Radial distribution polymer liquids

Figure 7(c). Normalized steady state radial distribution of difference stress contributed by Lennard-Jones interactions, for a simple liquid Nb = 0 (solid line) and polymer melts with Nb — 1,3 and 11 (dashed lines). Reduced density p — 0.9 for all cases. The results for the polymer melts are indistinguishable to within the statistical scatter of the simulations. 

With the two Li ions in as widely separated positions as the simulation box allows, the dynamics were initiated and equilibrated for the same thermodynamic conditions as in pristine PAc. By artificially constraining the diffusive motion of polymer chains, the motion of Li ions was monitored through the radial distribution function g(r) [Eq. (28)]. The gcxiC ) 9h,u( ) were found to describe a liquid-like motion for Li ions, while Qu,u( ) was almost featureless except for a very small range of dopant separations. This indicates that in agreement with the results of the static lattice calculations (Section 5.3), the motion of the ions was rapid along the chain directions, but correlated by the dominant interionic Coulomb repulsions, so that they maintained (on average) the maximum distance apart permitted by the simulation box. 

The theoretical approach we take to describe amorphous polymer liquids is based on integral equation theory which has ite rcxrts in the theory of monatomic liquids [5]. Corner for a mouKnt a uniform system of n spherical particles of density p = n/V. A convenient measure of the d ree of order in such a system is the radial distribution function g(r) defined as... 

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. 

Having determined the structure of the polymer liquid, it is in principle possible to compute most thermodynamic properties of interest. Whereas the structure or radial distribution functions at liquid density are primarily controlled by the repulsive part of the intersite potentials, thermodynamic quantities will also be sensitive to the attractive potentials. In the case of a one-component melt, thermodynamic quantities of interest include the pressure P, isothermal compressibility k, and the internal or cohesive energy U. Since in general one theoretically knows g(r) only approximately, the thermodynamic properties derived from structure will be approximate. Moreover, integral equation theory leads to thermodynamically inconsistent results in the sense that the predictions depend on the particular thermodynamic route used to relate the thermodynamic quantity to the structure. ... 

Herman Mark was also one of the leading liquid physicists, along with Prins, Debye, Frenkel, Lennard-Jones and Zermke. He included a thorough discussion of the interpretation of X-ray scattering from liquids in terms of the radial distribution function. In keeping with the views of his community, he modeled the liquid as a collection of small crystals. While this paradigm is now anathema in the liquid physics community, it stiU lives in the backwaters of German and Russian polymer science. Mark also included a discussion of liquid crystals. 

We will consider that each molecule of a liquid or an amorphotis organic polymer is centered on a normal site in a face centered cubic lattice and that all interactions may be described in terms of van der Waals interactions. The total potential of a molec ile will be calculated by summing over a large number of nei bors. Similarly, calcxilations of various properties of such a system will be peiv formed in terms of the fee lattice array. There is, of course, no physical basis for the selection of any one of the isotropic crystalline models except from the standpoint of simplifying the subsequent calculations. One cotild also perform calculations using a radial distribution function and integrating rather than performing lattice sums. Clearly these calculations are of an approximate na-ttare and are used only to demonstrate that several properties of liquids and polymers may be described in teims of van der Waals interactions. ... 

For amorphous systems with cylindrical symmetry, such as aligned uniaxial nematic liquid crystals or oriented amorphous polymers, the equivalent of the radial distribution function is the cylindrical distribution function (CDF). This function can be written as [2]... 

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