Long-chain branching in polyethylene has a striking effect on the zero shear viscosity and other rheological characteristics for polyethylene.  For example, three branches per 100,000 carbon atoms increases the zero shear viscosity by a factor of 5 in polyethylene [Ramachandran (2005)].  There exists no analytic technique that can quantitatively determine such a small number of branch sites in a polyethylene chain.  NMR can determine branched structure but only when branches are much more common.  Similarly, IR can be used to measure branching at much higher concentrations.  A combination of light scattering and intrinsic viscosity has been proposes as has a method using neutron scattering [Ramachandran (2005)].

In order to predict the rheology from structure it is necessary to determine certain topological features of a branched polymer.  Following the theory of Read and McLeish [Read and McLeish (2001)] chain segments between end-points and branch points are categorized as free-ends (1), inner segments (2)(3)(etc.) depending on the complexity of the branch-on-branch structure, see R&M(2001) Figure 1.  In order to calculate the chain relaxation and reptation, relaxation times for the different levels of chain structure are calculated with the shortest times for the lowest index numbers.  Read and McLeish, and later Larson, have successfully calculated the dynamic rheological behavior of polyethylene using this approach.

The approach of Read and McLeish has lead to the need to characterize the topological distribution for polyethylene.  The simplest case is that of a single site metallocene catalyst where the final distribution of polymer chain topology can be modeled using just two rate parameters, the relative probability of termination (compared to propagation, say 0.0005) of a growing chain and the relative probability of a macromonomer addition (compared to propagation, say 0.0001).  A macromonomer is another growing chain that adds to the chain by hydrogen abstraction forming a branch point at a random location on the growing chain.  The macromonomer is selected from the pool of growing chains.  This procedure was described by Soares & Hamielec (1996) and later by Costeux, Wood-Adams & Beigzadeh (2002), and Costeux (2003).  For copolymers such as mLLDPE see Beigzadeh, Soares & Duever (1999).  The approach is summarized in a book by John Dealy of McGill University and Ron Larson of the University of Michigan, Dealy & Larson (2006).

Problem:
For a population of 100,000 polyethylene chains us a simulation to calculate the fraction of chain segments that are linear chains, inner segments (Dealy & Larson (2006)), and free ends.  A chain segment is a section of chain between an end or a branch point and another end or branch point.  You also need to determine the fraction of chains that are linear, three arm stars and higher branch content (hyperbranched chains).  (It is believed that hyperbranched chains dominate the rheology of branched polyethylene.)  You also need to calculate the number average molecular weight for the linear, 3-arm stars, arms of 3 arm stars as well as the overall Mn, Mw, Mz and Mw/Mn, Mz/Mw for the 10,000 chains so that this can be compared with GPC results. 

Bin the resulting chains into number of chains versus molecular weight for 100 bins. Also calculate the total mass of chains (number * molecular weight) and plot versus the average molecular weight in a bin.

Method: A flow chart for the program is provided in the following references:  Dealy & Larson (2006), Soares & Hamielec (1996), Costeux, Wood-Adams & Beigzadeh (2002), and Costeux (2003).