Charged interfacesThe high dielectric constant of water makes it a very good solvent for ions,with the result that most surfaces in aqueous solutions are charged!The mechanisms causing charging are mainly these:

• Ionization of acids or bases on the surfaceDissociation of carboxylic acidsProtonation of amines

• Adsorption of ions from the solutionNegative ions (anions) typically adsorb to neutral interfacesto a greater extent than positive (cations).Aqueous interfaces against air or oils are often negativelycharged due to preferential adsorption of OH–-ions.

• Asymmetric dissociation from the surface.An AgI crystal in water becomes negatively charged in watersince more Ag+ ions than I- ions dissolve into the solvent.

The electrostatic double layer

The surface charge creates an excess of counter-ions nearthe surface; more strongly bound the nearer the surface.

Co-ions: ions of the same charge as the surface chargesCounter-ions: ions of opposite charge.

Counterion compensation models

Surface charges are compensated by adsorption of counterions.

+

_

_

_

_

_

__

+++

+++

+

Helmholtz

Counterions bind directlyto the surface andneutralize all charge.

+

_

_

_

_

_

__

++

+

+++

+

(Electricaldouble layer)

Gouy-Chapman

Takes thermal motionof the ions into account,which drives themaway from the surface.

Diffuselayer

Compact layer, ”Stern layer”,

~ 5 Å

+

_

_

_

_

_

__ +

++

++

+

+

+

Diffuselayer

Stern

Taking ion sizes and hydration into account, the layer is split in twocounterion layers.

Potential variation from a

charged surface

Ion distribution from the Stern plane,ψ = -30 mV, 1 mM NaCl

−= −+

xekT

zenn κψ 00 1

+= −−

xekT

zenn κψ 00 1

1/κ is the Debye-length; the distance from the surface where the potential has decreased to 1/e of the surface potential.

The Poisson-Boltzmann equation

Determines the electric potential for a given spatial charge distribution

Assumptions:

Ions are point charges, and do not interact with each other(but only with the averaged ionic background).

The Poisson–Boltzmann equation is a good approximation undermost physiological conditions, especially for monovalent ions andfor surface potentials which are not too large (< 50 mV).

( )2

0

2

0

( ) 1 iz e x

kTi i

r

d xz en e

dx

ψψε ε

−= −

( )rρ

~

Analytical solutions to the

PB-equationAssuming a symmetric electrolyte,

the Poisson-Boltzmann equation can be written

iz z z z+ −= = =

2

0

2

2sinh

n zed ze

dx kT

ψ ψε

=

Multiply both sides by and then integrate twice over x from ∞ to x = d

(where the diffuse layer starts), and let

Using the boundary conditions

we obtain

2d

dx

ψ

( ) 0xxψ →∞ = 0x

d

dx

ψ→∞

=

( ) dx dψ ψ= =

( )tanh tanh

4 4

x ddzezee

kT kT

κψψ − −=1 tanh

2 4( ) ln

1 tanh4

x d

x d

zee

kT kTxzeze

ekT

κ

κ

ψ

ψ ψ

−

−

+ =

−

Check: for small potentials … Debye-Hückel again!( )tanh

x d

dx x e κψ ψ − −≈ ≈

The Debye-Hückel approximation

If the electrical energy is small relative to the thermal energy, i.e.

we can use the Taylor series to simplify the PB equation:

The equation has solutions

iz e kTψ <2 3

1 ...2! 3!

x x xe x− = − + − +

0 0 2 2 0...

iz e

kTi i i i i iz en e z en z e n

kT

ψ ψ−≈ − +

2 2 0

0

i i

r

z e n

kTκ

ε ε= 1

κ

2

2

2

d

dx

ψ κ ψ= ( )x

dx e κψ ψ −=

= 0 due to electroneutrality of the solution

The Debye-Hückelapproximation, or the’linear PB equation’

[m-1] is the Debye-Hückel parameter, and the Debye length.

2 2 02

2 2 0 2

2

0 0

1 i i

i i

r r

z e ndz e n

dx kT kT

ψ ψ ψ κ ψε ε ε ε

≈ − − = =

Solutions to Poisson-

Boltzmann’s equation

ψ = -100 mV

ψ = -50 mV

Valency is very important for

ion adsorption!Adsorption of ions onto an SiO2-surface

Li+ Ca2+ Al3+

Charge reversal byovercompensation!

Langmuir 33, 10473 (2017), 10.1021/acs.langmuir.7b02487

Different surfactants Cationic surfactant at different concentrations

Surface properties - including the charge - canbe tuned by adsorption to the interface!

ζ-potentialThe surface potential cannot be determined experimentally!

Flow

Slip plane

Immobile ions, including hydration

Using different electrokinetic methods atleast the ζ-potential can be determined!

- Control the electric field parallel to thesurface, and measure the velocity of theions in the double layer (electroosmosis).

- Create a flow over the surface, andmeasure the current that the ions in thesolution (double layer) cause (streamingcurrent).

- ...many more methods and examples!ζ-potential

“True” surface potential

Electrokinetic phenomena

Potential Stationary wall Particles in motion

Applied ElectroosmosisCounterions in the diffuse layer migrate under an applied field,

inducing a flow as the liquid is dragged along.

ElectrophoresisParticles migrate in an applied field.

The velocity of each particle is determined by the field strength, solution viscosity, particle charge,

shape and size.

Induced Streaming potentialExternal pressure causes liquid to flow through a capillary. Ions in the diffuse

layer follow, creating a potential difference.

Sedimentation potentialCounterions sediment slower than

large particles, forming a dipole around every particle. The sum of all these dipoles results in a measurable potential difference between the top

and bottom of the suspension.

Electrophoresis

Particles migrate under the influence ofan applied field. In the Smoluchowskimodel, the mobility is independent ofshape and concentration:

Counterions sediment slower than large particles, forming a dipole at every particle. The sum of all these dipoles results in a measurable potential difference between the top and bottom of the suspension.

Sedimentation potential

--

--- -

-

++

+

+

+

+

+

+

d_

v_

Sedimentation under gravity or centrifugationEs

0 0( )

s

r

E

g

ηλζε ε ρ ρ

=−

0re D

ε ε ζµη

=D = 1 for ’thin’ double layers; κr < 1

D = 2/3 for ’thick’ double layers; κr > 1

--

---

- - ++

+

+

+

+

+

+

Applied electric field

Electrostaticforce

Friction force(electrophoreticretardation)

Science 344, 1138 (2014). DOI: 10.1126/science.1253793

Indicates re-orientationof interfacial water molecules