1. Introduction

A variety of nanomaterials have been studied including optical / mechanical metamaterials, metasurfaces, photonic crystals, nanocomposites, memory materials, and catalysts. While they enable unprecedented optical, mechanical, and chemical functionalities, inherently large surface-to-volume ratios of these materials often cause durability problems that hinder them from being used in many practical applications. Nevertheless, biological systems have made use of such nanostructured materials for their desired functionality over hundreds of millions of years. Thus, it would be beneficial to study how the nanostructures in biological systems are designed to achieve both desired functionality and durability. One such example is the nanostructure in Cyphochilus white beetle scales, which exhibit extremely strong light scattering while maintaining the structural integrity under various environmental conditions that have existed since the beetles’ habitation on Earth.

Cyphochilus white beetles [Fig. 1(a)] display whiteness with relatively flat scales of only 5–10 µm in thickness [Fig. 1(b)] on their black exocuticle [1,2]. Studies revealed that, among known biological materials, the scales exhibit the strongest light scattering power which is greater than common white papers by an order of magnitude [1]. The light scattering is broadband ranging from visible to near-infrared spectrum [3,4]. Remarkably, the strong broadband scattering is achieved with a structure made of chitin whose refractive index is relatively low at ∼1.56 [5,6]. The key to the intense scattering is in the internal nanostructure of the scales. The nanostructure consists of anisotropic random network of smooth fibrils that are ∼0.23 µm in diameter and ∼1.1 µm in length [4,7], with a fill fraction of f = 0.31–0.32 [Fig. 1(c)] [3,8]. (For detailed discussion on the fill fraction and its consistency with Ref. [4], see Section II in Ref. [8] and Section 1 in Supplement 1.)

 

Fig. 1. (a) A Cyphochilus white beetle, (b) a microscope image of a scale taken from the back of the beetle, and (c) an electron micrograph of a vertical cross section of the scale.

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The intricate internal nanostructure is believed to have originated from spinodal decomposition [3]. This hypothesis is based on the fact that arthropod cuticular nanostructures, which consist of spheres, cylinders, or saddle surfaces, are similar to morphologies resulting from self-organizing block copolymers and surfactants, even though the biological nanostructures are larger than the synthetic systems by an order of magnitude or greater [9,10]. After or while a lipid membrane template is developed, an extracellular polymer, chitin, may be deposited into a phase that is continuous with extracellular space, and subsequent desiccation may result in solid nanostructures [9–11]. In a late stage of spinodal decomposition [12], phase separation can lead to two intertwined continuous domains with their morphologies determined by minimization of the interfacial free energy [13–15]. The internal interface thus obtained in the bicontinuous structure has a constant mean curvature (CMC) [14,15]. Solutions to the Cahn-Hilliard equation, which describe spinodal decomposition [13,16,17], present highly regular random / periodic structures with a CMC interface and a characteristic length [3,13,16].

However, the late-stage [12] solutions to the Cahn-Hilliard equation do not capture some key features of the Cyphochilus beetle scales structure. The beetle scales structure [Fig. 2(a,b)] is anisotropic, consisting of long fibrils with an aspect ratio of ∼5, defined by a non-CMC interface (note the mean curvature difference between fibril struts and fibrils joints), and having a broad Fourier power spectrum [Fig. 2(c)] [1–4,7]. In comparison, the Cahn-Hilliard structures [Fig. 2(d,e)] are isotropic, consisting of relatively short fibrils, defined by a CMC interface, and having a bell-shaped Fourier power spectrum [Fig. 2(f)]. In fact, absence of a hydrodynamic convection term in the Cahn-Hilliard equation may oversimplify the physics of the process employed in the formation of the beetle scales [13,18]. Interactions between proteins and the lipid membrane can also complicate the process [19]. In particular, as manifest from non-CMC surfaces of the Cyphochilus beetle scales, their internal surface area is not minimized as in the late-stage Cahn-Hilliard solutions [14,15]. Pronounced non-CMC surfaces are found in other arthropod cuticles. For example, the nanostructures in Anoplophora zonatrix [9] and Anoplophora graafi [9,20] beetle scales are based on nanospheres of highly rough surfaces, and the network nanostructure in Thyreus pictus cuckoo bee setae [9] has bumpy surfaces (Fig. S2 in Supplement 1).

