Producing CrFeCoNiSi-Based High Entropy Alloy by Spark Plasma Sintering

Abstract

A high-entropy alloy (HEA) is a multi-component alloy obtained by blending at least five metal elements at compositions of 5–35%. Contrary to expectations, HEAs have a simple microstructure and exhibit unique properties such as excellent high-temperature strength, high tensile strength, and extremely slow diffusion rate. They are mainly produced by casting method, and heterogeneous microstructures have been found in the as-cast structure. Powder metallurgy has many advantages over ingot-metallurgical methods such as casting, including excellent material efficiency and ease of conversion to complex shapes. In this study, we produced a CrFeCoNiSi HEA and evaluated its properties. The Si content was changed within the range of 5–15 at% to investigate the characteristics of the corresponding HEAs. A Vickers hardness test and a corrosion test were performed. The Vickers hardness test revealed that the hardness of the samples of (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀ was approximately 550 HV, whereas that of (CrFeCoNi)₈₅Si₁₅ was approximately 700 HV. (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀ showed significant improvement in corrosion resistance. In constant, deterioration was observed for the (CrFeCoNi)₈₅Si₁₅ sample due to the presence of excess Si, which formed a stable silicide with metal elements.

Fig. 2 Result of X-ray diffraction test for CrFeCoNiMn and (CrFeCoNi)_100−xSi_x sintered samples.

1. Introduction

High entropy alloys (HEAs) have no main components, and are multi-component alloys obtained by blending at least five or more kinds of metal elements at equiatomic fractions (5–35%).¹⁾ However, many elements in HEAs lead to the formation of a single phase solid solution, and not intermetallic phases. In contrast conventional alloys, where the base metal atoms are likely to be surrounded by the same atoms, all atoms are surrounded by different atoms in HEAs.²⁾ This causes lattice distortion due to different atomic sizes of these surrounding atoms. Therefore, HEAs exhibits unique properties such as excellent high-temperature strength, high tensile strength, and extremely slow diffusion speed.^3,4) In addition, there are reports that the surface engineering has been further enhanced,^5–7) and HEAs have been attracting attention in recent years.

The definitions of HEAs are broad as they include face-centered cubic (fcc),^8–10) body-centered cubic (bcc),^11,12) fcc + bcc,^13,14) and the more recently developed hexagonal-close-packed (hcp) systems.¹⁵⁾ These HEAs are mainly produced by the ingot metallurgy (IM) method such as arc melting or casting, and it has been confirmed that the as-cast structure has microstructure separation or non-uniform microstructure.^16,17) To overcome the limitations of the IM method, the powder metallurgy (PM) method is used in this study. Compared to the conventional IM method, the PM method has the advantages of being superior in material efficiency, requiring almost no processing after molding, and being able to be sintered in a form close to the final product. Therefore, it can be used in applications where accuracy and precision are paramount. It was also reported that the mechanical properties and chemical properties of HEAs prepared by the PM method were different from those prepared by the IM method.^18,19)

In this study, CrFeCoNiSi based HEAs were used for increased hardness. It has been reported that HEAs containing Si show improved hardness^20,21) and corrosion resistance.²²⁾ As a sample, an alloy powder was prepared by mechanical alloying (MA) using a planetary ball mill, and sintered by spark plasma sintering (SPS). To compare HEAs characteristics by adding Si, sintered samples were prepared with three compositions of (CrFeCoNi)₉₅Si₅, (CrFeCoNi)₉₀Si₁₀, (CrFeCoNi)₈₅Si₁₅, and CrFeCoNiMn, and their characteristics were compared.

