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Hybrid Continuous-Discrete Repetitive Process Modeling of Meniscus Dynamics in Electrohydrodynamic Jet Printing
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International Federation of The International of Automatic Automatic Control Control Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 13414–13419
Hybrid Continuous-Discrete Repetitive Hybrid Continuous-Discrete Repetitive HybridModeling Continuous-Discrete Repetitive in Process of Meniscus Dynamics Process Modeling of Meniscus Dynamics Process Modeling of Meniscus Dynamics in in Electrohydrodynamic Jet Printing Electrohydrodynamic Jet Printing Electrohydrodynamic Jet Printing
Patrick M. Sammons ∗∗ David Hoelzle ∗∗∗ ∗∗ ∗∗ ∗∗ David Hoelzle ∗∗∗ ∗∗∗ Patrick M. Sammons Kira L. Barton ∗∗ David Hoelzle ∗∗∗ Patrick M. Sammons Patrick M. Sammons David Hoelzle ∗∗ ∗∗ Kira L. Barton Kira L. L. Barton Barton ∗∗ Kira ∗ Rensselaer Polytechnic University, Department of Mechanical, ∗ ∗ Rensselaer Polytechnic University, Department of Mechanical, Aerospace, and Nuclear Engineering, Troy, NY (email: ∗ Rensselaer Polytechnic University, Department of Rensselaer Polytechnic University, Department of Mechanical, Mechanical, Aerospace, and Nuclear Engineering, Troy, NY (email: [email protected]) Aerospace, and Nuclear Engineering, Troy, NY Aerospace, and Nuclear Engineering, Troy, NY (email: (email: ∗∗ University of Michigan,[email protected]) Mechanical Engineering Department, Ann [email protected]) [email protected]) ∗∗ ∗∗ University ofMI Michigan, Mechanical Engineering Department, Ann Ann Arbor, (email: psammons, bartonkl @umich.edu) ∗∗ University of Michigan, Mechanical Engineering Department, ofMI Michigan, Engineering Department, Ann ∗∗∗ University Arbor, (email: Mechanical psammons, bartonkl @umich.edu) The Ohio State University, Mechanical and Aerospace Engineering Arbor, MI (email: psammons, bartonkl @umich.edu) Arbor, MI University, (email: psammons, bartonkl Aerospace @umich.edu) ∗∗∗ ∗∗∗ The Department, State Engineering Columbus, Mechanical OH (email:and [email protected]) ∗∗∗ The Ohio Ohio Mechanical and Aerospace Engineering The Department, Ohio State State University, University, Mechanical and Aerospace Engineering Columbus, OH (email: [email protected]) Department, Columbus, OH (email: [email protected]) Department, Columbus, OH (email: [email protected]) Abstract: Electrohydrodynamic jet (E-Jet) printing is a micro-/nano-manufacturing process Abstract: Electrohydrodynamic jet (E-Jet) printing is a through micro-/nano-manufacturing in which a liquid ink is jetted from micron-scale nozzle the actuation of an process electric Abstract: Electrohydrodynamic jet printing is micro-/nano-manufacturing process Abstract: Electrohydrodynamic jetaa(E-Jet) (E-Jet) printing is aa through micro-/nano-manufacturing process in which a liquid ink is jetted from micron-scale nozzle the an field applied between the nozzle and a grounded substrate. Because the jettingof is in which which a a liquid liquid ink ink is is jetted jetted from from a a micron-scale micron-scale nozzle nozzle through through the the actuation actuation ofbehavior an electric electric in actuation of an electric field applied between the nozzle and aacapable grounded substrate. Because the jetting behavior is driven by electrohydrodynamics and is of producing printed features with length scales field applied between the nozzle and grounded substrate. Because the jetting behavior is field applied between the nozzleand andis acapable grounded substrate.printed Because the jetting behavior is driven electrohydrodynamics of producing features with length scales rangingby from nanometers to microns, E-Jet printing is a promising manufacturing tool for driven by electrohydrodynamics and is capable of producing printed features with length scales driven by electrohydrodynamics and is capable of producing printed features with length scales ranging from nanometers to microns, applications. E-Jet printing is a promising manufacturing tool for printed electronics and bioengineering However, a significant hurdle in adoption ranging from to E-Jet is manufacturing tool ranging from nanometers nanometers to microns, microns, applications. E-Jet printing printing is aa promising promising manufacturing tool for for printed electronics and bioengineering However, a significant significant hurdle in in adoption of E-Jet printing for these applications is the lack of appropriate process modeling and control printed electronics and bioengineering applications. However, a hurdle adoption printed electronics and bioengineering applications. However, a significant hurdle in adoption of E-Jet printing printing for theseprinting applications is the the lack ofrobustness appropriate process modeling and Here, control methodologies to for improve performance andof to process process modeling disturbances. a of E-Jet these applications is appropriate and control of E-Jet printing for theseprinting applications is the lack lack ofrobustness appropriate process modeling and Here, control methodologies to improve performance and to process disturbances. aa model describing the dynamics of the meniscus and the onset of jetting in E-Jet printing is given. methodologies to improve printing performance and robustness to process disturbances. Here, methodologies to the improve printing performance andthe robustness to process disturbances.isHere, a model describing dynamics of the meniscus and onset of jetting in E-Jet given. The model is posed as a repetitive hybrid continuous-discrete process whereprinting the continuous model describing the dynamics of the meniscus and the onset of jetting in E-Jet printing is given. model describing the dynamics of the meniscus and the onset of jetting in E-Jet printing is given. The modeldynamics is posed posed are as a ainterrupted repetitive hybrid hybrid continuous-discrete processand where the continuous meniscus by the continuous-discrete discrete jetting behavior are the dependent on The model is as repetitive process where continuous The modeldynamics is posed are as ainterrupted repetitive hybrid continuous-discrete processand where the continuous meniscus by the discrete jetting behavior are dependent on previous trial information. The developed model is compared to a set of experimental results meniscus dynamics are interrupted by the discrete jetting behavior and are dependent on meniscus dynamics are interrupted by themodel discrete jetting behavior andexperimental are dependent on previous trial information. The developed is compared to aa set of results and the model exhibits good quantitative agreement with time-to-jetting from a step voltage previous trial information. The developed model is compared to set of experimental results previous trial information. The developed model is compared to a set of experimental results and good quantitative agreement time-to-jetting aa step voltage inputthe andmodel good exhibits qualitative agreement in the dynamic with response after jettingfrom has ceased. and the model exhibits good quantitative agreement with time-to-jetting from and good quantitative agreement time-to-jetting a step step voltage voltage inputthe andmodel good exhibits qualitative agreement in the the dynamic with response after jetting jettingfrom has ceased. ceased. input and good qualitative agreement in dynamic response after has input good qualitativeFederation agreement in the dynamic after jetting © 2017,and IFAC (International of Automatic Control)response Hosting by Elsevier Ltd.has Allceased. rights reserved. Keywords: Modeling of manufacturing operations, Hybrid and switched systems modeling, Keywords: manufacturing operations, Electrohydrodynamic Repetitive processes Hybrid Keywords: Modeling Modeling of ofjetting, manufacturing operations, Hybrid and and switched switched systems systems modeling, modeling, Keywords: Modeling of manufacturing operations, Hybrid and switched systems modeling, Electrohydrodynamic jetting, Repetitive processes Electrohydrodynamic jetting, Repetitive processes Electrohydrodynamic jetting, Repetitive processes 1. INTRODUCTION grounded substrate by a vertical distance. A voltage dif1. grounded by a vertical distance. voltage difference is substrate applied between the nozzle and A substrate 1. INTRODUCTION INTRODUCTION grounded substrate by distance. A voltage dif1. INTRODUCTION grounded substrate by aa vertical vertical distance. Athe voltage difis applied between the nozzle and the substrate and a liquid meniscus forms at the outlet of the microMicro-/Nano-scale Additive Manufacturing (µ/nAM) is a ference ference is applied between the nozzle and the substrate ference is applied between the nozzle and the substrate and a liquid meniscus forms at the outlet of the microMicro-/Nano-scale Manufacturing (µ/nAM) is nozzle. When forms a critical voltage the class of processes inAdditive which parts, features, and structures and aa liquid meniscus at outlet of the Micro-/Nano-scale Additive Manufacturing (µ/nAM) is aaa capillary and liquid meniscus forms at the the outletis ofreached, the micromicroMicro-/Nano-scale Additive Manufacturing (µ/nAM) is capillary nozzle. When aaa critical voltage is reached, the class of processes in which parts, features, and structures meniscus deforms into so-called Taylor cone [Taylor can be fabricated by repeated addition of material and capillary nozzle. When critical voltage is reached, the class of processes in which parts, features, and structures capillary nozzle. When aa critical voltage is cone reached, the class of fabricated processes inbywhich parts,addition features,ofand structures meniscus deforms into so-called Taylor [Taylor can be repeated material and and Dyke (1969)] and a jet of material issues from the whose characteristic length scales range from nanometers meniscus deforms into a so-called Taylor cone [Taylor can be fabricated by repeated addition of material and meniscus deforms into a so-called Taylor cone [Taylor can be fabricated by repeated addition of material and and depositing Dyke (1969)] (1969)] and aa on jetthe of material material issues from the whose characteristic length from material substrate.issues As the nozzle to microns. µ/nAM processes are range emerging attractive tip, and Dyke and jet of from the whose characteristic length scales scales range fromasnanometers nanometers and Dyke (1969)] and a on jetthe of material issues from the whose characteristic length scales range from nanometers tip, depositing material substrate. As the nozzle to microns. µ/nAM processes are emerging as attractive traverses across the substrate, two-dimensional patterns methods for fabricating high-resolution sensors [Kim et al. tip, depositing material on the substrate. As the nozzle to microns. µ/nAM processes are emerging as attractive tip, depositing material on the substrate. As the nozzle to microns. µ/nAM processes are emerging as attractive across the substrate, two-dimensional methods for [Kim et can be created. process can then be patterns indexed (2016)], electronics [Parkhigh-resolution et al. (2007)], sensors and in the field of traverses traverses across the substrate, two-dimensional patterns methods for fabricating fabricating high-resolution sensors [Kim et al. al. traverses across The the printing substrate, two-dimensional patterns methods for fabricating high-resolution sensors [Kim et al. can be created. The printing process can then indexed (2016)], electronics [Park et al. (2007)], and in the field of to a new layer and jetting continues on top of be previously bioengineering [Poellmann et al. (2011)]. However, significan be created. The printing process can then be indexed (2016)], electronics [Park et al. (2007)], and in the field of can be created. The printing process can then be indexed (2016)], electronics [Park etetal.al.