review article electrochemical impedance spectra of...

Review ArticleElectrochemical Impedance Spectra of Dye-Sensitized SolarCells Fundamentals and Spreadsheet Calculation

Subrata Sarker1 A J Saleh Ahammad2 Hyun Woo Seo1 and Dong Min Kim1

1 Department of Materials Science and Engineering Hongik University Sejong 339-701 Republic of Korea2Department of Chemistry Jagannath University Dhaka 1100 Bangladesh

Correspondence should be addressed to Dong Min Kim dmkimhongikackr

Received 14 July 2014 Accepted 29 August 2014 Published 27 November 2014

Academic Editor Rajaram S Mane

Copyright copy 2014 Subrata Sarker et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Electrochemical impedance spectroscopy (EIS) is one of the most important tools to elucidate the charge transfer and transportprocesses in various electrochemical systems including dye-sensitized solar cells (DSSCs) Even though there are many books andreports on EIS it is often very difficult to explain the EIS spectra of DSSCs Understanding EIS through calculating EIS spectraon spreadsheet can be a powerful approach as the user without having any programming knowledge can go through each step ofcalculation on a spreadsheet and get instant feedback by visualizing the calculated results or plot on the same spreadsheet Here abrief account of the EIS of DSSCs is given with fundamental aspects and their spreadsheet calculation The review should help oneto develop a basic understanding about EIS of DSSCs through interacting with spreadsheet

1 Introduction

Impedance spectroscopy is a powerful method for charact-erizing the electrical properties of materials and their inter-faces [1ndash4] When applied to an electrochemical system itis often termed as electrochemical impedance spectroscopy(EIS) examples of such systems are electrochemical cellssuch as fuel cells rechargeable batteries corrosion anddye-sensitized solar cells (DSSCs) [2 3] Recently EIS hasbecome an essential tool for characterizing DSSCs [5ndash17]Typically a dye-sensitized solar cell (DSSC) is composed ofa ruthenium dye loaded mesoporous film of nanocrystallineTiO2on fluorine-doped tin oxide (FTO) glass substrate as

photoelectrode (PE) an iodidetriiodide (IminusI3

minus) based redoxelectrolyte solution and a Pt coated FTO glass substrate ascounter electrode (CE) [14 15 18ndash20] Analysis of EIS spec-trum of a DSSC provides information about several impor-tant charge transport transfer and accumulation processesin the cell These are (i) charge transport due to electrondiffusion through TiO

2and ionic diffusion in the electrolyte

solution (ii) charge transfer due to electron back reaction atthe FTOelectrolyte interface and recombination at theTiO2electrolyte interface and the regeneration of the redox

species at CEelectrolyte interfaces and (iii) charging of the

capacitive elements in the cells including the interfaces theconduction band and surface states of the porous network ofTiO2[2 7 15 21ndash23] Even though there are many books and

reports on EIS it is often very difficult to explain the EIS spec-tra of DSSCs Moreover the details of EIS calculation alwaysremain under several layers of programming abstraction andthus cannot be accessed by the user

Calculating EIS spectra on spreadsheet can be a powerfulapproach as the user without having any programming know-ledge can go through each step of calculation on a spread-sheet and get instant feedback by visualizing the calculatedresults or plot on the same spreadsheet From our experienceof learning EIS of DSSCs from scratch we found that it wasfar more easy and fun to learn EIS through spreadsheetcalculation than trying to decipher the abstract ideas of EISon books or papers

Here a brief account of the general aspects of EIS isgiven with mathematical expressions and their calculationon spreadsheet (see the interactiveMicrosoftExcel 2010 file inthe Supplementary Material available online at httpdxdoiorg1011552014851705) Most importantly we summarizethe fundamental charge transfer processes that take place inworking DSSCs and how those processes give rise to EISspectra

Hindawi Publishing CorporationInternational Journal of PhotoenergyVolume 2014 Article ID 851705 17 pageshttpdxdoiorg1011552014851705

2 International Journal of Photoenergy

2 Electrochemical Impedance Spectroscopy

21 Fundamentals Let us begin with the notion of an idealresistor having resistance 119877 According to Ohmrsquos law current(119868)flowing through the resistor and voltage (119881) across the twoterminals of the resistor is expressed by the following relation

119881 = 119868119877 (1)

On the other hand impedance is amore general concept thanresistance because it involves phase difference [4] Duringimpedancemeasurement a small-amplitudemodulated volt-age 119881(120596 119905) is applied over a wide range of frequency (119891 =

1205962120587) and the corresponding currents 119868(120596 119905) are recordedor vice versa The resultant impedance 119885(120596) of the system iscalculated as [1 2 4]

119885 (120596) =119881 (120596 119905)

119868 (120596 119905)(2)

provided that 119868(120596 119905) is small enough to be linear with respectto 119881(120596 119905) or vice versa At a certain frequency 120596 119881(120596 119905)may have different amplitude and phase than that of 119868(120596 119905)depending on the nature of the charge transfer processes inthe system that results in impedance of the correspondingcharge transfer process When the frequency of the appliedperturbation is very low the system is said to be driven withdc current and the impedance of the system coincides withits dc resistance (119877dc) that is impedance with zero phasedifference [2 10]

119885 (0) =119881 (0)

119868 (0)= 119877dc (3)

It is to be noted here that there are other response quantitiesrelated to impedance such as admittance (119884) modulusfunction (119872) and complex dielectric constant or dielectricpermittivity (120576) [2 4]

In complex number a small-amplitude AC voltage canbe described as 119881(120596 119905) = 119881

0exp(119895120596119905) and response to this

potential is the AC current 119868(120596 119905) = 1198680exp119895(120596119905 minus 120579) where

120579 is the phase difference between 119881(120596 119905) and 119868(120596 119905) and 119895 =

radicminus1 Therefore (2) can be written as [1 4]

119885 (119895120596) =1198810

1198680

exp (119895120579) (4)

Again (4) can be rewritten in terms of magnitude (1198850) as

[1 4]

119885 (119895120596) = 1198850exp (119895120579) (5)

Applying Eulerrsquos relationship and replacing 1198850with |119885| (5)

can be expressed as [1 4]

119885 (119895120596) = |119885| (cos 120579 + 119895 sin 120579) (6)

In general impedance is expressed as [1 4]

119885 (120596) = 119885Re + 119895119885Im (7)

or more simply as [1 4]

119885 = 1198851015840+ 11989511988510158401015840 (8)

where 119885Re = 1198851015840 = |119885| cos 120579 and 119885Im = 11988510158401015840 = |119885| sin 120579 are thereal and the imaginary parts of the impedance respectivelyThe real and imaginary parts of the impedance are related tothe phase angle 120579 as

120579 = tanminus1 (11988510158401015840

1198851015840) (9)

and the magnitude |119885| as

|119885| = radic11988510158402 + 119885101584010158402 (10)

EIS data can be displayed in differentways In the complexplane 11988510158401015840 is plotted against 1198851015840 The complex plane plots areoften termed as Nyquist plots [4] In Bode plot both log |119885|and 120579 are plotted against log119891 Sometimes it is helpful to plotlog11988510158401015840 against log119891 [1 2]

In the frequency domain current-voltage relations canbe rearranged as (2) If a purely sinusoidal voltage 119881(120596 119905) =1198810sin(120596119905) is applied across a resistor with resistance 119877 then

the current that flows through the resistor will be 119868(120596 119905) =

119881(120596 119905)119877 = 1198810sin(120596119905)119877 which can be written as 119868(120596 119905) =

1198680sin(120596119905) So the impedance of the resistor 119885

119877(120596) is [1]

119885119877(120596) =

119881 (120596 119905)

119868 (120596 119905)= 119877 (11)

In this case the applied voltage and the resultant current arein phase If the voltage is applied to a capacitor having capaci-tance119862 then the resultant current is 119868(120596 119905) = 119862119889119881(120596 119905)119889119905 =

1205961198621198810cos(120596119905) where 119868 = 119889119902119889119905 and 119902 = 119862119881 The above

expression for the current passing through the capacitor canbe written as 119868(120596 119905) = 120596119862119881

0cos(120596119905 minus 1205872) or 119868(120596 119905) =

1198680sin(120596119905) where 119868

0= 120596119862119881

0 The impedance of the capacitor

119885119862(120596) is thus [1]

119885119862(120596) =

119881 (120596 119905)

119868 (120596 119905)=

1

120596119862 (12)

where 1120596119862 or in complex notation 1119895120596119862 is the reactanceof a capacitor and minus1205872 is the phase difference According tothe above description reactance for any electrical element canbe deduced using fundamental relation between current andvoltage for that element as summarized in Table 1 [2 4]

Analysis of EIS data is central to the study of EIS of anelectrochemical system An overview of the system of interestfacilitates the translation of the charge transfer transport andaccumulation processes in the system to an electrical circuitcomposed of a lump of series and parallel combination ofresistors capacitors inductors and so forth The equivalentmodel is used to deduce the physically meaningful propertiesof the system Any equivalent circuit model can be con-structed using Kirchoff rsquos rules [1 2] For example if two ele-ments are in series then the current passing through them arethe same and if two elements are in parallel then the voltagesacross them are the same

International Journal of Photoenergy 3

Table 1 Basic electrical elements and their current-voltage relation

Component Symbol Fundamental relation Impedance 119885(120596)

Resistor 119877 119881 = 119868119877 119877

Capacitor 119862 119868 = 119862119889119881

119889119905

1

119895120596119862

Constant phase element 119876119899

119868 = 119876119899

119889119881

119889119905

1

(119895120596)119899

119876119899

Inductor 119871 119881 = 119871119889119868

119889119905119895120596119871

Figure 1 Screenshots of the spreadsheet calculation of impedance of a capacitor (119862dl) with capacitance of 100120583F at frequencies 10mHz and100 kHz showing formulas and corresponding results in MS Excel

In spreadsheet a complex number can be constructedusing built-in function and the number can be operated withall the basic mathematical operators available in the spread-sheet as functions for complex numbers Figure 1 showssuch calculation implemented for impedance of a capacitor(119862dl) Thus spreadsheet enables one to calculate EIS in itsuser friendly interface Based on the above concept all theEIS plots discussed in the present paper are calculated onspreadsheet (see the Microsoft Excel 2010 file in the Supple-mentary Material) unless otherwise mentioned

