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.

-- 1: Introduction, Two-atom Interactions --

Introduction to the Topic

What forces cause agglomeration or repulsion of particles? How are they influenced by the surrounding liquid and solutes? How do surface reactions and adsorbed species modify the forces? This chapter provides an introductory discussion at the atom-atom and particle-particle level, as though we were similar in size to a water molecule and could walk on the surface of a particle or swim through the liquid surrounding it to observe the surrounding ions, measure the forces, and sense the electric potentials.

Several distinct approaches are used in developing models for particle interactions.
-- For the interaction of two particles it is customary to present equations showing how the potential energy of interaction u [J] varies with some length dimension x [m] that provides the simplest representation of the potential energy. For atomic (or molecular) interactions the selected length dimension is the distance between their centers, while for particles it is the distance between particle surfaces.
-- For the response of a particle to an external field it is customary to present equations in terms of the force f [N] rather than the potential energy, and the selected length dimension is the position of the particle along the direction of the field gradient. Since force and potential energy are related as f = du/dx, f may be derived from the slope of the curve when u is plotted against x. A positive slope indicates an attraction force, while a negative slope indicates a repulsion force.
-- For the thermodynamic properties of a macroscopic system (as we do in Chapter 5) it is customary to use statistical mechanics to determine the collective effect of many billions of two-body interactions and to characterize the bulk behavior using thermodynamic parameters rather than molecular or particulate parameters.

Section 1 describes the types of force that arise between two atoms or molecules in a vacuum. Section 2 extends this discussion to (larger size) particles in a fluid. Section 3 describes how particles move through a fluid in reponse to external fields. Section 4 describes how the surrounding solution can affect both the surface charge density on particles and the distribution of counter-ions and electrical potential as a function of distance from the particle surface. Specific sources are cited only for derivations that are not often presented in introductory texts on physical chemistry.

Comments on Units: The move from the cgs-esu to SI as a system of units has created some problems in comparing equations in various textbooks. If you plan to compare equations in this chapter to equations in textbooks that use the cgs-esu system, be sure to read the discussion at the start of Section 7.

The Interaction of Two Atoms

Superposition and Resonance

A collision between two atoms causes overlap or superposition of their electron orbitals. This produces a potential energy that varies inversely as the 12-th power of x,

[1a]       usuper = Ksuper / x12

Since du/dx is negative this causes a repulsion force. The above expression may also be used for simple, compact molecules for which the assumption of a centralized repulsion force is reasonable. For more complex molecules, bond stiffness and rotational flexibility become important, and the effect, often called steric repulsion, becomes a complicated function of the specific conformation during a collision.

Quantum mechanical resonance interactions cause an attraction between atoms, sometimes called the polarizability attraction or dispersion attraction or London force attraction. This energy varies as the 6-th power of the separation of centers of the electron orbitals,

[1b]       ureso = Kreso / x6

The value of Kreso is NEGATIVE, and since du/dx is positive this causes an attraction force.

The sum of superposition and resonance energies is often called the Lennard-Jones 6-12 potential. The parameters are reformulated in terms of two scaling factors that are based on the location of the minimum in the potential energy curve (xmin, umin) -- see Figure 1--1.

[1c]       uLJ = - umin [(xmin / x)12 - 2 (xmin / x )6]

Note that the value for umin is NEGATIVE (it is below the zero line) so the minus sign in front makes the overall scaling factor positive. The expression in square brackets is positive (indicating repulsion) at small x and negative (indicating attraction) at large x.

For a simple substance such as argon one can estimate the position of the minimum in the energy-separation curve from the boiling temperature and density as follows:
-- At the boiling point the separation of atomic centers, xmin, is approximately the cube root of the volume per atom
-- At the boiling point the thermal jostling energy, ujost = kT just balances the attraction energy, umin (so their sum is 0).
This leads to the following equations:

[1d]       xmin [M /(N0 boil)]1/3 ,       umin -kTboil.

where M [kg/mol] is the molar mass and boil [kg/m3] is the density of the liquid at the boiling point, Tboil [K]

We may then use these to compute the constants in the superposition and resonance energy equations as

[1e]       Ksuper = - umin xmin12

[1f]       Kreso = 2 umin xmin6

Remember that umin has a negative value, so Ksuper will have a positive value and Kreso will have a negative value.


Determining Kreso from Spectral Data

Resonance interactions are related to transitions between quantum states, and these transitions are the basis for the electromagnetic adsorption spectrum of the material. The full spectrum for a liquid or solid can be quite complicated, but the polarizability is often dominated by a single large adsorption band in the visible region. If we assume that all of the polarizability comes from a single adsorption band we can approximate the dependence of refractive index, nRI [dimensionless], on frequency [Hz] using a simple dispersion formula

[1g]        (nRI02 - 1) / (nRI2 - 1) 1 - (/e)2

Measured values of refractive index at two frequencies on the side of the major adsorption band may be used to determine nRI0, the refractive index at zero frequency, and e [Hz], the principal electronic transition frequency NOTE: This oversimplified formula predicts negative values of refractive index in the spectral region just above the resonance frequency, so the points chosen for analysis should be above the resonance frequency.

