Analytic Expressions for the Statistics of the Primitive-Path Length in Entangled Polymers

Khaliullin, Renat N.; Schieber, Jay D.

doi:10.1103/physrevlett.100.188302

Cited by 43 publications

(59 citation statements)

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“…For ' ! 0:10 the modified parameter Ã oscillates around a plateau value of 1:07 AE 0:15 (data not visualized) which is very close to Ã ' 1 predicted in [18], and well below the random walk result, ¼ 3=2. This successful comparison demonstrates further the predictive power of the proposed slip-link model for chain systems from semidilute fluids to polymeric solids near the MRJ state.…”

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“…Consequently, the theoretical prediction of L PP statistics has drawn considerable scientific attention. Very recently, Khaliullin and Schieber [18] proposed an analytic expression for the cumulative probability of the primitive path length P c ðL PP Þ, based on a thermodynamically consistent slip-link model with the parameters being the number of Kuhn steps in the chain, N K , and which is approximately equal to the average number of Kuhn steps per entanglement strand, hN e i. Accordingly, P c ðL PP Þ is defined as [18] …”

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“…There is an interesting side result which deserves being mentioned: The shown knotting probabilities can be cast into the form P knotting ðNÞ ¼ 1 À e ÀðN=N 0 Þ a , where (i) the parameter N 0 increases with increasing characteristic ratio C n (combined Figs. 1-3) but is independent of N, (ii) N 0 % 10 5 at lowest volume fraction, and (iii) a ' 1:33 is an empirical factor suggesting a stronger dependence of P knotting on N for linear chains with respect full lines: exponential formula Tzoumanekas/Theodorou [11] full lines: slip-link model Khaliullin/Schieber [18] FIG. 2 (color online).…”

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Universal Scaling, Entanglements, and Knots of Model Chain Molecules

et al. 2008

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By identifying the maximally random jammed state of freely jointed chains of tangent hard spheres we are able to determine the distinct scaling regimes characterizing the dependence of chain dimensions and topology on volume fraction. Calculated distributions of (i) the contour length of the primitive paths and (ii) the number of entanglements per chain agree remarkably well with recent theoretical predictions in all scaling regimes. Furthermore, our simulations reveal a hitherto unsuspected connection between purely intramolecular (knots) and intermolecular (entanglements) topological constraints. DOI: 10.1103/PhysRevLett.101.265702 PACS numbers: 61.20.Ja, 02.10.Kn, 64.70.QÀ, 64.75.Gh Since Bernal's pioneering work almost half a century ago [1], a great deal of experimental, theoretical, and simulation effort has been devoted to the investigation of random packings of single spheres and other nonsymmetric hard-body objects, especially in the vicinity of the configuration referred to nowadays as the maximally random jammed (MRJ) state [2][3][4]. MRJ is defined as the state in which the most sensitive measure of order parameter is minimized among all statistically homogeneous and isotropic jammed structures [3]. In practice, as the MRJ state is approached the ability of hard spheres to move (''rattle'' [3] and ''flip' ' [5] for monoatomic and chain systems, respectively) without incurring overlaps declines precipitously. Regarding random assemblies of chains of hard spheres, connectivity endows them with a rich physical behavior. The problem of densely packed polymers is of vital importance in thermodynamics, biology, phase transitions, glassy state, colloids, granular media, combinatorics, and perturbation theory. Recently, the computational challenge of determining the MRJ state could be solved, albeit for short chains only [5]. Thus, universal scaling, and asymptotic behavior in the infinite chain length (N) limit, as predicted by Edwards [6] and de Gennes [6] could only be conjectured.In this Letter we determine, through extensive simulations, the MRJ state for strongly entangled freely jointed chains of tangent hard spheres (where bond length is fixed and equal to sphere diameter), deep in the asymptotic regime. This MRJ state is determined by means of large scale [Oð10 11 Þ steps], off-lattice, Monte Carlo simulations using a suite of powerful importance sampling, chainconnectivity altering algorithms [7] on a number of bulk systems ranging from 100 chains of N ¼ 12 to 54 chains of N ¼ 1000. Preliminary tests on cells of different sizes revealed the absence of system size effects on all calculated properties.We show that, within statistical uncertainty, hard-sphere chains reach their MRJ state at the same volume fraction (') as single spheres, ' MRJ % 0:638 AE 0:004, irrespective of chain length. The question is thus settled that neither chain length nor the tangency constraint of connectivity hinder random packing of chains with respect to assemblies of single spheres. This alone is an important resul...

