1. Introduction

Optical properties of micro/nano-structured materials have strong dependence on the arrangement of their structural building blocks. This dependence is observed in a large class of materials, ranging from photonic crystals to photonic glasses, that display varying degrees of order. In photonic crystals where the periodicity is on the order of optical length scale, the Bloch modes developed in the crystals result in unique optical effects, such as photonic bandgap [1], directional light emission [2], and negative refraction [3]. In the periodic structures, these optical effects are highly sensitive to structural defects [4], therefore, efficient quantitative characterization and reduction of defects are essential to induce the aforementioned effects. The importance of structural characterization is also evident in photonic glasses, where monodisperse microspheres are randomly packed. A short range order in the random media can significantly alter their scattering properties [5,6]. The scattering efficiencies and photon modes have shown their importance in many applications, including radiative cooling and random lasing [5,7,8].

Between periodic and random limits, recent discoveries illustrate that the intermediate order observed in quasi-random media can yield unexpected optical effects [9,10]. For example, the optical scattering strength can be enhanced significantly in quasi-random structures compared to perfect random media [9]. In another example, the light trapping efficiency in solar cells can be improved by manipulating the spatial correlation in the surface corrugation features [10]. The interplay of order and disorder is even more vividly displayed by numerous photonic structures found in nature. Many animals and plants combine periodic and random micro-arrangements for optimal optical performance, and the role of order/disorder arrangements in these structures is yet to be fully understood [11−16]. For a full spectrum of materials ranging from photonic crystals to photonic glasses, it is critical to develop quantitative methods to accurately characterize the level of order or disorder in their microstructures.

In search of an efficient method to extract structural information on photonic media, various characterization techniques have been developed including x-ray tomography [17], confocal microscopy [18−20], and optical coherence tomography [21,22]. These techniques are highly advantageous due to their non-destructive nature. However, the image resolution in these techniques is typically limited to 200 nm – 1 µm, and their application to micro/nano-structures is rather challenging as highly sophisticated versions of these methods are required [23]. Moreover, these techniques involve a three-dimensional (3D) reconstruction from many images taken at different sample depths, which is a complicated and time-consuming procedure. In some cases, the imaging techniques require chemical functionalization, including fluorescence labeling, to attain adequate image contrast. These difficulties limit the subject materials that can be analyzed and add complexity to the characterization.

In photonic glasses, the randomness is typically quantified by radial distribution function that represents spatial correlation between microsphere positions. The radial distribution function in turn is related to structure factor that governs optical scattering properties [6]. However, the difficulty of obtaining the radial distribution function from conventional characterization methods is that the position of each microsphere has to be registered in the 3D image reconstructed from many 2D images taken at fine depth intervals [18,19].

To overcome the complexity of conventional techniques, in this work, we develop a method to quantify the degree of randomness in photonic glasses from 2D images taken at a single cross-sectional plane. Because of the randomness and isotropy in photonic glasses, their key 3D structural information, such as fill fraction, specific surface, and autocorrelation function, is contained in a single 2D cross-sectional image as long as the image is much larger than the microsphere size [24]. Recognizing this connection, we derive a mathematical relation between the autocorrelation function of 2D images and the radial distribution function of 3D structures. Based on the derived relation, the autocorrelation function in the ideal random limit is calculated and compared to that of experimental 2D images. The 2D images are obtained by scanning electron microscopy (SEM), which provides a much higher resolution than the conventional non-destructive techniques.

