Modelling and characterization of a relative permittivity sensor

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May 27, 2019

BACHELOR ASSIGNMENT

MODELLING AND CHARACTERIZATION

OF 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

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Chapter 1 Abstract

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

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List of Figures

3.1 Schematic of a dielectric placed inside a capacitor.[10] . . . 8

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

3.3 Graph of the molar conductivity with respect to the square root of the concentration, for aqeuous KCl (a) and NaCl solutions (b).[5] . . . 10

3.4 Drawing of a parallel plate capacitor filled with a liquid as dielectric. . . 11

3.5 Model of a parallel plate capacitor with a non-electrolytic liquid dielectric.. . . 11

3.6 Model of a parallel plate capacitor with an electrolytic liquid dielectric. . . 12

3.7 Drawing of a parallel plate capacitor with the electrodes covered by a glass layer, filled with a liquid as dielectric. . . 12

3.8 Model of a parallel plate capacitor with a glass layer and a non-electrolytic liquid dielectric. 13 3.9 Model of a parallel plate capacitor with an electrolytic liquid dielectric. . . 13

3.10 Schematic of the measurement circuit . . . 13

3.11 Gain and phase plot of the transfer function for the measurement circuit with a non- electrolytic liquid, where Rf = 34M Ω, Rx= 10M Ωand Cf = 500pF. . . 14

3.12 Gain and phase plot of the transfer function for the measurement circuit with an electrolytic liquid and varying liquid resistance, where Rf = 1200Ω, Cx= 800pF, Cf= 820pF and Cdl= 30µF . . . 15

3.13 Gain and phase plot of the transfer function for the measurement circuit with an electrolytic 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 electrolytic liquid and varying liquid capacitance, where Rf = 1200Ω, Rx= 100Ω, Cf= 820pF and 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 covered by 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. . . 19

5.2 Plot of the theoretical and measured gain for different capacitor values. . . 20

5.3 Bode diagram of the measured gain and phase for different non-electrolytic liquids. . . 21

5.4 Plot of the measured and fitted capacitance with respect to the relative permittivity. . . 21

5.5 Gain and phase plot of the measurement with different concentrations of KCl solution, where Rf = 1200Ω . . . 22

5.6 Plot of the resistance with respect to concentration for different KCl solutions. . . 22

5.7 Plot of the theoretical and measured electric double layer capacitance for different KCl concentrations. . . 23

5.8 Bode diagram for different concentrations of KCl solutions in the nano molar range, where Rf = 1200Ωand Cf = 820pF. . . 24

5.9 Bode diagram of the measured gain and phase for different non-electrolytic liquids with the parallel plates covered with a glass layer, where Rf = 13.6M Ωand Cf = 100pF. . . . 25

5.10 Plot of the measured in series capacitance with respect to the relative permittivity. . . 25

5.11 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. . . 26

5.12 Plot of the measured resistance for different concentrations of KCl solution. . . 26

5.13 Plot of the measured equivalent capacitance for different concentrations of KCl solution. . 27

5.14 Bode diagram for different concentrations of KCl solutions in the nano molar range with the parallel plates with a glass layer, where Rf= 120kΩand Cf= 100pF. . . 28

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List of Tables

5.1 Relative permittivity of several liquids. . . 20

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Chapter 2 Introduction

This bachelor assignment is about modelling and characterization of a relative permittivity sensor, to understand the behaviour of different liquids inside such a sensor. The permittivity of a material is a measure for the polarization of that material within an external electric field, where the relative permittivity is the permittivity with respect to the permittivity of free space. The relative permittivity of a material depends on the material itself, so it is a material property. Researchers of the University of Twente have build a fully integrated microfluidic measurement system to measure different properties of a liquid, the density, viscosity, mass flow, heat capacity and the relative permittivity. [3] This device was designed to measure the composition of liquid mixtures, which can be used to determine the quality of oil or determine the flow and composition of drugs delivered by intravenous therapy.

