I

Technische Universität Hamburg-Harburg

Characterization of PEDOT

Electrodes

Master Thesis

by

Trisha Sadhu

January 2014

Supervisors: Grit Blume

Prof. Dr. Manfred Kasper

Prof. Dr. -Ing. W. Krautschneider

TUHH

II

ABSTRACT

The objective of this work is to model and simulate the electrical field around PEDOT

electrodes in an electrolytic medium in order to understand the variation in the electrical field

due to electrode polarization. The PEDOT electrodes were developed at the Institute of

Microsystem Technology at the Hamburg University of Technology for the purpose of

electrical stimulation of cell lines (fibroblasts) in a bioreactor. The impedance measurements

of the PEDOT electrodes in the electrolytic medium which were taken in the absence of

fibroblast cells are used to develop an electrical circuit. The equivalent electrical circuit is the

physical representation of the electrolytic medium and the electrode / electrolyte interface.

The simulation of electrical double layer forming at the electrodes was performed in one-

dimension.

III

STATEMENT

I, Trisha Sadhu, solemnly declare that I have written this master thesis independently, and that

I have not made use of any aid other than those acknowledged in this thesis. Neither this

master thesis report, nor any other similar work, has been previously submitted to any

examination board.

Trisha Sadhu Hamburg, January 2014.

IV

ACKNOWLEDGEMENTS

I would like to thank Professor Dr. Manfred Kasper for giving me the opportunity to develop

my master thesis at the Institute of Microsystems Technology. I do appreciate his taking time

out of his busy schedule to supervise the work as well as to provide detailed guidance towards

improving the work.

Also, I would like to thank my supervisor, Grit Blume, who has provided me with invaluable

advice and constant support.

I would like to thank all members from the department of Microsystems Technology of the

Technical University of Hamburg-Harburg for their assistance during the progression of this

master thesis.

And lastly, I would like to thank my family and friends for their support.

V

Contents

ABSTRACT ................................................................................................................................................ 2

STATEMENT ............................................................................................................................................. 3

ACKNOWLEDGEMENTS ........................................................................................................................... 4

LIST OF FIGURES ...................................................................................................................................... 5

1.0 INTRODUCTION ................................................................................................................................. 8

1.1 MOTIVATION ................................................................................................................................. 9

2.0 THEORY ............................................................................................................................................ 10

2.1 THE ELECTRODES ......................................................................................................................... 10

2.1.1 THE DIFFUSE DOUBLE LAYER THEORY .................................................................................. 11

2.2 THE ELECTROLYTIC MEDIUM ....................................................................................................... 14

2.2.1. CALCULATION OF THE CONDUCTIVITY OF THE ELECTROLYTE ............................................ 14

2.2.2. MASS TRANSPORT OF SPECIES IN THE ELECTROLYTE ......................................................... 16

2.2.3 DIELECTRIC RELAXATIONS IN THE ELECTROLYTE .................................................................. 17

2.3. THE ELECTRODE KINETICS .......................................................................................................... 21

2.4 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY ...................................................................... 23

2.4.1 EQUIVALENT CIRCUITS ......................................................................................................... 23

2.5 OPTIMIZATION OF VALUES .......................................................................................................... 27

2.5.1 SENSITIVITY ANALYSIS OF THE EQUIVALENT CIRCUITS ........................................................ 28

3.0 RESULTS AND DISCUSSION ........................................................................................................... 30

3.1. MODELLING OF THE EQUIVALENT ELECTRICAL CIRCUIT ............................................................ 30

3.1.1 ELECTRICAL CIRCUIT I ........................................................................................................... 32

3.1.2 ELECTRICAL CIRCUIT II .......................................................................................................... 35

3.1.3 ELECTRICAL CIRCUIT III ......................................................................................................... 37

3.2 SIMULATION OF THE ELECTROCHEMCAL SYSTEM ...................................................................... 40

3.2.1 SIMULATION OF THE ELECTRICAL FIELD IN THE ABSENSE OF AN ELECTROLYTE ................. 40

3.2.2 CALCULATION OF THE IMPEDANCE OF THE SOLUTION ....................................................... 41

3.2.3 SIMULATION OF THE DIFFUSE DOUBLE LAYER ..................................................................... 43

4.0 CONCLUSION AND OUTLOOK .......................................................................................................... 51

5.0 REFERENCES .................................................................................................................................... 52

6.0 APPENDIX ........................................................................................................................................ 54

6.1 OPTIMIZATION OF ELECTRICAL CIRCUIT PARAMETERS ............................................................... 54

VI

6.1.1 THE OPTIMIZER ..................................................................................................................... 54

6.1.2 CALCULATION OF THE QUALITY FUNCTION ......................................................................... 55

6.1.3 THE SCALING FUNCTION ...................................................................................................... 56

6.1.4 THE UNSCALING FUNCTION ................................................................................................. 56

6.1.5 DISPLAYING OF THE PLOTS ................................................................................................... 56

6.1.6 OPTIMIZATION PROCEDURE ................................................................................................. 57

6.1.7 SENSITIVITY OF THE OPTIMIZED PARAMETERS .................................................................... 58

6.1.8 THE SENSITIVITY FUNCTION ................................................................................................. 59

VII

LIST OF FIGURES

Figure 1 THE ELECTROCHEMICAL SYSTEM .............................................................................................. 9

Figure 2 DIAGRAM OF THE PEDOT INTERDIGITAL MICROELECTRODE WITH THE SPECIFICATIONS..... 10

Figure 3 TWO NEGATIVELY CHARGED SURFACES OF SURFACE CHARGE DENSITY σ IN A MEDIUM OF

PERMITTIVITY ε AND SEPARATED BY DISTANCE d ................................................................................ 11

Figure 4 STRUCTURE OF THE DIFFUSE DOUBLE LAYER [2] .................................................................... 12

Figure 5 POLARISATION MECHANISMS IN DIELECTRIC MEDIUM [8] .................................................... 21

Figure 6 THE PHYSICAL REPRESENTATION OF THE ELECTRIFIED INTERFACE ON THE LEFT IS THE R||C

CIRCUIT .................................................................................................................................................. 24

Figure 7 THE WARBURG ELEMENT ........................................................................................................ 24

Figure 8 THE CONSTANT PHASE ELEMENT ............................................................................................ 25

Figure 9 PHYSICAL REPRESENTATION OF THE ELECTRODE POLARIZATION .......................................... 25

Figure 10 PHYSICAL REPRESENTATION OF THE RANDLES MODEL ........................................................ 26

Figure 11 THE EQUIVALENT CIRCUIT FOR THE COMPLEX IMPEDANCE ................................................. 26

Figure 12 MAGNITUDE OF THE MEASURED IMPEDANCE (OHM).......................................................... 30

Figure 13 PHASE OF THE MEASURED IMPEDANCE OF THE PEDOT ELECTRODE (DEGREE) ................... 31

Figure 14 NYQUIST PLOT OF THE MEASURED IMPEDANCE .................................................................. 31

Figure 15 DIAGRAM OF EQUIVALENT ELECTRICAL CIRCUIT 1 ............................................................... 32

Figure 16 NYQUIST PLOT FOR CIRCUIT I ................................................................................................ 34

Figure 17 DIAGRAM OF EQUIVALENT CIRCUIT II ................................................................................... 35

Figure 18 NYQUIST PLOT FOR EQUIVALENT CIRCUIT 3 ......................................................................... 36

Figure 19 DIAGRAM FOR EQUIVALENT CIRCUIT 2 ................................................................................. 37

Figure 20 NYQUIST PLOT FOR CIRCUIT III .............................................................................................. 38

Figure 21 MAGNITUDE PLOT OF THE CIRCUIT III ................................................................................... 39

Figure 22 THE PHASE PLOT OF CIRCUIT III ............................................................................................. 39

Figure 23 THE 2D PLOT FOR THE MAGNITUDE OF THE ELECTRICAL FIELD ........................................... 41

Figure 24 THE PLOT OF THE VARYING MAGNITUDE CURRENT DENSITY .............................................. 42

Figure 25 DIMENSIONLESS CHARGE CONCENTRATION PROFILE .......................................................... 47

Figure 26 DIMENSIONLESS CHARGE DENSITY PROFILE ......................................................................... 48

Figure 27 DIMENSIONLESS POTENTIAL PROFILE ................................................................................... 49

8

1.0 INTRODUCTION

In wound healing stimulation electrodes are used to give rise to physiological electrical fields

to direct cell growth and migration [1, 3]. The electrical stimulation of biological tissues

requires the flow of ionic charge through them [7]. The use of microelectrodes allows for

close and selective contact with the fibroblast cells. When a current of small amplitude is

applied at the electrodes, electrochemical reactions take place at the interface of the electrode

and the biological medium. The reactions must be reversible (i.e. the reaction can be reversed

by reversing the current) to prevent corrosion of the electrode and damage of the surrounding

tissue due to the reaction. Irreversible reactions at the electrode interface thus have to be

avoided [3].