(2007)], and in the field of to a new layer and jetting continues on top of previously bioengineering [Poellmann (2011)]. However, signifideposited material to form three-dimensional, functional cant modeling and control challenges currently prevent the to a new layer and jetting continues on top of previously bioengineering [Poellmann [Poellmann et et al. al. (2011)]. (2011)]. However, However, signifisignifi- to a new layer and jetting continues on top of previously bioengineering material to form functional cant modeling control challenges prevent the structures. While there exist three-dimensional, many specific modes of matewidespread useand of these processes for currently repeatable, accurate deposited material to three-dimensional, functional cant modeling and control challenges currently prevent the deposited deposited material to form form three-dimensional, functional cant modeling and control challenges currently prevent the structures. While there exist many specific modes ofjetting matewidespread use of these processes for repeatable, accurate rial ejection that are termed electrohydrodynamic production of components. structures. While there exist many specific modes matewidespread use of these processes for repeatable, accurate While there exist many specific modes of ofjetting matewidespread usecomponents. of these processes for repeatable, accurate structures. rial ejection that are termed electrohydrodynamic production of [Cloupeau and Prunet-Foch (1994)], e.g. micro-dripping, rial ejection that are termed electrohydrodynamic jetting production of components. rial ejection that are termed electrohydrodynamic jetting production of components. A specific µ/nAM process which is capable of printing [Cloupeau andparticular Prunet-Foch (1994)], e.g. micro-dripping, micro-dripping, multi-jet, the regime of interest in this work is [Cloupeau and Prunet-Foch (1994)], e.g. andparticular Prunet-Foch (1994)], e.g. micro-dripping, A specific µ/nAM process which capable of printing a wide range of inks with feature on the of [Cloupeau A specific µ/nAM process which is issizes capable of order printing multi-jet, the regime of interest in thisjetwork is termed cone-jet. In the cone-jet regime, a single issues A specific µ/nAM process which is capable of printing multi-jet, the particular regime of interest in is multi-jet, the particular regime of interest in this thisjetwork work is aahundreds wide range of inks with feature sizes on the order of of nanometers up to microns – more than an wide range of inks with feature sizes on the order of termed cone-jet. In the cone-jet regime, a single issues from the Taylor cone which enables operation in both DCa wide range of inks with feature sizes on the order of termed cone-jet. In the cone-jet regime, a single jet issues termed cone-jet. In the cone-jet regime, a single jet issues hundreds of nanometers up to microns –– more than an order of magnitude smaller than traditional ink jetting hundreds of nanometers up to microns more than an Taylor cone which enables operation both DCmode (constant voltage difference thein and hundreds of nanometers up to microns – more than an from from the the Taylor which enables operation innozzle both Taylor cone cone which enablesbetween operation both DCDCorder of magnitude than ink jetting printing is Electrohydrodynamic order of processes magnitude– smaller smaller than traditional traditional Jet ink (E-Jet) jetting from modethe (constant voltage difference between theinnozzle nozzle and substrate) and Drop-on-Demand (DoD) or pulsed printing order of magnitude smaller than traditional ink jetting mode (constant voltage difference between the and (constant voltage difference(DoD) between the nozzle and printing processes –– is Electrohydrodynamic Jet et al. (2007)]. In E-Jet, a micro-capillary printing [Park processes is Electrohydrodynamic Jet (E-Jet) (E-Jet) mode substrate) and Drop-on-Demand or pulsed printing mode (the voltage is pulsed between a low and high value). printing processes – is Electrohydrodynamic Jet (E-Jet) substrate) and Drop-on-Demand (DoD) or pulsed printing substrate) and Drop-on-Demand (DoD) or pulsed printing printing [Park et al. (2007)]. In E-Jet, a micro-capillary nozzle is filled with a liquid ink and separated from a printing [Park [Park et et al. al. (2007)]. (2007)]. In In E-Jet, E-Jet, aa micro-capillary micro-capillary mode (the voltage is pulsed between aa low and high value). printing (the is between and high mode (the voltage voltage is pulsed pulsedE-Jet between a low lowcan andgenerally high value). value). nozzle is with liquid ink and from aa mode The dynamics in cone-jet printing be nozzle is filled filled with a liquid ink Science and separated separated from nozzle is filled with aa by liquid ink and separated a The dynamics in cone-jet E-Jet printing can generally be This work was supported National Foundation from through divided into two processes: the meniscus dynamics and The dynamics in cone-jet E-Jet printing can generally be work was supported by National Foundation through The dynamics in cone-jet E-Jet printing can generallyand be the Division of Civil, Mechanical, andScience Manufacturing Innovation This work supported by Science Foundation through This divided into two processes: the meniscus dynamics This work was was supported by National National Science Foundation through the droplet dynamics. The meniscus dynamics describe the divided into two processes: the meniscus dynamics and the Division of Civil, Mechanical, and Manufacturing Innovation under Grant 1434660 and Grant 1434693. The authors would like divided into two processes: the meniscus dynamics and the Division of Civil, Mechanical, and Manufacturing Innovation the droplet dynamics. meniscus dynamics the the Division of Civil, and Mechanical, and Manufacturing Innovation temporal evolution of The the meniscus e.g.,describe the height the droplet dynamics. The meniscus dynamics describe the under Grant 1434660 Grant 1434693. authors would like to acknowledge Christopher Pannier for his The technical assistance. the droplet dynamics. The meniscus shape, dynamics describe the under Grant and 1434693. The authors would under Grant 1434660 1434660 and Grant Grant 1434693. The authors would like like temporal evolution of the meniscus shape, e.g., the height to acknowledge Christopher Pannier for his technical assistance. temporal evolution of the meniscus shape, e.g., the height to acknowledge Christopher Pannier for his technical assistance. temporal evolution of the meniscus shape, e.g., the height to acknowledge Christopher Pannier for his technical assistance. Mamadou Mamadou Mamadou Mamadou
Diagne ∗ ∗ ∗ Diagne Diagne Diagne ∗
Copyright 13956Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 13956 Copyright © 2017 IFAC 13956 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 13956Control. 10.1016/j.ifacol.2017.08.2285
Proceedings of the 20th IFAC World Congress Mamadou Diagne et al. / IFAC PapersOnLine 50-1 (2017) 13414–13419 Toulouse, France, July 9-14, 2017
Fig. 1. Sequential still image captures of high speed video taken during DoD E-Jet printing of deionized water. Black lines above each image indicates the height of the meniscus. of the meniscus as measured from the nozzle orifice datum or the curvature of the surface, as a function of process inputs. Alternatively, the droplet dynamics describe both the temporal and spatial evolution of the droplet shape that results from the jetting of material onto the substrate after a jetting event. At the end of the jetting event, the jet retracts and the meniscus transitions towards a static equilibrium while the droplet dynamics evolve concurrently. During this transition phase, the meniscus can exhibit an oscillatory response (see Figure 1 for a timelapse of deionized (DI) water). In typical operation, a dwell time is explicitly included in the printing trajectory to ensure these oscillations have decayed sufficiently before attempting to continue fabrication. Printing in this regime can lead to unwanted or unexpected meniscus behavior. Therefore, it is necessary to understand and model the evolution of the meniscus dynamics during the pre- and post-jetting phases. Models describing the meniscus dynamics in E-Jet printing typically rely on numerical solutions to the NavierStokes equations, e.g., [Wright et al. (1995)]. While these models are capable of capturing the complex interactions between the meniscus and the electric field induced by the voltage potential between the substrate and the nozzle, in addition to the modeling the full meniscus surface shape, they are not appropriate for process control. Additionally, for those models that are appropriate for process control, [Wright et al. (1993); Yang et al. (2014)] for example, only dynamics of the meniscus in the pre-jetting phase are described. Thus, in order to describe the jetting onset and post-jetting dynamics, further modeling efforts are needed. While each sub-process could be modeled separately, process control requires a model of the process that is able to capture the dynamics of each sub-process and the transitions between them. Because the continuousdiscrete-continuous regime is inherent in the E-Jet material ejection process, a hybrid continuous-discrete model is a logical choice for describing the meniscus dynamics. Further, in both DC and DoD modes, the material ejection process is repetitive; the continuous-discrete-continuous paradigm repeats for each droplet ejection event and the time evolution of the meniscus dynamics is dependent on the trajectory of the previous ejection event. A schematic of this operating paradigm is shown in Figure 3 (description details are given below in Section 3). Therefore, the meniscus dynamic process can posed further as a hybrid repetitive process. This work examines the dynamic evo-
