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The sedimentation method of particle size analysis applies to dispersions in which the effect of the Brownian motion is negligible. This condition can be expressed by using the dimensionless Péclet number (for example, Ramaswamy S 2001), defined as follows:
| Pe = vS a / DBr | (1) |
where vS [m s-1] is the terminal (Stokes) settling velocity of a particle with half-size a [m], and DBr is the diffusion coefficient (see Dynamic light scattering). The settling velocity, vS, of a solid spherical particle in still fluid is expressed by:
| vS = [g / (18 ηf )] D2 (ρ - ρf ) | (2) |
where g [m s-2] is the acceleration due to gravity, ηf is the dynamic viscosity [kg m-1 s-1] of the dispersion fluid, ρ [kg m-3] is the density of the particle, and ρf [kg m-3] is the density of the dispersion fluid. It thus follows from Eq. 1 and the definition of the diffusion coefficient (Eq. 1 in Dynamic light scattering) that the Péclet number for a spherical particle equals:
| Pe = 4π Δρ g a4 / ( 3 kBT ) | (3) |
where Δρ [kg m3] = ρ - ρf is the differential density of the particle (a difference of the particle density, ρ, and that of the surrounding fluid, ρf ), g [m s-2] is the gravity acceleration, a [m] is the particle radius, kB [J K-1] is the Boltzmann constant, and T [K] is the absolute temperature of the dispersion.
The Péclet number, Pe, can be interpreted as a ratio of energy, Esed, gained by a particle with buoyant mass, mb, settling at a distance equal to its radius, a, in the gravity field, i.e.:
| Esed | = mbga | |
| = (4/3)π Δρ a3 ga | (4) |
to the thermal energy, Etherm = kBT. Thus, if Pe >> 1, the effect of the Brownian motion is negligible. It follows from Eq. 3 that the critical value of the sphere radius, a, i.e. a value that yields Pe = 1, is:
| a = [3 kBT / ( 4π Δρ g )]1/4 | (5) |
For a polystyrene sphere in water (Δρ ~ 50 kg m3, see Polystyrene) at a temperature of 23°C, the critical radius, a ≈ 1.2 µm. We have Pe ~ 5 x 10-5, for an a = 0.1 µm polystyrene sphere (hence, Brownian diffusion dominates) and Pe ~ 5030 for an a = 10 µm sphere (hence, sedimentation dominates).
Note that the Péclet number criterion (Eq. 5) applies to particles under the influence of any force, not only that exerted by gravity. In order to evaluate the relative importance of the Brownian motion in another force field, one simply needs to replace the acceleration due to gravity, g, in Eq. 5 with the acceleration representative of the force field in question.
| CITATION: Jonasz M. 2006. Sedimentation vs. Brownian motion (www.tpdsci.com/Tpc/SdmBrMtn.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 23-Feb-2006 Modified: 25-Oct-2008 Peer-reviewed: 09-Sep-2008 |
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