Silk is emerging as a promising new class of photonic materials. Silk is a protein-based nanocomposite with a well-known capacity to respond structurally to its chemical environment, such as pH and water content [1, 2]. Silk is also a very stable material that forms at room temperature, enabling complex biomolecules to be incorporated into the silk matrix . Silk-based materials are, therefore, very promising candidates for optically based chemical and biological sensors that are biocompatible, biodegradable, and highly sensitive. Silk-based photonic devices fabricated, to date, have mostly involved working with silk fibroin solution cast into planar photonic structures such as diffraction gratings . There is emerging interest now in employing silks’ intrinsic chemical response for micro-fiber-based sensor geometries.
Spider silks from orb-weaver spiders are naturally suited to this application, as they possess the required optical and geometric properties; they are optically transparent, spun as micro-fibers (with diameters typically between 0.5 and 5.0 μm), with a double-cylinder structure, and are produced naturally by spiders with no further post-processing requirements. Spider silks have also been proposed as interconnects between integrated photonic components , taking advantage of silks’ well-known mechanical robustness .
To date, much is unknown about the transverse optical (refractive index) structure of spider silks. Transmission electron microscopy has previously revealed the presence of layers within the silks, with this layered structure being more pronounced in some species than others [7, 8]. It is also possible that the nanocrystalline domains within the silks’ nanocomposite structure are not spread uniformly throughout the silk, but may vary radially within the central core of the silk. These structural elements could potentially manifest as discrete or graded refractive index layers. In addition to affecting the light-guiding properties of spider silks, transverse optical structure is a potential proxy for studying the distribution of the silks’ nanocrystalline domain elements and other mechanical qualities.
To probe the optical structure of spider silks, a variant of the inverse-scattering technique; where analysis of scattered electromagnetic fields are used to infer the properties of the scatterer, was employed. The experiment was contained within a standard bright-field microscope, which was used to illuminate the silk and image the scattered fields. Silks were immersed in a liquid with a traceable calibrated refractive index to maximize the sensitivity of the scattered field to the silks’ optical properties. Dragline silks are birefringent double cylinders, with the birefringent axis oriented parallel to the silk axis, and so rotating the incident linear polarization effectively “tunes” the refractive index of the silk. This effect has previously been utilized to measure the principal refractive indices of these birefringent silks .
The analytical procedure presented in this paper is based on the key observation that, even with careful index matching, the silks cannot be rendered completely invisible; there is always a “residual” visibility (or contrast) that emerges, due to a) the mixing of p- and s-polarized scattered fields, which possess slightly different transverse profiles, b) mismatch in the imaginary refractive index (or optical absorption), and c) transverse optical structure in the silks. The electromagnetic scattering theory can then be used to link the observed residual visibility to the optical characteristics of the silks, enabling the measurement of the principal refractive indices, the optical absorption (if the cylinder is assumed to be optically homogeneous), and the detection of transverse optical structure (when the absorption is independently determined). This procedure can be applied, in principle, at any wavelength where the silk and immersion liquid are transparent.
Section 2 of this paper outlines the theoretical basis of this analytical approach in detail. The procedure for calculating the optical properties of spider silks from images of the scattered field at different polarizations is outlined, as well as the procedure used to evaluate uncertainties. Section 3 details the experimental implementation in the specific case of dragline silks from Plebs eburnus and Argiope keyserlingi. Measurement of the silks’ optical properties is presented in Section 4, assuming a homogeneous silk. The strong contradiction between the calculated optical absorption of the silk with existing measurements is held as evidence that a transverse optical structure is almost certainly present within these silks. Quantitative analysis of the internal refractive index contrast is also presented for step index and graded index refractive index profiles. The main conclusions of this work are presented in Section 5.