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Fig. 2. Sequential still image captures of high speed video taken during DoD E-Jet printing of NOA 81 with prejetting, jetting, and recovery phases labeled. All times are in milliseconds. lution of the meniscus in both the pre- and post-jetting conditions, but does not examine the process dynamics of the jetting regime. The rest of the paper is structured as follows. First, the continuous dynamics of the meniscus are given in addition to criterion for the onset and duration of the jetting mode. Then, the continuous model and the switching conditions are presented as a hybrid process and finally posed as hybrid repetitive process. In the section following, details regarding the dynamic modes of the continuous model are given along with some experimental evidence of these modes. Finally, a summary of the work and conclusions are given. 2. CONTINUOUS MENISCUS MODEL As mentioned above, the meniscus dynamics in E-Jet printing can be divided into three phases; pre-jetting, jetting, and post-jetting. In the pre-jetting phase, the pressure in the meniscus increases due to the electric field and the meniscus deforms until jetting occurs (top set of photographs in Figure 2). During jetting, the ink flows through the jet towards the substrate to form a discrete droplet (middle set of photographs in Figure 2). After the duration of the jetting phenomenon, the jet retracts from the substrate, and the post-jetting phase begins (bottom set of photographs shown in Figure 2). Here, the preand post-jetting behaviors are described by continuous dynamics while the jetting behavior is indicated by a discrete event, the onset and termination of which is determined by conditions given at the end of this section. 2.1 Flow Rate Dynamics In the following, the ink is incompressible and viscous and the flow laminar. Additionally, it is assumed that the inner radius of the microcapillary nozzle is constant along the length of the microcapillary, the length of the microcapillary is much greater than the inner radius, and the distance between the nozzle and the substrate remains fixed. Then, the ink flow rate through the nozzle can be described as Hagen-Poiseuille flow where the pressure drop across the length of the nozzle is proportional to the flow rate,
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8µ πr2 ˙ Q(t) = N ∆P (t) − 2 Q(t) ρL ρrN
(1)
with Q the volumetric flow rate of ink through the nozzle (m3 /s), rN is the nozzle radius (m), ρ the ink density (kg/m3 ), µ the ink viscosity (Pa-s), L the length of the nozzle (m), and ∆P the total pressure drop across the nozzle (N/m2 ). The pressure drop across the nozzle is given as ∆P (t) = PB (t) + Pγ (t) + PE (t) 1 4γh(t) + E 2 (t) = PB (t) − 2 rN + h2 (t) 2
(2)
Fig. 3. Schematic of the switching process in the hybrid meniscus model.
where PB is backpressure supplied to the nozzle (N/m2 ), Pγ is the surface tension pressure (N/m2 ), PE is the pressure in the meniscus generated by the electric field and termed the electric pressure (N/m2 ), γ is the ink surface tension coefficient (N/m), h is the meniscus height (m), is the permittivity of air (F/m), and E is the electric field (V/m). Backpressure PB is typically zero for all t in ejet printing. This corresponds to the ink being exposed to ambient conditions. A schematic of the nozzle and substrate interaction zone is shown in Figure 4. For modeling and control purposes, it is advantageous to express the electric field between the nozzle and the substrate as a function of process inputs and states. While an exact expression relating the applied voltage U (process input) and meniscus height h (process state) to the electric field requires a solution to the Navier-Stokes equations, a common approximation [Kim et al. (2010)] of the electric field for electrohydrodynamic jetting is that of a metal hyperboloid revolution above a flat plate electrode, E(t) = fE (U (t), h(t)) =
2U (t) 4hN R(t) ln R(t)
(3)
where hN is the distance between the nozzle and the 2 substrate (m) and R(t) = (rN +h2 (t)(2h(t))−1 is the radius of the spherical cap formed by the average meniscus height and the nozzle radius (m).
Fig. 4. Schematic of the nozzle-substrate interaction zone in E-Jet Printing ˙ h(t) =
υ(t) =
π 2 h(t)(3rN + h2 (t)) 6
(4)
where υ is the meniscus volume (m3 ). Because volume is conserved within the meniscus pre-jetting, the volume rate of change is equal to the volume flow rate through the d nozzle dt υ(t) = Q(t). Thus, an expression for the average meniscus height dynamics is obtained by differentiating (4) with respect to time and rearranging to give,
(5)
The model ((1) and (5)) thus far is able to capture the pre- and post-jetting dynamics in specific cases, i.e., no post-jetting oscillations. However, it is not able to capture the post-jetting dynamics in the case when oscillatory behavior occurs. To model this oscillatory phenomenon, a second order linear model is used [Yang et al. (2014)]. The height oscillation dynamics are given as, ¨ = − 2ζωn δ h(t) ˙ δ h(t) − ωn2 δh 2 4γh(t) πrN 1 2 fE (U (t), h(t)) − 2 + 2meff 2 rN + h2 (t)
2.2 Meniscus Height Dynamics In the absence of jetting, ink flows from the nozzle and forms a meniscus at the nozzle outlet. It is assumed this meniscus forms a spherical cap parametrized by the nozzle radius rN , the average meniscus height h, and the radius of the spherical cap R. Thus, the volume of a meniscus of height h anchored to a nozzle orifice of radius rN is
2Q(t) 2 + h2 (t)) . π(rN
(6)
where δh is the small change in meniscus height (m), ζ and ωn are the damping coefficient and natural frequency of the response, respectively, and meff is the effective mass of the meniscus (kg). The difference between the electric pressure and the surface tension pressure are considered a forcing term on the second order system. The resonant frequency and damping coefficient of a sessile droplet can be determined from scaling laws, e.g., [Sharp (2012)]. Note, because the model of the meniscus dynamics in (6) is a linear approximation of the actual dynamics, the frequency and damping relationships must be identified for a specific ink. Combining (1) (with (2) and (3)), (5), and (6) gives the continuous dynamic meniscus model for E-Jet printing pre- and post-jetting where Q, h, and δh are the process
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states and PB and U are process inputs, and the total meniscus height is H(t) = h(t) + δh(t). 2.3 Jetting Onset and Duration In the E-Jet process, the onset of jetting defines a distinctly different dynamic regime than that of non-jetting. Thus, while the model above describes the flow rate and height of the meniscus pre-jetting, it does not describe the dynamics of the process during jetting and thus, a discontinuity in the process dynamics occurs. Physically, the onset of jetting occurs in ambient conditions (when PB = 0, as is typical in e-jet printing) when the pressure in the meniscus due to the electric field exceeds the pressure due to surface tension [Kim et al. (2010)]. That is, jetting occurs when the following condition is satisfied, 1 2 fE (t) 2
S1 :
>
Electric Pressure
4γh(t) . + h2 (t) 2 rN
(7)
Surface Tension Pressure
where S1 denotes the switching condition from meniscus dyanmics into jetting. After the condition in (7) is satisfied and the process dynamics enter the jetting regime, the internal pressure difference does not govern the transition into pre-jetting dynamics. Instead, the duration of the jetting regime is governed by the time required for the jet to shed the charge accumulated during the pre-jetting phase and reach a new electrostatic equilibrium. The jetting lifetime is generally reported in literature as a scaling law, e.g., [Choi et al. (2008); Barton et al. (2011),]. Thus, here, the lifetime of the jet is given by [Choi et al. (2008)],
S2 :
tjet =
ρ2 γ 3
1/4
(2rN )3/4 3/2
E0
(8)
where S2 denotes the switching condition from jetting into post-jetting dynamics, tjet is the temporal duration of the jet (s), E0 = 4Umax (2rN ln(8hN (2rN )−1 ))−1 is the characteristic electric field (V/m), and Umax is the maximum voltage applied during jetting (V).
T T where x = [Q h] is the state vector, u = PB U 2 is the input vector. Define the jetting initiation time t0 as the time instant when the condition in (7) is satisfied and denote x˙ o = Ao xo + go (x)u as the oscillatory dynamics (6). Then, the hybrid model is written as ˙ = f (x(t)) + g(x(t))u(t) x(t) x˙ (t) = Ao xo (t) + go (x(t))u(t) o Jetting
The continuous pre-jetting dynamics of the meniscus, in the case when the oscillatory dynamics are not excited, are given by (1) and (5). Compactly, these are written as ˙ x(t) = f (x(t)) + g(x(t))u(t)
(9)
h(x, u) < 0 µ ∈ Σµ t0 ≤ t ≤ t0 + tjet (10)
where h is the pre-jetting to jetting transition function defined as h(x(t), u(t)) = sgn
4γh(t) 1 2 fE (t) − 2 2 rN + h2 (t)
(11)
sgn(·) is the signum function, and Σµ is the set of ink viscosities such that post-jetting oscillations occur. Note, the jetting dynamics are not modeled here, only that the process enters the jetting condition or state when h(x, u) > 0 for a duration equal to tjet . Figure 3 gives a schematic of the switching process. The switch into the jetting dynamics is governed by the pressure difference inside the meniscus. The reverse route either experiences oscillatory behavior brought on by material properties or returns to the pre-jetting dynamics. If the oscillatory dynamics are triggered, they decay before transition into the pre-jetting dynamics. 4. MENISCUS HYBRID REPETITIVE PROCESS MODEL In both the DC and DoD modes of E-Jet printing, the end of the post-jetting phase initializes the start of a new jetting sequence, i.e., pre-jetting, jetting, post-jetting, the initial conditions of which are related to the final conditions of the previous jetting sequence. Because of this paradigm, the meniscus dynamics in either DC or DoD E-Jet printing can be viewed as a repeating process. Mathematically, the meniscus hybrid repetitive process model is given by (for compactness, the material property dynamics are omitted here),
3. MENISCUS HYBRID PROCESS MODEL As mentioned above, the continuous meniscus dynamics are disrupted by the discrete jetting behavior when the condition in (7) is satisfied. At the end of the jetting behavior, determined by the jetting duration in (8), the continuous meniscus dynamics resume, albeit with potentially different initial conditions. Because the continuous meniscus dynamics are interrupted by the discrete jetting behavior and resume again at the end of jetting, the process can be modeled as a continuous-discrete hybrid process, specifically, a switched system.
