Sedimentation equilibrium

Colloid

In a colloid, the colloidal particles are said to be in sedimentation equilibrium if the rate of sedimentation is equal to the rate of movement from Brownian motion. For dilute colloids, this is described using the Laplace-Perrin distribution law:

\Phi(z) = \Phi_0\exp\biggl(-\frac{m^*g}{k_BT}z\biggr)=\Phi_0e^{-z/l_g}

where

\Phi(z) is the colloidal particle volume fraction as a function of vertical distance z above reference point z=0,

\Phi_0 is the colloidal particle volume fraction at reference point z=0,

m^* is the buoyant mass of the colloidal particles,

g is the standard acceleration due to gravity,

k_B is the Boltzmann constant,

T is the absolute temperature,

and l_g is the sedimentation length.

The buoyant mass is calculated using m^*=\Delta\rho V_P=\frac{4}{3}\pi\Delta\rho R^3

where \Delta\rho is the difference in mass density between the colloidal particles and the suspension medium, and V_P is the colloidal particle volume found using the volume of a sphere (R is the radius of the colloidal particle).

Sedimentation length

The Laplace-Perrin distribution law can be rearranged to give the sedimentation length l_g. The sedimentation length describes the probability of finding a colloidal particle at a height z above the point of reference z=0. At the length l_g above the reference point, the concentration of colloidal particles decreases by a factor of e.

l_g=\frac{k_B T}{m^* g}

If the sedimentation length is much greater than the diameter d of the colloidal particles (l_gd), the particles can diffuse a distance greater than this diameter, and the substance remains a suspension. However, if the sedimentation length is less than the diameter (l_g), the particles can only diffuse by a much shorter length. They will sediment under the influence of gravity and settle to the bottom of the container. The substance can no longer be considered a colloidal suspension. It may become a colloidal suspension again if an action to undertaken to suspend the colloidal particles again, such as stirring the colloid.

Example

The difference in mass density \Delta\rho between the colloidal particles of mass density \rho_1 and the medium of suspension of mass density \rho_2, and the diameter of the particles, have an influence on the value of l_g. As an example, consider a colloidal suspension of polyethylene particles in water, and three different values for the diameter of the particles: 0.1 μm, 1 μm and 10 μm. The volume of a colloidal particles can be calculated using the volume of a sphere V=\frac{4}{3}\pi R^3.

\rho_1 is the mass density of polyethylene, which is approximately on average 920 kg/m3 and \rho_2 is the mass density of water, which is approximately 1000 kg/m3 at room temperature (293K). Therefore \Delta\rho=\rho_1-\rho_2 is -80 kg/m3.

Generally, l_g decreases with d^3. For the 0.1 μm diameter particle, l_g is larger than the diameter, and the particles will be able to diffuse. For the 10 μm diameter particle, l_g is much smaller than the diameter. As l_g is negative the particles will cream, and the substance will no longer be a colloidal suspension.

In this example, the difference is mass density \Delta\rho is relatively small. Consider a colloid with particles much denser than polyethylene, for example silicon with a mass density of approximately 2330 kg/m3. If these particles are suspended in water, \Delta\rho will be 1330 kg/m3. l_g will decrease as \Delta\rho increases. For example, if the particles had a diameter of 10 μm the sedimentation length would be 5.92×10−4 μm, one order of magnitude smaller than for polyethylene particles. Also, because the particles are more dense than water, l_g is positive and the particles will sediment.

Ultracentrifuge

Modern applications use the analytical ultracentrifuge. The theoretical basis for the measurements is developed from the Mason-Weaver equation. The advantage of using analytical sedimentation equilibrium analysis for Molecular Weight of proteins and their interacting mixtures is the avoidance of need for derivation of a frictional coefficient, otherwise required for interpretation of dynamic sedimentation.

Sedimentation equilibrium can be used to determine molecular mass. It forms the basis for an analytical ultracentrifugation method for measuring molecular masses, such as those of proteins, in solution.

References

References

1. "The Nobel Prize in Physics 1926".
2. (2012-06-27). "The unbearable heaviness of colloids: facts, surprises, and puzzles in sedimentation". Journal of Physics: Condensed Matter.
3. Batra, Kamal. "Role of Additives in Linear Low Density Polyethylene (LLDPE) Films".
4. (2014). "CRC handbook of chemistry and physics: a ready-reference book of chemical and physical data.".