NPTEL  Chemical Engineering  Interfacial Engineering
Module 1: Lecture 3
Colloidal Materials: Part II
Dr. Pallab Ghosh
Associate Professor
Department of Chemical Engineering
IIT Guwahati, Guwahati–781039
India
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Table of Contents
Section/Subsection
1.3.1 Brownian motion
Page No.
3
1.3.2 Osmotic pressure
5–12
1.3.2.1 Determination of number average molecular weight
8
1.3.2.2 Donnan equilibrium
10
1.3.3 Optical properties of colloids
12
Exercise
16
Suggested reading
17
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1.3.1 Brownian motion
 In 1827, the English botanist Robert Brown observed under a microscope that
pollen grains suspended in water had a ceaseless chaotic movement. Pollen grains
which had been stored for a century moved in the same way. Colloid particles
when examined under an ultramicroscope through light scattering were also
found to execute ceaseless random motion. This motion is known as Brownian
motion (Fig. 1.3.1).
Fig. 1.3.1 Brownian movement.
 Even when all possible causes which may lead to motion (such as heat, light,
mechanical vibrations and other disturbances) are carefully eliminated, the
particles still move about ceaselessly. Therefore, it can be concluded that the
Brownian motion is due to the molecular impacts from the medium on all sides of
these dispersed particles.
 Brownian motion becomes less vigorous with the increase in size of the particles.
It is a fundamental property of the colloidal systems. Increase in viscosity of the
medium can reduce the vigor of the movement.
 Brownian motion is often cited as an indirect evidence of the existence of the
molecules and their incessant thermal motion.
 At any instant, the numerous impacts suffered by a colloid particle on all sides are
not evenly matched. This results in a net displacement. This displacement is,
however, random. The random movements result in self-diffusion of the particles
in the fluid. The works of Albert Einstein and Marian Smoluchowski about a
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century ago have formed the basis of the theory of Brownian motion. They
derived a relationship between the distance moved by a particle (as a result of
Brownian motion) and its diffusion coefficient
D
by treating Brownian
movement as a random-walk process. They obtained,
x 2  2 Dt
(1.3.1)
where x 2 is the mean square displacement of the particle during a period t.

Equation (1.3.1) is known as EinsteinSmoluchowski equation. This equation
provides us with a means to calculate the diffusion coefficient of colloid particles
which can be viewed under a microscope. The actual displacement of a particle in
time t is measured by a microscope. From a large number of observations, the
mean square displacement is calculated statistically.

The diffusion coefficient of a spherical dispersed particle can be related to its
diameter (d) and the viscosity of the liquid () as,
D
RT
3πμN Ad
(1.3.2)
where R is gas constant, NA is Avogadro’s number and T is temperature. Equation
(1.3.2) is known as StokesEinstein equation. This equation has been found to be
quite successful for describing the diffusion of large spherical molecules.
Example 1.3.1: The diffusion coefficient of a colloid particle in water at 293 K is
4.51010 m2/s. Estimate its diffusion coefficient in ethylene glycol at 313 K. Given:
viscosity of ethylene glycol at 313 K is 12.5 mPa s.
Solution: From StokesEinstein equation, D  T μ . Therefore, we can write,
D2  T2   μ1   313   1  103 
     