2 Measuring the optical properties of silk: theory and uncertainties
The analytical approach used in this paper is designed for application to uniaxial birefringent cylinders in a double-cylinder configuration, with the birefringent axis lying parallel to the silk axis. The double-cylinder configuration arises naturally from the silk-spinning process, where two spinnerets each yield a single cylinder, which then adhere together. The principal refractive indices of the double cylinder are labeled;
where np and ns are the complex refractive indices for light polarized parallel and perpendicular to the birefringent axis, respectively. If the double cylinders are homogeneous, full characterization of the refractive indices requires evaluation of four quantities: np, ns, κp, and κs. As silks generally do not exhibit any obvious linear dichroism, the optical absorption is assumed to be isotropic, hence,
As with all immersion-based analysis techniques, double cylinders are immersed in a liquid with refractive index nL =nL−iκL such that ns<nL<np. As the double cylinders’ refractive index at some polarization orientation, θ, is given by
there exists a polarization orientation, θ0, such that n(θ0)=nL.
The double cylinder was positioned a distance z0 behind the focal (object) plane of the imaging objective, where z0 was defined as the position that yielded the highest irradiance contrast when imaging the scattered field with a parallel-polarized incident field at normal incidence. Modeled scattered fields on the X−Y plane intersecting the z-position where the on-axis irradiance was highest, could then be directly compared with experimental images and circumvent the need to accurately measure the position of the double cylinder with respect to the focal plane.
Silks were then imaged at both p- and s-polarizations. The visibility of the double cylinder under illumination was quantified by three contrast parameters, Cp, Cs, and C0, defined as
where Imax and Imin denote the maximum and minimum irradiances, respectively, in the measured irradiance profiles, and I0 is the incident (or background) irradiance. Uncertainties in Cp, Cs, and C0 (labeled δCp, δCs, and δC0) were calculated as twice the standard deviation of the normalized irradiance.
As the polarizations are separable in the case of a cylindrical scatterer, the theoretical relationship between normalized irradiance in the object volume (Ip, Is) and the optical properties of the double cylinder can be summarized as
where i=p, s; a is the radii of each cylinder in the double-cylinder structure, and k is the incident wave vector. The functions fp and fs are derived from electromagnetic scattering theory [10, 11]. It was assumed, based on previous TEM studies of spider dragline silk, that the double cylinders were in contact, with the same radius and refractive index, and with both cylinder axes in a plane parallel to the focal plane of the imaging objective.
To obtain a similar relation for the irradiance in the image volume, the functions fp and fs were convolved with the point-spread function of the imaging objective, p.
where the prime denotes the image volume, ∗ denotes a 2-D convolution, and k′ denotes the central illumination wave vector. Fi represents the same thing as fi, except with the numerical aperture of the incident field and imaging objective accounted for. As the double cylinder is closely refractive index matched with the surrounding medium, and k′ is usually fixed normal to the focal plane with magnitude defined by the measurement wavelength, Eq. (9) can be approximated as
where Δni=ni−nL and Δκ=κ−κL (i=p, s).
Optical properties of double cylinders were inferred by determining the optical constants that minimized the following error terms:
where max and min denote the maximum and minimum values in the specified quantities. While the irradiance is imaged over a 2-D plane, the symmetry of the scatterer means that the profile can be reduced to a 1-D cross-section.
To minimize the error terms, ΔCp and ΔCs were optimized for several estimates of κ. As the remaining value of ΔC0 was approximately linear with κ, new estimates for κ that simultaneously minimized all three error terms could be computed reasonably quickly with a Newton’s method-type approach.
Uncertainties in np, ns, and κ were derived by linearizing fp and fs about the measured values for these quantities. For brevity, perfect positive correlation was assumed (yielding the largest uncertainty estimates).
where δΔnp, δΔns, and δΔκ represent uncertainties in Δnp, Δns, and Δκ, respectively. Partial derivatives in Eq. (14) were computed numerically in the process of optimizing the error terms using the procedure outlined above. Given the optical properties of the double cylinder and liquid are uncorrelated, the final uncertainties for np, ns, and κ were calculated as
where δnL and δκL represent uncertainties in nL and κL, respectively, and were determined using specifications provided by the manufacturer of the refractive index liquid.
