bachelor assignment modelling and …

May 27, 2019

BACHELOR ASSIGNMENT

MODELLING ANDCHARACTERIZATIONOF A RELATIVE PER-MITTIVITY SENSOR

Jannes Bloemert – s1720554

Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS)Chair

Exam committee:dr.ir R.J Wiegerink, dr.ir R.A.R van der Zee, T.V.P Schut MSc

Department:Integrated Devices and Systems

Chapter 1

Abstract

The relative permittivity is an important liquid property which can play a big role in determining thecomposition of liquid mixtures, for example in determining the quality of oil or the compositions of drugsdelivered by intravenous therapy. Researchers of the University of Twente have build a fully integratedmicrofluidic measurement system, from which the relative permittivity sensor is a part.[3] This paperfocuses on the physical behaviour of this sensor when measuring with non-electrolytic en electrolyticliquids. In this paper impedance models are proposed from which the gain is analytically calculated andcompared to measurements. The measurements show the potential of determining the liquid capaci-tance, and thus the relative permittivity, for both non-electrolytic and electrolytic liquids.

Contents

1 Abstract 2

2 Introduction 6

3 Analysis 73.1 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1 Without a dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.2 With a dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Electric double layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Ionic conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Impedance model of liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4.1 Parallel plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.2 Parallel plates with a glass layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5 Measurement circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5.1 Parallel plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5.2 Parallel plates with a glass layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Simulations 17

5 Results and Discussion 195.1 Measurement circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Parallel plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Non-electrolytic liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2.2 Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Parallel plates with glass layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3.1 Non-electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3.2 Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Conclusion 29

7 Outlook 31

List of Figures

3.1 Schematic of a dielectric placed inside a capacitor.[10] . . . . . . . . . . . . . . . . . . . . 83.2 Schematic of the electric double layer.[9] . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Graph of the molar conductivity with respect to the square root of the concentration, for

aqeuous KCl (a) and NaCl solutions (b).[5] . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Drawing of a parallel plate capacitor filled with a liquid as dielectric. . . . . . . . . . . . . . 113.5 Model of a parallel plate capacitor with a non-electrolytic liquid dielectric. . . . . . . . . . . 113.6 Model of a parallel plate capacitor with an electrolytic liquid dielectric. . . . . . . . . . . . 123.7 Drawing of a parallel plate capacitor with the electrodes covered by a glass layer, filled

with a liquid as dielectric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.8 Model of a parallel plate capacitor with a glass layer and a non-electrolytic liquid dielectric. 133.9 Model of a parallel plate capacitor with an electrolytic liquid dielectric. . . . . . . . . . . . 133.10 Schematic of the measurement circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.11 Gain and phase plot of the transfer function for the measurement circuit with a non-

electrolytic liquid, where Rf = 34MΩ, Rx = 10MΩ and Cf = 500pF . . . . . . . . . . . . . 143.12 Gain and phase plot of the transfer function for the measurement circuit with an elec-

trolytic liquid and varying liquid resistance, where Rf = 1200Ω, Cx = 800pF , Cf = 820pFand Cdl = 30µF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.13 Gain and phase plot of the transfer function for the measurement circuit with an elec-trolytic liquid and varying electric double layer capacitance, where Rf = 1200Ω, Rx =100Ω, Cx = 800pF and Cf = 820pF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.14 Gain and phase plot of the transfer function for the measurement circuit with an elec-trolytic liquid and varying liquid capacitance, where Rf = 1200Ω, Rx = 100Ω, Cf = 820pFand Cdl = 30µF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1 Simulation of the gain and phase in case of a parallel plate setup filled with an electrolyte,where Rf = 1200Ω, Cf = 820pF , Rx = 500Ω, Cx = 870pF and Cdl = 1µF . . . . . . . . . . 17

4.2 Simulation of the gain and phase in case of a parallel plate, with the elctrodes coveredby a glass layer, setup filled with an electrolyte, where Rf = 120kΩ, Cf = 100pF , Rx =60kΩ, Cx = 50pF and Cdl = 1µF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.1 Bode diagram of the measured gain and phase for different capacitor values. . . . . . . . 195.2 Plot of the theoretical and measured gain for different capacitor values. . . . . . . . . . . 205.3 Bode diagram of the measured gain and phase for different non-electrolytic liquids. . . . . 215.4 Plot of the measured and fitted capacitance with respect to the relative permittivity. . . . . 215.5 Gain and phase plot of the measurement with different concentrations of KCl solution,

where Rf = 1200Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.6 Plot of the resistance with respect to concentration for different KCl solutions. . . . . . . . 225.7 Plot of the theoretical and measured electric double layer capacitance for different KCl

concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.8 Bode diagram for different concentrations of KCl solutions in the nano molar range, where

Rf = 1200Ω and Cf = 820pF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.9 Bode diagram of the measured gain and phase for different non-electrolytic liquids with

the parallel plates covered with a glass layer, where Rf = 13.6MΩ and Cf = 100pF . . . . 255.10 Plot of the measured in series capacitance with respect to the relative permittivity. . . . . 255.11 Bode diagram of the measured gain and phase for different concentrations of KCl solution

with the parallel plates covered with a glass layer, where Rf = 1200Ω. . . . . . . . . . . . 265.12 Plot of the measured resistance for different concentrations of KCl solution. . . . . . . . . 265.13 Plot of the measured equivalent capacitance for different concentrations of KCl solution. . 275.14 Bode diagram for different concentrations of KCl solutions in the nano molar range with

the parallel plates with a glass layer, where Rf = 120kΩ and Cf = 100pF . . . . . . . . . . 28

