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Abstract
Scales of Cyphochilus white beetles present one of the strongest optical scattering materials in nature. However, the intricate optical fibrillar network nanostructure inside the scales has been difficult to mimic. Here, characteristic structural parameters inside Cyphochilus scales – mean fiber diameter, diameter distribution, filling fraction, and structural anisotropy – are replicated in synthetic nanofibrous materials to functionally mimic the biological material. To fabricate the synthetic nanostructure, electrospinning is chosen because this conventional technique is amenable to nanomanufacturing. The optimized parameters in electrospun structures are found to be only slightly different from those in Cyphochilus scales. At the optimum, electrospun structures exhibit even stronger optical scattering than Cyphochilus scales. An electrospun film with the similar characteristic structural parameters as those in Cyphochilus scales gives two resonance peaks in visible reflectance spectrum in the limit of a uniform fiber diameter, giving a purple structural color. As the distribution of diameter increases appreciably to experimentally achievable degrees, the resonance peaks broaden and the reflectance spectrum becomes relatively flat, resulting in disappearance of the structural color. These results support that controllable fibrous nanostructures that exceed the exceptionally strong broadband optical scattering found among living organisms can be volume-produced.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
Bo Kyung Park, Sang M. Han, and Sang Eon Han, "Surpassing Cyphochilus scales in optical scattering strength by well-controlled electrospun nanostructures: errata," Opt. Mater. Express 13, 1726-1726 (2023)https://opg.optica.org/ome/abstract.cfm?uri=ome-13-6-1726
1. Introduction
Visual perception of white color stems from multiple scattering of broadband light in random structures. Such structures are found in diverse organisms including beetles [1–4], butterflies [5,6], spiders [7,8], cuttlefish [9], squids [10], and clams [11]. These organisms share common structural features: low-index scatterers of optical length scales are randomly distributed to render white color. However, white beetles are distinguished from the others by their unique structure. White beetle scales contain an interconnected fibrillar nanostructure, while the others possess structures that are based on particles or compartmentalized units such as spheres [9], ellipsoids [5,6], prisms [8], and multilayer aggregates [10,11]. The fibrillar nanostructure in the scales of a white beetle genus, Cyphochilus [ Fig. 1(A)], is strongly anisotropic [Fig. 1(B)], which makes it one of the most efficient light-scattering biological structures known to us today [1,2]. The fibrils in the nanostructure are made of a low-index chitin [12] and mostly oriented in transverse directions with near isotropy in the lateral plane [2,4]. Combined with the anisotropy originating from the fibril orientations, a mean fibril diameter [1,2] of ∼0.25 µm (Supplement 1) and a filling fraction [4,13] of ∼31.5% achieve the exceptionally strong optical scattering. Prior studies computationally reconstructed approximations of this nanostructure, using the Cahn-Hilliard equation [4], descriptor functions [14], and a branching random walk algorithm [15], to establish the structure-scattering relationship. A rather unrealistic but simple model of suspended spheroid nanoparticles [16] was also used to elucidate the relationship. These studies have revealed that the strong scattering depends on the interplay between key structural parameters such as fiber mean diameter [16], filling fraction [4], anisotropy [15,16], and regularity [14].
While the key structural parameters have been identified in these studies, mass production of such computationally reconstructed structures, especially the anisotropic structures, has been challenging due to their formidable complexity. If one sacrifices anisotropy, which is an important structural parameter for the strong scattering in Cyphochilus scales [1–3,13,15,16], nanostructures that approximately follow the solutions to the Cahn-Hilliard equation can be obtained by slow phase separation [4,14,17]. However, previous studies [4,18–20] used kinetically-controlled fast phase separation of polymeric mixtures to create porous nanostructures. In this case, the separation process does not lead to a thermodynamically stable state and produces structures of multiple lengths scales with a large surface-to-volume ratio. The large surface free energy makes the structures less stable against various environmental factors such as temperature variations, chemical exposures, mechanical stresses, and high energy irradiations. In comparison, Cyphochilus scales preserve structural regularity with relatively uniform length scales, while not strictly minimizing the surface free energy [14]. Within this scientific context, one of the remaining challenges is how to create strongly scattering anisotropic nanostructures of relatively low surface free energy, such that they share the key structural parameters with Cyphochilus scales, using fabrication techniques that are amenable to mass production.
