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We review the methodology, algorithmic implementation and performance characteristics of a hierarchical modeling scheme for the generation, equilibration and topological analysis of polymer systems at various levels of molecular description: from atomistic polyethylene samples to random packings of freely-jointed chains of tangent hard spheres of uniform size. Our analysis focuses on hitherto less discussed algorithmic details of the implementation of both, the Monte Carlo (MC) procedure for the system generation and equilibration, and a postprocessing step, where we identify the underlying topological structure of the simulated systems in the form of primitive paths. In order to demonstrate our arguments, we study how molecular length and packing density (volume fraction) affect the performance of the MC scheme built around chain-connectivity altering moves. In parallel, we quantify the effect of finite system size, of polydispersity, and of the definition of the number of entanglements (and related entanglement molecular weight) on the results about the primitive path network. Along these lines we approve main concepts which had been previously proposed in the literature.

During the last decades numerous and foremost advances in various technological areas constantly render computer processors faster and more powerful at an affordable cost allowing the formation of super-computers of even myriads of processors. In parallel, scientific breakthroughs in the various fields of computer simulations have lead to the development of novel algorithms that are capable of taking full advantage of the allocated resources. Thus, it is not surprising that nowadays modeling and simulations are widely accepted as valuable companions to the more “mature” experimental and theoretical studies. Computer-generated specimen of varied molecular detail and diverse chemical constitution can be simulated under “idealized” and well-controlled conditions addressing “what-if” questions on the structure-property relation that otherwise would require the execution of a series of cost-demanding and time-consuming experiments. However, severe limitations often hinder the effectiveness and plague the performance of conventional simulation techniques necessitating the fabrication of novel algorithms or even their combined employment through hierarchical and multi-scale modeling schemes.

In particular, regarding macromolecular (polymer) systems the complex chemical constitution of the monomer (repeat) units along with the large spectrum of characteristic length and time scales require the development and employment of highly-efficient, system-specific methodological approaches. For example, in a typical polymer melt the shortest characteristic distance is that of the bond length ^{0}) Å). For conceptual purposes a chain can be further divided into equivalent freely-jointed Kuhn segments of uniform size. For flexible chain molecules each segment typically spans up to a dozen of successive monomers [^{2}_{N}_{∞}^{2}^{2} [_{1000} (where the number in the subscript denotes the carbon atoms along the chain) the average end-to-end distance ^{2}^{0.5} is by almost two orders of magnitude higher than the carbon-carbon bond length (^{−15}) s – moving to the de-correlation time of torsional (dihedral) angles, usually on the order of picoseconds, ^{−12}) s [_{G}, and the corresponding zero shear-rate viscosity, _{0}, scale with molecular length _{G} ^{−b}_{0} ^{c}_{d}_{1000} PE melt, _{d}^{−5}) s)), which is about ten orders of magnitude longer than the vibrational relaxation of bonds. The terminal relaxation mechanism is even more complicated for polymers exhibiting highly non-linear chain architecture like long-chain branching, stars or rings [

Building on the original tube model by de Gennes [

The challenge of polymer equilibration can be addressed by mapping the atomistic reference system into a less-detailed “coarse-grained” configuration which is characterized by significantly fewer degrees of freedom allowing for simulations which extend over time and lengths scales that are not accessible in the atomistic level of description. Numerous schemes exist for the systematic “coarse-graining” of the monomer units of various polymers resulting into supersites (

Alternatively, the equilibration of macromolecular systems can be achieved through stochastic Monte Carlo (MC) [_{6000} [_{1000} [

We should note that the efficient application of MC schemes, especially the ones built around CCAMs, is not straightforward for chemically complex atomistic macromolecular systems where it may happen that the performance of the underlying moves is so poor that the whole MC scheme is rendered practically useless. For example, the ability of MC mixtures, based on the configurational bias algorithm [

During the last years well equilibrated, representative polymer configurations obtained by different methods and at diverse levels of molecular detail have been subjected to extensive topological analyses for the extraction of the primitive paths (PP) and the corresponding network of intermolecular topological constraints (entanglements). The original algorithms for the construction of the PP network were based on secant areas [_{pp}) can be readily correlated with key parameters of the reptation theory [