During previous research the relative permittivity sensor of the fully integrated microfluidic measurement 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 physics of that divergent behaviour will be investigated. It was expected that this divergent behaviour, which was 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 an electric double layer. These aspects will be investigated in depth during this assignment. This will be done by proposing impedance models for two different setups, one for a parallel plate setup without a glass layer and one with a glass layer on the electrodes. These impedance models will be investigated analytically and then validated by measurements, to show the accuracy of the impedance models and to get an understanding of the behaviour. When this behaviour can be understood completely, a setup more similar to the actual sensor could be investigated, so that eventually the behaviour seen when measuring with the sensor itself could be explained and a measurement circuit capable of measuring with non-electrolytic and electrolytic liquids could be designed.

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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 in a higher capacitance. But first lets consider a parallel plate capacitor without a dielectric between the plates. The divergence of the electric field is described by the Maxwell equation shown in equation3.1, where Q is the charge enclosed.

∇ · E = Q

₀ (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 parallel plates the electric field can be written as shown in equation3.3.

‹

S

E · ˆn dS = Q

0

(3.2)

E = Q

0A (3.3)

The electric field is the divergence of the potential, shown in equation 3.4. Then to calculate the potential from the electric field the electric field should be integrated along its path, shown in equation 3.5.

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 be described by the product of the electric field with the distance between the plates. Then the capacitance can be calculated by dividing the charge by the potential, shown in equation3.6.

C = Q V = Q

Ed = 0A

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. However it does influence the capacitance when placed between the plates of a capacitor. In this section the behaviour of a dielectric placed in an electric field will be discussed. In figure 3.1a dielectric placed between 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 when only air is present between the plates, the electric field inside the dielectric must be reduced. This is caused by the polarization of the dielectric, meaning that positive and negative charges get aligned inside the electric field. As a result a net charge can be observed at the edges of the dielectric, which can be seen from figure 3.1. From equation3.3it can be seen that the electric field depends on the charge enclosed, in case of air between the plates, this is only the charge present at the plates. But in case of a dielectric the edges of the dielectric also have a net charge which counteracts the charge at the plates. Then the electric field can be described by equation3.7, where polis the net charge at

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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 freely through the dielectric, but dipoles only can rotate, and the polarization is uniform inside the dielectric.

E = Q − Qpol

₀A (3.7)

To obtain an insight of the electric field inside the dielectric the polarization charge is chosen as vector P. Then the vector P can be described by equation3.8, where χ is the electric susceptibility of the dielectric. Then when equation3.8is filled in equation3.7and rewritten equation3.9is obtained.

From equation3.9it can be seen that the electric field is reduced by a factor _1+χ¹ when a dielectric is placed between the plates of a capacitor, depending on the type of dielectric.

P = χ0E (3.8)

E = Q

0A(1 + χ) (3.9)

The capacitance when a dielectric is present can be calculated the same way as described above in case of only air. This results in equation3.10, where _1+χ¹ is substituted by r which is the relative permittivity.

C = r0A

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 case it is assumed that charges are not able to move through the material. But when using a liquid dielectric containing ions, the ions in the liquid are not immobilized, they can move. In this case an additional effect is taking place, the forming of an electric double layer. An electric double layer is formed at the interface of a charged surface and an electrolytic liquid, meaning that it contains ions. The surface charge is partly counteracted by the Stern layer and the diffusive layer. The Stern layer is a layer of immobilized ions, as can be seen from figure3.2, which get attracted by the charges at the surface.

The capacitance of the Stern layer can be approximated by the Helmholtz model. According to the Helmholtz model the capacitance of the Stern layer can be seen as a parallel plate capacitor, which is described by equation3.7where κ is the relative permittivity of the solvent. However since the ions in the Stern layer are immobilized this influences the polarization of the ions, as result the relative permittivity is 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 is not linear, which is expected in case of a parallel plate capacitor. To obtain a relation for the capacitance of the diffusive layer first a relation for the potential inside of the diffusive layer should be found. The potential is related to space charge by equation3.11.

