Nonlinear control of high-frequency phonons in spider silk

Transcription

1 Nonlinear control of high-frequency phonons in spider silk Dirk Schneider 1, Nikolaos Gomopoulos 1, Cheong Y. Koh, Periklis Papadopoulos 1,3, Friedrich Kremer 4, Edwin L. Thomas 5, George Fytas* 1, Max Planck Institute for Polymer Research, Ackermannweg 10, 5518 Mainz, Germany. Functional and Smart Materials Laboratory, Emerging Systems, DSO National Laboratories, S117605, Singapore. 3 Department of Physics, University of Ioannina, Ioannina, Greece. 4 Institute of Experimental Physics I, Linnéstr. 5, University of Leipzig, Leipzig, Germany. 5 Department of Materials Science and Nano-Engineering, Rice University, Houston, Texas USA. 6 Department of Materials Science University of Crete and IESL-F.O.R.T.H, Heraklion, Greece * fytas@mpip-mainz.mpg.de This document contains: Supplementary data and eleven supplementary figures with legends NATURE MATERIALS 1

2 Contents S1. Experimental BLS data... 3 S1a. Brillouin light scattering... 3 S1b. BLS of a single fiber and bundles of fibers... 3 S1c. BLS spectra of major ampullate spider silk... 5 S. Dispersion relations... 6 Sa. Normal direction to the spider silk fiber... 6 Sb. Along the axial direction of the spider silk fiber... 9 S3. Absorption spectra for determination of crystallinity... 9 S4. Mechanical anisotropy of [ CO (CH ) 5 NH ] n (Polyamide 6)... 1 S5. Single embedded fiber S6. Stretching effects S6a. Morphological changes upon stretching S6b. Stress-strain curves of supercontracted and native spider silk S6c. Negative group velocity and the physical origin of non-locality... 0 S6d. Weak elastic nonlinearity... 3 References... 8 NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION S1. Experimental BLS data S1a. Brillouin light scattering BLS is a unique non-destructive and non-contact optical technique to probe propagation of thermally excited acoustic waves (phonons) at hypersonic frequencies (and hence wavelengths at the sub-micrometer length scale) in microstructures 1. It utilizes the interaction between incident photons with phonons propagating in the medium along a selected direction determined by the scattering geometry. The scattering wavevector q = k s k i defined by the wavevectors of the incident (k i ) and scattered (k s ) photons denotes the phonon propagation direction, or more precisely, the phonon wave vector of the material structure. The polarization of the laser beam (vertical, V) and the scattered light (V or horizontal, H) relative to the scattering plane (k i, k s ) defines the polarized (VV) and depolarized (VH) BLS spectra. In the simplest case of homogeneous media, momentum conservation requires the wavevector of the phonon k and q be equal, and the BLS spectrum at a given q consists of a doublet with a Doppler frequency shift ω = ± c q, where c is the phase velocity of sound with longitudinal or transverse polarization in the VV and VH BLS spectra, respectively. For inhomogeneous systems over length scales spanning the phonon wavelength (π / q), the phononic behavior can be revealed by recording the dispersion relation ω(q). It yields insight into the mechanical anisotropy of both the structure and the intrinsic material properties S1b. BLS of a single fiber and bundles of fibers Two special scattering geometries were adopted to select the direction of q either parallel (q in Fig. 1g) or normal (q in Fig. 1h) to the fibers; the magnitude of q and q are given in the caption of Fig. 1 and only the value of q depends on the refractive index (n = 1.55). The polarized BLS measurements of spider silk were performed on a bundle of fibers to have sufficient intensity. In order to verify that bundles of fibers do not change the observed spectrum, we used as a model system a single glass fiber (d 5 µm) as well a collection of fibers (5 7 pieces) in a parallel arrangement. Figure S1a displays the corresponding spectra with no difference other than in overall intensity. Hence the BLS NATURE MATERIALS 3

4 measures the dispersion relation of the fiber, i.e. there is no mechanical coupling between different spider silk fibers. In addition, we observe only the single acoustic modes as expected, since the glass fiber does not possess a hierarchical structure; a cutoff due to its structural feature (i.e. the diameter) would appear at too low frequency (< 1GHz) for its detection Figure S1a: Single glass fiber and a bundle of glass fibers in a parallel arrangement. Only the acoustic phonon is observed, whose intensity is increased for several fibers in a bundle. 4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION S1c. BLS spectra of major ampullate spider silk The polarized (VV) BLS spectra of major ampullate spider silk for phonon propagation along and perpendicular to the axis are shown at seven selected q-values in Fig.S1b. The peak positions of the BLS spectra are obtained from double Lorentzian line fits the peak positions and together with the corresponding q-values construct the phononic dispersion Figure S1b: Polarized BLS spectra (The anti-stokes side only is shown) at different scattering wave vectors: one q (normal to the fiber axis, upper panel with one peak ) and six q (parallel to the fiber long axis, lower panel with two peaks 1 and 3) corrsponding to the indicated scattering angles, θ. NATURE MATERIALS 5

