Electrostatic Circular Membrane MEMS: An Approach to the Optimal Control

In the recent years, micro-electro-mechanical systems (MEMS) with circular membrane have come to be increasingly exploited in various fields, such as thermo-elasticity [1,2,3], microfluidics [4,5], electroelasticity [6,7,8,9,10,11], and, of course, biomedical applications [12,13,14]. This is mainly due to the ease of construction of these devices as well as the versatility of their usage [15,16]. In the last decade, physical-mathematical modeling has produced highly sophisticated models handling these devices under the most varied operating conditions [17,18,19]. However, these models do not provide explicit solutions, so that we must be content with obtaining analytical and/or algebraic conditions ensuring the existence, uniqueness, and regularity of the solutions, so that any solutions obtained from any numerical procedures do not represent ghost solutions (i.e., numerical solutions that do not satisfy the conditions of existence, uniqueness) [2,20,21,22,23,24]. In [21], in the 1D-MEMS membranes framework, a formalization of the problem of the existence and uniqueness of the solution related to the profile of the membrane of an electrostatic actuator in steady conditions has been proposed, putting forth an innovative differential semilinear elliptic model. Once V is applied, the membrane deforms, as the electrostatic field $\mathbf{E}$ in the device creates an electrostatic pressure $p_{el}$ generating a mechanical pressure p, which acts on the membrane. Being $\mathbf{E}$ locally orthogonal to the membrane tangent straight line [25], $\left|\mathbf{E}\right|$ depends locally on the distance between the membrane and the upper disk. Starting from the fact that $\left|\mathbf{E}\right|$ can be locally considered to be proportional to the curvature of the membrane and, for this model, the result of existence was obtained by means of the Schauder–Tychonoff’s fixed point theorem establishing, subsequently, conditions of uniqueness exploiting both Poicaré inequality and Gronwall lemma. If, on the one hand, this work presented the merit of the innovativeness of the proposed approach, on the other hand, it highlighted the defect of offering a condition of uniqueness independent of the electromechanical properties of the material constituting the membrane. This represented an evident defect, since it is evident that the electromechanical behavior of the device is strongly influenced by the intrinsic properties of the material constituting the membrane. Subsequently, this drawback has been referred to in [22], where the condition of uniqueness obtained depended on the electromechanical properties of the material constituting the membrane. Furthermore, in [23], the elliptic semilinear model that was proposed in [21,22] has been numerically studied by means of shooting techniques, reconstructing the profile of the membrane in different operating conditions and in compliance with the conditions of existence and uniqueness achieved in [21,22], respectively obtaining numerical solutions not representing ghost solutions. A 2D generalization with circular symmetry was proposed in [20] in which $\left|\mathbf{E}\right|$ can be considered to be proportional to the mean curvature $H\left(r\right)$ of the membrane (with r radial coordinate). The analytical model obtained is a non-linear ordinary second-order differential equation with radial symmetry, which, in dimensionless conditions, presents a singularity $1/r$, where the independent variable is the profile of the membrane $u\left(r\right)$ [20]. For this model, an algebraic condition of existence of the solution obtained by highlighting the fact that the uniqueness of the solution is not guaranteed has been proposed. Moreover, only in 1D framework, the stability and relative equilibrium positions have been studied (see [2]), while no indications were given regarding the presence of possibly stable equilibrium positions. In the present paper, we focus our attention on the 2D electrostatic circular membrane MEMS device that was studied in [20], with the aim of studying the critical point and stability (important, as they avoid the risk of any instability that could cause contact of the membrane with the upper disk), also determining the range of possible values of the external electrical voltage applied while taking both the mechanical inertia of the membrane and the risk that the membrane accidentally touches the upper disk into account, because, as highlighted in [24], it has only been dealt with in 1D geometry. Finally, from an engineering point of view, it is advisable to obtain optimal control conditions starting from energy evaluations, which, at present, are an open problem for the model under study. The remainder of the paper is structured, as follows. After a brief description of the simplified geometry of the device under study, the pertinent scientific literature is analyzed in Section 2. Subsequently, in Section 3, the only equilibrium configuration obtained is studied by evaluating its stability. Section 4, then, analyzes the model in order to determine the range of possible values for V, while taking into account the fact that the inertia of the membrane must be overcome and the upper disk of the device cannot be reached during deformation. Finally, some interesting conditions pertaining to the optimal control of the device are achieved, starting with energy considerations (Section 5).

