Structure in the Liquid near a Surface

Molecular polarizability [C m² / V] is the parameter used to characterize the extent to which charges in a molecule can shift in response to an elecrtic field. The net polarizability includes contributions due to the ability of electron orbital populations to shift into excited polar states, vibrational modes populations to bend the molecule into more polar shapes, and dipolar groups to rotate to oppose an electric field.

[4a]      _net = _disp + _vib + u² / (kT)

where _elec [C m] is the electric dipole moment.

The experimental measure characterizing a material's response to an electric field is the dielectric permittivity relative to a vacuum . The equations which predict the rate at which electric field strength decreases with distance from the particle are generally expressed in terms of _l, but it is useful to know how this is related to the polarizability of the molecules in the liquid. If _elec is small, is related to by

[4b]      ( - 1) / ( + 2) = N_{0 net} / (3 ₀ M)

_disp is related to the refractive index n_RI measured in the visible region, substituting n_RI² for in the above equation. The value of _vib is not easy to determine, but it is usually small, so it is often crudely approximated as 0.1 _disp.

The multiplicative factor for converting values of and from cgs-esu units to SI units are not simple powers of ten. An of 1 Angstrom³ becomes 1.11 * 10^-40 C m² V and a of 1 Debye becomes 3.336 * 10^-30 C m.

The electric field near the particle's surface can hold polar molecules from the liquid as part of a relatively immobile, oriented sheath. If the particle is negatively charged, the positive end of a dipolar group will be held nearest to the particle. This reduces the translational and orientational freedom of the dipolar group, so the sheath has a higher viscosity and a lower dielectric constant than the bulk liquid has. The sheath increases the effective size of the particle and causes the electric potential to fall off more slowly than would be predicted using the bulk dielectric constant.

Effects of Non-Surfactant Solutes

• changing the pH to shift the ionization equilibrium
• chemically creating ionizable surface groups
• chemically deactivating ionizable surface groups
• changing the concentration of surfactant to shift the adsorption equilibrium
• changing to a different surfactant
• using a more strongly adsorbed surfactant to displace an unwanted adsorbate

The ions at the surface or solution fall into one of four categories:

• chemically bound, potential-determining ions -- H⁺ and OH^- from hydrolysis or ions from the particle's ionic lattice

• physically adsorbed (Stern layer) surfactant species --usually multiply charged non-surfactant counter-ions such as Ca⁺⁺ and SO₄⁼, but may be neutral species or co-ions

• locally concentrated counter-ions (Guoy-Chapman layer) --singly charged non-surfactant spectator ions such as Na⁺ and Cl^-

• locally depleted co-ions (also part of the Guoy Chapman layer) -- singly or multiply charged ions

pH and the Isoelectric Points of Metal Oxides

The electrophoretic velocity which a particle attains in an electric field gradient depends, NOT on the potential at the particle's surface ₀ [V], but on the potential at the shear plane (the zeta potential, [V]). The shear plane is an imaginary egg-shaped surface that represents the boundary between solvation-sheath liquid (which moves with the particle) and bulk liquid (through which the particle-plus-sheath is moving). In many cases the solvation sheath has a good balance of positive and negative ions, so ₀. It is common practice to assume that this approximation holds true and to use and ₀ interchangeably.

Fig. 4-1. The Shear Plane

An oxide or hydroxide surface can become charged by reacting with H⁺ or OH^- ions. The isoelectric pH is the pH at which = 0. To a first approximation this occurs when the surface charge density _charge [C/m²] is zero. At pH < pH_isoel the surface reacts with H⁺ ions to become positively charged, and at pH < pH_isoel the surface reacts with OH^- ions to become negatively charged. The isoelectric pH's of some common oxides and hydroxides are given in Appendix A.

may drop by as much as 50 mV per pH unit near the isoelectric pH. When is between +30 and -30 mV, the ionic repulsion barrier is so low that thermal jostling can push particles over the top and into the deep primary well, so within about half a pH unit of the pH_isoel, the particles are quite likely to coagulate.

The reaction of a hydroxide surface with water may be expressed as a series of hydrolyses (water-splitting reactions) in which the OH^- ions bind to the metal site and the H⁺ ions remain in solution. The series starts with the equation for the first OH^- ion added (j=1) to the bare (and positively charged) metal site and ends with the equation for the maximum of J OH^- groups at the site.

