Saddle-shaped surfaces

In the example above, a maximum point was found within the explored domain. This is, however, not often encountered. Most frequently, the response surface is either monotonous in the explored domain or describes saddle-shaped surfaces or ridge systems. In such cases, it is not easy to comprehend the shape of the response surface from the algebraic expression of the model. A transformation to... 

Instead of being concave, the water surface extending between adjacent soil particles may assume a semicylindrical shape, i.e., like a trough or channel. One of the radii of curvature then becomes infinite for example, r2 may be infinite (= °°) in such a case, the pressure is —alri by Equation 9.6. If the air-liquid surface is convex when viewed from the air side, the radii are negative we would then have a positive hydrostatic pressure in the water (see Eq. 9.6). In the intermediate case —one radius positive and one radius negative (a so-called saddle-shaped surface)—whether the pressure is positive or negative depends on the relative sizes of the two radii of curvature. 

Figure 1.6 The Gauss map of a surface. The normal vectors in the triangular ABC region of the saddle-shaped surface define a region on the unit sphere, A B C, given by the intersection of the unit sphere with the collection of normal vectors (each placed at the centre of a unit sphere) within the ABC region. Notice that for the example illustrated the bounding curve on the surface and on the uiut sphere are traversed in opposite senses. This is a necessary feature of saddle-shaped surfaces, with negative Gaussian curvature.

Curved surfaces can also be saddle-shaped. Figure 10.20 shows an example. Suppose that a surfactant film is made between the two frames. Surface tension causes the film to assume the smallest surface area possible. In the situation depicted, this surface is saddle-shaped. Moreover, the surface has zero curvature. As drawn for the middle cross section of the film, the principal radii of curvature are equal, but of opposite sign, since the tangent circles are at opposite sides of the film (which is, actually, the definition of a saddle-shaped surface). In other words, = 0 because l/i i + l/(—R2) = 0. This is true for every part of the film surface. 

Inspection of Fig. 2.7 shows that the transition state linking the two minima represents amaximum along the direction ofthe IRC, but along all other directions it is a minimum. This is a characteristic of a saddle-shaped surface, and the transition state is called a saddle point (Fig. 2.8). The saddle point lies at the center of the saddle-shaped region and is, like a minimum, a stationary point, since the PES at that point is parallel to the plane defined by the geometry parameter axes we can see that a marble placed (precisely) there will balance. Mathematically, minima and saddle points differ in that although both are stationary points (they have zero first derivatives Eq. (2.1)), a minimum is a minimum in all directions, but a saddle point is a maximum along the... 

Figure 6B shows the demixing pressures obtained in this cell for a sample of PDMS in supercritical carbon dioxide at several concentrations in the range from 0.06 to 6 % by mass. Above each curve represents the miscible region. These demixing pressures are of the type shown in Figure 4A displaying UCST and LOST branches, that may also lead to hour-glass shaped region of immiscibility as discussed in Figure 3. The demixing pressures at different compositions and temperatures define a saddle-shaped surface from which constant-temperature cuts or constant-pressure cuts lead to pressure-composition or...

For the case of droplet microemulsions, tq = ri = r2, and so the mean curvature H = 1/ro. According to convention, H is positive for oil-in-water (o/w) droplets, and negative for water-in-oil (w/o) droplets. For bicontinuous microemulsions, which according to freeze-fracture electron microscopy have saddle-shaped surfaces of negative and positive curvature (13), c —C2, and so the mean curvature // 0. Note that lamellar liquid crystalline phases (Lc,), which are planar layers of oil and water, also have zero mean curvature (c = c 2 = 0), and are often located at higher surfactant concentrations nearby bicontinuous microemulsion phases (19). 

The concave and saddle-shaped surface traced out by the inward facing part of a probe sphere when it is touching more than one atom. 

Surfactant monolayers have two sides which are not identical. Therefore, all the curvatures have a sign, and the states with positive and negative curvatures are physically different. According to the sign convention, the curvature is counted as positive if the monolayer is curved towards oil e.g. in O/W micelles). Thus, for an OAV spherical micelle with radius R, H = H, = H2 = I/R. For an infinite O/W cylinder with the radius R, Hi = l/R and H2 = 0, and H = I/2R. For a plane. Hi = H2 = H = 0. For saddle-shaped surfaces, the principal curvatures have opposite signs and the mean curvature can be equal to zero, even though both Hi and H2 are non-zero i.e. when Hi = —H2, see Figure 7.3). 

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