Simple model for chain packing and crystallization of soft colloidal polymers

Local and Global Order in Dense Packings of Semi-Flexible Polymers of Hard Spheres

The local and global order in dense packings of linear, semi-flexible polymers of tangent hard spheres are studied by employing extensive Monte Carlo simulations at increasing volume fractions. The chain stiffness is controlled by a tunable harmonic potential for the bending angle, whose intensity dictates the rigidity of the polymer backbone as a function of the bending constant and equilibrium angle. The studied angles range between acute and obtuse ones, reaching the limit of rod-like polymers. We analyze how the packing density and chain stiffness affect the chains' ability to self-organize at the local and global levels. The former corresponds to crystallinity, as quantified by the Characteristic Crystallographic Element (CCE) norm descriptor, while the latter is computed through the scalar orientational order parameter. In all cases, we identify the critical volume fraction for the phase transition and gauge the established crystal morphologies, developing a complete phase diagram as a function of packing density and equilibrium bending angle. A plethora of structures are obtained, ranging between random hexagonal closed packed morphologies of mixed character and almost perfect face centered cubic (FCC) and hexagonal close-packed (HCP) crystals at the level of monomers, and nematic mesophases, with prolate and oblate mesogens at the level of chains. For rod-like chains, a delay is observed between the establishment of the long-range nematic order and crystallization as a function of the packing density, while for right-angle chains, both transitions are synchronized. A comparison is also provided against the analogous packings of monomeric and fully flexible chains of hard spheres.

Polymorphism and Perfection in Crystallization of Hard Sphere Polymers

We present results on polymorphism and perfection, as observed in the spontaneous crystallization of freely jointed polymers of hard spheres, obtained in an unprecedentedly long Monte Carlo (MC) simulation on a system of 54 chains of 1000 monomers. Starting from a purely amorphous configuration, after an initial dominance of the hexagonal closed packed (HCP) polymorph and a transitory random hexagonal close packed (rHCP) morphology, the system crystallizes in a final, stable, face centered cubic (FCC) crystal of very high perfection. An analysis of chain conformational characteristics, of the spatial distribution of monomers and of the volume accessible to them shows that the phase transition is caused by an increase in translational entropy that is larger than the loss of conformational entropy of the chains in the crystal, compared to the amorphous state. In spite of the significant local re-arrangements, as reflected in the bending and torsion angle distributions, the average chain size remains unaltered during crystallization. Polymers in the crystal adopt ideal random walk statistics as their great length renders local conformational details, imposed by the geometry of the FCC crystal, irrelevant.

Crystallization of Flexible Chains of Tangent Hard Spheres under Full Confinement

We present results from extensive Monte Carlo simulations on the crystallization of athermal polymers under full confinement. Polymers are represented as freely jointed chains of tangent hard spheres of uniform size. Confinement is applied through the presence of flat, parallel, and impenetrable walls in all dimensions. We analyze crystallization as the summation of two contributions: one that occurs in the bulk volume of the system (bulk crystallization), and one on the wall surfaces (surface crystallization). Depending on volume fraction initially amorphous (disordered) hard-sphere chain packings transit to the stable crystal phase. The established ordered morphologies consist primarily of hexagonal close-packed (HCP) crystals in the bulk volume and of triangular (TRI) crystals on the surface. As in the case of athermal packings in the bulk (without confinement), a structural competition is observed between the 5-fold local symmetry and the formation of close-packed crystallites. Effectively, the full confinement inside a cube favors the growth of the HCP crystal, as the FCC one is quite incompatible with the imposed spatial constraints. Consequently, we observe the formation of noncompact ordered motifs which grow from the surface to the inner volume of the simulation cell. We further compare the 2D and 3D crystals formed by monomeric hard spheres under the same simulation conditions. Significant differences are observed at low densities that tend to diminish as concentration increases.