List of Tables

5.1 Relative permittivity of several liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Chapter 2

Introduction

This bachelor assignment is about modelling and characterization of a relative permittivity sensor, tounderstand the behaviour of different liquids inside such a sensor. The permittivity of a material is ameasure for the polarization of that material within an external electric field, where the relative permittiv-ity is the permittivity with respect to the permittivity of free space. The relative permittivity of a materialdepends on the material itself, so it is a material property. Researchers of the University of Twente havebuild a fully integrated microfluidic measurement system to measure different properties of a liquid, thedensity, viscosity, mass flow, heat capacity and the relative permittivity. [3] This device was designed tomeasure the composition of liquid mixtures, which can be used to determine the quality of oil or deter-mine the flow and composition of drugs delivered by intravenous therapy.During previous research the relative permittivity sensor of the fully integrated microfluidic measure-ment system was tested. [4] This research showed divergent behaviour when measuring with water,the capacitance measured was much lower than expected. In this assignment the underlying physicsof that divergent behaviour will be investigated. It was expected that this divergent behaviour, whichwas seen when measuring with water, is due to the presence of ions in water. These ions could af-fect the measurement of the relative permittivity by the increase of conductivity and the formation of anelectric double layer. These aspects will be investigated in depth during this assignment. This will bedone by proposing impedance models for two different setups, one for a parallel plate setup without aglass layer and one with a glass layer on the electrodes. These impedance models will be investigatedanalytically and then validated by measurements, to show the accuracy of the impedance models andto get an understanding of the behaviour. When this behaviour can be understood completely, a setupmore similar to the actual sensor could be investigated, so that eventually the behaviour seen whenmeasuring with the sensor itself could be explained and a measurement circuit capable of measuringwith non-electrolytic and electrolytic liquids could be designed.

Chapter 3

Analysis

3.1 Dielectrics

3.1.1 Without a dielectric

A dielectric is a material placed in an electric field. This causes a decrease in electric field, resulting ina higher capacitance. But first lets consider a parallel plate capacitor without a dielectric between theplates. The divergence of the electric field is described by the Maxwell equation shown in equation 3.1,where Q is the charge enclosed.

∇ · E =Q

ε0(3.1)

The Maxwell equation above for the divergence of the electric field can be written in integral form,Gauss law, shown in equation 3.2. When the electric field is assumed perpendicular to the parallelplates the electric field can be written as shown in equation 3.3.

‹S

E · ndS =Q

ε0(3.2)

E =Q

ε0A(3.3)

The electric field is the divergence of the potential, shown in equation 3.4. Then to calculate thepotential from the electric field the electric field should be integrated along its path, shown in equation3.5.

E = −∇ · φ (3.4)

φ = −ˆ b

a

E · ds (3.5)

When assuming that the electric field lines are perpendicular to the plates the potential can bedescribed by the product of the electric field with the distance between the plates. Then the capacitancecan be calculated by dividing the charge by the potential, shown in equation 3.6.

C =Q

V=

Q

Ed=ε0A

d(3.6)

3.1.2 With a dielectric

A dielectric, in an ideal case, does not conduct electricity when placed inside an electric field. Howeverit does influence the capacitance when placed between the plates of a capacitor. In this section thebehaviour of a dielectric placed in an electric field will be discussed. In figure 3.1 a dielectric placedbetween the plates of a parallel plate capacitor can be seen.

When the charge at the plates in case of a dielectric between the plates is equal for the case whenonly air is present between the plates, the electric field inside the dielectric must be reduced. This iscaused by the polarization of the dielectric, meaning that positive and negative charges get alignedinside the electric field. As a result a net charge can be observed at the edges of the dielectric, whichcan be seen from figure 3.1. From equation 3.3 it can be seen that the electric field depends on thecharge enclosed, in case of air between the plates, this is only the charge present at the plates. Butin case of a dielectric the edges of the dielectric also have a net charge which counteracts the chargeat the plates. Then the electric field can be described by equation 3.7, where pol is the net charge at

Figure 3.1: Schematic of a dielectric placed inside a capacitor.[10]

the edge of the dielectric due to its polarization. But this only holds when charge can not move freelythrough the dielectric, but dipoles only can rotate, and the polarization is uniform inside the dielectric.

E =Q−Qpolε0A

(3.7)

To obtain an insight of the electric field inside the dielectric the polarization charge is chosen asvector P. Then the vector P can be described by equation 3.8, where χ is the electric susceptibility ofthe dielectric. Then when equation 3.8 is filled in equation 3.7 and rewritten equation 3.9 is obtained.From equation 3.9 it can be seen that the electric field is reduced by a factor 1

1+χ when a dielectric isplaced between the plates of a capacitor, depending on the type of dielectric.

P = χε0E (3.8)

E =Q

ε0A(1 + χ)(3.9)

The capacitance when a dielectric is present can be calculated the same way as described abovein case of only air. This results in equation 3.10, where 1

1+χ is substituted by εr which is the relativepermittivity.

C =εrε0A

d(3.10)

3.2 Electric double layer

In the section above the influence of a dielectric on the capacitance is discussed. However for that caseit is assumed that charges are not able to move through the material. But when using a liquid dielectriccontaining ions, the ions in the liquid are not immobilized, they can move. In this case an additionaleffect is taking place, the forming of an electric double layer. An electric double layer is formed at theinterface of a charged surface and an electrolytic liquid, meaning that it contains ions. The surfacecharge is partly counteracted by the Stern layer and the diffusive layer. The Stern layer is a layer ofimmobilized ions, as can be seen from figure 3.2, which get attracted by the charges at the surface.

The capacitance of the Stern layer can be approximated by the Helmholtz model. According to theHelmholtz model the capacitance of the Stern layer can be seen as a parallel plate capacitor, which isdescribed by equation 3.7 where κ is the relative permittivity of the solvent. However since the ions in theStern layer are immobilized this influences the polarization of the ions, as result the relative permittivityis influenced. The relative permittivity is often assumed constant in the Stern layer.[1]

However the diffusive layer can not be seen as a parallel plate capacitor since its potential profile isnot linear, which is expected in case of a parallel plate capacitor. To obtain a relation for the capacitanceof the diffusive layer first a relation for the potential inside of the diffusive layer should be found. Thepotential is related to space charge by equation 3.11.