2. Theory

In this section, we outline our theory and confirm its validity by computer simulations. Consider that N solid monodisperse spheres of radius R are randomly positioned in a space of volume V without any overlap with each other at a fill fraction of f [ Fig. 1(a)]. We assume that N >> 1, and the system is isotropic. Two intensities I = 0 [black in Fig. 1(a)] and 1 [white in Fig. 1(a)] are assigned to regions occupied by voids and spheres, respectively. An autocorrelation function C_I(r) for the 0–1 intensity field is, by definition,

 

Fig. 1. Illustration of an (a) original system and (b) translated system of randomly distributed spheres, with an intensity assignment of I = 0 and 1 for voids and spheres, respectively. (b) is translated from (a) by r. Definitions of vectors (r and $\rho $) and an angle (α) for (c) $r \ge 2R$ and (d) $r < 2R$. Gray areas in (c) and (d) indicate an overlap volume V_l(ρ) between two different spheres in the original and the translated system. A purple area in (d) is an overlap volume V_l(r) between two identical spheres in the original and the translated system.

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The radial distribution function g(r) is defined such that the local number density of spheres at r from the center of a sphere is $fg({\boldsymbol r})/{V_s}$. Suppose that the origin of the original sphere system is chosen at the center of a sphere. When $r \ge 2R$ [Fig. 1(c)], the average overlap volume per sphere is

For Eq. (6) to be useful for experimental characterization, it still remains to be shown that C_I(r) obtained from a 3D structure is the same as that from a 2D cross-sectional plane. Consider that the system is a cube and the integrals in Eq. (1) are taken over an xy-plane. Any plane parallel to the xy-plane of the system contains many cross-sections of the spheres. Because of the large size and isotropy of the system, the 2D integrals are independent of the z-location of the plane. This means that a 3D integral in Eq. (1) is the same as a 2D integral times the z-direction thickness of the system. Thus, the C_I(r) found from a 3D distribution of intensity I(r) is the same as the C_I(r) obtained from any plane that intersects the 3D system. This fact allows us to find C_I(r) from a single cross-sectional plane of photonic glasses.

The g(r) can be obtained by solving Eq. (6). The double integral equation can be written in a dimensionless form as

To confirm the validity of Eq. (6) and its inverse problem in Eq. (12), we performed Monte Carlo (MC) simulations. Using the simulations, we rendered realizations of ideal random arrangements of spheres for f = 0.45 [26]. From the computer-generated structures, we obtained C_I(r) using Eq. (1) with the integrals taken on a 2D plane and g(r) using the location of each sphere. For an ideal random arrangement of spheres, g(r) is given by the Percus-Yevick (PY) approximation. An analytical solution of the approximation exists [27]. Using the PY solution for g(r), we obtained the autocorrelation function of an ideal random structure from Eq. (6). Figure 2(a) shows that C_I(r)’s from PY solution and our MC simulation agree with each other indistinguishably. We also found g(r) from the inverse transform of Eq. (12) using the C_I(r) of our MC simulations. This g(r) is identical to the PY solution as shown in Fig. 2(b). Thus, Fig. 2 confirms the validity of our theoretical results in Eqs. (6) and (12).

 

Fig. 2. Comparison of (a) the autocorrelation function C_I(r) and (b) the radial distribution function g(r) between the Percus-Yevick approximation and the results from Eq. (6) based on MC simulated structures for f = 0.45.

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We see in Fig. 2(a) that the correlation in the microsphere position decays fast over a distance of ∼2R. As the distance increases further, the autocorrelation shows an oscillating feature that slowly decays to a plateau after approximately 8R. The plateau reaches f = 0.45 as predicted in the limiting case of Eq. (6). The peaks of the oscillation are due to the short-range correlations induced by the non-zero microsphere size. These peaks are located when r/(2R) is roughly comparable to positive integers.

To estimate the deviation of randomness in the experimental structures from the ideal limit, the C_I(r) from the PY solution can be compared to that obtained from 2D images of the experimental structures by Eq. (1). Conversely, the g(r) of the PY approximation could be compared to that of experimental structures, similarly to Fig. 2(b). However, in practice, we find that the solution to Eq. (6) is highly sensitive to small experimental errors in C_I(r) due to the double integral in Eq. (6) (Another method of calculating g(r) from 2D images is given in Appendix A). Thus, for experimental structures, we make comparisons with the ideal random limit on C_I(r) rather than on g(r). The degree of randomness in an experimental structure can then be characterized by how close the C_I(r) in the structure is to that of the PY solution.

