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Porous supercapacitors: theory and simulations Svyatoslav Kondrat & Alexei Kornyshev

Imperial College London

London, March 31, 2011


Bob Evans: ions are an invention of the Devil


Bob Evans: ions are an invention of the Devil

q φ ∼ q/r


Bob Evans: ions are an invention of the Devil

L q φ ∼ q/r

q φ ∼ q (RL)−1/2 e−πR/L


Outline Intro Charge/energy storage devices Super-capacitors with nano-porous electrodes Anomalous capacitance behaviour Model & theory Model Results Conclusions & critique Monte Carlo simulations Model Results Fini: open questions/outline


Charge/energy storage devices

Conventional (dielectric) capacitors Conventional batteries Electrolytic capacitor


Charge/energy storage devices

Conventional (dielectric) capacitors Conventional batteries Electrolytic capacitor Fuel cells


Charge/energy storage devices

Conventional (dielectric) capacitors Conventional batteries Electrolytic capacitor Fuel cells Electrical double-layer capacitors, or super-capacitors, or ultra-capacitors, or supercondenser, or pseudocapacitor, or electrochemical double-layer capacitor


Charge/energy storage devices Ragone chart (from wikipedia)


Electric double-layer capacitors (EDLCs) Industrial applications

VGA Nuremberg uses hybrid diesel-electric battery engine with EDLCs since 2001


Electric double-layer capacitors (EDLCs) Industrial applications

VGA Nuremberg uses hybrid diesel-electric battery engine with EDLCs since 2001 Mannheim Stadtbahn operates light-rail vehicle with EDLCs for breaking energy since 2003


Electric double-layer capacitors (EDLCs) Industrial applications

VGA Nuremberg uses hybrid diesel-electric battery engine with EDLCs since 2001 Mannheim Stadtbahn operates light-rail vehicle with EDLCs for breaking energy since 2003 ‘Capabus’ in China (Shanghai) with electric umbrellas at bus stops since 2005


Electric double-layer capacitor (supercapacitor)

Electrolyte/Ionic liquid +

+

+ −

+

electrode

V =0

One electrode of porous super-capacitor.


Electric double-layer capacitor (supercapacitor)

Electrolyte/Ionic liquid + −

+ −

+

+ −

+ −

electrode

V >0

One electrode of porous super-capacitor.


Electric double-layer capacitor (supercapacitor) Use porous electrodes to increase energy density

Electrolyte/Ionic liquid +

+ + −

+ − − −

− −

+

+

electrode − − −

− −

− −

V >0

− −

One electrode of porous super-capacitor.


Capacitance of porous super-capacitors: experiments ‘Ordinary’ electrolyte (Chmiola et al, Science 2006)


Capacitance of porous super-capacitors: experiments Room temperature ionic liquid (Largeot, et al, JACS, 2008)


Capacitance of porous super-capacitors MD simulations (Shim & Kim, ASC Nano, 2010)


Model and theory


Model and theory Model of a single, slit-like, metallic pore

R = (R1 , R2 )

Ionic Liquid V

V

V

0

L

z


Model and theory Potential of a point charge in a single, slit-like metallic pore

Electrostatic potential created by a point charge

R = (R1 , R2 )

φ(z, z1 , R) =

4 X sin(πnz/L) sin(πnz1 /L) εL n=1

× K0 (πnR/L),

(R1 , z1 ) 0

L

z


Model and theory Potential of a point charge in a single, slit-like metallic pore

Electrostatic potential created by a point charge

R = (R1 , R2 )

φ(z, z1 , R) =

4 X sin(πnz/L) sin(πnz1 /L) εL n=1

× K0 (πnR/L), For large R  L:

(R1 , z1 ) 0

L

z

φ(z, z1 , R) ≈

exp {−πR/L} √ 2LRε2 × sin(πz1 /L) sin(πz/L),

Packing of counter-ions in a pore becomes much easier


Model and theory (1) Total internal energy due to electrostatics: assumptions

Single two-dimensional layer of ionic liquid inside a slit-like pore Improved mean-field (cut-out disk) approximation


Model and theory (1) Total internal energy due to electrostatics

Total internal energy per surface area: ∞ X sin2 (πm/2) K1 (πmRc (ρ)/L) βUel (ρ, c) = βeV c+4c LB Rc (ρ) m 2

m=1

two-dimensional total density ρ = ρ+ + ρ− and charge density c = Z+ ρ+ − Z− ρ− (Z± are valencies)


