Crystallization of short aliphatic polymer chains. I. General chain-folding behavior

Short aliphatic polymer chains of different lengths were prepared by degrading polyethylene samples of appropriately chosen initial fold lengths to the chain lengths which correspond to a single chain traverse through the lamella. The resulting dicarboxylic acids were either used as such for further crystallization experiments or were first converted into diiodides to remove polar endgroups. The resulting short polymers all crystallized by chain folding even if the chains (peak of distribution) were only 1.5–4 times the length of a traverse through the lamella. In the diiodides the fold length varied continuously with crystallization temperature, as is usual in high molecular weight material, but with the dicarboxylic acids such variation, while observable, was only small. The effect of the molecular weight on the fold length due to its influence on supercooling at a given crystallization temperature has become apparent. Renewed degradation with nitric acid and subsequent GPC analysis of the degradation products confirmed the folded nature of the chains in the above crystals. This analysis combined with experiments on the reactivity of chain ends has led to the picture that each chain folds completely, once, twice etc. so that both folds and ends are in the surface zone but are located at varying heights, as appropriate to the overall layer thickness for the molecular weight distribution in question. This picture is consistent with other concurrent work.     

Molecular dynamics simulation of polyethylene on single wall carbon nanotube

Molecular dynamics simulations are carried out on the isothermal crystallization process of single polyethylene chains with different chain lengths on the single wall carbon nanotube. This process is summarized as two steps, i.e., adsorption and orientation, and the bond-orientational order parameter is used to show the details of this process. The results show that the attractive van der Waals interactions control the adsorption and orientation of polyethylene on single wall carbon nanotube, and as the chain length increases, more microstructures appear in the last ordered structure. The stems of the ordered structure align parallel to the single wall carbon nanotube axis.     

分子动力学方法研究单链聚乙烯的结晶过程

用分子动力学方法（ＭＤ）研究了单链聚乙烯在不同温度（１００Ｋ、２００Ｋ、３００Ｋ、４００Ｋ、５００Ｋ）下的结晶过程,并用能量和结构参数进行了描述．结果表明伸直分子链的结晶过程都经历了三个阶段,首先是伸直链的卷曲与聚集,然后通过链段的排列形成规整的片晶结构,最后是结晶形成的片晶在结构与能量上的涨落变化。研究表明,结晶温度越高,分子链的内聚速度越快．研究发现,分子链在内聚阶段经历一个局部凝聚的中间状态,在结晶温度很低（１００Ｋ）时,局部凝聚的结构是有序的．而在５００Ｋ时,该结构为无规线团．结晶温度的差异,一般来说,将导致得到的片晶厚度的不同．对于模拟的单链,随着结晶温度的降低而形成了较厚的片晶．该行为与聚乙烯本体结晶中片晶厚度对结晶温度的依赖性相反．在有序化阶段和之后的片晶调整运动阶段,分子链线团的回转半径基本保持不变．这与宏观多链体系的结果相同．另外,在模拟中发现,尺寸微小的单链聚乙烯晶片的扭转运动     

Molecular segregation and nucleation of poly(ethylene oxide) crystallized from the melt. IV. Computer modeling

A crystal surface is modelled and Monte Carlo simulations are attempted to learn about conformations and motion of a macromolecule that could lead to crystallization. Chains with up to 200 mobile segments have been analyzed. The diffusion distance of a given chain segment increases with positioning closer to the chain end, and completion of crystallization accelerates with decreasing length of remaining amorphous chain ends and increasing temperature.     

Simulation of multiple ordered phases in C23 n-alkane

Normal alkanes display multiple ordered phases, including an orthorhombic crystal (X) and two partially ordered rotator phases (RI and RII). The rotator phase transitions X-RI and RI-RII are of interest because they are weakly first-order, and because experiments suggest that crystalline polyethylene may nucleate via a metastable rotator phase. We have performed heating and cooling scans of all-atom NσT (isothermal, isostress) simulations of a pure C(23) solid. We find a sequence of phases, transition temperatures, structural and thermodynamic properties, all reasonably consistent with experiment, except that a monoclinic crystal is more stable in our simulations than the experimental orthorhombic structure. We find that the RI phase is well described as an orthorhombic crystal disordered by random ±90° rotations of molecules about their stem axis, and the RII phase can be represented as a loose hexagonal packing of parallel chain stems, which tend to orient with the in-plane projection of C-C bonds pointing between neighbors. To measure local orthorhombic, RI, or RII order, we define Potts- and Ising-like order parameters, from which global order parameters and correlation functions can be computed. We observe modest pretransitional fluctuations of local RI order in the RII phase near T(RI-RII), characteristic of this weakly first-order transition.     

