Helfand model

Figure 3. Time-dependent anisotropy for anthracene-labeled polyisoprene in dilute hexane solution. The experimental anisotropy was obtained by setting the delay between the excitation and probe pulses to a given position and then varying the polarization of the probe beam. In the bottom portion of the figure, the smooth curve through the data is the best fit to the Hall-Helfand model(Ti=236 ps, t2=909 ps, and r(0)=0.250). Unweighted residuals for the best fit to this model are shown along with the experimental error bars in the top portion of the figure. Note that the residuals are shown on an expanded scale (lOx). The instrument response function is indicated at the left.

Figure 5. Time-dependent anisotropies for labeled polyisoprene chains in dilute 2-pentanone solutions. The smooth curves through the data points are the best fits to the Hall-Helfand model for 22.8 C, -8.6 C, and -26.5 °C (bottom to top). The data at 35.1 °C is omitted for clarity. Semilog plots of the best fit correlation functions are shown in the inset. Note that all the correlation functions are quite non-exponential.

Figure 6. Arrhenius plot for dilute solutions of labeled polyisoprene in 2-pentanone. The activation energy calculated from the slope of the best fit line is 7.4 kJ/mole. On the vertical scale. T represents the 1/e point of the best fit correlation functions using the Hall-Helfand model. The data points represent results of independent experiments. The units for t and n are ps and centipoise, respectively.

The constant shape of the correlation function in various solvents at different temperatures implies that the same mechanisms are involved in local motions under all conditions investigated. In terms of the Hall-Helfand model, the ratio of correlated to uncorrelated transitions is constant. Analysis of the temperature dependence of the labeled polyisoprene yields an activation energy of 7.4 kJ/mole for local segmental motions. 

To apply the models to the interpretation of the data, the approach developed for the polycarbonates will be followed. The phenyl proton Tj s are interpreted first in terms of segmental motion. For these protons, the dipole-dipole interaction is parallel to the chain backbone and therefore relaxed only by segmental motion. In the three bond jump model the parameters tjj and m are adjusted to account for phenyl proton data, and in the Weber-Helfand model the parameters tq and tj are adjusted. Table II contains the three bond jump parameters, and Table III, the Weber-Helfand model parameters. Both models can simulate the data within 10% which is equivalent to the experimental error. 

Table III Phenyl Group Motion Simulation Parameters Weber-Helfand Model Using the...

Table IV Formal Group Simulation Parameters Helfand Model(a) Using the Weber-...

The second maimer in which a model can provide some predictive power is if it can predict multiple observables. The Hall-Helfand model [80] for conformational dynamics contains two adjustable parameters. It has been shown, however, that if these parameters are adjusted to fit the conformational autocorrelation function from BD simulations, the same parameters can reasonably predict conformational cross correlation functions [28]. 

Weber-Helfand model the primary parameter is the correlation time for cooperative backbone transitions, X]. At the lower temperatures studied, xq plays an increasing role in the Weber-Helfand model but xj is still the major factor. This is an interesting point in itself since cooperative transitions were also found to predominate when the Weber-Helfand model was applied to the polycarbonates. Here in the polyformal, single bond conformational transitions do play a larger role and this can be seen in the three bond jump model as well by the drop of m to 1 at lower temperatures. Since xj and x are both measures of the time scale for cooperative motions, it is interesting to note that the Arrhenius summaries of the two correlation times in Tables II and III are very similar. This similarity, taken together with the domination of cooperative transitions in the Interpretations, supports the utility of both models though the Weber-Helfand model is developed from a more detailed analysis of chain motion. 

Two models have been developed independently by Hall and Helfand [105] and by Monnerie at al. [106]. For the Hall-Helfand model (HH), the anisotropy decays according to the equation... 

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