Article pubs.acs.org/JPCB
Structural Properties of Dense Hard Sphere Packings Boris A. Klumov,*,†,‡,⊥ Yuliang Jin,§ and Hernán A. Makse§ †
Joint Institute for High Temperatures RAS, Izhorskaya str. 13/2, Moscow, 125412, Russia Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow, 141700, Russia ⊥ Landau Institute for Theoretical Physics, Russian Academy of Sciences, Kosygina ul. 2, Moscow, 119334, Russia § Physics Department and Levich Institute, 140th Street and Convent Avenue, New York, New York 10031, United States ‡
ABSTRACT: We numerically study structural properties of mechanically stable packings of hard spheres (HS), in a wide range of packing fractions 0.53 ≤ ϕ ≤ 0.72. Detailed structural information is obtained from the analysis of orientational order parameters, which clearly reveals a disorder−order phase transition at the random close packing (RCP) density, ϕc ≃ 0.64. Above ϕc, the crystalline nuclei form 3D-like clusters, which upon further desification transform into alternating planar-like layers. We also find that particles with icosahedral symmetry survive only in a narrow density range in the vicinity of the RCP transition.
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INTRODUCTION As a fundamental model in condensed matter physics and material science, packings of hard spheres (HS) reproduce many essential structural properties of glassy and granular media.1−3 Structural changes of the HS system have been observed when a packing is densified above the density of random close packing (RCP4), ϕc ≃ 0.64 (see, e.g., refs 5 and 6). A new order parameter, which is based on the cumulative distribution of the rotational invariant w6, was proposed in ref 5 to identify these structural changes. Random arrangement is transferred to partial crystallization across RCP, and two lattice types, the face centered cubic (fcc) and the hexagonal close-packed (hcp), are observed in the crystalline clusters. However, more detailed information has not been obtained due to insufficient system size. In this study, we aim to explore more detailed structural changes of the transition at RCP in comparison with previous studies,5,6 based on larger scale simulations. We numerically generate a large set of HS packings composed of N = 64 × 103 monodispersed spheres located in the cubic box with periodic boundary conditions, using the modified Lubachevsky−Stillinger (LS) algorithm.7
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Figure 1. Radial distribution function g(r) of hard spheres (solid lines) and its cumulative function Z(r) (dashed lines) at different packing fractions ϕ. The values of ϕ, and the positions of peaks, are indicated on the plot.
additional peaks appear at (2)1/2, (8/3)1/2, and (11/3)1/2, which are typical distances in the fcc and hcp lattices. The observation indicates the onset of crystallization when the packings are densified above ϕc. More detailed structural information can be obtained from the well-known bond order parameter method,9 which is widely used to study the structural properties of condensed matter,9−11 hard spheres,12−19 Lennard-Jones systems,20−24 complex plasmas,25−34 colloidal and patchy systems,35−37 granular media,38 metallic glasses,39 repulsive shoulder systems,40 etc. Each particle
RESULTS
To examine the structural properties of jammed spheres, we first study their spatial correlations. Figure 1 shows the radial distribution function g(r) of hard spheres for several different packing fractions in the range [0.60, 0.68]. The cumulative function Z(r) ≡ 4πρ∫ r0 r′2g(r′) dr′, which is the mean number of particles inside a sphere of radius r, is also plotted in Figure 1. When ϕ ≤ ϕc, addition to the strong peaks at the contact distances r = 1, 2, ... (we set unit diameter), a weak secondary peak at r = (3)1/2 is visible. This peak corresponds to a contact angle θ = 2π/3 but not necessarily a lattice structure.8 Above ϕc, © XXXX American Chemical Society
Received: May 8, 2014 Revised: August 3, 2014
