Effective curvature

Bates DM, Hamilton DC, Watts DG. Calculation of intrinsic and parameter-effects curvatures for nonlinear regression models. Commun Stat Simul Comput 1983 12 469-77. 

At least for weak strain and weak curvature, the influences of these strain and curvature phenomena on the flame structure can be characterized in terms of a single quantity, an effective curvature of the flame with respect to the flow or the total stretch of the flame surface produced by the flow with respect to the moving, curved flame. By evaluating b from the rate-of-strain tensor in the products just behind the flame, the last of these quantities may be expressed nondimensionally as... 

Measure an area as homogeneous as possible. Shadow effects, curvatures and texture effects could significantly alter the measurement. 

As long as cq remains close to 2/Zq, a spherical shape is the best compromise. Monotonically decreasing Co> the radius R of the sphere must in principle increase correlatively, in order to maintain an effective curvature as close as possible to cq. But R is not allowed to exceed the length Iq of the amphiphilic molecule. When cq becomes comparable with I/Iq, the type of aggregation changes. It now favours that shape which, with core radius R close to /q, best allows for a lower effective curvature than the sphere. The result is a cylinder... 

Figure 7.8 Schematic illustration of the effects of anharmonicity on the reaction probability (a) the anharmonicity of the barrier shape along the reaction direction reduces the tunneling probability and (b) excitation of the vibrational mode changes the effective curvature felt by the reaction mode and affects the sharpness of the reaction probability.

Now, if at any temperature Tc the solid phase confined by the interface with the effective curvature Ks is in the equilibrium with the liquid phase confined by the interface with the curvature Kl, the equality should be held. By equating... 

FbwO) is the Fourier transformation of effective beam width as a function of spatial frequency / Fuff) is the MTF of the XRll. Because of the XRll windows curvature, projection data must be transformed to obtain uniform pixel spacing, described by Errors in object centre... 

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. 

C. The Effect of Curvature on Vapor Pressure and Surface Tension... 

A very important thermodynamic relationship is that giving the effect of surface curvature on the molar free energy of a substance. This is perhaps best understood in terms of the pressure drop AP across an interface, as given by Young and Laplace in Eq. II-7. From thermodynamics, the effect of a change in mechanical pressure at constant temperature on the molar h ee energy of a substance is... 

Here, r is positive and there is thus an increased vapor pressure. In the case of water, P/ is about 1.001 if r is 10" cm, 1.011 if r is 10" cm, and 1.114 if r is 10 cm or 100 A. The effect has been verified experimentally for several liquids [20], down to radii of the order of 0.1 m, and indirect measurements have verified the Kelvin equation for R values down to about 30 A [19]. The phenomenon provides a ready explanation for the ability of vapors to supersaturate. The formation of a new liquid phase begins with small clusters that may grow or aggregate into droplets. In the absence of dust or other foreign surfaces, there will be an activation energy for the formation of these small clusters corresponding to the increased free energy due to the curvature of the surface (see Section IX-2). 

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

Since shallow-level impurities have energy eigenvalues very near Arose of tire perfect crystal, tliey can be described using a perturbative approach first developed in tire 1950s and known as effective mass theoiy (EMT). The idea is to approximate tire band nearest to tire shallow level by a parabola, tire curvature of which is characterized by an effective mass parameter m. ... 

Kofman R ef a/1989 Solid-liquid transition of metallic clusters occurrence of surface melting Physica A 157 631 Kofman R ef al 1994 Surface melting enhanced by curvature effects Surf. Sc/. 303 231... 

In general there are two factors capable of bringing about the reduction in chemical potential of the adsorbate, which is responsible for capillary condensation the proximity of the solid surface on the one hand (adsorption effect) and the curvature of the liquid meniscus on the other (Kelvin effect). From considerations advanced in Chapter 1 the adsorption effect should be limited to a distance of a few molecular diameters from the surface of the solid. Only at distances in excess of this would the film acquire the completely liquid-like properties which would enable its angle of contact with the bulk liquid to become zero thinner films would differ in structure from the bulk liquid and should therefore display a finite angle of contact with it. 

Fig. 3.21 The effect of meniscus curvature on surface tension. Plot of /) against r . y is the surface tension of the meniscus having the mean radius of curvature and y that of a plane surface of liquid, according to Melrose. The value of y/y was calculated by the equation V = /x(l - with a . = 3 a.

If the actual response is that represented by the dashed curve, then the empirical model is in error. To fit an empirical model that includes curvature, a minimum of three levels must be included for each factor. The 3 factorial design shown in Figure 14.13b, for example, can be fit to an empirical model that includes second-order effects for the factor. 

Curvature and rotation lenses and their effects on the ion beam. 

Any effects of centrifugal distortion will show up as slight curvature of the A2F(J) versus... 

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