Curvature distribution measurement

Curvature Distribution Measurements of Phase-Separating Bicontinuous Structures... 

Interfacial curvature distributions, F(H,X), were evaluated from the 3D morphologies in Fig. 27 according to the SFM. The value of RI computed for the TEMT data analyzed here is 0.12. If RI is less than 0.2 in the curvature distribution measurements, a 5% error is expected [98]. 

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]. 

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). 

As the SFM provides local surface curvatures at a given point on the surface, the curvature distributions of H and K can be obtained by conducting the above measurements at many points on the surface. A joint probability density of H and K, is calculated as... 

Direct Measurement of Interfacial Curvature Distributions in a Bicontinuous Block Copolymer Nanostructure... 

D visualization of bicontinuous morphologies in block copolymer systems has been achieved [26-27] by TEMT (see Sect 2.2). This technique affords the real-space structural analysis of complex nanoscale morphologies without a priori synunetry or surface assumptions [97]. Application of numerical methods developed [39, 98] to measure interfacial curvatures from 3D LSCM images of SD polymer blends (see Sects. 3.2.3 and 4.3.3) to a TEMT reconstruction of the G morphology yields the first experimental measurements of interfacial curvature distributions, as well as (H) and an, in a complex block copolymer nanostructure. 

The curvature sensor first proposed by Roddier (Roddier, 1988), does not make measurements at the focal plane. Instead measurements are taken at two planes symmetrically distributed around the focal plane as shown in Fig. 10. These measurements are best thought of as blurred images of the aperture and consequently our ability to measure the tilt is affected since as we move from the focal plane the hght is spread over a wider angle. Consequently we would expect the tilt signal to degrade as we move away from the aperture. This can be formalized by the CRLB and Fig. 11 shows the best attainable tilt performance (Marcos, 2002). 

As a quantification of the amount of nonlinearity, we see that when we compare the values of the nonlinearity measure between Tables 67-1 and 67-3, they differ. This indicates that the test is sensitive to the distribution of the data. Furthermore, the disparity increases as the amount of curvature increases. Thus this test, as it stands, is not completely satisfactory since the test value does not depend solely on the amount of nonlinearity, but also on the data distribution. 

It is probable that numerous interfacial parameters are involved (surface tension, spontaneous curvature, Gibbs elasticity, surface forces) and differ from one system to the other, according the nature of the surfactants and of the dispersed phase. Only systematic measurements of > will allow going beyond empirics. Besides the numerous fundamental questions, it is also necessary to measure practical reason, which is predicting the emulsion lifetime. This remains a serious challenge for anyone working in the field of emulsions because of the polydisperse and complex evolution of the droplet size distribution. Finally, it is clear that the mean-field approaches adopted to measure > are acceptable as long as the droplet polydispersity remains quite low (P < 50%) and that more elaborate models are required for very polydisperse systems to account for the spatial fiuctuations in the droplet distribution. 

The nanotube network was successfully investigated in our laboratory by Dalmas et al. (63), in terms of distribution of radius of curvature. Indeed, the 2D apparent nanotube segment curvature radius distribution was then measured on TEM bright field images acquired on the composites (see Figure 3.8). The experimental curvature... 

Figure 3.8. (a) Measurement of the 2D apparent distribution of the nanotube curvature radius in polymer/nanotube composites (63). (b) example of 3D nanotube network (64). 

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