Random Packing of Hard Spherocylinders - Journal of Chemical

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Random Packing of Hard Spherocylinders Claudia Ferreiro-Córdova* and Jeroen S. van Duijneveldt* School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, United Kingdom ABSTRACT: Random packings have been studied using computer simulations for some time, but there is no agreement on the best way to obtain them, especially for nonspherical particles. The present work focuses on random packing of hard spherocylinders. Starting from the mechanical contraction method (Williams and Philipse. Phys. Rev. E 2003, 67, 051301), a modification is introduced that makes it easier to obtain reproducible values, by simplifying the selection of some parameters. Furthermore, it is found that the final packings can be compressed further using Monte Carlo style particle moves. Random packings were generated for a wide range of aspect ratios, 0 ≤ L/D ≤ 40, and it was verified that the final packings do not show positional nor orientational ordering.

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INTRODUCTION A random packing is, in a broad sense, a compact configuration in which the particles that form the system are disordered. Random packings of macroscopic objects can be obtained using simple experiments, by pouring, shaking, and compressing a container where such particles are held. This has been done for different particle shapes, and it has been found that for each kind of particle a reproducible value (within a certain range) can be obtained.1−3 Because of this one might expect that colloidal suspensions and macroscopic particles of the same shape would give the same random packing volume fraction, but this does not always seem to happen. Sacanna et al. studied packing densities of ellipsoidal silica colloids for aspect ratios 1 ≤ α < 4.5 and compared these with simulation results.4 They report slightly higher packing fractions than the values reported in simulations and no long-range positional or orientational order. Buitenhuis and Philipse studied the sedimentation of colloidal rods with aspect ratios α > 10 via centrifugation, and in this work the formation of ordered sediments was reported.5 Another study also reported nematic order in sediments obtained using centrifugation of colloidal suspensions of rods with aspect ratios 3.6 ≤ α ≤ 8.6 These results seem to indicate that for relatively high aspect ratios compact random configurations are not easy to obtain. There have been several attempts at a theoretical account of random packing, in particular for packings of hard spheres. The reproducibility of random packing (within a certain range depending on the limitations of the method used) and a formal definition of random packing have been investigated. Truskett et al. introduced the concept ”maximally random jammed state” for spheres, defined as the configuration which maximizes disorder among all jammed hard-sphere arrangements and that have minimum values for a set of typical order parameters.7 Kamien and Liu explained the random close packing density of spheres as a special well-defined divergent end point of a set of metastable branches of the pressure.8 This concept could © 2014 American Chemical Society

explain the reproducibility of such a value obtained using a wide variety of methods, including experiments as well as simulations. Chaikin et al. pointed out that at high density the densest packed state is the one that is thermodynamically stable. So far for all known systems, the crystalline state is denser than the random packing.1 For hard particles, the entropy difference between the disordered and ordered phases, which is related to packing fractions, is the driving force to crystallization. A large difference between these two phases makes both states accessible and relatively easy to observe.1 For colloidal suspensions of spherical particles a random sediment can be obtained via centrifugation, but as mentioned above, this is not as easy for rod-like particles. This might be explained considering the mechanism of the isotropic−nematic phase transition. Whereas nucleation of crystals composed of spheres uniquely proceeds via a nucleation and growth mechanism, an alternative is present for formation of a nematic phase in rod-like particles. There are indications that, for sufficiently long rods and at a sufficiently high concentration, this may proceed via a spinodal decomposition-type mechanism. This was shown using both experiments9 as well as computer simulations.10 In the latter paper, Ni et al. studied phase behavior of supersaturated isotropic fluids of hard rods with aspect ratios L/σ = 2 and 3.4. For the short rods they found nucleation from a saturated isotropic fluid into crystal and smectic phases. However, for the longer rods they found that, after quenching the system, spinodal-like orientation fluctuations occur, and large nematic domains form. This may help to explain why fully disordered colloidal packings are difficult to realize in practice. Typically, a dense packing is generated using centrifugation. As particles are being Special Issue: Modeling and Simulation of Real Systems Received: February 1, 2014 Accepted: May 21, 2014 Published: June 5, 2014 3055

