Curvature corrections

Although the Rockwell test is intended to be used on flat parallel-sided specimens, its use can be extended to rounded surfaces by using a curvature correction factor. Compound surfaces such as gear teeth can be tested but the results must be corrected for curvature. 

It is usual to record data in absorbance units and, although a straight line relationship is theoretically valid, the effective linear range does not usually exceed 1.0 absorbance unit and in many cases may be only up to 0.5 absorbance units. In order to increase the versatility of an instrument, some manufacturers incorporate a concentration mode in which it is possible to alter the sensitivity of the instrument and so work over various concentration ranges. If a direct read-out of concentration is used, the problem of non-linearity becomes more serious and in order to try to overcome this, some instruments incorporate a curvature correction device. 

Note that instmment function and curvature corrections have a greater effect on narrow peaks (e.g. the substrate) than on broad peaks. 

This expression includes the curvature corrections to the Gaussian function, which play an important role in the averaging procedure they ensure the smoothed spectrum g(E) to be approximated, through the above definition, by its own truncated Taylor expansion. This smoothing procedure of the one-electron spectrum is an application of the method of moments, also used in other systems [23]. 

The form of Eq. (5-25) was suggested by noting that the first order curvature corrections to Eqs. (3-47) and (5-35) are near unity and by matching the expression to the creeping flow result, Eq. (3-49), at Re = E Equation (5-25) also represents the results of the application of the thin concentration boundary layer approach (Sc oo) through Eq. (3-46), using numerically calculated surface vorticities. Thus the Schmidt number dependence is reliable for any Sc > 0.25. 

This expression includes the so-called curvature corrections to the Gaussian function which ensures that the smooth density g E) does not change after its own averaging through the same procedure. 

The curvature correction can be important when K is small, R is small, and/or a is large. With regard to this aspect, Hirotsu has recently found strong size-dependence of the first-order transition temperature and proposed that a can be much larger in ionized gels than in neutral gels due to the presence of an electric double layer at the interface [31]. 

Zhao J. Image curvature correction and cosmic removal for high-throughput dispersive Raman spectroscopy. Applied Spectroscopy 2003, 57, 1368-1375. 

Fig. 9.8 Curvature correction factor for drag flow for a Power Law model fluid. /v,. fl 0 denotes the ratio of drag flow between parallel plates to drag (Couette) flow between concentric cylinders at equal gaps and moving surface velocities. The subscript 6 — 0 indicates that in screw extrusion this correction factor is rigorously valid only in the limit of a zero helix angle. [Reprinted by permission from Z. Tadmor and I. Klein, Engineering Principles of Plasticating Extrusion, Van Nostrand Reinhold, New York, 1970.]...

The area per molecule as which appears in the preceding equation is evaluated at a distance d from the surface in contact with water and is curvature dependent. Expressions for ag are given by eq 7.15. The distance <5 is estimated as the distance from the surface in contact with water to the surface where the center of the counterion is located. k is the reciprocal Debye length, na is the number of counterions in solution per cubic centimeter, Cuon is the molar concentration of the singly dispersed ionic surfactant molecules in water, Cadd is the molar concentration of the salt added to the surfactant solution, andA Av is Avogadro s number. The last term in the right-hand side of eq 7.17 provides a curvature correction to the ionic interaction energy. For normal droplets, Cg = 2/(7 w + d) for reverse droplets, Cg = —2/(7 w — <5) and for flat layers, Cg = 0. 

Also we see from Eq. (14.59) that the next-order curvature correction to Deija-guin s approximation [2] is of the order of l/y/ica) (1=1, 2). This has been suggested first by Dukhin and Lyklema [20] and discussed by Kijlstra [21]. In the case of the interaction between particles with constant surface potential, on the other hand, the next-order curvature correction to Derjaguin s approximation is of the order of l/jca, (i = 1, 2), since in this case no electric fields are induced within the interacting particles. 