2. Experimental Method

2.1 Alloy design

Parameters such as mixed entropy (ΔS_mix), mixed enthalpy (ΔH_mix), atomic radius ratio (δ), omega parameter (Ω), and valence electron concentration (VEC) value were calculated to determine their effect on the formation of a solid solution and type of crystal structure. Each value was obtained from eqs. (1) to (5),²³⁾ where x_i is the composition of element i, ΔH_ij is the mixed enthalpy of elements i and j, r_i is the atomic radius of element i and T_m,i is the melting point of element i. VEC_i is the valence electron concentration of element i, it is empirically known that fcc single-phase occurs when VEC > 8.0, bcc single-phase occurs when VEC ≤ 6.78, and bcc + fcc coexists when 6.78 < VEC < 8.0.²⁴⁾ For ΔH_ij, r_i refer to Ref. 25), and VEC_i to Ref. 26).   

\begin{equation} \Delta S_{\textit{mix}} = -R\sum\nolimits_{i=1}^{n} x_{i}\ln x_{i} \end{equation} (1)

  

\begin{equation} \Delta H_{\textit{mix}} = 4\sum\nolimits_{j\neq i}^{n} \sum\nolimits_{i=1}^{n} x_{j}x_{i}\Delta H_{ij} \end{equation} (2)

  

\begin{equation} \delta = \sqrt{\sum\nolimits_{i=1}^{n} x_{i}\left[1 - \frac{r_{i}}{\bar r} \right]}\times 100 \quad \bar{r} = \sum\nolimits_{i=1}^{n} x_{i}r_{i} \end{equation} (3)

  

\begin{equation} \varOmega = \frac{T_{m}\times \Delta S_{\textit{mix}}}{|\Delta H_{\textit{mix}}|} \quad T_{m} = \sum\nolimits_{i=1}^{n} x_{i} \times T_{m,i} \end{equation} (4)

  

\begin{equation} \mathit{VEC} = \sum\nolimits_{i=1}^{n} x_{i} \times \mathit{VEC}_{i} \end{equation} (5)

2.2 Sample

2.2.1 Powder

As starting materials, Ni powder (particle size: 53 µm), Si powder (137 µm) (The Nilaco Co., Ltd., Tokyo, Japan), Cr powder (45 µm), Co powder (45 µm) and Fe powder (45 µm) (Fuji Film Wako Pure Chemical Co., Ltd., Osaka, Japan) were used.

2.2.2 Mechanical alloying

The powders were subjected to MA. First, Cr steel balls were placed in a stainless steel ball mill container (volume: 250 mL). Then, hand milled powder was placed in the container and 75 mL of heptane was added as solvent. After adding heptane and sealing the container, the inside of the container was filled with argon (Ar) atmosphere in order to prevent the oxidation of the powder during MA treatment. The evacuation of air inside the container and filling of Ar gas was completed in approximately 5 min. The container was then set in a planetary ball mill device (pulverisette 6, Fritsch Japan Co., Ltd.). The balancer scale was adjusted to the total weight of the ball, sample, and container, before initiating the comminution. In this method, the milling time was 22.5 h, revolution was 300 rpm, and ball to powder ratio was 10:1.

2.2.3 SPS processing

The mechanically-alloyed powder was placed in a graphite die (ϕ20 mm) into which a graphite punch was subsequently inserted. The powder was sintered using an SPS apparatus (SPS-1020, Sumitomo Coal Mining Co., Tokyo, Japan) at a pressure of 50 MPa and a vacuum pressure of 10⁻² Pa by passing a large pulse current. Sintering was performed at a heating rate of 80 Kmin⁻¹ and at a sintering temperature of 1073 K for 10 min. When the sintering process was completed, the sample was cooled in the furnace. After sintering, the sample surface was ground to #2000 with wet emery paper, polished using Al₂O₃ powder (1.0 µm in diameter), ultrasonically degreased, and dried in the air.

2.3 Analysis

2.3.1 X-ray diffraction

The phases of the mechanically-alloyed powder and the surface of the sintered sample were characterized using an X-ray diffractometer (XRD; RINT-250V, Rigaku Co., Ltd., Tokyo, Japan). The XRD conditions were as follows: the sample is fixed to a holder, CuKα was used as the X-ray source, tube voltage was 40 kV, tube current was 300 mA, scan speed was 10 deg.min⁻¹, scan step was 0.02 deg, and scan range was 20 to 90 deg. Further, the lattice constant of each composition was calculated from the Bragg’s conditional expression from the detected diffraction line. Bragg’s conditional expression was expressed as in (6).   

\begin{equation} 2d_{\textit{hkl}}\sin \theta = \lambda \end{equation} (6)

