The Elastic Capacitor and its Unusual Properties

Transcription

1 1 The Elastic Capacitor an its Unusual Properties Michael B. Partensky, Department of Chemistry, Braneis University, Waltham, MA 453 The elastic capacitor (EC) moel was first introuce in stuies of lipi bilayers (the major components of biological membranes). This electro-elastic moel accounts for the compression of a membrane uner applie voltage V an allows one to obtain information about the membrane s elastic properties from the measurements of its capacitance. Later on, ECs were use to analyze the electrical breakown of biological membranes. This effect is use nowaays in various meical applications, such as targete rug elivery. The EC moel was also helpful in stuies of microscopic capacitors electric ouble layers in various electrifie interfaces (of which the electroe/ electrolyte interface is the most common example). This comparatively simple moel, the analysis of which requires only high-school physics, has a close relationship to some real-life problems in physics, chemistry an biology. We will be examining a version of an EC moel, which is a parallel-plate capacitor with an elastically suspene upper plate an a lower plate, which is fixe an groune (Fig. 1). The separation between the uncharge plates is h, the area of each plate is A, an the effective spring constant is KA (K is the spring constant per unit area). As it is usually assume, the istance h between the plates is much smaller than A 1/ an the fringe effects can be neglecte. Builing up charge on the plates causes the spring to stretch, until the attractive electric force an its opposing elastic force become equal, an an equilibrium position is establishe. The mathematical evelopment that follows will escribe the equilibrium position of the plates an the electrical parameters of the EC, an will eal with possible instabilities in the EC. The relationships that result from this seemingly simple electro-mechanical system are quite interesting, an affor one an opportunity to better unerstan the unerlying physical processes. I hope that both teachers an stuents will fin this iscussion instructive an challenging. 1. Capacity anomalies in the isolate EC Statement of the Problem To stuy the electric properties of an isolate EC, the charge Q on the upper plate is incremente by small portions Q. Every time a Q is ae, the new value of charge stays fixe (this is what we mean by isolate) while h an V assume new equilibrium values corresponing to the new value of Q. Now, stuents may be challenge by following questions: 1. Describe the electro-compression in EC, i.e. hq ( ). At what charge Q max will the plates come into contact?. Draw the curve V( Q ) for Q< Q max. Try to escribe this result in terms of ifferential capacitance C = Q/ V Q / V. Di you fin anything unusual? Try to explain the anomaly, if any.

2 3. How will V(Q) change if a stop is put in the gap at the istances h /3 or h /3above the lower plate, preventing the gap from further contraction? Suggestion: try casting the problem in imensionless units, such as z = h/ h. Figure 1. Isolate Elastic Capacitor Solution The energy of the isolate elastic capacitor (per unit area) consists of elastic an electrostatic contributions. W EC Kh ( h) σ = + h (1) εε where σ = Q/ A is the charge ensity an ε is the ielectric constant in the gap of EC Orienting the x-axis as the upwar normal to the groune plate, we fin the resultant force (per unit area) acting on the top plate, σ Fx = K( h h) () εε where the first term is the elastic force cause by the stretche spring, an the secon term is the opposite force ue to the electrostatic attraction of the plates. Note that the weight of the plate oes not appear in Eqs. (1) an () because it is fixe an compensate by the elastic force. The potential ifference between the plates is σ V = h (3) εε It is convenient to use imensionless units,

3 3 W F Energy : w= ; Force : f = ; Kh Kh 1/ Gap with : z = ; Voltage : v = ( ) ; 3 h Kh 1 Charge ensity : s = ( ) εε Kh h 1/ σ εε V (4) In these units, imensionless potential, energy, an force are: (1 z) s s v = sz, wec = + z, f = 1 z (5) The equilibrium istance between the plates is erive from the force balance conition f = : z = 1 s / (6) We now see that the gap closes ( z ) at s smax =. This result answers the first question of the problem. Using equations (5) an (6), we fin the potential ifference as v = s s (1 / ) (7) The epenencies z(s) an v(s) are shown in Fig.. At small charges ( s.3) EC behaves similarly to a regular (fixe) capacitor (escribe in Fig. by v () s ): the potential is practically a linear function of the charge. However, as s grows, vs () progressively flattens, an it finally reaches its maximum at scr =.81, zcr =, vcr =.54 (8) Stuents can verify these critical values either numerically (using Eq. 7), or by fining the maximum from the ifferential conition v / s =. For s > s, v( s) becomes the escening function of charge. In other wors, an cr increase of electric charge on the plates is accompanie by a ecrease of the potential rop across the capacitor 1 1 To better unerstan this unusual behavior, a kinematical analogy may be useful. As you know, an automobile moving with constant velocity v covers a istance L= v t in time t. Suppose that you rive a "super car" that can run as fast as you wish, an your final goal is to cover a maximum istance. However, in orer to make your life more ifficult, a restriction is impose. The higher the velocity you chose, the shorter becomes the time you are allowe to move. In fact, the time is explicitly escribe as a escening function of v: t = t (1 -.5 v ), where t is a given constants. Try to fin the optimal velocity v=v max by rawing the function L= v t(v) an fining at what v=v max it reaches maximum. You will iscover that v max = 1/3. After this problem is solve, note that it exactly relates to the EC problem through the substitution v s, t(v) z(s): the reuction of time epening on v is analogous to the contraction of the capacitor gap epening on charge.

