CONTIN algorithm

If the CONTIN algorithm was running in equal spacing on the logarithmic scale and the linewidth distribution was normalized by the area, then the intensity of scattered light for each fraction P can be expressed as G(lnP ) which is related to G(Pj ) by the relation... 

Operator independence. The GEX method is virtually operator independent. The only inputs required before fitting are parameters concerning the precision of the numerical integration and exit criteria for the Marquardt algorithm. The same set of Inputs was used for all five MWDs. Currently the method also requires upper and lower limits on the GEX parameters but a simple modification of the code can eliminate this need. The CONTIN algorithm has several operator and case dependent parameters that have to be chosen before analysis. However, it is fairly stable with respect to bad choices for some of these inputs. The GEX fit method cannot fit multimodal MWDs without prior knowledge of the number of peaks in the distribution. While CONTIN does not impose this limitation it 1s recommended that the number of peaks be specified before analysis. In our experience with CONTIN if this condition is not met the algorithm tends to compute unimodal solutions for multimodal MWDs and sometimes visa versa. 

Figure 5 shows an example of NNLS and CONTIN analyses of PCS data for a Stober 68) silica suspension (Fig. 4). In this example, the NNLS and CONTIN algorithms of Brookhaven s light scattering software (70) were used.

A method for obtaining G(T, 9) from ACF, gn)(tcorr,0), suitable fortreating broad and multi-modal distributions, was developed by S. Provencher [47,48]. This method, based on regularized non-negative least-squares technique, also known as the CONTIN algorithm, has won common acceptance and is utilized in commercial PCS instruments. While the detailed mathematical description of this method is rather cumbersome and is beyond the scope of this book, we will still briefly explain the essence of it. 

The p (f) vector corresponds to the maximum entropy principle of the/-order. This procedure was used to modify the CONTIN algorithm (CONTIN/MEM-/, where / denotes the order of p (f)). A self-consistent regularization procedure (starting calculations were done without application of MEM) with an unfixed regularization parameter (for better fitting) was used on CONTIN/MEM-/ calculations, which were applied for mesoporous ordered silicas (vide infra). 

Equations 1.71 and 1.72 can be solved using the regularization procedure based on the CONTIN algorithm (Provencher 1982). 

Equations 10.69 and 10.70 as Fredholm integral equations of the first kind were solved using a regularization procedure based on the CONTIN algorithm. The use of Equation 10.70 allows us to describe the sum relaxation, which can demonstrate certain deviation from the Arrhenius law because of, for example, the cooperative effects characteristic for such supramolecular systems as PVA or PVA/nanosilica. 

By the cumulant analysis the intensity weighted hydrodynamic diameter (z-average) and a polydispersity index (PDI, a parameter for the width of the size distribution) are obtained. These values give, however, just an "overview" over the whole sample. For samples with a very broad size distribution (PDI > 0.5), the z-average does not reliably represent the size of the nanoparticles and interpretation is often difficult or even meaningless. SLN dispersions normally possess relatively broad size distributions with PDI values between 0.15-0.30. By complex calculations (e.g. using the Contin algorithm), the size distribution (by intensity, volume or number) can be estimated. For calculations of the size distribution, the optical properties of the particles and the dispersant (refractive indices, absorption) have to be known or estimated. 

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