Modeling of aqueous poly(oxyethylene) solutions. 2. Mesoscale simulations

We extend our work on aqueous solutions of poly(oxyethylene) oligomers H-(CH2-O-CH2)n -H (POEn). On the basis of atomistic simulations of trimer and decamer solutions (first part of this series of papers), different sets of coarse-grained implicit-solvent potentials have been constructed using the iterative Boltzmann inversion technique. The comparison of structures obtained from coarse-grained simulations (gyration radii, end-to-end distances, radial distribution functions) with atomistic reference simulations and experiments shows that the state-specific potentials are transferable both to a wide concentration range, if the same molecule size is considered, and to at least 2 orders of magnitude larger molecules (in terms of molecular mass). Comparing the performance of different mesoscale potentials, we find different applicability ranges in terms of molecule sizes. The experimental gyration radii for chains comprising up to 1500 monomers are reproduced almost quantitatively by the decamer-fitted potentials with dihedral interactions included. The trimer-fitted potentials reproduce experimental chain dimensions of up to some hundred monomers but seem to become metastable beyond a certain chain length, as we evidenced some chain collapses. Relaxation of large-scale features is 1-2 orders of magnitude faster in the mesoscale simulations than in the atomistic simulations. The diffusion behavior in dependence of concentration is captured correctly when the decamer potential is applied to the decamer itself. For all other chain lengths, we find that time mapping from coarse-grained to atomistic trajectories has to be determined separately for each concentration. Overall, diffusion is 1-2 orders of magnitude faster on the mesoscale, depending considerably on the Lowe-Andersen thermostat parameters. The CG simulations provide an overall speed-up of about 3 orders of magnitude.     

Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states

Molecular dynamics simulation generates large quantities of data that must be interpreted using physically meaningful analysis. A common approach is to describe the system dynamics in terms of transitions between coarse partitions of conformational space. In contrast to previous work that partitions the space according to geometric proximity, the authors examine here clustering based on kinetics, merging configurational microstates together so as to identify long-lived, i.e., dynamically metastable, states. As test systems microsecond molecular dynamics simulations of the polyalanines Ala(8) and Ala(12) are analyzed. Both systems clearly exhibit metastability, with some kinetically distinct metastable states being geometrically very similar. Using the backbone torsion rotamer pattern to define the microstates, a definition is obtained of metastable states whose lifetimes considerably exceed the memory associated with interstate dynamics, thus allowing the kinetics to be described by a Markov model. This model is shown to be valid by comparison of its predictions with the kinetics obtained directly from the molecular dynamics simulations. In contrast, clustering based on the hydrogen-bonding pattern fails to identify long-lived metastable states or a reliable Markov model. Finally, an approach is proposed to generate a hierarchical model of networks, each having a different number of metastable states. The model hierarchy yields a qualitative understanding of the multiple time and length scales in the dynamics of biomolecules.     

A study on the structural transition of a single polymer chain by parallel tempering molecular dynamics simulation

The structural transition of a single polymer chain with chain length of 100,200 and 300 beads was investigated by parallel tempering MD simulation.Our simulation results can capture the structural change from random coil to orientationally ordered structure with decreasing temperature.The clear transition was observed on the curves of radius of gyration and global orientational order parameter P as the function of temperature,which demonstrated structural formation of a single polymer chain.The linear relationships between three components of square radius of gyration R_gx²,R_gx²,R_gz² and global orientational order P can be obtained under the structurally transformational process.The slope of the linear relationship between x（or y-axis） component R_gx²(or R_gy²) and P is negative,while that of RL as the function of P is positive.The absolute value of slope is proportional to the chain length.Once the single polymer chain takes the random coil or ordered configuration,the linear relationship is invalid.The conformational change was also analyzed on microscopic scale.The polymer chain can be treated as the construction of rigid stems connecting by flexible loops.The deviation from exponentially decreased behavior of stem length distribution becomes prominent,indicating a stiffening of the chain arises leading to more and more segments ending up in the trans state with decreasing temperature.The stem length N_tr is about 21 bonds indicating the polymer chain is ordered with the specific fold length.So,the simulation results,which show the prototype of a liquid-crystalline polymer chain,are helpful to understand the crystallization process of crystalline polymers.     