The PEDOT electrode has a high reversible charge injection capacity (i.e. the amount of

charge that is reversibly injected during a single stimulation pulse) [3]. The PEDOT material

is biocompatible as it does not release any toxic chemical species in the human tissue and cell

cultures respectively. The PEDOT coated electrode is also flexible and has its mechanical

integrity intact after there is movement between the implantable electrode and tissue [7]. The

PEDOT electrode is made flexible by using thin film technology for the realization of the

electrode on a thin PET film which acts as the carrier material.

The characterization of the PEDOT electrode is required for its future use in cell culture

mediums. Low electrode-electrolyte impedance at the interface of the electrode and the

electrolytic medium is an important factor in the design of microelectrodes for biomedical

applications [5]. This thesis work has been done to model and simulate the electric field

around the PEDOT electrodes in the electrolytic medium. Impedance measurements of the

PEDOT ring electrode were done in an electrolytic medium (i.e. isotonic salt solution)

without fibroblast cells [14]. Using electrochemical impedance spectroscopy, the equivalent

electrical circuit to represent the electrode / electrolyte interface was developed. The

electrochemical system used for the impedance measurement is described below.

The electrochemical system consists of the cathode | electrolyte | anode placed in an

electrolytic cell. Measurements were taken with PEDOT and gold electrodes. This work deals

with only PEDOT electrodes. The system described in this work is -

| |

where

The other electrolytes used are PBS (phosphate buffered saline) and isotonic salt solution.

9

Figure 1 THE ELECTROCHEMICAL SYSTEM

1.1 MOTIVATION The performance of the PEDOT electrode in cell culture mediums depends on the electrical

properties of the electrode material. The electrical properties of the electrode affect the

fibroblast cells, their growth and the electric field in the cell culture medium. The

characterization of the PEDOT electrode is necessary for the analysis of the PEDOT electrode

and its stimulation effect.

Present day methods for monitoring wound healing are inadequate as they require the early

opening of the dressing after surgery [14]. Impedance characterization of the electrode /

electrolyte interface is required for biosensing in which changes in impedance are related to

cell growth and to their movement [12, 13]. The measurement of the changes in the AC

impedance of the PEDOT electrode allows for monitoring the cell growth using a non-

invasive method [14].

The purpose of this thesis is the development of the electrical circuit for the physical

representation of the electrode / electrolyte interface. The sensitivity analysis of the circuit

parameters of the electrical circuits is also performed. Finally, the simulation of the electrical

field around the electrode is performed. This is done for both the absence and presence of the

electrolytic medium in order to account for the electrode polarization. Isotonic salt solution is

the electrolytic medium used in the COMSOL simulation of the diffuse double layer.

PEDOT Stainless

Steel

10

2.0 THEORY

The electrochemical system is used for the measurement of impedance of the electrode in the

electrolytic solution in the absence of fibroblast cells. The first section describes the

properties of electrodes. A subsection deals with the diffuse double layer theory. The second

section deals with the electrolytic medium. In the third section the electrode kinetics is

described. In the fourth section the process of electrochemical impedance spectroscopy is

described. Finally, the process of optimization of values is discussed with a subsection on the

sensitivity analysis of the equivalent circuits.

2.1 THE ELECTRODES The PEDOT stimulation foil electrodes were used to stimulate the cells. The foil electrodes

were connected to the stimulation electronics which direct voltage and a unipolar pulsed

voltage [15]. The amplitude of the applied voltage pulse is 50 mV. A low electrode /

electrolyte impedance for stimulation electrodes is required to ensure there is effective

stimulation of cells and the applied voltage pulse is not lost in the electrolytic media. The

interface impedance has to be characterized for understanding of the processes occurring at

the interface, leading to an optimized electrode / electrolyte interface design [5].

The PEDOT measurement electrode is an interdigital microelectrode on a PET foil. The

diagram of the electrode is given below. Interdigital microelectrodes are designed to have the

length of the individual electrode much greater than the width of the electrode. Interdigital

microelectrodes are used for electrochemical reactions as the geometric structure allows for

the reversible redox reactions to occur easily as the gap between the electrodes is very small

(50 µm) [16].

Figure 2 DIAGRAM OF THE PEDOT INTERDIGITAL MICROELECTRODE WITH THE SPECIFICATIONS

L = 15 µm

w = 50 µm

d = 50 µm

L

w d

11

At the electrode / electrolyte interface, electron transfer reactions take place giving rise to

potential difference across the boundary. At the interface of the electrolyte there is a change in

forces due to presence of the electrode material. The application of a potential difference at

the electrodes gives rise to forces which bring about the change in the arrangement of the

dissolved molecules and ions. There is a formation of the diffuse double layer at the electrode

/ electrolyte interface. The diffuse double layer theory is described below.

2.1.1 THE DIFFUSE DOUBLE LAYER THEORY

Figure 3 TWO NEGATIVELY CHARGED SURFACES OF SURFACE CHARGE DENSITY σ IN A MEDIUM OF PERMITTIVITY ε AND SEPARATED BY DISTANCE d

When an uncharged solid electrode is placed in a liquid electrolytic medium, there is an

accumulation of charges on the surface of the electrode. Due to reorientation of the water

molecules and redistribution of dissolved ions there is a change in the electroneutrality at the

phase boundary giving rise to an electrical field. The free electrons in the electrode which is a

metal (i.e. a conductor) move towards the phase boundary and a charge is induced on the

surface of the metal electrode. The induced charge is opposite to the charge of the adsorbed

ions on the electrode surface from the electrolyte side of the phase boundary. As a result of

the induced charge, there is separation of charge at the phase boundary but the whole

interphase is electrically neutral [17].

The position or center of all ions adsorbed on the surface of the electrode form the IHP (Inner

Helmholtz plane). This gives the electrode a surface charge density of . The ions bound to

the surface of the electrode attract ions of opposite charge and are thus counterbalanced and

form the double layer.

When the ion surface charge densities are lower, there is a layer of oppositely charged ions

further away from the surface and hence it is the diffuse double layer. The locus of the

oppositely charged ions at their closest distance to the surface of the electrode forms the OHP

(Outer Helmholtz plane). The distance between the bound ions and the oppositely charged

+

+

+

+ +

+

+

+

-

- -

-

-

σ - - -

-

-

σ

d

Ԑ

Electrolytic medium

Electrode surfaces

12

attracted ions is given by the Debye length which is in the order of nanometers. It is defined

below –

√(

∑ ( )

)

(2.1)

where

( )

The concentration of oppositely charged ions depends on the surface charge density and the

concentration in the central plane of the electrochemical cell. The region beyond the OHP is

the diffuse layer which consists of the ions dispersed in the electrolytic solution. The net

charge on the IHP, the OHP and the diffuse layer is zero [10].

