Kinetic Properties of Colloids

The kinetic properties of colloids include all the motions of solute particles in the dispersion medium. The motions can be thermally induced (Brownian motion, diffusion, osmosis), gravitationally induced (sedimentation), or applied externally (viscosity).

We consider Electrically induced motion as well but it is well covered in the Electrical Properties of Colloids.

Let’s examine each one in detail.

Thermally-induced motion

Brownian movement

Brownian motion is a random zig-zag movement of colloidal particles arising due to the uneven distribution and collisions between solute-solute, solute-solvent.

But we cannot see the Brownian motion with our naked eyes because the colloidal particles are too small.

The velocity of particles depends on the following factors:

1. Particle size and molecular weight – Velocity increases as the particle size decreases.
2. The viscosity of the dispersion medium – We can increase the viscosity of the dispersion medium by adding agents like glycerin. The velocity decreases by increasing the viscosity.

Diffusion

It is the direct effect of the Brownian movement of colloidal particles.

A particle diffuses from a region of higher concentration to a region of lower concentration until the concentration of the system is uniform. This is called Diffusion.

Fick’s first law exactly defines the diffusion phenomena in terms of a mathematical equation.

By Fick’s first law, the change in concentration, dc, with distance traveled, dx, is directly proportional to the amount, dq, of substance that diffuses in time, dt.

We can write it as:

dq = -DS (dc/dx) * dt

where,

1. In terms of area per unit time, D is the diffusion coefficient.
2. dc/dx is the concentration gradient.

How to obtain diffusion coefficient D?

1. Perform diffusion experiment – Allow the material to pass through a porous disk, remove the sample and analyze periodically
2. Measure the change – When we bring solvent and colloidal solution together and allow them to diffuse, it forms a free boundry. Measure the change in concentration or refractive index of that free boundry.

Sutherland & Einstein equation

Sutherland and Einstein considered the colloidal particles as spherical and suggested an equation. We can use this equation to find out the radius and the molecular weight of the particle.

D = kT / 6 π ƞ r

But k = R / N

Therefore,

D = RT / 6 π ƞ r N

where,

1. D is the diffusion coefficient
2. k is the Boltzman constant
3. R is the molar gas constant
4. T is the absolute temperature
5. ƞ refers to the viscosity of the dispersion medium
6. r is the radius of spherical particle
7. N is the Avagadro’s number

As we discussed earlier, we can determine the diffusion coefficient in two ways. We can use this measured D to determine the molecular weight or radius of the spherical molecule by using the following formula.

The volume of a sphere V is,

V = 4/3 π r³ ———-(1)

Also, V = MƲ / N_A ———-(2)

1. where, Ʋ is the partial specific volume

Equate equation (1) & (2)

4/3 π r³ = MƲ / N_A

r³ = MƲ / π N_A * 3/4

By rearranging the equation,

r = ∛ 3M Ʋ / 4 π N_A

Substituting the value of r in the previous formula of D

D = R T / 6 π ƞ N . ∛ 4 π N_A / 3 M . Ʋ

We can evaluate three main rules of diffusion using the formulas mentioned above:

1. Velocity of molecules decreases with the increase in the particle size.
2. Velocity of the molecules increases with the increase in the temperature.
3. With the decreasing viscosity, velocity of the molecules increases.

Osmotic Pressure

Osmotic pressure is one of the most important kinetic properties of colloids because of its large applications in the biological and pharmaceutical fields.

Van’t Hoff equation describes the osmotic pressure, π of the colloidal solution. However, the equation is valid only for low concentrations of solute.

The equation is:

π = c R T

where,

1. c is the molar concentration of solute

By doing some modifications to this formula, we can find out the molecular weight of the colloidal particle.

We can replace c with c_g / M term where c_g is the grams of solute per liter of solution and M is the molecular weight.

The equation becomes,

π = c_g / M RT

Therefore,

π / c_g = RT / M ——For low concentration and ideal solution ——(1)

But for the high concentration of solute or real solution like linear lyophilic molecules, we need to induce one more term in our formula.

π / c_g = RT (1 / M + B c_g) ——- For high concentration and real solution —–(2)

Where B is the interaction constant. We have already discussed this term in our previous article.

So, what do we mean by a real colloidal solution?

