Educ. Reso. for Part. Techn. 024Q-Nelson
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Copyright © 2002 Ralph Nelson, Licensed to ERPT

Dispersing Powders in Liquids, Part 2, by Ralph D. Nelson, Jr.

-- 2: Interaction between Two Particles --

Comparing Atom and Particle Interactions

The same phenomena that control the interactions between atoms also control the interactions between particles and also control the adsorption of surfactants or ions on particle surfaces. Steric (superposition) repulsion, resonance attraction, ionic repulsion, counterion atmosphere shielding, and thermal jostling are concepts common to both the atom (nanometer) and particle (micrometer) size ranges. The major differences are that
  • Isolated atoms are separated by a (non-polarizable) vacuum rather than a polarizable fluid. Shielding is not a factor in atom-atom interactions. But the atoms or ions on the far side of one particle are shielded from interacting with a second particle not only by the first particle's bulk, but also by the intervening liquid, whose electrons can shift (become polarized) to partially offset the electrostatic fields produced by the particles, so for particles the properties of the fluis are important

  • Atoms have a single center of action for electromagnetic interactions and we express their interactions in terms of the distance between the atoms' centers x [m]. But particles consists of many atoms, each of which acts as a separate center for electromagnetic interaction, so we express particle interactions using both the separation between the surfaces along the line-of-centers s [m] and also the diameters of the two particles dp and dq [m]. The center-to-center separation x [m] is related to these parameters by x = s + (dp + dq)/2.

  • Ionic atmospheres cannot form about isolated ions in a vacuum because there is no intervening medium to prevent ions from coming into direct contact, thus neutralizing the charge. But ions in a solution will diffuse through a liquid so as to concentrate in the vicinity of an oppositely-charged particle and to avoid the vicinity of a similarly-charged particle. Tightly-bound solvent molecules and thermal jostling may prevent oppositely-charged ions from coming into contact with and thus neutralizing surface ions. The result of this response of solution ions to a charged particle is that the electric potential caused by a particle's surface charge will drop more rapidly with distance from the particle than if the solution contained no ions.

Steric Repulsion (Superposition)

Several simple models have been developed to help us understand and predict the steric repulsion properties of particles.

The hard spheres model is used for large particles (over 10 micrometers in diameter). Here we ignore attraction and repulsion and set u = 0 when the particles are not in contact (s > 0) and u = (infinite repulsion) when they come into contact (s 0). The particles do not attract or repel; they simply bounce off one another without deformation or loss of kinetic energy. This model is used for dry particles flowing in chutes.

The sticky sphere model adds a new parameter -- the sticking coefficient. A small fraction of the collisions between particles (chosen at random) cause the two colliding particles to stick together. Conversely, a small fraction of collisions that involve bound particles (chosen at random according to the sticking coefficient) result in breakup.This model is used for fluidized powders that are being pneumatically conveyed in a humid environment.

For small particles attraction gradually increases as the particles approach one another, but at close distances there is a rapid buildup of steric repulsion.

The elastic sphere model is useful for small particles in which the atoms on the surface of a particle are displaced backward elastically to distort the lattice of other atoms in the solid and T. The steric repulsion rises less sharply for particles than it does for atoms. An exponent of eight is convenient, but somewhat arbitrary. We may write the equation as

[2a]      up,super = Kp,reso / s8

Note that since the repulsion energy goes to infinity as the separation between surfaces goes to zero, the surfaces can never come in contact, just as one would expect from superposition repulsion forces. This model should not be applied to interactions of loosely-bound floc particles, for which longer-range repulsions produce a considerably weaker bonding that can deform significantly or breakup during a collision.

The hairy sphere model is usefiul for cases where surfactant molecules that are partly anchored to a particle surface and partly solvated produce a long-range steric repulsion. For such a situation the dependence of inter-particle potential energy on surface separation depends on surfactant-liquid interactions and polymer chain configurations. These topics are best treated by thermodynamics, and they are discussed in a later section.

