Download - Electrochemical Impedance Spectra of Dye-Sensitized Solar Cells Fundamentals and Spreadsheet Calculation

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

Subrata Sarker,1 A. J. Saleh Ahammad,2 Hyun Woo Seo,1 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; [email protected]

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

Academic Editor: Rajaram S. Mane

Copyright 2014 Subrata Sarker et al. This is an open access article distributed under the Creative Commons Attribution License,which 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 [14]. 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 [517].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 iodide/triiodide (I/I3

) based redoxelectrolyte solution, and a Pt coated FTO glass substrate ascounter electrode (CE) [14, 15, 1820]. 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 FTO/electrolyte interface and recombination at theTiO2/electrolyte interface and the regeneration of the redox

species at CE/electrolyte 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, 2123]. 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 http://dx.doi.org/10.1155/2014/851705). 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 pageshttp://dx.doi.org/10.1155/2014/851705

2 International Journal of Photoenergy

2. Electrochemical Impedance Spectroscopy

2.1. Fundamentals. Let us begin with the notion of an idealresistor having resistance . According to Ohms law, current()flowing through the resistor and voltage () across the twoterminals of the resistor is expressed by the following relation:

= . (1)

On the other hand, impedance is amore general concept thanresistance because it involves phase difference [4]. Duringimpedancemeasurement, a small-amplitudemodulated volt-age (, ) is applied over a wide range of frequency ( =/2) and the corresponding currents (, ) are recorded,or vice versa. The resultant impedance () of the system iscalculated as [1, 2, 4]

() = (, )

(, )(2)

provided that (, ) is small enough to be linear with respectto (, ), or vice versa. At a certain frequency , (, )may have different amplitude and phase than that of (, )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 (dc), that is, impedance with zero phasedifference [2, 10]:

(0) = (0)

(0)= dc. (3)

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

In complex number, a small-amplitude AC voltage canbe described as (, ) =

0exp() and response to this

potential is the AC current (, ) = 0exp{( )}, where

is the phase difference between (, ) and (, ) and =1. Therefore, (2) can be written as [1, 4]

() =0

0

exp () . (4)

Again, (4) can be rewritten in terms of magnitude (0) as

[1, 4]

() = 0exp () . (5)

Applying Eulers relationship and replacing 0with ||, (5)

can be expressed as [1, 4]

() = || (cos + sin ) . (6)

In general, impedance is expressed as [1, 4]

() = Re + Im (7)

or more simply as [1, 4]

= + , (8)

where Re = = || cos and Im =

= || sin are thereal and the imaginary parts of the impedance, respectively.The real and imaginary parts of the impedance are related tothe phase angle as

= tan1 (

) (9)

and the magnitude || as

|| = 2 + 2. (10)

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

In the frequency domain, current-voltage relations canbe rearranged as (2). If a purely sinusoidal voltage (, ) =0sin() is applied across a resistor with resistance then

the current that flows through the resistor will be (, ) =(, )/ =

0sin()/, which can be written as (, ) =

0sin(). So, the impedance of the resistor,

(), is [1]

() =

(, )

(, )= . (11)

In this case, the applied voltage and the resultant current arein phase. If the voltage is applied to a capacitor having capaci-tance then the resultant current is (, ) = (, )/ =0cos(), where = / and = . The above

expression for the current passing through the capacitor canbe written as (, ) =

0cos( /2) or (, ) =

0sin(), where

0=

0. The impedance of the capacitor,

(), is thus [1]

() =

(, )

(, )=1

, (12)

where 1/ or in complex notation 1/ is the reactanceof a capacitor and /2 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 s 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, ()

Resistor =

Capacitor =

1

Constant phase element

=

1

()

Inductor =

Figure 1: Screenshots of the spreadsheet calculation of impedance of a capacitor (dl) with capacitance of 100F 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(dl). 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.

2.2. Equivalent Circuit of Some Electrochemical Systems andTheir Impedance

2.2.1. Ideally Polarizable Electrode in Contact with Electrolyte.An ideally polarizable electrode behaves as an ideal capac-itor because there is no charge transfer across the solu-tion/electrode 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). If

is the solution resist-

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

() =

() + dl() , (13)

where

() and dl() are the impedance for

and

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

() = +1

dl. (14)

Rearranging (14), one gets

() =

dl. (15)

Here, the real () and the imaginary () parts of theimpedance are

and 1/dl, respectively. Figure 2(a)

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

, in this case

= 50, while the capacitive impedance contributes to the

negative imaginary part of the impedance. At the low fre-quency limit ( 0) the capacitive impedance is so largethat the total impedance is infinity. Therefore, the dc resis-tance,(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 ( ), the capacitor becomes short-circuited andthere remains the resistance

only. However, complex plane

plot does not tell us about the corresponding frequency of theimpedance explicitly. In the Bode plot (Figure 2(b)), log ||and are plotted against log. The plot of impedance (redcircle) versus frequency has a breakpoint, which correspondsto the characteristic frequency = 1/

dl or characteristic

time constant = 1/ = dl = 0.005 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 90 atlow frequency.

2.2.2. Nonpolarizable Electrode in Contact with Electrolyte.If the electrode is nonpolarizable, then the system can bemodeled by introducing a resistance ct parallel to thecapacitance dl as shown in the inset of Figure 2(c), which


0 50 100 150 200 250 3000

100

200

300Z(

)

Z ()

CdlRs

(a)

105

104

103

102

101

100

f (Hz)

0105104103102101100

30

60

90

101

|Z|(

)

()

(b)

0 40 80 120 1600

Z(

)

Z ()

40

80

120

160

Cdl

Rct

Rs

(c)

50

75

100

125

150

f (Hz)105104103102101100101

0

10

20

30

|Z|(

)

()

(d)

Z(

)

f (Hz)105104103102101100101

0

20

40

60

0

10

20

30

40()

(e)

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

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

= 50, ct = 100, and dl = 100 F as shown in the inset; (d) Bode magnitude and phase; and (e) Bode

imaginary and phase plot of the impedance.

is known as simplified Randles circuit [1, 24]. Eventually, thecircuit consists of a series connection of a solution resistancewith a parallel combination of a charge transfer resistancect and a double layer capacitance dl. The impedance of thesystem can be written as

() = () + pl () , (16)

where() =

andpl() is the impedance of the parallel

combination of the ct and the dl.

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

() = +

ct1 + 22ct

2

dl 2ctdl

1 + 22ct2

dl. (17)

Here, +ct/(1+

22ct2

dl) and 2

ctdl/(1+22ct

2

dl) arethe real and imaginary parts of the impedance, respectively.Figure 2(c) shows the impedance of the system in complexplane plot. The plot has a semicircle, which is typical for akinetic control system. When , the capacitive


impedance is short-circuited, and this eventually shunts thect. Therefore, only the remains 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 thect. So, the dc resistance(0) of this system is(0) = +ct.It can be noticed from (17) that the maximum of the occurs at =

+ ct/2, which corresponds to the

characteristic frequency of the charge transfer process (max).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

1:

1=1

1

=1

21

=ctdl+ ct, (18)

and the second breakpoint corresponds to the time constant2:

2=1

max=1

2max= ctdl. (19)

Here, the frequency1in the Bodemagnitude plot (red circle,

Figure 2(d)) can be calculated from (18) as 1= 1/2 =

1/21= 47.75Hz. On the other hand, max is calculated to

be 15.92Hz forct = 100 anddl = 100 F.The Bode phaseplot (blue square, Figure 2(d)) has a maximum at around thefrequency max 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 max,which 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 ct. The semicircle progressively increasedas the value of ct increased from 50 to 100 while dlremained the same. The Bode magnitude and phase plotsdepicted in Figures 3(b) and 3(c) clearly show increase ofmagnitude and decrease of characteristic frequency (max)with the increase ofct. On the other hand, the complex planeplot (Figure 4(a)) remained unchanged for a fixed value ofct and different values of dl. Thus, (0) changes as the ctchanges while it remains fixed for all values of dl. Forboth cases, the Bode magnitude and phase plots depicted inFigures 4(b) and 4(c) clearly show thatmax shifts towards thelow frequency edge for increasing either ct or dl. It is to benoted here that phase angle at the maxima decreases with thedecrease of ct while it is the same for different values of dl.

2.2.3. 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. However,the 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 hand,impedances 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].

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

() =1

()

, (20)

whereand are the CPE prefactor and index, respectively.

If the index is equal to 1.0 the CPE coincides with a purecapacitor. Generally, varies from 1.0 to 0.5 to fit an experi-mental data. The impedance corresponding to the simplifiedRandles circuit with CPE (Figure 6(a)) can be expressed as

() = +

ct

1 + ()

ct. (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 . As the value of decreases from 1.0 to 0.5 the semicircle deviates to a depressedsemicircle. In this case, the characteristic frequency max isexpressed as [2]

max =1

(ct)1/. (22)

From (22), we can see that the CPE response decelerates withthe decrease of , 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 (dl) of the electrochemical interfacecorresponding to the parallel combination of ct and ofFigure 6(a) can be calculated by comparing (22) with (19) as

dl = 1/

1/1

ct . (23)

2.2.5. 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]

() =

2

. (24)

Since 1/ = (1 )/2, (24) can be written as

() =

(1 ) . (25)

The coefficient is defined as [1, 24]

=

222(1

+1

) , (26)

where and

are the bulk concentration of oxidant and

reductant, respectively; and

are the diffusion coef-

ficients of the oxidant and reductant, respectively; is the


0 40 80 120 1600

Z(

)40

80

120

160

Z ()

(a)

50

70

90

110

130

150

|Z|(

)

f (Hz)105104103102101100101

Rct ()

100 709080

6050

(b)

0

10

20

30

40

f (Hz)105104103102101100101

Rct ()

100 709080

6050

()

(c)

Figure 3: (a) Complex plane plot for the impedance corresponding to the simplified Randles circuit with = 50, dl = 100 F, and

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

surface area of the electrode; and is the number of electronsinvolved. The semi-infinite diffusion impedance cannot bemodeled by simply connecting resistor and capacitor becauseof square root of frequency () [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.

2.2.6. Randles Circuit. If the kinetic control process as dis-cussed in Section 2.2.2 is coupled with mass transfer process


0 40 80 120 1600

40

80

120

160

Z(

)

Z ()

(a)

60

80

100

120

140

160

f (Hz)105104103102101100101

|Z|(

)

100

10

5150

Cdl (F)

(b)

f (Hz)105104103102101100101

0

10

20

30

40

100

10

5150

Cdl (F)()

(c)

Figure 4: (a) Complex plane plot for the impedance corresponding to the simplified Randles circuit with= 50,ct = 100 and different

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

then the simplified Randles circuit can be modified byintroducingWarburg impedance () 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 Randlescircuit. 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 simplifiedRandles 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 Randles circuitexcept at the low frequency region where phase angle gradu-ally increases and at the limit of low frequency it reaches 45due to diffusion process. If the time constant (

= 1/max =

ctdl) of the faradic or charge transfer kinetics is toofast compared to the time constant (

= 2ct/2

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


0 40 80 12040

0

40

80

120

Z(

)

Z ()

Cdl

Rs

Rct

L1

(a)

0 40 80 12040

0

40

80

120

Z ()

Cdl

R1

Rct

L1

Z(

)

Rs

(b)

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

= 10, ct = 100, 1 = 160, dl = 100 F, and 1 = 10 H for (a) and 100H

for (b).

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

2.2.7. 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

, , and

as shown in Figures 8(a)

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

= =1

. (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 in a thin film of thickness ,where the characteristic frequency

is [26]

=

2. (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 blocked.This diffusionphenomenon can be modeled as a FTL with short-circuit atthe terminus similar to that in Figure 8(a), however, without

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

hand, in an absorbing boundary condition, electrons areinjected at - junction and are collected at the outer edge ofthe neutral region of a semiconductor.Thediffusion processcan be modelled as a FTL with open-circuit at the terminussimilar to that in Figure 8(b), of course, without

.

The diffusion impedance (,) for a reflecting boundary

condition is expressed as [26]

,() =

coth

, (29)

where (= ) and

(= 1/

) are the diffusion resis-

tance and characteristic frequency of diffusion, respectively.Complex 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

. When

, the system

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

,() =

. (30)

At the low frequency region, the impedance becomes [26]

,() =

3+

. (31)


Rs

Rct

Qn

(a)

0 40 80 120 1600

Z(

)

40

80

120

160

Z ()

(b)

50

70

90

110

130

150

f (Hz)105 106104103102101100101

|Z|(

)

1.00.90.8

0.70.60.5

n

(c)

f (Hz)105 106104103102101100101

0

10

20

30

40

1.00.90.8

0.70.60.5

n

()

(d)

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

to the equivalent circuit with = 50, ct = 100, and = 100 Fs

1 and different values of CPE index as mentioned.

For absorbing boundary condition, the diffusionimpedance (

,) can be expressed as [26]

,() =

tanh

. (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 (dr,) is expressed as [26]

dr, () = ()

(1 + /)coth(

)(1 +

) (33)

and the impedance for the absorbing boundary condition(dr,) is expressed as [26]

dr, () = ()

(1 + /)tanh(

)(1 +

), (34)

where and

are the diffusion resistance and character-

istic frequency for diffusion, respectively, as in (29) and (32).The additional terms

and

are 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,

, , and

. The relation among the physicochemical parameters is

expressed as [26]

=

= (

)2

, (35)


rm rm rm rm

cmcmcm

(a)

0 20 40 60

Z(

)

0

20

40

60

Z ()

(b)

f (Hz)105104103

103

102

102

101

101

100

100

101101

|Z|(

)

0

15

30

45

60

75

90

()

(c)

0 50 100 150 200 250

Z(

)

Z ()

0

50

100

150

200

250

Rs

Rct

Cdl

w

(d)

f (Hz)105104103102101100101

0

10

20

30

40103

102

101

|Z|(

)

()

(e)

Figure 7: (a) Semi-infinite transmission line depicting diffusion process. (b)Complex plane and (c) Bode plot for theWarburg diffusionwherethe coefficient = 150 s0.5. (d) Complex plane and (e) Bode plot for the impedance corresponding to Randles circuit with

= 50,

ct = 100, and dl = 100 F and theWarburg coefficient = 150 s0.5. Inset (d) shows Randles circuit and magnitude and phase plot for

Randles circuit.

where and are the film thickness and the diffusion

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

=

. (36)

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

/. When

is very large (red

circles, Figure 8(e)), (33) reduces to (30) of simple diffusion.In this case, the reaction resistor

in 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 10Z ()

Z(

)

0

2

4

6

8

10

(c)

0 2 4 6 8 10Z ()

Z(

)

0

2

4

6

8

10

(d)

0 20 40 60 80 100 120

0 2 4 6 8 10

0

Z(

)

Z(

)

40

20

80

100

60

120

Z ()

Z ()

0

2

4

6

8

10

10010 0.1

Rk/Rd

(e)

0 2 4 6 8 10

10010 0.1

Z(

)

Z ()

0

2

4

6

8

10

Rk/Rd

(f)

Figure 8: Finite-length transmission line models of diffusion-reaction impedance with (a) reflective and (b) absorbing boundary condition.Complex 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 ,

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

> (blue squares, Figure 8(e)), the

impedance at high frequency region ( ) follows (30)

and at the low frequency region ( ) the expression is

dr, () =1

3+

(1 + /). (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

dc = (0) =1

3+ . (38)

When < (green triangles, in the inset of Figure 8(e)),

(33) gives the expression

dr, () = ()

(1 + /), (39)

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

). This is the case when diffusing

species are lost before they reach the outer edge of the film.The model corresponding to (39) is called Gerischersimpedance and the dc resistance has the form

dc = (0) = . (40)

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

/. For a very large value of

(red

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

.

If > (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

due to additional contribution

of s as in Figure 8(d). When

< (green triangles,

Figure 8(f)), (34) reduces to Gerischers 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

/I redox couple,are shown in Figure 9(a) with plausible time constants [19,27, 28]. Within the frequency range of EIS measurement,several 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 (oc) 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, 13,29]. 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 Pt/electrolyte and uncovered FTO/electrolyte interfaceswith a characteristic frequency CE, the second or middlesemicircle corresponds to the electron diffusion in the TiO

2

film and electron back reaction with oxidized redox speciesat the TiO

2/electrolyte interface, and the third semicircle at

the low frequency region corresponds to the diffusion of I3

in the electrolyte solution with a characteristic frequency .

The characteristic frequency for electron transport or diffu-sion (

) appears at the high frequency region of the middle

semicircle while the peak frequency () of that semicircle

corresponds to the electron back reaction. Similarly, the Bodeplots (Figure 9(c)) show all characteristic frequencies except, 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 ct is the charge transfer resistance of thecharge recombination process at the TiO

2/I3

in electrolyte;is the chemical capacitance of the TiO

2film;

is the

transport resistance of electrons in TiO2film;

is the

Warburg element showing the Nernst diffusion of I3

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

2contact;

is the series

resistance; and 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

)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; however,proper 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.

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

OS = OS. (41)


Pt

(1)(6)

(3) (4)

(5)

(7)

(2)

(9)

(8)

ee

103 s1

100102 s1

TiO2

10101012 s1 Dye

106 s1FTO

108 s1

102 s1

I

I3

100101 s1

102 s1

103 s1

(a)

0 10 20 300

10

20

30

Z(

)

Z ()

CEd

kD

(b)

Z(

)

CE

k

D

0

1

2

3

4

5

0

4

8

12

16

f (Hz)105 106104103102101100101

()

(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

/I), (5) diffusion of I3

, (6) diffusionof I, (7) back reaction of TiO

2conduction band electrons with I

3

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

3

. (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 CE, , , and correspond to thecharge transfer processes at the Pt/electrolyte interface, electron diffusion in the TiO

2film, 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

c cc

CTCO

rctrctrctCCORPt

TiO2

Solution

TCO

Zd

CPt

TCO + Pt

RctTCO

RsTCO

(a)

RCO

CTCO

CCORPt

TiO2

Solution

TCOCPt

TCO + Pt

RctTCO

RctTiO2

C

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.

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

3

. The exchange current density (0) of the

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

Pt =

0

, (42)

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

Pt =Pt

1 + ()CE PtPt

. (43)

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

CE =1

(PtPt)1/Pt (44)

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

Pt = 1/PtPt

1/Pt1Pt . (45)

3.3. 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

2/electrolyte 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 (PE) appears in the middle semi-circle of EIS spectra of DSSCs (Figure 9(b)) with charac-teristic frequencies

and

. 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

is replaced with distributed CPE

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

can be

expressed in terms of CPE as [32, 33]

=

1

()1/

. (46)

Similarly, the characteristic frequency can be written as

[32, 33]

=

1

()1/

(47)

and the impedance can be expressed as [32, 33]

PE = ()

{1 + (/)}

coth(

){1 + (

)

},

(48)

where is the electron transport or diffusion resistance,

is the electron recombination resistance, and

and

are the CPE prefactor and index, respectively. The chemicalcapacitance (

) of the TiO

2film permeated with electrolyte

can be calculated from CPE as

= 1/

1/1

. (49)


0 5 10 15 20 25 300

5

10

15

20

25

30

Z(

)

Z ()

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 (PE).

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

PE = ()

{1 + () }

coth(

) {1 + () }.

(50)

3.4. Diffusion of 3

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

3

. As a result, only I3

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

3

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

(Figure 8(b)) and the

corresponding finite-length Warburg impedance (I) can

be expressed as

I= II/2

tanh

I/2, (51)

whereIis the diffusion resistance,I is the diffusion coeffi-

cient of I3

, and 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

= I/

2 where is the characteristic

frequency of the diffusion.The frequency maxima (max) arerelated to

as max = 2.5.

3.5. 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 impedancesOS,PE,I , andPt [7, 13].Thus, the impedance of complete DSSCs (DSSC) can becalculated by summing up (41), (43), (50), and (51) as

DSSC = OS + PE + I + Pt. (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, I/I

3

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 3.1, 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 other.Any 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 OS 10.0 Charge transfer resistance at the Pt CE Pt 3.5 CPE for capacitance at the Pt CE/electrolyte interface Pt 2.6 10

5 Fs1

CPE index for capacitance at the CE/electrolyte interface Pt 0.90 N/AElectron diffusion resistance through TiO2 0.8 Electron recombination resistance at the TiO2/electrolyte interface 9.0 CPE prefactor corresponding to the chemical capacitance (

) of TiO2 film 1.0 10

3 Fs1

CPE index corresponding to the chemical capacitance () of TiO2 film 0.95 N/A

Ionic diffusion resistance in the electrolyte DI 5.0 Characteristic frequency of ionic diffusion DI 2.0 rad/s

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, Electrochemical impedance spectroscopy and itsapplications, inModern Aspects of Electrochemistry, B. E. Con-way, J. O. M. Bockris, and R.White, Eds., pp. 143248, Springer,New York, NY, USA, 2002.

[2] J. Bisquert and F. Fabreagat-Santiago, Impedance spectro-scopy: a general introduction and application to dye-sensitizedsolar cells, in Dye-Sensitized Solar Cells, K. Kalyanasundaram,Ed., p. 457, CRC; Taylor & Francis, Boca Raton, Fla, USA, 2010.

[3] J. R. Macdonald, Impedance spectroscopy, Annals of Biomed-ical Engineering, vol. 20, no. 3, pp. 289305, 1992.

[4] J. R. Macdonald and W. B. Johnson, Impedance Spectroscopy,John Wiley & Sons, New York, NY, USA, 2005.

[5] F. Fabregat-Santiago, J. Bisquert, G. Garcia-Belmonte, G. Bos-chloo, and A. Hagfeldt, Influence of electrolyte in transportand recombination in dye-sensitized solar cells studied byimpedance spectroscopy, Solar Energy Materials and SolarCells, vol. 87, no. 14, pp. 117131, 2005.

[6] M. Adachi, M. Sakamoto, J. Jiu, Y. Ogata, and S. Isoda, Deter-mination of parameters of electron transport in dye-sensitizedsolar cells using electrochemical impedance spectroscopy,TheJournal of Physical Chemistry B, vol. 110, no. 28, pp. 1387213880,2006.

[7] F. Fabregat-Santiago, J. Bisquert, E. Palomares et al., Correla-tion between photovoltaic performance and impedance spec-troscopy of dye-sensitized solar cells based on ionic liquids,TheJournal of Physical Chemistry C, vol. 111, no. 17, pp. 65506560,2007.

[8] J. Bisquert, F. Fabregat-Santiago, I. Mora-Sero, G. Garcia-Belmonte, and S. Gimenez, Electron lifetime in dye-sensitizedsolar cells: theory and interpretation of measurements, TheJournal of Physical Chemistry C, vol. 113, no. 40, pp. 1727817290,2009.

[9] E.M. Barea, J. Ortiz, F. J. Paya et al., Energetic factors governinginjection, regeneration and recombination in dye solar cellswith phthalocyanine sensitizers, Energy & Environmental Sci-ence, vol. 3, no. 12, pp. 19851994, 2010.

[10] J. Halme, P. Vahermaa, K. Miettunen, and P. Lund, Devicephysics of dye solar cells, Advanced Materials, vol. 22, no. 35,pp. E210E234, 2010.

[11] S. R. Raga, E. M. Barea, and F. Fabregat-Santiago, Analysis ofthe origin of open circuit voltage in dye solar cells, Journal ofPhysical Chemistry Letters, vol. 3, no. 12, pp. 16291634, 2012.

[12] R. Kern, R. Sastrawan, J. Ferber, R. Stangl, and J. Luther, Model-ing and interpretation of electrical impedance spectra ofdye solar cells operated under open-circuit conditions, Elec-trochimica Acta, vol. 47, no. 26, pp. 42134225, 2002.

[13] Q.Wang, J.-E. Moser, andM. Gratzel, Electrochemical imped-ance spectroscopic analysis of dye-sensitized solar cells, Journalof Physical Chemistry B, vol. 109, no. 31, pp. 1494514953, 2005.

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

2, Chemical Physics Letters, vol. 585,

pp. 193197, 2013.[15] S. Sarker, H. W. Seo, and D. M. Kim, Calculating current

density-voltage curves of dye-sensitized solar cells: a straight-forward approach, Journal of Power Sources, vol. 248, pp. 739744, 2014.

[16] S. R. Raga and F. Fabregat-Santiago, Temperature effects indye-sensitized solar cells, Physical Chemistry Chemical Physics,vol. 15, no. 7, pp. 23282336, 2013.

[17] J. Bisquert, L. Bertoluzzi, I.Mora-Sero, andG.Garcia-Belmonte,Theory of impedance and capacitance spectroscopy of solarcells with dielectric relaxation, drift-diffusion transport, andrecombination, The Journal of Physical Chemistry C, vol. 118,no. 33, pp. 1898318991, 2014.


[18] B. O'Regan and M. Graetzel, Low-cost, high-efficiency solarcell based on dye-sensitized colloidal TiO

2films, Nature, vol.

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

crystalline dye-sensitized solar cells, Journal of Photochemistryand Photobiology A: Chemistry, vol. 164, no. 13, pp. 314, 2004.

[20] M. K. Nazeeruddin, F. de Angelis, S. Fantacci et al., Com-bined experimental and DFT-TDDFT computational study ofphotoelectrochemical cell ruthenium sensitizers, Journal of theAmerican Chemical Society, vol. 127, no. 48, pp. 1683516847,2005.

[21] E. M. Barea, C. Zafer, B. Gultekin et al., Quantification of theeffects of recombination and injection in the performance ofdye-sensitized solar cells based on N-substituted carbazoledyes, Journal of Physical ChemistryC, vol. 114, no. 46, pp. 1984019848, 2010.

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

2in a room

temperature ionic liquid, Physical Chemistry Chemical Physics,vol. 8, no. 15, pp. 18271833, 2006.

[23] F. Fabregat-Santiago, G. Garcia-Belmonte, J. Bisquert, A. Zaban,and P. Salvador, Decoupling of transport, charge storage,and interfacial charge transfer in the nanocrystalline TiO

2/

electrolyte system by impedance methods, The Journal ofPhysical Chemistry B, vol. 106, no. 2, pp. 334339, 2002.

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

[25] J. Halme, Linking optical and electrical small amplitude per-turbation techniques for dynamic performance characteriza-tion of dye solar cells, Physical Chemistry Chemical Physics, vol.13, no. 27, pp. 1243512446, 2011.

[26] J. Bisquert, Theory of the impedance of electron diffusion andrecombination in a thin layer, Journal of Physical Chemistry B,vol. 106, no. 2, pp. 325333, 2002.

[27] K. Hara and H. Arakawa,Handbook of Photovoltaic Science andEngineering, John Wiley & Sons, New York, NY, USA, 2005.

[28] A. B. F. Martinson, T. W. Hamann, M. J. Pellin, and J. T. Hupp,New architectures for dye-sensitized solar cells, Chemistry: AEuropean Journal, vol. 14, no. 15, pp. 44584467, 2008.

[29] A. Hauch and A. Georg, Diffusion in the electrolyte andcharge-transfer reaction at the platinum electrode in dye-sensitized solar cells, Electrochimica Acta, vol. 46, no. 22, pp.34573466, 2001.

[30] L.Han,N.Koide, Y. Chiba, A. Islam, andT.Mitate, Modeling ofan equivalent circuit for dye-sensitized solar cells: improvementof efficiency of dye-sensitized solar cells by reducing internalresistance,Comptes Rendus Chimie, vol. 9, no. 5-6, pp. 645651,2006.

[31] J. Bisquert, Theory of the impedance of electron diffusion andrecombination in a thin layer,The Journal of Physical ChemistryB, vol. 106, no. 2, pp. 325333, 2002.

[32] J. Bisquert, G. Garcia-Belmonte, F. Fabregat-Santiago, and A.Compte, Anomalous transport effects in the impedance ofporous film electrodes, Electrochemistry Communications, vol.1, no. 9, pp. 429435, 1999.

[33] J. Bisquert, G. Garcia-Belmonte, F. Fabregat-Santiago, N. S.Ferriols, P. Bogdanoff, and E. C. Pereira, Doubling exponentmodels for the analysis of porous film electrodes by impedance.Relaxation of TiO

2nanoporous in aqueous solution, Journal of

Physical Chemistry B, vol. 104, no. 10, pp. 22872298, 2000.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttp://www.hindawi.com

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com 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

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014


CatalystsJournal of

Download - Electrochemical Impedance Spectra of Dye-Sensitized Solar Cells Fundamentals and Spreadsheet Calculation

Top Related