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Figure 2 shows how the PDFs vary at different l (l = 4, 6) and ϕ (covering both amorphous and solid-like states). Densification of
i is connected via vectors (bonds) with its Nnn(i) nearest neighbors (NN), and the rotational invariants (RIs) of rank l of second ql(i) and third wl(i) orders are calculated as ⎛ 4π ql(i) = ⎜⎜ ⎝ (2l + 1) wl(i) =
m=l
∑ m =−l
∑ m1, m2 , m3 m1 + m2 + m3 = 0
⎞1/2 |qlm(i)| ⎟ ⎠ 2⎟
(1)
⎡l l l ⎤ ⎢ m m m ⎥qlm1(i)qlm2(i)qlm3(i) ⎣ 1 2 3⎦
(2)
−1
∑Nj=1nn(i) Ylm(rij),
where qlm(i) = Nnn(i) Ylm are the spherical harmonics, and rij = ri − rj are vectors connecting centers of ⎡ ⎤ particles i and j. In eq 2, ⎣ lm1 lm2 lm3 ⎦ are the Wigner 3jsymbols, and the summations performed over all the indexes mi = −l, ..., l satisfying the condition m1 + m2 + m3 = 0. As shown in the pioneer paper,9 the bond order parameters ql and wl can be used as measures, to characterize the local orientational order and the phase state of considered systems. Because each lattice type has a unique set of bond order parameters, the method of RIs can also be used to identify lattice structures in mixed systems. The values of ql and wl for a few common lattice types (including the liquidlike HS state) are presented in Table 1. Some of the values were Table 1. Bond Order Parameters ql and wl (l = 4, 6) for Several Typical Lattices and Liquid-Like HS Random Packings (rp) at ϕ = 0.55, Calculated via the Fixed Number of Nearest Neighbors (NN) lattice type hcp (12 NN) fcc (12 NN) ico (12 NN) bcc (8 NN) bcc (14 NN) rp at ϕ = 0.55 (12 NN)
q4
q6
w4
w6
0.097 0.19 1.4 × 10−4 0.5 0.036 ≈0.16
0.485 0.575 0.663 0.628 0.51 ≈0.34
0.134 −0.159 −0.159 −0.159 0.159 ≈ −0.019
−0.012 −0.013 −0.169 0.013 0.013 ≈ −0.032
reported previously (see, e.g., refs 9 and 41−44). From eq 2, it is easy to see that wl ∝ ql3, and therefore, we expect wl’s to be more sensitive measures compared to ql’s. In three dimensions (3D), the densest possible packings of identical hard spheres are the fcc and hcp lattices (with the densest packing fraction (pf) ϕfcc = ϕhcp = (2)1/2π/6 ≃ 0.74). Dense HS systems may also include particles having the icosahedral (ico) type of symmetry. Icosahedral, fcc, and hcp spheres have 12 nearest neighbors (the spheres located in the first coordination shell). To identify lattice-like particles, we calculate the bond order parameters ql and wl (l = 4, 6) for each particle with a fixed number of the nearest neighbors Nnn = 12.45 A particle is called fcc-like (hcp-like, ico-like) if its coordinates in the fourdimensional space (q4, q6, w4, w6) are sufficiently close to those of the perfect fcc (hcp, ico) lattice type. An amorphous (liquidlike) particle is identified if its order parameters are sufficiently −1/2 fcc/hcp/ico small, for example, qliq . Note that, by 6 ≃ Nnn ≃ 0.29 ≪ q6 varying the NN number and rank l of parameters wl and ql, in principle, it is possible to find any lattice type. For instance, the first and second (next nearest neighbors) shells of body centered cubic (bcc) lattices correspond to Nnn = 8 and 14, respectively. To quantify the local orientational order, it is convenient to use the probability distribution functions (PDFs) P(ql) and P(wl).
Figure 2. Probability distribution functions (PDFs) of the rotational invariants qi (a and c) and wi (b and d) at various packing fractions ϕ. Values of the bond order parameters for ideal lattice types (fcc and hcp) are also indicated. Cumulative distributions (normalized to unity) of these PDFs are shown by dashed lines, which are much less noisy than PDFs. By using these distributions, it is easy to estimate the abundance of crystalline and liquid-like particles. Insets (from top to bottom) show spatial distributions of N = 64 × 103 HS in the cubic box at different packing fractions of ϕ ≃ 0.65, 0.66, and 0.68, respectively. Hard spheres are color-coded by their q6 values, in order to visualize liquid-like (blue color) and crystalline hcp-like (green color) and fcc-like (red color) structures.