Resonance, or Polarizability Attraction

The resonance attraction energy up,reso [J] arises from interactions between not only the electron orbitals in the two particles but also the electron orbitals in the liquid between the particles. The general equation for the interaction may be written as

[2b]      up,reso = - AHplq S{s, dp, dq} / 12

The scaling factor, AHpdq [J], is commonly called the Hamaker constant . It carries three subscripts because its value depends on the properties of the two particles (p and q) and of the liquid (l).

The distance-dependence function S [dimensionless] is a function of the surface separation s [m] and of both particle sizes, dp and dq. When the particles are so close together that s dp we can use the approximation S dp / s,

For the case of two identical particles Fowkes developed an approximation for the Hamaker constant

[2c]      AHplp = (AHp0p0.5 - AHl0l0.5)2

See Section 8, which discusses interactions between unlike particles, provides several methods for estimating AHplq from experimental data, and describes the dependence of S on surface separation and particle properties in more complex situations.

Combining steric repulsion and polarizability attraction


The particle version of the Lennard-Jones potential for atoms is the "Elastic Sphere Hamaker" equation. For the simple case of two same-diameter particles made of the same material

[2d]      uESH = Kp,reso (spit / s )8 - AHplp dp / (12 s)

The potential goes through a minimum or maximum at a separation for which du/ds = 0. In this case it is a minimum that we shall call a pit. Taking the derivative of the above equation and setting the left side to zero gives

[2e]      Kp,reso = AHplp dp / (12 spit)

Substituting this back into uESH we get

[2f]      uESH = [AHplp dp / (12 spit)] [0.125 (spit / s)7 - 1]

It is difficult to estimate or to measure the value of spit in practical cases due to surface roughness, adsorbed materials, and inadequate theoretical models, so many people simply set it at a "reasonable value" such as 0.4 nm.

In most real cases we must consider additional repulsion due either to zeta potential (see following sub-section) or to steric repulsion of adsorbed molecules (see a later section). These wil cause the minimum of the total energy between the two particles to move to a different position -- which cannot be determined from a simple equation.

Example: Two Neutral TiO2 Particles

For TiO2: M = 0.0799 kg/mol. At 298K s = 4,200 kg/m3, disp,s = 0.090 J/m2. For comparison with the charged case (in the next sub-section we shall use particles with dp = 150 nm. We shall assume that spit = 0.4 nm.
For water: M = 0.018 kg/mol, l = 1,000 kg/m3, disp,l = 0.021 J/m2, l = 80

You may download and use the computer program Nelsbp01.tru, which allows you to plot the same graph with these or other parameters values

Using these values, we get AH1l2 = 3.49 * 10-21 J, tc = 1.22 nm, Kp,super = 1.36 * 10-20 J, and the potential energy curves shown below.


Figure 2--1 Potential Energy of a Pair of Neutral TiO2 Particles
in water at room temperature

The characteristic collision or vibrational energy at thermal equilibrium is kT. At room temperature (298K) this is 4.11 * 10-21 J. Note that for our example two particles would be bound much more strongly than two argon atoms, but the range of motion during vibration of the TiO2 particles at 298K would be comparable to that of the argon atoms at 58K -- about 0.2 nm.

Ionic Repulsion

The best formulation for the ionic repulsion of particles is the DLVO theory, named after its developers -- Derjaguin, Landau, Verwey, and Overbeek (Parfitt, pages 30-37). This theory incorporates the shielding effects of the counterion atmosphere formed from ions in solution. The governing differential equations are not simple analytic forms, so we generally use approximations that cover specific typical cases.

The ionic repulsion scaling factor is expressed in terms of the particle's diameter d [m] and surface potential 0 [V], or zeta potential [V]. Section 3 discusses the relationship between0, , and surface charge density [C/m2]. The dimensionless separation dependence is expressed in terms of the separation of the surfaces s [m] and the counterion atmosphere thickness tc [m]. Section 3 also discusses the physical meaning of tc and gives the equations relating it to the concentrations of ions in solution.

