搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

胶体聚合物弹性模量的微观理论: 键长的效应

张博凯

引用本文:
Citation:

胶体聚合物弹性模量的微观理论: 键长的效应

张博凯

Microscopic theory for elastic modulus of colloidal polymers: Effect of bond length

Zhang Bo-Kai
PDF
HTML
导出引用
  • 使用模耦合理论, 并结合非线性朗之万方程理论中的动力学自由能、局域尺寸和玻璃化转变点的概念, 研究胶体聚合物的弹性问题. 以微观的静态结构为基础, 在理论上推导出剪切弹性模量的显式表达. 该表达式包含了单链结构因子、链间单体的静态结构与动力学局域尺寸. 首先报道了键长的增加对过冷液体转变体积分数有降低作用. 之后, 将胶体链的静态结构作为输入函数, 基于键长对局域尺寸的关系, 重点探索了键长对剪切弹性模量和体积弹性模量的影响. 研究发现: 当使用过冷深度作为自变量时, 同一键长的链的局域尺寸和体积弹性模量能被一条普适曲线刻画, 而剪切弹性模量则不能塌缩到一条普适曲线上. 基于零波矢静态结构因子的普适曲线, 我们猜想这来自于键长对长波矢静态结构的影响. 该工作为日后对聚合物材料的弹性性质的调控提供了理论指导.
    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.
      通信作者: 张博凯, bkzhang@zstu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11904320, 11847115)和浙江理工大学科研启动基金(批准号: 18062243-Y)资助的课题
      Corresponding author: Zhang Bo-Kai, bkzhang@zstu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11904320, 11847115) and the Scientific Research Staring Foundation of Zhejiang Sci-Tech University (Grant No. 18062243-Y)
    [1]

    Berthier L, Biroli G 2011 Rev. Mod. Phys. 83 587Google Scholar

    [2]

    Langer J S 2014 Rep. Prog. Phys. 77 042501Google Scholar

    [3]

    Stillinger F H, Debenedetti P G 2013 Annu. Rev. Condens. Matter Phys. 4 263Google Scholar

    [4]

    Biroli G, Urbani P 2016 Nat. Phys. 12 1130Google Scholar

    [5]

    Hill L J, Pinna N, Char K, Pyun J 2015 Prog. Polym. Sci. 40 85Google Scholar

    [6]

    Yang M, Chen G, Zhao Y, Silber G, Wang Y, Xing S, Han Y, Chen H 2010 Phys. Chem. Chem. Phys. 12 11850Google 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 641Google Scholar

    [8]

    Hunter G L, Weeks E R 2012 Rep. Prog. Phys. 75 066501Google 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 1181Google Scholar

    [12]

    Schweizer K S, Saltzman E J 2004 J. Chem. Phys. 121 1984Google Scholar

    [13]

    Chen K, Saltzman E J, Schweizer K S 2010 Annu. Rev. Condens. Matter Phys. 1 277Google Scholar

    [14]

    Kobelev V, Schweizer K S 2005 Phys. Rev. E 71 021401Google Scholar

    [15]

    Chen K, Schweizer K S 2009 Phys. Rev. Lett. 102 038301Google Scholar

    [16]

    Schweizer K S, Curro J G 1997 Adv. Chem. Phys. 98 1Google Scholar

    [17]

    Chen K, Schweizer K S 2007 J. Chem. Phys. 126 014904Google Scholar

    [18]

    Chen K, Saltzman E J, Schweizer K S 2009 J. Phys. Condens. Matter 21 503101Google Scholar

    [19]

    Martin T B, Gartner T E, Jones R L, Snylder C R, Jayaraman A 2018 Macromolecules 51 2906Google Scholar

    [20]

    Kulshreshtha A, Jayaraman A 2020 Macromolecules 53 4 014904Google Scholar

    [21]

    Dell Z E, Schweizer K S 2018 Soft Matter 14 9132Google Scholar

    [22]

    Gartner T E, Haque F M, Gomi A M, Grayson S M, Hore M J, Jayaraman A 2019 Macromolecules 52 4579Google Scholar

    [23]

    Zhou Y, Schweizer K S 2020 Macromolecules 53 22Google Scholar

    [24]

    Hooper J, Schweizer K S 2006 Macromolecules 39 5133Google Scholar

    [25]