In the present contribution we review the salient technical characteristics of a new MC scheme [

In the present section we will describe how the existing chain-connectivity altering moves [

In the original end-bridging (EB) algorithm [_{ch}) and spheres (

Double bridging (DB) alleviates the disadvantages and limitations of the EB move (mainly the requirements of chain length polydispersity and the presence of chain-ends) by performing two trimer bridgings in backbone segments that lie in the internal parts of the participating chains. Consequently, it is generally applicable to simulations of macromolecular systems with no free ends (cyclic peptides), with long branches (H-shaped macromolecules) or with specific constrained geometries (terminally-grafted brushes) [_{1000} PE melts at

The non-negligible deterioration of the acceptance rate for the class of chain-connectivity altering moves is especially apparent in simulations of dense packings of freely-jointed, hard-sphere chains primarily as a consequence of the reduced free volume available for the reconstruction of a whole trimer of hard spheres (or of two trimers for DB). Consequently, in the original implementation of the EB algorithm for dense systems of hard-sphere chains, for the vast majority of the attempted moves the reconstruction of the triplet of spheres of all candidate geometrical configurations leads to energetic overlaps with the fixed sites. A second source for the poor performance of chain-connectivity altering moves is that for freely-jointed chains the absence of any kind of imposed bending hindrance in the form of a potential function allows for the bond angles to fluctuate uniformly in the closed interval [0,120°] (except for the existence of characteristic peaks at high volume fractions as a result of the excluded volume interactions [

The schematic representation of the sEB move is shown in _{+1} is cut so that the whole sequence (_{1}_{+1}_{2}). In pattern (c) the bond between spheres _{−}_{1} is removed leaving the sequence (_{1},..._{−}_{1}) as _{2}

In the final step of its application, the sEB move is accepted or rejected according to the following criterion:
_{sEB} denotes the weighting factor for the attempted transition according to:
_{sEB} being the number of the neighbors with whom spheres _{end} and _{end} of _{+1} (pattern (b)) or to site _{−1} (pattern (c)).

The simplified intramolecular end-bridging (sIEB) move, by being the intramolecular variant of the sEB algorithm, is executed in a very similar fashion, the only difference being that all alternations occur in a single chain. The schematic representation of sIEB is depicted in _{1}_{2} breaks and a new one is formed between _{1}_{2}) (as seen in _{2} becomes the new terminal site of chain

The acceptance criterion of the sIEB move is very similar to the corresponding one of sEB of _{sIEB}(_{2}) and _{sIEB}(_{2} and

In the course of a typical MC simulation based on the chain-connectivity altering moves special lists are kept and are constantly updated for both sEB and sIEB. For every chain-end

Both moves commence by selecting randomly one of the pairs included in their special lists. If an sEB (or sIEB) is attempted and its list of initiators is empty then the move is automatically rejected. For sEB if both combinations (b) and (c) are permitted then one is randomly picked. In the reverse transition (new

The Monte Carlo scheme used for the simulation of the freely-jointed chains of tangent hard spheres is built around the sEB and sIEB moves and it further consists of local moves that undertake the task of providing short-range relaxation. Additionally, by combining chain-connectivity altering algorithms and a set of varied localized ones we are able to significantly reduce the “shuttling effect”, _{dis} in the reverse and forward transitions being strongly dependent on the volume fraction (packing density) _{dis} is equal to 10 and 80 at _{dis} with _{dis} is identified through preliminary trial simulations at the specified volume fraction.

The simulation of dense random packings of freely-jointed chains of tangent hard spheres splits into two phases: the first step of the generation for the simulation box filled with the non-overlapping chain monomers at the desired volume fraction ^{M}

For the creation of the initial structures, we start from very dilute non-overlapping configurations (^{−7}) as we approach the MRJ state [_{ch}

The following systems of freely-jointed chains of tangent hard spheres have been studied: (i) 100 chains of average length ^{−4}^{−8}^{10} at dilute conditions (^{11} MC steps at densities close to the MRJ state. System configurations including sphere coordinates and chain connectivity along with MC statistics were recorded every 1 × 10^{7} resulting in simulation trajectories consisting of thousands of uncorrelated MC frames.

