Interdigitated capacitance sensors in the mm scale with ... · The simulations were carried out in...

INSTRUMENTATION Revista Mexicana de Fısica60 (2014) 451–459 NOVEMBER-DECEMBER 2014

Interdigitated capacitance sensors in the mm scale with sub-femtoFaradresolution suitable for monitoring processes in liquid films

A. Guadarrama-Santanaa,∗, A. Garcıa-Valenzuelaa, F. Perez-Jimeneza and L. Polo-ParadabaCentro de Ciencias Aplicadas y Desarrollo Tecnologico, Universidad Nacional Autonoma de Mexico,

Apartado Postal 70-186, Distrito Federal 04510, Mexico,∗Phone: +52-55-56228602 ext. 1116

e-mail: [email protected] of Missouri Department of Medical Pharmacology and Physiology, Medical School,

Dalton Cardiovascular Research Center 134 Research Park Drive Rd. Columbia, MO 65211 USA,e-mail: [email protected]

Received 28 April 2014; accepted 19 September 2014

We propose and analyze a high resolution capacitive sensor appropriate for monitoring physical or chemical processes in liquid films. Theproposed sensor is based on a planar interdigitated capacitor with planar electrodes of dimensions in the milimeter scale. The electric fieldbetween electrodes extend above the plane of the electrodes up to a few milimeters and thus permits placing a dielectric container with a liquidfilm on top and still be sensitive to changes in the liquid film. First, we present numerical calculations of the capacitance of an interdigitatedsensor as a function of the thickness and dielectric constant of a film placed on top of it using a finite element method. Then, we describe asuitable electronic system to sense with very high-resolution capacitance variations of the sensor by measuring the phase lag and amplitudechange of a sinusoidal current signal passing through it. This is accomplished by subtracting a stable sinusoidal current of the same frequencygoing through a reference device. Initially the system is balanced by adjusting the reference current to cancel out the net output current. Inthis way, we compensate parasitic capacitances due to electronics, wiring and system hardware as well. When the capacitance of the sensorelement varies the system gets unbalanced and a net current appears. The resulting current is measured with a locking-amplifier. To illustratethe sensitivity and resolution achieved by the sensing system, we present results of monitoring the capacitance of the sensor during theevaporation of liquid solvent films and discuss the time-evolution of the registered signals. The floor noise throughout the measurements wasin the order of 50 atto-Farads while the signal varied in the range of several femto-Farads.

Keywords: Capacitance; interdigitated; electrodes; dielectric; solvent; evaporation.

PACS: 84.37.+q; 77.22.-d; 64.70.F-; 07.07.Df

1. Introduction

Interdigitated Capacitance Sensors (IDCS) have been usedin different fields of industry as well as in physical, chemi-cal, biochemical and medical research [1]. They offer prac-tical advantages for material characterization by means ofelectrical capacitance measurements due to frequency depen-dent dielectric properties of polymeric layers, liquids, micro-granular and macro-granular matter.

Often, the time evolution of a physical property in a Ma-terial Under Test (MUT) can be obtained by means of a di-electric constant measuring-system [2]. In general, both therelative permittivity (which can be complex at high frequen-cies) and conductivity of a material contribute to the dielec-tric constant of a material. In an IDCS the sensitivity to vari-ations of the conductivity or permittivity depends on the pen-etration of the electric field inside a MUT contributing to thecapacitance and impedance between electrodes [3]. Depend-ing on the geometric configuration of the electrodes the elec-tric field lines can penetrate deeper or less deep. The capaci-tance of an IDCS always depends on the dielectric propertiesof the MUT and the geometry of the electrodes. The elec-trode pattern needs to be repeated several times in order tostrengthen the measured signal. In these cases, the measured

fringing field capacitance is usually considerably low. IDCShave widespread applications. Biochemical sensors for DNAdetection [4-6] and capacitive chemical sensors [7-10] con-stitute a major portion of all IDC sensors.

