Thesis for the degree of Doctor of Technology, Sundsvall 2010

NIP MECHANICS, HYDRODYNAMICS AND PRINT QUALITY INFLEXO POST-PRINTING

Martin Holmvall

Supervisors:Tetsu UesakaMyat Htun

Magnus Norgren

FSCN - Fibre Science and Communication NetworkDepartment of Natural Sciences, Engineering and Mathematics

Mid Sweden University, SE-851 70 Sundsvall, Sweden

ISSN 1652-893X,Mid Sweden University Doctoral Thesis 85

ISBN 978-91-86073-74-9

Akademisk avhandling sommed tillstand avMittuniversitetet i Sundsvall framlaggstill offentlig granskning for avlaggande av teknologie doktorsexamen, torsdag, 17:ejuni, 2010, klockan 10:00 i SCA-salen O102, Mittuniversitetet Sundsvall. Seminarietkommer att hallas pa engelska.

NIP MECHANICS, HYDRODYNAMICS AND PRINT QUALITY INFLEXO POST-PRINTING

Martin Holmvall

c© Martin Holmvall, 2010

FSCN - Fibre Science and Communication NetworkDepartment of Natural Sciences, Engineering and MathematicsMid Sweden University, SE-851 70 SundsvallSweden

Telephone: +46 (0)771-975 000

Printed by Kopieringen Mittuniversitetet, Sundsvall, Sweden, 2010

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NIP MECHANICS, HYDRODYNAMICS AND PRINT QUALITY INFLEXO POST-PRINTING

Martin Holmvall

FSCN - Fibre Science and Communication Network, Department of Natural Sci-ences, Engineering and Mathematics, Mid Sweden University, SE-851 70 Sundsvall,SwedenISSN 1652-893X, Mid Sweden University Doctoral Thesis 85; ISBN 978-91-86073-74-9

ABSTRACT

Corrugated board is used extensively for transporting goods. However, overthe last two decades, the corrugated board packaging has also been used more andmore for displaying and marketing purposes. Accordingly, the print quality de-mands for flexographic post-printing, which is the predominant printing methodfor corrugated board, has increased greatly. The most common print non-uniformityproblem of flexo post-printing is striping, i.e., periodic print density and/or printgloss variations parallel to the flutes. In spite of its long research history, the basicmechanisms of striping have not been fully understood, and no concrete solution hasbeen provided. The objective of this thesis is to obtain a fundamental understandingof the striping mechanisms, which is required for developing practical solutions.

To achieve this objective both experimental and numerical approaches have beentaken. Non-linear finite element models have been constructed in both corrugatedboard and halftone dot scales to study the nip mechanics. Parametric studies havebeen performed to investigate the effects of printing system variables and deforma-tions. Ink transfer experiments have been conducted to determine the print densityvs. nip pressure relations. A hydrodynamic two-phase model have been developedto study fluid transfer from a printing plate to a fibre network.

The results have shown that striping is predominantly print density variationscaused by pressure variations in the printing nip. The pressure variations are in-herent to the corrugated board structure with its fluted medium. Washboarding,defined as a geometrical liner board imperfection, was shown to play a minor partin causing print density variations, but might contribute to gloss striping. A newprinting plate design has been proposed to eliminate the pressure variations andhence the print density striping.

The ink transfer simulations have shown that, once contact between ink andboard is made, ink–fibre network interactions occur only at the topmost surface,with a penetration depth much less than the fibre thickness. This is because at a re-alistic printing time/length scale, ink behaves like an elastic gel, rather than a fluid.Therefore, the actual contact area between fibre network and ink is the dominantfactor that controls the ink transfer amount in flexo post-printing.

Keywords: corrugated board, flexography, ink transfer, nip pressure, printing, strip-ing

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SAMMANDRAG

Wellpapp anvands i stor utstrackning vid transporter av alla typer av varor.Under de senaste tva decennierna har dock wellpapp ocksa mer och mer borjatanvandas i presentations- och marknadsforingssyften. Foljaktligen har ocksa kravenpa tryckkvaliteten for flexografi, vilket ar den vanligaste tryckmetoden for direkt-tryckt wellpapp, okat markant. Den vanligaste tryckdefekten for direkttryckt well-papp ar “striping”, dvs periodiska tryckdensitets- och/eller glansvariationer paral-lellt med flutingpiporna. Trots dess langa forskningshistoria har inte en fundamentalforstaelse for de bakomliggande mekanismerna for striping uppnatts. Ingen konkretpraktisk losning har heller presenterats. Avsikten med denna avhandling ar att fajust denna grundlaggande forstaelse for stripingmekanismerna, vilken kravs for attutveckla en praktisk losning.

For att na dettamal har bade experimentella och numeriska angreppssatt anvants.Ickelinjara finita-element-modeller har utvecklats i badewellpapp- och rasterpunkts-skala for att studera nypmekaniken. Med dessa modeller har parameterstudier ge-nomforts for att forsta effekten av variationer i tryckvariabler och deformationer.Experimentella fargoverforingsstudier har genomforts for att utreda hur tryckden-siteten varierar med nyptrycket. En hydrodynamisk tvafasmodell har ocksa skapatsfor att studera fargoverforingen fran kliche till fibernatverk.

Resultaten visar att striping ar till storsta delen tryckdensitetsvariationer som or-sakats av tryckvariationer i nypet. Dessa tryckvariationer ar direkt associerade tillstrukturen genom att flutingpiporna bara ar forankrade till linern langs flutingtop-parna. Washboarding, vilket definieras som en geometrisk defekt hos linern, har vi-sat sig ha mindre betydelse for tryckdensitetsvariationerna, men skulle kunna geupphov till glansvariationer. En losning pa stripingproblemet har foreslagits genomatt ge ett materialdesignforslag for en ny kliche som skulle minimera nyptrycksvari-ationerna, och darmed tryckdensitetsvariationerna.

Fargoverforingssimuleringarna har visat att nar val farg och substrat har fatt kon-takt, sker interaktionen bara i det oversta skiktet. Penetrationsdjupet ar mindre anfibertjockleken. Detta beror pa att for realistiska tids- och langdskalor for tryckningbeter sig fargen snarare som en elastisk gel an som en vatska. Foljaktligen ar kontak-tytan den dominerande parametern for fargoverforingsmangden vid flexografiskttryck pa wellpapp.

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TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

SAMMANDRAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF PAPERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. FLEXOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. CORRUGATED BOARD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3. PRINT QUALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4. WASHBOARDING AND STRIPING TERMINOLOGY . . . . . . . . . . . . . 4

1.5. CORRUGATED BOARD DEFORMATIONS . . . . . . . . . . . . . . . . . . . 5

1.6. INK TRANSFER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.7. PROBLEM FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. STRIPINGOFCORRUGATEDBOARD INFULL-TONE FLEXOPOST-PRINTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1. MEASUREMENTS OF PRINT DENSITY AND PRINT GLOSS VARIATIONS . 8

2.2. DETERMINING INK TRANSFER VS. PRESSURE RELATIONSHIPS . . . . . 9

3. MODELLING OF NIP MECHANICS IN FLEXO POST-PRINTING . . . 12

3.1. NIP MECHANICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.1. Linear elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.2. Hyperelastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3. Non-linear finite element formulation . . . . . . . . . . . . . . . . 15

3.2. FINITE ELEMENT MODELLING . . . . . . . . . . . . . . . . . . . . . . . . 183.2.1. Nip mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2. Halftone deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4. PRESSURE VARIATIONS AND PRINT UNIFORMITY . . . . . . . . . . . 21

4.1. FACTORS AFFECTING PRESSURE VARIATIONS IN THE PRINTING NIP . . 21

4.2. EFFECTS OF DOT GAIN ON STRIPING IN HALFTONE PRINTING . . . . . 23

4.3. CORRUGATED BOARD DEFORMATIONS . . . . . . . . . . . . . . . . . . . 24

4.4. MINIMISING PRESSURE VARIATIONS IN THE NIP . . . . . . . . . . . . . 24

5. MODELLING HYDRODYNAMICS OF INK TRANSFER . . . . . . . . . . 28

5.1. NIP HYDRODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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5.1.1. Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.2. Diffuse interface method . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.3. Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1.4. Fibre network model . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2. INK TRANSFER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3. SURFACE PARAMETERS THAT CONTROL INK TRANSFER . . . . . . . . . 38

6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

ACKNOWLEDGENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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LIST OF PAPERS

This thesis is based on the following five papers, herein referred to by their Romannumerals:

Paper I Nip mechanics of flexo post-printing on corrugated board,Martin Holmvall and Tetsu Uesaka,Journal of Composite Materials, 41(17):2129-2145, 2007.

Paper II Striping of corrugated board in full-tone flexo post-printing,Martin Holmvall and Tetsu Uesaka,Appita Journal, 61(1):35-40, 2008.

Paper III Print uniformity of corrugated board in flexo printing: Effects of cor-rugated board and halftone dot deformations,Martin Holmvall and Tetsu Uesaka,Packaging Technology and Science, 21:385-394.

Paper IV Simulation of two-phase flow with moving immersed boundaries,Martin Holmvall, Stefan B. Lindstrom and Tetsu Uesaka,submitted to International Journal for Numerical Methods in Fluids.

Paper V Transfer of a microfluid to a stochastic fibre network,Martin Holmvall, Tetsu Uesaka, Francois Droletand Stefan B. Lindstrom,submitted to Journal of Fluids and Structures.

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1. INTRODUCTION

In the printing industry the achieved print quality, as a function of productioncost, is the main competitive parameter. For contact printing processes, like letter-press or offset lithography, the quality of the printed image is, to a large extent, de-termined in the printing nip. The ink is transferred when the printing plate andsubstrate make contact and then retract from each other. Therefore, a fundamen-tal understanding of the nip mechanics and hydrodynamics in the printing nip isof great importance to optimise the printing process, in turn to optimise the printquality and to minimise the production cost and environmental impact.

