American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020460

Dimension reduction of thermistor models for large-area organic light-emitting diodes

Annegret Glitzky , Matthias Liero , and  Grigor Nika

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

* Corresponding author: Matthias Liero

Received  April 2020 Revised  September 2020 Published  November 2020

An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional $ p(x) $-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter $ \varepsilon>0 $, which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.

Keywords: Thermistor system, dimension reduction, Joule heat, organic light-emitting diode, thin-film devices, multi-scale limit.
Mathematics Subject Classification: Primary:35J92, 35Q79, 35J57, 80A20, 35B30, 35B20.
Citation: Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020460
References:
[1]

E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity, 25 (1991), 137-148.  doi: 10.1007/BF00042462.  Google Scholar

[2]

M. Bulíček, A. Glitzky and M. Liero, Systems describing electrothermal effects with $p(x)$-Laplace like structure for discontinuous variable exponents, SIAM J. Math. Analysis, 48 (2016), 3496-3514.  doi: 10.1137/16M1062211.  Google Scholar

[3]

M. Bulíček, A. Glitzky and M. Liero, Thermistor systems of $p(x)$-Laplace-type with discontinuous exponents via entropy solutions, Discr. Cont. Dynam. Systems Ser. S, 10 (2017), 697-713.  doi: 10.3934/dcdss.2017035.  Google Scholar

[4]

P. G. Ciarlet and P. Destuynder, A justification of a nonlinear model in plate theory, Computer Methods in Applied Mechanics and Engineering, 17/18 (1979), 227-258.  doi: 10.1016/0045-7825(79)90089-6.  Google Scholar

[5]

L. Diening, P. Harjulehto, P. Hästö and M. Rủžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[6]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$, Journal of Mathematical Analysis and Applications, 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[7]

A. Fischer, T. Koprucki, K. Gärtner, M. L. Tietze, J. Brückner, B. Lüssem, K. Leo, A. Glitzky and R. Scholz, Feel the heat: Nonlinear electrothermal feedback in organic LEDs, Adv. Funct. Mater., 24 (2014), 3367-3374.  doi: 10.1002/adfm.201303066.  Google Scholar

[8]

A. Fischer, P. Pahner, B. Lüssem, K. Leo, R. Scholz, T. Koprucki, K. Gärtner and A. Glitzky, Self-heating, bistability and thermal switching in organic semiconductors, Phys. Rev. Lett., 110 (2013), 126601. doi: 10.1103/PhysRevLett.110.126601.  Google Scholar

[9]

A. Fischer, M. Pfalz, K. Vandewal, S. Lenk, M. Liero, A. Glitzky and S. Reineke, Full electrothermal OLED model including nonlinear self-heating effects, Phys. Rev. Applied, 10 (2018), 014023. doi: 10.1103/PhysRevApplied.10.014023.  Google Scholar

[10]

T. Frenzel and M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDP-convergence, Discr. Cont. Dynam. Systems Ser. S, (2020), Online first doi: 10.3934/dcdss.2020345.  Google Scholar

[11]

G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Archive for Rational Mechanics and Analysis, 180 (2006), 183-236.  doi: 10.1007/s00205-005-0400-7.  Google Scholar

[12]

A. Kirch, A. Fischer, M. Liero, J. Fuhrmann, A. Glitzky and S. Reineke, Experimental proof of {J}oule heating-induced switched-back regions in OLEDs, Light: Science & Applications, 9 (2020), 5. doi: 10.1038/s41377-019-0236-9.  Google Scholar

[13]

P. Kordt, J. J. M. van der Holst, M. Al Helwi, W. Kowalsky, F. May, A. Badinski, C. Lennartz and D. Andrienko, Modeling of organic light emitting diodes: From molecular to device properties, Adv. Func. Mater., 25 (2015), 1955-1971.  doi: 10.1002/adfm.201403004.  Google Scholar

[14]

O. Kováčik and J. Rákosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618.   Google Scholar

[15]

M. Liero, J. Fuhrmann, A. Glitzky, T. Koprucki, A. Fischer and S. Reineke, 3{D} electrothermal simulations of organic LEDs showing negative differential resistance, in Opt. Quantum Electron., 49 (2017), 330/1–330/8. doi: 10.1109/NUSOD.2017.8010013.  Google Scholar

[16]

M. Liero, T. Koprucki, A. Fischer, R. Scholz and A. Glitzky, $p$-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices, Z. Angew. Math. Phys., 66 (2015), 2957-2977.  doi: 10.1007/s00033-015-0560-8.  Google Scholar

[17]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM Journal on Mathematical Analysis, 39 (2007), 687-720.  doi: 10.1137/060665452.  Google Scholar

[18]

K. Schmidt and S. Tordeux, Asymptotic modelling of conductive thin sheets, Zeitschrift für Angewandte Mathematik und Physik, 61 (2010), 603–626. doi: 10.1007/s00033-009-0043-x.  Google Scholar

show all references

References:
[1]

E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity, 25 (1991), 137-148.  doi: 10.1007/BF00042462.  Google Scholar

[2]

M. Bulíček, A. Glitzky and M. Liero, Systems describing electrothermal effects with $p(x)$-Laplace like structure for discontinuous variable exponents, SIAM J. Math. Analysis, 48 (2016), 3496-3514.  doi: 10.1137/16M1062211.  Google Scholar

[3]