the HS, as clearly seen in Figure 2, results in appearance of the crystalline fcc-like and hcp-like spheres in the vicinity of the RCP. Corresponding cumulative distributions Clq and Clw (shown by dashed lines of the same color) are also plotted in Figure 2. For instance, the cumulative function Clq associated with the P(ql) is defined as Cql(x) ≡ B
x
∫−∞ P(ql) dql
(3)
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Evidently, Clq(x) is the abundance of particles, having values of ql < x and Clq(∞) = 1. Figure 3 shows two-dimensional PDFs
Figure 4. Cumulant Wc6 (blue and cyan circles) and ⟨q6⟩ (green and orange circles) versus packing fraction ϕ. Blue and green circles correspond to the HS system with N = 104 spheres; cyan and orange circles correspond to N = 64 × 103 spheres. The minimum of Wc6 can be used to locate the RCP transition, ϕc ≃ 0.645. The explosive-like growth of Wc6 above ϕc is a clear indicator of the appearance of ordered, crystalline-like (hcp-like and/or fcc-like) spheres. The growth of the ⟨q6⟩ value near ϕc is rather monotonous and relatively slow. The possibility for the dense HS system to be packed via different abundance of fcc-like and hcp-like particles is the key reason behind the fluctuation of ⟨q6⟩ when ϕ ≳ 0.68. The inset shows the number of ico-like spheres versus ϕ: ico-like spheres survive only in the narrow range of parameters ϕ near the RCP transition.
freezing phase transitions of Lennard-Jones23,24 and Yukawa systems.32−34 On the other side, the density dependence of ⟨q6⟩ also reveals fascinating properties. The relatively slow increase of ⟨q6⟩ at ϕ ≲ ϕc can be attributed to single hcp-like particles,5 while at ϕ > ϕc the bond order parameter ⟨q6⟩ increases more rapidly due to the emergence of crystalline clusters. Further densification of the HS leads to strong fluctuations (“oscillations”) of the global parameter ⟨q6⟩, which start at ϕ ≃ 0.68 and suggest another transition.5,6 This oscillatory behavior is due to the fact that the hcp and fcc crystalline structures are characterized by quite different values of the bond order parameter q6 (see Table 1), and at a given ϕ, dense partially crystallized HS packings can be realized with quite different abundances of hcp-like and fcclike spheres. Another important parameter characterizing the local structure is the abundance Nico of spheres having icosahedral-type (5-fold) symmetry (see Figure 4, inset). Here, ico-like spheres are defined as spheres having q6 ≥ 0.61. The density dependence of Nico shows that ico-like spheres can exist only in a narrow range of pf, and its abundance maximizes at ϕc. The above analysis provides detailed information about the local and global orders. Next, we study the abundance of crystalline particles and their spatial arrangement. We use the standard friends-of-friends algorithm46 to find crystalline clusters. Figure 5 presents the spatial distribution and the shape of crystalline clusters composed of hcp- and fcc-like spheres, at several different ϕ. When ϕ < 0.65 (Figure 5a,b), the crystalline particles are mostly isolated; increase of packing fraction ϕ leads to an appearance of several 3D (containing tens of spheres, both hcp- and fcc-type) clusters (Figure 5c,d at ϕ ≈ 0.66). Upon further densification, these 3D clusters transform into 2D layers spanning system-wide (see Figure 5e,f at ϕ ≈ 0.68).5 The abundances of hcp-like (nhcp) and fcc-like (nfcc) spheres are shown in Figure 6 as functions of packing fraction ϕ. At ϕ ≤
Figure 3. Probability distributions on the order parameter plane of q4− q6 in the vicinity of the RCP at ϕ = 0.64 (a) and ϕ = 0.65 (b). Loci of the perfect fcc, hcp, and ico are also indicated on the plot. An fcc peak appears at ϕ = 0.65. Note that the hcp lattice may have a large overlap with the liquid-like structure in this plot.
