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 Laplace-Perrin distribution law:
Φ ( z ) = Φ 0 exp ( − m ∗ g k B T z ) = Φ 0 e − z / l g {\displaystyle \Phi (z)=\Phi _{0}\exp {\biggl (}-{\frac {m^{*}g}{k_{B}T}}z{\biggr )}=\Phi _{0}e^{-z/l_{g}}}
where
Φ ( z ) {\displaystyle \Phi (z)} is the colloidal particle volume fraction as a function of vertical distance z {\displaystyle z} above reference point z = 0 {\displaystyle z=0} ,
Φ 0 {\displaystyle \Phi _{0}} is the colloidal particle volume fraction at reference point z = 0 {\displaystyle z=0} ,
m ∗ {\displaystyle m^{*}} is the buoyant mass of the colloidal particles,
g {\displaystyle g} is the standard acceleration due to gravity,
k B {\displaystyle k_{B}} is the Boltzmann constant,
T {\displaystyle T} is the absolute temperature, and l g {\displaystyle l_{g}} is the sedimentation length. The buoyant mass is calculated using m ∗ = Δ ρ V P = 4 3 π Δ ρ R 3 {\displaystyle m^{*}=\Delta \rho V_{P}={\frac {4}{3}}\pi \Delta \rho R^{3}}
where Δ ρ {\displaystyle \Delta \rho } is the difference in mass density between the colloidal particles and the suspension medium, and V P {\displaystyle V_{P}} is the colloidal particle volume found using the volume of a sphere ( R {\displaystyle 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 {\displaystyle l_{g}} . The sedimentation length describes the probability of finding a colloidal particle at a height z {\displaystyle z} above the point of reference z = 0 {\displaystyle z=0} . At the length l g {\displaystyle l_{g}} above the reference point, the concentration of colloidal particles decreases by a factor of e {\displaystyle e} .
l g = k B T m ∗ g {\displaystyle l_{g}={\frac {k_{B}T}{m^{*}g}}}
If the sedimentation length is much greater than the diameter d {\displaystyle d} of the colloidal particles ( l g >> d {\displaystyle l_{g}>>d} ), 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 < d {\displaystyle l_{g}<d} ), 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.
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