FOMC simulations of 256 soft spheres interacting with a soft repulsion given in Eq. (6) at a packing fraction . (a) The ensemble-averaged excess pressure and the number of overlapping particle pairs . For the two cases where (i.e., ), there was no particle overlapping during the course of our simulations. (b) The statistical inefficiency computed from the data and the maximum displacement obtained at an acceptance rate of 93% for trial moves of random particle displacement. measures, on average, the of number Monte Carlo steps needed to generate one statistically uncorrelated sample in the simulations, and larger corresponds to statistically less correlated samples collected in the simulations. In both plots, the ideal-gas (“IG”, where ) case is shown on the left axis, while the hard-sphere (“HS”, where ) case is on the right axis.
log-log plots of the mean-square chain radius of gyration and end-to-end distance with the number of bonds , obtained from single-chain FOMC simulations using (a) Eq. (6), and (b) and with either Eq. (6) (“no grids”) or an anisotropic and position-dependent pair potential. In this latter case, the space is divided into cubic cells of size by grids that are either fixed (“fixed grids”) or randomly shifted after each trial move (“shifted grids”). “HS” in (a) denotes the single-chain FOMC simulations with . In each case, the straight line through symbols represents the unweighed least-squares fit using data points for , with the scaling exponent given in the legend. The straight line without symbol represents the commonly accepted exponent in the long-chain limit.
(a) log-log plot of the ensemble-averaged nonbonded interaction energy with the number of bonds , (b) semilogarithmic plot of the average bond length with , and (c) semilogarithmic plot of the single-chain structure factor for , obtained from single-chain FOMC simulations. “DGC” in (c) denotes the ideal case of discrete Gaussian chain with . Refer to the caption of Fig. 2 for more details.
Semilogarithmic plots of the mean-square chain end-to-end distance and radius of gyration , ensemble-averaged bonding energy , and average bond length obtained in FOMC simulations of compressible homopolymer melts of chains each having segments in a box of length : (a) and , where the filled symbols are obtained with and the open ones with . (b) , , and . In both (c) and (d), , , and .
(a) Single-chain structure factor and (b) the total structure factor obtained in FOMC simulations of compressible homopolymer melts with and . “DGC” in (a) denotes the ideal case of discrete Gaussian chains (i.e., ), and “RPA” in (b) represents the prediction under the random phase approximation with . Also in (b) each symbol at takes the value of ; see Table I for the corresponding number of chains in each case.
Effects of in the DGC model on the value of (a) and (b) bulk lamellar period (in units of ) at the ODT of symmetric diblock copolymers obtained under the RPA. Different symbols correspond to the different functional forms of given in the text, with denoting the interaction range.
The mean-square chain end-to-end distance and radius of gyration , ensemble-averaged bonding energy , and average bond length obtained in FOMC simulations of compressible homopolymer melts of chains each having segments in a box of length at various dimensionless chain number density . The case corresponds to the single-chain simulation without the PBCs. Note that we set and the dimensionless excluded-volume parameter .
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