Lecture 05.

Lecture 05. Molecular-kinetic and optical properties of dispersed systems

 

Optical Properties of Colloids

 

Light Scattering

When a beam of light is directed at a colloid al solution or dispersion, some of the light is absorbed (colour is produced when light of certain  wavelengths is selectively absorbed), some is scattered and the remainder is transmitted undisturbed through the sample. Light scattering results from the electric field associated with the inci dent light including periodic oscillations of the electron clouds of the atoms of the material  in question—these then act as secondary sources and radiate scattered light.

 

The Tyndall effect-turbidity

 

All colloidal solutions are capable of scattering light (Tyndall effect) to some extent. The noticeable turbidity associated with many colloidal dispersions is a consequence of intense light scattering. A beam of sunlight is often visible from the side because of light scattered by dust particles. Solutions of certain macromolecular materials may appear to be clear but in fact they are slightly turbid because of weak light scattering. Only a perfectly homogeneous system would not scatter light; therefore, even pure liquids and dust-free gases are very slightly turbid. The turbidity of a material is defined by the expression where I₀  is the intensity of the incident light beam,  I_t  is the intensity of the transmitted light beam, l  is the length of the sample and  τ  is the turbidity.

There is a wide variation of the intensity of Tyndall cone for colloidal particles. Greater the difference between the refractive index of the dispersed phase and dispersing medium, more intense the cone is. When the difference between  the two phases is almost the same, the Tyndall cone will be very faint or entirely absent. This is  why lyophobic sols give more clearer cone than lyophilic systems. The Tyndall cone can be observed through ultra microscope as shown below.

A is an arc light whose beams are condensed by a lens B. This condensed beam passes through a slit C placed near the point of convergence to make the beam narrow to the desired extent. The beam is then passed through another converging le ns D in front of which is placed an objective lens  O/E  whose function is to reduce the image of the slit. The beam is focused by the lens D into the objective lens E before entering cell F. The light  scattered by the colloidal particle (kept in F) is observed by a microscope G at an angle of 90 °  with the direction of the beam. The whole thing is so adjusted that a sharp Tyndall cone is pr oduce inside F immediately below the microscope objective. The colloidal particles will appear as pin points twinkling and moving against the dark background of the dispersion medium. An ultra microscope can resolve particles as small as 3 mµ (3  ×  10-7 cm).

 

Particle size:  The size of the colloidal particle in sol may be calculated by counting the volume in a giveumber by means of ultra-microscope  and analyzing for the concentration of dispersed phase. If  n  be the number of particles per mL of sol,  m  be the total amount of dispersed phase per mL and d  be the density of dispersed phase then the volume of a single particle is  

Assuming that particles are cubes i.e. V =  13 (1 = length of one side or a cube)

Alternatively if we assume that particles are spherical then

 (r= radius of particle)

The radius of particle can be calculated.

 

Measurement of Scattered light

 

As we shall see, the intensity, polarisation and angular distribution of the light scattered from a colloidal system depends on the size and shape of the scattering particles, the interactions between them, and the difference between the refractive indices of the particles and the dispersion medium.

Light scattering offers the following advantage over some of the alternative techniques of particle-size analysis:

(1) It is absolute and no calibration is required.

(2) Measurements are made almost instantaneously, which makes it suitable for rate studies.

(3) There is no significant perturbation of the system.

(4) The number of particles involved is very large, which permits representative sampling of polydispersed samples.

The intensity of the light scattered by colloidal  solutions or dispersions of low turbidity is measured directly. A detecting photocell is mounted on a rotating arm to permit measurement of the light scattered at several angles, and fitted with a polaroid for observing the polarisation of the scattered light.

Weakening of the scattered beam itself as it passes through the slightly turbid sample can be neglected, and its  intensity can be compared with that of the transmitted beam.

Although simple in principle, light-scattering measurements present a number of experimental difficulties, the most notable being the necessity  to free the sample from impurities such as dust the relatively large particles of which would scatter light strongly and introduce serious errors.

Light-scattering theory

It is convenient to divide the scattering of light by independent particles into three classes:

 (1) Rayleigh scattering (where the scattering particles are small enough to act as point sources of scattered light).

