One Hundred Years of Electrified Interfaces: The Poisson-Boltzmann theory and some recent developments
Soft Matter - Theoretical and Industrial Challenges
Celebrating the Pioneering Work of Sir Sam Edwards
One Hundred Years of Electrified Interfaces: Poisson-Boltzmann theory
and some recent developments
T. Markovich, A. Levy & D. Ben-Yaakov (Tel Aviv)
H. Orland (Saclay); R. Podgornik (Ljubljana & Amherst)
Y. Nakayama (Kyushu Univ); D. Harries (Hebrew Univ)
• The 100-year old Poisson-Boltzmann Theory
– Introduction and motivation
• Finite ion size
• Hydration shells & dielectric decrement
• Surface tension beyond Onsager-Samaras
Biology Cell membrane: DPPC/DPPS[+],
ion channels
DNA/charged polymers
Industry/chemistry Stabilization of colloids - DLVO
Charged micelles
Water based paints, aerosols, emulsions
Anionic/cationic detergents
Why electrostatics?
micelle
head
tail
lipid
DNA
colloids
membrane
water
Multi-scale interactions Collapsed vs. Swollen Chains
2m
virus Exploded DNA
Added multivalent counter-ions
Collapsed DNA
100 nm
Electrostatics in BioSoft Matter
Point Ions Ionic solutions & Electrolytes
d=0
Charged Polymers Polyelectrolytes Monica’s talk
d=1 DNA
Charged Surfaces Membranes, Interfaces
d=2 Charged lipids, surfactants
Charged Macromolecules Proteins, Polypeptides, Colloids
d=3
, ., , , ., .Cl Ca MNa Mng
• Entropy of ions ~ kT
• Dielectric solvent (water)
• Mobile ions around macro & charged objects
• Screening of Elec. Int.
• Coupling between conformation & ionic degrees of freedom
The “Grand” Problem Statistical Mechanics of Coulombic Systems
water 80
The Poisson-Boltzmann Theory
Simeon Denis Poisson 1781-1840 Ludwig Boltzmann 1844-1906
The Poisson-Boltzmann Theory
z water
80
+
+ +
- +
-
+
- +
-
+ +
surface
- - - - - - - - - - -
electrolyte
• Only electrostatic interactions
• Point-like charges
• Continuum dielectric media
• Thermodynamical equilibrium -> electrochemistry
• Mean-field densities of ions & electric potentials
Poisson-Boltzmann Equation
• Boltzmann distribution for mobile ions:
• Poisson equation:
• Poisson-Boltzmann equation (1:1 salt)
• Boundary conditions: Dirichlet (CP), Newman (CC),
Charge regulation, …
0
/e
ez kTn n
2
0
8sinh( / )
en e kT
2 4ee nn
Adaptive Poisson-Boltzmann Solver (APBS) Software for evaluating the electrostatic properties
of nanoscale biomolecular systems
https://sites.google.com/a/poissonboltzmann.org/software/apbs
z water
80
+
+ +
+
+
+
+ +
surface
- - - - - - - - - - -
electrolyte
( )n z
(0)n
z
( ) 0n
ion densities
One flat surface – neutral system Gouy 1910; Chapman 1913
• Negative surface charge • Positive mobile ions
One flat surface – neutral system Gouy 1910; Chapman 1913
z
+
-
- +
+
-
-
+
1
2 2
2( ) ln( )
~ Gouy-Chapman length 2
1( )
potential profle
ionic prof 2 ( )
e
il
kTz z b
e
kTb
e
kTn z
e z b
0
( ) ( ) / 2
b
Q b z dz
+ -
+
+ +
+
+
Diffusive “double-layer”
b
Debye-Huckel screening - 1923
+
-
- +
+
-
-
+
+ -
-
- +
+
-
z
0
Salt
1/2
0
2
0 Å / [M]
Å 1
83
: 3 m
D
D
ne n
kT
2 2 ; sma
( ) exp( / )
ll
D
D
z z
• Electric potential
• Debye-Huckel screening length
2
0
/ 7Å
Bjerrum Length
1 / 8
B
d B
e kT
l n
l
Beyond Poisson-Boltzmann: Ion-specific effects
• finite ionic size • solvent mixtures
• hydration shell • ion-surface Int.