Model and theory (1) Total internal energy due to electrostatics

Total internal energy per surface area: ∞ X sin2 (πm/2) K1 (πmRc (ρ)/L) βUel (ρ, c) = βeV c+4c LB Rc (ρ) m 2

m=1

two-dimensional total density ρ = ρ+ + ρ− and charge density c = Z+ ρ+ − Z− ρ− (Z± are valencies) Voltage V acts as an external potential


Model and theory (1) Total internal energy due to electrostatics

Total internal energy per surface area: ∞ X sin2 (πm/2) K1 (πmRc (ρ)/L) βUel (ρ, c) = βeV c+4c LB Rc (ρ) m 2

m=1

two-dimensional total density ρ = ρ+ + ρ− and charge density c = Z+ ρ+ − Z− ρ− (Z± are valencies) Voltage V acts as an external potential Bjerrum length LB = βe2 /εp ≈ 11 ÷ 28 nm (εp ≈ 2 ÷ 5)


Model and theory (1) Total internal energy due to electrostatics

Total internal energy per surface area: ∞ X sin2 (πm/2) K1 (πmRc (ρ)/L) βUel (ρ, c) = βeV c+4c LB Rc (ρ) m 2

m=1

two-dimensional total density ρ = ρ+ + ρ− and charge density c = Z+ ρ+ − Z− ρ− (Z± are valencies) Voltage V acts as an external potential Bjerrum length LB = βe2 /εp ≈ 11 ÷ 28 nm (εp ≈ 2 ÷ 5) Rc (ρ) = (πρ)−1/2 is cut-out radius


Model and theory Self-energy of a point charge

Energy of charging a point charge: Z q q2 q φ(r, r1 )dq = lim φ(r, r1 ) us = lim r→r1 0 2 r→r1 Self energy: qα2 f (z/L) εp L  Z ∞ 1 sinh(Q(1 − z)) sinh(Qz) f (z) = − dQ. 2 sinh(Q) 0 vα (z) = us (L) − us (L = ∞) = −

f (z) is positive for 0 < z < 1


Model and theory Self-energy of a point charge

Energy of charging a point charge: Z q q2 q φ(r, r1 )dq = lim φ(r, r1 ) us = lim r→r1 0 2 r→r1 Self energy: qα2 f (z/L) εp L  Z ∞ 1 sinh(Q(1 − z)) sinh(Qz) f (z) = − dQ. 2 sinh(Q) 0 vα (z) = us (L) − us (L = ∞) = −

f (z) is positive for 0 < z < 1


Model and theory (2) ‘Re-solvation’ and self-energy (Gedankenexperiment)

‘re-solvation’ energy δEα

‘self energy’ vα (z)

εb

εp

εp

bulk

pore (L = ∞)

pore (L < ∞)


Model and theory (2) ‘Re-solvation’ and self-energy

‘re-solvation’ energy δEα

‘self energy’ vα (z)

εb

εp

εp

bulk

pore (L = ∞)

pore (L < ∞)

Free energy of transfer (per surface area):  X  Zα2 LB βEs (ρ± ) = βδEα − f (1/2) ρα L α={±}


Model and theory (3) Model of (ionic) liquid: ´ a la entropy-only approach

Free energy due to â&#x20AC;&#x2DC;everything elseâ&#x20AC;&#x2122;: ( Fil (Ď Âą ) = â&#x2C6;&#x2019;T S(Ď Âą ) = kB T

Ď v  Îą 0 L Îą=Âą  L Ď v0   Ď v0  + 1â&#x2C6;&#x2019; ln 1 â&#x2C6;&#x2019; v0 L L X

Ď Îą ln

v0 = Ď&#x20AC;d3 /6Ρm is the minimal volume per ion Ρm the maximum packing fraction (Ρm = Ď&#x20AC;/6)


Model and theory (3) Model of (ionic) liquid: ´ a la entropy-only approach

Free energy due to â&#x20AC;&#x2DC;everything elseâ&#x20AC;&#x2122;: ( Fil (Ď Âą ) = â&#x2C6;&#x2019;T S(Ď Âą ) = kB T

Ď v  Îą 0 L Îą=Âą  L Ď v0   Ď v0  + 1â&#x2C6;&#x2019; ln 1 â&#x2C6;&#x2019; v0 L L X

Ď Îą ln

v0 = Ď&#x20AC;d3 /6Ρm is the minimal volume per ion Ρm the maximum packing fraction (Ρm = Ď&#x20AC;/6) Second term accounts for â&#x20AC;&#x2122;saturation effectsâ&#x20AC;&#x2122;