An important advantage of planar sensors is that only oneof the surfaces of the Material Under Test (MUT) is directlyin contact with the electrodes, leaving the other surface ofthe MUT interacting with the environment and allowing theIDCS to sense physical or chemical variables of the environ-ment. For instance, for a chemical sensor a chemically sen-sitive layer is deposited on top of IDC electrodes in order tosense the absorption of a gas, concentration of a particularchemical in the environment, moisture, heat exchange withthe surrounding medium, organic impurities or electric bio-signals in the case of biological samples [11].

When the sensitive layer (commonly a polymer) interactswith the chemicals present in the environment, the chemicallysensitive layer changes its conductivity (σ), dielectric con-stant (ε) or its effective thickness (d). The interdigitated ca-pacitance (IDC) chemical sensor can then detect a change incapacitance and/or impedance due to a change of the dielec-tric constant and thickness of the layer. IDC chemical sensorshave been investigated by many researchers because they are

452 A. GUADARRAMA-SANTANA, A. GARCIA-VALENZUELA, F. PEREZ-JIMENEZ AND L. POLO-PARADA

inexpensive to manufacture and easily integrated with othersensing components and signal processing electronics [3].

There are other applications in which IDCS are useful,such as microfluidic devices [2], Surface Acoustic Wave de-vices for consumer electronics and telecommunication ap-plications [12] and microwave integrated circuits [13-15].Also, IDCS are suitable devices for Non-Destructive Testing(NDT) [16]. As already mentioned, a physical parameter ofinterest of the MUT can be obtained indirectly by means ofthe relation with the effective dielectric function. Thus, thesedevices can also be used advantageously to sense temporalchanges of physical or chemical variables of a MUT [17].

Recently, polymer nanoparticles have been applied in awide range of fields as electronics, photonics, conducting ma-terials, sensors, medicine, biotechnology, pollution controland environmental technology. Physico-chemicals factors assolvent composition, evaporation velocity, humidity, temper-ature, etc., are involved in the solvent evaporation processwhich is a very important method for the preparation tech-nique of polymer nanoparticles, thin structured micro andnanoparticles coatings, [18,19].

Our aim is to study the sensitivity of IDCS theoreticallyand propose and implement a high-resolution electronic sys-tem to measure capacitance variations of the IDCS. Also inthis work, we are interested in understanding the sensitiv-ity and resolutions achievable with IDCS for monitoring theevaporation process of liquid films.

In Sec. 2 we present an analysis of the sensitivity of ID-CSs to dielectric layers by means of Finite Element Methodsimulations. 3D simulations are presented in order to obtainits response curves of capacitanceC as a function of the di-electric constantk and thickness d of the dielectric layer. Itis not difficult to understand that for layers thicker than someminimum valuedmin the capacitance depends only on the di-electric constant of the layer and not on its thickness. Thesimulations allow us to estimate the value ofdmin for the ex-ample that was simulated, as well as the maximum sensitivityof the interdigitated sensor. Also, the dynamic range of thesensor and the maximum capacitance value are obtained fromthe simulated curves in order to characterize the response ofthe sensor. Then in Sec. 3 we propose and describe an ex-perimental setup of a low noise Temporal Capacitive Mea-suring System (TCMS) for characterizing physico-chemicalprocesses by sensing temporal capacitance changes with in-terdigitated sensors. The signal of the TCMS is conditionedelectronically before it gets into a Lock-in Amplifier whichextracts the contribution of the MUT from the surroundingnoise, with a reference signal of 10 KHz central frequency.The capacitance values during a physico-chemical processare proportional to the imaginary part of the current signalpassing through the MUT. In order to illustrate the sensitivityand resolution of the system in Sec. 4 we present examplesof monitoring the evaporation of a solvent film on top of theIDCS. An attempt to explain the capacitance signal over timeduring the experiments is also presented. Finally, in Sec. 5we provide our conclusions along with some discussion.

