Use of grey relational analysis for multi-objective optimisation of NiTiCu shape memory alloy produced by powder metallurgy process

Abstract

Powder metallurgy process was used in this work to produce NiTi specimens. Uniaxial compression, X-ray diffraction and differential scanning calorimetry were used for characterising the produced samples. L9 orthogonal arrays were chosen based on the Taguchi method for conducting the experiments. In order to optimise the processing parameters and also to determine the level of importance of each parameter, experiments were performed based on grey relational method. Our goal was to optimise recoverable strain (ε) and the finishing temperature of austenitic transformation (A_f). Sintering time, compaction pressure, milling time and the atomic percentage of Cu were all selected as controllable parameters. Our results reveal that the sintering time and the atomic percentage of Cu are the most significant parameters. The findings were verified through a confirmatory test.

Keywords

Powder metallurgy nitinol shape memory grey relational analysis optimisation

Introduction

Currently, NiTi alloys are known as effective biomaterials for orthopaedic, dental and cardiovascular applications. These alloys have valuable characteristics, such as shape memory effect, super-elasticity, corrosion resistance, wear resistance and biocompatibility (Arciniegas et al., 2007; Chu et al., 2004; Maziarz et al., 2004).

Some current methods for producing the practically used NiTi alloys are vacuum induction melting and electric arc melting. Segregation formation, crucible contamination absorption and gas absorption are conventional problems of producing NiTi alloys by these methods; in addition, producing complex shapes is difficult with these methods (Elahinia et al., 2012).

In recent years, there have been numerous investigations on the use of conventional powder metallurgy (PM) techniques to produce porous NiTi alloys. The PM techniques are good candidates for producing near net-shape components due to the potential to utilise lesser volume of precursor materials (Bram et al., 2002). Moreover, the PM methods allow for the exact control of the chemical composition (Chung et al., 2004).

Formation of NiTi in PM processing is not favoured in primary reactions between Ni and Ti due to prevailing thermodynamic conditions. Thus, NiTi is obtained as a product of secondary reactions involving primary reaction products of NiTi₂ and Ni₃Ti (Neves et al., 2011).

Some of the PM methods are as follows: self-propagating high temperature synthesis (SHS) (Chu et al., 2004; Chung et al., 2004), conventional pressing and sintering of the powder (Li et al., 1998; Zhang et al., 1992), hot isostatic pressing (HIP) (Lagoudas and Vandygriff, 2002; McNeese et al., 2000), metal injection moulding (Guoxin et al., 2008; Schöller et al., 2005), sintering in reduction atmosphere (Zhu et al., 2004), vacuum sintering (Khalifehzadeh et al., 2007), spark plasma sintering (Zhao et al., 2005) and mechanical alloying (MA) (Pilarczyk et al., 2011; Sadrnezhaad and Selahi, 2004). Among these, researchers have focused mainly on SHS, HIP and sintering in vacuum or a reduction atmosphere for producing NiTi shape memory alloys. Inability to control the intermetallic phases is one drawback of SHS method, and usually Ti₂Ni, Ni₃Ti and Ni₄Ti₃ precipitates are present in the matrix of SHS products (Lagoudas and Vandygriff, 2002); costly equipment and probability of creation of secondary phases are some difficulties of HIP method (Elahinia et al., 2012) and limitation in shape and pore size of samples, long heating times and undesirable secondary phases are some difficulties of conventional sintering (Elahinia et al., 2012).

For biomaterial applications, the elastic modulus of implants produced via PM can be made closer to the modulus of bone, and this can eliminate the problems associated with stress-shielding phenomena in conventional metallic implants. Moreover, enhanced fixation can be achieved by improving the bone tissue growth throughout the porous matrix of these implants (Ryan et al., 2006).

Use of copper as a ternary alloying element results in increase in the martensitic transformation temperature. Also, copper has good corrosion resistance, less composition sensitivity of martensitic start temperature (M_s) and narrow transformation hysteresis, but an addition of more than 10 at % reduces alloy formability (Goryczka and Van Humbeeck, 2008).