3. Experiment

Figure 3 illustrates the experimental procedure for obtaining a cross-sectional 2D image of photonic glasses. We prepared photonic glass films by inducing instability in a colloidal microsphere solution [Fig. 3(a)] [5,28]. Specifically, we added CaCl₂ to a 2 vol % aqueous solution of ∼0.87-µm-diameter silica microspheres (Fiber Optic Center Inc.) until the resulting concentration of CaCl₂ becomes 0.01 M. The electrolyte addition induces colloidal instability and flocculation of microspheres that precipitate out onto a substrate in a random fashion. Sample thickness varied from 30 to 60 µm in our experiment. We infiltrated the photonic glass with a polymer (Spurr’s resin) that is stable under electron beam irradiation [Fig. 3(b)] [29]. A cross-section of the film was exposed and polished by a focused ion beam (FIB, FEI Quanta 3D) [Fig. 3(c)]. From our SEM images of the cross-section [Fig. 3(d)], we confirmed that the polished surface was flat without dimples, cracks, or gaps at silica-polymer interfaces. To avoid surface charging during SEM imaging (30 keV beam energy, we coated a 20-nm-thick gold layer on the cross-section and used Spurr’s resin.

 

Fig. 3. Illustration of sample preparation process for acquiring cross-sectional images of photonic glasses. The structure (a) is infiltrated by a polymer (b) and cross-sectioned by focused ion beam (c). In the SEM image (d), contrast is adjusted to give a black-and-white image (e).

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We performed the Hough transform [30] using MATLAB to detect all the circles in the image. Using the detected circles, we adjusted the image brightness, such that I = 0 and 1 for the polymer and the silica spheres, respectively [Fig. 3(e)]. C_I(r) was obtained from the binarized images using Eq. (1). In the Hough transform, the boundary of each circle was accurately determined using Otsu’s method [31]. By careful comparison between original and binarized images, we confirmed that the circle detection method worked well in all our SEM images. Image resolution affects the accuracy in circle size determination which, in turn, dictates the accuracy in f. We set the image size to be greater than 2000 µm², which included ∼3000 microspheres, to obtain C_I(r). This image size included cross sectional planes in multiple photonic glass films to increase the accuracy in C_I(r). With an image resolution of 25 pixels/µm, the Hough transform renders the absolute error in f much less than ∼0.06.

4. Results and discussion

The C_I(r) obtained from an experimental 2D image possesses circular symmetry because of the isotropy in microsphere arrangement. The strong peak at the center and absence of bright spots surrounding the peak in Fig. 4(a) right inset are a clear signature of randomly arranged microspheres. We take a circular average of the 2D autocorrelation function as a function of r/(2R) and display it in Fig. 4(a) in comparison to the PY solution for f = 0.55 and 0.6. Note that the C_I(r) was obtained from a much larger area than the image shown in Fig. 4(a) left inset. From the C_I value when r is appreciably large, we determine that the fill fraction of our structure is f ∼ 0.6 while a previous study [32] estimated it as ∼0.55 for photonic glasses prepared by a method similar to ours.

 

Fig. 4. (a-b) Comparison of autocorrelation function C_I(r) of (a) an experimental photonic glass with the PY approximation for f = 0.55 and 0.6; (b) a crystalline domain with a (111) plane of f.c.c. structure. The insets in Figs. 4(a) and 4(b) show small representative images and their 2D autocorrelation functions from experimental structures. (c) Effect of microsphere polydispersity on the C_I(r) of photonic glasses. The C_I(r)’s are calculated from MC simulated structures. The polydispersity is represented by the standard deviation σ in the microsphere size normalized by the average size.