22 Equivalent Circuit of Some Electrochemical Systems andTheir Impedance

221 Ideally Polarizable Electrode in Contact with ElectrolyteAn ideally polarizable electrode behaves as an ideal capac-itor because there is no charge transfer across the solu-tionelectrode interface [1] Impedance of such system can bemodeled as a series combination of a resistor and a capacitoras shown in the inset of Figure 2(a) If119877

119904is the solution resist-

ance and 119862dl is the double layer capacitance then the totalimpedance of the system becomes

119885 (120596) = 119885119877119904

(120596) + 119885119862dl

(120596) (13)

where 119885119877119904

(120596) and 119885119862dl(120596) are the impedance for 119877

119904and

119862dl respectively Equation (13) can be written in terms ofreactance as [1 24]

119885 (120596) = 119877119904+

1

119895120596119862dl (14)

Rearranging (14) one gets

119885 (120596) = 119877119904minus

119895

120596119862dl (15)

Here the real (1198851015840) and the imaginary (11988510158401015840) parts of theimpedance are 119877

119904and minus1120596119862dl respectively Figure 2(a)

shows complex plane plot of the impedance as a straightline perpendicular to the real or 119909-axis at 119877

119904 in this case

119877119904= 50Ω while the capacitive impedance contributes to the

negative imaginary part of the impedance At the low fre-quency limit (120596 rarr 0) the capacitive impedance is so largethat the total impedance is infinity Therefore the dc resis-tance119885(0) of the system is infinity and there is no dc currentto flow through the system As the frequency increases thecapacitive impedance decreases At the limit of very high fre-quency (120596 rarr infin) the capacitor becomes short-circuited andthere remains the resistance119877

119904only However complex plane

plot does not tell us about the corresponding frequency of theimpedance explicitly In the Bode plot (Figure 2(b)) log |119885|and 120579 are plotted against log119891 The plot of impedance (redcircle) versus frequency has a breakpoint which correspondsto the characteristic frequency 120596 = 1119877

119904119862dl or characteristic

time constant 120591 = 1120596 = 119877119904119862dl = 0005 s of the system On

the other hand the Bode phase plot (blue square) shows thatthe phase angle changes from 0∘ at high frequency to minus90∘ atlow frequency

222 Nonpolarizable Electrode in Contact with ElectrolyteIf the electrode is nonpolarizable then the system can bemodeled by introducing a resistance 119877ct parallel to thecapacitance 119862dl as shown in the inset of Figure 2(c) which


0 50 100 150 200 250 3000

minus100

minus200

minus300Z998400998400(Ω

)

Z998400 (Ω)

CdlRs

(a)

105

104

103

102

101

100

f (Hz)

0105104103102101100

minus30

minus60

minus90

10minus1

|Z|(Ω

)

120579(∘)

(b)

0 40 80 120 1600

Z998400998400(Ω

)

Z998400 (Ω)

minus40

minus80

minus120

minus160

Cdl

Rct

Rs

(c)

50

75

100

125

150

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

|Z|(Ω

)

120579(∘)

(d)

Z998400998400(Ω

)

f (Hz)10510410310210110010minus1

0

minus20

minus40

minus60

0

minus10

minus20

minus30

minus40120579(∘)

(e)

Figure 2 (a) Complex plane plot for the impedance corresponding to the equivalent circuit as shown in the inset with 119877119904= 50Ω and

119862dl = 100 120583F and (b) Bode magnitude and phase plot of the impedance (c) Complex plane plot for the impedance corresponding to thesimplified Randlersquos circuit with 119877

119904= 50Ω 119877ct = 100Ω and 119862dl = 100 120583F as shown in the inset (d) Bode magnitude and phase and (e) Bode

imaginary and phase plot of the impedance

is known as simplified Randlersquos circuit [1 24] Eventually thecircuit consists of a series connection of a solution resistance119877119904with a parallel combination of a charge transfer resistance

119877ct and a double layer capacitance 119862dl The impedance of thesystem can be written as

119885 (120596) = 119885119904(120596) + 119885pl (120596) (16)

where119885119904(120596) = 119877

119904and119885pl(120596) is the impedance of the parallel

combination of the 119877ct and the 119862dl

Thus (16) can be written in terms of reactance as [1]

119885 (120596) = 119877119904+

119877ct1 + 12059621198772ct119862

2

dlminus 119895

1205961198772ct119862dl

1 + 12059621198772ct1198622

dl (17)

Here 119877119904+119877ct(1+120596

21198772ct1198622

dl) and minus1205961198772

ct119862dl(1+12059621198772ct119862

2

dl) arethe real and imaginary parts of the impedance respectivelyFigure 2(c) shows the impedance of the system in complexplane plot The plot has a semicircle which is typical for akinetic control system When 120596 rarr infin the capacitive


impedance is short-circuited and this eventually shunts the119877ct Therefore only the 119877

119904remains at the high fre-

quency intercept As the frequency decreases the capacitiveimpedance increases At the low frequency intercept thecapacitive impedance is infinitely large but still there is the119877ct So the dc resistance119885(0) of this system is119885(0) = 119877

119904+119877ct

It can be noticed from (17) that the maximum of the 11988510158401015840

occurs at 1198851015840

= 119877119904+ 119877ct2 which corresponds to the

characteristic frequency of the charge transfer process (120596max)In Figure 2(d) the Bode magnitude plot (red circle) of

the system has two breakpoints [1] From the high frequencyedge the first breakpoint corresponds to the time constant 120591

1

1205911=

1

1205961

=1

21205871198911

=119877119904119877ct119862dl

119877119904+ 119877ct

(18)

and the second breakpoint corresponds to the time constant1205912

1205912=

1

120596max=

1

2120587119891max= 119877ct119862dl (19)

Here the frequency1198911in the Bodemagnitude plot (red circle

Figure 2(d)) can be calculated from (18) as 1198911= 12059612120587 =

121205871205911= 4775Hz On the other hand 119891max is calculated to

be 1592Hz for119877ct = 100Ω and119862dl = 100 120583FThe Bode phaseplot (blue square Figure 2(d)) has a maximum at around thefrequency 120596max and 0∘ phase shift at both the high and lowfrequency limit However the maximum of the phase angleappears at somewhat higher frequency than the actual 120596maxwhich appears at the maxima of the Bode imaginary plot(Figure 2(e)) [1]

Figure 3(a) shows EIS spectra in complex plane fordifferent values of 119877ct The semicircle progressively increasedas the value of 119877ct increased from 50 to 100Ω while 119862dlremained the same The Bode magnitude and phase plotsdepicted in Figures 3(b) and 3(c) clearly show increase ofmagnitude and decrease of characteristic frequency (120596max)with the increase of119877ct On the other hand the complex planeplot (Figure 4(a)) remained unchanged for a fixed value of119877ct and different values of 119862dl Thus 119885(0) changes as the 119877ctchanges while it remains fixed for all values of 119862dl Forboth cases the Bode magnitude and phase plots depicted inFigures 4(b) and 4(c) clearly show that120596max shifts towards thelow frequency edge for increasing either 119877ct or 119862dl It is to benoted here that phase angle at the maxima decreases with thedecrease of 119877ct while it is the same for different values of 119862dl

223 Inductance So far we have seen that the imaginarypart of the impedances for different combination of resis-tances and capacitors showed negative values and the spectraappeared in the first quadrant of the complex plane Howeverthe imaginary parts sometimes take positive values and thusthe spectra appear in both first and forth quadrants due to theinductance of the contact wire which often produces a tail athigh frequencies (Figure 5(a)) [2] On the other handimpedances of several types of solar cells show similarphenomenon however at low frequency region as a loop thatforms an arc in the fourth quadrant (Figure 5(b)) which isattributed to specific adsorption and electrocrystallizationprocesses at the electrode [2 4]

224 Constant Phase Element In equivalent circuit model ofan electrochemical system the capacitance 119862dl is oftenreplaced by a constant phase element (CPE) to account for thedeviation of the 119862dl from an ideal capacitor The impedanceof the CPE is expressed as [2 4 25]

119885119876119899

(120596) =1

(119895120596)119899

119876119899

(20)

where119876119899and 119899 are the CPE prefactor and index respectively

If the index 119899 is equal to 10 the CPE coincides with a purecapacitor Generally 119899 varies from 10 to 05 to fit an experi-mental data The impedance corresponding to the simplifiedRandlersquos circuit with CPE (Figure 6(a)) can be expressed as

119885 (120596) = 119877119904+

119877ct

1 + (119895120596)119899

119877ct119876119899 (21)

Figure 6(b) shows EIS spectra for the impedance corre-sponding to the equivalent circuit (Figure 6(a)) in complexplane for different values of CPE index 119899 As the value of 119899decreases from 10 to 05 the semicircle deviates to a depressedsemicircle In this case the characteristic frequency 120596max isexpressed as [2]

120596max =1

(119877ct119876119899)1119899

(22)

From (22) we can see that the CPE response decelerates withthe decrease of 119899 which is evident at the second breakpointfrom high frequency end of Figure 6(c) Moreover the phaseangle at the maxima decreases as well (Figure 6(d)) Theequivalent capacitance (119862dl) of the electrochemical interfacecorresponding to the parallel combination of 119877ct and 119876

119899of

Figure 6(a) can be calculated by comparing (22) with (19) as

119862dl = 1198761119899

1198991198771119899minus1

ct (23)

225 Semi-Infinite Diffusion There is another importantimpedance element that accounts for the impedance of redoxspecies diffuse to and from the electrode surface Theimpedance is known as semi-infinite Warburg impedanceand is expressed as [1]

119885119882(120596) = radic

2

119895120596120590 (24)

Since 1radic119895 = (1 minus 119895)radic2 (24) can be written as

119885119882(120596) =

120590

radic120596(1 minus 119895) (25)

The coefficient 120590 is defined as [1 24]