Model and theory (4) Chemical potential

Solve the same ‘base’ Fil in the ‘bulk’ (zero electrostatic potential) to get chemical potentials µ± !   (∞) v0 ρ¯± γ Z± −βµ± = ln = ln 1 − v0 ρ¯∞ Z(1 − γ) - Z = Z+ + Z− (‘total’ valency) (∞)

- ρ¯∞ = ρ¯+ the bulk

(∞)

+ ρ¯−

is the three-dimensional total ion density in

- γ = v0 ρ¯∞ (0 < γ ≤ 1)

Set the chemical potentials in the pore to µ± .


Model and theory Total free energy and differential capacitance

The total free energy (per surface area) of ionic liquid in a single, metallic, slit-like pore: X µα ρα F (ρ± ) = Uel (ρ, c) + Es (ρ± ) − T S(ρ± ) + α=±

F (ρ± ) is minimized numerically to obtain equilibrium ρ±


Model and theory Total free energy and differential capacitance

The total free energy (per surface area) of ionic liquid in a single, metallic, slit-like pore: X µα ρα F (ρ± ) = Uel (ρ, c) + Es (ρ± ) − T S(ρ± ) + α=±

F (ρ± ) is minimized numerically to obtain equilibrium ρ± Differential capacitance per surface area C=−

1 dQ , 2 dV

Q = ρ+ − ρ−


Improved mean-field theory: results

apa itan e,

C/CH

Differential capacitance vs pore width (VT = 1/βe ≈ 26 mV)

3

V /VT = 0 V /VT = 20 V /VT = 50

2 1 0 1

1.2

1.4

pore width,

1.6 L/d

1.8

Model parameters: d = 0.7 nm, δE = 10 kB T , LB = 20 nm.


Improved mean-field theory: results

apa itan e,

C/CH

Differential capacitance vs voltage (VT = 1/βe ≈ 26 mV)

3

L/d = 1 L/d = 1.2

2 1 0

0

20

40 voltage, V /VT

60

Model parameters: d = 0.7 nm, δE = 10 kB T , LB = 20 nm.


Improved mean-field theory: results

ρ+ /ρmax

ρ/ρmax

Total ion density and cation density (V > 0)

1 0.8 0.6 0.4

L/d ≈ 1.08 L/d = 1.2

0.2 0 20

30 voltage,

V /VT

40

Model parameters: d = 0.7 nm, δE = 10 kB T , LB = 20 nm.


Improved mean-field theory: results Two phases – two ‘charging regimes’

low voltage −

+ − +

V

+

V

− − +

cation-rich phase


Improved mean-field theory: results Two phases – two ‘charging regimes’

low voltage

high voltage −

+ −

− +

V

+

V

V

+

+

cation-rich phase

V

cation-poor phase


Improved mean-field theory: results Phase diagram

voltage,

V /VT

30 Cation-poor phase

25 20

Cation-ri h phase

15 1

1.2

1.4

pore width,

L/d

1.6

Model parameters: d = 0.7 nm, δE = 10 kB T , LB = 20 nm.


Model and theory: r´esum´e Main results

Exponential pore-shielding seems to be responsible for an anomalous increase of capacitance


Model and theory: r´esum´e Main results

Exponential pore-shielding seems to be responsible for an anomalous increase of capacitance There is an interesting jump in capacitance, manifesting a first-order phase thransition


Model and theory: r´esum´e Main results

Exponential pore-shielding seems to be responsible for an anomalous increase of capacitance There is an interesting jump in capacitance, manifesting a first-order phase thransition Two phases correspond to two ‘charging regimes‘


Model and theory: r´esum´e An attempt of self-critique

Carbon vs metallic pores


Model and theory: r´esum´e An attempt of self-critique

Carbon vs metallic pores Opening/closing of a pore, and inter-pore interactions


Model and theory: r´esum´e An attempt of self-critique

Carbon vs metallic pores Opening/closing of a pore, and inter-pore interactions Single two-dimensional layer of ionic liquid inside a slit-like pore


Model and theory: r´esum´e An attempt of self-critique

Carbon vs metallic pores Opening/closing of a pore, and inter-pore interactions Single two-dimensional layer of ionic liquid inside a slit-like pore Mean-field like approximation: no fluctuations/correlations


