Investigation and calculation of filling factor of SnO₂ inverse opal

Since Yablonovitch [1] and John [2] theoretically showed the ability of three-dimensional periodic dielectric materials possessing photonic band gap, in 1987, photonic band-gap materials have become an important and popular field of research. Inverse opal structure, as an important three-dimensional photonic band-gap material, has promising applications in optical communication, photonic computing, switching, lasing, solar cells, etc [3–5].

One of the most important characteristics of opal and inverse opal is band-gap position, which can confine and control the propagation of light with minimal losses. Band-gap positions of opal and inverse opal can be described with Bragg's law [6, 7]:

$$\begin{eqnarray}&&{\lambda }_{{\rm{max}}}=2{d}_{111}\sqrt{{n}_{{\rm{a}}{\rm{v}}{\rm{g}}}^{2}-{\mathrm{sin}}^{2}\theta }\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{d}_{111=}{\left(\displaystyle \frac{2}{3}\right)}^{\mathrm{1/2}}D\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{n}_{{\rm{a}}{\rm{v}}{\rm{g}}}=\sqrt{{n}_{1}^{2}f+{n}_{2}^{2}(1-f)}\end{eqnarray} \tag{ 3 }$$

Where ${\lambda }_{{\rm{max}}}$ is the wavelength of the band-gap position, ${d}_{111}\;$ is the lattice constant, ${n}_{{\rm{a}}{\rm{v}}{\rm{g}}}$ is the average refractive index of the materials, and $\theta$ is the angle between the incident light and the surface normal of the sample. As the surface of the sample is predominated by the (111) face, the lattice constant can be calculated with formula (2). Where $D$ is the diameter of the PS microspheres. ${n}_{{\rm{a}}{\rm{v}}{\rm{g}}}\;$ can be calculated with formula (3). Where ${n}_{1}\;$ and ${n}_{2}$ are the refractive index of different materials, ${\rm{f}}$ is the volume fraction of the compositional material.

In the past, inverse opal materials have been prepared with many methods [8–11]. In general, the preparation of inverse opal materials can be conducted in the following sequential steps: assembly of an opal template on microslide; infiltration and deposition of a matrix phase, or a precursor to a solid matrix phase; selective removal of the opal template to yield an inverse opal structure. In fact, in the process of preparing inverse opal, the filling factor of inverse opal is affected by the concentration of the precursor solution, capillary force, infiltration time, operation times, annealing temperature, heating rate of annealing, and so on. The framework of inverse opal cannot occupy all the residual space, and the true filling factor $\;f$ of air in inverse opal is difficult to directly test. However, the true filling factor $\;f$ of inverse opal is an important parameter for application in optical communication, and photonic computing. Therefore, it is necessary to research the filling factor $\;f$ of inverse opal.

As far as we know, in the work of Willen Vos et al [12], the true filling factor $f$ of TiO₂ inverse opal was tested with small-angle x-ray scattering (SAXS) and calculated with the Beer–Lambert law. However, in their result, the filling factor $f$ was not consistent with the observed result in the scanning electron microscopy (SEM) images. In Yang et al [13], the filling factor $f$ of ZnO inverse opal was investigated and calculated by changing the ethanol/water ratio of the precursor on one opal template. In Zhao et al [14], the filling factor $f$ of inverse opal was investigated by changing the liquid surface dropping velocity on a 580 nm opal template. In Ruda et al [15], the filling factor $f$ of ferroelectric inverse opal film was introduced by using sol–gel infiltration on one opal template, but the detailed calculation process was not explained. In Zhou et al [16–18] and Jianbei et al [19], the ${{\rm{n}}}_{{\rm{avg}}}\;$ was investigated and calculated by changing the incident angle, then the filling factor $\;f$ of inverse opal was calculated with formula (3).

However, in our opinion, we do not need to calculate the ${n}_{{\rm{a}}{\rm{v}}{\rm{g}}}.$ If the ${\lambda }_{{\rm{max}}},$ ${d}_{111},$ $\theta$ are fixed, the filling factor $f$ of air in inverse opal can be calculated with the following formula (4) which is derived from formulas (1) and (3).

$$\begin{eqnarray}&&f=\left(\displaystyle \frac{{\lambda }_{{\rm{max}}}^{2}}{4{d}_{111}^{2}}+{\mathrm{sin}}^{2}\theta -{n}_{2}^{2}\right)/({n}_{1}^{2}-{n}_{2}^{2})\end{eqnarray} \tag{ 4 }$$

As an important semiconductor material, SnO₂ with the optical band gap between the energy band and conduction band of 3.7 eV [20] is applied in chemistry electrodes, solar cells, gas sensors, etc. SnO₂ inverse opal material has been prepared by the opal template cooperating with the sol–gel method [21–24]. However, the filling factor $f$ of air in SnO₂ inverse opal has not been investigated and calculated. In this paper, the filling factor $f$ of air in SnO₂ inverse opal was investigated and calculated with formula (4).

