Sedimentation equilibrium

Sedimentation equilibrium in a suspension of different particles, such as molecules, exists when the rate of transport of each material in any one direction due to sedimentation equals the rate of transport in the opposite direction due to diffusion. Sedimentation is due to an external force, such as gravity or centrifugal force in a centrifuge.
It was discovered for colloids by Jean Baptiste Perrin for which he received the Nobel Prize in Physics in 1926.
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 La Place-Perrin distribution law:
formula_1
where
formula_2 is the colloidal particle volume fraction as a function of vertical distance formula_3 above reference point formula_4,
formula_5 is the colloidal particle volume fraction at reference point formula_4,
formula_7 is the buoyant mass of the colloidal particles,
formula_8 is the standard acceleration due to gravity,
formula_9is the Boltzmann constant,
formula_10 is the absolute temperature,
and formula_11 is the sedimentation length.
The buoyant mass is calculated using formula_12
where formula_13 is the difference in mass density between the colloidal particles and the suspension medium, and formula_14 is the colloidal particle volume found using the volume of a sphere (formula_15 is the radius of the colloidal particle).
Sedimentation length.
The La Place-Perrin distribution law can be rearranged to give the sedimentation length formula_11. The sedimentation length describes the probability of finding a colloidal particle at a height formula_3 above the point of reference formula_4. At the length formula_11 above the reference point, the concentration of colloidal particles decreases by a factor of formula_20.
formula_21
If the sedimentation length is much greater than the diameter formula_22 of the colloidal particles (formula_23), 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 (formula_24), 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 formula_13 between the colloidal particles of mass density formula_26 and the medium of suspension of mass density formula_27, and the diameter of the particles, have an influence on the value of formula_11. 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 formula_29.
formula_26 is the mass density of polyethylene, which is approximately on average 920 kg/m3 and formula_27 is the mass density of water, which is approximately 1000 kg/m3 at room temperature (293K). Therefore formula_32 is -80 kg/m3.
Generally, formula_11 decreases with formula_34. For the 0.1 μm diameter particle, formula_11 is larger than the diameter, and the particles will be able to diffuse. For the 10 μm diameter particle, formula_11 is much smaller than the diameter. As formula_11 is negative the particles will cream, and the substance will no longer be a colloidal suspension.
In this example, the difference is mass density formula_13 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, formula_13 will be 1330 kg/m3. formula_11 will decrease as formula_13 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, formula_11is 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.