Colloidal Materials: Part III - NPTELnptel.ac.in/courses/103103033/module1/lecture4.pdf ·...

NPTEL Chemical Engineering Interfacial Engineering Module 1: Lecture 4

Joint Initiative of IITs and IISc Funded by MHRD 1/22

Colloidal Materials: Part III

Dr. Pallab Ghosh

Associate Professor

Department of Chemical Engineering

IIT Guwahati, Guwahati–781039

India



Table of Contents

Section/Subsection Page No. 1.4.1 Determination of molecular weight by light scattering 3

1.4.2 Experimental characterization of colloids 6–9

1.4.2.1 Transmission electron microscopy 6

1.4.2.2 Scanning electron microscopy 7

1.4.2.3 Dynamic light scattering 8

1.4.2.4 Small-angle neutron scattering 8

1.4.3 Electrical properties of colloids 10–19

1.4.3.1 Electrostatic double layer 10

1.4.3.2 Electrokinetic phenomena 12

1.4.3.2.1 Zeta potential 14

1.4.3.2.2 Reciprocal relationship 19

Exercise 20

Suggested reading 22



1.4.1 Determination of molecular weight by light scattering

In liquid dispersions, the scattering of light is due to fluctuations in the solvent

density and fluctuations in the particle concentration. The total intensity of the scattered

light in all directions is given by,

20

002 sins

si

I I r dI

(1.4.1)

The second term inside the integral in Eq. (1.4.1) represents an area on the surface of a

sphere, where r is the radius of the sphere and is the angle with the horizontal axis.

From the theory of Rayleigh scattering in a solution, the quantity 0si I can be

determined as shown below. The Rayleigh scattering equation for a solution (at constant

temperature) is expressed by,

2

2

2

2 40

2

1 cos

rr

s

o

dnn kTc

dcidI

rdc

(1.4.2)

where si is the intensity of the light scattered per unit volume of solution. The gradient

of refractive index of the solution rn , is given by rdn dc . The osmotic pressure is given

by,

2 2o

c B cRT c RT Bc

M RT M

, B

BRT

(1.4.3)

Where M is the molecular weight of the solute, c is the concentration, and B is the

second virial coefficient. Therefore, the osmotic pressure gradient is given by,

12od

RT Bcdc M

(1.4.4)

From Eq. (1.4.2), we can write,

2

20

1 cos

1 2s

Kci

I r M Bc

(1.4.5)

The quantity K is a constant, which is given by,



22

4

2 rr

A

dnn

dcK

N

(1.4.6)

From Eqs. (1.4.1) and (1.4.6) we obtain,

2

0 0

2 1 cos sin

1 2s

Kc dI

I M Bc

(1.4.7)

The value of the integral: 2

0sin 1 cos d

is 8 3 . Therefore, Eq. (1.4.7) becomes,

0

16

3 1 2 1 2sI Kc Hc

I M Bc M Bc

(1.4.8)

where 16 3H K is a constant. It is related to the refractive index, its gradient and

the wavelength of light in the medium by the following equation.

23 2

4

32

3

r r

A

n dn dcH

N

(1.4.9)

where is the wavelength of light in the solution and AN is Avogadro’s number. The

quantity 0sI I is the turbidity, . Therefore, from Eq. (1.4.8) we have,

12

HcBc

M (1.4.10)

Equation (1.4.10) is known as Debye equation. It predicts that the plot of Hc versus c

should be a straight line. From the intercept, the molecular weight M can be determined.

Example 1.4.1: The variation of Hc with concentration for a polymeric colloid in

benzene is given below.

c (kg/m3) 2.5 4.0 6.5 8.0 10.0

310Hc (mol/kg) 5.6 6.1 6.6 7.2 8.0

From these data, calculate the molecular weight of the polymer.



Solution: The plot of Hc vs. c is shown in the following figure. The data were fitted

by a straight line, as shown in Fig. 1.4.1.

Fig. 1.4.1 Variation of Hc with concentration.

The intercept is,

1

0.0048M

mol/kg

208.333M kg/mol

If the system is polydispersed, Eq. (1.4.10) is applicable for each molecular weight

fraction. For dilute solutions, we can neglect the second term on the right side of Eq.