D. Ben-Yaakov et al, COCIS (2009)
The Modified Poisson-Boltzmann finite-ion size
Saturation at charged surfaces
Modified Poisson-Boltzmann
Finite size of ions
+ + + + + + + + +
+ + +
Saturation !
+ +
+
+
Borukhov, Orland + DA ; Iglic ‘90s, Bikermann ’40s; Eigen ’50s
solion vent ententropy ropy
log log (1 ) log(1 )
el
S k n n n n n n n n
F U TS
a
Modified Poisson-Boltzmann
0
0
3
small
1 cosh( )1 large
n en e
n
a
3
02 close packingn a
Fermi-Dirac distribution
MPB 2 2 sinh( )( )
1 co
saturation
sh( )D
/e kT
rescaled
Modified PB
2o
0
o
1 / 25A ; 0.75M
8A large ions
e n
a
-
Strong effect for
Large size ions
X-ray reflectivity at water/air interface saturation, beyond PB
Rondelez & Cuvillier ‘98
3-
electron density
amine surfactant
very large counterions
tungstic acid 3 12 4H PW O
Comparison with experiments
Area per charge
MPB Exp.
(0)n
Exp: Rondelez & Cuvillier ‘98
Ion hydration shell
Dielectric decrement
• Ion surrounded by dipolar water molecules
• Response to the E field is reduced: dielectric decrement
Hydration shell around ions
Hydration shell
E
water
0( )n
Hasted, Ritson and Collie, 1948
0( )n n
Salt
HCl 10
LiCl 7
NaCl 5.5
KCl 5
RbCl 5
KF 6.5
NaI 8
Dielectric decrement
1[M ]
[M]n
)(n
Salt conc.
The Dipolar Poisson-Boltzmann (DPB)
A mixture of:
– permanent dipoles (water) bulk conc.
– Ions +, - bulk conc.
pdn
0n
The Dipolar PB
• The free energy of ions and dipoles
• Mean-Field: the Dioplar PB equation
2
0 2
cosh[ ] s
dipo
inh[ ]2 si
io
nh ·
les;
4 ( )
ns /
d
pE pEn e e n p
pE pE
E
e e
e E
3 2
02sinh[ ]
d r cos[ ( )] ( )h ]8
[ d
pEE e n
kT pEF n
r r
Elec. fi( ) | el| dE r
Hydration layer
• Solve the Dipolar PB around a point ion
hydration layer thickness
~ [1 2]Åh B
pl l
e
hl
hl
profile around ion
2 / 7Å BjerrumB el kT
Hasted, Ritson and Collie, 1948
0( )n n
Salt
HCl 10
LiCl 7
NaCl 5.5
KCl 5
RbCl 5
KF 6.5
NaI 8
Dielectric decrement
1[M ]
[M]n
)(n
Salt conc.
Dipolar PB: one-loop expansion
• Field-theory expansion of the free energy
• Fluctuations in dipole and ion densities
• One-loop produces a closed formula for dielectric decrement (non-linear)
31D D D
2 3 3
D
tan2 3
ww
dn a a a
Levy, DA & Orland, PRL ‘12, JCP ‘13
0~ 1/ Debye length
minimal distance ~ f Åew
D n
a
Dielectric const.