Let’s take an example of a linear lyophilic polymer as a solute. These solute molecules get solvated in the solution eventually reducing the concentration of free solvent and increasing the solute concentration.

Therefore, we use the term B which estimates the asymmetry of particles and their interactions with solute.

We can determine the molecular weight of the molecule graphically using the osmotic pressure method.

As you can see in this graph, we have plotted π / c_g against c_g.

The results are one of the three lines, depending on whether the system is ideal (I) or real (II, III).

Equation (1) applies to the ideal system (I) and equation (2) applies to the real system (II, III).

Now let’s examine the three lines:

The extrapolated intercept in the graph is called the term RT / M. And if we know the temperature during the experimentation, we can determine the value of molecular weight.

Interaction constant (B): The slope is the interaction constant that is B. In II and III the slopes have some value because of high concentration and real system.

But in the first line (I), the slope B is zero because of low concentration and ideality.

Line III – Line III represents a typical linear colloid in a solvent having a high affinity towards the dispersed particles.

We consider such solvent as “GOOD” solvent for a particular colloid.

As it is not an ideal system and there is a significant deviation from ideality as the concentration increases, the B value also increases and so does the slope.

In such a case, type III lines can become non-linear. And we need to expand our second formula (2) to a power series:

π / c_g = RT ( 1/M + B c_g + C c_g² + ………)

where C is another interaction constant.

Line II – Here, the slope is lesser than that of line III.

Line II represents the same colloid as in line III but a different solvent having less affinity towards colloid and is referred to as “RELATIVELY POOR” solvent than line III.

Note –

The extrapolated line in both II and III originate from the same point on Y-axis.

This means the molecular weight of the colloid is independent of the solvent used.

But the interaction with the solvent will however differ which is represented by the different slopes.

Gravitationally induced

Sedimentation

Colloidal particles settle down under the influence of forces like a gravitational force. This is called sedimentation in colloids.

Sedimentation is one of the most important kinetic properties of colloids because pharmaceutical colloidal solutions require sedimentation studies for durability and various other reasons.

Stoke’s law explains the velocity at which colloidal particles (considering particles as spherical) will settle down with the following equation:

v = 2 (radius)² (ρ – ρ_o) g / 9 η_o

Where,

1. v is the velocity of sedimentation
2. g is the acceleration due to gravity
3. ρ is the density of colloidal particles
4. ρ_o is the density of the medium
5. η_o is the viscosity of the medium

Why do we use ultracentrifuge for sedimentation?

When the colloidal particles are under the gravitational force, the lower limit of particles obeying Stoke’s law is about 0.5 μm. This means below 0.5 μm, particles will not obey the Stokes equation. And this will result in the false estimation of results.

This happens because below 0.5 μm, the Brownian motion is dominant and gravitational force alone is not enough to overcome that motion.

Therefore, a stronger force is required which is measurable and quantitative to maintain the accuracy of the procedure.

That is why we use ultracentrifuge developed by Svedberg in 1925.

Modified Stokes equation for centrifugation sedimentation:

In the modified equation, acceleration due to gravity is replaced by ω²x, where ω is the angular velocity and x is the distance of the particles from the center of rotation.

The equation becomes,

v = dx / dt = 2r2 (ρ – ρ_o) ω²x / 9 η_o

Centrifugation speed is expressed in terms of revolution per minute or rpm.

Svedberg sedimentation coefficient s is used to express the instantaneous velocity, v = dx / dt.

s = sx / dt / ω²x

As of now, we have seen why and how we use the centrifugation technique to find out the molecular weight of the colloidal particle. However, there are two methods by which we can determine the molecular weight using centrifugation.

Sedimentation velocity technique / Schlieren pattern

As we discussed earlier, we are familiar with the Svedberg sedimentation coefficient. Now consider during the centrifugation process, a particle is moving from the position x₁ at time t₁ to the position x₂ at time t₂.

To find the sedimentation coefficient in this case, we need to integrate our previous formula.

So, the integrated formula is,

s = ln (x₂ / x₁) / ω² (t₂ – t₁)

The distances x₁ and x₂ indicate the positions of the solvent-high molecular-weight component interface in the centrifuge cell.

We can locate this boundary by the change in the refractive index attained at any time during the run.

This is then translated into a peak on a photographic plate.

We can take the photographs at definite time intervals so we will know the peak (that is x) at a particular time t.

