Supplementary Material for ”Hard convex lens-shaped particles: Densest-known
packings and phase behavior”
Giorgio Cinacchi1, a) and Salvatore Torquato2, b)
1)
Departamento de Fı́sica Teórica de la Materia Condensada,
Instituto de Fı́sica de la Materia Condensada (IFIMAC),
Instituto de Ciencias de Materiales “Nicolás Cabrera”,
Universidad Autónoma de Madrid, Campus de Cantoblanco,
E-28049 Madrid, Spain
2)
Department of Chemistry, Department of Physics,
Institute for the Science and Technology of Materials,
Program for Applied and Computational Mathematics,
Princeton University, Princeton, New Jersey 08544, USA
(Dated: 12 November 2015)
a)
Electronic mail: [email protected]
b)
Electronic mail: [email protected]
1
In this supplementary material, we provide additional details on the dense packing constructions as well as on our results on the phase behavior, including positional and orientational
correlation functions, of hard convex lens-shaped particles (lenses).
I.
MORE DETAILS ON DENSE PACKING CONSTRUCTIONS
This section provides first the unit cell parameters for the lattice packings of Sec.II that
were derived directly from the hard-sphere crystal: the hcp-like, the fcc-like and the bco-like
crystals. Next, the fundamental cell parameters for the non-lattice packings of Sec.II are
also provided: the elo-like crystal, derived from the elo crystal of hard oblate ellipsoids, and
the bco2-like crystal, whose derivation from the bco-like crystal is motivated by the results
known for hard oblate ellipsoids.
1. hcp-like crystal. This crystal is derived from the hard-sphere hcp crystal. Equivalently,
it can also be derived from the latter (Barlow) stacking variants, including the fcc
crystal once this is viewed as composed of stacked triangular-lattice layers of particles.
Each hard sphere is replaced by a lens. These are aligned along the same direction
perpendicular to the triangular-lattice layers. Referring to Fig. 1(a), the quantities
labeled Lx , Ly and Lz are respectively given by:
Lx = a;
√
(1)
3
;
Ly = a
2
"r
Lz =
(2)
#
2
a
b
4R2 −
.
−2 R−
3
2
(3)
The packing fraction results to be φ = v/(Lx Ly Lz ).
2. fcc-like crystal. This crystal is derived from the hard-sphere fcc crystal. Each hard
sphere is replaced by a lens. These are aligned along the same direction coincident,
e.g., with the [001] direction. Referring to Fig. 1(b), the quantities labeled Lx , Ly and
Lz are respectively given by:
2
(a)
Lz
Ly
Lx
Lx
(b)
Lz
Ly
Lx
Lx
(c)
Ly
Lz
Lx
q
Lx 2 + Ly 2
FIG. 1. (a) Schematic top (left) and front (right) views of the hcp-like crystal. (b) Schematic top
(left) and front (right) views of the fcc-like crystal. (c) Schematic top (left) and diagonal (right)
views of the bco-like crystal.
3
Lx =
√
Ly =
√
Lz =
2a;
(4)
2a;
"r
(5)
4R2 −
a2
b
−2 R−
2
2
#
.
(6)
The packing fraction results to be φ = 2v/(Lx Ly Lz ).
3. bco-like crystal. This crystal is derived from the hard-sphere fcc crystal once it is
viewed as a special body-centered-orthorhombic crystal (Bain transformation). Each
hard sphere is replaced by a lens. These are aligned along the same direction coincident with the vertical axis of the aforementioned body-centered-orthorhombic crystal.
Referring to Fig. 1(c), the quantities labeled Lx , Ly and Lz are respectively given by:
Lx = a;
√
Ly = 8Rb − a2 − b2 ;
Lz = b/2.
(7)
(8)
(9)
The packing fraction results to be φ = 2v/(Lx Ly Lz ).
The above-listed crystals are all lattice packings. Two-particle-basis non-lattice packings for
lenses can be constructed resorting to the results known for hard oblate ellipsoids.
4. elo-like crystal. This crystal derives from the elo crystal of hard (oblate) ellipsoids. The
elo crystal is the densest-known arrangement for hard ellipsoids and it too derives from
the hard-sphere fcc crystal once this is viewed as a laminate of square-centered layers.