(2) Debye scattering (where the particles are relatively large, but the difference between their refractive index and that of the dispersion medium is small).

(3) Mie scattering (where the particles are relatively large and have a refractive index significantly different from that of the dispersion medium.

Kinetic Properties of Colloids

Brownian motion and translational diffusion

Brownian motion: A fundamental consequence of the kinetic theory is that, in the absence of external forces, all suspended particles, rega rdless of their size, have the same average translational kinetic energy. The average transl ational kinetic energy for any particle is  3 ⁄ 2 kT or 1 ⁄ 2 kT along a given axis–i.e.1 ⁄ 2  m  ( d x/ dt )2  =  1 ⁄ 2 kT , etc.; in other words, the average particle velocity increases with decreasing particle mass.

The motion of individual particles is continually changing direction as a result of random collisions with the molecules of the suspended medium, other particles and the walls of the containing vessel. Each particle pursues a complicated and irregular zig-zag path. When the particles are large enough for observation, this random motion is referred to as  Brownian motion, after the botanist who first observed this phenomenon  with pollen grains suspended in water. The smaller the particles, the more evident is Brownian motion.  

Treating Brownian motion as a three-dimensional ‘random walk’, the mean Brownian displacement  x–    of a particle from its original position along a given axis after a time  t  is given by Einstein’s equation. 

where D  is the diffusion coefficient.

The diffusion coefficient of a suspended material  is related to the frictional coefficient of the particles by Einstein’s law of diffusion:

Therefore, for spherical particles,

where NA  is Avogadro constant, and 

The equilibrium distribution for particles of gamboge shows the tendency for granules to move to regions of lower concentration when affected by gravity.

Dynamic equilibrium is established because the more that particles are pulled down by gravity, the greater is the tendency for the particles to migrate to regions of lower concentration. The flux is given by Fick’s law,

where J = ρv. Introducing the formula for ρ, we find that

In a state of dynamical equilibrium, this speed must also be equal to v = μmg. Notice that both expressions for v are proportional to mg, reflecting how the derivation is independent of the type of forces considered. Equating these two expressions yields a formula for the diffusivity:

Here the first equality follows from the first part of Einstein’s theory, the third equality follows from the definition of Boltzmann’s constant as k_B = R_N, and the fourth equality follows from Stokes’ formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant R, the temperature T, the viscosity η, and the particle radius r, Avogadro’s number N can be determined.

The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by J. J. Thomson in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a concentration gradient given by Fick’s law and the velocity due to the variation of the partial pressure caused when ions are set in motion “gives us a method of determining Avogadro’s Constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other”.

An identical expression to Einstein’s formula for the diffusion coefficient was also found by Walther Nernst in 1888 in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the frictional force and the velocity to which it gives rise. The former was equated to the law of van ‘t Hoff while the latter was given by Stokes’s law. He writes for the diffusion coefficient k′, where is the osmotic pressure and k is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes’s formula for the viscosity. Introducing the ideal gas law per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein’s. The use of Stokes’s law in Nernst’s case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the mean free path.

Smoluchowski model

Smoluchowski‘s theory of Brownian motion starts from the same premise as that of Einstein and derives the same probability distribution ρ(x, t) for the displacement of a Brownian particle along the x in time t. He therefore gets the same expression for the mean squared displacement: . However, when he relates it to a particle of mass m moving at a velocity u which is the result of a frictional force governed by Stokes’s law, he finds

where μ is the viscosity coefficient, and a is the radius of the particle. Associating the kinetic energy with the thermal energy RT/N, the expression for the mean squared displacement is 64/27 times that found by Einstein. The fraction 27/64 was commented on by Arnold Sommerfeld in his necrology on Smoluchowski: “The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt.”

Smoluchowski attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal. In order to do so, he uses, unknowingly, the ballot theorem, first proved by W.A. Whitworth in 1887. The ballot theorem states that if a candidate A scores m votes and candidate B scores n−m that the probability throughout the counting that A will have more votes than B is

no matter how large the total number of votes n may be. In other words, if one candidate has an edge on the other candidate he will tend to keep that edge even though there is nothing favoring either candidate on a ballot extraction.

