A 2D Non-Linear Second-Order Differential Model for Electrostatic Circular Membrane MEMS Devices: A Result of Existence and Uniqueness

In recent years, a growing demand for embedded engineering applications has convinced researchers to develop low cost micro- to nano-sized components, in which actuators and transducers play important roles. This presents a link between the physical nature of the problem under study and the need to use machine languages to manage interfaces with other devices [1]. The interest of the scientific community in MEMS (Micro-Electro-Mechanical Systems) technology, which was born when Nathanson et al. produced the first batch device in 1964 [1,2], has been growing exponentially. At present, MEMS technology is fully part of the multi-disciplinary field of mathematical physics, allowing for highly varied engineering applications [1,3]. This is mainly due to the fact that it has been supported by sophisticated theoretical models, both static and dynamic [4,5]. However, even if these models appear to adhere to reality, they are often structured in an implicit form that does not provide explicit solutions (except in particular cases), for which numerical solutions must be necessarily sought [6], or analytical conditions, which ensure the existence, uniqueness, and regularity (up to the desired order) of the solution must be derived [7,8]. Analytical and numerical approaches can be used to obtain numerical solutions that do not represent ghost solutions [9,10,11]. In the field of MEMS technologies, the scientific community has been actively engaged both in the development of theoretical models and in technology transfer. In particular, some important models for coupled problems have been developed: ranging from magnetically actuated systems [1,12,13] to thermo-elastic models [14], and from electro-elastic models [15] to micro-fluid models [16], featuring highly complex formulations for the management of different MEMS devices (with plates, membranes, and so on). These theoretical models have had excellent feedback in technology transfer, through the design, realization, and distribution of the MEMS devices used in the various applications [17]. Biomedical diagnostics increasingly require efficient, low-cost, and reliable micro-components, which can be of help to health-care personnel both in on-line and off-line modes [18,19]. Many mathematical models have been theoretically conceived of in special functional spaces, in order to provide the conditions of the existence and uniqueness of the solutions, which are otherwise difficult to detect [20,21]. Cassani et al. [22] built a sophisticated mathematical model of a MEMS device constituted of two metallic plates (one fixed and one deformable), which was clumped at the edges and subjected to a drop voltage, which deformed the deformable plate towards the other one. Obviously, determining a solution for this model is very difficult, and some simplifications were necessarily required. In particular, neglecting the inertial and non-local effects, Cassani et al. obtained a simplified model in [23], which was studied using Steklov boundary conditions to achieve Dirichlet and Navier boundary conditions. Starting from this simplified model, the authors studied a new elliptical semi-linear dimensionless model for a 1D membrane MEMS, based on the proportionality between the electric field magnitude $\left|\mathbf{E}\right|$ and the curvature of the membrane, achieving results of the existence and uniqueness for the solution. In particular, in [7], the algebraic condition ensuring the uniqueness of the solution did not depend on the electro-mechanical properties of the material constituting the membrane. Consequently, in [10], the authors achieved a new algebraic condition depending on these properties, ensuring the uniqueness of the solution.

In this work, for applicative reasons, the authors focus their attention on a 2D circular membrane MEMS device, which is useful in several industrial and/or biomedical applications [1,19]. In addition, the authors consider $\left|\mathbf{E}\right|$ to be proportional to the mean curvature of the membrane of the device, in order to achieve a non-linear second-order differential model with singularity in the steady-state case. Finally, an algebraic condition, depending on both the mechanical and electrical properties of the membrane, guarantees the existence of at least one solution to the proposed model. However, the uniqueness of the solution is not ensured.

The paper is structured as follows. After presenting how the proposed model was achieved in Section 2, Section 3 describes the 2D circular membrane MEMS from two points of view: as an actuator and as a transducer. In the latter case, starting from the theory of the elasticity of the plates, the most important elasticity formulas of the membrane are presented, which will be useful in the following. Moreover, an important link between mechanical pressure and electrostatic pressure is detailed for our purposes. The formulation of the problem under study is detailed in Section 4. From the geometry of the problem, the model is formulated in terms of the mean curvature of the membrane, obtaining a non-linear second-order differential equation with singularity (Section 5). Once the problem under study is formulated in a general manner in Section 6, and after presenting some preliminary Lemmas in Section 7, a result of the existence of at least one solution to the problem (5) is detailed in Section 8. However, although the existence of at least one solution is ensured, uniqueness is not guaranteed, as is proven by means of two different approaches in Section 9. Finally, some conclusions and perspectives complete this work.