Entropy-Driven Heterogeneous Crystallization of Hard-Sphere Chains under Unidimensional Confinement

We investigate, through Monte Carlo simulations, the heterogeneous crystallization of linear chains of tangent hard spheres under confinement in one dimension. Confinement is realized through flat, impenetrable, and parallel walls. A wide range of systems is studied with respect to their average chain lengths (N = 12 to 100) and packing densities (ϕ = 0.50 to 0.61). The local structure is quantified through the Characteristic Crystallographic Element (CCE) norm descriptor. Here, we split the phenomenon into the bulk crystallization, far from the walls, and the projected surface crystallization in layers adjacent to the confining surfaces. Once a critical volume fraction is met, the chains show a phase transition, starting from regions near the hard walls. The established crystal morphologies consist of alternating hexagonal close-packed or face-centered cubic layers with a stacking direction perpendicular to the confining walls. Crystal layer perfection is observed with an increasing concentration. As in the case of the unconstrained phase transition of athermal polymers at high densities, crystal nucleation and growth compete with the formation of sites of a fivefold local symmetry. While surface crystallites show perfection with a predominantly triangular character, the morphologies of square crystals or of a mixed type are also formed. The simulation results show that the rate of perfection of the surface crystallization is not significantly faster than that of the bulk crystallization.

Identification of Local Structure in 2-D and 3-D Atomic Systems through Crystallographic Analysis

In the present work, we revise and extend the Characteristic Crystallographic Element (CCE) norm, an algorithm used to simultaneously detect radial and orientational similarity of computer-generated structures with respect to specific reference crystals and local symmetries. Based on the identification of point group symmetry elements, the CCE descriptor is able to gauge local structure with high precision and finely distinguish between competing morphologies. As test cases we use computer-generated monomeric and polymer systems of spherical particles interacting with the hard-sphere and square-well attractive potentials. We demonstrate that the CCE norm is able to detect and differentiate, between others, among: hexagonal close packed (HCP), face centered cubic (FCC), hexagonal (HEX) and body centered cubic (BCC) crystals as well as non-crystallographic fivefold (FIV) local symmetry in bulk 3-D systems; triangular (TRI), square (SQU) and honeycomb (HON) crystals, as well as pentagonal (PEN) local symmetry in thin films of one-layer thickness (2-D systems). The descriptor is general and can be applied to identify the symmetry elements of any point group for arbitrary atomic or particulate system in two or three dimensions, in the bulk or under confinement.

Crystal, Fivefold and Glass Formation in Clusters of Polymers Interacting with the Square Well Potential

We present results, from Monte Carlo (MC) simulations, on polymer systems of freely jointed chains with spherical monomers interacting through the square well potential. Starting from athermal packings of chains of tangent hard spheres, we activate the square well potential under constant volume and temperature corresponding effectively to instantaneous quenching. We investigate how the intensity and range of pair-wise interactions affected the final morphologies by fixing polymer characteristics such as average chain length and tolerance in bond gaps. Due to attraction chains are brought closer together and they form clusters with distinct morphologies. A wide variety of structures is obtained as the model parameters are systematically varied: weak interactions lead to purely amorphous clusters followed by well-ordered ones. The latter include the whole spectrum of crystal morphologies: from virtually perfect hexagonal close packed (HCP) and face centered cubic (FCC) crystals, to random hexagonal close packed layers of single stacking direction of alternating HCP and FCC layers, to structures of mixed HCP/FCC character with multiple stacking directions and defects in the form of twins. Once critical values of interaction are met, fivefold-rich glassy clusters are formed. We discuss the similarities and differences between energy-driven crystal nucleation in thermal polymer systems as opposed to entropy-driven phase transition in athermal polymer packings. We further calculate the local density of each site, its dependence on the distance from the center of the cluster and its correlation with the crystallographic characteristics of the local environment. The short- and long-range conformations of chains are analyzed as a function of the established cluster morphologies.

Effects of spherical confinement and backbone stiffness on flexible polymer jamming

We use molecular simulations to study jamming of a crumpled bead-spring model polymer in a finite container and compare to jamming of repulsive spheres. After proper constraint counting, the onset of rigidity is seen to occur isostatically as in the case of repulsive spheres. Despite this commonality, the presence of the curved container wall and polymer backbone bonds introduce new mechanical properties. Notably, these include additional bands in the vibrational density of states that reflect the material structure as well as oscillations in local contact number and density near the wall but with lower amplitude for polymers. Polymers have fewer boundary contacts, and this low-density surface layer strongly reduces the global bulk modulus. We further show that bulk-modulus dependence on backbone stiffness can be described by a model of stiffnesses in series and discuss potential experimental and biological applications.