120590 =119877119879

11989921198652119860radic2(

1

119862lowast119874radic119863119874

+1

119862lowast119877radic119863119877

) (26)

where 119862lowast119874and 119862lowast

119877are the bulk concentration of oxidant and

reductant respectively 119863119874and 119863

119877are the diffusion coef-

ficients of the oxidant and reductant respectively 119860 is the


0 40 80 120 1600

Z998400998400(Ω

)minus40

minus80

minus120

minus160

Z998400 (Ω)

(a)

50

70

90

110

130

150

|Z|(Ω

)

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

(b)

0

minus10

minus20

minus30

minus40

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

120579(∘)

(c)

Figure 3 (a) Complex plane plot for the impedance corresponding to the simplified Randlersquos circuit with 119877119904= 50Ω 119862dl = 100 120583F and

different values of 119877ct as mentioned (b) Bode magnitude and (c) phase plot for the impedance

surface area of the electrode and 119899 is the number of electronsinvolved The semi-infinite diffusion impedance cannot bemodeled by simply connecting resistor and capacitor becauseof square root of frequency (radic120596) [1 24] A semi-infinitetransmission line (TL) composed of resistors and capaci-tors (Figure 7(a)) describes the impedance as a distributedelement This impedance appears as a diagonal line with aslope of 45∘ in complex plane plot (Figure 7(b)) In the Bodeplot (Figure 7(c)) the magnitude of the impedance (red

circle) increases linearly from a very low value at highfrequency limit to a high value at low frequency limit and thephase angle (blue square) always remains at 45∘ which is thecharacteristic of a diffusion process This kind of diffusionphenomenon is seen where diffusion layer has infinite thick-ness

226 Randlersquos Circuit If the kinetic control process as dis-cussed in Section 222 is coupled with mass transfer process


0 40 80 120 1600

minus40

minus80

minus120

minus160

Z998400998400(Ω

)

Z998400 (Ω)

(a)

60

80

100

120

140

160

f (Hz)10510410310210110010minus1

|Z|(Ω

)

100

10

5150

Cdl (120583F)

(b)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40

100

10

5150

Cdl (120583F)120579(∘)

(c)

Figure 4 (a) Complex plane plot for the impedance corresponding to the simplified Randlersquos circuit with119877119904= 50Ω119877ct = 100Ω and different

values of 119862dl as mentioned (b) Bode magnitude and (c) phase plot for the impedance

then the simplified Randlersquos circuit can be modified byintroducingWarburg impedance (119882) as shown in the inset ofFigure 7(d) to model the mixed control process [1 24] Themodel of this mixed control system is known as Randlersquoscircuit The complex plane plot (Figure 7(d)) shows that theimpedance of faradic process appears as a semicircle at highfrequency edge and the diffusion process appears as a diag-onal line with a slope of 45∘ at the low frequency edge TheBode magnitude plot (red circles Figure 7(e)) of the samesystem has three breakpoints in the order of decreasing fre-quency the first two breakpoints are similar to that of the case

for kinetic control process which is modeled as simplifiedRandlersquos circuit and the last one corresponds to the diffusionprocess The Bode phase plot (blue squares Figure 7(e)) issimilar to the Bode phase plot for simplified Randlersquos circuitexcept at the low frequency region where phase angle gradu-ally increases and at the limit of low frequency it reaches 45∘due to diffusion process If the time constant (120591

119865= 1120596max =

119877ct119862dl) of the faradic or charge transfer kinetics is toofast compared to the time constant (120591

119889= 1198772ct2120590

2) of diffusionprocess then the system is said to be under diffusion controlOn the other hand the system will be under kinetic control


0 40 80 12040

0

minus40

minus80

minus120

Z998400998400(Ω

)

Z998400 (Ω)

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

minus40

minus80

minus120

Z998400 (Ω)

Cdl

R1

Rct

L1

Z998400998400(Ω

)

Rs

(b)

Figure 5 Complex plane plots for the impedances that show inductive effects at (a) the high frequency and (b) the low frequency regionsInset shows the corresponding equivalent circuits with 119877

119904= 10Ω 119877ct = 100Ω 119877

1= 160Ω 119862dl = 100 120583F and 119871

1= 10 120583H for (a) and 100H

for (b)

if the time constant associated with the kinetics is relativelyslower than that of diffusion [1 24]

227 Diffusion in aThin Film Diffusion occurs in a thin filmalso for example triiodide diffusion in the electrolyte solu-tion of DSSCs Moreover diffusion can be coupled with reac-tion such as the electron diffusion-recombination at the PE ofDSSCs Impedance of such diffusion is known as finite-lengthdiffusion impedance The impedance of the diffusion andrecombination or diffusion and coupled reaction can bemodeled as a finite-length transmission line (FTL) composedof distributed elements 119903

119898 119903119896 and 119888

119898as shown in Figures 8(a)

and 8(b) where 119903119896is given by [26]

119903119896= 119877119896119871 =

1

120596119896119888119898

(27)

In thin film diffusion the diffusion layer is bounded and theimpedance at lower frequencies no longer obeys the equationfor semi-infinite Warburg diffusion [1 2 4] Professor Bis-quert has modeled various aspects of diffusion of particleswith diffusion coefficient 119863 in a thin film of thickness 119871where the characteristic frequency 120596

119889is [26]

120596119889=

119863

1198712 (28)

In a reflecting boundary condition electrons being injectedat the interface between a conducting substrate and a poroussemiconductor film diffuse through the film to the outer edgeof the film where electron transport is blockedThis diffusionphenomenon can be modeled as a FTL with short-circuit atthe terminus similar to that in Figure 8(a) however without

119903119896as the diffusion is not coupled with reaction On the other

hand in an absorbing boundary condition electrons areinjected at 119901-119899 junction and are collected at the outer edge ofthe neutral119901 region of a semiconductorThediffusion processcan be modelled as a FTL with open-circuit at the terminussimilar to that in Figure 8(b) of course without 119903

119896

The diffusion impedance (119885119889119900) for a reflecting boundary

condition is expressed as [26]

119885119889119900

(120596) = 119877119889radic

120596119889

119895120596cothradic

119895120596

120596119889

(29)

where 119877119889(= 119903119898119871) and 120596

119889(= 1119888

119898119903119898) are the diffusion resis-

tance and characteristic frequency of diffusion respectivelyComplex plane plot of this impedance shows a straight linewith 45∘ at high frequency and then vertically goes up at thelow frequency (Figure 8(c)) The high and the low frequencyregions clearly show two distinct features separated by thecharacteristic frequency 120596

119889 When 120596 ≫ 120596

119889 the system

behaves as a semi-infinite and (29) coincides with (24) as [26]

119885119889119900

(120596) = 119877119889radic

120596119889

119895120596 (30)

At the low frequency region the impedance becomes [26]

119885119889119900

(120596) =119877119889

3+119877119889120596119889

119895120596 (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)

Figure 6 (a) Equivalent circuit with119876119899as CPE (b) Complex plane (c) Bodemagnitude and (d) phase plot for the impedance corresponding

to the equivalent circuit with 119877119904= 50Ω 119877ct = 100Ω and 119876

119899= 100 120583Fsdots119899minus1 and different values of CPE index 119899 as mentioned

For absorbing boundary condition the diffusionimpedance (119885

119889119888) can be expressed as [26]

119885119889119888

(120596) = 119877119889radic

120596119889

119895120596tanhradic

119895120596

120596119889

(32)

The impedance in complex plane plot appears as an arc atthe low frequency region and a straight line with 45∘ showingsemi-infinite behavior at high frequency region that follows(30) as shown in Figure 8(d)

The impedance of the diffusion and recombination for thereflective boundary condition (119885dr119900) is expressed as [26]

119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)

and the impedance for the absorbing boundary condition(119885dr119888) is expressed as [26]

119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596

119889are the diffusion resistance and character-

istic frequency for diffusion respectively as in (29) and (32)The additional terms119877

119896and120596

119896are the resistance correspond-

ing to homogeneous reactions and the characteristic fre-quency of the reaction respectively Equations (33) and (34)have three independent parameters for example 119877

119889 120596119889 and

120596119896 The relation among the physicochemical parameters is

expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)

Figure 7 (a) Semi-infinite transmission line depicting diffusion process (b)Complex plane and (c) Bode plot for theWarburg diffusionwherethe coefficient 120590 = 150Ω sminus05 (d) Complex plane and (e) Bode plot for the impedance corresponding to Randlersquos circuit with 119877

119904= 50Ω

119877ct = 100Ω and 119862dl = 100 120583F and theWarburg coefficient 120590 = 150Ω sminus05 Inset (d) shows Randlersquos circuit and magnitude and phase plot forRandlersquos circuit

where 119871 and 119871119899are the film thickness and the diffusion

length respectively Comparing (28) and (35) one can write

119871119899= radic

119863

120596119896

(36)

Figure 8(e) shows EIS spectra for impedance of diffusion-reaction with reflective boundary condition in complex planeplot for different ratio of 119877

119896119877119889 When 119877

119896is very large (red

circles Figure 8(e)) (33) reduces to (30) of simple diffusionIn this case the reaction resistor 119903

119896in the transmission


rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)

Figure 8 Finite-length transmission line models of diffusion-reaction impedance with (a) reflective and (b) absorbing boundary conditionComplex plane plots of the impedance model for diffusion with the (c) reflective boundary condition and (d) the absorbing boundarycondition Complex plane plot of the impedance model for diffusion coupled with a homogeneous reaction with the (e) reflective boundarycondition (inset shows magnified view of the high frequency region of the plot) and (f) the absorbing boundary condition


line model (Figure 8(a)) is open circuit For a finite 119877119896

the impedance takes two different shapes depending on thequotient of (35) If 119877

119896gt 119877119889(blue squares Figure 8(e)) the

impedance at high frequency region (120596 ≫ 120596119889) follows (30)

and at the low frequency region (120596 ≪ 120596119889) the expression is

119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)

Thus the complex plane plot of the impedance has a smallWarburg part at high frequency and a large arc at lowfrequency In this case the dc resistance is expressed as

119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)

When 119877119896lt 119877119889(green triangles in the inset of Figure 8(e))

(33) gives the expression

119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)

where the reaction time is shorter than the time for diffusionacross the layer (120596

