 0.159 | 0.6237 |

20 | (plates ${P}_{soft}^{\mathrm{comp}}$) | 0.7025 |

60 | 0.7057 | |

100 | 0.6627 | |

160 | 0.5981 | |

average: | $0.642\pm 0.044$ |

**Table 4.**Experimental vs. theoretical scaling exponents of the viscous finger wavelength $\lambda (x)\sim {x}^{{\alpha}_{x}}$, as a function of the printing parameters x: printing velocity v, ink viscosity $\eta $, surface tension $\sigma $, gap widening coefficient $\kappa $, and cylinder radius ${r}_{\mathrm{n}}$. The theoretical exponents for rigid, i. e. non-deformable plates, are shown for comparison.

Exponent | Theory | Experiment | |
---|---|---|---|

${\mathit{\alpha}}_{\mathit{x}}=\frac{\partial log\mathit{\lambda}}{\partial log\mathit{x}}$ | Rigid ($\mathit{\kappa}=\mathbf{0}$) | Elastic ($\mathit{\kappa}>\mathbf{0}$) | Elastic Plates |

${\alpha}_{v}$ | $-0.5$ | $-0.1$ | $-0.07\pm 0.02$ |

${\alpha}_{\eta}$ | $-0.5$ | $-0.1$ | $-0.20\pm 0.06$ |

${\alpha}_{\sigma}$ | $+0.5$ | $+0.5$ | $+0.21\pm 0.23$ |

${\alpha}_{\kappa}$ | — | $+0.4$ | $+0.74\pm 0.49$ |

${\alpha}_{{r}_{n}}$ | 0 | $+0.2$ | (not measured) |

**Table 5.**Estimates of scaling exponents of further parameter as a function of the printing velocity v, derived from the exponents in Table 4: meniscus height ${D}_{0}$, length of the wetting zone ℓ, pressure amplitude ${p}_{0}$, and shear rate $\dot{\gamma}$.

Parameter | Exponent | Theory | Derived from Exper. | |
---|---|---|---|---|

${\mathit{\beta}}_{\mathit{y}}=\frac{\partial log\mathit{y}}{\partial log\mathit{v}}$ | Rigid ($\mathit{\kappa}=\mathbf{0}$) | Elastic ($\mathit{\kappa}>\mathbf{0}$) | Elastic Plates | |

Nip meniscus height | ${\beta}_{{D}_{0}}$ | 0 | $+0.4$ | $+0.57\pm 0.02$ |

Wetting zone length | ${\beta}_{\ell}$ | 0 | $+0.2$ | $+0.285\pm 0.01$ |

Pressure amplitude | ${\beta}_{p}$ | 1 | $+0.4$ | $+0.57\pm 0.02$ |

Nip shear rate | ${\beta}_{\gamma}$ | 1 | $+0.6$ | $+0.43\pm 0.02$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Brumm, P.; Sauer, H.M.; Dörsam, E. Scaling Behavior of Pattern Formation in the Flexographic Ink Splitting Process. *Colloids Interfaces* **2019**, *3*, 37.
https://doi.org/10.3390/colloids3010037

**AMA Style**

Brumm P, Sauer HM, Dörsam E. Scaling Behavior of Pattern Formation in the Flexographic Ink Splitting Process. *Colloids and Interfaces*. 2019; 3(1):37.
https://doi.org/10.3390/colloids3010037

**Chicago/Turabian Style**

Brumm, Pauline, Hans Martin Sauer, and Edgar Dörsam. 2019. "Scaling Behavior of Pattern Formation in the Flexographic Ink Splitting Process" *Colloids and Interfaces* 3, no. 1: 37.
https://doi.org/10.3390/colloids3010037