There are limited works about producing ternary NiTiCu shape memory alloys by PM methods. For instance, Goryczka and Van Humbeeck (2008) produced Ni_50−xTi₅₀Cu_x (wherex= 2, 3, 5, 10, 15, 20, 25 at %) by powder technology. They tried various conditions for sintering and concluded that a homogeneous alloy can be obtained only by correct combination of sintering temperature and time (for copper less than 5 at%, it was 940 °C for 7 h; for copper more than 10 at%, it was 850 °C for 20 h). Terayama and Kyogoku (2010) fabricated Ni_50.2−xTiCu_x (x = 0, 5, 10, 15, 20 mol%) alloy by MA of elemental powders; they investigated shape memory characteristics, phase transformation behaviour and also thermo-mechanical properties of the produced alloys.

The PM process is affected by many different parameters, such as the material composition, size and purity of the primary powders, milling time and milling speed, sintering time and sintering temperature and compaction pressure. Due to the large number of input parameters, design of experiments (DOEs) based on the Taguchi method can be very useful for saving time and cost. This issue has gained little attention in other works. Additionally, in medical applications, it is ideal to have a NiTi alloy with transformation temperature around the body temperature for good shape memory function. Therefore, the objective of this work was to maximise both A_f and ε to determine the optimal parameter combination and the level of importance of each parameter for processing. Since the traditional Taguchi method cannot solve a multi-objective optimisation, a grey relational analysis (GRA) was used to overcome this problem.

Materials and methods

Raw materials and powder processing

Elemental powders of titanium (mean particle size: ∼650 µm, purity: 99.5%), nickel (mean particle size: ∼10 µm, purity: 99.8%) and copper (mean particle size: ∼60 µm, purity: 98%) were mixed at room temperature for producing Ni_50−xTi₅₀Cu_x specimens with different atomic percentage of Cu (0, 5, 10). Mixing was performed on a planetary ball mill with stainless steel vial (1500 mL in volume) and two different sizes of stainless steel balls (8 and 10 mm in diameter) without the addition of a process control agent. The ball-to-powder weight ratio and the milling speed were kept at 40:1 and 300 r/min, respectively. The maximum milling time was 54 h. The vial was evacuated and filled with high-purity argon gas (99.99%) during the mixing. The cylindrical preforms with a size of 10 mm in diameter and 20 mm in height were obtained by a uniaxial cold compaction under different pressures (600, 750 and 900 MPa). Then, sintering was carried out under pure argon atmosphere (99.99%) at 1050 °C for various times (4, 7 and 10 h), and finally after sintering, the specimens aged at 500 °C for 0.5 h in an argon atmosphere and then quenched in water.

Characterisation

The phase constituents of the specimens were determined by X-ray diffraction (XRD). X-ray diffractometer Philips PW1140 was used at room temperature with Cu K_α radiation (λ = 0.1541874 nm) at a voltage and electrical current of 40 kV and 30 mA, respectively. To study the stress–strain behaviour of the samples, uniaxial compression tests were performed at room temperature at a rate of 0.2 mm/min on a Zwick (Z 250) testing machine. The strain is measured in this machine by a contact type extensometer (Type B066552). The characteristic temperatures of transformation were determined by differential scanning calorimetry (DSC) with a preheating temperature of 250 °C and a heating/cooling rate of 10 °C/min.

Experimental design

For PM processing, a high number of experiments must be carried out to determine the optimal processing conditions, as there are many parameters to consider. Use of experimental design in such cases can be very useful. Conventional approaches for DOEs are classical (full factorial, fractional factorial, response surface methodology) and Taguchi method.

Basically, classical approaches are complicated, ineffective and frustrating for managers, engineers and workers and so those methods tended to be preferred by only those with a statistical or mathematical inclination (Tay and Butler, 1999).

In full factorial approach, all paired interactions can be studied. However, the number of runs goes up exponentially as additional factors are added. Fractional factorial design can be used to reduce the number of runs by evaluating only a subset of all possible combinations of the factors. In a large system, it usually produces an experimental design that is desired. However, random design works poorly for systems with a small number of variables (Tay and Butler, 1999).

The Taguchi method (Roy, 2010) allows for the analysis of many different parameters without a prohibitively high amount of experimentation. Taguchi method emphasises a mean performance characteristic value close to the target value rather than a value within certain specification limits, thus improving the product quality. One limitation is that the Taguchi methods are offline, and therefore inappropriate for a dynamically changing process such as a simulation study. Also, for more than three levels, the selection is limited to a maximum of six factors. Another limitation is that Taguchi method cannot solve a multi-objective optimisation.

In this work, experiments were designed and conducted based on Taguchi’s orthogonal array; since the traditional Taguchi method cannot solve a multi-objective optimisation, a GRA was used to determine the optimum process parameters for multiple responses. The grey system theory (Deng, 1982, 1989) is useful for dealing with poor, incomplete and uncertain information. Optimisation of the complicated multiple performance characteristics can be converted into optimisation of a single grey relational grade.

