Theoretical Upper-Bound Estimate

In order to estimate the theoretical upper bounds for the electrical conductivities of polymer nanotube composites, equation (15.1) was used to calculate conductivity values for model composites based on both SWNT and MWNT. The values are included in Table 15.2 together with the electrical conductivities of individual CNTs as reported in the literature. Although arc-synthesized MWNTs are likely to possess higher conductivities than CVD-grown ones, no distinction is made in the present analysis between the two types due to the unavailability of reliable data. An electrical conductivity of IE-9 S/m is taken to represent the conductivity of a typical polymer matrix. 

Bois et al. (1990) used Monte Carlo simulations of a coupled PBPK, multistage model to predict a median cancer risk estimate for humans. With their model, they predicted that a concentration of0.00015 ppm was associated with a theoretical upper-bound risk of 1.6 new cases of cancer per million exposed individuals. The 5, 25, 75, and 95 percentile risk estimates are 0, 0.04,2.8, and 6.8 per million, respectively. The kinetic parameters defining the metabolie rate were the most influential parameters in the model. Without PBPK modeling, the theoretieal upper-bound human eancer risk for exposure to 1 pg tetrachloroethylene/m was estimated to be 5.1 x 10 (Travis et al. 1989). With PBPK modeling using a model similar to those described by Hattis et al. (1990), the theoretieal upper-bound cancer risk was estimated to be 3.1 x 10 (Travis et al. 1989). 

The experimental and theoretical results for E., are shown in Figure 3-41 for a resin content by weight ranging from 10% to 100%. Because E. is not a function of C, only k was varied — two values were chosen k = 1 and k =. 9. Some experimental results in Figure 3-41 lie above the curve for k=1 (i.e., above the upper bound ) some results lie below k =. 9. However, most results lie between k =. 9 and k = 1 with k =. 9 being a conservative estimate of the behavior. The actual specimens were handmade, so the resin content might not be precise, and fiber misalignment is not unexpected. Thus, the results above the upper bound are not unusual nor is the basic fact of variation in E. ... 

Figure 3. Theoretical estimate of the average anisotropy for a system of randomly oriented crystallites of bcc Fe8oSi2o (Ai = 8.2 kj/m3) with grain size D and embedded in an amorphous matrix with a volume fraction x = 0.75 [29]. The atomic-scale anisotropy constant of the amorphous phase was taken as K = 430 kJ/m3, which is the value for Fe2B and which can be looked upon as an upper bound. Further materials parameters are found in Ref. [28].

The metal ion-acetylene complexes are calculated in the bridging structure geometry. No asymmetric structures were examined. Theoretical studies on symmetric Cu" - and Ag -acetylene complexes have recently been reportedThe only experhnental value of the dissociation energy for the three systems studied here is for CUC2H2 where an upper bound of 30.5 kcalmoL has been estimated. The QCISD(T)/PP//MP2/PV calcu-... 

Figure 5J9 Estimated thermal conductivity of Nafion 1100 EW at different humidity ratios. The thermal conductivity variation of pme water is shown as an upper bound for the theoretical moist Nafion thermal conductivity [49].

To determine the second virial coefficients to the required acciu acy of about 100 ppm, one needs to know the helium dimer potential to a few mK at the minimum. Consider first the interaction of two hehum atoms in the nonrelativistic Born-Oppenheimer (BO) approximation. The best published calculations using the supermolecular method have estimated error bars of 8 mK [ 150,151]. More recently, improved calculations of this type reached an accuracy of 5 mK [ 158]. Also, the SAPT calculations have been repeated with increased accuracy, resulting in an agreement to 5 mK with the supermolecular method. Furthermore, the upper bound from four-electron exphcitly correlated calculations is 5 mK above the new supermolecular value [159]. All this evidence from three different theoretical models seems to show convinc-... 

The theoretical results provided by the large basis sets II-V are much smaller than those from previous references [15-18] the present findings confirm that the second-hyperpolarizability is largely affected by the basis set characteristics. It is very difficult to assess the accuracy of a given CHF calculation of 2(ap iS, and it may well happen that smaller basis sets provide theoretical values of apparently better quality. Whereas the diagonal eomponents of the eleetrie dipole polarizability are quadratic properties for which the Hartree-Fock limit can be estimated with relative accuracy a posteriori, e.g., via extended calculations [38], it does not seem possible to establish a variational principle for, and/or upper and lower bounds to, either and atris-... 

It is useful to be able to estimate diffusion coefficients either to supplement mass transport data or to compare with experimentally determined values. A theoretically based method to estimate the diffusion coefficient includes upper and lower bounds for small molecules and large diffusants, respectively [40], The equation... 

Methyl methacrylate Figure 4 presents values obtained by relaxation experiments for k tor methyl methacrylate at 50 C at Wp values in the range 0.3 - 0.95 (7). Also presented in this figure are the theoretical estimates for the upper and lower bounds for residual termination. It should first be noted that the absolute magnitude of the measured values of k (lO -lO dm mol S ) is several orders of magnitude smaller than the literature value (- 10 dm3 mol- s- reported for low conversions (11). In the Wp range studied, it seems reasonable to infer that residual termination is the major radical annihiliation mechanism operative. The flexible limit for residual termination seems to be the more appropriate in these systems, at least at the lower values of Wp studied. This limit seems physically reasonable. 

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