2. Numerical simulations with the Finite Ele-ment Method

Interdigitated capacitors consist of conductive coplanar elec-trodes or fingers placed parallel to each other with a gap gbetween them. They are deposited on a plane surface of adielectric substrate. The surface where the electrodes are de-posited is called the sensitive surface [11]. The finger elec-trodes connected to the source voltage are called the drivingelectrodes whereas those connected to ground are referred toas the sensing electrodes.

In order to gain some insight into the performance ofIDCS to sense a dielectric layer on top of it we performed3D simulations using a Finite Element Method (FEM) of aninterdigitated capacitor with and without a dielectric layer ontop of it. We calculated the capacitance for different valuesof the dielectric constant and thickness of the dielectric layer(that is, the MUT).

The geometry of the simulated IDCS is shown inFigs. 1(a) and 1(b). We considered a six finger interdigitatedsensor deposited on the surface of a homogeneous substrateof dielectric constantks = 4.8 variable with thicknesshs,and surface area of1.6 × 1.5 cm2. The fingers electrodeswere assumed to have a thickness ofhe = 25 µm, a widthw = 1 mm, and lengthl = 9 mm. The gap between elec-trodes was assumed to beg = 1 mm. The MUT surface areawas1.4 cm2 with thickness d and dielectric constantk, bothvariables.

FIGURE 1. a) Cross section of six finger interdigitated capacitorsensor with fringing electric fields and constant parameters usedfor 3D Finite Element Method simulations. b) Complete six fingerinterdigitated capacitor sensor used for 3D Finite Element Methodsimulations.

Rev. Mex. Fis.60 (2014) 451–459

INTERDIGITATED CAPACITANCE SENSORS IN THE mm SCALE WITH SUB-FEMTOFARAD RESOLUTION SUITABLE. . . 453

The simulations were carried out in the electrostatic ap-proximation with a potential difference ofV = 1 volt be-tween driving and sensing electrodes. We calculated the netcharge on the driving electrodes,Q, and obtained the capaci-tance simply as,

C =Q

V(1)

The capacitanceC was calculated for different values of thedielectric constantk and thicknessd of the MUT to obtainthe performance parameters of the IDCS [20,21].

The charge on the driving electrodes was calculated withthe following integrals over the entire surface of the drivingelectrodes, including the electrode connecting the fingers (seeFig. 1b),

Q = εS

∫

ΣS

E · nds + εM

∫

ΣM

E · nds, (2)

whereΣS is the surface of the electrodes in contact with sub-strate of electric permittivityεS andΣM is the fraction ofthe electrode’s surface in contact with the upper medium (theMUT) of electric permittivityεM = kε0.

Graphs ofC vs k and C vs d were obtained from theFEM simulations for thickness values fromd = 100 µm tod = 2 mm and dielectric constant values fromk = 2 tok = 20. These are shown in Fig. 2 and 3 respectively. Theblack star mark represents the capacitance valueC for theinterdigitated capacitor without MUT.

In Fig. 2 we can appreciate a linear variation of the ca-pacitanceC with the dielectric constantk for different thick-nessesd of the MUT film. The minimum capacitance value,corresponding tok = 1, that is when there is no MUT overthe interdigitated capacitor. When the dielectric constantk is

FIGURE 2. C vsk curves with different thicknessesd of the MUT.The black star mark represents the capacitance valueC without aMUT, that is, fork = 1.

FIGURE 3. C vs d curves with different dielectric constantk ofthe MUT. The black star mark represents the capacitance valueCwithout a MUT, that is, fork = 1.

increased the capacitance value is also increased for anythicknessd. However, for a MUT film of thickness be-tween 1.5 mm and 2 mm and for larger values, the capaci-tance variation versusk is nearly the same. Thus, theC vsk curves show that the capacitance of the MUT tends to beindependent ofd for d ≥ 2 mm in this specific geometry.Furthermore, the sensitivitySmax for a thickness larger thandmin = 2 mm can be determined from the slope of theC vsk curves, that is

Sk,max =∂C

∂k

∣∣∣∣∣

∣∣∣∣∣dmin

. (3)