 

Fig. 2. (a,b) Anisotropic nanostructure in Cyphochilus white beetle scale [4] and (c) its Fourier spectral power density with f (=0.315) and 1 – f assigned to solid and void phases, respectively. (d – f) are for isotropic nanostructure in late-stage spinodal decomposition and correspond to (a – c). Side lengths of the cubes in (a,d) and (b,e) are 5 µm and 1.2 µm, respectively. For (c), azimuthal average was taken for each polar angle θ. Both (a,b) and (d,e) have the same internal surface area per unit volume (see Fig. S3 for their S₂ functions).

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While the in vivo process in Cyphochilus beetle scales is largely unknown [9,10], it has long been believed that the nanostructure has been optimized through adaptive evolution to maximize light scattering power [1,2,4]. However, recent computational studies indicate that anisotropy, fill fraction, and fibril size could be further optimized for the regular random structure [21,22]. These studies have arrived at the same conclusion that light scattering power would be enhanced by increasing anisotropy such that the fibrils directions are closer to the scale plane directions. This finding agrees with our analysis of anisotropic optical diffusion in Cyphochilus beetle scales [8]. Specifically, the thickness direction component of anisotropy tensor, K_zz, which quantifies the degree of anisotropy in optical diffusion and is unity for isotropic diffusion [23], is appreciably less than unity in the scales to reflect incident light more strongly. An opposite argument has been made based on an observation that a Cahn-Hilliard structure, which is isotropic, can exhibit stronger light scattering power than the beetle scales [3]. (It can be noted that periodic Cahn-Hilliard structures can have a large photonic band gap [24].) The theoretical predictions in these studies [3,21,22], which show possibilities of exceeding light scattering power in Cyphochilus beetle scales, are all based on regular random structures with highly uniform fibril dimensions or a CMC. On the contrary, in experiments [3,25,26], strongly irregular random structures with multiple length scales have been shown to surpass the beetle scales’ optical performance. Enhancement of light scattering power by multiple length scales has been shown in particle-based systems [27,28], but not in anisotropic fibrillar network structures. Thus, it has not yet been elucidated whether irregularity would be desirable to enhance light scattering power in the Cyphochilus beetle scales structure.

In principle, irregularity can be computationally introduced into the structures by randomly distributing a solid phase over the space such that bicontinuity is preserved. However, in this case, an extremely large parameter space would be involved. Thus, in this work, we introduce random noise gradually into the Cyphochilus beetle scales structure by keeping constant two descriptor functions – two-point probability function (S₂) and lineal-path function (L_m). S₂ and L_m describe spatial correlation between two points and connectedness in the material phase, respectively, as will be defined in Section 2. These two descriptors have been conventionally used to characterize porous media [29–31]. The constraint of the fixed descriptors implies that the total internal surface area and structural anisotropy are preserved upon introducing irregularity [32–34]. Irregularity of the structures caused by the random noise is manifest by multiple length scales that are smaller than a typical fibril diameter (∼0.23 µm). This small-scale irregularity is similar to that of the nanostructures in Anoplophora zonatrix [9], Anoplophora graafi [9,20], and Thyreus pictus [9] (Fig. S2 in Supplement 1). We investigate how the irregularity at the small length scales affects light scattering properties of the structures. Investigation of this effect is completely absent in past studies on arthropod cuticular nanostructures, presumably due to an expectation that such small-scale irregularity compared to optical wavelengths would not significantly affect optical scattering. However, we show that optical scattering power of Cyphochilus beetle scales structure is highly sensitive to such irregularity and even decreases as multiple length scales are introduced. The reduction in optical scattering power by irregularity is primarily due to transition from anisotropic to isotropic diffusion, as quantitatively manifested by an increase in K_zz toward unity. When the noise level is not very high, the noise effect on the structure is to increase roughness of the internal surfaces under a constraint of a constant surface area. Thus, our results demonstrate that the in vivo process in Cyphochilus beetle scales makes use of smooth non-CMC surfaces and anisotropy at the same surface free energy cost to achieve their strong light scattering power. Moreover, our results may have implications for enhancing durability in synthetic scattering structures by using anisotropic network of smooth fibrils, because they can have moderate surface free energy without excessively thin morphologies compared to the structures of multiple length scales in the recent experiments [3,25,26].

