A schematic of the linearly self-assembling system under study and the morphology of the structures formed. The two species of self assembler give rise to two activated states with associated free energies a 1 and a 2, accounting for conformational changes necessary for binding, and three binding free energy gains b 11, b 12 = b 21, and b 22. Depending on the relative values of binding free energies, the two species arrange themselves in blocky, random, or alternating ordering.
The fraction of self-assemblies f as a function of the overall molar fraction of dissolved material Φ = Φ1 + Φ2 at different stoichiometric ratios α = Φ1/Φ2 of the two components 1 and 2. (a) and (b) show the results for weak chemical bidispersity (b 11 = 10, b 12 = b 22 = 8, a 1 = 2.3 and a 2 = 1.6). The squares on the curves in (a) signify the fraction of self-assemblies at the critical concentration, associated with each α, as discussed in the main text. (c) and (d) graphs show the case of a strong degree of chemical bidispersity (b 11 = 20, b 12 = b 22 = 8, a 1 = 6, and a 2 = 1.6). The squares on each α curve in (c) present the fraction of self-assemblies at their respective critical concentrations. The solid lines show the slopes for each α calculated based on the Eq. (38) and show a good agreement with numerical results at sufficiently low values of Φ. In the insets, we depict the slope of f at low concentrations as a function of α according to Eq. (38).
The critical molar fraction Φ* as a function of ratio of the two components α = Φ1/Φ2 plotted for (a) weak chemical bidispersity: b 11 = 10, b 12 = b 22 = 8, a 1 = 2.3, and a 2 = 1.6, corresponding to J = 0.5, and . The lines show the approximate functions valid for very low and very high α values according to Eq. (35); (b) strong chemical bidispersity: b 11 = 20, b 12 = b 22 = 8, a 1 = 6 and a 2 = 1.6 corresponding to a large value of J = 3, , and . The line shows the analytical results obtained for the critical concentration in the large J limit based on Eqs. (29) and (30).
(a) Ratio of the concentrations of the free monomer of species 1 and 2 α f divided by the ratio of total density of monomers present in the solution α, as a function of total concentration shown for different values of α = Φ1/Φ2. The corresponding α values are depicted in the legends. (a) Weak chemical bidispersity, with an equivalent J = 0.5. (b) strong chemical bidispersity corresponding to J = 3. The dotted lines correspond to concentrations and , as indicated in the figure. The free energy parameters used here are the same as those of Figures 2 and 3.
The average fraction of monomers of type 1 for a composition of α = 1 as a function of N shown for different concentrations. The upper graphs shows a weakly bidisperse case, with an equivalent J = 0.5, while the lower graph shows a strongly bidisperse case, corresponding to J = 3. The free energy parameters used here are the same as those of Figures 2 and 3.
The number-averaged degree of polymerization of linear assemblies as a function of the overall concentration for different stoichiometric ratios α. α = 0 and α = ∞ correspond to monodisperse cases of types 1 and 2, respectively. (a) Weakly bidisperse case, with an equivalent J = 0.5. (b) Strongly bidisperse case corresponding to J = 3. The free energy parameters used here are the same as those of Figs. 2 and 3. The vertical dotted lines in each figure show the concentrations and , respectively.
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