Donnan theory for hetero-interaction

The interaction forces between colloidal particles

PB 1 Donnan

6.5 Donnan theory for hetero-interaction

In practical situations we often encounter problems that depend on the interaction between dissimilar materials, thus between particles or surfaces that have a different surface chemistry, for instance in the capture of charged nanoparticles by (charged) porous media.

If after reading the previous sections the reader came to the conclusion that interaction of equal surfaces is complicated (and fascinating), then the reader will also appreciate the complexities that arise in the study of the interaction between different surfaces, i.e., hetero- interaction. In this broad class of problems one encounters many special and unexpected

148 The interaction forces between colloidal particles

features. For instance, when two materials have an equal sign of the surface charge they are always repulsive, but when they have an equal sign of the surface potential, they repel when further apart but closer together they attract one another. Or, when at least one surface is charge regulating (with the charge dependent on local pH), we can have a sequence of repulsion, attraction, and again repulsion when particles approach one another. And this is only the electrostatic part of the interaction, not yet involving the Van der Waals force. This Van der Waals force can also be attractive and repulsive, and can change sign dependent on the separation. All of these elements together results in what can be a very intricate problem.

In this section we neglect the Van der Waals force, and focus on the electrostatic part of the interaction between particles and materials. We demonstrate that even when we only consider electrostatics, in many problems we readily go from attractive to repulsive and vice- versa. This is different from homo-interaction: in that case the electrostatic contribution to the forces and energies is always repulsive, with or without charge regulation.

The complete problem of hetero-interaction is best addressed by numerically solving the PB-equation in the gap between the (charge regulating) surfaces, with appropriate boundary conditions. The electrostatic component to the disjoining pressure, 𝑃_e, follows from evaluating at some point in the gap the sum of the repulsive osmotic part,𝑅𝑇 𝑐

T−𝑐

T,∞

(for az:z-salt), and the attractive Maxwell contribution,−1/2𝜀 𝐸². The result of this calculation will be the same irrespective of where in the gap (at which positionx) these pressures are evaluated. This numerical calculation is done at many values of separation𝐷and calculated values of 𝑃_e are numerically integrated to obtain the force and energy between curved surfaces, see Eqs. (6.1) and (6.2).

As an example, we consider the interaction of two amphoteric materials, alumina and titania, of which the charge is described by Eq. (3.73) (𝑗={a,t}),

Σ𝑗 =𝐹 𝑁_𝑗 1

2

− 1

1+10^pK𝑗−pH𝑒^−𝜙

where pH refers to bulk solution. For these materials we have a pK of resp. pKa=8.7 and pKt=4.4. For a pH in between these two pK-values, we can expect an attraction between these oppositely charged interfaces. This is indeed correct for distances that are a few times the Debye length and further apart, but what happens when the surfaces come closer?

In the following calculation we only evaluate the pressure at contact. If this is positive (repulsive) we know that the electrostatic pressure went from attractive to repulsive (because at large distances it is attractive). In the calculation we include that the two surfaces together are charge neutral because there is no space left for ions to contribute to the charge balance, thus

Σ_t+Σ_a=0. (6.25)

Donnan theory for hetero-interaction 149 This calculation can be done for any pH-value, to calculate the charge of each surface,Σ𝑗, and the common value of potential in the vanishingly thin gap,𝜙. The charge that follows from this calculation does not depend on salt concentration, and –intriguingly enough–

neither depends on external, bulk, pH as we discuss further on. Having now calculated the surface charge (opposite for the two surfaces), we can calculate the Maxwell attraction according to

𝑃e,Maxwell=−1 2

𝜀 𝐸²=− 1 2𝜀

Σ². (6.26)

Thus this attractive term directly follows from Eqs. (3.73) and (6.25) and is independent of salt concentration and bulk pH. The osmotic pressure, 𝑃

e, can also be directly calculated from Eq. (6.21) because it is a function of𝜙and of𝑐_∞. Thus the osmotic pressure, which is zero or repulsive, will not be very prominent when𝑐_∞is low, but will have a larger effect at a higher𝑐_∞. Thus we can have the situation that for low𝑐_∞the total electrostatic interaction (osmotic plus Maxwell) at contact is attractive, but becomes repulsive at contact for a higher 𝑐_∞. This prediction is illustrated in Fig. 6.4.

