Generalizations of Huggins–Guggenheim–Miller-type theories to describe the architecture of branched lattice chains

Abstract

We introduce two methods for extending Huggins-Guggenheim-Miller ͑HGM͒-type theories for lattice model polymer chains to describe the dependence of polymer thermodynamic properties on chain architectures ͑e.g., linear, branched, comb, structured monomer chains͒, thereby rectifying a half-century old deficiency of these venerable theories. The first approach is based upon a mathematically precise definition of the ''surface fractions'' that appear in the final HGM random mixing theory. These surface fractions are determined from exact enumerations for short chains, which are found to converge rather rapidly. The approach is illustrated for linear chains, but is readily applied for branched systems. The resultant ''improved'' HGM theory is tested by parameter-free comparisons with Monte Carlo simulations as well as with Flory-Huggins theory, the original HGM theory, and the lattice cluster theory ͑LCT͒. A second improved HGM theory is generated by providing more accurate treatments of the nearest-neighbor pair probabilities that form the basic assumptions and ingredients in, for instance, Guggenheim's derivation of the HGM theory. The more accurate pair probabilities are obtained from the LCT for branched polymer systems ͑or chains with structured monomers͒, and comparisons are again provided with Monte Carlo simulations and previous theories. These comparisons serve to underscore inherent limitations of fundamental assumptions invoked by HGM theories and possible methods for their alleviation. Unfortunately, all simple ''improvements'' of the HGM theory diminish its accuracy, thereby demonstrating that the apparent successes of the HGM theory emerge from a cancellation of errors.

Lattice Cluster Theory for Pedestrians: The Incompressible Limit and the Miscibility of Polyolefin Blends

“…This last term contains both composition-dependent and -independent portions. The composition-dependent part provides a correction to the random mixing approximation (in the strict probabilistic sense) . A portion of the composition-dependent order ε/ z terms in eq 11 reduces for linear chains to the order 1/ z contributions from Guggenheim counting (in the high molecular weight limit).…”

Single-Chain Conformations in Symmetric Binary Polymer Blends: Quantitative Comparison between Self-Consistent Field Calculations and Monte Carlo Simulations

“…Hence powerful self-consistent field, − density functional, − and liquid state theories , have been developed to describe the bulk and interfacial thermodynamics of macromolecular mixtures. In addition, the influence of the particular chain architecture − on various properties has been investigated. However, the same intermolecular forces which determine the thermodynamic properties alter the conformations of the extended macromolecules.…”

Influence of Short Chain Branching on the Miscibility of Binary Polymer Blends: Application to Polyolefin Mixtures

“…The importance of enthalpic interactions in a system with equal attractive interaction potentials may be understood to emerge because the PRISM calculations contain the realistic feature of the presence of excess free volume. 8,9 This excess free volume acts as a noninteracting, nonselective solvent whose affinity for the blend components is influenced by what has loosely been associated with the "polymer surface fraction", 10,11 which must be different for the two blend components. 11 Both the Bates-Fredrickson theory and the solubility parameter analyses consider incompressible blend models and, therefore, do not contain the extra contribution, due to the presence of excess free volume, that is obtained from the PRISM calculations and that has been demonstrated using the lattice cluster X Abstract published in Advance ACS Abstracts, December 1, 1995. theory 12 (LCT) to exert a significant influence upon the temperature, composition, molecular weight, and pressure dependence of χ.…”