Further, the formula of the cubic spacing is expressed as:   

\begin{equation} \frac{1}{d_{\textit{hkl}}} = \frac{\sqrt{(h^{2} + k^{2} + l^{2})}}{a} \end{equation} (7)

2.3.2 Elemental analysis

Scanning electron microscope (SEM) (JSM-6060LV, JEOL Ltd., Tokyo, Japan) and with an attached energy dispersive X-ray spectroscope (EDX) was used for observing the structure and analyzing the elemental composition in the alloyed powder and the sintered sample. The observation was run at an acceleration voltage of 15 kV. In addition to observing the structure, the attached EDX detected the characteristic X-rays of Cr, Fe, Co, Ni, Si and Mn elements and performed surface analysis to investigate the homogeneity of the elements in the sample. Additionally, to investigate the composition of the sintered samples, elemental analysis was conducted out using X-ray fluorescence analysis (XRF) (JSX-1000S, JEOL Ltd., Tokyo, Japan) to quantify the composition of the sintered samples.

2.3.3 Density measurement

The density of the sintered samples was measured by the Archimedes method. The specific procedure is to first measure the mass (m) of the sintered sample in the atmosphere. Next, the mass (m′) of the sintered sample in water was measured, and the density (ρ) was determined by (8). Here, ρ_w indicates the density of pure water used at the time of the measurement.   

\begin{equation} \rho = \frac{m}{m- m'} \rho_{w} \end{equation} (8)

The theoretical density ρ_mix of each composition was calculated by (9) based on the composition obtained by elemental analysis.   

\begin{equation} \rho_{\textit{mix}} = \frac{\displaystyle\sum x_{i}A_{i}}{\displaystyle\sum x_{i}A_{i}/\rho_{i}} \end{equation} (9)

x_i is the atomic concentration of element i, A_i is the atomic weight of the component i, and ρ_i is the density of the element i.

2.3.4 Hardness test

For measuring the hardness of the sintered samples, a hardness test was conducted using microVickers hardness tester (PMT-X7A, Matsuzawa Co., Ltd., Akita, Japan). The test conditions included a load of 0.98 N (100 gf) and a load time of 15 s. The hardness value was obtained by averaging three hardness values measured five times.

2.3.5 Corrosion test

A corrosion test was conducted for investigating the differences in corrosion resistance among the sintered samples. The sample was spot-welded with a SUS304 lead wire and covered with Teflon tape with a ϕ 6 mm hole to limit the wetted area of the sample. A 3.5 mass% NaCl aqueous solution was used as the test solution. Ag/AgCl was used as the reference electrode and Pt was used for the counter electrode. The measurement was performed by degassing with nitrogen gas for 30 min to remove dissolved oxygen in the solution. After immersing the sample, and voltage was applied by a potentiostat (HA-501G, Hokuto Denko Co., Tokyo, Japan). Polarization test was performed potentiodynamically from −1.0 to +1.5 V_Ag/AgCl, and the polarization curves were recorded at a sweep speed of 1.6 mV/s. On the basis of these polarization curves, the potential corresponding to a current of 10 Am⁻² was selected as the pitting potential. In addition, the corrosion marks after the polarization test were examined with a profilometer (SE-300, Kosaka Laboratory Ltd., Osaka, Japan) and optical microscope and elemental analyzed by SEM-EDX.

3. Results and Discussion

3.1 Alloy design

Table 1 shows the results for each parameter. Additionally, if δ ≤ 6.6, the difference in atomic radius ratio of the constituent elements is small, and if Ω ≥ 1.1, ΔS_mix is as large as possible, and the absolute value of ΔH_mix is close to zero. Previous reports show a high possibility of occurrence of such a solid solution.²³⁾ It has also been reported that if VEC > 8.0, fcc single-phase systems are formed. Similarly, if VEC ≤ 6.87 and 6.87 ≤ VEC < 8.0, bcc-single phase, fcc + bcc two-phases are formed, respectively. Considering the results of this calculation under these conditions, the following can be concluded.

(CrFeCoNi)₉₅Si₅

• •    The possibility of forming a solid solution is higher than Ω ≥ 1.1.
• •    Since VEC > 8.0, it becomes fcc single phase.

