Surface, equations principal curvatures

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 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 ... 

As a consequence of surface tension, there is a balancing pressure difference across any curved surface, the pressure being greater on the concave side. For a curved surface with principal radii of curvature rj and r2 this pressure difference is given by the Young-Laplace equation, Ap = y(llrx + l/r2), which reduces to Ap = 2y/r for a spherical surface. 

Equation 5.163 can be generalized for smooth surfaces of arbitrary shape (not necessarily spheres). Eor that purpose, the surfaces of the two particles are approximated with paraboloids in the vicinity of the point of closest approach Qi = h. Let the principal curvatures at this point be Cl and c[ for the hrst particle, and C2 and c for the second particle. Then the generalization of Equation 5.163 reads ... 

Considering capillary dynamics, the pressure drop term is often described by the Laplace equation, AP = 2yH, where y represents liquid surface tension and H represents the mean curvature of the liquid-gas interface associated with aU curves, C, passing through the surface. Furthermore, the character of a sufficiently smooth surface is through the invariant from differential geometry, the principal curvature of each curve, kj. The radii of curvature are the inverse of each principal curvature, A , = 1/r,. Considering the maximum and minimum radii of curvature at a point on a three-dimensional surface, the mean curvature can be calculated explicitly (see Appendix for more thorough derivation of the mean curvature parameter) ... 

Figure 5.8 shows a drop of liquid contained between two horizontal plates with an angle of contact less than njl. The principal curvature, (1// ), of the surface of the liquid in a plane perpendicular to the plates is shown. The other principal curvature can be neglected as the radius of curvature is much greater than R. The Laplace-Young equation gives the excess pressure, p, across the surface as... 

In Appendix IV the Euler-Lagrange equation is derived for the general two dimensional problem. It is applied to the minimum area contained by a simple closed line boundary. It is shown analytically that the surface area will be minimized if every point on the surface, with principal radii of curvature R and / 2, satisfies the equation,... 

It differs from Equation (10) for the motion of curves over a flat surface by the presence of the additional term in the right-hand side of this equation. This term is proportional to the local Gaussian curvature F of the surface (the Gaussian curvature is defined as a product T = k ki of the two principal curvatures of the surface in a given point). 

In conclusion, classical lamination theory enables us to calculate forces and moments if we know the strains and curvatures of the middle surface (or vice versa). Then, we can calculate the laminae stresses in laminate coordinates. Next, we can transform the laminae stresses from laminate coordinates to lamina principal material directions. Finally, we would expect to apply a failure criterion to each lamina in its own principal material directions. This process seems straightfonward in principle, but the force-strain-curvature and moment-strain-curvature relations in Equations (4.22) and (4.23) are difficult to completely understand. Thus, we attempt some simplifications in the next section in order to enhance our understanding of classical lamination theory. 

When the volume dV2 of the liquid evaporates, the volume of the vapor increases by dVt the two partial differentials refer to the same mass of substance. Thus (3 V2/d Vl)P2 = —Pi ip2, Pi and p2 being the densities of the two phases. Integration of the equation (3p1/3p2)K1 = P1/P2 affords- p0 = (p,/p2) (p2 - Pa)-The pressure p0 is that on both sides of a plane liquid surface. Pressure p2 is different from p0 whenever the liquid surface is curved. If its two principal radii of curvature are/ and/ 2, then... 

Young-Laplace Equation. Interfacial tension causes a pressure difference to exist across a curved surface, the pressure being greater on the concave side (i.e., on the inside of a droplet). In an interface between phase A in a droplet and phase B surrounding the droplet, the phases will have pressures and If the principal radii of curvature are Ri and R2, then... 

Young-Laplace Equation The fundamental relationship giving the pressure difference across a curved interface in terms of the surface or interfacial tension and the principal radii of curvature. In the special case of a spherical interface, the pressure difference is equal to twice the surface (or interfacial) tension divided by the radius of curvature. Also referred to as the equation of capillarity. 

For polymers interfacial and surface tensions are more practically obtainable from analysing the shapes of pendant or sessile drops or bubbles, all of which are examples of axisymmetrical drops. Bubbles may be used to obtain surface tensions at liquid/vapour interfaces over a range of temperatures and for vapours other than air. Drops can also be used to obtain vapour/liquid surface tensions but they are particularly suited to determination of liquid/liquid interfacial tensions, for example for polymer/polymer interfaces. All the methods are based on the application of equation (2.2.1). The principles are illustrated in figure 2.4, in which a sessile drop is used as the specific example. Just like for the capillary meniscus, the drop has two principal radii of curvature, R in the plane of the axis of symmetry and / 2 normal to the plane of the paper. At the apex, O, the drop is spherically symmetrical and R = Rz = b and equation (2.2.12) becomes... 

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