Model and theory: r´esum´e An attempt of self-critique

Carbon vs metallic pores Opening/closing of a pore, and inter-pore interactions Single two-dimensional layer of ionic liquid inside a slit-like pore Mean-field like approximation: no fluctuations/correlations Short-range repulsive (hard-sphere type) interaction are not properly accounted for


Model and theory: r´esum´e An attempt of self-critique

Carbon vs metallic pores Opening/closing of a pore, and inter-pore interactions X Single two-dimensional layer of ionic liquid inside a slit-like pore X Mean-field like approximation: no fluctuations/correlations X Short-range repulsive (hard-sphere type) interaction are not properly accounted for


Monte Carlo simulations


Monte Carlo simulations Two co-writers

M. V. Fedorov, MPI f¨ ur Mathematik in den Naturwissenschaften, Leipzig

N. Georgi, MPI f¨ ur Mathematik in den Naturwissenschaften, Leipzig


Monte Carlo simulations: model Single, slit-like, metallic pore d

(HS)

Interaction potential: Uij = Uij

(C)

+ Uij

+

Interaction due to electrostatics:

− −

V

V

+

∞ 4qi qj X sin(πnzi /L) εL n=1

− 0

(C)

Uij (zi , zj , R) =

z L

T = 400◦ K, ε = 2

× sin(πnzj /L) K0 (πnR/L)


Monte Carlo simulations: model Single, slit-like, metallic pore d

(HS)

Interaction potential: Uij = Uij

(C)

+ Uij

+

Interaction due to electrostatics:

− −

V

V

+

∞ 4qi qj X sin(πnzi /L) εL n=1

− 0

(C)

Uij (zi , zj , R) =

× sin(πnzj /L) K0 (πnR/L)

z L

T = 400◦ K, ε = 2

‘External’ potential due to ion’s self-energy: (s)

Ui (zi ) = −

qi2 f (zi /L) εL


Monte Carlo simulations: model Single, slit-like, metallic pore d

(HS)

Interaction potential: Uij = Uij

(C)

+ Uij

+

Interaction due to electrostatics:

− −

V

V

+

∞ 4qi qj X sin(πnzi /L) εL n=1

− 0

(C)

Uij (zi , zj , R) =

× sin(πnzj /L) K0 (πnR/L)

z L

T = 400◦ K, ε = 2 µbulk = −2.5 kB T

‘External’ potential due to ion’s self-energy: (s)

Ui (zi ) = −

qi2 f (zi /L) εL

Electro-chemical potential: µbulk + qi V


Monte Carlo simulations: model Single, slit-like, metallic pore and pure ionic liquid d

(HS)

Interaction potential: Uij = Uij

(C)

+ Uij

+

Interaction due to electrostatics:

− −

V

V

+

∞ 4qi qj X sin(πnzi /L) εL n=1

− 0

(C)

Uij (zi , zj , R) =

× sin(πnzj /L) K0 (πnR/L)

z L

T = 400◦ K, ε = 2 µbulk = −2.5 kB T no ‘re-solvation’: δE± = 0

‘External’ potential due to ion’s self-energy: (s)

Ui (zi ) = −

qi2 f (zi /L) εL

Electro-chemical potential: µbulk + qi V


Monte Carlo simulations: results

total harge (Q),

ÂľC/ m2

Total charge accumulated in a pore per surface area

20 15 10 5 0

0

L = 0.75 L = 0.8 L=1 L = 1.2 1 2 voltage, V

nm nm nm nm

3


Monte Carlo simulations: results Total charge accumulated in a pore per surface area

Fitting function:   n o Q(V ) = Q0 tanh V A + exp α(V − V0 ) with only four fitting parameters: Q0 , V0 , and A and α


Monte Carlo simulations: results

total harge (Q),

ÂľC/ m2

Total charge accumulated in a pore per surface area

20 15 10 5 0

0

L = 0.75 L = 0.8 L=1 L = 1.2 1 2 voltage, V

nm nm nm nm

3


Monte Carlo simulations: results

apa itan e,

ÂľF/ m2

Differential capacitance vs voltage

30 L = 0.75 L = 0.8 L=1 L = 1.2

20

nm nm nm nm

10 0

0

1

2 voltage, V

3


Monte Carlo simulations: results

apa itan e, ÂľF/ m2

Differential capacitance vs pore-width at zero voltage

simulations experiment mean-eld

20 15 10 0.7

0.8

0.9

pore width, nm

1

1.1


Monte Carlo simulations: results

pa king fra tion,

η

Total ion density vs pore-width at zero voltage (η = πd3 ρ/6)

