Curvature differences

Figure 3.7 An undulating surface possessing regions of positive and negative curvature. The curvature differences lead to diffusion-potential gradients that result in surface smoothing by diffusional transport.

The activity coefficients of the polymers are much lower than those of the monomer. Theories predict, within their limits, a concentration, molar mass, and chemical structure independence of the counterion activity. As seen from Fig. 14, the experimental curvatures differ from the theoretical predictions. The concentration dependence and the absolute values of fa change with the molar mass. Further, the activity coefficient has been found to be reciprocally related to the molar mass [38]. To obtain reliable results a minimization of the salt out-... 

The initial stage of sintering [20—23] is fiequently referred to as the neck formation stage, as is shown in Figure 16.1. The sintering driving force for the initial powder compact is due to the curvature difference between particle surface and that of the neck, see Figure 16.5. The six... 

According to Fig. 1.5, the conduction as well as the valence band consists of several bands. Some valence bands are degenerated around k = 0 (the F point). Since the curvature differs from one band to another, each band is associated with a different effective mass (see also Appendix D). Rather flat energy profiles correspond to heavy holes... 

The equations in Table 4.2 were derived for systems where the driving force of sintering is the capillary pressure difference due to curvature difference. However, when an external pressure Pappi. is applied, the total sintering pressure Pt is the sum of the capillary and external pressures,... 

Particle Sphericity. The two-detector-pair arrangement has another useful feature, and that is to give information with regard to the curvature over a certain arc of the particle surface. If the curvature measured at two different locations on the surface (phase difference) is identical, the particle is said to be spherical. If the two local curvatures differ, d>i2 and i3 will point at diameter values differing by AD. Consequently, a measure of the deviation from sphericity is available, and if AD exceeds a certain limit set by the user, the particle is said to be invalid. The underlying equations of size determination using the PDA technique assume that the particle is spherical, and hence any deviation from this assumption will introduce errors in the absolute determination of the particle size. 

Recently a number of experiments have been carried out to clear up the physical nature of the phenomenon [8-11]. A lot of experimental data describing the kinetics of cone s filling with various liquids have been obtained. One of the principal features of the phenomenon is that it takes place only if a gas inside a channel is bounded by liquid surfaces of different curvatures. 

There are two approaches to explain physical mechanism of the phenomenon. The first model is based on the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

If the first plane is rotated through a full circle, the first radius of curvature will go through a minimum, and its value at this minimum is called the principal radius of curvature. The second principal radius of curvature is then that in the second plane, kept at right angles to the first. Because Fig. II-3 and Eq. II-7 are obtained by quite arbitrary orientation of the first plane, the radii R and R2 are not necessarily the principal radii of curvature. The pressure difference AP, cannot depend upon the manner in which and R2 are chosen, however, and it follows that the sum ( /R + l/f 2) is independent of how the first plane is oriented (although, of course, the second plane is always at right angles to it). 

The basic device is very simple. A tip of refractory metal, such as tungsten, is electrically heat-polished to yield a nearly hemispherical end of about 10" cm radius. A potential of about 10 kV is applied between the tip and a hemispherical fluorescent screen. The field, F, falls off with distance as kr, and if the two radii of curvature are a and b, the total potential difference V is then... 

An important further consequence of curvature of the interaction region and a late barrier is tliat molecules that fail to dissociate can return to the gas-phase in vibrational states different from the initial, as has been observed experunentally in the H2/CU system [53, ]. To undergo vibrational (de-)excitation, the molecules must round the elbow part way, but fail to go over the barrier, eitlier because it is too high, or because the combination of vibrational and translational motions is such that the molecule moves across rather than over the barrier. Such vibrational excitation and de-excitation constrains the PES in that we require the elbow to have high curvature. Dissociation is not necessary, however, for as we have pointed out, vibrational excitation is observed in the scattering of NO from Ag(l 11) [55]. 

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. 

The recommended design procedure uses the values of (E /and m from Figures 7 and 8 in equation 56 and yields a very good estimation of Alp despite the curvature of the operating and the equilibrium lines. This value differs from A/q obtained by equation 49 because of the /(I — y) term in the latter equation. A convenient approach for purposes of approximate design is to define a correction term AA/q which can be added to equation 55 ... 

A real foam has further degrees of freedom available for estabHshing local mechanical equiHbrium the films and Plateau borders may curve. In fact, curvature can be readily seen in the borders of Figure 1. In order to maintain such curvature, there must be a pressure difference between adjacent bubbles given by Laplace s law according to the surface free energy of the film and the principle radii of curvature of the film AP = ) Note that the... 

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