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x˙ j (t) = f (xj (t)) + g(xj (t))uj (t) Jetting
h(xj , uj ) < 0 t0 ≤ t ≤ t0 + tjet (12)
with xj (0) = ∆(xj−1 (T )) where j is the trial index, T < ∞ is the trial duration (s), and ∆ is a mapping from the previous trial final condition to the current trial initial condition. An open question for the meniscus process in E-Jet is whether a description with trial-to-trial internal memory is a more apt description than the description given in (12). While experimental evidence indicates the initial condition of the current trial is indeed a function of the previous trial terminal condition, it is not clear whether or not the whole previous trial trajectory acts as a forcing function on the current trial trajectory, as is typically the case in repetitive or multipass processes.
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In the former case, i.e., (12), the control problem resembles that of iterative learning or repetitive control with a ”noreset” initial condition. While this violates a classical assumption in these control methodologies, there exists a body of work addressing problems of this class, e.g., [Steinbuch (2002)]. In both cases, however, an extension of the theory is required to treat nonlinear hybrid processes. 5. MODEL INVESTIGATION In this section, a series of experimental trials are detailed to examine the quantitative accuracy of the model presented in the section above in addition to a brief investigation into the cause of oscillatory dynamics. 5.1 Pre- and Post-Jetting Dynamics The tests are performed on an E-Jet testbed located at the University of Michigan. The meniscus motion is measured using a Phantom v9.0 high speed camera with a frame rate of 250 kHz and an image size of 96 × 96 pixels. The tests were performed in the drop-on-demand (DoD) regime with Norland Optical Adhesive (NOA) 81 ink in a 30 µm inner diameter micro-capillary nozzle set off from a silicon wafer substrate by 150 µm. In this regime, the voltage between the nozzle and substrate is pulsed between a baseline (low) voltage Ul and a high voltage Uh for a duration of Tp seconds. The values for high voltage, low voltage, and pulse width are experimentally determined to produce a single droplet during the pulse width. A small, constant backpressure is applied to ensure the meniscus is visible below the nozzle outlet. The three test conditions are detailed in Table 1. Table 1. Process parameters used for DoD experiments.
Low Voltage Ul (V) High Voltage Uh (V) Pulse Width Tp (ms)
A 500 1200 1.0
Trial B 500 1400 1.0
C 500 1000 2.5
During each trial, the high speed camera and the voltage pulse are triggered simultaneously and the meniscus behavior is recorded. For each frame in each recording, an edge finding algorithm (MATLAB’s edge command) is used to track the meniscus shape. The beginning and end of jetting is determined manually by inspecting the high speed video to find the first and last frame when a jet is observed. Finally, the calculated position of the meniscus is normalized to a range of [0,1]. Figure 5 shows the resulting time history for each of the three cases listed in Table 1 where time t = 0 indicates the beginning of the pulse and high speed video recording. Simulation results of the trials given in Table 1 are given in Figure 6. Due to mismatch between modeled electric field and the actual electric field [Kim et al. (2010)], a correction factor of 1.2 is applied to the high voltage level in simulation. Three general trends can be gleaned from Figure 5. First, as the high voltage Uh increases, the time-to-jetting decreases. The onset of jetting is observed to occur at t = 0.92, 0.64, and 1.68 ms for Trial A, B, and C, respectively. Secondly, the length of jetting generally scales with pulse
Fig. 5. Normalized meniscus height determined from high speed video of DoD E-Jet for the three different EJet process parameter sets listed in Table 1. Vertical dashed lines indicate boundaries between pre-jetting (Region 1 in bottom plot), jetting (Region 2 in bottom plot), and post-jetting (Region 3 in bottom plot). width - longer pulse width, longer jetting duration. The observed length of the jetting phenomenon was 0.60 ms and 0.72 ms for Trials A and B, respectively, and 1.20 ms for Trial C. Finally, the pre-jetting phase response is significantly faster than the response of the post-jetting phase. Qualitatively, the simulated jetting onset results in Figure 6 agree well with the observed experimental behavior; the jetting onset times were t = 0.81, 0.52, and 1.87 ms for Trials A, B, and C, respectively. However, the jetting duration times do not exhibit the same trend as described above. The simulated jetting durations were 0.24, 0.51, and 0.77 ms, for Trials A, B, and C, respectively. This discrepancy is attributable to the jetting duration condition in (8). More experimental work is needed to properly characterize the true jetting time. Finally, the simulated post-jetting experiences the same characteristically slow decay as observed in the experimental trials. In the simulations, the post-jetting initial conditions are assumed to be the final conditions of the pre-jetting phase. This assumption adds significant error to the response of the post-jetting phase and thuse more experimental work is needed to characterize the mapping between the end of the pre-jetting phase and the beginning of the post-jetting phase. 5.2 Oscillatory Response Experimental evidence, see Figures 1 and 2, indicate that inks with lower viscosity exhibit oscillatory responses, i.e., the damping coefficient is proportional to viscosity and inversely proportional to the square root of both density and surface tension [Sharp (2012)]. For typical inks (water, optical adhesives, etc.), ink-air surface tension values vary by approximately a factor of 2 (γ ≈ 70 × 10−3 N/m for DI water and γ ≈ 40 × 10−3 N/m for NOA 81), while viscosity values vary by at least an order of magnitude (µ ≈ 1.00 × 10−3 Pa-s for DI water and µ ≈ 3.00 × 10−1 Pa-s for NOA 81).
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dynamics before and after jetting was presented, in addition to a condition for the onset and duration of jetting. Simulation and experimental results agree well quantitatively with the time-to-jetting, but the jetting durations show less agreement. A better estimation scheme is needed for the switching from jetting to post-jetting. Additionally, the qualitative post-jetting behavior in both the simulations and experiments agree well. The model framework presented here allows further accuracy refinement of the individual dynamic regimes while remaining tractable for control design and applicable to the continuous-discretecontinuous dynamic paradigm in E-Jet printing.
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Fig. 7. Simulated normalized meniscus height for DI water (top) and pole location trends of the oscillation dynamics as a function of ink viscosity (µ ∈ [1×10−3 , 1× 10−1 ] (bottom). Figure 7 shows the pole locations of the oscillation dynamics as a function of ink viscosity along with a DoD mode simulation result for deionized water. A pulse width of height 900 V is commanded for a duration of 1 ms, starting at time t = 0. The post jetting response exhibits an oscillatory phenomenon with the meniscus undershooting the equilibrium height value before reaching a steady state height. This response is significantly different from that observed in the experimental and simulation results for NOA 81 (Figures 5 and 6) and agrees well qualitatively with the DI water still images shown in Figure 1. Further work is needed to fully characterize these dynamics, however. 6. CONCLUSION The meniscus dynamics in E-Jet printing evolve continuously until the onset of jetting, at which point a discontinuity occurs. Following the jetting phase, the dynamics either evolve in an oscillatory manner or decay similarly to a first-order system. A model describing the the continuous
Barton, K., Mishra, S., Alleyne, A., Ferreira, P., and Rogers, J. (2011). Control of high-resolution electrohydrodynamic jet printing. Control Engineering Practice, 19, 1266–1273. Choi, H., Park, J.U., Park, O., Ferreira, P., Georgiadis, J., and Rogers, J. (2008). Scaling laws for jet pulsations associated with high-resolution electrohydrodynamic printing. Applied Physics Letters, 92, 123109:1–3. Cloupeau, M. and Prunet-Foch, B. (1994). Electrohydrodynamic spraying functioning modes: A critical review. Journal of Aerosol Science, 25, 1021–1036. Kim, C., Jung, H., Choi, H., and Choi, D. (2016). Synthesis of one-dimensional sno2 lines by using electrohydrodynamic jet printing for a no gas sensor. Journal of the Korean Physical Society, 68, 357–362. Kim, H., Song, J., Chung, J., and Hong, D. (2010). Onset condition of pulsating cone-jet mode of electrohydrodynamic jetting for plane, hole, and pin type electrodes. Journal of Applied Physics, 108, 102804:1–10. Park, J., Hardy, M., Kang, S., Barton, K., Adair, K., Mukhopadhyay, D., Lee, C., Strano, M., ALleyne, A., Georgiadis, J., Ferreira, P., and Rogers, J. (2007). Highresolution electrohydrodynamic jet printing. Nature Materials, 6, 782–789. Poellmann, M., Barton, K., Mishra, S., and WagonerJohnson, A. (2011). Patterned hydrogel substrates for cell culture with electrohydrodynamic jet printing. Macromolecular Bioscience, 11, 1164–1168. Sharp, J. (2012). Resonant properties of sessile droplets; contact angle dependence of the resonant frequency and width in glycerol/water mixtures. Soft Matter, 8, 399– 407. Steinbuch, M. (2002). Repetitive control for systems with uncertain period-time. Automatica, 38(12), 2103–2109. Taylor, G. and Dyke, M. (1969). Electrically driven jets. Proceedings of the Royal Society of London A, 313, 453– 475. Wright, G., Krein, P., and Chato, J. (1993). Factors affecting dynamic electrical manipulation of menisci. IEEE Transactions on Industry Applications, 29(1), 103–112. Wright, G., Krein, P., and Chato, J. (1995). Self-consistent modeling of the electrohydrodynamics of a conductive meniscus. IEEE Transactions on Industry Applications, 31, 768–777. Yang, J., Kim, H., Cho, B., and Chung, J. (2014). Modeling of sessile droplet oscillation on electrohydrodynamic jetting nozzle at constant back pressure. Journal of Mechanical Science and Technology, 28, 2815–2823.