∇ · ∇ · φ = −ρ

0r

(3.11) The ions in the liquid can be seen as space charges, which obey the Boltzmann distribution, shown in equation3.12, where n is the number density of the ions. The number density of ions depends on the distance from the surface since the potential decays, which can also seen in figure3.2.

n_a= n^∞_a exp

−q_a∆φ kBT

(3.12)

From equation3.12the ionic distribution in the x direction can be obtained, by multiplying the number density of ions with the charge number of the ions, shown in equation3.13.

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

ρ(x) =X

a

qa· na(x) (3.13)

When equation3.13is filled in equation3.11, for the case of a monovalent electrolyte, the solution becomes as shown in equation 3.14. If the first and the second term in the exponential satisfy the condition _k^e

BT < 1, equation3.14can be reduced to equation3.15.

d²φ(x)

dx² = −en^∞

0r

exp

−eφ(x) kBT

− expeφ(x) kBT

(3.14)

d²φ(x)

dx² = 2en^∞

0r

·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 shown in equation3.17.

φ(x) = φ(0)exp(κ) (3.16)

κ⁻¹ =r r₀k_BT

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 equation3.18. [2]

CD= r0

κ⁻¹ cosh zeφ

2k_BT (3.18)

3.3 Ionic conductivity

When ions are present in a solvent, for example water, the conductivity will be increased due to the presence of ions. The conductivity due to ions depends on the amount of ions present and the ability of ions to move through a liquid. So the conductivity in an ideal case can be described by equation 3.19, where Λ⁰_i is the molar conductivity of each specie. The molar conductivity of each specie can be described by equation,3.20, where z is the charge number, F Faradays constant, D the diffusion coefficient, R the gas constant and T the temperature of the liquid.

σ =X

i

Λ⁰_iCi (3.19)

Λ⁰=z²F²D

RT (3.20)

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As stated before the conductivity of an electrolyte can be described by equation 3.19. But in real electrolytes 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 be counteracted by ions with the opposite charge, resulting in a lower conductivity, which can be seen in figure3.3.

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

So for low concentrations equation 3.19 does not hold due to the interionic attraction causing a decrease in molar conductivity,Λ, when increasing concentration. A relation for the decrease in molar conductivity due to interionic attraction has been found by Debye and Huckel, when assuming total dissociation, shown in equation3.21.[6] The term ^K¹

D³2

w¹represents the electrical retardation. Because of the time needed for the ions to be redistributed while there are ions with opposite sign in the rear of the ions. Another effect which causes a decrease in molar conductivity is due to the term ^K²

D¹2

w²b.

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

Λ⁰− Λ

Λ⁰ = K₁

D³²w¹+ K₂

D¹²w²b!√

zC (3.21)

When rewriting the equation above, equation3.21, equation3.22is obtained.

Λ = Λ⁰ 1 − K1

D³²w¹+ K2

D¹²w²b!√

zC

!

(3.22) Then equation3.22can be further simplified in equation3.23, where Λ⁰is the molar conductivity at infinite dilution, K the Kohlrauschs constant, z the charge number of the ion and C the concentration of the electrolyte.

Λ = Λ⁰

1 − K√ zC

(3.23) So when taking into account the interionic attraction the conductivity of an electrolyte can be written by equation3.24.

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σ =X

i

Λ⁰_i

1 − K√ zC

C_i (3.24)

When the conductivity of the liquid is known the resistance of the liquid in case of a certain setup can be calculated with use of equation3.25, where d is the distance of the conductive path, A the area of 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 discussed and their impedance models for electrolytic and non-electrolytic liquids. First the impedance models in case of a parallel plate setup will be discussed. In that case the liquid is in direct contact with the electrodes. Second the impedance models for a parallel plate setup with electrodes covered by a glass 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 seen in figure3.4. 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 present when used as a dielectric between two parallel plates. In that case the impedance due to the liquid can be described by a capacitance in parallel with a resistor, shown in figure3.5. The capacitance is linearly related to the relative permittivity of the liquid between the plates, as can be seen from equation3.7.

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.