(1.4.10). For the jth fraction, we can write,

1j

j j

Hc

M (1.4.11)

The experimentally measured concentration, turbidity and the average molecular weight

are correlated by Equation (1.4.11) as,

exp

exp

1Hc

M (1.4.12)

where,

exp jc c (1.4.13)

and



exp j (1.4.14)

From Eqs. (1.4.11)–(1.4.14), we obtain,

exp

exp

j j j j j

j j j

H c M c MM

Hc H c H c c

(1.4.15)

Since j j j dc n M V , we have,

2j j

j j

n MM

n M

(1.4.16)

This average molecular weight (i.e., M ) is known as the weight-average molecular

weight.

1.4.2 Experimental characterization of colloids

Colloids are characterized by various methods. Some of these are, transmission

electron microscopy (TEM), scanning electron microscopy (SEM), dynamic light

scattering (DLS), and small-angle neutron scattering (SANS). Depending on the

properties of the colloidal matter, the appropriate characterization method is

selected.

1.4.2.1 Transmission electron microcopy

Many colloid particles are too small to be viewed in an optical microscope. The

numerical aperture of an optical microscope is generally less than unity, which

can be increased up to 1.5 with oil-immersion objectives. Therefore, for light of

wavelength 600 nm, the resolution limit is of the order of 200 nm.

To increase the resolving power of a microscope so that colloidal dimensions can

be directly observed, the wavelength of the radiation must be reduced

considerably below that of visible light. Electron beams can be produced which

have wavelengths of the order of 0.01 nm. These are focused by electric or

magnetic fields, which act as the equivalent of lenses. A resolution of the order of

0.2 nm can be attained after smoothing the noise. Single atoms appear to be



blurred irrespective of the resolution, owing to the rapid fluctuation of their

location.

The TEM can be used for measuring particle size between 1 nm and 5 m. Due to

the complexity of calculating the degree of magnification directly, calibration is

done using pre-characterized polystyrene latex particles.

The use of electron microscopy for studying colloid systems is limited by the fact

that electrons can travel without any hindrance only in high vacuum. Therefore,

the samples need to be dried before observation.

A small amount of the sample is deposited on an electron-transparent plastic or

carbon film (10–20 nm thick) supported on a fine copper mesh grid. The sample

scatters electrons out of the field of view, and the final image can be viewed on a

fluorescent screen. The amount of scattering depends on the thickness and the

atomic number of the atoms of the sample.

The organic materials are relatively transparent for electrons whereas, the heavy

metals make ideal samples. To enhance contrast and obtain three-dimensional

effect, various techniques (e.g., shadow-casting) are generally employed.

1.4.2.2 Scanning electron microcopy

In scanning electron microscopy, a fine beam of medium-energy electrons scans

across the sample in a series of parallel tracks. These electrons interact with the

sample to produce various types of signals such as secondary electron emission,

back-scattered-electrons, cathodoluminescence and X-rays. These are detected,

displayed on a fluorescence screen and photographed.

In the secondary-electron-emission mode, the particles appear to be diffusely

illuminated. Their size can be measured and the aggregation behavior can be

studied.

In the back-scattered-electron mode, the particles appear to be illuminated from a

point source, and the resulting shadows can provide good impressions of height.

The resolution limit in SEM is about 5 nm, and the magnification achieved is

generally less than that in a TEM. However, the depth of focus is large, which is



important for studying the contours of solid surfaces, particle shape and

orientation.

1.4.2.3 Dynamic light scattering

If the light is coherent and monochromatic (e.g., a laser), it is possible to observe

time-dependent fluctuations in the scattered intensity using a suitable detector

such as a photomultiplier capable of operating in photon-counting mode.

These fluctuations arise because the particles are small, and they undergo

Brownian movement. The distance between them varies continuously.

Constructive and destructive interference of light scattered by the neighboring

particles within the illuminated zone gives rise to the intensity fluctuation at the

detector plane.

From the analysis of the time-dependence of the intensity fluctuation, it is

possible to determine the diffusion coefficient of the particles. Then, by using the

StokesEinstein equation (see Lecture 3, Module 1), the hydrodynamic radius of

the particles can be determined.

An accurately known temperature is necessary for DLS because knowledge of the

viscosity is required (because the viscosity of a liquid is related to its

temperature). The temperature also needs to be stable, otherwise convection

currents in the sample will cause non-random movements that will ruin the

correct interpretation of size.

Dynamic light scattering is also known as photon correlation spectroscopy (PCS)

or quasi-elastic light scattering (QELS). It is a well-established technique for the

measurement of the size distribution of proteins, polymers, micelles,

carbohydrates, nanoparticles, colloidal dispersions, emulsions and

microemulsions.

