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We review recent results from extensive simulations of the crystallization of athermal polymer packings. It is shown that above a certain packing density, and for sufficiently long simulations, all random assemblies of freely-jointed chains of tangent hard spheres of uniform size show a spontaneous transition into a crystalline phase. These polymer crystals adopt predominantly random hexagonal close packed morphologies. An analysis of the local environment around monomers based on the shape and size of the Voronoi polyhedra clearly shows that Voronoi cells become more spherical and more symmetric as the system transits to the ordered state. The change in the local environment leads to an increase in the monomer translational contribution to the entropy of the system, which acts as the driving force for the phase transition. A comparison of the crystallization of hard-sphere polymers and monomers highlights similarities and differences resulting from the constraints imposed by chain connectivity.

Crystallization and phase transitions in general play a key role in many processes related, among others, to material engineering, physics, chemistry and biology. Advances in crystallography, mainly through X-ray diffraction measurements, have provided significant information on crystal structures. However, how such crystals nucleate and grow and how processing history further affects the corresponding ordered morphologies remain open topics of intense scientific debate. While experimental, theoretical and modeling advances constantly enrich our fundamental understanding of the phenomenon in a wide range of physical systems [

Deep in the heart of numerical simulations lays the molecular model, which determines the level of detail and the corresponding approximations with respect to the way atoms and molecules are represented. Atomistic models incorporate highly detailed force fields to describe interactions between atoms, while coarse-grained ones sacrifice detailed information in favor of computational efficiency. In one of the simplest possible representations, atoms, either as monomeric entities or as part of molecular species, are treated as non-overlapping hard spheres. The hard-sphere model is obviously void of any kind of chemical information. However, because of its simplicity it stands as an invaluable simulation tool: It is accessible to analytical approaches, requires minimal computational resources and time, and can thus be employed under conditions which remain inaccessible to more detailed molecular models. Furthermore, the hard-sphere model allows us to discriminate and accurately identify the different governing factors (for example density or entropy) that affect various phenomena and physical processes. Ideally the knowledge gleaned from such simplified models could shed light onto the fundamental role of analogous mechanisms in much more complex physical and biological applications. Thus, it is not surprising that during the last decades an ever-growing body of simulations has successfully employed the hard-sphere model in studies of systems that range from colloids, microgels and granular materials to synthetic and biological polymers.

The study of how objects of different shapes and sizes arrange in a multidimensional space and of the corresponding packing morphologies has been in the spotlight of research since early historical times. During the last decades pioneering scientific contributions have been achieved in general packing with modeling studies having greatly benefited by the continuous advances in computer hardware and software.

Almost four centuries ago, Kepler conjectured that in three dimensional space the densest hard-sphere packing is that of a face centered cubic (fcc) lattice, a long-standing geometrical problem that has been addressed only recently in a series of papers by Hales and coworkers [

Regarding phase transition in athermal packings, Onsager was the first to predict an anisotropic-nematic transition in hard rods as a result of the increase in entropy caused by ordering [_{F} = 0.494 and ϕ_{M} = 0.545, respectively [

Random hard-sphere packings are further characterized by short-range order in the form of polytetrahedral structures with fivefold symmetry [

Recent advances in experimental and simulation techniques have contributed to the detailed analysis and characterization of the phase behavior and self-assembly in packings of objects with highly complex shapes [

In the present manuscript we review our latest results from extensive Monte Carlo (MC) simulations on the crystallization in dense packings of freely-jointed chains of tangent hard spheres of uniform size [

Athermal polymer packings consist of freely jointed, linear chains of tangent hard spheres. Monomers are treated as non-overlapping spheres of diameter σ. Tangency implies that the bond length ^{−4}] [^{−8}] showed no difference in the crystal growth and nucleation as well as in the self-assembly of the ordered morphologies. The freely-jointed model allows for full flexibility in the conformations as there are no constraints in bond bending and torsion (dihedral) angles. However, it has been shown that, due to strong excluded-volume interactions, bond bending angles and torsion angles tend to adopt specific geometric arrangements, which become increasingly more favorable as packing density increases [

