^{1,a)}, Jacek Dudowicz

^{2,b)}and Karl F. Freed

^{2,c)}

### Abstract

Cooperativity is an emergent many-body phenomenon related to the degree to which elementary entities (particles, molecules, organisms) collectively interact to form larger scale structures. From the standpoint of a formal mean field description of chemical reactions, the cooperativity index , describing the number of elements involved in this structural self-organization, is the order of the reaction. Thus, for molecular self-assembly is the number of molecules in the final organized structure, e.g., spherical micelles. Although cooperativity is crucial for regulating the thermodynamics and dynamics of self-assembly, there is a limited understanding of this aspect of self-assembly. We analyze the cooperativity by calculating essential thermodynamic properties of the classical order reactionmodel of self-assembly (model), including universal scaling functions describing the temperature and concentration dependence of the order parameter and average cluster size. The competition between self-assembly and phase separation is also described. We demonstrate that a sequential model of thermally activated equilibrium polymerization can quantitatively be related to the model. Our analysis indicates that the essential requirement for “cooperative” self-assembly is the introduction of constraints (often nonlocal) acting on the individual assembly events to regulate the thermodynamic free energy landscape and, thus, the thermodynamic sharpness of the assembly transition. An effective value of is defined for general self-assembly transitions, and we find a general tendency for self-assembly to become a true phase transition as . Finally, various quantitative measures of self-assembly cooperativity are discussed in order to identify experimental signatures of cooperativity in self-assembling systems and to provide a reliable metric for the degree of transition cooperativity.

This work is supported, in part, by the Joint Theory Institute which is funded by Argonne National Laboratory and the University of Chicago and by NSF Grant No. CHE-0749788. We thank Ka Yee Lee and Peter Zapol for helpful discussions.

I. INTRODUCTION

II. FLORY–HUGGINS THEORY OF EQUILIBRIUM POLYMERIZATION

A. Characterization of the models of equilibrium polymerization

B. Comparison of models of equilibrium polymerization

C. Basic thermodynamic relations for the model

D. Simplified models of cooperative transitions

III. ANALYSIS OF SIMILARITIES BETWEEN AND MODELS

A. Average degree of aggregation as a measure of the sharpness of the self-assembly transition

B. Extent of association : Universal reduced temperature and concentration variables for self-assembly

C. Specific heat and nature of self-assembly transition in the high cooperativity limit

D. Regularities in the magnitude of for self-assembly

E. Quantitative mapping between the and models

F. Measures of transition cooperativity and transition rounding

G. Phase boundaries in the and models

IV. SOME BASIC ASPECTS OF COOPERATIVE SELF-ASSEMBLY

A. Signatures of sharp cooperative transitions in osmotic and transport properties

B. Concentration dependence of as an indicator of cooperativity

C. Signatures of cooperativity from the phase boundaries of self-assembling systems

D. Unusual trends in the virial coefficients of cooperatively assembling fluids

E. High cooperativity and the dynamics of self-assembly

F. Cooperativity effects on protein folding

V. DISCUSSION

### Key Topics

- Self assembly
- 145.0
- Polymerization
- 58.0
- Polymers
- 41.0
- Enthalpy
- 24.0
- Thermodynamic properties
- 23.0

## Figures

Average aggregate size as a function of temperature for fixed initial monomer concentration , for the model of equilibrium polymerization, and for a series of models with varying cluster size . The free energy parameters and and the stiffness parameter are given in the text. The same values of , , and are used in the calculations illustrated in Figs. 2–13 and 16–19.

Average aggregate size as a function of temperature for fixed initial monomer concentration , for the model of equilibrium polymerization, and for a series of models with varying cluster size . The free energy parameters and and the stiffness parameter are given in the text. The same values of , , and are used in the calculations illustrated in Figs. 2–13 and 16–19.

Extent of association as a function of temperature for fixed initial monomer concentration and for the model. Different curves correspond to different in the model. The curve for the model is included for comparison.

Extent of association as a function of temperature for fixed initial monomer concentration and for the model. Different curves correspond to different in the model. The curve for the model is included for comparison.

Extent of association as a function of the initial monomer concentration at fixed temperature . Different curves correspond to different in the model. The curve for the model is included for comparison.

Extent of association as a function of the initial monomer concentration at fixed temperature . Different curves correspond to different in the model. The curve for the model is included for comparison.

Extent of association as a function of the reduced temperature [with defined by the condition ] for the model with . Different symbols correspond to the different initial monomer concentrations specified in the figure. Note that varies with .

Extent of association as a function of the reduced temperature [with defined by the condition ] for the model with . Different symbols correspond to the different initial monomer concentrations specified in the figure. Note that varies with .

Extent of association as a function of the reduced concentration [with defined by the condition ] for the model with . Different symbols correspond to the different temperatures indicated in the figure. Note that varies with .

Extent of association as a function of the reduced concentration [with defined by the condition ] for the model with . Different symbols correspond to the different temperatures indicated in the figure. Note that varies with .

Derivative as a function of the initial monomer concentration for the model and for a series of models with varying cluster size .

Derivative as a function of the initial monomer concentration for the model and for a series of models with varying cluster size .

Specific heat as a function of temperature at fixed initial monomer concentration for the model and for a series models with varying cluster size .

Specific heat as a function of temperature at fixed initial monomer concentration for the model and for a series models with varying cluster size .

Specific heat as a function of temperature at fixed initial monomer concentration for the model with .

