Deviations from Gaussian theory

Although at the moment no adequate theory exists which relates the structure of a network to its elastic behaviour, we may venture to hypothize that the deviation from Gaussian behaviour, as e.g. measured by C2/Cj, is indeed related to the structure. It is of interest to note that... 

More elaborate versions of the molecular theory may account for entanglement, loose ends or loose loops, deviations from Gaussian statistics, the statistics of actual chains, excluded-volume effects (real chains with finite thickness cannot interpenetrate each other), the movement of junction points, the contribution of the internal energy to elasticity, changes in volume, etc. 

There are several reasons why the behaviour of the shorter chains deviate from the original formulation of the Jacobson and Stockmayer theory (Flory, 1969). First, if the ring size is small enough to induce strain, the enthalpy change for cyclisation (16) will differ from that for the intermolecu-lar process (17). In terms of the 0 operator (39), 0AH° will differ from zero and, presumably, be positive. Secondly, (57) is based on the implicit assumption that the relative orientation of the reacting bonds, when they come in close proximity in the cylisation reaction, is random. This independence of orientation and proximity, which leads to the absence of any factor referring to orientation in (57), must fail for short chains. Thirdly, short chains may not follow Gaussian statistics. When this occurs, an appropriate expression for the density of end-to-end vectors is required. 

However, deviations from the parabolic profile become progressively important as the length of the polymers N or the grafting density pa decreases. In a systematic derivation of the mean-field theory for Gaussian brushes [52] it was shown that the mean-field theory is characterized by a single parameter, namely the stretching parameter fi. In the limit p oo, the difference between the classical approximation and the mean-field theory vanishes, and one obtains the parabolic density profile. For finite /3 the full mean-field the-... 

Theories based on these concepts all have to take into account the phenomenology of the stress-strain behaviour of networks. In unilateral extension as well as compression one observes, even at moderate extension (1.1 deviations from the Gaussian behaviour, which can be empirically described by the so-called Mooney-Rivlin equation ... 

The WSL theory developed by Leibler has been shown to be incorrect because of deviations from the fundamental underlying mean-field assumption. Figure 13.14 shows experimental results for a poly(ethylene-propylene/ethylethylene) (PEP-PEE) diblock copolymer that has been fit to the predictions of the Leibler theory without any adjustable parameters, since the ODT and / were calculated from rheological measurements (Bates et al., 1990). This mean-field theory does not qualitatively describe the behavior of this material. Other experiments have indicated that the RPA approximation (Sttihn and Stickel, 1992) and the Gaussian coil assumption (Bates and Hartney, 1985 Holzer et al., 1991) are inaccurate near the ODT. 

In Gaussian theory of diffusion (the most commonly used), x is a function of wind speed (inverse proportionality), the values of the standard deviation of the Gaussian distribution, the height of release and the position of a point in space with reference to the release point. The relevant formulae and diagrams, valid within several tens of kilometres from the release point, are given in Section 6-4. For simplified evaluations that usefully support quick decisions, they are not strictly necessary. For now it is sufficient to know that the formulae and diagrams... 

The theories of gas chromatography use the following approximations (a) the deviation from equilibrium is small (b) the retention time is high compared to the time needed for the equilibrium to be reached (c) the distribution of a component in a band is Gaussian. 

---