CCoollllooiiddaal l SSttaabbiilliittyy Introduction Interparticle Repulsion Interparticle Attraction

Hamaker constantMeasurement techniquesSolvent Effects

Electrostatic StabilisationCritical Coagulation

Concentration Kinetics of Coagulation

IntroductionIntroductionColloid stability: ability of a colloidal dispersion to avoid coagulation.

KineticKinetic vs thermodynamicthermodynamic parameters.

Two kinds of induced stability:

(1) ElectrostaticElectrostatic induced stability:• (like) charges, repel• van der Waal’s forces, attract

V

+verepulsive

stable

-veattractiveunstable

0

H=particle separation

(2) Polymer induced or Steric StabilitySteric Stability:

Stability is a result of a steric effect,where the two polymer layers on interacting particles overlap and

repel one another.

Interparticle RepulsionInterparticle Repulsion

Goal is to calculate repulsive potentialVR between two particles

H

d

Two possibilities for :

Due to adsorption of charged species remains constant, decreases

Due to intrinsic charge on the particles constrained to remain constant,

increases as overlap increases

o

Derjaguin ApproximationDerjaguin Approximation

•Approximate sphere by a set of “rings”•Assumes:

•Constant potential case.•Sphere radius much larger than

double layer thickness, a>10.•NO assumptions on potentials.

Ha1

dH

a2

1

1

2kTze

2kTze

d

d

e

e H

dR

H

R

eaV

ezeaa

TkaaV

2

22

21

21

22

21

2

64

low potentials (D-H approx.)both particles the same.

SummarySummary

Simplest form of repulsive interaction: spherical like particles low potentials large interparticle distances.

As increases, repulsion decreases, destabilisation occurs:

increase in electrolyte concentration increase in counter-ion charge.

Like charged particles stabilise, unlike charges destabilise.

H2dR ea2V

Interparticle AttractionInterparticle Attraction

Van der Waal’s forces: exist for all particlesatom-sized and up.

permanent dipole-permanent dipole KeesomKeesom interaction

permanent dipole-induced dipole DebyeDebye interaction

induced dipole-induced dipole LondonLondon or dispersion interaction ALWAYS PRESENT

always attractive (?) long range (0.2 - 10 nm)

6rV

Form of van der Waal’s InteractionsForm of van der Waal’s Interactions

includes contributions from London,Keesom and Debye forces.

= f(polarizability, dipole moment)

Relative contributions:

CCl4 0 10.7 4.41 0 0 100Ethanol 1.73 5.49 3.4 42.6 9.7 47.6Benzene 0 10.5 4.29 0 0 100Phenol 1.55 11.6 6.48 14.5 8.6 76.9Toluene 0.43 11.8 5.16 0.1 0.9 99Water 1.82 1.44 2.1 84.8 4.5 10.5

Compound % % % Debye x1030 m3 x1077 Jm6Keesom Debye London

(single particle)

Van der Waal’s interactions betweenVan der Waal’s interactions betweentwo particlestwo particles

Must sum over each volume elementof a large particle -- introduces error!

For two spheres close together (H<<a):

2

21

21

)(6

12

MN

A

aaHaAa

V

HAa

V

A

A

A Equal Spheres

Unequal Spheres

Hamaker Constant!

where...

units of Joules

Hamaker constant determined by bothpolarizability and dipole moment ofmaterial in question...

Acetone 4.2Alumina 15.4Gold 45.3Magnesia 10.5Metals 16-45Rubber 8.58polystyrene 7.8-9.8Silver 39.8Toulene 5.4Water 4.35

Material A (x 1020 J)

Means of measuring

determine from and (approximate and not always possible to get values)

Measure using bulk properties:

Surface tension is an obvious one

224 odA

Direct Measurement of forces

This is a difficult thing to do...

Insert Fig. 1.27 here

Solvent EffectsSolvent Effects

Previous results were in vacuum.

Presence of a solvent between particleswill affect the overall Hamaker constant:

3solvent

3solvent

3solvent

3solvent1

1 2

2

AVR

13233312132

132

AAAAA

AAAA initialfinaleffective

...,2/1

221112 etcAAA

22/1

33

2/1

11131

2/1

33

2/1

22

2/1

33

2/1

11132

AAA

AAAAA

Net result:

If particles are the same reduces to...

If particles are the same… Aeff is always positive -- i.e attractive. If A’s are similar, attraction is weak.

If particles are different… Aeff is positive if A33>A11,A22 or A33<

A11,A22 attractive. Aeff is negative if A11<A33<A22 i.e.

repulsive interaction if the solvent Hamaker constant is intermediate to those of the particles.

Electrostatic StabilisationElectrostatic Stabilisation

We may combine the two expressions forthe potential experienced as follows…

HAa

ezeTak

VVV H

ar 1232

22

222

0 20 40 60 80 100120140

-40

-20

0

20

40

Vne

t/kT

H (nm)

= 100 mV= 1x108 m-1

a = 100 nm

A=2x10-20 J

5x10-20 J

1x10-19 J2x10-19 J

Effects of changing AEffects of changing A

Least control, setby system.

Effective over long range.

Effects of changing Effects of changing (i.e. (i.e. ):):

HAa

ezeTak

VVV H

ar 1232

22

222

Much shorter rangeeffect.