But the situation can be even more interesting. The contact condition, Eq. (6.25), is independent of external pH. The system will always ‘find the pH’ in the gap that is needed to make sure the two surfaces have an opposite charge. In this case this is pHgap∼8.1. And to achieve that value of pH a certain potential in the gap is required, only a function of external pH, namely𝜙=ln(10) · pHgap−pHext

.

Interestingly, this result shows that also when we are at a pH below the iso-electric point of titania (pH<pKt), or likewise, pH is above the pK of alumina, we can also push the surfaces in full contact with the two materials becoming oppositely charged. Thus one of the materials then reverses its sign of charge upon being pushed towards the other material.

And for all values of pH and𝑐_∞ at contact the Maxwell attractive term is the same, in this case equal to an attractive force of 41.8 MPa.

At pHext∼8.1 the osmotic repulsive force is always zero whatever the salt concentration (because in the gap we always have𝜙=0 in this case), but away from this pH-value we can make the surfaces less attractive, even repulsive, by adding salt. Fig. 6.4 shows calculation results of the electrostatic pressure at contact for hetero-interaction of alumina and titania.

What will be immediately observed is the very high values of pressure, in the 100s of MPa- range! Clearly at a pH between the pK-values of the two materials, when they are overall attractive, this attraction is extremely strong, while if we go sufficiently above pK of alumina, or below pK of titania, especially if we add a sufficient amount of salt, the two materials become highly repulsive. Under these latter conditions they can be mixed without any risk of agglomeration.

150 The interaction forces between colloidal particles

Furthermore, note how in Fig. 6.4 the entire curve is symmetric around pHgap, which is close to the pK of alumina. Alumina in this calculation is the material with the higher density of surface groups,𝑁. On this side of the curve, thus for instance at pH 9, pH 10 and even pH 11 (when𝑐_∞=10 mM), both materials are negatively charged when they are far apart, but when pushed together into contact, they are very strongly attractive! This is because the alumina material will reverse its charge when titania and alumina are pushed together, to become positive.

The full interaction curve, for instance at pH 11 and𝑐_∞=10 mM will go from repulsive when the surfaces are far apart to very attractive at contact. Increasing salt concentration to 100 mM, or increasing pH to 12, will disperse the particles again.

But we can also have the reverse situation. For instance at pH 5 and𝑐_∞=100 mM. In this case the two materials will be attractive when far apart, while at contact the total pressure is repulsive! The dashed line in Fig. 6.4 describes the minimum salt concentration to make the interaction at contact repulsive. This curve predicts that between 4.4<pH<6 the surfaces –which are attractive at large distances– become repulsive at contact, even for moderate𝑐_∞. Between pH 8.7 and pH 11, even pH 12, the surfaces are repulsive far apart, but can be turned to attractive at contact when the salt concentration is low enough.

In the last sections, the Stern layer was neglected in the electrostatic description, but in a full description should be included. This layer creates a difference between the surface potential𝜙

0, and the diffuse layer potential𝜙

D. It is possible to include the Stern capacitance correctly in analytical expressions for the energy between flat surfaces,V (thus forF/R), for which examples are given in Biesheuvel (2004). But it is easier to include them in the numerical 1D Poisson-Boltzmann calculation that creates an output of the function𝑃

e vs.

D, and this function is subsequently integrated toV(F/R) and other energies. In conclusion, the problem of hetero-interaction has many intriguing aspects, even when only considering the electrostatic pressure 𝑃_𝑒 at various conditions. It will be interesting to study the full interaction curve (𝐹/𝑅vs. D), certainly in combination with the Van der Waals force, and understand its dependence on pH,𝑐_∞, etc.

Donnan theory for hetero-interaction 151

-50 0 50 100 150

2 4 6 8 10 12 14

P

e

(MP a)

pH

attractive repulsive

c^*(mM)

Fig. 6.4:Contact pressure for hetero-interaction between alumina (a) and titania (t) as function of pH and salt concentration,𝑐_∞. For pK_t<pH<pK_a, the two materials are always attractive at large separations, but as can be seen, for high enough𝑐_∞at contact the interaction can be made repulsive.

The dashed line gives the minimum salt concentration𝑐^∗(for a 1:1 salt solution) to make the interaction at contact repulsive (pKt=4.4,𝑁

t=3 nm⁻², pKa=8.7,𝑁

a=5 nm⁻²).

Part II