Polyethylene bond length

Fig. 6. A primary means of regulating polyethylene chain length is through control of reactor temperature. Increasing the temperature enhances termination, probably by destabilizing the Cr-chain bond, resulting in shorter chains.

An interesting application of the molecular dynamics technique on single chains is found in the work of Mattice et al. One paper by these authors is cited here because it is relevant to both RIS and DRIS studies and deals with the isomerization kinetics of alkane chains. The authors have computed the trajectories for linear polyethylene chains of sizes C,o to Cioo- The simulation was fully atomistic, with bond lengths, bond angles, and rotational states all being variable. Analysis of the results shows that for very short times, correlations between rotational isomeric transitions at bonds i and i 2 exist, which is something a Brownian dynamics simulation had shown earlier. 

In order to understand the multitude of conformations available for a polymer chain, consider an example of a polyethylene molecule. The distance between carbon atoms in the molecule is almost constant 1.54 A. The fluctuations in the bond length (typically 0.05 A) do not affect chain conformations. The angle between neighbouring bonds, called the tetra-hedral angle 9 = 68° is also almost constant. 

Example Calculate the Kuhn length of a polyethylene chain with Coo = 2.4, main-chain bond length / = 1.54 A, and bond angle 0= 68°. 

Fig. 15. Approximate mapping of a chemically realistic polymer (polyethylene in this example) to the bond fluctuation model on the (simple cubic) lattice. In this coarse-graining one integrates n successive chemical monomers (e.g. n = 3) into one effective monomer which blocks 8 adjacent sites on the simple cubic lattice (or 4 on the square lattice in d = 2 dimensions) from occupation by other monomers. The chemical bonds 1, 2, 3 then correspond to effective bond I, bonds 4, 5, 6 to effective bond II. Some information on the chemical structure can be kept indirectly by using suitable distributions P (9) for the angle between subsequent effective bonds, but so far this has been done for homopolymer melts only [94-99]. In the simplest version of the bond fluctuation model [84-88] studied for blends in d = 3 dimensions [88, 91, 92, 99], bond lengths t are allowed to fluctuate freely from i = 2 to t = v/l0, with t = being excluded to maintain that chains do not cut through each other in the course of the random hops of the effective monomers. From Binder [95]...

Example 12.2 A polyethylene molecule has a degree of polymerization of 2000. Calculate (a) the total length of the chain and (b) the contour length of the planar zigzag if the bond length and valence angle are 1.54 A and 110°, respectively. 

By assuming the standard bond lengths, angles and preferred orientations around bonds, estimate the most likely chain repeat length for polyethylene. 

The distance between H atoms on adjacent carbon atoms in the T conformation of polyethylene can be calculated as 0.25 nm from the bond length of 0.154 nm and valence angle of 109.6. This distance is greater than the sum of the van der Waals radii of 0.24 nm for the two hydrogen atoms. Consequently, crystalline poly(ethylene) occurs in the T conformation. 

Fig. 7.20. Backbone of a 200-bond polyethylene chain projected onto the (x, y) plane. Generated by computer on the basis of bond lengths, valence angles, internal rotation angles and energies of the t, g and g positions...

Fig. 8.3. Schematic drawing of a polyethylene chain backbone in all-trans conformation, showing the bond length 6,the bond angle rand the structural composition of a monomer unit with the molar mass Mu.The torsion angle 6 between the first bond of a monomer unit to the first bond of the next monomer unit is shown along the connecting bond of the monomers...

With the bond length b (0.154 nm for a simple polyethylene backbone) the average end-to-end distance of a polymer coils in its unperturbed dimensions (so-called theta-conditions) can be calculated directly from Eq. (8.15). 

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