M. Bulíček, A. Glitzky and M. Liero, Thermistor systems of $p(x)$-Laplace-type with discontinuous exponents via entropy solutions, Discr. Cont. Dynam. Systems Ser. S, 10 (2017), 697-713.  doi: 10.3934/dcdss.2017035.  Google Scholar

[4]

P. G. Ciarlet and P. Destuynder, A justification of a nonlinear model in plate theory, Computer Methods in Applied Mechanics and Engineering, 17/18 (1979), 227-258.  doi: 10.1016/0045-7825(79)90089-6.  Google Scholar

[5]

L. Diening, P. Harjulehto, P. Hästö and M. Rủžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[6]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$, Journal of Mathematical Analysis and Applications, 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.  Google Scholar

[7]

A. Fischer, T. Koprucki, K. Gärtner, M. L. Tietze, J. Brückner, B. Lüssem, K. Leo, A. Glitzky and R. Scholz, Feel the heat: Nonlinear electrothermal feedback in organic LEDs, Adv. Funct. Mater., 24 (2014), 3367-3374.  doi: 10.1002/adfm.201303066.  Google Scholar

[8]

A. Fischer, P. Pahner, B. Lüssem, K. Leo, R. Scholz, T. Koprucki, K. Gärtner and A. Glitzky, Self-heating, bistability and thermal switching in organic semiconductors, Phys. Rev. Lett., 110 (2013), 126601. doi: 10.1103/PhysRevLett.110.126601.  Google Scholar

[9]

A. Fischer, M. Pfalz, K. Vandewal, S. Lenk, M. Liero, A. Glitzky and S. Reineke, Full electrothermal OLED model including nonlinear self-heating effects, Phys. Rev. Applied, 10 (2018), 014023. doi: 10.1103/PhysRevApplied.10.014023.  Google Scholar

[10]

T. Frenzel and M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDP-convergence, Discr. Cont. Dynam. Systems Ser. S, (2020), Online first doi: 10.3934/dcdss.2020345.  Google Scholar

[11]

G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Archive for Rational Mechanics and Analysis, 180 (2006), 183-236.  doi: 10.1007/s00205-005-0400-7.  Google Scholar

[12]

A. Kirch, A. Fischer, M. Liero, J. Fuhrmann, A. Glitzky and S. Reineke, Experimental proof of {J}oule heating-induced switched-back regions in OLEDs, Light: Science & Applications, 9 (2020), 5. doi: 10.1038/s41377-019-0236-9.  Google Scholar

[13]

P. Kordt, J. J. M. van der Holst, M. Al Helwi, W. Kowalsky, F. May, A. Badinski, C. Lennartz and D. Andrienko, Modeling of organic light emitting diodes: From molecular to device properties, Adv. Func. Mater., 25 (2015), 1955-1971.  doi: 10.1002/adfm.201403004.  Google Scholar

[14]

O. Kováčik and J. Rákosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618.   Google Scholar

[15]

M. Liero, J. Fuhrmann, A. Glitzky, T. Koprucki, A. Fischer and S. Reineke, 3{D} electrothermal simulations of organic LEDs showing negative differential resistance, in Opt. Quantum Electron., 49 (2017), 330/1–330/8. doi: 10.1109/NUSOD.2017.8010013.  Google Scholar

[16]

M. Liero, T. Koprucki, A. Fischer, R. Scholz and A. Glitzky, $p$-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices, Z. Angew. Math. Phys., 66 (2015), 2957-2977.  doi: 10.1007/s00033-015-0560-8.  Google Scholar

[17]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM Journal on Mathematical Analysis, 39 (2007), 687-720.  doi: 10.1137/060665452.  Google Scholar

[18]

K. Schmidt and S. Tordeux, Asymptotic modelling of conductive thin sheets, Zeitschrift für Angewandte Mathematik und Physik, 61 (2010), 603–626. doi: 10.1007/s00033-009-0043-x.  Google Scholar

Figure 1.  Sketch of the domain $ \Omega_ \varepsilon $ consisting of the glass substrate $ \Omega^\mathrm{sub} $ and the OLED $ \Omega_ \varepsilon^\mathrm{oled} $. The latter consists of $ N $ layers (with $ N = 5 $ in the figure). The bottom and top layer $ \Omega_ \varepsilon^1 $ and $ \Omega_ \varepsilon^N $ describe the electrodes with Dirichlet boundaries $ \Gamma_ \varepsilon^- $ and $ \Gamma_ \varepsilon^+ $ (green) for the potential where the voltage is applied. In the effective limit, the current-flow equation reduces to coupled equations on the two-dimensional domain $ \Gamma_0 $ (red) and the heat equation is solved only in $ \Omega^\mathrm{sub} $ with an additional boundary source term on $ \Gamma_0 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004
[2]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
[3]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024
[4]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190
[5]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
[6]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[7]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003
[8]

Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037
[9]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021041
[10]

Qian Cao, Yongli Cai, Yong Luo. Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021095
[11]

Shoufeng Ji, Jinhuan Tang, Minghe Sun, Rongjuan Luo. Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021051
[12]

Sumon Sarkar, Bibhas C. Giri. Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021048
[13]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[14]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008
[15]

Tao Wang. Variational relations for metric mean dimension and rate distortion dimension. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021050
[16]

Thomas Barthelmé, Andrey Gogolev. Centralizers of partially hyperbolic diffeomorphisms in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021044
[17]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[18]

Fuzhi Li, Dongmei Xu. Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3517-3542. doi: 10.3934/dcdsb.2020244
[19]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365
[20]

Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294

2019 Impact Factor: 1.233

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences