carbon aero gel (cag) super-capacitor a fractional order

Carbon Aero Gel (CAG) Super-Capacitor a Fractional Order Element

in reality and why?

Shantanu DasScientist H+, RCSDS BARC Mumbai,

Senior Research Professor Dept. of Phys, Jadavpur Univ (JU),Adjunct Professor DIAT-Pune

UGC-Visiting Fellow Dept. of Appl. Math Calcutta [email protected]

http://scholar.google.co.uk/citations?user=9ix9YS8AAAAJ&hl=enwww.shantanudaslecture.com

VECC-Kolkata12 May 2015

mailto:[email protected]

http://scholar.google.co.uk/citations?user=9ix9YS8AAAAJ&hl=en

http://www.shantanudaslecture.com/

Research and development team & Funding organizations

CMET ThrissurDr. N. C. Pramanik (Scientist Aerogel Div. CMET); [email protected]

IIT BombayProf. Vivek Agarwal (Dept. of EE) ; [email protected]

VNIT NagpurProf. Mohan. V. Aware (Dept. of EE); [email protected]

NIT RaipurProf. Subhojit Ghosh (Dept. of EE) ; [email protected]

ECILSri. M. R. K. Naidu (Head Corporate R & D) ; [email protected]

Funding Organizations: DeitY, DST, BRNS

Keltron

ECIL

Tata Motors,

SPEL Technologies

Mahindra Reva Electric Vehicle Pvt. Ltd.

Chheda Electrical & Electronics Pvt. Ltd.

Power Systems Ltd. Aartech Solution Ltd.

SAMEER

CSIR-CMERI Durgapur

Probable end users

Various of CAG super capacitors cells made

Values of the CAG super capacitors are 1 Farad to 50 Farad 2.5V made by CAG electrodes

BRNS Sanction No.2009/34/31/BRNS dated 22.10.2009 CMET & BARC Joint Project

Parameter Imported Super capacitor

In-house made CAG super-capacitor

Electrode size (mm) 10 X 180 10 X 110Material Carbon Foam

Graphine, Gold Foam, Activated Carbon

Carbon Aero gel

Capacitance 1.00 Farad 1.00 FaradTolerance +/-20 % +/- 10 %Operating Voltage 0.5-2.50 Volts 0.5- 2.50 VoltsEquivalent Series Resistance ESR

0.4-0.6 Ohms 0.35-0.45 Ohms

Comparing imported & in house 1Farad super-capacitors

CAG super capacitors Requires 0.6 times electrode material compared toimported ones


From Organic Gel RF Gel i.e. Resorcinol Phenol sol-gel polymer is made.

Then at inert atmosphere pyrolysis is done to get Carbon Aero gel

The process from RF Gel to Farad Capacitors


Electrolyte : TEATFBTetra Ethyl Ammonium Tetra Fluorite Borate

2.1 cm

1.2 cmdia

2.4 cmdia

4.1 cm

Size-1; 5 F-20 F Size-2; 1F-3 F

Packing size for CAG Capacitor

Packed in two sizesBRNS Sanction No.2009/34/31/BRNS dated 22.10.2009 CMET & BARC Joint Project

The circuit for bicycle lighting system

6V Generator

Tail Lamp

1N4001

SC 1F

8.2M

6

7

0.1uF1 5

1K

2

0.1uF

390K

4 83

0.1uF

1K2N3906Timer IC

555

TailLight


The tail light flashes even while cycle stops

The circuit driving toy car with geared wheel

1N4007 1N4007

27 ohm

100 ohm

100 ohm

LED

27 ohmSC 18 F

Motor

SW SW

5V DC


The circuit LED backup

1N4007

1N4007

LED-1

LED-2SC 18F

100 ohm

100 ohm

100 ohm

SW

SW5V DC


Photograph of CAG supercapacitor powered fast charging Emergency Lighthaving two 25F/ 2.5 V, 1-3W Power LED, which takes ~90 second for fullcharging and provide energy to light power LED for 30 min to 90 min,depending on LED lamp power(1-3W)

The emergency light


Emergency Lamp Circuit

230V AC

10K

LED

2000 uF25V

0.22 uF

Voltage Regulator7805

1N4007

1N40070.22 ohm

220 uF25V SC2

25 F

SC125 F

1K

1K

5V DC

7

8

RT

56K

1

GND

3 4

L110 uH

6

LEDLamp3Watt

9

Vc

180K

47K

150K5

1uF 15KBuck/BoostDC-DC Convertor

LTC34405V DC

Output VoltageRegulator

Constant Voltage Charging Circuit for super-capacitor (passive charge balance scheme)

10

Vin Vout

SW2SW1

SHDN/SS

2 MODE/SYNC


Active charge Balance Circuit

+

−

2F 1000 2000F ESR 0.5 1000 500μ× = = Ω÷ = Ω

This floating current sink circuit is across each cell. Here one cell interface is shown to schemethe working logic. That is if any of the stacked series connected cells of these unit cells gets over--voltage (more than or equal to 2.73 V the 300-400 mA current sink turns on till the cell voltage islessened to safe value. Practically there ought to be unequal voltage distribution while connecting eachcell of 2.5V say to get 50V super-capacitor power pack

Use these 2000F cells of 2.5V each to get100F 50V super-capacitor power packSystem, by series connection of 20 such cells

Here module to module balancing is eliminated

Under development

cell +

cell-

300-400mA

Trip

Trip

2.2V

cell +

cell-Integrated Reference Comparator and current sink

Detailed circuit active cell balance

Under development

Cell+

Cell-

NTR4101P

243K

1.15M

.01uF GND

V+ OUT

1M

100KU1

10K

100K

2N2222

2N2907

A

B

C

Optional R

TRIG 2.8V NOM TRIG 2.86V MAX TRIG 2.73V NOMU1: Integrated Reference Comparator TRIG 2.3V, 2% Accuracy, 110mV Hysteresis. Open Drain output , pulls low until thresh hold 2.2V reached

A

B

C

Stop charge

2N2907

1N4148

100K

10K

2N2222100K

MUD112

2 ohm ½ W2N2222100K

100 ohm

100 ohm

LEDOptional

TP

Stacking of cells to get power pack

Power Pack 100F, 50V ESR 10 milli-ohm

Under development

Stop chargeNegative Bus Bar Positive Bus Bar

ActiveBalanceCircuit



cell-1

cell-19

cell-20

Overall assembled Power Pack 100F, 50V ESR 10 milli-Ohm

DeutschDTM connector

Monitor Cable for signalsLength 350mm

Negative TerminalM10

PositiveTerminal M8

Voltage Management System(active charge balance circuits)

Monitor Signal s:Over voltage Alarm, Temperature (Analog)State of Charge (Analog) & Facility WLL connectivityStop Charge Signal (Digital)

Weight 10kg

420mm190mm

100mm

20mm175mm

120mm

Under development

Other active cell balance approach

+

-+-

-+

cell n

cell n+1

cell n+2

cell n+3+

+

+

+

REFcell n

cell n+1+

+ The scheme transfers the excess energy of cell voltage toadjacent cell in order to equalize the voltage.