  0.0855

D1  T1   μ2   293   12.5  103 

D2  4.5  1010  0.0855  3.85  1011 m2/s
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 Equation (1.3.2) can be used to determine the value of Avogadro’s number. In
1908, Jean Perrin and his collaborators performed studies with monodisperse
spheres of natural colloids (e.g., gamboge). The mean square displacement was
measured as follows. The particles of the dispersion were focused under a
microscope, which carried the design of a previously-calibrated square paper
within its eye-piece. The Brownian displacement along the x-axis was read by
noting its position on the square design. A large number of observations were
made on a single particle, noting its displacement at every 30 s interval. This was
repeated for a large number of particles. The magnitude of x 2 in 30 s was thus
determined.
 The value of Avogadro’s number obtained by Perrin and collaborators varied
between 5.5  1023 mol 1 and 8  1023 mol 1 . Later, Theodor Svedberg studied
monodispersed gold sols of known particle size in ultramicroscope and obtained
N A  6.09  1023 mol 1 . These results are considered as strong evidences in favor
of the kinetic theory.
1.3.2 Osmotic pressure
 French physicist Jean Antoine Nollet discovered in 1748 that when alcohol and
water were separated by a pig’s bladder membrane, the water passed through the
membrane into the alcohol, causing an increase of pressure. However, the alcohol
was not able to pass out into the water. This happened because the bladder was
semipermeable. It allowed the water to pass through it, but the alcohol was not
allowed to pass.
 The phenomenon of transport of a solvent through a semipermeable membrane
from the solvent to a solution, or from a dilute solution to a concentrated solution
is known as osmosis. In biology, there are many instances of semipermeable
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membrane, e.g., inner walls of egg-shells, potato-skin, and intestinal walls of
some animals.
 To demonstrate the origin of osmotic pressure, let us perform a thought
experiment in the setup shown in Fig. 1.3.2.
Fig. 1.3.2 Osmosis and osmotic pressure.

A semipermeable membrane is placed at the center of the U-tube as shown in the
figure. The right part of the tube is filled with a solution and the left part is filled
with the pure solvent. In absence of any external field (e.g., electric field) the
solvent spontaneously passes through the membrane into the solution (i.e.,
osmosis). The transport of the solvent into the solution can be counteracted by
applying a pressure  πo  on the solution. This excess pressure on the solution
which would just prevent osmosis is called the osmotic pressure of the solution.
Therefore, the osmotic pressure of a solution is the pressure required to prevent
osmosis when the solution is separated from the pure solvent by a semipermeable
membrane.
 If the pressure on the solution side is greater than πo , the solvent will flow in the
reverse direction (i.e., from the solution to the solvent). In this case, the
semipermeable membrane functions like a filter that separates the solvent from
the solution. This process is known as reverse osmosis, which has large-scale
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industrial applications (e.g., desalination, concentration of fruit juice, and removal
of pollutants from water).
 It is important, however, to note that the concept of osmotic pressure is more
general than that discussed above. In fact, one does not have to invoke the
presence of a membrane to define osmotic pressure. The osmotic pressure is a
property of the solution. For example, in electrostatic double layer (Lecture 4,
Module 1), the concentration of the counterions in the vicinity of the surface is
larger than their concentration in the bulk solution. This difference in
concentration generates osmotic pressure, which maintains the double layer.
 The van’t Hoff’s law of osmotic pressure is: πo  cRT , where c is the
concentration of the solute in the solution, R is gas constant and T is temperature.
This relationship is very similar to the ideal gas law. Therefore, van’t Hoff stated
that the osmotic pressure of a solution is the same as the solute would exert if it
existed as a gas in the same volume as that occupied by the solution at the same
temperature.
 It was also found by van’t Hoff that the dilute aqueous solutions of electrolytes
such as NaCl showed considerable departure from the above law. The observed
osmotic pressure of salt solutions was found to be much higher than that
predicted from the equation: πo  cRT . To account for this deviation from
ideality, van’t Hoff introduced a factor i, which is defined as the ratio of the
observed and ideal osmotic pressures. Therefore, the modified van’t Hoff’s law
is: πo  icRT .
 The origin of van’t Hoff factor greater than unity is the dissociation of the
electrolyte in solution. In dilute solution, the dissociation is almost complete.
Thus, one NaCl molecule generates two ions in a dilute solution and the value of
i approaches 2. Similarly, for H2SO4, the value of i approaches 3 in dilute
solution.
 The osmotic pressure of a non-electrolyte solution may be represented as,
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 RT
πo  
 M

2
3
 c  Bc  C c  

Module 1: Lecture 3
(1.3.3)
where c is the concentration, M is the molecular weight of the solute, and B and
C  are constants related to the second and third virial coefficients, respectively. In
dilute solutions (small c), we can drop the terms beyond the second term on the
right side of Eq. (1.3.3). Therefore,
o
c
RT
 Bc
M

(1.3.4)
When  o c is plotted against c , a straight line is obtained. From the slope and
intercept, we can calculate B and M .
1.3.2.1 Determination of number average molecular weight
 The colloid dispersions are usually polydispersed. For such a system, the average
molecular weight can be determined by osmometry. Let us assume that the
dispersion behaves ideally. If the experimentally observed values of concentration
and osmotic pressure are cexp and  oexp , respectively, then,
 oexp
cexp