1.1. Flexography

A letterpress printing technology, which is commonly used in packaging, is flex-ography, or flexo for short. In flexo, ink is transferred from a printing plate to asubstrate with printing elements raised from the surface of the printing plate. Inrelation to other letterpress printing techniques, the flexo printing plates are moreflexible. This is the origin of the name itself; flexography.

Ink trough

Corrugated board

Impression cylinder

Printing cylinder

Printing plate

Anilox cylinder

Figure 1. A flexo printing press with four printing stations.

A characteristic feature of flexo printing is the low printing pressure; so called kissprinting [1]. The low printing pressure and flexible printing plates make flexo the pri-mary printingmethod for corrugated boards (see Fig. 1). The term flexo post-printingis used for printing on combined corrugated boards (see Fig. 2), as opposed to pre-printing of the liner board, which is done before assembling the corrugated board.There is a substantial risk that the corrugated medium of corrugated boards getsdamaged during printing if the pressure is too high. The consequence of damagedcorrugated medium is the loss of stiffness and structural integrity, which severelydecreases the packaging performance of the corrugated board boxes [2, 3].

1.2. Corrugated board

Corrugated board was first patented in 1856 in England and was originally usedas sweatbands in hats. As a packaging material, single faced corrugated board wasfirst used in the United States in 1871 to protect bottles [4]. Since then, corrugated

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boards have been extensively used to contain and to protect merchandise from dam-ages during transportation. Initially the demands on print quality on corrugatedboard was low. However, the flexo print quality today is superior to the quality justone or two decades ago. This has enabled the corrugated box to shift from just beinga container of merchandise, to also being a part of the display in the stores, i.e., shelf-ready. With this increased exposure to customers, the demands for even better printquality is constantly increasing.

Single wall corrugated board

Corrugating medium

Double backer linerboard

Single facer linerboard

Figure 2. Combining linerboards and medium into a corrugated board inthe corrugating machine.

1.3. Print quality

The prints done by flexo post-printing exhibit the same print defects as otherletterpress printing techniques, e.g., mottling, missing dots, streaks, etc. However,prints on corrugated boards particularly show visible stripes in the printed images,which is a phenomenon denoted as striping or banding [5] (see Fig. 3).

The scientific literature available about striping is rather limited. The most no-ticeable are works done by Netz [6] and Hallberg et al. [7, 8, 9]. Netz investigatedthe mechanisms and the effects of washboarding on print quality. Netz defined“washboarding” as a wavy deformation pattern of the top liner coinciding withthe flutes of the corrugated medium board (see Fig. 4). The study was motivatedby the fact that washboarding is routinely pointed out as the cause for the stripypatterns in flexo post-prints, because of the non-uniform contact conditions it maycreate [10]. To measure the magnitude of washboarding, Netz developed an opti-cal, non-contacting, topography measurement equipment. Using this equipment, hestudied, among other parameters, the effect of washboarding on striping. Netz drewthe conclusion that washboarding does not have a dominant influence on striping.This was in contradiction to the general view within the industry [11, 12, 13]. Netz,instead, concluded that the flute structure, causing pressure variations, was the mostinfluential factor on striping. Hallberg et al. showed, by using a pressure sensor, thatpressure differences between ridge and valley areas correlate with striping [8]. They

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claimed that striping was indeed caused by washboarding [7, 8], but the direct rela-tionship between washboarding and pressure variations were not shown.

Figure 3. A photo of striping on a full-toneprinted corrugated board.

Figure 4. Washboarding of the double backer linerboard of the corru-gated board. No washboarding (top) and washboarding (bottom).

Netz [6] studied striping for both halftone and full-tone flexo post-printing.“Halftone” and “full-tone” refers to how the imaging areas on the printing plate isconstructed (see Fig. 5). A full-tone area is a continuously inked area over the en-tire image, whereas a halftone image area is made up of small dots (see Fig. 6). Thehalftone dots can have different sizes, depending on how dark or light the imageshould be printed. Netz put forward some recommendations on how to avoid strip-ing in the two cases based on his studies. For halftone printing he recommendedless print squeeze and harder printing plate. For full-tone printing he recommendedmore print squeeze and softer printing plate. In practise this measure is difficult toimplement, because frequently halftone and full-tone areas exist on the same print-ing plate simultaneously. Accordingly, Netz suggested that halftone and full-toneshould be printed on separate printing plates. For the difference between the twoprinting methods, Netz speculated that halftone striping might come from dot gainvariations between ridge and valley areas, caused by pressure variations.

The sensitivity of dot gain to print squeeze was detailed by Bould [14, 15] by us-ing experimental and numerical (finite element) methods. Dot gain increases withprint squeeze, but levels out for a higher print squeeze because of a limited ink sup-ply on the halftone dot. For lower coverage dot gain becomes more sensitive to print

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Figure 5. Full-tone (left ) and halftone printing (right) element.

squeeze. Dot gain (physical) generally consists of deformation of dots on the print-ing plate and ink spreading. They showed that ink spreading is the main part of dotgain.

Then the next question is how striping observed in half-tone areas can be ex-plained by the above dot gain mechanism.

α

Dot height

Dot distance

Dot top diameter

Plate

Figure 6. Halftone dots and characteristicdimensions.

1.4. Washboarding and striping terminology

In this work, the term striping is defined as any type of periodic reflectance varia-tions in the printed area [6], and the term washboarding refers to the periodic defor-mations (geometrical imperfections) of the top liner board coinciding with the flutesof the corrugated medium board [4, 10] as in Fig. 4. However, within the area offlexo post-printing the terms “washboarding” and “striping” are often used inter-changeably. Netz called washboarding for unprinted stripes, which is defined as thewaviness of the liner board that can be felt with the fingertips [6]. The term stripingwas used for “common” printed stripes that are seen in final prints. Cusdin, how-ever, used the term “washboard” for the deflection of the liner board under printingpressure, and “washboarding” as a term to refer to a specific printed stripy patternas well [5].

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1.5. Corrugated board deformations

There are corrugated board deformations, other than washboarding, that mightcause print non-uniformities. These deformations might arise due to, e.g., produc-tion defects, machine failures or moisture variations in the structural part of the cor-rugated board. Warp has been mentioned as a candidate to cause printing problems[13]. Warp is a deformation of the entire corrugated board that makes the edges ofthe board curl up or down relative to the centre of the board [10]. The impact ofgeneral corrugated board deformations on the printing pressure (warp, and smallerwavelength deformations) has not been studied before in the literature.

1.6. Ink transfer

Another important aspect of printing, which affects print quality, is ink transfer.The print quality and print density depends on how much ink is transferred, andwhere it is deposited in and on the fibre network structure of the liner board.

Transferring ink precisely on the prescribed spot without much penetration intothe porous sheet structure is the target in high quality printing. In the case of stochas-tic fibre network structures [16], such transfer is controlled by local topography, com-monly referred to as surface roughness. There have been a number of studies of in-vestigating the relationship between ink transfer and surface roughness. To list afew [17, 18, 19, 20, 21, 22, 23, 24], the literature includes a phenomenological for-mulation of ink transfer, the examination of various concepts of surface roughness,and mechanistic studies using novel experimental techniques. However, the mostbasic challenge in printing research is to predict the amount and the location of thetransferred ink before the substrate is actually printed. It has been shown that sur-face properties such as pore size and local topography has an impact on ink transfer.Small pores retain the ink, whereas large pores empty, when the printing plate re-cedes from the substrate [25]. Also, too deep and/or steep recessions in the surfacecan prevent a uniform ink transfer [24]. For halftone printing the printing plate andink interact at a length scale that approximately corresponds to the halftone dot size,or the fibre width, i.e., below 100 µm. At this moment, the micro ink transfer vs.surface structure relationships are not well established.

1.7. Problem formulation

It is possible to break down print quality problems into a well defined scientific prob-lem, as shown in Fig. 7. In Fig. 7 the problem structure is shown for the main printquality problem in flexo post-printing, i.e., striping. On a fundamental level, stripingis observed as reflectance variations in the printed image. These reflectance varia-tions can have a diffuse or specular origin. In the diffuse case, the variations caneither originate from ink transfer variations due to local pressure variations in thelength scale of the variations, i.e, in the millimetre range, or from interaction be-tween the ink and the structure surface, i.e., in the fibre dimension length scale. In thespecular case, the variations might be from either variations in amount of reflected

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variations

variations

(Striping etc.)Print quality

Non-printing issue

Nip mechanics of

Reflectance

Variations inVariations in

Surface distortion(Washboarding etc.)

Ink transfernon-uniformity(micro scale)

non-uniformity(macro scale)

Ink transfer

Hydrodynamics ofmicrofluid transfercorrugated structures

Pressure sensitivityof ink transfer

Gloss variations inthe ink film

diffuse reflectance specular reflectance

Local pressure

Figure 7. A holistic view of flexo post-printing.

light by microroughness variations in the length scale of the light wavelength, or byvariations in the specular angle due to global distortion of the substrate, i.e., wash-boarding. Gloss non-uniformity caused by microroughness variations must comefrom ink transfer mechanisms in a corresponding length scale. The global distortionthat induces specular reflectance variations can be easily detected on the unprintedsample. Therefore, variations in gloss by a globally distorted surface is not a printingproblem, but a board manufacturing/converting problem.

The scientific problems that are connected to the problem structure in Fig. 7 areshown in Table 1. Mechanistic modelling of a flexo post-printing nip for both full-tone and halftone printing situations must handle several different length scales.Halftone dots can have a diameter below 100 µm, one corrugated medium wave-length for corrugated board of B-type is approximately 6.5 mm and structural de-formations, such as warp, is in the range of centimetres or larger. The problem alsoencompasses contact mechanics between printing cylinders and liner boards, as wellas between liner boards and the medium board. The printing plate is commonlytreated as hyperelastic material and the liner and medium boards as orthotropic ma-terials. Additionally, both the printing plate structures and the corrugated board can,during compression, experience large deformations (geometrical non-linearity).

