Polyethylene distribution function

Figure 14 shows the displacement of the distribution function towards high / , i.e. the uncoiling of molecules under the influence of stretching for polyethylene (A = 3 x 10-9 m, N = 100 and T = 420 K). This displacement will be characterized by the position of the maximum of the distribution curve, the most probable value of / , i.e. j3m, as a function of x (Fig. 15). Figure 15 also shows the values of stresses a that should be applied to the melt to attain the corresponding values of x (o = xkT/SL, where S is the transverse cross-section of the molecule). 

The present theoretical approach to rubberlike elasticity is novel in that it utilizes the wealth of information which RiS theory provides on the spatial configurations of chain molecules. Specifically, Monte Carlo calculations based on the RIS approximation are used to simulate spatial configurations, and thus distribution functions for end-to-end separation r of the chains. Results are presented for polyethylene and polydimethylsiloxane chains most of which are quite short, in order to elucidate non-Gaussian effects due to limited chain extensibility. 

The importance of the carrier was also emphasized by Bdhm who found that a supported system, of the Mg(OC2H5)2 —TiCl. and A1(C2H5)3 type, yields polyethylene having a narrower MWD (Q = 7) than that obtained with the traditional TiClj— AlCCjHjlj catalytic system (Q = 10). However, in both cases, MWD can be described by the normal logarithmic distribution function Bdhm subsequently also showed that 1-butene/ethylene copolymers, obtained with the same supported catalytic system, present a polydispersity which can be compared to that of polyethylene. 

Wide angle X-ray scattering from an aligned sample of copolyester prepared from kO mol % polyethylene terephthalate and 60 mol % p.aceto benzoic acid was reported together with an lysis of the derived cylindrical distribution function (CDF)... 

The distribution function just derived is an expression for the probability that an end-to-end vector terminates in the volume element centered about the coordinates x,y,z (at the end of the vector ). Figure 1.31 is drawn for a random flight of 10,000 steps, each step being 0.25 nm, almost double the length of a chain segment of polyethylene. Because of bond restrictions that will be discussed later in this section. 

The polydispersity of polyethylene can often be described with the help of the Wesslau distribution function ... 

Fig. 11a, b. Inter-chain pair distribution function g(r). Sotid curve is the result of bulk NPT Monte Carlo simulation. DasM curve is the PRISM prediction using as input the exact intra-diain pair density prediction using as input the approximate co(r) obtained from fast sampling of continuous unperturbed chains a 32-chain 24-tnsr (n-tetracosane) system at 450 K and 0.1 MPa b 10-chain 78-mer (polyethylene) system at 450 K and ai MPa... 

Figures 9b and 11b show tte intra-chain pair density function and the interchain pair distribution function, respectively, for a system of 10 chains of 78 units each ( polyethylene ) at 450 K and 0.1 MPa. We see in Fig. 9b the increase in the size and density of the segment doud with increase in chain length relative to the n-tetracosane system. Fig. 9a. In Fig. 11b the correlation hole now appears much stronger than for the n-tetracosane system.

If go(r), g CrX and g (r) are known exactly, then all three routes should yield the same pressure. Since liquid state integral equation theories are approximate descriptions of pair correlation functions, and not of the effective Hamiltonian or partition function, it is well known that they are thermodynamically inconsistent [5]. This is understandable since each route is sensitive to different parts of the radial distribution function. In particular, g(r) in polymer fluids is controlled at large distance by the correlation hole which scales with the radius of gyration or /N. Thus it is perhaps surprising that the hard core equation-of-state computed from PRISM theory was recently found by Yethiraj et aL [38,39] to become more thermodynamically inconsistent as N increases from the diatomic to polyethylene. The uncertainty in the pressure is manifested in Fig. 7 where the insert shows the equation-of-state of polyethylene computed [38] from PRISM theory for hard core interactions between sites. In this calculation, the hard core diameter d was fixed at 3.90 A in order to maintain agreement with the experimental structure factor in Fig. 5. 

The above considerations suggest that the hard core radial distribution function obtained for polyethylene was sufficiently accurate on short length scales to predict the perturbative contribution (Eq. (4.9)) to the attractive branch... 