6 The polarized spectra perpendicular and parallel to the spider silk axis are quite distinct, implying mechanical anisotropy. For propagation along the fiber, peak 1 is acoustic at low q s (Fig. b) and hence its frequency increases with q in contrast to peak, which is a quasi-localized peak. These two modes have predominantly longitudinal polarization and appear in the BLS VV spectra. Modes with transverse polarization are better observed in the VH BLS spectra 3. However, in the present case, the inevitably small multiple scattering is a source of VH scattering that can obscure the low inherent VH scattering. In this case, the VH BLS spectra appear replica of the corresponding VV spectra. We therefore focus in this study only on VV BLS spectra. S. Dispersion relations Sa. Normal direction to the spider silk fiber Along the q direction only an acoustic-like mode is observed (Fig. S1b and Fig. b). The high values of the axial Young s modulus (E about 10 GPa) have been well documented by mechanical techniques 4 at low measuring frequencies (quasi-static testing long wavelength deformation). The elastic modulus E normal to the fiber axis is, however not accessible by such techniques due to the finite fiber size. AFM experiments on native fibers embedded in a resin have claimed a somewhat lower transverse modulus (about 6.4 GPa), while fibers exhibited non-uniformities 5. We first examine the dispersion for the elastic wave (phonon) propagation normal to the fiber axis. As the accessible frequency and q range is experimentally limited, we are primarily probing acoustic wavelengths, Λ (= π / q ) < 00 nm. The probed high frequency phonons propagating in the radial directions sample the internal structure of the fibers. For phonon wavelengths, Λ < 00 nm (or q > 0.03 nm -1 ), the fiber diameter is larger than 0 λ, and hence the quasi-bulk modes of the fiber in the transverse direction are probed. Indeed, the experimental BLS spectrum displays a single doublet (the lower panel of Fig. a) at q = nm 1 and the frequency of peak increases linearly with q (red dashed line in the shaded region of Fig. b). This allows determination of the effective medium sound velocity c = ω / q = 3140 ± 40 ms 1 and the radial longitudinal modulus, M = ρc = 6 NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION ± 0. GPa (using the reported bulk density, ρ = 1.5 cm 3 ) 6. This modulus, inaccessible through conventional testing techniques, is reported here for the first time. To show that we are indeed recovering the linear (acoustic-like) dispersion, we consider a periodic composite, similar in structure to the spider silk. This azimuthally longitudinal-like dispersion is quasi-linear throughout the first three Brillouin zones (BZ), except for strong interactions vis-a-vis band bending at the BZ boundaries (ka / π = 1), as seen in Fig. S. Since this is not generic to all periodic systems, it needs to be checked for different impedance contrasts. We have used experimentally obtained values of the axial and azimuthal modulus and assumed reasonable values of the Poisson s ratio of the constituent fibrils and the noncrystalline matrix (see caption of Fig. S). For the wavelengths and frequencies of interest, the phonons propagate as in an effective medium (red dashed line), except at q values near the approximated BZ boundaries. For an estimated effective unit cell with a = 00 nm and q a / π >, we focus between the second and third Brillouin zones, as seen in Fig. S. We have further overlaid the experimental data over the theoretical dispersion to reveal the physical q-values of the anticipated quasi-linear dispersion of the longitudinal mode (orange dashed line in Fig. S). The linear dispersion of the longitudinal mode is quite robust for a much wider class of material elastic parameters which were tested (not shown here). The orange dashed line serves as the guide for the experimental (red circles) linear dispersion, following between the nd and 3 rd BZ. The disparity, between the computed dispersion relation and the experimental data points (red circles) occurs as our modelled system assumes a periodic system with a transverse lattice spacing between the fibrils in a square lattice, while the nanofibril bundles are certainly not arranged strictly on a periodic lattice. NATURE MATERIALS 7

8 Figure S: Computed normalized dispersion relations fa / c vs. ka / π in the [100] direction but assuming periodic boundary conditions of a single unit cell consisting of the fibril a = 00 nm, c=3070 ms -1 (see S3) embedded in the noncrystalline matrix. In this configuration, the parameters used were: Young s moduli, E = 0 GPa, E = 60 GPa, Poissons s ratios, ν = 0.33, ν = 0.5 and shear moduli μ = 3.85 GPa, μ = 3 GPa, for the fibril and E = 10 GPa, E = 0 GPa, ν = 0.30, ν = 0.5, μ = 3.9 GPa, μ = 7 GPa for the noncrystalline matrix. The green dotted circles outline where the strong Bragg scatterung occurs near the Brillouin zone boundaries and the orange dashed line is the longitudinal acoustic phonion along the [100] direction. The red circles denote the overlaid experimental data points Importantly, however, the range of the radial impedance (ρc 11 ) sampled displays the acoustic-like behavior even beyond the first BZ (Fig. S) except for strong band bending at the BZ boundaries, which is a characteristic of mechanical composites of cermet topology 7,8. In addition, at wavelengths of about 00 nm and geometric diameter and hence boundary of the fiber (5 µm), the sampled phonons do not interact with the 8 NATURE MATERIALS

9 SUPPLEMENTARY INFORMATION boundary, i.e. we are sampling bulk phonons by virtue of wavelength selection 9,10. We clearly observe (Fig. S) the quasi-linear dispersive trend (in yellow), at the experimental q- values of interest, allowing the estimation of the radial longitudinal modulus 7,8. Thus the effective linear modulus M can reliably be extracted from such measurements. Sb. Along the axial direction of the spider silk fiber The BLS spectra for q (Fig. S1b) display two distinctive peaks leading to the full dispersion relation of Fig b. As shown in Fig. 1e,f, the hierarchical structure consists of bundles of nanofibrils oriented parallel within a single spider silk fiber, imparting the overall uniaxial mechanical anisotropy and hence affecting the phonon propagation. Each nanofibril is comprised of beta-crystals linked by amorphous chains (see Fig. S5a). We note that both the presence of the beta-crystal platelets as well as their deliberate orientation are unique features of the spider silk structure. To verify their effect on the BLS spectra, we next estimate the degree of crystallinity S3. Absorption spectra for determination of crystallinity The oriented native nanofibril structure was destroyed by dissolving the native fiber in hexafluoroisopropanol and subsequently by spin-coating a film and then and drying at 100 C under vacuum for days. The absence of crystallinity was documented using infrared spectroscopy, as shown in Figure S3a and is in accord with literature reports 11. Infrared spectroscopy can determine the volume fraction of crystalline β-sheet poly(l-alanine) from the absorption at 960 cm -1 (Fig. S3a, inset) relative to the absorption of pure b-poly (lalanine) synthetic polypeptide 1. In native major ampullate silk this crystallinity ratio amounts to In silk dissolved in hexafluoroisopropanol (HFIP) and spin coated on a CaF substrate (black), the absence of this absorption band implies the β-sheet poly (L- NATURE MATERIALS 9