Figure 4 STRUCTURE OF THE DIFFUSE DOUBLE LAYER [2]

Metal IHP OHP Diffuse

Layer

Diffusion

Layer

Bulk

0.2 nm 0.2 nm

1 to 10 nm 0.3 mm 10 to 100 mm

13

Charge transfer at the electrode / electrolyte interface takes place in two ways –

Faradaic reactions

Non-Faradaic reactions

In Non-Faradaic reactions there is no charge transfer between the electrode and electrolyte.

There is only the redistribution of ions in the electrolyte. The metal electrode in an electrolyte

attracts ions present in the electrolyte. This causes the surface of the electrode to be charged

surrounded by ions in the electrolyte of the opposite charge.

Faradaic reactions involve the charge exchange between the electrode and the electrolyte.

These reactions are of two types –

Reversible reactions

Irreversible reactions

Faradaic reactions with very fast kinetics relative to the rate of mass transport are reversible.

Irreversible Faradaic reactions result in the net change of the chemical environment of the cell

and may give rise to harmful chemical species which are damaging to tissue or the electrode.

The polarizability of an interface depends on the rates of transfer of charges across the

interface [16]. Thus, based on the ease of passage of charged components across the electrode

/ electrolyte interface there are two classifications –

Ideally polarizable interface

Ideally non-polarizable interface

Ideally non-polarizable / non-blocking interphases allow the exchange of charged chemical

species. This means the potential drop across the interphase does not vary with the change in

the current. These interphases are modeled as Faradaic impedance with zero resistance (i.e. as

a solution resistance) [6]. For ideally polarizable / blocking interphase the exchange of

common charged species does not occur easily. For a given voltage, the interphase does not

allow the flow of faradaic current. This is modeled as Faradaic impedance with infinite

resistance in the electrical circuits (i.e. as a pure capacitor in series with the solution

resistance). All electrode / electrolyte interfaces display a behavior in-between ideally

polarizable and non-polarizable interphase.

PEDOT electrodes have both reversible faradaic processes and non-faradaic process due to

the reversible adsorption of the charged species on the PEDOT electrode surface. The AC

impedance of the electrode is also reduced by increasing the surface area of the interphase by

introducing roughness [7].

14

2.2 THE ELECTROLYTIC MEDIUM The electrolytic mediums used during our measurements are DMEM, PBS and isotonic salt

solution [9]. DMEM is a medium providing nutrients for cells, PBS and isotonic salt solution

are used in cell cultures as well. PBS was used which is a phosphate buffered salt solution.

The salts of PBS are needed to provide physiological environment for the cells. The

supporting electrolyte also by the large number of ions shields the bulk electrolyte from the

electrical field and a potential drop occurs at the interface [6]. The composition of the

electrolytic mediums is described below.

Table 1 COMPOSITION OF THE PBS MEDIUM

PBS

Salt Buffered molecular

weight

Concentration (mg/l) Concentration

(mM)

KH2PO4 136 144 1.06

NaCl 58 9000 155.17

Na2HPO4. 7H2O 268 795 2.97

Table 2 COMPOSITION OF THE DMEM HAM’S F12 GLUTAMAX MEDIUM

DMEM Ham’s F12 Glutamax

Salt Buffered molecular

weight

Concentration

(mg/l)

Concentration

(mM)

CaCl2 111 116.3 1.05

CuSO4. 5H2O 250 0.0013 0.0000052

Fe(NO3)3.9H2O 404 0.05 0.000124

FeSO4.7H2O 278 0.417 0.0015

MgCl2.6H2O 203 61 0.3

MgSO4.7H2O 246 100 0.407

KCl 75 311.8 4.16

NaHCO3 84 1200 14.29

NaCl 58 6995.5 120.61

Na2HPO4.7H2O 268 134 0.5

NaH2PO4.H2O 138 62.5 0.453

ZnSO4.7H2O 288 0.432 0.0015

2.2.1. CALCULATION OF THE CONDUCTIVITY OF THE ELECTROLYTE

The conductivity of the three electrolytic mediums was experimentally measured. The

measured values were compared with the theoretical values at 298 K. The method for

calculation of the conductivity is described below [11, 4].

The concentration of the mixture is ( ). The net molar conductivity

( ) for an unsymmetrical electrolyte is given by the formula below.

15

( ) ( )

(2.2)

where

The molar conductivity ( ) is given by the Kohlrausch formula below.

( ( √ ))

(2.3)

where

( )

( )

(2.4)

( )

√(

)

(2.5)

where

( )

After substitution of values –

( )

√(

)

(2.6)

After substitution of values –

(2.7)

The Kohlrausch formulae for water reduces to the following –

16

( ) ( )

( ) ( )

The conductivity of the electrolyte is calculated by the formula below.

(2.8)

where

( )

Table 3 COMPARISON OF MEASURED AND CALCULATED CONDUCTIVITY VALUES

Electrolytic Medium Measured conductivity

(mS cm-1

) at 298 K

Calculated conductivity

(mS cm-1

) at 298 K

PBS 15.29 21.20

DMEM 14.53 37.00

Isotonic salt solution (0.9%) 15.88 14.10

The difference in the calculated value of the conductivity for PBS and DMEM is observed

due to the many salts present in the solution in each case giving rise to greater differences in

the values of the viscosity of the salt at 298 K.

2.2.2. MASS TRANSPORT OF SPECIES IN THE ELECTROLYTE

The mass transport in an electrolytic medium defines the movement of mobile ionic species

by migration, diffusion or convection. As the electrolytic medium is kept stationary, mass

transport by convection does not occur. The following equation below describes the flux

density of dissolved species in a dilute solution due to migration and diffusion [2].

(2.9)

where

( )

The movement of the charged species under the driving force of the electrical field is called

migration. The electrical field drives the cations towards the cathode and the anions to the

anode with the cations moving against the gradient of the potential applied. The migration

velocity with which the ions move under the influence of the electric field is dependent on the

mobility of the ion in the electrolytic medium. It is described below.

(2.10)

Where

( )

17

( )

( )

The flux density of the dissolved species in the electrolyte by migration is

(2.11)

On substituting values from equation 2.10 we get –

( ) (2.12)

Where

( )

The mass transport of ions also takes place due to the presence of concentration gradients in

the solution by diffusion. The flux density of species under diffusion is given by the equation

below.

( ) (2.13)

where

( )

2.2.3 DIELECTRIC RELAXATIONS IN THE ELECTROLYTE

As biological tissue has dielectric properties [18], thus the electrolyte conducts current but

shows dielectric properties as well. A dielectric behaves as an insulator in absence of an

electrical field. There is an alignment of charges in the material in the direction of the applied

electric field. In presence of the applied electrical field the charges in the dielectric medium

undergo polarization which brings about a change in the net electric field. The dielectric has

an electrical conductivity and the electrical permittivity . The frequency dependent

polarization occurs in a large range of frequencies giving rise to the complex dielectric

permittivity. The imaginary part of the complex dielectric permittivity gives the energy loss of

the dielectric while the real part gives the storage of energy by the dielectric [8].

(2.14)

( ) (

)

(2.15)

where

( )

In polar liquids, the dipoles take finite time to orient themselves in the direction of the electric

field. This is the relaxation time given by . Due to the complex structure and composition of

the biological medium, there is broadening of the dispersion region which is given by the

distribution parameter . The modeling of an electrical circuit given by equation 2.16 is

described in section 2.4.1.5 where the dispersion mechanism is given by the constant phase

element. The relative permittivity of the medium varies within the frequency range and is

described below [24, 19].

18

( )

( ( )( ))

(2.16)

where

( ) ( )

substituting in equation 2.15

( ( )

( ( )))

(2.17)

The dispersion is given below

(2.18)

where

The electrical field equations used to arrive at the complex permittivity and the characteristic

frequency for the dielectric medium is given below [14]. The charge relaxation frequency is

also described below [8].

