DLVO

The plot below shows some plots of a simple version of the DLVO Potential for the energy $(V_{\text{DLVO}})$ between two spherical colloidal particles separated by a distance $r$. Intended primarily as a teaching tool, it is definitely not numerically accurate, but does illustrate the physics well enough to give you a feel what's happening.

The DLVO potential is treated as a simple sum of hard sphere repulsion (v. short range), electrical double layer repulsion (intermediate range), and van der Waals attraction (long range).

$$ V_{\text{DLVO}}(r) \sim V_{\text{Hard Sphere}}(r) + V_{\text{Double Layer}}(r) + V_{\text{van der Waals}}(r) $$

This model works well enough to illustrate the key points: a primary minimum (coagulation), secondary minimum (flocculation), and a barrier between them. The precise form for the terms is:

$$ V_{\text{DLVO}}(r) \sim \frac{D}{r^{12}} + \frac{B\,\,L_D^2 \exp\left(-\frac{r}{L_D}\right)}{r^Y} - \frac{A}{r^{n}} $$

Parameters are defined as:

$L_D$ is the Debye-Hückel screening length.
$A$ is roughly proportional to the Hamaker constant for the particles (attractions)
$B$ is a scaling factor that controls the amount of double layer repulsion.
$n$ sets the length scale of the van der Waals interactions (typically 1 or 2 for colloidal particles, 6 for molecules).
$Y$ is set to 1 if the repulsion potential is set to 'Yukawa', otherwise set to zero to give a simple exponential.
$D$ is set to be $10^{-8}$.

The energy zero point is defined as the energy of the particles at infinite separation.

Attractions ($A$) = 0.2 DL Repulsion ($B$) = 0.2 Screening Length ($L_D$) = 0.2 Degree of vdW Attraction ($n$): Type of Repulsion Potential: Presets:

Notes on the presets:

Coagulated Suspension shows the case where there is no barrier, attractions win, and the suspension irreversibly falls into the primary deep minimum and coagulates. Soy milk turns to tofu. Milk turns to butter.
Flocculated Suspension shows the case where there is a barrier that produces a secondary minimum, where the suspension will reversibly flocculate (provided that $k_BT$ is lower than the depth of the minimum).
Stable Dispersion shows the case where the repulsion is so strong that there is no secondary minimum, and the particles are kept away from each other: kinetically trapped as a colloidal dispersion.