More effective at lowvalues of

Experimentally,we measure thezeta potential.

0 20 40 60 80 100120140

0

200

400

= 50 mV

= 75 mV

= 150 mV

= 200 mV

Vne

t/kT

H (nm)A = 2x10-19 J= 1x108 m-1

a = 100 nm

Effects of changing Effects of changing (i.e. electrolyte (i.e. electrolyteconcentration):concentration):

0 20 40 60 80 100120140-40

-20

0

20

40

= 1 x 107m-1

3 x 107m-1

1 x 108m-1

3 x 108m-1

Vne

t/kT

H (nm)

A = 2x10-19 J= 25 mVa = 100 nm

This is the itemwe have mostcontrol over!

Affects potentialsat short distances.

For a 1:1 electro-lyte, the transitionis about 10-2 - 10-3

molar.

HAa

ezeTak

VVV H

ar 1232

22

222

22

A ecz2NkT 1

Critical Coagulation ConcentrationCritical Coagulation ConcentrationThe Schulze-Hardy RuleThe Schulze-Hardy Rule

C.C.C. is fairly ill-defined:

The concentration of electrolyte which is just sufficient to coagulate a dispersion to an arbitrarily chosen extent in an arbitrarily defined time.

0

V

H

At the C.C.C:

dV/dH = 0 at V= 0

012

32

012

32

222

222

22

222

HAa

ezeTak

HAa

ezeTak

H

H

Assuming a symmetrical electrolyte(i.e. z+ = z-):

626

455329980...

zAeNTk

cccA

As becomes large 1 small ze /4kT

Thus:

c.c.c.c.c.c. 1/z 1/z66 at high potentials

c.c.c. c.c.c. 1/z 1/z22 at low potentials

Effect is independent of particle size!Strongly dependent on temperature!

As2S3 (negative) AgI (negative) Al2O3 (positive)

LiCl 58 LiNO3 165 NaCl 43.5NaCl 51 NaNO3 140 KCl 46KCl 49.5 KNO3 136 KNO3 60KNO3 50 RbNO3 126KC2H3O2 110 AgNO3 0.01

CaCl2 0.65 Ca(NO3)2 2.40 K2SO4 0.30MgCl2 0.72 Mg(NO3)2 2.60 K2Cr2O7 0.63MgSO4 0.81 Pb(NO3)2 2.43 K2oxalate 0.69

AlCl3 0.093 Al(NO3)3 0.067 K3[Fe(CN)6] 0.08Al2(SO4)3 0.096 La(NO3)3 0.069Al(NO3)3 0.095 Ce(NO3)3 0.69

Stronger dependency is typical ofadsorption in the Stern layer: softerspecies tend to adsorb better (morepolarizable) so have a slightlystronger effect.

Any potential determining ion willhave a significant effect.

Critical Coagulation ConcentrationsCritical Coagulation Concentrations(mmol/L)

Kinetics of CoagulationKinetics of Coagulation No dispersion is stable thermodynamically. Always a potential well.

Two steps in mechanism:

(1) Colloids approach one another diffusion controlled: perikinetic. externally imposed velocity

gradient: orthokinetic (e.g. sedimentation, stirring, etc.).

(2) Colloids stick to one another (assume probability of unity).

Two forces then controlling approach:

(1) Rapid diffusion controlled.(2) Interaction-force controlled (potential barrier, slows approach).

The Stability RatioThe Stability Ratio

W= Rate of diffusion-controlled collision Rate of interaction-force controlled collision

W = large : particles are relatively stable.

W = 1 : rate unhindered, particles unstable.

Diffusion-controlled (Rapid) Rate:Diffusion-controlled (Rapid) Rate:

RR1

R2

R1+R2

2

2

drNd

DdtdN

Fick’s Second law can now be used:

Which can be used to show that foridentical particles, the collision rate:

DRNZ 16

Since 2 particles are involved, the reactionfollows second order kinetics:

2

2Nk

dtdN

Thus, the rate constant is given by:

34

82

kT

DRk o

•Only binary collisionsoccur (dilute solution).•Neglect solvent flow outof gap.•For second relationshipStokes-Einstein is used.

slow

o

kk

W 2

The stability ratio can thus be given by:

kslow will depend upon the potentialaround the particles.

Can acquire an expression for kslow bymodifying Fick’s second law with an“activation energy”, V(R), where V(R)is the potential barrier previously dicussed.

dRRe

aWa

kTV

2

2

/

2

Assume a (very simple) barrier suchas the following...

V

Vmax

2a -10

particlestouch

Then…

kTVea

W /max

21

Critical Coagulation ConcentrationCritical Coagulation Concentration

Can solve previous simple expressionfor W in terms of Vmax, determined fromwhen dV/dH = 0

For water as dispersion medium

cza

xKW log1006.2log2

29

AgI Particle CoagulationAgI Particle Coagulation

Plot is linear

When log W =0 we are at the CCC, breaks in the curve appear as coagulation occurs at a rapid rate.

Coagulation rates cannot be measured in this system beyond about log W = 4. Corresponds to an energy barrier of about 15 kT.

Can use the slopes to analyze for o, if the particle size is known.