Curvature distribution

So far, Santos has been able to express the relation between a set of coefficients af, aj J 6 / describing a vector field and the overall curvature of the stream lines of this vector field. Based on the curvature field, they constructed the measure E of the curvature distribution in the simulation box. Provided that the homogeneous curvature field of curvature c0 is the one that minimizes E, the problem of packing has been recast as a minimization problem. However, the lack of information about the gradient of the error function to be minimized does not facilitate the search. Fortunately, appropriate computer simulation schemes for similar minimization problems have been proposed in the literature [105-109]. 

Fig. 3.3. Final variance of the distribution of curvature for the target values 0.1, 0.2, 0.5, 1, 5 and 10 for the curvature. The minimal width of the curvature distribution always lies just above 0.2co...

Fig. 3.2.B). For small goal curvatures it is difficult to eliminate the long-tail part of the curvature distribution.

The final vector fields have a curvature field, the stream lines of which exhibit a fluctuating curvature around the goal curvature c0. Santos and Suter reported that they had not been able to reduce the width of the curvature distribution below the limit of 0.2c0. This lower limit is reached at the end of the cooling process for each of the goal curvatures as shown in Fig. 3.3. The final vector fields obtained have a curvature field, the stream lines of which are characterized by a fluctuating curvature around the target c0. 

The formulas (120) and (127) define a set of the local curvature variables that can be used for the digital pattern analysis in the case of an arbitrary lattice. Calculation of the curvature distribution is, in principle, impossible within the digital pattern methods. 

Figure 34. At the early stages of the phase separation, the scaling of the curvature distributions is in accordance with the dynamic scaling hypothesis [Eqs. (4)—(7)] (the order parameter is nonconserved).

Figure 36. The scaled distributions of mean, P(H/Y ) (a), and Gaussian, P(K/Y]2) (b), curvatures scaled with the inteface area density, computed at several time intervals of the spindal decomposition of a symmetric blend. There is no scaling at the late times because the amplitude of the thermal undulations does not depend on the average growth of the domains, and therefore the scaled curvature distributions functions broaden with rescaled time.

Figure 37. The maximum of the mean curvature distribution scaled with the interface density increases very rapidly (up 2.5 times) within a short time interval, x , after the noise term has been switched off in the simulation. The Euler characteristic and the average domain size, / o, remain constant, and the surfaces area decreases by 3%. This illustrates that the curvature distributions are very sensitive to the thermal undulations of the interface. The times are x = 0.0, 0.032, 0.085, 0.225, 0.896, 2.05 from bottom to top at /// ] 0.

Surface curvature Distribution of carriers and active atoms on surface and thus type and rate of reactions... 

Figure 14.6 Illustration of the use of four rollers to form a component with arbitrary curvature distribution. The process takes advantage of the inextensible nature of the plies, which are only required to slide over each other in the roller region...

Curvature distribution for the whole surface as well as the condensing and subliming parts. The histogram of the condensing (respectively subliming) surface is typical from faceted (rounded) shape s. ... 

Local geometrical features are essential for understanding binding properties, catalytic behavior, and molecular recognition. Many of the descriptors used for global analysis can be adapted to study local features. For instance, mean and Gaussian curvature distributions of a surface, curvature and torsion of a molecular space curves, and the variation of the fractal index Df(r) over a molecular model serve this purpose. 

H. Jinnai, Y. Nishikawa, R J. Spontak, S. D. Smith, D. A Agard, and T. Hashimoto, "Direct Measurement of Interfacial Curvature Distributions in a Bicontlnuous Block Copolymer Morphology," Phys Rev. Lett 84, 518-521 (2000). 

Figure 43 The brush molecules adopt either a straight conformation (a) or a curved conformation (b) with weii-defined curvature. The iatter was evidenced by comparing the monomodai (c) vs. the bimodai curvature distribution function in (d). Reprinted from Potemkin, i. i. Khokhiov, A. R. Prokhorova, S. etal. Macromolecules20M, 37(10), 3918-3923, with permission from ACS. ...

As the normalized mismatch strain Cm is increased above the value 0.3, the curvature distribution becomes increasingly nonuniform. The general trend is that the curvature assumes values substantially below the average curvature for portions of the substrate near its center, and it takes on values substantially above the average value near the periphery of the substrate. For example, for a normalized mismatch strain of Cm = 2, the normalized curvature varies from about 0.7 at the substrate center to a value of about... 

Figure 5.14 Characteristics of a sponge-Uke morphology produced in a binary blend of a lamellar A(A/B)B triblock copol5mer (40 wt % A/B midblock) with homopolymer A [175,176], This morphology is observed in (a) to coexist with swollen lamellae at an overall blend composition of 90 wt % A, and a TEMT image of the morphology [177] is provided in (b). The mean (H) and Gaussian (K) curvature distributions computed directly from the TEMT image are included in (c) and (d), respectively.

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