probability distributions on the plane q4−q6in the vicinity of the RCP at ϕ ≃ 0.64 and ϕ ≃ 0.65. At ϕ ≃ 0.64, structurally HS is amorphous, while at ϕ ≃ 0.65 the formation of crystalline fcc-like spheres is clearly seen. The result suggests a disordered−ordered transition in the density range [0.64, 0.65], consistent with previous studies.5,6,16 The global order parameters, i.e., the cumulant Wc6 (defined as l Cw(Wcl ) = 1/2) and the mean value of q6 (defined from the global averaging ⟨q6⟩ = (1/N)∑Ni=1 q6(i)), are shown in Figure 4. No significant finite size effects are observed between N = 104 and N = 64 × 103 systems. For the purpose of detecting crystallization onset, we find Wc6 to be a more sensitive measure than ⟨q6⟩. This is because (1) liquid-like and lattice-like spheres have quite different values of w6 and (2) w6 is nearly the same for perfect fcc and hcp lattices (see Table 1). Because of this reason, the measure Wc6 has been recently used to describe the melting and C
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Figure 5. Distributions of hcp and fcc clusters over the space at different ϕ values: ϕ ≃ 0.65 (a, b), ϕ ≃ 0.66 (c, d), and ϕ ≃ 0.68 (e, f). Particles are colorcoded by the mass of the cluster (in the unit of the mass of a single sphere). At ϕ ≃ 0.65, single hcp-like spheres dominate. At ϕ ≃ 0.65, three-dimensional local clusters of both hcp- and fcc-like particles are observed. Further densification of HS results in a structural transition: the 3D local clusters transform into global and planar layers (e, f).
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ϕc, the relatively slow increase of ntot = nhcp + nfcc is mainly due to
DISCUSSION AND CONCLUSION The above results illustrate a sharp first-order-like transition6 at ϕc between disordered (frictional) packings and partially ordered packings. Amorphous frictionless packings only exist at a single density ϕc: below ϕc, the packings are unstable unless they are frictional (and the frictional packings are disordered);6 above ϕc, the packings are partially ordered. On the other hand, multiple amorphous frictionless states have been found in other simulations,47−50 which supports the mean-field theoretical prediction that amorphous jammed packings should exist in the interval (so-called J-line) ϕ ∈ [ϕth, ϕGCP].1 We stress that this apparent contrast is due to the presence of crystallization: If the nucleation rate is slow enough, such as in polydisperse47 or large
the emergence of randomly distributed single hcp-like spheres. Around ϕ ≃ ϕc, the slopes of nhcp(ϕ) and nfcc(ϕ) increase significantly. At ϕ ≃ 0.66, the abundance of the fcc-like spheres is equal to that of the hcp-like spheres (nfcc ≈ nhcp). Additionally, the abundance of spheres having number of contacts Nc = 12 is plotted in Figure 6, as one more sensitive indicator of the crystallization of a dense HS. The relative cumulative spectrum of cluster mass is plotted in the inset of Figure 6, showing that this distribution becomes wider as the density increases. D
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Figure 6. Abundance of hcp-like (red line points) and fcc-like (green line points) spheres versus the packing fraction ϕ, with N = 104. Results of larger scale simulations with N = 64 × 103 spheres are shown by squares (orange and green symbols correspond to hcp- and fcc-like spheres). Additionally, the abundance of spheres having number of contacts Nc = 12 is plotted by blue squares. The inset shows the relative cumulative spectrum of hcp (fcc)-like (red and green lines, respectively) clusters versus the cluster mass at different ϕ values: ϕ ≃ 0.68 (solid, triangles), ϕ ≃ 0.66 (dashed, gradients), and ϕ ≃ 0.65 (solid, circles).
dimensional systems,48 then crystallization is suppressed and dense amorphous packings can be obtained by slow compressions; while in 3D monodisperse systems, the nucleation process is so fast such that any slow compression would result in partial crystallization. Combined with the mean-field prediction, our results suggest two possible scenarios for 3D monodisperse packings: (i) ϕc < ϕth, therefore, dense amorphous packings ϕ > ϕth are hidden by crystallization; (ii) ϕth ≲ ϕc, however, the difference between ϕth and ϕc is very small, such that the amorphous packings in the range [ϕth, ϕc] are difficult to detect on the basis of the present numerical accuracy. Further studies are required to test these two scenarios. To conclude, we have studied structural properties of dense hard spheres in the vicinity of the Bernal limit. The spatial and bound order parameters and the abundances of hcp-, fcc-, and ico-like particles all indicate a sharp disorder−order phase transition at ϕc. Finally, we find that at packing fraction ϕ ≃ 0.68 an additional structural transition occurs: the 3D-like shape of solid-like crystalline clusters changes to a planar-like structure.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank F. Zamponi and P. Charbonneau for helpful discussions. This study was supported by NSF-CMMT and DOE Geosciences Division. B.A.K. was supported by the Russian Science Foundation, Project No. 14-12-01185. E
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dx.doi.org/10.1021/jp504537n | J. Phys. Chem. B XXXX, XXX, XXX−XXX
Structural Properties of Dense Hard Sphere Packings Boris A. Klumov,*,†,‡,⊥ Yuliang Jin,§ and Hernán A. Makse§ †
Joint Institute for High Temperatures RAS, Izhorskaya str. 13/2, Moscow, 125412, Russia Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow, 141700, Russia ⊥ Landau Institute for Theoretical Physics, Russian Academy of Sciences, Kosygina ul. 2, Moscow, 119334, Russia § Physics Department and Levich Institute, 140th Street and Convent Avenue, New York, New York 10031, United States ‡
ABSTRACT: We numerically study structural properties of mechanically stable packings of hard spheres (HS), in a wide range of packing fractions 0.53 ≤ ϕ ≤ 0.72. Detailed structural information is obtained from the analysis of orientational order parameters, which clearly reveals a disorder−order phase transition at the random close packing (RCP) density, ϕc ≃ 0.64. Above ϕc, the crystalline nuclei form 3D-like clusters, which upon further desification transform into alternating planar-like layers. We also find that particles with icosahedral symmetry survive only in a narrow density range in the vicinity of the RCP transition.