119896≫ 120596119889) This is the case when diffusing

species are lost before they reach the outer edge of the filmThe model corresponding to (39) is called Gerischerrsquosimpedance and the dc resistance has the form

119877dc = 119885 (0) = radic119877119889119877119896 (40)

Figure 8(f) shows the complex plane plot of the impedancefor diffusion-reactionwith the absorbing boundary conditionfor different cases of 119877

119896119877119889 For a very large value of 119877

119896(red

circles Figure 8(f)) (34) turns into (32) of simple diffusion asin Figure 8(d) The dc resistance of the impedance equals 119877

119889

If 119877119896gt 119877119889(blue squares Figure 8(f)) (34) approximates to

(32) however the dc resistance is slightly less than that of thecase for very large value of 119877

119896due to additional contribution

of 119903119896rsquos as in Figure 8(d) When 119877

119896lt 119877119889(green triangles

Figure 8(f)) (34) reduces to Gerischerrsquos impedance of (39)and the dc resistance of the impedance is given by (40)

3 EIS Spectra of DSSCs

The charge transfer kinetics involved in working DSSCsbased on liquid electrolyte containing I

3

minusIminus redox coupleare shown in Figure 9(a) with plausible time constants [1927 28] Within the frequency range of EIS measurementseveral time constants are well dispersed in the frequencydomain and they give rise to three distinct semicircles incomplex plane plot (Figure 9(b)) or three distinct peaks inBode plot (Figure 9(c)) of EIS of a DSSC at a certain steady-state at around open-circuit voltage (119881oc) under illuminationor at high potential under dark attained by applying a voltageand illumination These semicircles in the EIS spectra havebeen assigned to corresponding charge transfer processes bymeans of theoretical and experimental approach [5 6 12 1329] Among the three semicircles of the complex plane plot(Figure 9(b)) in the order of decreasing frequency the first

semicircle corresponds to the charge transfer processes atthe Ptelectrolyte and uncovered FTOelectrolyte interfaceswith a characteristic frequency 120596CE the second or middlesemicircle corresponds to the electron diffusion in the TiO

2

film and electron back reaction with oxidized redox speciesat the TiO

2electrolyte interface and the third semicircle at

the low frequency region corresponds to the diffusion of I3

minus

in the electrolyte solution with a characteristic frequency 120596119863

The characteristic frequency for electron transport or diffu-sion (120596

119889) appears at the high frequency region of the middle

semicircle while the peak frequency (120596119896) of that semicircle

corresponds to the electron back reaction Similarly the Bodeplots (Figure 9(c)) show all characteristic frequencies except120596119889 which may appear as a break point at the high frequency

limit of second semicircle in complex plane plot at certainsteady-states but not in Bode plot The above description isconsistent with the time constants shown in Figure 9(a)

Several research groups have already demonstrated sys-tematic approach to characterize EIS of DSSCs [7 12 13 30]Determination of physical parameters from EIS spectra ofDSSCs is often done by fitting the spectra to an equivalentcircuit The most widely used equivalent circuit of thecomplete DSSCs is a transmission line model as shown inFigure 10(a) where 119903ct is the charge transfer resistance of thecharge recombination process at the TiO

2I3

minus in electrolyte119888120583is the chemical capacitance of the TiO

2film 119903

119905is the

transport resistance of electrons in TiO2film 119885

119889is the

Warburg element showing the Nernst diffusion of I3

minus inelectrolyte 119877Pt and 119862Pt are the charge transfer resistance anddouble-layer capacitance at the Pt CE 119877TCO and 119862TCO arethe charge transfer resistance and the corresponding double-layer capacitance at exposed transparent conducting oxide(TCO)electrolyte interface 119877CO and 119862CO are the resistanceand the capacitance at TCOTiO

2contact 119877

119904is the series

resistance and 119871 is the thickness of the mesoscopic TiO2

film [7] At high illumination the equivalent circuit may besimplified to Figure 10(b) In addition to selecting an appro-priate equivalent circuit one must be able to estimate theparameters to a good approximation from the EIS spectrato initiate the fitting on a program that usually comes withevery EIS workstation Adachi et al showed how to deter-mine the parameters relating to charge (electrons and I

3

minus)transport in a DSSC from EIS spectra [6] The EIS spectraof DSSCs do not necessarily show three distinct arcs in thecomplex plane plot or three peaks in Bode plot howeverproper inspection of the experimental data may help toextract the important parameters efficiently Even though thecharge transfer processes in a working DSSCs are morecomplicated than the above description we will mainlydiscuss most significant processes and how the impedance ofthose individual processes shapes the EIS spectra of completeDSSCs

31 Ohmic Series Resistance The sheet resistance of electrodesubstrate and the resistance of electrolyte solution are themain contributor to the Ohmic series resistance (119877OS) inDSSCs The impedance (119885OS) for the 119877OS is

119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2

1010sim1012 sminus1 Dye

sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)

Figure 9 (a) Charge transfer kinetics involved in dye-sensitized solar cells where dark arrow shows lossmechanism (1) injection of electrons(2) diffusion of electrons in the TiO

2 (3) regeneration of dye (4) regeneration of redox mediator (I

3

minusIminus) (5) diffusion of I3

minus (6) diffusionof Iminus (7) back reaction of TiO

2conduction band electrons with I

3

minus (8) recombination of electrons with oxidized dye and (9) back reactionof electrons from FTO to I

3

minus (b) Typical impedance spectra of a DSSC presented in complex plane and (c) Bode imaginary (blue solid line)and phase (green solid line) plot In the order of decreasing frequency the characteristic frequencies 120596CE 120596119889 120596119896 and 120596

119863correspond to the

charge transfer processes at the Ptelectrolyte interface electron diffusion in the TiO2film electron back reaction with oxidized redox species

in the electrolyte and diffusion of redox species in the electrolyte solution respectively


RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd

middot middot middot


CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)

Figure 10 (a) General transmission line model and (b) simplified model at high illumination intensities of DSSCs Reprinted (adapted) withpermission from [7] Copyright (2014) American Chemical Society

32 Charge Transfer at the CE The charge transfer resistance(119877Pt) at the Pt CE is associated with the redox reaction involv-ing Iminus and I

3

minus The exchange current density (1198940) of the

reaction is related to 119877Pt by Buttler-Volmer equation as [13]

119877Pt =119877119879

1198991198651198940

(42)

where 119877 is the ideal gas constant 119865 is the Faraday constant 119879is the temperature and 119899 is the number of electrons involvedin the reaction The charge transfer process at the CE can bemodeled as a 119877-119862 parallel circuit and the correspondingimpedance (119885Pt) can be expressed in terms of CPE as

119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)

So the characteristic frequency of the charge transfer process(120596CE) can be calculated as

120596CE =1

(119877Pt119876Pt)1119899Pt (44)

and the equivalent capacitance of 119876Pt(119862Pt) can be calculatedas

119862Pt = 1198761119899PtPt 1198771119899Ptminus1Pt (45)

33 Electron Diffusion and Recombination at the PE InDSSCs electron transport through diffusion in the TiO

2is

coupled with electron back reaction generally termedas recombination at the TiO

2electrolyte interface The

impedance of diffusion and recombination of electrons atthe PE of DSSCs has been extensively studied by several

research groups [5 12 26 31] Impedance of this diffusion-recombination process (119885PE) appears in the middle semi-circle of EIS spectra of DSSCs (Figure 9(b)) with charac-teristic frequencies 120596

119889and 120596

119896 The PE permeated with liq-

uid electrolyte clearly resembles the electrochemical systemwith reflecting boundary as shown in Figure 8(a) Thus theimpedance of diffusion and recombination of electrons atthe PE of DSSCs can be expressed by (33) In practice thedistributed capacitance 119888

119898is replaced with distributed CPE

to account for the nonideality in the diffusion-recombinationprocesses In this case the characteristic frequency 120596

119896can be

expressed in terms of CPE as [32 33]

120596119896=

1

(119877119896119876119896)1119899119896

(46)

Similarly the characteristic frequency 120596119889can be written as

[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)

and the impedance can be expressed as [32 33]

119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896

cothradic(120596119896

120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)

where 119877119889is the electron transport or diffusion resistance

119877119896is the electron recombination resistance and 119876

119896and 119899

119896

are the CPE prefactor and index respectively The chemicalcapacitance (119862

120583) of the TiO

2film permeated with electrolyte

can be calculated from CPE as

119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI

Figure 11 Complex plane plot for the impedance of a DSSC showing calculated impedance of individual components and complete DSSCusing parameters as summarized in Table 2 The blue circle shows the EIS spectra simulated on Zview software using the same parametersand according to the equivalent circuit as shown in the inset where DX is the extended element 11 Bisquert number 2 that corresponds tothe impedance of the diffusion-recombination process at the PE of DSSCs (119885PE)

According to (35) and (46) (48) can be rearranged as

119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896

times cothradic(119877119889

119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)

34 Diffusion of 1198683

minus in the Electrolyte Solution In the elec-trolyte solution concentration of Iminus is much higher thanthat of I

3

minus As a result only I3

minus contributes to the diffusionimpedance that appears at the low frequency region [6 12 13]The impedance of I

3

minus diffusion can be modelled as FTL withshort-circuit terminus and without 119903

119896(Figure 8(b)) and the

corresponding finite-length Warburg impedance (119885119863I) can

be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)

where119877119863Iis the diffusion resistance119863I is the diffusion coeffi-

cient of I3

minus and 120575 is the diffusion length which is consideredas half of the thickness of the electrolyte film [12] Equation(51) is the same as (32) which expresses the impedance offinite-length diffusion with absorbing boundary conditionprovided that 120596

119863= 119863I120575

2 where 120596119863is the characteristic

frequency of the diffusionThe frequency maxima (120596max) arerelated to 120596

119863as 120596max = 25120596

119863

35 Constructing EIS Spectra of Complete DSSCs Accordingto Figures 9(a) and 10(a) a simple electrical equivalent circuit

of DSSCs can be constructed by combining the elements thatare involved in the impedances119885OS119885PE119885119863I and119885Pt [7 13]Thus the impedance of complete DSSCs (119885DSSC) can becalculated by summing up (41) (43) (50) and (51) as

119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)

