Scaling scattering

Fig. 9a,b. The intermoleailar scattering patterns calculated for a the smectic A b the smectic B phases of the mesogen GB(4.4, 20.0, 1, 1) with the scattering vector parallel to the layer normal. The scaled scattering vectors Q and Q range from -8n to 8n... 

Fig. 10-17. Time change of scaled scattering intensity J(7 , )[7 m(0] critical PS-l/PVME-1 blend...

Fig. 10-18. Normalized scaled scattering intensity I k, (= - ( )) indicated Tq for a critical PS-...

When structural-rheological simplicity does not hold and, for example, fi in equation (7) changes with physical aging, exact reduction of the creep or stress relaxation cnrves cannot be accomplished. However, it is estimated by the author that a change in the KWW parameter of approximately 5-10% would result in a deviation of the logarithm of the aging time shift factor (log ate) of 0.05 over two decades in time scale. Scatter in reduced curves of this order of magnitude is often observed because of the scatter in the creep or stress relaxation data (23,39,40). Researchers have not systematically looked for such small deviations in their rednced curves. 

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]. 

One nice thing about H in mass-scaled coordinates is that it is identical to the Hamiltonian of a mass point movmg in two dimensions. This is convenient for visualizing trajectory motions or wavepackets, so the mass-scaled coordinates are commonly used for plotting data from scattering calculations. 

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

When we draw a scatter plot of all X versus Y data, we see that some sort of shape can be described by the data points. From the scatter plot we can take a basic guess as to which type of curve will best describe the X—Y relationship. To aid in the decision process, it is helpful to obtain scatter plots of transformed variables. For example, if a scatter plot of log Y versus X shows a linear relationship, the equation has the form of number 6 above, while if log Y versus log X shows a linear relationship, the equation has the form of number 7. To facilitate this we frequently employ special graph paper for which one or both scales are calibrated logarithmically. These are referred to as semilog or log-log graph paper, respectively. 

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