DLVO

DLVO (Derjaguin and Landau, Verwey and Overbeek) theory ? is obligatory for everyone wanting some insights into stability of nanoparticles, emulsions and colloids. The fact that it often requires input values (and there are lots of them!) that are unknown to the users, and the fact that other factors can completely overwhelm DLVO effects is an unfortunate truth. But all the same, any formulation team should share the common language of DLVO and should make at least some attempt to place their formulations within the context. The model here takes the minimum number of relevant inputs and plots the three key effects plus the resulting balance. Any standard text on DLVO (even Wikipedia) will provide much more detail. The equations are provided below for those who are interested. The model used here is based on one created by Dr Robert Lee of Particle Sciences and his assistance is warmly acknowledged. A companion model explores the meaning of Zeta potential

This is all very fine for theoreticians, but how does one get hold of all those input values when you just know that you have some particles you want to disperse? The first advice is "Don't Panic". Now think things through one at a time.

• You know the radius of your particle, you know T, you know the Density of your liquid (1 is good enough for most purposes) and its Molar Volume is simply MWt/Density (so MWt is close enough).
• You can look up Dielectric Constants (and Molar Volumes) here ε .
• The A₁₂ Hamaker constants don't vary all that much, so 1 is good enough, but you can look up some values here A12 .
• We're almost there. Your particle is either neutral (Zeta=0) or strongly charged (Zeta=+50 or -50) or has a modest charge somewhere in between. The effectiveness of the charge repulsion depends on your background concentration of salts and their charges, i.e. their ionic strength (shown as an output value)
• Your dispersant is either a small molecule (δ=0.5nm), an oligomer (δ=1nm) or a polymer (δ=10nm). It is either absorbed to a small (Γ=0.5) or large amount (Γ=2).
• And finally your dispersant is either highly compatible with the solvent (Χ=<0.1) or highly incompatible with the solvent (Χ>0.5).

Once you get used to making rough estimates of the parameters using such thinking, you can quickly work out for your specific system which (if any) parameters need to be defined more carefully. You might also work out that other effects (solubility via HSP thinking, bridging or depletion flocculation, Smoluchowski exceptions) are more important, making DLVO largely irrelevant.

A specific application of DLVO is discussed in the Polymers section of Practical Surfactants. To stabilise an emulsion a polymeric surfactant can be very effective provided:

• It can reach the surface fast enough during creation of the emulsion
• The polymer is a di-block with strong attachment of the hydrophobic part to the oil
• The other part of the di-block is highly water soluble so Χ is low
• The length of the water soluble chain is high - so δ=5-10nm
• The coverage (Γ) is high

The Stability Ratio, W

Particles, in the absence of a barrier, will come into contact with each other at a rate that depends on the diffusion coefficients of the particles. Given a barrier they will come together more slowly. The ratio of the barrier-free and barrier values is W, the Stability Ratio. A high value of W (typically 5000 or more) indicates that the barrier is adequate for stability over "normal" timescales.

There are many contested ways to calculate W, involving ever more esoteric numerical tricks such as Bessel functions and exponential integrals. A paper, Hiroyuki Ohshima, Approximate analytic expression for the stability ratio of colloidal dispersions, Colloid Polym Sci (2014) 292:2269–2274, has a super-accurate calculation that has defeated me. However, the paper includes a simpler approximation (though not as simple as the Fuchs approximation that neglects van der Waals attractions!) that differs only in regions of zero practical interest. So that's what I've implemented.

The method only takes into account the Hamaker and Debye terms, Steric is not included (and is so large anyway that we don't need W). Like so much of DLVO theory, when you implement it, the results are somewhat disappointing - W rises to huge values (I simply stop displaying them and say "Large - Stable") once the zeta potential gets into the conventional range of ~25-30mV, confirming the rule of thumb that that's the range required for stability. However, that rule assumes that your Hamaker constant is the typical small value. Once A12 gets large then the rule of thumb breaks down and W is more important.

Coulombic Stabilization

There is continuing controversy about charge stabilization in low dielectric constant solvents. If we assume that somehow particles can aquire a zeta potential, then the effect of it extends over a long range because we don't have the electrical double layer effects behind the standard V_D calculations. If you turn on the Coulombic option (and have a low dielectric constant) then you will see some funny V_D curves that are especially significant for larger particles.

The calculations are based on a paper from Ian D. Morrison, Criterion for Electrostatic Stability of Dispersions at Low Ionic Strength, Langmuir 1991, 7, 1920-1922. They include a calculation of W, the Stability Ratio (rate constant for diffusion-controlled coagulation/second-order coagulation rate constant), where a large value (say 5000) means that the particles are stable against coagulation.

I welcome feedback on this new (Nov 2019) feature in the app, given its controversial nature.

The DLVO equations

We have dielectric constant ε, charge on the electron e, the permittivity of free space e₀, Boltzmann's constant k, Avogadro's number N_A, the Hamaker constant A₁₂ and finally the interparticle distance h and particle radius r.

The Hamaker term, V_H, is given by:

`V_H=-(A_(12)r)/(12hkT)`

The Debye term, V_D, is given by:

`V_D=(2πe_0εrφ^2ln(1+e^(-h/(k^-1))))/(kT)`

where k^-1 is given by:

`k^(-1)=sqrt((e_0εkT)/(2N_Ae^2I))`

and the ionic strength I is given by:

`I=0.5Σc_iz_i^2`

Note that up to Feb 2023 there was an error in the ionic strength calculator - it underestimated values for, say, divalent + monovalent salts.

For the steric replusion V_S we need the length of the molecule or polymer sticking out, δ, the surface coverage Γ, a density ρ and molar volume MVol, plus the Flory-Huggins chi parameter, χ.

`V_S=(30N_AπrΓ^2)/(ρ^2MVol)(0.5-χ)(1-h/(2δ))^2`

Shear-induced Aggregation

It's well-known that shear can break apart particles but that at high-enough shear rates, particles can be forced to agglomerate. I had never known the theory but met Prof Janet Preston of Imerys and Åbo Akademi University who not only knew the theory but had validated it in an impressive undergraduate thesis in 1992, Measurement and theoretical treatment of the colloidal forces acting on a model colloid - shear induced aggregation at Plymouth U. Similar results were obtained in the same year by Husband and Adams, Shear-induced aggregation of carboxylated polymer latices, Colloid Polym. Sci. 270, 1194–1200 (1992). To simplify greatly, the force, F, in pN at interparticle distance h is deduced from differentiation of the curve of V_Th, so we can determine F_max at the peak of the V_T curve. Via the Goren equation, the force in pN between two particles at a shear rate γ with a viscosity (at that shear rate) of η is given by:

`F=6π1.02r^2γη10^12`

where the 1.02 is half of a Gorens factor 2.04 for identical spheres, with the half being a geometric factor. Obviously the 1.02 can be ignored, but it's provided in order to relate to the original equation.

You input your known shear viscosity and shear rate (for convenience as Log₁₀, with the real value shown) and the maximum DLVO force is compared to the force from the shear. If the shear force is larger (the background colour changes to pink) then you are likely to get agglomeration. It was a delight to see in the thesis that as the NaCl concentration increased, decreasing the DLVO barrier for the charged polystyrene particles being used, the predicted and measured onset of agglomeration were closely matched. Put in a particle of ~200nm radius, a salt concentration of 0.01M, a zeta potential of ~40 mV and you will find a peak DLVO force of ~110 pN so that with a 5 mPa.s viscosity at a shear rate ~ 3.10₄ you exceed the safe limit.