Research ArticlePHYSICS

How crystals form: A theory of nucleation pathways

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Science Advances  05 Apr 2019:
Vol. 5, no. 4, eaav7399
DOI: 10.1126/sciadv.aav7399
  • Fig. 1 Typical structures obtained from three-dimensional cDFT calculations, as reported in (50).

    Each figure is a view of the local density obtained by free-energy minimization of the Lennard-Jones system at a reduced temperature of T* = 0.4. The leftmost figure is a slice through a dense-solution, liquid-like, droplet; the center figure is a contour representation of an amorphous, glass-like cluster; and the rightmost figure is a contour representation of a face-centered cubic (fcc) cluster. The droplet shows the packing into shells separated by low-density regions, which is typical of confined liquids. In the other two structures, the density is localized into “atoms” separated by very low density regions.

  • Fig. 2 Droplet free-energies as a function of size as determined from CNT and the present theory.

    Nucleation pathway for a cluster of 286 molecules at kBT = 0.475ε and supersaturation S = 2.2, displayed as the excess grand canonical free energy (i.e., relative to the weak-solution free energy) versus the excess number (relative to the weak solution) of particles in the simulation cell (which is effectively the number of particles in the cluster). The blue line is the prediction of CNT, and the red line is the result of allowing the surface tension (γ) to depend on the radius (R) of the cluster [γ = γ0(1 + l/R)] and fitting to the larger clusters (those with ΔN > 100) with the result that the Tolman length l = 0.232σ. The images marked in red are shown in subsequent figures. The inset shows the cluster radius (based on the radius of gyration, as described in the text) and the average density for particles within the sphere of radius R. The arrows indicate the direction of movement along the MLP. The broken line is the CNT path.

  • Fig. 3 Snapshots of slices through the strong-solution clusters along the nucleation pathway, leading to the critical cluster composed of about 286 molecules.

    The images show the density on a logarithmic color scale ranging from black [ln(nσ3) = −10 or lower] to red to yellow to white [ln(nσ3) = 7 or higher]. (A) Point at the origin of Fig. 2, i.e., the pure weak solution. Counting from that one, (B) to (F) correspond to points 1, 10, 20, 35, and 49, respectively, with the latter being the final point, i.e., the critical cluster.

  • Fig. 4 Snapshots of slices through the strong-solution clusters, as in Fig. 3 but showing the density on a linear color scale ranging from black (nσ3 = 0) to dark blue and light blue to white (nσ3 ≥ 2).

    The packing structure, which manifests as alternating spherical shells of high and low density, is clearly visible in this view.

  • Fig. 5 Nucleation pathway for a cluster of approximately 286 growth units: The lines and points plot the excess free energy along the path as a function of an abstract reaction coordinate.

    Inset images show the log of the density on a slice through the center of the cluster for the free energy points outlined in red. The inset graph on the left shows the mass of the cluster (black symbols) and the number of density peaks above a threshold of nσ3 > 5 (red symbols), which is a measure of the number of “solid” molecules; the one on the right is a magnified look at the free energy in the latter stages of the process. The path displays the two-step mechanism consisting of the formation of a droplet followed by freezing at constant mass. As shown in the magnified view, the free energy also contains a shallow minimum after the initial barrier. An examination of the images shows that the minimum is due to the completion of a crystalline shell. The free energy barrier is about 175 kBT.

  • Fig. 6 Snapshots of the density on a linear scale during the onset of crystallization.

    Panels (A) to (C) correspond to points 19, 20, and 21, respectively, on Fig. 5. The later stages of crystallization are shown in panels (D) to (F) corresponding to points 25, 30, and 49, respectively, on Fig. 5.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/4/eaav7399/DC1

    Supplementary Text

    Fig. S1. Phase diagram as presented in (50).

    Fig. S2. Supersaturation as a function of the excess number of particles for liquid critical droplets.

    Fig. S3. Initial and final paths for crystallization with a final cluster of 286 molecules using 50 images and a final path using 20 images showing the excess free energy as a function of Euclidean distance along the path (arbitrarily scaled so that the distance between images is equal to one).

    Fig. S4. Log of the density for slices of images from the initial guess to the pathway.

    Fig. S5. Nucleation pathways for several systems displayed in terms of the free energy along the paths.

    Fig. S6. Nucleation pathways for several systems displayed in terms of the number of molecules in the clusters along the paths.

    Fig. S7. Excess number of particles in the critical clusters as a function of supersaturation.

    Fig. S8. Excess free energies in the critical clusters as a function of supersaturation.

    Table S1. Thermodynamic quantities at weak-solution/dense-solution liquid-liquid coexistence.

    Table S2. Thermodynamic quantities used in liquid-liquid calculations.

    Movie S1. Cross section of liquid droplet density as it evolves along the nucleation pathway for the example presented in the main text showing the log of the density.

    Movie S2. Cross section of liquid droplet density as it evolves along the nucleation pathway for the example presented in the main text showing the density on a linear scale.

    Movie S3. Cross section of solid cluster density as it evolves along the nucleation pathway for the example presented in the main text showing the log of the density.

    Movie S4. Cross section of solid cluster density as it evolves along the nucleation pathway for the example presented in the main text showing the density on a linear scale.

    References (6265)

  • Supplementary Materials

    The PDF file includes:

    • Supplementary Text
    • Fig. S1. Phase diagram as presented in ( 50).
    • Fig. S2. Supersaturation as a function of the excess number of particles for liquid critical droplets.
    • Fig. S3. Initial and final paths for crystallization with a final cluster of 286 molecules using 50 images and a final path using 20 images showing the excess free energy as a function of Euclidean distance along the path (arbitrarily scaled so that the distance between images is equal to one).
    • Fig. S4. Log of the density for slices of images from the initial guess to the pathway.
    • Fig. S5. Nucleation pathways for several systems displayed in terms of the free energy along the paths.
    • Fig. S6. Nucleation pathways for several systems displayed in terms of the number of molecules in the clusters along the paths.
    • Fig. S7. Excess number of particles in the critical clusters as a function of supersaturation.
    • Fig. S8. Excess free energies in the critical clusters as a function of supersaturation.
    • Table S1. Thermodynamic quantities at weak-solution/dense-solution liquid-liquid coexistence.
    • Table S2. Thermodynamic quantities used in liquid-liquid calculations.
    • References (6265)

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    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.mov format). Cross section of liquid droplet density as it evolves along the nucleation pathway for the example presented in the main text showing the log of the density.
    • Movie S2 (.mov format). Cross section of liquid droplet density as it evolves along the nucleation pathway for the example presented in the main text showing the density on a linear scale.
    • Movie S3 (.mov format). Cross section of solid cluster density as it evolves along the nucleation pathway for the example presented in the main text showing the log of the density.
    • Movie S4 (.mov format). Cross section of solid cluster density as it evolves along the nucleation pathway for the example presented in the main text showing the density on a linear scale.

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