The Maxwell equations for electric field in free space are given below [12, 15] –

( )

(2.19)

For static electric fields

(2.20)

( )

(2.21)

(2.22)

where

( )

For conductive media, the following relation Maxwell applies

( )

(2.23)

Taking divergence on both sides

19

( ) (

)

(2.24)

The divergence of the curl of a vector is zero

( ) (2.25)

For conductive media, the following relations apply

(2.26)

(2.27)

where

( )

( )

(2.28)

Substituting values in equation 2.6

( )

(2.29)

( )

(2.30)

where

( )

For a medium with dielectric permittivity Ԑ and conductivity σ having a width d

The capacitance C (farad) of the parallel plate capacitor formed by the two electrodes with a

dielectric medium in between is given by equation 2.13

( )

(2.31)

The resistance to the flow of the current through the medium is given by equation 2.32

Ф = 0 Ф = 1 d

Ԑ, σ

20

(2.32)

where

( ) ( )

(2.33)

where

( ) The admittance ( ) is given by the equation 2.34

( )

(2.34)

Substituting values from equation 2.31 and 2.33 in equation 2.35

( )

(2.35)

(2.36)

(2.37)

(2.38)

where

( )

( )

(2.39)

where

( )

The charge relaxation time is defined as the time taken by the ion to move the distance (i.e.

the Debye length) by diffusion. The dipoles in the dielectric medium (i.e. the electrolyte)

vibrate with the characteristic frequency when an external electric field is applied. Due to

the alternating electric field changing its polarity, all the charged particles move in the

direction of the electric field.

The dielectric interactions with the electric field give rise to different polarization

mechanisms. The polarization mechanisms are distinguished by the characteristic frequency

response in the dielectric medium. As the impedance measurements are carried out in the

frequency range 20 Hz to 1 MHz, interfacial polarization is present. Due to the high relaxation

21

frequency of water (20 GHz), orientational polarization is also present. The diagram below

shows the frequency regions in which they are observed.

Figure 5 POLARISATION MECHANISMS IN DIELECTRIC MEDIUM [8]

2.3. THE ELECTRODE KINETICS

The electrochemical reaction due to the passage of the Faradaic current is related to the

electrolysis rate given by the following equation-

( )

(2.40)

where

( )

Substituting values from equation 2.37 in the equation below

(2.41)

( )

(2.42)

22

( )

(2.43)

where

The electrochemical reaction has a reference potential defined as the equilibrium potential.

The departure of the cell potential from the equilibrium value upon passage of an electric

current is called polarization. The difference in the potential is given as the overpotential

which is denoted by and the unit is volt.

(2.44)

where

( )

( )

The electrochemical system consists of the PEDOT electrode and the stainless steel beaker.

The current passes through the cathode and the anode which completes the electrical circuit.

For an electrochemical reaction to take place a driving force is required. The Butler - Volmer

rate equation relates the surface overpotential with the current density at the electrodes [2].

(

) (

)

(2.45)

where

These kinetic parameters relate the choice of the forward or backward direction of the

reaction based on the applied potential. The values lie between [2].

( )

The forward and reverse electrochemical reactions are each associated with a different current

density. At equilibrium the current density for the forward reaction is equal to that for the

reverse reaction. At equilibrium the net current density is called the exchange current density.

Generally the electrochemical cell is not in equilibrium and the net current density has a finite

value.

23

The reactions taking place at the electrodes are of importance for the defining of the

electrochemical system. The rate of the electrode reaction at steady state depends on the

current density across the electrode surface, the diffuse layer in the electrolyte and the

electrode potential.

2.4 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY The electrical stimulation of cells causes many processes to occur throughout the cell which

brings about changes in the overall electrical response of the system. Impedance spectroscopy

is the method utilized to characterize materials and their interfaces with conducting

electrodes. A unipolar voltage pulse is applied at the electrode / electrolyte interface to

measure the magnitude and phase shift of the resulting current at that frequency is used to

calculate the impedance at the interface. This process is repeated over a wide frequency range.

For most electrodes the voltage pulse is of small-signal magnitude such that the AC

(alternating current) response of the overall system is electrically linear. The electrochemical

system is approximated to be linear as long as the amplitude of the maximum voltage is less

than thermal voltage for a given electrode design [14, 19].

( ) ( ) (2.46)

where

( ) ( )

(

)

(2.47)

( )

( ) (( ) ) (2.48)

where

( ) ( )

( )

( )

( )

(2.49)

( ) (2.50)

where

( )

2.4.1 EQUIVALENT CIRCUITS

The impedance spectroscopy measurements are fitted to an equivalent electrical circuit which

is the physical representation of the processes occurring in the electrochemical cell. The

equivalent circuits and circuit elements which are used as building blocks for the

electrochemical system are described below.

2.4.1.1 THE ELECTRIFIED INTERFACE

The electrified interface (i.e. the metal / solution interface) can be represented by an

equivalent circuit with a resistor in parallel with a capacitor [16].

24

Figure 6 THE PHYSICAL REPRESENTATION OF THE ELECTRIFIED INTERFACE ON THE LEFT IS THE R||C CIRCUIT

The Nyquist plot of the complex impedance of || circuit is a semi-circle with the center

on the real impedance axis. The equivalent impedance equation for the electrified interface.

( )

(2.51)

where ( ) ( )

2.4.1.2 THE WARBURG ELEMENT

The Warburg element is the physical representation of the occurrence of the process of

diffusion in the electrochemical system. It is the solution of the Fick’s equation for linear

diffusion. It is given the equation below [6, 25].

Figure 7 THE WARBURG ELEMENT

( ) ( )

(2.52)

where

( )

( )

(( ) √ )

(2.53)

where

( )

W

ELECTRODE ELECTROLYTE

C

R

25

( )

( )

( )

2.4.1.3 THE CONSTANT PHASE ELEMENT

The constant phase element is the physical representation of the straight line behavior seen in

the lower frequency range of the Nyquist plot. The constant phase element has a phase angle

which is independent of the frequency. The equation is given below [6].

Figure 8 THE CONSTANT PHASE ELEMENT

( ) (2.54)

where

( )

2.4.1.4 CIRCUIT REPRESENTING ELECTRODE POLARIZATION

Electrode polarization at low frequency is due to the excess charge at the electrode /

electrolyte interface due to presence of water molecules and hydrated ions. It is modeled as

series connection of a resistor with a capacitor [26, 18].

The equation of the equivalent impedance is given below –

( )

(2.55)

Figure 9 PHYSICAL REPRESENTATION OF THE ELECTRODE POLARIZATION

2.4.1.5 RANDLES MODEL

The equivalent circuit represents a single step charge transfer reaction in the presence of

diffusion. The decoupling of the faradaic current and non-faradaic current is modeled as the

double layer capacitance in parallel to the faradaic impedance. The electrolyte resistance is

represented as in the figure below [6].

CPE

CP RP

26

Figure 10 PHYSICAL REPRESENTATION OF THE RANDLES MODEL

The equation for the equivalent impedance of the model is given below –

( )

(2.56)

( )

(2.57)

(

( ) (

)

)

(2.58)

2.4.1.6 CIRCUIT REPRESENTING COMPLEX IMPEDANCE IN BIOLOGICAL TISSUE

The Nyquist plot of the complex impedance of biological tissues and some electrode /

electrolyte interfaces display a locus having a depression in the half circle with the center of

the semi-circle lying below the real impedance axis. The complex impedance in case of the

biological tissues where there is a distribution of relaxation time was described by the circuit

and the equation described below [6].

