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A magnetically controlled electrostatic-elastic membrane system.

Abstract.--Electrostatic-elastic membrane systems have an inherent instability where no stable configuration exists when the applied voltage exceeds a critical value. In this paper the effects of a magnetic field on a para-magnetic membrane in a capacitive system is investigated as a means of controlling this instability. The effects of a magnetic field due to a large current loop and that of a point dipole on the membrane are modeled. Both models show that an increase in magnetic field strength results in an increase in stable operating voltage and a decrease in the maximum deflection of the membrane. Laboratory experiments are performed and qualitative agreement with both models is observed.

In recent years electrostatic-elastic membrane systems have been the subject of much study. This is mainly due to the design and study of microelectromechanical systems (MEMS) devices. In many of these devices, electrostatic actuation is used to provide locomotion within the system. It has been observed, even in the earliest days of MEMS design (Nathanson et al. 1967), that although electrostatic actuation is a desirable property to exploit, it suffers from an inherent flaw. When the applied voltage between components becomes too large the system becomes unstable and no steady state configuration for the device exists which maintains the separation of the components. This is often referred to as the "pull-in" instability. Since these earliest observations, attempts have been made to help control this instability, to allow for wider operating ranges for these devices. Some attempts include capacitive control schemes (Pelesko & Triolo 2001), tailored dielectric properties of the components (Pelesko 2002; Guo & Ward 2005; Beckham & Pelesko 2008), voltage control schemes (Chu & Pister 1994), external pressures (Beckham & Pelesko 2011), and others (Brubaker et al. 2013). In addition to electrostatic forces, thermal, biological, and magnetic forces have also been used in these devices as a means of locomotion and manipulation of components (Pelesko & Bernstein 2003).

Recently several papers have looked at the use of magnetic actuators to control elastic structures (Khoo & Liu 2001; Esquivel-Sirvent et al. 2009; Petrescu et al. 2009; Shields et al. 2010; Tsumori & Brunne 2011). This paper explores the use of a magnetic field to help control and stabilize an electrostatically actuated elastic membrane. The interaction of both electrostatic and magnetic forces on an elastic structure are considered. Although the interactions of these forces on elastic structures have been considered, (Shijie et al. 2012) this particular system of interest has not been explored. A capacitive system where one of the components is created by an elastic structure, a membrane, is studied. This system is subject to an external magnetic field and the membrane is made so as to react with this field. It has been observed that one drawback of magnetic actuation is the extreme dependence on size, due to magnetic field strength dropping off with distance. It is for this reason that the magnetic field is not considered as the driving force of the system but instead acting as a stabilizer. Also, much of the modeling of magnetically actuated systems involves a membrane with a permanent magnet attached. Although permanent thin film magnets have been created (Chin 2000), this paper will explore micro-scale membranes that contain nano-sized magnetite particles. Due to the paramagnetic nature of the particles, it will align with the magnetic field present. Both types of magnetic interactions are present in the literature as cited above. This paper considers ferrofilm membranes instead of permanent magnetic membranes.


For two specific field configurations, the large current loop and the dipole, the system is simplified, obtaining a leading order model. In each case the resulting differential equation is solved numerically and the qualitative behavior of the system is described in terms of bifurcation diagrams relating the voltage and central deflection of the membrane as the magnetic field strength is changed. For the large current loop model, existence and nonexistence results are established in order to obtain a better understanding of the pull-in phenomenon. Experiments are done to determine the validity of the model and its qualitative features. It is observed that the qualitative features predicted by both models are present in the system created. Exploration of this system gives an added design option for controlling the pull-in instability in MEMS devices which employ these components. A depiction of this system is given in Figure 1. That is, we consider a circular elastic membrane, held at a potential V, which is fixed at the boundary. It is separated from a ground plate, by a distance [d.sub.1]. Outside (above) this capacitor is a magnet, and the membrane is a ferro-membrane, so as to be affected by the magnet. We assume that the gap distance, [d.sub.1], is small compared to the characteristic length of the membrane which we denote by R, [d.sub.1]/R= [epsilon] << 1. The deflection of our membrane will be given by z' = u'(r', [theta]'). Here our coordinate system has the origin at the center of the undefleeted membrane with the negative z' direction toward the ground plate and the r' [theta]'-plane is in the plane of the undeflected membrane. We must consider the three dominate features affecting the behavior of the membrane: (i) the elasticity of the membrane, (ii) the electric field and (iii) the magnetic field. The consideration of these three effects leads to:



