Flexibility parameter

This decrease in effective flexibility can be attained by extending the macromolecules in a mechanical or stationary hydrodynamic field3). The extending fields decrease the freedom of internal rotation of the chain units, and the new effective flexibility parameter now becomes ... 

The thermodynamic analysis of conformational and structural transformations in the melt at high pressures34 showed that the free volume and free energy minimum required for hydrostatic compression is attained as a result of the transition of the molecules in the melt into a more extended conformation (gauche —> trans transitions) since the extended molecules ensure a more compact packing of the chains at compression. Chain uncoiling leads to a decrease in their flexibility parameter f with increasing pressure p ... 

Noting the influence of temperature on the intrinsic viscosity is given by the parameter of chain flexibility (dln[ j]/d7), which gives information about the conformation of the macromolecule chain in solution (Kasaii 2007, Chen and Tsaih 1998). The chain flexibility parameter in the temperature range of 20-29°C is dln[t]]/dT = 4,404.11K-i, 0.9993 and in... 

These dimensionless residence time aistributions E(0) can be obtained by inverting the fully normalised form of the system transfer function G(s). This is given by eqn. (45), which emphasises more clearly than eqn. (43) that the model in question contains only one flexible parameter, N. 

For the purpose of estimating conformational parameters of silaiylene carboorganocyclosiloxanes of the structure XV, two approaches were used [117] computerized mathematical simulation using the Monte-Carlo method and experimental estimation of the flexibility parameters in solution under natural conditions. In the first case, mathematical simulation has determined the skeletal flexibility of the molecule in the absence of substituting agents at atoms forming the backbone. In the second case, all fragments of the chain, possible interactions with the solvent and temperature effect have been taken into account in the flexibility estimation. 

True value of macromolecule flexibility was calculated from [t ], A2 and Mco of the fractions using some theoretical equations. Calculation of flexibility parameters at free rotation by the Monte-Carlo method indicates that free rotation of the molecule is possible in four cases only (12, 14, I5, 1,) in three cases it is hindered if. 13, 17), which is observed from Table 21 of the schematic presentation of the chain unit (scheme 1). 

In accordance with calculations, current flexibility parameter (( h2) fr/M), where... 

The interference of cyclization should be, however, considered for any new system and examined experimentally. The spanning-tree approximation with properly selected chain flexibility parameters may be suitable. One can also use the simulation in three-dimensional space mentioned in Section 3.6. 

This system of rigid blocks with flexible spacers may serve as a model of polymers with a limited flexibih ty. In his early work Flory considered the behavior of semiflexible chain polymers by introducing the flexibility parameter / which represents the fraction of bonds which are not in a colhnear position in the... 

Fig. 6. The effect of the variation of the flexibility parameter for a semirigid-chain polymer (at X = 100) on the composition of coexisting phases (according to...

These factors were mentioned soon after the first experimental works on liquid crystalline state of polymers were published. For example, Frenkel considered the effect of these factors by studying the change in the statistical flexibility parameter/ introduced by Flory in his analysis of flexible chain polymers Frenkel proposed the following equation to describe the effect of the type of the solvent on the flexibility of a polymer chain ... 

In a similar way, the effect of an external field extending a macromolecule may be expressed in terms of the effective flexibility parameter in a stationary external field ... 

We next provide an approximate method to estimate the effects of particle flexibility (the fact that the elastic moduli of the dispersed particles are not infinite). Assuming that the effects of flexibility can be described completely by making 4>m0 a function of flexibility, calculated so that as y—the flexibility effects vanish in the same manner as < > —> d>ocp, we use Equation 13.43 where P is a flexibility parameter ranging from 0 in the limit of infinite rigidity to 1 in the limit of complete flexibility, with Equation 13.44 where E is Young s modulus (in MPa, for consistency with how the equation was calibrated to obtain the factor of 22.5) which is valid for both cylindrical fibers of aspect ratio Af>l and cylindrical platelets of aspect ratio (1/Af)[Pg.574]

An explanation of the erroneous estimate of stoichiometric number may be that Equations 6 and 7 contain a number of flexible parameters such as v, Kbutane, Kbutenes and the exponents of partial pressures of both butane and butenes. In addition the equation is nonlinear with respect to these parameters, and the parameters are highly correlated. In those situations it is extremely difficult to obtain precise estimates of parameters, even with well-developed nonlinear regression computer programs. Uncertainty associated with the estimate of stoichiometric number may arise from the fact that the assumptions made when Equations 6 and 7... 

This value is very close to 0.740A which has been obtained for high molecular weight dextran (7 ). By using this value, the flexibility parameter, a (= [/ ]l/2j estimated to be... 

The chemical nature of the IL is introduced by the activity coefficients in Eq. (9.13). By determining the orders of the reaction, kg and the activation energy, taking into account the activity coefficients, are no more flexible parameters but keep a physical meaning. As long as the reaction and scheme does not change, the kinetic parameters k, E, and the individual orders will be independent of the chemical nature of the IL in order to show the applicability of this approach to real data, the hydrogenation of propene at a Wilkinson catalyst is assessed as a test reaction [27]. It is not of commercial interest but has no side or consecutive reactions and therefore is a suitable test system. 

Flexibility specifications are imperative because they drive the design of supply chain processes and shape collaboration with supply chain partners. Static specifications are not acceptable the Upton method makes possible definition of ranges of operations and expectations for customer service. Also, many CEOs seek some kind of visual cockpit for their operations. One based on defined flexibility parameters such as those in the example is an excellent start. 

I he interaction x, and flexibility parameters are determined from [ ) and M measured on a series of polymer fractions in a good solvent by means of Flory-Fox (1951) equation. The latter results from substitution of a from Equation 72 into Equation 3.1-113... 

The molecular parameters characterizing the pure components and the mixtures in the S-S theory, are taken from reference [6], The pure component parameters were estimated from equation of state data [13,14]. Values for the mixing parameters e i2 and v i2 were adjusted to give quantitative agreement between the computed and experimental critical conditions. Since all the model parameters are available, we are in a position to predict other thermodynamic properties. As an example, spinodal conditions are considered. Details concerning the computational methods have been presented elsewhere [5]. It can be observed in Figure 1 that, in comparison to the experimental spinodals, the predicted spinodals become too narrow with decreasing molar mass. If the flexibility parameter c is allowed to vary with molar mass in a manner dictated by the experimental spinodal data, a quantitative description of these data can be obtained [6]. 

The denominator is proportional to the total fluorescence intensity, r is in the main a measure of the rotational motion or dynamics of the molecule, the latter being very useful in obtaining flexibility parameters of local segments of a macromolecule. 

A simple (though not very efficient) solution is to use simple convergence control schemes which only admix the new solutions to the starting orbital set via a flexible parameter. Much more efficient schemes have been developed such as the direct inversion of the iterative subspace (DllS) by Pulay et al. [380-382], in which the new guess spinors for the next iteration step are expanded into a couple of previous solutions. This expansion can even very conveniently be done with the (approximate) Fock operators. 

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