1

d

d

ReferenceParticle

Coagulation of Particles

Aerosol particles collide due to their random motions and coalesce to form largerchains of flocs made up of many particles. The Brownian (thermal) motion of particles,turbulence, presence of a shear field, and external forces such as gravity and electricalforces could cause coagulation.

Coagulation of Monodisperse Spheres

Smoluchowski was the first to develop a model for the coagulation ofmonodispersed spherical aerosols. He considered the reference particle to be fixed asshown in Figure 1. The other particles would then diffuse to the reference particle by theaction of the Brownian motions. The concentration then satisfies the following diffusionequation:

˜¯

ˆÁÁË

Ê

∂

∂+

∂

∂=—=

∂

∂

r

c

r

2

r

cDcD

t

c2

22 (1)

subject to boundary conditions

0c = at dr = , (2)

0cc = at 0t = , or •Ær . (3)

Here, d is the diameter of the particle and 0c is the concentration far away. Equation (2)

assumes sticking of particles without rebound upon contact.

Figure 1. Schematic of Brownian coagulation of monodispersed particles.

Equation (1) may be restated as

2

( ) ( )2

2

r

crD

t

cr

∂

∂=

∂

∂. (4)

The concentration field then is given by

˙˚

˘ÍÎ

Ș¯

ˆÁÁË

Ê -+-=

Dt2

drerf

r

d

r

d1cc 0 . (5)

The flux to surface of the reference particle is given by

dr2 |

r

cDd4JAI =∂

∂p== , (6)

where r

cDJ

∂

∂= is the flux and I is the total flux to the surface of the reference particle

per unit time. Using (5), it follows that

˙˚

˘ÍÎ

È

p+p=

Dt

d1dDc4I . (7)

For large D

dt

2

p>> ,

dDc4I p= . (8)

In reality, the reference particle is not fixed and is diffusing itself. The relativediffusivity of two particles is sum of their diffusivities. That is according to Einstein'sequation

( )2

x

t2

xx2

t2

x

t2

xxD

2jji

2i

2

jiij +-=

-= ,

jiij DDD += , (9)

In the derivation of (9) it is assumed that 0xx ji = , because the motions of the two

particles are independent. Therefore, in a time dt , the reference particle collides with

cdtdDcdt8 b=p (10)

3

particles. Here, b is the collision frequency function. With c particles per unit volume,

there will be 2

c collisions if all the particles collide once. Assuming that the particles

stick to each other upon collision, it follows that

2c2dt

dc b-= , (11)

where

cc3

kT8dD8

m=p=b , (12)

is the collision frequency function of coagulation constant.

Equation (11) may be solved. i.e.,

t2c

1

c

1

0

b=- , (13)

or

h

0

0

0

t

t1

c

2

tc1

cc

+=

b+

= . (14)

Here, ht is the half-value time, which is the time that the concentration becomes half ofits original value.

Coagulation of Many Sizes

Consider an aerosol, which initially consists of particles of different sizes. Theconcentration of the kth size particle then satisfies the following equation:

( ) •

=

-

=-- b-b=

1jjkkj

1k

1jjkjjkj

k cccc2

1

dt

dc, (15)

where the first term on the right hand side is the generation rate of the kth particle bycombination and the second term is the loss of kth particle due to coalescence. Here, kjb

is the collision frequency function (coagulation parameter)for particles with diameters

jd and kd shown in Figure 2 and is given by

4

dj dk

( )( )kjkjkj DDdd2 ++p=b (16)

The corresponding values of ( )kjkj d,db are listed in Table 1.

Figure 2. Brownian coagulation of dissimilar particles.

Table 1. Collision frequency function ( ) s/cm10d,d 310kj ¥b .

)m(d2 m

)m(d1 m

3102 -¥ 3104 -¥ .01 .02 .04 0.1 0.2 0.4 1 2 4 10 20

0.002 2.250.004 3.75 3.01 15 7.5 4.5.02 45 20 7.5 6.04 150 55 17.5 8.5 5.50.1 800 275 60 20 7.5 3.60.2 2500 700 135 40 12.5 4 2.60.4 6500 2050 300 85 23.5 5.5 2.65 21 18500 4750 800 210 57.5 12 4.5 2.35 1.72 38500 10000 1600 420 115 22.5 8 3.55 1.85 1.64 80000 20000 3300 850 225 45 15 6 2.6 1.75 1.510 200000 50000 8000 2150 550 110 36 14 5.15 2.8 1.85 1.520 400000 100000 15000 4250 1100 215 70 27 8.5 4.8 2.75 1.65 1.5

Effect of Particle Force Field

The particle force field will modify the collision frequency. It may be shown thatthe collision distribution function is given as (Friedlander, 2000)