In this study, we make use of electrospinning [21–23], which is a well-established technique for mass production of nanoscale fibers [24], to create highly anisotropic fibrous network where fiber diameter is relatively uniform around a quarter micrometer and filling fraction is near 30-40% in agreement with those in Cyphochilus scales. Compared to the phase separation techniques [4,18–20], the electrospinning method enables precise control of the nanostructural parameters. Table 1 compares strength and weakness of these techniques. In our straightforward approach, however, it is not yet clear whether the exceptionally strong scattering can be achieved in an electrospun structure by simply replicating the key structural parameters found in Cyphochilus scales. The optical scattering in electrospun structures may not be sufficiently strong since the fibers are much longer than their diameter. These long fibers support mostly 2-dimensional (2D) scattering which is known to be weak as manifested by a strong peak in the forward direction in the phase function [25]. Despite the uncertainty, our study reveals that electrospun films of nonwoven silk nanofibers, when appropriately structured, can surpass Cyphochilus scales in scattering strength for the entire visible spectrum. Furthermore, our detailed modeling study shows how the key structural parameters affect scattering properties in the electrospun films. This detailed investigation, as well as optimization, is facilitated by the fact that, in comparison to highly irregular structures in previous studies [4,18–20], electrospun structures are simple such that their optical modeling is straightforward.
Table 1. Comparison of various techniques for fabricating white films of highly strong scattering.
2. Materials and methods
2.1. Electrospinning of regenerated silk fibroin
Silk solutions were prepared by our previously reported method [23] using Bombyx mori (Geumokjam) white cocoons. For electrospinning, the solutions were loaded into a syringe with a 21-gauge stainless steel needle (0.495 mm inner diameter). A high voltage (21 kV) was applied to the tip of the needle and a stainless steel collector was grounded. Distance between the tip and the collector was kept at 15 cm. Filling fraction was controlled by changing the sizes of square-shaped collectors (3, 3.5, 4, 4.5, and 5 cm).
2.2. Optical and structural characterization
Transmissivity spectra of electrospun silk films were measured by a spectrophotometer (USB4000VIS-NIR, Ocean Optics) with an integrating sphere (ISP-50-8R, Ocean Optics). Nanostructures in electrospun silk films were characterized by scanning electron microscopy (SEM) (FEI, Helios Nanolab 660) after Au-Pd coating. Mean values and standard deviations in fiber diameters were calculated from measurements over 1100–1200 fibers based on the SEM images.
3. Results and discussion
3.1. Optical modeling
Based on the high length-to-diameter aspect ratio and low curvature of electrospun fibers, we approximate them as a collection of infinitely long cylinders to model their optical properties. In the limit of negligible interaction between the cylinders, the Mie solutions for single cylinders can be used to predict the optical properties of the media. For dense fibrous media, the optical behavior becomes complicated due to interactions between the cylinders. In particular, as the cylinder diameter increases toward optical wavelengths, resonant modes tend to spread out of the cylinder, increasing and complicating the interactions [26]. In our approach, we assume weak interactions, so that the distant region surrounding the cylinder is regarded as being occupied by a uniform medium with an effective refractive index represented by the Maxwell-Garnett mixing rule [27]. For a medium filling fraction f, the cylinder with a diameter d is assumed to be surrounded by a concentric air cylinder with a diameter $d/\sqrt f $ so that the filling fraction in the space occupied by the two concentric cylinders is f [ Fig. 2(A)]. The region outside the air cylinder is treated as an effective medium. The Mie solutions for the concentric cylinders [28] in an effective medium are integrated over their orientations and diameter distributions to give the scattering cross section and the phase function [29]. A more accurate model, while not used in this study, can also be used where the effective index is determined self-consistently by imposing a condition that an average optical energy density in the two concentric cylinders is the same as that in the effective medium [30].