RT
M
(1.3.5)
where M is the average molecular weight. For each of the molecular weight
fractions (e.g., the jth fraction) we have,
 oj
cj

RT
Mj
(1.3.6)
The experimental osmotic pressure is the sum of the pressures exerted by each of
the fractions, i.e.,
 oexp    oj
(1.3.7)
Also, the experimental concentration is the sum of the concentrations of the
fractions, i.e.,
cexp   c j
(1.3.8)
Therefore, from Eqs. (1.3.5)(1.3.8) we get,
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M
Module 1: Lecture 3
cj
(1.3.9)
cj M j
The concentration, c j , is defined as,
cj 
n jM j
(1.3.10)
Vd
From Eqs. (1.3.9) and (1.3.10) we obtain,
M
 n jM j
(1.3.11)
nj
Therefore, the average molecular weight determined from osmometry is the
number-average molecular weight.
Example 1.3.2: The variation of osmotic pressure with the concentration of
nitrocellulose in methanol is given below.
c (kg/m3)
o
RTc
(mol/kg)
0.7
1.9
6.7
12.1
0.011
0.012
0.015
0.021
Determine the molecular weight of nitrocellulose from these data.
Solution: From Eq. (1.3.4) we can write,
o
RTc

1  B

M  RT

c

The given data are plotted as shown in Fig. 1.3.3.
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Fig. 1.3.3 Plot of  o
Intercept 

 RTc 
Module 1: Lecture 3
versus concentration.
1
 0.0102 mol/kg
M
M  98.04 kg/mol
1.3.2.2 Donnan equilibrium
 Sometimes the molecular weight determined by osmometry of a colloid
containing macroion is found to be significantly lower than that obtained by other
methods. Since the macroion cannot pass through the membrane, its counterion(s)
also do not pass through the membrane to maintain the electroneutrality of the
solution.
 Osmotic pressure depends upon the number of solute particles. For a molecule
that dissociates as Na+ and I, the osmotic pressure is associated with two
particles. If the colloidal electrolyte were A +z I z , the osmotic pressure would be
associated with
 z  1
particles. The observed molecular weight from
osmometric measurement is the number average molecular weight of the
macroion and the ions dissociated from it. So, if we do not consider the presence
of the counterions, the molecular weight calculated from Eq. (1.3.4) would be
lower than the actual molecular weight.
 It has been observed that the osmotic pressure of the solution of a substance
containing the membrane-impermeable macroion is lowered if the other side of
the membrane contains the solution of a salt that has the same counterion as that
of the macroion. The reason for this behavior was explained by Irish chemist
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Frederick Donnan in 1911. Suppose that the charge of the macroion, I z  , is
balanced by Na+ ions. When the solution of I z  is kept separated from a NaCl
solution by a semipermeable membrane, it is expected that only the Na+ and Cl
ions would diffuse through the membrane because the membrane acts as an
impermeable barrier towards I z  . This perturbs the concentration distributions of
the small ions and gives rise to an ionic equilibrium which is different from the
equilibrium that would result if I z  were absent. The resulting equilibrium is
known as Donnan equilibrium. This equilibrium is very important for the
biological membranes.
 Suppose that a semipermeable membrane separates an aqueous NaCl solution
from an aqueous solution of an organic salt, Na  I  (Fig. 1.3.4). All ions except
I are permeable through the membrane. Both NaI and NaCl are fully dissociated
in the solutions. It can be easily shown [see Ghosh (2009)] that the ratio of the


observed osmotic pressure  oobs to the true osmotic pressure  o  is given by,
 oobs  c1  c2 


