Barrett formulae

It could be expected that the presence of built-in electric field in incipient ferroelectric films generates the ferroelectric phase in them despite the fact that latter phase is absent in corresponding bulk samples up to zero Kelvins. The ferroelectric phase has actually been observed in SrTiOs thin films (see [17] and references therein). However the theoretical description in Ref. [17] is laeking consideration of the surface effects and depolarization field, which are extremely important for nanomaterials description [18]. All these factors have been taken into account in Ref. [11] in the framework of aforementioned theory. Namely, it was taken into account there, that as low temperature region plays an important role in the properties of incipient ferroelectrics, the parameter a T) in the free energy (3.5a) has to be written with respect to Barrett formula [19] a T) = quantum vibration temperature. It is seen that at... 

T TqOUT coth — To) = CUT (T — Tq) so that Barrett formula transforms... 

Barrett, J.R. 2002. Soy and children s health A formula for trouble. Environ Health Perspect 110 A294—A296. 

Barrett [1984] has further proposed TP expressions for and an, formulated within the Kirkwood-Riseman hydrodynamic theory in the nondraining limit and using approximate formulas, based on numerical simulation, for the requisite statistical averages, (/f >and Rf/), where Ry refers to the distance between the ith andyth chain segments ... 

With the known expression for > 1 from the above-mentioned computer simulations and that for z perturbation calculations, Domb and Barrett [29] made an interpolation formula for or. The latest one [30] proposed after some modifications of the original reads... 

Domb and Barrett [29] made an interpolation formula for as /ck. which agrees with eq 1.27 for small 2 and converges to the limiting value of 0.933 for large 2. It was corrected by Suzuki [34] to give... 

If eq 1.18 is used with eq 1.28, it is possible to calculate as as a function of 2 over the entire range of positive 2. We call the combined equation (not written explicitly here) the Domb-Barrett interpolation formula for the radius expansion factor. 

The above method cannot be applied when the available data for as are limited to those for as < 2, as in the case where we deal with low-molecular-weight polymers or poor solvent systems. In such a case, only curve-fitting is practical for the estimation of B. Thus we prepare two log-log graphs, drawing on one the as vs. z curve graphing the Domb-Barrett interpolation formula, and plotting on the other the measured data (under fixed solvent conditions) of as against Then, we shift the data points on the latter horizontally... 

On the basis of the data shown in Figure 2-4 combined with the perturbation calculations for a(2 x,y) and o(—l x, y) Barrett made the following interpolation formulas ... 

The line TS in Figure 2-6 shows I as a function of as computed from eq 2.8 and the Domb-Barrett interpolation formula for as. It gives a curve which is convex upward and approaches a (oo) value of 0.456 from below. This limiting value is twice as laige as the experimental estimates of 0.22 — 0.25 (see the next section). 

Recently, by combining the perturbation expansion with Monte Carlo simulation data, Barrett [66] constmcted the following inteipolation formula ... 

Expansion on the right-hand side yields D = 2.86 and D2 = 13.1, in good agreement with the correct values. For z > 1 this equation gives h = 4A5z -. The line B in Figure 2-6 has been calculated from eq 2.12 and the Domb-Barrett interpolation formula for as . It levels off more rapidly than the line TS does and approaches 0.235 from below. This limiting value is in excellent accord with the experimental values quoted above. 

Fig. 3-3. Calculated penetration functions (2). Solid line, Douglas-Freed theory (eq 3.63). Dashed line B, Barrett interpolation formula. 

Fig. 4-2. Line WC, calculated from eq 1.6 with v = 0.6. Line DB, calculated from the Domb-Barrett interpolation formula combined with eq 1.9.

Contrary to binary oxides the exchange integral depends on the particle radius and temperature via the size and temperature dependence of the dielectric permittivity e TJt). In particular, one has to substitute s T,R) for 82 in Eq. (4.6) for exchange integral, where r is average nanoparticle radius. The calculations of dielectric permittivity dependence on temperature and particles size have been performed analogously to those described in Sects. 3.2.2.2 and 3.2.2.3 and lead to Barrett-type formula, which could be found in Ref. [48]. In Fig. 4.18 two roots (rji fixed temperature and particle... 

Halmos EP, Muir JG, Barrett JS, Deng M, Shepherd SJ, Gibson PR. Diarrhoea during enteral nutrition is predicted by the poorly absorbed short-chain carbohydrate (FOD-MAP) content of the formula. Aliment Pharmacol Ther 2010 32(7) 925-33. 

Strictly speaking, this formula is not directly comparable. As stated very clearly by Domb and Barrett, only in a special limit does the Comb-Joyce model become the two-parameter model whereas the formula (92) is determined from series in another limit, viz. the self-avoiding walk limit.)... 

These asymptotic relations indicate that both (R ) and (S ) are proportional to in the good solvent limit. This exponent 1.2, predicted early by Flory, is now established experimentally for (S ) the toluene data in Figure 3 indeed have this asymptotic slope for M> 10 . With the above expressions for small and large z, Domb and Barrett constructed interpolation formulas for the expansion factors applicable to the entire positive z range. The result for reads... 

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