The elo-like crystal is obtained by replacing each ellipsoids with a lens. These are
assumed to be aligned as ellipsoids are in the elo crystal. Based on particle orientations,
two cases can thus be distinguished.
√
(a) 1/ 3 ≤ κ ≤ 1. Particles of two adjacent layers are mutually perpendicular, as
shown in Fig. 2(a). Referring to this figure, the quantities labeled Lx , Ly and Lz
4
(a)
Ly
Lx
(b)
Ly
Ψ
Lx
√
FIG. 2. (a) Schematic top (left) and front (right) views of the elo-like crystal for 1/ 3 ≤ κ ≤ 1.
√
(b) Schematic top view of the elo-like crystal for 0 < κ ≤ 1/ 3.
5
Lz
are respectively given by:
Lx = 2ζ+ ;
(10)
Ly = 2ζ+ ;
s
2
b
a 2
− R + ζ+ −
.
R+
Lz =
2
2
(11)
(12)
b
where ζ+ is the positive solution of the quadratic equation ζ + 2 R −
ζ+
2
2
b
− 2R2 = 0. This value of ζ+ is valid for Φ − 1 = 1/Φ ≤ κ ≤ 1,
2 R−
2
a
with Φ the golden section. In fact, for κ = Φ − 1=1/Φ, ζ+ = . Provided that
2
the constraint Lx =Ly is conserved and particles belonging to two adjacent layers
2
are constrained to be perpendicular, as it happens for hard oblate ellipsoids for
√
√
κ ≥ 1/ 3, this is the value that ζ+ takes on also in the interval 1/ 3 ≤ κ ≤
Φ − 1 = 1/Φ.
√
(b) 0 < κ < 1/ 3. Particles of two adjacent layers become progressively more aligned
while the fundamental cell stretches along the x̂ direction and flattens along the
ŷ direction. Fig. 2(b) illustrates the meaning of the quantities labeled Lx and
Ly . Concerning the quantity labeled Lz not shown in this figure, it is analogous
to that shown in Fig. 2(a), i.e. it is half the length of the fundamental cell
dimension perpendicular to the xy plane. The three quantities are respectively
given by:
s
b
R−
2
2
b
Lx =
cos Ψ − 2 R −
sin Ψ;
−4
2
s
2
b
b
2
Ly = 4R − 4 R −
sin Ψ − 2 R −
cos Ψ;
2
2
s
2
b
bz
ax 2
2
−
+2 R−
cos Ψ .
Lz = 4R −
4
2
2
4R2
(13)
(14)
(15)
with Ψ the angle that û axis of one lens of the basis forms with the x̂ axis, while
the angle that the other lens û axis forms with x̂ axis is −Ψ. For hard oblate
√
ellipsoids Ψ=π/4 for κ = 1/ 3 and decreases until vanishing as κ → 0.?
In either case (a) and (b), the packing fraction results to be φ = 2v/(Lx Ly Lz ).
6
5. bco2-like crystal. This crystal consists in a modification of the bco-like crystal inspired
by the results known for hard (oblate) ellipsoids. The central lens of the bco-like crystal
is rotated in such a way its C∞ axis lies
q along x̂. The value of Lx and Lz remains
the same while that of Ly changes to 2 3R2 − (R + a/2 − b/2)2 . It is a reasonable
crystal structure only for quasi-spherical lenses, whose packing fraction results to be
φ = 2v/(Lx Ly Lz ).
II.
MORE DETAILS ON THE PHASE BEHAVIOR
In Fig. 3, an example of an equilibration run starting from an elo-like structure for lenses
with an intermediate value of the aspect ratio κ is given. In these cases, that structure
spontaneously evolved toward a bco-like structure, as discussed in the main text.
Figures 4, 5 and 6 provide the equation of state and the value of the nematic order parameter
as a function of packing fraction for the same lenses as in Fig. 7 of the main text, namely
√
√
R∗ = 0.47 (κ ≃ 0.75), R∗ = 1/ π (κ = 1/ 3) and R∗ = 1 (κ ≃ 0.29). In these graphs,
compression runs obtained by starting from a sufficiently low pressure in the isotropic phase
are also included. For globular lenses, the formation of a plastic solid phase from an isotropic
melt is relatively fast. Its full formation required a number of MC cycles of the order of 106 .