If the probability of m gains and n−m losses follows a binomial distribution,

with equal a priori probabilities of 1/2, the mean total gain is

If n is large enough so that Stirling’s approximation can be used in the form

then the expected total gain will be

showing that it increases as the square root of the total population.

Suppose that a Brownian particle of mass M is surrounded by lighter particles of mass m which are traveling at a speed u. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be mu/M. This ratio is of the order of 10⁻⁷ cm/sec. But we also have to take into consideration that in a gas there will be more than 10¹⁶ collisions in a second, and even greater in a liquid where we expect that there will be 10²⁰ collision in one second. Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the order of 10⁸ to 10¹⁰ collisions in one second, then velocity of the Brownian particle may be anywhere between 10 to a 1000 cm/sec. Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.

Gravitational motion

In stellar dynamics, a massive body (star, black hole, etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars. The rms velocity V of the massive object, of mass M, is related to the rms velocity of the background stars by

where is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both and V.^[22] The Brownian velocity of Sgr A*, the supermassive black hole at the center of the Milky Way galaxy, is predicted from this formula to be less than 1 km s⁻¹.^[

Colloidal suspensions are dispersions of small particles, ranging in size from 1 nanometer to 1 micrometer, in a solvent. The following image shows a colloid suspension of coenzyme 10:

Colloidal suspensions differ from solutions in that the substance used as a solute is appreciably larger than the solute of a solution, as the photo shows. In addition, the components are mixed manually since physical interactions are not strong enough to cause the solute to dissolve. A phenomenon known as Brownian motion keeps the particles suspended in the solvent. Brownian motion is the constant bombardment of the colloid particles by the solvent molecules. This action prevents the colloid particles from settling out of the solvent. More importantly, Brownian motion contributes to the diffusion of colloid particles through the solution. In 1905, Albert Einstein investigated this phenomenon and derived a general expression for the diffusivity of colloid particles in solvents. This derivation differs significantly from that given by Bird, Stewart, and Lightfoot, but the end result is the same.

Derivation of Diffusivity:

Suppose a colloidal suspension is contained in a cylinder with a unit cross-sectional area. Diffusion of the colloid particles is occurring throughout the entire length of the cylinder. Let’s examine a small volume element of the cylinder, dx. Pressure is being applied on the volume element at x and x+dx. The net pressure in the positive direction can be given by:

 This pressure is equivalent to the osmotic pressure. If this net pressure is divided by the length of the volume element, dx, the result is the net force on the element per unit volume, K:

 In the limit that the volume element has an infinitesimal length, dx–> 0 and:

The osmotic pressure is given by the van’t Hoff equation:

where c is the concentration of the solute in the solvent, R is the gas constant, and T is the absolute temperature of the solution. Due to diffusion in the cylinder, the concentration of the solute depends upon the position, x. Differentiating the van’t Hoff equation with respect to x gives:

 The osmotic pressure force will impart a velocity on the volume element. Since the colloid particles are fairly large, they will experience a drag force, decreasing their velocity. Assuming that the colloid particles can be modeled as hard spheres of a set radius, the velocity of the suspension is scaled by the Stokes friction coefficient:

 where mu is the viscosity of the solvent and Ra is the radius of the colloid spheres. This leads to an expression for the velocity of the solution per unit volume:

 This expression applies to the entire volume element, so it must be divided by the number of particles in the volume element to be applicable to a single particle:

 

If each side of the equation is divided by the concentration gradient term, the resulting equation gives the expression for the diffusivity:

 If the Boltzmann constant is defined as R/N, the expression is identical to that given by Bird, Stewart, and Lightfoot:

 Using this equation, the diffusivity of any colloidal suspension can be determined if the temperature of the suspension, viscosity of the solvent, and radius of the colloid particle are known as the following example shows.

Numerical Example:

Determine the diffusivity of a suspension of water and coenzyme 10 at 20 C. Assume that the average radius of a coenzyme 10 particle is 75 nm.