Requires even number of series cells.

Requires connecting cables to connect to adjacent cells

This is module to module balancing schemeUnder development

Multi-phase drive with bi-directional DC-DC converter for EV usingindigenous super-capacitor/battery energy storage

HVDCMain Energy System Variable Frequency Motor

Drive for vehicle electricPropulsion

Five Phase InverterHigh specific energyStorage (High EnergyBattery, Fuel Cell, Engine)

Multiphase Bi-DirectionalDC-DC Converter

POWER-PACKSuper-Capacitor

Fractional Order PIDi

p dkk k ss

βα+ +

Under development

Constant current charging of capacitor and super-capacitor-the difference

Vz = 5.6V

10V

100 ohm

10K

10V

100 ohm

10K

Vz = 5.6V

C

SC

50mA 50mA

CRO CRO

Vc ( t )

timeSW ON

SW SW

V max V max

Vsc( t )

SW ON

BRNS Sanction No.2009/34/31/BRNS dated 22.10.2009

time

Constant current (50 mA) charge-discharge pattern of 10F, 20 F and 25 F aerogel super-capacitors,studied by using Super Capacitor Test System.(Courtesy CMET Thrissur)

Actual observed voltage profile for constant current charge discharge of fractional capacitor-supercapacitor

BRNS Sanction No.2009/34/31/BRNS dated 22.10.2009

Im Z

R e Z1 .9 mΩ 2 .1mΩ 2 .3mΩ

0 .2 m− Ω

0 .4 m− Ω

0 .6 m− Ω 150 m H z

70 H z

1 1( )j

d( ) ( )d

ZsC C

i t C v tt

ωω

≈ =

= 1 1( )( j )

d( ) ( )d

0.5

n nn n

n

n n

Zs C C

i t C v tt

n

ωω

≈ =

=

∼

super-capacitor

Ideal capacitor

Impedance spectroscopy Nyquist plot of super-capacitorin comparison with ideal capacitor

ESR is about 1.7m Ohm

Fractional order element is super-capacitor

1 j( )j1( )

ZC C

Z ss C

ωω ω

≡ = −

=

Impedance of capacitor

Voltage-Current Relation of capacitord ( )( )

dv ti t C

t=

Impedance of super-capacitor 1( ) 0 1( j )

1( )

n nn

n nn

Z nC

Z ss C

ωω

≡ < <

=

Voltage-Current Relation of super-capacitor

d ( )( ) 0 1d

n

n n

v ti t C nt

= < <

Expressions are having fractional derivative

Impedance plots for market available super-capacitor

Frequency domain analysis

Test circuit -

+

110 ohm

Variable frequencySignal generator

CRO( )I s

( )cV s

( ) 0.5 0.5sin( ) ( )( ) ( ) Re ( ) jIm ( )

( )

s

s c

V t tV s V sI s Z s Z Z

R I s

ω

ω ω

= +

= = = −

Impedance plot Re (z) and -Im (z) with frequencyNyquist Plot

Magnitude and Phase PlotUnder development

1F 10 ohmNational P

ImZ−

R e Z

Impedance plot of super-capacitor with Graphine electrode andCAG Electrode

1( ) 0 1

0.5

nn

Z s nC s

n

= < <

∼

Courtesy CMETGraphine electrode

Similar behavior for market available carbon foam capacitors, CNT based capacitors and activated carbon capacitors

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.0

Re Z

-Im Z


•Graphine based systems under development at CMETUnder DeitY funding

Courtesy CMETCAG Electrode

ESR is subtracted in the plots

Expressing fractional impedance via Curie relaxation lawAs in di-electric relaxation, the relaxation of current to an impressed constant voltage stress

an step voltage applied at , to an initial uncharged system 0U 0t =0( ) 0 0 1n

n

Ui t t nh t

= > < <

-4

-5

-6

-7

-8

-1 0 1 2

Slope = - n

We now get Transfer Function of a capacitor via Laplace Transform

{ } 0( ) ( ) nn

UI s i th t

⎧ ⎫= = ⎨ ⎬

⎩ ⎭L L

Use Laplace pair 1

! nn

n ts + ↔ and ! (1 )n n= Γ −

To get 00 1

(1 ) (1 )( ) n nn n

Un nI s Uh s h s s− + −

⎛ ⎞Γ − Γ − ⎛ ⎞= = ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

Note that the voltage excitation is a constant step input at time zero

0 0( )

0 0U t

u tt>⎧

= ⎨ ≤⎩Laplace is 0( ) UU s

s=

( ) (1 ) (1 )( )( )

(1 ) ( ) 1( )( )

1( )( j )

n nnn

n n

n nn n

nn

I s n nH s s C sU s h s h

n U sC Z sh I s C s

ZC

ωω

−

Γ − Γ −= = = =

Γ −= = =

=

For unit step we have

This is Curie law

2 22 2

2 2

1 11

1 1

1 12 21 1

2 21 1

2 2

1 12 2

1 1 1 12 2 2 2

d d( ) ( )d d

d d( ) ( )d d

d 1 dd o r ( ) ( )d o r ( )d d

d d( ) ( )d d

1 d 1 d( ) ( )d d

s sV s v tt t

s s V s v tt t

s t V s v t t v tt s t

s s V s v tt t

V s v ts t s t

− −−

− −

− −

− −

→ ↔

→ ↔

→ ↔

→ ↔

→ ↔

∫ ∫

1( ) 0 1

( ) ( )

d( ) ( )d

nn

nn

n

n n

Z s nC s

s C V s I s

i t C v tt

= < <

=

=

Fractional derivative and Fractional Integral-very simple extension from standard Laplace pair

One whole derivative

Two whole derivative Double differentiation

One whole Integration

Half derivative

Half integration

Fractional impedance is

New capacitor relation is Assuming ESR zero

Fractional integration & Fractional differentiation in integral formDefined as

Riemann-Fractional Integration

10

0

1( ) ( ) ( )d 0( )

tn n

tD u t t u nn

τ τ τ− −= − >Γ ∫

Riemann-Liouvelli Fractional Differentiation

( ) 11 d( ) ( ) ( )d 0 ( 1)( ) d

tmn m n

a t ma

D u t t u n m m n mm n t

τ τ τ− − += − > ∈ − < <Γ − ∫

For 0 1n< <1 d( ) ( ) ( )d 0 1

(1 ) d

tn n

a ta

D u t t u nn t

τ τ τ−= − < <Γ − ∫

Fractional derivative of a constant function

[ ] [ ]0 00 0 0 0 0

0

d d ( )d (1 ) d (1 )

t tn nn n n nt a tn n

a

U UD U U t D U U t at n t n

− −= = = = −Γ − Γ −

Unlike our normal thinking the fractional derivative of constant is non-zero

The above expression are convolution integrals, implying these arememory based operations-capacitor having memory!!

