Scattering invariant

Fig. 5.6 Small-angle scattering invariant (line), domain spacing (d, open circles) and PE lamellar thickness (/, filled circles) versus time during a series of quenches for a PE-PEE diblock (M = 20kgmol, /pE = 0.55) (Hamley etal. 1997a). The sample was successively quenched from 140°C to 104 C, 100°C, 98°C, 106°C. (Tm (PE) = 108°C for this sample.)...

As we have already discussed, a lossless FDN results when the feedback matrix is chosen to be unitary. Smith and Rocchesso have shown that the waveguide interpretation leads to a more general class of lossless scattering matrices [Smith and Rocchesso, 1994], This is due to the fact that each waveguide may have a different characteristic admittance. A scattering matrix is lossless if and only if the active complex power is scattering-invariant, i.e., if and only if... 

Figure 2. Small-angle x-ray scattering invariant versus scattering angle for 1100 equivalent weight precursor and Its acid form showing the effect of hydration. Reproduced with permission from Ref. 25. Copyright 1981 John Wiley Sons, Inc.

The latter assumption has been verified within experimental error from an analysis of the total scattering invariant which has been calculated from the absolute intensity of scattering. The results for n listed in Table II show an apparent increase at low water contents and then a slight decrease at large water contents. It is noted that this decrease in H implying particle coalescence is in apparent contradiction to the hard sphere model used above. 

The values of the reduced scattering invariant calculated fi-om the SANS data collected during the first freezing step of PVA/D2O solutions are reported in Fig. 17 as a function of the permanence time of the solutions at — 13 °C. In Fig. 17a (inset), the values of Piow(0 of pure D2O are also shown for comparison. 

Fig. 18 Avrami plot of the reduced scattering invariant IPhighW normalized to the extrapolated value of the scattering invariant at infinite time IPhighCt °o) for solutions with PVA volume fraction pva of 0.086 circles) and 0.042 squares). The Avrami exponents n are indicated. (Reproduced with permission from [77]. Copyright 2008 by the American Chemical Society)...

The scattering invariant, which can be experimentally obtained as the integrated total intensity, describes the mean square fluctuations of the scattering length density (SLD), Qinv = f I Q)Q dQ = which is independent... 

Table 2.1 Chemical and physical structures of the PUs studied in this work. Q is the relative SAXS scattering invariant obtained from the intensity distribution, d is the dominant long period obtained from q the position of the peak SAXS intensity, AJV is the particle surface-to-volume ratio obtained from SAXS, and y is the degree of crystallinity as determined from WAXS [135],...

Fig. 4.27 and 4.28 show the cycle work input plotted versus scattering invariant Q and degree of crystallinity if. is a measure of the mean flow stress during... 

From (11.8), the normalized scattering invariant has been calculated as the ratio between the invariant at a deformation e to the value of the invariant in the undeformed state ... 

Fig. 11.11 Small angle scattering invariant of the samples 1PP4 with [rr] = 5.9 mol% (a, a ) and iPP5 with [rr] = 11.0 mol% (b, b ) normalized for the invariant of the unoiiented sample (e = 0) as a function of strain (R, 11.9). Curves a and b are corrected for the decrease in thickness t by assuming that t decreases according to the power law t = to loll)", with to the thickness of the undeformed specimen and i/ = 0.5 as expected for an ideal rubbery material. The correction has been applied for deformations >200%...

More precisely, the scattering invariant for an ideal binary phase system without nano-voids is expressed by Eq. (11.10) (Glatter and Kratky 1982) ... 

However, in the stretching experiments, we have to consider also cavitation. Cavitation creates an additional phase, characterized by zero electron density. Therefore, in the deformation range between 200% and 400-500% our systems consist of four phases, y form, mesophase, amorphous compraient and nanocavities. At strain higher than 400-500 % the y form is almost completely transformed into mesophase, and only three phases (mesophase, amorphous component and nanocavities) should be considered. The scattering invariant for this three-phase system (that is, for deformations e above the threshold limit at which transformation of y form is almost complete) can be written as (Glatter and Kratky 1982) ... 

In this paper, we calculate quantities which are independent of the one dimensional model assumption. Here, the scattering invariant, Q, is found fitxn. 

Figure 6. Scattering invariant, Q, and Bragg long period, Lb, of m-iPP a.) vs. time during isothermal crystallization at 117°C b.) vs. temperature during subsequent melting at heating rate of 1 °C/min,...

Porod s Law and the scattering invariant,both of which are readily accessible characteristics of the scattering curves in the Ultra SAXS region, to test our mechanism. Porod s Law constant, Ci, is obtained from the region to the right of the low-q Scattering peak where intensities decrease with a q dependence. 

Fig. 37 SANS spectra of flexible phthalic (A) and rigid naphthalenic ( ) SPI membranes soaked in deuterated water. The SANS spectrum of a water-swollen Nafion membrane is given as reference ( ). The data were normalized by the scattering invariant for comparison...

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