Colloidal interaction of particles with a fixed surface chargecharge

The interaction forces between colloidal particles

6.2 Colloidal interaction of particles with a fixed surface chargecharge

Colloidal interaction of particles with a fixed surface charge 135

6.2 Colloidal interaction of particles with a fixed surface

136 The interaction forces between colloidal particles

and this refers to a distance between the two Stern planes, i.e., it is the thickness of the electrolyte (ion containing) region. When we introduce the Van der Waals force, we will specifically refer to the ‘distance between the hard surfaces’, which are a distance 2𝛿further apart than the thickness of the electrolyte region, i.e., this distance is 2𝛿 larger than the distance between the Stern planes.

For interaction beteen two equal surfaces, both flat and placed in parallel, the electrostatic contribution to the disjoining pressure,𝑃

e, can best be evaluated at the midplane, ‘m’, in the gap between the two surfaces, and then we obtain

𝑃_e=2𝑅𝑇 𝑐_∞(cosh(|𝑧|𝜙_m) −1) . (6.3) which assumes the PB-framework based on ideal point charges for a symmetric salt (1:1, 2:2, etc.) with |𝑧| the magnitude of the charge of the ions (|𝑧|={1,2,etc.}). Note that in this chapter we include the term ‘𝑅𝑇’ for pressures and other energy units. The pressure in Eq. (6.3) is the osmotic pressure due to the ions at the midplane in the gap,𝑅𝑇 Í

𝑖𝑐

m,𝑖

minus that far away, 2𝑐_∞, and we used the Boltzmann equation to relate𝑐

m,𝑖to𝑐_∞and𝜙

m. Instead of evaluating Eq. (6.3) at the midplane, it can also be evaluated at any other position in the gap, but then we must add the attractive Maxwell pressure as well,−1/₂𝜀 𝐸2. This term does not have to be considered at the midplane for equally charged surfaces because there it is zero. For hetero-interaction, this extra term must always be evaluated besides the osmotic pressure of Eq. (6.3). The background to these equations is further discussed in Ch. 8.

To arrive at a simple analytical expression for𝑃

e, we use the result derived by Gregory (1973) to take the low-potential limit of the Poisson-Boltzmann equation and integrate across the gap between two equally charged planar surfaces, which results in

𝑄= Σ𝜆

D

𝜀 𝑉T

=𝜙

m sinh(𝜅 ℎ) (6.4)

and

𝜙D=𝜙

m cosh(𝜅 ℎ) . (6.5)

These two equations can be used for a 1:1 salt as well as for any asymmetric salt or salt mixture, as long as𝜅 is calculated by Eq. (3.18). Note that𝑄 is a function of the Debye length, 𝜆_D = 𝜅⁻¹, and thus it depends on ion concentrations and valencies. The same definition ofQwas used in §3.3.

We assume now a symmetric salt solution (1:1, 2:2, etc.), and combine Eq. (6.3) with Eq. (6.4) to arrive at (Biesheuvel, 2001)

𝑃_e=2𝑅𝑇 𝑐_∞

cosh

|𝑧|𝑄 sinh(𝜅 ℎ)

−1

. (6.6)

Colloidal interaction of particles with a fixed surface charge 137 A useful simplification can be made forℎ≫𝜆

Dand then we obtain 𝑃e= 2

𝜀

Σ² exp(−𝜅 𝐷) (6.7)

which indicates that𝑐_∞ and𝑧only have an impact on the pressure via the inverse Debye length,𝜅. Thus with higher𝑐_∞or higher𝑧, the Debye length decreases and thus the repulsion goes down. This is a general result for the ‘constant charge’ (CC) approach, also for surfaces that are close. Later on we explain that for ionizable surfaces, at short distances the situation reverses and repulsion can go up with salt concentration.

Based on Eqs. (6.1) and (6.7) we derive an expression for the interaction energy𝑉between flat surfaces (in J/m²)

𝑉e=

∫ ∞

𝐷

𝑃ed𝐷= 2 𝜀 𝜅

Σ² exp(−𝜅 𝐷). (6.8)

Having integrated 𝑃

e over 𝐷 to arrive at the interaction energy 𝑉 in J/m², we can now implement the Derjaguin approximation which is to multiply𝑉by a factor 2𝜋to obtain the

‘force/radius’ (F/R) as often reported in force studies using the atomic force microscope (AFM) and the surface force apparatus (SFA). Or, to find the force, 𝐹_e, that acts between two equal spherical particles, each of radius𝑎, we multiply𝑉 by𝜋 𝑎(unit of force is N),

𝐹e= 2𝜋 𝑎 𝜀 𝜅

Σ² exp(−𝜅 𝐷) (6.9)

which can be integrated to the energy𝐸between particles (unit J), see Eq. (6.2) 𝐸e=

∫ ∞

𝐷

𝐹ed𝐷= 2𝜋 𝑎

𝜀 𝜅2Σ² exp(−𝜅 𝐷) . (6.10) In Eq. (6.10), the prefactor in𝐸_edepends on 1/|𝑧|²(within𝜅), but an even more significant dependence on𝑧goes via the exponential term.