The employed Monte Carlo scheme consisted of the following mix of moves: (i) reptation (10%); (ii) end-mer rotation (10%); (iii) configurational bias (20%); (iv) inter-chain reptation (25%); (v) internal libration (34.98%); (vi) simplified end-bridging (sEB, 0.1%) and (vii) simplified intramolecular end-bridging (sIEB, 0.1%), where the percentages in parenthesis denote the attempt probabilities of each move. All local moves (i–v) are executed in a configurational bias pattern [_{at}_{ch}_{at} is the total number of spheres, _{ch} is the number of chains, _{av} (1 − Δ), _{av} (1 + Δ)], where _{av} is the average chain length and Δ is the reduced half width of the distribution divided by _{av}, and a most probable (Flory) one with the shortest allowed chain length set at _{min}.

All simulations were executed in cubic cells with periodic boundary conditions applied in all dimensions. Three different polymer systems were modeled, each one containing a total of 1200 hard spheres: (i) _{av} = 12, Δ = 0.5; (ii) _{av} = 12, _{min} = 3 and (iii) _{av} = 24, Δ = 0.5. Additional simulations conducted with simulation cells of 3000 sites to investigate the effect of system size on crystallization and on the formation of ordered morphologies showed no appreciable qualitative and quantitative differences. Initial configurations were generated at very low volume fractions using fully equilibrated, atomistic polyethylene structures [_{av} = 12 system were further generated at selected packing densities by splitting all chains of a _{av} = 24 configuration in half to guarantee that the structural characteristics of the initial random packings and the phase transition were not affected by the generation protocol of the modeling procedure. In production simulations system snapshots and ensemble statistics were recorded every 2 × 10^{5} MC steps, while the total simulation time exceeded 1 × 10^{10} steps at the higher densities. Due to very long runs required to observe crystallization at volume fractions in the vicinity of the MRJ state, modeling studies were necessarily limited to packing densities of ϕ = 0.56, 0.58, 0.60 and 0.61 above the melting point. More details on the algorithm and the procedure to generate and equilibrate random packings of athermal polymer chains can be found in [

For comparison purposes parallel sets of simulations for analogous monomeric systems were carried out by event-driven Molecular Dynamics (edMD). The edMD algorithm used was a minor modification of the conventional edMD technique, which proceeds on a simple collision-by-collision basis until a preset number of collisions is reached [

Once a large number of system configurations (frames) is collected, the analysis proceeds by a detailed characterization of the local environment around each site. An accurate and highly discriminating descriptor is required to quantitatively describe the degree of randomness as well as the appearance and propagation of ordered nuclei corresponding to specific crystal structures. Existing descriptors of local structure include the widely used pair radial distribution function,

Recently, we qualitatively and quantitatively analyzed the local structure of athermal packings through a novel scheme that consists of two main steps: (i) Identification of the local environment around each sphere through a Voronoi tessellation and by measuring the shape and size of the corresponding Voronoi cell, and (ii) application of a novel structural descriptor based on the concept of the characteristic crystallographic element (CCE), as used in crystallography [

In structural characterization via Voronoi tessellation the set of neighbors closer to a reference site than to any other sites is identified. This task was performed with the

with _{ver} being the number of vertices of the polyhedron, _{i}_{1}, _{2,}_{3}). Once the Voronoi tessellation is completed, the internal principal axes system is determined for each Voronoi cell from the coordinates of its vertices. The three real eigenvalues of the intertia tensor _{1}, _{2} and _{3} (_{1} ≥ _{2} ≥ _{3}) correspond to the principal moments of inertia. The inertia tensor provides useful information on the shape and size of the Voronoi cell and accordingly on the local environment around each hard sphere. Based on the eigenvalues, a coarse-grained ellipsoid can be constructed with the lengths of the semiaxes being calculated as:

with semiaxis lengths _{2} and _{3} being calculated in an analogous fashion as in

asphericity:

acylindricity:

and relative shape anisotropy:

These measures are defined so that the lower the values of ^{2} the closer the resemblance to spherical, cylindrical and isotropic shapes, respectively.

