Intramolecular long-range correlations in polymer melts: The segmental size distribution and its moments

J. P. Wittmer, P. Beckrich, H. Meyer, A. Cavallo, A. Johner, and J. Baschnagel

Phys. Rev. E 76, 011803 – Published 12 July 2007

Abstract

We present theoretical arguments and numerical results to demonstrate long-range intrachain correlations in concentrated solutions and melts of long flexible polymers, which cause a systematic swelling of short chain segments. They can be traced back to the incompressibility of the melt leading to an effective repulsion $u (s) \approx s ∕ ρ R^{3} (s) \approx c_{e} ∕ \sqrt{s}$ when connecting two segments together where $s$ denotes the curvilinear length of a segment, $R (s)$ its typical size, $c_{e} \approx 1 ∕ ρ b_{e}^{3}$ the “swelling coefficient,” $b_{e}$ the effective bond length, and $ρ$ the monomer density. The relative deviation of the segmental size distribution from the ideal Gaussian chain behavior is found to be proportional to $u (s)$ . The analysis of different moments of this distribution allows for a precise determination of the effective bond length $b_{e}$ and the swelling coefficient $c_{e}$ of asymptotically long chains. At striking variance to the short-range decay suggested by Flory’s ideality hypothesis the bond-bond correlation function of two bonds separated by $s$ monomers along the chain is found to decay algebraically as $1 ∕ s^{3 ∕ 2}$ . Effects of finite chain length are briefly considered.

