Abstract

In order to address the problem with titania distribution and recovery, series of Ti0.91O2/CFA photocatalysts (Ti0.91O2/CFA-, and ) were fabricated by assembling Ti0.91O2 nanosheets on coal fly ash (CFA) microspheres via the layer-by-layer assembly (LBLA) process and characterized by scanning electron microscopy (SEM), X-ray diffraction analysis (XRD), N2-sorption, and ultraviolet-visible absorption (UV-vis) techniques. The SEM images and UV-vis spectra illustrated that Ti0.91O2 nanosheets were immobilized successfully on the CFA by the LBLA approach and changed the characteristics of CFA noticeably. The photocatalytic activity of Ti0.91O2/CFA was evaluated by the photodegradation of methylene blue (MB) under UV irradiation. The results demonstrated that Ti0.91O2/CFA-6 showed the best photocatalytic activity among the series of Ti0.91O2/CFA irradiated for 60 min, with a decoloration rate above 43%. After photocatalysis, the Ti0.91O2/CFA could be easily separated and recycled from aqueous solution and Ti0.91O2 nanosheets were still anchored on the CFA.

1. Introduction

Fabrication of multilayers of colloidal particles through electrostatic attraction has been studied since 1966 [1]. Generally, the composites consist of organic or inorganic particles as cores and inorganic nanocoatings as shells. For the great potentials of core-shell composites in photonics and catalysis areas, many researches have been focused on the preparation of core-shell structure via the layer-by-layer assembly (LBLA) approach in recent years [24]. The LBLA method has been applied to assemble charged thin films of various materials on oppositely charged templates. The inorganic coatings are deposited on the templating cores via electrostatic attraction, so the shell coatings can be adsorbed on the cores uniformly and firmly. Titanium oxide is popular with researchers for its excellent performance of high photocatalytic activity, chemical inertness, stability against photocorrosion, and cost-effectiveness [5]. Titania nanosheet (Ti1-x), synthesized by chemical delamination and exfoliation of precursor, is of typical 2D structure and the lateral size of it ranges from hundreds of nanometers to a few micrometers and the thickness of the individual layer is approximately 1 nm [6]. The titania nanosheets after exfoliation are negatively charged, so titania nanosheets can be assembled on substrates via the LBLA approach coupled with cationic polyelectrolyte [7]. Multilayer titania nanosheets possess unique properties besides photocatalytic activity [8], such as optical absorption properties [7, 9], high anisotropy [10], photoelectrochemical properties [11], and high thermal resistance [12].

Titania nanosheet hollow shells and hollow spheres consisting of titania and graphene nanosheets have been fabricated by the layer-by-layer assembly method in previous studies [13, 14]. However, the titania nanosheet hollow spheres are hard to separate spontaneously and recycle from aqueous suspension after photocatalysis, which limits their practical application in wastewater treatment. In order to improve the convenience of separation and recovery of nano-TiO2, photocatalysts by immobilizing nano-TiO2 on some substrates were recently prepared, such as on glass, polymer, and active carbon [1517]. However, most substrate materials are expensive, and cheap and stable substrates are desired. In our early work, photocatalysts of TiO2 immobilized on coal fly ash (CFA) were prepared [1820]. CFA is one of the solid wastes generated from thermal power plants. CFA chosen as a supporter has advantages as follows: (1) CFA particles are microspheres and easy to precipitate in water, so photocatalysts supported by CFA are easy to recycle from aqueous solution after reaction; (2) CFA, consisting primarily of Al2O3 and SiO2, as a supporter can inhibit recombination of electron and hole effectively [21, 22]; (3) the cost of preparation and a source of environment pollution can be reduced. However, it is difficult to control uniform distribution of TiO2 on CFA [17, 23], which greatly restrains the activity of the photocatalyst and the availability of substrate. Because the lateral size of titania nanosheets is in the micron range and CFA particle size is less than 100 μm, the problem can be solved by loading titania nanosheets on CFA through electrostatic attraction via layer-by-layer assembly method at ambient temperature [24]. The LBLA method involves electrostatic sequential deposition of negatively charged titania nanosheets onto substrate along with an oppositely charged polymer. Based on this principle, an assembly of layered titanate on CFA is expected to achieve very well.