Let us consider the usual Euclidean space ${\mathbb{R}}^3$, and in it, let us consider a system of Cartesian axes $Oxyz$, where the z axis is the vertical one, as highlighted in Figure 1. The device studied in this work consists of two parallel and circular metal disks with radius R, placed at the distance d from each other. In particular, the lower disk is located on the $xy$ plane, so its center coincides with the origin of the Cartesian axis system. The device is subjected to an external electric voltage V in such a way that the lower disk is potential $V=0$. A membrane (whose radius is equal to the radius of the disks, if it is at rest), constrained to the edges of the lower disk, deforms towards the upper disk when V is applied. It is immediately apparent that the device, being circular, is endowed with radial symmetry, and therefore, by indicating the radial coordinate with r, $0<r\le R$, the profile of the membrane can only be expressed as a function of r (i.e., $u\left(r\right)$). Obviously, V must be such that it can overcome the mechanical inertia of the membrane and the corresponding value of $\mathbf{E}$ inside the device must generate the following electrostatic pressure [1,25] where ${ϵ}_0$ is the permittivity of the free space. $p_{el}$ translates into an equivalent electrostatic force $f_{el}$, as given by [1,25] so that a corresponding mechanical pressure p deflects the membrane towards the upper disk, obtaining a displacement in the center of the membrane, $u_0$, equal to $\frac{R^2}{4T}p$ [1,26], where T is the radial mechanical tension of the membrane. Because the membrane must not touch the upper disk for obvious electrostatic reasons, it is necessary to keep a critical security distance, $d^{*}$, such that $u\left(0\right)=u_0\le d-d^{*}$, which, in dimensionless conditions, becomes $u\left(0\right)=u_0\le 1-d^{*}$ [1,25].

Historically, the first complete mathematical model of an electrostatic MEMS device was presented in [28], whose device consisted of two metal plates, one fixed and the other deformable (but anchored to the edges of a region $\mathsf{\Omega }\in {\mathbb{R}}^3$ and subject to drop voltage, which would deform the lower plate). This model, which assumed the following structure ($f_1$, bounded function, depending on the dielectric properties of the material constituting the deformable plate; $\alpha$, $\beta$, $\gamma$, $\chi$, parameters that are related to the mechanical and electrical properties), although complete and exhaustive from a physical-mathematical point of view, presented certain difficulties in resolution, so that the solution was only obtainable if some simplifications were implemented [1]. Subsequently, neglecting the inertial and non-local effects, the following simplified model was obtained [28,29]:

(4) was studied using the Steklov boundary conditions in order to obtain Dirichlet and Navier boundary conditions when $\widehat{d}=0$ or $\widehat{d}=+\infty$. In [30], starting from (4), the following elliptical semi-linear dimensionless model for a one-dimensional (1D) membrane MEMS has been studied, where the deformable lower plate was replaced by a membrane and, moreover, $\left|\mathbf{E}\right|$ was considered to be proportional to the curvature K of the membrane [20,21,22,30,31]. In these works, the conditions of existence, uniqueness, and regularity of the solution were obtained, which allowed for the obtaining of non-ghost numerical solutions whose convergence was ensured by particular ranges of $\theta {\lambda }^2$ values [20,21,30]. We highlight that $\theta$ takes the applied voltage V into account and ${\lambda }^2$ takes the electro-mechanical properties of the material constituting the membrane into account [1]:

From (6), ${\lambda }^2\propto V^2$, so ${\lambda }^2$ is bounded below, because a minimum value of V is required to overcome the mechanical inertia of the membrane. On the other hand, V cannot indefinitely increase, because its maximum value is fixed by the intended use of the device. Therefore, ${\lambda }^2$ is also limited by the intended use of the device.