o
 c1  2c2 
(1.3.12)
Fig. 1.3.4 Partitioning of ions by membrane in presence of membrane impermeable
ion.
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If c1  c2 ,  oobs  o  1 . On the other hand, if c2  c1 , then  oobs  o  1 2 . If
c1  c2 , it is easy to see that  oobs  o  2 3 . Therefore, due to the unequal
distribution, addition of an electrolyte with common counterion will reduce the
observed osmotic pressure of the salt which is completely dissociated in the
solution, but contains a non-permeating ion. Thus, a large amount of salt can
‘swamp out’ the effect associated with the macroion.
1.3.3 Optical properties of colloids
 When a beam of light falls on a colloid dispersion, some of the light may be
absorbed, some part may be scattered, and the remaining part is transmitted
undisturbed through the sample. If light of certain wavelengths is selectively
absorbed by the particles, the dispersion appears to be colored. The wavelength of
visible light lies between 400 nm (violet) and 700 nm (red). That is the reason
why a gold sol is red when the particles are very fine, but the sol is blue when the
dispersed particles are bigger.
 When light impinges on matter, the electric field of the light induces an
oscillating polarization of the electrons in the molecules. The molecules then
serve as secondary sources of light, and subsequently radiate light. The shifts in
frequency, angular distribution, polarization, and the intensity of the scattered
light are determined by the size, shape and molecular interactions in the scattering
material. The most ubiquitous manifestation of light-scattering is observed when
a beam of light passes through a dark room. The air-borne dust particles are
responsible for this scattering.
 When a strong narrow beam is allowed to pass through a colloid system placed
against a dark background, bright specks or flashes of light can be observed
against the darkness when viewed from above through a microscope. The specks
of light change continuously in the dark field. This proves the heterogeneous
nature of the colloid dispersions, light-scattering ability of the colloid particles,
and their continuous motion. This effect is known as Tyndall effect (named after
the Irish physicist John Tyndall). The scattered beam is polarized, and its
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intensity depends upon factors such as the position of the observer, properties of
the system, and the wavelength of light.
 Quantitave studies of Tyndall effect and other optical properties of colloid
dispersions have become possible after the development of ultramicroscope.
Particles smaller than about 200 nm cannot be seen directly in ordinary
microscopes. However, these particles can scatter light which is visible in the
ultramicroscope. The presence of particles as small as 5–10 nm can be discerned
by ultramicroscope. A narrow beam of parallel or slightly convergent light (from
a powerful source such as an arc-lamp) is passed at right angles to the direction of
the microscope through a cell on which the instrument is focused. If no particle is
present in the cell, the field will appear completely dark. On the other hand, if the
cell contains a colloidal dispersion, the particles will scatter light, some of which
will pass vertically into the microscope. Each particle will appear as a small disc
of light on a dark background. A schematic of slit ultramicroscope is shown in
Fig. 1.3.5.
Fig. 1.3.5 Slit ultramicroscope.
The light-scattering by colloid particles can be divided into three categories.
Rayleigh scattering:
In Rayleigh scattering, the particles are small so that they
can act as point sources of scattered light. The size of the
particles is much smaller than the wavelength of light.
Debye scattering:
In Debye scattering, the particles are relatively large,
however, the refractive indices of the dispersed phase and
the dispersion medium are similar.
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Mie scattering:
Module 1: Lecture 3
In Mie scattering, the particles are relatively large
(diameter/wavelength ratio is close to unity), and the
refractive index of the dispersed phase is significantly
different from the dispersion medium.
 Rayleigh developed the light-scattering theory by applying the electromagnetic
theory of light to the scattering by small, non-absorbing spherical particles in a
gaseous medium. According to the Rayleigh theory, when an electromagnetic
wave falls on a small particle, oscillating dipoles are induced in the particle. The
particle then serves as a secondary source for the emission of scattered radiation
of the same wavelength as the incident light. According to the Rayleigh theory,
the scattering intensity is inversely proportional to the fourth power of the
wavelength. Therefore, blue light is scattered more than red light. If the incident
light is white, a colloid will appear to be blue if viewed at right angles to the