A major advantage of the set of the sEB and sIEB moves compared to the original algorithms (EB and IEB) but also to the localized algorithms steams from their particular form of application in combination with the applied hard-core potential: they are able to provide robust re-arrangements in chain connectivities through bond swapping without displacing any sites as explained in detail in sections 2.1. and 2.2.. Therefore, it is expected that the new set of moves would be tailored to function at very dense random packings, even in the close vicinity of the MRJ state. The validity of the above speculation is verified by the data shown in

At very low concentrations the chain-connectivity altering moves, especially the EB and sEB moves which require the close proximity of segments belonging to different macromolecules, exhibit very low acceptance rates since in a highly diluted environment chains do not “feel” the presence of each other as they are far apart. Consequently, there exist very few to zero pairs of intermolecular neighbors that lie close within the bridgeable distance of 4_{acc} ^{−7})) that the inclusion of the move in the MC scheme becomes practically fruitless. In sharp contrast, the acceptance probabilities of both sEB and sIEB get continuously augmented with increasing packing density. Given that the moves do not entail site displacements their performance depends solely on the wealth and the update rate of the lists of the initiator pairs for each move (see also related discussion in sections 2.1. and 2.2.). As density increases, and especially close to the random close packing, chains collapse in size [

While the data on the acceptance rate reveal significant limitations and advantages of the employed simulation algorithms their ability in equilibrating the long-range characteristics of polymer systems is typically quantified by the evolution in (computational) time of the orientational autocorrelation function of the end-to-end unit vector

From the time evolution of the _{c}_{c}_{c}^{7} MC steps at _{c}^{10} steps at _{c}

In the remaining subsection we will investigate the effect of system size on the structural properties of model polymer systems of freely-jointed hard-sphere chains. As a test case we employ extensive MC simulations on two different realizations of the ^{2}^{0.5} of chains is smaller by a factor of approximately 2 and larger by a factor of more than 1.5 for the large and the small system, respectively, at both densities. We should note that in the continuation all reported values of the end-to-end distance (and of the contour length of the primitive path, see next section) are rendered dimensionless by dividing with the collision diameter

The evolution of the instantaneous value of the mean square end-to-end distance ^{2}^{2}^{2}^{2}

Identical conclusions are drawn for all measures of chain size and for the radial distribution functions in the level of individual spheres and for the whole range of concentrations even for nearly jammed structures. It is thus established that there exist no system size effects on the MC results that could potentially affect key findings regarding local packing [

A remaining effect of system size on the final configuration, which has to be taken into account into a more detailed analysis of our data, is the degree of polydispersity. Statistical properties of chain molecules eventually depend on the shape of the distribution of chain lengths in a rather nontrivial manner. Polydispersity does not affect those single chain statistical properties which are strictly proportional to

For the melt configurations the reduction to primitive paths was performed using a procedure described in Refs. [_{pp}, and paths have zero thickness. In addition to the set of lengths of the primitive paths, and the configuration of the entanglement network, the Z1 analysis also yields the number of interior “kinks” [

Within the Z1 code periodic images of the same chains are treated as different chains, while all parts of a physically connected chain (which may cross the border of the simulation box) are treated as belonging to the same chain. The minimization procedure terminates as soon as the mean length of the primitive paths, _{pp}_{pp}

The alternate CReTA method shares many similarities with Z1, and the conclusions reached here for Z1 analysis should apply similarly to CReTA results [

Within the Z1 algorithm the primitive path is a connected (mathematical) path of straight segments. Contour length of the multiple disconnected path is monotonically reduced by iteratively applying basic moves of the type shown in

Obviously, a number of kinks then corresponds to the number of changes in direction along the PP, but does not inform us on how many chains contributed to their existence, see