2. Results and discussion

2.1. Stochastic reconstruction of random media

Random structures can be generated computationally according to reference descriptor functions [34]. For statistically homogeneous random media, the two-point probability function is defined as [30,31]

 

Fig. 3. Stochastic reconstruction of a random structure according to (a) S₂ and (b) L_m of a Cyphochilus white beetle scale beginning from a random configuration (c), through intermediate structures (d) and (e), to the final structure (f). Only cross sections of the structures in a lateral plane are shown. The dimensions of (c – f) are 2.4 µm × 2.4 µm.

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We computer-generated random structures according to the two descriptor functions of the Cyphochilus beetle scale structure. Figures 3(c)–3(f) show a cross section of a structure developing from an initial to a desired configuration. For the structure generation, we divided a simulation space of dimensions 2.4 µm × 2.4 µm × 4.0 µm into 48 × 48 × 80 voxels. In this example, initially, black (I = 0, void) and white (I = 1, material) were randomly assigned to the voxels with a fraction of white voxels equal to the material fill fraction of f = 0.315 for the beetle scales [Fig. 3(c)] [8]. The structure was made to develop by selecting a black and a white voxel randomly and interchanging them to decrease an “energy” of the configuration. The “energy” was defined by

We modified the initial configuration of this example to generate random structures that have varying degrees of irregularity while sharing the same descriptor functions as the Cyphochilus beetle scale structure. To modify the initial configuration, an equal number of black and white voxels were randomly selected from the beetle scale structure. The number of the selected voxels were a fraction of the total number of voxels. Random pairs of black and white voxels were made from the selected voxels and the voxels of each pair were interchanged resulting in an initial configuration. From the initial configuration, the structure developed by minimizing E. Depending on the fraction of the randomly selected voxels in preparing the initial configuration, the degree of irregularity in the final structure varied. Because quantifying the irregularity in such a complicated network is not trivial, we represent it by

Figure 4(a) and Fig. 4(b) show 3D and xy-plane cross-section images, respectively, of random structures with varying F obtained by our method. Only cubic parts of the simulated structures are shown in Fig. 4(a). These structures share the same S_2,xy, S_2,z, L_m,xy, and L_m,z shown in Fig. 3(a) and Fig. 3(b), which are extracted from a Cyphochilus beetle scale structure. The cross-section image for F = 1 in Fig. 4(b) is the same as Fig. 3(f). The images in Fig. 4(a) and Fig. 4(b) show that irregularity increases as F increases. As irregularity is introduced, small material islands appear in the structures. However, the volume fraction of these islands is only 0.2% at maximum and the structure is continuous otherwise (it turns out that S_2,xy, S_2,z, L_m,xy, and L_m,z in the Cyphochilus beetle scale structure constrains the stochastically reconstructed structures to be a single continuous body except the negligible islands). We have confirmed by numerical calculations that, when these islands are removed, change in the optical properties of the structures is negligible.

 

Fig. 4. (a) 3D images of a cube side 2.4 µm and (b) their lateral cross sections of random structures that exhibit varying degrees of irregularity represented by F and share the same descriptor functions given in Fig. 3(a) and Fig. 3(b).