[4c]      M(OH)_j-1^J-j + H₂O M(OH)_j^J-j-1 + H⁺

The charge density due to surface hydrolysis _hydro [C/m²] is related to the surface density of metal sites _site [mol/m²] and the Faraday constant F = 96,487 C/mol by an equation with the general form

[4d]      _hydro = K_num F _site / K_den

For surfaces having a single hydroxyl per metal site (in the uncharged state) and a maximum of two hydroxyls per site,

      K_num = 10^-2pH - K₁ K₂
      K_den = 10^-2pH + K₁ 10^-pH + K₁ K₂

where K_j is the equilibrium constant for reaction with the j-th OH^- ion. Using the notation pK₁ = -log K₁ and pK₂ = -log K₂, we can compute

[4e]      pH_isoel = 0.5 (pK₁ + pK₂)

For surfaces with two hydroxyls per (uncharged) site and a maximum of three hydroxyls per site,

      K_num = 2 * 10^-3pH + K₁ 10^-2pH - K₁ K₃ K₃
      K_den = 10^-3pH + K₁ 10^-2pH + K₁ K₂ 10^-pH + K₁ K₂ K₃

For surfaces with three hydroxyls per (uncharged) site and a maximum of four hydroxyls,

      K_num = 3 * 10^-4pH + 2 K₁ 10^{-3 pH} + K₁ K₂ 10^{-2 pH} - K₁ K₂ K₃ K₄
      K_den = 10^-4pH + K₁ 10^-3pH + K₁ K₂ 10^-2pH
            + K₁ K₂ K₃ 10^-pH + K₁ K₂ K₃ K₄

The surface equilibrium constants for a real industrial material may depend on the history of the particle -- heat treatment, impurities, grinding, and aging under water. Therefore, although you may use a material with the same chemical name as was used in another study, you will likely get a different dependence of on pH unless both the particle synthesis and the slurry preparation are carefully repeated.

EXAMPLE

Figure 4-2 illustrates how (computed from _hydro) changes as a function of pH for a metal hydroxide with a single hydroxyl per neutral site and _site = 5 mol/m², for two sets of pK values, one set produces a smooth drop in surface potential with pH; the other set shows two drops separated by 3 units of pH.

Physical Adsorption of Ions

A later section on Surface Adsorption will discuss how the fraction of surface sites covered by adsorbed material is related to its concentration in solution. The contribution to charge density due to adsorbed ions _ads [C/m²] is related to the adsorbed ion's charge number and sign z_j and fractional surface coverage _j by

[4f]      _ads = F _site z_{j j}

The isoelectric pX is defined as the negative log-base-ten of the ion activity ratio, a_rat,X, at which = 0. The activity ratio is the effective concentration divided by the reference concentration. The reference concentration for defining pH and pX_isoel is 1 mol/L, which is 1000 mol/m³. To remind readers that a_rat,X is a dimensionless ratio and not a concentration (either mol/L or mol/m³) I have written the next two equations with the reference concentrations -- and their the units -- stated explicitly. In SI units we define pH and pX_isoel as

[4g]

where the activity (effective concentration) of X is that at which = 0. For dilute solutions, a_rat,X C_rat,X, so concentration may be used in place of activity.

Several cases are treated in a later section on Surface Adsorption:
-- the case where BOTH the surface and the adsorbing species are charged
-- the case where EITHER the surface OR the adsorbing species (but not both) are charged
Johnson (see reference list) compared theoretical predictions with measurements of as a function of both pH and salt concentration for the adsorption of inorganic ions on metal hydroxides.

The adsorption of a single layer of ions with a SINGLE CHARGE on surface sites with a SINGLE CHARGE (of the opposite sign) will neutralize (but will not reverse) the particle's charge. However, but the adsorption of a single layer of ions with MULTIPLE-CHARGES or MULTIPLE LAYERS of ions with a SINGLE CHARGE (which can occur for highly polarizable ions) CAN produce surface charge reversal. A later section on Surface Adsorption discusses multilayer adsorption.

Small adsorbed molecules may exhibit two-dimensional solid, liquid, or gas behavior. If the energy of moving from one site to another on the surface is comparable to thermal energy, thermal jostling will cause adsorbed molecules to hop from one site to another, so that the adsorbed molecules are mobile rather than fixed. If adsorbed molecules on neighboring sites attract one another strongly, the surfactant will adsorb in patches rather than randomly on the surface. If the forces are weak enough to permit some unfilled sites and surfactant movement within the patch, the behavior is a two-dimensional liquid; otherwise the patch is a two-dimensional solid.