A model of the hydrodynamics in a flexo post-printing nip, in a length scale be-low 100 µm, includes complex fluid flow when the fluid interacts with a porousstructure. The two fluid phases, ink and air, are separated by a free surface whichneeds to be simulated in an efficient way by, e.g., a diffuse interface approach. Dur-ing compression, the fibre boundaries are moving through the ink and air while si-multaneously changing their relative positions by compression. These fluid–structureinteraction problems are in the central theme of recent microfluidics.

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Nip mechanics of Hydrodynamics ofcorrugated structures microfluid transfer

• Multiscale modelling of • Complex fluid dynamicscorrugated structures • Two-phase fluid flow

• Contact mechanics • Moving immersed boundaries• Hyperelasticity • Diffuse interfaces• Orthotropic materials • Compression of• Large deformations stochastic structures

Table 1. The scientific problems involved in this thesis.

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2. STRIPING OF CORRUGATED BOARD IN FULL-TONE FLEXOPOST-PRINTING

2.1. Measurements of print density and print gloss variatio ns

To be able to discuss striping in a precise way, the nature of the print non-uniformitymust be determined. Therefore, an investigation of the striping phenomenon is nec-essary. Striping can be either print density variations or print gloss variations, orboth. In order to determine the nature of striping, special reflectance measurementswere done on the prints of two corrugated board samples, one coated and one un-coated (Fig. 8). Since the spatial resolution of conventional print density/print glossmeasurements are too low to resolve striping, a He-Ne laser and two detectors, asshown in Fig. 9, were used. This equipment was developed at PAPRICAN [26].The laser beam falls on the sample with an incidence angle of 45 degrees, and thedetectors at 0 degrees and 45 degrees receive the reflected light from the surface si-multaneously. The 0 degree detector measures diffuse reflectance, representing localprint density, and the 45 degree detector measures specular reflectance, representinglocal print gloss.

The reflectance was measured over an area of 51.2 by 51.2 mm with measuringpoints separated by 50 µm in a rectangular grid. The two-dimensional reflectancemaps from themeasurements were averaged along the crossmachine direction (alongthe flutes) to remove non-periodic signals, giving a one-dimensional curve. The datawas processed with Fourier analysis, to extract specific periodic components of thevariations, such as striping. The details for this study are given in Paper II.

Figure 8. Two flexo post-printed corru-gated boards with striping. One uncoated(left) and one coated (right).

Detec

torHeNe−laser

Det

ecto

r

Figure 9. Schematics of the He-Ne lasertopography measuring device.

Fig. 10 shows the normalised power spectrum for the reflectance measurementsof the printed corrugated board samples shown in Fig. 8. The normalisation wasdone by dividing the power at each frequency by the sum of the total power. Thetop left spectrum shows the diffuse reflectance result for the uncoated liner board.A clear peak corresponding to the visible striping pattern dominates the entire spec-trum. The top right spectrum shows the specular reflectance for the same uncoatedboard. No distinct peak is seen except two small peaks at periods 3.2 and 6.4 mm.

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These small peaks are likely due to washboarding, because almost all corrugatedboards exhibit some degree of washboarding and the specular reflectance is expectedto be higher both at the ridge areas and in the middle of the valley areas. (The linerhas an approximately flat reflecting surface at those positions).

The bottom left spectrum shows the diffuse reflectance result for the coated linerboard. In this spectrum there is no significant peak, corresponding well to the visi-ble appearance of the sample (no print density striping). The bottom right spectrumshows the result from the specular reflectance result for the same coated liner board.There is only one peak that is somewhat larger than the others; the one correspond-ing to the fluting wavelength. If we presume that diffuse reflectance is associated

0 3.2 6.4 9.6 12.80

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Figure 10. Power spectra for the reflectance measurements. Un-coated 0 degree detector (top left), uncoated 45 degree detector(top right), coated 0 degree detector (bottom left) and coated 45degree detector (bottom right).

to print density and specular reflectance to print gloss, then it is clear from theseresults that striping is predominantly caused by print density variations, with a pos-sible small influence of print gloss variations.

2.2. Determining ink transfer vs. pressure relationships

An underlying assumption for stripe formation is that ink transfer characteristicsare pressure-dependent, causing print density/print gloss variations. To validatethis assumption it is necessary to determine the pressure sensitivity of print densityand print gloss. The results from a pressure sensitivity experiment will also be nec-essary to evaluate the results of numerical simulations discussed in chapter 4. Thepressure dependence of ink transfer was determined by using an IGT F1 printabilitytester at different printing forces (30 to 500 N) [27]. Four different types of singleliner boards were printed; a test liner, a brown kraft liner, a white top kraft liner and

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a coated liner. Measuring the average print density and print gloss on the samplesgives a relationship between print density and force (or print gloss and force). Print-ing force was converted into printing pressure by measuring the pressure in the nipwith a pressure sensitive sensor (Tekscan, model 5051). It was placed in the nip, andfor each printing force the corresponding static pressure distribution was recorded.The pressure distribution in the nip is approximately a bell shape [28] with the max-imum in the centre of the nip. The maximum pressure (peak pressure) was used asthe pressure measure. The result from this study is presented in detail Paper II.

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1.25

Pressure [MPa]

Prin

t den

sity

[−]

Figure 11. Print density as a function of pressure for the four dif-ferent liner board grades. Top left: white top test liner. Top right:brown kraft liner. Bottom left: white top kraft liner. Bottom right:coated liner. In each figure the upper and lower curves representthe two different ink levels transferred to the printing plate by theanilox cylinder.

In Fig. 11 the relationship between print density and pressure is shown for thefour different liner board grades. All curves have a similar shape. Print densityrapidly increases with pressure until they reach a certain pressure level. The amountof ink placed on the printing plate does not have a significant influence on the be-haviour except for the print density values at the plateau level. A rougher linerboard, such as the test liner board (top left), is more sensitive to the pressure than asmoother liner board such as the coated liner (bottom right). In Fig. 12 the relation-ships between print gloss and pressure are shown for the same four different linerboard grades. For the print gloss, however, there is almost no significant dependenceon pressure. These results indicate that print density variations can be explained bypressure variations in the printing nip, whereas print gloss variations can not be ex-plained by the pressure variations. This suggests that the specular reflectance peaksin Fig. 10 did not come from local pressure variations but from the washboarding,i.e., the geometry of the top liner board. They might also originate from rubbing or

10

wear of the surface after printing, causing a microroughness smoothening of the topliner in the ridge areas.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Pressure [MPa]

Prin

t glo

ss [%

]

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Pressure [MPa]

Prin

t glo

ss [%

]

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Pressure [MPa]

Prin

t glo

ss [%

]

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

Pressure [MPa]

Prin

t glo

ss [%

]

Figure 12. Print gloss measured in the MD as a function of pres-sure for the four different liner board grades. Top left: white toptest liner. Top right: brown kraft liner. Bottom left: white top kraftliner. Bottom right: coated liner.

11

3. MODELLING OF NIP MECHANICS IN FLEXO POST-PRINTING

3.1. Nip mechanics

Aflexo post-printing nip is a two cylinder systemwith a corrugated board printedbetween the two cylinders, as shown in Fig. 13. This system contains several differ-ent materials. The flexo impression cylinder and the printing cylinder core are madeof steel. A corrugated board is constructed with a liner board on each side of a cor-rugated medium board. Around the steel core of the printing cylinder a printingform is wrapped. It has three layers: the plate cushion, the mounting plastic and theprinting plate. The plate cushion is a cellular solid (foam) of open-cell urethane, aMylar polyester film is used for the mounting plastic, and the printing plate is madeof a light sensitive photopolymer, which is similar to an elastomer.

Printing cylinder core

Corrugated board

Impression cylinder

Plate cushion

Photopolymer

Mounting plastic

Figure 13. Materials used in the flexo postprinting nip

In this work all materials have been assumed to be elastic materials. An elasticmaterial has a unique relationship between stress and strain [29], and returns to itsoriginal shape after all forces are removed. Using the first law of thermodynamics itcan be shown that a strain energy density function W can be defined that will yieldthe constitutive equation through [30]:

σij =∂W

∂eij, (3.1)

where σij is the stress tensor, and eij is the corresponding strain tensor, which willbe defined later.

Although flexo printing uses low printing pressures, the compression might stillinflict finite (large) strains, mainly in the photopolymer. This fact imposes a require-ment for finite stress and strainmeasures in the non-linear finite element formulationof the problem.

12

3.1.1. Linear elastic materials

We assume that the liner/medium boards, the steel and the mounting plastic arelinear elastic materials. For linear elastic materials the stress depends linearly onthe strain [31]. The strain energy density function for linear elastic materials is thusgiven by [30]:

W =1

2Dijkleijekl, (3.2)

where Dijkl is a fourth order elastic stiffness tensor. With Eq. (3.1) the generalisedtensor form of Hook’s law is derived as:

σij = Dijklekl. (3.3)

Paper can be treated as having three symmetry planes; the machine direction (MD),the cross machine direction (CD) and the thickness direction (ZD) [32]. Such ma-terials are called orthotropic materials and has nine independent elastic constants inthe elastic stiffness tensor, Dijkl. In this work it is assumed that the boards are or-thotropic linear elastic materials [33, 34]. Inverting the elastic stiffness tensor Dijkl

yields the compliance tensor, (Dijkl)−1. For orthotropic materials (Dijkl)−1 is writ-ten out on matrix form in Cartesian coordinates as:

(Dijkl)−1 =

1Ex

−νxy

Ex

−νxz

Ex0 0 0

−νyx

Ey

1Ey

−νyz

Ey0 0 0

−νzx

Ez

−νzy

Ez

1Ez

0 0 0

0 0 0 1Gxy

0 0

0 0 0 0 1Gyz

0

0 0 0 0 0 1Gxz

, (3.4)

where Ei is the Young’s moduli, νij , the Poisson’s ratios, and Gij the shear moduli.Also the compliance tensor is symmetric so that:

νxy

Ex=

νyx

Ey,

νxz

Ex= νzx

Ez,

νyz

Ey=

νzy

Ez.

