High-resolution SEM secondary electron images (SEI) and related models showing the influence of the length-to-diameter ratio (L/D) on the organization of the octapods. 8 (a) and (c) For L/D = 4.8 only simple square-lattice crystals were formed, while (b) and (d) for L/D = 5.9 binary-lattice square crystals were occasionally found, as indicated by the outline in (b). The scale bars are 100 nm.
(a) An example of the hard octapod model, which consists of four interpenetrating spherocylinders with a length-to-diameter ratio L/D = 6.0. The orange arrows indicate the length L and diameter D definition we used for our model. (b) The model for several of the values of L/D that we considered in this paper. (c) An illustration of the octapod model in the quasi-2D geometry that we used. We constrained the bottom four tips to be in contact with the substrate; the octapods therefore effectively behave as if they are trapped between two frictionless walls.
Top views of quasi-2D densely packed structures obtained for different length-to-diameter ratios (L/D) of the hard octapod model. (a) A rhombic crystal (RC) for L/D = 1.0. (b) A square crystal (SC) for L/D = 4.0, which is not interlocking. (c) A binary-lattice square crystal (BSC) for L/D = 6.0. The different orientations of the particles in the two sublattices are illustrated by the use of color. Note that the total lattice is again square, hence the name binary-lattice square crystal. (d) Another SC, for which the octapods are interlocking (L/D = 7.0). (e) A 3D image showing four octapod models in an interlocking configuration, the octapods are indicated with different colors for clarity. The inset shows a top view of the 3D image, in which the arms of the interlocking octapods appear to overlap.
(a) Solid blue circles show the “maximum” packing fraction η c as a function of L/D < 2.0. Open red circles show the angle θ (in degrees) between the two lattice vectors in xy-plane that span the unit cell of the crystal structure. (b) Solid blue circles show the difference in the packing fraction η2 − η1 between the square crystal (SC, one particle in the unit cell, η1) and the binary-lattice square crystal (BSC, two particles in the unit cell, η2) as a function of shape factor L/D. We only observed the BSC phase for L/D ∈ [5.0, 6.3], i.e., η2 − η1 ≠ 0 in this region. Open red circles show the angular difference Δα (in degrees) between the orientation of the particles in the two sublattices of the BSC.
(a) The equation of state (EOS) near the coexistence region for octapods with L/D = 4.0 in the planar quasi-2D geometry. Here P* = PD 2/k B T is the reduced pressure, with P the (2D) pressure, k B Boltzmann's constant, T the temperature, and D the diameter of the pods. We also define η = ρV p /h, where is the height of the octapods and V p is their volume. The green line and black points show the coexistence pressure and densities of the square-lattice crystal (SC) and isotropic-liquid (IL) phase. (b) The reduced free energy f − ρμ c + P c as a function of volume fraction η. Here f is the Helmholtz free energy per volume, μ c is the coexistence chemical potential, and P c is the coexistence pressure. This choice of representation ensures that the η-axis acts as a common tangent to the free energy.
The phase diagram for hard spherocylinder-based octapods in a planar quasi-2D system. We show the volume fraction η as a function of the length-to-diameter ratio L/D. The light-grey area indicates the coexistence region and the dark-grey area indicates the forbidden region above the maximum packing fraction (thick black line) of the densest-known crystal. “SC” denotes the stable square-lattice crystal, “BSC” denotes the binary-lattice square crystal, “RC” denotes the rhombic crystal, and “HR” denotes the stable hexagonal plastic crystal (rotator) phase. The blue circles indicate the isotropic-liquid (IL) phase-coexistence volume fraction, the blue squares the HR and SC coexistence volume fractions. The solid blue lines are a guide to the eye. The SC-BSC transition indicated by green stars and thin dotted lines, the RC-HR transition is indicated by light-blue triangles and thin solid line, and the RC-SC transition is indicated by red squares and thin dotted line.
Snapshots of an NVT Monte Carlo simulation for which there is phase coexistence between the isotropic (right) and square crystal (left) phase for octapods with L/D = 4.0. (a) Initial configuration for the coexistence simulation with ηI = 0.344 and ηSC = 0.385 and (b) final configuration for the coexistence simulation after 3 × 106 Monte Carlo cycles. The color indicates the orientation of the octapods.
(Left column) The angle distribution function (ADF) of the difference in orientation θ (in degrees) between neighboring octapods with L/D = 6.0 for several values of the reduced pressure P* = PD 2/k B T. (Right column) We also show snapshots and structure factors based on the centres of the particles (insets) for the systems to illustrate their state: (a) P* = 0.230, (b) P* = 0.260, (c) P* = 0.280, and (d) P* = 0.450. The blue dots show measured values for the distribution and the blue lines show a single or double-Gaussian fit to the simulation results. The dashed green lines in (b) give the distribution function obtained by a double-Gaussian fit.
