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Calculating the free energy of self-assembled structures by thermodynamic integration
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View: Figures


Image of FIG. 1.
FIG. 1.

Sketch of the reversible path that connects the homogeneous, disordered state, the externally ordered, and the self-assembled state. Configurational snapshots of a symmetric diblock melt illustrate the different states. In the snapshots three-dimensional contour plots of the composition are shown. The -rich component is removed for clarity and the interface between the different components is colored blue. , , and the maximal strength of the ordering field is . The linear extension of the simulation cell is and the lamellar spacing is .

Image of FIG. 2.
FIG. 2.

Variation of and along the integration path. The error bars (comparable to the symbol size) mark the variance of the two quantities for selected points, , along both branches. Both branches are discretized into about 200 points. The insets present the variation of the integrands of Eqs. (9) and (10).

Image of FIG. 3.
FIG. 3.

Evolution of the ordering field, , in the course of the expanded ensemble simulation along both branches. The “time” is measured in units of the Rouse time of the macromolecules. The inset presents the probability distribution, .

Image of FIG. 4.
FIG. 4.

Changes of the free energy along the two branches of the integration path.

Image of FIG. 5.
FIG. 5.

Free energy as a function of incompatibility, . Data from SCF calculations with optimized and fixed lamellar spacing and thermodynamic integration with fixed lamellar spacing are presented. The shaded region indicates the location of the ODT that has been estimated by the observation of hysteresis. The inset highlights the free energy in the vicinity of the ODT as extracted from SCMF simulations. The thin lines mark the extrapolation of the free energy into the region where the lamellar structure with fixed spacing is unstable.

Image of FIG. 6.
FIG. 6.

Locating the ODT via hysteresis: The main panel presents the results of Monte Carlo simulations by Vassiliev and Matsen (Ref. 9) EPD simulations of Alexander-Katz and Fredrickson (Ref. 11), and our result as a function of the invariant degree of polymerization, . The inset illustrates the long-lived metastability and the ordering observed in SCMF simulations at , 13.65, and . For each value of the incompatibility, running averages of the ordering energy during the course of the simulations are shown for five independent simulation runs.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Calculating the free energy of self-assembled structures by thermodynamic integration