Figure 11 shows complex plane plot for the impedance of aDSSC showing individual components calculated through(52) using the parameters obtained from an EIS spectrum of aDSSCwithN719 loadedTiO

2as a PE IminusI

3

minus based liquid elec-trolyte and a platinized CE measured at open-circuit voltageunder 1 sun condition (Table 2) To compare the EIS spec-trum calculated on spreadsheet (green solid line Figure 11)with that obtained by commercially available software EISspectrum of DSSC (blue circle Figure 11) was also calculatedon Zview software (Zview version 31 Scribner AssociatesInc USA) according to the equivalent circuit shown in theinset It is found that both spreadsheet calculation and Zviewsimulation generate exactly the same EIS spectrum of DSSC

4 Conclusions

Spreadsheet calculation can successfully simulate EIS spectraof DSSCs Calculation of EIS on spreadsheet allows one toget overall idea of how EIS spectra of DSSCs evolve fromimpedance response of individual components of DSSCs andhow the properties of the EIS spectra are related to each otherAny kind of EIS spectra can be calculated on spreadsheetusing the built-in function available in the spreadsheetprovided that the corresponding impedance expression isknown This review should help one to learn EIS of DSSCs


Table 2 Parameters used to calculate EIS spectra of DSSC

Description Parameters Value UnitOhmic series resistance 119877OS 100 Ω

Charge transfer resistance at the Pt CE 119877Pt 35 Ω

CPE for capacitance at the Pt CEelectrolyte interface 119876Pt 26 times 10minus5 Fsdots119899minus1

CPE index for capacitance at the CEelectrolyte interface 119899Pt 090 NAElectron diffusion resistance through TiO2 119877

11988908 Ω

Electron recombination resistance at the TiO2electrolyte interface 119877119896

90 Ω

CPE prefactor corresponding to the chemical capacitance (119862120583) of TiO2 film 119876

11989610 times 10minus3 Fsdots119899minus1

CPE index corresponding to the chemical capacitance (119862120583) of TiO2 film 119899

119896095 NA

Ionic diffusion resistance in the electrolyte 119877DI 50 Ω

Characteristic frequency of ionic diffusion 120596DI 20 rads

as well as to develop a basic understanding of EIS in generalfrom scratch

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas supported byNational Research Foundation ofKorea (NRF) Grants (NRF-2009-C1AAA001-2009-0093168and 2012-014844) funded by the Ministry of Education Sci-ence and Technology (MEST) Also this work was partiallysupported by the NRF Grant 2011-0024237 funded by MESTthrough the Basic Science Research Program

References

[1] B A Lasia ldquoElectrochemical impedance spectroscopy and itsapplicationsrdquo inModern Aspects of Electrochemistry B E Con-way J O M Bockris and RWhite Eds pp 143ndash248 SpringerNew York NY USA 2002

[2] J Bisquert and F Fabreagat-Santiago ldquoImpedance spectro-scopy a general introduction and application to dye-sensitizedsolar cellsrdquo in Dye-Sensitized Solar Cells K KalyanasundaramEd p 457 CRC Taylor amp Francis Boca Raton Fla USA 2010

[3] J R Macdonald ldquoImpedance spectroscopyrdquo Annals of Biomed-ical Engineering vol 20 no 3 pp 289ndash305 1992

[4] J R Macdonald and W B Johnson Impedance SpectroscopyJohn Wiley amp Sons New York NY USA 2005

[5] F Fabregat-Santiago J Bisquert G Garcia-Belmonte G Bos-chloo and A Hagfeldt ldquoInfluence of electrolyte in transportand recombination in dye-sensitized solar cells studied byimpedance spectroscopyrdquo Solar Energy Materials and SolarCells vol 87 no 1ndash4 pp 117ndash131 2005

[6] M Adachi M Sakamoto J Jiu Y Ogata and S Isoda ldquoDeter-mination of parameters of electron transport in dye-sensitizedsolar cells using electrochemical impedance spectroscopyrdquoTheJournal of Physical Chemistry B vol 110 no 28 pp 13872ndash138802006

[7] F Fabregat-Santiago J Bisquert E Palomares et al ldquoCorrela-tion between photovoltaic performance and impedance spec-troscopy of dye-sensitized solar cells based on ionic liquidsrdquoTheJournal of Physical Chemistry C vol 111 no 17 pp 6550ndash65602007

[8] J Bisquert F Fabregat-Santiago I Mora-Sero G Garcia-Belmonte and S Gimenez ldquoElectron lifetime in dye-sensitizedsolar cells theory and interpretation of measurementsrdquo TheJournal of Physical Chemistry C vol 113 no 40 pp 17278ndash172902009

[9] EM Barea J Ortiz F J Paya et al ldquoEnergetic factors governinginjection regeneration and recombination in dye solar cellswith phthalocyanine sensitizersrdquo Energy amp Environmental Sci-ence vol 3 no 12 pp 1985ndash1994 2010

[10] J Halme P Vahermaa K Miettunen and P Lund ldquoDevicephysics of dye solar cellsrdquo Advanced Materials vol 22 no 35pp E210ndashE234 2010

[11] S R Raga E M Barea and F Fabregat-Santiago ldquoAnalysis ofthe origin of open circuit voltage in dye solar cellsrdquo Journal ofPhysical Chemistry Letters vol 3 no 12 pp 1629ndash1634 2012

[12] R Kern R Sastrawan J Ferber R Stangl and J Luther ldquoModel-ing and interpretation of electrical impedance spectra ofdye solar cells operated under open-circuit conditionsrdquo Elec-trochimica Acta vol 47 no 26 pp 4213ndash4225 2002

[13] QWang J-E Moser andM Gratzel ldquoElectrochemical imped-ance spectroscopic analysis of dye-sensitized solar cellsrdquo Journalof Physical Chemistry B vol 109 no 31 pp 14945ndash14953 2005

[14] S Sarker H W Seo and D M Kim ldquoElectrochemical imped-ance spectroscopy of dye-sensitized solar cells with thermallydegraded N719 loaded TiO

2rdquo Chemical Physics Letters vol 585

pp 193ndash197 2013[15] S Sarker H W Seo and D M Kim ldquoCalculating current

density-voltage curves of dye-sensitized solar cells a straight-forward approachrdquo Journal of Power Sources vol 248 pp 739ndash744 2014

[16] S R Raga and F Fabregat-Santiago ldquoTemperature effects indye-sensitized solar cellsrdquo Physical Chemistry Chemical Physicsvol 15 no 7 pp 2328ndash2336 2013

[17] J Bisquert L Bertoluzzi IMora-Sero andGGarcia-BelmonteldquoTheory of impedance and capacitance spectroscopy of solarcells with dielectric relaxation drift-diffusion transport andrecombinationrdquo The Journal of Physical Chemistry C vol 118no 33 pp 18983ndash18991 2014


[18] B ORegan and M Graetzel ldquoLow-cost high-efficiency solarcell based on dye-sensitized colloidal TiO

2filmsrdquo Nature vol

353 no 6346 p 737 1991[19] M Gratzel ldquoConversion of sunlight to electric power by nano-

crystalline dye-sensitized solar cellsrdquo Journal of Photochemistryand Photobiology A Chemistry vol 164 no 1ndash3 pp 3ndash14 2004

[20] M K Nazeeruddin F de Angelis S Fantacci et al ldquoCom-bined experimental and DFT-TDDFT computational study ofphotoelectrochemical cell ruthenium sensitizersrdquo Journal of theAmerican Chemical Society vol 127 no 48 pp 16835ndash168472005

[21] E M Barea C Zafer B Gultekin et al ldquoQuantification of theeffects of recombination and injection in the performance ofdye-sensitized solar cells based on N-substituted carbazoledyesrdquo Journal of Physical ChemistryC vol 114 no 46 pp 19840ndash19848 2010

[22] F Fabregat-Santiago H Randriamahazaka A Zaban J Garcia-Canadas G Garcia-Belmonte and J Bisquert ldquoChemicalcapacitance of nanoporous-nanocrystalline TiO

2in a room

temperature ionic liquidrdquo Physical Chemistry Chemical Physicsvol 8 no 15 pp 1827ndash1833 2006

[23] F Fabregat-Santiago G Garcia-Belmonte J Bisquert A Zabanand P Salvador ldquoDecoupling of transport charge storageand interfacial charge transfer in the nanocrystalline TiO

2

electrolyte system by impedance methodsrdquo The Journal ofPhysical Chemistry B vol 106 no 2 pp 334ndash339 2002

[24] L R F Allen and J Bard Electrochemical Methods Fundamen-tals and Applications Wiley 2nd edition 2000

[25] J Halme ldquoLinking optical and electrical small amplitude per-turbation techniques for dynamic performance characteriza-tion of dye solar cellsrdquo Physical Chemistry Chemical Physics vol13 no 27 pp 12435ndash12446 2011

[26] J Bisquert ldquoTheory of the impedance of electron diffusion andrecombination in a thin layerrdquo Journal of Physical Chemistry Bvol 106 no 2 pp 325ndash333 2002

[27] K Hara and H ArakawaHandbook of Photovoltaic Science andEngineering John Wiley amp Sons New York NY USA 2005

[28] A B F Martinson T W Hamann M J Pellin and J T HuppldquoNew architectures for dye-sensitized solar cellsrdquo Chemistry AEuropean Journal vol 14 no 15 pp 4458ndash4467 2008

[29] A Hauch and A Georg ldquoDiffusion in the electrolyte andcharge-transfer reaction at the platinum electrode in dye-sensitized solar cellsrdquo Electrochimica Acta vol 46 no 22 pp3457ndash3466 2001

[30] LHanNKoide Y Chiba A Islam andTMitate ldquoModeling ofan equivalent circuit for dye-sensitized solar cells improvementof efficiency of dye-sensitized solar cells by reducing internalresistancerdquoComptes Rendus Chimie vol 9 no 5-6 pp 645ndash6512006

[31] J Bisquert ldquoTheory of the impedance of electron diffusion andrecombination in a thin layerrdquoThe Journal of Physical ChemistryB vol 106 no 2 pp 325ndash333 2002