0t =

( )inv t

R

1/ nns C

0 ( )v t

0 0

0 0

d ( ) ( ) ( )d

d ( ) ( ) ( )d

0 1

i n

n

n i nn

R C v t v t v tt

R C v t v t v tt

n

+ =

+ =

< <

The fractional differential equation

A circuit with fractional capacitor gives fractional differential equation

0t =

RV

1( )Z s

2( )Z s0( )V s

20

1 2

2 1

0

0

( )( ) ( )( ) ( )

1( ) ; 0 1; ( )

(1 / ) 1( ) ( ) ( ) ,(1 / )

( ) ( ) ; ( ) 1 for 0 else ( ) 0

( ) ( )( )

in

nn

nn

in inn nn n

in R

R Rin n

Z sV s V sZ s Z s

Z s n Z s Rs C

s C kV s V s V s kR s C s k RC

V t V u t u t t u t

V V kV s V ss s s k

=+

= < < =

= = =+ +

= = ≥ =

= =+

10 ( )

( )R

n

V kv ts s k

− ⎧ ⎫= ⎨ ⎬+⎩ ⎭L

Constant voltage charging of fractional capacitor

01

2 3

1 2 3

2 3

1 2 1 3 1 1

2 2 3 3

0

2 2

( )( ) 1

1 . . .

1. . . u s e( 1)

( ) . . .( 1) ( 2 1) ( 3 1)

1 1( 1) ( 2

R Rn

nn

Rn n n n

n

R n n n n

n n n

R

n n

R

V k V kV sks s k ss

V k k k ks s s s

k k k tVs s s s n

k t k t k tv t Vn n n

k t k tVn n

+

+

+ + + +

= =+ ⎛ ⎞+⎜ ⎟

⎝ ⎠⎡ ⎤

= − + − +⎢ ⎥⎣ ⎦

⎡ ⎤= − + − ↔⎢ ⎥ Γ +⎣ ⎦

⎡ ⎤= − + −⎢ ⎥Γ + Γ + Γ +⎣ ⎦

= − − +Γ + Γ

3 3

0

. . .1) ( 3 1)

( )1 1 ( )( 1)

1

n

n mn

R R nm

n

R nn

k tn

k tV V E k tm n

tV ER C

∞

=

⎛ ⎞⎡ ⎤− +⎜ ⎟⎢ ⎥⎜ ⎟+ Γ +⎣ ⎦⎝ ⎠

⎛ ⎞− ⎡ ⎤= − = − −⎜ ⎟ ⎣ ⎦Γ +⎝ ⎠⎡ ⎤⎛ ⎞

= − −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∑

( )nE x is one parameter Mittag-Leffler function

, , ( 1 )0 0

( ) ( )( ) , ( )( ) ( 1 )

l n ln

n nl l

x k tE x E k tl n l nα β α β

∞ ∞

+= =

−− =

Γ + Γ + +∑ ∑

Solution to obtain Laplace inverse from series

10 ( )

( )R

n

V kv ts s k

− ⎧ ⎫= ⎨ ⎬+⎩ ⎭L

We use for{ }1 ( ),

!( )p p p st E a ts a

α βα β α

α β α

−+ − =

−L 0 1p n nα β= = = +

to have . With this we obtain the following 1

1, 1 ( )n n

n nn

s t E ats a

−−

+

⎧ ⎫=⎨ ⎬−⎩ ⎭

L

10 , 1

, 1

( ) ( )( )

n nRR n nn

nnR

n nn n

V kv t V kt E kts s k

V tt ER C R C

−+

+

⎧ ⎫= = −⎨ ⎬+⎩ ⎭

⎛ ⎞= −⎜ ⎟

⎝ ⎠

L

, , ( 1 )0 0

( ) ( )( ) , ( )( ) ( 1)

l n ln

n nl l

x k tE x E k tl n l nα β α β

∞ ∞

+= =

−− =

Γ + Γ + +∑ ∑Mittag-Leffler function is defined as

1 ( ) xE x e=

Via using Mittag-Leffler function to get solution

Mittag-Leffler function

( )E tα −

0 , 1

0

( ) 1

F o r 1

( ) (1 )

n nnR

R n n nn n n

tR C

R

Vt tv t V E t ER C R C R C

n

v t V e

+

−

⎡ ⎤⎛ ⎞ ⎛ ⎞= − − = −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

=

= −

For charging current of circuit

For charging voltage of capacitor

{ }1 1

( ) u sin g ( )1( ) 1

( )

For 1

( )

R n nnR R

n nn

nnn

nR

nn

tR RC

VV V s ssI s E at

Z s R s ass R R Cs C

V ti t ER RC

nVi t eR

− −

−

⎛ ⎞⎜ ⎟⎜ ⎟= = = =

−⎛ ⎞ ⎜ ⎟++ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎛ ⎞

= −⎜ ⎟⎝ ⎠

=

=

L

Charge discharge comparison of classical capacitor and fractional capacitor

RV

tcT0

cT

( )inv t

0( )v t

t

0 , 1 ,1

0 ,1 , 1

( ) ; 0( ) ( ) ( )

(0)( ) (0) ( ) ;( ) ( ) ( )

n nn R sR

n n n cn s n s s n s

n nnc R c c

n c c c n n cs n s n s n s

V RV t tv t t E E t TC R R C R R R R C R R

v R V T Ttv t E v v T E t TR R C R R C R R C R R

+

+

⎛ ⎞ ⎛ ⎞= − + − < <⎜ ⎟ ⎜ ⎟+ + + +⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= − = = − >⎜ ⎟ ⎜ ⎟+ + + +⎝ ⎠ ⎝ ⎠

0( )cv T

0

( )c cv T ( ) (0)c c cv T v=

Constant voltage charging and discharging voltage profile at super-capacitor

R

sR

nC

( )inv t

0 ( )v t

0t =

S

XC

+

A−

−V

cons tan t current /cons tan t voltagepower sup ply

CCCV

cons tan t currentdisch arg er

3 0 m i n

2U

2t

1U

RU

1t

3UΔ

V o l t a g e ( V )

T i m e ( s )

3 : d r o pU I RΔ −

IEC-62931 Standard to test super capacitor-2007

I

I f−

tdT

cT0

I R

R

I R KU+

≤(1 )IR f+

cT dT

( )i t

( )v t

t

Excitation current profile for charging and discharging andmeasured voltage across the super-capacitor

1 1

1( ) ( ) ( ) ( ) ( ) ( )

0

(1 )( )

1 (1 )( ) ( ) ( )