Isostaticity and the solidification of semiflexible polymer melts

Using molecular dynamics simulations of a tangent-soft-sphere bead-spring polymer model, we examine the degree to which semiflexible polymer melts solidify at isostaticity. Flexible and stiff chains crystallize when they are isostatic as defined by appropriate degree-of-freedom-counting arguments. Semiflexible chains also solidify when isostatic if a generalized isostaticity criterion that accounts for the slow freezing out of configurational freedom as chain stiffness increases is employed. The configurational freedom associated with bond angles (θ) can be associated with the characteristic ratio C_∞ = (1 + 〈cos(θ)〉)/(1 - 〈cos(θ)〉). We find that the dependence of the average coordination number at solidification [Z(T_s)] on chains' characteristic ratio C_∞ has the same functional form [Z ≃ a - b ln(C_∞)] as the dependence of the average coordination number at jamming [Z(ϕ_J)] on C_∞ in athermal systems, suggesting that jamming-related phenomena play a significant role in thermal polymer solidification.

Crystallization of finite-extensible nonlinear elastic Lennard-Jones coarse-grained polymers

The ability of a simple coarse-grained finite-extensible nonlinear elastic (FENE) Lennard-Jones (LJ) polymer model to be crystallized is investigated by molecular dynamics simulations. The optimal FENE Lennard-Jones parameter combinations (ratio between FENE and LJ equilibrium distances) and the optimal lattice parameters are calculated for five different perfect crystallite structures: simple tetragonal, body-centered tetragonal, body-centered orthorhombic, hexagonal primitive, and hexagonal close packed. It was found that the most energetically favorable structure is the body-centered orthorhombic. Starting with an equilibrated polymer liquid and with the optimal parameters found for the body-centered orthorhombic, an isothermal treatment led to the formation of large lamellar crystallites with a typical chain topology: folded, loop, and tie chains, and with a crystallinity of about 60%-70%, similar to real semicrystalline polymers. This simple coarse-grained Lennard-Jones model provides a qualitative tool to study semicrystalline microstructures for polymers.

Jamming of Semiflexible Polymers

We study jamming in model freely rotating polymers as a function of chain length N and bond angle θ_{0}. The volume fraction at jamming ϕ_{J}(θ_{0}) is minimal for rigid-rodlike chains (θ_{0}=0), and increases monotonically with increasing θ_{0}≤π/2. In contrast to flexible polymers, marginally jammed states of freely rotating polymers are highly hypostatic, even when bond and angle constraints are accounted for. Large-aspect-ratio (small θ_{0}) chains behave comparably to stiff fibers: resistance to large-scale bending plays a major role in their jamming phenomenology. Low-aspect-ratio (large θ_{0}) chains behave more like flexible polymers, but still jam at much lower densities due to the presence of frozen-in three-body correlations corresponding to the fixed bond angles. Long-chain systems jam at lower ϕ and are more hypostatic at jamming than short-chain systems. Implications of these findings for polymer solidification are discussed.

Effect of chain stiffness and temperature on the dynamics and microstructure of crystallizable bead-spring polymer melts

We contrast the dynamics in model unentangled polymer melts of chains of three different stiffnesses: flexible, intermediate, and rodlike. Flexible and rodlike chains, which readily solidify into close-packed crystals (respectively, with randomly oriented and nematically aligned chains), display simple melt dynamics with Arrhenius temperature dependence and a discontinuous change upon solidification. Intermediate-stiffness chains, however, are fragile glass-formers displaying Vogel-Fulcher dynamical arrest, despite the fact that they also possess a nematic-close-packed crystalline ground state. To connect this difference in dynamics to the differing microstructure of the melts, we examine how various measures of structure, including cluster-level metrics recently introduced in studies of colloidal systems, vary with chain stiffness and temperature. No clear static-structural cause of the dynamical arrest is found. However, we find that the intermediate-stiffness chains display qualitatively different dynamical heterogeneity. Specifically, their stringlike motion (cooperative rearrangement) is correlated along chain backbones in a way not found for either flexible or rodlike chains. This activated "crawling" motion is clearly associated with the dynamical arrest observed in these systems, and illustrates one way in which factors controlling the crystallization versus glass formation competition in polymers can depend nonmonotonically on chain stiffness.