Once a system configuration was recorded in the course of MC or edMD simulations, a Voronoi tessellation was performed to identify the characteristics of the corresponding polyhedra including a shape analysis based on asphericity, acylindicity and relative shape anisotropy. These global shape measures for each individual Voronoi cell can be directly compared with the analogous measures for the trapezo-rhombic dodecahedron and the rhombic dodecahedron, which are the characteristic Voronoi polyhedra arising from tessellation of hcp and fcc lattices, respectively. Thus, a systematic analysis of the shape measures of the Voronoi cells at each instance provide a reliable estimate of the current state of the athermal packing as well as of possible phase transition (crystallization). In addition, a change in the local environment around each sphere, quantified by the Voronoi global shape measures, can be directly related to changes in translational entropy, quantified in turn by sphere mobility.

The second descriptor of local structure, the characteristic crystallographic element (CCE) norm for a given configuration of point-like atoms around a reference atom _{j}^{X}_{j}^{X}_{j}^{X}_{j}^{X}_{j}^{Y}

Once the minimum CCE norm is calculated for each site in system, an order parameter with respect to perfect order

where ^{X}^{X}^{thres} is a threshold value below which a site is considered to possess ^{thres} = 0.245 of is adequately small to discriminate between different crystal types but also large enough to correctly identify the disorder-order transition in initially random packings and the emergence of specific crystal morphologies.

The hcp and fcc crystals are the two competing structures that arise when dense hard-sphere packings crystallize. Thus, the CCE norms (ɛ^{hcp}, ɛ^{fcc} ) and the corresponding order parameters (^{hcp}, ^{fcc}) for each were calculated with respect to these ordered structures. As the CCE-based analysis is highly discriminating between different crystal lattices, the degree of ordering τ^{c} can be estimated as the total number of sites with either hcp or fcc structural similarity (τ^{c} = ^{hcp} + ^{fcc}). Additional measurements were conducted to detect sites with fivefold local symmetry, a structural motif which is favored at high packing densities and constitutes an alternative local arrangement to hcp and fcc crystals [

The analysis of local structure based on the concepts of the Voronoi cells and of the CCE-based norm was performed at equally spaced frames of the long MC (or edMD) trajectories over all computer-generated samples and at all packing densities. ^{thres} = 0.245) below which a site is assigned to one of the reference structures (hcp, fcc or fivefold). According to ^{X}

According to the data reported in

Information obtained from the CCE-norm distribution also allows the calculation of crystallinity (degree of ordering),τ^{c}, as a function of steps (or number of collisions) from MC and edMD simulations on athermal chains and on hard-sphere monomers, respectively. Panels (a) and (b) in _{M} = 0.545), hard sphere monomers of uniform size show a clear disorder-order transition while the corresponding athermal chains remain in the original amorphous state throughout the simulation. Evidently, at this packing density, chain connectivity suppresses crystallization. However, as concentration increases, polymer packings spontaneously evolve into a stable crystal phase. This trend is clearly shown in ^{c} = 0.05) for both chains and monomers as expected for random (amorphous) packings. However, as MC (MD) simulations evolve, a sharp ordering transition occurs as crystallinity adopts high values (τ^{c} = 0.83), which are very similar for chain and monomeric packings. In the final stable crystal phase, the majority of sites adopt a highly ordered structure of either hcp or fcc character.

From the simulation results presented in Section 3.1 it is evident that once a critical volume fraction is reached, which lies at higher concentrations compared to monomers, athermal packings of freely-jointed chains of hard spheres transit from the initial amorphous (random) to the final crystal (ordered) phase. In the present section, we study in detail the structural features of the characteristic ordered morphologies that arise during athermal polymer crystallization.

