Scattering structural parameters from correlation

In order to eliminate parameters that are correlated to each other, we calculate their Pearson correlation coefficients (25). Linearly uncorrelated parameters have Pearson correlation coefficients close to zero and likely describe different aspects of the phenotype under study (exception for non-linearly correlated parameters which cannot be scored using Pearson s coefficient). We have developed an R template in KNIME to calculate Pearson correlation coefficients between parameters. Redundant parameters that yield Pearson correlation coefficients above 0.4 are eliminated. It is important to visually inspect the structure of the data using scatter matrices. A Scatter Plot and a Scatter Matrix node from KNIME exist that allow color-coding the controls for ease of viewing. 

Nonetheless, by using the Fourier transform and two other mathematical operations, convolution and correlation, we can obtain several important structural parameters fiom the experimentally measured scattered intensity. The usual assumption is that the system can be represented by two-phases, with either sharp or difiuse boundaries between the phases. Some structural parametas of interest include average separation distance between the phases specific surface, Og average thicknesses cf the two phases and width of the boundary between phases, if diffose. These parameters are obtained from the "auto-correlation" function of p(x), which is a specific type of convolution. 

Once the fitting parameters A, B, C are determined from the scattering profiles, one can determine the structural parameters such as the repeat distance D between oil and water, and the structural correlation length and the area Ah per head of the surfactant molecule in the interface by the following equations [1,3] ... 

A framework is introduced, within which we may begin to discuss usefully the local molecular arrangements of noncrystalline polymers. The statistical structure is partitioned into intrachain or conformational, orientational, and spatial interchain parameters. The procedures which initially employ the comparison of the experimental intensity data with scattering functions derived from molecular models, are described with reference to natural rubber. This is seen as a "typical" polymer system. More complex chemical configurations are considered. Both poly(a-methylstyrene) and the phenylene range of polymers appear to exhibit distinct and additional local correlations. The role of these special correlations within the general framework of noncrystalline polymers is discussed. 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

The polymerizability of R-(EO)n-VB macromonomers has its maximum (Rp) around n=15-20 [51]. This finding was related to the micelle formation which is expected to be unfavored for either too long or too short chain length of PEO. The macromonomers and their polymacromonomers with very short R are soluble in water and therefore they lose their amphiphilic nature. The parameters of R and n of macromonomer (R-(EO)n-VB) were found to correlate with the formation of micelles and their structure. In the aqueous phase the scattering intensity increased with the concentration of macromonomer above the CMC. The critical micellar concentration in water was found to be in the range from 3.3 xl0 5to 7.1xl0 5 mol dm-3 for several R-(EO)n-VB macromonomers. 

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