The expression (3) represents the ratio of changes incapacitanceC to changes in dielectric constantk of theMUT for a thicknessdmin and larger. For instance, withdmin = 2 mm and fork1 = 6, k2 = 8 we obtain their respec-tive capacitance valuesC1 = 2.343 pF andC2 = 2.797 pF,from curves of Fig. 1. With this data, the sensitivitySk,max

for dmin can be calculated form the slopeSk,max ≈ 0.23 pF.From Fig. 3 we see that the sensitivity to thickness vari-

ations of the MUT,Sd = ∂C/∂d, depends on its thick-ness and dielectric constant. It is larger for smaller thick-nesses. In fact it is relatively constant for any value ofk forthicknessesd smaller than about 0.5 mm. For larger thick-nesses the sensitivity decays approaching zero at a thicknessof dmin = 2 mm. The sensitivity to thickness variations isaboutSd ≈ 4 pF/mm for a dielectric constant ofk = 20 anda thickness less than 0.5 mm. This value decreases about tenfold for a dielectric constant ofk = 4.

In the case analyzed numerically here we see that thethicknessdmin is twice the gap g between electrodes,

dmin∼= 2g. (4)

The values just stated give us an idea of the sensitivity ofan interdigitated sensor to a dielectric layer on top of it. Of

Rev. Mex. Fis.60 (2014) 451–459


course these values will change for different geometries, thatis, for different values ofg andw and for larger number offingers. Nevertheless, if the geometrical aspects of the IDCSare kept constant the values just given can be easily translatedto other scales in view of what we call the isotropic scalinglaw of the electrical capacitance [22]. For instance, as longasw = g in the device,l = 9w and he is much smaller thanother dimensions Eq. (4) will be true at any scale for a sixfinger IDCS.

The value ofdmin also tells us how deep the electric fieldlines penetrate into the MUT [3]. In many applications it isnecessary to isolate the electrodes with a dielectric coating. Itis convenient that its thicknessd be as thin as possible com-pared todmin in order to use the maximum dynamic rangeof the capacitive sensor for a sample of interest placed abovethe sensitive surface of the sensor.

3. Temporal Capacitive Measuring System(TCMS)

The methods commonly used to measure electrical capaci-tance and interrogate interdigitated sensors are based on Volt-age Sensing Circuits, Current Sensing Circuits and Phase De-tection Circuits [17,21]. In all sensing methods an AC drivingvoltage is required to generate an electric field that penetratesthe MUT and the dielectric substrate as well. Perturbationsof the electric field by the MUT change the capacitance be-tween the interdigitated electrodes. In general it is possible tomonitor physical or chemical changes in the MUT from mea-suring the capacitance variations on time. The time scale inwhich one can monitor changes in the MUT will depend onthe frequency used to measure the capacitance of the IDCS.If one is using a frequencyf to measure the capacitance theIDCS will be limited to sense changes of the MUT over timesmuch larger than1/f .

As in most capacitance sensors, it will be necessary tocompensate for stray capacitances in the measuring systemin order to extract the contribution to the capacitance of theMUT placed on the sensitive surface of the IDCS [23].

The capacitance variations due to changes in a thin di-electric film on top of a IDCS will in general be very smallcompared to the capacitance of the IDCS by itself. It willalso be very small compared to the typical parasitic capaci-tances in the wiring. In order to extract the small contributionof the MUT to the capacitance it is necessary to subtract themuch larger capacitances of the IDCS, associated electron-ics and wiring. To this end we can use the method originallyproposed in Ref. [24]. This method was used successfullyby our research group to measure capacitance variations ofa sharp pointer electrode due to a dielectric surface in closeproximity [25]. Basically, the method consists of using an-other interdigitated device, ideally identical to the IDCS as areference. A driving voltage of fixed frequencyfd is appliedto both, the sensor and the reference devices. The output volt-

ages are then subtracted. The idea is that in the absence of aMUT the output signal is ideally zero.