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Note that, in Fig. 4, irregularity is introduced such that the total internal surface area, hence the surface free energy, of the structure is kept almost constant. This can be seen from the fact that the surface area is determined by an angle average of dS₂/dr at r = 0 [32] and all of the structures in Fig. 4(a) share the same S_2,xy and the same S_2,z. In this case, S₂’s of these structures are not very different from each other in directions other than x, y, and z. When F is small (less than ∼0.3), a major structural change as F increases is that the fibril roughness increases and the fibrils partially merge with each other. Both processes of the roughness increase and the partial merging happen in such a way that the total internal surface area remains almost unchanged. When F is sufficiently large, the correlation between two computer-generated structures at the same F is weak, so that it is difficult to describe how the structure evolves as F increases.

In addition to the nearly constant internal surface area, the fibril surface area projected onto a plane, whose normal is in a lateral or vertical direction, remains constant. These surface area properties are deduced from the fact that |dS₂/dr|_r=0 is equal to the projected area of internal surface per unit volume onto a plane whose normal is in the r direction. A proof of this theorem is given in Appendix. When we model the fibrils as cylinders randomly oriented in lateral directions, the ratio of |dS₂/dr|_r=0 between vertical and lateral directions is obtained as π / 2 (see Appendix). Figure 3(a) gives the ratio as ∼1.7, which is comparable to π / 2.

2.2. Optical transport properties

Optical transport in the random structures can be approximately described by the diffusion theory. Within this theory, optical transport in lossless isotropic random media is defined by two length scales: the scattering mean free path (l) and the transport mean free path (l^*). The scattering mean free path is the average distance between scattering events, and the transport mean free path is the length that light travels before its propagation direction is randomized. For anisotropic media, these two lengths take tensorial forms $\mathcal{L}$ and $\mathcal{L}^*$, respectively. However, $\mathcal{L}^*$ cannot be interpreted as the length for randomization of light propagation direction. Rather, the effective transport mean free path tensor defined by $\mathcal{L}^{* \prime}=\mathcal{K} \cdot \mathcal{L}^*$ has this physical meaning, where $\mathcal{K}$ is the anisotropy tensor introduced in our previous work in Ref. [23]. For isotropic media, $\mathcal{K}$ becomes an identity tensor.

We extracted the zz component of these optical transport tensors from the simulated random structures using the method that we reported in Ref. [8]. To solve Maxwell’s equations, the finite element method was used. For each F, we computer-generated structures with lateral dimensions of 2.4 µm × 2.4 µm and eight different thicknesses of 0.5 µm, 1.0 µm, 1.5 µm, $\cdots$, 4.0 µm. For all the structures, the voxels were a cube of a side length 0.05 µm (see Section 4 in Supplement 1 on structure smoothing for finite element calculations). These structures were embedded in an effective medium determined by the Maxwell-Garnett mixing rule. For photonic glasses, a sophisticated calculation by the mean field theory resulted in an effective medium index that differs from the Maxwell-Garnett index by only 1.7% at a fill fraction of f = 0.315 [36]. At this low fill fraction, the effective index difference between the Bruggeman theory, which may be more appropriate for fibrillar network, and the Maxwell-Garnett theory is negligible (0.6%). For the eight different structure thicknesses, ballistic and diffuse transmittance values were calculated at a light wavelength of 0.9 µm. In previous studies [1,2], the reflectance spectrum of a Cyphochilus white beetle scale was almost constant in a wavelength range from 0.45 µm to 0.85 µm. Because calculation accuracy decreases as the wavelength decreases, we chose the wavelength of 0.9 µm which is close to the upper end (0.85 µm) and ensures a high accuracy in the calculations. Refractive index of chitin was set to 1.56 [5,6]. Optical interaction at a boundary of random media is commonly modeled by Fresnel’s law [37]. In our case, to increase accuracy, Fresnel transmission/reflection at a boundary of the structures was replaced by scattering within a finite region near the boundary defined as the optical boundary layer [38]. From the transmittances at eight different thicknesses, we extracted the optical transport parameters using linear regression [8] (see Section 5 in Supplement 1). The coefficient of determination, which measures goodness of fit, for the regression in obtaining $\mathcal{L}$ and the other parameters was 0.9840 and 0.9942, respectively, on the average over the F values (Fig. S5 in Supplement 1). To ensure high accuracy, we took an average transmittance of more than three computer-generated structures at each thickness and F.