As mentioned above, Cassani et al. [22] proposed a detailed mathematical model of a MEMS device consisting of two metallic plates, where one is fixed (the lower plate) and the other is deformable (the upper plate), but anchored at the boundary of a region $\Omega \in {\mathbb{R}}^N$ and subjected to a drop voltage, which deforms the lower plate (at $u=0$) towards the upper plate (at $u=1$). This steady-state model assumes the following structure [22]: where $f_1$ is a bounded function depending on the dielectric properties of the material constituting the deformable plate and the positive parameters $\alpha$, $\beta$, $\gamma$, and $\chi$ are related to the mechanical and electrical properties of the system. In addition, ${\sigma }_1$ is the Coulomb exponent, such that ${\sigma }_1\ge 2$ takes into account higher order Coulombian behaviors. Obviously, obtaining a solution to the problem (1) is a difficult task, as the choice of boundary conditions is an extremely delicate task [22,23]. From the model (1), by neglecting the inertial and non-local effects and setting $\alpha =1$ and $\beta =\gamma =0$, one obtains the following simplified model:

Model (2) has also been studied by Cassani et al. [23], in which the existence of a solution was studied, using Steklov boundary conditions, to obtain Dirichlet and Navier boundary conditions when $\widehat{d}=0$ or $\widehat{d}=+\infty$. In the past, starting from (2), the authors studied the following new elliptical semi-linear dimensionless model for a 1D membrane MEMS: where the membrane replaces the deformable lower plate and $\left|\mathbf{E}\right|$ is considered to be proportional to the curvature, K, of the membrane. Interesting results of the existence and uniqueness for the solution to the problem (3) were achieved in [7]. In particular, the uniqueness condition was independent of the electro-mechanical properties of the material constituting the membrane; thus, in [10], the authors obtained a new condition of the uniqueness for the solution to Problem (3), which took into account these properties. Moreover, Model (3) was solved numerically (using the shooting approach) in both [9,24], highlighting the range of $\theta {\lambda }^2$, which ensured convergence of the numerical procedure in the absence of ghost solutions. Finally, in [25], the shooting procedure and the Keller-box scheme were compared to achieve an optimal range of $\theta {\lambda }^2$ without ghost solutions. It is worth noting that $\theta$ takes into account the applied voltage V and that ${\lambda }^2$ takes into account the electro-mechanical properties of the material constituting the membrane [9,10].

In this paper, we focus our attention on 2D circular membrane MEMS actuators, which are useful in industrial and/or biomedical applications [1,19]. First, we observe an axial symmetry in the geometry of the membrane. Then, considering the z axis as the rotation axis, the profile u of the circular membrane can be considered as the surface generated by rotating a curve C around z on the vertical plane $rz$ located in the first quadrant, with $0\le r\le R$. Then, u is only dependent on the radial co-ordinate r, such that the 2D problem here can be considered as a 1D problem in which the independent variable x is replaced by r. Then, only considering the radial part of the Laplace operator, the model (2) can be written as follows: where ${\lambda }^2\left(r\right)={\lambda }_1\left(r\right)f_1\left(r\right)$ and $1/r$ is a singularity. In addition, in (4), $\left|\mathbf{E}\right|$ is represented by $\frac{\lambda \left(r\right)}{(1-u(r\left)\right)}$, as in [7]. Then, we consider $\left|\mathbf{E}\right|$ to be proportional to the curvature of the membrane itself (with $u\left(r\right)\in C^2(\Omega )$). Finally, we exploit the expression of mean curvature to achieve, from Model (4), the following model: where $d^{*}$ is the critical security distance, which ensures that the deflection of the membrane does not produce contact between the membrane itself and the fixed upper plate. Then, we prove a theorem of the existence of the solution to the model (5). However, as we will prove below, the uniqueness of the solution to the problem (5) is not guaranteed.

