Gaussian coil assumption

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. 

The thermodynamic affinity of cyclohexane to polystyrene is known to increase with temperature and, naturally, increasing the temperature must further raise the volume of the polystyrene networks in cyclohexane. There is, however, an additional point we should consider. The plot of Q vs. temperature exhibits a steplike discontinuity at around 30°C (Fig. 1.14). This discontinuity, resemhling very much a -transition, is located 3-5°C below the -temperature for linear polystyrene in cyclohexane and about 8°C above the -point for star-shaped polystyrene macromolecules. This phenomenon is outside the scope of the questions discussed here, but, naturally, the first assumption of the authors [143] seems to be very logical, according to which the discontinuity reflects a transition from Gaussian coil to a supercoiled compact structure on cooling the swollen gel below that temperature zone. 

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

So far the development of the theory is very general - the fact that we deal with polymers has not yet been used. Now the important point is that the RPA [2,114] allows an explicit calculation of the r s, using gaussian chain statistics. The central assumption that the interactions, Eq. (173), do not affect the coil configurations, will be critically examined in the light of computer simulation results in Sect. 5.4. We shall not give the details of this RPA calculation here, but just quote its most relevant results [43] ... 

---