1.4.2.4 Small-angle neutron scattering

Neutrons from reactors or accelerators are slowed down in a moderator to kinetic

energies corresponding to room temperature or less. The wavelength probed by

SANS is quite different from the visible light. A typical range of wavelength is



0.33 nm. This range is much smaller than that of visible light (i.e., 400700

nm).

The usefulness of SANS to colloid and polymer science becomes evident when

one considers the length scales and energy involved in neutron radiation. Light

scattering is indispensable for studying particles having size in the micrometer

range. For very small particles, neutrons are useful. The energy of a neutron with

0.1 nm wavelength is 201.3 10 J. Due to such low energy, neutron scattering is

useful for sensitive materials. For colloidal dispersions, where the distance

between the particles is large, the scattering angle is small (~ 20 rad or less).

Light and X-rays are both scattered by the electrons surrounding atomic nuclei,

but neutrons are scattered by the nucleus itself. There is, however, no systematic

variation of the interaction with the atomic number. The isotopes of the same

element can show significant difference in scattering. For example, neutrons can

differentiate between hydrogen and deuterium. This has useful applications in

biological science in the technique known as contrast matching.

The interaction of neutrons with most substances is weak, and the absorption of

neutrons by most materials is very small. Neutron radiation, therefore, can be

very penetrating.

Neutrons can be used to study the bulk properties of samples with path-length of

several centimeters. They can also be used to study samples with somewhat

shorter path-lengths but contained inside an apparatus (e.g., cryostat, furnace,

pressure cell or shear apparatus).

The SANS technique provides valuable information over a wide variety of

scientific and technological applications such as chemical aggregation, defects in

materials, surfactant assemblies, polymers, proteins, biological membranes, and

viruses.



1.4.3 Electrical properties of colloids

The electrical properties of colloidal materials lead to some of the most important

phenomena in interfacial engineering. The presence of electrostatic double layer

surrounding the particles results in their mutual repulsion so that they do not

approach each other closely enough to coagulate. An increase in the size of the

particles by coagulation would lead to a decrease of total area, and hence to a

decrease of free energy of the system. Therefore, union of colloid particles would

be expected to occur, were it not for the repulsion caused by the electrostatic

double layer. The stability of the charged colloid particles depends on the

presence of electrolytes in the dispersion.

1.4.3.1 Electrostatic double layer

Let us consider a solidliquid interface. Suppose that the interface is positively

charged and the atmosphere of the negatively charged counterions is around it.

This visualization of the ionic atmosphere near a charged interface originated the

term electrostatic double layer.

The Coulomb attraction by the charged surface groups pulls the counterions back

towards the surface, but the osmotic pressure forces the counterions away from

the interface. This results in a diffuse double layer.

The double layer very near to the interface is divided into two parts: the Stern

layer and the GouyChapman diffuse layer. The compact layer of adsorbed ions

is known as Stern layer. This layer has a very small thickness (say, 1 nm). The

counterions specifically adsorb on the interface in the inner part of the Stern

layer, which is known as inner Helmholtz plane (IHP) (see Fig. 1.4.2). The

potential drop in this layer is quite sharp, and it depends on the occupancy of the

ions.



Fig. 1.4.2 Electrostatic double layer.

The outer Helmholtz plane (OHP) is located on the plane of the centers of the

next layer of non-specifically adsorbed ions. These two parts of the Stern layer

are named so because the Helmholtz condenser model was used as a first

approximation of the double layer very close to the interface.

The diffuse layer begins at the OHP. The potential drop in each of the two layers

is assumed to be linear. The dielectric constant of water inside the Stern layer is

believed to be much lower (e.g., one-tenth) than its value in the bulk. The value is

lowest near the IHP.

The diffuse part of the electrostatic double layer is known as GouyChapman

layer. The thickness of the diffuse layer is known as Debye length (represented by

1 ). This length indicates the distance from the OHP into the solution up to the

point where the effect of the surface is felt by the ions. is known as

DebyeHückel parameter. The Debye length is highly influenced by the

concentration of electrolyte in the solution. The extent of the double layer

decreases with increase in electrolyte concentration due to the shielding of charge



at the solidsolution interface. The ions of higher valence are more effective in

screening the charge.

1.4.3.2 Electrokinetic phenomena

The phenomena associated with the movement of charged particles through a

continuous medium, or with the movement of a continuous medium over a

charged surface are known as electrokinetic phenomena. There are four major

types of electrokinetic phenomena, viz. electrophoresis, electroosmosis,

streaming potential and sedimentation potential. There is a common origin for all

the electrokinetic phenomena, i.e., the electrostatic double layer.