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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 inside the liquid. In that case an additional capacitance is created by the electric double layer at the surfaces of the plates. In figure3.6the impedance model can be seen in case of ions present in the liquid, where C_x is the capacitance of the solvent, Rx the resistance of the liquid and Cdl the electric double layer capacitance. The capacitance due to the solvent remains the same, assuming that the ions present in the solution do not influence the relative permittivity of the liquid. But the resistance of the liquid is not only due to the conductivity of the solvent but also due to the ions present in the solution. The resistance of 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 with a liquid as dielectric.

The setup with parallel plates with a glass layer consists of two parallel plates covered by a glass layer, which can be seen in figure3.7. The impedance models for electrolytic and non-electrolytic liquids for the parallel plates without a glass layer only differ from this setup by the in series capacitance of the glass layer. So both electrolytic and non-electrolytic liquid impedance models have to be extended with an 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 in figure3.10is measured. The measurement circuit is an inverting amplifier with a gain shown in equation 3.26. When the feedback impedance is known the impedance of the liquid can be obtained. Which can be described according to the impedance models above, depending of the presence of ions.

A = −Zf

Z_m (3.26)

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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, the gain of the measurement circuit can be described by equation3.27.

A = −Cx

Cf

s +_R¹

xCx

s +_R¹

fC_f

(3.27)

From equation3.27it can be seen that there will be two cutoff frequencies, one will be due to the resistance and capacitance of the liquid itself and one will be due to the feedback impedance. Also it can be seen that the passband gain is the relation between the capacitance of the liquid and the feedback capacitance. Figure3.11shows the gain and phase of the transfer function shown in equation 3.27, 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 relation between the capacitance and the relative permittivity is linear, which can be seen from equation3.10.

When considering electrolytic liquids the capacitance of the electric double layer should also be taken 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, considering an parallel plate setup without a glass layer, is filled in equation3.26 and the feedback impedance is chosen to be a resistor in parallel with a capacitor the gain of the measurement circuit is shown in equation3.28, where Ceq is described by equation3.29.

A = −Ceq

Cf

s(s +_R¹

xCx) (s +_R¹

fC_f)(s +_R ²

x(C_dl+2C_x)) (3.28)

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

Ceq = CdlCx

Cdl+ 2Cx

(3.29) The gain and phase of the transfer function shown in equation3.28depends on multiple properties of the liquid, the capacitance of the liquid, the resistance of the liquid and the capacitance of the electric double layer. The resistance of the liquid depends on the concentration of ions, which is described in section Ionic conductivity. Also the capacitance due to the electric double layer depends on the concentration 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 the transfer function of the measurement circuit will be investigated. From equation3.28it can be seen that two cutoff frequencies depend on the resistance of the liquid. And at low frequencies is the gain the relation between the resistance of the liquid and the resistance of the feedback impedance, which can be seen in figure3.12.

Another property of an electrolytic liquid is the electric double layer capacitance, which influence on the transfer function of the measurement circuit is shown in figure3.13. From the transfer function, equation3.28it can be seen that the only influence of the electric double layer capacitance is on one of the cutoff frequencies, this shift in cutoff frequency can be seen in figure3.13.

Then the property of interest is the capacitance of the liquid itself, which is linearly related to the relative permittivity according to equation3.10. To be able to measure this capacitance when measuring with an electrolytic liquid, the influence on the transfer function of the measurement circuit should be investigated. From equation3.28and3.29it can be seen that the passband gain is the relation between the 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 be seen in figure3.14. However the cutoff frequency which depends on the sum of the electric double layer and the liquid capacitance is not much influenced by the liquid capacitance, since the electric double layer 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 model changes, as shown in figure3.8. This results in a different gain and phase. The gain in this case can be described by equation3.30, with Csshown in equation3.31

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Figure 3.12: Gain and phase plot of the transfer function for the measurement circuit with an electrolytic liquid 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 electrolytic liquid and varying electric double layer capacitance, where Rf = 1200Ω, Rx= 100Ω, Cx= 800pF and Cf = 820pF.

