Overlap parameter

Here, is an effective overlap parameter that characterizes the tunneling of chaiges from one site to the other (it has the same meaning as a in Eq. (14.60)). T0 is the characteristic temperature of the exponential distribution and a0 and Be are adjustable parameters connected to the percolation theory. Bc is the critical number of bonds reached at percolation onset. For a three-dimensional amorphous system, Bc rs 2.8. Note that the model predicts a power law dependence of the mobility with gate voltage. 

Fig.2. Specific viscosity T]sp vs. the overlap parameter (c [r)]) for various narrowly distributed polystyrene samples in toluene [18]... 

It is worth noting that for the limiting case of a=l, the X0-Mw-c relationship exhibits the same proportionalities as obtained for the Rouse equation [Eq. (18)]. On the basis of Eq. (25) it is possible to obtain a standardisation of the relaxation time if the product i0 c1+1/a is plotted against the overlap parameter, c-[r ],as shown in Fig. 12. 

Fig. 14. Slope in the power-law region of the flow curve as a function of the overlap parameter for narrowly distributed polystyrene in toluene...

In Eq. (34) K is a constant that approximates to unity. Substitution of (t 0-r s) by the r 0-Mw-c relationship, by the A0-Mw-c relationship and the exponent n, as mentioned above, leads to the results that r ( j) is solely a function of the overlap parameter, the term c1+1/a and the rate of deformation, i.e. the shear rate dependence of T) can be satisfactorily described by application of a three-parameter approach. 

The molar mass of the degraded sample can easily be calculated by assuming that the overlap parameter is equal to the intersection point of the horizontal with the fine of slope 1. 

Below a critical concentration, c, in a thermodynamically good solvent, r 0 can be standardised against the overlap parameter c [r)]. However, for c>c, and in the case of a 0-solvent for parameter c-[r ]>0.7, r 0 is a function of the Bueche parameter, cMw The critical concentration c is found to be Mw and solvent independent, as predicted by Graessley. In the case of semi-dilute polymer solutions the relaxation time and slope in the linear region of the flow are found to be strongly influenced by the nature of polymer-solvent interactions. Taking this into account, it is possible to predict the shear viscosity and the critical shear rate at which shear-induced degradation occurs as a function of Mw c and the solvent power. 

Table 1. Angular overlap parameters in tetragonal Cr(III) systems, taken from Refs. 57 and 58. All energies are in kK...

This definition of c is the same as that given in Section 2.4.2 with m = 1 and different units. A variation in the units and symbols used is quite common in the literature and one should be wary of this. Combining Equations (5.65) and (5.72) we obtain an expression related to the overlap parameter ... 

The various zones in Figure 5.23 define the areas where we expect different rheological responses. These transitions are not sharp and the actual location of the phase boundaries can differ slightly depending upon the assumptions used for defining the overlap parameters, or the presence of specificity between the solvent and the chain. In the following sections we will consider the rheological properties associated with some of these phases. 

There have been many studies of hydrophobic crosslinking. For example, Flynn40 produced a series of poly (acrylamides) (PAM) and recorded the low shear rate viscosity as a function of the chain overlap parameter. This was performed for a range of molecular weights and concentrations. This procedure was then repeated with the same polymer backbone but with the addition of differing concentrations of alkyl side chains which give rise to hydrophobic association (HPAM). A comparison between hydrophobe and non-hydrophobe polymers is shown in Figure 5.30. 

The chain overlap parameter has been very successful at superimposing the data from the systems without hydrophobic modification, producing the continuous curve. However, it is clear from Flynn s work that once the hydrophobes are introduced into the polymer the viscosity rapidly increases at lower values of the chain overlap parameter. Increasing the mole percentage of hydrophobes also increases the viscosity at lower values of the chain overlap parameter. The position and number of the hydrophobes on a chain are important in determining the structure that forms and the onset of the increase in viscosity. The addition of side chains to hydroxyethyl cellulose modifies the network modulus as a function of concentration. This is discussed further in Section 2.3.4. 

