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Lattice model of equilibrium polymerization. VII. Understanding the role of “cooperativity” in self-assembly
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10.1063/1.2909195
/content/aip/journal/jcp/128/22/10.1063/1.2909195
http://aip.metastore.ingenta.com/content/aip/journal/jcp/128/22/10.1063/1.2909195

Figures

Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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 .

Image of FIG. 5.
FIG. 5.

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 .

Image of FIG. 6.
FIG. 6.

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

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

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

Image of FIG. 9.
FIG. 9.

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 .

Image of FIG. 10.
FIG. 10.

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 .

Image of FIG. 11.
FIG. 11.

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%).

Image of FIG. 12.
FIG. 12.

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%).

Image of FIG. 13.
FIG. 13.

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%).

Image of FIG. 14.
FIG. 14.

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.

Image of FIG. 15.
FIG. 15.

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).

Image of FIG. 16.
FIG. 16.

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).

Image of FIG. 17.
FIG. 17.

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.

Image of FIG. 18.
FIG. 18.

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.

Image of FIG. 19.
FIG. 19.

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 .

Image of FIG. 20.
FIG. 20.

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

Tables

Generic image for table
Table I.

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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2008-06-09
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Lattice model of equilibrium polymerization. VII. Understanding the role of “cooperativity” in self-assembly
http://aip.metastore.ingenta.com/content/aip/journal/jcp/128/22/10.1063/1.2909195
10.1063/1.2909195
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