Fund Project: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)
Received Date:19 January 2021
Accepted Date:20 February 2021
Available Online:17 June 2021
Published Online:20 June 2021
Abstract: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 na?ve 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. Keywords:colloidal polymer/ elastic modulus/ nonlinear Langevin equation/ polymer glass
首先, PRISM理论将聚合物中的单体假设为一系列位点所组成的系统[16]. 如图1所示的聚合物系统, 系统静态的结构由两个单体$ a $和$ b $的总关联函数${\hat{h}}_{ab}$、单链结构因子${\hat{\omega }}_{ab}$和直接关联函数${\hat{C}}_{ab}$刻画. 它们之间的关系由Ornstein-Zernike方程描述[30]: 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 Figure1. Schematic of colloidal polymers. Blue spheres represent three consecutive monomers with diameter $ \sigma $, bond angle $ \theta $ and bond length $ l $.
在本文中, 长度单位取为单体的直径$ \sigma $, 弯曲能的单位为$ {k}_{\mathrm{B}}T $, 弹性模量的单位为$ {k}_{\mathrm{B}}T/{\sigma }^{-3} $. 3.计算结果与讨论首先, 研究了离散的蠕虫链的单链结构因子$ \omega \left({k}\right) $, 重点关注键长在不同波矢下对单链结构因子的影响. 如图2(a)所示, 单链结构因子$ \omega \left({k}\right) $在波矢${k}\to 0$趋近于链的单体数目N, 在${k}\to +\infty$时, 振荡衰减到0. 数据显示: 在不同键长下, 在${{k}} < 1$区域内, 单链结构因子几乎没有差别. 在中间波矢区域, 即${k}=1—10$, 不同键长的单链结构因子开始展现出差异. 具体地, 键长越大, 在此区域内, 归一化的单链结构因子具有更大的数值. 更定量地, 在这个区域, 本文计算结果展示单链结构因子满足幂律衰减$ \omega \left({k}\right)\sim {{k}}^{-\mu } $且$ \mu \approx 2 $, 这个衰减指数不依赖于键长. 以往的模拟和理论都揭示了衰减指数$ \mu $反映了不同的拓扑结构的分子, 如硬棒分子($ \mu =1 $)、环性聚合物($ \mu =3 $)和线性柔性分子($ \mu =2 $)[32]. 图 2 静态结构 (a) 在不同键长下的单链结构因子, 虚点线显示在中级波矢范围满足幂律衰减$\sim {{k}}^{-2}$; (b) 不同键长下的径向分布函数 Figure2. 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2(b)进一步展示了不同键长下单体分布的径向分布函数$ g\left(r\right) $, 可以看出, 键长主要影响$r\approx $$ 1.25\sigma — 1.3\sigma$区域内, 这个位置对应了径向分布函数极小的位置, 即单体第一壳层的位置. 因此, 理论计算猜想键长对于结构的影响主要在第一壳层粒子的分布, 该分布对应着示踪粒子的临近粒子的分布, 从而改变了由周围粒子形成的笼效应. 将上述计算的静态结构代入动力学自由能(7)式, 关注键长对于过冷液体转变的临界体积分数$ {\phi }_{\mathrm{c}} $的影响. 图3展示了在不同链内弯曲能$\varepsilon$下, ${\phi }_{\mathrm{c}}\text{-}l$的关系: 临界体积分数随着键长增加而单调的下降, 并在$ l\geqslant 1.5\sigma $处趋于饱和. 这意味着键长增加使得系统的玻璃化转变更容易了. $ {\phi }_{\mathrm{c}} $饱和的位置弱依赖于链内弯曲能. 当弯曲能由$\varepsilon=0—1.0$时, $ {\phi }_{\mathrm{c}}\approx 0.43—0.422 $. 图3中虚线显示出硬球的临界转变体积分数$ {\phi }_{\mathrm{c}}^{\mathrm{H}\mathrm{S}}=0.432 $. 在$l\geqslant 1.5\sigma$时, 柔性的胶体聚合物($\varepsilon=0$)的转变点更接近于$ {\phi }_{\mathrm{c}}^{\mathrm{H}\mathrm{S}} $. 图 3 不同链内弯曲能下, 玻璃化转变体积分数随着链长的变化 Figure3. Crossover volume fraction as a function of bond length for different bending energies.
