Electrostatic Forces & The Electrical Double...

Electrostatic Forces &The Electrical Double Layer

Dry Clay Swollen Clay

Repulsive electrostatics control swelling of clays in water

LiquidLiquid--Solid Interface; ColloidsSolid Interface; Colloids

Separation techniques such as :Separation techniques such as :column chromatography, HPLC, Paper column chromatography, HPLC, Paper Chromatography, TLCChromatography, TLCThey are examples of the adsorption of They are examples of the adsorption of solutes at the liquid solid interface.solutes at the liquid solid interface.Liquid solid interfaces can be found Liquid solid interfaces can be found everywhere and in any form and size, from everywhere and in any form and size, from electrode surfaces to ship hulls.electrode surfaces to ship hulls.


Not so obvious examples for a solidNot so obvious examples for a solid--liquid liquid interface is a colloid particle.interface is a colloid particle.Colloids are widely spread in daily use:Colloids are widely spread in daily use:PaintPaintBloodBloodAir pollutionAir pollution


A very sensitive method to measure the A very sensitive method to measure the amount of adsorbed material is by using a amount of adsorbed material is by using a QCM (quartz crystal microbalance)QCM (quartz crystal microbalance)


Adsorption at low solute concentration:Adsorption at low solute concentration:These isotherms can be either fitted by the These isotherms can be either fitted by the langmuir isothermlangmuir isothermOr the freundlich isothermOr the freundlich isotherm


Stearic acid is adsorbed onto carbon black Stearic acid is adsorbed onto carbon black differently in different solvents:differently in different solvents:


Different chain lengthes will show Different chain lengthes will show differences in adsorption behaviour:differences in adsorption behaviour:


TraubeTraube´́s rule:s rule:


Composite adsorption isotherms:Composite adsorption isotherms:

Intermolecular Forces

Separation Distance (nm)0 10 20 30 40 50

Inte

ract

ion

Forc

e/R

adiu

s (m

N/m

)

-2

-1

0

1

2

3

4

5

6van der WaalsElectrostaticStericDepletionHydrophobicSolvation

Repulsive Forces (Above X-axis)

Attractive Forces (Below X-axis)


Flagella

E-Coli demonstrate tumbling & locomotive modes of motion in the cell to align themselves with the cell’s rear portion

Flagella motion is propelled by a molecular motor made of proteins – Influencing proton release through protein is a key molecular approach to prevent E-Coli induced diarrhea

Mingming Wu, Cornell University (animation)Berg, Howard, C. Nature 249: 78-79, 1974.

Electrostatic Forces & van der Waals Forces jointly influence Flocculation / Coagulation

Suspension of Al2O3 at different solution pH

Critical for Water Treatment Processes


1) Sources of interfacial charge2) Electrostatic theory: The electrical double layer3) Electro-kinetic Phenomena4) Electrostatic forces

SOURCES OF INTERFACIAL CHARGE• Immersion of some materials in an electrolyte

solution. Two mechanisms can operate.

(1) Direct Ionization of surface groups.

(2) Specific ion adsorption

M

OH

OO O+ + +

O

M

H HO

M

HO

MO O O OO O OO O

H HO -+ H2O

SURFACE CHARGE GENERATION (cont.)

(3) Differential ion solubilitySome ionic crystals have a slight imbalance in number of lattice cations or anions onsurface, eg. AgI, BaSO4, CaF2, NaCl, KCl

(4) Substitution of surface ionseg. lattice substitution in kaolin

HOSi

HO

O

OAl

O

OSi

O

OSi

OH

OH

ELECTRICAL DOUBLE LAYERS

COUNTER IONS

+

+

+

+

-

-

-

+-

--------

SOLVENT MOLECULES

Ψ0

x

OHP CO IONS

Helmholtz (100+ years ago) proposed that surface charge is balanced by a layer of oppositely charged ions

Gouy-Chapman Model (1910-1913)

+

+ +

++

-

-

-

+-

Ψ0

x---------

Diffusion plane• Assumed Poisson-Boltzmann distribution of ions from surface • ions are point charges• ions do not interact with each other

• Assumed that diffuse layer begins at some distance from the surface

Stern (1924) / Grahame (1947) ModelGouy/Chapman diffuse double layer and layer of adsorbed charge.Linear decay until the Stern plane.