In this paper, the titania nanosheet (Ti1-x) referred to is Ti0.91 and is abbreviated as Ti0.91O2. This study describes the fabrication of Ti0.91O2/CFA by the LBLA approach for controlling the uniform distribution of Ti0.91O2. The layered Ti0.91O2 nanosheets were obtained by swelling and exfoliation of layered protonic titanate (-FeOOH type) in (C4H9)4NOH (TBAOH) solution and were negatively charged. The surface of CFA was modified by cationic polyelectrolyte beforehand. Ti0.91O2 nanosheets were adsorbed on the surface of CFA due to the electrostatic attraction and immobilized continually by repeating the LBLA procedure. Ti0.91O2 nanosheets anchored served as a shell and CFA could be regarded as core. The photocatalytic activity of Ti0.91O2/CFA was also evaluated by the decoloration of methylene blue (MB).

2. Experimental Section

2.1. Reagent and Materials

Cs2CO3 (99.99% metals basis), TiO2 (AR), polyethylenimine (PEI) (99% purity) and poly(diallyldimethylammonium chloride) (PDDA) (35 wt.% aqueous solution), and TBAOH (~0.8 M) were purchased from Aladdin Reagent Company. Ti0.91O2 nanosheets were synthesized by a previously reported method [25]. Briefly, the protonic titanate (H0.7Ti1.825O4·H2O) was prepared from cesium titanate (Cs0.7Ti1.825O4) by acid exchange for 3 days. And then, H0.7Ti1.825O4·H2O (0.4 g) was shaken vigorously (300 rpm) in 100 mL aqueous solution of TBAOH (0.004 M) for 2 weeks, so the layered hosts underwent osmotic swelling and exfoliating into nanosheets. The colloid produced stood for storage. CFA, calcined at 800°C, was immersed in a bath of 1/1 methanol/HCl and concentrated H2SO4 with stirring for 20 min each and then washed with copious water to remove acid residue. The pretreated CFA was dried for use later. Deionized water was used throughout the experiments.

2.2. Fabrication Procedure

Core/shell composites (CFA was employed as a substrate core and Ti0.91O2 nanosheets were inorganic shell) were fabricated by electrostatic sequential deposition approach [13, 14]. For Ti0.91O2 nanosheets were negatively charged, in order to load Ti0.91O2 nanosheets onto the CFA by electrostatic deposition, the surface of CFA was modified by cationic polyelectrolytes (PEI and PDDA). The detailed fabrication procedure of layer-by-layer assembly (LBLA) method is as follows: 5 g CFA was dispersed in 200 mL PEI solution (2.5 g·L−1, pH = 9) and further stirred for 60 min to ensure PEI was adsorbed on the CFA surface for introducing positive charges. After the mixture was centrifuged at 3000 rpm for 20 min, the supernatant was removed and then the CFA was washed dried in a drying oven under 60°C. Subsequently, the PEI-coated CFA (CFA/PEI) was dispersed in Ti0.91O2 colloidal suspension (200 mL) and further stirring was carried out for 60 min. Small flocculated aggregates formed in the solution during this process for electrostatic attraction. The resulting CFA/PEI/Ti0.91O2 was washed by water for 2 cycles to remove excess Ti0.91O2 nanosheets. After drying, it was dispersed in 200 mL PDDA solution (20 g·L−1, pH = 9), as the same as the PEI modifying process, and CFA/PEI/Ti0.91O2/PDDA was fabricated. The procedure for PDDA/Ti0.91O2 alternative deposition was repeated frequencies to synthesize a multilayer assembly-PEI/Ti0.91O2/(PDDA/Ti0.91O2)m (, and 7). In this study, the photocatalysts as-prepared were abbreviated as Ti0.91O2/CFA- (, and 8), where n represented the loading frequency of Ti0.91O2 nanosheets by LBLA.

2.3. Characterizations

The morphologies of the samples were observed by a field emission scanning electron microscope (SEM) and the Ti elemental mapping was detected with an energy dispersive X-ray spectrometer (EDX) (S-4800, Japan). X-ray diffraction (XRD) data of the samples were collected using diffractometer (X’Pert PRO, Holland) with Cu Kα irradiation. The morphology of the exfoliated nanosheets was observed by transmission electron microscope (TEM, H-7650, Japan). UV-vis absorption spectra of all samples were recorded by a Shimadzu spectrophotometer (UV-2450, Japan). The chemical composition of CFA was characterized by quantitative X-ray fluorescence spectrum (XRF) analysis (Axios mAX, Holland). The N2 adsorption-desorption isotherms, Brunauer-Emmett-Teller (BET) surface area, and Barrett-Joyner-Halenda (BJH) pore size distribution were obtained by surface area and porosity analyzer (ASAP 2020 M+C, America).