The loss of positional and radial randomness and the formation of well-ordered morphologies during crystallization are depicted more vividly once we adopt a visualization scheme based on the information obtained by the CCE analysis. In

Analogous snapshots of the final crystal phases as obtained from MC simulations on the

According to the results presented here, crystal morphologies of dense assemblies of athermal polymers correspond to randomly stacked hexagonal close packings (rhcp), and no ordered structures were found of exclusive fcc (or hcp) character. As mentioned in the introduction, this trend is in accordance with the Ostwald rule of stages [

In the introduction we proposed a second method to identify crystallization by analyzing the changes in the local environment through a systematic study of the shape and size of the Voronoi cell around each sphere site. This approach is presented through a series of illustrations starting in

An estimate of the local density around each sphere site can be obtained as the reciprocal of the volume of the enclosing Voronoi polyhedron. Since the volume of the simulation cell remains constant during the simulation and the Voronoi tessellation is a space-filling geometrical procedure, the average local density of the system does not change during the whole simulation time and consequently during the phase transition. However, significant qualitative and quantitative changes occur in the shape of the Voronoi cells as the chain assembly crystallizes spontaneously and ordered morphologies are formed.

Visual comparison of the shapes of the Voronoi cells depicted in

All shape measures of deviation from isotropy, averaged over all Voronoi cells for each system configuration, decrease monotonically as the MC simulation advances. According to the data shown in ^{10} MC steps. It is exactly the same regime where the values of asphericity, acylindricity and relative shape anisotropy of the Voronoi polyhedra show a precipitous decline. In the final stable crystal phase all values of shape measures are significantly lower than the initial ones of the random phase. Based on the above, it can be safely concluded that during crystallization of athermal polymer packings, on average, the local environment around each sphere site becomes more symmetric and more spherical. Thus, a detailed geometrical analysis of the Voronoi polyhedra can shed light on the structural changes that occur during the phase transition. Such a methodological approach could be complementary to more refined structural descriptors like the CCE-based norm. In addition, such shape transformations of the local environment around each sphere site can be directly connected with local dynamics and consequently with translational entropy.

The present analysis based on the Voronoi polyhedra can further serve as the basis for a descriptor, which would potentially identify shape similarities with respect to specific Voronoi cells of reference crystal structures.

In isolated athermal systems, a phase transition can only be driven by an increase in entropy. Accordingly, in athermal packings of chain molecules, entropy is the driving force for crystal nucleation and growth. The conformational contribution entropy is actually reduced as a result of sphere arrangements adopting specific conformations both in bonded and non-bonded terms. This trend is easily identified by comparing the pair radial distribution function,

Characteristic peaks appear in the ordered phase, especially near contact. At the same time, long-range correlations features in the pair distribution unambiguously point towards the emergence of the ordered phase. Even if a more refined measure of pair correlation would be required to capture the anisotropic features of layered crystals,

The two previous sources of entropy loss must be more than compensated for by an independent mechanism of entropy gain. In Section 3.3 we have reported that the local environment around each hard sphere becomes more isotropic as the crystal phase appears. In order to establish a connection between shape transformation of local structure and entropy increase, we first studied local sphere dynamics. In this direction, and given that MC simulation provide no dynamical information, we resort to the concept of “flipper” originally employed to identify the jamming transition in polymer packings [^{10} MC steps, a regime which marks the disorder-order transition. Thus, as the local environment becomes more spherical and more symmetric monomers are able to explore more efficiently the free volume that surrounds them. Consequently, the translational entropy of the athermal polymer system increases during crystallization. This increase in translational entropy is large enough to compensate for the losses in conformational and orientational entropy, and is thus responsible for crystallization.

In order to better understand the strong correlation between the shape transformation of the local environment and the increase in translational entropy during crystallization,

We have reviewed recent studies on the phase transition and self-assembly of crystal morphologies from extensive simulations of packings of freely-jointed chains of tangent hard spheres of uniform size. The key finding is that once a critical packing density is reached, athermal polymer chains crystallize, just as monomers do, in spite of the additional constraint set by chain connectivity. However, at volume fractions very close to the melting point, chain connectivity does indeed frustrate crystallization; for the systems studied at ϕ = 0.61, polymer packings remain amorphous while monomeric analogs show a clear phase transition. The exact origins of the frustration along with the extent of the effect of connectivity on crystallization are under investigation.

