Selecting the Appropriate Carbon Source in the Synthesis of SiC Nano-Powders Using an Optimized Fuzzy Model

In this study, different sources of carbon in the synthesis of silicon carbide were evaluated using a multi-attribute group decision-making fuzzy model including IF-MAGDM method. In this model, the aim was to find the carbon precursor which has the minimum price, highest carbon content, good water solubility, lowest synthesis temperature, and the optimum crystallite size. Based on the results obtained from mentioned method, sugar was the best candidate. Therefore, sugar was selected as the efficient carbon source in the synthesis of SiC, also tetraethyl ortho-silicate was used as the source of silicon. The synthesis temperature was 800 °C, which is a relatively low temperature for the synthesis of crystalline SiC. To optimize the carbothermal process, microwave heating and just 15 min were enough to form crystalline SiC nanostructures. Various analyses were used for efficient characterize of the products. TGA-DTA were used to investigate thermal behaviours of the samples. XRD results showed that the SiC powder prepared by microwave heating was fully crystalline phases. FE-SEM and TEM analyses showed the nanometric nature in average particle size distributions of products with a flake-like morphology.


Introduction
Structural properties of SiC ceramics and coatings largely depend on the characteristic of the starting raw materials and precursors. Generally, the production of high-performance SiC ceramics requires ultrafine and pure SiC powders [1]. To produce such an ultrafine and pure SiC powder, several alternative techniques, such as the sol-gel [2,3], thermal decomposition of silane compound [4,5], and chemical vapor deposition procedures [6], have been developed to eliminate the limitation of conventional Acheson process. However, the powders prepared by these methods have a lot of costs, due to expensive raw materials, low production rate of the methods, and their demand for expensive equipment. The sol-gel process has been established as a novel method for synthesis of nano-powders with several outstanding features including, high purity [7], high chemical activity, enhancing powder sinterability, and the possibility for materials mixing at the molecular scale. A sol-gel process using metal alkoxides as precursors has been widely used for the synthesis of fine powders with homogeneous size distribution and controlled shape [8].
Tetraethyl ortho-silicate (TEOS) is an interesting precursor as a source of Si for the production of SiC powder using the sol-gel method. Aelion et al. were among the first researchers to evaluate the kinetics of hydrolysis and condensation of TEOS. The rate and extent of the hydrolysis reaction were found to be influenced firstly by the type and concentration of the acid or base catalyst while the temperature and solvent were of secondary importance [9]. Although their groups investigate TEOS, there are only a few reports on the details of SiC powder preparation by a combination of sol-gel and carbothermal reduction route [10].
Selection of appropriate starting materials, including silicon and carbon sources as initial precursors, is vital for the successful synthesis of ultrafine and pure SiC by sol-gel method, especially the difference in nature of carbon that is provided from the diverse sources can have an important effect on the characteristics of as-prepared SiC nano-particles. As a result, optimum selection of the carbon sources plays an important role in the production process of SiC nano-powders by sol-gel technique [11].
TEOS is widely used as a silicon source for synthesizing nanosized SiC by sol-gel route while the preparation of SiC using the sol-gel process largely depends on the appropriate selection of the carbon source as a precursor. There are different factors to be considered in the selection of a suitable carbon source including; carbon yield, reactivity, cost, and nature of impurities. While various materials including polyphenylenes oxide-polystyrene [12], carbon black [13], phenolic resol [14], sucrose [15], and sugar [16] can be used as carbon source.
Traditionally, when selecting a material whose features are identified, experts usually apply previous experiences to find an optimized method of synthesis [17]. However, these methods do not cover all of the options for the synthesis process. This blind spot can be addressed by adopting a multi-factor decision-making model (MADM). On the other hand, the MADM approach can be used in the evaluation and selection problems, in which decisions include a set of performance features [18]. Previous study showed that while some researchers have proposed MADM methods for selecting materials to manufacture systems, none of them have chosen the MADM selection method [19].
Fuzzy set theory is useful when the purchase situation is full of uncertainty and imprecision because of the subjectivity of human judgments [20,21]. The following studies used fuzzy set theory to handle uncertainty in the material selection problem [22,23]. Rao et al. considering quantitative and qualitative characteristics, developed a MADM method to solve the material selection problem for an engineering design [24]. This method considers the objective weights of the importance of the traits as well as the mental preferences of the decision-maker to decide on the integrated weights of the importance of the traits. Also, they proposed a fuzzy conversion scale-rated value judgment to express the qualitative selection feature (e.g., little value not available). Choosing the right phase change material (PCM) leads to efficient application of latent thermal energy storage systems. Rathod et al. use two MADM methods to solve the PCM selection problem [25]. Two methods (e.g., TOPSIS and fuzzy TOPSIS) provide linguistic variables and fuzzy function [26,27]. Briefly, a review of the literature shows that although some researchers have proposed MADM methods in the selection of materials for manufacturing systems, none of them have considered the selection of precursors using MADM methods [28].
In this paper, for the first time, an intuitionistic fuzzy multiattribute group decision-making (IF-MAGDM) model was chosen to solve the problem of selection the best source of carbon for the synthesis of SiC. Then, different analyses including thermo-gravimetric analysis (TGA), differential thermal analysis (DTA), X-ray diffraction (XRD), field emission scanning electron microscopy (FE-SEM) and transmit electron microscopy (TEM) were used to characterize the final products.