(3.5)

The mounting plastic and steel is assumed to be isotropic linear elastic materials.Isotropic material symmetry imposes the following conditions:

E = Ex = Ey = Ez,ν = νxy = νxz = νyz,G = Gxy = Gxz = Gyz,G = E

2(1+ν) .

(3.6)

Therefore, isotropic materials have only two independent elastic constants, i.e. E

13

and ν giving:

(Dijkl)−1 =

1E

−νE

−νE 0 0 0

−νE

1E

−νE 0 0 0

−νE

−νE

1E 0 0 0

0 0 0 2(1+ν)E 0 0

0 0 0 0 2(1+ν)E 0

0 0 0 0 0 2(1+ν)E

. (3.7)

The elastic constants have to be determined through experiments. For materials likesteel, this is usually done with tensile testing. For paper, one method is to use ultra-sonic methods to determine the nine elastic constants [35, 36].

In this work moisture and heat induced deformations of the corrugated boardwere introduced by using thermal strain, (ekl)

th. In this case the elastic strain inEq. (3.3) must be replaced with:

ekl = (ekl)tot − (ekl)

th, (3.8)

where (ekl)tot is the total strain. Inverting Eq. (3.3) and using Eq. (3.8), we obtain:

(ekl)tot = (ekl)

th + (Dijkl)−1σij . (3.9)

The thermal strain vector, eth, in rectangular Cartesian coordinates is:

eth = ∆T [αsecx αsec

y αsecz 0 0 0]T , (3.10)

where αseci is the secant coefficient of thermal expansion in the direction i, ∆T =

T − Tref , T the current temperature and Tref the reference temperature at which thethermal strain is zero. In the case of hygro-strain, these terms are interpreted as thehygro-expansion coefficient and the moisture change, respectively.

3.1.2. Hyperelastic materials

The photopolymer printing plate and the plate cushion are assumed to be hy-perelastic materials. Hyperelastic materials are defined by a specific strain energydensity function. In this work two hyperelastic material models have been utilised.The Mooney-Rivlin material model [37, 38] and the Hill-Ogden compressible foammodel [39, 40]. The Mooney-Rivlin model is commonly used for incompressible ornearly incompressible materials, e.g., rubber. Flexo printing plates are often made ofrubber or photosensitive polymers [41] with similar stress-strain behaviour. Cellularfoams, on the other hand, can be highly compressible [42]. Foam materials can beused in many different applications to protect against damages by absorbing energy.In both cases the materials are assumed to be isotropic. The constitutive equationsfor hyperelastic materials are derived with Eq. (3.1) and a strain energy density func-tion W . The strain energy density function is, however, often given in terms of theprincipal stretch ratios, λi = 1 + ei, where ei is the principal strain.

With σi as the principal Cauchy’s stress components, assuming incompressiblematerials, Eq. (3.1) is expressed as a function of principal stretch ratios as [43, 44]:

σi = λi∂W

∂λi− p. (3.11)

14

The arbitrary constant p, identified as the pressure, is needed due to the incompress-ible constraint λ1λ2λ3 = 1. In the case of compressible materials, where the incom-pressibility constraint no longer holds, with J = λ1λ2λ3, Eq. (3.11) becomes:

σi = J−1λi∂W

∂λi. (3.12)

The strain energy density function for the Mooney-Rivlin material model is:

WM = C1(λ21 + λ2

2 + λ23 − 3) − C2(λ

−21 + λ−2

2 + λ−23 − 3), (3.13)

with C1 and C2 being material constants, yielding the constitutive equation:

σiM = 2C1λ

2i + 2C2λ

−2i − p. (3.14)

The strain energy density function for the Hill-Ogden compressible foam model ofthe first order is:

WO =µ

α

(

λα1 + λα

2 + λα3 − 3 +

1

β

(

J−αβ − 1)

)

, (3.15)

and thus the constitutive equation is given by:

σiO = J−1µ

[

λαi − J−αβ

]

, (3.16)

where µ and α are material constants and β is a material compressibility constant. Asmall β corresponds to a highly compressible material.

3.1.3. Non-linear finite element formulation

The finite element method is a general method for finding approximate solu-tions to partial differential equations as well as integral equations. These types ofequations are frequently found in, for example, structural mechanics. In modellingthe flexo post-printing nip, non-linearity exists both in materials and geometry. Theprinting plate photopolymer and plate cushion have non-linear material properties.In the printing nip, it is also possible that one or several parts of the nip experiencelarge (finite) strain. Flexo printing is a direct printing technique, it requires contactbetween printing plate and substrate for the ink transfer. During compression newcontacts are created between the medium and the liner board inside the corrugatedboard, as well as between the printing cylinders and the liner boards. Contact createsnon-linearity because boundary conditions change from traction free to constrained(or the opposite), when the load changes.

To derive the finite element formulation we begin with the equations of motion,derived by Cauchy in 1882, for an arbitrary region of a body [30]:

σij,j + bi = ρui, (3.17)

where σij,j is the the Cauchy stress tensor, bi the body force per unit volume in theregion of the body with volume V , ρ is the density and ui the acceleration vectorwhere a dot denotes a time derivative. The vector ui is the displacement vector.From Eq. (3.17) we can derive what is known as a weak formulation, and in solid

15

mechanics often referred to as the principle of virtual work. The equations of motionare multiplied with an arbitrary vector vi and integrated over the volume V :

∫

V

vi (σij,j + bi − ρui) dV = 0. (3.18)

By rewriting and applying the divergence theorem we can rewrite Eq. (3.18) as theweak form of the equations of motion as

∫

V

ρviuidV +

∫

V

ǫvijσijdV =

∫

S

vitidS +

∫

V

vibidV, (3.19)

where ti is the traction vector on the surface S of the volume V . The quantity ǫvij is

defined as

ǫvij =

1

2(vi,j + vj,i) , (3.20)

by using the arbitrary vector vi as the strain tensor, eij , is related to the displacementvector ui. At this stage it is convenient to put the weak formulation of the equationsof motion into matrix notation:

∫

V

ρvT udV +

∫

V

(ǫv)T

σdV =

∫

S

vT tdS +

∫

V

vT bdV, (3.21)

where T denotes the transpose and

ǫv =

ǫv11

ǫv22

ǫv33

2ǫv12

2ǫv13

2ǫv23

; σ =

σ11

σ22

σ33

σ12

σ13

σ23

;

u =

u1

u2

u3

; v =

v1

v2

v3

; t =

t1t2t3

; b =

b1

b2

b3

.

In the finite element method the displacement vector u is approximated by

u = Na, (3.22)

where N denotes the global shape functions and a is a column matrix including allnodal displacements of the body. The shape functions are interpolating functions forthe nodal displacements. Since the shape functions do not depend on time, we have

u = Na. (3.23)

With the displacements defined as in Eq. (3.22) the corresponding strains ǫ can beexpressed as

ǫ = Ba. (3.24)

16

The matrix B is derived frommatrix N . The arbitrary vector v is now approximatedin the same manner as the displacement vector u, which gives

v = Nc, (3.25)

andǫv = Bc, (3.26)

where the column matrix c is also arbitrary. Now, Eq. (3.23), Eq. (3.25) and Eq. (3.26)are inserted into Eq. (3.21), and with c independent of position we have:

cT

[(∫

V

ρNT NdV

)

a +

∫

V

BT σdV −

∫

S

NT tdS −

∫

V

NT bdV

]

= 0, (3.27)

Since Eq. (3.27) holds for arbitrary c, it is concluded that

Ma +

∫

V

BT σdV = f , (3.28)

where the mass matrix M is

M =

∫

V

ρNT NdV, (3.29)

and the external forces f

f =

∫

S

NT tdS +

∫

V

NT bdV. (3.30)

If we now focus on static problems, the nodal accelerations a are zero, and wehave

∫

V

BT σdV = f . (3.31)

These equations hold for any constitutive relation. In the linear elastic case the rela-tionship is simple, as discussed previously:

σ = Dǫ = DBa. (3.32)

If Eq. (3.32) is inserted into Eq. (3.31) we can write the equations on the familiar finiteelement form as

Ka = f , (3.33)

where the constant stiffness matrix K is defined as

K =

∫

V

BT DBdV. (3.34)

For general non-linear problems it is not possible to express the constitutive relation-ship between current stress σ and current strain ǫ directly. In general problems onlythe incremental relation between stress rate σ and strain rate ǫ is known. Therefore,Eq. (3.31) is differentiated to accommodate for insertion of the expression for thestress rate. This yields an incremental form of Eq. (3.33)

Kta = f , (3.35)

where Kt is the tangential stiffness matrix. Since the tangential stiffness matrix isnot a constant matrix, Eq. (3.35) is a system of non-linear equations. For a moredetailed discussion on the finite element method we refer to, for instance, Bathe [45]or Zienkiewicz [46].