A = −Ceq

Cf

s(s +_R¹

xCx) (s +_R¹

fC_f)(s +_R ²

x(C_g+2C_x)) (3.30)

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

C_eq = C_gC_x Cg+ 2Cx

(3.31) When the parallel plate setup with a glass layer on the electrodes is filled with an electrolyte, the impedance model has to be extended with an extra in series capacitance, as shown in figure3.9. Then the gain of the measurement circuit can be described by equation3.32, with Cs being the in series capacitance of the glass layer and the electric double layer, equation3.33

A = − C_sC_x Cf(Cs+ 2Cx)

s(s +_R¹

xC_x) (s +_R¹

fCf)(s +_R ²

x(Cs+2Cx)) (3.32)

Cs= CdlCg

C_dl+ C_g (3.33)

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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 figure4.1the simulation results of a parallel plate setup without a glass layer and filled with an electrolyte can be seen. It can be seen that approximately around 1MHz the measurement circuit shows unexpected behaviour, when compared with the analytical results, shown in figure3.14. The same holds when simulating with the parallel plate setup with a glass layer, which can be seen in figure 4.2. Due to this behaviour, the relation between the liquid and feedback 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 be taken into account when measuring with electrolytes. To minimize the effect of the limits of the operational amplifier, an operational amplifier capable of measuring at high frequencies should be chosen.

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

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Chapter 5 Results and Discussion

5.1 Measurement circuit

The measurement circuit used can be seen in figure3.10, with Zmthe impedance model, depending on the type of liquid and measurement setup, and Zf the feedback impedance. To test the working of the measurement setup it is tested with different capacitor values first. Then, according to equation3.27, the gain is a relation between the to be measured capacitance and the capacitance of the feedback impedance. The results with different capacitor values and a feedback capacitance of 470pF can be seen in figure 5.1. It can be seen that the gain increases when increasing the capacitance of the to be measured capacitance, which is in accordance with equation3.27. To validate the voltage gain with respect to the measured capacitance, the theoretical and measured voltage gain has been plotted in figure5.2. The equation for the theoretical voltage gain is the passband gain of equation3.27, which can be seen in equation 5.1. From figure 5.2 it can be seen that the measured voltage again is in accordance with the theoretical voltage gain, except from small deviations. These deviations could be caused by the deviation of the capacitor value.

A = Cx

C_f (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, the capacitance of a parallel plate capacitor filled with different non-electrolytic liquids can be measured.

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

In figure5.3the gain and phase of the measurements with air, isopropanol, acetone and ethanol are shown. The relative permittivities of these liquids can be found in table 5.1. From figure 5.3it can be seen that it shows similar behaviour as was expected according the proposed impedance model of parallel plates filled with a non-electrolytic liquid, which can also be seen in figure3.11.

Liquids _r Isopropanol 17.9 Acetone 20.7 Ethanol 24.5

Table 5.1: Relative permittivity of several liquids.

As can be seen from equation3.27, the passband gain is the relation between the liquid capacitance and the feedback capacitance. So with that equation the liquid capacitance can be calculated from the gain for each liquid. The calculated capacitances are plotted with respect to their relative permittivities, shown in figure5.4. It can be seen that the capacitance of the liquid increases linearly when the relative permittivity is increased, which is also in accordance with equation3.10. The calculated capacitances, for the liquid mentioned above, is linearly fitted. From this fit the capacitance in parallel and the area of the liquid capacitor, when the distance between the plates is known, can be obtained. The parallel capacitance is due to the glass layer used as separation between the two electrodes. Since capacitors in parallel can be summed, the offset of the the fit is the capacitance of the capacitor in parallel, which has been found to be 99 pF . The area of the liquid capacitor could be calculated from the slope of the fit 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 different concentrations of KCl. First the behaviour of the resistance of the liquid and the electric double layer will be 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 the gain and phase is dependent on the resistance and electric double layer capacitance, which can be seen from figure3.12and3.13. Since the capacitance of the feedback impedance has only influence at higher frequencies it can be left out, so the feedback impedance consists only of a resistor in this case.