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Hybrid Continuous-Discrete Repetitive Hybrid Continuous-Discrete Repetitive HybridModeling Continuous-Discrete Repetitive in Process of Meniscus Dynamics Process Modeling of Meniscus Dynamics Process Modeling of Meniscus Dynamics in in Electrohydrodynamic Jet Printing Electrohydrodynamic Jet Printing Electrohydrodynamic Jet Printing
Patrick M. Sammons ∗∗ David Hoelzle ∗∗∗ ∗∗ ∗∗ ∗∗ David Hoelzle ∗∗∗ ∗∗∗ Patrick M. Sammons Kira L. Barton ∗∗ David Hoelzle ∗∗∗ Patrick M. Sammons Patrick M. Sammons David Hoelzle ∗∗ ∗∗ Kira L. Barton Kira L. L. Barton Barton ∗∗ Kira ∗ Rensselaer Polytechnic University, Department of Mechanical, ∗ ∗ Rensselaer Polytechnic University, Department of Mechanical, Aerospace, and Nuclear Engineering, Troy, NY (email: ∗ Rensselaer Polytechnic University, Department of Rensselaer Polytechnic University, Department of Mechanical, Mechanical, Aerospace, and Nuclear Engineering, Troy, NY (email: [email protected]) Aerospace, and Nuclear Engineering, Troy, NY Aerospace, and Nuclear Engineering, Troy, NY (email: (email: ∗∗ University of Michigan,[email protected]) Mechanical Engineering Department, Ann [email protected]) [email protected]) ∗∗ ∗∗ University ofMI Michigan, Mechanical Engineering Department, Ann Ann Arbor, (email: psammons, bartonkl @umich.edu) ∗∗ University of Michigan, Mechanical Engineering Department, ofMI Michigan, Engineering Department, Ann ∗∗∗ University Arbor, (email: Mechanical psammons, bartonkl @umich.edu) The Ohio State University, Mechanical and Aerospace Engineering Arbor, MI (email: psammons, bartonkl @umich.edu) Arbor, MI University, (email: psammons, bartonkl Aerospace @umich.edu) ∗∗∗ ∗∗∗ The Department, State Engineering Columbus, Mechanical OH (email:and [email protected]) ∗∗∗ The Ohio Ohio Mechanical and Aerospace Engineering The Department, Ohio State State University, University, Mechanical and Aerospace Engineering Columbus, OH (email: [email protected]) Department, Columbus, OH (email: [email protected]) Department, Columbus, OH (email: [email protected]) Abstract: Electrohydrodynamic jet (E-Jet) printing is a micro-/nano-manufacturing process Abstract: Electrohydrodynamic jet (E-Jet) printing is a through micro-/nano-manufacturing in which a liquid ink is jetted from micron-scale nozzle the actuation of an process electric Abstract: Electrohydrodynamic jet printing is micro-/nano-manufacturing process Abstract: Electrohydrodynamic jetaa(E-Jet) (E-Jet) printing is aa through micro-/nano-manufacturing process in which a liquid ink is jetted from micron-scale nozzle the an field applied between the nozzle and a grounded substrate. Because the jettingof is in which which a a liquid liquid ink ink is is jetted jetted from from a a micron-scale micron-scale nozzle nozzle through through the the actuation actuation ofbehavior an electric electric in actuation of an electric field applied between the nozzle and aacapable grounded substrate. Because the jetting behavior is driven by electrohydrodynamics and is of producing printed features with length scales field applied between the nozzle and grounded substrate. Because the jetting behavior is field applied between the nozzleand andis acapable grounded substrate.printed Because the jetting behavior is driven electrohydrodynamics of producing features with length scales rangingby from nanometers to microns, E-Jet printing is a promising manufacturing tool for driven by electrohydrodynamics and is capable of producing printed features with length scales driven by electrohydrodynamics and is capable of producing printed features with length scales ranging from nanometers to microns, applications. E-Jet printing is a promising manufacturing tool for printed electronics and bioengineering However, a significant hurdle in adoption ranging from to E-Jet is manufacturing tool ranging from nanometers nanometers to microns, microns, applications. E-Jet printing printing is aa promising promising manufacturing tool for for printed electronics and bioengineering However, a significant significant hurdle in in adoption of E-Jet printing for these applications is the lack of appropriate process modeling and control printed electronics and bioengineering applications. However, a hurdle adoption printed electronics and bioengineering applications. However, a significant hurdle in adoption of E-Jet printing printing for theseprinting applications is the the lack ofrobustness appropriate process modeling and Here, control methodologies to for improve performance andof to process process modeling disturbances. a of E-Jet these applications is appropriate and control of E-Jet printing for theseprinting applications is the lack lack ofrobustness appropriate process modeling and Here, control methodologies to improve performance and to process disturbances. aa model describing the dynamics of the meniscus and the onset of jetting in E-Jet printing is given. methodologies to improve printing performance and robustness to process disturbances. Here, methodologies to the improve printing performance andthe robustness to process disturbances.isHere, a model describing dynamics of the meniscus and onset of jetting in E-Jet given. The model is posed as a repetitive hybrid continuous-discrete process whereprinting the continuous model describing the dynamics of the meniscus and the onset of jetting in E-Jet printing is given. model describing the dynamics of the meniscus and the onset of jetting in E-Jet printing is given. The modeldynamics is posed posed are as a ainterrupted repetitive hybrid hybrid continuous-discrete processand where the continuous meniscus by the continuous-discrete discrete jetting behavior are the dependent on The model is as repetitive process where continuous The modeldynamics is posed are as ainterrupted repetitive hybrid continuous-discrete processand where the continuous meniscus by the discrete jetting behavior are dependent on previous trial information. The developed model is compared to a set of experimental results meniscus dynamics are interrupted by the discrete jetting behavior and are dependent on meniscus dynamics are interrupted by themodel discrete jetting behavior andexperimental are dependent on previous trial information. The developed is compared to aa set of results and the model exhibits good quantitative agreement with time-to-jetting from a step voltage previous trial information. The developed model is compared to set of experimental results previous trial information. The developed model is compared to a set of experimental results and good quantitative agreement time-to-jetting aa step voltage inputthe andmodel good exhibits qualitative agreement in the dynamic with response after jettingfrom has ceased. and the model exhibits good quantitative agreement with time-to-jetting from and good quantitative agreement time-to-jetting a step step voltage voltage inputthe andmodel good exhibits qualitative agreement in the the dynamic with response after jetting jettingfrom has ceased. ceased. input and good qualitative agreement in dynamic response after has input good qualitativeFederation agreement in the dynamic after jetting © 2017,and IFAC (International of Automatic Control)response Hosting by Elsevier Ltd.has Allceased. rights reserved. Keywords: Modeling of manufacturing operations, Hybrid and switched systems modeling, Keywords: manufacturing operations, Electrohydrodynamic Repetitive processes Hybrid Keywords: Modeling Modeling of ofjetting, manufacturing operations, Hybrid and and switched switched systems systems modeling, modeling, Keywords: Modeling of manufacturing operations, Hybrid and switched systems modeling, Electrohydrodynamic jetting, Repetitive processes Electrohydrodynamic jetting, Repetitive processes Electrohydrodynamic jetting, Repetitive processes 1. INTRODUCTION grounded substrate by a vertical distance. A voltage dif1. grounded by a vertical distance. voltage difference is substrate applied between the nozzle and A substrate 1. INTRODUCTION INTRODUCTION grounded substrate by distance. A voltage dif1. INTRODUCTION grounded substrate by aa vertical vertical distance. Athe voltage difis applied between the nozzle and the substrate and a liquid meniscus forms at the outlet of the microMicro-/Nano-scale Additive Manufacturing (µ/nAM) is a ference ference is applied between the nozzle and the substrate ference is applied between the nozzle and the substrate and a liquid meniscus forms at the outlet of the microMicro-/Nano-scale Manufacturing (µ/nAM) is nozzle. When forms a critical voltage the class of processes inAdditive which parts, features, and structures and aa liquid meniscus at outlet of the Micro-/Nano-scale Additive Manufacturing (µ/nAM) is aaa capillary and liquid meniscus forms at the the outletis ofreached, the micromicroMicro-/Nano-scale Additive Manufacturing (µ/nAM) is capillary nozzle. When aaa critical voltage is reached, the class of processes in which parts, features, and structures meniscus deforms into so-called Taylor cone [Taylor can be fabricated by repeated addition of material and capillary nozzle. When critical voltage is reached, the class of processes in which parts, features, and structures capillary nozzle. When aa critical voltage is cone reached, the class of fabricated processes inbywhich parts,addition features,ofand structures meniscus deforms into so-called Taylor [Taylor can be repeated material and and Dyke (1969)] and a jet of material issues from the whose characteristic length scales range from nanometers meniscus deforms into a so-called Taylor cone [Taylor can be fabricated by repeated addition of material and meniscus deforms into a so-called Taylor cone [Taylor can be fabricated by repeated addition of material and and depositing Dyke (1969)] (1969)] and aa on jetthe of material material issues from the whose