A free boundary problem for MEMS with curvature

# Dynamics of a free boundary problem with curvature modeling electrostatic MEMS

Joachim Escher Leibniz Universität Hannover
Institut für Angewandte Mathematik
Welfengarten 1
D–30167 Hannover
Germany
Philippe Laurençot Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse
F–31062 Toulouse Cedex 9, France
and  Christoph Walker Leibniz Universität Hannover
Institut für Angewandte Mathematik
Welfengarten 1
D–30167 Hannover
Germany
July 17, 2019
###### Abstract.

The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltages values. Furthermore, existence of asymptotically stable steady-state configurations is proved in case of small voltage values as well as non-existence of steady-states if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.

###### Key words and phrases:
MEMS, free boundary problem, curvature, well-posedness, asymptotic stability, small aspect ratio limit
###### 2010 Mathematics Subject Classification:
35R35, 35M33, 35Q74, 35B25, 74M05

## 1. Introduction

The focus of this paper is the analysis of a model describing the dynamics of an electrostatically actuated microelectromechanical system (MEMS) when the deformation of the devices are not assumed to be small. More precisely, consider an elastic plate held at potential and suspended above a fixed ground plate held at zero potential. The potential difference between the two plates generates a Coulomb force and causes a deformation of the membrane, thereby converting electrostatic energy into mechanical energy, a feature used in the design of several MEMS-based devices such as micropumps or microswitches [26]. An ubiquitous phenomenon observed in such devices is the so called “pull-in” instability: a threshold value of the applied voltage above which the elastic response of the membrane cannot balance the Coulomb force and the deformable membrane smashes into the fixed plate. Since this effect might either be useful or, in contrast, could damage the device, its understanding is of utmost practical importance and several mathematical models have been set up for its investigation [7, 21, 26].

In the following subsection we give a brief description of an idealized device as depicted in Figure 1, where the state of the device is characterized by the electrostatic potential in the region between the two plates and the deformation of the membrane which is not assumed to be small from the outset, cf. [5].

### 1.1. The Model

To derive the model for electrostatic MEMS with curvature we proceed similarly to [5, 7, 22]. We consider a rectangular thin elastic membrane that is suspended above a rigid plate. The -coordinate system is chosen such that the ground plate of dimension in -direction is located at , while the undeflected membrane with the same dimension in -direction is located at . The membrane is held fixed along the edges in -direction while the edges in -direction are free. Assuming homogeneity in -direction, the membrane may thus be considered as an elastic strip and the -direction is omitted in the sequel. The mechanical deflection of the membrane is caused by a voltage difference that is applied across the device. The membrane is held at potential while the rigid plate is grounded. We denote the deflection of the membrane at position and time by and the electrostatic potential at position and time by . We do not indicate the time variable for the time being. The electrostatic potential is harmonic, i.e.

 Δ^ψ=0in^Ω(^u) (1.1)

and satisfies the boundary conditions

 ^ψ(^x,−H)=0 ,^ψ(^x,^u(^x))=V ,^x∈(−L,L) , (1.2)

where

 ^Ω(^u):={(^x,^z);−L<^x

is the region between the ground plate and the membrane. The total energy of the system constitutes of the electric potential and the elastic energy and reads

 E(^u)=Ee(^u)+Es(^u) .

The electrostatic energy in dependence of the deflection is given by

 Ee(^u)=−ϵ02∫L−L∫^u(^x)−H|∇^ψ(^x,^z)|2d^zd^x

with being the permittivity of free space while the elastic energy only retains the contribution due to stretching (in particular, bending is neglected) and is proportional to the tension and to the change of surface area of the membrane, i.e.

 Es(^u)=T∫L−L(√1+(∂^x^u(^x))2−1)d^x .

Introducing the dimensionless variables

 x=^xL ,z=^zH ,u=^uH ,ψ=^ψV

and denoting the aspect ratio of the device by , we may write the total energy in these variables in the form

 E(u)=TL∫1−1(√1+ε2(∂xu(x))2−1)dx−ϵ0V22ε∫Ω(u)(ε2|∂xψ(x,z)|2+|∂zψ(x,z)|2)d(x,z) , (1.3)

with

 Ω(u):={(x,z)∈(−1,1)×(−1,∞) : −1

so that, formally, the corresponding Euler-Lagrange equations are

 0=ε2∂x(∂xu√1+ε2(∂xu)2)−ε2λ(ε2|∂xψ(x,u(x))|2+|∂zψ(x,u(x))|2) (1.4)

for , where we have set

 λ=ϵ0V2TLε3 .