In the original method the subtraction was accomplishedby introducing a phase difference of 180◦ to the voltage ap-plied to the reference device. Here we use a slight variation ofthe method. We simply feed the output voltages of the sensorand reference device to the non-inverting and inverting inputsof an instrumentation operational amplifier (IOA). This mod-ification makes the system more stable over time. The outputvoltage from the IOA is analyzed with a lock-in amplifier.

We refer to this condition as the system being balanced.In practice an output voltage of exactly zero is not possible,but one seeks to make the output voltage as small as possible.The advantage of using a balancing scheme is two-fold. Onthe one hand, we have that when the IDCS and the referencedevice are as similar as possible the system becomes less sen-sitive to ambient noise, since these affect both, the sensor andreference equally, and the changes in the output voltages areequal for both devices and cancel out upon subtraction. Onthe other hand, when a MUT is placed on top of the IDCS thecapacitance change and the system becomes unbalanced anda small signal appears, but it does so from a null background,allowing one to amplify the signal by a large factor withoutsaturating the amplifier. In practice, to achieve a nearly nulloutput it is necessary to have a conditioning circuit to makefine adjustments to the reference voltage, since the capaci-tance of the reference device and the parasitic capacitancesassociated with it will never be identical to those of the sen-sor device. Therefore it is necessary to introduce a compen-sation stage to balance the system and make the output signalas close to zero as possible.

A block diagram of the TCMS proposed here is shown inFig. 4. It is formed by the IDCS, a compensation stage, adifferential stage and a Lock-in amplification stage.

The compensation stage is formed of three sub-stages,Phase Compensation, Amplitude Compensation and StrayCalibration Capacitor. The function of the Phase Compen-sation sub-stage is to compensate the signal phase variableVac(ω ± Φ) due to stray capacitances of the conditioningelectronics. The Amplitude Compensation sub-stage allowsus to compensate the amplitude voltage of the Phase Com-pensation output asGVac(ω ± Φ) whereG is the amplifica-tion factor. The output voltage of this sub-stage is suppliedto the Reference Interdigitated Capacitor (RIDC) in order toobtain the same amplitude and phase as the excitation voltageVac(ω) applied to the IDCS.

The output signal of the Differential Stage goes into aLock-in Amplifier (We used a SR850 Stanford Research) tomeasure and register the temporal differential changes of out-put voltage.

The internal oscillator of the Lock-in Amplifier generatesthe excitation voltageVac = 1 Vrms at fd = 10 KHz refer-ence frequency which is used to feed the system. The sameLock-in amplifier detects the output differential-signal in a10 KHz central frequency with a 1.2 Hz bandwidth in orderto reduce surrounding noise.

Rev. Mex. Fis.60 (2014) 451–459


FIGURE 4. Block diagram of the Temporal Capacitive Measure-ment System (TCMS).

FIGURE 5. Interdigitated Capacitors used in our TCMS as the sen-sor and reference device.

The capacitance change∆C when a MUT is presentabove the IDCS is obtained from the increment of the imag-inary part of the current entering the Lock-in amplifier,∆iLock-in. We have [26],

∆C =Im[∆iLock-in]

ωVac. (5)

whereω = 2πfd.We fabricated a TCMS following the block diagram in

Fig. 4. The IDCS used in the proposed TCMS, has ten planeconductive electrodes or fingers parallel each other, five com-mon electrodes form the driving electrode and the rest formthe sensing electrode as shown in Fig. 5. The electrodesare placed over a dielectric substrate with sensitive surface of20 × 14 mm2, gapg = 1 mm, widthw = g and electrodethicknesshe = 35 µm. The electrodes were fabricated on aFR-4 Printed Circuit Board (PCB) using conventional tech-niques. The sensitivity of the IDCS can be increased withthe number of fingers and the dimensions of the parametersmentioned before [11].

In order to test and characterize the performance of theconstructed TCMS we performed some monitoring experi-ments. One of our interests in this type of TCMS’s is to

monitor physico-chemical processes in liquids. Therefore,we chose for the purpose of illustrating the resolution of theTCMS to monitor the evaporation of liquid solvent films. Wealso attempt to explain the behavior of the temporal evolutionof the capacitance signal through qualitative modeling.