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INTRODUCTION As a fundamental model in condensed matter physics and material science, packings of hard spheres (HS) reproduce many essential structural properties of glassy and granular media.1−3 Structural changes of the HS system have been observed when a packing is densified above the density of random close packing (RCP4), ϕc ≃ 0.64 (see, e.g., refs 5 and 6). A new order parameter, which is based on the cumulative distribution of the rotational invariant w6, was proposed in ref 5 to identify these structural changes. Random arrangement is transferred to partial crystallization across RCP, and two lattice types, the face centered cubic (fcc) and the hexagonal close-packed (hcp), are observed in the crystalline clusters. However, more detailed information has not been obtained due to insufficient system size. In this study, we aim to explore more detailed structural changes of the transition at RCP in comparison with previous studies,5,6 based on larger scale simulations. We numerically generate a large set of HS packings composed of N = 64 × 103 monodispersed spheres located in the cubic box with periodic boundary conditions, using the modified Lubachevsky−Stillinger (LS) algorithm.7
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Figure 1. Radial distribution function g(r) of hard spheres (solid lines) and its cumulative function Z(r) (dashed lines) at different packing fractions ϕ. The values of ϕ, and the positions of peaks, are indicated on the plot.
additional peaks appear at (2)1/2, (8/3)1/2, and (11/3)1/2, which are typical distances in the fcc and hcp lattices. The observation indicates the onset of crystallization when the packings are densified above ϕc. More detailed structural information can be obtained from the well-known bond order parameter method,9 which is widely used to study the structural properties of condensed matter,9−11 hard spheres,12−19 Lennard-Jones systems,20−24 complex plasmas,25−34 colloidal and patchy systems,35−37 granular media,38 metallic glasses,39 repulsive shoulder systems,40 etc. Each particle
RESULTS
To examine the structural properties of jammed spheres, we first study their spatial correlations. Figure 1 shows the radial distribution function g(r) of hard spheres for several different packing fractions in the range [0.60, 0.68]. The cumulative function Z(r) ≡ 4πρ∫ r0 r′2g(r′) dr′, which is the mean number of particles inside a sphere of radius r, is also plotted in Figure 1. When ϕ ≤ ϕc, addition to the strong peaks at the contact distances r = 1, 2, ... (we set unit diameter), a weak secondary peak at r = (3)1/2 is visible. This peak corresponds to a contact angle θ = 2π/3 but not necessarily a lattice structure.8 Above ϕc, © XXXX American Chemical Society
Received: May 8, 2014 Revised: August 3, 2014
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Figure 2 shows how the PDFs vary at different l (l = 4, 6) and ϕ (covering both amorphous and solid-like states). Densification of
i is connected via vectors (bonds) with its Nnn(i) nearest neighbors (NN), and the rotational invariants (RIs) of rank l of second ql(i) and third wl(i) orders are calculated as ⎛ 4π ql(i) = ⎜⎜ ⎝ (2l + 1) wl(i) =
m=l
∑ m =−l
∑ m1, m2 , m3 m1 + m2 + m3 = 0
⎞1/2 |qlm(i)| ⎟ ⎠ 2⎟
(1)
⎡l l l ⎤ ⎢ m m m ⎥qlm1(i)qlm2(i)qlm3(i) ⎣ 1 2 3⎦
(2)
−1
∑Nj=1nn(i) Ylm(rij),
