Chain distribution function

Fig. 8.8 The bond fluctuation model. In this example three bcmds in the polymer arc incorporated into a singk effecti bond between effective moncmers . (Figure adapted from Baschnagel J, K Binder, W Paul, M Laso, U Sutcr, I Batouli [N ]ilge and T Burger 1991. On the Construction of Coarse-Grained Models for Linear Flexible Polymer-Chains -Distribution-Functions for Groups of Consecutive Monomers. Journal of Chemical Physics 93 6014-6025.)...

For investigation of the effect of the width, w, of the distribution on the ultimate fibre strength the uniform chain distribution function is chosen. As in experiments often only the molecular weight distribution fw(z)=zfi(z)z l is determined, we used a uniform molecular weight distribution with a width w defined as... 

Zimm s approximation [5] assumes that inter-chain interactions occur only through single contacts. Given two monomers i and j that belong to two different chains (say, called 1 and 2), the two-chain distribution function is ... 

For fast flow deformations of polymer fluids, a non-linear theory of chain deformation and orientation is considered. To account for non-hnear effects and finite chain extensibility, inverse Langevin chain statistics is assumed. Time evolution of chain distribution function in the systems with inverse Langevin chain statistics has been discussed in earlier papers [12,13] providing physically sensible stress-orientation behaviour in the entire range of the deformation rates and chain deformations. 

If two different chains with the interchain interaction parameter i o3 differ in their lengths Noi and N02 and their interaction parameters Uqi and t>02) then, on the basis of Equations 5.1-62,-85, 118, a two-chain distribution function with fixed ends... 

As for the random force due to thermal motion, we follow the conventional treatment of the Gaussian white noise [26] leading to a diffusion term in the time-development equation of the distribution function. We then finally have the coupled equations for the chain distribution functions in the forms... 

The amplitude e is assumed to be sufficiently small, and hence we can expand the chain distribution functions in powers of e. 

This theory, like any other molecular theory of mbber-like elasticity, is based on a chain-distribution function, which gives the probability of any end-to-end... 

J. Chem. Phys., 95, 6014 (1991). On the Construction of Coarse-Grained Models for Linear Flexible Polymer Chains—Distribution Functions for Groups of Consecutive Monomers. 

The dynamical model is that of convection-diffusion. By assumption, on the coarse-grained time scale the chain distribution function is relaxed constantly, since all internal modes are in equilibrium. The simplification is enormous, since now we do not have to consider memory effects associated with unrelaxed chain conformations. In fact, we simply combine the flux equations of linear nonequilibrium thermodynamics... 

For the chain (homogenous) consisting of one con-former, osmotic forces are similar to the ones stretching the molecule by the ends. Then, labor of the distance being estimated at constant temperature T , one can estimate 5ch value from the condition = F AR = T ASch)- If a more accurate estimation of the distance change valRe between the ends is required, one may calculate the R value, taking into account the distribution function of the distances between the ends R. The value of the mean-square distances between the ends of the chain, being stretched by forces, applied to the ends equals [14] ... 

Figure 4 (curve 1) shows that in the absence of extension the distribution function W(fi) lies in the range 0 < /S < 0.2 for relatively long chains. In other words, in the absence of external forces, crystallization of flexible-chain polymers always proceeds with the formation of FCC since in the unperturbed melt the values of /3 are lower than /3cr. For short chains, the function W(/3) is broader (at the same structural flexibility f) (Fig. 4, curve 2) and the chains are characterized by the values of > /3cr, i.e. they can crystallize with the formation of ECC. Hence, at the same crystallization temperature, a... 

Figure 11 shows that the molecular weight distribution in the melt (presence of short chains) can account for the coexistence of two types of crystals in the absence of molecular orientation or at a slight stretching of the melt. However, there is a purely thermodynamic reason for the appearance of this main structural feature of samples crystallized under conditions of molecular orientation, even at high degrees of orientation, when virtually the whole distribution function is displaced into the region of /S > /3cr. 

The results of the three-dimensional random walk, based on the freely-jointed chain, has permitted the derivation of the equilibrium statistical distribution function of the end-to-end vector of the chain (the underscript eq denotes the equilibrium configuration) [24] ... 

The average force f(r) in the chain when the ends are held a distance r apart could then be obtained from Eq. (10) providing the appropriate configuration distribution function p(r) is known. In the limit of a small extension ratio, p(r) is approximately proportional to peq(r) ... 

Applying the TABS model to the stress distribution function f(x), the probability of bond scission was calculated as a function of position along the chain, giving a Gaussian-like distribution function with a standard deviation a 6% for a perfectly extended chain. From the parabolic distribution of stress (Eq. 83), it was inferred that fH < fB near the chain extremities, and therefore, the polymer should remain coiled at its ends. When this fact is included into the calculations of f( [/) (Eq. 70), it was found that a is an increasing function of temperature whereas e( increases with chain flexibility [100],... 

A plausible assumption would be to suppose that the molecular coil starts to deform only if the fluid strain rate (s) is higher than the critical strain rate for the coil-to-stretch transition (ecs). From the strain rate distribution function (Fig. 59), it is possible to calculate the maximum strain (kmax) accumulated by the polymer coil in case of an affine deformation with the fluid element (efl = vsc/vcs v0/vcs). The values obtained at the onset of degradation at v0 35 m - s-1, actually go in a direction opposite to expectation. With the abrupt contraction configuration, kmax decreases from 19 with r0 = 0.0175 cm to 8.7 with r0 = 0.050 cm. Values of kmax are even lower with the conical nozzles (r0 = 0.025 cm), varying from 3.3 with the 14° inlet to a mere 1.6 with the 5° inlet. In any case, the values obtained are lower than the maximum stretch ratio for the 106 PS which is 40. It is then physically impossible for the chains to become fully stretched in this type of flow. 

Pulsed deuteron NMR is described, which has recently been developed to become a powerftd tool for studying molectdar order and dynamics in solid polymers. In drawn fibres the complete orientational distribution function for the polymer chains can be determined from the analysis of deuteron NMR line shapes. By analyzing the line shapes of 2H absorption spectra and spectra obtained via solid echo and spin alignment, respectively, both type and timescale of rotational motions can be determined over an extraordinary wide range of characteristic frequencies, approximately 10 MHz to 1 Hz. In addition, motional heterogeneities can be detected and the resulting distribution of correlation times can directly be determined. 

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