Sedimentation

[3a]      f_sedG = ( / 6) g (_s - _l) d_p³

We can create a larger sedimentation force using the centrifugal force in a centrifuge (in which the wall rotates about an axis and the fluid is relatively static with respect to the wall) or a cyclone (in which the walls are static and the fluid moves rapidly with respect to the walls). Centrifugal acceleration is related to the radial rotation rate = 2 R_RPM / 60 [rad/s], where R_RPM is the number of revolutions per minute, and to the distance from the axis of centrifugation to the particle r_cent [m]. The centrifugal sedimentation force is

[3b]      f_sed,cent = ( / 6) ² r_cent(_s - _l) d_p³

Not that if the liquid is more dense than the particle, the particles will rise (or move toward the axis of centrifugation) rather than sinking (or moving toward the outer wall).

Thermal Jostling

The energy distribution function (see discussion in Section 1) indicates that about 8 % of the pairs will collide with less than a quarter of the characteristic velocity (half the thermal energy) and 4 % will collide with more than four times the characteristic velocity (twice the thermal energy).

The thermal jostling due to numerous molecular impacts from the liquid moves the particle at random. This tends to break up weakly bonded flocs that have no primary minimum, but thermal jostling can cause a weakly flocculated system to form strong flocs if the height of the barrier is smaller than about 4kT. For such a low barrier, the thermal jostling can provide enough energy to push two weakly-bound particles over the barrier and close enough to fall into the primary well. A section in a later part will provide an equation relating the height of the barrier to the rate of flocculation.

Electrophoresis

      f_ion = d_p² _charge E_E

Since it is hard to measure _charge, you will often find that it has been replaced by an approximate relation to the zeta potential -- the electrical potential at the shear plane (Hiemenz, pages 738-757),

[3c]      _charge = _{0 l} K_ion ( / d_p)

where K_ion has a value between 1.8 and 6, depending on solution conditions. The shear plane is a (non-spherical) surface surrounding a moving particle, inside of which the fluid -- and the associated ionic atmosphere based on the particles surface charge -- tends to travel with the particle and outside of which the fluid tends to remain static with respect to the bulk fluid and the ions in that fluid are more uniformly distributed. It is not feasible to compute the exact boundary of the shear plane, but we know that it will move closer to the particle at high particle velocities through the liquid. It is a useful concept, but it is not easy to quantify. Combining equations we get

[3d]      f_ion = ² _{0 l} K_ion d_p E_E

Viscous Drag

[3e]      N_Re = v_p d_{p l} / _l

If the inertia of the particle is so small compared to drag that N_Re is less than 1 (the usual situation in dispersions of small particles), we are in the creeping regime, and the drag force, f_drag [N], may be computed using Stokes' Law,

[3f]      f_drag = 3 _l d_p v_p

Particle Motion and the Balance of Forces

[3g]      d_p = {18 _l v_sedG / [g (_s - _l)]}^0.5

so the distance moved in time t [s] may be computed from

[3h]      x_sedG = v_sedG t = g (_s - _l) d_p² t / (18 _l)

it is important to note that for these equations x and v are defined as being positive in the downward dirtection. This equation is a good approximation for dilute dispersions (in which the volume fraction of solids is below 0.01). I section in a later part will discuss settling at higher solids loadings.

The analysis of particle concentration during centrifugation is more complicated than the analysis of gravitational settling. In a centrifuge the particles follow radial (rather than parallel) paths, and as they move farther from the axis they experience higher centrifugal forces, making them move faster as they move outward.

In an electrical field gradient charged particles will accelerate until f_drag = f_ion. It is easiest to model this in zero gravity or in a case where the electrical field gradient is horizontal -- perpendicular to gravity). The electrophoretic mobility of a charged particle is the ratio of its terminal velocity v_ion [m/s] to the electric field gradient E_E [V/m]. The ratio can be used to determine using the equation based on this balance of forces,

[3i]      = 3 _l v_ion / ( _{0 l} K_ion E_E)

Electrophoretic motion is directed toward the charged plate, and the distance moved in time t [s] may be computed from

[3j]      x_ion = v_ion t = _{0 l} K_ion E_E t / (3 _l)

In the presence of thermal jostling particles wander randomly in the liquid, an effect called thermal diffusion or Brownian motion. Using a microscope you may observe thes in a good dispersion of particles with d_p 2m. The root-mean-square average distance x_therm [m] (based on many observations) that a particle will move away (in a random direction) from a position at time t = 0 in time t may be computed from

[3k]      x_therm = [_i=1ⁿ (x_i,t - x_i,0)² / n]^0.5 = [2 D_diff t]^0.5

where D_diff [m²/s] is the diffusion coefficient based on kinetic molecular theory. For spherical particles it is

[3l]      D_diff = kT / (3 _l d_p)

Example: the Relative Magnitude of External Forces

Gravitational sedimentation dominates the motion of large particles, while thermal jostling dominates for small particles. Particle charge can dominate only in the mid-size range. Figure 3-3 shows how far a particle would travel in one second as a function of particle diameter for three driving forces:
-- thermal jostling at 298K
-- sedimentation at Earth's gravity
-- electrophoresis in an electric field gradient of 25 V/m
for a particle with _s = 2,000 kg/m³ and = 0.1 V in a liquid with _l = 1,000 kg/m³, _l = 0.001 Pa s, and K_ion = 3.

Consider an organic powder with _s = 700 kg/m³ and d_p = 2 m dispersed in water with _l = 1 mPa s at T = 300K. This will have D_diff = 2.2 10^-13 s/m². For an observation interval of t = 1 s, x_RMS = 0.66 m. We can easily observe this degree of motion using a microscope and a magnification of 400. The thermal jostling would not be so obvious if the particle were 16 times larger (it would move only 1/4 as far) or if it were 9 times as dense (it would move only 1/3 as far). Although smaller particles move farther during the same time period, particles smaller than about 0.2 m are hard to see under a microscope because they scatter little light. However their motion can be measured using laser illumination and light scattering techniques.

Thermal Diffusion in a Gravitational Field

[3m]      C_m,h = B H C_m,ave e^-Bh / (1 - e^-BH)

at a rate that depends on the total distance from the bottom of the container to the surface of the liquid H [m] and on the ratio of net sedimentation force to thermal energy B [m^-1],

[3n]      B = g (_s - _l) d_p³ /(6 kT)

These same equations can be used to determine the compute the distribution of gas molecules as a function of height above a planet. The gravitational attraction for the molecules toward the planet's surface is balanced against thermal jostling in a small scale analog of the way that particulate settling is balanced against thermal jostling.

Example: Distribution of Silica Particles in a Drum

Note that the distribution shifts dramatically as the particle diameter changes by a factor of 17. The vertical line at 100 indicates the initial well-mixed and evenly-distributed state. A dispersion of 3 nm particles will remain relatively well-distributed throughout the height of the drum, a dispersion of 12 nm particles shows a clear buildup toward the bottom of the container, while a dispersion of 50 nm particles settles almost completely to the bottom.