Curvature Constants

Curvature in surfactants is of great importance as it contributes strongly to the energy balance of the system. Some curvatures require a lot of energy, γ, others are low energy. So what is curvature and how is the energy calculated?

Bending at first seems simple. Take a planar film and bend it like a sheet of paper. This requires energy and the bending constant k is intuitively straightforward. This can be considered as an intrinsic property of the interfacial film. Now wrap the sheet of paper into some shape such as a cylinder and try to bend it. The resistance depends strongly on the overall curvatures of the sheet of paper in two orthogonal directions, c1 and c2, as well as on the properties of the paper itself. These two curvatures are combined in two different ways. So we end up with a bending constant, kc, which is then applied to the mean curvature, H=0.5(c1+c2). The complication is that this curvature is not enough to describe the whole physics. The system also needs the “splay” or “Gaussian” curvature constant k̄c which is applied to the splay curvature K = c1c2. Finally, because surfactants have a natural curvature, H₀, from their shape any bending has to be seen as being away from that natural curve – i.e. a straight interface is bent! The surface energy γ for a mean curvature H starting with an unbent energy γ₀ is then given by:

γ = γ₀ + 2kc(H-H₀)²+k̄c.K

For simplicity in this app, H₀ and γ₀ are set to zero and the units are arbitrary.