17

3.2. Finite element modelling

3.2.1. Nip mechanics

Pressure distribution of flat corrugated boards

Since we have determined the print density pressure dependency in chapter 2,a pressure distribution analysis of the flexo post-printing nip will show if the pres-sure variations in the nip can explain striping. The pressure distribution was stud-ied by constructing a finite element model of a flexo post-printing nip with corru-gated board (see Fig. 14). The model included the printing plate photopolymer, theplate backing materials and the corrugated board structure with liner and mediumboards. Elastic constants for the photopolymer and the backing plate cushion weredetermined with tensile tests and for the liner and medium boards with ultrasonictesting. For the photopolymer and the plate cushion the Mooney-Rivlin hyperelasticmaterial model was used. For the liner and medium boards an orthotropic linearelastic material model was used. The model was built with three-dimensional eight-node structural solid elements (SOLID185 [47]) in ANSYS. Contact problems in thissystem were modelled with the contact elements CONTA174. For the liner-cylindercontact, the contact elements were set for standard unilateral contact. The liner-medium contact was also modelled with the same contact element type. In this casethey were set for bonded contact to simulate the glue. Symmetry boundary condi-tions were applied on the left, right and front side of themodel in Fig. 14. The bottomplane was held fix and the top plane was translated towards the corrugated boardto simulate printing. With this model a series of parametric studies were performedto determine which principal factors of the flexo printing nip that affect the pressurevariations most. Thickness and stiffness for all components were varied as well asprint squeeze and washboarding magnitude. This study is included in Paper I.

Printing direction

Mounting plastic

corrugated board

Photopolymer

B−flute

Plate cushion

Width direction

Figure 14. Finite element model used for the parameter study of the flexo post-printing nip.

18

Pressure distribution of deformed corrugated boards

Striping is a periodic print non-uniformity. One of the causes may be in the cor-rugated board medium structure. There are, however, other print non-uniformitieswhich are not periodic, but may be structure-related. Typical examples are warp [4]and other distortions at a scale which is larger than the fluting wavelength, whoseeffects may be superimposed on the striping effect. To study this, a finite elementmodel was created as seen in Fig. 15. The model uses the same material parametersas the flexo nip model in Paper I, but includes ten flutes. During manufacturing ofcorrugated boards, heat and moisture gradients are always present. These gradientscause deformations of the corrugated board by contracting and expanding the linerboards and the medium board. To mimic this, the deformation of the corrugatedboard was created by using virtual thermal strain on the top liner. The details fromthis study are presented in Paper III.

Figure 15. Finite element model used for the corrugated board deformationstudy.

3.2.2. Halftone deformation

Experimental evidence [6] has shown that there are differentmechanisms of strip-ing in full-tone and halftone prints. For the halftone striping, individual dot gainand/or dot print density are the factors that affect print density differences betweenthe fluting ridge and valley areas. A finite element model was created to determinethe amount of dot gain and local pressure in the individual dot (see Fig. 16). Due tosymmetry, only a quarter of a halftone dot was modelled. The material parametersused for the simulation were the same as for the flexo printing nip model in Paper

19

I. Only the photopolymer part of the printing form and the top liner board wereused in the model. The thickness of the printing plate and the stiff mounting plasticmake the deformation of the backing materials insignificant as compared with thehalftone dot deformation. The backing materials as well as the fluting structure onlyaffect how much print squeeze is applied to the photopolymer. This can be incorpo-rated into the translation of the halftone dot model. Since the variations of printingpressure due to the flute structure is already captured with the full-tone model, itis possible to compare the halftone dot deformation at a given pressure and a givenprint squeeze. This study is described in Paper III.

Liner board (top)

Photopolymer

Figure 16. Finite element model used for the halftone dot deformation study.

20

4. PRESSURE VARIATIONS AND PRINT UNIFORMITY

4.1. Factors affecting pressure variations in the printing nip

Figs. 17 through 19 show the contact pressure between the photopolymer andthe top liner for different parameter settings of the model in Fig. 14. The pressuredistributions are plotted over the width of the flute (left to right) and at the centre ofthe nip width (front side) of the model, in Fig. 14.

In Fig. 17 pressure variations across the flute width are shown for different pho-topolymer stiffnesses (left) and photopolymer thicknesses (right). Increasing thestiffness or decreasing the thickness of the photopolymer increased the pressure vari-ations. To decrease the pressure variations in full-tone printing, a thicker and/orsofter photopolymer is obviously better. However, the corresponding decrease inaverage pressure level, must be compensated for with increased print squeeze, andthis results in the same situation in the pressure variations as before, as will be dis-cussed later.

In Fig. 18 the effects of top liner stiffness (left) and thickness (right) on the pres-sure distribution are shown. In this case increasing the stiffness and/or the thicknessof the top liner board decreased the pressure variations. This result explains the fieldobservations that liner boards with a high content of recycled fibres sometimes havedifficulties with striping since they are usually thinner and/or less stiff.

0 1.3 2.6 3.9 5.2 6.50

0.1

0.2

0.3 50 % 75 % Control125 % 150 %

Position [mm]

Pre

ssur

e [M

Pa]

0 1.3 2.6 3.9 5.2 6.50

0.1

0.2

0.3 50 % 75 % Control125 % 150 %

Position [mm]

Pre

ssur

e [M

Pa]

Figure 17. Pressure distributions for different photopolymer stiffnesses (left) and thicknesses(right).

The pressure curves in Fig. 19 show the effects of stiffness and thickness of themedium board. A decrease in stiffness and/or thickness decreased the average pres-sure level. There were no impact on the pressure variations. However, the change inaverage pressure level might induce striping, because, in the lower pressure range,there is a steeper dependence of print density on printing pressure (see Fig. 11).

Fig. 20 shows the effect of print squeeze on the pressure difference between theridge and valley areas. The results showed that as the print squeeze increases (or thepressure increases), the pressure difference between the ridge and valley areas grad-ually increases. This result has a practical implication: in order to avoid striping, itis often recommended to use a softer photopolymer. However, this action decreasesthe average pressure level, requiring an increase in print squeeze to maintain the

21

0 1.3 2.6 3.9 5.2 6.50

0.1

0.2

0.3 50 % Control 150 %

Position [mm]

Pre

ssur

e [M

Pa]

0 1.3 2.6 3.9 5.2 6.50

0.1

0.2

0.3 50 % Control 150 %

Position [mm]P

ress

ure

[MP

a]

Figure 18. Pressure distributions for different top liner board stiffnesses (left) and thicknesses(right).

0 1.3 2.6 3.9 5.2 6.50

0.1

0.2

0.3 50 % Control 150 %

Position [mm]

Pre

ssur

e [M

Pa]

0 1.3 2.6 3.9 5.2 6.50

0.1

0.2

0.3 50 % Control 150 %

Position [mm]

Pre

ssur

e [M

Pa]

Figure 19. Pressure distributions for different medium board stiffnesses (left) and thicknesses(right).

22

average pressure. Therefore, because of the effect shown in Fig. 20, using a softerphotopolymer does not necessarily lead to smaller pressure variations.

Since washboarding has been blamed for causing striping, the effect of wash-boarding on the pressure variation was investigated. In Fig. 21 the pressure variationis shown for some realistic washboardingmagnitudes. The difference in the pressurevariation between the boards with no washboarding (control case) and with somewashboarding is small. In other words, most of the pressure variation originatesfrom the fluting structure that supports the top liner.

100 200 300 400 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7Peak pressureValley pressure

Print squeeze [µm]

Pre

ssur

e [M

Pa]

Figure 20. The relationship between thepressure at the ridges and in the valley.

0 1.3 2.6 3.9 5.2 6.50

0.1

0.2

0.3 Control 30 µm 50 µm

Position [mm]

Pre

ssur

e [M

Pa]

Figure 21. Pressure distributions for dif-ferent washboarding magnitudes (0, 30and 50 µm) of the top liner.

4.2. Effects of dot gain on striping in halftone printing

The pressure difference between ridge and valley areas in full-tone printing isalso present in halftone printing. In halftone printing these pressure variationsmightcause print density differences of individual dots but also dot expansion (dot gain)differences. A finite element model was constructed to evaluate the local pressurefor the dots and their expansion (dot gain). In Fig. 22 (left) the contact area betweena halftone dot and the liner board is shown as a function of average contact pressure.The initial dot area was 0.051 mm2. In both cases, the dots expanded with increasedpressure, but in comparison with the control case the stiffer halftone dot expandedless with increased pressure, as expected. Pressure differences between ridge andvalley areas can thus induce dot gain differences which are perceived as (average)print density differences due to different coverage.

Fig. 22 (right) shows the difference between halftone and full-tone printing re-garding print density as a function of print squeeze. Print density was calculated bythe pressure values obtained from the finite element analyses and the pressure-printdensity relation shown in Fig. 11. As can be seen in the figure, the halftone dot curvereaches a plateau print density at much lower print squeezes than the full-tone case.If there are both halftone and full-tone image areas on the same printing plate, thehalftone dots will reach their maximum print density level before the full-tone does.Further print squeeze on the halftone dots after reaching the plateau print densityonly produces dot gain. Striping in halftone printing is therefore caused by the dot

23

gain differences between ridge and valley areas, rather than the print density differ-ences of individual dots.

0 0.1 0.2 0.3 0.4 0.5 0.60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Control

Harder

Pressure [MPa]

Are

a [m

m2 ]

0 100 200 3000

0.2

0.40.6

0.81

1.21.4

1.6

Halftone dot

Full−tone

Print squeeze [µm]

Prin

t den

sity

[−]

Figure 22. Left: halftone dot contact area as a function of average pressure. “Harder” dot has50 % higher Mooney-Rivlin parameters, C1 and C2, than the control case. Right: Comparisonbetween halftone and full-tone print density as a function of print squeeze.

4.3. Corrugated board deformations

Corrugated board exhibits various types of deformations, other than washboard-ing, some of which are structure-related and may cause pressure variations. Someexamples are warp and smaller scale distortion of the board. A simulation wasperformed to see if corrugated board deformations create pressure variations thatare enough to cause print non-uniformities. Based on topography measurementsof corrugated board samples three finite element models were constructed. Virtualthermal strains were created on the top liner, causing board distortions in a con-trolled way, such as shown in Fig. 23. (The machine direction thermal strain was0.06 percent in the case of warp, corresponding to the measured warp magnitude ofa corrugated board sample in Paper III, and 0.18 percent for the smaller scale defor-mations, corresponding to deformations comparable to washboarding magnitudes).The simulated pressure variations corresponding to these deformations are shownin Fig. 24. The pressure variations for the warped corrugated board do not differ sig-nificantly from the control case. The smaller scale deformations, on the other hand,showed some noticeable disturbances in pressure. These results indicate that largescale deformations, such as warp, which is quite common, have a small effect onprint uniformity, whereas small scale deformations (over a few flute wavelengths orless) have more effect.