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The PY approximation deviates from the true solution at high f [33]. However, we have confirmed that the C_I(r) of the PY solution is only slightly different from the true solution even at a higher fill fraction of f = 0.62 than the ours (f = 0.55–0.6) [34]. Specifically, at f = 0.62, the PY solution of C_I(r) over 0.45 ≤ r/(2R) ≤ 1 is slightly lower than the true solution only by ∼0.01 and the two solutions are very close to each other in other regions. Thus, for our fill fraction, the PY solution can be regarded as an ideal random limit with negligible errors.

In Fig. 4(a), the experimental C_I(r) deviates slightly from the ideal random limit for f = 0.6. As r increases from zero, the experimental C_I decays more slowly than the PY approximation. From this observation, one might infer that the slow decay stems from greater short-range correlations in the experimental structure than in the ideal limit. To test this inference, we prepared a sample that included an appreciable fraction of crystalline domains. To make such domains to form, we added KCl instead of CaCl₂ to the colloidal solution. The screening of surface charge on microspheres is less pronounced in a monovalent KCl solution than in a divalent CaCl₂ solution, leading to particle agglomeration containing ordered structures.

The SEM images of the sample prepared using KCl show many crystalline regions with different orientations. In Fig. 4(b) left inset, we show, as an example, cross-sectional image of the crystalline domain and its autocorrelation function. This cross-section is similar to a (111) plane in the f.c.c. structure. Accordingly, the 2D autocorrelation function in Fig. 4(b) right inset exhibits a periodic pattern, but its intensity is not uniform over a triangular lattice as it would happen for a (111) plane.

While rotational symmetry is lacking, we take a circular average of the autocorrelation function with the spike at the center and obtain C_I(r). As a reference for comparison, we calculate the C_I(r) of a (111) plane that corresponds to a 2D fill fraction of 0.55 and 0.6 for a perfect crystal. In the reference C_I(r), the peaks appear near the intersphere distances at r/(2R) = 1, $\sqrt 3 $, 2, $\sqrt 7 $, 3, $\sqrt {13},\, \cdots$. Figure 4(b) shows the C_I(r) for the crystalline domain cross-section and the reference C_I(r)’s. Near r = 0, the experimental crystal C_I decays slower than the C_I of the perfect crystal as r increases. This observation makes clear that the initial decay of C_I does not always become slower as the short range correlations increase. This fact is confirmed by comparing the two experimental C_I(r)’s in Figs. 4(a) and 4(b). Namely, the C_I(r) of experimental random structure (red circles) in Fig. 4(a) shows slower initial decay than the C_I(r) of experimental crystalline structure (blue circles) in Fig. 4(b). Thus, the slower initial decay in the experimental C_I(r) than the ideal random limits in Fig. 4(a) is not necessarily due to the possible increase in short range correlations.

To explain the difference in C_I(r) between experiment and ideal limits in Fig. 4(a), we rely on the observation that the experimental C_I(r) exhibits a smaller oscillation amplitude than the PY solution in both random and crystalline cases. This suggests that the inevitable polydispersity in the size of experimentally used microspheres may decrease the oscillation amplitude. The initial decay rate is also affected by the polydispersity as we show in the discussion below.

To verify the above speculation, we simulate an ideal random structure of non-zero polydispersity by the MC method and investigate how the polydispersity affects C_I(r). The microsphere radius is assumed to have a normal distribution where the average diameter is R, and the relative standard deviation in R is σ. For non-overlapping polydisperse spheres, the fill fraction becomes f = f₀(1 + 3σ²) where f₀ is the fill fraction when the spheres become monodisperse without overlap with their positions fixed (Appendix B). In our simulation, f₀ = 0.45 so that f is slightly smaller than that in our experiment. When f is as high as 0.6, it took an extremely long time to simulate a random structure. The slope of C_I(r) at r = 0 for isotropic media is determined by the specific surface, which is the surface area per unit volume [35]. For σ = 0, the slope is calculated to be –3/2, which also results from Eq. (9) using g(x) = 0 for x < 1 (Appendix C). For polydisperse microspheres, dC_I / d(r/(2R)) → –3(σ²+1) / (6σ²+2) ≈ –3/2 + 3σ² as r/(2R) → 0 (Appendix D) so that, in principle, σ can be obtained from the slope. However, the accuracy of our experimental C_I(r) is not sufficient to determine σ from the slope. Instead, we inspect how C_I(r) is affected by σ in its overall behavior.