We now take again time into account and derive the dynamics of the dimensionless deflection by means of Newton’s second law. Letting and denote the mass density per unit volume of the membrane and the membrane thickness, respectively, the sum over all forces equals . The elastic and electrostatic forces, given by the right hand side of equation (1.4), are combined with a damping force of the form being linearly proportional to the velocity. This yields

 ρδ∂2^tu+a∂^tu=ε2∂x(∂xu√1+ε2(∂xu)2)−ε2λ(ε2|∂xψ(x,u(x))|2+|∂zψ(x,u(x))|2) .

Finally, scaling time based on the strength of damping according to and setting , we derive for the dimensionless deflection the evolution problem

 γ2∂2tu+∂tu=∂x(∂xu√1+ε2(∂xu)2)−λ(ε2|∂xψ(x,u(x))|2+|∂zψ(x,u(x))|2) (1.5)

for and . Instead of considering this hyperbolic equation, however, we assume in this paper that viscous or damping forces dominate over inertial forces, i.e. we assume that and thus neglect the second order time derivative term in (1.5). The membrane displacement then evolves according to

 ∂tu−∂x(∂xu√1+ε2(∂xu)2)=−λ (ε2 |∂xψ(t,x,u(x))|2+|∂zψ(t,x,u(x))|2) , (1.6)

for and with clamped boundary conditions

 u(t,±1)=0 ,t>0 , (1.7)

and initial condition

 u(0,x)=u0(x) ,x∈I . (1.8)

In dimensionless variables, equations (1.1)-(1.2) read

 ε2∂2xψ+∂2zψ=0 ,(x,z)∈Ω(u(t)) ,t>0 , (1.9)

subject to the boundary conditions (extended continuously to the lateral boundary)

 ψ(t,x,z)=1+z1+u(t,x) ,(x,z)∈∂Ω(u(t)) ,t>0 . (1.10)

In the following we shall focus our attention on (1.6)-(1.10), its situation being depicted in Figure 1.

### 1.2. Simplified Models

Besides assuming that damping forces dominate over inertial forces and thus reducing equation (1.5) to (1.6), other simplifications of the model above have been considered as well in the literature. For instance, restricting attention to small deformations of the membrane yields a linearized stretching term in (1.6). The corresponding semilinear evolution problem with (1.6) being replaced by

 ∂tu−∂2xu=−λ (ε2 |∂xψ(t,x,u(x))|2+|∂zψ(t,x,u(x))|2) ,x∈I ,t>0 , (1.11)

is investigated in [6]. It is shown therein that the problem (1.7)-(1.11) is well-posed locally in time. Moreover, solutions exist globally for small voltage values while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, as the parameter approaches zero, the solutions are shown to converge toward the solutions of the so-called small aspect ratio model, see (1.13) below. Indeed, letting (and applying a potential with as suggested in [5]), one can solve (1.9)-(1.10) explicitly for the potential , that is,

 ψ0(t,x,z)=1+z1+u0(t,x) ,(t,x,z)∈[0,∞)×I×(−1,0) , (1.12)

and the displacement satisfies

 ∂tu0−∂2xu0=−λ(1+u0)2,x∈I,t∈(0,∞),u0(t,±1)=0,t∈(0,∞),u0(0,x)=u0(x),x∈I. (1.13)

In the limit , the free boundary problem is thus reduced to the singular semilinear heat equation (1.13) which has been studied thoroughly in recent years, see [7] for a survey as well as e.g. [8, 9, 13, 14, 15, 16, 18, 22, 25]. It is noteworthy to remark that the picture regarding pull-in voltage for the small aspect ratio model (1.13) is rather complete.

Let us point out that [6] is apparently the first mathematical analysis of the parabolic free boundary problem (1.7)-(1.11) while the corresponding elliptic (i.e. steady-state) free boundary problem is investigated in [20]. Moreover, we shall emphasize that the inclusion of non-small deformations is a feature of great physical relevance and, even though the results presented herein are reminiscent of the ones in [6], the quasilinear structure of (1.6) is by no means a trivial mathematical extension of (1.11).