    Zhou Y, Schweizer K S 2020 J. Chem. Phys. 153 114901Google Scholar

    [26]

    Schweizer K S 2005 J. Chem. Phys. 123 244501Google 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 1Google Scholar

    [28]

    Zhang B K, Li H, Li J, Chen K, Tian W D, Ma Y Q 2016 Soft Matter 12 8104Google Scholar

    [29]

    Honnell K, Curro J G, Schweizer K S 1990 Macromolecules 23 3496Google 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 116101Google Scholar

    [32]

    Li J, Zhang B K 2020 Europhys. Lett. 130 56001Google Scholar

  • 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

    Fig. 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

    Fig. 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

    Fig. 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

    Fig. 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.

    Fig. 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 587Google Scholar

    [2]

    Langer J S 2014 Rep. Prog. Phys. 77 042501Google Scholar

    [3]

    Stillinger F H, Debenedetti P G 2013 Annu. Rev. Condens. Matter Phys. 4 263Google Scholar

    [4]

    Biroli G, Urbani P 2016 Nat. Phys. 12 1130Google Scholar

    [5]

    Hill L J, Pinna N, Char K, Pyun J 2015 Prog. Polym. Sci. 40 85Google Scholar

    [6]

    Yang M, Chen G, Zhao Y, Silber G, Wang Y, Xing S, Han Y, Chen H 2010 Phys. Chem. Chem. Phys. 12 11850Google 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 641Google Scholar

    [8]

    Hunter G L, Weeks E R 2012 Rep. Prog. Phys. 75 066501Google 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 1181Google Scholar

    [12]

    Schweizer K S, Saltzman E J 2004 J. Chem. Phys. 121 1984Google Scholar

    [13]

    Chen K, Saltzman E J, Schweizer K S 2010 Annu. Rev. Condens. Matter Phys. 1 277Google Scholar

    [14]

    Kobelev V, Schweizer K S 2005 Phys. Rev. E 71 021401Google Scholar

    [15]

    Chen K, Schweizer K S 2009 Phys. Rev. Lett. 102 038301Google Scholar

    [16]

    Schweizer K S, Curro J G 1997 Adv. Chem. Phys. 98 1Google Scholar

    [17]

    Chen K, Schweizer K S 2007 J. Chem. Phys. 126 014904Google Scholar

    [18]

    Chen K, Saltzman E J, Schweizer K S 2009 J. Phys. Condens. Matter 21 503101Google Scholar

    [19]

    Martin T B, Gartner T E, Jones R L, Snylder C R, Jayaraman A 2018 Macromolecules 51 2906Google Scholar

    [20]

    Kulshreshtha A, Jayaraman A 2020 Macromolecules 53 4 014904Google Scholar

    [21]

    Dell Z E, Schweizer K S 2018 Soft Matter 14 9132Google Scholar

    [22]

    Gartner T E, Haque F M, Gomi A M, Grayson S M, Hore M J, Jayaraman A 2019 Macromolecules 52 4579Google Scholar

    [23]

    Zhou Y, Schweizer K S 2020 Macromolecules 53 22Google Scholar

    [24]

    Hooper J, Schweizer K S 2006 Macromolecules 39 5133Google Scholar

    [25]

    Zhou Y, Schweizer K S 2020 J. Chem. Phys. 153 114901Google Scholar

    [26]

    Schweizer K S 2005 J. Chem. Phys. 123 244501Google 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 1Google Scholar

    [28]

    Zhang B K, Li H, Li J, Chen K, Tian W D, Ma Y Q 2016 Soft Matter 12 8104Google Scholar

    [29]

    Honnell K, Curro J G, Schweizer K S 1990 Macromolecules 23 3496Google 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 116101Google Scholar

    [32]