Our method of calculating optical transport parameters is valid when the structure thickness is even smaller than the transport mean free path, because the structure is embedded in an effective medium [8]. Note that our analysis does not assume a commonly-used simplified phase function (e.g., the Henyey-Greenstein phase function [39]), replaces the popular assumption [37] of Fresnel reflection by a rigorous boundary condition [38], and uses the exact solution [23] to the anisotropic diffusion equation. The calculation of zz components of optical transport tensors, rather than simply transmittance or reflectance, enabled a detailed analysis of bulk optical properties.

 

Fig. 5. The zz component of (a) scattering mean free path tensor, (b) anisotropy tensor, (c) transport mean free path tensor, and (d) effective transport mean free path tensor of the random structures shown in Fig. 4(a) as a funtion of F which represents a degree of irregularity. (e) Reflectance of structures in air at a thickness of 7 µm and (f) scattering unit thickness as a function of F.

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Figures 5(a)–5(d) show the zz component of $\mathcal{L}$ , $\mathcal{K}$ $\mathcal{L}^*$, and $\mathcal{L}^{* \prime}$, i.e., $L_{zz}$, $K_{zz}$, $L_{z z}^{*}$, and $L_{z z}^{*}{ }^{\prime}$, respectively, as a function of F. As irregularity increases from F = 0 to 1, $L_{zz}$ decreases from 1.7 µm to 1.4 µm [Fig. 5(a)]. This indicates that irregularity increases the number of scattering events over a light path length. However, oppositely, $L_{z z}^{*}{ }^{\prime}$ increases in its overall trend from 1.8 µm to 2.5 µm [Fig. 5(d)]. In comparison, the variation in $L_{z z}^{*}$ (2.4 µm – 2.6 µm) is relatively less significant [Fig. 5(c)]. The overall increasing trend in $L_{z z}^{*}{ }^{\prime}$ is mainly related to that in $K_{zz}$ which varies from 0.83 to 1.03 as F increases from 0 to 1 [Fig. 5(b)]. $K_{zz}$ is appreciably smaller than 1 for the original Cyphochilus beetle scale structure and approaches that of isotropic diffusion ($K_{zz} = 1$) as the irregularity increases. Thus, optical diffusion in the beetle scale structure changes gradually from anisotropic to isotropic character as irregularity increases. The sensitivity of $L_{z z}^{*}{ }^{\prime}$ to irregularity is remarkable when we consider that, from F = 0 to F = 0.12, the structure change appears to be only an increase in fibril roughness at a fixed surface area [Fig. 4(a,b)] and $L_{z z}^{*}{ }^{\prime}$ increases by as much as 21%. The roughness length scale, which ranges 50–100 nm as obtained from Fig. 4(b) or Fig. S4(d) in Supplement 1, is smaller than the light wavelength of 0.9 µm by an order of magnitude. The critical role of structural anisotropy in achieving high scattering strength was reported in past studies [8,21,22]. Our calculations show that, even with desired structural anisotropy, irregularity involving a small-scale roughness of 50–100 nm in the fibrils can significantly reduce scattering strength $1/L_{z z}^{*}{ }^{\prime}$. From the optical transport parameters, reflectance of the structures in air at a thickness of $\mathsf{L}$ = 7 µm, which is the average thickness of Cyphochilus beetle scales [2], was calculated from Eq. (S2) in Supplement 1. Figure 5(e) shows that reflectance decreases from 0.64 to 0.58 as irregularity increases from F = 0 to 1. The scattering unit thickness $\mathsf{L}_{su}$ can be defined as the thickness at which the average cosine of scattering angles is the same as $K_{z z}-L_{z z} / L_{z z}^{*}$ [8,23] (Section 6 in Supplement 1). Interestingly, Fig. 5(f) shows that the scattering unit thickness decreases from 3.6 µm to 1.7 µm as irregularity increases from F = 0 to 1.

