Determination of the Complete Elasticity of Nephila pilipes Spider Silk

Spider silks are remarkable materials designed by nature to have extraordinary elasticity. Their elasticity, however, remains poorly understood, as typical stress–strain experiments only allow access to the axial Young’s modulus. In this work, micro-Brillouin light spectroscopy (micro-BLS), a noncontact, nondestructive technique, is utilized to probe the direction-dependent phonon propagation in the Nephila pilipes spider silk and hence solve its full elasticity. To the best of our knowledge, this is the first demonstration on the determination of the anisotropic Young’s moduli, shear moduli, and Poisson’s ratios of a single spider fiber. The axial and lateral Young’s moduli are found to be 20.9 ± 0.8 and 9.2 ± 0.3 GPa, respectively, and the anisotropy of the Young’s moduli further increases upon stretching. In contrast, the shear moduli and Poisson’s ratios exhibit very weak anisotropy and are robust to stretching.


Section S2. Transversely isotropic elasticity model
The spider silk was assumed to be transversely isotropic, similar to previous studies. 41,43 To facilitate the analysis, a "123" coordinate system was constructed, with the "3"-axis parallel to the silk axis. For a transversely isotropic material, the elastic stiffness tensor contains 5 independent elastic constants (here chosen as C11, C13, C33, C44, C66), and has the following form (in the Voigt notation), The elastic stiffness constants are coupled with the direction-dependent sound velocities in the framework of the Christoffel's equation. [46][47][48][49] Given the latter, the former could be uniquely determined through least squares fitting. Because of the transverse isotropy, it is only necessary to consider phonon propagation in a quarter of the "23" plane, i.e., 0 o ≤ α ≤ 90 o (Figure 1e,f).
For a particular direction represented by an angle, α, there exist one quasi-longitudinal (Q-L) mode, one quasi-transverse (Q-T) mode, and one pure-transverse (P-T) mode. The directiondependent sound velocities of the Q-L, Q-T, and P-T modes can be expressed as follows. where,

Section S3. Brillouin light spectroscopy (BLS) experiments
The direction-dependent sound velocities were obtained by BLS, a non-contact, non-destructive technique. The experiments were conducted with a micro-BLS setup, which includes a six-pass tandem Fabry-Perot interferometer and a Nd/YAG laser (λ0 = 532 nm in air) mounted on a goniometer. "Nikon T Plan SLWD 50x" objectives with a numerical aperture (NA) of 0.4 and a super long working distance of 22 mm were used in the measurements to achieve a laser focal spot size 2.0 ± 0.5 μm (the 1/e 2 definition), which is much smaller than the diameter of the silk fiber (8.3 ± 0.6 μm). The laser input power is around 5 mW. Transmission, reflection, and backscattering geometries were employed to probe phonon propagation in directions parallel, normal, and oblique to the spider fiber axis, respectively. In the transmission, reflection, and backscattering geometries, we conducted 4 (scattering angle from 60 o to 130 o ), 5 (scattering angle from 90 o to 130 o ), and 9 (incident angle from 0 o to 80 o ) measurements, respectively. In the transmission and reflection geometries, the direction of the phonon wave vector remains unchanged, but its magnitude varies depending on the scattering angle. In the backscattering geometry, the magnitude of the phonon wave vector slightly changes from 0.00331 to 0.00334 nm -1 because the spider silk is slightly birefringent, but its direction varies depending on the laser incident angle. The polarization of the probed phonons (i.e., Q-L, Q-T, P-T) was selected by using different polarization combinations of the incident and scattered light beams (e.g., VV, VH, HH) and meanwhile taking into account the intensity of the scattered light. Considering the optical birefringence of the spider fiber, the magnitudes of the phonon wave vectors probed in the three scattering geometries (e.g., transmission, reflection, backscattering) with the different light polarization configurations (e.g., VV, VH, HH, HV) were calculated as follows.
In the transmission geometry: Here, β is the laser incident angle, and n is the spider fiber's refractive index. It should be noted that the optical birefringence affects both the magnitude, q, and direction, α, of the phonon wave vector, and that α is related to β by the Snell's law, sin β = n sin γ, where γ = 90 oα (Figures 1e and S1). The sound velocity was calculated as, where f is the phonon frequency obtained from the Lorentzian fits to the Brillouin peaks. In the experiments studying the strain effect on the spider silk's elastic properties, the silk fiber was stretched using a customized stretch meter by up to 20%.

Section S4. Optical birefringence
The spider silk was found to be optically birefringent from polarized optical microscopy measurements (inset to Figure S2). To account for the light-polarization-dependent refractive index, the following formula was adopted, S1 where vL is the sound velocity of the L mode. The frequency shift in Figure S2 indicates the optical anisotropy, i.e., o VV e HH nf nf  . Based on eqs S10 and S14, the vL and no were obtained by conducting BLS measurements in the reflection geometry at multiple  values. Finally, the principal refractive indices were determined to be no = 1.40  0.01 and ne = 1.46  0.02.