Much faster is the formation of a nematic liquid-crystalline phase from an isotropic melt for
flat lenses. Its formation was already completed after a number of MC cycles of the order
of 105 . Instead, in all cases, the formation of a crystalline solid phase was quite inhibited.
Systems stuck in plastic (for globular lenses), isotropic (for lenses that are neither globular
nor flat) or nematic (for flat lenses) states at least for a number of MC cycles of the order
of 107 .
Figures 7 and 8 give examples of positional and orientational correlation functions used in
the characterization of the various phases met.
In Fig. 9, the schematic phase diagram of lenses in the κ − φ plane is re-plotted with the
incorporation of curves that attempt to analytically describe the trend of the isotropic-tonematic phase transition and of the plastic-to-crystal phase transition respectively in the
hard-infinitesimally-thin-disk limit (κ → 0) and in the hard-sphere limit (κ → 1). In the
7
0
500
1000
1500
2000
0.66
φ
0.63
equilibration from elo-like
production from bco-like
0.60
1
0.8
0.6
S2
0.4
0.2
0
500
1000
1500
0
2000
thousand of MC cycles
FIG. 3. Evolution of packing fraction (top panel) and nematic order parameter (bottom panel)
√
√
during the equilibration run for the system with R∗ = 1/ π (κ = 1/ 3) at pσ 3 /kB T =70 started
from an elo-like configuration (green or lighter gray curves) compared to the evolution of the same
quantities during the production run for the same system at the same pressure whose corresponding
equilibration run started from a bco-like configuration (red or darker gray curves). The curves
corresponding to the latter case have been shifted of 1 million of MC cycles rightward so as to
make them superimpose the second half of the curves corresponding to the former case. Images
consist of a view of the initial elo-like configuration and two views of the final configuration obtained
after 2 millions of MC cycles, as indicated by the arrows. Images were created using the program
QMGA.1
8
1
500
0.8
400
0.6
pσ 3
kB T
S2
300
200
0.4
100
0.2
0
0.3
0.4
0.5
φ
0.6
0.7
0.8
0.3
0.4
0.5
φ
0.6
0.7
0
0.8
FIG. 4. Behavior of dimensionless pressure pσ 3 /kB T (left panel) and nematic order parameter S2
(right panel) as a function of packing fraction φ for a system of lenses with R∗ =0.47 (κ ≃ 0.75):
black circles are for the set of MC calculations starting from the bco-like structure, red squares for
the set of MC calculations starting from the elo-like structure and green triangles for the set of MC
calculations compressing from a sufficiently low pressure in the isotropic phase.
former limit, one can resort to the known numerical results for the isotropic-to-nematic
phase transition in a system of hard infinitesimally thin disks2 while in the latter limit,
the plastic-to-crystal phase transition boundary must end up to the hard-sphere densest
π
packing fraction, √ . To estimate how the relevant phase transition boundaries approach
18
the corresponding limit, the key assumption made is to take the density values at which the
relevant phase transition occurs at the corresponding limit value of κ as remaining constant,
once suitably scaled, in the close neighborhood of the corresponding limit value of κ. In this
respect, out of numerous possibilities, two are considered: (i) the density is multiplied by
a3 ; in both the hard-infinitesimally-thin-disk and hard-sphere limits, a coincides with the
particle diameter; (ii) particles with the same surface area are considered and the density is
multiplied by σ 3 , with σ such that 2σ 2 is the particle surface area. These assumed constant
values of the scaled density are then multiplied by the scaled particle volume to give the
corresponding values of the packing fraction.
9
1
100
0.8
80
0.6
pσ 3
kB T
S2
60
0.4
40
0.2
20
0
0.3
0.4
0.5
φ
0.6
0.7
0.8
0.3
0.4
0.5
φ
0.6
0.7
0
0.8
FIG. 5. Behavior of dimensionless pressure pσ 3 /kB T (left panel) and nematic order parameter S2
√
√
(right panel) as a function of packing fraction φ for a system of lenses with R∗ = 1/ π (κ = 1/ 3):
black circles are for the set of MC calculations starting from the bco-like structure, red squares for
the set of MC calculations starting from the elo-like structure and green triangles for the set of MC
calculations compressing from a sufficiently low pressure in the isotropic phase.