incident beam, and red when viewed from end-on. This phenomenon is
responsible for the blue color of the sky and the yellowish-red color of the rising
and setting sun.
 Some colloidal dispersions show very interesting light-scattering phenomena. For
example, Fig. 1.3.6 shows the Lycurgus Cup (in the British Museum) made of
ruby glass.
Fig. 1.3.6 The Lycurgus Cup (source: F. E. Wagner et al., Nature, 407, 691,
2000; reproduced by permission from Macmillan Publishers,  2000).
This cup dates from the Roman times. The glass appears green in daylight
(reflected light), but looks red when light is transmitted from the inside of the
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vessel. The red color of gold-ruby glass is caused by small particles of metallic
gold, which form when the gold-containing colorless glass is annealed. Modern
gold-ruby glass is usually made by adding gold to the raw materials as a solution
of tetrachloroauric acid or potassium dicyanoaurate. At about 1400°C, the gold
disperses in the melt, and the glass remains colorless after rapid cooling to
ambient temperature. Upon annealing between about 500–700°C, the red color
appears as a result of the nucleation and growth of small metallic gold particles.
The optimum size for yielding a strong ruby color is 5–60 nm.
 Colloidal dispersions sometimes appear turbid due to the scattering of light.
Solutions of some macromolecular materials may appear to be clear, but actually
they are slightly turbid due to weak scattering of light. When light is incident on a
perfectly homogeneous system, there is no scattering. Pure liquids, dust-free
gases and true solutions can approach this limit of no-scattering. The turbidity of
a material is defined by the expression,
I
 exp   l 
I0
(1.3.13)
where I 0 is the intensity of the incident light beam, I is the intensity of the
transmitted light beam, l is the length of the sample and  is the turbidity.
Equation (1.3.13) is valid for a non-absorbing system. The LambertBeer law
describes the intensity of the transmitted light when only absorption takes place,
but no scattering.
I
 exp   a l 
I0
(1.3.14)
where  a is the absorbance of the material. In a system where both absorption
and scattering are important, the following composite formula applies.
I
 exp     a    l 
I0
(1.3.15)
The experimental value of extinction is the sum of  a and  .
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Exercise
Exercise 1.3.1: The variation of osmotic pressure of a polystyrene solution in toluene
with its concentration at 298 K is given below.
c (kg/m3)
0.2
0.4
0.6
0.8
0.9
 o (cm of toluene)
0.04
0.09
0.16
0.22
0.28
Determine the molecular weight of the polymer from these data. Given: density of
toluene at 298 K = 860 kg/m3.
Exercise 1.3.2: Calculate the diffusion coefficient of a colloid particle of 5 nm diameter
in benzene at 300 K. What would be its mean square displacement based on Brownian
movement after 60 s? Estimate the diffusion coefficient of the particle at 310 K. Given:
viscosity of benzene at 300 K is 0.6 mPa s, and the same at 310 K is 0.5 mPa s.
Exercise 1.3.3: Answer the following questions.
a. What is Brownian movement? What type of particles exhibit such movement?
b. Explain EinsteinSmoluchowski equation.
c. How does the diffusion coefficient of a colloid particle depend upon the viscosity
of the liquid according to StokesEinstein equation?
d. Explain how you would determine Avogadro’s number from Brownian motion.
e. What is the origin of osmotic pressure? What is reverse osmosis?
f. Explain how you will determine the molecular weight of a macromolecular
colloid from osmotic pressure measurements.
g. What is number-average molecular weight?
h. Explain Donnan equilibrium.
i. What is the difference between Rayleigh scattering and Mie scattering?
j. What type of light scattering would you expect in pure liquid or air?
k. What is Tyndall effect?
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l. Explain the main features of ultramicroscope.
m. Explain why the blue light is more scattered than the red light.
n. Explain how the molecular weight of a macromolecule is related to the turbidity
of its solution.
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Suggested reading
Textbooks
 D. J. Shaw, Introduction to Colloid and Surface Chemistry, ButterworthHeinemann, Oxford, 1992, Chapter 2.
 P. C. Hiemenz and R. Rajagopalan, Principles of Colloid and Surface Chemistry,
Marcel Dekker, New York, 1997, Chapters 2, 3 & 5.
 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,
Chapter 1.
 G. J. M. Koper, An Introduction to Interfacial Engineering, VSSD, Delft, 2009,
Chapter 1.
 R. J. Hunter, Foundations of Colloid Science, Oxford University Press, New
York, 2005, Chapter 1.
Journal articles
 F. E. Wagner, S. Haslbeck, L. Stievano, S. Calogero, Q. A. Pankhurst, and K. -P.
Martinek, Nature, 407, 691 (2000).
 F. G. Donnan, J. Membr. Sci., 100, 45 (1995).
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