Contact network in nearly jammed disordered packings of hard-sphere chains

We present salient results of the analysis of the geometrical structure of a large fully equilibrated ensemble of nearly jammed packings of linear freely jointed chains of tangent hard spheres generated via extensive Monte Carlo simulations. In spite the expected differences due to chain connectivity, both the pair-correlation function and the contact network for chain packings are found to strongly resemble those in packings of monomeric hard spheres at the maximally random jammed (MRJ) state. A remarkable finding of the present work is the tendency of chains to form closed loops at the MRJ state as a consequence of chain collapse. Our simulations on disordered nearly jammed chain packings yield an average coordination number of 6, which fulfills the isostaticity condition and is in excellent agreement with the corresponding simulation [A. Donev, S. Torquato, and F. H. Stillinger, Phys. Rev. E 71, 011105 (2005)] and experimental [T. Aste, M. Saadatfar, and T. J. Senden, Phys. Rev. E 71, 061302 (2005)] findings for jammed packings of monatomic hard spheres. An exact correspondence between the statistical-mechanical ensembles of monomeric spheres and of hard-sphere chains offers insights regarding the structure and topology of the contact network of hard-sphere systems at the MRJ state.

Figure 1

(Color online) Evolution of the HCP $\left(s^{\text{HCP}}\right)$ and the FCC $\left(s^{\text{FCC}}\right)$ order parameters and of their sum $\left({\tau }_{\text{c}}=s^{\text{HCP}}+s^{\text{FCC}}\right)$ as functions of MC steps at packing densities $\phi =0.63$ (filled symbols) and $\phi ={\phi }^{\text{MRJ}}$ (open symbols). Since the CCE-based order parameters are highly discriminating, the obtained ${\tau }_{\text{c}}$ parameter can be interpreted as an accurate estimate for the total degree of crystallinity.

Figure 2

The ensembles ${\Xi }_{\text{chains}}^{\text{MRJ}}$ and ${\Xi }_{\text{ss}}^{\text{MRJ}}$ for hard-sphere chains and single (monatomic) hard spheres, respectively. Each member $\left({\mathbf{R}}_j,{\mathbf{C}}_k\right)∊{\Xi }_{\text{chains}}^{\text{MRJ}}$ is uniquely mappable on a member of ${\Xi }_{\text{ss}}^{\text{MRJ}}$ by simply deleting chain connectivity. ${\Xi }_{\text{chains}}^{\text{MRJ}}$ splits naturally in equivalent classes. Each class contains all chain configurations that differ in connectivity ${\mathbf{C}}_k$ but are mapped on the same ${\mathbf{R}}_j$.

Figure 3

(Color online) (a) Intermolecular pair radial distribution function $g_{\text{inter}}\left(r\right)$, (b) intramolecular pair density function $w_{\text{intra}}\left(r\right)$, and (c) total pair radial distribution function $g_{\text{total}}\left(r\right)$ as obtained from MC simulations on the $N=24$ hard-sphere chain system at the MRJ state. The plot is magnified so as to focus around the split-second peak. Also shown as guides for the eye are vertical lines to indicate the discontinuities at $r=\sqrt{3}\sigma$ and $r=2\sigma$. Noise in the distribution is due to the very narrow binning. In more standard plots, bin width is usually two orders of magnitude larger, resulting in very smooth $g\left(r\right)$, see for example [39, 40, 43, 44] .

Figure 4

(Color online) Inset: probability distribution for the supplement of bending angle $\theta$ from MC simulations for the $N=24$ system at volume fractions $\phi =0.63$ and $\phi ={\phi }^{\text{MRJ}}\simeq 0.639$. Bending angle is defined as the angle between two successive bonds along the chain. As in Fig. 3, noise in the distribution is due to the very narrow binning. Main: double-logarithmic plot of the probability distribution as a function of the divergence from the contact angle at ${\theta }_{\text{c}}=120°$ (since this value of the bending angle ensures contact for intramolecular neighbors that are separated by two bonds).

Figure 5

(Color online) Inset: probability distribution for the torsion angle $\varphi$ from MC simulations for the $N=24$ system at volume fractions $\phi =0.63$ and $\phi ={\phi }^{\text{MRJ}}\simeq 0.639$. Torsion angle is defined as the angle between the planes formed by successive triplets of spheres along the chain; all-trans conformation corresponds to 0°. As in Fig. 3, noise in the distribution is due to the very narrow binning. Main: double-logarithmic plot of the probability distribution as a function of the divergence from the characteristic torsion angle at ${\varphi }_{\text{c}}=109.5°$ for (top/right axis): $\varphi >{\varphi }_{\text{c}}$ and (bottom/left axis): $\varphi <{\varphi }_{\text{c}}$.

Figure 6

(Color online) Inset: end-to-end vector distribution $W\left(\mathbf{R}\right)$ for chains of lengths $N=6$, 12, and 18 at $\phi ={\phi }^{\text{MRJ}}\simeq 0.639$. Main: linear-logarithmic scale of $W\left(\mathbf{R}\right)$ versus $\mathbf{R}$ in the vicinity of the contact distance for chains of lengths $N=6$, 12, and 18 at $\phi =0.63$ and at $\phi ={\phi }^{\text{MRJ}}\simeq 0.639$.

Figure 7

(Color online) System configuration of a typical (randomly selected) frame from MC simulations on the $N=12$ system at the MRJ state. (a) coordinates of sphere centers are subjected to periodic boundary conditions (i.e., wrapped within the simulation cell) and sites of chains that form closed loops are colored differently, (b) coordinates of sphere centers are fully unwrapped in space and each site is colored according to its chain, and (c) only chains (unwrapped in space) that form closed loops are shown. In panel (d), the backbones of the same loop-forming chains are represented as a stereo pair. In this specific example 8 out of 100 polymers exhibit ringlike chain architecture $\left(\mathbf{R}=1\right)$. Images (a)–(c) are created with the VMD software [60, 61].

Figure 8

(Color online) Cumulative coordination number $Z$ as a function of the gap between spheres $\left(r-\sigma \right)$ in logarithmic-linear scale from MC simulations on the $N=24$ system at volume fractions $\phi =0.63$ and $\phi ={\phi }^{\text{MRJ}}\simeq 0.639$. Also shown by the horizontal dotted line is the theoretical expectation of $Z=2d$ from the isostatic condition where $d$ is the dimensionality of the system.

Figure 9

(Color online) System configurations at [left: (a) and (b)]: $\phi =0.63$ and [right: (c) and (d)]: $\phi ={\phi }^{\text{MRJ}}=0.639$. The ball&stick representation is adopted with reduced sphere radius for visual purposes. Sites are colored according to the identity of their parent chain (online version). Bonds are constructed according to [top: (a) and (c)] the “inherent coordination” as imposed by the double tangency condition as a result of chain connectivity and [bottom: (b) and (d)] the proximity condition independent for a pair being in the same or different chains. In the latter case (right panels) the threshold value for the identification of a bond is the value for which the plateau of $Z=6$ is obtained for the cumulative coordination at the MRJ state (see also Fig. 8). A bond that is affected by periodic boundary conditions appears as a dangling bond emanating from one site of the bonded pair. Image is created with the VMD software [60, 61].