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…