3 Experimental procedure
3.1 Sample collection and preparation
Spider silk samples were collected from the orb webs of two orb-web spider species, Plebs eburnus and Argiope keyserlingi. Spiders were collected from the grounds of Macquarie University (North Ryde, Sydney, Australia) and housed in a laboratory with controlled temperature, humidity, and day-night cycle. Spiders were watered each day and fed Drosophila (fruit) flies three to four times a week. Dragline silks were collected from webs using small Perspex frames backed with double-sided sticky tape, which helped preserve the native strain of the silk. A second layer of tape was then used to secure the silk and also acted as a spacer that enabled the silk to be suspended about 100 μm above a microscope coverslip when imaging.
3.2 Microscope system and image acquisition
An Olympus IX-81 inverted microscope (Tokyo, Japan) was used to image the silks. The instrument was operated in transmission mode, using a 40× objective (NA=0.6). Silks were aligned so that the plane containing the axes of both cylinders in the double cylinder structure were oriented parallel to the image plane of the microscope (Figure 1). The metal-halide (Olympus U-PS50MH) illumination source was filtered using an aperture stop set to minimum, corresponding to a numerical aperture of around 0.06. The bandwidth of the wavelength filters used to filter the incident illumination was around 10 nm. Filters with center wavelengths 450 nm, 486 nm, 546 nm, 589 nm, and 656 nm were used to investigate wavelength dependence.
Irradiance profiles at the image plane were recorded using an Olympus DP-72 digital camera. Images were collected as 8-bit grayscale and were averaged over 64 acquisitions to minimize random temporal noise. Because the background illumination field was not completely uniform, a background subtraction was performed using the formula;
where Ibc represents the background-corrected image, I and I̅ represent the original image and its average grayscale level, respectively, and Ib and I̅b represent the background image (i.e. with the spider silk removed) and its mean grayscale level, respectively. Performing the background subtraction in this way ensured that the correct contrast values were preserved through the operation.
After background subtraction, images were then normalized to the background level (i.e. the grayscale level far from the silk). Silks were oriented vertically in the field of view of the camera, which enabled a 20- to 30-fold reduction in random spatial noise by averaging the grayscale levels along the length of the silk to obtain a 1-D irradiance profile along the cross-section of the silk.
3.3 Silk immersion
Silks were immersed in a NIST traceable refractive index liquid from Cargille (Cedar Grove, NJ, USA). The condition that nL lays between np and ns was checked by rotating the illumination polarization orientation through 90°. If this condition was satisfied, the irradiance profile would undergo a characteristic “inversion” (Figure 2).
Liquids were temperature calibrated to the ambient temperature using the known thermo-optic coefficient. Liquids were wavelength calibrated using dispersion data supplied by Cargille. The imaginary refractive index of the liquid, κL, was calculated using;
where α is the absorption coefficient supplied by Cargille. Uncertainty in the real component of the refractive index was specified as ±0.0002 at 589 nm, and ±0.001 at 486 nm and 656 nm. Uncertainties at other wavelengths were inferred using a Monte Carlo approach in conjunction with the Hartmann dispersion formulae , using the uncertainties specified above to estimate probability distributions of the fit coefficients.
3.4 Measuring silk radii
Silk size was measured optically by measuring the separation of the diffraction fringes observed under parallel-polarized illumination with a wavelength of 546 nm and comparing them to theory . Here, the silk size is quantified as the radii of the silk cylinders, which are assumed to be identical. This method yielded a radius measurement of 0.740±0.055 μm for the Plebs eburnus silk and 1.09±0.06 μm for the Argiope keyserlingi silk. This is an improvement in precision over conventional microscopy. The ability to measure silk size radius in situ, with high precision, is advantageous in terms of consistency and simplifying the experimental procedure, as silks had a tendency to break under electron microscopy.