From figure5.5it can be seen that the resistance decreases when the KCl concentration increases, since the passband gain at lower frequencies is determined by ^R_R^f

x. The resistance with respect to their concentration can thus be calculated from the passband gain, which can be seen in figure5.6. It can be seen that the measured resistance does show expected behaviour. However it seems that the measured resistance is higher than the theoretical resistance, which has been calculated with use of equation 3.25, without taking into account the interionic attraction. So this deviation could be due to

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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 the gain-phase analyzer.

When the resistance of the electrolyte is known the electric double layer can be calculated from the cutoff frequency, from equation3.28it can be seen that one cutoff frequency depends on the resistance of the liquid and the electric double layer capacitance. Since the electric double layer capacitance is in the micro farad range and the liquid capacitance in the pico farad range, the liquid capacitance can be

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Figure 5.5: Gain and phase plot of the measurement with different concentrations of KCl solution, where R_f = 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 found at the -3 dB point, when the passband gain is subtracted. The electric double layer capacitance can then by calculated with use of equation5.2, where Rx is the resistance of the liquid and fc the cutoff frequency due to the resistance and the electric double layer capacitance.

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C_EDL= 1 πRxfc

(5.2) In figure5.7the measured and theoretical electric double layer capacitance for different concentrations 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 equation3.18, where φ is the potential after the Stern layer. In the Stern layer the potential decays linearly, meaning that the 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 at the beginning of the diffusive layer was assumed to be 60 mV , with respect to 100 mV applied to the electrodes. 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 be 160 pm. Then the capacitances of the two layers should be taken into series, which gives the theoretical result as shown in figure5.7.

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

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 with three different KCl concentrations, 100nM, 200nM and 300nM. But since these concentrations are very low the concentrations of H⁺ and OH⁻ should also be taken into account, which increases the ionic strength with 200nM, when assuming a pH of 7. In figure5.8the results of the measurement can be seen. It can be seen that the gain is higher for lower frequencies, meaning that the resistance is lower for low concentrations. This could be due to the interionic attraction playing a significant role at such low concentrations, which is discussed in the Ionic conductivity section. The gain does seem to tend to the same value after 100kHz for all three concentrations. But can not accurately be seen from the bode diagram, since the measurement circuit can only measure up to approximately 1MHz. This behaviour is due to the limitations of the measurement circuit, which can also be seen from the simulations, figure 4.1. Also the feedback impedance influences the measurement, since one of the cutoff frequencies depends on it. This can also be seen from figure 5.8, around 100kHz. Since the relation between the resistance of the feedback impedance and the liquid determines the gain at lower frequencies, the feedback resistance should be chosen such that it does not exceed the maximum amplification of approximately 40dB, otherwise it would be clipping. Also the feedback capacitance should be chosen such that the gain due to the relation between the capacitance of the feedback and the liquid does not exceed its maximum. So the ability of the measurement circuit to measure the liquid capacitance, in case of an electrolytic liquid, depends severely on the electrolyte concentration.

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Figure 5.8: Bode diagram for different concentrations of KCl solutions in the nano molar range, where R_f = 1200Ωand 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-electrolytic liquid the impedance model differs from the parallel plate setup without a glass layer, shown in figure 3.8. 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 equation3.30. A measurement has been done with the liquids shown in table5.1. The bode diagrams for the different liquids are shown in figure5.9.

From figure5.9it can be seen that after approximately 100kHz the passband gain according equation 3.30 is reached. The capacitance of the glass layer and the liquid, shown in equation 3.31, can be calculated from the passband gain, shown in figure5.10. The measurement data has been fitted with the fit described in equation5.3, where C0is the capacitance when the plates would be filled with air, Cg

the parallel capacitance due to the glass layer used as separator, gthe relative permittivity of the glass and rthe relative permittivity of the solvent. The relative permittiviy of the glass layer was assumed to be 7.