characteristic length from material substrate.issues As the nozzle to microns. µ/nAM processes are range emerging attractive tip, and Dyke and jet of from the whose characteristic length scales scales range fromasnanometers nanometers and Dyke (1969)] and a on jetthe of material issues from the whose characteristic length scales range from nanometers tip, depositing material substrate. As the nozzle to microns. µ/nAM processes are emerging as attractive traverses across the substrate, two-dimensional patterns methods for fabricating high-resolution sensors [Kim et al. tip, depositing material on the substrate. As the nozzle to microns. µ/nAM processes are emerging as attractive tip, depositing material on the substrate. As the nozzle to microns. µ/nAM processes are emerging as attractive across the substrate, two-dimensional methods for [Kim et can be created. process can then be patterns indexed (2016)], electronics [Parkhigh-resolution et al. (2007)], sensors and in the field of traverses traverses across the substrate, two-dimensional patterns methods for fabricating fabricating high-resolution sensors [Kim et al. al. traverses across The the printing substrate, two-dimensional patterns methods for fabricating high-resolution sensors [Kim et al. can be created. The printing process can then indexed (2016)], electronics [Park et al. (2007)], and in the field of to a new layer and jetting continues on top of be previously bioengineering [Poellmann et al. (2011)]. However, significan be created. The printing process can then be indexed (2016)], electronics [Park et al. (2007)], and in the field of can be created. The printing process can then be indexed (2016)], electronics [Park etetal.al.(2007)], and in the field of to a new layer and jetting continues on top of previously bioengineering [Poellmann (2011)]. However, signifideposited material to form three-dimensional, functional cant modeling and control challenges currently prevent the to a new layer and jetting continues on top of previously bioengineering [Poellmann [Poellmann et et al. al. (2011)]. (2011)]. However, However, signifisignifi- to a new layer and jetting continues on top of previously bioengineering material to form functional cant modeling control challenges prevent the structures. While there exist three-dimensional, many specific modes of matewidespread useand of these processes for currently repeatable, accurate deposited material to three-dimensional, functional cant modeling and control challenges currently prevent the deposited deposited material to form form three-dimensional, functional cant modeling and control challenges currently prevent the structures. While there exist many specific modes ofjetting matewidespread use of these processes for repeatable, accurate rial ejection that are termed electrohydrodynamic production of components. structures. While there exist many specific modes matewidespread use of these processes for repeatable, accurate While there exist many specific modes of ofjetting matewidespread usecomponents. of these processes for repeatable, accurate structures. rial ejection that are termed electrohydrodynamic production of [Cloupeau and Prunet-Foch (1994)], e.g. micro-dripping, rial ejection that are termed electrohydrodynamic jetting production of components. rial ejection that are termed electrohydrodynamic jetting production of components. A specific µ/nAM process which is capable of printing [Cloupeau andparticular Prunet-Foch (1994)], e.g. micro-dripping, micro-dripping, multi-jet, the regime of interest in this work is [Cloupeau and Prunet-Foch (1994)], e.g. andparticular Prunet-Foch (1994)], e.g. micro-dripping, A specific µ/nAM process which capable of printing a wide range of inks with feature on the of [Cloupeau A specific µ/nAM process which is issizes capable of order printing multi-jet, the regime of interest in thisjetwork is termed cone-jet. In the cone-jet regime, a single issues A specific µ/nAM process which is capable of printing multi-jet, the particular regime of interest in is multi-jet, the particular regime of interest in this thisjetwork work is aahundreds wide range of inks with feature sizes on the order of of nanometers up to microns – more than an wide range of inks with feature sizes on the order of termed cone-jet. In the cone-jet regime, a single issues from the Taylor cone which enables operation in both DCa wide range of inks with feature sizes on the order of termed cone-jet. In the cone-jet regime, a single jet issues termed cone-jet. In the cone-jet regime, a single jet issues hundreds of nanometers up to microns –– more than an order of magnitude smaller than traditional ink jetting hundreds of nanometers up to microns more than an Taylor cone which enables operation both DCmode (constant voltage difference thein and hundreds of nanometers up to microns – more than an from from the the Taylor which enables operation innozzle both Taylor cone cone which enablesbetween operation both DCDCorder of magnitude than ink jetting printing is Electrohydrodynamic order of processes magnitude– smaller smaller than traditional traditional Jet ink (E-Jet) jetting from modethe (constant voltage difference between theinnozzle nozzle and substrate) and Drop-on-Demand (DoD) or pulsed printing order of magnitude smaller than traditional ink jetting mode (constant voltage difference between the and (constant voltage difference(DoD) between the nozzle and printing processes –– is Electrohydrodynamic Jet et al. (2007)]. In E-Jet, a micro-capillary printing [Park processes is Electrohydrodynamic Jet (E-Jet) (E-Jet) mode substrate) and Drop-on-Demand or pulsed printing mode (the voltage is pulsed between a low and high value). printing processes – is Electrohydrodynamic Jet (E-Jet) substrate) and Drop-on-Demand (DoD) or pulsed printing substrate) and Drop-on-Demand (DoD) or pulsed printing printing [Park et al. (2007)]. In E-Jet, a micro-capillary nozzle is filled with a liquid ink and separated from a printing [Park [Park et et al. al. (2007)]. (2007)]. In In E-Jet, E-Jet, aa micro-capillary micro-capillary mode (the voltage is pulsed between aa low and high value). printing (the is between and high mode (the voltage voltage is pulsed pulsedE-Jet between a low lowcan andgenerally high value). value). nozzle is with liquid ink and from aa mode The dynamics in cone-jet printing be nozzle is filled filled with a liquid ink Science and separated separated from nozzle is filled with aa by liquid ink and separated a The dynamics in cone-jet E-Jet printing can generally be This work was supported National Foundation from through divided into two processes: the meniscus dynamics and The dynamics in cone-jet E-Jet printing can generally be work was supported by National Foundation through The dynamics in cone-jet E-Jet printing can generallyand be the Division of Civil, Mechanical, andScience Manufacturing Innovation This work supported by Science Foundation through This divided into two processes: the meniscus dynamics This work was was supported by National National Science Foundation through the droplet dynamics. The meniscus dynamics describe the divided into two processes: the meniscus dynamics and the Division of Civil, Mechanical, and Manufacturing Innovation under Grant 1434660 and Grant 1434693. The authors would like divided into two processes: the meniscus dynamics and the Division of Civil, Mechanical, and Manufacturing Innovation the droplet dynamics. meniscus dynamics the the Division of Civil, and Mechanical, and Manufacturing Innovation temporal evolution of The the meniscus e.g.,describe the height the droplet dynamics. The meniscus dynamics describe the under Grant 1434660 Grant 1434693. authors would like to acknowledge Christopher Pannier for his The technical assistance. the droplet dynamics. The meniscus shape, dynamics describe the under Grant and 1434693. The authors would under Grant 1434660 1434660 and Grant Grant 1434693. The authors would like like temporal evolution of the meniscus shape, e.g., the height to acknowledge Christopher Pannier for his technical assistance. temporal evolution of the meniscus shape, e.g., the height to acknowledge Christopher Pannier for his technical assistance. temporal evolution of the meniscus shape, e.g., the height to acknowledge Christopher Pannier for his technical assistance. Mamadou Mamadou Mamadou Mamadou
Diagne ∗ ∗ ∗ Diagne Diagne Diagne ∗
Copyright 13956Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 13956 Copyright © 2017 IFAC 13956 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 13956Control. 10.1016/j.ifacol.2017.08.2285
Proceedings of the 20th IFAC World Congress Mamadou Diagne et al. / IFAC PapersOnLine 50-1 (2017) 13414–13419 Toulouse, France, July 9-14, 2017