salt [M]
Linear Decrement:
0
1[9 15]M 1r Mfo n
0( )n n
The dielectric constant is space-dependent
0( ) ( )n z n z
electrostatic energy ion entropy
Ben-Yaakov, DA & Podgornik, 2011
The Free Energy
Ben-Yaakov, DA & Podgornik, 2011
Saturation layer at interfaces: Ions & hydration
Ion density
( )z
0( ) ( )z n z
Plateau width: ~ | | w Å~ fel
Augmented PB eqn with
( )zPB
APB
Sterically-Modified PB Differential Capacity
• PB: single min • smPB: steric saturation
& double hump (Kornychev’ 07)
• ionic specific dielectrophoetic & steric saturation (Nakayama + DA ’14)
KCl 10mM
/ sC
Cl
K
Ion-Surface Interaction
Surface tension of solutions
Ion-surface
Surface Tension of electrolytes: Onsager-Samaras (1934)
80 water
1 air
-
-
• Ion depletion from neutral interface
• Onsager-Samaras Dielectric discontinuity & Debye-Huckel theory
water air
Image charge
0 0 increase tension with salln t conc.kTn n
neutral repulsive
Non-specific
2
0 ~ Dn
air water
( )n zbn
Experiments on protein precipitation Many chemical and biological systems
Ionic specific
3- 2- 2- - - - - - -
3 4citrate >sulfate >phosphate >F >Cl >Br >I > NO >ClO
+ + + + + + 2+ 2+ 3+
3 4 4N(CH ) NH >Cs >Rb >K > Na >H >Ca >Mg >Al
most stabilizing strongly hydrated anions
weakly hydrated cations strongly hydrated cations
most destabilizing weakly hydrated anions
F. Hofmeister, Arch. Exp. Pathol. Pharmakol. (1888)
Hofmeister Series: Ionic specific effects (1888)
Ion-Specific
Surface Tension
Surface Tension: Ion Specificity
Air/water interface
Cl Br I F
Hofmeister series
Exp: Matubayasi et al (2001)
salt
OS
Ion-Specific
Surface Tension
Self-Consistent Theory: Surface-ion interaction
Markovich, DA, Podgornik (JCP ’15)
80 water
1 air
-
: adhesivity
MF OS ion spec.
• Mean-Field: Davies isotherm with
• One-loop expansion: OS & ion specific effects
analytical
neutral
22
2
( )1/ ( ) ln . ln
16 1 1
eDB e D Dk T a const a
Surface Tension of Electrolytes: analytical results
• a: distance of closest approach
• Analytical results for small a
• Effective surface charge small
Ion-specific Onsager-Samaras Mean-field small
0 ~ ( / )e n kT
Ion-Specific
Surface Tension
Self-Consistent Theory: Fit with experiments
/ (NaF) ions at air/water
ionic siz
0.18
e 7.1Å
0.02 (NaI
9
)
6.
kT
r r
air-water interface
I F BCl r
Strong fluctuation effect
OS
Ion-Specific
Surface Tension
Fit at oil/water: Ion specific surface-tension
Markovich, DA, Podgornik (JCP ’15)
dodecane-water interface
dispersion forces induce attractive
0.29 (KI ?); -0.03 (KBr)/ 0.08 (KCl); kT
oil 2
KI
KBr
KCl Br I Cl
Conclusions Going beyond PB theory
• Finite ion-size effects
– Saturation at interfaces
• Interaction of ions and dipoles
– Hydration shells & dielectric decrement
• Ion specific effects at interfaces:
– Surface tension of electrolytes • More microscopic model to predict surface
adhesively and its specificity • Disturbance of water structure
O H H
1 air 80 water
Levin, PRL (209)
O H H
Conclusions Going beyond PB theory
• Finite ion-size effects
– Saturation at interfaces
• Interaction of ions and dipoles – Hydration shells & dielectric decrement
• Ion specific effects at interfaces: – Surface tension of electrolytes
• Ion profiles
Na
I
Ionic profiles - NaI
Markovich, DA, Orland ‘16
100 years of the Poisson-Boltzmann theory: The future
• More formal understanding of long-range Coulombic liquid systems
• Narrow the gap between physics and biological complexity of proteins, membranes, DNA, …
• We still do “patch work”:
– Try to extract the global picture from the specific details…