Solution: Maple can be used to easily solve this problem

The diffusivity of the colloidal suspension is given by the Einstein equation derived above

Define the known constants in the equation

 k:=1.38066e-16*erg/K; T:=293.15*K; Ra:=75*nm;

The viscosity of water at 20 C is given in Table

Evaluate Dab

Now convert all units to C.G.S. units

Now evaluate Dab again with all units converted

This answer is reasonable for a colloidal suspension because the size of the colloid particles results in a small flux, as the small value of the diffusivity indicates.

 

References:

 1.The abstract of the lecture.

2. intranet.tdmu.edu.ua/auth.php

3. Atkins P.W. Physical chemistry. – New York. – 1994. – P.299-307.

4. en.wikipedia.org/wiki

5.Girolami, G. S.; Rauchfuss, T. B. and Angelici, R. J., Synthesis and Technique in Inorganic Chemistry, University Science Books: Mill Valley, CA, 1999

6.John B.Russell. General chemistry. New York.1992. – P. 550-599

7. Lawrence D. Didona. Analytical chemistry. – 1992: New York. – P. 218 – 224.

8. www.youtube.com/watch?v=iEJSOhFaHos

9. www.youtube.com/watch?v=BsClg6z_PSw

10. www.youtube.com/watch?v=O3_hpvYlavA

11. www.youtube.com/watch?v=E9rHSLUr3PU

Optional:

·       D.M. Ruthven Principles of Adsorption & Adsorption Processes John Wiley & Sons, New York, 1984.

·       Yiacoumi, S. and Tien, C. Kinetics of Metal Ion Adsorption from Aqueous Solutions: Models, Algorithms, and Applications.  Kluwer Academic Publishers, Norwell, 1995.

·       Cohen, Y. and Peters, R.W. Novel Adsorbents and Their Environmental Applications
AIChE, New York, 1995. 

·       Adsorption and its applications in industry and environmental protection”, Vol. I-Applications in Industry,1061 pages.

·       Adsorption by powders and porous solids F. Rouquerol, J. Rouquerol and K Sing Academic Press, 1999.

·       D.J. Shaw: Introduction to Colloid and Surface Chemistry, 3^rd Ed., Butterworths, London 1980, ISBN 0-408-71049-7

·       F. MacRitchie: Chemistry at Interfaces, Academic Press, San Diego, 1990, ISBN 0-12-464785-5

·       J.N. Israelachvili: Intermolecular and Surface Forces, 2^nd Ed., Academic Press, New York, 1991, ISBN 0-1237-5181-0

·       C.J. van Oss: Interfacial Forces in Aqueous Media, Marcel Dekker, New York, 1994, ISBN 0-8274-9168-1

·       S. Hartland, R.W. Hartley: Axissymmetric Fluid-Liquid Interfaces, Elsevier, Amsterdam, 1976, ISBN 0-4444-1396-0

·       K.L. Mittal (Ed): Adsorption at Interfaces, 5^th Ed., Wiley Interscience, New York, 1990, ISBN 0-8412-0249-4

·       D.K. Chattoraj, K.S. Birdi: Adsorption and the Gibbs Surface Excess, Plenum Press, New York, 1984, ISBN 0-306-41334-5

·       M.J. Rosen: Surfactants and Interfacial Phenomena, 2^nd Ed., John Wiley, New York, 1989, ISBN 0-471-83651-6

·       M.J. Rosen (Ed.): Structure performance relationships in surfactants, ACS, Washington, DC, 1984, ISBN 0-8412-0839-5

·       S.A. Safran: Statistical Thermodynamics of Surfaces, Interfaces, and Membranes, Addison-Wesley, Reading MA, 1994, ISBN 0-201-62633-0

·       L.E. Schramm: Dictionary of Colloid and Interface Science, 2^nd Ed., Wiley-Interscience, New York, 2001, ISBN 0-471-39406-8

·       Journal of Colloid and Interface Science, Academic Press, New York, ISSN 0021-9797

·       Colloids and Surfaces A – Physicochemical and engineering aspects, Elsevier, Amsterdam, ISSN 0927-7757

Prepared by PhD Falfushynska H.