Estimation of the True Crystallite Modulus of Silk Fibers from the Relation between the X-ray-Analyzed Apparent Crystallite Modulus and Heterogeneous Stress Distribution

To understand the mechanical property of silkworm silk fibers, it is indispensable to know the so-called crystallite modulus or the ultimate Young’s modulus of the crystalline phase along the chain axis. In many papers, the crystallite modulus of silk fiber was evaluated by using an X-ray diffraction method in which the shift of the X-ray diffraction peak induced by an externally applied force is estimated as the crystalline strain (εc). The external tensile stress of the bulk sample (σb) is assumed to work on the crystalline phase at an equal value (σc = σb). The Young’s modulus obtained under this assumption (or the homogeneous stress distribution model) is called the apparent crystallite modulus (Ecapp = σb/εc). Not only for the various kinds of synthetic polymers as reported in the previous paper (Macromolecules, 2021, 54, 6449) but also for the silk fibers targeted in the present report, the Ecapp changed remarkably depending on the annealing temperature. The thus-observed behavior of Ecapp (and the bulk Young’s modulus, Ebulk) was reproduced well by building up a mechanical series–parallel model with the heterogeneous stress distribution taken into account. As a result, the Ectrue, or the true crystallite modulus, was estimated to be 65 ± 5 GPa, which was common to both the native and regenerated silk fibers. Besides, the thus-estimated Ectrue was in good agreement with the theoretical value of 66 GPa calculated for one typical crystal structural model. The successful reproduction of the observed Ecapp and Ebulk allowed us to speculate that the existence of so-called taut tie chain segments plays an important role in the generation of heterogeneous stress distribution in these silk fibers. We are apt to imagine that the natural fibers should behave ideally with a homogeneous deformation behavior, but the actual silk fibers are not very ideal against our expectation. The difference in the mechanical property (Ecapp and Ebulk) between the native and regenerated silk samples was discussed from the viewpoint of a higher-order structure. The effect of absorbed water on the Ecapp value was also investigated experimentally.