10 Figure S3: a) Absorption coefficient spectra of native and regenerated spider silks, used for determination of crystallinity. (blue) native major ampullate silk, (black) silk film cast from hexafluoroisopropanol (HFIP) solution of native major ampullate silk, (red) the same film after treatment with methanol. Inset: The absorption band of β-poly(l-alanine) at ca. 965 cm -1 is prominent in native silk but is absent in the as-cast film, showing that the alanine block did not form crystals. The crystals form again after treatment with methanol. b) Amorphous silk protein chains in the as-cast from HFIP film. c) After treatment with MeOH nanocrystals form (yellow) without longrange anisotropy. d) Native spider silk with oriented nanocrystals and higher crystallinity. alanine) crystals have been destroyed. After treatment with methanol, β-sheet poly (Lalanine) crystals are again formed (red curve in Fig. S3a). No polarization dependence was observed, as the crystals in the film are isotropically distributed. To determine the crystallinity of the methanol treated film, its absorption coefficient is compared to the one of native silk fibers (blue). As native silk is strongly anisotropic, we used the absorption 10 NATURE MATERIALS

11 SUPPLEMENTARY INFORMATION coefficient, iso =, where and are the absorption coefficients for polarization parallel and perperndicular to the fiber axis, respectively. Hence, the volume fraction of β-sheet poly(l-alanine) crystals after dissolution in HFIP and treatment with methanol is about 0.5. While post treatment with methanol 13 induces the formation of β-sheet nanocrystals (Fig. S3c) randomly distributed in the film, the long range structural hierarchy and global orientation of the native fiber are, of course, not recovered. The dispersion diagram of the methanol treated films (Fig. d, triangles) exhibits an isotropic acoustic branch with c L = 3070 ± 30 ms -1, approximately 6% higher than the original fully amorphous regenerated film. In comparison, the experimental dispersion relations of the spider silk fibers clearly indicate the presence of structural inhomogeneities and most strikingly, the formation of an axial phononic band gap. To further check the unique requirement of the hierarchical superstructure, we considered in S4 a chemically similar polyamide 6 and compared the mechanical anisotropy against the spider silk, highlighting the much more significant mechanical anisotropy, which requires the hierarchical structure present only in the native silk. NATURE MATERIALS 11

12 S4. Mechanical anisotropy of [ CO (CH ) 5 NH ] n (Polyamide 6) The BLS spectrum of a 15 µm thick film of Nylon-6 consists of a single doublet for both inplane and normal to the film plane phonon propagation as seen in Fig. S4a. The dispersion diagram of Fig. S4b presents a purely acoustic behavior for both directions and the corresponding sound velocities correspond to c = 3700 ± 30 ms 1 and c = 830 ± 30 ms 1. The higher in-plane sound velocity is attributed to the stretching of the chains along this direction. The longitudinal modulus anisotropy M / M 1.7 (assuming isotropic density in the film) is clearly lower than either the native (M / M.6 for stretched (M / M 5) spider silk fiber (Fig. 3d) Figure S4: Nylon 6 film. a) BLS spectra at q =0.01 nm -1 (upper panel) and q =0.035 nm -1 (lower panel) parallel and normal to the film respectively, at 5 C. b) Linear dispersion relation for the acoustic modes normal ( ) and parallel to the plane of the film ( ). 1 NATURE MATERIALS

13 SUPPLEMENTARY INFORMATION S5. Single embedded fiber The selection rules under our experimental conditions imply the detection of the radialaxial mode R 0m polarized primarily in the axial direction. The displacement field at q = 0 nm -1 is shown in Fig. S5b (right panel). This eigenmode only occurs in waveguide structures that are embedded within another solid medium, i.e. matrix. One key criterion here is that the axial shear modulus of the nanofibrils and the oriented non-crystalline matrix need to be different; were the shear modulus identical in the two phases, the guided mode would have had some radial polarization 14 19, in addition to the axial polarization. We note that the selection rules do not require that the nanofibrils be perfectly circular in cross-section. Instead, a weaker condition that they approximately possess vertical mirrorplane symmetry, with the mirror plane lying along the axis of propagation (fiber axis), is sufficient. As we will see below, even in the absence of strict mirror-plane symmetry, the predominantly longitudinally polarized mode persists as a guided mode at the cutoff q 0 (Fig. S5d). We now formulate a simplified analytical model to arrive at the dispersion relation of the cut-off mode 3, which is ubiquitous to embedded waveguides We focus on the dominant elastic forces entering the equation of motion, deferring the exact details for the mechanical anisotropy of both the nanofibril and the surrounding partially oriented amorphous phase. Considering the axial direction, one may consider the longitudinal equation of motion of a stiff rod subject to shear forces from the softer matrix along its boundaries. Hence we are considering wave-guiding type of propagation, with the matrix acting as the confining medium. In this limit (valid near q = 0) 14 16,0, u u + ρ fiber d f = ( Efiber d f ) + τ ( z) τ ( z), (S5.1) t z v( x, z, t) τ ( z) = μ, (S5.) x where u(x,y,t) and v(x,y,t) are, respectively, the longitudinal displacement of the nanofibril and the matrix from equilibrium, ρ fiber, E fiber and d f are the mass density, the axial Young s modulus and the diameter of the nanofibril 15 17, while µ is the axial shear modulus of the NATURE MATERIALS 13