PS microspheres with a diameter of about 280 nm and a standard deviation of less than 5% were prepared by emulsifier-free emulsion polymerization technology [25]. SnCl₄ · 5H₂O, oxalic acid and ethanol are AR and were bought from Sinopharm Chemical Reagent Co. Ltd. All chemicals used in the experimental section were used as received without further purification.

Preparation of SnO₂ precursor solution

Preparation of SnO₂ inverse opal

Characterization

Opal templates made of different diameter PS microspheres have been reported in many papers. Figure 1 shows the SEM and TEM images of the opal templates. At macro-scale, the opal templates are a long-range ordered structure; at micro-scale, the PS microspheres in the opal template arrange as well-ordered hexagonal close-packed structure. Figures 1(a) and (b) show the SEM and TEM images of the 220 nm PS microsphere opal template. Figures 1(c) and (d) show the SEM and TEM images of the 390 nm PS microsphere opal template. They all show the well-ordered hexagonal close-packed structure of PS microspheres.

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Since PS microsphere is an organic compound, the precursor solution of SnO₂ was made up of SnCl₄ and the organic compound oxalic acid and ethanol. The strong interaction between the precursor solution of SnO₂ and polystyrene spheres tends to result in 'surface-templated' structures. F.2, 3, 4 all show the 'surface-templated' structures.

In the process of preparing SnO₂ inverse opal, the repetition time of the operation is very important. Figure 2 shows the SEM images of SnO₂ inverse opal which were prepared by infiltrating the precursor into the 220 nm PS microsphere opal with different time.

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Figures 2(A) and (B) are the SEM images which were prepared once (one time). In A, it can be seen that the top of the SnO₂ inverse opal is not a long-range ordered structure. In B, circle 1 represents SnO₂ occupying the space of the adjacent two PS microspheres, circle 2 represents SnO₂ occupying the space of the adjacent three PS microspheres, circle 3 exhibits the skeleton of SnO₂ formed around one PS microsphere. The skeleton was not completely formed, but the sublayer holes can be observed from the top holes.

Figures 2(C) and (D) are the SEM images which were prepared twice (two times). In C, it can be seen that the top of the SnO₂ inverse opal is a long-range ordered structure, although there are some defects e.g. circle 1, rectangle 2 existing. The circle 1 in D represents the perfect skeleton of SnO₂ formed around one PS microsphere, although a few of the skeletons are broken e.g. circle 2, 3, 4 existing, most of the skeletons in C and D are unbroken and ordered. Of course, we can consider that the defects such as circle 1 in C and circle 4 in D arise from the polystyrene template itself, such as PS microsphere vacancy.

Figures 2(E) and (F) are the SEM images which were prepared three times. F is the magnifying part of E. It is obvious that the top of the SnO₂ inverse opal is covered with SnO₂, and there is no interconnected exit hole. But in the crossing section, many long-range ordered holes exist, and inverse opal structure exists under the top of the SnO₂.

From figure 2, for the 220 nm PS microsphere opal, it can be seen that the infiltrating precursor for one time or three times cannot form perfect inverse opal structure, but the two times is appropriate.

Figure 3 shows the SEM images of the SnO₂ inverse opal which were prepared by infiltrating the precursor into the 390 nm PS microsphere opal with different time. Figures 3(A) and (B) are the SEM images which were prepared one time. In A, except circle 1, few perfect skeletons were formed at the top of the SnO₂ inverse opal. The most structure is shown as circle 2 and 3 which represent SnO₂ occupying the space of the adjacent three PS microspheres. If SnO₂ does not completely fill the spaces around the PS microspheres, different structures will be formed. In B, it shows different structures, such as circle 1, 3, 4 and rectangle 2. Figures 3(C) and (D) are the SEM images which were prepared two times. D is the magnified part of C. In C and D, most perfect skeletons on the top of the SnO₂ inverse opal were formed, although a few broken skeletons exist e.g. in circle 1 and 2. Figures 3(E) and (F) are the SEM images which were prepared three times. F is the magnified part of E. It is obvious that perfect skeletons on the top of the SnO₂ inverse opal were formed.

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From figure 3, for the 390 nm PS microsphere opal, it can be seen that infiltrating precursor one or two times is not enough to form perfect inverse opal structure, but three times is appropriate.

Figure 4 shows the SEM images of the SnO₂ inverse opal which were prepared by infiltrating precursor into the 530 nm PS microsphere opal three times. It is obvious that a perfect skeleton was seldom formed.