where prime denotes dimensional variables. The first term accounts for the elasticity of the membrane, where T is the surface tension in the membrane and the Laplace operator is the small deflection approximation to the more general mean curvature operator. This assumption is validated due to the small aspect ratio of the gap distance to membrane size. This assumption is based on the structure of the capacitive part of our system and not on the magnetic field. However, provided that the size of the magnetic field is not so large as to cause "large" deflections, this assumption still holds in the presence of the external magnetic field. The second term represents the electrostatic force, where [[epsilon].sub.0] is the permittivity of free space and [psi]' is the potential function in the region between the plates. To find [psi]' we note that it must solve Poisson's equation

[nabla]' * E = [nabla]' * [nabla]'[psi]' = [[nabla]'.sup.2][psi]' = [rho]/[[epsilon].sub.0] = 0, (1)

and we have


where Vis the applied voltage to the system, and u' gives the distance from the membrane to the ground plate for (r', [theta]'). Next, we rewrite our problem in dimensionless form by defining

[psi]=[psi]'/V, u=u'/[d.sub.1], z=z'/[d.sub.1], r=r'/R.

Here di is the undeformed gap length between the membrane and ground plate, and R is a characteristic scaling with respect to the size of the membrane. We have


Now assuming a small aspect ratio, [epsilon]=[d.sub.1]/R <<1, we solve for [psi] giving

[psi] (=) z/[1+u].

From this we have that

Finally, we consider the magnetic field term [F.sub.mag]. Since we have a ferro-membrane, the magnet will induce a pressure on the membrane given by


where M' is the magnetization due to the interaction of the membrane with the magnetic field and B' is the magnetic field in the region, (Rosensweig 1985). Assuming that the membrane is purely elastic, our force will be the normal component. That is,

[F.sub.mag] = n' * (M' * [nabla]')B'.

Here the normal vector is given by


Since our membrane is paramagnetic the ferro-particles will align with the applied field so that M'= [[chi].sub.m]H', where [[chi].sub.m] is the magnetic susceptibility. Further assuming a linear media, B'=[[mu].sub.0](H'+M'), we get


where [[mu].sub.0] is the magnetic permeability. To proceed in our derivation it is convenient at this point to specify the structure of our magnetic field. As a first attempt to explore the behavior of this system we will consider both the large current loop and the magnetic dipole. We choose these due to their nice symmetric structure with our current geometry and their difference on dependence on the radius. For the large current loop there will be no radial dependence but for the magnetic dipole there will be a strong radial dependence.


Large Current Loop.--We continue by describing our magnetic field as being induced by a large current loop. By large we mean in comparison with the size of the membrane so that we expect the z component of the magnetic field to be dominant and be given by the field along the centerline to leading order (see Fig. 2).

For a loop of radius [R.sub.1] and current I we have

B'=B'[??], (B'=[[mu].sub.0]I[R.sup.2.sub.1]/[2[(R.sup.2.sub.1)+[([d.sub.2]-z').sup.2].sup.3/2]])

Here [d.sub.2] is the undeflected distance from the plane of the membrane to the plane of the loop. We nondimensionalize B' by [B.sub.0] = [[mu].sub.0]I/[2[R.sub.1][(1 + [[alpha].sup.2]).sup.3/2]] so that


where a=[d.sub.2]/[R.sub.1] and [rho]=[d.sub.1]/[R.sub.1]. We have chosen our scaling so that B is unity at the center of the undeflected membrane (z=0). From this our force becomes


