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

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 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.