Proposed the Efficient Model
Atanassov extended traditional fuzzy set to IFS by adding an uncertainty or hesitation degree [29,30]. The flow diagram of the Atanassov's intuitive fuzzy decision method is shown in Fig. 1 [32]. Also, IF-MAGDM method has 11 steps, which should be done sequentially as follows (details of the method is included in Appendix): Step 1: Identifying the attributes to be required and candidates have been identified based on features that are needed.
Step 2: Fuzzy decision matrix, decision-makers create the intuition.
Step 3: Using an entropy weight method to determine the weight of each expert from the decision matrix.
Step 4: According to the expert opinion, an IF decision matrix is collected. After gaining weight values for experts, the assessment values that are provided by different experts can be obtained from the IFWG operator.
Step 5: Setting the entropy values of the traits, all traits may not be assumed to be of equal importance. W represents a set of degrees of importance. To obtain x, an intuitive fuzzy entropy is used [33].
Step 6: Building an IF weighting decision matrix. The decision-weight matrix of the IF weight is obtained based on the importance of characteristics.
Step 7: Determine intuitionistic fuzzy positive-ideal solutions (PIS) and negative-ideal solutions (NIS) as r þ j j and r − j in Appendix.
Step 8: Defining an IF positive-ideal separation matrix (D + ) and negative-ideal separation matrix (D − ) fined and calculated the grey relational coefficient of each candidate from the PIS and the grey relational coefficient of each candidate from the NIS.
Step 9: Calculation of the Gray relational coefficient for each of the candidates.
Step 10: Calculating the Total Collective Indicator (CI) based on the Gray relational coefficient.
Step 11: Determining the priority of candidates Þby the proposed If any of the candidates has the highest CI i value, then, it is the most important candidate.

Materials and Methods
For silicon carbide synthesis, tetraethyl ortho-silicate (TEOS) was used as the source of silicon. Hydrochloride acid (HCl) also was used to keep the pH of the solution at 4. The TEOS, carbon source, ethanol, and DI water were mixed to prepare the determined solution. This solution was stirred (speed of 250 rpm) at the temperature of 30°C for 4 h. The resultant gel was kept in the hot air oven at 60°C for 4 h. For the carbothermal process, two different methods were performed. The first one includes microwave heating at a power of 1200 W and a frequency of 2.45 GHz for 10 and 15 min. The second method was annealing in a furnace at the temperature of 700-800°C in flowing argon for 3 h. The flow diagram of the modeling and synthesis procedure of nanocrystalline SiC powders by the sol-gel path is shown in Fig. 2. Five carbon sources are provided for the sol-gel process, which is denoted as CS 1 , CS 2 , CS 3 , CS 4 , and CS 5 . The description of these carbon source candidates is given in Table 1.