14 matrix. In this model, the matrix interacts with the nanofibril mainly through shear at their common interface, similar to the case of an embedded axially stiff rod within a softer matrix medium 16,0. In addition, we do not need to assume a particular nanofibril geometrical shape, only that the fibril separation, Λ f (Fig. S5a), is large enough to prevent interactions between the nanofibrils. Hence, linearizing the matrix displacement, v( x, y, t) v m +1 v τ ( z) = μ μ m, (S5.3) x Λ with m denoting the transverse displacement between adjacent nanofibrils (m, m+1) which gives rise to the shear displacement of the matrix, the simple analytic dispersion relation in the paper: + Ω with Ω ~ f. (S5.4) is obtained. All parameters but the average separation between nanofibrils,,can be obtained experimentally. The longitudinal sound velocity c f is the effective propagation velocity within the guided eigenmode, to be distinguished from the effective medium longitudinal velocity, c (Sb). According to the theoretical dispersion derived in Section S6d, there are two regimes of approximately linear dispersion (Fig. S6c). The slope at very low frequencies corresponds to the initial modulus (quasi-static, low strain) and this speed of sound in major ampullate silk at low (khz) frequencies is about 300 ms -1. In fact, the present model does predict similar values for the initial (low frequency) slope as well (Fig. S6c) with the value smaller than c which corresponds to the second, at higher frequencies, linear dispersion regime. Utilizing a volume fraction of about 35% and the experimentally retrieved amorphous phase sound velocity (850 ms -1, S4), we compute from Wood s law (one dimensional phonon propagation) c f (q 0) = 350 ms -1 ; for a fibril volume fraction, ϕ, between 30% to 40% (absorption spectra in Fig. S3), the values of c f vary between 3350 ms -1 up to 3600 ms -1. The computed dispersion curve of mode 3 for the native spider silk sample (as well as the strained and super-contracted samples in Fig. S6c) utilizing eq. S5.4 matches well the experimentally observed dispersion. 14 NATURE MATERIALS

15 SUPPLEMENTARY INFORMATION The fibril spacing Λ f and the fibril diameter d f can be obtained from a knowledge of the areal fraction of the fibril and the cutoff frequency. Utilizing the expression for Ω in eq. S5.4, we obtain an estimate for the average nanofibril spacing Λ f, Λ f 1 = Ω cutoff ρ fiber μ r (1 Λ 0 0 )( ) f r Λ f 1. (S5.5) Fig. S5c depicts the fibril spacing as a function of the possible volume fraction ranges, for a given cutoff frequency and for reported values of shear moduli µ ranging from 1 to 4 GPa; Fig. S5c shows the experimentally obtained shear values. The impact of the matrix modulus on the fibril spacing is visualized in Figure S5d. Hence, the center to center nanifibril spacing, Λ f = 00 0 nm and the nanofibril diameter can be obtained from the choice of the volume fraction (30 40% from the absorption spectra Fig. S3). The latter value of 40% is consistent with experimental and theoretical results 1 6. For the present uniaxial anisotropic case, the contrast in the shear modulus is a necessary and sufficient condition in order to generate a guided mode that is predominantly axially polarized. We have observed this over a large range of varying shear moduli contrast combinations and have found this condition to be robust. Importantly, such a longitudinally polarized guided mode only exists in embedded geometries, i.e. it is absent in a free rod. In particular, the position of the cutoff frequency Ω is relatively insensitive to the changes in the longitudinal modulus of either the nanofibril or the matrix phase with full numerical calculations (at q ~ 0). We also computed the dispersion relation for embedded nanofibrils with various geometrical cross-section shapes, both with and without mirror symmetry along the axial direction (Fig. S5d) and we find that this axially polarized eigenmode persists quite robustly, hence our rationale in assigning this mode as the eigenmode. We find in each case that the prevailing contrast in shear modulus between the nanofibril and matrix phase consistently favors this axially polarized mode as one of the lowest three order modes (besides the torsional and the flexural mode) ,18 NATURE MATERIALS 15

16 331 a b Figure S5: a) Schematic of the proposed hierarchical mechanical structure of spider silk fiber with the relevant dimensions and parameters under investigation and two orientation of the q vector. The inset shows the increase in anisotropy of the nano-fibrils under varying amounts of pre-strain. b) An effective unit with a nanofibril diameter <d f > and spacing <Λ f > (left panel) and theoretical eigenmode of the confined radial-axial mode 3 at the cutoff frequency Ω(q ) (right panel), displaying the displacement fields from small (blue) to large (red). The black dots are arrows indicating the displacement vector field direction, which is into the plane, i.e. along the axial direction. 16 NATURE MATERIALS