2.4. Evaluation of Photocatalytic Activity

The photocatalytic reaction was carried out in a photochemical reaction system as described before [18]. Briefly, the initial concentration and volume of MB were 20 mg/L and 50 mL, respectively. The dosage of Ti0.91O2/CFA- was 0.2 g. A 500 W UV lamp with major emission at 365 nm was used as a light source, and the irradiation time was 60 min. After reaction and settling for a while, the upper solution was centrifuged at 3000 rpm for 20 min to eliminate fine particles. Then the absorbance of supernatant was analyzed at the wavelength of 664 nm by a UV-vis spectrophotometer.

3. Results and Discussion

3.1. SEM Analysis of Samples

SEM was used to define the morphology of the Ti0.91O2/CFA microstructure. The images present CFA before (Figures 1(a) and 1(b)) and after (Figures 1(c)1(h)) Ti0.91O2 nanosheets were loaded by the LBLA approach. From Figures 1(a) and 1(b), it can be found that the surface of the CFA is smooth and barren. There are no obvious attachments on the CFA surface after rinsing with H2SO4 and CH3OH, which indicates that the impurities on the surface had been removed. After Ti0.91O2 nanosheets were deposited on the CFA by LBLA, some aggregates consisting of Ti0.91O2 multilayer nanosheets were adhered to the CFA (Figures 1(c)1(h)). With the increase of loading frequency, the area covered by corrugation was enlarged, and multilayers can be seen. It can be observed in Figure 1(c) that only very few aggregates are anchored on the surface of the CFA after twice repeated layer-by-layer loading. When the loading took place 4 times, the area of agglomerates was obviously enhanced (Figure 1(e)). Under high magnification (Figure 1(f)), there are more nanosheet agglomerates than with two loadings (Figure 1(d)). When the layer-by-layer loading frequency was further raised to 8, the agglomerates almost covered the CFA surface (Figures 1(g) and 1(h)). There is an obvious difference between the Ti0.91O2 nanosheet agglomerates here and the TiO2 nanoparticle agglomerates loaded on CFA. The Ti0.91O2 agglomerates are of approximate rectangle sheet shape (Figure 2), and their counterparts (anatase TiO2) are nanosphere agglomerates of 3D structure [19]. This is ascribed to the different loading methods. The conventional loading methods for fabricating TiO2 include the sol-gel-adsorption method and the hydrothermal method [26, 27], and the TiO2 agglomerates prepared are usually granular after calcination. In this study, the Ti0.91O2/CFA was fabricated without calcination under high temperature, so Ti0.91O2 was able to keep its original crystalline phase and did not transform into anatase or rutile. Therefore, the Ti0.91O2 nanosheets were still lamellar after loading on CFA. From the images of the morphology of the Ti0.91O2/CFA, it is certain that the Ti0.91O2 nanosheets are successfully immobilized on the surface of CFA with layer-by-layer approach, although the amount of Ti0.91O2 is smaller compared to our earlier work [19].