4. Experimental evaluation of a TCMS

Interdigitated capacitance sensors have been used for sometime to monitor the drying kinetics of coating films [27-31].The capacitance signal is sensitive to the evaporation of thesolvent and the curing stages in the formation of a coating aswell as in adhesives and epoxies [32].

Characterization of coatings like polymer thin films is im-portant for capacitive sensors and microelectronics fabrica-tion. Curing process and evaporation solvent of the polymercoating can alter its electric and mechanical properties. So,it is important to measure changes in dielectric constant andthickness of the polymeric thin film especially in the curingprocess where the temperature is critical for the complete cur-ing of the polymer [33,34].

4.1. Monitoring the evaporation of a thin liquid film

Here we used the IDCS to monitor the evaporation of liq-uid solvents in order to characterize and evaluate the sensi-tivity and resolution of the proposed TCMS. We measuredthe time-evolution of the capacitance during the evaporationof a thin liquid film just above the interdigitated electrodes.To perform the experiments it was necessary to place a cu-vette on top of the IDCS to form and hold the liquid film ontop of the electrodes. We used a glass cuvette with 100µmthick walls. Thus, in the experiments the sensor consistedof the interdigitated electrodes with a glass slab on top. TheMUT is the liquid dispensed on the glass cuvette forming athin film on its bottom and the glass slab is the bottom wallof the cuvette as show schematically in Fig. 6.

The experimental procedure was as follows. The TCMSwas balanced without a cuvette on its top and obtained an av-erage base line of 30 aF for a 60 min period in our laboratoryat ambient temperature and humidity (23◦C and 21% Hr av-erage). The balancing was achieved by tuning the phase andamplitude of the reference signal at the input of the referencedevice. Next we placed a square glass cuvette on top of thesensing device and tuned the reference signal again to recovera minimal output current. We obtained a baseline signal with-out a noticeable drift and a rms noise around 50 aF/

√Hz. The

bottom of the cuvette was a 100µm thick glass slide of about

FIGURE 6. Cross section view of the Interdigitated CapacitanceSensor with glass cuvette and solvent sample used in experiments.

Rev. Mex. Fis.60 (2014) 451–459


FIGURE 7. Evaporation monitoring curves of Isopropyl Alcohol,Acetone and Ethanol.

FIGURE 8. Evaporation monitoring curve of Thinner.

330 mm2 sensitive area. Then a 30µl volume of solvent sam-ple was deposited in the cuvette forming a liquid film about100µm thick.

We monitored the evaporation of isopropyl alcohol, ace-tone, ethanol and thinner. The first three liquids are pure sub-stances whereas thinner is a mixture of different substances.In Figs. 7 and 8 we show the capacitance change versus timeas the liquid films evaporate. In Fig. 7 we show those for thepure liquids and in Fig. 8 that of thinner. In Fig. 7 we can seethat as a pure liquid film evaporates the capacitance increases,and it does so at an accelerating rate. At some point, theliquid film evaporates completely and the capacitance dropssharply returning to its base line. In Fig. 8 the same behavioris observed at the beginning but after a sharp drop of the ca-pacitance, the signal does not recover the base line, it starts toincrease again and then returns slowly to the base line. Thisbehavior is more complicated probably because the liquid is amixture of substances and these evaporates at different rates.

Clearly the behavior of the capacitance curves do notcorrespond simply to a dielectric film reducing its thicknesswhile the rest of the parameters remain constant, since in this

case one would see a signal decreasing in time monotonously(see Fig. 3). We monitored the weight reduction in time dur-ing the evaporation of the liquid layers presented in Fig. 7and verified it was constant during the entire process. Forcompleteness of this work, we try to provide a plausible ex-planation of the behavior of the capacitance signal in Fig. 7.