[32] J Bisquert G Garcia-Belmonte F Fabregat-Santiago and ACompte ldquoAnomalous transport effects in the impedance ofporous film electrodesrdquo Electrochemistry Communications vol1 no 9 pp 429ndash435 1999

[33] J Bisquert G Garcia-Belmonte F Fabregat-Santiago N SFerriols P Bogdanoff and E C Pereira ldquoDoubling exponentmodels for the analysis of porous film electrodes by impedanceRelaxation of TiO

2nanoporous in aqueous solutionrdquo Journal of

Physical Chemistry B vol 104 no 10 pp 2287ndash2298 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


2 Electrochemical Impedance Spectroscopy

21 Fundamentals Let us begin with the notion of an idealresistor having resistance 119877 According to Ohmrsquos law current(119868)flowing through the resistor and voltage (119881) across the twoterminals of the resistor is expressed by the following relation

119881 = 119868119877 (1)

On the other hand impedance is amore general concept thanresistance because it involves phase difference [4] Duringimpedancemeasurement a small-amplitudemodulated volt-age 119881(120596 119905) is applied over a wide range of frequency (119891 =

1205962120587) and the corresponding currents 119868(120596 119905) are recordedor vice versa The resultant impedance 119885(120596) of the system iscalculated as [1 2 4]

119885 (120596) =119881 (120596 119905)

119868 (120596 119905)(2)

provided that 119868(120596 119905) is small enough to be linear with respectto 119881(120596 119905) or vice versa At a certain frequency 120596 119881(120596 119905)may have different amplitude and phase than that of 119868(120596 119905)depending on the nature of the charge transfer processes inthe system that results in impedance of the correspondingcharge transfer process When the frequency of the appliedperturbation is very low the system is said to be driven withdc current and the impedance of the system coincides withits dc resistance (119877dc) that is impedance with zero phasedifference [2 10]

119885 (0) =119881 (0)

119868 (0)= 119877dc (3)

It is to be noted here that there are other response quantitiesrelated to impedance such as admittance (119884) modulusfunction (119872) and complex dielectric constant or dielectricpermittivity (120576) [2 4]

In complex number a small-amplitude AC voltage canbe described as 119881(120596 119905) = 119881

0exp(119895120596119905) and response to this

potential is the AC current 119868(120596 119905) = 1198680exp119895(120596119905 minus 120579) where

120579 is the phase difference between 119881(120596 119905) and 119868(120596 119905) and 119895 =

radicminus1 Therefore (2) can be written as [1 4]

119885 (119895120596) =1198810

1198680

exp (119895120579) (4)

Again (4) can be rewritten in terms of magnitude (1198850) as

[1 4]

119885 (119895120596) = 1198850exp (119895120579) (5)

Applying Eulerrsquos relationship and replacing 1198850with |119885| (5)

can be expressed as [1 4]

119885 (119895120596) = |119885| (cos 120579 + 119895 sin 120579) (6)

In general impedance is expressed as [1 4]

119885 (120596) = 119885Re + 119895119885Im (7)

or more simply as [1 4]

119885 = 1198851015840+ 11989511988510158401015840 (8)

where 119885Re = 1198851015840 = |119885| cos 120579 and 119885Im = 11988510158401015840 = |119885| sin 120579 are thereal and the imaginary parts of the impedance respectivelyThe real and imaginary parts of the impedance are related tothe phase angle 120579 as

120579 = tanminus1 (11988510158401015840

1198851015840) (9)

and the magnitude |119885| as

|119885| = radic11988510158402 + 119885101584010158402 (10)

EIS data can be displayed in differentways In the complexplane 11988510158401015840 is plotted against 1198851015840 The complex plane plots areoften termed as Nyquist plots [4] In Bode plot both log |119885|and 120579 are plotted against log119891 Sometimes it is helpful to plotlog11988510158401015840 against log119891 [1 2]

In the frequency domain current-voltage relations canbe rearranged as (2) If a purely sinusoidal voltage 119881(120596 119905) =1198810sin(120596119905) is applied across a resistor with resistance 119877 then

the current that flows through the resistor will be 119868(120596 119905) =

119881(120596 119905)119877 = 1198810sin(120596119905)119877 which can be written as 119868(120596 119905) =

1198680sin(120596119905) So the impedance of the resistor 119885

119877(120596) is [1]

119885119877(120596) =

119881 (120596 119905)

119868 (120596 119905)= 119877 (11)

In this case the applied voltage and the resultant current arein phase If the voltage is applied to a capacitor having capaci-tance119862 then the resultant current is 119868(120596 119905) = 119862119889119881(120596 119905)119889119905 =

1205961198621198810cos(120596119905) where 119868 = 119889119902119889119905 and 119902 = 119862119881 The above

expression for the current passing through the capacitor canbe written as 119868(120596 119905) = 120596119862119881

0cos(120596119905 minus 1205872) or 119868(120596 119905) =

1198680sin(120596119905) where 119868

0= 120596119862119881

0 The impedance of the capacitor

119885119862(120596) is thus [1]

119885119862(120596) =

119881 (120596 119905)

119868 (120596 119905)=

1

120596119862 (12)

where 1120596119862 or in complex notation 1119895120596119862 is the reactanceof a capacitor and minus1205872 is the phase difference According tothe above description reactance for any electrical element canbe deduced using fundamental relation between current andvoltage for that element as summarized in Table 1 [2 4]

Analysis of EIS data is central to the study of EIS of anelectrochemical system An overview of the system of interestfacilitates the translation of the charge transfer transport andaccumulation processes in the system to an electrical circuitcomposed of a lump of series and parallel combination ofresistors capacitors inductors and so forth The equivalentmodel is used to deduce the physically meaningful propertiesof the system Any equivalent circuit model can be con-structed using Kirchoff rsquos rules [1 2] For example if two ele-ments are in series then the current passing through them arethe same and if two elements are in parallel then the voltagesacross them are the same




Resistor 119877 119881 = 119868119877 119877

Capacitor 119862 119868 = 119862119889119881

119889119905

1

119895120596119862


119868 = 119876119899

119889119881

119889119905

1

(119895120596)119899

119876119899

Inductor 119871 119881 = 119871119889119868

119889119905119895120596119871







119885 (120596) = 119885119877119904

(120596) + 119885119862dl

(120596) (13)

where 119885119877119904


119904and


119885 (120596) = 119877119904+

1

119895120596119862dl (14)


119885 (120596) = 119877119904minus

119895

120596119862dl (15)




119904 in this case










0 50 100 150 200 250 3000

minus100

minus200

minus300Z998400998400(Ω

)

Z998400 (Ω)

CdlRs

(a)

105

104

103

102

101

100

f (Hz)

0105104103102101100

minus30

minus60

minus90

10minus1

|Z|(Ω

)

120579(∘)

(b)

0 40 80 120 1600

Z998400998400(Ω

)

Z998400 (Ω)

minus40

minus80

minus120

minus160

Cdl

Rct

Rs

(c)

50

75

100

125

150

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

|Z|(Ω

)

120579(∘)

(d)

Z998400998400(Ω

)

f (Hz)10510410310210110010minus1

0

minus20

minus40

minus60

0

minus10

minus20

minus30

minus40120579(∘)

(e)







119885 (120596) = 119885119904(120596) + 119885pl (120596) (16)

where119885119904(120596) = 119877




119885 (120596) = 119877119904+

119877ct1 + 12059621198772ct119862

2

dlminus 119895

1205961198772ct119862dl

1 + 12059621198772ct1198622

dl (17)

Here 119877119904+119877ct(1+120596

21198772ct1198622

dl) and minus1205961198772

ct119862dl(1+12059621198772ct119862

2






119904+119877ct


occurs at 1198851015840




1

1205911=

1

1205961

=1

21205871198911

=119877119904119877ct119862dl

119877119904+ 119877ct

(18)


1205912=

1

120596max=

1

2120587119891max= 119877ct119862dl (19)








119885119876119899

(120596) =1

(119895120596)119899

119876119899

(20)



119885 (120596) = 119877119904+

119877ct

1 + (119895120596)119899

119877ct119876119899 (21)


120596max =1

(119877ct119876119899)1119899

(22)


119899of


119862dl = 1198761119899

1198991198771119899minus1

ct (23)


119885119882(120596) = radic

2

119895120596120590 (24)


119885119882(120596) =

120590

radic120596(1 minus 119895) (25)


120590 =119877119879

11989921198652119860radic2(

1

119862lowast119874radic119863119874

+1

119862lowast119877radic119863119877

) (26)







0 40 80 120 1600

Z998400998400(Ω

)minus40

minus80

minus120

minus160

Z998400 (Ω)

(a)

50

70

90

110

130

150

|Z|(Ω

)

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

(b)

0

minus10

minus20

minus30

minus40

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

120579(∘)

(c)







0 40 80 120 1600

minus40

minus80

minus120

minus160

Z998400998400(Ω

)

Z998400 (Ω)

(a)

60

80

100

120

140

160

f (Hz)10510410310210110010minus1

|Z|(Ω

)

100

10

5150

Cdl (120583F)

(b)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40

100

10

5150

Cdl (120583F)120579(∘)

(c)





119865= 1120596max =


119889= 1198772ct2120590



0 40 80 12040

0

minus40

minus80

minus120

Z998400998400(Ω

)

Z998400 (Ω)

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

minus40

minus80

minus120

Z998400 (Ω)

Cdl

R1

Rct

L1

Z998400998400(Ω

)

Rs

(b)


119904= 10Ω 119877ct = 100Ω 119877

1= 160Ω 119862dl = 100 120583F and 119871

1= 10 120583H for (a) and 100H

for (b)



119898 119903119896 and 119888



119903119896= 119877119896119871 =

1

120596119896119888119898

(27)


119889is [26]

120596119889=

119863

1198712 (28)




119896



119885119889119900

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(29)

where 119877119889(= 119903119898119871) and 120596

119889(= 1119888



119889 When 120596 ≫ 120596

119889 the system


119885119889119900

(120596) = 119877119889radic

120596119889

119895120596 (30)


119885119889119900

(120596) =119877119889

3+119877119889120596119889

119895120596 (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)






119885119889119888

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(32)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)


