Weak-link statistics

When the analytical laboratory is not responsible for sampling, the quality management system often does not even take these weak links in the analytical process into account. Furthermore, if sample preparation (extraction, cleanup, etc.) has not been carried out carefully, even the most advanced, quality-controlled analytical instruments and sophisticated computer techniques cannot prevent the results of the analysis from being called into question. Finally, unless the interpretation and evaluation of results are underpinned by solid statistical data, the significance of these results is unclear, which in turn greatly undermines their merit. We therefore believe that quality control and quality assurance should involve all the steps of chemical analysis as an integral process, of which the validation of the analytical methods is merely one step, albeit an important one. In laboratory practice, quality criteria should address the rationality of the sampling plan, validation of methods, instruments and laboratory procedures, the reliability of identifications, the accuracy and precision of measured concentrations, and the comparability of laboratory results with relevant information produced earlier or elsewhere. 

We can regard a fiber as consisting of a chain of links. We assume that fiber failure occurs when the weakest link fails. This is called the weakest-link assumption. It turns out that such a weak-link material is well described by the statistical distribution known as the Weibull distribution (Weibull, 1939,1951). We first describe the general Weibull treatment for brittle materials and then describe its application for fibers. 

Chain cleavage within a polymer chain preferably takes place at weak bonds ( weak links ) present in statistical distribution over the entire molecule. As a result of chain cleavage, molecular weight is significantly reduced, and cleavage of volatile components is minimal. Figure 1.10 [20]. 

Skewed distributions have been seldom reported or even noticed. It is more common to report results in terms of Gumbel or Weibull distributions, which take account of weak-link aspects of voltage breakdown. Standard statistical texts can be consulted for the details of such analysis. An extreme-value distribution for the breakdown voltage of two films is shown in Figure 17. 

For an example of a control chart see Fig. 1.31 and Sections 4.1 and 4.8. Control charts have a grave weakness the number of available data points must be relatively high in order to be able to claim statistical control . As is often the case in this age of increasingly shorter product life cyeles, decisions will have to be made on the basis of a few batch release measurements the link between them and the more numerous in-process controls is not necessarily straight-forward, especially if IPC uses simple tests (e.g. absorption, conductivity) and release tests are complex (e.g. HPLC, crystal size). 

Telechelic polymers rank among the oldest designed precursors. The position of reactive groups at the ends of a sequence of repeating units makes it possible to incorporate various chemical structures into the network (polyether, polyester, polyamide, aliphatic, cycloaliphatic or aromatic hydrocarbon, etc.). The cross-linking density can be controlled by the length of precursor chain and functionality of the crosslinker, by molar ratio of functional groups, or by addition of a monofunctional component. Formation of elastically inactive loops is usually weak. Typical polyurethane systems composed of a macromolecular triol and a diisocyanate are statistically simple and when different theories listed above are... 

The synthetic chemist can see beauty in an approach to a chemical object, be it in the form of the perceived elegance, efficiency, directness, or combination of approaches. Covalent bond forming, coordination chemistry and purely non-covalent possibilities are all important of course, but each one on its own is weak. Take the statistical approach to link formation A flexible linear molecule in random motion is not likely to form a knot, and it is even less likely that we could separate and characterise it even if it did. There is no direction in the reactivity. Covalent bonds can be used to hold fragments together, and then those bonds used to orient the fragments are removed to leave the linked molecule. With today s control over molecular conformation and covalent bond making and breaking, this has to be considered a viable approach. But, those approaches based on coordination chemistry and non-covalent bonds are more direct and efficient for the moment. 

The values of 3.3 to 3.6 obtained for the pK pp value to be assigned to the C-25 gel is close to the pK value of 3.31 (I = O.lCNaClO )) published for methoxyacetic acid [58], the weak carboxylic acid that most closely resembles the weak-acid functionality repeated in the cross-linked carboxymethyldextran. It can be concluded, on the basis of this observation, that Eq. (25) provides an adequate resolution of pKjn,. Indeed, the resemblance between pKjpp and pKjp, has led to the use of statistical arguments for the rationali2ation of the absence of nonideality in the a/(l - a) term. The pKjpp values for the highly swollen gel of C-50 at Cs = 0.01 mol dm, on the other hand, increases with increasing a. The extrapolated pK pp values of 3.3-3.6 at a = 0 are consistent with the values of 3.3-3.6 obtained for the C-25 gel over the a range studied. 

The factors inhibiting macro-flow in vulcanised rubber are the chemical cross links due to vulcanisation (and probably also molecular entanglements), considerably assisted by the phenomenon of crystallisation. In unvulcanised raw rubber macro-flow may occur to a certain extend. In cellulose crystalline junction points of a high degree of stability are responsible for the exclusion of macro-flow. In rubber the chains are very flexible and consist of a large number of statistical chain sections in cellulose the chains are stiffer and consist of a small number of chain elements. In the former case the intermolecular forces are weak, in the latter case they are strong. 

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