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Universal Scaling, Entanglements, and Knots of Model Chain Molecules

et al. 2008

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“…Independent efforts by Kröger and co-workers 15,16 and Tzoumanekas and Theodorou 17 employing geometrical rather than dynamical operations to calculate the PP helped gain additional insight into the statistical properties of the PP mesh ͑entanglement length, PP length, PP potential, etc.͒ for a number of linear polymers and compare against analytic expressions derived from statistical mechanics for a chain that is random walk with randomly scattered entanglements. 18 We provide here an answer to question ͑b͒, namely the systematic calculation of the function ͑s , t͒ which makes the connection between primitive chain dynamics on one hand and macroscopic viscoelastic properties and theoretical models on the other hand, for an entangled polymeric liquid. We achieve this by introducing a methodology that reduces trajectories from detailed ͑and very long͒ atomistic molecular dynamics ͑MD͒ simulations to trajectories of PPs, followed by a geometric and dynamical mapping onto the tube model.…”

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Quantifying chain reptation in entangled polymer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model

Stephanou

Baig

Tsolou

et al. 2010

The Journal of Chemical Physics

107

153

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Articles you may be interested inTemperature dependent micro-rheology of a glass-forming polymer melt studied by molecular dynamics simulation J. Chem. Phys. 141, 124907 (2014) The topological state of entangled polymers has been analyzed recently in terms of primitive paths which allowed obtaining reliable predictions of the static ͑statistical͒ properties of the underlying entanglement network for a number of polymer melts. Through a systematic methodology that first maps atomistic molecular dynamics ͑MD͒ trajectories onto time trajectories of primitive chains and then documents primitive chain motion in terms of a curvilinear diffusion in a tubelike region around the coarse-grained chain contour, we are extending these static approaches here even further by computing the most fundamental function of the reptation theory, namely, the probability ͑s , t͒ that a segment s of the primitive chain remains inside the initial tube after time t, accounting directly for contour length fluctuations and constraint release. The effective diameter of the tube is independently evaluated by observing tube constraints either on atomistic displacements or on the displacement of primitive chain segments orthogonal to the initial primitive path. Having computed the tube diameter, the tube itself around each primitive path is constructed by visiting each entanglement strand along the primitive path one after the other and approximating it by the space of a small cylinder having the same axis as the entanglement strand itself and a diameter equal to the estimated effective tube diameter. Reptation of the primitive chain longitudinally inside the effective constraining tube as well as local transverse fluctuations of the chain driven mainly from constraint release and regeneration mechanisms are evident in the simulation results; the latter causes parts of the chains to venture outside their average tube surface for certain periods of time. The computed ͑s , t͒ curves account directly for both of these phenomena, as well as for contour length fluctuations, since all of them are automatically captured in the atomistic simulations. Linear viscoelastic properties such as the zero shear rate viscosity and the spectra of storage and loss moduli obtained on the basis of the obtained ͑s , t͒ curves for three different polymer melts ͑polyethylene, cis-1,4-polybutadiene, and trans-1,4-polybutadiene͒ are consistent with experimental rheological data and in qualitative agreement with the double reptation and dual constraint models. The new methodology is general and can be routinely applied to analyze primitive path dynamics and chain reptation in atomistic trajectories ͑accumulated through long MD simulations͒ of other model polymers or polymeric systems ͑e.g., bidisperse, branched, grafted, etc.͒; it is thus believed to be particularly useful in the future in evaluating proposed tube models and developing more accurate theories for entangled systems.

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“…The tube model provides a basis for working out analytically the linear and nonlinear viscoelasticty of polymer melts 9,[16][17][18][19][20] and networks 10,15,[21][22][23][24] at a molecular level. Slip link models 19,20,24 also allow for analytical 8,[11][12][13]15,25,26 or stochastic simulation 14,[27][28][29][30][31][32][33][34][35][36][37] treatments with a molecular representation, at the single or many-chain level.…”

Section: Introductionmentioning

confidence: 99%

Microscopic Description of Entanglements in Polyethylene Networks and Melts: Strong, Weak, Pairwise, and Collective Attributes

2012

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Abstract:We present atomistic molecular dynamics simulations of two Polyethylene systems where all entanglements are trapped: a perfect network, and a melt with grafted chain ends. We examine microscopically at what level topological constraints can be considered as a collective entanglement effect, as in tube model theories, or as certain pairwise uncrossability interactions, as in slip-link models.A pairwise parameter, which varies between these limiting cases, shows that, for the systems studied, the character of the entanglement environment is more pairwise than collective. We employ a novel methodology, which analyzes entanglement constraints into a complete set of pairwise interactions, similar to slip links. Entanglement confinement is assembled by a plethora of links, with a spectrum of confinement strengths, from strong to weak. The strength of interactions is quantified through a link 'persistence', which is the fraction of time for which the links are active. By weighting links according to their strength, we show that confinement is imposed mainly by the strong ones, and that the weak, trapped, uncrossability interactions cannot contribute to the low frequency modulus of an elastomer, or the plateau modulus of a melt. A selfconsistent scheme for mapping topological constraints to specific, strong binary links, according to a given entanglement density, is proposed and validated. Our results demonstrate that slip links can be viewed as the strongest pairwise interactions of a collective entanglement environment. The methodology developed provides a basis for bridging the gap between atomistic simulations and mesoscopic slip link models.

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Analytic Expressions for the Statistics of the Primitive-Path Length in Entangled Polymers

Cited by 43 publications

References 20 publications

Universal Scaling, Entanglements, and Knots of Model Chain Molecules

Universal Scaling, Entanglements, and Knots of Model Chain Molecules

Quantifying chain reptation in entangled polymer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model

Microscopic Description of Entanglements in Polyethylene Networks and Melts: Strong, Weak, Pairwise, and Collective Attributes

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