The interaction forces between colloidal particles
6.3 Theory of colloidal interaction for ionizable materials
Theory of colloidal interaction for ionizable materials 141
-10 -5 0 5 10
0.01 0.1 1 10
-10 -5 0 5 10
0 5 10 15 20
F/R(mN/m)
D (nm)
c=10 mM
=0 mC/m2
=25 mC/m2
10 20
F/R(mN/m)
D (nm)
15
=0 mC/m2
=25 mC/m2
10 20 15
Fig. 6.2:Interaction curves using the Ettelaie-Buscall DVDW theory, Eq. (6.16), for several values of the fixed charge (𝑐∞=10 mM). (For other parameters see Fig. 6.1.)
separations up to 10 nm), and the coagulated particles will disperse again and we obtain a stable dispersion. Thus changing the charge to higher than approx. 17 mC/m2(for instance by a pH change) will disperse particles that were coagulated. So interestingly, there is hysteresis: when we start out with a stable dispersion and now decrease the charge, then this dispersion stays stable down to a charge of∼5 mC/m2and only below this charge coagulates.
On increasing the charge again, we must go beyond 17 mC/m2 to stabilize the dispersion.
So between the numbers of 5 and 17 mC/m2(these numbers are examples, for the specific calculation made here), the dispersion can be stable or flocculated, dependent on the history of the sample.
142 The interaction forces between colloidal particles
charge; thus the surface charge should go to zero. And this is exactly what CR theory predicts.iv
Until this point, assuming a fixed surface charge density, we did not have to consider the Stern layer (capacitance). This is because with a fixed surface charge, the gradient
𝜕 𝜙/𝜕 𝑥|Dfollows directly from the charge density, irrespective of the Stern capacitance.
However, because from this point onward we are including surface ionization, the Stern layer will play a role. But to simplify the discussion from this point forward, we assume an infinitely high Stern capacitance, thus𝜙
0 =𝜙
D, and thus the potential at the 0-plane,𝜙
0, is the same as𝜙
D, which is the potential where the diffuse layer starts.
As an example of a charge-regulating material, we choose for an amphoteric material, which is a material that can charge both positively and negatively. Examples are alumina and titania, see §3.7, but also many biological materials and protein molecules are amphoteric.
For alumina and titania, pK (e.g., pK=4.4 for titania) is equal to the pH at which the material is uncharged (and this pH is called the point of zero charge, PZC, or equivalently, the iso- electric point, IEP, see p. 507). When pH is higher than PZC, the material is negatively charged, and for a lower pH it is positively charged.v
For this problem of the overlap of two EDLs and an amphoteric charge-regulating surface, there are no analytical solutions that work. We thus show numerical results using the full 1D Poisson-Boltzmann equation, for a 1:1 solution given by Eq. (3.8), evaluated at a range of separationsDand subsequently numerically integrated to obtain𝐹/𝑅vs. D, in the same way as discussed for the EB calculation. Thus, we put aside the EB-approach because it does not give very accurate results at high charge, especially when the surfaces are far apart.
But even without solving the equations, we can already predict what happens when two equal charge-regulating materials interact across a thin gap. Let us assume we are at a pH above the iso-electric point of titania, thus titania is negatively charged at this pH>pK. We thus have cations as counterions in the gap between the surfaces at a larger concentration than outside the gap. Thus pH in the gap is lower because we have more H+-ions here than outside (the relative increase in H+concentration equals the relative increase in cation concentration, similar to Eq. (2.16) in Ch. 2). When the gap is further narrowed, pH in the gap goes down
ivInstead, for hetero-interaction the charge can both go up and down with separation, but at contact the two materials will have an exactly opposite charge (together they must be charge-neutral because at that point there are no longer ions in the gap between the surfaces to compensate for any charge mismatch).
vThough this discussion relates to pH right at the surface, let us stress that pH in equations is the pH in bulk solution, outside the overlapping diffuse layers, and is not a pH at the 0- or D-plane.
Theory of colloidal interaction for ionizable materials 143 more, until at contact pH is such that the material has discharged completely, i.e., pH in the gap becomes equal to pK of the material. Thus we know the ‘final’ pH in the gap (i.e., when there is almost or full contact), and thus we can calculate the potential in the gap𝜙when there is contact, and this tells us the electrostatic pressure at contact. This pressure is the maximum value, i.e., with separation𝐷going down,𝑃
emonotonically increases from zero when𝐷=∞, to a maximum value at contact. This increase in pressure occurs even though the surface charge continues to decrease the closer the surfaces come together, down to zero charge at contact, when the repulsive pressure is the highest. Concentration and potential profiles across the gap become more and more ‘flat’ the smaller the gap, because the charge density goes down, and because the gap gets more narrow. [By ‘flat’ we mean that the difference between the potential (concentration) at the surface and in the center of the gap, becomes very small.]