Figure 4(c) displays C_I(r) from the simulated structures at σ = 0, 10, and 20%. The results show that, as σ increases, the initial decay becomes slower, the oscillation amplitude decreases, and the C_I(r) converges to a higher value at f = f₀(1 + 3σ²). The microsphere manufacturer (Fiber Optic Center Inc.) reports that σ < 10% for the microspheres used in our experiment. When σ increases from 0 to 10%, the C_I(r) in Fig. 4(c) increases slightly, which is similar to the increase observed in our experiment from the PY solution at f = 0.6 in Fig. 4(a). However, in this case, the oscillation amplitude does not change significantly [Fig. 4(c)], while our experimental C_I(r) shows an appreciable weakening of the oscillation compared to the PY solution at f = 0.6. When σ increases to 20%, the deviation from the PY solution (σ = 0) in Fig. 4(c), including the oscillation weakening, is similar to that of the experimental curve deviating from the PY solution at f = 0.55 in Fig. 4(a). That is, the polydispersity in our microspheres is expected to approach σ = 20%, counter to σ < 10% reported by the manufacturer. Thus, the difference in C_I(r) between our photonic glass and the ideal random structure can be explained in part by the microsphere polydispersity.

Figure 4(c) shows that, as σ increases, the peak at r/(2R) ∼ 1.2 shifts to the right. In comparison, in Fig. 4(a), the peak in our experimental C_I(r) (red circles) is located slightly on the left side of the peak in the calculations for σ = 0 (black solid line and green dashed line). Thus, this peak location in our experimental C_I(r) is not explained by polydispersity and more detailed study is required to fully explain the difference in C_I(r) between the experimental structure and the ideal random structure.

5. Conclusion

In conclusion, we have developed a method to characterize randomness in photonic glasses with respect to an ideal random limit without using complicated imaging techniques for inner structures, 3D reconstruction procedures from multiple images, and particle position tracking. In our method, the autocorrelation function of 2D cross-sectional images of photonic glasses is compared to that of the ideal limit. To obtain the autocorrelation function of the limiting case, we have derived a mathematical relation between the autocorrelation function and the radial distribution function. The relation enables us to obtain the ideal random limit of the autocorrelation function from the Percus-Yevick radial distribution function. The autocorrelation function of our experimental photonic glasses shows only a slight deviation from the ideal limit. The deviation can be explained in part by the microsphere polydispersity.

Our method of characterization is broadly applicable to the study of randomness in various optical materials. In particular, our method would be useful for investigating how the interplay between random and periodic potentials impacts optical properties of quasi-random media. The results of this investigation would elucidate the structure-property relationships in biological photonic materials made of quasi-random structures [11−16] and help optimize randomness in artificial materials, such as radiative cooling coatings [5] and random lasing photonic glasses [7,8], in a quantitative manner.

Appendix A: Calculation of g(r) from 2D images

Consider a reference sphere in a 3D random structure of isotropic monodisperse spheres and the distance r is measured from the center of the reference sphere. According to the definition of g(r), the number of spheres dN between r and r + dr is

(13)$$dN(r) = g(r)\frac{f}{{{V_s}}}\frac{{N - 1}}{N}4\pi {r^2}dr$$

When N >> 1, Eq. (13) becomes

(14)$$g(r) = \frac{{{V_s}}}{f}\frac{{dN(r)}}{{4\pi {r^2}dr}}$$

Now we consider a 2D image of a cross-sectional plane of the random structure. In this case, a reference sphere is chosen to be one whose center is right at the cross-sectional plane. In the 2D image, only the spheres whose centers are located within R from the 2D plane are manifest as circles. The differential volume between r and r + dr where these spheres are found is $4\pi rR\;dr$. Thus, the number of these spheres dM in this differential volume is related to g(r) by modifying Eq. (14) as

(15)$$g(r) = \frac{{{V_s}}}{f}\frac{{dM(r)}}{{4\pi rRdr}}.$$

In the 2D image, the distance ${r_\parallel }$ between a circle and the reference circle is measured. r is calculated from ${r_\parallel }$ by $$ r=\sqrt{r_{\|}^{2}+R^{2}-R^{\prime 2}} $$ where $R^{\prime}$ is radius of a circle in the 2D image.

