# American Institute of Mathematical Sciences

November  2021, 14(11): 3953-3971. doi: 10.3934/dcdss.2020460

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

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

* Corresponding author: Matthias Liero

Received  April 2020 Revised  September 2020 Published  November 2021 Early access  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.

Citation: Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 3953-3971. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] O. Kováčik and J. Rákosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] O. Kováčik and J. Rákosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618. [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. [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. [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. [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.
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 $\Gamma_0$
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