(CrFeCoNi)₉₀Si₁₀ and (CrFeCoNi)₈₅Si₁₅

• •    The possibility of forming a solid solution is higher than δ ≤ 6.6 and Ω ≥ 1.1.
• •    Since 6.87 ≤ VEC < 8.0, the two phases are fcc + bcc.

Table 1 ΔS_mix, ΔH_mix, δ, Ω and VEC for (CrFeCoNi)_100−xSi_x.

3.2 MA powder

Figure 1 shows the result of XRD to identify the phase formed in MA powder. From Fig. 1, the diffraction line of each element of the starting material was detected during hand milling; however, in the MA powder, only the diffraction line of the fcc phase in (CrFeCoNi)₉₅Si₅, (CrFeCoNi)₉₀Si₁₀, and (CrFeCoNi)₈₅Si₁₅ were detected, suggesting that a single-phase solid solution in (CrFeCoNi)₉₅Si₅, (CrFeCoNi)₉₀Si₁₀, and (CrFeCoNi)₈₅Si₁₅ was formed by MA. Additionally, broadening of the diffraction lines of MA powder was observed. The following three factors are responsible for the broadening of diffraction lines.^27–29) First is the miniaturization of the crystal grain size. Scherrer’s formula is one of the methods to calculate the crystallite size. Scherrer’s equation is expressed as (10).   

\begin{equation} L = \frac{K \lambda}{(\beta \cos \theta)} \end{equation} (10)

Where L is the crystallite size, K is the Scherrer constant, λ is the wavelength of the X-ray source, β is the full width at half maximum, and θ is the diffraction angle. From the (10), it is considered that the half-width β increases as the crystallite size L, i.e., the crystal grain size decreases, and the diffraction line becomes broad. Second is the introduction of lattice strain. HEAs are characterized by the formation of a single-phase solid solution even when five or more elements are present. Therefore, it is believed that, compared to the case where only the pure element is used, the lattice is distorted by the amount of the solid solution of the other elements and that the lattice constant thereby increases. Third is the decrease in crystallinity. It is considered that the MA treatment formed an amorphous region in the powder and did not satisfy Bragg’s conditional expression, which caused broadening of the diffraction line.

Fig. 1

Result of X-ray diffraction test for powders.

3.3 Sintered sample

3.3.1 XRD

Figure 2 shows the XRD results for the sintered samples. From Fig. 2, the fcc phase and slight Ni silicide were identified in (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀. In (CrFeCoNi)₈₅Si₁₅, the fcc phase and silicides of Ni, Cr, and Co were detected.

Fig. 2

Result of X-ray diffraction test for CrFeCoNiMn and (CrFeCoNi)_100−xSi_x sintered samples.

Table 2 shows the results of calculating the lattice parameter. From Table 2, it was found that the lattice constant decreased as the Si content increased. Moreover, the lattice constant of CrFeCoNi–Si was smaller than that of CrFeCoNiMn. This is thought to be due to the difference in atomic radius between Si and Mn. Since the atomic radii of Si and Mn are 1.17 and 1.27 Å, respectively, it can be considered that the lattice constant becomes smaller by replacing Mn with Si.

Table 2 Lattice parameters for CrFeCoNiMn and (CrFeCoNi)_100−xSi_x sintered samples.

3.3.2 Elemental analysis

Figure 3 shows result of EDX element mapping for (CrFeCoNi)₈₅Si₁₅ and CrFeCoNiMn sintered samples. From Fig. 3, (CrFeCoNi)₈₅Si₁₅ did not confirm the enrichment of Si confirmed as a compound. This result shows that the silicide detected by XRD is very fine and uniformly dispersed. Figure 4 shows a result of XRF elemental analysis for CrFeCoNi–Si sintered samples. It is evident from Fig. 4, that for all three samples, the composition was roughly close to the target.

Fig. 3

Result of EDX elemental mapping CrFeCoNiMn and (CrFeCoNi)₈₅Si₁₅ sintered samples.

Fig. 4

Elemental analysis for (CrFeCoNi)_100-xSi_x sintered samples by XRF.