A later section on Surface Adsorption discusses the factors that affect the configuration and multipoint adsorption of polymers (which may or may not have charged segments).

The Counter-ion Atmosphere, Ionic Strength

Debye and Huckel developed a theory for predicting the distribution of ions in solution about a central ion. Guoy and Chapman developed a similar theory for the distribution of ions in the solution surrounding a charged particle. The counter-ion atmosphere about a ion is called the Debye-Huckel layer, and the counter-ion atmosphere about a particle is called the diffuse Guoy-Chapman layer. Many texts describe the atmosphere in terms of an exponential decay factor, . I prefer using the reciprocal of this, the counter-ion atmosphere thickness, t_c [m], since it provides a more concrete image of the atmosphere. t_c is the distance at which shielding has reduced the effective field to 37% of its unshielded value.

A high concentration of salt permits a large concentration of counter-ions near the surface, so as the ionic strength, C_IS [mol/m³], increases, t_c decreases.

[4h]      t_c = [_{0 l} RT / (4 F² C_IS)]^0.5

The ionic strength C_IS [mol/m³ is formulated to take into account the relative electrical effectiveness (dependent on the square of their charge) of ions of different charge. Solutions of equal ionic strength (but different ionic composition) will affect the field surrounding charged particle in the same way.

[4i]      C_IS = 0.5 _j=1ⁿ z_j² C_j

where z_j is the number and sign of charges on species j, present at concentration C_j.

Fig. 4-3. Terms and Dependencies near Charged Particles

Several things are evident from these equations:

• t_c will decrease if we increase the salt concentration. We can destroy the stability of a charge-stabilized dispersion by a adding concentrated salt solution to reduce t_c and the height of the electrostatic repulsion barrier to the point that thermal jostling can push the particles into a attraction well.

• Multiply charged ions are be more strongly attracted to the charged surface than singly charged ions are. The t_c for a 0.01 molar MgSO₄ solution is only half as large as the t_c for an 0.02 molar NaCl solution (which has the same volume charge density), so we cannot replace NaCl by MgSO₄ in a formulation and expect to have similar electrostatic repulsion. Trace amounts of multiply charged ions can significantly diminish the effectiveness of ionic stabilization. A change from distilled to tap water or from tap water to river water can have disastrous consequences on a process if the change introduces small quantities of multiply charged ions into a slurry which is stabilized by particle charge.

• The dielectric permittivity of the liquid (related to its polarizability) affects t_c. The counter-ion atmosphere extends farther in liquids with low _l than in water, so the addition of a soluble alcohol to water will reduce the dielectric permittivity, increasing t_c and thus increasing the effective range and energy of electrostatic repulsion between particles. Such changes of liquid composition are rarely done because they usually shift other slurry properties (viscosity, flammability, salt solubility, pH) outside of acceptable limits.

Relating Surface Charge Density to Surface Potential

[4j]      _charge = _hydro + _ads

The Guoy-Chapman theory relates _charge to the surface potential ₀ [V] through (Hiemenz, page 700)

[4k]      ₀ = [2 RT / (z_j F)] sinh^-1 [_charge / (8 _{0 l} C RT)^0.5]

Here C is the concentration [mol/m³] of salt. For this particular model, the salt must consist of ions of equal and opposite charge. z_j should be limited to +1 or -1, since multiply charged ions tend to adsorb on the surface.

For small values of the surface charge (corresponding to ₀ < 26 mV), sinh^-1 y y, and

[4l]      _0,small [_charge / F] [RT /(2 _{0 l} C)]^0.5

Because ₀ and are often used interchangeably, the equations above are often used to predict the value of .

Mobility of Ions in the Ionic Atmosphere

[4m]      G_{charge atmos} = z_j F ₀

The charged particle will repel ions having the same charge as the surface (co-ions); it will attract counter-ions. The change in free energy that would occur if we could "turn off" the surface potential and permit the ions to diffuse back to their bulk homogeneous concentration C [mol/m³] is (Adamson, page 196)

[4n]      G_diff = -8 RT C t_c {cosh [z_j F ₀ / (2 RT)] - 1}

At small values of y, cosh y 1 + y² / 2. Using the definition of t_c we can write for small y,

[4o]      G_diff,small - F ² [_{0 l} C /(2 RT)]^0.5

This shows that G_diff is generally comparable to the thermal energy, RT, so the singly charged ions are NOT rigidly held, and can readily diffuse into and out of the counter-ion atmosphere.