Electrophoresis refers to the movement of particles relative to a stationary liquid

under the influence of an applied electric field. If a dispersion of positively

charged particles is subjected to an electric field, the particles move towards the

cathode. Electrophoresis is perhaps the most important electrokinetic

phenomenon. Three types of electrophoresis are usually used, viz.

microelectrophoresis, moving-boundary electrophoresis and zone electrophoresis.

Electrophoresis is widely used in biochemical analysis for separation of proteins.

Another very important application of electrophoresis is electrodeposition. A

cylindrical microelectrophoresis cell for studying the movement of air bubbles

under electric field is shown in the Fig. 1.4.3.

Fig. 1.4.3 Setup for microelectrophoresis (source: A. Phianmongkhol and J. Varley, J. Coll. Int. Sci., 260, 332, 2003; reproduced by permission from Elsevier

Ltd., 2003).



Electroosmosis refers to the movement of the liquid of an electrolyte solution past

a charged surface (e.g., a capillary tube or a porous plug) under the influence of

an electric field. The pressure necessary to balance the electroosmotic flow is

known as electroosmotic pressure. To understand how electroosmosis occurs,

consider a glass capillary containing an aqueous electrolyte solution. The charge

on the wall of the tube can develop from either the dissociation of the surface

SiOH groups or adsorption of the OH ions on the wall. This charge is balanced

by an equal and opposite charge in the solution, as shown in Fig. 1.4.4.

Fig. 1.4.4 Electroosmosis.

When the electric field is applied, the ions in the diffuse part of the double layer

move towards one of the electrodes depending on their charge. The motion of

these hydrated ions imparts a body-force on the liquid in the double layer. This

force sets the liquid in motion. Electroosmosis has been used in many

applications related to environmental pollution abatement.

Suppose that the electrolyte solution is forced to pass through a capillary under

pressure applied from outside. An electrical potential is generated between the

ends of the capillary. This is called streaming potential. The same phenomenon

can be observed when the solution is forced through a porous medium. If a

dispersion of charged particles is allowed to settle, the resulting motion of the

particles causes the development of a potential difference between the upper and

lower parts of the dispersion. It is known as Dorn effect and the potential is

known as sedimentation potential.

Therefore, the situations which give rise to streaming potential and sedimentation

potential are opposite to those of electroosmosis and electrophoresis, respectively.



1.4.3.2.1 Zeta () potential

The surface charge of the colloid particles is expressed in terms of the zeta

potential . It is the potential at the ‘surface of shear’.

When a particle moves in an electric field, the liquid layer immediately

adjacent to the particle moves with the same velocity as the surface, i.e.,

the relative velocity between the particle and the fluid is zero at the

surface. The boundary located at a very short distance from the surface at

which the relative motion sets in is known as the surface of shear.

The precise location of this surface cannot be determined exactly, but it is

presumed that it is located very close to the surface of the particle, may be

a few molecular-diameters apart.

The magnitude of -potential provides an indication of the stability of

the colloid system. The pH of the medium strongly affects the -

potential.

1.4.3.2.1.1 Determination of -potential from electrophoresis

Let us consider the motion of a small spherical colloid particle moving with

velocity u in an electric field E . In a dilute dispersion, the mobility is given by

the Hückel equation,

02

3

u

E

, 0.1sR (1.4.17)

where is the DebyeHückel parameter, is the dielectric constant of the

medium, 0 is permittivity of the free space, is the viscosity of the liquid and

sR is the radius of the sphere.

For large values of sR , the relationship between the electrophoretic mobility

and -potential is given by Smoluchowski equation (also known as

HelmholtzSmoluchowski equation),



0u

E

, 100sR (1.4.18)

Equation (1.4.18) is applicable for relatively high salt concentrations for which

is large.

Apart from these two limiting conditions, the zeta potantials for the other values

of sR can be calculated by the following equation.

02

3 su

f RE

(1.4.19)

where sf R can be calculated from either Henry’s or Ohshima’s equation.

The latter equation is more convenient for computation.

31

1

2.52 1

1 2exp

s

s s

f R

R R

(1.4.20)

This equation is valid for any value of sR with maximum relative error less

than 1%.

1.4.3.2.1.2 Determination of -potential from electroosmosis

When an electric field is applied across a capillary containing electrolyte solution,

the double layer ions begin to migrate. After some time, a steady state is reached

when the electrical and viscous forces balance each other, i.e., the force exerted

on the medium by the ions is balanced by the force exerted by the medium on the

ions.

