Surface-Curvature Argument

K = If these curvatures are small enough, the free energy of a unit area of 

---

So what about the cubic phase In polycatenar systems, it is possible to rationalize the formation of cubic phases on the basis of surface curvature alone, which will be considered in subsequent sections. However, it can be argued that, for calamitic systems, these arguments do not hold—at least on their own—and that other factors are important. For example, if cubic-phase formation is due to surface curvature, it is not possible to explain why an Sa phase (lamellar and with no surface curvature) is seen at higher temperatures. An important factor is the presence of specific intermolecular interactions and in the case of the silver systems, these are the intermolecular electrostatic interactions resulting from the presence of formally ionic groups. This is consistent with the observation of cubic phases in the biphenylcarboxylic acids and hydrazines (Fig. 29), as well as with other materials. However, it is also evident that this is not the only factor, as no cubic phase is seen with anion chains shorter than DOS, while other studies with fluorinated alkoxystilbazoles showed that the position of fluorine substitution could determine the presence or absence of the mesophase observed in the unsubstituted derivatives (56). Thus, structural factors are clearly not negligible. 

A. L. Mackay. Yes, if you use the same materials to build C60 it is clearly strained. But I should say that the definition of a minimal surface is that it has zero mean curvature i.e. you can say that either the divergence of the normal is zero, or that it has zero splay. Thus if you take the p-orbitals in one direction they are spread out, whereas in another they are compressed or folded in. So the whole evens out. Minimal surfaces have, characteristically, zero splay energy. This is an argument in its favour, whereas spheres are splayed in both directions. 

The argument can perhaps be put forward In the following way A step of radius r centered on the dislocation maintains in its neighborhood a concentration c(r) determined by the curvature of the step. Under steady state conditions the concentration in the solution decreases in all directions approaching the value c maintained at a macroscopic distance from the crystal surface. It follows then that the concentration at the dislocation c(0) must obey the condition c(r) > c(0) > c. In fact, a brief calculation shows that the relation among these three quantities is... 

If the shape domains are defined by relative local convexity, then the notation HP j (a,b), p = 0, 1,2, is u.sed for the shape groups of MIDCO surfaces G(a), where besides the dimension p of the homology group, the truncation type p, the charge density contour parameter a, and the reference curvature parameter b are also specified. For the special case of ordinary local convexity, b=0, the second argument in the parentheses can be omitted and one may simply write HP (a). Usually, we are interested in the Betti numbers of the groups HP (a,b) and HP (a) for these numbers the bp x(a,b) and bpp(a) notations are used, respectively. 

None of the arguments on the micelle structure are conclusive, but all the evidence points to an ultrathin helical cylinder with at least two turns ( = 12) or at most 16 turns. This is also in accordance with the curvature of the NaDCA molecule and related A/B-cw steroids. Contrary to the molecules of classical detergents, bile salts have a lipophilic outer surface which is the convex side of the steroid nucleus and a hydrophilic inner surface, which is the polyhydroxy-lated concave side of the nucleus. It is a structurally satisfactory finding that the concave side of the cholic acid molecule also forms the concave inside of the micelle. 

In writing down Eq. (37) it is assumed that the surface charge density does not depend on curvature. In Ref 28 arguments are given that the surface charge density does depend on the curvature. There, for an interface bent around the water side, it is assumed that... 

As previously mentioned, the analysis of microfluidic systems can be rather difficult for a variety of reasons. The direct implementation of the Navier-Stokes equations toward surface-directed microfluidic systems requires careful attention when considering the advection of the free surface and the associated curvature of this surface. Consequently, sophisticated computational fluid dynamics software packages are required for a comprehensive three-dimensional analysis of the fluid transport within surface-directed microfluidic devices. However, a time-consuming comprehensive analysis may be beyond the requirements of designing and manufacturing functional surface-directed microfluidic platforms. Consequently, empirical approximations and scaling arguments are commonly used in the characterization of microfluidic physics. 

This means that if the half time is calculated based on the unit of volume to external surface area, the non-dimensional times defined are closer for all three shapes of particle. The small difference is attributed to the curvature effect during the course of adsorption, which can not be accounted for by the simple argument of volume to surface area. Eq. (9.2-37) is useful to calculate the half time of solid having an arbitrary shape, that is by simply measuring volume and external surface area of the solid object the half time can be estimated from eq. (9.2-37). 

The shape of the meniscus is determined by the equilibrium between the capillary forces (responsible for the curvature) and gravity forces (which oppose it). One can invoke the following pressure argument Immediately underneath the surface, Laplace s pressure [equation (1.6)] is equal to the hydrostatic pressure. This can be written as... 

Next, we put this argument into a more quantitative form. We argue that the effect of surface tension can be captured by assigning stiffness equal to the Laplace pressure to the interior of the bubble. Consider the deformed bubble depicted in Fig. 8.5. The curvature of the interface has decreased on the left-hand side, while it has increased on the right. In the limit of small amplitude, shear transforms a spherical bubble into an ellipsoid. 

The thermodynamics of curved surfaces is more subtle than that of planar, and we discuss only the most important case, the spherical surface, for which the two principal radii of curvature are equal. Hie extension of the argument to other curved surfaces is beset with difficulties into which we do not enter. The spherical surface, the bubble or the drop, is the only one that is stable in ffie absence of an external field. The original analysis of Gibbs was clarified and its consequences worked out by Tolman, whose work Koenig extended to multicomponent systems. Buff, Hill, and Kondo describe explicitly how the surface tension depends on the position of the dividing surface to which it is referred, or at which it is calculated. 

The second route the surface tension can be found through the direct correlation function by an adaptation of the argument curved interface near the z-axis of a Cartesian coordinate system with Its orq at the centre of curvature. As before, apply a weak external potential whidi increases the radius of... 

The normalization factor in the denominator is the same as before. When expanding the argument of the S function, one generally has to take into account the curvature of the dividing surface so that the integral in the numerator is proportional to... 

---