Important to note, even though at contact the surfaces have discharged completely, 𝑃
e
will not go to zero, not at all! Instead, for equal materials, 𝑃
e monotonically increases when we bring the surfaces closer together, not only for a fixed surface charge but also for surfaces that are ionizable. Thus, while the two surfaces are pushed together, and the surfaces discharge,𝑃emonotonically increases, with a maximum at contact for CR (for CC 𝑃eincreases indefinitely, i.e., ‘it is unbounded’).
When the surfaces are far apart, we can calculate chargeΣand surface potential𝜙
0based on Eqs. (3.69) and (3.70). When far apart, the charge is at its maximum value, while potential 𝜙0is at a minimum value. When we now push the two materials together, the charge goes down, all the way to zero at contact, while 𝜙
0 reaches a maximum value. At that point the potential across the (by now very narrow) gap is almost constant (gradients in potential become negligible).vi This potential at contact is obtained by setting𝛼=0 in Eq. (3.73), which results in
𝜙c=ln 10· (pK−pH) . (6.17)
As this equation shows, the potential in the gap when𝐷→0 is negative for pH>pK, and positive for pH<pK. We will refer to this potential as𝜙
c, where index ‘c’ refers to being in contact. Note that this high potential is only reached at the point where the two materials are pushed together and there is no room left for ions in the gap between the surfaces. For an area on a particle that is still at a slight distance from other surfaces, the surface potential is lower in magnitude than𝜙
c.
At (points of) contact, the electrostatic contribution to the disjoining pressure,𝑃e,c, follows
viMinimum and maximum refer here to the magnitude of these variables, neglecting their sign.
144 The interaction forces between colloidal particles from inserting Eq. (6.17) in Eq. (6.3), which results in
𝑃e,c=𝑐∞
10pK−pH+10pH−pK−2
. (6.18)
Eq. (6.18) shows a very relevant phenomenon, that now with charge regulation (CR) included, at contact𝑃
e,cincreases with salt concentration𝑐∞(the two are exactly proportional). This is the exact opposite of what was predicted for the fixed charge case, where repulsion was always lower when𝑐∞was increased (for the same surface charge), see Fig. 6.1b. The very high repulsive pressure between amphoteric materials that occurs at contact in case of a high 𝑐∞, provides a method to keep very concentrated colloidal suspensions in a fluid-like state, with the pressure between particles at contact functioning as a very short-range repulsive force (Yuet al., 2002).
Even though now the repulsion is higher at contact when𝑐∞is higher, the pressure also decreases faster when the distance goes up, and thus there is a cross-over point, i.e., beyond a certain separation, the repulsion is larger for a lower𝑐∞. Thus with CR, the influence of 𝑐∞is not as straightforward as how it was for the CC situation.
We show in Fig. 6.3 as function of pH curves for the interaction between two equal titania surfaces, for which surface ionization is described by Eq. (3.73). As might be expected, the further pH is away from the iso-electric point (point of zero charge, when pH=pK), the higher is the repulsion. For a pH only 1 point away from pK, the theory predicts only a shallow repulsive barrier, which is likely not enough to inhibit aggregation. However, for a two- and three-point difference between pH and pK, there is repulsion at all distances, and we can expect a dispersion of titania particles to be stable.
Stability ratio𝑊. In this section we analyzed force curves in the ‘F/R’ property.
To describe the interaction force 𝐹 between equally sized particles with radius𝑎, we multiplyF/Rwith𝑎 and divide by 2. We can then (numerically) integrate𝐹to obtain the interaction energy𝐸in J, see Eq. (6.2) and a footnote on p. 140. Often this number is divided by kT to obtain the energy expressed in so many kT. The energy 𝐸 can again be integrated to obtain the stability ratio𝑊 which describes the stability of a dispersion. For𝑊 very large, e.g.,𝑊 >104, a dispersion is expected to be stable, and not to coagulate. But when𝑊is low, e.g. 𝑊=100 or less, or even lower than unity, the dispersion (‘sol’) will flocculate/coagulate.
The stability ratio𝑊 is given by 𝑊 =2𝑎
∫ ∞
0
exp(𝐸/𝑘 𝑇)
(2𝑎+𝐷)2 d𝐷 (6.19)
Theory of colloidal interaction for ionizable materials 145
-20 0 20 40
0.1 1 10
F/R (mN/m)
D (nm)
|pH-pK|=3
|pH-pK|=2
|pH-pK|=1
PB1