Fig. 2. (A) Effective medium model for fibrous random media. (B) Calculated phase function at λ = 0.555 µm and (C) effective transport mean free path (${L_{zz}}^{\ast \prime}$) spectra for anisotropic (random orientation only in the plane perpendicular to the incident light) and isotropic (random orientation in all directions) fibrous media for the same key structural parameters (except anisotropy) and the refractive index of the fibers as those in Cyphochilus scales. In (C), ${L_{zz}}^{\ast \prime}$ for Cyphochilus scales adapted from Ref. [15] is shown for comparison.
Optical scattering in dense electrospun fibrous media can be calculated following a radiative transfer approach. In this approach, the scattering cross section and the phase function are used in the radiative transfer equation (RTE) to calculate specific intensity as a function of spatial position and direction. RTE can be simplified into the diffusion equation when the angular dependence of specific intensity is assumed to be small [31,32]. In the diffusion approximation, for a lossless anisotropic medium, transmissivity and reflectivity of a thick film for normal incidence are determined by a single characteristic length. This length is called the effective transport mean free path ${L_{zz}}^{\ast \prime}$, which is the vertical thickness direction component (zz) of its tensorial form $\mathcal{L}^{\ast \prime}$ [31]. ${L_{zz}}^{\ast \prime}$ is the z-length over which the direction of light propagation becomes randomized, so that $1/{L_{zz}}^{\ast \prime}$ represents scattering strength in the z-direction [31]. The assumption of small angular dependence for the diffusion approximation can be problematic for fibrous media because of their strongly forward-scattering nature [25]. Nevertheless, transmissivity and reflectivity for electrospun media are calculated with negligible errors by the diffusion model [23]. Thus, we use $1/{L_{zz}}^{\ast \prime}$ to characterize the scattering strength of our electrospun films but solve RTE to calculate angular dependence of their transmitted intensity.
A collection of infinitely long cylinders supports mostly 2D scattering. Because the phase function for the 2D scattering is in general strongly peaked in the forward direction, electrospun structures might result in only weak scattering [25]. On an opposite side, an advantageous point for electrospun structures may be that the fibers are oriented mostly in the transverse directions, so that the orientation-induced anisotropy can be exploited for strong optical scattering. Indeed, previous studies have shown that optical scattering in Cyphochilus scales may be enhanced by increasing the anisotropy of the nanostructure [15,16]. However, it is not clear whether the same property would hold true for a medium of long fibers. Thus, we have tested by modeling calculations how optical scattering is affected by the anisotropy and how strong the 2D scattering in electrospun structures is in comparison to that in Cyphochilus scales.
Figure 2(B) shows calculated phase functions at a wavelength of λ = 0.555 µm for both isotropic and anisotropic electrospun structures that have the same refractive index [12], filling fraction [4,13], and diameter distribution [2] as those in Cyphochilus scales. The fibers are randomly oriented over all directions in the isotropic structure, whereas they are oriented only in transverse directions in the anisotropic structure. For the anisotropic structure, the phase function is for normal incidence. Forward scattering near the scattering angle of θ = 0° is strong [note that a logarithm is taken of the phase function in Fig. 2(B)] for both structures but, for the anisotropic structure, backward scattering is also significant. Accordingly, the phase function in the anisotropic structure contributes to increasing scattering strength or decreasing ${L_{zz}}^{\ast \prime}$ compared to that in the isotropic structure. Figure 2(C) shows that this behavior is maintained throughout the visible spectrum. Interestingly, ${L_{zz}}^{\ast \prime}$ in the Cyphochilus structure is located between ${L_{zz}}^{\ast \prime}$’s of the two structures. This is reminiscent of the previous study [15] where it is argued that, in the biological structure, scattering would become stronger as its anisotropy increases. The anisotropic structure considered in Fig. 2(B) and Fig. 2(C) has the maximum desired anisotropy among rotationally symmetric random fibrous structures because its fibers are oriented strictly in the transverse directions. In addition, the spectral dependence of ${L_{zz}}^{\ast \prime}$ for the electrospun structures is similar to that for Cyphochilus scales, as ${L_{zz}}^{\ast \prime}$ increases as λ increases for all structures considered in Fig. 2. Therefore, our modeling shows that anisotropic electrospun structures [turquoise line in Fig. 2(C)] may be able to achieve even stronger scattering than Cyphochilus scales throughout the visible spectrum, provided that the model for maximized anisotropy accurately represents optical behavior of experimental structures and that electrospinning of materials of similar refractive index to chitin can achieve the structural parameters used in the calculations. These two points are addressed below.