In this study, four parameters were used as control factors with three levels of variations for each parameter (Table 1).

Table 1.

Input parameters and their levels.

Processing parameter	Symbol	Level 1	Level 2	Level 3
Milling time (h)	A	36	20	54
Compaction pressure (MPa)	B	600	750	900
Sintering time (h)	C	7	10	4
Cu (at.%)	D	0	5	10

The control factor levels were selected according to our primary experiences, and some parameters, such as purity, the particle size of the primary elemental powders, ball milling speed and sintering temperature, were considered as noise factors and kept constant during experimentation. As shown in Table 2, L₉ orthogonal arrays were chosen based on the Taguchi method.

Table 2.

Experimental design using a L₉ orthogonal array.

Test no.	Milling time (h)	Compaction pressure (MPa)	Sintering time (h)	Cu (at.%)	ε (%)	A_f
1	36	600	7	0	0.95	17.2
2	36	750	10	5	1.07	21.1
3	36	900	4	10	1.22	27
4	20	600	10	10	1.45	32
5	20	750	4	0	0.75	8
6	20	900	7	5	1.2	33
7	54	600	4	5	0.78	21.9
8	54	750	7	10	1.1	39.6
9	54	900	10	0	0.85	24.3

Optimisation

In this method, the following steps are followed for optimisation using GRA (Deng, 1989):

Grey relational generation

In the first step, the experimental results are normalised in a range from 0 to 1 (Table 3).

Table 3.

Data preprocessing for each performance characteristic.

Experiment no.	ε (%)	A_f (°C)
Ideal sequence	1	1
1	0.2857	0.2911
2	0.4571	0.4145
3	0.6714	0.6012
4	1	0.7594
5	0	0
6	0.6428	0.7911
7	0.0428	0.4398
8	0.5	1
9	0.1428	0.5158

In this study, normalisation was computed for the ‘larger-the-better’ type of response according to equation (1)

x_{i} (k) = \frac{y_{i} (k) - \min y_{i} (k)}{\max y_{i} (k) - \min y_{i} (k)}

(1)

In the above equation, $x_{i} (k)$ are the normalised data (Table 3), $y_{i} (k)$ is the kth response of the ith experiment, $\min y_{i} (k)$ is the minimum value in the original sequence and $\max y_{i} (k)$ is the maximum value in the original sequence.

2. Grey relational coefficient

The grey relational coefficient is calculated according to equation (2)

ξ_{i} (k) = \frac{Δ_{\min} + ξ Δ_{\max}}{Δ_{oi} (k) + ξ Δ_{\max}}

(2)

Here, $Δ_{oi} (k)$ is the deviation sequence of the reference sequence according to equation (3)

Δ_{oi} (k) = ∥ x_{o} (k) - x_{i} (k) ∥

(3)

In the above equation, $∥ x_{o} (k) - x_{i} (k) ∥$ is the absolute of the difference between $x_{o} (k)$ and $x_{i} (k)$ , and $x_{o} (k)$ is the ideal or reference sequence. Also, Δ_min and Δ_max are defined as

Δ_{\min} = mi n_{. \forall j \in i} mi n_{. \forall k} ∥ x_{0} (k) - x_{i} (k) ∥

(4)

Δ_{\max} = ma x_{. \forall j \in i} ma x_{. \forall k} ∥ x_{0} (k) - x_{i} (k) ∥

(5)

The symbol ξ is the distinguishing coefficient ((ξ∈ [0, 1]).

ξ may be adjusted based on the practical needs of the system. Since both the object responses are of equal weight in this work, the value of ξ is taken as 0.5.

Results for grey relational coefficient can be seen in Table 4.

Table 4.

Calculated grey relational coefficient for each output parameter.

Experiment y	ε (%)	A_f (°C)
Ideal sequence	1	1
1	0.4117	0.4135
2	0.4794	0.4606
3	0.6034	0.5562
4	1	0.6751
5	0.3334	0.3334
6	0.5832	0.7053
7	0.3431	0.4716
8	0.3334	1
9	0.3684	0.5080

3. Grey relational grade

The grey relational grade $γ_{i}$ was obtained by ‘equation (6)’

γ_{i} = \frac{1}{n} \sum_{k = 1}^{n} ξ_{i} (k)

(6)

Here, $γ_{i}$ is the ith grey relational grade of each experiment and was obtained by averaging the grey relational coefficients. Additionally, n is the number of responses, where in this study, n = 2. As seen in Table 5, the results are ordered by value. Because $x_{o} (k)$ represents the best sequence, the higher value of the grey relational grade shows that the corresponding parameter combination is closer to the optimal setting. In this study, the fourth experiment has the highest value, indicating the best multiple performance characteristics using the combination A2B1C2D3.

