Elliptic

A quite different means for the experimental determination of surface excess quantities is ellipsometry. The technique is discussed in Section IV-3D, and it is sufficient to note here that the method allows the calculation of the thickness of an adsorbed film from the ellipticity produced in light reflected from the film covered surface. If this thickness, t, is known, F may be calculated from the relationship F = t/V, where V is the molecular volume. This last may be estimated either from molecular models or from the bulk liquid density. 

In ellipsometry monochromatic light such as from a He-Ne laser, is passed through a polarizer, rotated by passing through a compensator before it impinges on the interface to be studied [142]. The reflected beam will be elliptically polarized and is measured by a polarization analyzer. In null ellipsometry, the polarizer, compensator, and analyzer are rotated to produce maximum extinction. The phase shift between the parallel and perpendicular components A and the ratio of the amplitudes of these components, tan are related to the polarizer and analyzer angles p and a, respectively. The changes in A and when a film is present can be related in an implicit form to the complex index of refraction and thickness of the film. 

Figure B2.1.4 Fluorescence upconversion spectrometer based on the use of off-axis elliptical reflectors for the collection and focusing of fluorescence. Symbols used el, c2, off-axis elliptical reflectors s, sample x, nonlinear crystal. (After Jimenez and Fleming [21].)...

Here B is again a compressional elastic constant, is a bend elastic constant and tire elastic constant C results from an elliptical defonnation of tire rods (tliis tenn is absent if tire column is liquid). 

Regarded as an equation for e, this is a member of the class of elliptic partial differential equations for which a maximum principle is satisfied [76], SO e is required to take its greatest and least values on the... 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

Theoretical analysis of convergence in non-linear problems is incomplete and in most instances does not yield clear results. Conclusions drawn from the analyses of linear elliptic problems, however, provide basic guidelines for solving non-linear or non-elliptic equations. 

The standard Galerkin technique provides a flexible and powerful method for the solution of problems in areas such as solid mechanics and heat conduction where the model equations arc of elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of... 

Ciarlet, P.G., 1978, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam. 

Differential methods - in these techniques the internal grid coordinates are found via the solution of appropriate elliptic, parabolic or hyperbolic partial differential equations. 

The best way to avoid losing the physics of these procedures is to think of a particle descr ibing an elliptical path about an or igin. If we choose our coordinate system in an arbitrary way, the result might look like Fig. 2-1 (left). 

In general, the equation def cribing an elliptical path a.v 2hxy cy Q... 

The hydrogen molecule ion is best set up in confocal elliptical coordinates with the two protons at the foci of the ellipse and one electron moving in their combined potential field. Solution follows in mueh the same way as it did for the hydrogen atom but with considerably more algebraic detail (Pauling and Wilson, 1935 Grivet, 2002). The solution is exact for this system (Hanna, 1981). 

We shall concenPate on the potential energy term of the nuclear Hamiltonian and adopt a sPategy similar to the one used in simplifying the equation of an ellipse in Chapter 2. There we found that an arbiPary elliptical orbit can be described with an arbiParily oriented pair of coordinates (for two degrees of freedom) but that we must expect cross terms like 8xy in Eq. (2-40)... 

A visual inspection of a two-sample chart provides an effective means for qualitatively evaluating the results obtained by each analyst and of the capabilities of a proposed standard method. If no random errors are present, then all points will be found on the 45° line. The length of a perpendicular line from any point to the 45° line, therefore, is proportional to the effect of random error on that analyst s results (Figure 14.18). The distance from the intersection of the lines for the mean values of samples X and Y, to the perpendicular projection of a point on the 45° line, is proportional to the analyst s systematic error (Figure 14.18). An ideal standard method is characterized by small random errors and small systematic errors due to the analysts and should show a compact clustering of points that is more circular than elliptical. 

We designate the length of the ellipsoid along the axis of rotation as 2a and the equatorial diameter as 2b to define the axial ratio a/b which characterizes the ellipticity of the particle. By this definition, a/b > 1 corresponds to prolate ellipsoids, and a/b < 1 to oblate ellipsoids. 

The effect of ellipticity also increases [77] above the 2.5 value obtained for spheres. Analytical functions as well as graphical representations like Fig. 9.3 are available to describe this effect in terms of the axial ratios of the particles. In principle, therefore, a/b values for nonsolvated, rigid particles can be estimated from experimental [77] values. 

It is a frustrating aspect of Eq. (9.20) that the observed intrinsic viscosities contain the effects of ellipticity and solvation such that the two cannot be resolved by viscosity experiments alone. That is, for any value of [77], there is a whole array of solvation-ellipticity values which are consistent with the observed intrinsic viscosity. 

This state of affairs is summarized in Fig. 9.4a, which plots contours for different values of [77] in terms of compatible combinations of mj /nij and a/b. For the aqueous serum albumin described in Example 9.1 as an illustration, any solvation-ellipticity combination which corresponds roughly to [77] = 5 is possible for this system. Data from some other source are needed to pin down a more specific characterization. 

We shall see in Sec. 9.10 that sedimentation and diffusion data yield experimental friction factors which may also be described-by the ratio of the experimental f to fQ, the friction factor of a sphere of the same mass-as contours in solvation-ellipticity plots. The two different kinds of contours differ in detailed shape, as illustrated in Fig. 9.4b, so the location at which they cross provides the desired characterization. For the hypothetical system shown in Fig. 9.4b, the axial ratio is about 2.5 and the protein is hydrated to the extent of about 1.0 g water (g polymer)". ... 

Rigid particles other than unsolvated spheres. It is easy to conclude qualitatively that either solvation or ellipticity (or both) produces a friction factor which is larger than that obtained for a nonsolvated sphere of the same mass. This conclusion is illustrated in Fig. 9.10, which shows the swelling of a sphere due to solvation and also the spherical excluded volume that an ellipsoidal particle requires to rotate through all possible orientations. 

Figure 9.4b shows a theoretical f/fQ contour for a value of this ratio equal to 1.45. As noted in the discussion of this figure in Sec. 9.3, the intersection of the f/fQ and [77] contours permits the state of solvation and ellipticity of such a protein molecule to be characterized uniquely. 

We have emphasized biopolymers in this discussion of the ultracentrifuge and in the discussion of diffusion in the preceding sections, because these two complementary experimental approaches have been most widely applied to this type of polymer. Remember that from the combination of the two phenomena, it is possible to evaluate M, f, and the ratio f/fo. From the latter, various possible combinations of ellipticity and solvation can be deduced. Although these methods can also be applied to synthetic polymers to determine M, they are less widely used, because the following complications are more severe with the synthetic polymers ... 

Protein molecules extracted from Escherichia coli ribosomes were examined by viscosity, sedimentation, and diffusion experiments for characterization with respect to molecular weight, hydration, and ellipticity. These dataf are examined in this and the following problem. Use Fig. 9.4a to estimate the axial ratio of the molecules, assuming a solvation of 0.26 g water (g protein)"V At 20°C, [r ] = 27.7 cm g" and P2 = 1.36 for aqueous solutions of this polymer. 

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