Electrostatic Forces & The Electrical Double Layer

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6 van der Waals 5 Electrostatic Steric 4 Depletion Hydrophobic 3 Solvation 2 Repulsive Forces 1 (Above X-axis) 0 -1 Attractive Forces

Interaction Force/Radius (mN/m) (Below X-axis) -2 0 1020304050 Separation Distance (nm) Electrostatic Forces & The Electrical Double Layer

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 Electrostatic Forces & The Electrical Double Layer

1) Sources of interfacial charge 2) Electrostatic theory: The electrical double layer 3) Electro-kinetic Phenomena 4) Electrostatic forces SOURCES OF INTERFACIAL CHARGE • Immersion of some materials in an electrolyte solution. Two mechanisms can operate.

(1) Direct Ionization of surface groups.

HH H O O - O H OH M M M + H2O OO OO O O O O O

(2) Specific ion adsorption OH

M OO +++ O SURFACE CHARGE GENERATION (cont.)

(3) Differential ion solubility Some ionic crystals have a slight imbalance in number of lattice cations or anions on

surface, eg. AgI, BaSO4, CaF2, NaCl, KCl

(4) Substitution of surface ions HO O O O OH Si Al Si Si eg. lattice substitution in kaolin HO O O O OH ELECTRICAL DOUBLE LAYERS

+ x - SOLVENT MOLECULES - - - + - - + - + - - + - - - Ψ0 COUNTER IONS 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) x - + - + + - - - - - + - + + - - - - - Ψ0 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) Model Gouy/Chapman diffuse double layer and layer of adsorbed charge. Linear decay until the Stern plane. x - Difusion layer - - - + Ψζ + - + - - - - + - + + - - - - - + - - Ψ0- Stern Plane Shear Plane Gouy Plane Bulk Solution - Stern (1924) / Grahame (1947) Model

In 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. x - Difusion layer - - - + + - + Ψζ - + - - - + - + + - + - - + - - - Ψ0 - OHP Shear Plane Gouy Plane Bulk Solution POISSON-BOLTZMANN DISTRIBUTION 1st Maxwell law (Gauss law): “The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity” → → ρ(r) ∇ ⋅ E = Definitions ε 0ε r E: Electric filed Electric field is the differential of the electric potential E→(r) = −∇ ⋅ Ψ→ ()r Ψ: Electric potential ρ: Charge density Combining the two equations we get: E : Energy of the ion ρ(r) Q ∇ 2 ⋅ Ψ()r = − x, r : Distance ε 0ε r Boltzmann ion distribution Which for one dimension becomes: E eZΨ − Q − 2 kT kT d Ψ(x) ρ()x ρ()x = ρ0e = n0Zee 2 = − dx εε 0 Assuming Boltzmann ion distribution: d 2Ψ(x) ρ()x 1 n Zen Z nee−ZieΨ kT ()e−ZΨe kT eZΨe kT 2 = − = − ∑ i = − − dx εε 0 εε 0 i εε 0 POISSON-BOLTZMANN DISTRIBUTION 2 d ψ 2Zen ⎛ Zeψ ⎞ 2 = sinh⎜ ⎟ dx εrε0 ⎝ kT ⎠ Z = electrolyte valence, e = elementary charge (C) n = electrolyte concentration(#/m3)

εr = dielectric constant of medium ε0 = 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 DISTRIBUTION Debye-Hückel approximation ZeΨ For << 1 then: kT 2 2 d ψ 2Zen ⎛ Zeψ ⎞ 2Zen Zeψ 2n(Ze) 2 2 = sinh⎜ ⎟ ≈ = Ψ()x = κ Ψ()x dx ε rε 0 ⎝ kT ⎠ ε rε 0 kT ε rε 0kT

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

−κx Ψ(x)= Ψ0e Debye-Hückel parameter (κ) describes the decay length

2n()Ze 2 κ = ε rε 0kT DOUBLE LAYER FOR MULTIVALENT ELECTROLYTE: DEBYE LENGTH Debye-Hückel parameter (κ) describes the decay length 1/ 2 ⎛ e 2 n ⎞ ⎜ 2 ⎟ κ = ⎜ ∑Ci Z i ⎟ ⎝ ε rε 0 kT i=1 ⎠ Zi = electrolyte valence e = elementary charge (C) 3 Ci = ion concentration (#/m ) n = number of ions

εr = dielectric constant of medium ε0 = permittivity of a vacuum (F/m) k = Boltzmann constant (J/K) T = temperature (K)