We can now combine the above equations for electrostatic repulsion, with the attractive Van der Waals interaction. This attraction depends on the Hamaker constant, 𝐴, which for instance for two titania layers interacting across water is 𝐴∼6·10⁻²⁰J. We can also express this energy in ‘units of kT’, by dividing by𝑘 𝑇=4.12·10⁻²¹J, resulting in this case in𝐴=14.6 ‘kT’. [This is strictly speaking a non-dimensional number, and ‘kT’ should be omitted.]

For close surfaces (shortest distance between the surfaces much lower than the particle size), the Van der Waals attraction between flat surfaces is given by

𝑃vdw=− 𝐴 6𝜋 (𝐷+2𝛿)³

. (6.11)

138 The interaction forces between colloidal particles We can integrate 𝑃

vdw from Eq. (6.11) over 𝐷, and multiply by 𝜋 𝑎 to obtain the force between two particles of radius𝑎

𝐹vdw=− 𝐴 𝑎

12 (𝐷+2𝛿)² (6.12)

which we can integrate again, to obtain the interaction energy 𝐸vdw=− 𝐴 𝑎

12 (𝐷+2𝛿). (6.13)

—

We first analyze results for the force between curved surfaces, focusing on the total interaction force, 𝐹 = 𝐹

e+𝐹

vdw. Because often results are reported of experiments with AFM and SFA methods, we present results for the factorF/R. This implies that the force contributions,𝐹_𝑗, in the equations above, are multiplied by a factor 2 and divided by𝑎.

Using Eqs. (6.9) and (6.12), the total interaction force becomes 𝐹/𝑅= 4𝜋

𝜀 𝜅

Σ² exp(−𝜅 𝐷) − 𝐴 6 (𝐷+2𝛿)²

. (6.14)

In Fig. 6.1 we analyze the behavior of Eq. (6.14) for various surface charge densities Σ and salt concentrations 𝑐_∞ for a 1:1 salt. For zero surface charge the particles (surfaces) attract, and the more so the closer they get, until a limiting value that is reached when the Stern planes touch. With increasing surface charge, a repulsion develops, which for 𝑐_∞=10 mM starts at distances of 5–10 nm, and note how a twice larger surface charge means a four times larger electrostatic repulsion. At some point a maximum is reached in the total interaction curve, and the force decreases again when we push the particles closer.

For the conditions in Fig. 6.1 at contact an attraction remains, but for a surface charge larger thanΣ_c1=√︁

𝐴 𝜀 𝜅/48𝜋 𝛿2, which at𝑐_∞=10 mM isΣ_c1=22.4 mC/m², this is different. Even though forΣ>Σ_cthe repulsion goes down just before the surfaces touch, the repulsion stays positive until contact. At an even higher charge, the curve not even comes down again. This happens beyond a charge ofΣ_c2= Σ_c1/√

𝜅 𝛿, which in this case is at 71.2 mC/m².

Though all of this analysis is interesting, at near-contact the Gregory equation, Eq. (6.14), can be very much off, because in reality the electrostatic force is very different (as found by a full Poisson-Boltzmann calculation). The deviation already starts at a distance several times the Debye length. Though at a low charge (say 10 mC/m²) the correctly calculated𝑃

efollows Eq. (6.14) until𝐷/𝜆

D∼2, nevertheless for lower distances the correct 𝑃

e is significantly higher than predicted by the Gregory equation. For instance, Eq. (1.1) underestimates the

Colloidal interaction of particles with a fixed surface charge 139

0.01 0.1 1 10 100

0 5 10 15 20

-10 -5 0 5 10

0 5 10 15 20

F/R(mN/m)

D (nm)

c_=10 mM

=0 mC/m²

=20 mC/m²

10 15

F/R(mN/m)

D (nm)

=20 mC/m²

Fig. 6.1:Interaction curves using the analytical DVDW theory, Eq. (6.14), using the Gregory-function that is valid in the limit ofℎ≫ 𝜆

D(𝛿=0.3 nm, 𝐴=14.6 kT, 1:1 salt solution). a) As function of surface charge (𝑐_∞=10 mM). b) As function of salt concentration (Σ =20 mC/m²).

electrostatic pressure𝑃

eby a factor of 5 at𝑐_∞=100 mM and𝜎=10 mC/m². But intriguingly, at higher charge (already at 20 mC/m²) the correct𝑃

eis no longer larger than predicted by Eq. (6.14), but now is much lower! For instance there is a factor ∼25 overestimate by Eq. (6.9) compared to the correct pressure at 𝑐_∞=10 mM,Σ =0.1 C/m², and𝐷=5 nm separation. In conclusion, the Gregory equation based on this simple analytical solution can be very much off, and must ideally only be used when other calculations have ascertained it is an accurate simplification of the full PB-based solution.