Material Characterization
TGA (SCINCO thermal gravimeter S-100), and DTA (QMS403C, Netzsch) were turned from room temperature to 800°C with a heating rate of 20°C min −1 under an argon atmosphere. The grain size of the reaction products was measured from XRD patterns (Philips X-pert) by Cu-kα radiation. For the evaluation of agglomerates size and qualitative analysis of synthesized powders, TEM (HITACHI H-7650 electron microscope) and FE-SEM (Philips xl30) equipped with EDX analyzer were used. Investigation of phase transformations at corresponded temperatures was carried out using thermal analysis (TG/DTA). In this case, samples were heated at a rate of 20°C/min.

Result of IF-MAGDM Model
Grain size [36,37], cost [38], carbon content [39,40], water solubility [41], and temperature [11,42] of SiC formation can be attributed to the difference like carbon obtained from the various sources. Therefore, the attributes for the selection of the carbon source were obtained as cost (A 1 ) of carbon source that has an important role in the sol-gel process of solubility SiC [43]. An interesting strategy for efficient sol-gel processing of SiC is using a small amount of carbon source, so having  CS 5 phenolic resol [14] high carbon content (A 2 ) is another important attribute. The system needs less water to reach optimum when the carbon source has a high-water solubility (A 3 ). Increasing the water content leads to a corresponding increase in hydrolysis rate and the time to achieve stable sol is decreased at a higher water/precursor rate. Therefore, the rate of polymerization reaction may be impaired under such conditions. The primary advantage of the sol-gel process is that nano-powders can be synthesized at considerably lower temperatures plus time and energy saving [44]. Formation of SiC at low temperature (A 4 ) is necessary to obtain SiC nano-powders because at high temperature, the growth rate of nano-crystals is highly accelerated and large size of SiC powders is obtained. The most important properties of the as-synthesis SiC is the optimization of the grain size (A 5 ) [22,45]. The increased contact area between carbon source and silicon should make the reaction suitable for decreasing the grain size of SiC. In this study, the IF-MAGDM model was employed to evaluate and select the best carbon source to synthesize SiC by a sol-gel process. At first, the attributes and carbon sources as candidates identified, a committee of three professional experts is formed to conduct the evaluation and to select the most suitable carbon source. The IF-value ratings of these five-carbon source candidates by the linguistic variables and their respective IF-value are evaluated by the experts concerning the attributes. In the next step, the IFperformance matrix is formed for each of the three experts. The aggregated IF-decision matrix is constructed based on the opinions of experts by step 4. The obtained results are provided in Table 2. The weighted aggregated intuitionistic fuzzy set decision matrix (Table 3) is calculated by step 5. The positive-ideal and negative-ideal solutions are computed according to the concept of IF values by step 7 that are given in Table 3. The positive-ideal separation matrix is computed by step 8 (Eq. 12 in Appendix) and the negative-ideal separation matrix is computed by step 8 (Eq. 13 in Appendix). The grey relational coefficient of each candidate from PIS and NIS is calculated by step 9. The degree of the grey relational coefficient of each candidate is computed by.
Step 10 and the results are listed in Table 4. Proposed ranking index CI i for all carbon, source candidates are calculated by step 11 and are given in Table 4. The larger CI i , means the higher the priority of the candidate. Therefore, the final ranking based on the IF-MAGDM model is as follows:

Sensitive Analysis
In this section, each identification coefficient value ρ is employed to see whether each identification coefficient value has an impact on the results of the ranking order of the candidates using the proposed model. These variant identification coefficient values are employed to investigate the proposed model. The ranking order of the candidates and the values of the ranking index CI i based on variant ρ are depicted in Fig. 3. The results illustrate the variation of the ranking index value of each candidate using various identification coefficient values and also that the ranking orders of the five candidates are the same despite the changes of a resolving coefficient value from ρ = 0.1 to ρ = 1. Hence, this article can confirm that the results of the ranking orders of all candidates using the proposed model are reliable. These results can help decisionmakers to evaluate and select a suitable candidate. Moreover, the proposed model figures out that the gaps between CI i values of various candidates become larger when the resolving coefficient value is reduced from 0.1 to 1. Through the gap between each CI i value of each candidate, the decision-maker can distinguish the differences among the candidates more efficiently.
As noted above, it should be noted that the proposed model can simultaneously achieve the ideal candidate gap, ranking candidates, and priority points for improving each candidate. Regarding the sensitivity analysis, it has been shown that this model can measure any value of the identification coefficient to evaluate the distance between the different CI values of the candidates, which can help the decision-makers. In this case, the entropy method for direct use of information from the intuitional fuzzy decision-making matrix is used as a reasonable way for the features and experts in weight as well as for robust decision-making.
According to the results of IF-MAGDM model and sensitive analysis, it can be considering the attributes the best carbon source is sugar. Wou et al. used carbon black as a carbon source for the synthesis of SiC [46]. In another study, Raman et al. used the polyphenylene oxide-polystyrene as a carbon source and TEOS mixture was stirred well for about 2 h. The sol-containing polymer was then allowed to gel at room temperature and dried at 60°C to obtain polymer incorporated with sol-gel derived silica and the SiC precursor prepared as above was carbonized at 1000 C. The carbonized product was further heated at 1400 C under an argon atmosphere and the grain was distributed in macro/ micro sized nature.
Ping Lu et al. used polyphenylene oxide-polystyrene raw materials to make nano-porous SiC wires [47]. Their results showed that the nanowires prepared in different shapes are straight, bead and bamboo-like and that most of the wires are of the straight type with a diameter of about 105 nm. They have found that all of these nanowires are made of coarse particles with nanosized distributions. On the other hand, they stated that this range has different dimensions and forms of pyrolytic behaviors. Shunlong Pan and colleagues used carbon black precursors to synthesize β-SiC nanocrystalline powder particles [13]. These nanoparticles were synthesized by spraying a mixture of sodium silicate and carbon black. Completion of the carbothermal reaction in this study was by storing the spray and dried material at temperatures of 1200 to 1700°C. The observations showed that the appropriate temperature for the process in which the BET parameter has the highest value is 1550°C for 2 h.
Najafi et al. also fabricated β-SiC powder nanoparticles using phenolic resol carbon precursor. The researchers also studied different temperatures to complete the synthesis process, and their results showed that the germination of β-SiC particles ends at 1400°C. The nanoparticles created in this study were less than 100 nm in size and had non-uniform morphology. However, observations made by the TEM microscope showed that these particles are mostly round and their size is between 30 and 50 nm. The researchers did not study other properties for the particles present, and since the resulting particles are very small, they may agglomerate and make the properties uncontrollable [14]. Figure 4 illustrates the thermal analysis (DTA-TG) of the gel samples. The first stage of weight loss occurs at 160°C which is corresponded to endothermic dehydration and removal of structural water, which is considered to be a physical process. This endothermic reaction is followed by two exothermic reactions at 370 and 731°C that leads to SiC formation. According to DTA-TG analysis, the reactions related to the formation of SiC nanostructures was happened at 731°C so we have chosen 700 and 800°C for the annealing temperature of the dried gel [48,49]. Figure 5 illustrates the XRD pattens of an annealed sample at both of these temperatures. In the next set of experiments, the dried gel was put on an alumina crucible and exposed to microwave heating for 10 and 15 min for the carbothermal synthesis of SiC (Fig. 6). According to the XRD patterns, the products were indexed in cubic crystalline phase behaviour with JCPD No. 29-1129 [50].