If the steady state volumetric flow rate in a fine capillary due to electroosmosis is

V , then the -potential is given by,

0

V

EA

, 1cR (1.4.21)

where cR is the radius and A is the cross-sectional area of the capillary.

Therefore, by measuring the volumetric flow rate through the capillary, the -

potential can be determined.



Example 1.4.2: An aqueous solution of sodium chloride is placed inside a capillary in an

electroosmosis apparatus and subjected to an electric field of 100 V/m. The

electroosmotic velocity in the capillary is observed to be 10 m/s. Calculate -potential

from these data.

Solution: The electroosmotic velocity is given by,

610 10V

A m/s

For water, 78.5 , 31 10 Pa s , and the permittivity of free space is,

120 8.854 10 C2 J1 m1. From Eq. (1.4.21) we obtain,

6 3

120

10 10 1 100.1439

78.5 8.854 10 100

V

EA

V = 143.9 mV

1.4.3.2.1.3 Determination of -potential from streaming potential

The -potential can be correlated with streaming potential as follows. A pressure

difference p across a capillary is applied which sets the liquid in motion

inside it. The charge of the double layer moves with the surrounding liquid

generating an electric current, which is known as the streaming current. On the

other hand, the charge transferred downstream generates an electric field in the

opposite direction. After a short time, the two currents due to pressure gradient

and reverse electric field balance each other. The streaming potential sE is the

potential drop associated with this electric field.

The following equation gives the -potential.

0

s sk E

p

, 1cR (1.4.22)

where sk is the conductivity of the electrolyte solution. Equation (1.4.22) is valid

for the large values of cR (where cR is the radius of the capillary). Therefore, it

is likely to give erroneous results when the concentration of salt is low (which

would result in a low value of ).



1.4.3.2.1.4 Determination of -potential from sedimentation potential

An electric field is developed during the settling of charged particles. This is

known as sedimentation potential. Smoluchowski presented the first theoretical

estimate of the magnitude of the field. For a dispersion of solid non-conducting

spheres of radius sR , immersed in an electrolyte solution of conductivity sk ,

dielectric constant , and viscosity , the sedimentation potential is predicted to

be,

30

sed4

3s

s d

gnRE

k V

(1.4.23)

where n is the number of particles and dV is the volume of dispersion. This

equation is valid in those situations where the thickness of electrostatic double

layer is small with respect to the radius of the particles 1sR . It can be

observed from Eq. (1.4.23) that the sedimentation potential is proportional to the

amount of the dispersed phase.

The settling of fine droplets can generate high sedimentation potentials. For

example, the settling of water drops in the gasoline storage tanks can produce a

very high sedimentation potential owing to the low conductivity of the oil phase,

which can be dangerous. The value of sedimentation potential can be as high as

1000 V/m or above, even for a moderate value of the -potential (e.g., 25 mV).

If the diameter of the droplets is larger than 100 m, they settle down completely

and the sedimentation potential becomes zero. If they are smaller than 1 m, the

sedimentation potential gradient reduces the rate of settling and a haze of water

drops floats in the electric field.

The sedimentation velocity is reduced by the sedimentation potential gradient. If

the steady state settling velocity (i.e., terminal velocity) of the particle is tv when

the particle is uncharged, and cv is the velocity when the particle carries a surface

charge, then for a single sedimenting particle,



201

1c ts s

v vk R

, 1sR (1.4.24)

The experimental data agree with this equation within an order of magnitude. If

0.1sR m, 25 mV and 41 10sk 1 m1, it can be shown that the

velocity of the particle in water will be reduced by 30%. This causes a haze of

fine water drops in oil.

1.4.3.2.1.5 A comment on the -potential determined by various methods

The -potential depends only on the properties of the phases in contact.

Therefore, its value must be independent of the experimental method employed

for its determination. Several scientists, using the Smoluchowski theory, have

found that -potential obtained from streaming potential or electroosmotic

measurements is quite smaller than the value obtained from electrophoretic

mobility or sedimentation potential measurements.

These discrepancies can be due to the influence of surface conductivity on

streaming potential and electroosmotic flow.

If the condition 1R (where R is the radius of sphere, or the radius of

capillary) is not satisfied, then the Smoluchowski equation can yield inaccurate

values of the -potential.