Since we are concerned with the field at the membrane surface we set z=u and obtain the force balance equation


Multiplying through by [R.sup.2]/[Td.sub.1] gives




To complete our model we note that our membrane is fixed on the circular boundary. Due to the symmetry in our problem we expect u=u(r) and requiring smoothness of the solutions gives


with boundary conditions

u(1) = 0 and du/dr (0) = 0. (3)

This completes our model for the large loop. The two main parameters of interest are [lambda] and [[beta].sub.1]. Here [lambda] is the ratio of a reference electrostatic force and a reference elastic force and [[beta].sub.1] is the ratio of a reference magnetic force and a reference elastic force. Our problem of interest is being able to use the external magnetic field to stabilize and control the pull-in instability due to the electrostatic force. To illustrate how this occurs we solve our problem for different (fixed) values of [[beta].sub.1] and determine the solution dependence on [lambda]. First we will establish existence and nonexistence results for the problem. That is, even with the presence of the magnetic field, the pull-in instability still persists.

Theorem 1. For every fixed value of [[beta].sub.1], [alpha], and [rho], there exists a [lambda]* such that (2)-(3) has no solution for [lambda] > [lambda]*.

Proof. Consider the eigenvalue problem


Let [[gamma].sub.1] be the lowest eigenvalue with corresponding eigenfunction [w.sub.1]. We know that [[gamma].sub.1] is simple and we may choose [w.sub.1] to be strictly positive in [OMEGA]. Now consider


By the Fredholm Alternative Theorem,


must equal zero for a solution to exist. Since we have chosen [w.sub.1] to be strictly positive, we need


to be identically zero, or it must change sign. It will be identically zero if [lambda]=[[[beta].sub.1][alpha]]/(1+[[alpha].sub.2]) and u=0 on [OMEGA]. Otherwise this quantity is not identically zero because [lambda], [beta], [alpha], [rho] and [[gamma].sub.1] are non-zero constants and u is a continuous function in [OMEGA]. From this we can conclude that our quantity must change sign. We proceed by showing that [lambda] can always be chosen so that this will not occur. That is, we wish to show that, for a proper choice of [lambda] we have


We first consider the right hand side and note


From this we see that it is enough to require

C1[alpha]-([C.sub.1][rho] + [[gamma].sub.1])u<[lambda]/[(1+u).sup.2].

Since the left hand side is linear in u we find the point of tangency and this will give a threshold value for [lambda]. This occurs when

[lambda]* = 4/27 [(([C.sub.1][rho]+[[gamma].sub.1])+[C.sub.1][alpha]).sup.3]/[([C.sub.1][rho]+[[gamma].sub.1]).sup.2]

For [lambda] >[lambda]* our quantity will not change sign and thus we have no solution.

This establishes that the pull-in instability will persist regardless of the field generated by the current loop. To establish existence we follow the procedure of constructing upper and lower solutions. For a detailed look at upper and lower solutions of elliptic equations to establish existence see (Pao 1992) and for their use in related problems see (Flores et al. 2006), (Beckham & Pelesko 2011). We begin with a definition.

Definition 1. The function in [bar.u][member of][C.sup.2] ([bar.[OMEGA]]) is called an upper solution if


The function [][member of][C.sup.2]([bar.[OMEGA]]) is called a lower solution if the opposite inequalities are satisfied.

Construction of upper and lower solutions for our problem will establish existence of a solution. We first construct an upper solution to our problem. We simply set [bar.u] =c, where c is a positive constant, and require

-[[lambda]/[[(1+c).sup.2]]] + [[[beta].sub.1][(1+[[alpha].sup.2]).sup.3]([alpha] - [rho]c)]/[[(1+[([alpha] - [rho]c).sup.2]).sup.4]]] [less than or equal to] 0.

Choosing c > [alpha]/[rho] will guarantee that the inequality holds and we have an upper solution.