在(8)式中, 剪切弹性模量依赖于局域尺寸和静态结构, 因此首先报道键长对局域尺寸的影响. 在以往的工作中[28], 我们已经发现: 只要将过冷深度$ \phi -{\phi }_{\mathrm{c}} $作为自变量, 局域尺寸和动力学自由能势垒均能坍缩到一条普适的曲线上, 而并不依赖于链的单体数目和链内的弯曲自由能. 如图4(a)所示, 展示了3个不同键长下3个不同链内弯曲能的局域尺寸的数据. 局域尺寸随着过冷深度而单调下降, 并在大的过冷深度参数下, 满足指数衰减$ {r}_{\mathrm{l}\mathrm{o}\mathrm{c}}\sim {\mathrm{e}}^{-\alpha \left(\phi -{\phi }_{\mathrm{c}}\right)} $. 随着键长的增加, 这个指数上系数$ \alpha $的变化范围为$ \alpha =20.056—13.247 $, 在大的键长下, 接近于硬球的结果$ \alpha =14.658 $ (图4(a)中的绿色实线). 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 Figure4. (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.
将得到的局域尺寸和静态结构代入(8)式中, 计算了不同过冷深度的剪切弹性模量. 如图4(b)所示, 发现剪切弹性模量随着过冷深度的增加而迅速增加, 从$ {G}'\approx 20 $到$ {G}'\approx 3000 $, 增加了大约两个数量级. 并且在大的过冷深度下, 以指数的形式增加, $ {G}'\sim {\mathrm{e}}^{\beta (\phi -{\phi }_{\mathrm{c}})} $. 当$ l=1.1 $时, 柔性聚合物($\varepsilon=0$)在大的过冷深度下满足${G}'\!\sim\! {\mathrm{e}}^{27.027(\phi -{\phi }_{\mathrm{c}})}$, 而$ l=1.7 $时, ${G}'\!\sim\! {\mathrm{e}}^{25.707(\phi -{\phi }_{\mathrm{c}})}$. 对于硬球系统, ${G}'\!\sim\! {\mathrm{e}}^{26.581(\phi -{\phi }_{\mathrm{c}})}$, 发现其指数$ \beta $变化不大. 但与局域尺寸不同的是, 虽然对于同一个键长, 不同的链内弯曲能的剪切弹性模量数据更接近, 但是我们的理论揭示: 它们并不能坍缩到一条普适的指数增长曲线上. 我们转而研究体积弹性模量. 将长波长的直接关联函数代入(9)式进行计算, 图5(a)显示了不同键长下不同的链内弯曲能对胶体聚合物系统的体积弹性模量的影响. 研究发现体积弹性模量随着过冷深度迅速地增加, $ {K}_{\mathrm{B}}\approx 20—200 $, 约一个数量级. 在本文理论研究的整个范围内, 体积弹性模量的数据都展示出其随着过冷深度而发生指数增长的趋势, 在长键长下($ l=1.7\sigma $), $ {K}_{\mathrm{B}}\sim {\mathrm{e}}^{13.821\left(\phi -{\phi }_{\mathrm{c}}\right)} $, 接近于硬球的体积弹性模量的行为: $ {K}_{\mathrm{B}}\sim {\mathrm{e}}^{13.768\left(\phi -{\phi }_{\mathrm{c}}\right)} $. 有趣的是, 和局域尺寸一样, 体积弹性模量-过冷深度曲线也展现出独特的键长依赖关系: 同一键长而不同链内弯曲能的$ {K}_{\mathrm{B}}(\phi -{\phi }_{\mathrm{c}}) $曲线坍缩到一条普适的指数增长曲线. 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. Figure5. (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.