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Colloidal polymers have attracted increasing attention in condensed physics, statistical mechanics and polymer science and engineering due to their advances in synthesis and visualization. Many useful properties and applications of colloidal polymers make them an ideal model to explore fundamental problems in slow dynamics and rheology of chain-like molecules in supercooled regime. With temperature decreasing or density rapidly increasing, amorphous materials often exhibit nonzero shear moduli. In this article, we are to investigate the nonzero shear modulus and bulk modulus of colloidal polymer in supercooled regime based on recent microscopic theoretical development. At the segmental-scale level, an analytical derivation for elastic modulus of colloidal polymer is constructed based on the standard approximation in naïve mode-coupling theory (NMCT). In the framework of nonlinear Langevin equation theory (NLET), the derivation combines the concept of dynamic free energy, localization and NMCT crossover volume fraction. Taking the chain connectivity into account, an explicit expression for shear modulus including intrachain structure factor, interchain correlation and localized length is formulated. Bulk modulus can be obtained by relating it to long wavelength part of static structure factor. Firstly, our calculation for discrete wormlike chain shows that intrachain structure factor has a power law decay at intermediate wavevector which is similar to flexible linear chain. Secondly, we find that colloidal polymer with long bond length has a lower NMCT crossover volume fraction. Furthermore, inputting the localized length, long wavelength density fluctuation and static intrachain and interchain structures into the theoretical expression, the effect of bond length on shear modulus and bulk modulus are investigated. Interestingly, we find the bond length plays a critical and unique role in localized length and bulk modulus. For instance, when the supercooling degree is used as an independent variable, the local length and bulk elastic modulus of the chain with the same bond length can be collapsed onto a universal curve, which is independent of chain length and local bending energy. However, in the aspect of shear modulus, the bond length is not a unique quantity and the above universal curve cannot be found. The shear modulus depends on other parameters of chain, such as chain length and rigidity. According to the universal behavior of zero-wavevector static structure factor versus bond length, we guess that the nonuniversal curve of shear modulus is due to the bond length effect on long wavevector static structure factor. This work provides a theoretical foundation for controlling various properties of chain-like supercooled materials in the future.
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Keywords:
- colloidal polymer /
- elastic modulus /
- nonlinear Langevin equation /
- polymer glass
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Google Scholar
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图 1 胶体聚合物模型的示意图, 包含了模型中3个连续单体(蓝球)和关键的尺度(半径和键长)与键角
Figure 1. Schematic of colloidal polymers. Blue spheres represent three consecutive monomers with diameter
$ \sigma $ , bond angle$ \theta $ and bond length$ l $ .
图 2 静态结构 (a) 在不同键长下的单链结构因子, 虚点线显示在中级波矢范围满足幂律衰减
$\sim {{k}}^{-2}$ ; (b) 不同键长下的径向分布函数Figure 2. Static Structure functions: (a) Intrachain structure factor for different bond lengths (dashed-dotted line shows a power law decay
$\sim\!{{k}}^{-2}$ at intermediate wavevector); (b) the radial distribution functions for different bond lengths.
图 3 不同链内弯曲能下, 玻璃化转变体积分数随着链长的变化
Figure 3. Crossover volume fraction as a function of bond length for different bending energies.
图 4 (a) 不同链内弯曲能和键长的局域尺寸随着玻璃化深度的变化, 绿线是硬球液体的局域尺寸; (b) 不同链内弯曲能和键长的剪切弹性模量随着玻璃化深度
$ \phi -{\phi }_{\mathrm{c}} $ 的变化, 绿线是硬球的数据Figure 4. (a) Localization length as a function of
$ \phi -{\phi }_{\mathrm{c}} $ for different bending energies and bond lengths. Green line represents localization length for hard sphere liquids. (b) shear modulus as a function of$ \phi -{\phi }_{\mathrm{c}} $ for different bending energies and bond lengths. Green line represents shear modulus for hard sphere liquids.
图 5 (a) 不同链内弯曲能和键长下, 体积弹性模量随着玻璃化转变深度的变化. 绿色线代表硬球液体的体积弹性模量. (a)和(b)的图例是一致的. (b) 不同链内弯曲能和键长下, 静态结构因子的零波矢数值随着玻璃化转变深度的变化.