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packing fraction, yet without any evidence of ordering, and that these values are easily reproducible.

centrifuged down, an increase in concentration can take the system into the spinodal decomposition regime, and this can induce some degree of order in the final sediment. In experimental studies of rod packings it is straightforward to determine the volume fraction at close packing but considerably more challenging to obtain detailed structural characterization. This makes simulation an appealing tool to study random close packing, and it can be applied to different particle morphologies. Several methods have been developed for spherical and nonspherical particles. Such methods can be, in a broad sense, divided into two main categories: sequential generation models and collective rearrangement methods (CRM). In the sequential generation model particles are added in an iterative way to a simulation box until it reaches a maximum packing fraction. The CRM can be used starting with either a highly packed or a diluted system; in both cases, the initial configurations are randomly generated, and there is no correlation between the particles. In the high density initial configuration there are already overlaps which are removed (or this is attempted) in an iterative way, and after a number of trials the size of the particles is reduced; this is carried out until all overlaps are removed. For the low density configuration the system is compressed, and any overlaps this generates are removed after each compression by an iterative series of moves. One such CRM is the mechanical contraction method (MCM)11 which has been used to calculate packing fractions of spheres, spherocylinders, and mixtures of spherocylinders and spheres.11−15 Zhao et al. studied the random packing fraction of spherocylinders with aspect ratios 0 ≤ L/D ≤ 6 using a geometrical based relaxation algorithm, which starts with a highly dense initial configuration with overlaps present.16 The system is relaxed to reduce the overlaps in an iterative way until the overlap rate of spherocylinders is below a preset value (the paper does not specify quite how low this value is). Several studies have reported simulations of random packings of spherocylinders, most of them for short aspect ratios (L/D ≤ 10). There does not seem to be a consensus on the precise value of the random packing fractions for different aspect ratios. Nonetheless the behavior of the packing fraction as a function of the aspect ratio follows the same trend in all of these studies. It initially increases as the aspect ratio increases (0 < L/D < 0.5), and after that it decreases. Philipse showed that the packing fraction decreases in an asymptotic way proportional to the inverse of the aspect ratio η ≈ cD/L,2 where c is the average number of contacts per particle. Such an expression provides a good model for thin rods but is not valid for low aspect ratios. The first attempt in this work to simulate random packing was using an algorithm that compresses the simulation box and “shakes” the particles in a Monte Carlo (MC) simulation, without allowing any expansions of the simulation box (equivalent to MC at infinite pressure). Here it was found that the system does not always get arrested into a random configuration; instead of that, short-range order appears, which gives rise to local alignment of the spherocylinders. The next method studied here was the MCM, but it proved difficult to reproduce packings as dense as the ones reported previously.11,13,15,17 This was mainly because the method uses some parameters that need to be adjusted during the simulation and when this is done properly the system reaches high packing fractions. Here it was decided to implement this method in a slightly different way that we found easier to fine-tune. The aim is to achieve configurations with the highest possible high