Table 5.

Calculated grey relational grades and orders.

Experiment no.	Grey relational grade	Order
1	0.4126	7
2	0.47	5
3	0.5798	4
4	0.83755	1
5	0.3334	9
6	0.64425	3
7	0.40735	8
8	0.6667	2
9	0.4382	6

The average grey relational grades for each parameter level correspond to their orthogonal array table (Table 2), as listed in Table 6. As mentioned previously, the higher value of the grey relational grade shows that the corresponding parameter combination is closer to the optimal setting, and it can be seen from Table 6 that the combination A2B3C2D3 represents the largest average response. This is the optimal parameter combination.

Table 6.

Response table for the grey relational grades.

Processing parameters	Level 1	Level 2	Level 3	Max–min
Milling time	0.4874	0.6050	0.5040	0.1176
Compaction pressure	0.5525	0.4900	0.5540	0.064
Sintering time	0.5745	0.5819	0.4401	0.1418
Cu (at.%)	0.3947	0.5072	0.6946	0.2999

The difference between the maximum and minimum of the grey relational grades for each parameter is also listed in Table 6. The magnitude of this difference is related to the importance of the control parameters; more amount of this difference represents the stronger effect of parameter on the output response.

Results and discussion

The goal of this study was to optimise the recoverable strain (ε) and austenitic finish temperature (A_f). The DSC curves of nine experiments during cooling cycle can be divided into three categories: curves with one peak, two peaks and three peaks. Graphs that are shown in Figure 1 include all these three types of curves, and for brevity, all graphs of experiments are not presented here.

For the first experiment, during the heating cycle, a two-stage transformation (B19′-R-B2) can be seen where the B19′-R transformation peak is higher than R-B2 because the energy of the R phase is closer to the austenite phase.

Also during the cooling cycle, three peaks can be seen in this curve. The aggregation of Ni₄Ti₃ precipitates in sites with disorders results in a two-stage transformation. By comparison, in locations without precipitates, a typical one-stage transformation is observed. Additionally, differing compositions at each site lead to different transformation temperatures.

In the XRD patterns in Figure 2 for specimens 1, 5 and 9 (0% Cu), the Ni₄Ti₃ phase is recognisable. By increasing the sintering time (4, 7 and 10 h) in these specimens (5, 1 and 9, respectively), the height of the Ni₃Ti peaks was reduced, the amount of NiTi phase was increased, and thus, the amount of Ni in the matrix was increased. Moreover, the existence of more Ni in the matrix results in a drastic reduction in the transformation temperature (Frenzel et al., 2007) and increased amount of Ni₄Ti₃ precipitates, which facilitates the martensitic transformation. Meanwhile, it should be mentioned that according to the results of this work (listed in previous section; Table 6), the milling time and compaction pressure are insignificant processing parameters and so the effect of these parameters is not discussed here.

In Figure 1, for the second experiment, the specimen experiences a one-stage transformation. The presence of less than 5% Cu does not have any special effect on the transformation process, but it can facilitate twinning in austenite and increase the transformation temperature. Moreover, the presence of Cu prevents the formation of Ni₄Ti₃ (Goryczka and Van Humbeeck, 2008). For the third specimen (specimen with 10% Cu), a two-stage transformation of B2-B19-B19′ is observed, and the peak height of the first stage is smaller than that of the second stage, consistent with our expectations.

As mentioned previously, the existence of 10 at% Cu in the chemical composition prevents the formation of the Ni₄Ti₃ phase. During the ageing process, formation of Ti₂Cu precipitates facilitates the martensitic transformation and is considered as a proper site for the nucleation of martensite.

By increasing sintering time (Figure 1) in the ninth experiment, as it can be seen, the temperature distance between the sites with two-stage transformation and sites without precipitates increases. Besides the type of transformation, the phase distribution and transformation temperature can influence the recoverable strain amount.