(1 ) (1 )

c d

c d

c d c

c d

sT sT

sT sTn

sT sT sT sT

n n

t Ti t Iu t I If u t T Ifu t T u t T

t T

I f I IfI s e es s s

F I I f IfV s Z s I s R e es C s s s s

IR IR f e IR fe IF IF f e IF fes s s s s

− −

− −

− − − −

+ +

>⎧= − + − + − − = ⎨ <⎩

+= − +

+⎛ ⎞ ⎛ ⎞= = + + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ += − + + − + 1 2 2 2

(1 )

( ) ( ) (1 ) ( ) ( )

(1 )( ) ( )( ) ( ) ( )( 1) ( 1) ( 1)

(1 )( ) ( )( ) ( ) ( )

( )(

d c dsT sT

n

c dn nn

c dc d

c dc d

I I f e Ifes C s C s C s

v t IR u t IR f u t T IR fu t T

IF f t T IF f t TIF t u t u t T u t Tn n n

I f t T If t TIt u t u t T u t TC C C

IFv t IR

− −

+

++ − +

= − + − + − +

+ − −+ − − + − +Γ + Γ + Γ +

+ − −+ − − + −

= +Γ

(1 ) (1 )( ) (1 ) ( ) ( ) ( )1) ( 1)

( ) ( ) ( )( 1)

n nc c c

nd d d

I IF f I ft t u t IR f t T t T u t Tn C n C

IF f IfIR f t T t T u t Tn C

⎡ ⎤ ⎡ ⎤+ ++ − + + − + − − +⎢ ⎥ ⎢ ⎥+ Γ +⎣ ⎦ ⎣ ⎦

⎡ ⎤+ + − + − −⎢ ⎥Γ +⎣ ⎦

1( ) n

FZ s Rs C s

= + +

Detailed calculations of voltage profile for constant current charge discharge cycleImpedance of super-capacitor

RF C

( )i t

( )v t

Revised Test Procedure

( ) ( ) ( ) ( ) ( )

(1 ) (1 )( ) ( ) (1 ) ( ) ( ) ( )( 1) ( 1)

( ) ( ) ( )( 1)

c d

n nc c c

nd d d

i t Iu t I If u t T Ifu t T

IF I IF f I fv t IR t t u t IR f t T t T u t Tn C n C

IF f IfIR f t T t T u t Tn C

= − + − + −

⎡ ⎤ ⎡ ⎤+ += + + − + + − + − − +⎢ ⎥ ⎢ ⎥Γ + Γ +⎣ ⎦ ⎣ ⎦

⎡ ⎤+ + − + − −⎢ ⎥Γ +⎣ ⎦

Use the current excitation and voltage profile equations to fit the curve of voltage profileto extract the parameters from following set of expressions, , (or ) ,nR C F C n

Under development

Self discharge with memory via fractional derivative for super-capacitor

A capacitor is charged from time –T to t with a constant voltage Uo; the charging current is

0 0 0

0

d (1 ) (1 )( ) ( ( ))d (1 )

0 1 ( ) 0( )

t tn nn

c n n nn nT T

nn

U d U Un ni t C t Tt h dt h n

U n t Th t T

−

− −

⎛ ⎞Γ − Γ −= = = − −⎜ ⎟Γ −⎝ ⎠

= < < + >+

At t = 0 it is kept in open-circuit; there will be self discharge thus a discharge current willappear depending on decaying terminal voltage u ( t )

0 0

d ( ) (1 ) d ( )( )d d

t tn n

d n n nn

u t n u ti t Ct h t

Γ −= =

We combine the charge and self discharge together and write ( ) ( ) 0c di t i t+ =

0 (1 ) d ( ) 0( ) d

n

n nn n

U n u th t T h t

Γ −+ =

+

Do fractional integration of order n for above expression, to write the following

[ ]00 0

(1 ) ( ) 0( )

nt n

n n

U nD u t Uh t T h

− ⎡ ⎤ Γ −+ − =⎢ ⎥+⎣ ⎦

We apply the formula for fractional integration i.e. to get[ ]0 1-0

1 ( )d( )( ) ( - )

tnt n

f x xD f tn t x

− =Γ ∫

Contd…

Contd…0

010

1 d (1 ) (1 )( ) 0( ) ( ) ( )

t

n nn n n

U x n nu t Uh n T x t x h h−

⎡ ⎤Γ − Γ −+ − =⎢ ⎥Γ + − ⎣ ⎦

∫

Rearranging the above, we write0

0 10

d( )( ) (1 ) ( ) ( )

t

n n

U xu t Un n T x t x −= −

Γ Γ − + −∫

Put T x τ+ = d dx τ= so for 0x = Tτ = and x t= T tτ = + We have thus

0 00 01

d( ) ( )d( ) (1 ) ( ) ( ) (1 )

T t T t

n nT T

U Uu t U U Fn n T t n n

τ τ ττ τ

+ +

−= − = −Γ Γ − + − Γ Γ −∫ ∫

Now we break as and call the second term as ( )dT t

TF τ τ

+

∫0

0

( )d ( )d ( )dT t T t

T T

F F Fτ τ τ τ τ τ+ +

= +∫ ∫ ∫ ( )tI

1 1 10 0 0

d d 1 1( ) ( )d *( ) ( )

T t T t t

n n n n n nt FT t t t t

τ ττ ττ τ τ τ

+ +

− − −⎛ ⎞ ⎛ ⎞= = = = ⎜ ⎟ ⎜ ⎟+ − − ⎝ ⎠ ⎝ ⎠∫ ∫ ∫I

We write in terms of convolution of two functions

Using Laplace pair we write as { } { } { }(1 )( ) ( ) n nt s t t− − −= = ×L I I L L

( )[ ]1 (1 ) 1

(1 ) 1( 1) (1 ) ( )( ) n n

nn n nss ss− + − − +

Γ − − +Γ − + Γ − Γ= × =I

1

( 1)nn

nts +

Γ +↔

Extracting by inverse Laplace of obtained we get( )tI ( )sI

{ }1 1 1( ) ( ) (1 ) ( ) (1 ) ( )t I s n n n ns

− − ⎧ ⎫⎛ ⎞= = Γ − Γ = Γ − Γ⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭

I L L

We used Laplace pair 11s

↔ Contd…

Contd…

Using just derived expression i.e. we write theexpression for open circuit voltage as follows

100

d( )d (1 ) ( )( )

T tT t

n nF n nT t

ττ ττ τ

++

−= = Γ − Γ+ −∫ ∫

( )u t

[ ]0 0

0 0 00 0

0

00 0

0

01

0

( ) ( )d ( )d (1 ) ( ) ( )d(1 ) ( ) (1 ) ( ) (1 ) ( )

( )d ( )d(1 ) ( ) (1 ) ( )

d(1 ) ( ) ( )

T t

T T

T

TT

n n

U U Uu t U F F U n n Fn n n n n n

U UF Fn n n n

Un n T t

τ τ τ τ τ τ

τ τ τ τ

ττ τ

+

−

⎡ ⎤= − + = − Γ − Γ −⎢ ⎥Γ − Γ Γ − Γ Γ − Γ⎣ ⎦

= − =Γ − Γ Γ − Γ

=Γ − Γ + −

∫ ∫ ∫

∫ ∫

∫

Therefore

01

0

d( )(1 ) ( ) ( )

T

n n

Uu tn n T t

ττ τ −=

Γ − Γ + −∫

Is the voltage over open capacitor at self discharge. This function of time, depends on the total time T the capacitor has been on the voltage source.

