Hall-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. 

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]. 

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... 

Viovy, Monnerie, and Brochon have performed fluorescence anisotropy decay measurements on the nanosecond time scale on dilute solutions of anthracene-labeled polystyrene( ). In contrast to our results on labeled polyisoprene, Viovy, et al. reported that their Generalized Diffusion and Loss model (see Table I) fit their results better than the Hall-Helfand or Bendler-Yaris models. This conclusion is similar to that recently reached by Sasaki, Yamamoto, and Nishijima 3 ) after performing fluorescence measurements on anthracene-labeled polyCmethyl methacrylate). These differences in the observed correlation function shapes could be taken either to reflect the non-universal character of local motions, or to indicate a significant difference between chains of moderate flexibility and high flexibility. Further investigations will shed light on this point. 

Many NMR dynamics studies attempt to correlate the observed data with models of molecular motion. Some of the more successful are the Hall-Helfand [189,190], the Dejean-Laupretre-Monnerie (DEM) [189,191, 192], and the Williams-Watts [189,194,195] models. They predict the relaxation times and NOEs expected from specific types of motion such as unrestricted diffusion or discrete jumps. Very often, distinct parts of the molecule are be best modeled by different functions. For a poly(isobutyl methacrylate) solu-... 

FIG. 7.—Hall-Weber-Helfand (HWH) motional model. A[Pg.81]

In contrast with these models, which start from a rather crude representation of the chain, but allow a direct computation of the OACF, Hall and Helfand proposed recently a model able to predict conformational correlation functions (CCF) for rather realistic molecular potentials. The OACF cannot be derived from these CCF, at least at the present time. However, Hall and Helfand suggested that the CCF for a chain of two-state elements ... 

Our experimental measurements of the orientation autocorrelation function on sub-nanosecond time scales are consistent with the theoretical models for backbone motions proposed by Hall and Helfand(ll) and by Bendler and Yaris(12). The correlation functions observed in three different solvents at various temperatures have the same shape within experimental error. This implies that the fundamental character of the local segmental dynamics is the same in the different environments investigated. Analysis of the temperature dependence of the correlation function yields an activation energy of 7 kJ/mole for local segmental motions. 

In a previous publication 17, we compared the experimental anisotropies for dilute solutions of labeled polyisoprene in hexane and cyclohexane to several theoretical models. These results are shown in Table II. The major conclusions of the previous study are 1) The theoretical models proposed by Hall and Helfand, and by Bendler and Yaris provide good fits to the experimentally measured correlation function for both hexane and cyclohexane. The model suggested by Viovy, et al. does not fit as well as the other two models. 2) Within experimental error, the shape of the correlation function is the same in the two solvents (i.e, the ratio of t2/ti is constant). 3) The time scale of the correlation function decay scales roughly with the solvent viscosity. 

Among the various expressions that are based on a conformational jump model and have been proposed for the orientation autocorrelation function of a polymer chain, G t), the formula derived by Hall and Helfand (HH) [4] leads to a very good agreement with fluorescence anisotropy decay data. It is written as... 

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