Perimeter length and form factor in two-dimensional polymer melts

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Perimeter length and form factor in two-dimensional polymer melts

H. Meyer, T. Kreer, M. Aichele, A. Cavallo, A. Johner, J. Baschnagel, and J. P. Wittmer

Phys. Rev. E 79, 050802(R) – Published 21 May 2009

Abstract

Self-avoiding polymers in two-dimensional $(d = 2)$ melts are known to adopt compact configurations of typical size $R (N) \sim N^{1 / d}$ , with $N$ being the chain length. Using molecular-dynamics simulations we show that the irregular shapes of these chains are characterized by a perimeter length $L (N) \sim R {(N)}^{d_{p}}$ of fractal dimension $d_{p} = d - Θ_{2} = 5 / 4$ , with $Θ_{2} = 3 / 4$ being a well-known contact exponent. Due to the self-similar structure of the chains, compactness and perimeter fractality repeat for subchains of all arclengths $s$ down to a few monomers. The Kratky representation of the intramolecular form factor $F (q)$ reveals a strong nonmonotonous behavior with $q^{2} F (q) \sim 1 / {(q N^{1 / d})}^{Θ_{2}}$ in the intermediate regime of the wave vector $q$ . Measuring the scattering of labeled subchains the form factor may allow to test our predictions in real experiments.

Received 24 June 2008

DOI:https://doi.org/10.1103/PhysRevE.79.050802

Authors & Affiliations

H. Meyer, T. Kreer, M. Aichele, A. Cavallo, A. Johner, J. Baschnagel, and J. P. Wittmer

Institut Charles Sadron, 23 rue du Loess, BP 84047, 67034 Strasbourg Cedex 2, France

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Vol. 79, Iss. 5 — May 2009

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Images

Figure 1
(Color online) A snapshot of a melt configuration with chain length $N = 1024$ and monomer density $ρ = 7 / 8$ can be seen in the left panel. Only the perimeter monomers interacting with other chains are indicated. The numbers refer to an arbitrary chain index used for computational purposes. The chains are compact, i.e., they fill space densely, but compactness does not imply a disklike shape (see, e.g., chains 3 and 8). The main figure presents the end-to-end distance $R (s) = {⟨ r^{2} ⟩}^{1 / 2}$ , the radius of gyration $R_{g} (s)$ , and the mean perimeter length $L (s)$ of segments containing $s = m - n + 1$ monomers as indicated by the sketch on the right. The full symbols refer to overall chain properties $(s = N)$ . The solid lines confirm the exponent $ν = 1 / 2$ for the segment size, the dashed line confirms the scaling of $L (s)$ suggested by Eq. (1).Reuse & Permissions
Figure 2
(Color online) Scaling plot of various distributions of the end-to-end vector $r = r_{m} - r_{n}$ of chain segments of chains of length $N = 1024$ (stars) and $N = 2048$ (triangle): $G_{0} (r, N)$ for $n = 1$ and $m = N$ , $G_{1} (r, N)$ for $n = 1$ and $m = N / 2$ , $G_{2} (r, N)$ for $n = N / 4$ and $m = 3 N / 4$ . $G_{0} (r, N)$ and $G_{1} (r, N)$ are shifted vertically for clarity. The segmental size distribution $G (r, s)$ averaging over all pairs $(n, m)$ given for $N = 1024$ with $s = 256$ (squares) and $s = 512$ (spheres) scales as $G_{2} (r, N)$ . The thin lines indicate the Gaussian distribution $y = exp (- x^{2}) / π$ expected for ideal chains in two dimensions. The power laws $y \approx x^{Θ_{i}}$ (dashed lines) observed for $x ⪡ 1$ have been predicted by Duplantier [2]. The Redner–des Cloizeaux formula for $G_{2} (r, N)$ is indicated by a solid line.Reuse & Permissions
Figure 3
(Color online) Bond-bond correlation functions $P_{1} (s) = ⟨ e_{n} \cdot e_{m} ⟩$ and $P_{2} (s) = ⟨ {(e_{n} \cdot e_{m})}^{2} ⟩ - 1 / 2$ vs curvilinear distance $s - 1$ between normalized bond vectors $e_{n}$ and $e_{m}$ . The first Legendre polynomial $P_{1} (s)$ (inset) shows an anticorrelation at $s \approx 12$ . The second Legendre polynomial $P_{2} (s)$ decays over 2 orders in magnitude as a power law (dashed line) with an exponent $1 + ν Θ_{2} = 11 / 8$ in agreement with the return probability calculated from Eq. (2).Reuse & Permissions
Figure 4
(Color online) Kratky representation of the intramolecular form factor $F (q)$ as a function of $x = q R_{g} (N)$ for different chain lengths $N$ using the same symbols as in Fig. 3. The Debye formula (thin line) corresponds to a chain length independent plateau for $x ⪢ 1$ . By contrast to this, a strong nonmonotonous behavior is revealed by our data which approaches with increasing $N$ a power-law exponent $- Θ_{2} = - 3 / 4$ (dashed line) corresponding to a compact object of fractal line dimension $d_{p} = d - Θ_{2} = 5 / 4$ . Also included is the Porod scattering expected for a compact 2D object with smooth perimeter (dash-dotted line) and the Fourier transform of the Redner–des Cloizeaux approximation (solid line). The increase in the scattering for large $q$ (Bragg peak) is due to the packing of the beads on local scale. Only in this limit does $F (q)$ become chain length independent.Reuse & Permissions

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