∇ ·∇ · φ =−ρε0εr

(3.11)

The ions in the liquid can be seen as space charges, which obey the Boltzmann distribution, shownin equation 3.12, where n is the number density of the ions. The number density of ions depends onthe distance from the surface since the potential decays, which can also seen in figure 3.2.

na = n∞a exp(− qa∆φ

kBT

)(3.12)

From equation 3.12 the ionic distribution in the x direction can be obtained, by multiplying the numberdensity of ions with the charge number of the ions, shown in equation 3.13.

Figure 3.2: Schematic of the electric double layer.[9]

ρ(x) =∑a

qa · na(x) (3.13)

When equation 3.13 is filled in equation 3.11, for the case of a monovalent electrolyte, the solutionbecomes as shown in equation 3.14. If the first and the second term in the exponential satisfy thecondition e

kBT< 1, equation 3.14 can be reduced to equation 3.15.

d2φ(x)

dx2= −en

∞

ε0εr

(exp(− eφ(x)

kBT

)− exp

(eφ(x)

kBT

))(3.14)

d2φ(x)

dx2=

2en∞

ε0εr· eφ(x)

kBT(3.15)

When equation 3.15 is solved, the potential of the diffusive layer is described by equation 3.16,where κ is the Debye parameter, the inverse of the Debye parameter is the Debye length and is shownin equation 3.17.

φ(x) = φ(0)exp(κ) (3.16)

κ−1 =

√εrε0kBT

2e2NaC(3.17)

The capacitance of the diffusive layer of the electric double layer can be found when the differentiat-ing the charge in the diffusive layer with respect to the potential, resulting in equation 3.18. [2]

CD =εrε0κ−1

coshzeφ

2kBT(3.18)

3.3 Ionic conductivity

When ions are present in a solvent, for example water, the conductivity will be increased due to thepresence of ions. The conductivity due to ions depends on the amount of ions present and the abilityof ions to move through a liquid. So the conductivity in an ideal case can be described by equation3.19, where Λ0

i is the molar conductivity of each specie. The molar conductivity of each specie canbe described by equation, 3.20, where z is the charge number, F Faradays constant, D the diffusioncoefficient, R the gas constant and T the temperature of the liquid.

σ =∑i

Λ0iCi (3.19)

Λ0 =z2F 2D

RT(3.20)

As stated before the conductivity of an electrolyte can be described by equation 3.19. But in realelectrolytes the conductivity becomes more complicated. When the concentration of ions gets suffi-ciently low, interionic attractions will take place. Meaning that the movement of ions, the flux, will becounteracted by ions with the opposite charge, resulting in a lower conductivity, which can be seen infigure 3.3.

Figure 3.3: Graph of the molar conductivity with respect to the square root of the concentration, foraqeuous KCl (a) and NaCl solutions (b).[5]

So for low concentrations equation 3.19 does not hold due to the interionic attraction causing adecrease in molar conductivity,Λ, when increasing concentration. A relation for the decrease in molarconductivity due to interionic attraction has been found by Debye and Huckel, when assuming totaldissociation, shown in equation 3.21.[6] The term K1

D32w1 represents the electrical retardation. Because

of the time needed for the ions to be redistributed while there are ions with opposite sign in the rearof the ions. Another effect which causes a decrease in molar conductivity is due to the term K2

D12w2b.

When ions are with opposite signs are moving with respect to each other and both are carrying acertain amount of solvent with them, there will be more hydrodynamic resistance due to the movementof solvent in opposite directions.

Λ0 − Λ

Λ0=

(K1

D32

w1 +K2

D12

w2b

)√zC (3.21)

When rewriting the equation above, equation 3.21, equation 3.22 is obtained.

Λ = Λ0

(1− K1

D32

w1 +K2

D12

w2b

)√zC

)(3.22)

Then equation 3.22 can be further simplified in equation 3.23, where Λ0 is the molar conductivity atinfinite dilution, K the Kohlrauschs constant, z the charge number of the ion and C the concentration ofthe electrolyte.

Λ = Λ0(

1−K√zC)

(3.23)

So when taking into account the interionic attraction the conductivity of an electrolyte can be writtenby equation 3.24.

σ =∑i

Λ0i

(1−K

√zC)Ci (3.24)

When the conductivity of the liquid is known the resistance of the liquid in case of a certain setupcan be calculated with use of equation 3.25, where d is the distance of the conductive path, A the areaof the electrodes and σ the conductivity of the liquid.

R =d

Aσ(3.25)

3.4 Impedance model of liquids

The impedance model of a liquid depends on the setup used. Below two different setups will be dis-cussed and their impedance models for electrolytic and non-electrolytic liquids. First the impedancemodels in case of a parallel plate setup will be discussed. In that case the liquid is in direct contact withthe electrodes. Second the impedance models for a parallel plate setup with electrodes covered by aglass layer will be discussed.

3.4.1 Parallel plates

The parallel plate setup consists of two electrodes in direct contact with the liquid, which can be seenin figure 3.4. Below the impedance models for electrolytic and non-electrolytic liquids will be described.First lets consider the impedance model for a non-electrolytic liquid.

Figure 3.4: Drawing of a parallel plate capacitor filled with a liquid as dielectric.

A non-electrolytic fluid does not contain ions, meaning that there is no electric double layer presentwhen used as a dielectric between two parallel plates. In that case the impedance due to the liquid canbe described by a capacitance in parallel with a resistor, shown in figure 3.5. The capacitance is linearlyrelated to the relative permittivity of the liquid between the plates, as can be seen from equation 3.7.The resistance is the inverse of the conductivity of the liquid.

Figure 3.5: Model of a parallel plate capacitor with a non-electrolytic liquid dielectric.

The impedance of an electrolytic liquid becomes more complex in case ions are present in the liquid.When ions are present in the liquid an electric double layer is formed, since ions can freely move insidethe liquid. In that case an additional capacitance is created by the electric double layer at the surfacesof the plates. In figure 3.6 the impedance model can be seen in case of ions present in the liquid, whereCx is the capacitance of the solvent, Rx the resistance of the liquid and Cdl the electric double layercapacitance. The capacitance due to the solvent remains the same, assuming that the ions present inthe solution do not influence the relative permittivity of the liquid. But the resistance of the liquid is notonly due to the conductivity of the solvent but also due to the ions present in the solution. The resistanceof the solution can be calculated as discussed in the Ionic conductivity section.