For the lower solution let us consider u* to be a solution to [DELTA]u*= 1 in [OMEGA] with u*=0 on [partial derivative][OMEGA]. Now from the maximum principle we have that u*<0 in [OMEGA]. We consider the infimum of u* on [OMEGA] which we define to be m*=inf{u*(x)|x[member of][OMEGA]}. Next we define [c.sub.1]=-1/(2m*) and consider the function [c.sub.1]m*.


Since m* is the infimum of u* in [OMEGA] and [DELTA]u*= 1, we have


This will be satisfied if we require


and we conclude that a solution to our problem exists. Notice that [alpha] depends on m* which depends on the function u that solves [[nabla].sup.2]u* = 1 in [OMEGA], u=0 on [partial derivative][OMEGA]. In our particular case [OMEGA] [member of] [R.sup.2] is a disk and we can choose [c.sub.1] = 4. We have established the following result.

Theorem 2. For any fixed [[beta].sub.1], [alpha] and [rho], there exists a solution to (2)- (3) for all


Now that we have established existence and nonexistence results we continue our exploration of the problem though a series of bifurcation diagrams. For a given set-up we can envision a change in [[beta].sub.1] occurring by changing the current in the loop as well as by changing the concentration of ferrofluid in the membrane. Changing [lambda] is done by changing the input voltage to the system. To understand our system and the effect the loop has we first solve our problem for varying [[beta].sub.1] in the absence of an electric field, [lambda]=0. Next we consider [[beta].sub.1] fixed and vary [lambda]. In Figure 3 we give a bifurcation diagram of [[beta].sub.1] vs. u(0) with [lambda] =0 as well as a bifurcation diagram of [lambda] vs. u(0) for different values of [[beta].sub.1]. We see a simple fold in the middle of the diagram, as well as an infinite fold structure as the deflection at the origin nears the ground plate. This fold structure is due to the circular geometry and the structure of the electrostatic-elastic interaction, (Pelesko & Bernstein 2003).

The [lambda] value at the first bifurcation point corresponds to the pull-in voltage to the system, [lambda]*. We observe here that as the parameter [beta] increases the value of [lambda]* increases and thus a higher input voltage can be obtained before the system becomes unstable. We see that once [lambda] increases beyond a certain value, the bifurcation diagram predicts multiple solution. However, all branches other than the initial branch are unstable. We also observe that the maximum stable deflection of the membrane appears to decrease as [beta] is increased. A similar relationship has been observed when studying electrostatic-elastic systems in the presence of a constant external pressure, (Beckham & Pelesko 2011).

Magnetic Dipole.--For our second field we consider a magnetic dipole which is centered above the circular membrane a distance [d.sub.3] above the plane of the undeflected membrane, see Figure 4. For this field we will have both r and z dependence, but note that the field still has radial symmetry. That is, there will be no angular component. For the magnetic dipole we have, for the r and z components respectively,



where m is the magnetic moment. At this point we nondimensionalize B' by [B.sub.0]=3m[[mu].sub.0]/4[pi][[d.sub.3].sup.3], again based on the strength of the field at the center of the undeflected membrane. After nondimensionalizing, we have


where [delta]=[d.sub.3]/R. For our exploration we will consider the situation where [d.sub.3]/[d.sub.1]=O(1) and set [delta]=[epsilon]. Expanding [F.sub.mag] gives


Dividing through by [T.sub.[epsilon]/R] gives, to leading order,


where [[beta].sub.2] = ([9.sub.[micro]][[chi].sub.m][m.sup.2.R.sup.2])/([l6[pi].sup.2 T](1 + [[chi].sub.m])[d.sup.2.sub.1][d.sup.6.sub.3]). From this, along with radial symmetry and smoothness, we have obtained our leading order model consisting of the governing equation


along with the boundary conditions

u(1) = 0 and du/dr(0) = 0.