Equation of state (EOS) and the packing-fraction (η) dependence of several order parameters (ϕ4, Ψ4, Ψ6) and their susceptibilities for octapods with L/D = 1.0. (a) The EOS for this system, i.e., reduced pressure P* = PD 2/k B T as a function of η. The isotropic-liquid (IL) phase is denoted by blue circles, the hexagonal-rotator phase (HR) is denoted by red triangles, and rhombic crystal (RC) is denoted by green squares. (b) The global 6-fold bond orientational order parameter Ψ6 (blue circles), the global 4-fold bond orientational order parameter Ψ4 (red triangles), and the global orientational order parameter ϕ4 (green squares). (c) The susceptibility of the 6-fold bond orientational order parameter χ6 (blue circles), the susceptibility of the 4-fold bond orientational order parameter χ4 (red triangles), and the susceptibility of the global orientational order parameter (green squares). We have added dashed vertical lines to indicate the location of the phase boundaries. The solid lines in (b) and (c) are guides to the eye.
The packing-fraction (η) dependence of several order parameters and their susceptibilities for octapods with L/D = 2.0. (a) The EOS for this system with P* = PD 2/k B T the reduced pressure. We labelled the square crystal phase using “SC,” the rhombic crystal phase using “RC,” the hexagonal rotator phase using “HR,” and the fluid phase using “IL.” (b) The global 6-fold and 4-fold bond orientational order parameter Ψ6 (blue circles) and Ψ4 (red triangles), respectively. (c) The susceptibility χ6 (blue circles) and χ4 (red triangles) as a function of the packing fraction. The packing fractions corresponding to peaks in the susceptibilities (dashed vertical lines) give the location of the phase transitions. The solid lines in (b) and (c) are guides to the eye.
The 6-fold bond orientational correlation function g 6 as a function of the radial distance r from the centre of an octapod for several packing fractions η and L/D = 2 (a) and L/D = 1 (b), respectively. Results for different values of η are indicated by different colors. The thick red dashed line has a slope of 1/4 and corresponds to the power-law decay predicted by KTHNY theory.
Illustration of the definition of the minimum angle ψ i ∈ [−π/4, π/4]. The grey octapod indicates the Einstein-crystal reference frame and the red octapod shows the orientation of the octapod of interest. The blue lines indicate the cylinder centres. We consider two clockwise rotations of the octapod, a small one by less than π/4 (a) and a larger one by slightly more than π/4 (b). For the former we obtain a positive ψ i < π/4 value, whereas for the latter we use the symmetry to map the rotation on a negative ψ i > −π/4 value.
Finite size scaling for the free energy per particle βF/N obtained by Einstein integration for a system of hard octapods with a packing fraction η = 0.40 and a pod length-to-diameter ratio of L/D = 4.0. The blue dots show the results of Monte Carlo simulations for N = 64, 100, 121, and 400 particles. The dashed red line shows a linear fit to the data, by which it is possible to determine the free energy of this phase. In the limit N → ∞ we obtain βF/N = 5.073.
Top view of the cell model for a central octapod (red) that is only allowed to translate in the xy-plane, surrounded by four neighboring positionally and orientationally fixed octapods (grey), which are arranged according to the square-lattice crystal (SC) structure. The SC structure has been expanded uniformly to achieve a desired volume fraction η. The length-to-diameter ratio L/D = 4.0 in this case. The centre-to-centre vector between the central octapod and its top-left neighbor is given by , with AB = l x and OB = l y . The parameter Δ gives the size of the gap between neighboring particles. The grey square identifies the area in which the centre of the central octapod is free to move, i.e., its free area (volume).
The reduced pressure P* = PA/k B T as a function of the packing fraction η for the rotating and non-rotating octapod free-volume model with length-to-diameter ratio L* = 4.0 that follow from our theoretical calculations and from our NPT Monte Carlo (MC) simulations, respectively. Solid lines give the free-volume theory results and the dots indicate the results from our MC simulations. The inset shows 2P tr /3P t − 1 (red line) as a function of η, where P tr is the reduced pressure for the NPT Monte Carlo system, in which the octapods can rotate, and P t is the reduced pressure for the NPT Monte Carlo system, in which the octapods can only translate. The blue dashed line indicates the ideal situation for which P tr ≡ 3P t /2; the red line indicates the fractional deviation with respect to this scaling.
A top view of the cell model, in which the central octapod (red) is allowed to rotate and translate with respect to its neighbors (black). We used a length-to-diameter ratio L/D of 4.0 here. The angle which the central octapod makes with its fixed neighbors is given by ϕ.
The length of the square that delimits the free area available to the central octapod (L/D = 4), when it makes an angle ϕ with its neighbors. Note that for ϕ < ϕ m and ϕ > ϕ M Δ f (ϕ) = 0. The value of ϕ for which Δ f (ϕ) assumes its maximum is denoted by ϕ t .
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