Figure 11 THE EQUIVALENT CIRCUIT FOR THE COMPLEX IMPEDANCE

CPE

R0

R0 - R∞

C1

R1

W

RS

27

( )

( ( ) )

(2.59)

where

( )

The measured impedance is plotted on the Nyquist plot and an equivalent circuit according to

the relaxation behavior seen in the Nyquist plot. The equivalent circuit is a physical

representation of the electrochemical cell and the processes occurring on passage of an

electric current through it. It consists of the circuit representing the electrode / electrolyte

interface and the bulk electrolyte. The equivalent circuit is arrived at by comparing the

relative norm 2 error of the impedance of the circuit with the measured impedance values at

the same frequency. The electrical circuit with the minimum percentage error is the circuit

that best represents the behavior of the electrochemical system. The formula of the calculation

of the relative L2 norm error is described below.

√(∑( |( )| )

∑| | )

(2.60)

where

( )

( )

2.5 OPTIMIZATION OF VALUES The optimization of a quality function requires improving its design depending on its design

parameters [20]. The optimization is carried out on the impedance function to find the design

parameters when the function has minimum relative norm 2 error. This was done by an

optimizer written in MATLAB. A subsection describes the sensitivity analysis of the

electrical circuit.

The quality function is given by the impedance of the equivalent circuit of the electrochemical

system at a given frequency. Multiple parameter optimizations require an unconstrained

optimization method which calculates the design parameters. The MATLAB inbuilt functions

based on the simulated annealing method and Nelder-Mead-Simplex algorithm were used to

find the global minimum value of the impedance function. An optimizer based on the

evolution strategy was also used.

Simulated annealing is a method for solving unconstrained optimization problems. The

method models the process of heating up a material and then slowly reducing its temperature

to reduce any defects [22]. The temperature represents a variable denoting the distance of the

random point from the current point. It also affects the function’s choice of a random point,

28

with the higher value of the function to be optimized at each iteration. The method generates a

random point based on a probability distribution which is directly proportional to the

temperature of the annealing. The extent of the search of the method is reduced with the

reduction in the temperature as the system becomes stable (i.e. no further changes occur). The

MATLAB function “simulannealbnd” was used for the optimization of values. It was found

that the function took a lot of time to converge to a solution and the region of search was

narrow.

The Nelder-Mead-Simplex method develops the ( ) dimensional polygon with points

in the polygon along with the initial guess. The minimum value of the quality function is

calculated for the points in the polygon and worst valued function is discarded. A new point

is added to the polygon and the iterations continue till the stopping criterion is reached [21].

The function “fminsearch” in MATLAB is based on this algorithm. The optimization method

was not used as the minimum value reached in each run of the program showed large

variations from the previous value for the same initial guess.

The optimizer used is based on the Evolution strategy. The algorithm is based on the natural

processes of mutation and selection in nature. The optimizer assumes the plane of the

equation to have contour lines. The plane contains the global minimum of the function. The

first point selected is a random point on the plane. The quality factor for this point is

calculated. Then another random point is found within the circle of range of the step

size . If the quality factor of the new point is greater than that of the previous point, it is

discarded. This process is followed till the optimizer finds a point with a quality factor better

than the previous point saved in the variable and . The success rate is defined for

most optimizers to be around .

The calculation of the random points is done to reach the global minimum in minimum

iterations. However a certain minimum number of iterations is required for the optimizer to

generate enough random points to define the plane having the contour lines of the function.

The point at which the tangent cuts the contour, the probability of occurrence of the point

is . As the plane has two sides so the probability of occurrence of the point on one side is

half. If the random point occurs closer to the global minimum, the angle of approach

decreases. The random point is calculated and the quality factor is checked. If the quality

factor is smaller than that of the previous number the step size is multiplied with the

variable . The variable is defined as the value which determines how fast the function

converges to its final value. The variable is defined to determine the factor ( ) by

which the circle in which the random point lies is increased. As the point converges to the

global minimum its step size is reduced by multiplication with the assumed in

the optimizer.

2.5.1 SENSITIVITY ANALYSIS OF THE EQUIVALENT CIRCUITS

Each element in an equivalent circuit has a predefined tolerance. The tolerance of the circuit

element defines its maximum and minimum values without taking any effects due to

temperature into consideration. The sensitivity analysis of an electric circuit is its sensitivity

in the electrochemical system i.e. the ability of the electrical circuit to react to change in its

29

design parameters due to stimulation, either external or internal. These changes in the circuit

elements will occur within the tolerance range of each circuit element. Sensitivity is defined

as the limiting ratio of the fractional change in the measured value to the fractional change in

the value of a certain parameter [23]. It is defined by the equation below.

(2.61)

where

Thus for a large change in the fractional value of a circuit element if the net change measured

is too small, the circuit is then less sensitive to changes. An electrical circuit of high

sensitivity is required for measurement applications [23].

In the equivalent circuits, the sensitivity has been defined as the fractional change in the

quality factor of the impedance function given by the error in L2-norm and that of a 1%

variation in the calculated circuit parameters.

30

3.0 RESULTS AND DISCUSSION

In this chapter, in the first section the modeling of the equivalent electrical circuit

representation of the PEDOT electrode / electrolyte interface is described. In the next section,

the simulation of the double layer is described.

3.1. MODELLING OF THE EQUIVALENT ELECTRICAL CIRCUIT

The impedance measurement of the PEDOT electrode was done with ringer solution as the

electrolytic medium and without any cells in the medium. The measurements were conducted

with the PEDOT electrode for three different RMS voltages of 50 mV, 150 mV, and 335 mV.

The impedance measurements were taken for the frequency range 20 Hz to 1 MHz. The

different plots of the magnitude vs. log (frequency), phase vs. log (frequency) and the Nyquist

plot is given below.

Figure 12 MAGNITUDE OF THE MEASURED IMPEDANCE (OHM)

31

Figure 13 PHASE OF THE MEASURED IMPEDANCE OF THE PEDOT ELECTRODE (DEGREE)

Figure 14 NYQUIST PLOT OF THE MEASURED IMPEDANCE

32

The Nyquist plot shows three distinct relaxation regions – a straight rising line behavior in the

low frequency region, an inclined line in the region between and finally the rise in the curve

in the high frequency region. Each relaxation region corresponds to a polarization mechanism

with a characteristic time constant.

For the optimization of the values, the impedance equation of the calculated electrical circuit

was written as the quality function. When the optimization program was run, the initial guess

was taken as random equivalent values (i.e. 1 was assumed for each of the circuit elements). If

the initial guess of the values of the circuit element were unequal in their range (i.e.

capacitance in µF and resistance in kΩ), scaling (i.e. division by a scale factor) was performed

to have a homogenous set of numbers. The scale factors were chosen such that they were

defined in the range of the initial guess (i.e. for a capacitance in µF a scale factor of

was chosen). The scaled values were then passed through the optimizer. The optimizer

used is based on evolution strategy. After the optimization, the optimized values are

multiplied by the scale factor. The Nyquist plot along with the Bode plot was used for curve

fitting.

The sensitivity analysis of the electrical circuits was performed using their optimized values.

The optimized values were scaled to bring them to have a homogeneous set of values. A

change was introduced in each of the circuit elements and the relative error was calculated

with the changed element. The ratio of the absolute difference of the error values with the

change in element value is the sensitivity of the circuit element. The different electrical

circuits modeled to represent the system are described below.

3.1.1 ELECTRICAL CIRCUIT I

The electrode / electrolyte interface was modeled to have the CPE element to represent the

electrical double layer at the interface. The bulk electrolyte was modeled to have DC (direct

current) conductance.