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Received 12 April 2007

DOI:https://doi.org/10.1103/PhysRevE.76.011803

Authors & Affiliations

J. P. Wittmer^*, P. Beckrich, H. Meyer, A. Cavallo, A. Johner, and J. Baschnagel^†

Institut Charles Sadron, CNRS, 23 rue du Loess, 67037 Strasbourg Cédex, France

^*jwittmer@ics.u-strasbg.fr
^†URL: http://www-ics.u-strasbg.fr/∼etsp/welcome.php

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Vol. 76, Iss. 1 — July 2007

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Figure 1
(Color online) Sketch of a polymer chain of length $N$ in a dense melt in $d = 3$ dimensions. As notations we use $r_{i}$ for the position vector of a monomer $i$ , $l_{i} = r_{i + 1} - r_{i}$ for its bond vector $r = r_{m} - r_{n}$ for the end-to-end vector of the chain segment between the monomers $n$ and $m = n + s$ and $r = ∥ r ∥$ for its length. Segment properties, such as the $2 p$ -th moments $⟨ r^{2 p} ⟩$ , are averaged over all possible pairs of monomers $(n, m)$ of a chain and over all chains. The second moment $(p = 1)$ is denoted $R_{e} (s) = {⟨ r^{2} ⟩}^{1 ∕ 2}$ , the total chain end-to-end distance is $R_{e} (s = N - 1)$ . The dashed lines show the relevant graphs of the analytical perturbation calculation outlined in Sec. 2B. The numerical factors indicate for infinite chains (without chain end effects) the relative weights contributing to the $1 ∕ \sqrt{s}$ swelling of $R_{e} (s)$ indicated in Eq. (10).Reuse & Permissions
Figure 2
(Color online) Role of incompressibility and chain connectivity in dense polymer solutions and melts. (a) Sketch of the segmental correlation hole of a marked chain segment of curvilinear length $s$ . Density fluctuations of chain segments must be correlated, since the total density fluctuations (dashed line) are small. Consequently, a second chain segment feels an entropic repulsion when both correlation holes start to overlap. (b) Self-similar pattern of nested segmental correlation holes of decreasing strength $u (s) \approx s ∕ ρ R {(s)}^{3} \approx c_{e} ∕ \sqrt{s}$ aligned along the backbone of a reference chain. The large dashed circle represents the classical correlation hole of the total chain $(s \approx N)$ [5]. This is the input of recent approaches to model polymer chains as soft spheres [50, 52]. We argue that incompressibility on all scales and chain connectivity leads to a short distance repulsion of the segmental correlation holes, which increases with decreasing $s$ .Reuse & Permissions
Figure 3
(Color online) Diffusion time $τ_{e}$ over the (root-mean-squared) chain end-to-end distance $R_{e} (N - 1)$ as a function of chain length $N$ for different versions of the bond fluctuation model (BFM). All data indicated are for the high number density $(ρ = 0.5 ∕ 8)$ corresponding to a polymer melt with half the lattice sites being occupied. We have obtained $τ_{e} = R_{e}^{2} (N - 1) ∕ 6 D_{s}$ from the self-diffusion coefficient $D_{s}$ measured from the free diffusion limit of the mean-squared displacement of all monomers $⟨ δ r {(t)}^{2} ⟩ = 6 D_{s} t$ . Data from the classical BFM with topology conserving local Monte Carlo (MC) moves in six spatial directions (L06) [26] are represented as stars. All other data sets use topology violating local MC moves in 26 lattice directions (L26). If only local moves are used, L26 dynamics is even at relatively short times perfectly Rouse-like which allows the accurate determination of $D_{s}$ although the monomers possibly have not yet moved over $R_{e} (N - 1)$ for the largest chain lengths considered. Additional slithering snake (SS) moves increase the efficiency of the algorithm by approximately an order of magnitude (squares, bold line). The power law exponent is changed from 2 to an empirical 1.62 (dashed line) if in addition we perform double-bridging (DB) moves.Reuse & Permissions
Figure 4
(Color online) Mean-squared segment size $R_{e} {(s)}^{2} ∕ s$ vs curvilinear distance $s$ . We present BFM data for different chain length $N$ at number density $ρ = 0.5 ∕ 8$ . The averages are taken over all possible monomer pairs $(n, m = n + s)$ . The statistics deteriorates, hence, for large $s$ . Log-linear coordinates are used to emphasize the power law swelling over several orders of magnitude of $s$ . The data approach the asymptotic limit (horizontal line) from below, i.e., the chains are swollen. This behavior is well fitted by Eq. (5) for $1 ⪡ s ⪡ N$ (bold line). Nonmonotonic behavior is found for $s \to N$ , especially for small $N$ . The dashed line indicates the measured total chain end-to-end distances $b_{e} {(N)}^{2} \equiv R_{e} {(N - 1)}^{2} ∕ (N - 1)$ from Table , showing even more pronounced deviations from the asymptotic limit. The dash-dotted line compares this data with Eq. (19).Reuse & Permissions
Figure 5
(Color online) Replot of the mean-squared segment size as $y = K_{1} (s) = 1 - R_{e} {(s)}^{2} ∕ b_{e}^{2} s$ vs $x = c_{1} ∕ \sqrt{s}$ , as suggested by Eq. (5), for different trial effective bond lengths $b_{e}$ as indicated. Only BFM chains of length $N = 2048$ are considered for clarity. This procedure is very sensitive to the value chosen and allows for a precise determination. It assumes, however, that higher order terms in the expansion of $K_{1} (s)$ may be neglected. The value $b_{e}$ is confirmed from a similar test for higher moments (Fig. 6).Reuse & Permissions
Figure 6