17 SUPPLEMENTARY INFORMATION 345 a b Figure S5c: a) Normalized radius (r o / Λ f as a function of the nanofibril volume fraction for square and hexagonal two packing geometries and values of the shear modulus of the matrix µ = 1 4 GPa and cut-off frequency Ω = 17.5 GHz (see manuscript and eq. S5.5). Here we assume an effective periodic unit cell. b) Nanofibril spacing as a function of nanofibril volume fraction for various shear moduli of the matrix phase. µ = 1,,3 and 4 GPa from bottom to the top Figure S5d: Numerical calculations of the axially polarized eigemodes (at q = 0) for the same set of elastic constants but different nanofibril cross sectional shapes, ranging from (a) square crosssection shape (hence mirror symmetric) (black outline), (b) non-symmetric closed curved cross section, to (c) P-shaped, all on a square lattice. NATURE MATERIALS 17

18 S6. Stretching effects S6a. Morphological changes upon stretching We next discuss the unique features of mode 1 arising from the strain-dependent nonlinearity, a consequence of the spider silk structure. In spider silk there are essentially two types of β-sheets: the ones that contain the repetitive alanine sequence (Ala) n and others that form from other residues, including glycine units 7,8. Under super-contraction, the breaking of hydrogen bonds in the predominantly non-crystalline matrix region, allows the chains to relax and reach conformations closer to equilibrium, while the distribution and orientation of β-sheet crystals within the nanofibrils remains minimally affected 9 3. Since the orientation of nanocrystals along the fiber axis remains high 33 35, the nanofibril domains must be less elongated than in the native fiber, reflecting the axial shrinkage and radial thickening of the of the nanofibers during supercontraction. We note here that our model regards short β-sheet structures not belonging to the (Ala) n sequence as a part of the amorphous region. S6b. Stress-strain curves of supercontracted and native spider silk Depending on the species of spider and the particular silking conditions, native dragline silk can be stretched up to 30% (here only up to 18%). Hysteresis is always observed, when silk is stretched beyond about %. The stress-strain curve with fixed displacement speed (black curve) is not repeatable (Fig. S6a). If dynamic mechanical analysis is performed at some strain, a different and repeatable straight line behavior is observed (red segment). This yields the dynamic elastic modulus of silk (Table S1), which is nearly constant in the frequency range Hz. Here it has a value of 43.8 GPa for the native fiber at a strain of 0.1. With additional stretching, the stress-strain curve continues onto the non-repeatable black line, as shown in Fig. S6a. 18 NATURE MATERIALS

19 SUPPLEMENTARY INFORMATION Figure S6a: Mechanical properties of dragline silk of Nephila edulis spider in native (black) and supercontracted (blue) state. Strain is calculated taking the length of the unstretched native fiber as reference. As spider silk shows strong hysteresis above % strain, the stress-strain curve corresponding to application of low amplitude sinusoidal strain function is shown for comparison (red, inset). Table S1: Elastic modulus extracted from these measurements in Fig. S6a 9,33. Strain Elastic modulus / GPa native supercontracted Native dragline silk can shrink, or supercontract up to 50%, depending on silking conditions, use of liquid water or water vapor, and applied load during the shrinking process. Here the stress-strain curve of a sample supercontracted by 40% is shown (blue in Fig.S6a). Similar to native silk, this curve is not repeatable. The dynamic modulus was measured in a similar way to native silk. We note here that even at supercontractions of down to -0%, we are still in the strain-hardening regime, indicating that some pre-strain is still present in the spider silk and that the chains have not equilibrated. NATURE MATERIALS 19

20 S6c. Negative group velocity and the physical origin of non-locality The first anomalous feature in the dispersion relation lies in the presence of negative group velocity for axial phonon propagation. This clearly indicates the presence of spatial non- locality in the medium, as the spatial dispersion is not linear (i.e. frequency independent as in a typical linearly elastic medium). We first discuss the main features. Typically, a q* value of zero group velocity =0 demarcates a region of strong repulsion between bands >0 36 or the presence of periodicity along the propagation axis <0, i.e. a Bragg-type band gap as has been observed for example in colloidal crystals 1 in the linear elastic regime. The latter prerequisite is, however, absent in this remarkable natural construct. First, the spider silk fiber consisting of bundles of stiff nanofibrils surrounded by an amorphous matrix (Fig. 1f) does not possess a discrete periodicity at a value of a ~ 180 nm (q* = π / a ~ nm 1 ) along the propagation axis which is a precondition for a Bragg-like band gap. Moreover, we do not detect the corresponding upper branch which would be of the same symmetry and hence detectable 1,37,38. Another key observation ruling Bragg scattering out is the behavior under supercontraction and restretching. Under supercontraction, when the chains are relaxing, one would anticipate that if the spider silk had strict discrete periodicity, this would be reduced and hence q*= π / a shift to higher values. This however, does not occur, and the q* does not change under supercontraction. Clearly, the shift in the q* is also not monotonic, as one may expect in going from supercontraction to stretching, if the periodicity was responsible for the dispersion. In addition, the lower band edges of mode 1 do not change monotonically either, which one would expect with decreased (supercontraction) and increased (stretched) scattering strengths. Instead, under supercontraction, the band edge is lowered (in frequency), while under stretching, the band edge remains at the same frequency. All these factors, taken together, categorically exclude Bragg scattering as the mechanism for the phononic dispersion behavior. Second, the possible assignment of branch 1 dispersion to a hybridization mechanism 0 NATURE MATERIALS