4 4 This unusual feature becomes even more remarkable when escribe in terms of (imensionless) ifferential capacitance per unit area, C s/ v where s s is a small variation of charge ensity an v is a corresponing variation of potential. Those familiar with erivatives can use a precise efinition of : 1-1 C = s/ v = ( v/ s) = [1-( s/ sc r) ]. As s approaches s cr, C becomes infinite only in the limit z. In contrary, in an elastic capacitor at the point s cr C. Note, that in a fixe capacitor C = scr where C becomes infinite, gap with z is finite ( z = /3). The most intriguing consequence of our equations, however, is that C () s becomes negative for s > s. This unusual feature also appears in stuies of various elaborate moels of the microscopic capacitors (electrical ouble layers). As in ECs, it is generally cause by some sort of electro-compression, although the role of a spring is usually playe by a combination of molecular, electrostatic an entopic forces [1]. The iscussion of these effects an reality behin them might be a topic for a stuent s physics project. Suggestions for further stuy 1. We i not answer the question about the influence of a stop inserte in the gap of the EC. This question might be offere for inepenent stuy.. Our analysis was base on the equilibrium conition f =. It is always a goo iea to verify if a iscovere equilibrium is stable. This is equivalent to the requirement that the upper plate resies in a minimum of the energy curve W( z ), not in a maximum. If the contrary were true, then all our previous results incluing the anomalies of C woul become invali. The stuents can be aske to epict the epenencies values of charge, s s s an corresponing respectively to the regions with C >, C an < cr, = scr s > scr C <. They will fin that each curve has a single minimum, which means that the equilibrium is stable. w EC for at least three characteristic 3. The EC moel that we use oes not cover one very important feature of membranes or ouble layers. In reality, their plates are not rigi. For example, a membrane s flexibility allows a lateral variation of local variation of charge ensity σ (neee to maintain the conuctive plates as equipotentials). How can such flexibility alter our results? Once again, the EC can be helpful in answering this question [1]. h = an a corresponing The simplest example, is to use two ientical ECs connecte in parallel. It turns out that in a range of charges where the capacitance of a rigi plate EC woul become negative, the flexible plate EC (moele by two ECs in When the properties of a capacitor epen on its charge (or Voltage), the ifferential capacitance becomes a far more appropriate tool than the regular (integral) capacitance, C = s/ v (total charge by total Voltage). In a sense, using v / s instea of v/s is similar to using the instantaneous (v) rather than the average ( v ) velocity to escribe an accelerate motion. Obviously, instantaneous ( ifferential ) characteristics provie much more complete an unambiguous information about the system than its average properties. For instance, knowing that v= one woul still not be able to istinguish between a resting sate an a state of accelerate motion with v changing its irection.

5 5 parallel) loses its stability. Such instabilities an phase transitions in membranes an charge interfaces are well known, very important an wiely stuie. 4. The charge-controlle conitions iscusse in this chapter are very rare in the stuies of real microscopic capacitors, such as electrical ouble layers an membranes. It is much more practical to connect the electroes of a measuring cell to a potential source (battery) so that voltage V (not the charge) is uner control. The stuents coul be challenge to stuy the EC s properties in such an extene system. Particularly, they can be aske to analyze how the EC s stability epens on V. These questions are iscusse below. See also Ref. [], the problem 5-6. Figure. Depenencies of the gap with z, voltage v an its components, v an v 1 (with the opposite sign) on the charge ensity s (imensionless units). The meaning of the points a an b will be explaine in Section.. The Elastic Capacitor uner Potential Control We now consier the EC connecte to a battery, when the potential ifference V is controlle. In such an extene system, charge can be exchange between the EC an the battery. Using equations () an (3), the electric force. electr F x between the plates can be expresse in terms of V as electr F = V /εε h x where the minus inicates that the force is attractive. The charging energy for EC connecte to the battery is W = V /εεh. To stuy the electr ext properties of the extene system, we use the imensionless units of Eq. 4. The last two equations in (5) shoul be replace respectively by the equations

