Crossing length distribution

At this early stage we present a modification of Yoon s model by introducing the crossing length distribution. This probability function h u) of the lengths of chains which pass through a cross section of the filament is defined by... 

The number of chains of length u which cross a certain fibre cross section is determined by the crossing length distribution h(u) defined in Sect. 3.1.1. The tensile stress in a chain passing through a fibre cross section is calculated in Sect. 3.1.2. The force S per unit area in a cross section can be found from... 

An even more difficult problem occurs if the bar is not prismatic, i.e., if the moment of inertia is not constant along the length. We must then solve for l(x) where x is the axial coordinate of the bar. This optimization of cross-section distribution has been addressed only for problems with fixed E and L, not the general column design problem. 

Cylindrical Filaments. If cylindrical filaments are studied, the last-mentioned assumption is not be fulfilled. In this case it is more suitable to cross an elongated24 primary beam and the fiber in such a way that the irradiated height of the beam, Bgo, is constant. Then the irradiated volume becomes V = Bbo nrj, with /y being the radius of the fiber. In many practical applications the studied fiber is very thin, and the effective absorption coefficient can be approximated by exp (—fiteff) 1. If this approximation is not feasible, the absorption factor exp (—ji ) must be integrated for the chord length distribution gc ( ) of a circle with radius ry (cf. p 168) in order to yield the effective absorption. 

The behaviour of chain-length distribution is non-characteristic during thermal-oxidative degradation of isotropic PP films the degradation shifts towards decrease in molecular mass, while during oxidation in deformed films it shifts towards increase in molecular mass. It means that with oxidation in isotropic samples in the induction period the destruction of molecules prevails. On the other hand, with oxidation in deformed samples attachement and cross-linking dominate. 

In Chap. C we have discussed in some detail the application of the cascade theory to polycondensates in their unperturbed state. In Chap. D some experimental results were already given for cross-linked or vulcanisated linear chains. In this chapter we shall now outline in brief how cross-linking chain reactions or the vulcanization of preformed chains of an arbitrary length distribution can be treated by cascade theory. Second, we shall discuss how heterogeneities in branching or a rigidity of a certain domain can be taken into account. 

FIGURE 5.2.14 Solution AFM topographs of PHT (0.01 g/L) adsorbed onto hydrophobic Si substrates from chloroform/hexane mixture (1 5 v/v) (a, b) immediately after addition of hexane into chloroform solution of PHT (c) 15 min after addition of hexane (d) cross-section taken along the particle marked by aHTow. (e) Height distribution of spherical particles adsorbed from PHT solution (0.001 g/L) (f) length distribution of one-dimensional aggregates formed at 0.01 g/L of PHT. (Kiriy, N. et ah, Nano. Lett., 3(6), 707, 2003. With permission.)... 

Figure 2 The length distribution of TMS in 168 proteins expected (Figure 2A) and predicted as TMH by us (Figure 2B). Two-times cross validation procedure (Methods) was used.

---