6. Summary and Conclusions
6.4. Modeling of Hematite Stability Ratios
Hematite stability ratios as a function of pH are modeled by applying the DLVO theory of colloid stability (Sec. 2.3). The stability ratio, W, given by Eq. (2.31), has been calculated, numerically. For convenience the relevant equations from Chapter 2 are given below. The stability ratio is calculated from,
w=Γ (Γ∏⅛exp <v^kτ> d"'∕Γ (⅛ “p <ι^kτ> a*' <6∙2>
where, u' = H'∕(d + a), and H, — H — 2δ. The function β is given by,
= 6⅛3<z + 2
v , 6u2 + 4u ' j
The total interaction energy, Vp, is evaluated as the sum of the van der Waals attraction energy, V4, and the repulsive electrostatic energy, Vr. If s = R/a and A is the Hamaker constant of the particle,
va = -Δ
a 6
2 2 ∕32-4
H—V + In I---- 5—
>2 - 4 (6∙4)
and,
—<π⅜> J
1 + R 1 -al eXP ~ 2a'^
(6.5)where a, = a + <5, and Y = 4tanh(^FΦ⅞∕4J⅞T,).
The Hamaker constant, A, for hematite in water is reported to lie in the range 2 to 6×10^^2° Joules (see Table 6.4). In our model calculations, a value of 6×10-2°
Joules is used for hematite in water.
The potential Φ15 is taken as equal to the ((-potential, which is derived from mobility data by applying Wiersema’s approach (Appendix B). This substitution is
Reference Hamaker Constant (J)
Medium Visser (1972) 10.6-15.5×10-2θ Vacuum
Fowkes (1965) 4.5 × 10-2° Water
Fowkes (1967) 3.4 × 10~2° Water
Dumont and 15.2 × 10~2θ Vacuum
Watillon (1971) 6.3 × 10"2θ Water
Matijevic (1981) 6.2 × 10~2θ Vacuum
4.4 × 10~2θ Water
Penners (1985) 4.0 × 10-2° Water
Breeuwsma (1973) 2-4×10-2θ Water
Table 6.4: Hamaker constants for hematite.
justified if the d-plane coincides, or nearly coincides, with the slipping plane. The distance, δ, is defined as the Stern layer thickness. It is often taken as the radius of the adsorbed species. Since chloride is the counter ion in our acidic media (pH<pHzpc) for stability ratio determinations, δ is assumed to equal the chloride ion radius, i.e.
1.81Â. The distance from the surface to the slipping-plane must lie at least a distance equivalent to the size of the adsorbed ions away from the surface, because mobility is measured at a physical surface. Hence, the distance from the surface to the slipping- plane is not the same as that to the Stern plane. To distinguish these distances, we use t to denote the distance from the surface to the slipping plane. Mpandou and Siffert (1984) studied the adsorption of propionate on T1O2 and measured the mobility in the presence of propionate. They found that the slipping plane is 4Â (the length of the tail group: -CH2CH3) away from the Stern plane. The value of 4Â is probably reasonable for the molecule they considered, but is too large for our case.
Lyklema and de Wit (1978) used i=5.4Â for K+ to model their iron oxide potential at the critical coagulation concentration. As an approximation, we assume that the slipping plane is 3.6Â away from the surface.
Thus, Φ∕ was used in place of Φ⅛, and a value of t = 3.6Â replaced δ in calculating k⅛∙ Η was assumed that Φ⅛ = ζ, where ζ was evaluated from mobility data (Appendix
B). The result of the model is compared with the experimental values in Fig. 6.2, where the modeled stability ratio as a function of pH is drawn as a solid line and the experimental results are plotted as triangles. There is a good agreement between the experiment and model calculation. Since the surface potential is dependent on proton concentration, log IF as a function of pH can be viewed as reflecting the variation in the stability ratio as a function of surface potential.
At a constant pH of 10.5, hematite surfaces maintain approximately a constant potential, and we can model the stability ratio as a function of Na+ concentration.
Substituting Φi = —32 mV, and t = 4.0Â, the stability ratios are calculated and experimental and model results are shown in Fig. 6.3. It can be seen that a good agreement is obtained between the results of the experiments and the model. The reduction in the stability ratio illustrated in Fig. 6.3 is due to the compression of the diffuse layer as the sodium concentration increases.
On the basis of model calculations we can characterize the relationship between stability ratio and aqueous species concentration as follows.
(1) Specifically adsorbed species affect surface potential. Hence the variation of sta
bility with concentration reflects the dependence of the stability ratio on surface potential.
(2) Non-specifically adsorbed species do not affect surface potential, but influence the stability ratio by compressing the diffuse layer.
(3) When the concentration of specifically absorbed species exceeds the background ionic strength the stability ratio is influenced by both changes in the surface potential and compression of the diffuse layer.
In the presence of the specifically adsorbed species, such as phosphate, the sta
bility ratio as a function of ion concentration exhibits type (1) characteristics, if the background ionic strength is constant. If the potential determining ion concentra
tion is raised above the background electrolyte concentration, then the stability ratio
pH
Figure 6.2: Stability ratio of a hematite suspension as a function of pH. Filled trian
gles are experimental values, and the solid line represents a DLVO model calculation.
Φ⅛ equals the ^-potential given in Appendix B. Ionic strength=0.05M, t = 3.6Â, Hamaker constant, A = 6 × 10-2° Joules.
will exhibit type (3) behavior. For example, the dependency of the stability ratio on phosphate concentration at pH 6.55 (see Fig. 4.10) reflects a strong dependence of the stability ratio on surface potential. The change in surface potential caused by the adsorption of phosphate was calculated by applying the diffuse layer model (Fig. 5.5) and the relation of surface potential to ζ--potential (or Φ⅛) was discussed in
Figure 6.3: Stability ratio of a hematite suspension as a function of sodium concentra
tion at pH 10.5. Open triangles are experimental values, and the solid line represents a DLVO model calculation. Φ∕ = —32mV, t = 4.0Â, Hamaker constant, A = 6×10-2°
Joules.
Sec. 5.1.3. The value of the stability ratio is dependent on the potential, Φi, which can be estimated from the mobility data in the presence of the corresponding amount of phosphate.
It is assumed in the calculation of V⅛ that the ions within the diffuse layer ad
just themselves fast during particle encounters, so that the potential Φ< is unaltered.
Although the input parameters of the model are not precisely known, reasonable stability ratios are predicted when the potential Φi is derived from mobility measure
ments, and t is estimated from the size of the adsorbed ions.
The mobility and stability ratio of hematite particles are dependent on adsorption.
When strongly adsorbed species are present in the hematite suspension, the surface chemical properties are influenced by adsorption; consequently a shift of pHiep is observed in the mobility. The adsorption of a strongly adsorbed ion is able to reverse the surface potential (e.g., at pH 6.5, phosphate adsorption can change positive surface potential to negative). The stability ratio as a function of ion concentration shows a characteristic “V” shape, where the stability is due to a positive potential on one side of the “V” and a negative potential on the other side. Adsorption plays a key role in determining the observed coagulation rates and mobility of hematite particles.