3.2. Model validation and optical characterization
To test whether our modeling can reproduce experimental results, we examined angular dependence of transmitted light. For experimental examination, we produced two electrospun silk samples: a free-standing fibrous film [ Fig. 3(A) inset] and a fibrous coating on a 150-µm-thick borosilicate glass slide [Fig. 3(B) inset]. The angular distribution of transmitted light intensity at normal incidence of a He-Ne laser at λ = 0.654 µm was measured for the two samples and compared with predictions by RTE for fibers oriented only in transverse directions. For the coating sample, the transmitted light emerged from the glass slide. To eliminate Fresnel refraction from the bottom surface of the slide, a hemispherical borosilicate glass (N-BK7 Half-Ball Lens, Edmund Optics) was glued to the slide surface with an index matching gel [Fig. 3(B) inset]. For both samples, we chose a mean fiber diameter ${d_0} = \; $0.20 µm and a filling fraction f = 19.6% with a film/coating thickness of 26 µm. Figures 3(A) and Fig. 3(B) show P(${\mu _e}$)/${\mu _e}$ for the free-standing and coating samples, respectively, where P is the transmitted light power and ${\cos ^{ - 1}}{\mu _e}$ is the angle from the normal as the light exits the samples. P(${\mu _e}$) is normalized by $\mathop \smallint \nolimits_0^1 P({{\mu_e}} )d{\mu _e} = 1$ and P(${\mu _e}$)/${\mu _e}$ becomes 2 for a Lambertian surface. In general, the calculated P(${\mu _e}$)/${\mu _e}$ spectra match well with experiments. The exact reasons for small deviations between the two are difficult to identify. However, a likely reason is that our model is limited in describing optical scattering at interfaces, as it assumes a unit in Fig. 2(A) derived from bulk structures and the fibers near interfaces have a different environment from the bulk. In addition, the calculated curve exhibits a sharp feature near ${\mu _e}$ $= $ 0.68 in Fig. 3(B) because Fresnel’s law is assumed for internal reflection. The smooth experimental response near ${\mu _e}$ $= $ 0.68 can be better described by an improved model such as that we developed previously [33]. Outside these minor subtleties at the interfaces, however, the current model can predict properties in the bulk, such as ${L_{zz}}^{\ast \prime}$, with better accuracy than those that are directly affected by the interfaces. We will show later that the model calculations of ${L_{zz}}^{\ast \prime}$ spectrua match well with experiments on electrospun structures.
Fig. 3. Angular intensity distribution of light transmitted through an electrospun film and scattered into (A) air and (B) borosilicate glass, obtained from experiment (red solid circle) and solutions to RTE (blue line), at λ = 0.654 µm. Mean fiber diameter is ${d_0} = \; $0.20 µm and filling fraction is $f = \; $ 19.6%.
To see how closely the key structural parameters of Cyphochilus scales [i.e., (i) anisotropy, (ii) fiber mean diameter, (iii) regularity, and (iv) filling fraction] can be realized in electrospun structures, we analyzed images of electrospun fibers obtained by scanning electron microscopy (SEM) and measured density [23] of the structures. (i) SEM images of top and cross section in Fig. 4(A) show that the fibers are randomly oriented mostly in the transverse plane, so that the anisotropy is strong. (ii) The fiber mean diameter, ${d_0}$, varies approximately from 0.20 to 0.30 µm in our experimental conditions. The mean diameter in Cyphochilus scales (${d_0}$ = 0.25 µm) is located in the middle. (iii) We define regularity as a structural property represented by diameter uniformity and surface smoothness of the fibers [14]. The diameter varies minimally along the fiber axis and the fiber surface is smooth. The diameter distribution has a relative standard deviation of $\sigma $ = 0.32–0.35, which is narrower than that ($\sigma $ = 0.49) in Cyphochilus scales (Supplement 1). It follows from the observations that, according to our definition of regularity, our electrospun structures are more regular than the structure in the beetle scales. (iv) Matching the filling fraction observed in Cyphochilus scales (f = 31.5%) is not trivial, since electrospinning typically produces fibrous structures with a low filling fraction [34]. To overcome this difficulty, we decreased the collector size in the electrospinning system, which greatly increased the filling fraction approximately from 10% to 60%. As the filling fraction was varied by changing the collector size, the fiber mean diameter ${d_0}$ remained fairly constant. For the two ${d_0}$’s tested in this study, the variation in ${d_0}$ upon filling fraction change is 0.20 ± 0.01 µm and 0.30 ± 0.02 µm.