Figure 3.6: Model of a parallel plate capacitor with an electrolytic liquid dielectric.

3.4.2 Parallel plates with a glass layer

Figure 3.7: Drawing of a parallel plate capacitor with the electrodes covered by a glass layer, filled witha liquid as dielectric.

The setup with parallel plates with a glass layer consists of two parallel plates covered by a glasslayer, which can be seen in figure 3.7. The impedance models for electrolytic and non-electrolytic liquidsfor the parallel plates without a glass layer only differ from this setup by the in series capacitance of theglass layer. So both electrolytic and non-electrolytic liquid impedance models have to be extended withan in series capacitance, resulting from the glass layer, which can be seen in figure and .

3.5 Measurement circuit

In order to measure the relative permittivity of a liquid, the gain of the measurement circuit shown infigure 3.10 is measured. The measurement circuit is an inverting amplifier with a gain shown in equation3.26. When the feedback impedance is known the impedance of the liquid can be obtained. Which canbe described according to the impedance models above, depending of the presence of ions.

A = − ZfZm

(3.26)

Figure 3.8: Model of a parallel plate capacitor with a glass layer and a non-electrolytic liquid dielectric.

Figure 3.9: Model of a parallel plate capacitor with an electrolytic liquid dielectric.

Figure 3.10: Schematic of the measurement circuit

3.5.1 Parallel plates

In case of a non-electrolytic liquid and a capacitor and a resistor in parallel as feedback impedance, thegain of the measurement circuit can be described by equation 3.27.

A = −CxCf

s+ 1RxCx

s+ 1RfCf

(3.27)

From equation 3.27 it can be seen that there will be two cutoff frequencies, one will be due to theresistance and capacitance of the liquid itself and one will be due to the feedback impedance. Alsoit can be seen that the passband gain is the relation between the capacitance of the liquid and thefeedback capacitance. Figure 3.11 shows the gain and phase of the transfer function shown in equation3.27, for different values of Cx.

So for a non-electrolytic liquid the relative permittivity can be easily obtained from the passband gain,which is the relation between the liquid capacitance and the feedback capacitance, since the relationbetween the capacitance and the relative permittivity is linear, which can be seen from equation 3.10.

When considering electrolytic liquids the capacitance of the electric double layer should also betaken into account. Also the conductivity of the liquid gets higher when measuring an electrolytic liquid,due to the ions present in the liquid. When the impedance model for an electrolytic liquid, consideringan parallel plate setup without a glass layer, is filled in equation 3.26 and the feedback impedance ischosen to be a resistor in parallel with a capacitor the gain of the measurement circuit is shown inequation 3.28, where Ceq is described by equation 3.29.

A = −CeqCf

s(s+ 1RxCx

)

(s+ 1RfCf

)(s+ 2Rx(Cdl+2Cx)

)(3.28)

Figure 3.11: Gain and phase plot of the transfer function for the measurement circuit with a non-electrolytic liquid, where Rf = 34MΩ, Rx = 10MΩ and Cf = 500pF .

Ceq =CdlCx

Cdl + 2Cx(3.29)

The gain and phase of the transfer function shown in equation 3.28 depends on multiple propertiesof the liquid, the capacitance of the liquid, the resistance of the liquid and the capacitance of the electricdouble layer. The resistance of the liquid depends on the concentration of ions, which is described insection Ionic conductivity. Also the capacitance due to the electric double layer depends on the con-centration of ions. Below the influence of all the liquid parameters will be discussed separately.

First the influence of the electric conductivity of the electrolytic liquid on the gain and phase of thetransfer function of the measurement circuit will be investigated. From equation 3.28 it can be seen thattwo cutoff frequencies depend on the resistance of the liquid. And at low frequencies is the gain therelation between the resistance of the liquid and the resistance of the feedback impedance, which canbe seen in figure 3.12.

Another property of an electrolytic liquid is the electric double layer capacitance, which influenceon the transfer function of the measurement circuit is shown in figure 3.13. From the transfer function,equation 3.28 it can be seen that the only influence of the electric double layer capacitance is on one ofthe cutoff frequencies, this shift in cutoff frequency can be seen in figure 3.13.

Then the property of interest is the capacitance of the liquid itself, which is linearly related to therelative permittivity according to equation 3.10. To be able to measure this capacitance when measuringwith an electrolytic liquid, the influence on the transfer function of the measurement circuit should beinvestigated. From equation 3.28 and 3.29 it can be seen that the passband gain is the relation betweenthe feedback capacitance and the capacitance of the electric double layer and liquid itself in series.Another influence of the liquid capacitance can be seen on two cutoff frequencies, which can also beseen in figure 3.14. However the cutoff frequency which depends on the sum of the electric double layerand the liquid capacitance is not much influenced by the liquid capacitance, since the electric doublelayer capacitance can be assumed much higher.

3.5.2 Parallel plates with a glass layer

In case the electrodes of the parallel plate setup are covered with a glass layer the impedance modelchanges, as shown in figure 3.8. This results in a different gain and phase. The gain in this case canbe described by equation 3.30, with Cs shown in equation 3.31

Figure 3.12: Gain and phase plot of the transfer function for the measurement circuit with an electrolyticliquid and varying liquid resistance, where Rf = 1200Ω, Cx = 800pF , Cf = 820pF and Cdl = 30µF

Figure 3.13: Gain and phase plot of the transfer function for the measurement circuit with an electrolyticliquid and varying electric double layer capacitance, where Rf = 1200Ω, Rx = 100Ω, Cx = 800pF andCf = 820pF .

A = −CeqCf

s(s+ 1RxCx

)

(s+ 1RfCf

)(s+ 2Rx(Cg+2Cx)

)(3.30)

Figure 3.14: Gain and phase plot of the transfer function for the measurement circuit with an electrolyticliquid and varying liquid capacitance, where Rf = 1200Ω, Rx = 100Ω, Cf = 820pF and Cdl = 30µF .

