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August  2017, 10(4): 745-771. doi: 10.3934/dcdss.2017038

## A survey on second order free boundary value problems modelling MEMS with general permittivity profile

 Institute of Applied Mathematics, Leibniz Universität Hannover, Germany

* Corresponding author: Christina Lienstromberg

Received  July 2016 Revised  October 2016 Published  April 2017

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi-or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles.

The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account -the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

Citation: Joachim Escher, Christina Lienstromberg. A survey on second order free boundary value problems modelling MEMS with general permittivity profile. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 745-771. doi: 10.3934/dcdss.2017038
##### References:
 [1] M. G. Allen and A. B. Frazier, High aspect ratio electroplated microstructures using a photosensitive polyimide process, Proc. IEEE Micro Electro Mechanical Syst., (1992), 87-92. [2] H. Amann and J. Escher, Analysis Ⅲ, Birkhäuser, Basel, 2008. [3] F. Baillieu, J. P. Berry, P. Caillat, B. Diem, J. Ph. Ebersohl, F. Le Hung, S. Renard, P. Rey and L. Zimmermann, Airbag application: A microsystem including a silicon capacitive accelerometer, CMOS switched capacitor electronics and true self-test capability, Sensors and Actuators A, 46/, 47 (1995), 190-195. [4] D. Braess, Finite Elements: Theory, Fast Solvers and Applications in Elasticity Theory, Cambridge University Press, Cambridge, 2007. [5] S. Büttgenbach, A. Burisch and J. Hesselbach, Design and Manufacturing of Active Microsystems, Springer, Berlin, 2011. [6] B. Erdem Alaca, M. T. A. Saif and H. Sehitoglu, Analytical modeling of electrostatic membrane actuator for micro pumps, IEEE Journal of Microelectromechanical Systems, 8 (1999), 335-345. [7] J. Escher, P. Gosselet and C. Lienstromberg, A note on model reduction for microelectromechanical systems, Nonlinearity, 30 (2017), 454-465. [8] J. Escher, P. Laurençot and C. Walker, Finite time singularity in a free boundary problem modeling MEMS, C. R. Acad. Sci. Paris, Ser. I, 351 (2013), 807-812. [9] J. Escher, P. Laurençot and C. Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Rational Mech. Anal., 211 (2014), 389-417. [10] J. Escher, P. Laurençot and C. Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS, Trans. Amer. Math. Soc., 367 (2015), 5693-5719.  doi: 10.1090/S0002-9947-2014-06320-4. [11] J. Escher and C. Lienstromberg, A qualitative analysis of solutions to microelectromechanical systems with curvature and nonlinear permittivity profiles, Commun. Part. Diff. Eq., 41 (2016), 134-149. [12] J. Escher and C. Lienstromberg, Finite-time singularities of solutions to microelectromechanical systems with general permittivity, Submitted, 2015. [13] P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal., 15 (2008), 341-354. [14] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, volume 20 of Courant Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, New York, 2010. [15] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 2010. [16] G. Flores, G. Mercado, J. A. Pelesko and N. Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math., 67 (2007), 434-446. [17] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅱ: Dynamic case, Nonlinear Diff. Eqns. Appl., 15 (2008), 115-145. [18] Y. Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Diff. Eqns., 245 (2008), 809-844. [19] Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., 66 (2005), 309-338. [20] J. -S. Guo and P. Souplet, No touchdown at zero points of the permittivity profile for the MEMS problem, SIAM J. Math. Anal., 47 (2014), 614-625. [21] A. Henrot and M. Pierre, Variation et Optimisation de Formes: une Analyse Géométrique, Springer, Berlin, 2005. [22] K. M. Hui, The existence and dynamic properties of a parabolic nonlocal MEMS equation, Nonlinear Analysis: TMA, 74 (2011), 298-316. [23] O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, Rhode Island, 1968. [24] P. Laurençot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS, Arch. Rational Mech. Anal., 207 (2013), 139-158. [25] P. Laurençot and C. Walker, A fourth-order model for MEMS with clamped boundary conditions, Proc. Lond. Math. Soc., 109 (2014), 1435-1464. [26] P. Laurençot and C. Walker, A free boundary problem modeling electrostatic MEMS: Ⅰ. Linear bending effects, Math. Ann., 316 (2014), 307-349. [27] P. Laurençot and C. Walker, A free boundary problem modeling electrostatic MEMS: Ⅱ. Nonlinear bending effects, Math. Models Methods Appl. Sci., 24 (2014), 2549-2568. [28] P. Laurençot and C. Walker, A time singular limit for a fourth-order damped wave equation for MEMS, In: Springer Proceedings in Mathematics & Statistics, 233-246, Springer, Berlin, 2015. [29] P. Laurençot and C. Walker, A variational approach to a stationary free boundary problem modeling MEMS, ESAIM Control Optim. Calc. Var. , to appear 2015. [30] P. Laurençot and C. Walker, On a three-dimensional free boundary problem modeling electrostatic MEMS, Submitted, 2015. [31] J. Li and C. Liang, Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS, Discrete Contin. Dyn. Syst., 36 (2016), 833-849. [32] C. Lienstromberg, A free boundary value problem modelling microelectromechanical systems with general permittivity, Nonlinear Analysis: RWA, 25 (2015), 190-218. [33] C. Lienstromberg, On qualitative properties of solutions to microelectromechanical systems with general permittivity, Monatsh. Math., 179 (2016), 581-602. [34] C. Lienstromberg, Well-posedness of a quasilinear evolution problem modelling MEMS with general permittivity, submitted, 2016. [35] J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908. [36] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, Florida, 2003. [37] J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, Journal of Engineering Mathematics, 41 (2001), 345-366. [38] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser, Basel, 2007. [39] J. I. Seeger and B. E. Boser, Dynamics and control of parallel-plate actuators beyond the electrostatic instability, Transducers '99, The 10th International Conference on Solid-State Sensors and Actuators, (1999), 444-447.

