Sedimentation methods are based on the application of Stokes' Law, which describes the terminal velocity for an isolated sphere settling in a viscous fluid under the influence of a gravitational field (i.e., free falling). For low Reynolds numbers (i.e., laminar flow conditions), the terminal velocity depends on the density contrast between the particle and medium, the viscosity, and the particle size.
Applications of sedimentation can be grouped into cumulative and incremental techniques. In the cumulative method, the rate at which the particles settle is determined, typically by weighing the mass of settled particles at a certain depth over time. In the incremental method, the change in concentration or density of the material with time is measured at known depths, typically using optical or X-ray sensing. Because of the faster analysis time and better suitability for automation, incremental methods have found greater commercial application. In particular, the X-ray sedimentation (XGS) method is well established in several industries. In XGS, the absorption of a thin horizontally collimated X-ray beam is determined at a known depth and time, and referenced to the absorption of the pure solvent. The absorption caused by the presence of the particles at a specified depth at a given time is proportional to the concentration (by mass) of particles at that depth having a diameter smaller than dS, where dS is the calculated Stokes diameter. By measuring the particle density at different depths and times, a cumulative mass-averaged distribution is generated.
Stokes' Law is valid only if the Reynolds number (Re) does not exceed about 0.25 (in order for the error in the Stokes' diameter not to exceed about 3 %) . Since Re is a function of particle size, this permits calculation of a well-defined upper size limit for sedimentation methods. Portland cement is commonly assigned a density of 3.2 g/cm3. This value does not consider that different particles will have different density values depending on their composition, it simply applies a single mean density value for the powder. It also does not consider that variations in cement formulations will undoubtedly impact the average density. For purposes of estimating the upper limit, we will use a value of 3.2 g/cm3. The largest (spherical) particle diameter that can be sized accurately using XGS is therefore about 95 µm in isopropyl alcohol at 25 ºC. Particles larger than this value will settle more slowly than the velocity predicted by Stokes' Law. The upper size limit can be increased by using a suspending fluid with a higher viscosity. Irregularly shaped particles should settle according to their equivalent spherical volume at very low Re values. At higher Re values, the drag force is greater for asymmetric particles compared to spherical particles of an equivalent volume, and so the settling rate will decrease relative to Stokes' Law. For fine particles, the effect of Brownian motion exerts a significant influence on settling at diameters below about 1 µm in water and about 0.7 µm in isopropyl alcohol. Convection currents in the settling suspension may further limit the lower size range. Although instrument manufacturers frequently claim a lower limit of 0.2 µm, XGS results for particles smaller than a micrometer must be viewed with a critical eye.
The only parameters required for XGS are the density of the solid and liquid phases, and the viscosity of the pure liquid. Particle concentrations required for XGS analysis depend on the X-ray absorption properties of the solid, but for ceramics are typically around 1 % to 3 % in terms of volume fraction. The correct sample concentration will reduce the X-ray beam intensity by roughly 20 % to 30 % . The particles must remain stable against agglomeration during settling. An unstable suspension will exhibit a distribution skewed toward larger sizes. Dispersing agents, coupled with pH adjustment, are commonly utilized to increase stability in aqueous systems.
Gravitational sedimentation has limited practical value for particles under a few micrometers in diameter due to the prohibitively long settling times. It is not uncommon for a single size distribution analysis to require six hours or more, depending on the finest size fraction present. By replacing the gravity field with a centrifugal field, smaller particle sizes can be analyzed in much shorter periods of time. The underlying principles for centrifugal sedimentation are largely the same as in the gravitational case, but the calculations and measurement geometry are more complex because the particle velocity increases with distance from the center of rotation. Since most commercial instruments based on the centrifugal method are designed principally for the analysis of very fine particulates, the upper size range tends to be rather low (typically between 2 µm and 10 µm). On the low size end, the analysis range can extend down to about 0.01 µm, depending on the speed of the centrifuge and the sensitivity of the detection system. Centrifugal systems use either light or X-ray detection to determine particle concentration as a function of distance.