Among all the stress–strain curves of Figure 3, it can be seen that almost the least amount of upper stress is related to third specimen, which can be due to easier transformation of B2-B19 in comparison with B2-B19′. Also, creation of Ti precipitates through ageing process provides preferred sites for martensite nucleation, which in turn, facilitates the austenite transformation.

Furthermore, for the third experiment of Figure 1, finishing temperature of austenitic transformation (A_f) is 27 °C, while the temperature of the testing environment is 25 °C; it means that the austenite phase stability is rather low in this temperature and so least amount of stress is required for austenite transformation, as it can be seen in Figure 3.

By evaluating the XRD results using X’Pert software (Table 7), different approximate weight percents of NiTi and (Ni, Cu)₃Ti for nine experiments can be obtained. It can be seen that by increasing the ratio of NiTi to (Ni, Cu)₃Ti, the recoverable strain has increased (Figure 4).

Table 7.

X’Pert results for approximate amount of unwanted (Ni, Cu)₃ Ti phase.

No.	(Ni, Cu)₃Ti	NiTiC	NiTi/(Ni, Cu)₃Ti	ε
1	22	73	3.32	0.95
2	19	72	3.79	1.07
3	12	66	5.50	1.22
4	10	75	7.50	1.45
5	35	61	1.71	0.75
6	17	74	4.35	1.20
7	33	62	1.88	0.78
8	19	68	3.58	1.10
9	23	66	2.87	0.85

As it mentioned earlier in the optimisation section, more amount of difference between the maximum and minimum of the grey relational grades for each parameter, as listed in Table 6, represents the stronger effect of that parameter on the output response. So from this point of view, the Cu (at.%) and sintering time are more significant than compaction pressure and milling time which accommodates with our expectations. As mentioned previously, Cu plays different roles in composition. It seems that formation of (Cu, Ni)₃Ti causes local change in the chemical composition, and by decreasing the Ni and Cu content in austenite phase, the martensitic transformation temperature increases. Also for sintering time, as mentioned by Zhu et al. (2004) and Green et al. (1997), in the early stages of sintering, formation of fine pores occurs due to the faster diffusion rate of Ni and Cu than Ti according to the Kirkendall effect, and in the next stages, collection and shrinkage of original pores occur. It seems that for low sintering time (4 h), formation of new fine pores is dominant and for long sintering time (7 h, 10 h), shrinkage of original pores determines the final porosity of microstructure, which in turn impacts recoverable strain of samples. On the other hand, with regard to compaction pressure, as Zhu et al. (2005) concluded in their work, probably due to the work hardening, deformation of powders was difficult and so it is expected that the use of higher pressures may not have much impact on output responses. Although milling time is expected to be one of the important factors in this process, probably due to the selected range of time for milling (20–54), changing of milling time in this area shows no significant effect on the output responses.

To verify the influence of the optimal parameter combination on the output parameters (A_f and ε), a confirmation experiment was carried out. The parameter A_f was 38 °C and ε was 1.7, whereas the combination of experiment 4 (most amount in Table 5) had A_f as 32 °C and ε at 1.45, demonstrating 18.7% and 17.2% improvements for A_f and ε, respectively. The DSC and stress–strain curves for confirmation experiment can be seen in Figure 5.

Conclusion

This research focused on the optimisation of parameters for PM processing to produce NiTiCu specimens. For good shape memory function in biomedical applications, it is ideal to have an alloy with transformation temperature around the body temperature. So due to the results of this work for A_f and ε, the object of this optimisation was to maximise both A_f and ε.

Since various parameters are involved in production of specimens by PM methods, use of experimental design in such cases can be very useful, but usually classical experimental design methods are complicated, time-consuming and costly. Therefore, in this work, by using Taguchi orthogonal arrays, the number of experiments reduced, and due to limitation of Taguchi method for multi-objective optimisation, a GRA was used for optimisation of object responses. This approach converts a multiple response optimisation problem into a single response optimisation called grey relational grade. Using this method, the optimisation process can be significantly simplified. The results of our analysis revealed that the best levels for optimal performance can be obtained by setting the milling time at 20 h, the compaction pressure at 900 MPa, the sintering time at 10 h and Cu at 10%. Our results showed that the sintering time and the atomic percentage of Cu are the most significant parameters and have stronger effect on responses than milling time and compaction pressure. The results of our confirmation experiment revealed an 18.7% and 17.2% improvement for A_f and ε, respectively.

Footnotes

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was funded by the central office of ACECR (Academic Centre for Education, Culture and Research).

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