Electrokinetics of
correlated electrolytes and ionic liquids
Brian D. Storey
Olin College
Martin Z. Bazant
Departments of Chemical Engineering and Mathematics
Massachusetts Institute of Technology
Ionic liquids
• Molten salts (T~1000oC)
• Room temperature IL
• Supercapacitors
• Batteries
• Actuators
BMIM
wikipedia
– Large ions (~1 nm)
– No solvent. What is permittivity?
– Ion-ion correlations (+-+-+-+-)
– Ion size = 10 x Debye length
– Capacitance data often interpreted through
classic electrolyte model.
Classical double layer theory
i  kT ln ci  zi e
Chemical potential of dilute point ions:
At equilibrium:
ci  ci , e
Applied voltage =.025 V
3
zi e
kT
Applied voltage =0.75 V
20
10
2.5
10
10
1.5
C
C
2
0
10
1
-10
10
0.5
0
0
Would need ions to be 0.01 angstrom
-20
1
2
3
X
4
5
10
0
1
2
3
X
4
5
Finite sized ions
Stern (1924) Bikerman (1942)
i  kT ln ci  zi e  kT ln(1  )
Volume fraction
Bazant, Kilic, Storey, Ajdari – ACIS 2009
All mean-field theories
1. Electrochemistry
2. Electrostatics
mi = kT lnci + zi ef + m
ex
i
mi = 0 Þ r = å zi eci =r (f )
eq
-eÑ f = req (f )
2
3. Flow
Ñp = hÑ u - req (f )Ñf
2
• Same “mean electric field” in all equations
“Ginzburg-Landau” theory for
ionic liquids
“Ginzburg-Landau” theory for
ionic liquids
“In physics, Ginzburg–Landau theory, named after
Vitaly Lazarevich Ginzburg and Lev Landau, is a
mathematical theory used to model superconductivity. It
does not purport to explain the microscopic
mechanisms giving rise to superconductivity. Instead, it
examines the macroscopic properties of a
superconductor with the aid of general thermodynamic
arguments.” --- wikipedia
“Ginzburg-Landau” theory for
ionic liquids
e
æ
2
2
2 2 ö
G = ò dV ç g + rf - éë| Ñf | + c (Ñ f ) ùû÷
è
ø
2
chemical free energy
mean electrostatic energy
self energy of E field
electrostatic correlations (new)
Require
dG
mi =
=0
d ci
dG
=0
df
(
Ñ -1) eÑ f = req (f )
2
c
2
2
4th order modified Poisson-Boltzmann eqn
Is this crazy? Maybe not…
“Intermediate coupling” in one-component plasma
(Santangelo 2006; Hatlo, Lue 2010 --- statistical mechanics of point-like counterions near a wall)
𝜀 = 𝜀(1 − ℓ2𝑐 𝛻 2 )
Wavelength-dependent permittivity (Tosi 1986, molten salts)
D = eE
𝜀 = 𝜀(1 + ℓ2𝑐 𝑘 2 )
Nonlocal dielectric response (Kornyshev et al 1978, Hildebrandt et al 2004)
Nonlocal ion-ion correlations (this work)
r = -eÑ2f
Dgcorr = ò K(r,r ')r (r)r (r ')dr dr ' ~ - 2c r (r)2
RTIL double-layer structure
Set ℓ𝑐 to ion size, 𝛿𝑐 = ℓ𝑐 /𝜆𝐷
charge density at V=1,10,100 kT/e
This model vs. MD simulations
Fedorov, Kornyshev 2009
Solid: this model, Open: MD
si/|s|
10
|s|=0.8 mC/cm2
0
1.5
0
−1
1 2 3 4 5 6 7
1 2 3 4 5 6 7
|s|=32.0 mC/cm2
|s|=16.0 mC/cm2
0.5
1
i
|s|=8.0 mC/cm2
1
−10
s /|s|
2
0.5
0
0
−0.5
1 2 3 4 5 6 7
i
−0.5
1 2 3 4 5 6 7
i
RTIL differential capacitance
No correlations, but includes
size effects
(Fedorov & Kornyshev, 2008)
MD Simulations
(Fedorov & Kornyshev, 2008)
This model
Correlated electrolytes
high valence, high concentration
1M 2:1 salt
5
g(x)
4
Boda et al 2002 MC simulations
3
2
-
1
0
0
1
2
x/a
3
4
5
Comparison to DFT
2:1 salt
0
V= -1
2
Q (C/m )
-0.2
No corr.
V= -2
-0.4
V= -4
V= -6
-0.6
-0.8
-1 -3
10
This model
V= -8
-2
10
10
C+ (Molar)
-1
10
0
Comparison to DFT
2:1 salt
0
V= -1
2
Q (C/m )
-0.2
No corr.
V= -2
-0.4
V= -4
DFT
-0.6
V= -6
-0.8
This model
V= -8
-1 -3
10
10
-2
C+ (Molar)
10
-1
10
0
DFT of Gillespie et al, 2011
Slip velocity
2:1 salt
1
C=0.1M
C=0.01M
U/UHs
0
-1
-2
C=1M
-20
-10
0
V
10
20
Comparison to experiment
2:1 salt
80
pA/bar
60
40
20
0
-4
10
C (M)
10
-2
10
0
Van der Heyden 2006 nanochannel experiments
Conclusions
• Electrostatic correlations lead to overscreening,
which competes with crowding in ionic liquids and
concentrated, multivalent electrolytes
• Correlations may explain reduced/reversed
electro-osmotic flow at high concentration and
enhanced capacitance of nanopores
• A simple continuum model is proposed
Capacitance
2:1 salt
15
Size effects
Included, no
corr.
C=0.01M
C/CDH
This model
10
C=0.1M
5
C=1M
0
-20
-10
0
V
10
20
Correlations might explain why mean field theories need large ions to fit exp.
Overscreening vs. crowding
MZ Bazant, BD Storey, AA Kornyshev, Phys. Rev. Lett. (2011)
overscreening
- +
+
+
+ + + - +
+ - - +
(a)
+
+
+
V= 10
crowding
+ +
+
+
+
+
+
+
+
+
+
+
+
-
+
+
-
+
kT
e
+
(b)
V= 100
-
+
+
+
+
-
-
+
+
-
kT
e
Boundary conditions
• Electrostatic BC (no correlations)
[n × e E] = [n ×( Ñ -1)eÑf ] = qs
2
c
2
f = fs
• Neglect “bulk” correlations (finite size ions)
-[n × eÑf ] = qs Û n ×Ñr = n ×ÑÑ f = 0
2
Concentration profiles
2:1 salt, 1M, a=0.3 nm
V=1 kT/e
V=5 kT/e
80
6
60
4
40
2
20
c(x)/c

8
0
0
2
4
0
0
6
2
6
V=20 kT/e
V=10 kT/e
80
60
60
c(x)/c

80

c(x)/c
4
40
20
0
0
40
20
2
4
6
0
0
2
4
6
RTIL double-layer structure
4
3
2
1
1
2
3
4
5