Ceq =CgCx

Cg + 2Cx(3.31)

When the parallel plate setup with a glass layer on the electrodes is filled with an electrolyte, theimpedance model has to be extended with an extra in series capacitance, as shown in figure 3.9. Thenthe gain of the measurement circuit can be described by equation 3.32, with Cs being the in seriescapacitance of the glass layer and the electric double layer, equation 3.33

A = − CsCxCf (Cs + 2Cx)

s(s+ 1RxCx

)

(s+ 1RfCf

)(s+ 2Rx(Cs+2Cx)

)(3.32)

Cs =CdlCgCdl + Cg

(3.33)

Chapter 4

Simulations

The working of the measurement circuit has been validated by use of simulations, with use of LTspice,to show the limitations of the measurement circuit. In figure 4.1 the simulation results of a parallel platesetup without a glass layer and filled with an electrolyte can be seen. It can be seen that approximatelyaround 1MHz the measurement circuit shows unexpected behaviour, when compared with the analyticalresults, shown in figure 3.14. The same holds when simulating with the parallel plate setup with a glasslayer, which can be seen in figure 4.2. Due to this behaviour, the relation between the liquid andfeedback capacitance can not be measured accurately.

Figure 4.1: Simulation of the gain and phase in case of a parallel plate setup filled with an electrolyte,where Rf = 1200Ω, Cf = 820pF , Rx = 500Ω, Cx = 870pF and Cdl = 1µF .

This unexpected behaviour is due to the limitations of the operational amplifier, which should betaken into account when measuring with electrolytes. To minimize the effect of the limits of the opera-tional amplifier, an operational amplifier capable of measuring at high frequencies should be chosen.

Figure 4.2: Simulation of the gain and phase in case of a parallel plate, with the elctrodes covered bya glass layer, setup filled with an electrolyte, where Rf = 120kΩ, Cf = 100pF , Rx = 60kΩ, Cx = 50pFand Cdl = 1µF .

Chapter 5

Results and Discussion

5.1 Measurement circuit

The measurement circuit used can be seen in figure 3.10, with Zm the impedance model, depending onthe type of liquid and measurement setup, and Zf the feedback impedance. To test the working of themeasurement setup it is tested with different capacitor values first. Then, according to equation 3.27,the gain is a relation between the to be measured capacitance and the capacitance of the feedbackimpedance. The results with different capacitor values and a feedback capacitance of 470pF can beseen in figure 5.1. It can be seen that the gain increases when increasing the capacitance of the tobe measured capacitance, which is in accordance with equation 3.27. To validate the voltage gain withrespect to the measured capacitance, the theoretical and measured voltage gain has been plotted infigure 5.2. The equation for the theoretical voltage gain is the passband gain of equation 3.27, whichcan be seen in equation 5.1. From figure 5.2 it can be seen that the measured voltage again is inaccordance with the theoretical voltage gain, except from small deviations. These deviations could becaused by the deviation of the capacitor value.

A =CxCf

(5.1)

Figure 5.1: Bode diagram of the measured gain and phase for different capacitor values.

5.2 Parallel plates

5.2.1 Non-electrolytic liquids

Now the measurement circuit has been tested, and did show behaviour according expectation, thecapacitance of a parallel plate capacitor filled with different non-electrolytic liquids can be measured.

Figure 5.2: Plot of the theoretical and measured gain for different capacitor values.

In figure 5.3 the gain and phase of the measurements with air, isopropanol, acetone and ethanol areshown. The relative permittivities of these liquids can be found in table 5.1. From figure 5.3 it canbe seen that it shows similar behaviour as was expected according the proposed impedance model ofparallel plates filled with a non-electrolytic liquid, which can also be seen in figure 3.11.

Liquids εrIsopropanol 17.9Acetone 20.7Ethanol 24.5

Table 5.1: Relative permittivity of several liquids.

As can be seen from equation 3.27, the passband gain is the relation between the liquid capacitanceand the feedback capacitance. So with that equation the liquid capacitance can be calculated from thegain for each liquid. The calculated capacitances are plotted with respect to their relative permittivities,shown in figure 5.4. It can be seen that the capacitance of the liquid increases linearly when the relativepermittivity is increased, which is also in accordance with equation 3.10. The calculated capacitances,for the liquid mentioned above, is linearly fitted. From this fit the capacitance in parallel and the areaof the liquid capacitor, when the distance between the plates is known, can be obtained. The parallelcapacitance is due to the glass layer used as separation between the two electrodes. Since capacitorsin parallel can be summed, the offset of the the fit is the capacitance of the capacitor in parallel, whichhas been found to be 99 pF . The area of the liquid capacitor could be calculated from the slope of thefit and was found to be 4.54 cm2.

5.2.2 Electrolytes

To validate the impedance model for electrolytic liquids, measurements have been done with differentconcentrations of KCl. First the behaviour of the resistance of the liquid and the electric double layer willbe discussed. To see the effect of concentration on the resistance and electric double layer capacitance,measurements have been done with KCl solutions in the range of milli molar. At low frequencies thegain and phase is dependent on the resistance and electric double layer capacitance, which can beseen from figure 3.12 and 3.13. Since the capacitance of the feedback impedance has only influence athigher frequencies it can be left out, so the feedback impedance consists only of a resistor in this case.

From figure 5.5 it can be seen that the resistance decreases when the KCl concentration increases,since the passband gain at lower frequencies is determined by Rf

Rx. The resistance with respect to

their concentration can thus be calculated from the passband gain, which can be seen in figure 5.6. Itcan be seen that the measured resistance does show expected behaviour. However it seems that themeasured resistance is higher than the theoretical resistance, which has been calculated with use ofequation 3.25, without taking into account the interionic attraction. So this deviation could be due to

Figure 5.3: Bode diagram of the measured gain and phase for different non-electrolytic liquids.

Figure 5.4: Plot of the measured and fitted capacitance with respect to the relative permittivity.

interionic attraction causing a lower conductivity than expected or due to the output resistance of thegain-phase analyzer.