Both options are able to recover the limiting slope of the isotropic-to-nematic phase transition boundaries in the limit κ → 0, with the second option providing an overall better
representation since it is also able to capture the correct sign of the concavity of the curve
obtained by interpolating the MC numerical simulation data.
In the case of the plastic-to-crystal phase transition, the difference between the two options
is more marked. The first option significantly underestimates the stability of the plastic
solid phase while by adopting the second option this stability is a bit overestimated but the
MC numerical simulation trend in the very close proximity of κ=1 appears to be recovered.
10
1
500
0.8
400
0.6
pσ 3
kB T
S2
300
200
0.4
100
0.2
0
0.3
0.4
0.5
φ
0.6
0.7
0.3
0.4
0.5
φ
0.6
0.7
0
FIG. 6. Behavior of dimensionless pressure pσ 3 /kB T (left panel) and nematic order parameter S2
(right panel) as a function of packing fraction φ for a system of lenses with R∗ = 1 (κ ≃ 0.29): black
circles are for the set of MC calculations starting from the bco-like structure and green triangles
for the set of MC calculations compressing from a sufficiently low pressure in the isotropic phase.
11
r/σ
g(r)
4
r/σ
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(a)
(c)
12
3
8
g(r)
5
2
4
1
0
0
1
1
0.6
(b)
0.8
(d)
0.6
0.4
0.4
0.2
0.2
0
Gû2 (r)
Gû2 (r)
0.8
0
-0.2
-0.2
-0.4
-0.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
r/σ
r/σ
FIG. 7. Positional [ g(r), panels (a) and (c)] and orientational [Gû2 (r), panels (b) and (d)] correlation
functions for a system of lenses with R∗ =0.43 (κ ≃ 0.87) in the various phases: isotropic fluid
(amber lines, at pσ 3 /kB T =20), plastic solid [green lines, at pσ 3 /kB T =30 in panels (a) and (b) and
pσ 3 /kB T =300 in panels (c) and (d)] and crystalline solid (red lines, at pσ 3 /kB T =400).
12
r/σ
0
0.4 0.8 1.2 1.6
r/σ
2
2.4 2.8
0
0.4 0.8 1.2 1.6
2
2.4 2.8
1.4
1.5
1
0.8
1
g(r)
g(r)
1.2
2
0.6
(a)
0.4
(c)
0.5
0.2
0
0
1
1
0.8
0.6
Gû2 (r)
Gû2 (r)
0.8
0.4
0.6
0.2
0.4
0
(b)
-0.2
(d)
-0.4
0
0.4 0.8 1.2 1.6
r/σ
2
2.4 2.8
0
0.4 0.8 1.2 1.6
r/σ
2
2.4 2.8
0.2
0
FIG. 8. Positional [ g(r), panels (a) and (c)] and orientational [Gû2 (r), panels (b) and (d)] correlation
functions for a system of lenses with R∗ =0.75 (κ ≃ 0.41) in the various phases: isotropic fluid
(amber lines, at pσ 3 /kB T =30), nematic liquid-crystalline [blue lines, at pσ 3 /kB T =40 in panels (a)
and (b) and pσ 3 /kB T =60 in panels (c) and (d)] and crystalline solid (red lines, at pσ 3 /kB T =70).
13
1
0.8
crystal
plastic
0.6
φ
nematic
0.4
isotropic
0.2
0
0
0.2
0.4
κ
0.6
0.8
1
FIG. 9. Schematic phase diagram of lenses in the κ—φ plane in which the estimated limiting
curves for the isotropic-to-nematic and plastic-to-crystal phase transitions are also reported either
obtained by multiplying the number density by a3 (dashed lines) or by σ 3 (full lines).
14
REFERENCES
1
http://qmga.sourceforge.net; A. T. Gabriel, T. Meyer and G. Germano, J. Chem. Theory
Comput. 4, 468 (2008).
2
(a) D. Frenkel and R. Eppenga, Phys. Rev. Lett. 49, 1089 (1982). R. Eppenga and D.
Frenkel, Mol. Phys. 52, 1303 (1984). (b) G. Cinacchi and A. Tani, J. Phys. Chem. B 119,
5671 (2015).
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