where qlm(i) = Nnn(i) Ylm are the spherical harmonics, and rij = ri − rj are vectors connecting centers of ⎡ ⎤ particles i and j. In eq 2, ⎣ lm1 lm2 lm3 ⎦ are the Wigner 3jsymbols, and the summations performed over all the indexes mi = −l, ..., l satisfying the condition m1 + m2 + m3 = 0. As shown in the pioneer paper,9 the bond order parameters ql and wl can be used as measures, to characterize the local orientational order and the phase state of considered systems. Because each lattice type has a unique set of bond order parameters, the method of RIs can also be used to identify lattice structures in mixed systems. The values of ql and wl for a few common lattice types (including the liquidlike HS state) are presented in Table 1. Some of the values were Table 1. Bond Order Parameters ql and wl (l = 4, 6) for Several Typical Lattices and Liquid-Like HS Random Packings (rp) at ϕ = 0.55, Calculated via the Fixed Number of Nearest Neighbors (NN) lattice type hcp (12 NN) fcc (12 NN) ico (12 NN) bcc (8 NN) bcc (14 NN) rp at ϕ = 0.55 (12 NN)
q4
q6
w4
w6
0.097 0.19 1.4 × 10−4 0.5 0.036 ≈0.16
0.485 0.575 0.663 0.628 0.51 ≈0.34
0.134 −0.159 −0.159 −0.159 0.159 ≈ −0.019
−0.012 −0.013 −0.169 0.013 0.013 ≈ −0.032
reported previously (see, e.g., refs 9 and 41−44). From eq 2, it is easy to see that wl ∝ ql3, and therefore, we expect wl’s to be more sensitive measures compared to ql’s. In three dimensions (3D), the densest possible packings of identical hard spheres are the fcc and hcp lattices (with the densest packing fraction (pf) ϕfcc = ϕhcp = (2)1/2π/6 ≃ 0.74). Dense HS systems may also include particles having the icosahedral (ico) type of symmetry. Icosahedral, fcc, and hcp spheres have 12 nearest neighbors (the spheres located in the first coordination shell). To identify lattice-like particles, we calculate the bond order parameters ql and wl (l = 4, 6) for each particle with a fixed number of the nearest neighbors Nnn = 12.45 A particle is called fcc-like (hcp-like, ico-like) if its coordinates in the fourdimensional space (q4, q6, w4, w6) are sufficiently close to those of the perfect fcc (hcp, ico) lattice type. An amorphous (liquidlike) particle is identified if its order parameters are sufficiently −1/2 fcc/hcp/ico small, for example, qliq . Note that, by 6 ≃ Nnn ≃ 0.29 ≪ q6 varying the NN number and rank l of parameters wl and ql, in principle, it is possible to find any lattice type. For instance, the first and second (next nearest neighbors) shells of body centered cubic (bcc) lattices correspond to Nnn = 8 and 14, respectively. To quantify the local orientational order, it is convenient to use the probability distribution functions (PDFs) P(ql) and P(wl).
Figure 2. Probability distribution functions (PDFs) of the rotational invariants qi (a and c) and wi (b and d) at various packing fractions ϕ. Values of the bond order parameters for ideal lattice types (fcc and hcp) are also indicated. Cumulative distributions (normalized to unity) of these PDFs are shown by dashed lines, which are much less noisy than PDFs. By using these distributions, it is easy to estimate the abundance of crystalline and liquid-like particles. Insets (from top to bottom) show spatial distributions of N = 64 × 103 HS in the cubic box at different packing fractions of ϕ ≃ 0.65, 0.66, and 0.68, respectively. Hard spheres are color-coded by their q6 values, in order to visualize liquid-like (blue color) and crystalline hcp-like (green color) and fcc-like (red color) structures.