The form eqn (3.1) is to hold strictly for planar surfaces. We have already alluded to curvature corrections to the effective surface tension contribution. If the micellar surface is spherical, the effective surface tension y for both a drop and a hole in the liquid is less than that for a planar interface by a factor (1 —d/R), where d is of the order of one or two molecular radii and R is the position of the Gibbs dividing surface. Corresponding corrections to the electrostatic contributions are expected to be of much more importance, and can be handled within the framework of a capacitance description. Thus for a spherical capacitance the energy per unit area is from electrostatics... 

Except for two instances in the subsequent development, curvature corrections can be ignored. 

The first relation assumes that the density of hydrocarbon is constant in the interior of the micelle. If we ignore curvature corrections the free energy per amphiphile in a micelle is from eqn (3.1) and (4.2)... 

Fig. 2.—Concentration of amphiphiles in spherical micelles of aggregation number N. The curves are plotted for three assumed values of dielectric constant s. The curve for e = 0 is equivalent to ignoring curvature corrections to the electrostatic energy. The curves are normalised to the same maximum value. Xm is taken as 10 mole fraction (5 mmol dm ).

In comparing with experiment, the actual distributions are not known and are difficult to determine. What is known and available is the cmc under a range of conditions. In general, precise determinations of In (cmc) for a given choice of parameters involve a complicated numerical routine. However, as already remarked, the system is narrowly dispersed, and can be treated to a good approximation as a monodisperse system. Further, the area d corresponding to the mean aggregation number N differs by only 4 % from the optimal area ao. If curvature corrections to fiN are included d is shifted even closer to Oq. Thus eqn (4.3) becomes... 

Since D/R = 6v /Me (M/N), after a little algebra, repetition of our previous analysis gives, instead of eqn (4.6), a more complicated algebraic expression involving an additional parameter s. The corresponding distribution is plotted in fig. 2 fOT e = 80 and 40, and we see that the net effect of curvature corrections is to shift N towards M and d towards Qq. Thus, with little error, we can take d = Oq, N = M, and use eqn (2.9) which becomes... 

The best fit for M = 64 corresponds to an interfacial energy at the micelle-water interface of y = 52 erg cm in agreement with the measured value of y, at the bulk oil-water interface. The interfacial energies of liquid hydrocarbon-water interfaces vary from about 50 to 54 erg cm (from section 4 with the spherical approximation, the best fit y was 37 erg cm- ). With a 20 A, the value of K2 corresponds to a hydrophobic energy of 13 300 cal mol Curvature corrections (see below) reduce this to about 12 000 cal mol close to the expected value deduced in section 4. 

We first analyse the free energy of a one-component spherical vesicle bilayer (fig. 11), and investigate its stability in the absence of packing restrictions. It should be emphasised that without packing no sensible results emerge unless we include curvature corrections. Subsequently, when packing is built into the model, curvature corrections become of secondary importance, so that the principal results do not... 

Curvature corrections can be applied by assuming a specific form for the variation of AH " with temperature. In this manner we can calculate A (298.15 K) from A H T),... 

A dielectric sphere of dielectric coefficient e embedded in an infinite dielectric of permittivity 82 is an important case from many points of view. The idea of a cavity formed in a dielectric is routinely used in the classical theories of the dielectric constant [67-69], Such cavities are used in the studies of solvation of molecules in the framework of PCM [1-7] although the shape of the cavities mimic that of the molecule and are usually not spherical. Dielectric spheres are important in models of colloid particles, electrorheological fluids, and macromolecules just to mention a few. Of course, the ICC method is not restricted to a spherical sample, but, for this study, the main advantage of this geometry lies just in its spherical symmetry. This is one of the simplest examples where the dielectric boundary is curved and an analytic solution is available for this geometry in the form of Legendre polynomials [60], In the previous subsection, we showed an example where the SC approximation is important while the boundaries are not curved. As mentioned before, using the SC approximation is especially important if we consider curved dielectric boundaries. The dielectric sphere is an excellent example to demonstrate the importance of curvature corrections . 

---