In our experimental images (>2000 µm²) that resulted in Fig. 4(a), we regarded any circle with a diameter within ±10% of the average sphere diameter (0.87 µm) as the cross-section of a sphere with its center at the image plane. This 10% is the expected polydispersity based on the microsphere manufacturer (Fiber Optic Center Inc.). We calculated the radial distribution function using Eq. (15). Figure 5 shows g(r) both from the experiments and the PY approximation. The experimental g(r) is noisier than the corresponding C_I(r) in Fig. 4(a) but the general behavior of g(r) is discernable. A strong peak appears at r/(2R) = 1 and smaller peaks arise at larger distances. The peak amplitude decays as the distance increases to reach a plateau value of 1. Compared to the PY approximation, the peak at r/(2R) = 1 is broadened because of the polydispersity and the reference circle diameter range of ±10% introduced due to the uncertainty in finding spheres whose centers are at the image plane. These reasons are responsible for the disagreement in the rest of the g(r) peaks between the PY approximation and the experiment.

 

Fig. 5. Radial distribution function for f = 0.60 calculated by the PY approximation and Eq. (15) applied to the same experimental structure as that resulted in Fig. 4(a) red circles.

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As discussed above, a drawback of the method of calculating g(r) using Eq. (15) is that polydispersity inevitably introduces errors, which are avoided when a 3D image is available because the locations of polydisperse spheres are known from the image. Moreover, even with highly monodisperse samples, the number of spheres whose centers are right at the image plane is limited so that a large cross-sectional area has to be analyzed to ensure good accuracy when Eq. (15) is used. Thus, to accurately analyze our experimental structures, we used the autocorrelation function in Fig. 4(a) instead of the radial distribution function in Fig. 5 in this work.

Appendix B: Fill fraction for polydisperse spheres

When the total number of spheres in a system of a volume V is N, the average number of spheres with a radius between r_a and r_a + dr_a is

(16)$$n({r_a}) = \frac{N}{{R\sqrt {2\pi {\sigma ^2}} }}\exp \left[ { - \frac{{{{({r_a}/R - 1)}^2}}}{{2{\sigma^2}}}} \right],$$

where R is the average radius and σ is the relative standard deviation in r_a. The fill fraction of the spheres in the system is

(17)$$f = \frac{1}{V}\int_0^\infty {{V_s}({r_a})n({r_a})d{r_a}} \cong \frac{1}{V}\int_{ - \infty }^\infty {{V_s}({r_a})n({r_a})\;d{r_a}} ,$$

where ${V_s}({r_a}) = 4\pi {r_a}^3/3$ is the volume of a single sphere of radius r_a. Substituting Eq. (16) into Eq. (17), we have

(18)$$f = \frac{N}{V}{V_s}(R)({1 + 3{\sigma^2}} )= {f_0}({1 + 3{\sigma^2}} ).$$

Appendix C: Slope of autocorrelation function at x = 0 for monodisperse spheres

For $x < 1$, Eq. (9) can be written as

(19)$$12f\int_0^1 {({x + y} )g({x + y} )\left( {1 - \frac{3}{2}y + \frac{1}{2}{y^3}} \right)y\;dy} = {\left[ {xp(x) - \left( {x - \frac{3}{2}{x^2} + \frac{1}{2}{x^4}} \right)} \right]^{\prime}}.$$