Figure 3. Energy of electron emission (EE) [29] of high density polyethylene film. Solid line total emission as function of the applied retarding field voltage. Each point (circles) is determined by five measurements at least. Broken line energy distribution function the derivative of the solid line function Rate of deformation 10 %/s p= 10 Pa T = 298 K sample dimensions 20 x 10 x 0.098 mm...

The zero point of the chemical shift is arbitrarily fixed. The relative ratios of the principal values are fixed as for the principal values for crystalline polyethylene listed in Table 1. (B) Ou = c 22< 0 33- The thinner lines indicate the theoretical lineshapes according to eqns (14) and (15) and the solid lines are obtained from the theoretical lines by convoluting the Lorentzian distribution function with a linewidth of 2 ppm. 

Plot the radial distribution function 47ir P(r)against r/p fora linear polyethylene molecule of RMM = 63 000. taking / = 381 pm, and n = j (number of C-C bonds). Hence find the value of i- (r > p)at which the distribution function falls to a tenth of its maximum value. 

Takala M, Ranta H, Nevalainen P, Pakonen P, Pelto J, Karttunen M, Virtanen S, Koivu V, Pettersson M, Sonerud B, Kannus K (2010) Dielectric properties and partial discharge endurance of polypropylene-silica nanocomposite. IEEE Trans Diel Electr Insul 17 1259-1267 Tanaka T, Kozako M, Fuse N, Ohki Y (2005) Proposal of a multi-core model for polymer nanocomposite dielectrics. IEEE Trans Diel Electr Insul 12 669-681 Vaughan AS, Swingler SG, Zhang Y (2006) Polyethylene nanodielectrics the influence of nanoclays on structure formation and dielectric breakdown. Trans lEE Jpn 126 1057-1063 Venkatesulu B, Thomas MJ (2010) Erosion resistance of alumina-filled silicone rubber nanocomposites. IEEE Trans Diel Electr fiisul 17 615-624 Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech Trans ASME 18 293-297... 

Fig. 1.28. Comparisons among the rotational isomeric (RIS) radial distribution functions at 413 K for polyethylene (o) and PDMS ( ) chains having n — 20 skeletal bonds, and the Gaussian approximation ( — ) to the distribution for PDMS [45]. The RIS curves represent cubic-spline fits to the discrete Monte Carlo data, for 80 000 chains, and each curve is normalized with respect to an area of unity (with / being the skeletal bond length).

Figure 6. Predicted interchain radial distribution function for a hard-core polyethylene melt described by three single-chain models atomistic RIS at 430 K, overlapping (lid = 0.5) SFC model with appropriately chosen aspect ratio and site number density (see text), and the Gaussian thread model (shifted horizontally to align the hard core diameter with the value of rld = l).

Figure II. PRISM predictions for hard-core atomistic RIS models of polyolefins, (a) The three diagonal radial distribution functions of isotactic polypropylene. (b) A com parison of chain averaged site-site radial distribution functions at 473 K for = 400 models of polyethylene, isotactic polypropylene, and syndiotactic polypropylene. The charactcris tic ratio. C, of the RIS models employed for PP arc shown in parentheses.

Figure 16. The radial distribution functions for a blend of polyethylene and isotactic polypropylene = N = 200) at a volume fraction of polyethylene of /= 0.5. The four diagonal correlations are shown.

Different from s-PP, crystallites of polyethylene thicken with time. This was clearly demonstrated by time-dependent SAXS experiments carried out during an isothermal crystallization [6]. Fig. 4 shows a typical result, in a comparison of the interface distribution function K" z) obtained at the beginning and the final stage of the crystallization process. 

McBrierty, V.J. and Ward, I.M. (1968) Investigation of the orientation distribution functions in drawn polyethylene by broad line nuclear magnetic resonance. J Phys D Appl. Phys., 1, 1529. 

Figure 13.15 Cumulative distribution function W n), i.e. the mass fraction of linear chains shorter than n for polyethylene with 1.5% of chain defects.

---