Estimation\nof the True Crystallite Modulus of Silk\nFibers from the Relation between the X‑ray-Analyzed Apparent\nCrystallite Modulus and Heterogeneous Stress Distribution

To understand the mechanical property of silkworm silk\nfibers,\nit is indispensable to know the so-called crystallite modulus or the\nultimate Young’s modulus of the crystalline phase along the\nchain axis. In many papers, the crystallite modulus of silk fiber\nwas evaluated by using an X-ray diffraction method in which the shift\nof the X-ray diffraction peak induced by an externally applied force\nis estimated as the crystalline strain (ε<sub>c</sub>). The\nexternal tensile stress of the bulk sample (σ<sub>b</sub>) is\nassumed to work on the crystalline phase at an equal value (σ<sub>c</sub> = σ<sub>b</sub>). The Young’s modulus obtained\nunder this assumption (or the homogeneous stress distribution model)\nis called the apparent crystallite modulus (<i>E</i><sub>c</sub><sup>app</sup> = σ<sub>b</sub>/ε<sub>c</sub>). Not only for the various kinds of synthetic\npolymers as reported in the previous paper (<i>Macromolecules</i>, <b>2021</b>, <i>54</i>, 6449) but also for the\nsilk fibers targeted in the present report, the <i>E</i><sub>c</sub><sup>app</sup> changed\nremarkably depending on the annealing temperature. The thus-observed\nbehavior of <i>E</i><sub>c</sub><sup>app</sup> (and the bulk Young’s modulus, <i>E</i><sub>bulk</sub>) was reproduced well by building up a mechanical\nseries–parallel model with the heterogeneous stress distribution\ntaken into account. As a result, the <i>E</i><sub>c</sub><sup>true</sup>, or the true\ncrystallite modulus, was estimated to be 65 ± 5 GPa, which was\ncommon to both the native and regenerated silk fibers. Besides, the\nthus-estimated <i>E</i><sub>c</sub><sup>true</sup> was in good agreement with the theoretical\nvalue of 66 GPa calculated for one typical crystal structural model.\nThe successful reproduction of the observed <i>E</i><sub>c</sub><sup>app</sup> and <i>E</i><sub>bulk</sub> allowed us to speculate that the existence\nof so-called taut tie chain segments plays an important role in the\ngeneration of heterogeneous stress distribution in these silk fibers.\nWe are apt to imagine that the natural fibers should behave ideally\nwith a homogeneous deformation behavior, but the actual silk fibers\nare not very ideal against our expectation. The difference in the\nmechanical property (<i>E</i><sub>c</sub><sup>app</sup> and <i>E</i><sub>bulk</sub>) between the native and regenerated silk samples was discussed from\nthe viewpoint of a higher-order structure. The effect of absorbed\nwater on the <i>E</i><sub>c</sub><sup>app</sup> value was also investigated experimentally.