When the resistance of the electrolyte is known the electric double layer can be calculated from thecutoff frequency, from equation 3.28 it can be seen that one cutoff frequency depends on the resistanceof the liquid and the electric double layer capacitance. Since the electric double layer capacitance is inthe micro farad range and the liquid capacitance in the pico farad range, the liquid capacitance can be

Figure 5.5: Gain and phase plot of the measurement with different concentrations of KCl solution, whereRf = 1200Ω

Figure 5.6: Plot of the resistance with respect to concentration for different KCl solutions.

ignored for the calculation of the electric double layer capacitance. The cutoff frequency can be foundat the -3 dB point, when the passband gain is subtracted. The electric double layer capacitance canthen by calculated with use of equation 5.2, where Rx is the resistance of the liquid and fc the cutofffrequency due to the resistance and the electric double layer capacitance.

CEDL =1

πRxfc(5.2)

In figure 5.7 the measured and theoretical electric double layer capacitance for different concentra-tions of KCl solution can be seen, calculated from the resistance of the liquid and the cutoff frequency.The theoretical electric double layer capacitance has been calculated with use of equation 3.18, whereφ is the potential after the Stern layer. In the Stern layer the potential decays linearly, meaning thatthe potential at the beginning of the diffusive layer is lower than the potential applied at the electrodes.Since no method could be found to calculate this potential, a potential was assumed. The potential atthe beginning of the diffusive layer was assumed to be 60 mV , with respect to 100 mV applied to theelectrodes. Also the capacitance due to the Helmholtz layer should be taken into account. The thick-ness of the Helmholtz layer is determined by the radius of the ions, which was in case of KCl found to be160 pm. Then the capacitances of the two layers should be taken into series, which gives the theoreticalresult as shown in figure 5.7.

Figure 5.7: Plot of the theoretical and measured electric double layer capacitance for different KClconcentrations.

According to figure 3.14 the liquid capacitance could be measured at high frequencies, indepen-dent of the resistance and electric double layer capacitance. So a measurement has been done withthree different KCl concentrations, 100nM, 200nM and 300nM. But since these concentrations are verylow the concentrations of H+ and OH− should also be taken into account, which increases the ionicstrength with 200nM, when assuming a pH of 7. In figure 5.8 the results of the measurement can beseen. It can be seen that the gain is higher for lower frequencies, meaning that the resistance is lowerfor low concentrations. This could be due to the interionic attraction playing a significant role at suchlow concentrations, which is discussed in the Ionic conductivity section. The gain does seem to tend tothe same value after 100kHz for all three concentrations. But can not accurately be seen from the bodediagram, since the measurement circuit can only measure up to approximately 1MHz. This behaviouris due to the limitations of the measurement circuit, which can also be seen from the simulations, figure4.1. Also the feedback impedance influences the measurement, since one of the cutoff frequenciesdepends on it. This can also be seen from figure 5.8, around 100kHz. Since the relation betweenthe resistance of the feedback impedance and the liquid determines the gain at lower frequencies, thefeedback resistance should be chosen such that it does not exceed the maximum amplification of ap-proximately 40dB, otherwise it would be clipping. Also the feedback capacitance should be chosen suchthat the gain due to the relation between the capacitance of the feedback and the liquid does not exceedits maximum. So the ability of the measurement circuit to measure the liquid capacitance, in case of anelectrolytic liquid, depends severely on the electrolyte concentration.

Figure 5.8: Bode diagram for different concentrations of KCl solutions in the nano molar range, whereRf = 1200Ω and Cf = 820pF .

5.3 Parallel plates with glass layer

5.3.1 Non-electrolytes

When considering a parallel plate setup with a glass layer on the electrodes filled with a non-electrolyticliquid the impedance model differs from the parallel plate setup without a glass layer, shown in figure3.8. Due to the capacitance of the glass layer in series the measured capacitance is not linear anymore,which can be seen from the passband gain of equation 3.30. A measurement has been done with theliquids shown in table 5.1. The bode diagrams for the different liquids are shown in figure 5.9.

From figure 5.9 it can be seen that after approximately 100kHz the passband gain according equation3.30 is reached. The capacitance of the glass layer and the liquid, shown in equation 3.31, can becalculated from the passband gain, shown in figure 5.10. The measurement data has been fitted withthe fit described in equation 5.3, where C0 is the capacitance when the plates would be filled with air, Cgthe parallel capacitance due to the glass layer used as separator, εg the relative permittivity of the glassand εr the relative permittivity of the solvent. The relative permittiviy of the glass layer was assumed tobe 7.

C =C0εgC0εr

C0εg + 2C0εr+ Cg (5.3)

From C0 the area of the liquid capacitor could be determined, since the area is related to the ca-pacitance of a parallel plate structure without a dielectric by equation 3.6. The area and parallel glasscapacitance have been found to be 5.03 cm2 and 15 pF ,respectively. The area of the liquid capacitorin this case is a bit lower than the area of the liquid capacitor in case of no glass layer. But this can bedue to the fact that each setup is constructed by hand or that the value of the relative permittivity differsfrom the value used. However to calculate the area more accurate, more data points are needed for thefit.

5.3.2 Electrolytes

To validate the impedance model for a parallel plate setup with electrodes covered by a glass layer,measurements have been done with different concentrations of KCl solution. According to equation3.32 and 3.33 the capacitance of the glass and electric double layer should be taken in series. Then thecutoff frequency is the sum of the capacitance of the glass layer and electric double layer in series andthe capacitance of the liquid twice. Since the electric double layer capacitance is much higher than the

Figure 5.9: Bode diagram of the measured gain and phase for different non-electrolytic liquids with theparallel plates covered with a glass layer, where Rf = 13.6MΩ and Cf = 100pF .

Figure 5.10: Plot of the measured in series capacitance with respect to the relative permittivity.

capacitance due to the glass layer, the in series capacitance should tend to the capacitance of the glasslayer, which is in the pico farad range. In figure 5.11 the bode diagrams for different concentrations ofKCl solution can be seen.

It can be seen that the gain, which is a measure for the resistance of the liquid, is in accordance withthe expectation, the gain increases when the concentration increases. From the gain the resistance of

Figure 5.11: Bode diagram of the measured gain and phase for different concentrations of KCl solutionwith the parallel plates covered with a glass layer, where Rf = 1200Ω.

the liquid can be calculated, shown in figure 5.12.