κ-1 (Debye length) has units of length POISSON-BOLTZMANN DISTRIBUTION Exact Solution For 0.001 M 1-1 electrolyte Surface Potential (mV) Surface Potential Surface Potential (mV)

ZeΨ ZeΨ <1 >1 kT kT DEBYE LENGTH AND VALENCY

100 90 1-1 electrolyte

Debye Length 2-2 electrolyte 80 3-3 electrolyte 70 60 50 κ

-1 40 , (nm) 30 20 10 0 10-5 10-4 10-3 10-2 10-1 100 101 Electrolyte Concentration (M)

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

Point of Zero Charge (PZC) - pH at which surface potential = 0 Isoelectric Point (IEP) - pH at which zeta potential = 0 Question: 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 --

50 40 Increasing I.S.

Zeta Potential(mV) 30 20 10 0 -10 -20 -30

-40 Alumina -50 1234567891011 pH SPECIFIC ADSORPTION •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 + +

+ + + + + + + + +2 + + + + ----- SPECIFIC ADSORPTION

Ca2+

3- PO4

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

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 MEASUREMENT Electrophoresis Smoluchowski Formula (1921) assumed κa >> 1 κ = Debye parameter a = particle radius - electrical double layer thickness much smaller than particle ε ε ζ v = r 0 Ε v = velocity, η εr = media dielectric constant ε0 = permittivity of free space εrε0ζ µΕ = ζ = zeta potential, E = electric field η η = medium viscosity

µE = electrophoretic mobility ZETA POTENTIAL MEASUREMENT Electrophoresis Henry Formula (1931) expanded for arbitrary κa, assumed E field does not alter surface charge

-low σ0

2εrε0ζ v = velocity v = f1(κa)Ε 3η εr = media dielectric constant 2ε ε ζ ε = permittivity of free space µ = r 0 f (κa) 0 Ε 3η 1 ζ = zeta potential η = medium viscosity E = electric field

µE = electrophoretic mobility

Hunter, Foundations of Colloid Science, p. 560 ZETA POTENTIAL MEASUREMENT Streaming Potential

exp(Zeζ / 2kT) κ = Debye parameter << 1 a = particle radius κa ζ = zeta potential ε ε ζ ∆E = r 0 ∆p ∆Ε = Potential over capillary (V) ηKE εr = media dielectric constant ε0 = permittivity of free space (F/m) η = medium viscosity (Pa·s)

KE= solution conductivity (S/m) ∆p = pressure drop across capillary (Pa) Combined Effects of van der Waals and Electrostatic Forces DLVO Theory

DLVO – 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: AR 64πkTRnγ 2 U (x) = − + exp()−κx 12x κ 2 Van der Waals Electrostatics (Attractive force) (Repulsive force) DLVO Theory

AR 64πkTRnγ 2 U (x) = − + exp()−κx 12x κ 2

⎛ zeψ ⎞ A = Hamakar’s constant γ = tanh⎜ ⎟ ⎝ 4kT ⎠ R = Radius of particle z = valency of ion x = Distance of Separation e = Charge of electron k = Boltzmann’s constant Ψ = Surface potential T = Temperature n = bulk ion concentration κ= Debye parameter DLVO Theory

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

For short distances of separation between particles DLVO Theory

Hard Sphere Repulsion (< 0.5 nm) No Salt added

J/m Energy Barrier

x (distance) Secondary Minimum (Flocculation)

Primary Minimum (Coagulation) Discussion: Flocculation vs. Coagulation The DLVO theory defines formally (and distinctly), the often inter-used terms flocculation and coagulation Flocculation: • 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 of separation upon overcoming the energy barrier • The energy minimum is deep (strong attractions) • Once coagulated, particle separation is almost impossible Effect of Salt Hard Sphere Repulsion (< 0.5 nm) No Salt added Upon Salt addition J/m Energy Barrier

x (distance) Secondary Minimum (Flocculation)

Primary Minimum (Coagulation) 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

AR 64πkTRnγ 2 − + exp()−κH = 0 12H κ 2 H: Distance of separation at critical salt concentration At critical salt concentration, κH = 1. n: Concentration Upon simplification, we get: Z: Valence 1 nα Schultz – Hardy Rule: z 6 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 pHpH andand SaltSalt ConcentrationConcentration EffectEffect

Stability diagram for

Si3N4(M11) particles as produced from Agglomerate calculations Dispersion (IEP 4.4) assuming 90% probability of

Dispersion 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.