N=const
=σ0−kT ρ (4.15)
that is the entropy of the ideal gas lowers the surface tension. The same effect follows from excluded volume and repulsive interactions like charged headgroup repulsion.
4.2 Differential geometry of surfaces
The Hamiltonian of the effective interface model looks deceptively simple:
H=σ Z
dA (4.16)
However, the partition sum is the path integral over all possible configura-tions of the interface. In general, these kinds of calcuaconfigura-tions are not simple.
In order to proceed, one now has to deal with the geometrical properties of mathematical surfaces. To this end, we have to introduce a few concepts from differential geometry.
R
Figure 4.5: Differential geometry of surfaces. (a) At any point on the surface, one can construct a normal vector n. Any plane containing this normal vector defines a directionθon the surface, for which one can determine the radius of curvatureR.
(b)R as a function ofθ has a minimumR1 and a maximum R2. Mean curvature is defined as H = (1/R1 + 1/R2)/2 and Gaussian curvature as K = 1/(R1R2), respectively.
A two-dimensional surface in three-dimensional space is defined by the position vector f(x, y) = (f1(x, y), f2(x, y), f3(x, y)), where x and y are the internal coordinates. The two tangential vectors ∂xf and ∂yf span the tan-gential plane, while the normal vector n = (∂xf × ∂yf)/|∂xf ×∂yf| points perpendicularly to it. It we rotate a plane containing n around it, com-pare Fig. 4.5a, for each angle of rotation θ a curve is cut out of the surface, to which we can fit a circle with radius R(θ). The curvature is defined as κ(θ) = 1/R(θ). Since κ(θ) is a smooth function on a compact carrier, it must have a minimum and a maximum, compare Fig. 4.5b. This defines the principal curvatures κ1 and κ2 at the given point on the surface. They cor-respond to the two principal radii of curvature, R1 = 1/κ1 and R2 = 1/κ2. The mean curvature H and theGaussian curvature K are defined as
H = κ1+κ2 The mean curvatureH describes how strongly the surface bends on average.
At least equally important, the Gaussian curvature K denotes the kind of geometry one is dealing with, as shown in Fig. 4.6:
• elliptic geometry represented by a sphere: R1 =R2 =R, H =−1/R, K = 1/R2 >0
Figure 4.6: Three fundamentally different kinds of geometries can be distinguished according to the Gaussian curvature K. (a) A sphere has K > 0. (b) A saddle hasK <0. (c) A cylinder has K = 0.
• hyperbolic geometry represented by a saddle: R1 = −R2, H = 0, K =−1/R21 <0
• parabolic geometry represented by a cylinder: R2 =∞, H =−1/2R, K = 0
Note that the two minus signs for H are conventions, because we define curvature to be negative if the surface bends away from the normal, which we take to point outwards. The theorema egregium by Gauss states that K depends only on the inner geometry of the surface, that is it does not depend on the definition of a normal vector pointing in the surrounding space. The Gauss-Bonnet theorem states that K integrated over a closed surface is a topological constant:
Z
dAK = 2πχ (4.18)
where χ = 2(1−G) and G are integer numbers called Euler characteristic andgenus, respectively. The genus denotes the number of holes in the object.
So for the sphere, we have G= 0 and χ= 2, and indeed Eq. (4.18) specified to a sphere gives (4πR2)(1/R2) = (2π)2.
In order to evaluate integrals like the one in Eq. (4.18) for arbitrary surfaces, we need formulae for the area element dA, the mean curvature H and the Gaussian curvature K as a function of the internal coordinates x and y. For this purpose, we introduce three 2x2-matrices:
gij =∂if·∂jf, hij =−∂in·∂jf, a=hg−1 (4.19) g is called the first fundamental form or the metric tensor, h is the second fundamental form anda is theWeingarten matrix. The area element follows from the metric tensor:
dA=|∂xf×∂yf|dxdy = (detg)1/2 dxdy (4.20)
Figure 4.7: A torus has saddle, cylinder and sphere-like parts, depending on the internal coordinates.
and the curvatures follow from the Weingarten matrix:
H = 1
2 tr a K = det a= deth
detg (4.21)
We now can apply these concepts for any geometry of interest. For the cylinder, we have x = ϕ and y = z with f = (Rcosϕ, Rsinϕ, z). This leads to dA = Rdϕdz, H = −1/2R and K = 0. For the sphere, we have x = ϕ and y = θ with f = R(sinθcosϕ,sinθsinϕ,cosθ). This leads to dA = R2sinθdϕdθ, H = −1/R and K = 1/R2. In these two examples, the curvatures are constant over the surface. Therefore cylinder and sphere are examples for surfaces of constant mean curvature (CMS-surfaces). There is a theorem which states that the sphere is the only CMC-surface which is compact.
As an example for a geometry with varying curvature, we consider the torus, for which we have x = ϕ and y = θ with f = (tcosϕ, tsinϕ, rsinθ), where t =a+rcosθ. Here r and a > r are the outer and inner radii of the torus, respectively. This leads to
g =
Because the torus is invariant under a rotation with ϕ, the results for the curvatures only depend on θ. The Gaussian curvature now varies over the surface. For θ = 0 we are on the outer side, which is sphere-like with K = 1/(ar+r2) >0. For θ = π/2 and θ = 3π/2 we are on the top and bottom sides, which are cylinder-like with K = 0. And for θ = π we are on the inner side, which is saddle-like with K = −1/(ar−r2) < 0. Overall area is R
dA = 4π2ar and R
dAK = 0, in accordance with the Gauss-Bonnet theorem for χ= 0 (G= 1).
An important geometry for the following is theMonge parametrization for a surface which can be described by a height function h(x, y) (that is there should be no overhangs). Starting with f = (x, y, h(x, y)), one can derive exact, but complicated formulae for dA, H and K. In the approximation of a nearly flat surface (hx, hy ≪1), they simplify to
dA≈q
1 +h2x+h2y dxdy ≈[1 + 1
2(h2x+h2y)]dxdy (4.24) H ≈ 1
2(hxx +hyy), K ≈hxx hyy−(hxy)2 (4.25) Finally it is instructive to consider small variations around a given surface:
ft=f+tϕn (4.26)
where t is a small parameter andϕ(x, y) an arbitrary scalar field. Using the concepts introduced above, we can calculate the resulting change in the area element to first order in t:
dAt−dA=−2tϕ(x, y)H(x, y)(detg)1/2dxdy (4.27) For this first variation to vanish, we must have H = (κ1 +κ2)/2 = 0 every-where. A surface with vanishing mean curvature H is called a minimal sur-face, because stationarity under surface variation is a necessary (but not suf-ficient) condition for minimal surface area. FromH = 0 we getκ1 =−κ2 and K = −κ21 <0, therefore minimal surfaces are saddle-like everywhere. From this property it follows directly that minimal surfaces without boundaries cannot be compact, because the saddle-like shape would cut any container in which one tries to enclose the surface. Examples for non-periodic minimal surfaces are the plane and the catenoid. Most minimal surfaces known how-ever are periodic (including the singly-periodic helicoid, the doubly-periodic
Scherk-surface and the triply-periodic gyroid). Another important conse-quence of Eq. (4.27) is that surface and volume are related by curvature:
because
dVt−dV =tϕ(x, y)(detg)1/2dxdy (4.28) we have dA/dV =−2H.