    Li J, Zhang B K 2020 Europhys. Lett. 130 56001Google Scholar

  • [1] 陈康, 沈煜年. 软体机器人用多孔聚合物水凝胶的摩擦接触非线性行为. 物理学报, 2021, 70(12): 120201. doi: 10.7498/aps.70.20202134
    [2] 严大东, 张兴华. 聚合物结晶理论进展. 物理学报, 2016, 65(18): 188201. doi: 10.7498/aps.65.188201
    [3] 张朝民, 江勇, 尹登峰, 陶辉锦, 孙顺平, 姚建刚. 点缺陷浓度对非化学计量比L12型结构的A13Sc弹性性能的影响. 物理学报, 2016, 65(7): 076101. doi: 10.7498/aps.65.076101
    [4] 孙波. 胶原纤维网络和癌细胞的力学微环境. 物理学报, 2015, 64(5): 058201. doi: 10.7498/aps.64.058201
    [5] 张维然, 李英姿, 王曦, 王伟, 钱建强. 原子力显微镜高次谐波幅度对样品弹性性质表征的研究. 物理学报, 2013, 62(14): 140704. doi: 10.7498/aps.62.140704
    [6] 张志强, 李丛鑫, 谢平, 王鹏业. 果蝇昼夜生理节律之细胞质PER-TIM间隔定时器的大聚合物模型. 物理学报, 2012, 61(19): 198701. doi: 10.7498/aps.61.198701
    [7] 张正罡, 他得安. 基于弹性模量检测骨疲劳的超声导波方法研究. 物理学报, 2012, 61(13): 134304. doi: 10.7498/aps.61.134304
    [8] 宋云飞, 于国洋, 殷合栋, 张明福, 刘玉强, 杨延强. 激光超声技术测量高温下蓝宝石单晶的弹性模量. 物理学报, 2012, 61(6): 064211. doi: 10.7498/aps.61.064211
    [9] 何智兵, 阳志林, 闫建成, 宋之敏, 卢铁城. 辉光放电聚合物结构及力学性质研究. 物理学报, 2011, 60(8): 086803. doi: 10.7498/aps.60.086803
    [10] 史晶, 高琨, 雷杰, 解士杰. 基态非简并导电聚合物——坐标空间研究. 物理学报, 2009, 58(1): 459-464. doi: 10.7498/aps.58.459
    [11] 王 权, 丁建宁, 何宇亮, 薛 伟, 范 真. 氢化硅薄膜介观力学行为及其与微结构内禀关联特性. 物理学报, 2007, 56(8): 4834-4840. doi: 10.7498/aps.56.4834
    [12] 倪向贵, 殷建伟. 拉伸条件下双壁碳纳米管弹性性能的原子模拟. 物理学报, 2006, 55(12): 6522-6525. doi: 10.7498/aps.55.6522
    [13] 王义平, 陈建平, 李新碗, 周俊鹤, 沈 浩, 施长海, 张晓红, 洪建勋, 叶爱伦. 快速可调谐电光聚合物波导光栅. 物理学报, 2005, 54(10): 4782-4788. doi: 10.7498/aps.54.4782
    [14] 梁忠诚, 明海, 王沛, 章江英, 龙云泽, 夏勇, 谢建平, 张其锦. 偶氮液晶聚合物中的非线性光致双折射. 物理学报, 2001, 50(12): 2482-2486. doi: 10.7498/aps.50.2482
    [15] 史伟, 房昌水, 潘奇伟, 孟凡青, 顾庆天, 许东, 陈钢进, 余金中. 简单反射法测量聚合物薄膜线性电光系数的研究. 物理学报, 2000, 49(2): 262-266. doi: 10.7498/aps.49.262
    [16] 陈钢进, 夏钟福, 张冶文, 张红焰. Teflon AF/非线性光学聚合物驻极体双层膜的极化稳定性. 物理学报, 1999, 48(9): 1676-1681. doi: 10.7498/aps.48.1676
    [17] 李景德, 曹万强, 王勇. 聚合物慢极化的唯象理论. 物理学报, 1997, 46(5): 986-993. doi: 10.7498/aps.46.986
    [18] 凌帆, 吴长勤, 孙鑫. 基态非简并的聚合物中的晶格振动频谱. 物理学报, 1990, 39(5): 802-808. doi: 10.7498/aps.39.802
    [19] 帅志刚, 孙鑫, 傅柔励. 导电聚合物中的非线性光学效应. 物理学报, 1990, 39(3): 375-380. doi: 10.7498/aps.39.375
    [20] 王积方, 李华丽, 唐汝明, 查济璇, 何寿安. 熔石英的弹性模量和超声状态方程. 物理学报, 1982, 31(10): 1423-1430. doi: 10.7498/aps.31.1423
计量
  • 文章访问数:  5693
  • PDF下载量:  75
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-01-19
  • 修回日期:  2021-02-20
  • 上网日期:  2021-06-17
  • 刊出日期:  2021-06-20

/

返回文章
返回