Figure 5.12: Plot of the measured resistance for different concentrations of KCl solution.

From figure 5.12 it can be seen that the resistance does show the same shape as the resistancemeasured with the parallel plate setup without a glass layer, shown in figure 5.6. But the resistance incase of the glass layer is much higher than without a glass layer. In practice the setup is totally dippedin the liquid, causing a conductive path from the back of the electrodes, since the resistance of the glass

layer is in the range of mega ohms, which are not covered with a glass layer. Since the path from theback of the electrodes is longer than the path from the front of the electrodes, the conductivity is lower,resulting in a higher resistance, which can be seen in equation 3.25.When the resistance of the liquid is known the cutoff frequency, which depends also on the electricdouble layer capacitance, should be determined. This can be done by determining the frequency at the-3dB point. Then the in series capacitance of the glass layer and the electric double layer in parallel withthe liquid capacitance, the equivalent capacitance, shown in equation 5.4, can be determined, shown infigure 5.13.

Ceq =CdlCgCdl + Cg

+ 2Cx (5.4)

Figure 5.13: Plot of the measured equivalent capacitance for different concentrations of KCl solution.

According to expectation the measured capacitance should be in the pico farad range, due to thein series capacitance of the glass layer. However from figure 5.13 it can be seen that the measuredcapacitance is in the micro farad range. This suggests that either the impedance model of the setupis not accurate or there is another contribution to the electric double layer capacitance, apart from thepotential applied to the electrodes. The most explainable reason for this unexpected behaviour couldbe due to an additional contribution of the glass layer to the electric double layer capacitance. Becausewhen the glass layer gets in contact with water hydrogen atoms get dissolved in water, causing surfacecharge on the glass layer. [7] Then this surface charge causes an additional potential, which can beseen from equation 3.11, resulting in a higher capacitance.

From figure 5.14 it can be seen that the cutoff frequency due to the feedback impedance is lowerthan in case of the parallel plate setup without a glass layer. Since the conductivity in case of a glasslayer is lower, because of the conductive path from the back, the feedback resistance is chosen tobe 120kΩ. This shift in cutoff frequency is desirable, since the relation between the liquid and thefeedback capacitance should be measured after that cutoff frequency. According to figure 3.12 and3.13 the passband gain, which is the relation between the liquid and feedback capacitance, should notbe influenced by the resistance of the liquid nor the electric double layer capacitance. But this passbandgain should be measured after the cutoff frequency of the liquid itself. From figure 5.14 it can be seenthat the gain in all three cases tends to the same value, after the cutoff frequency of the liquid itself.The cutoff frequency due to the electric double layer capacitance and the resistance of the liquid isnot visible in the range in which is measured. However the cutoff frequency is approximately a factorthousand smaller than in the case of a parallel plate setup without a glass layer, because of the higherresistance of the liquid. Also the resistance differs from the parallel plate setup without a glass layer.The resistance does not decrease when measuring with 300nM KCl solution, which is the case with the

Figure 5.14: Bode diagram for different concentrations of KCl solutions in the nano molar range with theparallel plates with a glass layer, where Rf = 120kΩ and Cf = 100pF .

parallel plate setup without a glass layer. A possible explanation for this behaviour could be that thedissolved hydrogen ions, from the glass layer, increase the total concentration, which could cause anincrease in conductivity after a certain concentration, which can also be seen from figure 3.3.

Chapter 6

Conclusion

During this assignment impedance models have been proposed for two different setups, a parallel platesetup with and without a glass layer on the electrodes and a measurement circuit has been made.Also different impedance models have been proposed for non-electrolytic and electrolytic liquids. Theimpedance model for electrolytic liquids takes into account the increased conductivity of the liquid, dueto the ions present in the liquid, the formation of an electric double layer and the capacitance of the liq-uid itself. For non-electrolytic liquids only the conductivity, which is much lower compared to electrolyticliquids, and the capacitance of the liquid itself should be taken into account. From the capacitance ofthe liquid itself the relative permittivity can be obtained, which is the main goal of the relative permittivitysensor. The transfer functions for all the impedance models have been analytically calculated. Alsosimulations have been made for both the parallel plate setup with and without a glass layer on the elec-trodes in case of an electrolytic liquid, to show the limitations of the measurement circuit to measure theliquid capacitance accurately in this case.

To test the working of the measurement circuit, first a measurement has been done with fixed ca-pacitor values, which can be seen from figure 5.1. From this measurement the voltage gain with respectto the capacitance has been calculated, shown in figure 5.2. The measurement has been compared tothe theoretical voltage gain, which is also plotted in figure 5.2. It can be said that the voltage gain forfixed capacitances is in accordance with the theoretical voltage gain. The small deviations could be dueto the deviation in capacitance of the capacitor itself. Thus it can be said that the measurement circuitworks properly.

Then measurements have been done with the two different setups, with non-electrolytic and elec-trolytic liquids. The measurements with the parallel plate setup are in agreement with the analyticalresults for both non-electrolytic and electrolytic liquids. The measurement with non-electrolytic liquidsshows linear behaviour, which can be seen from figure 5.4. Also from the fit the parallel capacitancedue to the glass layer, used as separator, and the area of the liquid capacitor have been calculated.The magnitude of these values do seem to be in the correct range. The measurements with the paral-lel plate setup and an electrolytic liquid, a KCl solution, do show expected behaviour according to thecalculated gain. However the theoretical resistance is lower than the measured resistance, which canbe seen from figure 5.6. But this can be due to the fact that the interionic attraction was not taken intoaccount when calculating the resistance. Also the output resistance of the gain-phase analyzer mightbe measured in series with the liquid resistance, resulting in a higher resistance. The electric doublelayer capacitance was obtained from the cutoff frequency containing the resistance of the liquid and theelectric double layer capacitance, which can be seen from equation 5.2. The theoretical and measuredelectric double layer capacitance are in accordance with each other, which can be seen from figure5.7. However the potential at the beginning of the diffusive layer has been assumed to be 60mV withrespect to 100mV applied to the electrodes. So the theoretical electric double layer capacitance candiffer a bit due to the assumption made, therefore no concrete conclusion can be drawn. To validate thetheoretical model with the measurements accurately, a way to calculate the potential at the beginningof the diffusive layer should be found. The measurement with 100nM, 200nM and 300nM KCl solution,figure 5.8, does show expected behaviour according figure 3.12 and 3.13. However the limit of themeasurement circuit can be seen around 1MHz. This can also be seen from the simulations, figure 4.1.So no accurate conclusion can be drawn if it is possible to measure the liquid capacitance accurately.However the measurement does show that for all three concentrations, the gain does tend to the samevalue, but can not be concluded. To be able to draw that conclusion, a measurement circuit capable ofmeasuring at higher frequencies should be designed.

