Curvature principal

The two directions called principal directions could be found at every point of the mid-surface, which satisfy the following property. Orthogonal sections of the mid-surface along these directions define the principal curvatures ki, k2 of the surface. By that, the curvature lines could be defined on the mid-surface with tangents to them coinciding with the principal directions. Let a and / be parameters such that the coordinate net a = const, / = const on the mid-surface is orthogonal, and it coincides with the curvature lines. Then a position of every point x located at the mid-surface is defined by the parameters a, (3 ... 

If we imagine a line drawn on the primitive surface dividing all parts of the surface which are convex downwards in all directions from those which are concave downwards in one or both directions of principal curvature, this curve will have the equation (26), and is known as the spinodal carve. It divides the surface into two parts, which represent respectively states of stable and unstable equilibrium. For on one side A is positive, and on the other it is negative. If we assume that the tie-line of corresponding points on the connodal curve is ultimately tangent to that the direction of equations ... 

The curvature at a given point on a surface is characterized by the maximum and minimum radii, R and R2, of the circles in mutually perpendicular planes perpendicular to the plane tangent to the surface at that point, best approximating the curves formed by the intersection of the surface with these planes [221], (Fig. 1). The principal curvatures, k and k2 are defined as... 

The mean, Gaussian, and principal curvatures are well-defined quantities from the standpoint of the differential geometry. If the surface is given by the... 

At equilibrium (dt/) v n. = 0, which leads to the equilibrium condition for pressure expressed in terms of the two principal curvatures or alternatively in terms of the two principal radii of curvature ... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

Here, pb is the bond critical point (saddle point in three dimensions, a minimum on the path of the maximum electron density). In Eq. (44), and A.2 are the principal curvatures perpendicular to the bond path. The parameters A and B in Eq. (45) determined using various basis sets are given in Bader et al. [83JA(105)5061]. Convenient parameters in the quantitative analysis of a conjugation effect are the relative 7r-character tj (in %) of the CC formal double or single bonds determined with reference to the bond of ethylene (90MI2) ... 

The slope of the potential is zero for all directions and only one of the 3N-6 principal curvatures is negative. As it is an important fact that only one curvature is negative we must define what is meant by principal curvatures. At any point on the surface we can establish a matrix of second derivatives of the potential (force constants)... 

When applying the equation of Young and Laplace to simple geometries it is usually obvious at which side the pressure is higher. For example, both inside a bubble and inside a drop, the pressure is higher than outside (Fig. 2.7). In other cases this is not so obvious because the curvature can have an opposite sign. One example is a drop hanging between the planar ends of two cylinders (Fig. 2.7). Then the two principal curvatures, defined by... 

Example 2.5. For a drop in a gaseous environment, the two principal curvatures are positive and given by C = C2 1 / It. The pressure difference is positive, which implies that the pressure inside the liquid is higher than outside. 

For a bubble in a liquid environment the two principal curvatures are negative C = C2 = —1/R. The pressure difference is negative and the pressure inside the liquid is lower than inside the bubble. 

Surfactants form semiflexible elastic films at interfaces. In general, the Gibbs free energy of a surfactant film depends on its curvature. Here we are not talking about the indirect effect of the Laplace pressure but a real mechanical effect. In fact, the interfacial tension of most microemulsions is very small so that the Laplace pressure is low. Since the curvature plays such an important role, it is useful to introduce two parameters, the principal curvatures... 

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