C = C0gC0r

C₀_g+ 2C₀_r + Cg (5.3)

From C0 the area of the liquid capacitor could be determined, since the area is related to the capacitance of a parallel plate structure without a dielectric by equation3.6. The area and parallel glass capacitance have been found to be 5.03 cm²and 15 pF ,respectively. The area of the liquid capacitor in this case is a bit lower than the area of the liquid capacitor in case of no glass layer. But this can be due to the fact that each setup is constructed by hand or that the value of the relative permittivity differs from the value used. However to calculate the area more accurate, more data points are needed for the fit.

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 equation 3.32and3.33the capacitance of the glass and electric double layer should be taken in series. Then the cutoff frequency is the sum of the capacitance of the glass layer and electric double layer in series and the capacitance of the liquid twice. Since the electric double layer capacitance is much higher than the

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Figure 5.9: Bode diagram of the measured gain and phase for different non-electrolytic liquids with the parallel plates covered with a glass layer, where Rf= 13.6M Ωand 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 glass layer, which is in the pico farad range. In figure5.11the bode diagrams for different concentrations of KCl solution can be seen.

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

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Figure 5.11: 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.

the liquid can be calculated, shown in figure5.12.

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

From figure 5.12it can be seen that the resistance does show the same shape as the resistance measured with the parallel plate setup without a glass layer, shown in figure5.6. But the resistance in case of the glass layer is much higher than without a glass layer. In practice the setup is totally dipped in the liquid, causing a conductive path from the back of the electrodes, since the resistance of the glass

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layer is in the range of mega ohms, which are not covered with a glass layer. Since the path from the back 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 equation3.25.

When the resistance of the liquid is known the cutoff frequency, which depends also on the electric double 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 with the liquid capacitance, the equivalent capacitance, shown in equation5.4, can be determined, shown in figure5.13.

Ceq = CdlCg

Cdl+ 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 the in series capacitance of the glass layer. However from figure5.13 it can be seen that the measured capacitance is in the micro farad range. This suggests that either the impedance model of the setup is not accurate or there is another contribution to the electric double layer capacitance, apart from the potential applied to the electrodes. The most explainable reason for this unexpected behaviour could be due to an additional contribution of the glass layer to the electric double layer capacitance. Because when the glass layer gets in contact with water hydrogen atoms get dissolved in water, causing surface charge on the glass layer. [7] Then this surface charge causes an additional potential, which can be seen from equation3.11, resulting in a higher capacitance.

From figure5.14 it can be seen that the cutoff frequency due to the feedback impedance is lower than in case of the parallel plate setup without a glass layer. Since the conductivity in case of a glass layer is lower, because of the conductive path from the back, the feedback resistance is chosen to be 120kΩ. This shift in cutoff frequency is desirable, since the relation between the liquid and the feedback capacitance should be measured after that cutoff frequency. According to figure 3.12 and 3.13the passband gain, which is the relation between the liquid and feedback capacitance, should not be influenced by the resistance of the liquid nor the electric double layer capacitance. But this passband gain should be measured after the cutoff frequency of the liquid itself. From figure5.14it can be seen that 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 is not visible in the range in which is measured. However the cutoff frequency is approximately a factor thousand smaller than in the case of a parallel plate setup without a glass layer, because of the higher resistance 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

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

parallel plate setup without a glass layer. A possible explanation for this behaviour could be that the dissolved hydrogen ions, from the glass layer, increase the total concentration, which could cause an increase in conductivity after a certain concentration, which can also be seen from figure3.3.

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Chapter 6 Conclusion

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

To test the working of the measurement circuit, first a measurement has been done with fixed capacitor values, which can be seen from figure5.1. From this measurement the voltage gain with respect to the capacitance has been calculated, shown in figure5.2. The measurement has been compared to the theoretical voltage gain, which is also plotted in figure5.2. It can be said that the voltage gain for fixed capacitances is in accordance with the theoretical voltage gain. The small deviations could be due to the deviation in capacitance of the capacitor itself. Thus it can be said that the measurement circuit works properly.