The measurement with different non-electrolytic liquids with the parallel plate setup with a glass layeron the electrodes do show non-linear behaviour, which is as expected due to the in series capacitance

of the glass layer, shown in figure 5.10. The measurement data has been fitted, from which the areaof the liquid capacitor and the parallel capacitance, due to the glass layer used as separator, could bedetermined. These values do seem to be in the expected range, also when compared to the area of theparallel plate setup without a glass layer. However the areas do differ from each other, but that couldbe due to the precision of construction, since the setup is build by hand. The resistance measured withthe parallel plate setup with a glass layer on the electrodes is higher compared to the setup without aglass layer on the electrodes. This was expected since the conductive path in case of the glass layeris longer, since the conductive path is from the back of the electrodes. Also the electric double layercapacitance has been measured with this setup. However the measurement is not in accordance withthe analytical results, the measured capacitance was expected to be in the pico farad range due to thein series capacitance of the glass, shown in figure 5.13. From this measurement it can be said that theremust be another contribution to the electric double layer capacitance than the potential applied to theelectrodes. Another contribution to the electric double layer capacitance could be the surface chargeof the glass layer, which is created when glass gets in contact with water. [7] To test if the capacitanceof the liquid itself can be measured at high frequencies a measurement with different KCl solutions hasbeen done, shown in figure 5.14.

From the measurement it can be seen that the conductivity does not show linear behaviour. It ishard to draw conclusions from this measurement, especially regarding the conductivity of the liquid.To obtain more insight in the conductivity in this case more data points should be chosen. It can alsobe seen that the gain for all three concentrations does tend to the same value, after approximately100kHz. So from this measurement it can be seen that the passband gain, shown in equation 3.32,can be measured at high frequencies. However to do so the electrolyte concentration should not beto high, or a measurement circuit capable of measuring at higher frequencies should be designed.When comparing the measurement with the simulation shown in figure 4.2, it can be seen that themeasurement circuit is limited at approximately 1MHz.

Chapter 7

Outlook

The problem encountered in previous research was that it showed divergent behaviour when measur-ing with an electrolytic liquid. During this assignment impedance models for different setups in caseof non-electrolytic and electrolytic liquids have been proposed. With these models better insight wasobtained in how to measure the capacitance of the liquid itself, which contains information about therelative permittivity of the liquid. From figure 5.8 it can be seen that the gain for different concentrationsof electrolyte do seem to tend to the same value at high frequencies, the passband gain which is the re-lation of the capacitance of the liquid with the feedback capacitance. However no concrete conclusionscan be drawn due to the limitations of the measurement circuit, around 1MHz. To be able to measurethe liquid capacitance accurately, a measurement circuit capable of measuring at higher frequenciesshould be build.Other unexpected behaviour encountered, when measuring with the parallel plate setup with a glasslayer, was the capacitance calculated from the cutoff frequency being to high when compared to equa-tion 3.31. This assumes that there must be another contribution to the electric double layer capacitance.Which could be due to the surface charge of the glass layer caused by the solving of protons in water. Toinvestigate if this unexpected behaviour is caused by the surface charge of glass, the solving of protonsin water should be counteracted. The protonation of glass surfaces could be influenced by the pH of theliquid in contact with the glass layer. [8]

References

[1] W. Schmickler and E. Santos, Interfacial Electrochemistry, Springer, Heidelberg (2010)

[2] Hou, Y. CONTROLLING VARIABLES OF ELECTRIC DOUBLE-LAYER CAPACITANCE.,www.core.ac.uk/download/pdf/61363805.pdf

[3] J. Groenestijn et al. FULLY INTEGRATED MICROFLUIDIC MEASUREMENT SYSTEM FOR REAL-TIME DETERMINATION OF GAS AND LIQUID MIXTURES COMPOSITION.

[4] J. Bhojwani. Modelling and Characterization of an Integrated Permittivity Sensor.

[5] Chandra, Amalendu Biswas, Ranjit Bagchi, Biman. (1999). Molecular Origin of the DebyeHuck-elOnsager Limiting Law of Ion Conductance and Its Extension to High Concentrations: Mode Cou-pling Theory Approach to Electrolyte Friction. Journal of The American Chemical Society - J AMCHEM SOC. 121. 10.1021/ja983581p.

[6] Hartley, H. Interionic Forces in a Completely Dissociated Electrolyte 1. Nature News, Nature Pub-lishing Group, www.nature.com/articles/119322a0.

[7] Behrens, S. H., Grier, D. G. (2001). The charge of glass and silica surfaces. The Journal of ChemicalPhysics, 115(14), 6716-6721. doi:10.1063/1.1404988

[8] Kirby, B. J., Hasselbrink, E. F. (2004). Zeta potential of microfluidic substrates: 1. The-ory, experimental techniques, and effects on separations. Electrophoresis, 25(2), 187-202.doi:10.1002/elps.200305754

[9] Luxbacher, T Pui, Tanja Bukek, Hermina Petrinic, Irena. (2016). THE ZETA POTENTIAL OFTEXTILE FABRICS: A REVIEW.

[10] Feynman, R. P., Leighton, R. B., Sands, M. (1972). Lectures on physics. Menlo Park: Addison-Wesley.

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