A much better estimate of𝑃_eis provided by the solution of the PB-equation by Ettelaie and Buscall (1995). As long as the charge is not too high (e.g., for a charge less than 20 mC/m²), it is highly accurate, especially for very close surfaces. [Still, ideally we solve the full PB-equation across the gap, either numerically or using the exact solution that is based on the use of elliptic functions, see for instance Biesheuvel, 2004; Eqs. (20) and (21).]

The Ettelaie-Buscall (EB) solution is obtained from the PB equation in planar coordinates for a 1:1 salt solution, Eq. (3.8), by linearizing the term sinh𝜙around its midplane value (thus around𝜙

m), leading to

sinh𝜙(𝑥) →sinh𝜙

m+ (𝜙(𝑥) −𝜙

m) ·cosh(𝜙

m) (6.15)

which can be inserted in the PB-equation, which can then be integrated to obtain for the surface charge,Σ

Σ =√︁

2𝑐_∞𝑅𝑇 𝜀·√︁

cosh𝜙

m·tanh𝜙

m·sinh

𝜅 ℎ

√︁

cosh𝜙

m

. (6.16)

140 The interaction forces between colloidal particles

This result can be simultaneously solved with Eq. (6.3) for the disjoining pressure 𝑃e=2𝑐_∞(cosh𝜙

m−1)

to solve𝑃_evs.hfor a given value ofΣand subsequently calculateF/Rversus separation,D.ⁱⁱⁱ A correct prediction of the EB solution is that for a fixed surface charge, that the pressure 𝑃edoes not have a finite limiting value, as wrongly predicted by Eq. (6.6), but the pressure diverges when the Stern layers are about to come into contact (𝐷 → 0). This is due to the fact that in the increasingly narrow gap the concentration of counterions must go to infinity to neutralize the fixed surface charge. (This is different when surfaces are ionizable, i.e., reduce their surface charge upon compression, as we discuss in the next section.) This divergence does not show up in the energyF/Rthat we will analyze next.

Fig. 6.2 shows results of the full EB-expression for the electrostatic contribution toF/R, in combination with the same Van der Waals attractive force as used in Fig. 6.1. Now the required charge forF/Rto stay above zero in the entire range 0< 𝐷(nm)<10 is slightly lower than before. But there is an especially large effect on the chargeΣ_c2, which is the charge beyond which there is no longer a maximum in the interaction curve, but now upon decreasing𝐷until contact, repulsion monotonically increases. This critical charge was 71 mC/m²in Fig. 6.1 using the Gregory expression, and now –based on the EB equation– drops to a value around 25 mC/m² in Fig. 6.2. So for a value beyond this relatively moderate charge of 25 mC/m², this new analysis predicts that the interaction not only is repulsive until contact, but the interaction force monotonically increases all the way until particles are in contact.

An interesting phenomenon is hysteresis in coagulation/repeptization: as Fig. 6.2 points out, we can coagulate a dispersion when the repulsion barrier is low enough (perhaps around 5 mC/m²) and particles then end up in the deep attractive minimum that we see in Fig. 6.2 located below 1 nm separation. But if we now increase surface charge, in this case to 17 mC/m² or larger, the entire attraction curve becomes repulsive (𝐹/𝑅 >0 at all

iiiIn spreadsheet software a useful method is to make a list of values of 𝜙

m, and for each entry calculate 𝑃 e as well ash, and after that plot 𝑃

e vs. h. In this method, Eq. (6.16) must be inverted to an explicit equation forh. If we make a list of such calculations with index ifrom 1 to N, where distanceh(or, D) increases with index number, then we can now numerically calculateF/Rby the iterative ‘summation’

procedure𝐹/𝑅𝑖−𝐹/𝑅_𝑖+1=𝜋(𝑃𝑖+𝑃_𝑖+

1) · (𝐷_𝑖+1−𝐷_𝑖)with𝐹/𝑅|_𝑁set to zero. We must make sure in this list the steps between theD-values are small enough (they don’t have to be equi-distant) and that𝐷_𝑁is large enough, such thatPat that separation is much smaller thanP’s at closer separations. This procedure can be used both for the electrostatic part of the force and for the Van der Waals contribution. However, for the latter contribution, analytical solutions are available for its contribution to𝐹/𝑅as function ofD, so the Van der Waals contribution can also be added subsequently without being included in the integration based on the summation method.

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.