Figure 15 DIAGRAM OF EQUIVALENT ELECTRICAL CIRCUIT 1

R1

R0

C1

CPE

R3 C3

C2

R4 Electrode / Electrolyte interface

Bulk electrolyte

33

Impedance Equations for the electrical circuit –

( )

(3.1)

(( ) ( ))

(3.2)

( ( )( ))

(3.3)

( )

(3.4)

(

)

(

)

(3.5)

The optimized circuit parameters –

Circuit parameter Optimized Value Initial Guess Scale Factor

R1 269.7098 Ω 284.4680 Ω 100

R2 276.4608 Ω 276.4713 Ω 100

R3 12.9611 Ω 12.9541 Ω 10

R4 102891.3 Ω 101447.81280 Ω 10000

C1 2.8175 e-6 F 2.8309 e-6 F 1.0 * e-5

C2 9.6737 e-9 F 9.6921 e-9 F 1.0 * e-8

C3 4.4542 e-5 F 4.4565 e-5 F 1.0 * e-5

A 1.7391 2.2255 1.0

α (CPE angle) 0.7105 rad 0.71218 rad 1.0

Relative error 0.001603789 * 100 = 0. 16 %

Iterations 10 000 000 10 000 000

Final tolerance value

reached

1.4291e-6 1.1148e-6

Physical Significance of the circuit –

The electrical circuit represents the double layer with the capacitor . It is placed in parallel

to the resistor parallel to the CPE element to represent the different faradaic reactions. The

capacitor represents the non-faradaic reactions. The resistor forms a part of the circuit for

the bulk electrolyte. The circuit was designed to have DC conductance. The electrical circuit

was found to display the minimum error. It was observed that the electrical circuit reached the

relatively same minimum value irrespective of the initial guess.

34

Figure 16 NYQUIST PLOT FOR CIRCUIT I

Sensitivity Analysis –

R1 R2 R3 R4 C1 (Ω/F) C2 (Ω/F) C3 (Ω/F) A(s0.5

) alpha

36.6839 35.7914 76.3363 9.6160 351.1603 102.2774 22.2129 56.89

32

139.2504

It is observed the sensitivity values of resistors did not change significantly during

optimization of the values. The sensitivity values are also low. This is observed across all the

circuit parameters. This means that the above electrical circuit has a high sensitivity.

35

3.1.2 ELECTRICAL CIRCUIT II

Figure 17 DIAGRAM OF EQUIVALENT CIRCUIT II

Impedance equations of the circuit-

(

( ) )

(3.6)

( )

(3.7)

( )

(3.8)

(

( ) )

(

)

(3.9)

The optimized circuit parameters –

Circuit parameter Optimized value Initial Value Scale factor

R1 1996759.1533 Ω 1999.2824 Ω 1000

R2 17.05206 Ω 39.995 Ω 100

R3 288.4163 Ω 289.7648 Ω 100

C1 8.0729 e-6 F 6.7143 e-6

1.0 * 10-6

C2 4.1536 e-5 F 3.1129 e-5

1.0 * 10-5

C3 1.3895 e-11 F 1.4599 e-11

1.0 * 10-11

A 565.989 2.163 1000.0

Relative error 0.00483848*100 = 0.48 % 0.01609

iterations 3017785

tolerance value 9.9983 e-13

Physical significance of the circuit –

The electrode / electrolyte interface is modelled by a resistor in series with a Warburg

element. These together are placed in parallel with a capacitor. The capacitor represents the

C2

R3

W

R1

C1

R2

Electrode / Electrolyte interface Bulk Electrolyte

C2

36

electrical double layer which is placed in parallel with a faradaic reaction given by a resistor.

The Warburg element represents the diffusion process. The bulk electrolyte is modelled to

have DC conductivity. It was observed in the Nyquist plot below that the curve fitted well in

the low frequency region. For the high frequency the curve did not fit well. This is also due to

non-linear behavior which cannot be modeled by electrical circuits.

Figure 18 NYQUIST PLOT FOR EQUIVALENT CIRCUIT 3

Sensitivity Analysis –

R1 R2 R3 C1 (Ω/F) C2 (Ω/F) C3 (Ω/F) A (s0.5

)

0.049839 583.6014 34.50431 12.32709 23.95842 71.61502 175.8268

The change in the value of is very large compared to the optimized value. However the

sensitivity value is quite small. Thus, the sensitivity analysis shows the dc conductivity is not

very sensitive to changes. The variations for the rest of the circuit elements are same for the

sensitivity analysis and in the optimization. These elements are sensitive to the change in

their values. Similarly, the Warburg constant shows a large variation compared to the

optimized value and also for the sensitivity value. Thus, the electrical circuit shows that the

electrochemical system is highly sensitive to changes.

It was observed that this circuit did not have small changes in the optimized values. The

variations in circuit parameters were seen on several runs of the optimization procedure.

Based on this the circuit 2 was found to be stable with respect to the optimization procedure.

The circuit II was found to have the lowest percentage relative norm 2 error. As the

measurements of the impedance were conducted in an electrolytic medium without cells, the

37

circuit should only have processes of electrode polarization, diffusion and effects due to

interfacial relaxation.

3.1.3 ELECTRICAL CIRCUIT III

Figure 19 DIAGRAM FOR EQUIVALENT CIRCUIT 2

Impedance Equations for the electrical circuit -

( )

(3.10)

(( ) ( ))

(3.11)

(( ) ( ))

(3.12)

(( ) ( ))

(3.13)

( )

(3.14)

The optimized circuit parameters –

Circuit parameters Optimized value Initial guess Scale Factor

R1 274.7988 Ω 275.3 Ω 100

R2 10.9051 Ω 10.63 Ω 10

R3 3.2511 Ω 3.622 Ω 10

R4 13.88577 Ω 13.48 Ω 10

C1 4.8337e-5 F 48.27e-6 F 1.0*10

-6

C2 1.8995e-5 F 20.90e-6 F 1.0*10

-6

C3 5.5213e-6 F 6.07e-6 F 1.0*10-6

C4 7.5307e-9 F 8.12e-9 F 1.0*10-9

A 506.9067 503.10 1000

relative error 0.0016298*100 = 0.16 %

Iterations 10 000 000

Final value of tolerance

reached

5.1944e-6

R1 C1

W

R2

C2

R3

C3

R4

C4

38

Physical significance of the circuit –

The electrical circuit was modeled along the relaxations regions in the Nyquist plot. The

electrode polarization occurring in the low frequency region was modeled as a series resistor

and capacitor. The diffusion region was taken in the plot as the straight line in the central

frequency range (variations possibly due to inexact frequency values). The rise in the high

frequency region was modeled with R || C circuits. It was found that the relative error reduced

with addition of further R || C circuits. These could be attributed to the relaxations occurring

due to the dipolar molecules in the electrolytic medium.

Figure 20 NYQUIST PLOT FOR CIRCUIT III

Sensitivity Analysis –

R1 R2 R3 R4 C1

(Ω/F)

C2

(Ω/F)

C3

(Ω/F)

C4

(Ω/F)

A (s0.5

)

34.4302 86.7933 29.1021 68.1438 1.9573 4.9808 17.136 12.5637 186.6493

The variation in each of the circuit elements was small. As the values of the sensitivity

analysis also give smaller values for each of the circuit elements, thus the sensitivity of the

electrical circuit is high.

The Magnitude Plot and Phase plot versus the frequency are given below. It is seen that the

curve fitting is good as both the curves overlap (i.e. for the measured |Z| and calculated |Z|).

Hence, this is the electrical circuit which best represents the PEDOT electrode.

39

Figure 21 MAGNITUDE PLOT OF THE CIRCUIT III

Figure 22 THE PHASE PLOT OF CIRCUIT III

40

3.2 SIMULATION OF THE ELECTROCHEMCAL SYSTEM As described in the earlier chapters, electrode polarization causes a decrease in the electrical

field. There is significant variation in the presence of an electrolyte. The electrochemical cell

was simulated without any electrolytic medium in the first subsection to calculate the

electrical field. The next subsection describes simulation of the non-uniform current density

across the electrodes to calculate the impedance of the solution. The third subsection deals

with the simulation of the electrical double layer in one dimensional space.

3.2.1 SIMULATION OF THE ELECTRICAL FIELD IN THE ABSENSE OF AN ELECTROLYTE

The electrochemical system was modeled in two dimensional space as two electrodes

separated by a distance of 50 µm. A potential of 600 mV was applied at one of the electrodes

and the other electrode was kept at ground potential. The electrolytic medium was assumed to

be air. The model was developed using electrostatics interface for stationary study.