When considering a 2D circular membrane MEMS device, it follows that the profile u of the membrane only depends on the radial coordinate r, such that, only considering the radial part of the Laplace operator [31], and $\left|\mathbf{E}\right|$ in this case, is considered to be proportional to the mean curvature of the membrane (for detail, see [20]). Therefore, (4) was reformulated as follows [20]:

We note that (7), apparently, does not present the singularity, which characterizes (4). However, if in (7) we set $u\left(r\right)=1-d^{*}$, it follows that $\frac{du\left(r\right)}{dr}=0$ so that $\frac{d^{2}u\left(r\right)}{d{r}^2}=0$. But, being $\frac{d^{2}u\left(r\right)}{d{r}^2}=0=-\theta {\left|\mathbf{E}\right|}^2$, it follows that $\left|\mathbf{E}\right|=0$, and this occurrence would produce a linear deflection of the membrane that represents a physically unacceptable condition.

Model (7) has been studied both analitically and numerically, obtaining the following existence condition for the solution [20,30] in which k is the coefficient of proportionality between p and $p_{el}$. (8) ensures the absence of ghost solutions if numerical solutions that are achieved by any numerical approach exploited to recover the profile of the membrane satisfied it. It is worth noting that, from (8), the greater the external V applied the lower the value that is assigned to $\theta {\lambda }^2$ can be. Subsequently, from the equation of the model (7), $\frac{d^{2}u\left(r\right)}{d{r}^2}$ will be smaller. In other words, the higher the V, the more the membrane will be curved.

To achieve any critical points for the membrane whose profile is $u\left(r\right)$, it is necessary to transform the model (7) into the corresponding system of two ordinary differential equations of the first order in normal form [32]. In particular,

Therefore, to achieve any critical points, it will be sufficient to impose [32]

If the membrane of the device is at rest, then the distance between the membrane and the upper disk is d. Therefore, the electrostatic capacitance of the device is $C={ϵ}_0\frac{\pi R^2}{d}$, so that the potential energy of the device can be evaluated as

If the membrane deforms, $C={ϵ}_0{\int }_0^{\pi }Z\left(\varphi \right){\int }_{-R}^R\frac{dr}{d-u\left(r\right)}d\varphi$, and the final potential energy becomes where $Z\left(\varphi \right)$ is a bounded and continuous function, depending on $\varphi$. Therefore, the total variation of the potential energy, $\mathsf{\Delta }W=W_{final}-W_{initial}$, becomes

In addition, observing that $d-u\left(r\right)\ge d-d^{*}$, we can write $\frac{1}{d-u\left(r\right)}\le \frac{1}{d-d^{*}}$, so that (51) becomes

$Z\left(\varphi \right)$ being a bounded and continuous function, it follows that ${\int }_0^{\pi }Z\left(\varphi \right)d\varphi \le D$, where D is a positive constant. Subsequently, (52) can be written as

From (43), we can write $\sqrt{\frac{d^3{d^{*}}^{2}T}{4{ϵ}_0^2\theta k}}<V^2<\frac{2(1-d^{*})d^{*}}{k{ϵ}_0}$ so that, while taking (53) into account, it easily follows that

Conversely, from condition (35), $V^2>{\left(V_{min}\right)}_{inertia}^2=\sqrt{\frac{d^3{d}^{*}T}{4{ϵ}_0^2\theta k}}$, from which

Finally, combining (54) and (55), we easily obtain the range of the admissible values for $\mathsf{\Delta }W$: from which, with the usual values for the parameters, we achieve $1.8·{10}^{-8}\frac{\sqrt{T}}{\sqrt{k}}<\mathsf{\Delta }W<2.33·{10}^{-7}$. Figure 4 shows the area of possible values for $\mathsf{\Delta }W$, which corresponds to the area that is bounded above by the red straight line, while it is bounded below by the less concave blue curve. Obviously, this area shrinks as k increases. It is worth noting that, from an energy point of view, the fact that the area of possible values is limited (as shown in Figure 4) guarantees us that the membrane MEMS device under study is subject to variations in potential energy of limited amplitude reducing the risk of any damage. However, as seen above, $\mathsf{\Delta }W$ strictly depends on V and V is closely related to $\left|\mathbf{E}\right|$. Therefore, it is imperative to know the values of V and $\left|\mathbf{E}\right|$ maximizing $\mathsf{\Delta }W$. The following Subsections answer this question.