Above a critical packing density, which is higher than for monomeric systems, and given sufficient simulation time, polymer configurations self-assemble into well characterized ordered morphologies of predominately rhcp character. Such crystals consist of stack-faulted alternating layers of hcp or fcc type with a single stacking direction and never show twinning. For all chain systems studied so far no transition to a pure fcc (or hcp) crystal was observed during the allowed simulation time. A detailed comparison between the crystal morphologies of polymers and of monomeric hard spheres is currently in progress.

We have also described in detail two new descriptors of local structure, the characteristic crystallographic element norm and a geometric analysis based on the Voronoi cells. It is established that during crystallization the local environment around each site becomes more spherical and more symmetric. In turn, this shape transformation allows the sphere sites more freedom to move locally. Thus, the entropy of the system increases and it is the driving force for the crystallization of chain packings.

Present efforts include the modeling study of phase transition in athermal polymer packings under varied conditions of confinement, mainly through the presence of a hard wall. The proposed simulation approach is further generalized to treat polymer packings with a finite degree of chain stiffness.

We are grateful to Martin Kröger (ETH, Switzerland), Cameron Abrams (Drexel University, USA), Juan J. de Pablo and Rohit Malshe (University of Wisconsin, USA) and Jun Shen and Chen Wu (Harbin Institute of Technology, China) for stimulating discussions. NCK acknowledges support by the Spanish Ministry of Economy and Competitiveness (MINECO) through projects “Ramon y Cajal” (RYC-2009-05413), “I3” and MAT2010-15482. KF acknowledges support by MINECO through projects “Ramon y Cajal” (RYC-2010-06804) and MAT2011-24834. Authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Centro de Supercomputacion y Visualizacion de Madrid (CeSViMa).

The authors declare no conflict of interest.

Characteristic crystallographic element (CCE)-based distribution with respect to hexagonal close packing (hcp), face center cubic (fcc) and fivefold symmetries as obtained from Monte Carlo (MC) simulations on the ^{thres} = 0.245).

Same as in

Crystallinity, τ^{c}, as a function of MC steps and Molecular Dynamics (MD) collisions as obtained from simulations of freely-jointed chains of tangent hard spheres. (_{av} = 12, uniform chain length distribution) and on monomeric hard spheres, respectively through application of the CCE norm, at (^{10} and 19 × 10^{10}, respectively, of the corresponding MC trajectory by deleting all existing bonds, (^{9} of the corresponding MC trajectory.

Crystallinity (degree of ordering), τ^{c}, as a function of packing density, ϕ, for freely-jointed chains of tangent hard spheres (_{F}) and melting points (ϕ_{M}), respectively, for hard sphere monomers of uniform size.

System configurations of the hard-sphere chain packing (

Same as in

System configurations corresponding to the final crystal morphologies as obtained from MC simulations on

Illustration of the procedure adopted for the construction of the Voronoi polyhedron around a reference site. In the specific example a sphere site is randomly selected from a system configuration obtained from MC simulations on

System visualizations showing: (^{hcp} = 0.6730, ɛ^{fcc} = 0.7207). Global shape measures for the Voronoi cell: asphericity ^{2} = 0.0472.

Same as in ^{hcp} = 0.0675). Global shape measures for the Voronoi cell: asphericity ^{2} = 0.0146.

Same as in ^{fcc} = 0.0845). Global shape measures for the Voronoi cell: asphericity ^{2} = 0.00508.

Average asphericity, ^{2}, as a function of MC steps from simulations of freely-jointed chains of tangent hard spheres (

Total pair radial distribution function,

Fraction of sites (flippers), which can perform a flip-like move clockwise and counter-clockwise of amplitudes dφ = 0.01, 0.10 and 1.00° as a function of MC steps from simulations on the

Asphericity, ^{c}, and fraction of flippers (dφ = 1.00°) as a function of MC steps from simulations on the