Sedimentation equilibrium

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

${\displaystyle l_{g}={\frac {k_{B}T}{m^{*}g}}}$

If the sedimentation length is much greater than the diameter ${\displaystyle d}$ of the colloidal particles (${\displaystyle l_{g}>>d}$), 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 (${\displaystyle l_{g}<d}$), 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.[2]

The difference in mass density ${\displaystyle \Delta \rho }$ between the colloidal particles of mass density ${\displaystyle \rho _{1}}$ and the medium of suspension of mass density ${\displaystyle \rho _{2}}$, and the diameter of the particles, have an influence on the value of ${\displaystyle l_{g}}$. 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 ${\displaystyle V={\frac {4}{3}}\pi R^{3}}$.

${\displaystyle \rho _{1}}$ is the mass density of polyethylene, which is approximately on average 920 kg/m³ [3] and ${\displaystyle \rho _{2}}$ is the mass density of water, which is approximately 1000 kg/m³ at room temperature (293K).[4] Therefore ${\displaystyle \Delta \rho =\rho _{1}-\rho _{2}}$ is -80 kg/m³.

${\displaystyle l_{g}}$ for different sizes of polyethylene and silicon particles

Diameter ${\displaystyle d}$ (μm)	${\displaystyle l_{g}}$ for polyethylene particles (μm)	${\displaystyle l_{g}}$ for silicon particles (μm)
0.01	-9.84×106	5.92×105
0.1	-9840	592
1	-9.84	0.592
10	-9.84×10−3	5.92×10−4

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

In this example, the difference is mass density ${\displaystyle \Delta \rho }$ is relatively small. Consider a colloid with particles much denser than polyethylene, for example silicon with a mass density of approximately 2330 kg/m³.[4] If these particles are suspended in water, ${\displaystyle \Delta \rho }$ will be 1330 kg/m³. ${\displaystyle l_{g}}$ will decrease as ${\displaystyle \Delta \rho }$ 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, ${\displaystyle l_{g}}$is positive and the particles will sediment.