(Color online) Critical test of Eq. (5) where the rescaled moments $y = K_{p} (s)$ of the segment size distribution [defined in Eq. (1)] are plotted vs $x = \frac{3 {(2^{p} p! p)}^{2}}{2 (2 p + 1)!} \frac{c_{p}}{\sqrt{s}}$ . We consider the first five even moments $(p = 1, \dots, 5)$ for the BFM with $N = 2048$ and the BSM with $N = 1024$ . Also indicated is the rescaled radius of gyration $y = 5 ∕ 8 [1 - 6 R_{g}^{2} (s) ∕ b_{e}^{2} (s + 1)]$ , as a function of $x = c_{1} ∕ \sqrt{s + 1}$ (filled circles). The BSM data has been shifted upwards for clarity. Without this shift a perfect data collapse is found for both models and all moments. Keeping the same effective bond length $b_{e}$ for all moments of each model we fit for the swelling coefficients $c_{p}$ by rescaling the horizontal axis. We find $b_{e} \approx 3.244$ for the BFM and 1.34 for the BSM. If $b_{e}$ is chosen correctly, all data sets extrapolate linearly to zero for large $s$ $(x \to 0)$ . The swelling coefficients found are close the theoretical prediction $c_{e}$ , as indicated in Table . The plot demonstrates that the non-Gaussian deviations scale as the segmental correlation hole, $u (s) \sim c_{e} ∕ \sqrt{s}$ and this for all moments as long as $x ⪡ 1$ . The saturation at large $x$ is due to the finite extensibility of short chain segments. Since this effect becomes more marked for larger moments, the fit of $b_{e}$ is best performed for $p = 1$ .Reuse & Permissions
Figure 7
(Color online) Plot of $K_{λ} (s)$ as a function of $u (s) c_{1} ∕ c_{e} \sim 1 ∕ \sqrt{s}$ using the measured $u (s) \equiv \sqrt{24 ∕ π^{3}} s ∕ ρ R_{e} {(s)}^{3}$ . For $λ = 2$ (corresponding to two segments being connected) BFM and BSM data are compared. Several $λ$ values are given for $N = 2048$ BFM chains. For chain segments with $1 ⪡ s ⪡ N$ all data sets collapse on the bisection line confirming the so-called “recursion relation” $K_{λ} \approx u$ proposed by Semenov and Johner [12]. The statistics becomes insufficient for large $s$ (left bottom corner). Systematic deviations arise for $s \to N$ due to additional finite- $N$ effects.Reuse & Permissions
Figure 8
(Color online) The bond-bond correlation function $P (s) ∕ c_{P}$ as a function of the curvilinear distance $s$ . Various chain lengths are given for BFM. Provided that $1 ⪡ s ⪡ N$ , all data sets collapse on the power law slope with exponent $ω = 3 ∕ 2$ (bold line) as predicted by Eq. (23). The dash-dotted curve $P (s) \approx \exp (- s ∕ 1.5)$ shows that exponential behavior is only compatible with very small chain lengths. The dashed lines correspond to the theoretical prediction, Eq. (24), for short chains with $N = 16$ , 32, 64, and 128 (from left to right).Reuse & Permissions
Figure 9
(Color online) Non-Gaussian parameter $α_{p} (s)$ computed for the end-to-end distance of chain segments as a function of $c_{e} ∕ \sqrt{s}$ . Perfect data collapse for all chain lengths and both simulation models is obtained for each $p$ . A linear relationship over nearly two orders of magnitude is found as theoretically expected. Data for three moments $(p = 2, 3, 4)$ are indicated showing a systematic increase of non-Gaussianity with $p$ . The data curvature for small $s$ becomes more pronounced for larger $p$ .Reuse & Permissions
Figure 10
(Color online) Plot of $K_{x y} (s) = 1 - ⟨ x^{2} y^{2} ⟩ ∕ ⟨ x^{2} ⟩ ⟨ y^{2} ⟩$ averaged over all pairs of monomers $(n, m = n + s)$ and three different direction pairs as a function of $c_{e} ∕ \sqrt{s}$ . As indicated by the sketch at the bottom of the figure, $K_{x y} (s)$ measures the correlation of the components of the segment vector $r$ . All data points collapse and show again a linear relationship $K_{x y} \approx u (s)$ . Different directions are therefore coupled! No curvature is observed over two orders of magnitude confirming that higher order perturbation corrections are negligible. Noise cannot be neglected for large $s > 100$ and finite segment-size effects are visible for $s \approx 1$ .Reuse & Permissions
Figure 11
(Color online) Segment size distribution $y = G (r, s) {(b_{e} s^{1 ∕ 2})}^{3}$ vs $n = r ∕ b_{e} s^{1 ∕ 2}$ for several $s$ as indicated in the figure. Only data for BFM with $N = 2048$ and BSM with $N = 1024$ are presented. [A similar plot can be achieved by renormalizing the axes using $R_{e} (s)$ instead of $b_{e} s^{1 ∕ 2}$ .] The bold line denotes the Gaussian behavior $y = {(3 ∕ 2 π)}^{3 ∕ 2} \exp (- 3 n^{2} ∕ 2)$ . One sees that compared to this reference the measured distributions are depleted for small $n ⪡ 1$ (where the data does not scale) and enhanced for $n \approx 1$ .Reuse & Permissions
Figure 12
(Color online) Deviation $δ G (r, s) = G (r, s) - G_{0} (r, s)$ of the measured segmental size distribution from the Gaussian behavior $G_{0} (r, s)$ expected from Flory’s hypothesis for several $s$ and both models as indicated in the figure. As suggested by Eq. (3), we have plotted $y = [δ G (r, s) ∕ G_{0} (r, s)] ∕ (c_{e} ∕ \sqrt{s})$ as a function of $n = r ∕ b_{e} \sqrt{s}$ . The Gaussian reference distribution has been computed according to Eq. (2) for the measured effective bond length $b_{e}$ . A close to perfect data collapse is found for both models. This shows that the deviation scales linearly with $u (s) \approx c_{e} ∕ \sqrt{s}$ , as expected. The bold line indicates the universal function of $f (n)$ predicted by Eq. (3).Reuse & Permissions
Figure 13
(Color online) Replot of the relative deviation of the measured segment size distribution $y = [δ G (r, s) ∕ G_{0} (r, s)] ∕ (c_{e} ∕ \sqrt{s})$ , as a function of $m = r ∕ R_{e} (s)$ . The figure highlights that the measured segment size is the only length scale relevant for describing the deviation from Flory’s hypothesis. The same data sets and symbols are used as in Fig. 12.Reuse & Permissions
Figure 14
Interaction diagrams used in reciprocal space for the calculation of $δ G (q, t)$ in the scale free limit. There exist three nonzero contributions to first-order perturbation, the first involving two points inside the segment [first two lines of Eq. (A4)], the second one point inside and one outside the segment [third line of Eq. (A4)] and the third one point on either side of the segment [last line of Eq. (A4)]. Momentum $q$ flows from one correlated point to the other. Integrals are performed over the momentum $k$ . Dotted lines denote the effective interactions $\tilde{v} (k)$ given by Eq. (9), bold lines the propagators which carry each a momentum $q$ or $q - k$ as indicated.Reuse & Permissions

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