In the previous section we began by establishing existence and nonexistence results for our problem. The techniques used in that case will not work here. The main reason is that our right hand side now contains u' along with u. We have bounds on u but, in general, we can make no claims on the bounds for u'. We could very well have the situation where the membrane will cusp as it reaches a touchdown state. For this reason, we proceed by constructing bifurcation diagrams in an attempt to understand the controlling properties the magnetic field has on our system, see Figure 5. We first consider the bifurcation in [[beta].sub.2] with [lambda]=0. We can see that this bifurcation diagram is different from Figure 3. For the current loop, a solution exists for all values of [[beta].sub.1] and no folding occurs. In this case we see a fold in the bifurcation diagram. From this we conclude that there exists a [[beta].sub.2]*of which no solution to the steady state problem exists for [[beta].sup.2] > [[beta].sub.2].* We also consider the bifurcations in X for different (fixed) values of [[beta].sub.2]. Based on the bifurcation diagram, we have chosen values of [[beta].sub.2] so that a steady state solution exists in the absence of an electric field ([lambda]=0). We note that one can find solutions for [[beta].sup.2]> P2* provided that [lambda]>0. We see from this collection of bifurcation curves that we still observe the qualitative features that the pull-in voltage increases with increasing [[beta].sub.2] while the pull-in distance decreases. We notice that the different bifurcation curves converge on one another as we approach a touchdown solution. This makes sense due to the rapid drop off in the magnetic field of a dipole. Near touchdown the membrane will feel little effect from the magnetic field.

At this point a model for a magnetic-electrostatic-elastic membrane system has been created. The model has been explored by considering two different magnetic field configurations, the current loop and the dipole. From the bifurcation diagrams constructed the following observations are made. The effect on the membrane due to the magnetic field alone ([lambda]=0) is strongly dependent on the field structure and changes in this structure can result in drastically different bifurcation diagrams. For the current loop there is no fold structure where as in the case of a dipole there are multiple folds. This is not surprising based on magnetic-elastic models which have already been studied. In each model considered the presence of the magnetic field acts as a stabilizer for the membrane in terms of the pull-in due to the electrostatic field. An increase in the magnetic field strength gives an increase in the pull-in voltage for the system. In addition, there is a decrease in the pull-in distance with the magnetic field present. It seems feasible to conclude that many magnetic field configurations, if not all, will ultimately display these properties when put in opposition to the electrostatic field.


To test the validity of the mathematical model the following experiments were performed. Since, in both cases, there were simplifications made and a leading order model constructed, one natural question is whether or not the reduced models capture the fundamental qualitative behaviors of physical systems of this nature. To explore this, macro scale experiments were conducted similar to the models discussed. The set-up consists of a ferrofilm solution made up of a standard soap solution in a 10:1:5 ratio of water:soap:glycerine (Dawn.[TM] was used) and a ferrofluid solution (EMG 705 obtained from Ferrotec Corporation). After the solution was mixed it was allowed to sit for several days to minimize drainage during the experiments. This solution was then used to suspend an elastic membrane across a circular frame. The frame is connected to a high voltage power supply (WR100P2.5 from Glassman High Voltage Inc.), and held at a potential V above a rigid ground plate. A cylindrical neodymium magnet with a diameter of 2 cm is centered above the membrane so as to react with the membrane, see Figure 4 but with a cylindrical magnet instead of a dipole (the surface field strength of the magnet is approximately 850 Gauss). In the experiments the distance from the magnet to the membrane (10 mm) and the distance from the membrane to the bottom plate of the capacitor was the same. This magnet will not simulate either of the above models exactly but should lie somewhere in between. This magnetic field will not be as uniform as the magnetic field of a large current loop but it will be more uniform than the field of a dipole. However, the same qualitative properties should be present. That is, an increase in pull-in voltage and a decrease in pull-in distance is expected. Pull-in distance is defined as the maximum deflection at the center of the radially symmetric membrane. To simulate changing the value of the parameter [beta] experiments were done with membranes having different concentrations of ferrofluid. Solutions were created with soap solution/ferrofluid ratios of 5:1, 4:1 and 3:1.


