Distribution Of Chain Ends Research Articles

Computer simulation of end-linked elastomers. II. Bulk cured tetrafunctional networks

Sol–gel distributions at high conversions, simulated by computer with spatial constraints on the distribution of chain ends, are reported for end-linked tetrafunctional systems. The composition of the sol is found to be uniquely dependent upon the functionality f of the cross linker: for f=4, it mainly consists of linear monomers and oligomers containing one saturated cross linker. The molecular weight distribution of the sol exhibits a shoulder at the trimer due to the relative abundance of the ‘‘bow-tie’’ molecule. The populations of various types of dangling ends and loop defects in the gel are given and discussed. Our simulations reveal empirically that the single-loop probability varies with the −3/8 power of the molecular weight of the prepolymers. The power-law dependence of the loop formation on chain length of the prepolymers and the nonrandom nature of the cross-linking reaction is discussed.

Distribution of conformations and chain ends near the surface of a melt of linear flexible macromolecules

In a melt of linear flexible macromolecules the conformations adopted by the chains follow the ideal Kuhn random walk distribution, because of the extremely high, but uniform, local conformational attrition factor. Long-range exclusion effects are overwhelmed and pseudoideal conditions prevail. One may assume that at an interface with a melt all but the uppermost segment layers are still ideal in the above sense. This assumption leads to a method allowing the enumeration of conformations according to their end-to-end distance and the determination of the mean square end-toend distance. The result is that coils in the interfacial layer tend to be compressed. The extent of the surface zone of altered end-to-end distribution is scaled by the root mean square end-to-end distance of the coils in the bulk of the melt.

Interphases of chain molecules: Monolayers and lipid bilayer membranes

Using the lattice model for a liquid, we treat the packing of short-chain molecules in interphases such as bilayer membranes. The constant density in the interphase imposes intermolecular constraints on the configurations of the flexible chains. The statistical theory here presented predicts a diffuse distribution of chain ends near the bilayer midplane; no adjustable parameters are required. Inasmuch as some of the chains terminate relatively near the polar interface, the number of chains reaching deeper planar layers is diminished. Consequently, configurational freedom increases with depth. This is the source of the well-known disorder gradient.

Calculation of macrocyclization equilibria for poly(ethylene terephthalate)

AbstractThe cyclization equilibria constant Kx is evaluated for the formation of cyclic poly(ethylene terephthalate)s (PET) in the range of degree of polymerization 2 ≤ x ≤ 10 on the basis of an elaborated form of the Jacobson‐Stockmayer theory. According to this theory, Kx = W(0)·2Γ0(1)/(σcx NA), where W(r) is the probability density function for the chain vector r; Γ0(γ) is the probability distribution, when r = 0 of γ = cos Δθ; Δθ being the deviation of the bond angle formed by cyclization from its unconstrained value; σcx corresponds to the symmetry number of the ring; NA is Avogadro's number. The evaluation of W(r) is performed in different approximations. The required configurational averages for computing the density distribution W(r) and the angular correlation factor Γ0(1) are obtained by Monte Carlo techniques. The density W(0) of chain vectors at r = 0 is overestimated by the Gaussian approximation. Values of W(0) calculated in higher approximation lower Kx for x < 10.By taking into account the oriental correlation between terminal bonds of the x‐meric acyclic sequence, Kx is lowered additionally by factors of ca. 2,2, 1,3 and 1,1 for x = 4, 5 and 6, respectively. Similar to other polymer chains treated so far, Kx is lowered for medium sized chain lengths by taking into account the deviation from Gaussian distribution and angular correlations of chain ends. This result is in contradiction to experimental data, which yield even higher Kx‐values than expected by the classical Jacobson‐Stockmayer theory for medium sized chain lengths. The inclusion of cis‐ester residues into the all trans‐ester PET chain only slightly reduces this discrepancy. Structural modifications of the PET chain during cyclization experiments are discussed as a rationale for the deviations from the present theory.

Termination rate constant in free‐radical polymerization

AbstractThe specific rate constant for the termination reaction between two flexible polymer molecules with active chain ends has been considered in relation to the segmental diffusion of chain ends in solution. The probability of reaction between two chain ends per unit time when the centers of gravity of two polymer molecules are at a distance of separation has been calculated by using the Smoluchowski equation and a Gaussian distribution of chain ends. The time during which two polymer molecules are in contact has also been calculated by using the diffusion equation and the potential energy function for intermolecular interaction. The rate constant may then be completely expressed as a complex function of the intramolecular linear expansion factor, molecular weight, and the frictional properties of the reacting polymers' segment. This expression predicts that the rate constant is inversely proportional to solvent viscosity, decreasing with increasing molecular weight to some extent, and is affected by the excluded volume effect and chain flexibility. The complete expression for the rate constant has been simplified and the result compared with experimental data. Close agreement is found between the calculated rate constants and those experimentally obtained.

Theory of the Interface between Immiscible Polymers. II

A theory is developed to predict the interfacial properties between two immiscible polymers. A self-consistent-field approach is employed to describe the configurational statistics of polymer molecules in the interfacial region. The density profiles and surface tension are calculated, and compare well with experiments. The interfaces are found to be much broader than for ordinary liquids. Also discussed are the distribution of chain ends and block-copolymer joints in the interface, and the effects of adding moderate amounts of solvent.

Asymptotic Behavior of the Light-Scattering Function of Coiled Molecules

This paper discusses the asymptotic solution of the scattering function P(θ) for large x, where x is proportional to the scattering angle, radius of gyration, and the wavenumber. The theoretical model employed for the calculation of the P(θ)-vs-x curves was presented in the preceding paper. The results are discussed with respect to the experimental data, after the results for P(θ) are corrected for various values of polydispersity. A general analysis of the theoretical P(θ)-vs-x curves is presented, and the effect of the shape of the chain-end distribution function on the asymptotic behavior of the light-scattering function is discussed.

Distribution Function of the End-to-End Distances of Linear Polymers With Excluded Volume Effects.

The distribution function of the absolute values of chain lengths of a polymer molecule which displays the excluded volume effect cannot assume a Gaussian form. This fact follows directly from theoretical considerations based on the application of the Central Limit Theorem to the theory of Markov chains. In order to determine the exact shape of the polymer chain-end distribution function we calculated its various moments taken about the origin, and their dependence on the number of polymer segments, using a Monte Carlo technique for generating polymer chains on a lattice. The results obtained from the extrapolation of various combinations of these moments of the general form are used to determine the shape of the polymer distribution function. It is found that the chain-end distribution function can be approximated by the following form: with t=3.2 and α being a parameter, determinable from the average mean square chain-end distances.