show all references

##### References:
 [1] M. G. Allen and A. B. Frazier, High aspect ratio electroplated microstructures using a photosensitive polyimide process, Proc. IEEE Micro Electro Mechanical Syst., (1992), 87-92. [2] H. Amann and J. Escher, Analysis Ⅲ, Birkhäuser, Basel, 2008. [3] F. Baillieu, J. P. Berry, P. Caillat, B. Diem, J. Ph. Ebersohl, F. Le Hung, S. Renard, P. Rey and L. Zimmermann, Airbag application: A microsystem including a silicon capacitive accelerometer, CMOS switched capacitor electronics and true self-test capability, Sensors and Actuators A, 46/, 47 (1995), 190-195. [4] D. Braess, Finite Elements: Theory, Fast Solvers and Applications in Elasticity Theory, Cambridge University Press, Cambridge, 2007. [5] S. Büttgenbach, A. Burisch and J. Hesselbach, Design and Manufacturing of Active Microsystems, Springer, Berlin, 2011. [6] B. Erdem Alaca, M. T. A. Saif and H. Sehitoglu, Analytical modeling of electrostatic membrane actuator for micro pumps, IEEE Journal of Microelectromechanical Systems, 8 (1999), 335-345. [7] J. Escher, P. Gosselet and C. Lienstromberg, A note on model reduction for microelectromechanical systems, Nonlinearity, 30 (2017), 454-465. [8] J. Escher, P. Laurençot and C. Walker, Finite time singularity in a free boundary problem modeling MEMS, C. R. Acad. Sci. Paris, Ser. I, 351 (2013), 807-812. [9] J. Escher, P. Laurençot and C. Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Rational Mech. Anal., 211 (2014), 389-417. [10] J. Escher, P. Laurençot and C. Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS, Trans. Amer. Math. Soc., 367 (2015), 5693-5719.  doi: 10.1090/S0002-9947-2014-06320-4. [11] J. Escher and C. Lienstromberg, A qualitative analysis of solutions to microelectromechanical systems with curvature and nonlinear permittivity profiles, Commun. Part. Diff. Eq., 41 (2016), 134-149. [12] J. Escher and C. Lienstromberg, Finite-time singularities of solutions to microelectromechanical systems with general permittivity, Submitted, 2015. [13] P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal., 15 (2008), 341-354. [14] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, volume 20 of Courant Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, New York, 2010. [15] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 2010. [16] G. Flores, G. Mercado, J. A. Pelesko and N. Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math., 67 (2007), 434-446. [17] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices Ⅱ: Dynamic case, Nonlinear Diff. Eqns. Appl., 15 (2008), 115-145. [18] Y. Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Diff. Eqns., 245 (2008), 809-844. [19] Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., 66 (2005), 309-338. [20] J. -S. Guo and P. Souplet, No touchdown at zero points of the permittivity profile for the MEMS problem, SIAM J. Math. Anal., 47 (2014), 614-625. [21] A. Henrot and M. Pierre, Variation et Optimisation de Formes: une Analyse Géométrique, Springer, Berlin, 2005. [22] K. M. Hui, The existence and dynamic properties