3.2. UV-Vis Absorption Spectra of Samples

The UV-vis absorption spectra demonstrate the optical absorption characteristics of the CFA and Ti0.91O2/CFA samples. In Figure 3, the absorbances of the CFA do not change much along the range of wavelengths (200 nm–800 nm), except for a weak and broad peak at around 370 nm. Compared to the spectrum of the CFA, the spectra of the Ti0.91O2/CFA illustrate a characteristic absorption change along the whole range of wavelengths, with two strong absorption peaks at around 266 nm and 375 nm. The peak at around 375 nm was dependent on the immobilized multilayer nanosheets which were not completely exfoliated into monolayer nanosheets. So the absorption peak has a micro red-shift. The peak of the absorption curve around 266 nm is in accord with the multilayer titania nanosheets which had been studied [7], but the absorption peak is broader than that in the study of Sasaki et al. [28]. This may be attributed to the fact that the selected substrates are different. The previous substrates were quartz glass platelets and silicon wafer chips (1 × 5 cm2), which had a flat surface and different size from CFA. However, the substrate in this study was CFA of microsphere shape and the particle size of CFA was below 100 μm. Therefore, the assembly style of titania nanosheets on the substrate was not exactly the same. This led to the discrepancy. It can be also observed that the absorption peak of Ti0.91O2/CFA-6 around 266 nm is stronger than the others. It may be explained that the peak around 266 nm was previously found to be unilamellar Ti0.91O2 nanosheets [7], and there were more unilamellar Ti0.91O2 nanosheets on the surface of the CFA than other types. On Ti0.91O2/CFA-2, 4, 8, the incompletely exfoliated nanosheets made up a larger proportion than Ti0.91O2/CFA-6, which weakened the intensity of the peak around 266 nm.

3.3. Crystal Structure of Samples

XRD was used to investigate phase structure changes of the CFA before and after Ti0.91O2 were immobilized by LBLA. CFA and Ti0.91O2/CFA crystalline patterns are shown in Figures 4(a)4(c). In Figure 4(a), CFA primarily contains mullite (Al4.5Si1.5O9.75) and quartz (SiO2), which is similar to the previous study [29]. After Ti0.91O2 were loaded, there was no obvious difference between CFA and Ti0.91O2/CFA. It could be two reasons: (1) Ti0.91O2 nanosheets were adhered to the surface of CFA through electrostatic force, not entering the crystal lattices of mullite and quartz; (2) the amount of deposited Ti0.91O2 nanosheets was small, so the influence on overall diffraction peak was little relatively. Therefore, no prominent changes in crystalline patterns of the CFA were observed clearly. However, there are still tiny differences between CFA and Ti0.91O2/CFA in the range of 2θ among 26°-27° (Figure 4(b)) and 39°-40° (Figure 4(c)), which are partially enlarged details of Figure 4(a). At 2θ = 26.6°, before Ti0.91O2 were loaded, the peak of the CFA is very weak; after LBLA was carried out for several frequencies, there is a small peak, which is higher than the CFA sample. At 2θ = 39.5°, the intensity of the CFA peak decreases remarkably after Ti0.91O2 were immobilized. Figure 4(d) depicts the XRD pattern of protonic titanate before exfoliation. The peak at 9.5° is a characteristic of protonic titanate, and it disappears on Ti0.91O2/CFA, which can prove that slight changes of Ti0.91O2/CFA XRD pattern at 2θ = 26.6° and 39.5° resulted from the loaded titania nanosheets via LBLA, not protonic titanate. This confirms that LBLA method is feasible to immobilize Ti0.91O2 nanosheets.

3.4. Adsorption-Desorption and Pore Distribution of Samples

The BET surface area of the Ti0.91O2/CFA as-prepared is summarized in Table 2. Before Ti0.91O2 were loaded, the BET surface area of CFA was 2.62 m2/g. After Ti0.91O2 nanosheets were loaded, the BET surface area of Ti0.91O2/CFA-2 declined instead, so did the surface area of Ti0.91O2/CFA-4. When the loading frequency was 6, the specific surface area of Ti0.91O2/CFA-6 reduced to the minimum which is about 1.26 m2/g; the specific surface area increases slightly while the loading frequency rose to 8, however, still below 2 m2/g. The specific surface area decreased gradually with the increase in loading Ti0.91O2 nanosheets, similar to the results reported previously [30, 31]. The reason may be that Ti0.91O2 nanosheets immobilized by LBLA on the surface of the CFA were a few and lamellar, so it was difficult to form porous structure, which could not change the surface area much. The specific surface area of Ti0.91O2/CFA-6 was the smallest among the samples. Combined with UV-vis absorption analysis, this can be attributed to the monolayer nanosheets. Unilamellar Ti0.91O2 nanosheets were loaded more regularly than multilayer (incompletely exfoliated nanosheets) on the surface of CFA, so the surface structure of Ti0.91O2/CFA-6 is less complicated than others with a smaller surface area.

