Research ArticlePHYSICS

Size-dependent thermodynamic structural selection in colloidal crystallization

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Science Advances  13 Sep 2019:
Vol. 5, no. 9, eaaw5912
DOI: 10.1126/sciadv.aaw5912
  • Fig. 1 Snapshots from MD simulations of self-assembly with interaction strengths EAA/EAB = EBB/EAB = 0.15, temperature kBT/EAB = 0.1, and density ρσ2 = 0.1.

    Examples of square-to-hexagonal transformations occurring because of (A) growth of a square cluster, (B) attachment of one square cluster to another, and (C) attachment of a square cluster to a hexagonal cluster.

  • Fig. 2 Results from MD simulations of individual crystallites.

    (A) Predominant structure versus EAA/EAB = EBB/EAB and N at kBT/EAB = 0.08, based on 10 replicates each of MD runs starting from square and from hexagonal crystallites. If less than 95% of the total time across all replicates was spent in the indicated state, an open symbol is shown. Conditions for the simulations of (B) to (D) are indicated. (B) N = 100 and (C) N = 64 examples of transformation at EAA/EAB = EBB/EAB = 0.15 and kBT/EAB = 0.08. Structural identification is achieved by the RDF (radial distribution function) similarity parameter S discussed in Materials and Methods. (D) Additional example at N = 36, EAA/EAB = EBB/EAB = 0.19, and kBT/EAB = 0.08.

  • Fig. 3 Results from MD simulations with a DNA-functionalized nanoparticle system having explicitly modeled DNA chains.

    (A and B) Snapshots showing transformations taking place within crystallites. (C) Average nearest neighbor count for explicit chain model systems compared with reference values for pure static square and hexagonal crystallites in the chosen shapes.

  • Fig. 4 Comparison of lattice energies with results from free energy calculations and MD simulation results.

    (A) Lattice energy in the T → 0 limit and (B) Helmholtz free energy at kBT/EAB = 0.06 for square and hexagonal lattices at EAA/EAB = EBB/EAB = 0.1. N → ∞ limiting values are derived from linear extrapolation in 1/N based on data points with N ≥ 100. Shapes used are illustrated in Fig. 2A. (C) Fraction of total time for which the two structures appear in replicate MD simulations of individual crystallites for these conditions. (D) Probabilities of observing structures as calculated from the free energies.

  • Fig. 5 Size, temperature, and interaction dependence of crystallite free energies.

    Square-hexagonal coexistence as a function of N (A) for kBT/EAB = 0.08 and (B) over a range of temperatures. The locations of the curves are determined on the basis of polynomial fits to the free energy values. (C) The T → 0 energy, (D) internal energy at finite T, and (E) free energy for the conditions used in Fig. 4. Horizontal axes are linear in 1/N. Extrapolations to the N → ∞ limit are linear fits to data for N ≥ 100.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/9/eaaw5912/DC1

    Supplementary Text

    Fig. S1. Temperature dependence of nucleation barriers and critical sizes at EAA/EAB = EBB/EAB = 0.15.

    Fig. S2. Mean lengths of time that the initial structures in MD simulations of individual crystallites persist before either transformation occurs or the simulation ends.

    Fig. S3. Predominant structures in transformation MD simulations for asymmetric interactions.

    Fig. S4. Shapes and RDFs for crystallites of different structures.

    Fig. S5. Predominant structures in transformation MD simulations for various crystallite shapes and aspect ratios.

    Fig. S6. Comparison of components of free energy for additional cases of interactions and temperatures.

    Fig. S7. Interaction potentials for different DFP models.

    Fig. S8. Interaction dependence of the system crystallization temperature.

    Fig. S9. Illustration of the thermodynamic pathway used for the Einstein molecule method applied to nonperiodic systems.

    Movie S1. Animation showing a trajectory of self-assembly from which the snapshots presented in Fig. 1 are taken.

    Movie S2. Animation corresponding to the transformation trajectory shown in Fig. 2B.

    Movie S3. Similarly, animation corresponding to Fig. 2C.

    Movie S4. Similarly, animation corresponding to Fig. 2D.

    Reference (32)

  • Supplementary Materials

    The PDF file includes:

    • Supplementary Text
    • Fig. S1. Temperature dependence of nucleation barriers and critical sizes at EAA/EAB = EBB/EAB = 0.15.
    • Fig. S2. Mean lengths of time that the initial structures in MD simulations of individual crystallites persist before either transformation occurs or the simulation ends.
    • Fig. S3. Predominant structures in transformation MD simulations for asymmetric interactions.
    • Fig. S4. Shapes and RDFs for crystallites of different structures.
    • Fig. S5. Predominant structures in transformation MD simulations for various crystallite shapes and aspect ratios.
    • Fig. S6. Comparison of components of free energy for additional cases of interactions and temperatures.
    • Fig. S7. Interaction potentials for different DFP models.
    • Fig. S8. Interaction dependence of the system crystallization temperature.
    • Fig. S9. Illustration of the thermodynamic pathway used for the Einstein molecule method applied to nonperiodic systems.
    • Reference (32)

    Download PDF

    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.mp4 format). Animation showing a trajectory of self-assembly from which the snapshots presented in Fig. 1 are taken.
    • Movie S2 (.mp4 format). Animation corresponding to the transformation trajectory shown in Fig. 2B.
    • Movie S3 (.mp4 format). Similarly, animation corresponding to Fig. 2C.
    • Movie S4 (.mp4 format). Similarly, animation corresponding to Fig. 2D.

    Files in this Data Supplement:

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