6 6 w z v ext = (1 ) / / z (9) an f = 1 z v / z It is worth noting that the equilibrium istance erive from the conition f = can still be expresse through the charge ensity using Eq. 6. Only now, instea of being an inepenent variable, s represents the equilibrium charge ensity corresponing to a fixe potential v, an must be expresse through v using Eq. 7: s(1 s / ) = v. The equilibrium values of s can be foun graphically as the s-coorinates of the intersection points between a horizontal v = const an the curve vs ()(Eq. 7). Fig. shows that there are two istinct regions of potential: (a) for any v< vcr there are two equilibrium solutions, such as the points a an b shown for v = v (the meaning of v will be clear from Fig. 3). (b) for v > v cr no such intersections exist, an the extene system oes not have any equilibrium state. The critical potential, v cr, separates these two regions. Now it is appropriate to ask which of the two equilibrium solutions, s a or s b, shoul be chosen. In other wors, which one correspons to a stable equilibrium? To answer this question we can stuy the profiles w () s for ifferent values of v, preliminary making in Eq. 1 a substitution ext z v/ s (see Eq. 4). Several such profiles are shown in Fig, 3. In the range < v< vcr each curve has a well (minimum) an a hump (maximum). As the potential increases, the minimum shifts towars a larger s, while the hump shifts in the opposite irection. The secon curve in Fig. 3 correspons to v = v of Fig. Comparing these two figures we fin that the smaller of two equilibrium charge ensities shown in Fig., s a, correspons to minimum of the energy, while s b correspons to maximum This fining actually answers our question: the equilibrium corresponing to the solution s= s b (Fig. ) is unstable. In other wors, the range of charges that correspons to the equilibrium of the expane system is s < s cr. In this range of charge C is positive. (1) Therefore we shoul conclue that the ifferential capacitance uner V -control is strictly positive. This result, which we obtaine specifically for the EC moel, is in fact universal an applicable to all sorts of capacitors, both macroscopic an microscopic. At critical potential (curve 3) the well an the hump merge, an the equilibrium isappears. As a result, the elastic capacitor collapses. This phenomenon is closely relate to the electric breakown in membranes. Let us now estimate a breakown voltage for a typical lipi membrane. The equilibrium thickness of the lipi bilayer (the basic component of the membranes) is h ~.5 nm, its effective elastic constant K~1-11 N/nm, an ielectric constant ε. Using Eqs. 4 an 8 we fin that Vcr =.54 K h / εε V. This value correctly represents the orer of 3 magnitue of the membrane breakown voltage, although the experimental values are typically -3 times lower. To explain those ifferences woul require a much more elaborate moel of the membrane which is beyon the scope of this paper [1]. Finally, it shoul be notice that the electric breakown escribe above is similar, but not ientical to a phenomenon wiely know as ielectric breakown [], where a material loses its insulating properties when a sufficiently strong electric fiel (exceeing the ielectric strength of the material) is applie 3. 3 It is interesting to notice that some microscopic capacitors can hol electric fiels far exceeing the typical ielectric strength values for insulators. For example, the charge ensity in contacts of metal electroes with electrolytes (liqui or soli) can reach 1 µc/ cm, which correspons to electric fiel strength F~1 9 V/m existing at microscopic istances (~1 nm) near the interface. Note that the highest ielectric strengths of macroscopic materials can harly reach 1/1 of this value.

7 7 Fig. 3: Energy profiles w(z) for the expane system for ifferent values of the potential. The curve numbere n correspons to the potential v n =n v cr /3. Acknowlegement I am grateful to Vitaly J. Felman an Peter C. Joran for their value insights, an to John Griffin, Barry Cohen an Joseph Cox for helpful comments. References [1] Partenskii M.B an Joran, P.C. Electroelastic instabilities in ouble layers an membranes, in Liqui interfaces in Chemical, Biological an Pharmaceutical Applications, eite by A.G. Volkov ( Marcel Dekker, NY, 1) pp [] Young, H. D. an Freeman, R. A. University Physics (Aison-Wesley, 1996).