3.3.3 Density measurement

The density of the sintered sample was measured using the Archimedes method. The results are shown in Table 3. From Table 3, the relative densities of (CrFeCoNi)₉₀Si₁₀ and (CrFeCoNi)₈₅Si₁₅ exceeded 1.0. This is possibly to be because the densities of the formed silicides of Co and Ni exceeded the ideal densities of (CrFeCoNi)₉₀Si₁₀ and (CrFeCoNi)₈₅Si₁₅, Ni₃₁Si₁₂ is 7.56 gcm⁻³ and Co₂Si is 7.51 gcm⁻³.

Table 3 Density for (CrFeCoNi)_100−xSi_x sintered samples.

3.3.4 Hardness test

Figure 5 shows the results of the hardness test for CrFeCoNi–Si and CrFeCoNiMn sintered samples. The hardness of (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀ is approximately 550 HV, whereas that of (CrFeCoNi)₈₅Si₁₅ is approximately 700 HV. This difference in hardness is mainly because of silicide formation. Si is an element that very easily forms silicide; silicide was detected as shown in Fig. 2. It is observed that the formation of such hard and brittle silicide phases improves the hardness as compared with the conventional CrFeCoNiMn HEAs. The (CrFeCoNi)₈₅Si₁₅, in which a large amount of silicide is formed has a significantly improved hardness as compared with (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀. As shown in Fig. 3, enrichment of Si was not observed for (CrFeCoNi)₈₅Si₁₅ sample. This result suggests that precipitates of fine-grained silicides were dispersed in the sample. Precipitation hardening by compounds and σ-phase in HEAs was also reported.^30,31) This study is the first report on the precipitation hardening for CrFeCoNiSi-based HEAs.

Fig. 5

Hardness test for CrFeCoNiMn and (CrFeCoNi)_100−xSi_x sintered samples.

3.3.5 Corrosion test

Figure 6 shows the polarization curves when the corrosion test was performed on the sintered samples of CrFeCoNi–Si and CrFeCoNiMn. The potential at which the current density was 10 Am⁻² was defined as the pitting potential, and the corrosion resistances of the alloys were compared. Figure 7 shows pitting potential of each alloy. It is evident from Fig. 7 that the pitting potential E_pit of CrFeCoNi–Si was significantly higher than that of CrFeCoNiMn, and the corrosion resistance of CrFeCoNi–Si improved. Figure 8 shows the surface profile of corrosion marks after corrosion test. As shown in Fig. 8, localized corrosion of the base material such as pitting corrosion was observed in (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀, while it was not observed in (CrFeCoNi)₈₅Si₁₅. From Figs. 6 and 8, it can be seen that pitting corrosion occurred in (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀, but not in (CrFeCoNi)₈₅Si₁₅.

Fig. 6

Polarization curve for CrFeCoNiMn and (CrFeCoNi)_100−xSi_x sintered samples.

Fig. 7

Result of corrosion test for CrFeCoNiMn and (CrFeCoNi)_100−xSi_x sintered samples.

Fig. 8

Surface profile of corrosion marks after corrosion test.

Figure 9 shows the microstructure after the corrosion test. As shown in Fig. 9, pitting corrosion was observed in (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀ in addition to the original pores, and the voids were expanded by corrosion. The structure of (CrFeCoNi)₈₅Si₁₅ did not show any expansion of voids compared to that of (CrFeCoNi)₉₀Si₁₀, and the gray area seemed to expand from the voids. The reasons for this phenomenon were discussed as follows. The difference among (CrFeCoNi)₉₅Si₅, (CrFeCoNi)₉₀Si₁₀, and (CrFeCoNi)₈₅Si₁₅ is whether a passive film is formed or not. The reason for this was discussed as follows. In stainless steels, it has been reported that the presence of Cr and Si makes the passive film stronger,³²⁾ and adding Si to HEA has been reported to increase corrosion resistance.³³⁾ In fact, an increase in pitting corrosion potential was observed in (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀. However, in (CrFeCoNi)₈₅Si₁₅, compounds of Cr and Si precipitated in the alloy and the decrease in their solid solution concentration caused a decrease in the pitting corrosion potential.

Fig. 9

Microstructure at the boundary and inner of corrosion marks after corrosion test.