Figure 5 shows the N2 adsorption-desorption isotherms of CFA and Ti0.91O2/CFA samples. It can be seen that all the samples can be assigned as an isotherm of type 2 in the IUPAC classification [32, 33]. The fact that the hysteresis loop is difficult to observe and volume adsorbed is very small reveals that the samples are typically nonporous characteristic by LBLA [34]. Moreover, it is shown that the adsorption capacity of the Ti0.91O2/CFA is lower than the CFA. With the loaded Ti0.91O2 nanosheets increasing, adsorption capacity and the hysteresis loop of Ti0.91O2/CFA became low and small gradually, respectively, which implies that immobilization of Ti0.91O2 nanosheets resulted in forming new structure (not 3D network), but did not increase the pore volume.

The pore size distribution of the CFA and Ti0.91O2/CFA in Figure 6 illustrates that pores of all samples are micropores and mesopores in a wide distribution range (<2 nm to 50 nm). Pores below 3 nm of all samples take a large amount with small pore volume. This can be explained by the fact that the specific surface area was very small and the samples were nearly nonporous. After Ti0.91O2 nanosheets were anchored by LBLA on CFA, the mesopore distribution of the Ti0.91O2/CFA samples slightly shifts to 18 nm compared with 15 nm of the CFA, and the specific surface area of Ti0.91O2/CFA drops correspondingly. However, Ti0.91O2 nanosheets did not change the pore size distribution much either.

The results above of BET surface area, N2 adsorption-desorption isotherms, and pore size distribution demonstrate that the loaded Ti0.91O2 by LBLA do not change the pore structure of the CFA. It is ascribed to Ti0.91O2 nanosheets which are lamellar with general lateral size of submicrons to microns and unilamellar thickness of ~1 nm. In this study, the lateral size of Ti0.91O2 nanosheets presented in Figure 2 is in the range of 200 nm–500 nm. The nanosheets are bigger than the pores of CFA and can cover them. However, the particle diameter of the CFA was above 5 μm and greatly larger than the size of Ti0.91O2 nanosheets, so Ti0.91O2 nanosheets stacked on the surface of CFA closely to form new structure but cannot fundamentally change the architecture of CFA. This conclusion is further supported by the image magnification of the Ti0.91O2/CFA in Figure 1.

3.5. Photocatalytic Activity

The photocatalytic activity of Ti0.91O2/CFA was investigated by the degradation of MB solution as a test reaction. Blank experiment (MB solution without CFA and Ti0.91O2/CFA) was served as a comparison. Before the photocatalytic reaction began, the adsorption experiment in the dark lasted for 60 min to reach adsorption equilibrium [35]. The absorbance of MB is proportional to its concentration, so the decoloration rate can be calculated by the equation as follows: , where and are the absorbances of MB solution at initial and time, respectively.

Figure 7(a) shows MB was eliminated only ~10% by adsorption of samples and did not degrade in the blank experiment when the UV light was switched off. When the UV light was on, the MB degraded rapidly. In the initial 10 min, the decoloration efficiencies of Ti0.91O2/CFA- are almost equivalent to that during 1 h adsorption without UV irradiation. The concentration of MB decreases further with the prolonged UV light. After 60 min of photocatalysis reaction, the concentration of MB reduces to half of the initial. Figure 7(b) displays the decoloration rates of all the samples after 60 min UV light illumination. It can be seen that the removal efficiency of blank is much lower than that of the CFA and Ti0.91O2/CFA. This indicates that MB is removed through photocatalysis primarily under UV irradiation, not photolysis. Meanwhile, all Ti0.91O2/CFA samples have higher decoloration rate above 24% than CFA which is about 15%. A poor photocatalytic characteristic of CFA originated from its chemical component (Table 1), which was the same as our early work [18], so Ti0.91O2 contributes to photocatalytic activity principally. This demonstrates again that it is successful to load Ti0.91O2 nanosheets on CFA via LBLA.