There is, in addition, an issue to be discussed which is relevant if the system size is small compared with the size of the chains. This issue had not been thoroughly addressed in the literature, and gives rise to confusion whenever computational resources are not sufficient to study a large enough system. As already mentioned, within the Z1 code periodic images of the same chains are treated as different chains, while all connected parts of a chain belong to the same chain, regardless of its size compared to simulation box size, see

With one of these measures, the contour length _{pp}^{2}_{e}

There are remarkable and important deviations between the statistical properties of weakly entangled and infinitely long, infinitely entangled, chains, _{e}_{pp}^{2}_{e}_{0} + _{e}_{0} (which should be negative) and _{e}_{N}_{e}_{pp} values. The entanglement molecular weight appears here because it is defined to be the ratio between _{0} reflects the fact, that a minimum amount of material is required to produce a single entanglement. We expect it to depend on the molecular details of the model and state variables such as temperature, which influence, e.g. the flexibility of chains. Upon applying the proposed procedure, we can evaluate a corrected value for ^{*}_{e}^{*}^{*}_{e}^{*}^{*}_{pp}

We have reviewed the salient methodological characteristics of an hierarchical modeling scheme for the molecular simulation of model polymer systems, and mentioned open questions. The present approach combines Monte Carlo simulations, based on chain-connectivity altering moves that undertake the task of system generation and equilibration and a direct geometrical algorithm that renders the parent polymer configurations into a primitive path mesh and fully identifies the topological constraints (entanglements) between different chains. While the proposed algorithmic approach is general and can be applied to any polymer system irrespective of the chemical constitution, repeat units or molecular architecture, we have placed particular emphasis on the application on model random packings of linear freely-jointed chains of tangent hard spheres of uniform size. We have clearly demonstrated that the MC-based long-range equilibration of these systems is affected by neither the molecular length of the chain nor by the volume fraction (packing density). In addition, we have analyzed the effect of system size in key structural properties like chain size, contour length of the primitive path, number of entanglements and entanglement molecular weight. Finally, we have commented about the issue of treating self-entanglements in the analysis, and the difficulties which prevent us from identifying entanglements as (a fixed number of) kinks of the primitive path, while the underlying goal is to avoid any assumptions about the statistics of the primitive path. Correlations between segment vectors along the PP rather than a number of kinks can be used to characterize the properties of the PP as returned by the Z1 code, but as long as the measured correlations produce a single number (like a persistence length), this number should be related to a number of kinks. We have demonstrated how, starting from state-of-the-art simulation algorithms for atomistic, chemically-simple polymer melts, we can arrive, mainly through simplifications of the original implementation, on new moves tailored to provide rapid and robust equilibration in the simulation of model random packings of hard-sphere chains. Through extensive Monte Carlo simulations based of the sEB and sIEB moves [^{12})) of steps, we were able to identify the maximally random jammed (MRJ) state for model polymer systems and show that hard-sphere chains can be as efficiently and as densely packed as the monoatomic analogs do [

The reviewed methodology can be applied to follow the dynamics of the primitive path, and to study the fluctuations of the primitive path characteristics, in order to extract quantities such as the tube survival probability and shear modulus [

Current efforts and future applications further include the molecular modeling of polymer samples of varied molecular architecture in the bulk and at nano-interfaces at various descriptions of detail. The proposed hierarchical scheme for the simulation of the hard-sphere chains is being presently expanded so as to include dynamical and rheological information at and beyond equilibrium.

Very fruitful discussions with Katerina Foteinoulou and Manuel Laso (UPM, Spain) on the MC algorithms and the topological analysis of the random chain packings and with Robert Hoy (UCSB, United States) on the estimators of the entanglement molecular weights are deeply appreciated. This work was supported by the MRSEC Program of the National Science Foundation under Award No. DMR05-20415, the SNF (grant no. IZ73Z0-128169) as well as through contracts NMP3-CT-2005-016375 and NMP3-CT-2006-033339 of the European Community. We fully acknowledge the extensive usage of computational resources in the “magerit” supercomputer of CeSViMa (UPM, Spain) and in the “rosa” supercomputer of CSCS (Switzerland).