The role of fivefold symmetry in suppressing crystallization

Although long assumed to have an important role in the suppression of crystallization and the development of glassformers, the effect of local fivefold symmetry has never been directly tested. Here we consider whether such suppression of crystallization has a kinetic or thermodynamic nature and investigate its mechanism. We introduce a model in which the degree of fivefold symmetry can be tuned by favouring arrangements of particles in pentagonal bipyramids. We thus show that fivefold symmetry has both kinetic and thermodynamic effects on the mechanism of crystallization to a face-centred cubic crystal. Our results suggest that the mechanism of crystallization suppression is related to the surface tension between fluid and crystal. Interestingly, the degree of fivefold symmetry has little effect on crystal growth rate, suggesting that growth may be only weakly coupled to fluid structure in hard sphere like systems. Upon increasing the fivefold symmetry, we find a first-order transition to an alternative icosahedra-rich phase. At intermediate bias strengths we find a one-component glassformer.

The suppression of crystallization due to the appearance of structures with fivefold symmetry is widely adopted, but its kinetic and thermodynamic origin remains elusive. Taffs et al. show that fivefold symmetry substantially slows down the nucleation rate but not the crystal growth rate as expected.

Effect of chain stiffness on the competition between crystallization and glass-formation in model unentangled polymers

We map out the solid-state morphologies formed by model soft-pearl-necklace polymers as a function of chain stiffness, spanning the range from fully flexible to rodlike chains. The ratio of Kuhn length to bead diameter (lK/r0) increases monotonically with increasing bending stiffness kb and yields a one-parameter model that relates chain shape to bulk morphology. In the flexible limit, monomers occupy the sites of close-packed crystallites while chains retain random-walk-like order. In the rodlike limit, nematic chain ordering typical of lamellar precursors coexists with close-packing. At intermediate values of bending stiffness, the competition between random-walk-like and nematic chain ordering produces glass-formation; the range of kb over which this occurs increases with the thermal cooling rate |Ṫ| implemented in our molecular dynamics simulations. Finally, values of kb between the glass-forming and rodlike ranges produce complex ordered phases such as close-packed spirals. Our results should provide a useful initial step in a coarse-grained modeling approach to systematically determining the effect of chain stiffness on the crystallization-vs-glass-formation competition in both synthetic and colloidal polymers.

Resolving Dynamic Properties of Polymers through Coarse-Grained Computational Studies

Coupled length and time scales determine the dynamic behavior of polymers and underlie their unique viscoelastic properties. To resolve the long-time dynamics it is imperative to determine which time and length scales must be correctly modeled. Here we probe the degree of coarse graining required to simultaneously retain significant atomistic details and access large length and time scales. The degree of coarse graining in turn sets the minimum length scale instrumental in defining polymer properties and dynamics. Using linear polyethylene as a model system, we probe how the coarse-graining scale affects the measured dynamics. Iterative Boltzmann inversion is used to derive coarse-grained potentials with 2-6 methylene groups per coarse-grained bead from a fully atomistic melt simulation. We show that atomistic detail is critical to capturing large-scale dynamics. Using these models we simulate polyethylene melts for times over 500  μs to study the viscoelastic properties of well-entangled polymer melts.

The role of bond tangency and bond gap in hard sphere crystallization of chains

We report results from Monte Carlo simulations on dense packings of linear, freely-jointed chains of hard spheres of uniform size. In contrast to our past studies where bonded spheres along the chain backbone were tangent, in the present work a finite tolerance in the bond is allowed. Bond lengths are allowed to fluctuate in the interval [σ, σ + dl], where σ is the sphere diameter. We find that bond tolerance affects the phase behaviour of hard-sphere chains, especially in the close vicinity of the melting transition. First, a critical dl(crit) exists marking the threshold for crystallization, whose value decreases with increasing volume fraction. Second, bond gaps enhance the onset of phase transition by accelerating crystal nucleation and growth. Finally, bond tolerance has an effect on crystal morphologies: in the tangent limit the majority of structures correspond to stack-faulted random hexagonal close packing (rhcp). However, as bond tolerance increases a wealth of diverse structures can be observed: from single fcc (or hcp) crystallites to random hcp/fcc stackings with multiple directions. By extending the simulations over trillions of MC steps (10(12)) we are able to observe crystal-crystal transitions and perfection even for entangled polymer chains in accordance to the Ostwald's rule of stages in crystal polymorphism. Through simple geometric arguments we explain how the presence of rigid or flexible constraints affects crystallization in general atomic and particulate systems. Based on the present results, it can be concluded that proper tuning of bond gaps and of the connectivity network can be a controlling factor for the phase behaviour of model, polymer-based colloidal and granular systems.