Section S5. χ 2 fitting
Based on the BLS-measured, direction-dependent sound velocities (i.e., vQ-L(α), vQ-T(α), vP-T(α)), nonlinear χ 2 fitting was conducted to obtain the elastic stiffness constants. 50 where vi, fit and vi, exp are the fitted and experimental sound velocities, respectively, Δvi, exp is the uncertainty of the measured sound velocity, and the summation is over all experimental sound velocities. By considering the measurement uncertainties of the angles, refractive indices, phonon frequencies, and so on, the relative uncertainty of (Δvi, exp)/(vi, exp) was estimated to be 2%. To ensure positive Young's and shear moduli, the χ 2 fitting was performed subject to the following constraints: (i) C11 > |C12|, (ii) C44 > 0, and (iii) The fitting was based on 8-10 data points for the Q-L modes, 3-6 data points for the Q-T modes, and 0-3 data points for the P-T modes. The final χ 2 values are 8.7073, 0.7172, 1.0634, 0.6167, and 1.5541, for the data sets corresponding to 0%, 5%, 10%, 15%, and 20% strains, respectively. These χ 2 values indicate that qualitatively the discrepancy between the experimental and predicted sound velocities is within (8.7073) 0.5 Δvexp = 2.95Δvexp. Since Δvexp ≈ 2%vexp, the discrepancy is then within 5.9% (standard deviation). The discrepancy for the other four data sets is even smaller (within 2.5%). The residuals for the 0% strain data set, which has the largest χ 2 value, are shown in Figure S5. Similar residual profiles exist for the other four data sets. In Figure S5, no clear pattern of the residuals is seen. Therefore, we consider the residuals to be randomly distributed. At 0% strain, the availability of the vQ-L(α), vQ-T(α), and vP-T(α) data at multiple α values allows unique determination of C11, C13, C33, C44, and C66. At higher strains, where no unambiguous P-T modes were detected, the determination of the elastic tensors was completed by assuming the Poisson's ratio ν31 to be equal to that at 0% strain (i.e., ν31 = 0.35 ± 0.02). The obtained elastic stiffness constants of the spider fiber at the five strains are summarized in Table S1.
The independent elastic stiffness constants were used to predict the theoretical vQ-L(α), vQ-T(α), and vP-T(α), according to eqs S2-S4. Furthermore, they were used to calculate the characteristic mechanical properties, 41 Table S2. Their values for the Nephila pilipes MA silk in the native and stretched states are summarized in Table S3.

Section S6. Uncertainty quantification
By considering the measurement uncertainties of the angles, refractive indices, phonon frequencies, and so on, the relative uncertainty of the BLS-measured sound velocities, (Δvi, exp)/(vi, exp), was estimated to be 2%. The uncertainties of the elastic stiffness constants were determined by constructing a matrix G. 50 To simplify the notations, C11, C13, C33, C44, and C66 were denoted as a1, a2, a3, a4, and a5, respectively. G has elements of the following form, where m and n (= 1, 2, 3) are matrix indices, and the summation is over all experimental sound velocities. Mathematically, G dictates how variations in the elastic constants affect the fitted sound velocities. The partial derivatives were approximated by using a central finite difference , where δam represents the change in am. From a convergence study, δam was determined to be 10 -6 am. The inverse of G was calculated to obtain a covariance matrix, i.e., M = G -1 .  ()

Section S7. Scattering intensity calculations
The appearance/ disappearance of the phonon modes in a BLS spectrum is determined by the light scattering selection rules. To understand the Pockels (elasto-optic) coefficients, Pij, involved in the different measurements, the polarization vectors of the scattered light corresponding to the phonon modes in the transmission, reflection, and backscattering measurements were derived, according to the theory by Hamaguchi, S3 as summarized in the Table S4. The intensity of the scattered light is proportional to It is worth considering the P-T mode in the backscattering measurements. This mode remained non-detectable for the stretched spider fibers. Since the spider fiber is only weakly birefringent ( Figure S2), it is reasonable to assume γs = 90 oα and γi = 90 oα in the backscattering VH and HV measurements, respectively. Therefore, the invisibility of the P-T mode implies , which is in good agreement with the data for the stretched spider fibers in Table S1. Figure S1.     strain. Similar residual profiles exist for the data at 5%, 10%, 15%, and 20% strains.  Supporting Tables   Table S1. Summary of the elastic stiffness constants of the Nephila pilipes MA silk at five strains (i.e., 0%, 5%, 10%, 15%, 20%). The zero and positive strains indicate spider fibers in the native and stretched states, respectively.