Schematic drawing summarizing the construction of the (three–dimensional) primitive paths (PP). (a) Microscopic configuration. For clarity reasons, two out of hundreds of chains are shown. (b) For the construction of the PP within the Z1 code, the backbone is considered infinitely thin, and chain ends are fixed in space. (c) The length of the multiple disconnected path is monotonically reduced, subject to chain-uncrossability, by introducing a smaller number of nodes. (d) Upon iterating the geometrical procedure one converges at a final state, the shortest path, shown in (d), together with the original chain, and alone in (e). Each chain carries a part of the multiple disconnected shortest path, called its PP. A single PP is often characterized by its conformational properties such as reflected by contour length _{pp} and number of kinks, denoted as

Schematic representation of the simplified end-bridging (sEB) move [_{+1} (pattern (b)) or sphere _{–}_{1} (pattern (c)). A new bond is formed between sites

Schematic representation of the simplified intramolecular end-bridging (sIEB) move [_{1}_{2} while a new bond is formed between sites _{1}_{2}). By construction, a successful sIEB move does not alter the molecular length of

Representation of the two participating chains before (left) and after (right) the successful application of a simplified end-bridging (sEB) move from MC simulations on a

Same as in

Logarithm of the percentage acceptance rates, log_{10}(_{acc}) for the simplified end-bridging (sEB), simplified intramolecular end-bridging (sIEB) and of the original end-bridging (EB) moves as a function of the logarithm of packing density, log_{10}(

Orientational autocorrelation function of the end-to-end unit vector

Total equilibration time _{c}_{c}^{7} MC steps.

(a) Instantaneous values of the mean square end-to-end distance, ^{2}^{2}

Snapshots made during runtime of the Z1 code [

Snapshots made during runtime of the Z1 code [

Probability distributions for various quantities resulting from the entanglement analysis (Z1) for the system at _{pp}, and (c) number of interior kinks,

Same as

Central routine of the Z1 code finds the shortest path for two adjacent segments (solid lines) in the presence of none, one (a), two (b), or more obstacles. Each obstacle correspond a point which belongs to the contour of a different chain (not shown), and which intersects with the plane spanned by the two segments. Z1 locates potentially intersecting points efficiently, and operates at a variable number of nodes characterizing the shortest path. The central routine is iteratively applied to all pairs of adjacent segments until the multiple disconnected path has converged to a minimum length. The algorithm scales linearly with the total number of particles.

The primitive path of an individual chain is mainly characterized by its contour length _{pp} and end–to–end distance

Two system size effects potentially occurring during the construction of the PP, when chain dimensions exceed box dimensions. (a) Shown is an unfolded single chain, whose ends are residing within a central simulation cell. The system is subjected to periodic boundary conditions. Because self–entanglements are excluded from the calculation of the PP, the information about connectivity has to be kept, to see, that the chain in (a) is unentan-gled. If this information is not used, self-entanglements will be underestimated, depending on system size. (b) Shown are two out of 26 cells surrounding the central cell shown in (b). Periodic images can however entangle and are treated as different chains as to minimize system size effects in the determination of the PP. Concerning the algorithm it is of importance to prevent using segments of the PP which exceed box size. The PP (not shown for this graph, cf. previous

Ratios _{pp}^{–}^{1}) corrections [_{pp}_{pp}

Comparison between results for small and large systems (with same _{pp}_{e}^{*}_{pp}^{*}

system | chains | ^{2}^{1}^{/}^{2} |
CPU time | _{pp} |
^{*} |
_{e} |
| ||
---|---|---|---|---|---|---|---|---|---|

small | 6 | 500 | 31.1 | 0.12 s | 83.5 | 11.8 | 13.2 | 42.4 | 37.9 |

large | 162 | 500 | 30.8 | 7.93 s | 84.0 | 12.2 | 12.2 | 40.9 | 40.9 |

small | 6 | 500 | 27.9 | 0.16 s | 75.5 | 11.6 | 13.0 | 43.1 | 38.5 |

large | 162 | 500 | 28.1 | 8.34 s | 78.3 | 12.6 | 12.6 | 39.7 | 39.7 |