Figure 6 shows a progression of images from the experiments conducted. Figure 6(i) shows the deflected membrane near pull-in in the absence of the magnetic field. All of the images near pull-in are within one percent of the actual pull-in voltage. Figure 6(ii-iv) show the 5:1, 4:1, and 3:1 soap solution/ferrofluid ratio membranes, respectively, with magnet present near pull-in. It is observed that the stable operating voltage increases as the concentration of ferrofluid is increased. This is to be expected based on the model as well as the physical aspect of having more magnetite particles to react with the field. It is also observed that with an increase in ferrofluid there is a decrease in the maximum deflection of the membrane before pull-in. This is also to be expected based on the model. In particular, notice that the 3:1 solution (Fig. 6iv) is only slightly below plane before pull-in occurs. One question, based on the two models explored, is whether or not any concentration of ferrofluid can be used. A 2:1 solution was attempted, but there was no stable configuration for the set-up in the absence of the electric field. That is, with the field off, the membrane deflected upwards and contacted the magnet. To further illustrate the behavior of the membrane, images are given for the different solutions at the pull-in voltages of the prior experiment (Fig. 6v-vii). That is, Figure 6(vi) is the 4:1 solution near the pull-in voltage of the 5:1 solution in Figure 6(ii). This again shows that the ferrofluid acts to stabilize the membrane and increase the stable operating voltage for the system. An image of the 3:1 membrane in the absence of the electric field is given (Fig. 6viii). There is a noticeable deflection upward towards the magnet. For the 4:1 solution a slight deflection was observed and the deflection for the 5:1 was hard to notice at all. One question was whether or not the surface tension remains the same for all mixtures used. Although it appeared that the surface tension increased as the ferrofluid concentration increased, there was less than a one percent difference between pull-in voltages without the magnet. For this reason the images shown should give a good representation of what happened when the effect of the magnetic field is increased. To further validate the trend, the ratio of near pull-in voltages with magnet to without magnet was considered for each soap solution/ferrofluid concentration: The ratios were 1.02, 1.10, and 1.19 for the 5:1, 4:1, and 3:1 soap solution/ferrofluid concentrations, respectively. This confirms that the magnetic interaction does have a significant effect, independent of the surface tension.



In this paper an electrostatic-elastic membrane system in the presence of a magnetic field is explored. The membrane contains a ferrofluid making it susceptible to magnetic fields. Magnetic fields due to a large current loop and a dipole are explored. The field due to the large current loop has no radial dependence however the field due to the dipole has a radial dependence and drops off at a much faster rate than the large current loop. Due to the differences in the magnetic field structure the electrostatic-elastic membrane system behaves differently for each field as expected. However, both systems have the same qualitative behavior. Both models predict that increasing the magnetic field strength will increase the pull-in voltage and decrease the maximum deflection. Experimentally a cylindrical magnet is used to produce a magnetic field and vary the concentration of ferrofluid to change the strength of the magnetic interaction. The magnet used is smaller than the membrane so the field will not be as uniform as the field of a large current loop. In particular, it will have a radial dependence unlike the large current loop. However, the field of the magnet will be more uniform than the field of a point dipole. Since both models predict the same qualitative behavior, an increase in pull-in voltage and a decrease in maximum deflection, this behavior should be seen for a wide variety of magnetic field configurations. The qualitative features present in the two models are seen experimentally. The experiments give validity to the models developed. This work sheds light on the interaction of these two forms of actuation and may lead to consideration of this system in future designs.


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JRB at:

Randy Back (1) and J. Regan Beckham (2)

(1) Department of Chemistry, University of Texas at Tyler, Tyler, TX 75799

(2) Department of Mathematics, University of Texas at Tyler, Tyler, TX 75799
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Author:Back, Randy; Beckham, J. Regan
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Date:Jun 1, 2017
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