Note that the presence of transverse optical structure does affect this measurement procedure. It can be seen from previous work, however, that radii measurements of Argiope keyserlingi silks were in good agreement with SEM data, suggesting that transverse optical structure, if present, is not sufficient to affect the uncertainty of these measurements.
4.1 Homogeneous silks
Here, silks are first analyzed assuming them to be homogeneous, with the intent of using these results to establish, by contradiction with independent measurements of spider silks’ optical absorption [5, 14], that silks possess transverse optical structure.
Measured principal real refractive indices (np, ns) and imaginary refractive indices (κ) of a Plebs eburnus dragline silk and an Argiope keyserlingi dragline silk are shown in Figure 3. An analysis of the P. eburnus silk estimated these imaginary refractive index components to range from (4.5±0.7)×10−4 at 450 nm to (1.0±1.1)×10−4 at 589 nm. At 589 nm, it is reasonable to suppose that κ cannot be negative (implying optical gain); hence, (1.0±1.1)×10−4 implies only an upper bound of 2.1×10−4. For the A. keyserlingi silk, the imaginary refractive index component was estimated to range from (9.3±1.0)×10−4 at 450 nm to (4.5±0.8)×10−4 at 546 nm.
The corresponding optical absorption as a function of wavelength for these imaginary refractive index estimates are shown in Figure 4. Comparison of these estimates with those of Huby et al. (1.0±0.4 dB/mm for Nephila clavipes dragline silk at 633 nm ) and Tow et al. (0.2±0.1 dB/mm for Nephila edulis dragline silk at 633 nm ) demonstrates that the estimated imaginary refractive indices are highly implausible. Therefore, it can be concluded with confidence that the assumption that the silks are optically homogenous is false and that they do, in fact, possess transverse optical structure.
4.2 Silks with transverse optical structure
The reason the method outlined in Section 2 overestimates the apparent absorption of homogeneous double cylinders is because the (real) refractive index mismatch present internally within each cylinder of the double-cylinder silk generates excess contrast, most evident in the calculation of C0. The excess contrast was calculated as the error term ΔC0, which is obtained using a “known” absorption. With some assumptions, the excess contrast can be used to infer aspects of the transverse optical structure of the silk double cylinders.
A uniform-layered double cylinder with inner radius, ain, and internal refractive index contrast, Δnin, can be modeled at an incident polarization orientation of θ0, as a homogeneous double cylinder with cylinder axes spaced by 2a when the outer layer is refractive-index matched with its surrounds. Here, the following empirical formula was found to be accurate for estimating Δnin from the scattering theory, provided this refractive index contrast does not have a strong polarization dependence;
where ain is in μm. Setting ain to be consistent with outer layer thicknesses observed with TEM (100 nm for Plebs eburnus dragline, 50 nm for Argiope keyserlingi dragline , yielding ain=0.64 μm and 1.04 μm, respectively), Δnin was estimated to be (1.6±0.2)×10−4 at 589 nm and (3.6±0.3)×10−4 at 656 nm for the P. eburnus dragline, and (5.5±0.5)×10−4 at 589 nm and (5.1±0.4)×10−4 at 656 nm for the A. keyerlingi dragline. An absorption of 1.0 dB/mm was assumed for both silks; however, it should be noted that the influence of this absorption was minimal. Note that there is currently insufficient evidence to formulate an estimate for the uncertainty of ain; however, the consequent uncertainty in Δnin based on any given uncertainty of ain can be worked out from Eq. (22).
This contrast in refractive index is indicative of a change in either the density or the mean polarizability of the silk over its cross-section. The relationship between density and refractive index is generally summarized by the Lorentz-Lorenz formula;
where α is the mean polarizability, M is the molar weight, and NA is the Avogadro’s constant. Taking the density of spider silk to be 1.3 g/cm3, the ratio α/M is calculated to be around 0.058/NA at 589 nm, giving ∂ρ/∂n=1.94 g/cm3 for the Plebs eburnus dragline. Assuming that the mean polarizability and molar weight remains constant, a refractive index change of 1.6×10−4 corresponds to a density change of about 0.31×10−3 g/cm3. For the Argiope keyserlingi dragline, ∂ρ/∂n was computed to be 1.89 g/cm3, giving a density change of 1.04×10−3 g/cm3. We note that these values of ∂ρ/∂n are similar to values that have been measured for a broad range of optical materials [15, 16].
