Crossover Formula

The limits of high salt and low salt have been addressed in Seetions 4.1.1 and 4.1.2, respeetively. In experiments, the salt coneentrations are not neeessarily in sueh extreme limits. Similarly, the dimensionless exeluded volume parameter wVn and the electrostatic excluded volume parameter defined in Equation 4.40 can assume intermediate values, instead of being either zero or very large. The crossover formula for the free energy that recovers the limits of Equations 4.30 and 4.38 is (Muthukumar 1987) 

With the assumption of uniform expansion used in deriving the above crossover 

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Although the theory of polyelectrolyte dynamics reviewed here provides approximate crossover formulas for the experimentally measured diffusion coefficients, electrophoretic mobility, and viscosity, the validity of the formulas remains to be established. In spite of the success of one unifying conceptual framework to provide valid asymptotic results, in qualitative agreement with experimental facts, it is desirable to establish quantitative validity. This requires (a) gathering of experimental data on well-characterized polyelectrolyte solutions and (b) obtaining the relationships between the various transport coefficients. Such data are not currently available, and experiments of this type are out of fashion. In addition to these experimental challenges, there are many theoretical issues that need further elaboration. A few of these are the following ... 

Muthukumar M. Double screening in polyelectrolyte solutions limiting laws and crossover formulas. J Chem Phys 1996 105 5183-5199. 

It must be noted however that the globule is not fully compact, despite its dimension being three, and there is enough room for rearrangement of various monomers inside the globule. A simple crossover formula that connects the globule and the swollen coil is (Grosberg and Khokhlov 1994)... 

Analogous to the derivation of Equations 2.73 and 2.74 for an uncharged polymer, a crossover formula for the radius of gyration of a flexible polyelectrolyte in a solution with high enough salt is obtained as follows. Substituting Equation 2.45 for the free energy of chain connectivity and Equation 2.68 (with w replaced by Weg) for the excluded volume effect in Equation 2.67, we get... 

The crossover formula for the chain length dependence of the average translocation time, (Equation 10.52) is plotted in Figure 10.12 for several values of the driving force = 0,0.1, and 0.5. In this figure, the initial number... 

Similar crossover formulae can be written down for the other quantities of interest, too. While Eqs. (12)-(19) also remain valid for f -> exo, the behavior of fcoex, f j (0) and y /ksT gets correspondingly modified by the crossover. A pronouneed... 

Kramers formula for classical escape out of a metastable well in the case of moderate and strong damping [Kramers 1940]. In accord with the multidimensional theory predictions, the crossover temperature should be equal to... 

Vitamin K is a fat-soluble vitamin cofactor for the activation of factors II, VII, IX, and X in the liver. Almost all neonates are vitamin K-deficient at as a result of (1) insignificant transplacental vitamin K crossover, (2) lack of colonization of the colon by vitamin K-producing bacteria, and (3) inadequate dietary vitamin K intake (especially in breast-fed infants because human milk contains less vitamin K than infant formula or cow s milk). Vitamin K-deficiency bleeding (VKDB) refers to bleeding attributable to vitamin K deficiency within first 6 months of life. It occurs in three general time frames early (0-24 hours), classic (1-7 days), and late (2-12 weeks). Early onset occurs rarely and usually is associated with maternal ingestion of anticonvulsants, rifampin, isoniazid, and warfarin. Classic vitamin K-dependent bleeding usually results from the lack of prophylactic vitamin K administration in... 

Temperature quenching of broad band emission is usually explained by a simple configuration coordinate model consisting of two parabolas that have been shifted with regard to each other (Fig. 6). This is called the Mott-Seitz model. Nonradiative return from the excited to the ground state is possible via the parabola crossover. Its rate can be described with an activation-energy formula Pnr = C where C is a constant of the order of 10 sec i and AE is the... 

Of special interest is the case where the barrier is parabolic, as in Eq. (1.5). Here, it is possible to examine the crossover between the classical and quantum regimes in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the transmission coefficient of the parabolic barrier [Landau and Lifshitz, 1981]... 

This formula, aside from the prefactor, is simply a one-dimensional Gamov factor for tunneling in the barrier shown in Figure 2.10. The temperature dependence of k, being Arrhenius at high temperatures, levels off to kc near the crossover temperature, which, for AE = 0, is equal to kBTQ = hai/4. 

This formula, however, tacitly supposes that the instanton period depends monotonically on its amplitude so that the zero-amplitude vibrations in the upside-down barrier possess the smallest possible period 2itIw. This is obvious for sufficiently nonpathological one-dimensional potentials, but in two dimensions this is not necessarily the case. Benderskii et al. [1993] found that there are certain cases of strongly bent two-dimensional PES s in which the instanton period has a minimum at a finite amplitude. Therefore, the crossover temperature, formally defined as the lowest temperature at which the instanton still exists, turns out to be higher than that predicted by (4.7). At T>TC the trivial solution Q = Q (Q is the saddle point coordinate) replaces the instanton the action is S = pV (where V is the barrier height at the saddle point) and the Arrhenius dependence k exp(-/3V ) holds. 

The structure of Eq. (7.22) is the same as that of Eq. (2.1). It determines the statistically averaged probability of incoherent transition at T>TC. Note that it is the interference of many waves that renders the transition incoherent. Formally below the crossover temperature this formula should describe the rate of the incoherent transitions from the ground state. However, the transition under these conditions should be coherent. This means that the incoherent hopping rate should vanish at 7 = 0. To take this fact into account Clough et al. [1982] calculated 1 statistically averaging (mn) - (m0) rather than mn). Their results show that this correction is significant only for low barriers and T TC. 

CAS CHEMICAL REGISTRY 8,000,000 compounds CHEMICAL ABSTRACTS SERVICE The world s largest file of substance information, including coordination compounds, polymers, incompletely defined substances, alloys, mixtures, and minerals. In each record, the registry number is linked to molecular structure diagram, molecular formula, CA index name, synonyms, and the ten most recent references in Chemical Abstracts. Easy crossover to the bibliographic file... 

Once again, the crossover technique is a quick way to write a formula with the same number of pluses as minuses. 

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