In a way this capacitor is memorizing its charging history.

Self Discharge Curve of Super-capacitor

10 100 1000

2.5

1.0

2.0

Open CircuitVoltage ( V )

time

log ( t )

At this instant open circuit is done

Self Discharge voltage after open circuit is u ( t ) is almost constant for about time T then it decays as (1 )nt − −

More time the capacitor is kept at charge the more time it takes to self-discharge

T = 10hr

T = 100hr

T = 1000hr

This interesting observation we described by use of fractional derivative

We take example of super-capacitor electrodea rough electrode

Rough electrode

++++++++++++++

-------------Smooth electrode

Collector

Collector

Rough and Porous Electrode

A porous electrode

Metal

CAG electrode

CAG electrode

CAG electrode

electrolyte

Rs : Solution Resistance

CD : Double Layer Capacity

Rs Rs Rs Rs

CD CD CD CD

Transmission line for porous electrode

dR xdC x

i di i+

d- i

de e+e

( d ) d d d d d 0

( d ) d ( d ) d d d 0

e ee e e e iR x e x iR x iRx x

e e i ei i i i C x i x C x Ct x t x t

∂ ∂− + = − = = = − + =

∂ ∂∂ ∂ ∂ ∂ ∂

− + = − = = = − + =∂ ∂ ∂ ∂ ∂

(1)

(2)

Differentiate (1) w.r.t t and differentiate (2) w.r.t x, and combine to get2

2 0i iR Cx t

∂ ∂− =

∂ ∂Differentiate (1) w.r.t x and differentiate (2) w.r.t t, and combine to get

2

2 0e eR Cx t

∂ ∂− =

∂ ∂

Rather we got Fickian diffusion equation for current and voltage

Impedance of the porous electrode system2

2 0e eR Cx t

∂ ∂− =

∂ ∂Let us take , where is actually . Take the time Laplaceand get

e ( , )e x t

2

2

d 0d

e R C s ex

− = with , i.e. Laplace of ( , )e e x s≡ ( , )e x t

Exciting with step input at t = 0 or( 0 , )e t E= ( 0 , ) Ee ss

=

The solution is ( , ) x R C sEe x s es

−=

0e iRx

∂+ =

∂From the equation (1) i.e. , we get current as

1 ( , ) 1( , )

(0, )

x RCs x RCse x s E Ei x s RCs e RC seR x R s Rs

Ei s RC sRs

− −∂ ⎛ ⎞⎡ ⎤= − = − − =⎜ ⎟⎣ ⎦∂ ⎝ ⎠

=

Therefore impedance is (0 , ) RZ sC s

=

From this we may write volt-current expression as

d( ) ( ) 0 .5d

n

n

Ci t e t nR t

= =

This is for infinite field diffusion, the transmission line extends to infinity

More complications

Ionic conduction in the pore electrolyte and electronic conduction in solid phase

Charging of double layer at the solid liquid interface

A simple charge transfer reaction (ct) at the interface

c t dg x dC x

1dR xElectrolyte Ionic Current

Solid Phase Electronic Current

2

ct2

1e eC g et R x

∂ ∂= +

∂ ∂0

ct( )c

s

A S j nFgR T

=

This ct reaction is for Faradic super-capacitors. Our CAG is non-Faradic system

Other improvements include finite field diffusion when we are truncating this infinitetransmission line, consider the finite diffusion of transport species etc.

On the gct we also add hindrance factor

2 dR x

Metal

Electrolyte

Paasch et al and R De Levie analysis

Thickness dArea A

dx

1ϕ

2ϕ

dG1dR

2dR

d ddd 1, 2i

i

G g A xxR i

Aρ

=

= =

( ) ( )1 21 2 1 2

1 2

d dd 1 d 1d d d d

g gx x x x

ϕ ϕϕ ϕ ϕ ϕρ ρ

− = − − − = −

The decay length is 1 2R e ( )gλ ρ ρ⎡ ⎤= +⎣ ⎦

Constituting equations are

The interconnection conductance g we assume charge transfer (ct) , which is hindered in series by finite diffusion, both in parallel with double layer capacity C. The C is double layer capacity perunit area and S is the pore area per unit volume. The contains the standard exchange currentdensity of the ct process. The finite diffusion hindrance is contained by Y.

( )0 00 0j j

s s

j n F j n Fg C S YS C S YR T C R T

ω ω ω ω= + + = + =

0ω0j

g ≡ C S0

s

j nF YSR T

Finite diffusion hindrance and Impedance function( ) 0

0 0j sj R Tg C S YC F

ω ω ω= + =1

22

2 3 23

j1 co thj p

kYl

ω ω ω ωω ω

−⎡ ⎤

= + = =⎢ ⎥⎢ ⎥⎣ ⎦

DD

D is the diffusion coefficient of the electro active species, is rate constant of redox electrodereaction equal to and is characteristic pore dimension. This defines diffusionas finite owing to small pore size.

k

red o xk k+ pl

The complex decay length is now2

11 2

0 1 2

1j ( )d Y CS d

ωλ ωω ω ρ ρ

⎛ ⎞ = =⎜ ⎟ + +⎝ ⎠

2 21 2 1 2 1 2

1 2 1 2 1 2

2 1co ths in h

A dZd d

d

ρ ρ ρ ρ ρ ρλλρ ρ λ ρ ρ ρ ρ

+⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠

The impedance function of porous electrode is

The Nyquist Plot of Paasch and R De Levie Impedance

0ω1ω

2ω

3ω

R e Z

Im Z−

0 .1dλ

=

1 0dλ

=

1 0 0 2 0 0 3 0 0 4 0 0 5 0 0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

Thin electrodeThick electrode

2 21 2 1 2 1 2

1 2 1 2 1 2

2 1co thsin h

A dZd d

d

ρ ρ ρ ρ ρ ρλλρ ρ λ ρ ρ ρ ρ

+⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠

Non-linear regression analysis of complex Nyquist plot byequivalent circuit with lumped parametersRandles circuit

1R adR WZ adC

C P E

C P E Constant Phase Element C P E1 0 1

( j ) nZ nC ω

= < <

1R High frequency series resistance R e Z ω → ∞

adR Charge transfer (ct) resistance

adC Adsorption capacitance

WZ Generalized Warburg Impedance (diffusion)2

W

co th ( j )( j )

D pR lZ

α

α

τωτ

τω

⎡ ⎤⎣ ⎦= =D

DR Limiting diffusion resistance

The phase angle vis-à-vis frequencyExplanation of this electro-chemical Nyquist plot by R De Levie

At low frequencies the electro-active contents of the pore are used up within a small fraction of acycle, the electrode reaction behaving as capacitance (RC Transmission line), with phase angle as45 degree. At increasing frequencies the rate of diffusion of electro-active species towards and fromthe electrode becomes over-all determining (the corresponding phase angle is 22.5 degree). At stillhigher frequency the finite reaction takes over (the transmission line is R-R system with 0 degreeas phase angle. Finally at the highest frequency the double layer capacitance will dominate (againRC transmission line) with phase angle 45 degree.