Transverse optical structure may also manifest as a graded refractive index contrast. An approximate formula for the peak refractive index contrast can be derived from Eq. (22) by assuming that the contribution to ΔC0 from each infinitesimal layer is additive,
where Δnp is the peak refractive index contrast, and P(r) is the radial shape of the refractive index profile, scaled to a maximum value of 1. For example, if the refractive index profile is Gaussian with standard deviation σa, then,
If σa is taken to be ain/2 as an example (making ain the 1/e2 width), then the peak refractive index contrast of the Argiope keyserlingi silk is estimated from the values quoted above to be (1.93±0.17)×10−3 and (1.79±0.14)×10−3 at 589 nm and 656 nm, respectively.
It should be re-emphasized that these values for Δnp and Δnin are for the specific estimates of ain, and these values would change with any change in ain. In the case of a step index change, the dependence of Eq. (22) means the predicted Δnin would increase by a factor of 4 if ain is halved.
Determining ain optically may be possible if the dispersion of the silk material is accurately known. Contrast measurements made at different wavelengths would enable ain to be narrowed down to a single consistent value. Accurate measurement of spider silk dispersion and optical measurement of ain are topics of future work.
We have investigated the transverse optical structure of Plebs eburnus and Argiope keyserlingi dragline spider silks using an inverse-scattering approach, facilitated through quantitative bright-field microscopy. Using the incident linear polarization to tune the refractive index matching between the birefringent silks and immersion liquid, it was observed that the image contrast was much higher than expected for homogeneous silks with an optical absorption around 1 dB/mm. We conclude on this basis that the silks possess transverse optical structure.
This excess contrast was used to estimate the internal refractive index contrasts, Δnin, between the outer layer of the silks and silk core in each cylinder of the double-cylinder arrangement (which were assumed to be identical). Here, Δnin was estimated to be (1.6±0.2)×10−4 for Plebs eburnus dragline silk with a uniform 100-nm-thick outer layer and (5.5±0.5)×10−4 for Argiope keyserlingi dragline silk with an outer layer thickness of 50 nm, both at a wavelength of 589 nm. This led to an estimated density contrast of 0.31×10−3 g/cm3 and 1.0×10−3 g/cm3 for the P. eburnus and A. keyserlingi draglines, respectively, assuming no changes to the mean polarizability. The method successfully differentiates these small differences, and this is the first estimate of the refractive index contrast between the outer layer and core of spider silks. Assuming different characteristics in the optical structure, such as different outer layer thicknesses and/or graded index profiles, changes these estimates.
Insights gained into the optical structure of spider silk using this technique will contribute to the eventual understanding of guided-mode behavior within the silk micro-fibers and how they evanescently interact with the surrounding medium. This understanding will be important for the use of spider silks in sensing applications moving forward.
It is also worth highlighting that the procedure described herein can be applied to optical micro-structures that have a cross-section that is invariant along their length. All this requires appropriate modification in the functions fi, describing the scattered field profile. Similarly, the functions Fi can be modified to account for different illumination and imaging conditions.
This research was supported by the Australian Research Council (ARC) Discovery Project Grant DP130102674 and Macquarie University. The authors thank Professor Mariella Herberstein, Department of Biological Sciences, Macquarie University, for expert advice on spider biology and for access to webs from her laboratory.
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About the article
Published Online: 2016-10-20
Citation Information: Nanophotonics, Volume 6, Issue 1, Pages 341–348, ISSN (Online) 2192-8614, ISSN (Print) 2192-8606, DOI: https://doi.org/10.1515/nanoph-2016-0125.
©2016, Douglas J. Little et al., published by De Gruyter.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0