In an extended version of the hopping concept, positional ( off-diagonal ) disorder in addition to energetic ( diagonal ) disorder has been introduced [54,63]. The simplest ansatz was to incorporate this by allowing the electronic overlap parameter 2ya to vary statistically. Operationally, one splits this parameter into two site contributions, each taken from a Gaussian probability density, and defines a positional disorder parameter I, in addition to the energetic disorder parameter cr. 

Fig. 8.9. Power law exponent d as a function of the coil overlap parameter c[ ] at low concentrations. The filled circles are narrow distribution polystyrene solutions (1 77, 316, 318), the open circles are poly(a-methyl styrene) (198, 318). Solvents are chlorinated di-phenyls except the intrinsic viscosity data which were obtained in toluene. Symbols are for polystyrene M= 13.6 x 106, 4 1-8 x 10 , and 0.86 x 106 for poly(a-methyl styrene) O M = 7.5 x 10 , 6 3.3 xlO6, Cr 1.82 xlO6, O- 1.14x10 , a. 0.694x10 , and...

R. J. Penney, R. D. G. Prasad, and S. P. Reddy. Collision induced absorption spectra of the fundamental band of gaseous deuterium overlap parameters of D2-D2. J. Chem. Phys., 77 131, 1982. 

S. P. Reddy, G. Varghese, and R. D. G. Prasad. Overlap parameters of H2-H2 molecular pairs from the absorption spectra of the collision induced fundamental band. Phys. Rev., A 15 975, 1977. 

TABLE 5 Calculated Overlap Parameters for the Diradical Intermediates During the di-tr Methane and tri-7r-Methane Rearrangements of 14 (see Scheme 9)°... 

These difficulties seem to stem from the definition of the overlap parameter r. This parameter is based on the next-neighbour p7t,p7t overlap value in benzene according to equation 5 ... 

Table 3 Angular Overlap Parameters (cm- ) for some Chromium(III) Complexes...

Equation (75) contains only five parameters, namely, the resonance parameters Ei, Ti, E2, and T2 and the overlap parameter f). Details of the vectors g, or the mixing ratios r are irrelevant to the Q-matrix eigenvalues for the case of two overlapping resonances with an energy-independent background. 

Another approach [18] is to compute f U2 and thence also H via a perturbative expansion in powers of the overlap parameter o. That is, one writes... 

The dimensionless product c[k]] is defined as the coil overlap parameter it provides information about the changing nature of the interactions in a dispersion (Blanshard and Mitchell, 1979 Morris et al., 1981). For dilute dispersions, i.e., below c, the slope of log( qsp/cI) vs log(c[T ]) universally approximates 1.4. At the upper practical extreme, with exceptions (especially the galactomannans Morris et al., 1981), the slope increases sharply to 3.3, illustrating wide deviations from Newtonian flow in the segment approaching elasticity. The deviations are significant when 5 < < 10 (Barnes... 

It is worth mentioning here that the Angular Overlap parameters for the NP3 ligand suggest a -back-bonding interaction with the metal ion, a result which has been later confirmed for other cobalt(II) and nickel(II) complexes109. ... 

Recently Ferguson112 re-examined the polarized spectra of CoCl2 6H20 which contains octahedral C0O4CI2 chromophores. A thorough analysis of the polarizations, and of their temperature-dependence showed that both electric and magnetic dipole mechanisms are operative. The analysis of the crystal field parameters yielded Dq = 810, Ds = -640, Dt = 30 cm-1. Translated into Angular Overlap parameters, these values show that water is a better a and 7t donor towards cobalt(II) than the chloride ion. 

FIG. 16.17 Power law exponent — ft — 1 — n as a function of the coil overlap parameter c[ 7], for solutions of polystyrene (filled symbols) and poly(a-methyl styrene) in chlorinated biphenyls (open symbols). The values of [77] were obtained in toluene. Molecular weights range from 860 to 13,600 kg/mol for polystyrene and from 440 to 7500 kg/mol for poly(a-methyl styrene). From Graessley (1974). Courtesy Springer Verlag. 

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