+

+

+

+

+

Shear Plane Gouy Plane

Difusion layer

+

-

-

-

+-

Bulk SolutionStern Plane

-------------

-

-Ψ0

x

Ψζ

Stern (1924) / Grahame (1947) ModelIn different approaches the linear decay is assumed to be until the shear plane, since there is the barrier where the charges considered static. In this courese however we will assume that the decay is linear until the Stern plane.

+

+

+

+

+

Shear Plane Gouy Plane

Difusion layer

+

-

-

-

+-

Bulk SolutionOHP

-----------

-

-

+

+

Ψ0

Ψζ

x

POISSON-BOLTZMANN DISTRIBUTION1st Maxwell law (Gauss law): “The total of the electric flux out of aclosed surface is equal to the charge enclosed divided by the permittivity”

( )r

rEεε

ρ

0

=⋅∇

( )rrE Ψ⋅−∇=)(Electric field is the differential of the electric potential

→ →

→→

Combining the two equations we get:( )

r

rrεε

ρ

0

2 )(−=Ψ⋅∇

Which for one dimension becomes:( ) ( )

02

2

εερ x

dxxd

−=Ψ

Assuming Boltzmann ion distribution:( ) ( ) ( )kTeZkTeZ

n

i

kTeZi eeZenneeZx

dxxd

i ΨΨ−Ψ− −−=−=−=Ψ ∑

0002

2 1εεεεεε

ρ

( ) kTeZ

kTE

ZeenexQ Ψ

−−== 00ρρ

Boltzmann ion distribution

DefinitionsE: Electric filedΨ: Electric potentialρ: Charge densityEQ: Energy of the ionx, r : Distance

POISSON-BOLTZMANN DISTRIBUTION

⎟⎠⎞

⎜⎝⎛ ψ

εε=ψ

kTZesinhZen2

dxd

0r2

2

Z = electrolyte valence, e = elementary charge (C)n = electrolyte concentration(#/m3)εr = dielectric constant of mediumε0 = permittivity of a vacuum (F/m)k = Boltzmann constant (J/K)T = temperature (K)

• Poisson-Boltzmann distribution describes the EDL• Defines potential as a function of distance from a surface

• ions are point charges• ions do not interact with each other

POISSON-BOLTZMANN DISTRIBUTIONDebye-Hückel approximation

For then:

The solution is a simple exponential decay (assuming Ψ(0)=Ψ0 and Ψ(∞)=0):

Debye-Hückel parameter (κ) describes the decay length

ZeΨkT

<<1

Ψ x( )= Ψ0e−κx

( ) ( ) ( )xxkT

ZenkT

ZeZenkT

ZeZen

rrr

Ψ=Ψ=≈⎟⎠⎞

⎜⎝⎛= 2

0

2

002

2 22sinh2 κεε

ψεε

ψεε

ψdxd

( )kT

Zen

r 0

22εε

κ =

DOUBLE LAYER FOR MULTIVALENTELECTROLYTE: DEBYE LENGTH

Debye-Hückel parameter (κ) describes the decay length

Zi = electrolyte valencee = elementary charge (C)Ci = ion concentration (#/m3)n = number of ionsεr = dielectric constant of mediumε0 = permittivity of a vacuum (F/m)k = Boltzmann constant (J/K)T = temperature (K)

κ-1 (Debye length) has units of length

2/1

1

2

0

2

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

=

n

iii

r

ZCkT

eεε

κ

POISSON-BOLTZMANN DISTRIBUTIONExact Solution

For 0.001 M 1-1 electrolyte

Surf

ace

Pote

ntia

l (m

V)

Surf

ace

Pote

ntia

l (m

V)

ZeΨkT

> 1ZeΨkT

< 1

Electrolyte Concentration (M)10-5 10-4 10-3 10-2 10-1 100 101

Debye Length κ

-1, (nm)

0102030405060708090

1001-1 electrolyte2-2 electrolyte3-3 electrolyte

DEBYE LENGTH AND VALENCY

• Ions of higher valence are more effective in screening surface charge.