5

( )( )W

DDdd2 jijiij

++p=b (17)

where

( ) ( )

Ú•

+

f+=

2

dd2

kT

xji

2ix

dxe

2

ddW . (18)

Here, ( )rf is the potential energy of the central force per unit mass with

( )dr

drF

f-= . (19)

Van der Waals Force

The potential energy for the attractive van der Waals force is given by

( )

˙˙˙˙˙

˚

˘

ÍÍÍÍÍ

Î

È

˜¯

ˆÁÁË

Ê --

˜¯

ˆÁÁË

Ê +-

+

˜¯

ˆÁÁË

Ê --

+

˜¯

ˆÁÁË

Ê +-

-=f2

ji2

2

ji2

2

ji2

ji

2

ji2

ji

2

ddr

2

ddr

ln

2

ddr

2

dd

2

ddr

2

dd

6

Ar , (20)

where r is the distance between the centers of the sphere and A is the Hamaker constant.

For monosize particles,

ÔÔ˛

ÔÔ˝

¸

ÔÔÓ

ÔÔÌ

Ï

˜¯

ˆÁÁË

Ê-+

-+˜

¯

ˆÁË

Ê-=f2

2

22

2

2

r

d1ln

dr2

d

r

d

2

1

6

A (21)

and

( )Ú ˛

˝¸

ÓÌÏ-=

1

0

dxkT6

xAfexpW , (22)

where

6

( ) ( ) ( )˚

˘ÍÎ

È-+

-+= 2

2

22

x1lnx12

x

2

xxf . (23)

The increase of collision frequency with kT

A

p is shown in the figure:

(Diagram Here)

Coulomb Forces

For charged particles, the Coulomb force potential is given as

r

ezz 2ji

e=f , (24)

where iz is the number of charges, e is the electronic charge, and e is the dielectricconstant of the medium.

The expression for W then is given by

( )1ey

1W y -= , (25)

where

( )ji

2ji

ddkT

ezz2y

+e= (26)

is the ratio of the electrostatic potential energy at contact to the thermal energy kT .

For 0y = , 1W = . For charges of opposite sign, y is negative and 1W0 << andhence collision rate increases. When charges have the same sign, 0y > and 1W > andthe collision rate decreases.

7

For large numbers of charged particles, the coulomb potential must be modified toaccount for Debye shielding effects. Accordingly, equation (26) is replaced by

( )

( )ji

2

dd2

ji

ddkT

eezz1y

ji

+e=

+k-

. (27)

Shear Flows

The collision frequency function due to shear flow is given as

( )dy

dUdd

6

1 3jiij +=b . (28)

For ddd ji == ,

dy

dud

3

4 3=b . (29)

Turbulence

Saffman and Turner suggested that

2

13

jiij 2

dd3.1 ˜

¯

ˆÁË

Êne

˜¯

ˆÁÁË

Ê +=b , (30)

where e is the dissipation rate and n is the kinematic viscosity. For pipe flows (coreregion),

32

3

V2

f

R

2˜¯

ˆÁË

Ê=e , (31)

where V is the mean velocity, f is the friction factor, and R is the pipe radius. Averagedissipation rate for the pipe is given as

R

V

4

f 3

=e . (32)

8

Differential Motion by Gravitational Force

For particles falling according to Stokes settling velocity, the collision frequencyfunction is given by

( )( ) ( )

( )Ô

Ô˝¸

ÔÓ

ÔÌÏ

a+a-˙˚

˘ÍÎ

È

a++

a+-a+

m

pr=b 22

3

24j

P

ij 112

1

12

311d

g (33)

where

1d

d

j

i <=a

(Diagram Here)

9

Electrical Double Layer Interactions for Colloidal Systems

Most colloidal particles carry electrostatic charges. Since the dispersion as awhole is neutral, there must be an excess of ions of opposite charge in the solution. Theseexcess ions are found near the surface of suspended colloidal particles and form what iscalled a "diffused electrical double layer." The double layer has a profound effect oninteractions of colloidal particles.

Ion Distribution Near a Colloidal Particle

Consider the diffusion of ions near a charged surface. The convective diffusionequation for the number concentration, in , of ion i is given as

( ) ( )iii nnD iu⋅—=—⋅— , (1)

where iD is the diffusivity and iu is the velocity due to the electric field. Force balanceof a single ion gives

eEzc

ud3i

c

ii =pm

, (2)

where m is the viscosity, id is the diameter, cc is the slip correction, e is the unit

electric charge, iz is the valency of ion i and E is the electric field strength given as

y-—=E (3)

and y is the electric field potential.