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METHOD The method used in this work is based on the mechanical contraction method (MCM). The implementation was modified slightly as detailed below, and this new version is referred to as MCM2. The method uses an initial dilute system of N spherocylinders that is in an isotropic state. The volume of the particles is increased by modifying the length and diameter of the spherocylinders by a fixed percentage. This effectively increases the packing fraction of the system, and this is repeated until an overlap is found. Then all of the overlapping particles are moved away from each other, followed by an overlap calculation for the entire system. If more overlaps are found then the particles are moved again in an iterative way until there are no overlaps left. After this the system is compressed again, and the entire procedure is repeated until it is no longer possible to remove the overlaps. The displacements that are applied to the particles have an important impact on the final values reached and can arrest the system in a state that does not have the maximum packing fraction possible. We will first explain how we calculate the translational and rotational movements that need to be applied to the particles, and we will then explain the implementation in more detail. For spheres the distance between two particles r ij corresponds to the distance between the centers of mass of those particles. Then if two spheres that have the same diameter (d) are overlapping the amount of overlap between them is defined as δij = d − rij. If rij > d then the two spheres are not overlapping. A spherocylinder is a cylinder of length L and diameter D, capped with two hemispheres of diameter D at both ends, and its anisotropy is described by the ratio L/D. The distance between two spherocylinders i and j corresponds to the magnitude of the vector rcij that connects the points on their axes of symmetry such that the distance between both particles is a minimum (see Figure 1). Then, for two

Figure 1. Diagram of the quantities used to calculate the displacements that need to be applied to each spherocylinder.

overlapping spherocylinders the amount of overlap is defined as δij = D − rcij. Again, if rcij > D then there is no overlap. During the simulation, for each particle i that has C overlapping particles j, a new position can be calculated using the following equation: C

r′i = ri + γ ∑ δij n̂ ij j=1

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(1)

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where γ is a parameter that controls the rate at which the particles are moved, and n̂ij = rcij/rcij. Each particle i is moved in the opposite direction to the one corresponding to the vector that results of the sum of the vectors perpendicular to the plane of contact between i and j, with a magnitude equal to the amount of overlap, for all C overlaps. The displacement given by eq 1 includes an adjustable parameterwe found that γ = 0.95 allowed us to achieve the highest density packings. So far we have only addressed the translational displacement needed to break the overlaps, which is all that is required for packing spheres. For spherocylinders the particles have two extra degrees of freedom, that arise from their orientations. Particles need to be rotated, to prevent overlaps recurring upon displacement. The particle i is rotated upon itself around the vector wi that results from the sum of all the vectors normal to the plane formed by rλi and n̂ij for each overlapping particle j:

of the displacement, not its direction, and in the orientation of the axis around which the particles are rotated, as well as the magnitude of this rotation. Nevertheless, the scope of the sampling of rearrangements should have similar limitations to the ones corresponding to the MCM. Because of that, for particles with L/D ≥ 8, we implemented a series of MC moves after the MCM2. In our case, the final packing fraction could so be reduced by at most 5%. After increasing the volume fraction of the system using the MCM2 we were able to reproduce the values reported by Williams and Philipse. However, we noticed that, apart from spheres, these systems are not totally arrested and can be compressed further, but not via this method. So we applied a further MC-like compression where we only allow the system to compress (neglecting expansions) and the particles were randomly displaced and rotated. We also found that the same final packing fraction can be obtained even if the MCM2 has not reached its highest packing fraction possible.

C

wi =

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∑ δij(rλi × n̂ ij) j=1

(2)

SIMULATION DETAILS The simulation starts with an isotropic dilute configuration in which the initial packing fraction is in the range 0.04 < η < 0.10. For aspect ratios L/D > 20 the initial packing fraction is chosen to be η < 0.06, in order to avoid the isotropic−nematic transition. The initial configuration is generated randomly, and the system is equilibrated using isothermal−isobaric Monte Carlo (MC-NPT), to ensure a dilute configuration in an isotropic phase. The number of particles was chosen in such a way that in the maximum packing state the size of the box was at least twice the length of the spherocylinders. The packing fraction is related to the minimum number of particles in the following way:

Here we have defined rλi as the vector along the particle axis that connects the particle center of mass with the point of closest approach with particle j. This only provides a vector around which we need to rotate the particle; in addition the amount of rotation needs to be chosen. We decided, after some trials, that the best way was to allow the program for each particle to randomly select the magnitude of this rotation, given a maximum value that was initially established. As the density increased this maximum value was decreased until it reached a minimum value that was also selected. For all aspect ratios studied the particles were rotated by an angle in the range (π/ 2) × 10−3 < θ < π/8. This rotation is made in the direction moving away from the overlap. In our method the volume fraction is increased by increasing the size of the particles, but the particle’s center of mass remains in the same relative position to the box. The rate of volume change ΔV/V was chosen to start at 1 × 10−4, following the work by Williams and Philipse. After the change in volume all of the overlapping particles are simultaneously displaced, following the above procedure, in an iterative way until no overlaps are left. These movements have a threshold value of 1 × 103 iterations after which ΔV/V and θ are reduced by a factor of either 10 or 2 (for some low density configurations). Williams and Philipse also report a scaling down of ΔV/V similar to what is done here. In the original MCM11,13,15,17 a procedure similar to eq 2 is used to define wi, however with two key differences: first, the definition of rλi is slightly different; second, an additional weighting is introduced in the calculation of the sum of particle pairs, based on moments of inertia. This favors the rotation in certain directions. In that case the calculation of the new vector r′i is done in the same way, but the value of γ is different. The main difference in the implementation is that, once ri and wi are calculated, particle i is moved by half the distance of the smallest overlap δij. The MCM does not sufficiently sample rearrangements for long particles (L/D > 8). For such cases Williams and Philipse employed a series of MC moves where all the movement attempts are in a direction parallel to the particle’s axis of symmetry.11 After this series of movements the MCM was applied again, and the entire procedure was repeated until the system could no longer be reduced in volume. They reported that this can allow further reduction of the volume by at least a factor 2.11 Our version differs from the MCM in the magnitude

η≈

Nπ ⎛⎜ D ⎞⎟ 48 ⎝ L ⎠

2

(3)

For aspect ratios L/D ≤ 20 we chose to use 2000 particles. For L/D = 30 and 40 we used 2500 and 3200 particles, respectively. To reduce the time of the simulations, a cell list was used for boxes where the size of the sides is more than four times L + D: this condition was met for aspect ratios L/D ≤ 8. To identify whether the configurations reached were disordered, the nematic order parameter, the pair radial distribution g(r), and the orientational pair radial distribution function g2(r) were calculated. In our simulations S was calculated using the ordering matrix Q tensor that is defined in terms of the orientations of the molecular axes as Q=

1 N

N

⎛3 I ⎞⎟ ⎜ uu − i i 2 2⎠

∑⎝ i=1

(4)

where I is the unit tensor. The nematic order parameter corresponds to the largest eigenvalue that arises from the orthogonalization of Q. The pair radial distribution function is defined as g (r ) =

n(r ) ρVshell(r )

(5)

where n(r) is the number of particles in an spherical shell of width Δr at a distance r from the center of mass of i, ρ is the particle number density, and Vshell(r) is the volume of the shell. The orientational pair distribution function g2(r) can be defined in therms of the second Legendre polynomial as 3057

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Table 1. MCM2+MC Simulation Results for Hard Spherocylinders for Aspect Ratios 0 ≤ L/D ≤ 40a

(6)

where cos θij = ui·uj, ui is the direction of the main axis of particle i, and θij is the angle between particle i and j. The values of g(r), S, and g2(r) are obtained once the system reaches the maximum packing fraction possible. For each aspect ratio studied the contraction method was applied to 10 different initial configurations that meet the conditions explained before. Then those quantities were averaged, and uncertainties were obtained from the data distribution. In the MCM2, between 1 × 102 and 4 × 102 cycles of particle moves were carried out. For aspect ratios L/D ≥ 8, 5 × 104 cycles of MC moves were applied during the MCM2. After the MCM2, between 1 × 105 and 6 × 105 MC cycles were applied.