4.4. Minimising pressure variations in the nip

The corrugated board structure, through the corrugated medium, creates peri-odic pressure variations. In addition if the corrugated board is deformed in someway, non-periodic pressure variations can also arise. This work has shown howpressure variations can cause print density variations. If the pressure variationswere eliminated from the nip, the pressure dependence of the ink transfer would

24

0 10 20 30 40 50 60

100200300400

Position [mm]

Hei

ght [

µm]

0 10 20 30 40 50 60−50

050

100150

Position [mm]

Hei

ght [

µm]

Figure 23. Topography of the top liner. Top: warped corrugated board. Bottom: smaller scaledeformation.

0 10 20 30 40 50 600

0.05

0.1

0.15

ControlControlWarp

Position [mm]

Pre

ssur

e [M

Pa]

0 10 20 30 40 50 600

0.05

0.1

0.15

ControlControlSmaller wavelength deformation

Position [mm]

Pre

ssur

e [M

Pa]

Figure 24. Contact pressure between top liner and photopolymer. Top: warped corrugatedboard. Bottom: smaller scale deformation.

25

have no impact. Unfortunately, the printing plates used today can not be optimisedto avoid the pressure variations. The main culprit is in the stress-strain behaviourof the photopolymer used in the printing plate, as shown in Fig. 25 (top left, solidline). As can be seen in the figure, the photopolymer material stiffens, non-linearly,with increased strain. The compressive strain for ridge areas is higher than for val-leys, giving greater stress (pressure) variations. To avoid such pressure variationsthe stress-strain behaviour, such as shown in the dotted line in Fig. 25 (top left), maybe desired. This type of stress-strain behaviour is typically seen in a cellular foam.The “Hill-Ogden” foam reaches a pseudo-plateau at a certain strain level. As long asthe strain variations in the nip is within this plateau region, no pressure variationswill be expected. To utilise this feature of the Hill-Ogden foam, a new printing platedesign was made. Most of the original thickness of the photopolymer was replacedwith the Hill-Ogden foam (Fig. 25; top right). A thin layer of photopolymer waskept to transfer the ink, since a foam would absorb ink into the structure. It wasimportant to keep the thickness of the photopolymer layer as thin as possible so thatthe foam would be able to absorb the strain differences. A comparison between thenew printing plate design and the original one is shown in Fig. 25 (bottom). The newprinting plate clearly produces a much more uniform pressure profile than the pho-topolymer alone. This shows that redesigning the printing plates as outlined here isa promising approach to eliminate print density striping. These results also implythat since halftone striping is caused by pressure variations, halftone striping couldalso be avoided with this approach.

26

−0.75 −0.5 −0.25 0−0.3

−0.2

−0.1

0

Strain [−]

Str

ess

[MP

a] Printing plate

Mounting plastic

Plate cushion

Photopolymer

Flexible foam

Mounting plastic

Plate cushion

Printing plate

Photopolymer

0 1.3 2.6 3.9 5.2 6.50

0.1

0.2

0.3 Control Optimized

Position [mm]

Pre

ssur

e [M

Pa]

Figure 25. Top left: Stress-strain curve for the Hill-Ogden foam (dotted) and the Mooney-Rivlin photopolymer (solid) used in the simulations. Constitutive parameters for the Hill-Ogdenfoam: α = 13, β = −0.066 and µ = 110 kPa. Constitutive parameters for the Mooney-Rivlin photopolymer: C1 = 0.052 MPa and C2 = 0.062 MPa. Top right: Schematics of thenew printing plate design. Bottom: Pressure distribution for the control case compared to apressure distribution of an optimised printing plate (800 µm print squeeze).

27

5. MODELLING HYDRODYNAMICS OF INK TRANSFER

Striping can be prevented if the ink transfer reaches its maximum quickly, i.e,for low a contact pressure, as seen in Fig. 11. However, which parameters controlink transfer at low pressures are not well-known. Ink transfer has not been shown tocorrelate well with neither surface topography, e.g., different surface roughness mea-surements, nor area density parameters. Most ink transfer models are phenomeno-logical, as, e.g., the well-knownWalker–Fetsko equation [17, 18]. Therefore, to deter-mine the parameters which directly control ink transfer, it is necessary to take the fullcomplexity of the ink transfer problem into consideration, including fibre networkdeformation and ink dynamics.

The printing plate and ink interact at a length scale corresponding to the halftonedot size, or the fibre width, i.e., below 100 µm. For these geometrical dimensions“microfluidics” are important, i.e., the fluid flow is associated with small (micro do-main) geometrical dimensions. At this length scale, the fluid flow might be domi-nated by other parameters than for macroscopical flows, e.g., the surface tension orviscosity. The Reynold’s number is generally low, as in the case of halftone printingwhere it can be roughly estimated to about 0.05.

These circumstances make it hard to perform experimental investigations of inktransfer in a systematic and controlled way. Therefore, with new improved numeri-cal methods, halftone ink transfer might be better understood with numerical simu-lations.

5.1. Nip hydrodynamics

The nip hydrodynamics was investigated by solving the Navier-Stokes equationscoupledwith the Cahn-Hilliard equation tomodel a two-phase fluid. In this way twofluids are treated as one binary continuous fluid, with varying density and viscosityover a diffuse interface separating the two phases. This approach avoids the prob-lems with a sharp interface model, which is often plagued by a singularity problem,while retaining the essential physics, such as creation and breaking up of interfaces.

5.1.1. Navier-Stokes equations

For a continuous fluid the macroscopical behaviour is governed by the Navier-Stokes equations. Navier-Stokes equations for an incompressible (∇·V = 0) fluid is:

ρ∂V

∂t+ ρ(V · ∇)V = −∇P + µ∇2V + F body, (5.1)

where ρ is the density, V is the velocity vector, t is time, P the pressure, µ the (dy-namic) viscosity and F body a body force. The two terms on the left hand side inEquation 5.1 are the inertia terms. The terms on the right hand side are the pressuregradient, the diffusion of momentum, and external forces.

5.1.2. Diffuse interface method

In diffuse interface methods a fluid–fluid interface is regarded as having a finitethickness, ε, in contrast to sharp interface methods [48]. Two fluids (or phases) are

28

replaced with one phase-field (composition-field), C, describing how much of eachphase is present in every point. In the bulk phases the phase-field assumes onespecific value for each phase. It is only over the interface between the phases that thephase-field changes (continuously) from one value to the other. All fluid properties,such as density, viscosity, etc., are interpolated from the phase-field value, so theyalso changes continuously over the interface.

Van der Waals hypothesised that equilibrium interface profiles are those whichminimise the free energy, F , in a volume V [49]

F =

∫

V

f dV, (5.2)

where we take f (C ) to be the simplest free energy density function rendering twoimmiscible phases at equilibrium:

f (C ) =1

2εσα|∇C|2 +

σβ

εΨ(C). (5.3)

The surface tension of the interface is denoted by σ, while α and β are dimensionlessconstants determined by ε and the choice of the potential Ψ(C). In this paper thepotential is chosen to be Ψ(C) = (C − 1/2)2(C + 1/2)2, same as in Ref. [50], whichgives the bulk phase values C = ±1/2, and the energy constants as:

α =3

2 tanh−1(0.9)(5.4)

andβ = 12 tanh−1(0.9), (5.5)

if the interface width is defined to be the distance through the interface betweenC=± 0.45. The equilibrium phase transition profile is found by variational calculuson F to satisfy

σβ

εΨ′(C) − εσα∇2C = φ = constant, (5.6)

with φ as the chemical potential. The extension of van der Waals’ hypothesis todynamics was proposed by Cahn and Hilliard [51] by approximating the interfacialflux as being proportional to the gradient of the chemical potential. Later, advectionwas included by Antanovskii [52] and Jacqmin [53] resulting in the Cahn-Hilliardequation:

DC

Dt= κ∇2φ. (5.7)

Here, D/Dt is thematerial derivative, t is time and κ is a diffusion parameter referredto as mobility. The incompressible Navier-Stokes equation, including the forcingfrom the interface [50], becomes:

ρ(C)DV

Dt= −∇S + ∇ · [µ(C)(∇V + ∇V T )] − C∇φ, (5.8)

where ρ is the density, V the velocity and µ the dynamic viscosity. S enforces incom-pressibility and differs from the pressure due to the potential form of the interfacialforcing. The pressure is given by [54]

P = S + Cφ −σβ

εΨ(C) +

1

2εσα|∇C|2. (5.9)

29

A stress formulation of the interfacial force density can also be used in Eq. (5.8) [50],which replaces S with the true pressure P in Eq. (5.8), but this introduces higherorder derivatives of C. The potential form is thus better suited from the point ofcomputations.

To prevent diffusive flux of C over a boundary, considering Eq. (5.7), the bound-ary condition for the chemical potential is:

∂φ

∂xN= 0, (5.10)

where xN denotes the normal direction to the wall.The boundary condition for C is derived by assuming that the interface is at

or near local equilibrium and by postulating that the wall free energy, FWall, is afunction of C only.