045

022 .5

00

ω

Pore Exhaustion

Diffusion

ElectrodeReaction

Double LayerCapacity Im Z−

R e Z

0ω →

1ω 2ω 3ω 4ω 5ω 6ω

ω→∞

1ω2ω3ω

4ω5ω6ω

04 502 2 . 5

04 5

0

Piece wise continuous diagram of Nyquist Plot-showing phase angles withfrequency

In actual cases these zones may extend into each other

02 2 . 504 500 00

In fact solid state Ionics people often do just that !!

To characterize electrical properties –We do impedance spectroscopy measurementsPlot IMAGINARY PART OF IMPEDANCE – Z’’ against REAL PART OF IMPEDANCE Z’On a COLE-COLE PLOT, and use some equivalent circuit, use CPE

We did a new approach based on fractional calculus for anomalous diffusionhttp://arxiv.org/abs/1503.07121Electrical Impedance Response of Gamma Irradiated Gelatin based Solid

Gelatin, with glycerol as plasticizer, formaldehydeas anti-fungal agent and LiClO4 (x weight fraction)Was cast into films of thickness ~ 300-450 μm

Impedance Spectroscopy was done using HIOKILCR meter for x = 0.25 at room temperatureIn frequency range 50 – 2 MHz (Tania Basu et al.Poster presentation NCSSI-9)

The results are analyzed in 2 different ways1.By fitting with an equivalent circuit using CPE2.Using an analytical fractional calculus approachto take anomalous diffusion in but smooth electrode

Courtesy CMPRC Dept of Phys Jadavpur Univ.

http://arxiv.org/abs/1503.07121

Fractal Electrode-a rough surface of interface

Electrode

Electrolyte

At some Angstrom scale (10nm to 0.1mm) the surface appears to be not smooth. The ideal smooth electrode is only mercury electrode-all others solid electrodes have some degree of roughness at the surface. Therefore the electrolyte interfaces at these irregular interface-the roughness of the electrode electrolyte interface manifests as fractional impedance of super-capacitors. This roughness (rugosity) is cause of frequency dispersion in impedance.

We may characterize these rough geometry of the surface via fractal dimension (which isnot integer number as 1, 2, 3), rather it is a real number like 2.39. This analysis does not requirewhat is structure of roughness

Mandelbrot's fractal geometry

Fractal “Koch” curve

Step-0

Step-1

Step-2

Step-3

; ( )l L l∈= ∈ =

14, ( ) 43 3 3l lL l⎛ ⎞ ⎛ ⎞∈= ∈ = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

24, ( ) 169 9 3l lL l⎛ ⎞ ⎛ ⎞∈= ∈ = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

34, ( ) 6427 27 3l lL l⎛ ⎞ ⎛ ⎞∈= ∈ = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

3l

∈=

9l

∈=

27l

∈=4( )3

n

L l⎛ ⎞∈ =⎜ ⎟⎝ ⎠

Mandelbrot's fractal geometry

Repeating it ad infinitum yields a mathematical monster: a continuous curve of infinite length whichis no where differentiable

ln 4 1.2619ln 3fd = = This fractal dimension is measure of rugosity

Scaling of CPE

As mentioned earlier a CPE (constant Phase Element) is a Fractional Power Frequency Dependence(FPFD) where we write admittance as , the exponent is fractional number 0.5 to 1( )1 jY

Zασ ω= = α

The ratio of complex admittance at frequency and is a real number for any scale factork

ω kω

( ) ( )

( )

j / 2( ) j cos jsin ( j) (0 j) cos jsin2 2 2 2

( ) j cos jsin2 2

( )( )

Y e

Y k k k

Y k kY

αα α α α π

α α α

α

απ απ απ απω σ ω σω

απ απω σ ω σ ω

ωω

⎡ ⎤= = + = + = = +⎢ ⎥⎣ ⎦⎡ ⎤= = +⎢ ⎥⎣ ⎦

=

ImY

ReY

•

•

1x

1y

1k xα

1k yα

1ω

1kω

( ) 1 11 1

1

( ) tan2

yY Y kx

απω ω −∠ = ∠ = =

The phase angle of CPE is constant at all frequencies.

Electrode

Electrolyte

The scaling logic The double layer capacity is formed at the interface, with charge separation of Angstrom distance, this distance remains constant even if we blow up the structure say m times. Let the surface be characterized by fractal dimension, that is rough surface where2fd > 2 3fd< <

ACl

∝

In fractal scaling in themicroscopic scale the areascales as

Thus capacitance scales as

fdm

fdm C

The resistance is of bulk nature because of electrolyte where scaling the system m timesmeans that resistance scales as

lRA

∝

12

1m

mlR Rm A m

∝ =

l is Helmholtz distance

The admittance is macroscopic quantity the electrode surface is macroscopically 2 dimension(though microscopically it is say 2.39) thus scaling of Y is

21mY m Y=

Nyikos and Pajkossy

The derivation of frequency scaled Y via spatial scaling logic 1 j j1

j j 1 jj

1 j

i i ii i i

i i i i

i

i i i

R C CZ R YC C R CCYR C

ω ωω ω ωωω

+= + = =

+

=+∑

We blow up spatially the element m folds and write the following2 1( .1, ) (1, ) ( .1) (1) ( .1) (1)fd

i i i iY m m Y R m R C m m Cm

ω ω= = =

( )( )

1

1

1

1

j ( . 1 )( . 1 , )1 j ( .1 ) ( . 1 )

j (1 )1 j (1 ) (1 )

j (1 )

1 j (1 ) (1 )

(1, )

f

f

f

f

f

i

i i i

di

di i i

di

di i i

d

C mY mR m C m

m Cm R m C

m Cm

m R C

m Y m

ωωω

ωω

ω

ω

ω

−

−

−

−

=+

=+

=+

=

∑

∑

∑

This above expression relates spatial scaling to frequency dependence (scaling)2( .1, ) (1, )Y m m Yω ω=With and above expression we have

1(1, )(1, )

fdY m mY

ωω

−

=Nyikos and Pajkossy

The fractional exponent in terms of rugosity (fractal dimension)1(1, )

(1, )

fdY m mY

ωω

−

=The spatial scaling gave us

The frequency scaling law for CPE or FPFD (with fractional exponent) is (1, )(1, )

Y k kY

αωω

=

Comparing the two as above we can write 1fdk m k mα−= =( 1) 1fdm mα − =

1( 1) 11f

f

dd

α α− = =−

Therefore a fractional admittance (or impedance) has fractional exponentIn other words the fractional exponent is a measure of surface irregularity

( j )Y αω≈ 11fd

α =−

2fd =The perfect smooth surface , the exponent is a pure capacitor (ideal) behavior1α =

3fd →In the limit the exponent result of a porous electrode (it is a 3D case) 0.5α →

The significance of this result explains the connection of surface roughness and frequency dispersionfor the intermediate values of fractional exponent (0.5 to 1), but also offers a simple comparisonbetween surfaces with widely different morphologies. It is remarkable that the microscopic effectivedimension describes the essential features determining frequency dispersion.