### 1.3. Main Results

To state our findings on (1.6)-(1.10), we introduce the spaces

 W2αq,D(I):=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩{u∈W2αq(I);u(±1)=0} for 2α∈(1q,2] ,W2αq(I) for 0≤2α<1q . (1.14)

We shall prove the following result regarding local and global existence of solutions:

###### Theorem 1.1 (Well-Posedness).

Let , , and consider an initial value such that for . Then, the following are true:

• For each voltage value , there is a unique maximal solution to (1.6)-(1.10) on the maximal interval of existence in the sense that

 u∈C1([0,Tεm),Lq(I))∩C([0,Tεm),W2q,D(I))

satisfies (1.6)-(1.8) together with

 u(t,x)>−1 ,(t,x)∈[0,Tεm)×I ,

and solves (1.9)-(1.10) on for each .

• If for each there is such that and for , then the solution exists globally, that is, .

• If for , then for . If is even with respect to , then, for all , and are even with respect to as well.

• Given , there are and such that with for provided that and . In that case, enjoys the following additional regularity properties:

 u∈BUCρ([0,∞),W2−ρq,D(I))∩L∞([0,∞),W2q,D(I))

for some small.

Note that part (iv) of Theorem 1.1 provides uniform estimates on in the -norm and ensures that never touches down on -1, not even in infinite time. In contrast to the semilinear case considered in [6], the global existence result for the quasilinear equation (1.6) requires initially a small deformation, see also Remark 3.3 below. The proof of Theorem 1.1 is the content of Section 3. It is based on interpreting (1.6) as a abstract quasilinear Cauchy problem which allows us to employ the powerful theory of evolution operators developed in [4]. Let us emphasize at this point that the regularity properties of the right-hand side of equation (1.6) established in [6] are not sufficient to handle the quasilinear character of the curvature operator and we consequently have to derive Lipschitz properties of the right-hand side of (1.6) in weaker topologies than in [6]. This is the purpose of Section 2.

Regarding existence and asymptotic stability of steady-state solutions to (1.6)-(1.10) we have a similar result as in [20, Thm. 1] and [6, Thm.1.3].

###### Theorem 1.2 (Asymptotically Stable Steady-State Solutions).

Let and .

• Let . There are and an analytic function

 [λ↦Uλ]:[0,δ)→W2q,D(I)

such that is the unique steady-state to (1.6)-(1.10) satisfying with and when . The steady-state possesses the additional regularity

 Uλ∈C2+α([−1,1]) ,Ψλ∈W22(Ω(Uλ))∩C(¯¯¯¯¯¯¯¯¯¯¯¯¯¯Ω(Uλ))∩C2+α(Ω(Uλ)∪Γ(Uλ)) , (1.15)

where is arbitrary and denotes the boundary of without corners. Moreover, is negative, convex, and even with and is even with respect to .

• Let . There are such that for each initial value with , (1.6)-(1.10) has a unique global solution with

 u∈C1([0,∞),Lq(I))∩C([0,∞),W2q,D(I)) ,ψ(t)∈W22(Ω(u(t))) ,t≥0 ,

and

 u(t,x)>−1 ,(t,x)∈[0,∞)×I .

Moreover,

 ∥u(t)−Uλ∥W2q,D(I)+∥∂tu(t)∥Lq(I)≤Re−ω0t∥u0−Uλ∥W2q,D(I) ,t≥0 . (1.16)

Part (ii) of Theorem 1.2 shows local exponential stability of the steady-states derived in part (i). We also point out that the potential converges exponentially to in the -norm as , see Remark 4.1 for a precise statement. The proof of Theorem 1.2 is given in Section 4 and relies on the Implicit Function Theorem for part (i) and the Principle of Linearized Stability for part (ii).

Clearly, Theorem 1.2 is just a local result with respect to values. However, we next show that there is an upper threshold for above which no steady-state solution exists. This is expected on physical grounds and is related to the “pull-in” instability already mentioned in the introduction.

###### Theorem 1.3 (Non-Existence of Steady-State Solutions).

Let and . There is such that, if , then there is no steady-state solution to (1.6)-(1.10) such that , , and for . In addition, as .

Similar results have already been obtained for related models, including the small aspect ratio model [5, 7] and for the stationary free boundary problem corresponding to (1.7)-(1.11), see [20]. The proof of Theorem 1.3 relies on a lower bound on established in the latter paper and is given in Section 5.

The final issue we address is the connection between the free boundary problem (1.6)-(1.10) and the small aspect ratio limit (1.13). More precisely, we show the following convergence result:

###### Theorem 1.4 (Small Aspect Ratio Limit).

Let , , and let with for . For we denote the unique solution to (1.6)-(1.10) on the maximal interval of existence by . There are , , and depending only on and such that with and for all . Moreover, as ,

 uε⟶u0inC1−θ([0,τ],W2θq(I)) ,0<θ<1 ,

and

 ψε(t)1Ω(uε(t))⟶ψ0(t)1Ω(u0(t))inL2(I×(−1,0)) ,t∈[0,τ] , (1.17)

where

 u0∈C1([0,τ],Lq(I))∩C([0,τ],W2q,D(I))

is the unique solution to the small aspect ratio equation (1.13) and is the potential given in (1.12). Furthermore, there is such that the results above hold true for each provided that .

