Schematic representation of ensemble . Polymer volumetric fraction increases from left to right . is split in equivalence classes represented by the single-sphere configuration . The dashed line marks the boundary of equivalence class . The dotted rectangle represents , a set of configurations with prescribed chain length distribution, that share but differ in the connectivity.
Connectivity reconstruction moves, classified according to the type of chain growth/shrinkage (“accretion”), and to the volumetric fraction fluctuation they cause. Selected sample transitions are shown.
Solvent/chain configurations on a grid. A given system configuration (labeled ⌾) can be reached from other configurations (labeled ①, ②,⋯) by mutation of a single chain site into a solvent site. The associated graph on the center, its Laplacian matrix , the eigenvalues of , and its eigenvector quantify whether or not these configurations are directly accessible to each other in connectivity space. In the semidilute regime (top) the graph can be partitioned deeply, which indicates high accessibility and rapid mixing in configuration space. In the concentrated regime (bottom), configurations mix weakly, which renders a purely local analysis exact (see main text for details).
Qualitative representation of conditions in semidilute, marginal, and concentrated regimes.
Flory scaling exponent as a function of logarithm of packing density from MC simulations. Lines mark (transition from dilute to semidilute) from Ref. 2, for different chain lengths.
Log-log plot of characteristic ratio as a function of density for several chain lengths. Straight lines with slopes given by the expected scaling exponents are drawn as an aide to the eye (except in the concentrated regime, where scaling is predicted, see Fig. 7). Arrows mark (transition from dilute to semidilute) from Ref. 2, and also cross-over densities and predicted by Eqs. (29) and (26) for , at which semidilute scaling changes to marginal scaling , and marginal changes to concentrated scaling , respectively.
Logarithm of the characteristic ratio as a function of packing density as obtained from MC simulations focusing on the marginal and concentrated regimes. Predicted scaling is clearly observable above . Also shown are the theoretically predicted cross-over densities (for ) for the transitions (i) and (ii) .
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