the HS, as clearly seen in Figure 2, results in appearance of the crystalline fcc-like and hcp-like spheres in the vicinity of the RCP. Corresponding cumulative distributions Clq and Clw (shown by dashed lines of the same color) are also plotted in Figure 2. For instance, the cumulative function Clq associated with the P(ql) is defined as Cql(x) ≡ B
x
∫−∞ P(ql) dql
(3)
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Evidently, Clq(x) is the abundance of particles, having values of ql < x and Clq(∞) = 1. Figure 3 shows two-dimensional PDFs
Figure 4. Cumulant Wc6 (blue and cyan circles) and ⟨q6⟩ (green and orange circles) versus packing fraction ϕ. Blue and green circles correspond to the HS system with N = 104 spheres; cyan and orange circles correspond to N = 64 × 103 spheres. The minimum of Wc6 can be used to locate the RCP transition, ϕc ≃ 0.645. The explosive-like growth of Wc6 above ϕc is a clear indicator of the appearance of ordered, crystalline-like (hcp-like and/or fcc-like) spheres. The growth of the ⟨q6⟩ value near ϕc is rather monotonous and relatively slow. The possibility for the dense HS system to be packed via different abundance of fcc-like and hcp-like particles is the key reason behind the fluctuation of ⟨q6⟩ when ϕ ≳ 0.68. The inset shows the number of ico-like spheres versus ϕ: ico-like spheres survive only in the narrow range of parameters ϕ near the RCP transition.
freezing phase transitions of Lennard-Jones23,24 and Yukawa systems.32−34 On the other side, the density dependence of ⟨q6⟩ also reveals fascinating properties. The relatively slow increase of ⟨q6⟩ at ϕ ≲ ϕc can be attributed to single hcp-like particles,5 while at ϕ > ϕc the bond order parameter ⟨q6⟩ increases more rapidly due to the emergence of crystalline clusters. Further densification of the HS leads to strong fluctuations (“oscillations”) of the global parameter ⟨q6⟩, which start at ϕ ≃ 0.68 and suggest another transition.5,6 This oscillatory behavior is due to the fact that the hcp and fcc crystalline structures are characterized by quite different values of the bond order parameter q6 (see Table 1), and at a given ϕ, dense partially crystallized HS packings can be realized with quite different abundances of hcp-like and fcclike spheres. Another important parameter characterizing the local structure is the abundance Nico of spheres having icosahedral-type (5-fold) symmetry (see Figure 4, inset). Here, ico-like spheres are defined as spheres having q6 ≥ 0.61. The density dependence of Nico shows that ico-like spheres can exist only in a narrow range of pf, and its abundance maximizes at ϕc. The above analysis provides detailed information about the local and global orders. Next, we study the abundance of crystalline particles and their spatial arrangement. We use the standard friends-of-friends algorithm46 to find crystalline clusters. Figure 5 presents the spatial distribution and the shape of crystalline clusters composed of hcp- and fcc-like spheres, at several different ϕ. When ϕ < 0.65 (Figure 5a,b), the crystalline particles are mostly isolated; increase of packing fraction ϕ leads to an appearance of several 3D (containing tens of spheres, both hcp- and fcc-type) clusters (Figure 5c,d at ϕ ≈ 0.66). Upon further densification, these 3D clusters transform into 2D layers spanning system-wide (see Figure 5e,f at ϕ ≈ 0.68).5 The abundances of hcp-like (nhcp) and fcc-like (nfcc) spheres are shown in Figure 6 as functions of packing fraction ϕ. At ϕ ≤
Figure 3. Probability distributions on the order parameter plane of q4− q6 in the vicinity of the RCP at ϕ = 0.64 (a) and ϕ = 0.65 (b). Loci of the perfect fcc, hcp, and ico are also indicated on the plot. An fcc peak appears at ϕ = 0.65. Note that the hcp lattice may have a large overlap with the liquid-like structure in this plot.