Then measurements have been done with the two different setups, with non-electrolytic and electrolytic liquids. The measurements with the parallel plate setup are in agreement with the analytical results for both non-electrolytic and electrolytic liquids. The measurement with non-electrolytic liquids shows linear behaviour, which can be seen from figure5.4. Also from the fit the parallel capacitance due 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 parallel plate setup and an electrolytic liquid, a KCl solution, do show expected behaviour according to the calculated gain. However the theoretical resistance is lower than the measured resistance, which can be seen from figure5.6. But this can be due to the fact that the interionic attraction was not taken into account when calculating the resistance. Also the output resistance of the gain-phase analyzer might be measured in series with the liquid resistance, resulting in a higher resistance. The electric double layer capacitance was obtained from the cutoff frequency containing the resistance of the liquid and the electric double layer capacitance, which can be seen from equation5.2. The theoretical and measured electric double layer capacitance are in accordance with each other, which can be seen from figure 5.7. However the potential at the beginning of the diffusive layer has been assumed to be 60mV with respect to 100mV applied to the electrodes. So the theoretical electric double layer capacitance can differ a bit due to the assumption made, therefore no concrete conclusion can be drawn. To validate the theoretical model with the measurements accurately, a way to calculate the potential at the beginning of 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 the measurement circuit can be seen around 1MHz. This can also be seen from the simulations, figure4.1.

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 same value, but can not be concluded. To be able to draw that conclusion, a measurement circuit capable of measuring at higher frequencies should be designed.

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

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of the glass layer, shown in figure5.10. The measurement data has been fitted, from which the area of the liquid capacitor and the parallel capacitance, due to the glass layer used as separator, could be determined. These values do seem to be in the expected range, also when compared to the area of the parallel plate setup without a glass layer. However the areas do differ from each other, but that could be due to the precision of construction, since the setup is build by hand. The resistance measured with the parallel plate setup with a glass layer on the electrodes is higher compared to the setup without a glass layer on the electrodes. This was expected since the conductive path in case of the glass layer is longer, since the conductive path is from the back of the electrodes. Also the electric double layer capacitance has been measured with this setup. However the measurement is not in accordance with the analytical results, the measured capacitance was expected to be in the pico farad range due to the in series capacitance of the glass, shown in figure5.13. From this measurement it can be said that there must be another contribution to the electric double layer capacitance than the potential applied to the electrodes. Another contribution to the electric double layer capacitance could be the surface charge of the glass layer, which is created when glass gets in contact with water. [7] To test if the capacitance of the liquid itself can be measured at high frequencies a measurement with different KCl solutions has been done, shown in figure5.14.

From the measurement it can be seen that the conductivity does not show linear behaviour. It is hard 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 also be seen that the gain for all three concentrations does tend to the same value, after approximately 100kHz. 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 be to 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 the measurement circuit is limited at approximately 1MHz.

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Chapter 7 Outlook

The problem encountered in previous research was that it showed divergent behaviour when measuring with an electrolytic liquid. During this assignment impedance models for different setups in case of non-electrolytic and electrolytic liquids have been proposed. With these models better insight was obtained in how to measure the capacitance of the liquid itself, which contains information about the relative permittivity of the liquid. From figure5.8it can be seen that the gain for different concentrations of electrolyte do seem to tend to the same value at high frequencies, the passband gain which is the relation of the capacitance of the liquid with the feedback capacitance. However no concrete conclusions can be drawn due to the limitations of the measurement circuit, around 1MHz. To be able to measure the liquid capacitance accurately, a measurement circuit capable of measuring at higher frequencies should be build.

Other unexpected behaviour encountered, when measuring with the parallel plate setup with a glass layer, was the capacitance calculated from the cutoff frequency being to high when compared to equa- tion3.31. 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. To investigate if this unexpected behaviour is caused by the surface charge of glass, the solving of protons in water should be counteracted. The protonation of glass surfaces could be influenced by the pH of the liquid in contact with the glass layer. [8]

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References

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[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 AM CHEM 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 Chemical Physics, 115(14), 6716-6721. doi:10.1063/1.1404988

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

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Modelling and characterization of a relative permittivity sensor