Gravitational Sedimentation Methods
In gravitational sedimentation methods, the particle size is determined from the settling velocity and the undersize fraction by changes of concentration in a settling suspension. The equation relating particle size to settling velocity is known as Stokes law [ISO 13317 Part 1: General principles and guidelines]:
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were xSt is the Stokes diameter, h is viscosity, u is the particle settling velocity under gravity, rs is the particle density, rf is the liquid density and g is the gravitational acceleration.
The Stokes diameter is defined as the diameter of a sphere having the same density and the same velocity as the particle settling in a liquid of the same density and viscosity under laminar flow conditions. Corrections for the deviation from Stokes’ law may be necessary at the coarse end of the size range. Sedimentation velocities acc. To Equ. 1 are limited to sizes above a µm. For smaller sizes the onset of thermal diffusion (Brownian motion) has to be corrected.
An experimental problem is to obtain adequate dispersion of the particles prior to a sedimentation analysis. For powders that are difficult to disperse, the addition of dispersing agents is necessary, along with ultrasonic probing. It is essential to examine a sample of the dispersion under a microscope to ensure that the sample is fully dispersed.
Equations to calculate size distributions from sedimentation data are based on the assumption that the particles sink freely in the suspension. In order to ensure that particle-particle interaction can be neglected, a volume concentration below 0.2% is recommended.
There are various procedures available to determine the changing solid concentration of a sedimenting suspension:
In the pipette method, concentration changes are monitored by extracting samples from a sedimenting suspension at known depths and predetermined times. The method is best known as Andreasen modification [Andreasen, Kolloid-Z., 39, 253 (1929)], shown in FIG. 1. Two 10ml samples are withdrawn from a fully dispersed, agitated suspension at zero time to corroborate the 100% concentration given by the known weight of powder and volume of liquid making up the suspension. The suspension is then allowed to settle in a temperature-controlled environment, and 10ml samples are taken at time intervals in geometric 2:1 time progression starting at 1 minute (i.e. 1, 2, 4, 8, 16, 32, 64 minutes). The amount of powder in the extracted samples is determined by drying, cooling in a desiccator, and weighing. Stokes diameters are determined from the predetermined times and the depth, with corrections for the changes in depth due to the extractions. The cumulative mass undersize distribution comprises a plot of the normalized concentration against the Stokes diameter. A reproducibility of ±2% is possible using this apparatus. The technique is versatile in that it is possible to analyze most powders dispersible in liquids; its disadvantages are that it is a labor-intensive procedure, and a high level of skill is needed [ISO 13317 Part 2: Fixed Pipette method].
The hydrometer method is simpler in that the density of the suspension, which is related to the concentration, is read directly from the stem of the hydrometer while the depth is determined by the distance of the hydrometer bulb from the surface (ASTM Spec. Pub. 234, 1959). The method has a low resolution but is widely used in soil science studies.
FIG. 1 Equipment used in the pipet method of size analysis
In gravitational photo sedimentation methods the change of the concentration with time and depth of sedimentation is monitored using a light point or line beam. These methods give a continuous record of changing optical density with time and depth and have the added advantage that the beam can be scanned to the surface to reduce the measurement time. A correction needs to be applied to compensate for a deviation from the laws of geometric optics (due to diffraction effects the particles cut off more light than geometric optics predicts). The normalized measurement is a Q2(x) distribution [coming ISO 1337 Part 4: photo gravitational method].
In gravitational X-ray sedimentation methods, the change of the concentration with time and depth of sedimentation is monitored using an X-ray beam. These methods give a continuous record of changing X-ray density with time and depth and have the added advantage that the beam can be scanned to the surface to reduce the measurement time. The methods are limited to materials having a high atomic mass (i.e. X-ray opaque material) and give a Q3(x) distribution directly [ISO 13317 Part 3: X-ray gravitational technique].
Sedimentation Balance Methods
In sedimentation balances the weight of sediment is measured as it accumulates on a balance pan suspended in an initial homogeneous suspension. The technique is slow due to the time required for the smallest particle to settle out over a given height. The relationship between settled weight (P), weight undersize (W) and time (t) is given by the following equation.
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