These peaks are collectively called Schlieren patterns.

Suppose the sample consists of molecules having the same molecular weights. In this case, the Schlieren pattern will show only one sharp peak at any moment during the run.

But if the molecular weights are different, the pattern will show several peaks.

So, we can clearly see the degree of homogeneity in the sample along with the molecular weight.

So, from the Schlieren pattern experiment, we know the value of s. The angular velocity ω will be equal to 2π time the speed of the rotor in rps. From the diffusion data, we can get the value of diffusion coefficient D.

After knowing all these terms apply this final formula to get the molecular formula of the desired molecule:

M = R T_s / D(1 – Ʋ ρ_o )

1. R is the molar gas constant
2. T is the absolute temperature
3. Ʋ measures the partial specific volume of the colloidal molecule.
4. ρ_o is the density of the solvent

Note: Obtain s and D at 20^oC or correct the values to 20^oC

Sedimentation equilibrium technique

This is another method for determining the molecular weight of desired molecules.

In this technique, we let the system achieve sedimentation equilibrium.

The system establishes the sedimentation equilibrium when the sedimentation force is balanced by the counteracting diffusional force. The boundary is steady at equilibrium.

We don’t need to determine the diffusion coefficient. However, it might take several weeks to attain equilibrium.

But because of the technological advancements, newer methods are established to achieve the equilibrium in less time.

Externally applied motion

Viscosity

Under stress, a system’s viscosity reflects its resistance to flow.

As a result, the greater the viscosity, the greater will be the resistance to flow, eventually, reducing the movement of colloidal particles as they encounter more resistance.

Using these kinetic properties of colloids, we are interested in determining the molecular weights of colloidal particles comprising the dispersed phase in the system.

Along with the molecular weight, we can also determine the shape of the colloidal particle using the viscosity method.

Einstein developed an equation of flow that is only applicable to the dilute system of spherical colloidal particles. The equation is,

η = η_o (1 + 2.5φ)

This equation is based on the hydrodynamic theory. We can find η and η_o using a capillary viscometer.

where,

1. η is the viscosity of the dispersion
2. η_o represents the viscosity of the dispersion medium
3. φ is the volume fraction of colloidal particles in the dispersion

Note: The volume fraction (φ) is defined as the volume of particles divided by the total volume of the dispersion. Therefore φ is equivalent to the concentration term. Thus, the equation is as follows:

η_sp / c = k

Assuming that η_sp is the viscosity of the dispersion and c is the number of colloidal particles per 100 mL.

But for highly polymeric material dispersed in a medium at moderate concentration, we need to express the equation in a power series as follows:

η_sp / c = k₁ + k₂c + k₂c + k₃c₂

We can determine several viscosity coefficients concerning the Einstein equation like relative viscosity (η_rel), specific viscosity (η_sp), and intrinsic viscosity (η).

Relative viscosity: η_rel = η / η_o = 1 + 2.5φ

Specific viscosity: η_sp = η / η_o -1 = η – η_o / η_o = 2.5φ OR η_sp / φ = 2.5

How to find molecular weight using viscosity?

As we discussed earlier, we can determine η and η_o. After knowing these terms, we can determine the specific viscosity using the equation mentioned above.

Then find η_sp / c term.

Now plot η_sp / c against c and extrapolate the line to infinite dilution as mention in the following graph.

As you can see in this graph, after extrapolation, we get the intercept that is k₁.

Using the following Mark–Houwink equation, we can find the molecular weight of our colloidal molecule:

[η] = KM^a

The constants K and a are obtained initially by experimental determination of [η] for a polymer fraction whose molecular weight is determined by other methods like osmotic pressure, sedimentation, light scattering.

Once we know K and a, we can easily find out the molecular weight of the desired molecule.

How to find the shape of colloidal particles using viscosity?

Sphereocolloids forms dispersion of relatively low viscosity, whereas, the linear particles are more viscous.

When we put a linear colloid in a solvent having a low affinity towards colloid, tends to Ball-up. This means the colloidal particles start becoming spherical eventually reducing the viscosity.

So, by keeping an eye on the viscosity of the system, we can estimate the shape of our colloidal molecule.

Related Articles:

Stability of Colloidal Solution

Optical Properties of Colloids

Electrical Properties of Colloids