Hunter (1981, Appendix 5) tells how to compute the ionic repulsion for a wide range of conditions. For two particles with the same d and in an aqueous solution at moderate ionic strength, both d and s are much larger than tc, and the ion repulsion energy may be approximated by

[2g]      uDLVO,aq 0 l d 2 ln [1 + exp(-s /tc)]

For particles in organic liquids, the counterion atmosphere is typically very large, so dp << tc and

[2h]      uDLVO,org 0 l d 2 ( dp / [ s + dp] ) exp(-s / tc)

is a factor between 0.6 and 1 and is based on numerical integration of the differential equations that define the repulsion.

Electric and Magnetic Dipole Interactions

Particles with a net electric dipole moment are uncommon. Differential charging of the crystal faces may occur if there are differences between those faces in the distribution of lattice ions, the adsorption of surfactant, or the hydrolysis of surface groups. In some cases, part of the surface of a positively charged particle is bare and part is covered by adsorbed polymer with a high negative charge (so that the covered region has a net negative charge), creating a charge-patch particle.

Permanent dipoles, differential face charges, or charge-patches produce strong electric fields that cause strong electrostatic interactions with liquid, solutes, surfactants, or other particles. Since particles rotate slowly compared to molecules, they are not quantum rotors, and the equation given for molecular dipoles does not apply. The potential energy of interaction for two identical particles of low polarizability in a liquid of low polarizability may be either an attraction or a repulsion. If the dipole-polarizability term is NOT negligible, the equation becomes much more complex.

[2i]      up,elec = Korient elec2 / (4 0 l x3)

Korient is a factor which varies from +2 when the dipole vectors are parallel (repelling) to -2 when they are anti-parallel (attracting). Dipole-dipole interaction provides a longer range attraction than resonance does (since up,elec depends on x-3 while up,reso depends on x-6).

Transition metal oxide particles commonly have unpaired electrons. These magnetic particles are usually suspended in a medium with a low magnetic permeability, so only the dipole-permeability interaction is negligible and only the magnetic dipole-dipole interaction is significant. The interaction energy of a pair of identical magnetic spheres with magnetic moment mag [Wb m] in a liquid with magnetic permeability l relative to free space and separated by distance x [m] is

[2j]      up,mag = Korient mag2 / (4 0 l x3)

Korient here is the same as for electric dipoles. For cubic particles so small that the entire particle is a single magnetic domain, the magnetic moment is related to the saturation magnetism Msat [T] and particle diameter as

[2k]      mag = (0.5/ 48) Msat dp3

For Fe3O4, Msat 0.45 T.

Hair and Croucher (their pages 543ff) discuss magnetic particles in more detail.

Hydrogen Bonding

For particles this is best treated as a specific interaction. It is too complex to be handled here.

Example: Two Charged TiO2 Particles

We can get a curve that illustrates a double minimum in the potential by adding to the parameters used for Fig 2-1 the values = 0.020 V for the TiO2 particles and have NaCl (with zc = 1) in the solution at a concentration of C = 10 mol/m3 NaCl. Note that 1 mol/m3 = 0.001 mol/L.

You may download and use the computer program Nelsbp01.tru, which allows you to plot the same graph with these or other parameters values


Figure 2--2 Potential Energy of a Pair of Charged TiO2 Particles
in a dilute salt solution at room temperature

The horizontal line labelled "298K thermal" indicates how close together
and how far the two particles typically move due to collisions with water
molecules with a thermal jostling energy of 298K (room temperature).


Figure 2--3 same as Fig. 2-2 with a larger scale for separation

Note the intermediate peak (caused by ionic repulsion) acts as a barrier that keeps particles that are bound in the shallow secondary well from getting close enough to fall into the deep primary well. Particles that are weakly bound might be broken free by an external force such as that applied by liquid shear (stirring). Particles in the primary well are much more tightly bound and thus much less likely to be de-flocculated by shear. Particle interaction can change significantly as the ionic repulsion is modified by changes in salt concentration (or ionic strength, which affects tc) or in zeta potential (often affected by pH or adsorption of multiply-charged ions). As the ionic repulsion increases first the primary well disappears and then the secondary well.

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