The proof is given in Section 6. A similar result has been established in [20, Thm. 2] for the stationary problem and in [6, Thm.1.4] for the semilinear parabolic version (1.11). As in the latter paper, the crucial step is to derive the -independent lower bound on , which is not guaranteed by the analysis leading to Theorem 1.1. The proof of Theorem 1.4 uses several properties of (1.9)-(1.10) with respect to the -dependence shown in [6].

## 2. On the Elliptic Equation (1.9)-(1.10)

We shall first derive properties of solutions to the elliptic equation (1.9)-(1.10) in dependence of a given (free) boundary. To do so, we transform the free boundary problem (1.9)-(1.10) to the fixed rectangle . More precisely, let be fixed and consider an arbitrary function taking values in . We then define a diffeomorphism by setting

 Tv(x,z):=(x,1+z1+v(x)) ,(x,z)∈¯¯¯¯¯¯¯¯¯¯¯Ω(v) , (2.1)

with . Clearly, its inverse is

 T−1v(x,η)=(x,(1+v(x))η−1) ,(x,η)∈¯¯¯¯Ω , (2.2)

and the Laplace operator from (1.9) is transformed to the -dependent differential operator

 Lvw:=ε2 ∂2xw−2ε2 η ∂xv(x)1+v(x) ∂x∂ηw+1+ε2η2(∂xv(x))2(1+v(x))2 ∂2ηw+ε2 η ⎡⎣2 (∂xv(x)1+v(x))2−∂2xv(x)1+v(x)⎤⎦ ∂ηw .

An alternative formulation of the boundary value problem (1.9)-(1.10) is then

 (Lu(t)ϕ)(t,x,η) =0 , (x,η)∈Ω , t>0 , (2.3) ϕ(t,x,η) =η , (x,η)∈∂Ω ,t>0 , (2.4)

for . With this notation, the quasilinear evolution equation (1.6) for becomes

 ∂tu−∂x(∂xu√1+ε2(∂xu)2) =−λ [1+ε2(∂xu)2(1+u)2] |∂ηϕ(⋅,1)|2 ,x∈I ,t>0 , (2.5)

where we have used for and due to by (2.4). The investigation of the dynamics of (2.5) involves the properties of its nonlinear right hand side as well as the properties of the quasilinear curvature term. We shall see that these two features of (2.5) are somewhat opposite as the treatment of the former requires a functional analytic setting in to handle the second order terms of in (2.3), while a slightly weaker setting has to be chosen to guarantee Hölder continuity of with respect to time which is required in quasilinear evolution equations (see Remark 3.3 for further details). To account for these features of (2.5) we have to refine the Lipschitz property of the right-hand side of (2.5) derived in [6] as stated in (2.8) below.

Defining for the open subset

 Sq(κ):={u∈W2q,D(I);∥u∥W2q,D(I)<1/κ and −1+κ

of (defined in (1.14)) with closure

 ¯¯¯¯Sq(κ)={u∈W2q,D(I);∥u∥W2q,D(I)≤1/κ and −1+κ≤u(x) for x∈I} ,

the crucial properties of the nonlinear right-hand side of (2.5) are collected in the following proposition:

###### Proposition 2.1.

Let , , and . For each there is a unique solution to

 (Lvϕv)(x,η) =0 , (x,η)∈Ω , (2.6) ϕv(x,η) =η , (x,η)∈∂Ω . (2.7)

If is defined by for , then for . Moreover, for , the mapping

 gε:Sq(κ)⟶W2σ2,D(I) ,v⟼1+ε2(∂xv)2(1+v)2 |∂ηϕv(⋅,1)|2

is analytic and bounded with . Finally, if and , then there exists a constant such that

 ∥gε(v)−gε(w)∥Wν2,D(I)≤c1(κ,ε)∥v−w∥W2−ξq,D(I) ,v,w∈¯¯¯¯Sq(κ) . (2.8)

According to [6, Prop. 2.1] we actually only have to prove (2.8). Notice that this global Lipschitz property is in the weaker topology of instead of and improves [6, Prop. 2.1] where it was established for . The property (2.8) will be a consequence of a sequence of lemmas. For the remainder of this section we fix , , and .

