What is the entanglement length in a polymer melt?

Abstract

We present the results of molecular dynamics simulations of very long model polymer chains analyzed by various experimentally relevant techniques. The segment motion of the chains is found to be in very good agreement with the reptation model. We also calculated the plateau modulus G 0 N . The predictions of the entanglement length Ne from G 0 N and from the mean square dispacement of the chain segments disagree by a factor of about 2.2(2), indicating an error in the prefactor in the standard formula for G 0 N . We show that recent neutron spin echo measurements were carried out for chain lengths which are too small to allow for a correct determination of Ne.How an entangled polymer chain moves in a dense melt of other chains has been a long standing problem. The most widely accepted picture is that the chains reptate like a snake [1,2]. For short times chains move isotropically until they feel the constraints of their neighboring chains. For intermediate times, the chain segments move along the path or tube created by the surrounding chains in a Rouse-like [3] motion. Only the ends explore new space. For the inner section of the chain, this Rouse-like motion on a contour which is also a random walk gives rise to the famous t 1/4 power law regime in the mean-square displacement of the beads g 1 (t). For very long time, the motion is diffusive, with a chain diffusion constant D that scales as N −2 for large N , where N is the chain length. For short chains, the motion is much simpler and can be approximately described by the Rouse model and D ∼ N −1 . To characterize the crossover between the Rouse and reptation regime, one can define an entanglement length N e . Within the reptation model, N e can be related to both the tube diameter d T and crossover time τ e from the early time Rouse regime where g 1 (t) ∼ t 1/2 to the t 1/4 regime as well as from the value of the plateau modulus G o N . Over the years there have been a number of experiments [4][5][6] and simulations [7][8][9][10][11] designed to test various aspects of the theory. Recent neutron spin echo (NSE) scattering experiments [6] which measure the dynamic structure factor S(k, t), suggest that N e as measured from d T is consistent with that determined from G 0 N . Our previous simulation results [7,9] suggested an inconsistency in that N e measured from g 1 (t) was about one half that measured from S(k, t), though both g 1 (t) and S(k, t) are single chain quantities and measure the same motion. However since the chains were only a few N e (N ≤ 200), our results were not conclusive. We have now extended our simulations to much longer chains, as long as N = 10000, and measured not only single chain quantities but also G 0 N . We find clear evidence for differences on the order two between N e measured from g 1 (t) or S(k, t) and that from G 0 N . The previous reported agreement in N e determined from NSE data for S(k, t) and G 0 N is shown to be largely due to finite chain length effects in d T as determined from S(k, t).To overcome the long time scale...

Entanglements in quiescent and sheared polymer melts

“…Our value of ζ is about twice larger than in the previous simulations with nσ 3 = 0.85 (see Ref. [13]). We then discuss the case of N = 250, for which our data give τ r /τ R ∼ 4.6 × 10 4 /(0.027 × 250 2 ) ∼ = 3.6 and the theoretical results in eq.9 give τ d /τ R = 750/N e .…”

“…At this temperature there was no glass-like enhancement of the structural relaxation time, but at T = 0.2ǫ/k B the present model with N = 10 became glassy in our previous simulation [24]. Note that our density value is higher than the widely used value n = 0.85/σ 3 in the previous simulations [10,13,[15][16][17][18]25] (With increasing n the free volume for particles decreases and hence N e in our case should be somewhat shorter than in the previous simulations). The consecutive beads on each chain are connected by an anharmonic spring potential of the form,…”

“…The reptation theory [2][3][4][5][6][7] is the most successful approach to date in describing the dynamics of entangled polymer chains in a surprisingly simple manner. It has been supported by a number of experimental [8,9] and numerical [10][11][12][13][14][15][16][17][18][19][20] papers, where experimentally accessible quantities such as the stress relaxation function G(t), the incoherent dynamic scattering function, the mean square displacement, and the viscosity have been compared with the theoretical predictions [3]. However, N e estimated from the mean square displacement and that from the plateau modulus were around 30 and 70, respectively [13], where the difference of these two estimations indicates inaccuracy of the prefactors in the predicted formulae [3,21].…”

Fast convergence to equilibrium for long-chain polymer melts using a MD/continuum hybrid method

“…For the case of initially stretched polymers, we derive the mean square internal distance R(n) 2 , averaged over all internal distances n = |i − j| along all the polymer chains, where i < j ∈ [1, N] are the monomer indices. It is shown in Fig.…”

“…1 The length of the chain polymer is, however, limited to the order of 10 4 monomers even with the CG approach. 2 A variety of method has been proposed for obtaining well-equilibrated CG polymer. Auhl et al used the initial configuration reducing the density fluctuation and a doublebridging algorithm 4 for a MD calculation.…”