η

L/D 0.0 0.5 2.0 4.0 5.0 8.0 10.0 20.0 30.0 40.0

0.634 0.699 0.658 0.593 0.556 0.482 0.440 0.288 0.209 0.172

± ± ± ± ± ± ± ± ± ±

S 0.005 0.002 0.001 0.008 0.004 0.007 0.004 0.007 0.003 0.004

0.021 0.023 0.024 0.019 0.023 0.018 0.019 0.021 0.018

± ± ± ± ± ± ± ± ±

0.013 0.011 0.009 0.011 0.011 0.008 0.014 0.012 0.007

a The packing fraction η and the order parameter S shown are obtained from the average of 10 different configurations; the uncertainties correspond to one standard deviation.

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RESULTS AND DISCUSSION Random close packing configurations were obtained for spherocylinders with aspect ratios 0 ≤ L/D ≤ 40 using the

Figure 3. Order parameters S of the random packings obtained with MCM2 and MCM2+MC for 0 ≤ L/D ≤ 40.

Figure 2. Packing fractions η of spherocylinders as a function of the aspect ratio L/D (symbols) and phase boundaries of HSC (dotted lines).18 (a) Packing fractions obtained with the MCM2 and MCM2+MC compared with the results reported by Williams and Philipse. (b) Comparison between MCM2+MC and previous studies for L/D ≤ 10: Kyrylyuk and Philipse,17 Williams and Philipse,11 Wouterse et al.,15 and Zhao et al.16

Figure 4. Snapshots of the final configurations for different aspect ratios L/D: (a) 0.5, (b) 5, (c) 10, and (d) 30.

hybrid compression method composed of MCM2 followed by an MC-like compression. Figure 2a shows the packing fraction for different aspect ratios for the two stages of the method. Using only the MCM2 similar results to the ones reported by Williams and Philipse were obtained, after the MC-like compression (MCM2+MC) an increase in the final packing fractions can be observed. For both stages the values obtained follow the expected trend, where there is a peak when the

particle shape deviates slightly from a sphere and then it decreases as the aspect ratio increases. In Figure 2a we also compare the packing fractions of these random configurations with the phase diagram of hard spherocylinders for the same range of aspect ratios.18 As can be observed, for aspect ratios L/ D > 4, the random configuration is always in a region were a liquid crystal phase is thermodynamically stable. 3058

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Figure 5. Pair radial distribution functions g(r) (a) and orientational pair radial distribution functions g2(r) (b) for aspect ratios 0 ≤ L/D ≤ 5. Same functions (c) and (d) for aspect ratios 8 ≤ L/D ≤ 40.

fraction by up to 40%. We note that this combination of methods can be used even when the MCM2 has not reached its highest value possible. This makes it relatively easy to implement and reduces the amount of variations needed in the input values for the MCM2. The corresponding packing fractions and order parameters obtained after MCM2+MC are shown in Table 1. Figure 3 displays the order parameter of the system for each stage of the method, and as can be observed after the second compression the order parameter does not change significantly, and it always stays below 0.04. In Figure 4 snapshots of the final configurations, made using POV-Ray, are displayed for different aspect ratios and to the naked eye these also look random. The colors of the particles reflect the angle each particle makes with respect to the positive z-axis of the simulation box. The low values of the order parameter seem to indicate that the systems are disordered. The definition used here for the order parameter is the largest positive eigenvalue of the ordering matrix tensor Q, which for an isotropic system has a value that scales as 1/√N.19 This gives S ≈ 0.022 for N = 2000, S ≈ 0.02 for N = 2500, and S ≈ 0.017 for N = 3200, which agrees with Table 1. However, because the order parameter gives information about the overall orientation, it does not provide enough information to say if the system is in fact disordered or not. To get more information the pair radial distribution function and the orientational pair radial distribution function were calculated for the different aspect ratios (Figure 5). For the pair radial distribution function it can be observed that, as the aspect ratio increases, starting at L/D = 0 the peaks that are present for spheres start shifting slightly to