Because the Cyphochilus beetle scale structure is defined by a non-CMC surface, its surface free energy is not minimized. However, the high sensitivity of $K_{zz}$ and $L_{z z}^{*}{ }^{\prime}$ to irregularity in the structure suggests that its irregularity has been minimized, at a fixed surface free energy cost and anisotropy, to maximize optical scattering strength. To date, fabrication techniques used to mimic the beetle structure include polymer phase separation [3,25,26,40,41], electrospinning [42–44], supercritical carbon dioxide foaming [45], and particle assembly [46]. The structures generated by these techniques were strong in scattering particularly when they were irregular with large surface areas involving a mix of thin and thick morphologies [3,25,26]. However, for long-term stability of a structure, a large surface free energy and thin morphologies are in general undesirable. In comparison, regular random nanostructures of smooth surfaces without excessively thin morphologies would be more durable due to a smaller surface area and uniformity in structure dimensions. Unlike our study where the surface area was fixed as irregularity was introduced, in many experimental fabrications, surface area typically increases as structure irregularity increases [40,47]. Thus, in general, regular random nanostructures can be more advantageous than irregular ones for durability. These regular structures can be strong in light scattering when their anisotropy is exploited properly as in the Cyphochilus beetle scale structure [3,21,22]. For example, electrospinning can easily generate highly anisotropic regular random structures with smooth surfaces, while achieving strong scattering [42–44]. It has not yet been demonstrated whether these structures, when optimized, can surpass the Cyphochilus beetle scale structure in light scattering power [44]. Theoretical studies [21,22] indicate that this is possible when the fibrils of right dimensions are oriented mostly in plane directions, as in electrospun structures.

3. Conclusion

In conclusion, we have investigated the effects of irregularity on optical scattering in the anisotropic random fibrillar network in Cyphochilus white beetle scales. This study built on previous studies which revealed the importance of structural anisotropy, in addition to fibril diameter and fill fraction, in achieving strong scattering power in the beetle scales [8,21,22]. Irregularity was introduced into the beetle scale structure in varying degrees at fixed structural descriptors such that structure anisotropy was preserved and surface area was kept constant. Optical calculations on these structures showed that optical scattering strength decreases sensitively as irregularity is introduced into them. The reduction in scattering strength was due to a decrease in optical diffusion anisotropy, as manifested in the thickness component of the anisotropy tensor. This result shows that the in vivo process in Cyphochilus white beetle scales makes use of regularity and anisotropy to maximize light scattering strength. While the nanostructure in the scales does not minimize surface free energy as in late-stage spinodal decomposition, the regularity and smooth surface are utilized for strong optical scattering. Unlike our study where the surface area was kept constant as irregularity was introduced, irregularity typically increases surface area in many experimental fabrications. So far, only irregular nanostructures with multiple length scales have experimentally demonstrated optical scattering superior to Cyphochilus white beetle scales [3,25,26]. In this case, the large surface free energy and the existence of thin morphologies would make the structure vulnerable to external disturbances such as temperature cycles, chemicals, stresses, and high energy irradiations. Our work points to an opposite direction where anisotropic regular random structures may be used to achieve such strong scattering. In this case, surface free energy cost can be lowered and thin morphologies can be eliminated in comparison to irregular structures, potentially increasing long-term stability. While fabrication of regular random structures by late-stage spinodal decomposition is challenging because it commonly produces structures of a characteristic length of only tens of nanometers, other techniques such as electrospinning, melt blowing, and nanowires assembly can generate such structures, desirably even with strong anisotropy.

Appendix

Debye et al. proved that, for statistically isotropic random media, the slope of S₂ at r = 0, which is independent of direction, is determined by the specific surface (internal surface area per unit volume) [33]. Later, Berryman proved that, for anisotropic random media, the specific surface determines the angle average slope of S₂ at r = 0 [32]. Here we prove that, for anisotropic random media, the slope dS₂/dr at r = 0 in a direction is determined by the projected area of internal surface onto a plane whose normal is in the same direction. Consider an infinitesimal area element dS_n at a surface as depicted in Fig. 6. $\hat{\boldsymbol{n}}$ is the unit outward normal vector of the surface element and r makes an angle α with $\hat{\boldsymbol{n}}$ . Letting $\hat{\boldsymbol{r}}$ be a unit vector in the r direction, we obtain from Eq. (1)