Efficient Features of SiC Nanoparticles
The best results are obtained from sugar precursors, so FE-SEM and MAP element analysis were carried out for this sample. Figure 7 shows the FE-SEM image of the microwave furnace at 15 min. The SiC is formed in the entire sample surface and there are no other impurities. The size of the agglomerates is quite larger than the size of the nanopowder measured by XRD analysis. As it is expected and reported elsewhere [51], it is because XRD measures the crystallite size but FE-SEM shows the agglomerate size [11,52]. Also, TEM image showed a spherical morphology and CI i by the proposed Intuitionistic fuzzymulti-attribute group decision-making model  [53]. The carbonization temperature is taken into consideration with a high priority. On the other hand, using microwave synthesis we reduced the time and energy needed for the synthesis of SiC. The results showed that this method can be used in a short period and also cost without needing any specific type of atmosphere, as is reported in some other literature. Using this method can save time and money and can be a big step for materials science.
The choice of the precursor is the main process in the synthesis of nanomaterials. The impact of choosing the right precursors can be critical, as it affects the cost of processes and operations and the quality of the product. It seems very difficult to identify the best precursor among them with some contradictory characteristics.
This study aims to improve the quality of group decision-making for material selection. Thus, a new IF-MAGDM model based on a combination of TOPSIS and GRA concepts in an intuitive fuzzy environment has been developed in the selection of precursors in the synthesis of SiC nanopowders via the sol-gel method.
A practical example for evaluating precursors is shown to examine the application of the proposed model. This study shows that the proposed model can simultaneously capture the gap between the ideal candidate and the other candidates, the order of ranking of the candidates and the priority of improving each candidate's weaknesses. Also, the results obtained using IF-MAGDM show that there is a good correlation with the results obtained from the opinions of professional experts, which specifically solves the global application of this model while this type of material and problems Solves precursor selection. Although the proposed model presented in this paper is illustrated by the issue of pre-selection, it can also be applied to problems such as material selection, project selection, and other areas of management decision-making problems. Our future development is to consider the IF-MAGDM model for the selection of other precursors in the synthesis of SiC nanopowders by the sol-gel method.

Conclusion
I) The optimum carbon source for the synthesis of silicon carbide nanoparticles was selected using an intuitional fuzzy decision-making matrix. II) Some SiC nanostructure samples were prepared experimentally using microwave and conventional mode of heating. III) The optimum condition for annealing was calculated based on thermal analysis (TG/DTA) and the results showed that 800°C is the minimum temperature required for the conventional mode of heating in an argon atmosphere. IV) By using microwave route, crystallization of SiC nanoparticles were formed in a short period of 15 min without any special type of atmosphere. V) It is believed that the fuzzy model and optimization studies developed in the present study can be considered as a novel protocol for their application in other nanostructures samples.
Constructing a weighted aggregated IF-decision matrix. The weighted aggregated IF-decision matrix is determined based on the different importance of attributes as follows: Determining intuitionistic fuzzy positive-ideal solutions (PIS) and negative-ideal solutions (NIS) as r þ j j and r − j by: ð10Þ Let J 1 and J 2 be benefit attribute and cost attribute, respectively.
Defining an IF positive-ideal separation matrix (D + ) and negative-ideal separation matrix (D − ) as follows: Calculate the grey relational coefficient of each candidate from the PIS as below: Similarly, the grey relational coefficient of each candidate from the NIS is given as below: Where the identification coefficient ρ = 0.5. Calculate the degree of the grey relational coefficient of each candidate using the following equation: Where Calculate collective index (CI) based on the degree of the grey relational coefficient of each candidate by: Determine the priority of candidates A i i ¼ 1; 2; …; m ð Þby the proposed CI i i ¼ 1; 2; …; m ð Þ : If any of the candidates has the highest CI i value, then, it is the most important one.