Specific heat as a function of temperature at fixed initial monomer concentration for the model with .

Polymerization transition temperature as a function of initial monomer concentation for the and models. While the temperatures and (defined in the text) differ for the model, they are identical for the model, providing a unique defintion of .

Polymerization transition temperature as a function of initial monomer concentation for the and models. While the temperatures and (defined in the text) differ for the model, they are identical for the model, providing a unique defintion of .

Average aggregate size at the polymerization temperature as a function of initial monomer concentration for the model and for a series of models with varying .

Average aggregate size at the polymerization temperature as a function of initial monomer concentration for the model and for a series of models with varying .

Comparison of the temperature variation of the average cluster size for fixed initial monomer concentration between the model with (solid line) and the model with and (dotted line). These two curves are indistinguishable by the naked eye (the relative difference is less than 1%).

Comparison of the temperature variation of the average cluster size for fixed initial monomer concentration between the model with (solid line) and the model with and (dotted line). These two curves are indistinguishable by the naked eye (the relative difference is less than 1%).

Comparison of the temperature variation of the extent of association for fixed initial monomer concentration between the model with (solid line) and the model with and (dotted line). These two curves are hardly distinguishable by the naked eye (the highest relative difference is about 4%).

Comparison of the temperature variation of the extent of association for fixed initial monomer concentration between the model with (solid line) and the model with and (dotted line). These two curves are hardly distinguishable by the naked eye (the highest relative difference is about 4%).

Comparison of the temperature variation of the specific heat for fixed initial monomer concentration between the model with (solid line) and the model with and (dotted line). These two curves are hardly distinguishable by the naked eye (the highest relative difference is about 6%).

Comparison of the temperature variation of the specific heat for fixed initial monomer concentration between the model with (solid line) and the model with and (dotted line). These two curves are hardly distinguishable by the naked eye (the highest relative difference is about 6%).

Cooperativity index as a function of the initial monomer concentration , obtained by demanding the superposition of the temperature variation of average aggregate size for the and models. Different curves correspond to the indicated entropies of activation for the model.

Cooperativity index as a function of the initial monomer concentration , obtained by demanding the superposition of the temperature variation of average aggregate size for the and models. Different curves correspond to the indicated entropies of activation for the model.

The ratio of the van’t Hoff enthalpy and the enthalpy of propagation as a function of the entropy of propagation for a series of models (with varying cluster size ) and for the perfectly “uncooperative” model. The parameter is completely insensitive to the enthalpy of propagation . The polymerization line used to estimate (see the text for more details) is determined from the maximum in the specific heat , based on Eqs. (23) and (24).

The ratio of the van’t Hoff enthalpy and the enthalpy of propagation as a function of the entropy of propagation for a series of models (with varying cluster size ) and for the perfectly “uncooperative” model. The parameter is completely insensitive to the enthalpy of propagation . The polymerization line used to estimate (see the text for more details) is determined from the maximum in the specific heat , based on Eqs. (23) and (24).

The ratio of the van’t Hoff enthalpy and the enthalpy of propagation as a function of the entropy of activation for the model. The parameter is completely insensitive to the enthalpy of propagation , which coincides with the enthalpy of activation for the model. The polymerization line used to estimate (see the text for more details) is determined from the maximum of the specific heat computed by using Eq. (25).

The ratio of the van’t Hoff enthalpy and the enthalpy of propagation as a function of the entropy of activation for the model. The parameter is completely insensitive to the enthalpy of propagation , which coincides with the enthalpy of activation for the model. The polymerization line used to estimate (see the text for more details) is determined from the maximum of the specific heat computed by using Eq. (25).

Comparison of the spinodal curves between the model with (dashed curve) and the model for which the curves are identical (see Fig. 11). The spinodal for the model is computed by using a common for all compositions. The additional spinodal curves for the model (with ) illustrate that changing does not remove discrepancies between the spinodal curves for the two models in the low concentration regime.

Comparison of the spinodal curves between the model with (dashed curve) and the model for which the curves are identical (see Fig. 11). The spinodal for the model is computed by using a common for all compositions. The additional spinodal curves for the model (with ) illustrate that changing does not remove discrepancies between the spinodal curves for the two models in the low concentration regime.

Comparison of the spinodal curves between the model with (dashed curve) and the model for which the self-assembly transition resembles a second order transition.

Comparison of the spinodal curves between the model with (dashed curve) and the model for which the self-assembly transition resembles a second order transition.

Composition variation of the average cluster size at fixed temperature for the model. The different curves correspond to the temperatures indicated in the figure. The same linear variation of with is obtained for the model .

Composition variation of the average cluster size at fixed temperature for the model. The different curves correspond to the temperatures indicated in the figure. The same linear variation of with is obtained for the model .

The logarithm of the factor describing the concentration dependence of as a function of inverse temperature for the model and the model .

The logarithm of the factor describing the concentration dependence of as a function of inverse temperature for the model and the model .

## Tables

The polymerization temperature , the transition width , the effective cooperativity index , and the low temperature limit of the average cluster size for the model of equilibrium self-assembly analyzed in the text. The properties considered are presented as functions of the ratio , where the entropy of activation varies and the entropy of propagation is fixed as .

The polymerization temperature , the transition width , the effective cooperativity index , and the low temperature limit of the average cluster size for the model of equilibrium self-assembly analyzed in the text. The properties considered are presented as functions of the ratio , where the entropy of activation varies and the entropy of propagation is fixed as .

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