Schematic diagram of the gel system consisting of the polymer backbone, condensed counterions and dissociated mobile ions in a divalent salt solution. Possible charge complexes for each monomer: (a) Monomer (−1), (b) monomer-monovalent (−1,+1), (c) monomer-divalent (−1,+2), (d) monomer-divalent-coion (−1,+2,−1), and (e) a bridging configuration of monomer-divalent-monomer.
Gel volume fraction (solid curves) as a function of temperature with (black) and without (red/blue/green) charge regularization for a χ = 20.0 and δ = 7.6. (a) For the case with charge regularization, the variation of the degree of ionization with temperature (black dashed curve) accompanies the volume transition (black solid curve). (b) In the absence of charge regularization, the transition temperature can vary widely depending on the choice of α, and the gel never reaches the truly collapsed state.
(a) Volume transition temperature (circles) and (b) width of the collapse (diamonds) for charged (red) and neutral (black) gels as a function of the effective crosslink density S, for a χ = 7.0, δ = 7.6. The trend for polyelectrolyte gels is opposite to that of neutral gels for both cases. Also, as opposed to a neutral gel, there is no critical end-point for polyelectrolyte gels.
Temperature vs. gel volume fraction plots for a fixed value of the dielectric mismatch parameter δ and different values of the Flory-Huggins interaction parameter a χ for (a) salt-free (δ = 12.0), (b) monovalent salt (c s = 0.3 M, δ = 7.6), and (c) divalent salt (c s = 0.25 M, δ = 3.0) cases. The variation of the degree of ionization with temperature is shown in the insets in all three cases. The crosslinking density is taken to be S = 400 in all cases.
Room temperature plots for gel volume fraction as a function of external salt concentration for different Flory-Huggins interaction strengths in the case of (a) monovalent salt (δ = 7.6) and (b) divalent salt (δ = 3.0). The crosslinking density is taken to be S = 400 in both cases. The corresponding degree of ionization is shown in the insets of both plots.
Room temperature plots for gel volume fraction as a function of external salt concentration for a fixed interaction strength and different values of δ corresponding to different ionic species for a polyelectrolyte gel in a (a) monovalent salt solution (a χ = 5.0), and (b) divalent salt solution (a χ = 4.3). This is plotted with inverse of the gel volume fraction to facilitate comparison with experimental data. The delta parameter for Na + ions was arbitrarily assigned a value of 7.6 and the values for all other ions were calculated with reference to this value using the known ionic radii.
(a) Degree of ionization as a function of temperature for different values of the divalent salt concentration for a fixed value of the interaction parameter (a χ = 4.3) and dielectric mismatch parameter (δ = 3.0). Charge reversal is observed for higher values of the salt concentration. (b) The different constituent degrees of ionization (α, α1, α2a , α2b , α3) for a case with overcharging (a χ = 3.3, δ = 4.0, c s = 0.25 M).
Contributions of the different free energy terms to the total free energy as functions of temperature for a charged gel in a monovalent salt solution.
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