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 (HGM)-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 chains), 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 (LCT). 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 monomers), 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.