119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596



119896and120596



119889 120596119889 and


expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




Resistor 119877 119881 = 119868119877 119877

Capacitor 119862 119868 = 119862119889119881

119889119905

1

119895120596119862


119868 = 119876119899

119889119881

119889119905

1

(119895120596)119899

119876119899

Inductor 119871 119881 = 119871119889119868

119889119905119895120596119871







119885 (120596) = 119885119877119904

(120596) + 119885119862dl

(120596) (13)

where 119885119877119904


119904and


119885 (120596) = 119877119904+

1

119895120596119862dl (14)


119885 (120596) = 119877119904minus

119895

120596119862dl (15)




119904 in this case










0 50 100 150 200 250 3000

minus100

minus200

minus300Z998400998400(Ω

)

Z998400 (Ω)

CdlRs

(a)

105

104

103

102

101

100

f (Hz)

0105104103102101100

minus30

minus60

minus90

10minus1

|Z|(Ω

)

120579(∘)

(b)

0 40 80 120 1600

Z998400998400(Ω

)

Z998400 (Ω)

minus40

minus80

minus120

minus160

Cdl

Rct

Rs

(c)

50

75

100

125

150

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

|Z|(Ω

)

120579(∘)

(d)

Z998400998400(Ω

)

f (Hz)10510410310210110010minus1

0

minus20

minus40

minus60

0

minus10

minus20

minus30

minus40120579(∘)

(e)







119885 (120596) = 119885119904(120596) + 119885pl (120596) (16)

where119885119904(120596) = 119877




119885 (120596) = 119877119904+

119877ct1 + 12059621198772ct119862

2

dlminus 119895

1205961198772ct119862dl

1 + 12059621198772ct1198622

dl (17)

Here 119877119904+119877ct(1+120596

21198772ct1198622

dl) and minus1205961198772

ct119862dl(1+12059621198772ct119862

2






119904+119877ct


occurs at 1198851015840




1

1205911=

1

1205961

=1

21205871198911

=119877119904119877ct119862dl

119877119904+ 119877ct

(18)


1205912=

1

120596max=

1

2120587119891max= 119877ct119862dl (19)








119885119876119899

(120596) =1

(119895120596)119899

119876119899

(20)



119885 (120596) = 119877119904+

119877ct

1 + (119895120596)119899

119877ct119876119899 (21)


120596max =1

(119877ct119876119899)1119899

(22)


119899of


119862dl = 1198761119899

1198991198771119899minus1

ct (23)


119885119882(120596) = radic

2

119895120596120590 (24)


119885119882(120596) =

120590

radic120596(1 minus 119895) (25)


120590 =119877119879

11989921198652119860radic2(

1

119862lowast119874radic119863119874

+1

119862lowast119877radic119863119877

) (26)







0 40 80 120 1600

Z998400998400(Ω

)minus40

minus80

minus120

minus160

Z998400 (Ω)

(a)

50

70

90

110

130

150

|Z|(Ω

)

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

(b)

0

minus10

minus20

minus30

minus40

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

120579(∘)

(c)







0 40 80 120 1600

minus40

minus80

minus120

minus160

Z998400998400(Ω

)

Z998400 (Ω)

(a)

60

80

100

120

140

160

f (Hz)10510410310210110010minus1

|Z|(Ω

)

100

10

5150

Cdl (120583F)

(b)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40

100

10

5150

Cdl (120583F)120579(∘)

(c)





119865= 1120596max =


119889= 1198772ct2120590



0 40 80 12040

0

minus40

minus80

minus120

Z998400998400(Ω

)

Z998400 (Ω)

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

minus40

minus80

minus120

Z998400 (Ω)

Cdl

R1

Rct

L1

Z998400998400(Ω

)

Rs

(b)


119904= 10Ω 119877ct = 100Ω 119877

1= 160Ω 119862dl = 100 120583F and 119871

1= 10 120583H for (a) and 100H

for (b)



119898 119903119896 and 119888



119903119896= 119877119896119871 =

1

120596119896119888119898

(27)


119889is [26]

120596119889=

119863

1198712 (28)




119896



119885119889119900

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(29)

where 119877119889(= 119903119898119871) and 120596

119889(= 1119888



119889 When 120596 ≫ 120596

119889 the system


119885119889119900

(120596) = 119877119889radic

120596119889

119895120596 (30)


119885119889119900

(120596) =119877119889

3+119877119889120596119889

119895120596 (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)






119885119889119888

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(32)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)


119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596



119896and120596



119889 120596119889 and


expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 50 100 150 200 250 3000

minus100

minus200

minus300Z998400998400(Ω

)

Z998400 (Ω)

CdlRs

(a)

105

104

103

102

101

100

f (Hz)

0105104103102101100

minus30

minus60

minus90

10minus1

|Z|(Ω

)

120579(∘)

(b)

0 40 80 120 1600

Z998400998400(Ω

)

Z998400 (Ω)

minus40

minus80

minus120

minus160

Cdl

Rct

Rs

(c)

50

75

100

125

150

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

|Z|(Ω

)

120579(∘)

(d)

Z998400998400(Ω

)

f (Hz)10510410310210110010minus1

0

minus20

minus40

minus60

0

minus10

minus20

minus30

minus40120579(∘)

(e)







119885 (120596) = 119885119904(120596) + 119885pl (120596) (16)

where119885119904(120596) = 119877




119885 (120596) = 119877119904+

119877ct1 + 12059621198772ct119862

2

dlminus 119895

1205961198772ct119862dl

1 + 12059621198772ct1198622

dl (17)

Here 119877119904+119877ct(1+120596

21198772ct1198622

dl) and minus1205961198772

ct119862dl(1+12059621198772ct119862

2






119904+119877ct


occurs at 1198851015840




1

1205911=

1

1205961

=1

21205871198911

=119877119904119877ct119862dl

119877119904+ 119877ct

(18)


1205912=

1

120596max=

1

2120587119891max= 119877ct119862dl (19)








119885119876119899

(120596) =1

(119895120596)119899

119876119899

(20)



119885 (120596) = 119877119904+

119877ct

1 + (119895120596)119899

119877ct119876119899 (21)


120596max =1

(119877ct119876119899)1119899

(22)


119899of


119862dl = 1198761119899

1198991198771119899minus1

ct (23)


119885119882(120596) = radic

2

119895120596120590 (24)


119885119882(120596) =

120590

radic120596(1 minus 119895) (25)


120590 =119877119879

11989921198652119860radic2(

1

119862lowast119874radic119863119874

+1

119862lowast119877radic119863119877

) (26)







0 40 80 120 1600

Z998400998400(Ω

)minus40

minus80

minus120

minus160

Z998400 (Ω)

(a)

50

70

90

110

130

150

|Z|(Ω

)

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

(b)

0

minus10

minus20

minus30

minus40

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

120579(∘)

(c)







0 40 80 120 1600

minus40

minus80

minus120

minus160

Z998400998400(Ω

)

Z998400 (Ω)

(a)

60

80

100

120

140

160

f (Hz)10510410310210110010minus1

|Z|(Ω

)

100

10

5150

Cdl (120583F)

(b)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40

100

10

5150

Cdl (120583F)120579(∘)

(c)





119865= 1120596max =


119889= 1198772ct2120590



0 40 80 12040

0

minus40

minus80

minus120

Z998400998400(Ω

)

Z998400 (Ω)

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

minus40

minus80

minus120

Z998400 (Ω)

Cdl

R1

Rct

L1

Z998400998400(Ω

)

Rs

(b)


119904= 10Ω 119877ct = 100Ω 119877

1= 160Ω 119862dl = 100 120583F and 119871

1= 10 120583H for (a) and 100H

for (b)



119898 119903119896 and 119888



119903119896= 119877119896119871 =

1

120596119896119888119898

(27)


119889is [26]

120596119889=

119863

1198712 (28)




119896



119885119889119900

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(29)

where 119877119889(= 119903119898119871) and 120596

119889(= 1119888



119889 When 120596 ≫ 120596

119889 the system


119885119889119900

(120596) = 119877119889radic

120596119889

119895120596 (30)


119885119889119900

(120596) =119877119889

3+119877119889120596119889

119895120596 (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)






119885119889119888

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(32)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)


119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596



119896and120596



119889 120596119889 and


expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





119904+119877ct


occurs at 1198851015840




1

1205911=

1

1205961

=1

21205871198911

=119877119904119877ct119862dl

119877119904+ 119877ct

(18)


1205912=

1

120596max=

1

2120587119891max= 119877ct119862dl (19)








119885119876119899

(120596) =1

(119895120596)119899

119876119899

(20)



119885 (120596) = 119877119904+

119877ct

1 + (119895120596)119899

119877ct119876119899 (21)


120596max =1

(119877ct119876119899)1119899

(22)


119899of


119862dl = 1198761119899

1198991198771119899minus1

ct (23)


119885119882(120596) = radic

2

119895120596120590 (24)


119885119882(120596) =

120590

radic120596(1 minus 119895) (25)


120590 =119877119879

11989921198652119860radic2(

1

119862lowast119874radic119863119874

+1

119862lowast119877radic119863119877

) (26)







0 40 80 120 1600

Z998400998400(Ω

)minus40

minus80

minus120

minus160

Z998400 (Ω)

(a)

50

70

90

110

130

150

|Z|(Ω

)

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

(b)

0

minus10

minus20

minus30

minus40

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

120579(∘)

(c)







0 40 80 120 1600

minus40

minus80

minus120

minus160

Z998400998400(Ω

)

Z998400 (Ω)

(a)

60

80

100

120

140

160

f (Hz)10510410310210110010minus1

|Z|(Ω

)

100

10

5150

Cdl (120583F)

(b)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40

100

10

5150

Cdl (120583F)120579(∘)

(c)





119865= 1120596max =


119889= 1198772ct2120590



0 40 80 12040

0

minus40

minus80

minus120

Z998400998400(Ω

)

Z998400 (Ω)

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

minus40

minus80

minus120

Z998400 (Ω)

Cdl

R1

Rct

L1

Z998400998400(Ω

)

Rs

(b)


119904= 10Ω 119877ct = 100Ω 119877

1= 160Ω 119862dl = 100 120583F and 119871

1= 10 120583H for (a) and 100H

for (b)