21 SUPPLEMENTARY INFORMATION (between the guided mode and the continuum), is unlikely due to the presence of the negative dispersion region, and especially the sharpness of the curvature about q*. We emphasize that this region of negative group velocity is distinct from previous reported observations 17,18,39,40, where the negative dispersion arises from strong repulsion from the guided modes 39,41, i.e. similar to mode 3 and exists around a region of positive curvature, i.e. >0. In contrast, the observed negatively dispersive region occurs in mode 1 around a region of negative curvature, i.e. <0. Referring to Fig. S6b, we note that native spider silk possesses significant pre-strain due to the spinning process This pre-strain is not uniformly distributed throughout the spider silk fiber 9 31, particularly in the amorphous matrix, which is dominated by the semi- oriented glycine-rich amorphous chains. This leads to the presence of both stress and strain gradients 4 45 in the matrix, which forms the origin of the spatial non-locality within the spider silk matrix This spatial non-locality in the strain can be parametrized by l c, representing the correlation length over which the strain gradients act 4,43,50. Following the approach of Eringen and others 4 45, one can then represent the nonlocal 443 strain, ε non-local as: ε = ε + ε non local local lc non local, while the corresponding nonlocal stress, σ non-local can be given by σ non local = E (( 1 ξ) ε non local + ξε) and ξ provides the corresponding scaling (average between the local and non-local portions of the strain) of the non-local effect of the strain gradients. As we are probing the longitudinally polarized mode, we may subsequently represent the equation of motion to obtain: u c t 0 u l x u x t 4 c + ξ l c c 0 4 u = 0 4 x NATURE MATERIALS 1

22 Figure S6b: Top) Schematic of the inhomogeneous microstructure of a nanofibril of spider silk. The pre-strain distribution in the spider silk, incurred during spinning presents itself strongly as a distribution of pre-strain in the amorphous matrix 34 36, containing a spatial distribution of semi-oriented amorphous hydrogen bonded chains. This gives rise to spatial stress and strain gradients 47 49, which translate into an effective non-local correlation length. Bottom) Schematic of the equivalent anharmonic linear chain, with the discretized lattice parameter α 0 utilized to capture this non-locality in the amorphous matrix. The dispersion relation of such an inhomogeneously strained medium chain (Fig. S6b) is given by: 46 ω = kc 0 1+ ξl k c c 1+ l k We note that this non-local continuum model contains two parameters, related to the strain gradients present in spider silk, both of which are difficult to extract experimentally. However, it has been shown 43 49, that one can further show that this non-local model can be mapped onto a compact discrete Born-Karman type model lattice chain 43,44 to capture the spatial non-locality, essentially by matching the band-edge and the position of zero group velocity of the discrete lattice chain model with the derived dispersive equation from the continuum model above. NATURE MATERIALS

23 SUPPLEMENTARY INFORMATION This is a fundamentally important point, as it provides a compact mapping of the length scales between the physical non-locality and the corresponding lattice chain that represents it. This ratio, also obtained by others π 8, is given by: = ~ α l c 0 π Interestingly, the physical non-local correlation length is much smaller than the lattice spacing in the discrete chain model utilized here to model the spatial non-locality. In addition, the experimentally estimated physical distance between β-sheet crystallites is often taken 9 31 to be between 14 and 0 nm. This sets the minimum physically reasonable value of α 0 > 90 nm. In addition, to ensure that the discrete effects imposed by the choice of our lattice chain does not affect the dynamics, we require α 0 < 180 nm, to ensure that the dispersion is not due to an artefact of Bragg-scattering (at the Brillouin zone edges), which we have ruled out previously. Thus we chose to use such a compact discrete chain model (Fig. S6b) that provides a simplified but physically meaningful non-local correlation length arising from the strain gradients, into a discrete lattice chain model with a lattice spacing that captures the prestrain effect correctly. Another advantage of this choice is that we may further utilize this lattice chain model to incorporate the nonlinear elastic effects by incorporating anharmonicity through the spring constants, which retains the same physical nonlocality in spider silk S6d. Weak elastic nonlinearity Together with the observed cut-off dispersion exhibited by mode 3 (Section S5), it is obvious that the dispersion relation of spider silk in the axial direction is far from linear, pointing towards the rich dynamics (anomalous dispersion) present in the hierarchical silk structure at these small length scales. The dispersion of mode 1 is anomalous, possessing a negative dispersion region that evolves non-monotonically under large super-contraction and positive stretching (Fig. 3a,b). As mentioned earlier, the formation of the negative dispersion region does not originate from structural periodicity either along the fiber axis (S5, S6c) or transverse periodicity 39,40. To capture the evolution of the dispersion under NATURE MATERIALS 3

24 large finite strains, we utilize a nonlinear mass-chain model of the fiber. This model involves a discretization of the spider silk medium into effective unit-cells with the lattice spacing capturing the spatial non-locality, as discussed in the preceding section. The nonlinear elastic response is captured through the introduction of anharmonicity and higher order spring constants. In this discretization each effective unit-cell experiences an averaged effective stress (in the static sense), which arises from the pre-strain that is generated during the spinning. As already mentioned, the pre-strain is non-uniform, hence strain gradients are present, leading to the non-locality within the spider silk, especially the amorphous matrix. In order to capture both the nonlocal and nonlinear effects compactly, we elect to formally map this non-locality into a discrete chain model, where the lattice parameter α 0 is about 5 times the non-local correlation length, l c (Fig. S6b). At the same time, this effective lattice separation needs to be small enough such that this discretization approximation holds; in our case, this effective lattice separation should be α 0 < 180 nm. Following the approach of Askar et al., we consider the following anharmonic potential energy 4 : V n n = l+ l n k n 1 l, (S6.1) ( u u ) where we retain up to the 4 th order spring constant (the 1 st order spring constant is zero and the harmonic approximation starts from the nd order for stability, the 3 rd and 4 th order spring constants are required for stability in the potential energy). We note that this is the minimal number of higher order constants required to model the anharmonicity at large strains. As we are primarily dealing with a single axially polarized (longitudinal) propagation mode, we can simplify the model to the one-dimensional nonlinear lattice chain (Fig. S6b). We finally arrive at the following equation of motion: m eff u 3 3 [ u u ) ( u u )] + k [( u u ) ( u u ) ] + k [( u u ) ( u u ] l = k ( l + 1 l l l 1 3 l + 1 l l l 1 4 l + 1 l l l 1 ) (S6.) Where m eff is the effective center of mass of the unit-cell, k is the harmonic spring constant, k 3 is the third order spring constant and k 4 is the 4 th order spring constant. 4 NATURE MATERIALS