Generally, the photocatalytic activity is enhanced with increasing the Ti0.91O2 layer-by-layer loading frequency. With two frequencies loading, the decoloration rate of the Ti0.91O2/CFA-2 achieved 24.7% correspondingly; when Ti0.91O2 loading accomplished 4 frequencies, the photocatalytic activity of the Ti0.91O2/CFA improved remarkably, with a decoloration rate of 41.8%, due to more Ti0.91O2 nanosheets immobilized. While the loading frequency was raised to 6 further, the activity improved a little, up to a maximum decoloration rate about 43.2%. Nevertheless, the activity of Ti0.91O2/CFA-8 was lower than Ti0.91O2/CFA-4, 6. It is understandable that the amount of Ti0.91O2 nanosheets anchored on CFA influences the photocatalytic activity. The more Ti0.91O2 nanosheets were loaded, the higher activity the sample had. However, the excessive Ti0.91O2 nanosheets would block the UV illumination into the interior of photocatalyst, with a depressed availability of UV irradiation, which was consistent with other studies [3638]. Meanwhile, the transfer of charge carrier may be also limited. It should be noted that Ti0.91O2/CFA-6 exhibited the highest photocatalytic activity whose specific surface area and pore volume are the smallest. It may be also explained that there are more Ti0.91O2 monolayer nanosheets (with high photocatalytic activity by themselves and not blocking UV illumination) on the surface of Ti0.91O2/CFA-6 than others for its UV absorbance spectral profile with a stronger peak at 266 nm, which had been discussed in UV-vis and BET analysis. Although the decoloration rate (43.2%) was not too high when Ti0.91O2/CFA- were used as photocatalysts, Ti0.91O2/CFA- are very easy to separate and recycle from aqueous suspension due to the weight of CFA, which facilitates the recycle and reuse, as mentioned in our previous publication [1820].

Figure 8 illustrates the SEM and EDX micrographs of the Ti0.91O2/CFA-6 before (Figures 8(a) and 8(b)) and after photocatalysis (Figures 8(c) and 8(d)). It is obvious that there are almost no changes in the appearance of Ti0.91O2/CFA-6, and Ti0.91O2 nanosheets can be still found on the surface of CFA microspheres with a considerable amount, which suggests that Ti0.91O2 nanosheets can stick to CFA firmly through the LBLA method. From Figure 8(d), the enlargement of Figure 8(c), it is observed that the morphology of Ti0.91O2 nanosheets changes a little. The edge of certain Ti0.91O2 nanosheets bends slightly, called “warping,” which may be ascribed to the illumination of UV-light [8, 39]. However, this did not influence the stability for their partial existences on the surface of substrate. Figure 8(f) is the Ti mapping of Ti0.91O2/CFA as presented in Figure 8(e) which shows the distribution of Ti-containing species on the surface of CFA microsphere. This indicates that there is another difference between Ti0.91O2/CFA and conventional TiO2 cluster immobilized on substrates. In conventional studies, it was difficult to control TiO2 cluster loaded on supporters uniformly [40, 41]. However, from Ti0.91O2/CFA as-prepared in our work, it can be seen that Ti element is well-distributed, which can improve the availability of substrate. It is a unique feature of Ti0.91O2/CFA. In a word, the unilamellar nanosheets can be easily and compactly immobilized on the CFA by LBLA, which is in favor of maintaining the stability of Ti0.91O2/CFA.

4. Conclusion

We fabricated a novel photocatalyst successfully via the layer-by-layer assembly method. Ti0.91O2 nanosheets were immobilized on CFA by using sequential modification of cationic polyelectrolyte and Ti0.91O2 nanosheets. Since Ti0.91O2 nanosheets have special properties, Ti0.91O2/CFA exhibits different characteristics, which can be concluded as follows.(1)Difference in morphology depends on the loading frequency. The amount of Ti0.91O2 nanosheets on the CFA surface increases with the layer-by-layer assembly frequency increasing. Ti0.91O2 nanosheets distribute uniformly on the surface of CFA whose availability can be improved.(2)CFA exhibits weak optical absorption in whole range of wavelength compared with Ti0.91O2/CFA, which proves that the immobilization of Ti0.91O2 nanosheets by LBLA can enhance the UV-absorption of the CFA. Ti0.91O2/CFA has a UV-absorption peak around 266 nm.(3)Ti0.91O2/CFA shows photocatalytic activity. Significant enhancement in decoloration of MB can be achieved by Ti0.91O2/CFA compared to barren CFA. Ti0.91O2/CFA is also easy to recycle and of mechanical stability, causing little secondary pollution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by Key Project of Science and Technology Plan of Fujian Province (no. 2012Y0066); National Key Technology Support Program (2012BAC25B04); Science and Technology Major Project of Fujian Province (2013YZ0001-1). We thank Professor M. L. Fu and Dr. J. W. Chen for their kind help.