probability distributions on the plane q4−q6in the vicinity of the RCP at ϕ ≃ 0.64 and ϕ ≃ 0.65. At ϕ ≃ 0.64, structurally HS is amorphous, while at ϕ ≃ 0.65 the formation of crystalline fcc-like spheres is clearly seen. The result suggests a disordered−ordered transition in the density range [0.64, 0.65], consistent with previous studies.5,6,16 The global order parameters, i.e., the cumulant Wc6 (defined as l Cw(Wcl ) = 1/2) and the mean value of q6 (defined from the global averaging ⟨q6⟩ = (1/N)∑Ni=1 q6(i)), are shown in Figure 4. No significant finite size effects are observed between N = 104 and N = 64 × 103 systems. For the purpose of detecting crystallization onset, we find Wc6 to be a more sensitive measure than ⟨q6⟩. This is because (1) liquid-like and lattice-like spheres have quite different values of w6 and (2) w6 is nearly the same for perfect fcc and hcp lattices (see Table 1). Because of this reason, the measure Wc6 has been recently used to describe the melting and C
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Figure 5. Distributions of hcp and fcc clusters over the space at different ϕ values: ϕ ≃ 0.65 (a, b), ϕ ≃ 0.66 (c, d), and ϕ ≃ 0.68 (e, f). Particles are colorcoded by the mass of the cluster (in the unit of the mass of a single sphere). At ϕ ≃ 0.65, single hcp-like spheres dominate. At ϕ ≃ 0.65, three-dimensional local clusters of both hcp- and fcc-like particles are observed. Further densification of HS results in a structural transition: the 3D local clusters transform into global and planar layers (e, f).
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ϕc, the relatively slow increase of ntot = nhcp + nfcc is mainly due to
DISCUSSION AND CONCLUSION The above results illustrate a sharp first-order-like transition6 at ϕc between disordered (frictional) packings and partially ordered packings. Amorphous frictionless packings only exist at a single density ϕc: below ϕc, the packings are unstable unless they are frictional (and the frictional packings are disordered);6 above ϕc, the packings are partially ordered. On the other hand, multiple amorphous frictionless states have been found in other simulations,47−50 which supports the mean-field theoretical prediction that amorphous jammed packings should exist in the interval (so-called J-line) ϕ ∈ [ϕth, ϕGCP].1 We stress that this apparent contrast is due to the presence of crystallization: If the nucleation rate is slow enough, such as in polydisperse47 or large
the emergence of randomly distributed single hcp-like spheres. Around ϕ ≃ ϕc, the slopes of nhcp(ϕ) and nfcc(ϕ) increase significantly. At ϕ ≃ 0.66, the abundance of the fcc-like spheres is equal to that of the hcp-like spheres (nfcc ≈ nhcp). Additionally, the abundance of spheres having number of contacts Nc = 12 is plotted in Figure 6, as one more sensitive indicator of the crystallization of a dense HS. The relative cumulative spectrum of cluster mass is plotted in the inset of Figure 6, showing that this distribution becomes wider as the density increases. D
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Figure 6. Abundance of hcp-like (red line points) and fcc-like (green line points) spheres versus the packing fraction ϕ, with N = 104. Results of larger scale simulations with N = 64 × 103 spheres are shown by squares (orange and green symbols correspond to hcp- and fcc-like spheres). Additionally, the abundance of spheres having number of contacts Nc = 12 is plotted by blue squares. The inset shows the relative cumulative spectrum of hcp (fcc)-like (red and green lines, respectively) clusters versus the cluster mass at different ϕ values: ϕ ≃ 0.68 (solid, triangles), ϕ ≃ 0.66 (dashed, gradients), and ϕ ≃ 0.65 (solid, circles).
dimensional systems,48 then crystallization is suppressed and dense amorphous packings can be obtained by slow compressions; while in 3D monodisperse systems, the nucleation process is so fast such that any slow compression would result in partial crystallization. Combined with the mean-field prediction, our results suggest two possible scenarios for 3D monodisperse packings: (i) ϕc < ϕth, therefore, dense amorphous packings ϕ > ϕth are hidden by crystallization; (ii) ϕth ≲ ϕc, however, the difference between ϕth and ϕc is very small, such that the amorphous packings in the range [ϕth, ϕc] are difficult to detect on the basis of the present numerical accuracy. Further studies are required to test these two scenarios. To conclude, we have studied structural properties of dense hard spheres in the vicinity of the Bernal limit. The spatial and bound order parameters and the abundances of hcp-, fcc-, and ico-like particles all indicate a sharp disorder−order phase transition at ϕc. Finally, we find that at packing fraction ϕ ≃ 0.68 an additional structural transition occurs: the 3D-like shape of solid-like crystalline clusters changes to a planar-like structure.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank F. Zamponi and P. Charbonneau for helpful discussions. This study was supported by NSF-CMMT and DOE Geosciences Division. B.A.K. was supported by the Russian Science Foundation, Project No. 14-12-01185. E
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