Scattering monotonic

The polymer concentration profile has been measured by small-angle neutron scattering from polymers adsorbed onto colloidal particles [70,71] or porous media [72] and from flat surfaces with neutron reflectivity [73] and optical reflectometry [74]. The fraction of segments bound to the solid surface is nicely revealed in NMR studies [75], infrared spectroscopy [76], and electron spin resonance [77]. An example of the concentration profile obtained by inverting neutron scattering measurements appears in Fig. XI-7, showing a typical surface volume fraction of 0.25 and layer thickness of 10-15 nm. The profile decays rapidly and monotonically but does not exhibit power-law scaling [70]. 

Figure 2 Variations in the neutron scattering amplitude or scattering length as a function of the atomic weight. The irregularities arise from the superposition of resonance scattering on a slowly increasing potential scattering. For comparison the scattering amplitudes for X rays under two different conditions are shown. Unlike neutrons, the X-ray case exhibits a monotonic increase as a function of atomic weight.

Monotonic Overlayers. In the absence of elastic scattering, the signal of the substrate B covered by a fractional monolayer < >A of A is... 

The geometric properties of highly denatured states appear to be consistent with those expected for a random-coil polymer. For example, proteins unfolded at high temperatures or in high concentrations of denaturant invariably produce Kratky scattering profiles exhibiting the monotonic increase indicative of an expanded, coil-like conformation (Fig. 1) (Hagihara et al., 1998 see also Doniach et al., 1995). Consistent... 

Guinier s law exhibits two parameters, I (0) and R2, which describe structural aspects of the sample. The experimentalist should consider their determination, if the recorded SAXS data show a monotonous decay that is indicative for the scattering from uncorrelated1 particles. Particularly useful is the evaluation of Guinier s law, if almost identical particles like proteins or latices are studied in dilute solution (cf. Pilz in [101], Chap. 8). The absolute value of 1(0) is only accessible, if the scattering intensity is calibrated in absolute units (Sect. 7.10.2). 

The effect of restricted junction fluctuations on S(x) is to change the scattering function monotonically from that exhibited by a phantom network to that of the fixed junction model. Network unfolding produces the reverse trend, the change of S(x) with x is even less than that exhibited by a phantom network. Figure 6 illustrates how the scattering function is modified by these two opposing influences. 

Nonlinear mapping (NLM) as described by Sammon (1969) and others (Sharaf et al. 1986) has been popular in chemometrics. Aim of NLM is a two-(eventually a one- or three-) dimensional scatter plot with a point for each of the n objects preserving optimally the relative distances in the high-dimensional variable space. Starting point is a distance matrix for the m-dimensional space applying the Euclidean distance or any other monotonic distance measure this matrix contains the distances of all pairs of objects, due. A two-dimensional representation requires two map coordinates for each object in total 2n numbers have to be determined. The starting map coordinates can be chosen randomly or can be, for instance, PC A scores. The distances in the map are denoted by d t. A mapping error ( stress, loss function) NLm can be defined as... 

The best equations to use for intensity are those of the dynamical theory, but kinematical theory gives some useful insights, based simply upon the idea that the scattering increases monotonically with the stractrrre factor, equation (4.1). 

Therefore, we expect the scattered intensity /(q, t), proportional to ( /q(t)) to be exponential in time, 7(q, t) exp(2fiqr), with the rate Oq = q AT — q with both lower and upper cutoffs in q. If these arguments are valid, flq/ should rise sharply with q, reach a maximum, and then decrease at higher q values. These predictions are fully consistent with the experimental [10] and simulation [57] observations on /(q, t) and Oq. If the mechanism is simply a spinodal decomposition into two liquid phases, then D. /q should show a monotonic linear decrease from a finite positive value at q 0 with a slope independent of quench depth, which is not experimentally observed during... 

The strongest evidence supporting small (< 0.1 jum) particles in noctilucent clouds is the high degree of measured linear polarization it increases monotonically with scattering angle to almost unity near 90°. It is difficult to reconcile these observations with any conclusion other than that the particles are small. 

Mori et al. s results for three PBLG samples in m-cresol. For every sample, Dr decreases monotonically with increasing c, in a way resembling the simulation results shown in Fig. 16b. These Dr data, as well as the dynamic light scattering data, will be compared with Eqs. (50)- (52) in Sect. 8 together with zero-shear viscosity data. 

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