Nyikos and Pajkossy

100 nm

Distribution ( ) of aggregate pores of several sizes, on the electrode surface

r α−∼The SEM image of super-capacitor electrode showing roughness & porous nature (Courtesy CMET Govt. of India Thrissur, Kerala)

CAG Electrode porous and rough

r α−∼ α +∈The disorder can be thus ordered via a ‘power law’ distribution as

This is depicted in the histogram of pore size approximated as power law.

The Fuller mix can be expressed in terms of the grain size ‘distribution function’ as follows

2 .5m in

m in m ax3 .5( ) 2 .5 rr r r rr

ϕ = ≤ ≤

( )rϕ represents the probability distribution function

( )dr rϕ is the fraction of grains with diameter in the interval [ ], dr r r+m ax

m in

( )d 1r

rr rϕ =∫

Fuller’s packing of pores

IJAMAS & Am Math Soc.; S. Das, Pramanik

Frequency

m inrm axr

r α−∼

P o r e - S iz e

Frequency

C ap acitymC

( )a

( )b

a) Showing distribution of pores size, b) Corresponding distribution of capacity

Pore size distribution and its capacity


d bxx

Charge distribution at cleavage of electrode crystal

Formation of double layer capacity

Electrode material we consider as simple case made of positive nuclei on fixed ‘regular’ grid points inside a sea of homogeneous distribution of negative charge. By cleaving the electrode one obtains two halves which can be considered as electrodes; the cleaving is at location, as depicted. Let us assume that cleavage has made interface of metal (electrode) and organic (electrolyte), and immediate picture of negative charge sea.

bx

bx x

Q−


bx

bx

bx

x

( )a

( )b

( )c

( )d

Q−Δ

Q−

Q−

Q+

Q Q+ Δ

Q Q− − Δ

Double layer capacity

This spatial charge separation forms “capacity”; the metal-electrolyte capacity-and formation of double layer capacity mC

( )

( )

[ ( )]( )d

( )d

[ ( )]( )d

( )d

b

b

b

b

x

bm x

bxm

x

Q x x x xx

Q x x

Q x x x xx

Q x x

+ −∞

−∞

∞

−∞

Δ −=

Δ

Δ −=

Δ

∫∫

∫∫

( ) ( )1 ( )4m m mC x xπ

+ −= −

( )mx −

( )mx +

Electric fieldPerpendicular to electrode


The distribution function is different for different cleavage. Some cleavage may be symmetrical, as ideal as shown some may have different nuclei distribution near cleavage, with different numbers as per crystal face cleavage of electrode. This distribution thus gives rise to several time-constant system, better described by fractional calculus, with unique disorder parameter .

Now if the above obtained capacity is same, assuming electrode surface is uniformlysmooth, then we do not have problem to model; the capacitor system. However, due torough nature the charge distribution function at each of the cleavage is different-the distribution does not and need not be a normal, Gaussian type. This fractal chargedistribution can lead to a capacity of ‘rough’ electrode other than normal or Gaussiandistribution-leading to power law distribution too! This is ‘disordered’ system.

Disordered system

IJAMAS; & Am Math Soc.; S. Das, Pramanik

( )1

( , ) ( , ) ( ) 0u t u t tt

αλ λ λ δ α∂+ = >

∂

Consider a partial differential equation (PDE)

The above PDE is having free parameter λ

Now if for the free parameter 1α = then we have single time constant system 1λ τ −=

( , ) exp( )u t tλ λ= −with solution as with initial condition ( , 0 ) 1u λ =

The several time constants (discharge rate) is taken as power law distribution as ( )qλ

1 / 0 1q α α= < <

The strong-discharge or exponential discharge with one time constant follow a normal distribution with well defined average that represents average time constant or discharge rate, and that normal distribution has well defined standard deviation. Unlike the normal distribution the ‘power-law’ distribution has no defined average or moments (standard deviation); and is representation of system which has variety. The heterogeneity or the disordered system thus has varieties of ways by which dissipation mechanism takes place.

A several time-constant system-no average


( )1

( , ) ( , ) ( )u t u t tt

αλ λ λ δ∂+ =

∂

( )1( , ) ( , ) e x pu t h t tαλ λ λ= = −

( , )h tλ denotes ‘impulse response function’. On integrating this ‘impulse response function’ for free variable we get the function of time and that is called ‘impulse response’ Green’s function

( )1

0 0

(1 )( ) ( , ) ex p dg t h t d tt

αα

αλ λ λ λ∞ ∞ Γ +

= = − =∫ ∫1 (1/ )( / ) d ( / )dx t t xα αλ λ λ α−= =

1( 1 / )

0 0

11

10 0

( ) d d

( )d d

(1 )

x x

x x

x xg t e x e xt t t t

xe x e x xt t t t

t

αα

αα

α α α

α

α αλ λ

α α α α

α

−∞ ∞− − −

∞ ∞−− − −

−

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Γ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠Γ +

=

∫ ∫

∫ ∫

Steps are

Using the Gamma definition

1

0

( ) d

( ) (1 )

ye y yαα

α α α

∞− −Γ

Γ = Γ +

∫

“Impulse response function” to “Impulse response”

To get above substitute


( )1

( , ) ( , ) ( )u t u t f tt

αλ λ λ∂ ′+ =∂

Then the response to this new excitation is convolution of Green’s function obtained above with the forcing function that is:

0 0

( )( ) ( ) * ( ) ( ) ( )d (1 ) d 0 1t t f tr t g t f t g f t α

ττ τ τ α τ ατ′ −′ ′= = − = Γ + < <∫ ∫

Multiplying and dividing the above expression with (1 )αΓ −

[ ]

( )

0(1 )

( )( ) (1 ) (1 ) ( )d(1 )

(1 ) (1 ) ( )

(1 ) (1 ) ( )

t

t

t

tr t f

D f t

D f t

α

α

α

τα α τ τα

α α

α α

−

− −

− ′= Γ + Γ −Γ −

′Γ + Γ −

= Γ + Γ −

∫

Implying the appearance of fractional derivative for cases where several time-constants define arelaxation process. Therefore a disordered relaxation (response) may well be formulated by fractionaldifferential equation, the order giving the ‘intermittency’ of relaxation disordered process!