Self-assembly Behavior of Symmetrical Linear ABCA Tetrablock Copolymer: A Self-consistent Field Theory Study

ABCA tetrablock copolymers offer new opportunities for design of materials with novel structures. Using real-space self-consistent field theory and simulation, we systematically examined the self-assembly behavior of linear ABCA tetrablock copolymers in a 2D space. The simulation was carried out under conditions of symmetrical compositions and interactions. We focus on the influence of chain length ratio of block A and interactions between block A and other blocks B and C on the self-assembly behavior of the copolymer system. The simulation results show that most of the structures self-assembled by the ABCA tetrablock copolymers are centrosymmetric, such as diblock-like lamella phase, two kinds of lamellae with beads at interface, two kinds of hierarchical lamella phase, hexagonal honeycomb-like phase, lamella phase with mixed BC and hexagonal spheres with mixed BC. Furthermore, we find that a novel noncentrosymmetric Janus spheres can be obtained when the interaction between blocks B and C is strong, whereas a noncentrosymmetric lamella phase was obtained at weak interaction between blocks B and C. Phase diagrams for the ABCA tetrablock copolymers with different interaction strength between blocks B and C are constructed by comparing free energies of candidate ordered structures. In addition, studies on the metastable behavior of the system reveal that enthalpy plays an important role in the metastable behavior of the ABCA tetrablock copolymer system. Our work can provide useful guide for structure control of such kind of tetrablock copolymers in experiments.     

Nascent crystallization of a growing chain on a catalyst surface: a nonequilibrium molecular dynamics simulation study

The growing chain molecular dynamics (GCMD) simulation method, a new nonequilibrium molecular dynamics code, is proposed to simulate the polymer chain aggregation behavior during polymerization on a catalyst surface. We found that the growing chain crystallizes on the surface in two stages: the nucleation stage and the crystal growth stage. In the first part of the nucleation period, the short polymerizing chain first absorbs on the surface and can be in either an ordered or disordered structure. Still in the nucleation period, when the chain reaches a degree of polymerization, about 100 bonds, the chain folds into a stable nucleus on the substrate with 3-5 stems. In the crystal growth stage where the polymerization also proceeds, we observed a stem elongation process in combination with a chain folding process. In the stem elongation step, the number of stems in the nucleus remains constant, and all the stems expand together to a length of ca. 5-25 ns. In the subsequent chain folding step, the stem length decreases about 20 bonds within a period of ca. 0.1-0.5 ns. During chain growth, the elongation process and the folding process occur in an alternating and repeated fashion. The crystallization mechanism of the polymerizing chain was discussed.     

高分子熔体结晶的两维计算机模拟

高分子熔体结晶的两维计算机模拟胡文兵，于同隐，卜海山（复旦大学高分子科学系，复旦大学材料科学系，上海，２００４３３）关键词计算机模拟，有序相变，高分子１９５６年Ｆｌｏｒｙ［１］从平均场近似的格子模型证明：高分子链的非柔顺性会导致长链的完全有序排列．此...     

Intramolecularly ordered globules and the collapse of short chains

We investigate the poor-solvent collapse of short chains having different stiffness through self-consistent minimization of the intramolecular free energy under the constraint of fixed segment lengths between adjacent beads. At first the chains form the Random Gaussian Globule, where the beads are distributed at random at the same average distance from the centre of mass, while the segments show very little correlation. At a larger attractive potential, this collapsed globule undergoes a transition to one or more ordered compact states, depending on the chain stiffness. Under very strong contraction, all chains are described as a Compact Ordered Globule: the beads are again at a constant average distance from the centre of mass, while the segments jump back and forth at the globule's wall with a very large correlation. At intermediate contraction, the thinner and stiffer chains form the Oscillating Ordered Globule wherein the beads are alternatively distributed on two concentric on two concentric shells. In this case, we also find metastable states with nonsymmetrical conformations of the chain with respect to its ends. We also briefly discuss the thermodynamics of the coil-globule and globule-globule transitions, showing that in long polymer chains these ordered conformations cannot involve the whole chain. However, we suggest that they might still be found as local globules that form for kinetic reasons.     