Equation (2) leads to

kT

Dez

c

d3ez ii

c

i

i EEui =

pm= (4)

where

c

ii

c

d3kT

Dpm

= (5)

is used. Using (3) and (4) in (1), we find

10

( ) ˜¯

ˆÁË

Ê y—⋅-—=—⋅—

kT

eDznnD iii

ii (6)

or

kT

ez

n

nnln i

i

ii

y—-=

—=— . (7)

Integrating (7), one finds

˛˝¸

ÓÌÏ y—

-=kT

ezexpnn i

0ii , (8)

the well-known Boltzmann distribution. Note that the potential ( )xy=y .

Using Coulomb's law and equation (3), we find

e

r-=y— e2 (9)

where e is the permittivity (dielectric constant times permittivity of free space), and er

is the charge density given as

Â=ri

ii ezn . (10)

Using (10) in (9), the Poisson-Baltzamann equation follows, i.e.

Ây

-

e-=y— kT

ez

i0i2

i

ezne

. (11)

Solution of (11) gives y and in . Equation (11) is restricted to low electrolyteconcentrations, since ions are treated as point charges.

Debye-Huckel Approximation

For low potential kTezi <<y and

kT

ez1e ikT

ezi y-ª

y-

(12)

Using (12) in (11) and noting the neutrality condition of the burk suspension,

11

0ezni

i0i =Â , (13)

it follows that

yk=y— 22 (14)

where

Âe=k 2

i0i

2

znkT

e (15)

is the Debye-Huckel parameter.

The solution of (14) for double layer adjacent to a flat plate is given as

x0e k-y=y . (16)

For double layer near a spherical particle of radius a ,

( )

a

re ar

0

-k-

y=y . (17)

In (16) and (17), 0y is the potential at the surface, and y decays with distance from the

surface. The characteristic length is 1-k (Debye length) double layer thickness. Theconcentration of electrolyte is related to 0in by

0iA0i cN1000n = , (18)

where 0ic is the bulk concentration of ions of species i (in mol/L) and 23A 1002.6N ¥=

is Avogadro's number. Typically, 1-k varies from less than 1 nm to about 100 nm.

Gouy-Chapman Theory

An exact solution of Poisson-Baltzmann equation for symmetric electrolytes forwhich

zzz =-= -+ (19)

and

12

00 nn -+ = (20)

was found by Gouy-Chapman for double layer near a place. In this case, equation (11)reduces to

˜¯

ˆÁË

Ê y

e=

y

kT

zesinh

zen2

dx

d 02

2

. (21)

The solution to (21) is

x0

x0

e1

e1ln

ze

kT2k-

k-

g-

g+=y , (22)

where

kT

zetanh 0

0

y=g . (23)

The surface charge density, s , may be related to surface potential 0y , i.e.

˜¯

ˆÁË

Ê yek=

ye-=r-=s =

•

Ú kT

zesinh

ze

kT2|

dx

ddy B

0x

0

, (24)

for y given by (22) and

0eky=s (25)

for y given by (16). This means that the double layer may be considered as a capacitor

consisting of two plates a distance of 1-k apart.

For spherical particles, the surface charge density is

0a

a1y

k+e=s , (26)

and the total surface charge of the sphere becomes

( ) 0e a1a4Q yk+pe= . (27)

13

Electrostatic Double Layer Interactions

Colloidal particles carrying the same surface charges cause double layerrepulsion.

Constant Potential Interactions

The interaction energy for two spheres of radii 1a and 2a and potentials 01y and

02y at infinite separation is given as

( )( )

( )( )

( )˛˝¸

ÓÌÏ

-+k--

k-+

y+y

yy

+

y+ye= -y kh2

202

201

0201

21

020121 e1lnhexp1

hexp1ln

2

aa4

aaV (28)

where e is the dielectric constant of the medium, h is the distance of minimum approachand 1-k is the double layer thickness. Equation (28) holds for mV250 <y and

11 a<<k- , 2a .