FWall =

∫

A

γg(C)dA, (5.11)

where A is the wall surface area and γg(C) is the wall energy density. The naturalboundary condition for C is then given by [50]:

εσα∂C

∂xN+ γg′(C) = 0. (5.12)

In this paper the function g(C) is:

g(C) = (3C − 4C3), (5.13)

with γ related to the equilibrium contact angle, θC , as

γ =σ cos(θc)

2. (5.14)

5.1.3. Numerical Method

The incompressible Navier-Stokes equation Eq. (5.8) was solved with a finite dif-ference projection method on a staggered grid [55, 56, 57, 58]. The method involvessolving a Poisson equation to obtain the incompressibility-enforcing field S. ThePoisson equation was solved with the successive over relaxation method (SOR). Spa-tial velocity derivatives were evaluated by central finite difference approximations.The Cahn-Hilliard equation was updated in time with an Euler step. The spatialchemical potential and phase derivatives were evaluated with stencils constructedto minimise the spurious velocities that might arise at the interface [59].

Discrete boundary condition interpolations have been developed for moving im-mersed boundaries. The boundary condition for the phase-field is based on min-imisation of the free energy of the discretised phase-field. The interpolations fora Dirichlet and Neumann boundary condition were chosen with specific focus onavoiding singularities and extreme gradients, which might cause numerical prob-lems.

In Fig. 26, Fig. 27 and Fig. 28 the interpolations for the Dirichlet, Neumann andphase boundary conditions are shown. In the figures, xi, i=0,1,2, are the grid pointsnormal to the boundary surface, which is situated a distance δ from the boundary Γ

30

x

δ ∆x

x1 x2

ξξ1

ξ2

ξ1/2

ξΓ

x0 xΓ

Γ

ξ0

Figure 26. A Dirichlet boundary conditionwith arbitrary distance, 0 ≤ δ < ∆x, tothe boundary Γ. The field value ξi is de-fined in grid point xi. The new ghost pointvalue ξ0 is calculated by an extrapolationof the straight line going through (xΓ,ξΓ),(x1/2,ξ1/2) and (x2,ξ2).

x

δ ∆x

x1 x2

ξ1

ξ

ξ2

x0

ξ0

Γ

Figure 27. A Neumann boundary condi-tion with arbitrary distance, 0 ≤ δ < ∆x,to the boundary Γ. The field value ξi isdefined in the grid point xi. The ghostpoint value ξ0 is calculated by solving forξ0 from a second order polynomial inter-polation through ξ0, ξ1 and ξ2 and the re-quirement on the derivative on the bound-ary to be, ξ′Γ.

at xΓ. The grid point distance is denoted as ∆x, and the field value ξi or Ci corre-sponds to grid point xi. The ghost point field value ξ0 or C0, situated inside or at theboundary of an object, is calculated from the field values normal to the boundary;ξ1, ξ2 and C1. A Dirichlet boundary condition, with ξΓ as the prescribed value onthe boundary, is fulfilled by modification of the field value ξ0 at the ghost point (seeFig. 26). The Dirichlet boundary condition is given by:

ξ0 = −ξ1 +

(

2 −3

2 − δ

)

ξ2 +3

2 − δξΓ, (5.15)

with δ as a dimensionless distance defined as

δ =δ

∆x, 0 ≤ δ < 1, (5.16)

where δ is the distance to the boundary, and ∆x the grid point interspacing.A Neumann boundary condition, with ξ′Γ as the imposed boundary derivative,

is given by a second order polynomial fitting (see Fig. 27). The additional constraintthat the first derivative of the second order polynomial should be equal to ξ′Γ, thengives

ξ0 =(2 − 2δ)ξ1 + (δ − 1

2 )ξ2 − ∆xξ′Γ32 − δ

. (5.17)

For the phase-field the free energy in the volume is given by Eq. (5.3) and on thesurface by Eq. (5.11), with Cδ as the phase value on the boundary and Cm as thephase value in the mid point between the boundary and the nearest grid point (seeFig. 28). Cδ , is approximated by

Cδ ≈ δC1 + (1 − δ)C0, (5.18)

31

C

x

C0

δ

C1

∆x

Cδ Cm

Figure 28. Phase field boundary condi-tion. The ghost point C0 is calculated byminimising the total energy close to theboundary. Cδ is the value on the bound-ary, C1 the value closest to the boundarypoint and Cm the value in the middle be-tween Cδ and C1.

and Cm as

Cm =1

2((1 + δ)C1 + (1 − δ)C0). (5.19)

The discrete energy at the wall, WS , according to Eqs. (5.11) and (5.13) is

WS(C0) = γg(C) ≈ γ(3Cδ − 4C3δ ). (5.20)

The total discrete free energy in the bulk, WB , is, from Eqs. (5.2) and (5.3),

WB(C0) ≈(

1 − δ)

∆x

(

εσα

2

(

C1 − C0

∆x

)2

+σβ

εΨ(Cm)

)

+∞∑

k=1

WBk, (5.21)

where k denotes the grid point number in the normal direction of the boundary withC0 being the ghost point field value. The terms WBk

for k > 0 are not dependent onC0. The boundary condition is obtained by minimising the sum of the energies WS

and WB with respect to C0, i.e.

∂(WS + WB)

∂C0= 0. (5.22)

This yields a third order polynomial equation which has to be solved to find eachghost point value C0.

The numerical model presented here was validated for three example problems:a stationary droplet, capillary flow [60] and Stefan’s problem [61], which are de-scribed in detail in Paper IV.

32

Top plate

Bottom plate

Figure 29. A schematic figure of a cross section of athree-dimensional network of rectangular fibres.

5.1.4. Fibre network model

The fibre networkmodel, used to create network cross sections for the fluid trans-fer simulations, was developed by Drolet and Uesaka [62]. A network is created byrandomly placing fibres one at the time onto a plane surface from above. The fibrescan bend over already existing fibres, relaxing on top of the fibre mat. A schematicfigure of a cross section from such a deposition is shown in Fig. 29. The fibres arediscretised with Hookean springs between the top and the bottom surfaces of thefibres, as seen in Fig. 30. The consolidation of the fibre network is done by firstcompressing the network to a certain thickness or load, then the spring stiffness isincreased before the fibre network is unloaded. The compression or decompressionis done by translating the top and bottom plates towards or away from the centre ofthe network by an amount ∆zp. The force acting on each node is calculated as

∆zp

∆zp

∆zp

∆zp

Figure 30. The cross section from Fig. 29 discretised with springs forcompressional computations.

F1 = k(∆z6 −∆z1) + g(∆z2 −∆z1) + g(∆z3 −∆z1) + g(∆z4 −∆z1) + g(∆z5 −∆z1),(5.23)

where Fi is the force on node i, k is the spring stiffness of the spring between topand bottom surface of the fibre, g is the spring stiffness of the shear spring elements(connecting neighbouring nodes in the same surface) and zi is the displacement of

33

node i (see Fig. 31). The unknown displacement, ∆zi, for all nodes is found bysolving the resulting sparse linear system for the network. The cross sections usedfor the simulations were created by an additional compressing step (i.e, printing).

Z

SS

S

S

14

5

3

6

2

Figure 31. A figure showing how the nodes ina fibre are connected to each other. Node 1 onthe top surface of the fibre is connected witha spring of stiffness k to node 6 on the bot-tom surface. Node 1 is also connected to itsneighbouring nodes 2, 3, 4, and 5 on the samesurface with shear springs, S, with stiffness g.Force on node 1 is transferred to the neigh-bouring nodes through the shear elements. Allnodes on the fibre surfaces are connected sim-ilarly.

5.2. Ink transfer

Fluid transfer simulations were conducted for a flat backing surface and two dif-ferent fibre networks with 5 µmand 10 µm root-mean-square surface roughness. Thefibre network cross sections were created with the fibre network generation methoddescribed in section 5.1.4. Fluid transfer was studied by compressing the dropletonto the fibre network, and then by decompressing, splitting the fluid. The initialradius, r, of the droplet was in all cases approximately 22 µm. Microfluid transfer ona 1 mm wide fibre network cross section was simulated by randomly placing sim-ulation windows of 100 µm width, along the fibre network surface, as can be seenin Fig. 32. Each simulation window was initially positioned so that the bottom ofthe droplet was at a distance of twice the interface thickness, 2ε, from the highestpoint of the fibre network within the simulation window (Fig. 32, lower figure). Theprinting motion was created by translating the fibre network upwards by a distanceof half the initial distance between the highest point of the fibre network within thesimulation window and the printing plate at the top of the simulation window, asdescribed in Fig. 33. The translation distance, approximately 13 µm, was chosen toyield a minimum plate–network distance in the range of the flexo printing wet inkfilm thickness of 2-15 µm. During the translation, the fibre networks were simulta-

34

2ε

Simulation window

Fibre network

Droplet

Figure 32. The simulation windows are chosen randomly along the fibre network surface andpositioned in height so that the bottom of the droplet is a distance 2ε (twice the interfacethickness) above the highest point of the network surface within the window. The distancebetween the droplet and the fibre network surface is greatly exaggerated in this figure to showthe principle.

r

d=(r+2ε)/2p=r/2-ε

Figure 33. Printing is simulated by translating the fibrenetwork upwards a distance, d=(r+2ε)/2, where r is thedroplet radius.

35

neously compressed by one percent, approximating flexo post-printing conditions(kissing contact). For further details see Paper V.

Figs. 34(a)-34(f) show snapshots of a typical fluid transfer process. Fig. 34(a) de-picts the initial configuration. Fig. 34(b) shows a snapshot at maximum compressiveforce between printing plate and fibre network. This occurs just before the compres-sion part ends, when the velocity is zero and the flat surface–network distance is atits minimum. Approximately when the network motion change direction, the con-tact area between the fluid and the fibre network reaches its maximum, as shownin Fig. 34(c). Fig. 34(d) shows a snapshot when the fibre network recedes from theprinting plate and the maximum tensile force (tack) is measured [63]. In Fig. 34(e),the filament breakup is imminent. Fig. 34(f) shows the fluid droplets settled on thesurfaces after filament splitting.

(a) Initial configuration. (b) The configuration at max-imum compressive force.

(c) Configuration at maxi-mum contact area betweendroplet and fibre network.