(3.15)

where

( )

( )

(3.16)

where

( )

The magnitude of the electric field was calculated for the plot using the formula below.

| | √( ) ( )

(3.17)

where

( )

( )

41

Figure 23 THE 2D PLOT FOR THE MAGNITUDE OF THE ELECTRICAL FIELD

3.2.2 CALCULATION OF THE IMPEDANCE OF THE SOLUTION

The simulation for the current density was performed taking the electrolytic medium to

isotonic salt solution. The conductivity of the electrolytic medium was taken at at

25°C. The potential of 70 mV was applied at each electrode. The electrical currents interface

for conductive media with stationary study was used to develop the model. The equations

governing this interface are given below.

(3.18)

where

( )

( )

( ) (3.19)

where

( )

( )

(3.20)

42

Figure 24 THE PLOT OF THE VARYING MAGNITUDE CURRENT DENSITY

The magnitude of the y-component of the current density crossing the interface was calculated

using line integration of the interface in between the electrodes.

Current Density , y-component (A/m) Current Density , x-component (A/m)

0.00126 0.1853

The impedance of the solution was calculated and simulated for isotonic salt solution having

the conductivity as given below. The impedance was calculated for the stimulation electrodes.

As the electrodes have a circular geometry the average length of the electrode was calculated.

The simulation of the current density was performed using the electric currents interface for

stationary study described in subsection 3.2.3 was used. As the y component of the current

density is very small it was not considered for the calculation.

( )

43

( )

(

)

(3.21)

where

( )

3.2.3 SIMULATION OF THE DIFFUSE DOUBLE LAYER

The model of the diffuse double layer is based on the model of the diffuse double layer from

the COMSOL model documentation site [27]. The formation of the diffuse double layer has

been explained above in chapter 2.1.1. The diffuse double layer is solved by solving for

equations Nernst Planck equation which is described in chapter 2.2.2. The diffuse double

layer was modeled by using the electrostatics interface with stationary study coupled with the

Transport of dilute species physics interface and the global ODEs and DAEs interface. The

following assumptions are made for the simulation –

There are no heterogeneous reactions taking place in the electrolyte.

The condition of electroneutrality is assumed across the electrode interface and the

diffuse double layer.

The boundary is the reaction plane which is placed between the IHP and the diffuse

double layer.

A Stern compact layer of thickness was assumed

The analysis was carried out using dimensionless quantities. A variable was defined to

decrease the length of the electrolytic cell with respect to the Debye length. The length was

divided by the parameter to decrease the size of the system. This helps to decrease the cell

voltage and also to increase the current value to a value more than the diffusion limiting

current [28]. The use of dimensionless numbers also helped the solver converge faster

(3.22)

44

where

( )

( )

The parameter is used as the cell length is much larger than the Debye length or the Stern

layer thickness.

( )

(3.23)

where

( )

The Einstein’s relation relates mobility with the diffusion coefficient.

( )

(3.24)

where

( )

( )

( ) The reference ion concentration for both species was taken as initial values.

For no heterogeneous reactions -

(3.25)

From the Poisson’s equation

( )

(3.26)

where

( )

( )

For the different ionic concentrations of the species

( )

(3.27)

where

The surface charge density is given below -

45

( )

(3.28)

where

At the electrode / electrolyte interface, electrochemical reactions take place for ions. The

reaction rate is given by the Butler-Volmer equation (2.45). For the boundary conditions the

rate of the reaction gives the flux for the positive ion.

(3.29)

where

( )

The following parameter values were taken for the Butler-Volmer reaction –

For the negative ion the flux is assumed to be zero

(3.30)

The cell potential equation is -

|

(3.31)

( )

The cell voltage (V) was calculated using equation 3.31. The initial voltage was taken to be

50 mV.

The potential at one electrode is taken was taken to be zero and the other electrode at cell

voltage V. This was used to calculate the overpotential.

A global constraint for the negative ion concentration was used to have the concentration of

negative ions calculated in the simulation to be less than the total concentration of the

solution. This is modeled as the integration

( ) ∫

(3.32)

where

( )

( )

The discretization for homogenous view of the plot was varied with the variable epsilon. For

46

the simulation the discretization of the concentration is made quadratic.

The dimensionless cathodic rate constant –

( )

( )

(3.33)

where

( )

The anodic rate constant Ka -

( )

( )

(3.34)

where

(3.35)

The cell current density was calculated from the limiting current density.

( )

(3.36)

( )

(3.37)

where

( )

The following dimensionless parameters were used for the plots

(3.38)

where

( )

( )

( )

(3.39)

where

47

( )

( )

(3.40)

where

( )

( )

(3.41)

where

The above values were to simulate the following –

The concentration profile in the electrochemical cell

The charge density profile

The dimensionless potential profile

Figure 25 DIMENSIONLESS CHARGE CONCENTRATION PROFILE

The dimensionless concentration profile shows the region of steepest profile is closest to the

electrodes [27].

48

Figure 26 DIMENSIONLESS CHARGE DENSITY PROFILE

The dimensionless charge density profile is given as the ratio of the difference of the positive

and negative ion concentration with the reference ion concentration. The dimensionless

charge density profile shows the charge separation that occurs near the electrode surface. This

is shown by the variation in the Debye length. As the value of is increased, the diffuse

double layer stretches further into the bulk electrolyte [27].

49

Figure 27 DIMENSIONLESS POTENTIAL PROFILE

The variation of the potential is greater for larger Debye lengths. This occurs as longer

distance between the electrodes gives rise to greater ion transport losses [27].

The potential in the bulk electrolyte was calculated from the graph above.

( ( ))

( )

(3.42)

Two points were marked on the graph and the reading for was taken

(

)

(3.43)

50

(

)

(3.44)

It was seen that the potential in the bulk was simulated to be 9.634 mV. From the equivalent

circuit 3, the bulk resistance is given by R1 as described in section

51

4.0 CONCLUSION AND OUTLOOK

Electrochemical impedance spectroscopy was used to develop the electrical circuit for

representing the PEDOT electrode / electrolyte interface. Three circuits were described in the

chapter before as they had the least minimum norm 2 error. For the process of curve fitting,

the circuit parameters were optimized using an optimizer based on evolution strategy. The

third electrical circuit was found to have the physical representation of the electrolyte without

fibroblast cells. It is the preferred circuit for representing the electrode / electrolyte interface.

The sensitivity analysis of the electrical circuit elements was performed. As the circuits have

to be used for medical applications, the sensitivity was measured for 1% variation in

parameters. The circuits were found to be sensitive to changes in the electrochemical system.

The simulation of the electrodes was also performed to simulate the effect of the electrical

field on the electrochemical system. First the electrical field in the absence of electrolyte was

simulated. The impedance of the solution in the absence of cells was also simulated. Finally,

the diffuse double layer was simulated in Comsol Multiphysics finite element software. The

simulated model was used to calculate the potential measured in the bulk electrolyte due to

the diffuse double layer.

Further work for the sensitivity analysis can be done. The sensitivity analysis was done

assuming the optimized values of the electrical circuit were lying on flat plane. For sensitivity

values lying in the same range, this was applicable. However, in case of large variation of the

sensitivity values between elements, principle axis transformation is needed to perform an

accurate sensitivity calculation. Also, the diffuse double layer model could be extended to two

– dimensional space giving an accurate estimation of the voltage drop occurring in the bulk of

the electrolyte.

52

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54

6.0 APPENDIX

6.1 OPTIMIZATION OF ELECTRICAL CIRCUIT PARAMETERS The various parts of the optimization program written in MATLAB are discussed below.

