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DOI: 10.1007/s12541-014-0570-z
Mathematical Modeling and Simulations for Machine Directional Register in Hybrid Roll-to-Roll Printing Systems Hyunkyoo Kang1,# and Reinhard R. Baumann1,2 1 Digital Printing and Imaging Technology, Institute for Print and Media Technology, Technische Universität Chemnitz, Reichenhainer Strasse 70, Chemnitz, Germany, 09126 2 Department Printed Functionalities, Fraunhofer Research Institute for Electronic Nano Systems (ENAS), Technologie-Campus 3, Chemnitz, Germany, 09126 # Corresponding Author / E-mail:
[email protected], TEL: +49-371-531-33428, FAX: +49-371-531-836619 KEYWORDS: Hybrid printing, Mathematical modeling, Printed electronics, Roll-to-roll, Register
Roll-to-roll (R2R) solution printing method has been attracting as a promising technology for a mass production of flexible functionalities, such as radio-frequency identification (RFID), photovoltaic, flexible batteries, etc. High resolution of register control is mandatory to prevent electrical leakages, and short circuitries in the multilayered flexible printed circuitries through R2R mass production. In this paper, a mathematical modeling of the register was derived for the hybrid printing systems, which consisted of contact printing and non-contact printing such as gravure and inkjet printing. Parameter sensitivity analysis was carried out in various operation conditions. And linear quadratic (LQ) regulator was proposed to control both the tension and register. The performance of the LQ regulation of the register was trade-off considering the permitted variation of tension and resolution of register. The proposed mathematical model will be used for the development of hybrid register controller. Manuscript received: February 5, 2014 / Revised: April 22, 2014 / Accepted: May 30, 2014
NOMENCLATURE A : Cross-sectional area of a substrate E : Tensile modulus of elasticity T : Tension of a subsrate ε : Strain of a subsrate ∈ : Variation of strain of a subsrate σ : Stress of a subsrate W : Width of a substrate v : Velocity of a substrate V : Variation of velocity of a substrate L : Span length xr(t) : Position of a pattern on the downstream roll at (t) x*(t) : Position of a substrate transferred from x*(t-τ) at (t) xw(t) : Position of a pattern on the substrate at (t) rxi(t) : MD Register between the (i-1)-th and the i-th printing roll at (t) rxai(t) : Absolute MD Register between the first and the i-th printing roll at (t)
© KSPE and Springer 2014
ri : Radius of an i-th roll φi : Phase of an i-th roll vi : Tangential velocity of an i-th roll τ : Time lag between printing rolls rx0 : Steady-state value of MD register Rxi : Variation of an i-th MD register from a steady-state value ∆x*0 : Steady-state value of transferred length of a substrate during time lag ∆X* : Variation of transferred length of substrate during time lag from a steady-state value φi0 : Steady-state value of phase of an i-th printing roll Θi : Variation of a phase of an i-th printing roll from a steady-state value
1. Introduction Recently, roll-to-roll (R2R) solution printing technology has been
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attracting much interest for manufacturing of flexible electronic devices, such as capacitors, photovoltaic, chemical sensor, tactile sensor, microsieve, battery, antenna, printed circuit board, and display.1-8 It is an additive process, which reduces discarded materials, therefore environment-friendly in contrast to conventional patterning methods like photolithography and vapor deposition. For the implementation of multilayered flexible printed circuitries through the R2R mass production, a high resolution of register control should be achieved to prevent electrical leakages, and short circuitries. Register of a moving substrate is defined in the multilayered printed structures as a relative distance between deposited patterns.10 A linear mathematical model of a machine directional (MD) register of a moving substrate and compensation method were developed.11 A non-interacting control method between the tension and cut-off register error was proposed.12 Cross direction register modeling and feedforward control method were suggested.13 In previous studies,10-13 the register model derived between contact printing rolls like gravure or flexography. The velocity of contact printing roll affects the tension of a moving substrate which leads to the register variations. Thus, the dynamics of register depends on the dynamics of tension. In