ZETA POTENTIAL

Point of Zero Charge (PZC) - pH at which surface potential = 0Isoelectric Point (IEP) - pH at which zeta potential = 0Question: What will happen to a mixed suspension of Alumina and Si3N4 particles in water at pH 4, 7 and 9?

ZETA POTENTIAL-- Effect of Ionic Strength --

p H1 2 3 4 5 6 7 8 9 1 0 1 1

Zeta Potential (mV

)

- 5 0- 4 0- 3 0- 2 0- 1 0

01 02 03 04 05 0

A l u m i n a

IncreasingI.S.

•Free energy decrease upon adsorption greater than predicted by electrostatics

• Have the ability to shift the isoelectric point v and reverse zeta potential

• Multivalent ions: Ca+2, Mg+2, La+3, hexametaphosphate, sodium silicate

• Self-assembling organic molecules: surfactants, polyelectrolytes

SPECIFIC ADSORPTION

+2+

- - - - -

+

+

+

+

+

+ +

+

+

++

+ +

SPECIFIC ADSORPTION

Multivalent cations shift IEP to right (calcite supernatant)Multivalent anions shift IEP to left (apatite supernatant)

pH

PO43-

Ca2+

Amankonah and Somasundaran, Colloids and Surfaces, 15, 335 (1985).

ELECTROKINETIC PHENOMENA

• Electrophoresis - Movement of particle in a stationary fluid by an applied electric field.

• Electro-osmosis - Movement of liquid past a surface by an applied electric field

• Streaming Potential - Creation of an electric field as a liquid moves past a stationary charged surface

• Sedimentation Potential - Creation of an electric field when a charged particle moves relative to stationary fluid

ZETA POTENTIAL MEASUREMENT• Electrophoresis - ζ determined by the rate of diffusion (electrophoretic mobility) of a charged particle in an applied DC electric field.

• PCS - ζ determined by diffusion of particles as measured by photon correlation spectroscopy (PCS) in applied field

• Acoustophoresis - ζ determined by the potential created by a particle vibrating in its double layer due to an acoustic wave

• Streaming Potential - ζ determined by measuring the potential created as a fluid moves past macroscopic surfaces or a porous plug

ZETA POTENTIAL MEASUREMENTElectrophoresis

Smoluchowski Formula (1921)assumed κa >> 1 κ = Debye parameter

a = particle radius- electrical double layer thickness much smaller than

particle

v = velocity, εr = media dielectric constantε0 = permittivity of free spaceζ = zeta potential, E = electric fieldη = medium viscosityµE = electrophoretic mobility

ηζεε=µ

Εηζεε=

Ε0r

0rv

ZETA POTENTIAL MEASUREMENTElectrophoresis

Henry Formula (1931)expanded for arbitrary κa, assumed E field does not alter surface charge

- low σ0v = velocityεr = media dielectric constantε0 = permittivity of free spaceζ = zeta potentialη = medium viscosity

E = electric fieldµE = electrophoretic mobility

)a(f3

2

)a(f3

2v

10r

10r

κηζεε=µ

Εκηζεε=

Ε

Hunter, Foundations of Colloid Science, p. 560

ZETA POTENTIAL MEASUREMENTStreaming Potential

κ = Debye parametera = particle radiusζ = zeta potential

∆Ε = Potential over capillary (V)εr = media dielectric constantε0 = permittivity of free space (F/m)η = medium viscosity (Pa·s)KE= solution conductivity (S/m)∆p = pressure drop across capillary (Pa)

pK

EE

0r ∆η

ζεε=∆

1a

)kT2/Zeexp( <<κζ

Combined Effects of van der Waals and Electrostatic Forces

DLVO TheoryDLVO – Derjaguin, Landau, Verwey and Overbeek

Based on the sum of van der Waals attractive potential and a screened electrostatic repulsion potential arising between the “double layer potential” screened by ions in solution. The total interaction energy U of the system is:

( )xkTRnx

ARxU κκ

γπ−+−= exp64

12)( 2

2

Van der Waals (Attractive force)

Electrostatics (Repulsive force)

DLVO Theory

( )xkTRnx

ARxU κκ

γπ−+−= exp64

12)( 2

2

⎟⎠⎞

⎜⎝⎛=

kTze4

tanh ψγA = Hamakar’s constant

R = Radius of particle

x = Distance of Separation

k = Boltzmann’s constant

T = Temperature

n = bulk ion concentration

κ= Debye parameter

z = valency of ion

e = Charge of electron

Ψ = Surface potential

DLVO Theory

For short distances of separation between particles

100 nm Alumina, 0.01 M NaCl, Ψzeta=-20 mV

DLVO Theory

Primary Minimum

Secondary Minimum

Energy Barrier

Hard Sphere Repulsion (< 0.5 nm)

J/m

x (distance)

(Flocculation)

(Coagulation)

No Salt added

Discussion: Flocculation vs. CoagulationThe DLVO theory defines formally (and distinctly), the often

inter-used terms flocculation and coagulationFlocculation:

• Corresponds to the secondary energy minimum at large distances of separation

• The energy minimum is shallow (weak attractions, 1-2 kT units)

• Attraction forces may be overcome by simple shaking Coagulation:

• Corresponds to the primary energy minimum at short distances ofseparation upon overcoming the energy barrier

• The energy minimum is deep (strong attractions)

• Once coagulated, particle separation is almost impossible

Primary Minimum

Secondary Minimum

Energy Barrier

Hard Sphere Repulsion (< 0.5 nm)

J/m

x (distance)

Effect of Salt

(Flocculation)

(Coagulation)

No Salt addedUpon Salt addition

Addition of salt reduces the energy barrier of repulsion. How?

Secondary Minimum: Real System

100 nm Alumina, Ψzeta=-30 mV

Discussion on the Effect of Salt

The salt reduces the EDL thickness by charge screening

Reduces the energy barrier (may induce coagulation)

Also increases the distance at which secondary minimum occurs (aids flocculation)

Since increased salt concentration decreases κ-1 (or decreases electrostatics), at the Critical Salt Concentration U(x) = 0

Effect of Salt Concentration and Type

( ) 0exp6412 2

2

=−+− HkTRnH

AR κκ

γπ

H: Distance of separation at critical salt concentrationAt critical salt concentration, κH = 1.

Upon simplification, we get:

6

1z

nα Schultz – Hardy Rule:

n: ConcentrationZ: Valence

Concentration to induce rapid coagulation varies inversely with charge on cation

Effect of Salt Concentration and Type

For As2S3 sol, KCl: MgCl2: AlCl3 required to induce flocculation and coagulation varies by a simple proportion 1: 0.014: 0.0018

The DLVO theory thus explains why alum (AlCl3) and polymers are effective (functionality and cost wise) to induce flocculation and coagulation

pH and Salt Concentration EffectpH and Salt Concentration Effect

Dispersion

Dispersion

Agglomerate

Stability diagram for Si3N4(M11) particles as produced from calculations (IEP 4.4) assuming 90% probability of coagulation for solid formation.

REMARKS-- hydrophobic and solvation forces --

• Due to the number of fitting parameters (ψ0, A132, spring constant, I.S.) and uncertainty in force laws (C.C. vs C.P., retardation) hydrophobic forces often invoked to explain differences between theory and experiment.

• Because hydrophobic forces involve the structure of the solvent, the number of molecules to be considered in the interaction is large and computer simulation has only begun to approach this problem.

• Widely accepted phenomenological models of hydrophobic forces still need to be developed.

Electrostatic Forces & The Electrical Double...

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