Snapshot depicting the diamond lattice topology of the polyelectrolyte network. From the tetra-functional nodes (for illustration purposes drawn as oversized gray spheres) polymer chains emerge, on each monomer charged (white spheres) and neutral (blue spheres). While they only occupy those four subboxes shaded in gray, the counterions (orange spheres) move freely through the entire unit cell (color references are to the online version of the paper).
Averaged form factor of the network strands with monomers per chain, plotted for (black lines), (dark lines), and (light lines) in good solvent, rescaled by ; different line style corresponds to different charge fractions (see the legend).
Integrated counterion distribution for , with (black lines), (dark lines), and (light lines) in good solvent, normalized by the end-to-end distance ; different line style corresponds to different charge fractions (see the legend). For comparison, the corresponding ideal gas distribution for and is given as well (solid black line). Note that each of the systems has its own , the graphs therefore reach the horizontal axis at different .
Comparison of the pressure of the gas-like cloud of counterions to the assumed from the model in Sec. III A. While the upper plot displays significant deviations for larger and although compensating for the occurrence of counterion condensation (see Sec. III C) by using , the lower plot shows that those are entirely by the electrostatic effects contained in , see the text (same symbol refers to same , same tone refers to same ).
Network restoring force measured from the elastic contribution to the pressure, the excluded volume interaction between the monomers, and the ideal component, as a function of the average end-to-end-distance normalized by the maximum elongation of a network chain; as reference the force–extension relation for Gaussian chains , for freely jointed chains (FJC), and for worm-like chains (WLC) are given as well as the model fit . The lower plot compensates for the inner chain repulsion due to the electrostatics, see the text (same symbols as in Fig. 4).
The equilibrium swelling behavior of salt-free polyelectrolyte networks in good solvent and close to the -point following the scaling law (17) successfully. The two insets provide confirmation for modeling the individual pressure components (lower right) and (upper left) by Eqs. (15) and (14), respectively, leading to the pressure balance from where the scaling prediction is then derived (symbols as in Fig. 4).
The non-FRH based scaling prediction (18) compared to the measured data, showing fundamental nonagreement even in the regime of low electrostatics and Gaussian elongation of the chain (symbols as in Fig. 4).
Selected network properties for a system with monomers per chain in good solvent (top) and close to the -point (bottom) for different values of Bjerrum length and charge fraction , namely the Manning parameter , the fraction of condensed counterions , the effective charge fraction , all discussed in Sec. III C, as well as the length of the simulation box after reaching the equilibrium volume , the average chain extension , its swelling ratio compared to a single neutral chain, its extension relative to the contour length , its radius of gyration , and its hydrodynamic radius , all discussed in Sec. III B.
Detailed pressure components for a system with monomers per chain in good solvent (top) and close to the -point (bottom) for different values of Bjerrum length and charge fraction , namely the contributions to the gas-like pressure (ideal part , excluded volume , total electrostatic pressure ) and to the one of the gel (ideal part , excluded volume , bonded virial ) for the fully charged gel at its swelling equilibrium volume . Switching off electrostatics changes the total pressure from zero to , and shifts and by and , respectively.
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