(4)$${\left. {\frac{{d{S_2}}}{{dr}}} \right|_{r = 0}} = \mathop {\lim }\limits_{r \to {0^ + }} \frac{1}{V}\int_{{V_m}} {\hat{{\boldsymbol r}} \cdot \nabla I({\boldsymbol r}^{\prime} + {\boldsymbol r})d{\boldsymbol r}^{\prime}}, $$

where r → 0⁺ means that r is approaching 0 from the material side and V_m is the volume occupied by the material. The integral in Eq. (4) can be performed, following Berryman’s approach [32], as

(5)$$\mathop {\lim }\limits_{r \to {0^ + }} \frac{1}{V}\int_{{V_m}} {\hat{{\boldsymbol r}} \cdot \nabla I({\boldsymbol r}^{\prime} + {\boldsymbol r})d{\boldsymbol r}^{\prime}} ={-} \frac{1}{V}\int_{0 \le \alpha \le \pi /2} {\cos \alpha d{S_n}}. $$

 

Fig. 6. Schematic of a region near internal surface of a random structure defining the surface normal unit vector $\hat{\boldsymbol{n}}$ and the angle α between $\hat{\boldsymbol{n}}$ and r.

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Because dS_n cosα is the projected area of a differential element dS_n onto a plane whose normal is in the r direction, it follows from Eq. (4) and Eq. (5) that –V dS₂/dr |_r=0 is equal to the projected area of internal surface onto a plane whose normal is in the r direction.

As an example, we consider cylinders randomly oriented in lateral direction. Let the unit vectors in x, y, and z directions be $\hat{\boldsymbol{x}}$, $\hat{\boldsymbol{y}}$, and $\hat{\boldsymbol{z}}$, respectively. For a cylinder whose unit axis vector is given by $\hat{\boldsymbol{u}}$_f = cosφ_f $\hat{\boldsymbol{x}}$ + sinφ_f ŷ, the unit surface normal vector is

(6)$${\hat{\boldsymbol n}} = - \sin \alpha \sin {\varphi _f}{\hat{\boldsymbol x}} + \sin \alpha \cos {\varphi _f}{\hat{\boldsymbol y}} + \cos \alpha {\hat{\boldsymbol z}}$$

For the x direction, we get

(7)$${\left. {\frac{{d{S_2}}}{{dx}}} \right|_{x = 0}} = {\left\langle { - \frac{1}{V}\int {|{\hat{{\boldsymbol n}} \cdot \hat{{\boldsymbol x}}} |d{S_n}} } \right\rangle _{{\varphi _f}}} ={-} \frac{1}{{\pi V}}\int_0^\pi {\int_0^{\pi /2} {\sin \alpha \sin {\varphi _f}} \frac{{Sd\alpha }}{{\pi /2}}d{\varphi _f}} ={-} \frac{{4S}}{{{\pi ^2}V}}, $$

where S is the total cylinder surface area and ${\left\langle \cdots \right\rangle _{{\varphi _f}}}$ is a statistical average over all cylinder orientations. The same result is obtained for the y direction. For the z direction, we get

(8)$${\left. {\frac{{d{S_2}}}{{dz}}} \right|_{z = 0}} = {\left\langle { - \frac{1}{V}\int {|{\hat{{\boldsymbol n}} \cdot \hat{{\boldsymbol z}}} |d{S_n}} } \right\rangle _{{\varphi _f}}} ={-} \frac{1}{{\pi V}}\int_0^\pi {\int_0^{\pi /2} {\cos \alpha } \frac{{Sd\alpha }}{{\pi /2}}d{\varphi _f}} ={-} \frac{{2S}}{{\pi V}}. $$

Thus, the ratio of Eq. (8) to Eq. (7) is π / 2.

Funding

National Science Foundation (DMR-1555290, ECCS-1231046).

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.

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