Fig. 1. Sequential still image captures of high speed video taken during DoD E-Jet printing of deionized water. Black lines above each image indicates the height of the meniscus. of the meniscus as measured from the nozzle orifice datum or the curvature of the surface, as a function of process inputs. Alternatively, the droplet dynamics describe both the temporal and spatial evolution of the droplet shape that results from the jetting of material onto the substrate after a jetting event. At the end of the jetting event, the jet retracts and the meniscus transitions towards a static equilibrium while the droplet dynamics evolve concurrently. During this transition phase, the meniscus can exhibit an oscillatory response (see Figure 1 for a timelapse of deionized (DI) water). In typical operation, a dwell time is explicitly included in the printing trajectory to ensure these oscillations have decayed sufficiently before attempting to continue fabrication. Printing in this regime can lead to unwanted or unexpected meniscus behavior. Therefore, it is necessary to understand and model the evolution of the meniscus dynamics during the pre- and post-jetting phases. Models describing the meniscus dynamics in E-Jet printing typically rely on numerical solutions to the NavierStokes equations, e.g., [Wright et al. (1995)]. While these models are capable of capturing the complex interactions between the meniscus and the electric field induced by the voltage potential between the substrate and the nozzle, in addition to the modeling the full meniscus surface shape, they are not appropriate for process control. Additionally, for those models that are appropriate for process control, [Wright et al. (1993); Yang et al. (2014)] for example, only dynamics of the meniscus in the pre-jetting phase are described. Thus, in order to describe the jetting onset and post-jetting dynamics, further modeling efforts are needed. While each sub-process could be modeled separately, process control requires a model of the process that is able to capture the dynamics of each sub-process and the transitions between them. Because the continuousdiscrete-continuous regime is inherent in the E-Jet material ejection process, a hybrid continuous-discrete model is a logical choice for describing the meniscus dynamics. Further, in both DC and DoD modes, the material ejection process is repetitive; the continuous-discrete-continuous paradigm repeats for each droplet ejection event and the time evolution of the meniscus dynamics is dependent on the trajectory of the previous ejection event. A schematic of this operating paradigm is shown in Figure 3 (description details are given below in Section 3). Therefore, the meniscus dynamic process can posed further as a hybrid repetitive process. This work examines the dynamic evo-
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Fig. 2. Sequential still image captures of high speed video taken during DoD E-Jet printing of NOA 81 with prejetting, jetting, and recovery phases labeled. All times are in milliseconds. lution of the meniscus in both the pre- and post-jetting conditions, but does not examine the process dynamics of the jetting regime. The rest of the paper is structured as follows. First, the continuous dynamics of the meniscus are given in addition to criterion for the onset and duration of the jetting mode. Then, the continuous model and the switching conditions are presented as a hybrid process and finally posed as hybrid repetitive process. In the section following, details regarding the dynamic modes of the continuous model are given along with some experimental evidence of these modes. Finally, a summary of the work and conclusions are given. 2. CONTINUOUS MENISCUS MODEL As mentioned above, the meniscus dynamics in E-Jet printing can be divided into three phases; pre-jetting, jetting, and post-jetting. In the pre-jetting phase, the pressure in the meniscus increases due to the electric field and the meniscus deforms until jetting occurs (top set of photographs in Figure 2). During jetting, the ink flows through the jet towards the substrate to form a discrete droplet (middle set of photographs in Figure 2). After the duration of the jetting phenomenon, the jet retracts from the substrate, and the post-jetting phase begins (bottom set of photographs shown in Figure 2). Here, the preand post-jetting behaviors are described by continuous dynamics while the jetting behavior is indicated by a discrete event, the onset and termination of which is determined by conditions given at the end of this section. 2.1 Flow Rate Dynamics In the following, the ink is incompressible and viscous and the flow laminar. Additionally, it is assumed that the inner radius of the microcapillary nozzle is constant along the length of the microcapillary, the length of the microcapillary is much greater than the inner radius, and the distance between the nozzle and the substrate remains fixed. Then, the ink flow rate through the nozzle can be described as Hagen-Poiseuille flow where the pressure drop across the length of the nozzle is proportional to the flow rate,
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8µ πr2 ˙ Q(t) = N ∆P (t) − 2 Q(t) ρL ρrN
(1)
with Q the volumetric flow rate of ink through the nozzle (m3 /s), rN is the nozzle radius (m), ρ the ink density (kg/m3 ), µ the ink viscosity (Pa-s), L the length of the nozzle (m), and ∆P the total pressure drop across the nozzle (N/m2 ). The pressure drop across the nozzle is given as ∆P (t) = PB (t) + Pγ (t) + PE (t) 1 4γh(t) + E 2 (t) = PB (t) − 2 rN + h2 (t) 2
(2)
Fig. 3. Schematic of the switching process in the hybrid meniscus model.
where PB is backpressure supplied to the nozzle (N/m2 ), Pγ is the surface tension pressure (N/m2 ), PE is the pressure in the meniscus generated by the electric field and termed the electric pressure (N/m2 ), γ is the ink surface tension coefficient (N/m), h is the meniscus height (m), is the permittivity of air (F/m), and E is the electric field (V/m). Backpressure PB is typically zero for all t in ejet printing. This corresponds to the ink being exposed to ambient conditions. A schematic of the nozzle and substrate interaction zone is shown in Figure 4. For modeling and control purposes, it is advantageous to express the electric field between the nozzle and the substrate as a function of process inputs and states. While an exact expression relating the applied voltage U (process input) and meniscus height h (process state) to the electric field requires a solution to the Navier-Stokes equations, a common approximation [Kim et al. (2010)] of the electric field for electrohydrodynamic jetting is that of a metal hyperboloid revolution above a flat plate electrode, E(t) = fE (U (t), h(t)) =
2U (t) 4hN R(t) ln R(t)
(3)
where hN is the distance between the nozzle and the 2 substrate (m) and R(t) = (rN +h2 (t)(2h(t))−1 is the radius of the spherical cap formed by the average meniscus height and the nozzle radius (m).
Fig. 4. Schematic of the nozzle-substrate interaction zone in E-Jet Printing ˙ h(t) =
υ(t) =
π 2 h(t)(3rN + h2 (t)) 6
(4)
where υ is the meniscus volume (m3 ). Because volume is conserved within the meniscus pre-jetting, the volume rate of change is equal to the volume flow rate through the d nozzle dt υ(t) = Q(t). Thus, an expression for the average meniscus height dynamics is obtained by differentiating (4) with respect to time and rearranging to give,
(5)
The model ((1) and (5)) thus far is able to capture the pre- and post-jetting dynamics in specific cases, i.e., no post-jetting oscillations. However, it is not able to capture the post-jetting dynamics in the case when oscillatory behavior occurs. To model this oscillatory phenomenon, a second order linear model is used [Yang et al. (2014)]. The height oscillation dynamics are given as, ¨ = − 2ζωn δ h(t) ˙ δ h(t) − ωn2 δh 2 4γh(t) πrN 1 2 fE (U (t), h(t)) − 2 + 2meff 2 rN + h2 (t)
2.2 Meniscus Height Dynamics In the absence of jetting, ink flows from the nozzle and forms a meniscus at the nozzle outlet. It is assumed this meniscus forms a spherical cap parametrized by the nozzle radius rN , the average meniscus height h, and the radius of the spherical cap R. Thus, the volume of a meniscus of height h anchored to a nozzle orifice of radius rN is
2Q(t) 2 + h2 (t)) . π(rN
(6)
where δh is the small change in meniscus height (m), ζ and ωn are the damping coefficient and natural frequency of the response, respectively, and meff is the effective mass of the meniscus (kg). The difference between the electric pressure and the surface tension pressure are considered a forcing term on the second order system. The resonant frequency and damping coefficient of a sessile droplet can be determined from scaling laws, e.g., [Sharp (2012)]. Note, because the model of the meniscus dynamics in (6) is a linear approximation of the actual dynamics, the frequency and damping relationships must be identified for a specific ink. Combining (1) (with (2) and (3)), (5), and (6) gives the continuous dynamic meniscus model for E-Jet printing pre- and post-jetting where Q, h, and δh are the process
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states and PB and U are process inputs, and the total meniscus height is H(t) = h(t) + δh(t). 2.3 Jetting Onset and Duration In the E-Jet process, the onset of jetting defines a distinctly different dynamic regime than that of non-jetting. Thus, while the model above describes the flow rate and height of the meniscus pre-jetting, it does not describe the dynamics of the process during jetting and thus, a discontinuity in the process dynamics occurs. Physically, the onset of jetting occurs in ambient conditions (when PB = 0, as is typical in e-jet printing) when the pressure in the meniscus due to the electric field exceeds the pressure due to surface tension [Kim et al. (2010)]. That is, jetting occurs when the following condition is satisfied, 1 2 fE (t) 2
S1 :
>
Electric Pressure
4γh(t) . + h2 (t) 2 rN
(7)
Surface Tension Pressure
where S1 denotes the switching condition from meniscus dyanmics into jetting. After the condition in (7) is satisfied and the process dynamics enter the jetting regime, the internal pressure difference does not govern the transition into pre-jetting dynamics. Instead, the duration of the jetting regime is governed by the time required for the jet to shed the charge accumulated during the pre-jetting phase and reach a new electrostatic equilibrium. The jetting lifetime is generally reported in literature as a scaling law, e.g., [Choi et al. (2008); Barton et al. (2011),]. Thus, here, the lifetime of the jet is given by [Choi et al. (2008)],
S2 :
tjet =
ρ2 γ 3
1/4
(2rN )3/4 3/2
E0
(8)
where S2 denotes the switching condition from jetting into post-jetting dynamics, tjet is the temporal duration of the jet (s), E0 = 4Umax (2rN ln(8hN (2rN )−1 ))−1 is the characteristic electric field (V/m), and Umax is the maximum voltage applied during jetting (V).