0.4

0.3

1.5

2

pore width/ion diameter

2.5


Monte Carlo simulations: results

apa itan e,

ÂľF/ m2

Origin of the peak in the capacitance vs voltage curve

30 L = 0.75 L = 0.8 L=1 L = 1.2

20

nm nm nm nm

10 0

0

1

2 voltage, V

3


Monte Carlo simulations: results

apa itan e,

ÂľF/ m2

Origin of the peak in the capacitance vs voltage curve

30 20 10 0

0

1

V0 2 voltage, V

3


Monte Carlo simulations: results Origin of the peak in the capacitance vs voltage curve (η = πd3 ρ/6)

pa king fra tion,

η

0.35

0.34 0

Vf = 1

V0 2

voltage, V

3


Monte Carlo simulations: results Two charging ‘regimes’ but no transition

low voltage −

+ −

V

− + − +

V < Vf

V


Monte Carlo simulations: results Two charging ‘regimes’ but no transition

low voltage −

high voltage −

+

+

V

− + − +

V < Vf

V

V

− + − −

V > Vf

V


Monte Carlo simulations: results Two charging ‘regimes’ and a transition in MFT

low voltage

high voltage −

+ −

− +

V

+

V

V

+

+

cation-rich phase

V

cation-poor phase


Monte Carlo simulations: results No transition as we are away from the transition

Rough estimate for the minimal voltage to observe a transition: V1 ≈ −µ/e + e ln(2)/εL − δE/e


Monte Carlo simulations: results No transition as we are away from the transition

Rough estimate for the minimal voltage to observe a transition: V1 ≈ −µ/e + e ln(2)/εL − δE/e

For parameters used: V1 ≈ 6.33 V, and so V1 > Vf , V0


Monte Carlo simulations: results No transition as we are away from the transition

Rough estimate for the minimal voltage to observe a transition: V1 ≈ −µ/e + e ln(2)/εL − δE/e

For parameters used: V1 ≈ 6.33 V, and so V1 > Vf , V0 Suggests that it is rather unlikely to observe a transition for pure room temperature ionic liquid


Fini: open questions Is there a phase transition?


Fini: open questions Is there a phase transition? Optimal pore size: capacitance vanishes for V > V0 , hence no increase in energy stored if the operating voltage U > V0


Fini: open questions Is there a phase transition? Optimal pore size: capacitance vanishes for V > V0 , hence no increase in energy stored if the operating voltage U > V0 Finite pore wall width and inter-pore interactions Carbon/graphite electrodes vs metallic electrodes


Fini: open questions Is there a phase transition? Optimal pore size: capacitance vanishes for V > V0 , hence no increase in energy stored if the operating voltage U > V0 Finite pore wall width and inter-pore interactions Carbon/graphite electrodes vs metallic electrodes Opening/closing of a pore


Fini: open questions Is there a phase transition? Optimal pore size: capacitance vanishes for V > V0 , hence no increase in energy stored if the operating voltage U > V0 Finite pore wall width and inter-pore interactions Carbon/graphite electrodes vs metallic electrodes Opening/closing of a pore Dynamics. Different dynamics in two â&#x20AC;&#x2DC;charging regimesâ&#x20AC;&#x2122; ?


Acknowledgement

Theory/simulations: M. Fedorov, MPI Leipzig N. Georgi, MPI Leipzig A. Frolov, MPI Leipzig


Acknowledgement

Theory/simulations: M. Fedorov, MPI Leipzig N. Georgi, MPI Leipzig A. Frolov, MPI Leipzig Markus Bier, MPI Stuttgart Ludger Harnau, MPI Stuttgart Gleb Ochanin, Universit´e Pierre et Marie Curie, Paris Oleg Vasilev, MPI Stuttgart


Acknowledgement

Theory/simulations: M. Fedorov, MPI Leipzig N. Georgi, MPI Leipzig A. Frolov, MPI Leipzig Markus Bier, MPI Stuttgart Ludger Harnau, MPI Stuttgart Gleb Ochanin, Universit´e Pierre et Marie Curie, Paris Oleg Vasilev, MPI Stuttgart

Experiments: Yura Gogotsi, Drexel University, Philadelphia Patrice Simon, Universit´e Paul Sabatier, Toulouse Some others from Drexel. . .

Ionic Liquids Topics  
Ionic Liquids Topics  

A presentation given by Dr. Syvatoslav Kondrat at Imperial College, London.

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