Two conditions must be satisfied to justify comparison between the values of -

potential obtained by different electrokinetic experiments: the effect of surface

conductivity must be taken care of, unless it is negligible, and the surface of shear

must divide comparable double layers in all the cases. Fulfillment of the second

requirement depends upon the experimental procedure. For example, if the same

capillary is used for electroosmosis and streaming potential studies, the second

condition can be satisfied. On the other hand, the surfaces of a capillary and a

migrating particle can be quite different. Sometimes, the surfaces are coated with

a protein, and the characteristics of both surfaces are governed by the adsorbed

protein.



1.4.3.2.2 Reciprocal relationship

The cause and effect are interchanged in streaming potential and electroosmosis.

Similarly, the situation in sedimentation potential is the opposite of electrophoresis. From

Eq. (1.4.21), putting c sE I Ak , we can write,

0

c s

V

I k

(1.4.25)

From Eqs. (1.4.22) and (1.4.25) we can write,

0s

c s

E V

p I k

(1.4.26)

The coupling of two different electrokinetic ratios, i.e., sE p and cV I through Eq.

(1.4.26) is an example of the law of reciprocity of Lars Onsager.



Exercise

Exercise 1.4.1: Calculate the sedimentation potential for 50 m radius water drops in an

oil (dielectric constant = 2, viscosity = 0.5 mPa s, and conductivity = 1109 1 m1) if

the -potential is 0.025 V. The density of the oil is 700 kg/m3 and the volume fraction of

the dispersed aqueous phase is 0.05.

Exercise 1.4.2: Calculate the electrophoretic mobility of a 50 nm diameter spherical

colloid particle in an aqueous solution of NaCl at 298 K. The -potential is 0.02 V. The

concentration of NaCl in the solution is 100 mol/m3. Given: The Debye length is ~1 nm at

this concentration of the salt.

Exercise 1.4.3: Application of 101.325 kPa pressure produces a streaming potential of 0.4

V in an experiment using aqueous NaCl solution. Calculate the -potential. Given:

1 10.01 msk and 1 mPa s.

Exercise 1.4.4: Answer the following questions clearly.

a. What are the advantages and limitations of transmission electron microscopy in

the characterization of colloidal materials?

b. What are the different types of signals produced in a scanning electron

microscope?

c. Explain how dynamic light scattering can be used to measure the size of a colloid

particle.

d. For what types of colloids does the dynamic light scattering have advantage over

TEM or SEM?

e. What are the advantages of neutron scattering over visible light scattering? What

is small-angle neutron scattering? Where is it used?

f. Explain what you understand by electrokinetic phenomena.

g. What are the four major electrokinetic phenomena?

h. Explain electrophoresis. What are the major uses of electrophoresis?



i. What is zeta potential? Explain its significance.

j. Discuss the applicability criteria of Hückel and Smoluchowski equations.

k. What is electroosmosis? How does it differ from osmosis? What is electroosmotic

pressure?

l. Explain how you would calculate the zeta potential from electroosmosis.

m. What is streaming potential? How is it developed?

n. What is sedimentation potential?

o. Explain Onsager’s reciprocal relationship.



Suggested reading

Textbooks

D. J. Shaw, Introduction to Colloid and Surface Chemistry, Butterworth-

Heinemann, Oxford, 1992, Chapters 3 & 7.

J. C. Berg, An Introduction to Interfaces and Colloids: The Bridge to

Nanoscience, World Scientific, Singapore, 2010, Chapters 5 & 6.

P. C. Hiemenz and R. Rajagopalan, Principles of Colloid and Surface Chemistry,

Marcel Dekker, New York, 1997, Chapters 5 & 12.

P. Ghosh, Colloid and Interface Science, PHI Learning, New Delhi, 2009,

Chapter 2.

Reference books

D. F. Evans and H. Wennerström, The Colloidal Domain: Where Physics,

Chemistry, Biology, and Technology Meet, Wiley-VCH, New York, 1994,

Chapters 4 & 8.

R. J. Hunter, Foundations of Colloid Science, Oxford University Press, New

York, 2005, Chapters 5 & 8.

Journal articles

A. Phianmongkhol and J. Varley, J. Colloid Interface Sci., 260, 332 (2003).

E. W. Anacker, J. Colloid Sci., 8, 402 (1953).

F. Booth, J. Chem. Phys., 22, 1956 (1954).

H. Ohshima, J. Colloid Interface Sci., 168, 269 (1994).

H. V. Tartar and A. L. M. Lelong, J. Phys. Chem., 59, 1185 (1955).

J. B. Peace and G. A. H. Elton, J. Chem. Soc., 2186 (1960).

L. Onsager, Phys. Rev., 37, 405 (1931).

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