6.1.1 THE OPTIMIZER

%------------------ optimizer using evolution strategy used in MATLAB --------------------

function [x, y] = optimizer(x)

Qbest = Q(x); up = 1.001;

stop_crit = 1e-12; % minimum error value

success_rate = 0.2;

down = (1/up) ^ (success_rate);

iterations = 100000;

size_in = size(x); % pre-allocate array

delta = 1;

xp = (x);

for i = 1: iterations

x = xp + (delta .*(rand (1, 7) - 0.5));

actualQ = Q(x);

if ( actualQ < Qbest)

delta = delta * up;

xp = x;

Qbest = actualQ;

else

delta = delta * down;

end

if (delta < stop_crit)

break

end

end

x = xp;

y = Qbest;

end

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6.1.2 CALCULATION OF THE QUALITY FUNCTION

%------------- Function to find the quality factor circuit 3 ------------------------

function [Q] = Q_ladderp3 (xin) %xin is sent as input the function and the optimized values

%stored in Q are returned.

global frequenz;

global phase1;

global Z1;

x = unscale (xin);

R1 = x (1); //assigning values to meaningful local variables

R2 = x (2);

R3 = x (3);

C1 = x (4);

C2 = x (5);

C3 = x (6);

A = x (7);

w1 = 2 * pi * frequenz; % only in the range of the measured values

% ------------------calculation of the impedance in rectangular form---------------

phase1_rad = (phase1/180)*pi;

real_Z1 = Z1.* cos (phase1_rad);

img_Z1 = Z1 .* sin(phase1_rad);

Z1_rect = real_Z1 + (1i*img_Z1); % measured values

% -------------------------The Quality function (circuit 3) ---------------------------

Z4 = R3 + (A./((1i.*w1).^(0.5)));

Z3 = (1./(1i.*w1*C3));

Z2 = (1./ (1i .* w1 * C2)) + R2 ;

Z = (1./ ((1./R1) + (1i.*w1*C1)+ (1./Z2))) + (1./ ((1./Z3)+(1./Z4))) ;

Zi = sqrt(sum((abs(Z - Z1_rect)). ^ 2)./sum(abs(Z). ^2));

Q = (Zi);

end

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6.1.3 THE SCALING FUNCTION

% function to scale the circuit 3

function [x] = scale (xin) %function with xin as input and sends x as output

x (1) = xin (1) / 100; % R1

x (2) = xin (2) / 10; % R2

x (3) = xin (3) / 10; % R3

x (4) = xin (4) / 10; % R3

x (5) = xin (5) / 1.0e-6; % C1

x (6) = xin (6) / 1.0e-6; % C2

x (7) = xin (7) / 1.0e-6; % C3

x (8) = xin (8) / 1.0e-9; %C4

x (9) = xin (9) / 1000.0; %A

end

6.1.4 THE UNSCALING FUNCTION

function [x] = unscale (xin) %function with xin as input and sends x as output

% multiplying the circuit elements with the scale factors

x (1) = xin (1) * 100; % R1

x (2) = xin (2) * 10; % R2

x (3) = xin (3) * 10; % R3

x (4) = xin (4) * 10; % R4

x (5) = xin (5) * 1.0e-6; % C1

x (6) = xin (6) * 1.0e-6; % C2

x (7) = xin (7) * 1.0e-6; % C3

x (8) = xin (8) * 1.0e-9; %C4

x (9) = xin(9) * 1000.0; %A

end

6.1.5 DISPLAYING OF THE PLOTS

%program to display the Nyquist and bode plots for circuit 3

w1 = 2* pi *frequenz;

%------------calculated RC circuit analysis-----------------------------

R1 = y (1);

R2 = y (2);

R3 = y (3);

C1 = y (4);

C2 = y (5);

C3 = y (6);

A = y (7);

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Z4 = R3 + (A. / ((1i.*w1). ^ (0.5)));

Z3 = (1. / (1i.*w1*C3));

Z2 = (1. / (1i.* w1 * C2)) + R2;

Z = (1./ ((1./R1) + (1i.*w1*C1)+ (1./Z2))) + (1./ ((1./Z3)+(1./Z4))) ;

real_Z= real (Z);

img_Z =imag (Z);

abs_Z= abs (Z);

angle_Z=angle (Z);

deg_Z= (180/pi)* angle_Z; %calculated values

phase1_rad = (phase1/180)*pi;

real_Z1 = Z1 * cos (phase1_rad);

img_Z1 = Z1 .* sin(phase1_rad);

Z1_rect = real_Z1 + (1i*img_Z1); %/measured values

Plot (real_Z1,-img_Z1, real_Z, -img_Z); //Nyquist plot

Axis ([150, 350, 0, 250]); //axis values

xlabel ('Real Z (ohm)');

ylabel ('Imaginary Z (ohm)');

legend ('Measured Z', 'Calculated Z');

6.1.6 OPTIMIZATION PROCEDURE

% function to run the entire optimization procedure. This functions calls the optimizer and the

scaling functions and returns the optimized values in variable y. The values stored in y are

displayed using 6.5

global frequenz; %allows global workspace variables to be used in a function

global Z1;

global phase1;

%x = [1 1 1 1 1 1 1 1 1]; % random inital guess

x = [275.3 10.63 3.622 13.48 48.27e-6 20.90e-6 6.070e-6 8.121e-9 503.1];

%Z1 initial guess from optimization

%x=[274.8096 10.2691 3.0668 13.98877 4.8560e-05 1.9079e-05 5.5293e-06 7.4886e-09

517.9392]; %Z2 starting values

%x= [274.42216 2.87665 9.764516 14.06203 4.9802e-05 5.69563e-06 1.9383e-05 7.34126e-09

526.24216]; %Z3 starting values

x = scale(x); %scale the initial guess by dividing with scale factor

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[y, fval] = optimizer_ladder(x); %optimized values returned in variable y and the optimized

%function value (i.e. the minimum error) in fval

y = unscale(y); %unscale the values by multiplying with the scale factor

6.1.7 SENSITIVITY OF THE OPTIMIZED PARAMETERS

% ---------------To measure the sensitivity of the circuit parameters -----------------------------

function [S] = sensitivity_prop (yin, fval) %function has yin and fval as the input, vector S as

output

global frequenz;

global phase1;

global Z1;

S = zeros (size (yin)); % preallocating the initial size of sensitivity array

change = 0.01; % defines the percentage change in circuit element

iterations = 9;

y = scale (yin); %local variable stores the scaled values

%---------------original values -------------------------------------------

w1 = 2 * pi * frequenz; % only in the range of the measured values

phase1_rad = (phase1/180)*pi;

real_Z1 = Z1 * cos (phase1_rad);

img_Z1 = Z1 .* sin(phase1_rad);

Z1_rect = real_Z1 + (1i*img_Z1); % measured values

for a = 1: iterations

y_org = y (a); %local variable to store initial value

y (a) = y(a) - (change*y(a));

R1 = y (1); %assigning the optimized values to local variables

R2 = y (2);

R3 = y (3);

R4 = y (4);

C1 = y (5);

C2 = y (6);

C3 = y (7);

A = y (8);

alpha = y (9);

% The quality function

Z5 = (1./(1i.*w1*C2));

Z4 = (1./ ((1i.*w1*C3) + (1./R4))) + R3;

Z3 = ((R1). / (1 + ((1i.*w1 * A). ^ (1-alpha))));

Z2 = 1./(1i.*w1*(C1));

Z = R2 + (1./((1./Z4) + (1./Z5))) + (1./ ((1./Z2) + (1./Z3)));

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% the error calculation

Q = sqrt (sum ((abs (Z - Z1_rect)). ^ 2)/sum (abs (Z1_rect). ^2));

S (a) = abs ((Q-fval)/(y (a)-y_org));

y (a) = y_org;

end

end

6.1.8 THE SENSITIVITY FUNCTION

global frequenz;

global Z1;

global phase1;

[S] = sensitivity_prop_ckt3(y, fval); % the function call to the sensitivity function