(d) Configuration at maxi-mum tensile force (ink tackforce).

(e) Immediately before split-ting of the filament.

(f) The transferred fluid hassettled on the fibre network.

Figure 34. Snapshots of a typical fluid transfer simulation with a fibre network with surfaceroughness of 5 µm.

36

In Figs. 35(a)-35(f) another fluid transfer process is shown. In contrast to theshallow recess in Fig. 34(b), which is completely filled by the fluid, the recess inFig. 35(c) directly below the initial droplet is “bridged” by the fluid during compres-sion. When the fluid advances toward the network surface, the recess is too deepto be completely invaded by the fluid. Moreover, with the continued compressionof the droplet, the fluid is prevented from entering the recess by the trapped low-density fluid (air). In a three-dimensional network, the corresponding isolated poresare less likely to occur due to a high connectivity of pores. When the fibre networkis receding from the flat surface, as in Fig. 35(d), the fluid which bridged the recessis pulled away from the surface (Figs. 35(d) and 35(e)).

(a) Initial configuration. (b) The recess immediatelybefore the bridging.

(c) The recess has beenbridged by the fluid.

(d) The fluid which hasbridged the recess is pulledlose from the surface duringthe receding of the fibrenetwork.

(e) No fluid remains on theright side of the recess.

(f) The transferred fluid hassettled on the fibre network.

Figure 35. Snapshots of a fluid transfer simulation with bridging of a recess in the surfacetopography of a fibre network with surface roughness of 5 µm.

37

These two snapshot series showed that the fluid mainly interacts with the fibrenetwork at the top surfaces, without much penetration into the fibre network. Thisis supported by the fact that for the fibre network with a surface roughness of 5 µm,average maximum penetration of the fluid into the network, as measured from thetopmost point of the fibre network surface, was only 1.8 µm. For the fibre networkwith a surface roughness of 10 µm the penetration was 3.3 µm. In other words,the average maximum penetration into the networks was much less than the initialfibre thickness of 10 µm. This indicates that the ink–substrate interactions are mainlycontrolled by the topmost surfaces, once contact is established.

5.3. Surface parameters that control ink transfer

The ultimate goal of this investigation is to predict which surface parameter thatcontrols the ink transfer in flexo post-printing. As has already been observed, theinteraction between the fluid and the fibre network is restricted to the topmost partsof the fibre network. The most straight forward assumption is that the ink transfer isgoverned by the contact area between the fluid and the fibre network. To investigatethis we first define the horizontal (nominal) area as the contact area (line) betweenthe leftmost and rightmost contact point of the fluid with the network, as depictedin Fig. 36. In Fig. 37 the transferred fluid is plotted as a function of the horizon-

Figure 36. Definition of the horizontalcontact area and the bridged areas.

tal contact area measured for each transfer simulation at maximum compression.Although, there is a general tendency of increasing fluid transfer with increasingcontact area, there are notable deviations. The largest deviation corresponds to the

38

simulation shown in Fig. 35, where the horizontal contact area was partially dis-connected by the bridge. In Fig. 38, the horizontal contact area were corrected bysubtracting the bridged areas. This correction clearly improved the correlation, and

0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Horizontal contact area [µm]

Tra

nsfe

rred

flui

d ra

tio [−

]

Flat5 µm10 µm

Figure 37. The maximum horizontal con-tact area between the droplet and the fibrenetwork.

0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Bridge corrected horizontal contact area [µm]

Tra

nsfe

rred

flui

d ra

tio [−

]

Flat5 µm10 µm

Figure 38. Fluid transfer as a function ofthe horizontal contact area as in Fig. 37,but corrected for bridging cases. The solidline is a least square fit of the data pointsto a line with a coefficient of determinationR2=0.90.

a linear fit to the data points gave a coefficient of determination R2=0.90.The result that ink transfer is approximately described by the simple horizontal

contact area may be surprising. First, there was no significant capillary penetrationeffect. Dube et al. [25] performed systematic numerical simulations of ink transferonto the surface with a regular array of vertical capillaries, and showed that onlyvery small diameter pores (e.g., < 3.0 µm) can retain ink after the printing plate re-tracts from the surface. Since there were very few such small surface pores in thesimulated networks of this study, we can expect little capillary penetration effect inthese simulations. Second, there was little ink flow into larger pores. This may beexplained by the time scale and length scale of flexo post-printing. Within the mil-lisecond time-scale, the ink loses its property as “fluid” and becomes more resistantto flow. In addition, with the help of its surface tension, ink behaves almost like anelastic gel in the length scale of 60 µm or less. Therefore, it is understandable thatink interacted only with the topmost surfaces in these simulations.

Of course these observationsmay be only for the case of flexo post-printingwherea low printing pressure (i.e., less than 0.5 MPa) is applied. From the point of prevent-ing striping in the flexo post-printing, it is therefore important to achieve maximumink transfer at as low as possible printing pressure in the ink transfer curves (e.g.,Fig. 11). This means that the topography of the topmost surface of linerboards isanother key parameter for controlling the striping tendency.

39

6. CONCLUSIONS

Print quality is, to a large extent, determined in the printing nip. In the printingnip a large number of parameters interacts in a complex way.

The work done in this study has yielded a number of new insights. Periodicstripe patterns (called striping) typically seen in flexo post-printing of corrugatedboards are mainly print density variations rather than print gloss variations. Theseprint density variations are pressure-dependent and occur in both full-tone and halftoneprinting. In the case of full-tone printing, the striping was found to be pressure-induced print density variations. In halftone printing, striping, which also origi-nates from the periodic pressure variations, was found to be caused by variationsof coverage (dot gain), rather than print density differences of individual dots. Theperiodic pressure variations are, therefore, the fundamental cause for striping, andalso inherent to the corrugated board structure (flute structure).

Washboarding, defined as a structural deformation, have been shown to play aminor role in causing print density striping.

Effects of changing dimensions andmaterial stiffnesses of the printing system, in-cluding the corrugated board and photopolymer, were shown to be limited in termsof eliminating striping. In order to alleviate the pressure variations, a new design forthe printing plate was proposed by optimising its structure and materials.

A new model have been proposed to enable simulations of ink transfer betweena printing plate and a fibre network. The ink transfer model was then applied toa flexo post-printing case. The ink transfer simulations showed that, once contactbetween ink and board is made, ink–fibre network interactions occur only at thetopmost surface, with a penetration depth much less than the fibre thickness. Thisis because at a realistic printing time/length scale, ink behaves like an elastic gel,rather than a fluid. Therefore, the actual contact area between fibre network and inkis the dominant factor that controls the ink transfer amount in flexo post-printing.

In the ink transfer model, there are still two outstanding challenges. One is theextension to three dimensions, which is essential to simulate real flow in a fibre net-work. Another challenge is the two-way coupling of fluid and solid structures. Thisis also crucial for investigating other printing processes, where much higher printingpressures are applied, compared to flexo post-printing. With these implementations,the ink transfer model presented here will become a useful tool for investigatingink–paper interactions in general printing processes.

40

ACKNOWLEDGENTS

This work was carried out at Fibre Science and Communication Network (FSCN)at Mid Sweden University. FSCN, SCA R&D Centre in Sundsvall, the European Re-gional Development Fund, the KK Foundation and Lansstyrelsen i Vasternorrlandare greatly acknowledged for their financial supports. SCA Packaging in Jarfallais also acknowledged for sharing practical knowledge and for supplying me withsamples.

Many thanks to the people at SCA R&D Centre who have helped me over theyears: Folke Osterberg and Ragnhild Dolling as managers and supporters of theproject, Anders From and Marie Nassbjer for taking their time to discuss printingquestions and problems, Anna Zetterquist and Christina Westerlind for helping mewith the experimental work and all other people at the SCA R&D Centre who I havemet during the years.

My gratitude to Dr. Lyne Cormier at FPinnovations–PAPRICAN for valuablediscussions on reflectance measurements. Thank you Dr. and Adjunct Prof. FrancoisDrolet for taking your time to explain fibre network compression modelling. Also,thank you for the dinner with your family!

I would like to thank my supervisors, Prof. Tetsu Uesaka, Prof. Myat Htun andProf. Magnus Norgren. Tetsu, your knowledge and scientific experience is outstand-ing, and a resource I have had the opportunity to share. Myat, you provide a goodresearch environment and your cheerful attitude is an inspiration. Magnus, youshow that it is possible to combine other interests with the work of a researcher.

For all colleagues at the FSCN, thank you. You have created a good atmospherefor me to work in. Some colleagues deserve a special acknowledgement. AnnaHaeggstrom, for all the practical matters and for directing me in the right direc-tion when I have had questions. Former colleague Dr. Artem Kulachenko, youreally helped me getting started with finite elements and ANSYS. M.Sc. ChristinaDahlstrom and M.Sc. Hanna Wiklund. Thank you for your time. You will have a lotmore time to do research and solve outstanding questions now... Dr. Stefan Lind-strom, it has been a privilege to share the office and discuss my work with you. Iwill continue to call upon you when I need to discuss scientific matters.

My parents and my sister with family. From now on I will hopefully have moretime for you all. Thank you for your support!

My wife Asa; you have shared my burden when I have had to work late nights,weekends and holidays. You have a great part in seeing this through to the end.Thank you!

Sundsvall, April 2010

41

REFERENCES

[1] KIPPHAN, H. (ED.). , Handbook of print media. Springer-Verlag, Berlin, Heidel-berg, 2001.

[2] CHALMERS, I. R. , A new method for determining the shear stiffness of corru-gated board. APPITA J., 60(5):357–361, 2007.

[3] POPIL, R. , HABEGER, C.C. , AND COFFIN, D. , Transverse shear measurementfor corrugated board and its significance. APPITA J., 61(4):307–, 2008.

[4] SAVOLAINEN, A. (ED.). , Paper and paperboard converting. Fapet Oy, Helsinki,Finland, 1998.

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