Nevertheless, the experimental results in Fig. 7 and 8serve well our purpose. We can estimate from the experi-mental graphs in the figures that the output signal TCMS hasa rms noise around 50 aF/

√Hz (equivalent to∼ Im [3pA]

of the output current). This later value was obtained fromthe output signal of the Lock-in amplifier used in our TCMS(Standford Research model SR850). This means that in a1.2 Hz bandwidth the minimum detectable change in capac-itance is in the order of 100 aF. Also, we can appreciate inFig. 7 that the base line remained basically constant for atleast 900 seconds. We must point out that this resolution andstability was achieved at ambient conditions without any spe-cial care in isolating the TCMS from the environment.

4.2. On the behavior of the capacitance signal

To explain the behavior of the capacitance curves during theevaporation of the solvent films it is necessary to considerthe contribution to the capacitance change of the vapor abovethe liquid. To this end we will derive an approximate for-mula containing the leading terms for the time derivative thecapacitance during the evaporation process using an equiva-lent circuit analysis. Here we will only consider the signalof the evaporation of pure liquids shown in Fig. 7, since itis simpler. Let us refer to the interdigitated electrodes withglass slab as the sensor. We can consider the capacitance ofthe sensor as the sum of a large number of elementary ca-pacitances placed in parallel. Each elementary capacitor isformed by a thin strip of space around an electric field linegoing from one electrode to another. The electric field ofmany of these elementary capacitors emanating from a givenelectrode go above the glass slab into the liquid film and airabove it and then go back towards another electrode. The ca-pacitance of any given elementary capacitor sensitive to theliquid consists of three capacitances in series,

1C

=1C1

+1

CL+

1C2

. (6)

WhereC1 is the contribution of the electric field linestraversing the glass,CL is due to their path through the liquidfilm, C2 is due to their path through the air above the liquid.Each of the capacitors may be considered as parallel platecapacitor with effective areas and thicknesses. We can write,

C1 =ε1A

d1, CL =

εLA

dL, and C2 =

ε2A

d2. (7)

When the liquid is dispensed on top of the sensor, thedielectric constantε2 is that of air. However, when the liq-uid starts to evaporate the air above the sensor contains vapormolecules and its dielectric constant increases. On the other

Rev. Mex. Fis.60 (2014) 451–459


hand, as the liquid layer evaporates its thickness decreasesand the effective thickness of the liquid layer decreases whilethe effective thickness of the air capacitor increases. That is,the effective thicknesses involved inCL andC2 capacitorsare not independent. Assuming the path of the electric fieldlines do not change considerably upon dispensing the liquidfilm and during its evaporation we can write,

dL(t) + d2(t) = d(0)2 , (8)

whered(0)2 is equal to the effective thickness of the air ca-

pacitor without a liquid layer. When the liquid is dispensedon top of the sensordL has its maximum,dL(0). For sim-plicity we can assume that the dielectric constant of air isε0,and while the liquid layer evaporates it increases due to theincorporation of vapor solvent molecules to

ε2(t) = ε0 + ε0αmρv(t), (9)

where αm is the molecular polarizability of the sol-vent molecules andρv(t) is the number density of vapormolecules. Since the number of molecules in the vapor isequal to the number of molecules lost from the liquid, we canwrite,

ρL[dL(0)− dL(t)] + ρv(t)d2(t) = 0, (10)

whereρL is the number density of molecules in the liquidphase and can be assumed to be independent of time. Thelatter equation simply expresses that the molecules leavingthe liquid film are incorporated to the vapor. We can used2(t) = d

(0)2 − dL(t) in the latter equation. We are assuming

that the density of molecules in the liquid remains constantbut the density of molecules in the vapor layer and its thick-ness can both change in time.

Now, to understand the rate of change in capacitance asthe liquid evaporates let us take the derivative with respect totime of Eq. (6). Taking the partial derivative of Eq. (6) andusing Eq. (7) yields,

∂

∂t

(1C

)=

∂

∂dL

(1C

)∂dL

∂t. (11)

Now, the rate of change of the liquid thickness can be as-sumed constant since the rate of weight reduction in time isconstant (as we could verify with an analytical balance) andthe area covered by the liquid was constant in time. Thus wecan use,∂dL/∂t ≡ −β, whereβ is a positive constant.