Comparisons of different materials is more meaningful if they are described by their molecular polarizability, [C m2/V] rather than nRI0. The relationship is

[1h]        = [3 0 M / (N0 ] [(nRI02 - 1) / (nRI02 + 2)]

Hiementz (his page 627) uses and e to estimate the resonance interaction constant as

[1i]       Kreso = (-3/4) h e ( /[4 0] )2

How can we determine e and nRI0 from measurements? In order to minimize the impact of experimental uncertainties we should measure the refractive index at two known frequencies somewhere on the side of the adsorption band where the refractive index is changing significantly with frequency. We can rearrange the equation to get

[1j]        e2 = 2 [ 1 + (nRI2 - 1) / (nRI02 - 1)]

The right-hand side at one frequency 1 equals the right-hand side at a second frequency 2. Solving this for nRI0 gives

[1k]        nRI02 = (B - A C) / (A - 1)

where A = (1 / 2)2 (nRI,22 - 1) / (nRI0,12 - 1)], B = nRI,22 - 2, and C = nRI,12 - 2

An alternative method: If refractive index measurements are not available, may be estimated as the sum of atom and molecular structural contributions [tables are in many texts on physical chemistry], while e may be estimated either from the ionization energy, as Eionize = he, or from the equation for harmonic oscillation of the electron, me (2 e)2 = e02, where me [kg] and e0 [C] are the mass and charge of an electron (Hiemenz, page 627).

Ionic Repulsion

The ionic repulsion energy between two equivalent ions, each of which has an excess of z fundamental charges, is

[1l]       ucharge = z2 e02 / (4 0 x)

In the case of ionized atoms and molecules the ionic repulsion is much larger than the resonance attraction, even for singly-charged ions with a high molecular polarizability, so ions bearing the same sign of charge never agglomerate.

Electric and Magnetic Dipole Interaction

Molecules whose centers of positive and negative charge are not co-incident are called polar and the molecule has an electric dipole moment, elec [C m], equal to the magnitude of the positive charge multiplied by the separation of centers of charge. In addition interaction energies arising from superposition and resonance dipolar molecules interaction energies airsing from both electric dipole-polarizability (first term below) and dipole-dipole interactions (second term below). In the case of two identical dipolar molecules that are free to rotate into the ideal orientation (anti-parallel, dipoles parallel to one another but pointing in opposite directions) and are in a vacuum, the interaction energy is negative (indicating attraction) and is defined (Hiemenz, pages 622ff) as

[1m]        ufreerot = - 2 [ + elec2 / (3 kT) ] (4 0 elec)2 / x6

Magnetic interactions are the result of interactions between magnetic dipoles, which are created by unpaired electrons. The magnetic field can interact with magnetic permeability or other magnetic dipoles. The equations are similar to those for electric dipole interactions. Few stable molecules have unpaired electrons, so this interaction is not considered further here.

Hydrogen Bonding

Hydrogen bonding is the result of a special quantum mechanical interaction between a hydrogen atom attached to a highly electronegative atom (such as oxygen or fluorine), X-H, and the unshared pair of electrons on a nearby atom, :Y. A hydrogen bond is much weaker than a covalent bond, but stronger and more dependent on the X-H...:Y angle than a mutual polarization attraction would be. Hydrogen bonding is important in many particle-surfactant and particle-liquid interactions. Since the magnitude of this interaction drops off rapidly with distance, its effect on the system is generally computed by treating it as a chemical complexing reaction or a surface adsorption reaction.

Collision Energy, Thermal Jostling Energy

The kinetic energy present in the collision of two atom is

[1n]       ucoll = M vloc2 / (2 N0)

Here vloc [m/s] is the relative velocity of the atoms toward one another along the line-of-centers. While the atoms may each have large absolute velocities, vloc will be small if they are moving in nearly parallel paths. The colliding atoms will stick together as a vibrating, rotating pair only if ucoll + umin < 0. The pair may be broken apart if hit by another atom which has a high ucoll.

The thermal jostling energy of a material is a measure of the statistical average kinetic motion (of the atoms that make up the system) with respect to the center of mass of the system. I chose the term jostling because it reminds us that the particles are moving within the system and bouncing off of their neighbors. Every atom does not have the same kinetic energy, so you might think it would be hard to characterize the energy of a collection of particles; but for many systems the kinetic energy is distributed among energy states according to the Maxwell-Boltzman distribution, allowing the system to be characterized simply by its temperature. The fraction of pairs having collision energies greater than any specified value (fGTx) follows the co-error function (erfc). Discussion of these is beyond the scope of this tutorial.

The characteristic one-dimensional energy of a single atom in a box at thermal equilibrium is characterized by ucoll,char = kT/2, so the characteristic collision energy of two atoms is characterized by twice that,

[1o]       ucoll,char = kT

Combining [1n] and [1o] and solving for velocity we get a measure of the average (over many collisions) velocity along the line of centers,

[1p]       vloc,char = (2 kT N0/ M)0.5

The energy distribution function 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).

Example: Two Argon Atoms

Argon is an atomic gas with M = 0.040 kg/mol. At one atmosphere pressure it boils at Tboil = 87 K and the liquid has boil = 1,400 kg/m3.
Substituting these in [1d] gives xmin = 0.36 nm, and umin = -1.20 * 10-21 J and the following graph:
Figure 1--1 Potential Energy of a Pair of Argon Atoms
showing the range of motion of a pair of atoms with vibrational energy comparable to thermal energy for 58K (see text below)

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

The horizontal line labelled "58K thermal" indicates how close and how far apart two atoms would go if a bound pair had a vibrational energy equal to the thermal jostling energy at 58K (2/3 of Tboil for argon at one atmosphere pressure).

The total energy curve for two singly-charged argon ions would be dominated by ionic repulsion, which is much higher than the sum of the superposition and resonance energies for x > xmin. The value of uion calculated at xmin would be 684 * 10-21 J, far above the range of -3 to 3 * 10-21 J shown in Figure 1--1.

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