contrast, the register is directly affected by the velocity of printing module as well as the velocity of a moving substrate in the non-contact printing method. However, there are no studies with regard to the register control for R2R non-contact printing methods, such as R2R inkjet printing, which has become the most promising printing technology for a printed electronic circuitry.14,15 To fabricate a functional electrical circuitry, noncontact patterning as well as contact patterning should be employed, e.g. electrode layers were formed by gravure printing, besides semiconductor layers were deposited by inkjet printing for the non-volatile memory.9 Therefore, it is necessary to investigate the register control for the noncontact pattering method where inks were transferred onto the substrate without a roll-contact mechanism as shown in Fig. 1(b). In this study, a mathematical model of MD register between contact printing and non-contact printing, so-called ‘hybrid printing’, was derived using the equation of mass equilibrium. The non-contact printing module was converted as a full-slipping printing roll to derive the mathematical model. The model included the effects of velocity of contract and noncontact printing methods, the velocity of substrate, and the tension of substrate. Parameter sensitivity analysis was carried out in various operation conditions. And a linear quadratic (LQ) controller was proposed to regulate both tension disturbance and register error. This mathematical model will be used for the development of register controller in the hybrid R2R printing systems.
Fig. 1 (a) Schematic of experimental setup of hybrid printing system, (b) R2R inkjet printing module, and (c) R2R gravure printing module
2. Mathematical Modeling of Register in Hybrid Printing System
1) Tension dependent elastic stain is determined by the Hook’s law (T=AEε). 2) Tension is a constant between driven rolls. 3) All of the radii of the printing rolls are the same. 4) At initial state, phases of the printing rolls are the same. 5) The span length, L, is multiple of the circumference of the printing roll. 6) Inkjet printing is synchronized by trigger marks which printed by the upstream printing roll. For the derivation of register model for hybrid printing, a general hybrid printing system was defined as shown in Fig. 2(a). The system consisted of two contact printing cylinders and non-contact printing module. In the non-contact printing, inks deposited onto the substrate were synchronized with the velocity of a moving substrate, thus the required image could be printed in the R2R process. The position of deposited inks can be regulated by controlling of the synchronization between the actual velocity of the moving substrate and the virtual velocity of the non-contact printing system. Therefore, the non-contact printing module was converted to an equivalent full-slipping printing roll as shown in Fig. 2(b). The velocity of full-slipping roll does not affect the strain of substrate (ε2), but it affects the register between the printing roll 1 and 3. Also, the variations of strains (ε1, ε2), which caused by the variations of velocities of contact printing rolls, can generate variations of register.
A schematic of experimental setup of hybrid printing system was depicted in Fig. 1(a). The system consisted of unwinding, lateral guider, infeeding, heating, gravure printing, inkjet printing, drying, outfeeding, and rewinding section. In Figs. 1(b) and 1(c), R2R inkjet printing module and gravure printing module were depicted, respectively. Some assumptions were used for the modeling of MD register in the hybrid printing system as follows:
2.1 Modeling from contact printing to non-contact printing In Fig. 2(b), the first contact printing roll printed the pattern of the white square on the substrate at instant (t-τu), and the square was transported to the second non-contact printing section. The pattern of a gray square was deposited by the non-contact printing module at instant (t). Also, the mark was transported to the downstream, and then finally black square was printed by the third contact printing roll at
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where xr(t-τu) = r1φ1(t-τu), xr(t) = r2φ2(t)+L, x· r (t-τu) = r1φ1(t-τu), x· r (t) = r2φ2(t). The nonlinear model can be linearized by using the perturbation method. The variables were assumed as a sum of steady state value and variation as Eq. (7). ∆x* ( t) = ∆x*0 + ∆X* ( t ) εi ( t) = εi0 + ∈i ( t )
(7)
vi ( t ) = vi0 + Vi ( t ) Since εi
