Differential Geometry of Surfaces

The phenomenon of phase separation naturally gives rise to the presence of interfaces whose study is the focus of the chapters that follow. In this section, we discuss the mathematical definition of an interface or surface (of zero thickness) and show how one can calculate the area and curvature of a general surface. These concepts are particularly useful in the statistical physics of surfaces, interfaces, and membranes since one often has terms in the energy that depend on the area e.g., surface tension) and/or curvature e.g., bending energy). Of course the physical problem is more complicated since often the interface shape and size is not known but is determined self-consistently by the system. Thus one often asks which surfaces have a given area or curvature. An additional complication is that the surfaces of interest are often not deterministic but rather stochastic thermal fluctuations of the surface 

A curve in space, as shown in Fig. 1.10, is described by a position vector r = R(u) where m is a parameter that denotes the points along the curve in a simple, one-dimensional manner — e.g., u might label monomers in a polymer chain. The infinitesimal arc-length, dUj is the distance along the contour of the chain between two points labeled by ui and U2 in the limit that ui U2. The actual distance in space between those points is denoted by s. For discrete points As = — R u2), which in the limit that u - U2 is 

One shows that t is indeed a unit vector by using Eq. (1.81). The rate of change of the tangent is related to the curvature, k, and the normal, He, by 

Another example are spherical coordinates on a unit sphere so that u and v are the angles 9 and f respectively. On the surface one defines the two tangent vectors = dr/du and r = dr/dv. These vectors are not necessarily unit vectors, nor are they necessarily orthogonal. The two vectors define a tangent plane. The equation of the plane is given by r n = 0 where h is the normal to the surface at positions (u,v). The normal is given by the cross product  

If the implicit form of the surface is used, F(x,y,z) = 0, and in the Monge representation F(x, y, z) = z — h(x, y) = 0. For the general implicit form, one obtains the normal, by realizing that on the surface, where F is a constant, the total derivative of F is zero  

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It is well-known that implicit solvent models use both discrete and continuum representations of molecular systems to reduce the number of degrees of freedom this philosophy and methodology of implicit solvent models can be extended to more general multiscale formulations. A variety of DG-based multiscale models have been introduced in an earlier paper of Wei [74]. Theory for the differential geometry of surfaces provides a natural means to separate the microscopic solute domain from the macroscopic solvent domain so that appropriate physical laws are applied to applicable domains. This portion of the chapter focuses specifically on the extension of the equilibrium electrostatics models described above to nonequilibrium transport problems that are relevant to a variety of chemical and biological S5 ems, such as molecular motors, ion channels, fuel cells, and nanofluidics, with chemically or biologically relevant behavior that occurs far from equilibrium [74-76]. 

As we have seen, the interfadal region is, in fact, three dimensional (i.e., it has a finite thickness). It is very convenient, however, to represent an interface as a mathematical surface of zero thickness because such properties as area and curvature are well defined and because the differential geometry of surfaces is well understood. How can a thermodynamic analysis be developed that reconciles the use of mathematical surfaces with die actual three-dimensional character of the interface ... 

Finally, a theorem from the differential geometry of surfaces (Weatherbum, 1955) can be used to transform the integral along C to an integral over S ... 

In the model for interfacial curvature of a continuous surfactant film, we use results from the differential geometry of surfaces. A surface can be described by two fundamental types of curvature at each point P in it mean and Gaussian curvatures. Both can be defined in terms of the principal curvatures c = /R and ci = 1/Ri, where Ri and Ri are the radii of curvature. The mean curvature is... 

M.P. de Carmo, Differential Geometry of Curves and Surfaces, 1976, Prentice-Hall, New Jersey. 

M. do Carmo, "Differential geometry of curves and surfaces . (1976), Eaglewood Cliffe, N.J. Prentice-Hall Inc. 

A. Gray, "Modem Differential Geometry of Curves and Surfaces". Studies in Advanced Mathematics, ed. S. Krantz. (1993), Boca Raton, FLA. CRC Press. 

M. P. do Carmo, in Differential geometry of cuiwes and surfaces, Prentice- Hall, Englewood Cliffs, New Jersey, 1976. 

Eisenhart, L.P. A Treatise on the Differential Geometry of Curves and Surfaces Ginn and Co., 1909 Dover Publications, 1960. 

From the retention volumes at different temperatures the heats of adsorption at small coverage and changes of differential standard entropy of adsorption were calculated (see Table 1 and 2). The heats of adsorption are less sensitive to the properties of fullerene molecules and to the distinction in geometry of surface of fullerene crystals and surface of Carbopack. 

Thus far we have seen that differential geometry of the T(0, ) surface can produce a metastable liquid catastrophe, not that it must. Demonstration of the latter... 

Gray A, Abbena E, Salamon S. Modem Differential Geometry of Curves and Surfaces with Mathematica. Boca Raton Chapman Hall, CRC 2006. 

Differential geometry of n-dimensional non-Euclidean space was developed by Riemann and is best known in its four-dimensional form that provided the basis of the general theory of relativity. Elementary examples of Riemann spaces include Euclidean space, spherical surfaces and hyperbolic spaces. 

Popelier PLA (1996) On the differential geometry of interatomic surfaces. Can J Chem... 

Fig. 1.48 Examples of differential aeration cells (a) and (b) Differential aeration cells formed by the geometry of a drop of NaCl solution on a steel surface (c) differential aeration cells formed by the geometry of a vertical steel plate partly immersed in a NaCl solution. Increasing concentrations of Na2 CO3 decrease the anodic area (d) until at a sufficient concentration attack is confined to the water line (e) (/) shows the membrane of corrosion products formed at water...

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

In this method the local curvatures are calculated by using the first and second fundamental forms of differential geometry [7]. The surface is parameterized near the point of interest (POI) as p(u, v) (see Fig. 32). The coordinates (u, v) are set arbitrary on the surface in such a way that POI is located at p(u, v) = (0,0). The first and the second forms of the differential geometry are expressed as... 

Each of the groups we introduce in this text is a Lie group. We give the formal definition in terms of manifolds however, readers unfamiliar with differential geometry may think of a manifold as analogous to a nicely parametrized surface embedded in R. More to the point for our purposes, a manifold is a set on which differentiability is well defined. Since all the mamfolds we will consider are nicely parameterized, we can define differentiability in terms of the parameters. 

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