119898 119903119896 and 119888



119903119896= 119877119896119871 =

1

120596119896119888119898

(27)


119889is [26]

120596119889=

119863

1198712 (28)




119896



119885119889119900

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(29)

where 119877119889(= 119903119898119871) and 120596

119889(= 1119888



119889 When 120596 ≫ 120596

119889 the system


119885119889119900

(120596) = 119877119889radic

120596119889

119895120596 (30)


119885119889119900

(120596) =119877119889

3+119877119889120596119889

119895120596 (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)






119885119889119888

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(32)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)


119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596



119896and120596



119889 120596119889 and


expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 40 80 120 1600

Z998400998400(Ω

)minus40

minus80

minus120

minus160

Z998400 (Ω)

(a)

50

70

90

110

130

150

|Z|(Ω

)

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

(b)

0

minus10

minus20

minus30

minus40

f (Hz)10510410310210110010minus1

Rct (Ω)

100 709080

6050

120579(∘)

(c)







0 40 80 120 1600

minus40

minus80

minus120

minus160

Z998400998400(Ω

)

Z998400 (Ω)

(a)

60

80

100

120

140

160

f (Hz)10510410310210110010minus1

|Z|(Ω

)

100

10

5150

Cdl (120583F)

(b)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40

100

10

5150

Cdl (120583F)120579(∘)

(c)





119865= 1120596max =


119889= 1198772ct2120590



0 40 80 12040

0

minus40

minus80

minus120

Z998400998400(Ω

)

Z998400 (Ω)

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

minus40

minus80

minus120

Z998400 (Ω)

Cdl

R1

Rct

L1

Z998400998400(Ω

)

Rs

(b)


119904= 10Ω 119877ct = 100Ω 119877

1= 160Ω 119862dl = 100 120583F and 119871

1= 10 120583H for (a) and 100H

for (b)



119898 119903119896 and 119888



119903119896= 119877119896119871 =

1

120596119896119888119898

(27)


119889is [26]

120596119889=

119863

1198712 (28)




119896



119885119889119900

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(29)

where 119877119889(= 119903119898119871) and 120596

119889(= 1119888



119889 When 120596 ≫ 120596

119889 the system


119885119889119900

(120596) = 119877119889radic

120596119889

119895120596 (30)


119885119889119900

(120596) =119877119889

3+119877119889120596119889

119895120596 (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)






119885119889119888

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(32)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)


119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596



119896and120596



119889 120596119889 and


expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 40 80 120 1600

minus40

minus80

minus120

minus160

Z998400998400(Ω

)

Z998400 (Ω)

(a)

60

80

100

120

140

160

f (Hz)10510410310210110010minus1

|Z|(Ω

)

100

10

5150

Cdl (120583F)

(b)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40

100

10

5150

Cdl (120583F)120579(∘)

(c)





119865= 1120596max =


119889= 1198772ct2120590



0 40 80 12040

0

minus40

minus80

minus120

Z998400998400(Ω

)

Z998400 (Ω)

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

minus40

minus80

minus120

Z998400 (Ω)

Cdl

R1

Rct

L1

Z998400998400(Ω

)

Rs

(b)


119904= 10Ω 119877ct = 100Ω 119877

1= 160Ω 119862dl = 100 120583F and 119871

1= 10 120583H for (a) and 100H

for (b)



119898 119903119896 and 119888



119903119896= 119877119896119871 =

1

120596119896119888119898

(27)


119889is [26]

120596119889=

119863

1198712 (28)




119896



119885119889119900

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(29)

where 119877119889(= 119903119898119871) and 120596

119889(= 1119888



119889 When 120596 ≫ 120596

119889 the system


119885119889119900

(120596) = 119877119889radic

120596119889

119895120596 (30)


119885119889119900

(120596) =119877119889

3+119877119889120596119889

119895120596 (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)






119885119889119888

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(32)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)


119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596



119896and120596



119889 120596119889 and


expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 40 80 12040

0

minus40

minus80

minus120

Z998400998400(Ω

)

Z998400 (Ω)

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

minus40

minus80

minus120

Z998400 (Ω)

Cdl

R1

Rct

L1

Z998400998400(Ω

)

Rs

(b)


119904= 10Ω 119877ct = 100Ω 119877

1= 160Ω 119862dl = 100 120583F and 119871

1= 10 120583H for (a) and 100H

for (b)



119898 119903119896 and 119888



119903119896= 119877119896119871 =

1

120596119896119888119898

(27)


119889is [26]

120596119889=

119863

1198712 (28)




119896



119885119889119900

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(29)

where 119877119889(= 119903119898119871) and 120596

119889(= 1119888



119889 When 120596 ≫ 120596

119889 the system


119885119889119900

(120596) = 119877119889radic

120596119889

119895120596 (30)


119885119889119900

(120596) =119877119889

3+119877119889120596119889

119895120596 (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)






119885119889119888

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(32)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)


119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596



119896and120596



119889 120596119889 and


expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z998400998400(Ω

)

minus40

minus80

minus120

minus160

Z998400 (Ω)

(b)

50

70

90

110

130

150

f (Hz)105 10610410310210110010minus1

|Z|(Ω

)

100908

070605

n

(c)

f (Hz)105 10610410310210110010minus1

0

minus10

minus20

minus30

minus40

100908

070605

n

120579(∘)

(d)






119885119889119888

(120596) = 119877119889radic

120596119889


119895120596

120596119889

(32)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)cothradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (33)


119885dr119888 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896)tanhradic(

120596119896

120596119889

)(1 +119895120596

120596119896

) (34)

where 119877119889and 120596



119896and120596



119889 120596119889 and


expressed as [26]

119877119896

119877119889

=120596119889

120596119896

= (119871119899

119871)2

(35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z998400998400(Ω

)

0

minus20

minus40

minus60

Z998400 (Ω)

(b)

f (Hz)105104103

103

102

102

101

101

100

100

10minus110minus1

|Z|(Ω

)

0

minus15

minus30

minus45

minus60

minus75

minus90

120579(∘)

(c)

0 50 100 150 200 250

Z998400998400(Ω

)

Z998400 (Ω)

0

minus50

minus100

minus150

minus200

minus250

Rs

Rct

Cdl

w

(d)

f (Hz)10510410310210110010minus1

0

minus10

minus20

minus30

minus40103

102

101

|Z|(Ω

)

120579(∘)

(e)


119904= 50Ω




119871119899= radic

119863

120596119896

(36)


119896119877119889 When 119877





rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


rm rm rm rm

cm cm cm cm

rk rk rk rk

L x0

(a)

rm rmrmrmrm

cm cm cm cm

rkrkrkrk

L x0

(b)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(c)

0 2 4 6 8 10Z998400 (Ω)

Z998400998400(Ω

)

0

minus2

minus4

minus6

minus8

minus10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z998400998400(Ω

)

Z998400998400(Ω

)

minus40

minus20

minus80

minus100

minus60

minus120

Z998400 (Ω)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

10010 01

RkRd

(e)

0 2 4 6 8 10

10010 01

Z998400998400(Ω

)

Z998400 (Ω)

0

minus2

minus4

minus6

minus8

minus10

RkRd

(f)








119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of







119885dr119900 (120596) =1

3119877119889+

119877119896

(1 + 119895120596120596119896) (37)


119877dc = 119885 (0) =1

3119877119889+ 119877119896 (38)



119885dr119900 (120596) = radic(119877119889119877119896)

(1 + 119895120596120596119896) (39)




119877dc = 119885 (0) = radic119877119889119877119896 (40)



119896(red


119889









3



2




minus








2I3


2film 119903

119905is the


119889is the



2contact 119877

119904is the series



3



119885OS = 119877OS (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

eminuseminus

sim103 sminus1

100sim102 sminus1

TiO2


sim106 sminus1FTO

sim108 sminus1

sim10minus2 sminus1

Iminus

I3minus

100sim101 sminus1

sim10minus2 sminus1

sim103 sminus1

(a)

0 10 20 300

minus10

minus20

minus30

Z998400998400(Ω

)

Z998400 (Ω)

120596CE120596d

120596k

120596D

(b)

Z998400998400(Ω

)

120596CE

120596k

120596D

0

minus1

minus2

minus3

minus4

minus5

0

minus4

minus8

minus12

minus16

f (Hz)105 10610410310210110010minus1

120579(∘)

(c)



3




3


3






RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


RCO rt rt rt rt

c120583 c120583c120583

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd



CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C120583

Zd(sol)

RsTCO

(b)



3



119877Pt =119877119879

1198991198651198940

(42)


119885Pt =119877Pt

1 + (119895120596)119899CE 119877Pt119876Pt

(43)


120596CE =1

(119877Pt119876Pt)1119899Pt (44)




2is





119889and 120596





119896can be


120596119896=

1

(119877119896119876119896)1119899119896

(46)


[32 33]

120596119889=

1

(119877119889119876119896)1119899119896

(47)


119885PE = radic(119877119889119877119896)

1 + (119895120596120596119896)119899119896


120596119889

)1 + (119895120596

120596119896

)

119899119896

(48)



119896and 119899

119896


120583) of the TiO



119862120583= 1198761119899119896

1198961198771119899119896minus1

119896 (49)


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 5 10 15 20 25 300

minus5

minus10

minus15

minus20

minus25

minus30

Z998400998400(Ω

)

Z998400 (Ω)

ZOS

ZOS

ZPt

ZPt

ZPE

ZPE

ZDSSCZDSSC Zview

DX Ws

ZDI

ZDI



119885PE = radic(119877119889119877119896)

1 + (119895120596)119899119896 119877119896119876119896


119877119896

) 1 + (119895120596)119899119896 119877119896119876119896

(50)



3



3




be expressed as

119885119863I

= 119877119863Iradic119863I1205752


119895120596

119863I1205752 (51)


cient of I3


119863= 119863I120575



119863as 120596max = 25120596

119863



119885DSSC = 119885OS + 119885PE + 119885119863I

+ 119885Pt (52)


2as a PE IminusI

3


4 Conclusions








11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of







11988908 Ω


90 Ω




119896095 NA






Acknowledgments


References




























2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of









2in a room



2























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

review article electrochemical impedance spectra of...

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