Global radius of curvature

A natural choice for parametrizing the thickness of a polymer conformation with TV monomers, X=(xi.xat), is the global radius of curvature rgc [244]. It is defined as the radius rc of the smallest circle connecting any three different monomer positions x xj, x (i,J,k= 1. N)  

In Fig. 10.1, two exemplified circles with their associated radii of curvature rc are depicted. 

The circumradius or local radius of curvature rc can be easily calculated by the geometric relationships of angles in similar triangles in a circle with center C and radius rc, as shown in Fig. 10.2. The triangles A Xi,Xj,Xk), A(Xy,XfcD), and A(Xy,XfcC) subtend the same 

Eventually, with the definition of the global radius of curvature (10.1), the polymer tube X can be assigned the thickness (or diameter) /(X) = 2rgc(X) [244,245]. 

---

Since we are interested in classifying conformational pseudophases of polymers with respect to their thickness, it is useful to introduce the restricted conformational space 7 = X rge(X) > p of all conformations X with a global radius of curvature larger... 

Ground-State energy per monomer mm//V of tubelike polymers with nine monomers as a function of the global radius of curvature constraint p (solid line). For comparison, the energy curve of the perfect a-helix is also plotted (dashed line). The inset shows that for a small interval around p 0.686, the ground-state structure is perfectly o -helical. Also depicted are side and top views of putative ground-state conformations for various exemplified values of p. For the purpose of clarity, the conformations are not shown with their natural thickness. From [243]. 

Group the CMP variables listed in Chapter 3 in order of their importance (or impact) in affecting planarization. Which ones do you think are independent variables Discuss the impact of the wafer-bow, produced during wafer fabrication or due to various films deposited on the surface, on the CMP process used to achieve global planarization. Assuming a given radius of curvature R in the wafer, calculate the load necessary to counter the forces producing the wafer-bow. 

Figure 15 Schematic representation of hierarchical structures developed in critical binary mixtures of A and B molecules (A/B) in a phase-separation process of the late stage SD. Note that the two components here have the dynamic symmetry (i.e., nearly equal mobilities) and equal volume fraction, (a) to (c) refers to (1) global, (2) interface, and (3) interphase structure and (4) local structure, respectively, where r, Am, Rm, h, int, fr, and Rg refer to the length scale of observation, the characteristic length of the phase-separating domain structures, the scattering mean radius of curvature, the thickness of the diffuse boundary (interphase), the thermal correlation length within the interphase, the thermal correlation length within the phase-separated domains, and the radius of gyration of polymers, respectively. From Hashimoto, T. J. Polym. Sci., Part B Polym. Phys. 2004, 42, 3207-3262.= ...

---