Fig. 4. (A) SEM images of top and cross-section of an electrospun silk film. (B) Light scattering strength as a function of silk filling fraction and mean fiber diameter at $\lambda = $ 0.555 µm when $\sigma = $ 0.49 as in Cyphochilus scales. Effective transport mean free path (${L_{zz}}^{\ast \prime}$) as a function of filling fraction for (C) ${d_0} = \; $ 0.20 µm and (D) ${d_0} = \; $ 0.30 µm at $\lambda = $ 0.654 µm obtained by experiment (red dashed line and filled circles) and model calculation (blue solid line and filled squares). (E), (F) Effective transport mean free path (${L_{zz}}^{\ast \prime}$) spectra obtained by (E) model calculation and (F) experiment. In (E) and (F), ${L_{zz}}^{\ast \prime}$ for Cyphochilus scales adapted from Ref. [15] is presented for comparison. In (C) – (F) model calculation assumed experimentally determined $\sigma $.
Among the key parameters, anisotropy and diameter distribution are not easily controlled in electrospinning. Thus, for structural optimization using our model, we varied only fiber mean diameter and filling fraction with a fixed normal diameter distribution at $\sigma $ = 0.49 as in Cyphochilus scales (Supplement 1). Figure 4(B) shows scattering strength $1/{L_{zz}}^{\ast \prime}$ as a function of ${d_0}$ and f at λ = 0.555 µm where human eye is most sensitive under daylight. The optimum point over 0.1 µm $\le {d_0} \le $ 0.4 µm and 10% $\le f \le $ 60% is determined to be ${d_0}$ = 0.32 µm and f = 38%, which are slightly different from those in Cyphochilus scales (${d_0}$ = 0.25 µm and f = 31.5%). The optimum f is located roughly in the middle because the density of scattering centers vanishes as f approaches 0 or 100%. Having found the rough estimates of the optimum parameters in Fig. 4(B), we set electrospinning conditions to obtain samples with two fiber mean diameters ${d_0} = \; $0.20 µm and 0.30 µm with filling fractions varying from 10% to 60%. To extract ${L_{zz}}^{\ast \prime}$ from the samples, we performed linear regression on plots of inverse transmissivity vs. film thickness [23]. Figures 4(C) and Fig. 4(D) show that, at a He-Ne laser wavelength of 0.654 µm, ${L_{zz}}^{\ast \prime}$ is minimized at f = 26% and 40% for ${d_0} = \; $0.20 µm and 0.30 µm, respectively. These optimum f values are close to the predictions of 30% and 38% at the same ${d_0}$’s in Fig. 4(B). In fact, when actual diameter distributions are considered in our model, the ${L_{zz}}^{\ast \prime}$ curves show good match between experiment and calculation as shown in Fig. 4(C) and Fig. 4(D). ${L_{zz}}^{\ast \prime}$ spectra at the optimum f values for calculation and experiment are shown in Fig. 4(E) and Fig. 4(F), respectively. The calculation and experiment match reasonably well with each other; experiment gives slightly shorter ${L_{zz}}^{\ast \prime}$ than calculation.
