^{1,2,3}and Paul van der Schoot

^{1,4}

### Abstract

As a first step to understand the role of molecular or chemical polydispersity in self-assembly, we put forward a coarse-grained model that describes the spontaneous formation of quasi-linear polymers in solutions containing two self-assembling species. Our theoretical framework is based on a two-component self-assembledIsing model in which the chemical bidispersity, i.e., the presence of two distinct chemical entities, is parameterized in terms of the strengths of the binding free energies that depend on the monomer species involved in the pairing interaction. Depending upon the relative values of the binding free energies involved, different morphologies of assemblies that include both components are formed, exhibiting random, blocky or alternating ordering of the two components in the assemblies. Analyzing the model for the case of blocky ordering, which is of most practical interest, we find that the transition from conditions of minimal assembly to those characterized by strong polymerization can be described by a critical concentration that depends on the concentration ratio of the two species. Interestingly, the distribution of monomers in the assemblies is different from that in the original distribution, i.e., the ratio of the concentrations of the two components put into the system. The monomers with a smaller binding free energy are more abundant in short assemblies and monomers with a larger binding affinity are more abundant in longer assemblies. Under certain conditions the two components congregate into separate supramolecular polymeric species and in that sense phase separate. We find strong deviations from the expected growth law for supramolecular polymers even for modest amounts of a second component, provided it is chemically sufficiently distinct from the main one.

This work was part of the Research Programme of the Dutch Polymer Institute (DPI), Eindhoven, The Netherlands, as Project No. 610. S.J-F. would like to thank the foundation of “Triangle de la Physique” for further support with this project. We would like to thank Amalia Aggeli, Sarah Harris, Cor Koning, Bin Bin Liu, Tom McLeish, and Peter Sollich for stimulating discussions.

I. INTRODUCTION

II. EQUILIBRIUM STATISTICS OF BIDISPERSE SELF-ASSEMBLINGMONOMERS

III. MAPPING ONTO THE ISING MODEL

IV. LIMITING CASES

A. The monodisperse case

B. The bidisperse case for Φ ≫ Φ*

C. Bidisperse case for Φ ≪ Φ*

V. FRACTION OF SELF-ASSEMBLIES AND CRITICAL CONCENTRATION

VI. DISTRIBUTION OF MONOMERS IN THE ASSEMBLIES

VII. CONCLUDING REMARKS AND OUTLOOK

### Key Topics

- Polymers
- 87.0
- Self assembly
- 72.0
- Free energy
- 36.0
- Ising model
- 17.0
- Chemical analysis
- 14.0

##### C08F2/00

## Figures

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.

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 = Φ_{1} + Φ_{2} at different stoichiometric ratios α = Φ_{1}/Φ_{2} 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 fraction of self-assemblies *f* as a function of the overall molar fraction of dissolved material Φ = Φ_{1} + Φ_{2} at different stoichiometric ratios α = Φ_{1}/Φ_{2} 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 = Φ_{1}/Φ_{2} 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).

The critical molar fraction Φ* as a function of ratio of the two components α = Φ_{1}/Φ_{2} 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 = Φ_{1}/Φ_{2}. 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.

(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 = Φ_{1}/Φ_{2}. 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 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.

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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