For sphere-plate the interaction energy is twice that of sphere-sphere, i.e.,

y-

y- = ssps V2|V . (29)

The corresponding force (repulsion) is

( ) ˙˚

˘ÍÎ

È-

y+y

yy

-

ky+y

e=-= k-

k-

k-y-y

-h

202

201

0201h

h202

201

psps e

2

e1

e

2

a

dh

dVF (32)

For materials with 00201 y=y=y , equation (30) becomes

h

h20

ps e1

xeaF

k-

k-y- +

ye= . (31)

Constant Charge Interactions

The repulsion interaction energy of two colloidal particles with constant surfacecharge density is given as

( )( )

[ ]h2

21

202

20121 e1ln

aa2

aaVV k-

ys -

+

y+ye-= (32)

The corresponding sphere-plate force is

14

( ) ˙˚

˘ÍÎ

È+

y+y

yy

-

ky+y

e= k-

k-

k-s-

h202

201

0201h2

h202

201ps e

2

e1

e

2

aF (33)

When 00201 y=y=y ,

h

h20

ps e1

eaF

k-

k-s- -

kye= . (34)

In practice, 0y is not known and usually it is assumed that

potentialticElecrokinepotential0 =-x=y . (35)

(Diagram Here)

Two Spheres Interactions

Constant Potential (Deryaguin)

( )h20ss e1lna2V k-y

- +ype= (36)

h

h20ss e1

ea2F

k-

k-y- +

ype= (37)

Linear Superposition (Debye-Huckel)

h20

2

ss eha2

a4V k-y

- y+

pe= (38)

( ) h2

20ss e

a2

h1

a2h1F k-

-

˜¯

ˆÁË

Ê +

+k+pey= (39)

Superposition (thin double layer)

h022

ss ekT4

zetanha

ze

kT32V k-

- ˜¯

ˆÁË

Ê y˜¯

ˆÁË

Êpe= (40)

h022

ss ekT4

zetanha

ze

kT32F k-

- ˜¯

ˆÁË

Ê yk˜

¯

ˆÁË

Êpe= (41)

15

Electrokinetic Phenomena

Electrokinetic phenomena occur when there is relative motion between thecharged interface and the adjacent electrolyte solution so that part of the double layercharge moves with the liquid. The "plane of shear" separates the mobile part of thedouble layer from the fixed part. The electrical potential at the shear plane is theelectrokinetic potential or the zeta potential ( x -potential).

Electro-Osmosis

Elecro-osmosis is the flow of liquid due to influence of an applied electric field.Assume an electric field E is applied parallel to a surface. If the charge density is er , it

follows that

Edy

ude2

2

r=m (42)

(Diagram Here)

Using (9) to replace er in term of ( )yy , (e

r-=y— e2 ), we find

2

2

2

2

dy

dE

dy

ud ye-=m (43)

or

m

ex=

Eu 0e (44)

where 0eu is the electro-osmosis velocity and we assumed that at 0y = ,

0u = , x=y . (45)

Particle Electrophoresis

Particle electrophoresis is the movement of charged particles in an electric field.For thin double layer ( 1a >>k ),

m

ex=

Eu P (46)

16

which may be restated in terms of the electrophoretic mobility, U (velocity for unit field)as

m

ex=U . (47)

This is known as Smoluchowski equation.

For small particles with 1a <<k , the viscous drag must be balanced by theCoulomb force, i.e.

( )a4EEQaU6 pex==pm

where equation (27) is used for total charge and 0y is replaced by x .

It then follows that

m

ex=

3

2U , (48)

which is known as the Huckel equation. More generally, then

( )af3

2U 1 k

m

ex= (49)

where ( )af1 k varies from 1 to 1.5 for ak varying between 0 to • .

Born Repulsion

At very short distances, the interpenetration of electron shells leads to the strongrepulsive force known as the Born repulsion. The corresponding interaction energy isgiven as

( ) ˙˚

˘ÍÎ

È -+

+

+s=

72

6c

B h

ha6

7a2

ha8

7560

AV (50)

for sphere-plate where cs is the collision diameter (typical of the order of o

A5 ).

DLVO Theory of Colloidal Stability

The theory of colloidal stability was developed by Deryaguin, Landau (1941),Verway, and Overbeak (1948) and is now known as the DLVO theory.

17

The interaction potential between particles is composed as the sum of van derWaals, AV , electrical double layer, RV , and Born, BV , i.e.

BRAT VVVV ++= (51)

Depending on the magnitude of van der Waals and electrical double layer potentialenergies, the suspension could be stable or could rapidly aggregate.

Figure (a) shows a stable suspension where a strong energy barrier (EB) is formed. Thereis a deep primary minimum (SM). The secondary minimum could lead to weakaggregation which will break easily.

(Diagram Here)

Figure (b) shows the total potential for a colloidal system for which the electrical doublelayer is weak or absent. The particles will attract each other and the suspension willaggregate quickly.

(Diagram Here)

Steric Interaction

A colloidal suspension could remain stable when the particles absorb polymetricchains.

Hydrophobic Interaction

There is an attraction between hydrophobic surfaces as a result of water moleculesmigrating from the gap to the bulk.

Hydration Effects

At very short distances, hydrophilic surfaces may experience hydration repulsion.This is because of the need for the surfaces to become dehydrated for the particles tocome in contact.