Computer modeling of the crystallization process of single‐chain ethylene/1‐hexene copolymers from dilute solutions

Langevin molecular dynamics (LMD) simulations have been performed to understand the role of the short chain branches (SCB) on the formation of ordered domains by cooling dilute solutions of ethylene/α‐olefins copolymer models. Three different long single‐chain models (C₂₀₀₀) with 0, 5, and 10 branches each 1000 carbons were selected. These models were equilibrated at high reduced temperature (T* = 13.3) and cooling in steps of 0.45 until the final temperature (T* = 6.2) by running a total of 35 × 10⁶ LMD steps. During the cooling process, global order parameter, torsion distribution, position of the branches, and local‐bond order parameter were calculated and monitored. The peaks of crystallization for each model were calculated by differentiating the global order parameter with temperature. The T_c (crystallization temperature) decreases as the number of branches increases as has been experimentally reported. The formation of order in the copolymers is affected by the amount of the SCB in the backbone of the polymer chain. Initially, the SCB move to the folding surface. Once the SCB are located near the folding surface the order starts to grow. In all cases here shown, the C₄ branches are excluded from the ordered domains. To take into account, the influence of the branch distribution, a different branch distribution model has been considered for the two‐branched systems. The crystallization fraction (α) and the density of the amorphous and ordered fractions was defined using the local‐bond order parameter. Both magnitudes decrease as the number of branches increases. These facts fairly agree with experimental literature data. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Spinodal patterns indicating unstable regime of polymer crystallization

It has been considered that crystallization of polymer from melt proceeds via the coexistence of molten matrix and growing crystals that have once overcome a nucleation barrier to a critical size. The nucleation process has often been explained analogously with so-called nucleation and growth (NG) behavior of the phase separation of a binary mixture in metastable conditions, although the crystallization in one-component polymer is not a real component separation but a phase transition. Among the mechanisms of polymer crystallization, the topic is whether a liquid–liquid transition between states of different densities within one-component polymers takes place before the aforementioned nucleation process. The liquid–liquid transition between states, which is probably driven by chain orientation, is also categorized into NG and the controversial spinodal decomposition (SD) type processes depending on the quenching depth. This article provides the optical microscopic observations that favor the occurrence of the SD-like process when a one-component polymer melt is very rapidly quenched below a stability limit, including a drastic morphological change from a spherulitic to a spinodal pattern at the critical (or spinodal) temperature. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1817–1822, 2004     

Molecular dynamics simulations of semicrystalline polymers: Crystallization,melting, and reorganization

Large‐scale molecular dynamics (MD) simulations of semicrystalline entangled polymers are carried out to explore crystallization and melting processes. Semicrystalline polymers are obtained from disordered melts via homogeneous nucleation. In the early stage of the crystallization process, the collective scattering does not show the emergence of nuclei seeds. Although the crystallization process is thermodynamically simple, the melting process is complex resulting in multiple‐peaked melting endotherms. The molecular origin is found to be the different thermal stabilities of microcrystalline domains (MCDs). Coexistence of melting and growth of different MCDs during sufficiently slow heating enlarges the difference of their thermal stabilities. An increase of stem length close to the melting point is assisted by disorder effects in particular in the surface regions of the MCDs. The number of trans–trans states is decreasing, which increases the flexibility and mobility of the crystalline stems. We have also investigated self‐seeding processes, and we show how these can be used to obtain single lamellar crystals in MD simulations. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2010     

Heterogeneous directional mobility in the early stages of polymer crystallization

Recently, we demonstrated via large-scale molecular dynamics simulations a "coexistence period" in polymer melt ordering before crystallization, where nucleation and growth mechanisms coexist with a phase-separation mechanism [Gee et al., Nat. Mater. 5, 39 (2006)]. Here, we present an extension of this work, where we analyze the directional displacements as a measure of the mobility of monomers as they order during crystallization over more than 100 ns of simulation time. It is found that the polymer melt, after quenching, rapidly separates into many ordered hexagonal domains separated by amorphous regions, where surprisingly, the magnitude of the monomer's displacement in the ordered state, parallel to the domain axial direction, is similar to its magnitude in the melt. The monomer displacements in the domain's lateral direction are found to decrease during the time of the simulation. The ordered hexagonal domains do not align into uniform lamellar structures during the timescales of our simulations.     

Nucleation of ordered solid phases of proteins via a disordered high-density state: phenomenological approach

Nucleation of ordered solid phases of proteins triggers numerous phenomena in laboratory, industry, and in healthy and sick organisms. Recent simulations and experiments with protein crystals suggest that the formation of an ordered crystalline nucleus is preceded by a disordered high-density cluster, akin to a droplet of high-density liquid that has been observed with some proteins; this mechanism allowed a qualitative explanation of recorded complex nucleation kinetics curves. Here, we present a simple phenomenological theory that takes into account intermediate high-density metastable states in the nucleation process. Nucleation rate data at varying temperature and protein concentration are reproduced with high fidelity using literature values of the thermodynamic and kinetic parameters of the system. Our calculations show that the growth rate of the near-critical and supercritical ordered clusters within the dense intermediate is a major factor for the overall nucleation rate. This highlights the role of viscosity within the dense intermediate for the formation of the ordered nucleus. The model provides an understanding of the action of additives that delay or accelerate nucleation and presents a framework within which the nucleation of other ordered protein solid phases, e.g., the sickle cell hemoglobin polymers, can be analyzed.     

