(a) Chemical structure of a repeat unit of Nafion (EW = 1143) polymer. (b) Bead representation of the polymer architectures studied in this work. A beads are hydrophobic, C beads are hydrophilic.
Pore morphologies generated at 2 × 104 Δt obtained for (Ax[AC]Ay[AC]) sequences at ϕ w = 0.16. Bead definition: A beads (red), C beads (yellow), and W beads (blue). Iso-surfaces of the W bead densities are drawn at an iso-value of 0.8.
Pair correlation functions of the W beads for the eight morphologies displayed in Fig. 2 . (a) Polymers (x, y) for which the side chains are highly non-uniformly distributed: (1,5) (filled diamonds), (1,7) (open squares), (1,9) (open triangles), and (1,11) (crosses). (b) Polymers with uniform side chain distribution (y/x = 1): (3,3) (filled diamonds), (4,4) (open squares), (5,5) (open triangles), and (6,6) (crosses).
(a) D Bragg and (b) D Cl–Cl versus asymmetry parameter y/x. Lines in (a) and (b) are linear fits through data for which x + y are the same.
MSD curves obtained for the morphologies shown in Fig. 2 . (a) Diffusion restricted to the W phase (b) diffusion restricted to the W+C phase. In (a) and (b), the data obtained for the architectures with equidistant branching distances (y = x) are represented by filled diamonds (x = y = 3), filled squares (x = y = 4), filled triangles (x = y = 5), and filled circles (x = y = 6). Non-uniform architectures are represented by open diamonds (x = 1; y = 5), open squares (x = 1; y = 7), open triangles (x = 1;y = 9), and open circles (x = 1; y = 11). The pure water case, for which ϕ w = 1.0 and D MC = 1.0, is also included (crosses). The horizontal lines drawn at MSD = 1.6 × 105 give an indication of the system size.
D MC plotted against y/x. (a) Diffusion is restricted to the W pore network. (b) Diffusion is restricted to the W + C pore network.
MC tracer diffusion coefficients D MC for the architectures listed in Table I plotted against D Cl–Cl. The water content ϕ w = 0.16. (a) Diffusion restricted to the W phase. (b) Diffusion restricted to the (W + C) phase.
(a) Distribution of the maximum displacement obtained for N tracer = 2000 trajectories (a) x + y = 12, (b) x + y = 10, (c) x + y = 8, (d) x + y = 6. The results were obtained for MC runs with a duration of 7.6 × 107 MCS. Dashed lines give an indication of the system size.
Fraction of trajectories with maximum displacement less than system size (Φisol.(MC)) for all 18 architectures plotted against y/x. Filled (open) symbols were obtained for MC runs of period 7.6 × 107 MCS (2.6 × 107 MCS). (b) Φisol.(DPD) plotted against Φisol.(MC). The error bars in Φisol.(MC) are estimated as (with N tracer = 2000). The drawn line is a linear fit though the data.
(a) Changes in D Cl–Cl (ΔD Cl–Cl = D Cl–Cl(y,x)-D Cl–Cl(y = x)) plotted against differences in inter branching distances along the polymer backbones. Curves are quadratic fits obtained for data points for which x + y = constant (b) ΔD Cl–Cl plotted against 2nd moment in the D topol distribution. The drawn line is a linear fit through the data points.
Snapshot of polymer chain conformation of (1,11) architecture at ϕ w = 0.16. C beads (yellow) are located near the W pores (illustrated by iso-surfaces) in a pair wise fashion. Dashed circles illustrate the pairing of C beads that are topologically close together.
Comparison between (a) Bragg spacing and (b) diffusion constants for the x + y = 8 membranes (ϕ w = 0.16) with those obtained from Nafion1200 membranes (ϕ w = 0.2). DPD time is 20 000Δt.
Pore morphologies obtained after 20 000Δt for Nafion1200 membranes at ϕ w = 0.2 (upper figures) and ϕ w = 0.3 (lower figures) for a mixture of polymers with random side chain attachments (left hand side figures) and equidistant side chain attachments (right hand side figures). Polymer A beads (red); B beads (green); C beads (yellow). Water beads are blue. Iso surfaces of the W bead densities are drawn at an iso-value of 0.9.
Distance between water clusters, DCl–Cl (4th column), Bragg spacing D Bragg (5th column), Monte Carlo diffusion constants in W network D MC(W) (6th column), and within W+C network D MC(W+C) (7th column) for 18 chain architectures at hydration level of ϕ w = 0.16. Column 8: Fraction of the W pore space Φisol.(MC) that does not contribute to long range water diffusion, values in parenthesis Φisol.(DPD) give fraction of beads not contained within the largest W cluster for R c rit = 1.025 DPD length units (0.58 nm).
DPD repulsions used here. The approximate Flory-Huggins χij-parameters according to Eq. (14) are given in parentheses. [Reproduced with permission from G. Dorenbos and K. Morohoshi, J. Mater. Chem. 21, 13503 (2011). Copyright 2011 Royal Society of Chemistry.]
Permeability of oxygen (K O2) and hydrogen (K H2) in Barrer calculated at ϕ w = 0.2 and ϕ w = 0.3 within Nafion1200 at T = 40 °C. Values were obtained using the exactly the same procedure described in Ref. 34 . The water diffusion constants at T = 30 °C given in the last column were obtained by assuming a local water mobility of 2.3 × 10−5 cm2/s within the W (W+C) pore networks. K O2 and K H2 values deduced from experiment for Nafion1200 were derived from Ref. 68 . The experimental values were obtained by averaging over available literature values from Refs. 2 and 9 , and 10 for Nafion1100.
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