of a parabolic nonlocal MEMS equation, Nonlinear Analysis: TMA, 74 (2011), 298-316. [23] O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, Rhode Island, 1968. [24] P. Laurençot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS, Arch. Rational Mech. Anal., 207 (2013), 139-158. [25] P. Laurençot and C. Walker, A fourth-order model for MEMS with clamped boundary conditions, Proc. Lond. Math. Soc., 109 (2014), 1435-1464. [26] P. Laurençot and C. Walker, A free boundary problem modeling electrostatic MEMS: Ⅰ. Linear bending effects, Math. Ann., 316 (2014), 307-349. [27] P. Laurençot and C. Walker, A free boundary problem modeling electrostatic MEMS: Ⅱ. Nonlinear bending effects, Math. Models Methods Appl. Sci., 24 (2014), 2549-2568. [28] P. Laurençot and C. Walker, A time singular limit for a fourth-order damped wave equation for MEMS, In: Springer Proceedings in Mathematics & Statistics, 233-246, Springer, Berlin, 2015. [29] P. Laurençot and C. Walker, A variational approach to a stationary free boundary problem modeling MEMS, ESAIM Control Optim. Calc. Var. , to appear 2015. [30] P. Laurençot and C. Walker, On a three-dimensional free boundary problem modeling electrostatic MEMS, Submitted, 2015. [31] J. Li and C. Liang, Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS, Discrete Contin. Dyn. Syst., 36 (2016), 833-849. [32] C. Lienstromberg, A free boundary value problem modelling microelectromechanical systems with general permittivity, Nonlinear Analysis: RWA, 25 (2015), 190-218. [33] C. Lienstromberg, On qualitative properties of solutions to microelectromechanical systems with general permittivity, Monatsh. Math., 179 (2016), 581-602. [34] C. Lienstromberg, Well-posedness of a quasilinear evolution problem modelling MEMS with general permittivity, submitted, 2016. [35] J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908. [36] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, Florida, 2003. [37] J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, Journal of Engineering Mathematics, 41 (2001), 345-366. [38] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser, Basel, 2007. [39] J. I. Seeger and B. E. Boser, Dynamics and control of parallel-plate actuators beyond the electrostatic instability, Transducers '99, The 10th International Conference on Solid-State Sensors and Actuators, (1999), 444-447.
Sketch of the investigated idealised MEMS device
Cross section of the investigated idealised MEMS device
Membrane's deflection $u$ of the coupled system for $p(x)=x^8 + 0.1$ with $u_{\ast} \equiv 0$, $\lambda = 1$, and $\varepsilon \in \{0.4, 0.6\}$
Membrane's deflection $u$ of the coupled system for $p(x)=x^8 + 0.1$ with $u_{\ast} \equiv 0$, $\lambda = 1$, and $\varepsilon \in \{0.1, 0.2\}$
Membrane's deflection $u$ of the coupled system for $p(x)=x^8 + 0.1$ with $u_{\ast} \equiv 0$, $\lambda = 1$, and $\varepsilon = 0.15$
Approximate solution $u$ to the small-aspect ratio model with $u_{\ast} \equiv 0$ for $p(x)=x^8+0.1$ and $\lambda = 1$
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