25 SUPPLEMENTARY INFORMATION Assuming translation invariance, m eff + k iqα 0l ul ( t) U ( t) exp =, we obtain: iqα 0 iqα 0 iqα 0 iqα 0 u l = k [( exp 1) ( 1 exp )] u l + k 3 [( exp 1 ) ( 1 exp ) ] iqα 0 3 iqα [( exp 1 ) ( 1 exp ) ] u 4 l u l (S6.3) We see that this equation is nonlinear up to 3 rd order in u l ; we now seek a time harmonic solution that holds for small but finite (not infinitesimal) amplitude. As this equation is nonlinear, we apply Poincare s Method to avoid divergent secular terms, and consider the 530 frequency as ω( λ) = ω + λω + λ ω +...; ω = k m eff qα 0 sin and the displacement λ function as η ( ωt, λ) = ul ( t) = η0( ωt) + λη1( ωt) + λ η( ωt) +... ; here the parameter λ sets α 0 the scaling parameter (and is set to 1 eventually). By grouping terms in order of the powers of λ, and applying boundary conditions of a pre- strain u l (0), we eventually solve for all terms of Ω m up to second order (because our potential energy is only up to the 4 th order term), and obtain the dispersion relation: F nonlinearity k3 k' 3 = ( k α ); k' clinear qα 0 ω ( q, u l (0)) = sin( ) Fnonlinearity ( q) (S6.4) α ul (0) 5 ( q) = {1 + ( ) [ k' 3 α k4 = ( α k 0 ) 0 3 sin ( qα 0) + k' 4 (cos(qα 0 ) cos( qα 0))]} 4 (S6.5) u l (0) is the normalized pre-strain, α 0 is the discretization length-scale that captures the effective structural nonlocality and nonlinearity (by setting the length scale over which k 3 and k 4 are defined as effective spring constants) in the medium, k 3 and k 4 are the ratios of the 3 rd and 4 th order modulus over the linear modulus, capturing both the velocity increase and the curvature in the dispersion relation, respectively. We note that for the present experimental cases of interest (supercontraction at -0% strain, native fiber to strained fiber to +18%), pre-strain is present within the spider silk, as evidenced from Fig. S6a. In our model, for each configuration in the strain hardening regime, the microstructure and the associated nonlinearity are captured through {α 0, u l (0), NATURE MATERIALS 5

26 k 3, k 4 }. Hence, the different strain states represent a particular non-linear and non-local configuration of the spider silk. Importantly, as the pre-strain structure is not known a priori, the native spider silk case serves as our reference state, with the super-contracted and strained states perturbations of this native state. We recall that in the fully equilibrated state, where the pre-strains have been completely relieved, there should not be any anomalous dispersion and this is exactly what we observed, in the regenerated silk sample in Fig. d. For the native spider silk, the following parameters were used: a 0 = 115 nm, u l (0)/α 0 = 0.05, k 3 = -3.5, k 4 =.0. As stated previously, 90 nm < α 0 < 00 nm for the lattice chain model to be valid to represent the nonlocal and nonlinear effects. While the prestrain, u l (0) / α 0 is used as a fitting parameter in the representation of the experimental dispersion, the obtained u l (0) / a 0 < 0.5 (weak nonlinearity) is meaningful as well 50. For the supercontracted silk, the following parameters were used: α 0 = 110 nm, u l (0) / α 0 = 0.18, k 3 = -3.0, k 4 =.0. We note there that u l (0) / α 0 = 0.18 represents a 1% reduction in pre-strain over the reference native case, which is reasonable and a reduction in the 3 rd order effective spring constant k 3, which is also reasonable. The reduction in the lattice spacing is further consistent with the reduction in the non-locality due to some breakage of the hydrogen bond network upon super-contraction, as well as chain relaxation. For the strained spider silk, the following parameters were used: α 0 = 140 nm, u l (0) / α 0 = 0.5, k 3 = -3.9, k 4 =.0. We note that u l (0) / α 0 = 0.5 represents about 10% increase in the pre-strain over the reference native fiber case and an increase in the 3 rd order effective spring constant k 3, which are also reasonable. In all cases, we have kept the 4 th order spring constant the same in all the three cases, and have utilized the experimentally obtained effective medium acoustic velocity of 3080 ms -1 for all three cases as well for consistency. The matrix is comprised primarily of disoriented amorphous chains which are bound to nanofibrils, aggregates of β-sheet crystals and partially oriented amorphous chains. BLS discriminates phonon propagation in mode 1 under various nonlinear strains, exemplified 6 NATURE MATERIALS