Figure 2b compares the results obtained with MCM2+MC with other studies for aspect ratios 0 ≤ L/D ≤ 10. While all series follow the same qualitative trend, there are significant quantitative differences between the various studies. Most of the data shown in Figure 2b was obtained using the MCM,11 and the fact that the various data sets deviate from each other illustrates the difficulties in implementing this approach reliably. Figure 2b also displays the data obtained by Zhao et al.16 They report the highest packing fractions, and their values are also higher than the values reported in the present work. The authors show various distributions to demonstrate their configurations are indeed random; in terms of orientational order, in addition to an overall order parameter S, they consider the two angles formed by the orientation vector of the particles with the x−y and x−z planes. They define an orientational frequency ratio as the number of angles in a specific section of orientations with the average number of angles in one section. Such ratios have values between 0.8 and 1.2, which they argue is a signal that the orientations are randomly distributed. While this does demonstrate a lack of preferred overall orientations, perhaps short-range orientational correlations may be present that do not show up in this way. The final packing fractions obtained here are overall higher than the other results shown in Figure 2b (apart from the results of Zhao et al.), and this is more evident for high aspect ratios. For spheres (L/D = 0) the extra compression applied after the MCM2 has no effect because the system is already arrested, and for aspect ratios L/D ≤ 2 the effect of this compression is lower than for higher aspect ratios. For higher aspect ratios this further compression increases the packing 3059

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the right and decrease, until for aspect ratios L/D ≥ 4 a clear peak does not appear to form (Figure 5a). Also, for short rods in Figure 5b it can be observed that the orientational pair radial distribution function presents a peculiarity for L/D ≤ 2, where the function gives negative values at short distances. For those distance values, the particles i and j tend to have orientations such that the angle between them range from π/4 to π/3. In overall g2(r) tends to zero quite rapidly, which confirms that there is no orientational order. For high aspect ratios g2(r) also decreases rapidly, and as was mentioned g(r) does not show a peak. To rule out the possibility of an isolated nematic droplet nevertheless being present, we also used the nematic cluster criterion of Cuetos and Dijkstra.20 For L/D = 4 this identified 0.85% of particles as nematic. For larger aspect ratios, no particles were classified as nematic at all. Finally we return to the results reported by Zhao et al.16 They report noticeably higher packing fractions than any other study, for all aspect ratios considered. By constructions their simulations only gradually reduce overlaps, and some may remain in the final state (no values are cited). The authors argue that all measurements indicate that their configurations are random. However, upon inspection of the snapshots included in their paper, several small clusters of aligned particles seem to be present. These may not show up using the measurements presented in that study; the orientational correlation function g2(r) should be a useful measure to compare with the present work, in order to resolve this matter.

ACKNOWLEDGMENTS This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol (http://www.bris.ac.uk/acrc/).