Let us consider a surface S generated by rotating, around the vertical axis z, a curve C located in a plane orthogonal to the $xy$ plane forming an angle t with the $zx$ plane (see Figure 1) [27]. To simplify our calculations, we suppose that C is parametrized with a generic parameter r differing from the curvilinear co-ordinate s, such that $P\left(r\right)=(\overline{f}\left(r\right),0,\overline{g}\left(r\right))$, $r\in I\subset \mathbb{R}$, where $\overline{f}\left(r\right)$ and $\overline{g}\left(r\right)$ are regular functions satisfying: for each $r\in I=[0,R]$. The parametrization of the surface S is then: we observe that $P\left(r\right)$, as a natural parametrization, ensures that C is regular everywhere, and so, by rotation, S is regular. Then, we easily obtain:

Thus, the coefficients of the first fundamental form are:

As $\mathrm{F}=0$ everywhere, the co-ordinate lines are everywhere orthogonal. Then, we can write:

Finally, we have: and the unit normal vector to S in $P(t,r)$ is:

Then, the coefficients of the second fundamental form are:

To achieve the principal curvatures ${\overline{k}}_1(t,r)$ and ${\overline{k}}_2(t,r)$, it is sufficient to solve the following algebraic equation: from which we can easily obtain:

Thus, the mean curvature $H(t,r)$ becomes:

In our case, assuming that C lies in the plane $y=0$, we set: where $\frac{d\overline{f}\left(r\right)}{dr}$ and $\frac{d\overline{g}\left(r\right)}{dr}$ satisfy (20). Thus,

Taking into account (32), the model (19) can be written as follows: we observe that, in (33), $\frac{d^{2}u\left(r\right)}{d{r}^2}+\frac{1}{r}\frac{du\left(r\right)}{dr}\ne 0$ if $u\left(r\right)\ne 0$. In fact, let us suppose that: then, the equation of Model (34) admits the trivial solution $u\left(r\right)=0$, which cannot be considered a solution to the problem (34). In fact, let us suppose absurdly that a solution to the model (33) also satisfies $\frac{dy\left(r\right)}{dr}+\frac{1}{r}y\left(r\right)=0;$ from which $\frac{dy\left(r\right)}{dr}=-\frac{1}{r}y\left(r\right)$, and again, $y=y\left(r\right)=-e^{lnr}=c_2{\left(\frac{1}{e}\right)}^{lnr}=\frac{c_2}{r}$. Taking into account that $y\left(r\right)=\frac{du\left(r\right)}{dr}$, we easily achieve: which is a solution for the equation of the model (33) that also satisfies the equation of the model (34). However, although both Equations (33) and (34) are satisfied by the solution (35), this does not satisfy the condition $\frac{du\left(0\right)}{dr}=0$. Then, in (34), $\frac{d^{2}u\left(r\right)}{d{r}^2}+\frac{1}{r}\frac{du\left(r\right)}{dr}\ne 0$. Thus, we can divide both sides of the differential equation of Model (33) by $\frac{d^{2}u\left(r\right)}{d{r}^2}+\frac{1}{r}\frac{du\left(r\right)}{dr}$, obtaining:

Finally, Problem (33) can be written as Problem (5).

Model (5) can be considered as a special case of the following general problem. Specifically, let us consider a closed interval $\Omega =[a,b]$ in which a singularity takes place at a. Let us consider $u\left(r\right):(a,b)\to \mathbb{R}$, such that $u\left(r\right)\in C^2\left(\overline{\Omega }\right)$. Then, (5) is a particular case of the general following model: where $\overline{F}\in C^0((a,b]\times \mathbb{R}\times \mathbb{R})$ and $B,m\in \mathbb{R}$. If we set:

$B=0$, and $m=0$, we obtain the model (5). We focus our attention on achieving conditions that ensure both the existence and uniqueness of the solution for Model (5). Section 8 describes a very interesting existence result for Problem (5). For this purpose, let us introduce some preliminary Lemmas.

We first present the following well known result, which will be exploited to prove the existence of a solution to the problem (5).

For details, see [28]. Furthermore, we will use the following Lemmas to prove that the uniqueness of the solution to the problem (5) is not ensured.

Again, see [28]. Before we present Lemma 3, we need to introduce the following definition:

We now reveal and prove our principal goal, regarding the existence of at least one solution for the problem (5).