27 SUPPLEMENTARY INFORMATION 576 clearly by the linear strain dependence of the moduli (Fig. 3c), which is a clear indication of 577 the elastic nonlinearity at play. The parameter u 0 tracks the nonlinearity with supercontraction or large positive strains in this microscopic model to the pre-strain generated in the spider silk during spinning, and as we can see from the simplified analytical model, forms the basis of the nonlinear dynamical behavior Figure S6c: Theoretically computed dispersion relations for native spider silk (blue) and silk under super-contraction (red) and native fiber stretched by 18% strain While this model is clearly oversimplifying the complex dynamics, it naturally tracks the leading contributions of the non-local and nonlinear components and importantly captures the acoustic velocity c linear (= 3070 m/s) was that obtained experimentally for the regenerated silk. Finally, we note that in our modeling of mode 1, we are modeling the dispersion of the effective medium, i.e. we ignore the effect of the 5 µm fiber diameter. The reason for this is because at hypersonic frequencies of our interest (> 8 GHz), we are dealing with phonons of wavelengths of < 500 nm, hence the phonons which we are probing experimentally do not sample the boundary. We note that the cutoff mode for the 5μm fiber, surrounded by air, is around 40 MHz, which is below the BLS detection limit (approx. 1 GHz resolution NATURE MATERIALS 7

28 dependent on to the Rayleigh wing); hence we are not concerned with the behavior in that regime. This point is also corroborated in the experiments conducted on the model glass fibers of a diameter of about 5μm in section S1. Another point which we noted earlier is that in this model, there is an initial linearly dispersive region that has a lower slope (Fig. S6d). This sound velocity is similar to that observed for the initial modulus measured by others 51, for the same spider silk and the sound velocity in the regenerated amorphous silk (Fig. d). Access to this low sound velocity would require very low frequencies and hence q-range that is non accessible experimentally by BLS. Our model very well matches the experimentally obtained data, even under super-contraction and strains to 18%. References 1. Cheng, W., Wang, J., Jonas, U., Fytas, G. & Stefanou, N. Observation and tuning of hypersonic bandgaps in colloidal crystals. Nat. Mater. 5, (006).. Little, D. J. & Kane, D. M. Image contrast immersion method for measuring refractive index applied to spider silks. Optics Express 19, 1918 (011). 3. Alonso-Redondo, E. et al. A new class of tunable hypersonic phononic crystals based on polymer-tethered colloids. Nat. Commun. 6, 8309 (015). 4. Yang, Y. et al. Toughness of Spider Silk at High and Low Temperatures. Adv. Mater. 17, (005). 5. Ebenstein, D. M. & Wahl, K. J. Anisotropic nanomechanical properties of Nephila clavipes dragline silk. J. Mater. Res. 1, (011). 6. Fossey, S., Kaplan, D. & Mark, E. J.. Polymer Data Handbook. (Oxford University Press, 1999). 7. Koh, C. Y. PhD thesis, MIT. (011). 8. Economou, E. N. & Sigalas, M. M. Classical wave propagation in periodic structures: Cermet versus network topology. Phys. Rev. B 48, (1993). 8 NATURE MATERIALS

29 SUPPLEMENTARY INFORMATION Gomopoulos, N., Cheng, W., Efremov, M., Nealey, P. F. & Fytas, G. Out-of-Plane Longitudinal Elastic Modulus of Supported Polymer Thin Films. Macromolecules 4, (009). 10. Achenbach, J. Wave Propagation in Elastic Solids. (Elsevier, 1984). 11. Seidel, A., Liivak, O. & Jelinski, L. W. Artificial Spinning of Spider Silk. Macromolecules 31, (1998). 1. Ene, R., Papadopoulos, P. & Kremer, F. Quantitative analysis of infrared absorption coefficient of spider silk fibers. Vib. Spectrosc. 57, 07 1 (011). 13. Tsukada, M. et al. Structural Changes of Silk Fibroin Membranes Induced by Immersion in Methanol Aqueous Solutions. J. Polym. Sci., Part B: Polym. Phys. 3, (1994). 14. Davydov, S. L., Zaretskii-Feoktistov, G. G. & Sudakov, V. V. Propagation of axially symmetric elastic vibrations along a waveguide in an elastic medium. Polym. Mech. 10, (1974). 15. Miklowitz, J. The Theory of Elastic Waves and Waveguides. (North-Holland, 1984). 16. Parnes, R. Dispersion relations of waves in a rod embedded in an elastic medium. J. Sound Vib. 76, (1981). 17. Landau, L. D., Pitaevskii, L. P., Lifshitz, E. M. & Kosevich, A. M. Theory of Elasticity. 7, (1986). 18. Meitzler, A. H. Backward-Wave Transmission of Stress Pulses in Elastic Cylinders and Plates. J. Acous. Soc. Am. 38, (005). 19. Marston, P. L. Negative group velocity Lamb waves on plates and applications to the scattering of sound by shells. J. Acous. Soc. Am. 113, (003). 0. Ayzenberg-Stepanenko MV. Non-steady state wave propagation in composite structures. in sec. 4.8., Sapede, D. et al. Nanofibrillar Structure and Molecular Mobility in Spider Dragline Silk. Macromolecules 38, (005).. Rousseau, M.-E., Hernández Cruz, D., West, M. M., Hitchcock, A. P. & Pézolet, M. Nephila clavipes Spider Dragline Silk Microstructure Studied by Scanning Transmission X-ray Microscopy. J. Am. Chem. Soc. 19, (007). NATURE MATERIALS 9