Figure 5. (a) Bulk modulus as a function of
$ \phi -{\phi }_{\mathrm{c}} $ for different bending energies and bond lengths. Green line represents bulk modulus for hard sphere liquid. The legend is the same as in panel (b). (b) Static structure factor at zero wavevector for different bending energies and bond lengths. Green line represents corresponding data for hard sphere liquid. -
[1] Berthier L, Biroli G 2011 Rev. Mod. Phys. 83 587
Google Scholar
[2] Langer J S 2014 Rep. Prog. Phys. 77 042501
Google Scholar
[3] Stillinger F H, Debenedetti P G 2013 Annu. Rev. Condens. Matter Phys. 4 263
Google Scholar
[4] Biroli G, Urbani P 2016 Nat. Phys. 12 1130
Google Scholar
[5] Hill L J, Pinna N, Char K, Pyun J 2015 Prog. Polym. Sci. 40 85
Google Scholar
[6] Yang M, Chen G, Zhao Y, Silber G, Wang Y, Xing S, Han Y, Chen H 2010 Phys. Chem. Chem. Phys. 12 11850
Google Scholar
[7] Zhao Y, Xu L, Liz-Marzán L M, Kuang H, Ma W, Asenjo-Garcıa A, García de Abajo F J, Kotov N A, Wang L, Xu C 2013 J. Phys. Chem. Lett. 4 641
Google Scholar
[8] Hunter G L, Weeks E R 2012 Rep. Prog. Phys. 75 066501
Google Scholar
[9] Gotze W 2008 Complex Dynamics of Glass-forming liquids: A Mode-coupling Theory (New York: Oxford University Press) pp177−209
[10] Zwanzig R 2001 Nonequilibrium Statistical Mechanics (New York: Oxford University Press) pp163−165
[11] Schweizer K S, Saltzman E J 2003 J. Chem. Phys. 119 1181
Google Scholar
[12] Schweizer K S, Saltzman E J 2004 J. Chem. Phys. 121 1984
Google Scholar
[13] Chen K, Saltzman E J, Schweizer K S 2010 Annu. Rev. Condens. Matter Phys. 1 277
Google Scholar
[14] Kobelev V, Schweizer K S 2005 Phys. Rev. E 71 021401
Google Scholar
[15] Chen K, Schweizer K S 2009 Phys. Rev. Lett. 102 038301
Google Scholar
[16] Schweizer K S, Curro J G 1997 Adv. Chem. Phys. 98 1
Google Scholar
[17] Chen K, Schweizer K S 2007 J. Chem. Phys. 126 014904
Google Scholar
[18] Chen K, Saltzman E J, Schweizer K S 2009 J. Phys. Condens. Matter 21 503101
Google Scholar
[19] Martin T B, Gartner T E, Jones R L, Snylder C R, Jayaraman A 2018 Macromolecules 51 2906
Google Scholar
[20] Kulshreshtha A, Jayaraman A 2020 Macromolecules 53 4 014904
Google Scholar
[21] Dell Z E, Schweizer K S 2018 Soft Matter 14 9132
Google Scholar
[22] Gartner T E, Haque F M, Gomi A M, Grayson S M, Hore M J, Jayaraman A 2019 Macromolecules 52 4579
Google Scholar
[23] Zhou Y, Schweizer K S 2020 Macromolecules 53 22
Google Scholar
[24] Hooper J, Schweizer K S 2006 Macromolecules 39 5133
Google Scholar
[25] Zhou Y, Schweizer K S 2020 J. Chem. Phys. 153 114901
Google Scholar
[26] Schweizer K S 2005 J. Chem. Phys. 123 244501
Google Scholar
[27] Cheng S, Xie S, Carrillo J Y, Carroll B, Martin H, Cao P, Dadmun M D, Sumpter B G, Novikov V N, Schweizer K S, Sokolov A P 2017 ACS Nano 11 1
Google Scholar
[28] Zhang B K, Li H, Li J, Chen K, Tian W D, Ma Y Q 2016 Soft Matter 12 8104
Google Scholar
[29] Honnell K, Curro J G, Schweizer K S 1990 Macromolecules 23 3496
Google Scholar
[30] Hansen J P, McDonald I R 2013 Theory of Simple Liquids (Elsevier: Academic Press) pp105−145
[31] Zhang B K, Li J, Chen K, Tian W D, Ma Y Q 2016 Chin. Phys. B 25 116101
Google Scholar
[32] Li J, Zhang B K 2020 Europhys. Lett. 130 56001
Google Scholar
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