In the following, if we let denote the subspace of elements in whose boundary trace is zero, and if we set . We equip with the norm

 ∥Φ∥W12,D(Ω):=(∥∂xΦ∥2L2(Ω)+∥∂ηΦ∥2L2(Ω))1/2 ,

and introduce the notation

 W−θ2,D(Ω):=(Wθ2,D(Ω))′ ,0≤θ≤1 .
###### Lemma 2.2.

For each and there is a unique solution to the boundary value problem

 −LvΦ =F in Ω , (2.9) Φ =0 on ∂Ω , (2.10)

and there is a constant such that

 ∥Φ∥W12,D(Ω)≤c2(κ,ε)∥F∥W−12,D(Ω) . (2.11)

Furthermore, if , then and

 ∥Φ∥W22,D(Ω)≤c2(κ,ε)∥F∥L2(Ω) . (2.12)
###### Proof.

According to [12, Def. 1.3.2.3, Eq. (1,3,2,3)], we may write any in the form with . Consequently, [10, Thm. 8.3] ensures that the boundary value problem (2.9)-(2.10) has a unique solution . Furthermore, taking as a test function in the weak formulation of (2.9)-(2.10) gives

and thus

 ∫Ω⎡⎣ε2(∂xΦ−η∂xv1+v∂ηΦ)2+(∂ηΦ1+v)2⎤⎦d(x,η)≤ε2∥∥∥∂xv1+v∥∥∥L∞(I)∥∥∥∂xΦ−η∂xv1+v∂ηΦ∥∥∥L2(Ω)∥Φ∥L2(Ω)+∥F∥W−12,D(Ω)∥Φ∥W12,D(Ω) . (2.13)

Note then that by definition of and Sobolev’s embedding, there is such that

 1+v(x)≥κ ,x∈I ,∥v∥C1([−1,1])≤c3κ (2.14)

for all . Also, if , Young’s inequality ensures that, for ,

 ε2ζ21≤2ε2(ζ1−η∂xv(x)1+v(x)ζ2)2+2ε2η2(∂xv(x)1+v(x))2ζ22≤2(1+2ε2∥∂xv∥2∞)⎛⎝ε2(ζ1−η∂xv(x)1+v(x)ζ2)2+12(ζ21+v(x))2⎞⎠ .

Therefore, introducing

 ν(κ,ε):=12min{ε2κ2κ2+2ε2c23,κ2(κ+c3)2} ,

we infer from (2.14) that

 ν(κ,ε)(ζ21+ζ22)≤ε2(ζ1−η∂xv(x)1+v(x)ζ2)2+(ζ21+v(x))2 . (2.15)

Consequently, (2.13), (2.14), and (2.15) give

 ε2∥∥∥∂xΦ−η∂xv1+v∂ηΦ∥∥∥2L2(Ω)+∥∥∥∂ηΦ1+v∥∥∥2L2(Ω)≤ε2c3κ2∥∥∥∂xΦ−η∂xv1+v∂ηΦ∥∥∥L2(Ω)∥Φ∥L2(Ω)+∥F∥W−12,D(Ω)√ν(κ,ε)(ε2∥∥∥∂xΦ−η∂xv1+v∂ηΦ∥∥∥2L2(Ω)+∥∥∥∂ηΦ1+v∥∥∥2L2(Ω))1/2 ,

whence, using again (2.15),

 ∥Φ∥W12,D(Ω)≤ε√ν(κ,ε)c3κ2∥Φ∥L2(Ω)+∥F∥W−12,D(Ω)ν(κ,ε) . (2.16)

We now proceed as in [10, Lem. 9.17] and argue by contradiction to show (2.11) (see also [6, Lem. 6.2] for a similar argument in a slightly different functional setting). The last statement of Lemma 2.2 is proved in [6, Lem. 6.2]. ∎

Now, introducing

 (2.17)

for given, we readily deduce that with

 ∥fv∥L2(Ω)≤c4(κ,ε) . (2.18)

Consequently, Lemma 2.2 provides a unique solution to

 −(LvΦv)(x,η) =fv , (x,η)∈Ω , Φv(x,η) =0 , (x,η)∈∂Ω .

Clearly, defining

 ϕv(x,η):=Φv(x,η)+η ,(x,η)∈¯Ω , (2.19)

gives then the unique solution to (2.6)-(2.7). To prove a Lipschitz dependence of on , we introduce a bounded linear operator

 A(v)∈L(W12,D(Ω),W−12,D(Ω))∩L(W