Using definition of fractional integral

Fractional derivative operator-in disordered system

10 0

0

1( ) ( ) ( ) ( )( )

t

t tI f t D f t t f dγ γ γτ τ τ γγ

− − += = − ∈Γ ∫

We get


The electrical double layer-its birth

The first model for the distribution of ions near the surface of a metal electrode was devised by Helmholtz in 1874. He envisaged two parallel sheets of charges of opposite sign located one on the metal surface and the other on the solution side, a few nanometers away, exactly as in the case of a parallel plate capacitor. The rigidity of such a model was allowed for by Gouy and Chapman independently, by considering that ions in solution are subject to thermal motion so that their distribution from the metal surface turns out diffuse. Stern recognized that ions in solution do not behave as point charges as in the Gouy-Chapman treatment, and let the center of the ion charges reside at some distance from the metal surface while the distribution was still governed by the Gouy-Chapman view. Finally, in 1947, D. C. Grahame transferred the knowledge of the structure of electrolyte solutions into the model of a metal/solution interface, by envisaging different planes of closest approach to the electrode surface depending on whether an ion is solvated or interacts directly with the solid wall. Thus, the Gouy-Chapman-Stern-Grahame model of the so-called electrical double layer was born, a model that is still qualitatively accepted, although theoreticians have introduced a number of new parameters of which people were not aware 50 years ago.

B E Conway

Unfortunately here we miss Lippmann’s electro-capillary experiments-giving the genesis of formationof Electric Double layer capacity

The electro-chemical double layer

Dates back to 1875 where Lippmann experimented with variable potential to Mercury and electrolyteHg2 Cl2 and measured surface tension of mercury via capillary phenomena. The curves he obtainedwere called Electro-Capillary curves. The Lippmann equation governing charge (rather excess charge)to change in surface tension is also called Electro capillary equations

dd

qEγ

= −

If C is defined as differential capacitance per unit area then

,

2def

2,el el TT

qCE Eμ μ

γ⎛ ⎞∂ ∂⎛ ⎞= = − ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠Therefore the double layer capacity is due to double rate of change of surface tension

If C is constant the Lippmann curve is parabola a convex one

Lippmann curves Lippmann experiment scheme

D C Grahme

Some remarks about Electro-capillary curvePZC Point of zero charge

Negative polarizationPositive polarization

The physical mechanism of electro-capillary effect is attributed to accumulation of surface chargeof equal magnitude of opposite sign on each side of an interfacial boundary. Any such narrow region (10-200 Angstrom) supporting such separation is termed as Electric Double Layer Capacity.

The voltage at which surface tension is maximum peak of electro-capillary curve is PZC and represents the point at which the net charge on each side of interface boundary is zero.

The mutual repulsion of the charges tangential to the interface leads to an effective decrease in thesurface tension

Up to 2-3 Volts the interface draws a very little current-behaves like ideal polarizable electrode.

Possible charge distribution at the interfacePZC Point of zero charge

Negative polarizationPositive polarization

Metal

+

+++++++

+

+

+

+

+

+

+

+

Electrolyte

Negative polarization

q−q

Metal Electrolyte− +− +− +− +− +− +− +− +− +− +− +− +− +− +− +− +

−−−−−−−−−−−−−−−−

0q=

PZC Positive polarization

++++++++

Metal

−

−

−

−

−

−

−

−

Electrolyte

Positive polarization: Excesspositive charge on metal side,and some negative charge areadsorbed giving a ‘compactdouble layer’PZC: Negative charges adsorbedat interface

Negative polarization: Excess negative charge on metal side, noadsorbed charges, giving a diffusedouble layer

D C Grahme

Though not very convincing!

ConclusionWe get frequency dispersion for impedance of super-capacitor when we carry outImpedance spectroscopy.

We get non ideal curves for circuits with super-capacitor charging discharging

The double layer formation is via changes in surface tension and thermodynamics

The internal of super-capacitors are heterogeneous: are porous, are rough are havingseveral time constants of relaxation

Classical electro-chemical laws give the distributed parameter R C cases

The fractal electrode also relate to non-ideality of super-capacitor vis-à-vis ideal one.

The FRACTIONAL DERIVATIVE embeds many of these observations, to representthis non-ideal capacitor, without knowing internal structure, electro-chemical detailsthe internal distributed R C structure or the fractal dimension of the electrode-butonly from two terminals of the devise of. super-capacitor

Perhaps a new way to represent the circuit element super-capacitor-and use fractionalorder circuit theory

ReferencesThe Electrical double layer & theory of electro-capillarity; D C Grahme.Electro-chemical super-capacitors: Scientific Fundamentals & Technological Applications; B E Conway.Theory of electrochemical impedance of macro homogeneous porous electrode; G Paasch et al.Fractals and rough electrodes; R De Levie.On porous electrodes in electrolyte solutions; R De Levie.Fractal dimension and fractional power frequency dependent blocking electrodes; Nyikos and Pajkossy.Diffusion to fractal surface; Nyikos and Pajkossy. Diffusion to fractal surfaces-II verification of theory; Nyikos and Pajkossy.Fractal Geometry of Nature; Mandelbrot.Self discharge and potential recovery phenomena at thermally & electrochemical prepared RUO2 super-capacitorElectrodes; Liu, Pell, Conway.Electromechanics of fluid double layer system; Joseph Charles Zuercher.Electric Double layer characteristics of nano-porous carbon derived from Titanium Carbide; Permann et al.Advanced nano-structured carbon material for electric double layer capacitor; A Janes, et al.Impedance modeling of porous electrode; H Gohr.Impedance of inhomogeneous porous electrode (a novel transfer matrix calculations); Paasch & Nguen.Indigenous development of Carbon Aerogel Super Capacitors and application in Electronic Circuit; S Das & PramanikBARC News Letter. Micro-structural Roughness of Electrodes Manifesting as Temporal Fractional Order Differential Equation in Super-Capacitor Transfer Characteristics; IJAMAS & Am. Math. Soc. S Das, N C Pramanik. DEVELOPMENT OF SUPERCAPACITOR & APPLICATIONS IN ELECTRONICS CIRCUITS PROJECT REPORT-BRNS; N C Pramanik.Electrical Impedance Response of Gamma Irradiated Gelatin based Solid, Tania Basu, Abhra Giri, Sujata Tarafdar, Shantanu Das Applying Fractional Calculus in Solid State Ionics, Sujata TarafdarFunctional Fractional calculus 2nd Edition (Spriger-Verlag Germany); S. Das.

Thank-you

carbon aero gel (cag) super-capacitor a fractional order

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