Coexistence of Crumpling and Flat Sheet Conformations in Two-Dimensional Polymer Networks: An Understanding of Aggrecan Self-Assembly

Alexandros Chremos and Ferenc Horkay

Phys. Rev. Lett. 131, 138101 – Published 28 September 2023

Abstract

We investigate the conformational properties of self-avoiding two-dimensional (2D) ideal polymer networks with tunable mesh sizes as a model of self-assembled structures formed by aggrecan. Polymer networks having few branching points and large enough mesh tend to crumple, resulting in a fractal dimension of $d_{f} \approx 2.7$ . The flat sheet behavior ( $d_{f} = 2$ ) emerges in 2D polymer networks having more branching points at large length scales; however, it coexists with crumpling conformations at intermediate length scales, a feature found in scattering profiles of aggrecan solutions. Our findings bridge the long-standing gap between theories and simulations of polymer sheets.

Received 11 January 2023
Revised 12 July 2023
Accepted 8 September 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.138101

Published by the American Physical Society

Physics Subject Headings (PhySH)

Aggregation Biological self-organization Clustering Polymer conformation changes

Polymers & Soft MatterPhysics of Living Systems

Authors & Affiliations

Alexandros Chremos ^*,† and Ferenc Horkay

Section on Quantitative Imaging and Tissue Sciences, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, Maryland 20892, USA

^*alexandros.chremos@nih.gov
^†Present address: Chemical Sciences Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899–8320, USA.

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Vol. 131, Iss. 13 — 29 September 2023

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Images

Figure 1
(a) Schematic of a regular 2D polymer network having $N_{b} = 4$ branches in each direction. Screenshots of typical equilibrated 2D polymer networks are also presented. For clarity, the repeating units located at the edge of the polymer network are in blue and for the units in the middle in red. (b) Radius of gyration $R_{g}$ of 2D polymer networks as a function of molecular mass $M_{w}$ . The highlighted regimes approximately outline the emergence of crumpling (pale yellow) and flat sheet regimes (pale blue). The dotted lines are power laws with an exponent of 0.588 and $1 / 2$ and the dashed line is a power law with an exponent of $1 / 2.7$ ; all lines are guides for the eye. The uncertainty estimates correspond to two standard deviations. Inset: $R_{g}$ normalized by $M_{w}^{1 / 2}$ and $l_{0}$ as a function of $N_{b}$ . The dashed line is $R_{g} \approx l_{0} M_{w}^{1 / 2} (1.3 / N_{b}^{1.3} + 1)$ .
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Figure 2
Form factor $P (q)$ of polymer networks having two different mesh sizes. The highlighted regimes approximately outline the regions at which the polymer networks exhibit common structural features and where they deviate due to differences in the internal structure of these polymer networks. The exponent $λ$ is a parameter to overlap the normalized $P (q)$ curves in the low $q$ regime.
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Figure 3
(a) Form factor $P (q)$ of polymer networks having chain length $M = 20$ based on our simulation model. The highlighted regimes approximately outline the different power-law regimes that the polymer networks exhibit. The dashed lines indicate power laws as guides to the eye. The drawings illustrate structural features probed at different length scales in 2D polymer networks. The vertical arrows approximately point to the size of the corresponding polymer network, $q σ = 2 π / R_{g}$ . (b) Combined static light scattering (SLS) and small angle neutron scattering (SANS) measurements of the scattering from 0.1% ( $w / w$ ) aggrecan solutions with no salt (circles) and 100 mM ${CaCl}_{2}$ (squares). The drawings illustrate aggrecan assemblies probed at different length scales; an outline of the self-assembled structures is drawn for the case of low $q$ to highlight the 2D polymer network formation. The experimental data are from Refs. [44, 45].
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Figure 4
Ratio of the hydrodynamic radius over the radius of gyration $R_{h} / R_{g}$ as a function of the number of branched points in each direction of the 2D polymer network $N_{b}$ . From top to bottom, the dotted lines correspond to the exact results of $R_{h} / R_{g}$ for uniform spheres, circular discs, and an estimate of the ratio for random walks and percolation clusters [69]. The dashed arrow points to the plateau reached by large molecular mass polymer sheets $R_{h} / R_{g} \approx 0.92$ . The uncertainty estimates correspond to one standard deviation.
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