Furthermore, for the corrosion mark of (CrFeCoNi)₈₅Si₁₅, we believe that the ionization of the material by pitting corrosion and the expansion of the voids did not occur because of the large amount of silicide formed. The change to gray color was also considered to be caused by the reaction of the compounds with oxygen starting from the surface voids, without ionization due to the formation of the compounds. In order to verify this consideration, elemental analysis was carried out on the corrosion marks of (CrFeCoNi)₈₅Si₁₅ using SEM-EDX. The results of the elemental analysis are shown in Fig. 10 and Table 4. From Table 4, Cr, Si, and O were detected more intensively in the white area of the corrosion mark than in other areas. The reason for color change to the gray in the area covered with the Teflon tape is also considered to be the progress of change to gray color due to the diffusion of oxygen from the gray area of the corrosion mark.

Fig. 10

Result of EDX elemental mapping at the corrosion mark of the (CrFCoNi)₈₅Si₁₅ after corrosion test.

Table 4 Elemental concentration at the corrosion mark of (CrFCoNi)₈₅Si₁₅ sample after corrosion test (at%).

4. Conclusions

A CoCrFeMnSi-based high entropy alloy was newly prepared by PM using MA and SPS, and its characteristics were evaluated. The results obtained were as follows.

1. (1)    The result of calculating the parameters δ, Ω, and VEC values with (CrFeCoNi)₉₅Si₅, (CrFeCoNi)₉₀Si₁₀, and (CrFeCoNi)₈₅Si₁₅ by alloy design, can be summarized as follows.(CrFeCoNi)₉₅Si₅:
   • •    The possibility of forming a solid solution is higher than Ω ≧ 1.1.
   • •    Since VEC > 8.0, fcc is a single-phase system.(CrFeCoNi)₉₀Si₁₀ and (CrFeCoNi)₈₅Si₁₅:
   • •    The possibility of forming a solid solution is higher than δ ≤ 6.6 and Ω ≥ 1.1.
   • •    From 6.87 ≤ VEC < 8.0, there are two phases of fcc + bcc.
2. (2)    From the XRD test of the mechanically-alloyed powder, the fcc phase in (CrFeCoNi)₉₅Si₅, (CrFeCoNi)₉₀Si₁₀, and (CrFeCoNi)₈₅Si₁₅ were detected, suggesting that a single-phase solid solution was formed by MA.
3. (3)    From the XRD test of the HEAs sintered sample, fcc phase and slight Ni silicide were identified in (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀, and fcc phase and Ni, Co, and Cr silicides were identified in (CrFeCoNi)₈₅Si₁₅. It was also confirmed that the lattice constant decreased with increase in the concentration of Si composition.
4. (4)    From the microstructure observation and elemental analysis of (CrFeCoNi)₈₅Si₁₅ and CrFeCoNiMn, no difference was found between the two elemental distributions. In addition, no compound phase could be confirmed in (CrFeCoNi)₈₅Si₁₅.
5. (5)    In the density measurement of the sintered samples, the density of (CrFeCoNi)₉₀Si₁₀ and (CrFeCoNi)₈₅Si₁₅ exceeded the theoretical density.
6. (6)    In the hardness test of the sintered samples, the hardness of (CrFeCoNi)₉₅Si₅ and (CrFeCoNi)₉₀Si₁₀ was approximately 550 HV, whereas that of (CrFeCoNi)₈₅Si₁₅ was approximately 700 HV.
7. (7)    In the corrosion test of the sintered samples, the pitting potential was high in the order of (CrFeCoNi)₉₀Si₁₀, (CrFeCoNi)₉₅Si₅, (CrFeCoNi)₈₅Si₁₅. In addition, the pitting potentials of the CrFeCoNiSi HEAs were all higher than that of CrFeCoNiMn.

It is a conventional high-entropy alloy in which the Mn in the CrFeCoNiMn-based alloy is replaced by Si. As a result, improvements in hardness and corrosion resistance were observed.

Acknowledgments

This research was funded by a grant for the high-entropy alloys research group of ORDIST, Kansai University. The authors wish to express their gratitude to Prof. T. Maruyama and Prof. Y. Hoshiyama, Kansai University, for their valuable comments and direction.

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