Evaluating the derivative with respect todL of 1/C re-quires using the relation,

∂ε2

∂dL= ε0αm

∂ρv∂dL

, (12)

which is obtained from Eq. (9), and

∂ρv(t)∂dL

≈ −ρL

d(0)2

, (13)

which is obtained from Eq. (10). Additionally we can as-sume thatd(0)

2 À dL(t). Then if we useεL À ε2 and keeponly the leading terms we get,

∂

∂dL

(1C

)≈ 1

ε2A

(1− ε0

ε2αm

d2

d(0)2

). (14)

Then, we can approximateε0/ε2 ≈ 1 andd2/d(0)2 ≈ 1 in

the latter equation. Using this result in Eq. (11) and solvingfor ∂C/∂t from the following equality,

∂

∂dL

(1C

)= − 1

c2

∂C

∂t, (15)

finally yields,

∂C

∂t≈ C2

ε0A(αmρL − 1)β. (16)

This equation tells us that the time derivative of the capaci-tance is positive wheneverαmρL > 1 (recall thatβ is posi-tive). This latter inequality is in general true when the dielec-tric constant of the liquid is larger than 2, as can be deducedfrom Clausius-Mossotti equation [35],

ρLαm =3(εL − 1)εL + 2

, (17)

whereεL is the dielectric constant of the liquid. Now, it isknown that the Clausius-Mossotti relation is accurate for non-polar liquids and no so accurate for polar liquids [36]. But,for polar liquids the molecular polarizability is larger thanthat predicted by the Clausius-Mossotti relation. Therefore,we conclude thatρLαm is in general larger than one when-ever the dielectric constant is larger than 2, and in these casesthe rate of change of the capacitance given by Eq. (16) ispositive.

Additionally, Eq. (20) tells us that the rate of change ofthe capacitance increases in time since it is proportional toC2

andC increases with time. This behavior is maintained untilall the liquid evaporates and the vapor layer diffuses away.At this point the capacitance signal should drop to its initialvalue before the liquid was dispensed over the sensor. Thatis, the capacitance signal returns to its base line. As alreadysaid this is the behavior of the capacitance signal shown inFig. 7.

In summary, our analysis indicates that the concentrationof vapor molecules above the evaporating liquid is respon-sible for the rather surprising behavior of capacitance signalobserved in the experiments.

5. Summary and Conclusions

We studied the sensitivity and resolution of interdigitated ca-pacitance sensors to sense a liquid film on top of it. In sen-sors with a number of electrodes between 5 and 10 with di-mensions in the range of millimeters, the capacitance willvary in the range of tens of femto-Farads due to changes in

Rev. Mex. Fis.60 (2014) 451–459


layer of thickness in the range of millimeters. To sense smallvariations of the thickness and dielectric constant of a liquidfilm and its surroundings, a resolution in the sub-femtoFaradrange is required.

We proposed and demonstrated experimentally a differ-ential capacitance measuring system providing a resolutionin capacitance variations in the 100 atto-Farad range and adynamic range in the femto-Farad range. This resolution anddynamic range is sufficient to sense variations in a dielectriclayer on top of an interdigiated sensor with ten electrodes ofwidth 1 mm and length 9 mm, separated by a gap of 1 mm.We illustrated the performance of the developed system bymonitoring the evaporation of a liquid layer in a glass cuvetteon top of the sensor. The results show experimentally that the

system provides sufficient resolution and stability to monitorthe process under ambient laboratory conditions. However,the behavior of the capacitance signal during the evapora-tion process was rather unexpected, and thus, we also ana-lyzed these particular experiments with an equivalent circuitapproach in order to provide a plausible explanation of thebehavior of the signal.

Acknowledgments

The authors want to thank Direccion General de Asuntos delPersonal Academico (DGAPA) from Universidad NacionalAutonoma de Mexico for financial support of AGS’s post-doctoral work.

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