Equation (19) can be rearranged to give

(20)$$p^{\prime}(x) = \frac{{12f}}{x}\int_0^{x + 1} {yg(y )\left( {1 - \frac{3}{2}(y - x) + \frac{1}{2}{{(y - x)}^3}} \right)(y - x)\;dy} + \frac{{1 - 3x + 2{x^3} - p(x)}}{x}.$$

In the first term in the right hand side of Eq. (20), we define

(21)$$F(x,y) \equiv yg(y )\left( {1 - \frac{3}{2}(y - x) + \frac{1}{2}{{(y - x)}^3}} \right)(y - x).$$

Because $g(y )= 0$ for $y < 1$, we have

(22)$$\mathop {\lim }\limits_{x \to 0} \frac{1}{x}\int_0^{x + 1} {F(x,y)dy} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{1}{{\Delta x}}\int_1^{\Delta x + 1} {F(\Delta x,y)dy} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{1}{{\Delta x}}\frac{{F(\Delta x,1) + F(\Delta x,1 + \Delta x)}}{2}\Delta x = 0.$$

Also, as $x \to 0$, the second term in the right hand side of Eq. (20) becomes

(23)$$\mathop {\lim }\limits_{x \to 0} \frac{{1 - 3x + 2{x^3} - p(x)}}{x} ={-} 3 - p^{\prime}(0).$$

Therefore, substituting Eqs. (22) and (23) into Eq. (20), we have

(24)$$p^{\prime}(0) ={-} \frac{3}{2}.$$

Appendix D: Slope of autocorrelation function at x = 0 for polydisperse spheres

Debye derived a general relation for isotropic media [35]

(25)$${\left. {\frac{{d{C_I}}}{{dr}}} \right|_{r = 0}} ={-} \frac{S}{{4fV}},$$

where S is the surface area of the spheres. For a special case of monodisperse spheres, $S = 4N\pi {R^2}$ and $fV = \frac{4}{3}N\pi {R^3}$ so that Eq. (25) becomes

(26)$${\left. {\frac{{d{C_I}}}{{d({{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r {2R}}} \right.}\!\lower0.7ex\hbox{${2R}$}}} )}}} \right|_{r = 0}} ={-} \frac{3}{2},$$

which is identical to Eq. (24). For polydisperse spheres, Eq. (25) becomes

(27)$${\left. {\frac{{d{C_I}}}{{dr}}} \right|_{r = 0}} ={-} \frac{1}{4}\frac{{\sum\limits_{j = 1}^N {4\pi {r_{a,}}{{_j}^2}} }}{{\sum\limits_{j = 1}^N {\frac{4}{3}\pi {r_{a,}}{{_j}^3}} }},$$

where ${r_{a,}}_j$ is the radius of the j^th sphere. When ${r_{a,}}_j$ follows the normal distribution, Eq. (27) becomes

(28)$${\left. {\frac{{d{C_I}}}{{dr}}} \right|_{r = 0}} ={-} \frac{3}{4}\frac{{\int {\frac{{{r_a}^2}}{{\sqrt {2\pi } \sigma R}}\exp \left[ { - \frac{{{{({r_a} - R)}^2}}}{{2{\sigma^2}{R^2}}}} \right]d{r_a}} }}{{\int {\frac{{{r_a}^3}}{{\sqrt {2\pi } \sigma R}}\exp \left[ { - \frac{{{{({r_a} - R)}^2}}}{{2{\sigma^2}{R^2}}}} \right]d{r_a}} }}.$$

When the integrals in Eq. (28) are evaluated, we obtain

(29)$${\left. {\frac{{d{C_I}}}{{d({{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r {2R}}} \right.}\!\lower0.7ex\hbox{${2R}$}}} )}}} \right|_{r = 0}} ={-} \frac{3}{2}\left( {\frac{{{\sigma^2} + 1}}{{3{\sigma^2} + 1}}} \right).$$

Funding

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

Disclosures

The authors declare no conflicts of interest.

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