T T where x = [Q h] is the state vector, u = PB U 2 is the input vector. Define the jetting initiation time t0 as the time instant when the condition in (7) is satisfied and denote x˙ o = Ao xo + go (x)u as the oscillatory dynamics (6). Then, the hybrid model is written as ˙ = f (x(t)) + g(x(t))u(t) x(t) x˙ (t) = Ao xo (t) + go (x(t))u(t) o Jetting
The continuous pre-jetting dynamics of the meniscus, in the case when the oscillatory dynamics are not excited, are given by (1) and (5). Compactly, these are written as ˙ x(t) = f (x(t)) + g(x(t))u(t)
(9)
h(x, u) < 0 µ ∈ Σµ t0 ≤ t ≤ t0 + tjet (10)
where h is the pre-jetting to jetting transition function defined as h(x(t), u(t)) = sgn
4γh(t) 1 2 fE (t) − 2 2 rN + h2 (t)
(11)
sgn(·) is the signum function, and Σµ is the set of ink viscosities such that post-jetting oscillations occur. Note, the jetting dynamics are not modeled here, only that the process enters the jetting condition or state when h(x, u) > 0 for a duration equal to tjet . Figure 3 gives a schematic of the switching process. The switch into the jetting dynamics is governed by the pressure difference inside the meniscus. The reverse route either experiences oscillatory behavior brought on by material properties or returns to the pre-jetting dynamics. If the oscillatory dynamics are triggered, they decay before transition into the pre-jetting dynamics. 4. MENISCUS HYBRID REPETITIVE PROCESS MODEL In both the DC and DoD modes of E-Jet printing, the end of the post-jetting phase initializes the start of a new jetting sequence, i.e., pre-jetting, jetting, post-jetting, the initial conditions of which are related to the final conditions of the previous jetting sequence. Because of this paradigm, the meniscus dynamics in either DC or DoD E-Jet printing can be viewed as a repeating process. Mathematically, the meniscus hybrid repetitive process model is given by (for compactness, the material property dynamics are omitted here),
3. MENISCUS HYBRID PROCESS MODEL As mentioned above, the continuous meniscus dynamics are disrupted by the discrete jetting behavior when the condition in (7) is satisfied. At the end of the jetting behavior, determined by the jetting duration in (8), the continuous meniscus dynamics resume, albeit with potentially different initial conditions. Because the continuous meniscus dynamics are interrupted by the discrete jetting behavior and resume again at the end of jetting, the process can be modeled as a continuous-discrete hybrid process, specifically, a switched system.
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x˙ j (t) = f (xj (t)) + g(xj (t))uj (t) Jetting
h(xj , uj ) < 0 t0 ≤ t ≤ t0 + tjet (12)
with xj (0) = ∆(xj−1 (T )) where j is the trial index, T < ∞ is the trial duration (s), and ∆ is a mapping from the previous trial final condition to the current trial initial condition. An open question for the meniscus process in E-Jet is whether a description with trial-to-trial internal memory is a more apt description than the description given in (12). While experimental evidence indicates the initial condition of the current trial is indeed a function of the previous trial terminal condition, it is not clear whether or not the whole previous trial trajectory acts as a forcing function on the current trial trajectory, as is typically the case in repetitive or multipass processes.
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In the former case, i.e., (12), the control problem resembles that of iterative learning or repetitive control with a ”noreset” initial condition. While this violates a classical assumption in these control methodologies, there exists a body of work addressing problems of this class, e.g., [Steinbuch (2002)]. In both cases, however, an extension of the theory is required to treat nonlinear hybrid processes. 5. MODEL INVESTIGATION In this section, a series of experimental trials are detailed to examine the quantitative accuracy of the model presented in the section above in addition to a brief investigation into the cause of oscillatory dynamics. 5.1 Pre- and Post-Jetting Dynamics The tests are performed on an E-Jet testbed located at the University of Michigan. The meniscus motion is measured using a Phantom v9.0 high speed camera with a frame rate of 250 kHz and an image size of 96 × 96 pixels. The tests were performed in the drop-on-demand (DoD) regime with Norland Optical Adhesive (NOA) 81 ink in a 30 µm inner diameter micro-capillary nozzle set off from a silicon wafer substrate by 150 µm. In this regime, the voltage between the nozzle and substrate is pulsed between a baseline (low) voltage Ul and a high voltage Uh for a duration of Tp seconds. The values for high voltage, low voltage, and pulse width are experimentally determined to produce a single droplet during the pulse width. A small, constant backpressure is applied to ensure the meniscus is visible below the nozzle outlet. The three test conditions are detailed in Table 1. Table 1. Process parameters used for DoD experiments.
Low Voltage Ul (V) High Voltage Uh (V) Pulse Width Tp (ms)
A 500 1200 1.0
Trial B 500 1400 1.0
C 500 1000 2.5
During each trial, the high speed camera and the voltage pulse are triggered simultaneously and the meniscus behavior is recorded. For each frame in each recording, an edge finding algorithm (MATLAB’s edge command) is used to track the meniscus shape. The beginning and end of jetting is determined manually by inspecting the high speed video to find the first and last frame when a jet is observed. Finally, the calculated position of the meniscus is normalized to a range of [0,1]. Figure 5 shows the resulting time history for each of the three cases listed in Table 1 where time t = 0 indicates the beginning of the pulse and high speed video recording. Simulation results of the trials given in Table 1 are given in Figure 6. Due to mismatch between modeled electric field and the actual electric field [Kim et al. (2010)], a correction factor of 1.2 is applied to the high voltage level in simulation. Three general trends can be gleaned from Figure 5. First, as the high voltage Uh increases, the time-to-jetting decreases. The onset of jetting is observed to occur at t = 0.92, 0.64, and 1.68 ms for Trial A, B, and C, respectively. Secondly, the length of jetting generally scales with pulse
Fig. 5. Normalized meniscus height determined from high speed video of DoD E-Jet for the three different EJet process parameter sets listed in Table 1. Vertical dashed lines indicate boundaries between pre-jetting (Region 1 in bottom plot), jetting (Region 2 in bottom plot), and post-jetting (Region 3 in bottom plot). width - longer pulse width, longer jetting duration. The observed length of the jetting phenomenon was 0.60 ms and 0.72 ms for Trials A and B, respectively, and 1.20 ms for Trial C. Finally, the pre-jetting phase response is significantly faster than the response of the post-jetting phase. Qualitatively, the simulated jetting onset results in Figure 6 agree well with the observed experimental behavior; the jetting onset times were t = 0.81, 0.52, and 1.87 ms for Trials A, B, and C, respectively. However, the jetting duration times do not exhibit the same trend as described above. The simulated jetting durations were 0.24, 0.51, and 0.77 ms, for Trials A, B, and C, respectively. This discrepancy is attributable to the jetting duration condition in (8). More experimental work is needed to properly characterize the true jetting time. Finally, the simulated post-jetting experiences the same characteristically slow decay as observed in the experimental trials. In the simulations, the post-jetting initial conditions are assumed to be the final conditions of the pre-jetting phase. This assumption adds significant error to the response of the post-jetting phase and thuse more experimental work is needed to characterize the mapping between the end of the pre-jetting phase and the beginning of the post-jetting phase. 5.2 Oscillatory Response Experimental evidence, see Figures 1 and 2, indicate that inks with lower viscosity exhibit oscillatory responses, i.e., the damping coefficient is proportional to viscosity and inversely proportional to the square root of both density and surface tension [Sharp (2012)]. For typical inks (water, optical adhesives, etc.), ink-air surface tension values vary by approximately a factor of 2 (γ ≈ 70 × 10−3 N/m for DI water and γ ≈ 40 × 10−3 N/m for NOA 81), while viscosity values vary by at least an order of magnitude (µ ≈ 1.00 × 10−3 Pa-s for DI water and µ ≈ 3.00 × 10−1 Pa-s for NOA 81).
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dynamics before and after jetting was presented, in addition to a condition for the onset and duration of jetting. Simulation and experimental results agree well quantitatively with the time-to-jetting, but the jetting durations show less agreement. A better estimation scheme is needed for the switching from jetting to post-jetting. Additionally, the qualitative post-jetting behavior in both the simulations and experiments agree well. The model framework presented here allows further accuracy refinement of the individual dynamic regimes while remaining tractable for control design and applicable to the continuous-discretecontinuous dynamic paradigm in E-Jet printing.
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Fig. 6. Simulated normalized meniscus height for the three different E-Jet process parameter sets listed in Table 1. Vertical dashed lines indicate boundaries between pre-jetting, jetting, and post-jetting. 1 0.8 0.6 0.4 0.2 0 -0.2 -1
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Fig. 7. Simulated normalized meniscus height for DI water (top) and pole location trends of the oscillation dynamics as a function of ink viscosity (µ ∈ [1×10−3 , 1× 10−1 ] (bottom). Figure 7 shows the pole locations of the oscillation dynamics as a function of ink viscosity along with a DoD mode simulation result for deionized water. A pulse width of height 900 V is commanded for a duration of 1 ms, starting at time t = 0. The post jetting response exhibits an oscillatory phenomenon with the meniscus undershooting the equilibrium height value before reaching a steady state height. This response is significantly different from that observed in the experimental and simulation results for NOA 81 (Figures 5 and 6) and agrees well qualitatively with the DI water still images shown in Figure 1. Further work is needed to fully characterize these dynamics, however. 6. CONCLUSION The meniscus dynamics in E-Jet printing evolve continuously until the onset of jetting, at which point a discontinuity occurs. Following the jetting phase, the dynamics either evolve in an oscillatory manner or decay similarly to a first-order system. A model describing the the continuous
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