Our electrospun structures outperform Cyphochilus scales in optical scattering as evidenced by lower ${L_{zz}}^{\ast \prime}$ curves in Fig. 4(F). Specifically, ${L_{zz}}^{\ast \prime}$ is approximately 0.9 µm – 2.1 µm for the two electrospun structures in Fig. 4(F), whereas it is 1.5 µm – 2.4 µm for Cyphochilus scales in the visible spectrum. Thus, the enhancement factor in the scattering strength by electrospun fibers from Cyphochilus scales is 1.1–1.7 in the visible spectrum. For all the structures, ${L_{zz}}^{\ast \prime}$ increases with $\lambda $. Because the refractive indices of silk (1.54–1.58) [35] and chitin (1.54–1.57) [12] are negligibly different from each other over the visible spectrum, the stronger scattering of electrospun silk is a structural effect. This result is remarkable when we consider that only a small number of key structural parameters were matched to those of Cyphochilus scales and that the visual queue of anisotropy was only qualitatively mimicked. More importantly, the result demonstrates that a mass-producible nonwoven fabric can exceed the exceptionally strong scattering observed in nature.
3.3. Scattering characterization
As the structure in an electrospun film is quite different from that in Cyphochilus scales, its scattering characteristics may also be different, despite similar magnitudes and spectral dependence in ${L_{zz}}^{\ast \prime}$. Scattering characteristics in an electrospun film can be investigated using our model based on infinitely long cylinders. In this case, the scattering characteristics can be represented by the differential scattering cross section, which gives the angular distribution of scattered intensity. However, this quantity is not trivial to obtain for the structure in Cyphochilus scales. At the minimum, a scattering unit has to be defined for such an intricate continuous structure to determine this quantity. Lee et al. developed a method to define a scattering unit in the structure within the diffusion theory [13]. In the work of Lee et al., instead of the differential scattering cross section, average cosine of the polar angles ($\theta $) of the scattered light, $\left\langle {\cos \theta } \right\rangle $, was calculated at $\lambda $ = 0.900 µm for varying incident directions represented by $\textrm{cos}\theta ^{\prime}$ where $\theta ^{\prime}$ is the incidence polar angle from the normal. Thus, we obtained $\left\langle {\cos \theta } \right\rangle ({\theta^{\prime}} )$ for our electrospun structure with the strongest scattering at the same wavelength. This structure was determined based on Fig. 5(A), where ${L_{zz}}^{\ast \prime}$ as a function of filling fraction at $\lambda $ = 0.900 µm shows a good match between experiment and calculation. Based on the model calculation result in Fig. 5(A), the electrospun structure with the strongest scattering has ${d_0} = \; $0.30 µm and f = 32%. At these parameters, $\left\langle {\cos \theta } \right\rangle ({\theta^{\prime}} )$ was calculated for an electrospun structure and compared with that for Cyphochilus scales [13].
Fig. 5. (A) Effective transport mean free path (${L_{zz}}^{\ast \prime}$) as a function of filling fraction for electrospun silk structures of ${d_0} = \; $ 0.30 µm at $\lambda = $ 0.900 µm obtained by experiment (red dashed line and filled circles) and model calculation (blue solid line and filled squares). Model calculation assumed experimentally determined $\sigma $. (B) Average cosine of scattering angle $\left\langle {\cos \theta } \right\rangle $ as a function of the cosine of incident angle $\cos \theta ^{\prime}$ for a scattering unit of Cyphochilus scales [13] and our model. The model calculation assumed the same ${d_0}$, f, $\sigma $, and refractive index as those in Cyphochilus scales. Inset shows the definition of $\theta $ and $\theta ^{\prime}$.
Figure 5(B) shows comparison between our results on the electrospun structure and the results on Cyphochilus scales by Lee et al. [13]. The sign of $\left\langle {\cos \theta } \right\rangle $ determines whether the scattering is overall in the positive or negative z-direction [see Fig. 5(B) inset]. The structures in the electrospun film and Cyphochilus scales exhibit different scattering behaviors from each other. Specifically, $\left\langle {\cos \theta } \right\rangle$ remains relatively independent of the incident angle for the electrospun structure, whereas Cyphochilus scales exhibit more pronounced dependence. However, both structures overall show $\left\langle {\cos \theta } \right\rangle $ not far from 0 for all incident angles, which would be observed for near isotropic scattering.