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REFERENCES

(1) Chaikin, P. M.; Donev, A.; Man, W.; Stillinger, F. H.; Torquato, S. Some Observations on the Random Packing of Hard Ellipsoids. Ind. Eng. Chem. Res. 2006, 45, 6960−6965. (2) Philipse, A. P. The Random Contact Equation and Its Implications for (Colloidal) Rods in Packings, Suspensions, and Anisotropic Powders. Langmuir 1996, 12, 1127−1133. (3) Donev, A.; Cisse, I.; Sachs, D.; Variano, E. A.; Stillinger, F. H.; Connelly, R.; Torquato, S.; Chaikin, P. Improving the Density of Jammed Disordered Packings Using Ellipsoids. Science 2004, 303, 990−993. (4) Sacanna, S.; Rossi, L.; Wouterse, A.; Philipse, A. Observation of a shape-dependent density maximum in random packings and glasses of colloidal silica ellipsoids. J. Phys.: Condens. Matter 2007, 19, 376108. (5) Buitenhuis, J.; Philipse, A. P. Orientational Order in Sediments of Colloidal Rods. J. Colloid Interface Sci. 1995, 176, 272−276. (6) Mukhija, D.; Solomon, M. J. Nematic order in suspensions of colloidal rods by application of a centrifugal field. Soft Matter 2011, 7, 540−545. (7) Truskett, T. M.; Torquato, S.; Debenedetti, P. G. Towards a quantification of disorder in materials: Distinguishing equilibrium and glassy sphere packings. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2000, 62, 993−1001. (8) Kamien, R. D.; Liu, A. J. Why is Random Close Packing Reproducible? Phys. Rev. Lett. 2007, 99, 155501. (9) Van Bruggen, M.; Lekkerkerker, H. Tunable Attractions Directing Nonequilibrium States in Dispersions of Hard Rods. Macromolecules 2000, 33, 5532−5535. (10) Ni, R.; Belli, S.; van Roij, R.; Dijkstra, M. Glassy Dynamics, Spinodal Fluctuations, and the Kinetic Limit of Nucleation in Suspensions of Colloidal Hard Rods. Phys. Rev. Lett. 2010, 105, 088302. (11) Williams, S. R.; Philipse, A. Random packings of spheres and spherocylinders simulated by mechanical contraction. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2003, 67, 051301. (12) Kyrylyuk, A. V.; Wouterse, A.; Philipse, A. P. Percolation and Jamming in Random Heterogeneous Materials with Competing Length Scales. Prog. Colloid Polym. Sci. 2010, 137, 29−33. (13) Kyrylyuk, A. V.; Wouterse, A.; Philipse, A. P. Random packings of rod-sphere mixtures simulated by mechanical contraction. AIP Conf. Proc. 2009, 1145, 211−214. (14) Kristiansen, K. D. L.; Wouterse, A.; Philipse, A. Simulation of random packing of binary sphere mixtures by mechanical contraction. Phys. A 2005, 358, 249−262. (15) Wouterse, A.; Williams, S. R.; Philipse, A. P. Effect of particle shape on the density and microstructure of random packings. J. Phys.: Condens. Matter 2007, 19, 406215. (16) Zhao, J.; Li, S.; Zou, R.; Yu, A. Dense random packings of spherocylinders. Soft Matter 2012, 8, 1003−1009. (17) Kyrylyuk, A. V.; Philipse, A. P. Effect of particle shape on the random packing density of amorphous solids. Phys. Status Solidi A 2011, 208, 2299−2302. (18) Bolhuis, P.; Frenkel, D. Tracing the phase boundaries of hard spherocylinders. J. Chem. Phys. 1997, 106, 666−687. (19) Eppenga, R.; Frenkel, D. Monte Carlo study of the isotropic and nematic phases of infinitely thin hard platelets. Mol. Phys. 1984, 52, 1303−1334. (20) Cuetos, A.; Dijkstra, M. Kinetic Pathways for the IsotropicNematic Phase Transition in a System of Colloidal Hard Rods: A Simulation Study. Phys. Rev. Lett. 2007, 98, 095701.

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CONCLUSIONS With the method detailed in this work a highly packed configuration for spheres was obtained with a volume fraction of 0.634 that agrees well with what has been reported before and a pair distribution function g(r) that looks like a dense liquid and indicates a disordered configuration. For spherocylinders, dense packed configurations were obtained with order parameters that agree with a disordered system. For aspect ratios L/D > 2, g(r) does not give information about correlation between the particles but g2(r) can tell if there is any degree of order, and for the aspect ratios studied it does not indicates the presence of any structure. In overall the systems obtained are highly packed and seem to be isotropic. The fact that g2(r) tends to zero suggests that good random packings are obtained. In our opinion this combination of methods gives a convenient and reproducible route to random close packing of rod-like particles. For aspect ratios up to L/D = 5 the final packing fractions agree well with those reported by Kyrylyuk and Philipse;17 however for longer rods, higher packing densities are reported here.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Funding

This work was funded by the Mexican Secretariat of Public Education, the Mexican government, and a Douglas Everett bursary. Notes

The authors declare no competing financial interest. 3060

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