Determination of fluid--solid transitions in model protein solutions using the histogram reweighting method and expanded ensemble simulations

Protein crystallization conditions are usually identified by empirical screening methods because of the complexity of the process, such as the existence of nonequilibrium phases and the different crystal forms that may result from changes in solution conditions. Here the crystallization of a model protein is studied using computer simulation. The model consists of spheres that have both an isotropic interaction of short range and anisotropic interactions between patch-antipatch pairs. The free energy of a protein crystal is calculated using expanded ensemble simulations of the Einstein crystal, and NpT-Monte Carlo simulations with histogram reweighting are used to determine the fluid-solid coexistence. The histogram reweighting method is also used to trace out the complete coexistence curve, including multiple crystal phases, with varying reduced temperature, which corresponds to changing solution conditions. At a patch-antipatch interaction strength five times that of the isotropic interaction, the protein molecules form a stable simple cubic structure near room temperature, whereas an orientationally disordered face-centered-cubic structure is favored at higher temperatures. The anisotropic attractions also lead to a weak first-order transition between orientationally disordered and ordered face-centered-cubic structures at low temperature, although this transition is metastable. A complete phase diagram, including a fluid phase, three solid phases, and two triple points, is found for the six-patch protein model. A 12-patch protein model, consistent with the face-centered-cubic structure, leads to greater thermodynamic stability of the ordered phase. Metastable liquid-liquid phase equilibria for isotropic models with varying attraction tails are also predicted from Gibbs ensemble simulations.     

Dissipative particle dynamics simulations of polymer melts. I. Building potential of mean force for polyethylene and cis-polybutadiene

We present the derivation of coarse-grained force fields for two types of polymers, polyethylene (PE), and cis-polybutadiene (cis-PB), using the concept of potential of mean force. Coarse-grained force fields were obtained from microscopic simulations for several coarse-graining levels, i.e., different number of monomers lambda per mesoscopic unit called "bead." These force fields are then used in dissipative particle dynamics (DPD) simulations to study structural and dynamical properties of polymer melts of PE and cis-PB. The radial distribution functions g(R), the end-to-end distance R0, the end-to-end vector relaxation time tau, and the chain center of mass self-diffusion D(CM), are computed for different chain lengths at different coarse-graining factor lambda. Scaling laws typical of the Rouse regime are obtained for both polymers for chain lengths ranging from 6 to 50 beads. It is found that the end-to-end distance R0 obtained from DPD simulations agree well with values obtained from both microscopic simulations and experiments. The dependence of the friction coefficient used in DPD simulations versus the coarse-graining level is discussed in view of the overall scaling of the dynamical properties.     

Following the Crystallization Process of Polyethylene Single Chain by Molecular Dynamics: The Role of Lateral Chain Defects

Summary: Langevin molecular dynamics (LMD) simulations have been performed in order to understand the role of the short chain branches (SCB) on the formation of ordered domains by cooling ethylene/α-olefins single chain models. Different long single-chain models (C₂₀₀₀) with 0, 5 and 10 branches each 1000 carbons were selected. The branches were randomly distributed along the backbone chain. Furthermore, C₁ (methyl) and C₄ (butyl) branches were taken into account. These models mimic the molecular architecture of ethylene/1-butene and ethylene/1-hexene random copolymers. The simulations are performed according to the following protocol: 20 random chain conformations for each model were equilibrated at high temperature (T* = 13.3) and then they were cooled in steps of 0.45 until the final temperature (T* = 6.2) by running a total of 35 × 10⁶ LMD steps. The distribution peaks of crystallization for each model were calculated by differentiating the global order parameter with respect to the temperature. The Tc* (crystallization temperature) decrease as the number of branches increases as it is experimentally observed. The formation of order in the copolymers is affected by the type and amount of the SCB in the backbone of the polymer chain. The stem lenght and crystallization fraction (α) were defined using the local-bond order parameter. Both parameters decrease as the number of branches increase. In all cases here shown, the C₄ branches are excluded from the ordered domains. However, we have observed that the methyl branch can be incorporated into the ordered regions. These facts satisfactorily agree with experimental data available in the literature.