While this feature is remarkable, it is not clear how individual fibers affect the scattering characteristics because the diameter distribution is rather broad for both structures. Moreover, $\left\langle {\cos \theta } \right\rangle $ does not fully represent $1/{L_{zz}}^{\ast \prime}$ without being considered in combination with the scattering efficiency ${Q_{sca}}$. For example, when $\sigma = 0$ and the zz component of the anisotropy tensor is assumed to be unity, they are related by [31]
Figures 6(A)–6(C) show the results at $\lambda $ = 0.900 µm and f = 31.5% (same as f in Cyphochilus scales). In this case, the general trend of ${d_0}/{L_{zz}}^{\ast \prime}$ is captured more closely by ${Q_{sca}}$ than by $\left\langle {\cos \theta } \right\rangle $ [see Figs. 6(A)–6(C) by referring to Eq. (2)]. A resonance appears at ${d_0}$ $= $ 0.33 µm as $\sigma \to 0$ based on the peak in ${Q_{sca}}$ [Fig. 6(B)]. The resonance translates to a peak in ${d_0}/{L_{zz}}^{\ast \prime}$ as $\sigma \to 0$ at ${d_0}$ $= $ 0.30 µm [not clearly seen with the color scheme in Fig. 6(C)] which is only slightly different from that in ${Q_{sca}}$. The resonance peak broadens as the fiber diameter distribution increases.
Fig. 6. (A) – (C) Calculated $\left\langle {\cos \theta } \right\rangle $, ${Q_{sca}}$, and ${d_0}/{L_{zz}}^{\ast \prime}$ as a function of mean diameter and diameter distribution for $f = $ 31.5% at $\lambda = $ 0.900 µm. (D) – (F) are the same as (A) – (C) but at $\lambda = $ 0.555 µm. (G) and (F) Calculated electric energy density for incident polarization parallel to the fiber axis at $d = \; $ 0.20 µm and 0.34 µm, respectively, corresponding to the two resonances seen in (E) at $\sigma = 0$. Inner and outer circles represent fiber and air, respectively, as indicated in Fig. 2(A). (I) Calculated reflectivity spectra of a 10-µm-thick fibrous film for ${d_0} = \; $ 0.25 µm and $f = $ 31.5% at $\sigma = $ 0 (black solid line) and $\sigma = $ 0.49 (dashed red line), showing disappearance of a purple structural color when $\sigma $ is large. The colors shown for the two spectra are those under CIE standard illuminant D65.
The effect of mean diameter and diameter distribution on the scattering properties at a visible wavelength $\lambda $ = 0.555 µm and f = 31.5% is shown in Figs. 6(D)–6(F) [see the figures by referring to Eq. (2)]. At this wavelength, $\left\langle {\cos \theta } \right\rangle $, ${Q_{sca}}$, and ${d_0}/{L_{zz}}^{\ast \prime}$ over 0.1 µm $\le {d_0} \le $ 0.24 µm are similar to those at $\lambda $ = 0.900 µm over 0.16 µm $\le {d_0} \le $ 0.4 µm, due to scale invariance (${d_0}/\lambda $ = constant) apart from a small refractive index dispersion. Because the wavelength is now shorter, two resonances appear in ${Q_{sca}}$ at ${d_0} \approx $ 0.20 µm and 0.34 µm as $\sigma \to 0$ [Fig. 6(E)]. The locations of the resonances are similar to those in ${d_0}/{L_{zz}}^{\ast \prime}$ [Fig. 6(F)]. These resonances are also manifested as dips in $\left\langle {\cos \theta } \right\rangle $ [the dip at ${d_0}$ ${\approx} $ 0.20 µm is sufficiently broad that its positive curvature is not clearly visible in Fig. 6(D)]. As the resonance locations are primarily determined by ${d_0}$, similar locations would be obtained when filling fractions are not very different from f = 31.5%.