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From figures 2, 3 and 4, we can conclude that, in order to prepare a perfect skeleton on the top of the SnO₂ inverse opal, in the same operational process, the diameter of the PS microsphere is bigger, and the repetition time of the infiltrating precursor is greater.

Figure 5 shows the TEM images of the SnO₂ inverse opal. A and B were prepared by infiltrating the precursor into the 220 nm PS microsphere opal two times, C and D were prepared by infiltrating the precursor into the 390 nm PS microsphere opal three times. All of the TEM images show that the SnO₂ inverse opal is long-range ordered and symmetrical.

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Figure 6 shows the electron diffraction pattern of the SnO₂ inverse opal based on the 390 nm PS microsphere opal and XRD patterns of the SnO₂ inverse opal. The testing results of the lattice distance of the electron diffraction pattern and labeling the peak positions show that the crystal form of the SnO₂ inverse opal is tetragonal phase, which corresponds with JCPDS41-1445. The ${n}_{{\rm{Sn}}{{\rm{O}}}_{2}}$ of the tetragonal phase is 2.006 [28].

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Optical property is the most important characteristic of opal and inverse opal. In theory, band-gap positions of opal and inverse opal can be calculated with formulas (1), (2) and (3). The calculating ${\lambda }_{{\rm{max}}}$ of opal is shown in table 1. However, for the SnO₂ inverse opal, the filling factor $f$ of air is not fixed since it is affected by the operation of precursor infiltration and calcination, and the ${\lambda }_{{\rm{max}}}$ of the SnO₂ inverse opal cannot be calculated.

Table 1.  Relative data of PS microsphere opal and SnO₂ inverse opal

Sample	D (nm)	Calculating ${\lambda }_{{\rm{max}}}$ (nm)	Testing ${\lambda }_{{\rm{max}}}$ (nm)	$f$	${f}_{Sn{O}_{2}}$
Opal	340	793	791	0.74	—
Inverse opal	225	—	437	0.86	0.14
Opal	390	909	920	0.74	—
Inverse opal	265	—	500	0.89	0.11
Opal	425	1014	1015	0.74	—
Inverse opal	295	—	535	0.92	0.08

For opal, D is the diameter of the PS microsphere; for inverse opal, D is the distance of the adjacent big void center.

The test results of the reflecting spectrum are shown in figure 7. The randomly selected positions of the opal have almost the same peak position and are very close to the calculating ${\lambda }_{{\rm{max}}}$ in table 1, which demonstrates the uniform structure of opal. The randomly selected positions of the SnO₂ inverse opal have almost the same peak position, which also demonstrates uniform structure.

In this paper, according to the geometry, for opal, the filling factor $f\;$ of colloid microspheres is 74% [9]. ${n}_{1}\;$ is the refractive index of PS microspheres, ${n}_{2}$ is the refractive index of air. For inverse opal, in theory, the filling factor $f\;$ of air is also 74%, ${n}_{1}\;$ is the refractive index of air, the framework of inverse opal occupies the residual space, and ${n}_{2}$ is the refractive index of SnO₂.

According to the formula (4) and testing λ_max, the true filling factor $f$ of air in SnO₂ inverse opal was calculated and shown in table 1. It can be seen that, for inverse opal, in the same operational process, the filling factor $\;f$ of air in SnO₂ inverse opal gradually enlarges as the diameter of the void increases, while the filling factor of SnO₂ decreases. In order to achieve perfect inverse opal structure, the filling quantity of the precursor should be appropriate. In other words, the diameter of the PS microsphere of opal should be bigger, and the filling quantity of precursor should be greater. On the other hand, in the process of preparing inverse opal, the filling factor of inverse opal is affected by the concentration of the precursor solution, capillary force, infiltration time, operational times, annealing temperature, heating rate of annealing, and so on, so the framework of inverse opal cannot occupy all the residual space of opal. So the true filling factor $f$ of air in SnO₂ inverse opal is always higher than the original opal.

In summary, SnO₂ inverse opal was prepared with an opal template cooperating with a sol–gel method. The repetition times of the infiltrating precursor into the opal templates were investigated in detail. In order to prepare perfect inverse opal structure, the diameter of the PS microsphere of opal should be bigger, and the filling quantity of the precursor should be greater. The filling factor $\;f$ of air in inverse opal can be calculated with the formula (4) derived from Bragg's law. For inverse opal, the filling factor $\;f$ of air is bigger as the diameter of the void increases.

This study was financially supported by the Foundation of Key Laboratory for Palygorskite Science and Applied Technology of Jiangsu Province (Project No. HPK201507), the Foundation of Jiangsu Provincial Engineering Laboratory for Advanced Materials of Salt Chemical Industry (No. SF201407) and the Huaiyin Institute of Technology Science Foundation (No. HGC1402).