Analysis of the Coarse-Grained Membrane Structure

The hydrophilic subphase of water, hydronium ions, and charged sidechain beads forms a three-dimensional network of irregular channels. Typical channel sizes are 1, 2, and 4 nm at water contents of A. = 4,9,15, respectively. 

The analysis of RDFs, gy (ry), provides valuable stmctural information of simulated many-particle systems. It allows rationalizing structural correlations between atoms or beads, aggregation behavior, and phase separation, as well as sizes, shapes, and coordination structure of distinct phase domains. Experimentally, the RDF is obtained from the structure factor S(q), which determines the intensity of x-ray or neutron scattering (Ashcroft and Mermin, 1976). The structure factor for scattering from phase domains formed by a single component, for example, apolar beads in fibrils of ionomer backbones, is given by 

Microscopically, the RDF is defined as the conditional probability density of finding a bead j at a position ry relative to the position of a reference bead i. 

The labels i and j could represent beads of the same type or of different types. The ratio of Nj to V corresponds to the particle density of beads of type j. Calculation of gij(r) involves ensemble averaging over a sufficient number of statistically independent configurations, along the simulated trajectory and over statistically independent choices of the reference bead. 

For an isotropic system, the RDF, g(r), is related to the potential of mean force between two particles via (Hansen and McDonald, 2006) 

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