Exponents, scaling

Sin cc til e basis set is oblairicd from atom ic calcii laliori s, it is still desirable to scale expon eti ts for the rn oleeular en viron tn eti t, Th is is accom piished by defiri in g an in ri er valen ce scale factor 1 and an outer valence scale factor C" ( doiihle zeta ) and multiplying the correspon din g in ri er an d otiler ct s by th e square of these factors. On ly the valen ce sh ells arc scaled. 

Since the basis set is obtained from atomic calculations, it is still desirable to scale exponents for the molecular environment. This is accomplished by defining an inner valence scale factor and an outer valence scale factor ( double zeta ) and multiplying the corresponding inner and outer a s by the square of these factors. Only the valence shells are scaled. 

Note that scaling exponents depend on preparative conditions. ... 

Size Factoring Exponents It is possible to estimate the cost of equipment from the cost of a similar piece of equipment with different size using size factoring or scaling exponents as follows ... 

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

Determination of the Scaling Exponent from the Griineisen Parameter.666... 

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

To obtain the scaling exponent from EOS data, we consider a conventional metric of the relative... 

Since the glass transition corresponds to a constant value of the relaxation time [15], dTjdP is just the pressure coefficient of Tg. Comparing Equations 24.10 and 24.13, we see that the scaling exponent is related to quantities—thermal pressure coefficient, thermal expansion coefficient, Tg, and its pressure coefficient—that can all be determined from PVT measurements... 

DETERMINATION OF THE SCALING EXPONENT FROM THE GRUNEISEN PARAMETER... 

The MW dependences of the normalized chain relaxation times in melts of linear and branched samples are compared in Fig. 12. Both can be represented by scaling power laws, but with remarkably different scaling exponents. For the melts of linear chains, the exponent 3.39 is observed close to the typical value of 3.4 for such systems. In contrast, for the fractions of the branched polymer, the exponent is considerably lower (2.61). It is interesting to note that the value of the normalized chain relaxation time for the feed polymer with the broad M WD fits nicely into the data for the fractions with narrow MWDs. This seems to indicate that conclusions can also be drawn from a series of hyperbranched polymers with broad MWDs. 

Similar relations between different scaling exponents were also developed by Stauffer [37] by combining two of the scaling relations at a time to eliminate Ip - Pel-... 

The scaling exponent a can be related to the particle shape. One finds a = 2,0, 0.5, and 0.8 for a thin rod, solid sphere, ideal chain, and swollen chain, respectively. For most polymers K and a have been tabulated [23]. For a monodisperse sample Equation (36) can be used for a crude determination of the molar mass ... 

For the symmetric system (< )0 = 0.5) the scaling exponent for the Euler characteristic has been found in accordance with the dynamic scaling hypothesis x L(t) 3 (see Section I.G). The homogeneity index, HI, of the interface defined as [222]... 

The exponents ft and a are the growth and scaling exponents respectively. In both the EW and LG models, the surface width saturates, and the roughening is stable. 

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