In Fig. 6(E), the peak in ${Q_{sca}}$ as $\sigma \to 0$ at ${d_0} \approx $ 0.34 µm is stronger than that at ${d_0} \approx $ 0.20 µm. This is related to electric energy density distribution in the fibers for light incident in the z-direction shown in Fig. 6(G) and Fig. 6(H) corresponding to $d = $ 0.20 µm and 0.34 µm, respectively. The incident light is polarized with its electric fields parallel to the fiber axis because this polarization dominates the resonances. The resonance at $d = $ 0.34 µm supports stronger local electric field in the fiber than the other. As d increases from 0.20 µm to 0.34 µm, resonant modes exhibit a greater number of nodal planes, and interference between the modes and the incident field induces stronger local electric fields. As $\sigma $ becomes large enough so that $\sigma {d_0}$ becomes comparable to half the distance between the two resonance peaks in Fig. 6(E) and Fig. 6(F), these peaks become sufficiently broadened that they disappear. This broadening is manifested as a color change in the sample. For example, the calculated colors based on reflectivity spectra for a 10-µm-thick film in Fig. 6(I) show that a purple structural color at $\sigma = 0$ tends to disappear when $\sigma $ increases to 0.49 for ${d_0} = $ 0.25 µm and f = 31.5%, which are the parameter values for Cyphochilus scales. Principle of scale invariance, $d/\lambda $ = constant, indicates that the resonant modes at ${d_0} = $ 0.20 µm and 0.34 µm in Fig. 6(E) would appear near $\lambda $ = 0.69 µm and 0.41 µm, respectively, for a uniform diameter of $d = $ 0.25 µm. This effect is seen in Fig. 6(I) as reflectivity peaks at the expected locations when $\sigma = 0$. The stronger peak in Fig. 6(F) results in a higher reflectivity at $\lambda $ = 0.41 µm than at $\lambda $ = 0.69 µm in Fig. 6(I). When $\sigma $ is large, these peaks broaden so that reflectivity decreases monotonically as the wavelength increases. For our electrospun structures, $\sigma $ is 0.32–0.35 and, in this range, spectral features in the visible are almost absent in the ${L_{zz}}^{\ast \prime}$ as seen in Fig. 4(E) and Fig. 4(F). In these spectra, ${L_{zz}}^{\ast \prime}$ increases as the wavelength increases. The reason for this behavior is that, because ${L_{zz}}^{\ast \prime}$ is roughly proportional to transmissivity, the wavelength dependence of ${L_{zz}}^{\ast \prime}$ in the electrospun structures shows an opposite behavior to that of reflectivity shown in Fig. 6(I) at $\sigma = $ 0.49.
4. Conclusion
We have achieved fibrous nanostructures that surpass Cyphochilus scales in light scattering strength by focusing on key structural parameters found in the scales, i.e., mean fiber diameter, diameter distribution, filling fraction, and anisotropy. These nanostructures were fabricated by electrospinning, which is amenable to mass production, and scattering characteristics in these structures were investigated by both experiment and optical modeling. Our modeling revealed that, despite large nanostructural difference between Cyphochilus scales and electrospun films, the mean diameter and filling fraction at the optimum point for electrospun films are similar to those in Cyphochilus scales. With the optimized parameters in electrospun films, their optical scattering is even stronger than that in Cyphochilus scales. However, scattering characteristics are different between the two. Our detailed modeling study showed two resonance peaks in reflectivity in the limit of uniform-diameter fibers where the diameter matches the mean diameter of Cyphochilus scales: a strong and a weak resonance in the blue and red spectral region, respectively. As the fiber diameter becomes more distributed, the resonance peaks broaden and the spectrum becomes relatively flat. Because stronger resonance is located in the blue spectral region, when the peaks broaden, scattering strength becomes stronger as the wavelength decreases. This spectral dependence is also observed in Cyphochilus scales. Our work suggests that well-controlled fibrous nanostructures can be fabricated by conventional manufacturing techniques and optimized for their optical properties by simple optical modeling. To illustrate, control over diameter distribution enables structural color in fibrous media and core-shell fiber structures provide effective control of scattering directions by the Kerker effect [36]. Moreover, fibrous optical films can be flexible with high curvatures for a variety of practical use, unlike other common scattering materials such as daytime radiative cooling fabrics [23], light extractors in optoelectronics [20], wearable sensors [19], whitening agents, optical diffusers, or solid foams.
Funding
National Science Foundation (DMR-1555290, ECCS-1231046).
Acknowledgments
We thank Prof. In Chul Um in Kyungpook National University for providing experimental equipment and technical advice.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
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