Influence of Milling Time on Structural and Microstructural Parameters of Ni 50 Ti 50 Prepared by Mechanical Alloying Using Rietveld Analysis

1Laboratoire de Physique des Solides, Department of Physics, Faculty of Science, University Badji Mokhtar, Annaba, Algeria 2Laboratoire d’Elaboration et d’Analyse des Materiaux LEAM, Department of Physics, Faculty of Science, University Badji Mokhtar, Annaba, Algeria 3Laboratoire de Microstructures et Defauts, Department of Physics, Faculty of Science, University of Mentouri, Constantine, Algeria 4Department of Physics, College of Science, University of Bahrain, P.O. Box 32038, Manama, Bahrain


Introduction
NiTi-based shape memory alloys have attracted great scientific and technological interests, due to their remarkable mechanical and chemical properties, such as superelasticity, shape memory behavior, good corrosion resistance [1,2], hydrogen storage ability, and good biocompatibility [3,4]; thereby they are very appropriate for biomedical and dentistry applications [5,6].The observed shape memory effect in NiTi alloy is mainly caused by the existence of NiTi-austenite (B2) and NiTi-martensite (B19  ) phases [7].
Among the above-cited methods, mechanical alloying (MA) was found to be very effective in the synthesis of Ni 50 Ti 50 -based nanocrystalline alloys or nanostructured powders.Through this technique, a large variety of materials are produced such as intermetallics, extended solid solutions, quasicrystals, and amorphous phases [17].Moreover, in this process, the reduction of particle size is observed, resulting in ultra-fine grained or nanocrystalline materials.Because of the very fine grain size, nanocrystalline materials exhibit diverse and interesting properties, which are different and often considerably improved in comparison with conventional coarse-grained polycrystalline materials [18].MA  elemental powders mixture can lead to amorphous phase, nanocrystalline solid solution [12], NiTi-austenite (B2), and NiTi-martensite (B19  ) phase [19].
Moreover, it is very important to characterize the powders obtained by MA in terms of phase composition in addition to structural and microstructural features.The crystallite size and microstrain in powdered particles can be determined through X-ray diffraction peak broadening.Herein, it is worth mentioning that the shape of peak can be described as a combination of two functions: Gaussian associated with microstrain and Lorentzian associated with crystallite size (Figures 1, 2, and 3) [20,21].Meanwhile, instrumental parameters have to be taken into consideration.Numerous models, like Williamson-Hall [22], Halder-Wagner [23], Warren-Averbach [24], and Rietveld method [25], have been widely used for the determination of structural and microstructural parameters.
The Rietveld analysis, a full-pattern fitting of X-ray diffraction patterns, has been reported as a powerful method for structural and microstructural characterization [26], including qualitative and quantitative phase analyses of multiphase nanocrystalline materials containing significant number of overlapping reflections [27].In the literature, Rietveld's method was successfully adopted for the determination of microstructural parameters of various systems [28][29][30].The aim of this research work is devoted to the investigation of milling time on structural and microstructural modifications occurring in mechanically alloyed Ni 50 Ti 50 powder mixture.In-depth analysis using the Rietveld method of X-ray data is carried out, in terms of phase composition, crystallite size, microstrain, lattice parameter, and lattice defects such as dislocation density and stacking faults.MA was performed under argon atmosphere using Fritsch Pulverisette 7 planetary ball mill equipped with hardened steel vial (80 mL) and balls (15 mm diameter).The disk speed rotation was Ω = 400 rpm.The ball-to-powder mass ratio was 23 : 1.To avoid excessive temperature increase, a milling time pause of 30 min was applied after each hour of milling.The powders were milled for several periods of time, that is, 0, 1, 3, 6, 24, 48, and 72 h.

Characterization.
The milled powders were examined by X-ray diffraction (XRD) method, using X-ray Philips X, Pert diffractometer equipped with Cu-K  radiation source ( Cu = 0.15418 nm).The phase analysis was performed by using ICDD (PDF-2, 2012) files.Both structural and microstructural parameters were determined from the refinements of XRD patterns using MAUD program (version 2.55) based on the Rietveld method [31].
Using the Rietveld refinement and Warren-Averbach method [24,25] in the frame of MAUD software [31], a detailed analysis of XRD profiles can be performed leading to the determination of phase composition in addition to structural and microstructural parameters for each phase, such as lattice parameters (, , ), the average crystallite size ⟨⟩, microstrain ⟨ 2 ⟩ 1/2 , and stacking fault probabilities (SFP).
The Rietveld method consists of simulating a pattern from simulated crystallographic model that is as close as possible to the measured pattern, which depends on analytical functions in order to characterize the microstructure of powders and determine ⟨⟩, ⟨ 2 ⟩ 1/2 , and SFP.
The structural refinement method uses the minimization technique of least squares making it possible to approach the experimental pattern from a starting structural model.The minimized function, or residue, is written: where   and   are, respectively, the observed and calculated intensities;   is the weight associated with the intensity   .
For refinements by least square method,   is taken equal to 1/  .From a structural model, each contribution   is obtained by the combination of different Bragg contributions and continuous background: where   is the integrated intensity of the reflection,   is the profile shape function, and   a polynomial function fitting the background.The integrated intensity of the reflection is calculated according to certain structural and geometric parameters: where   is the scale factor,   is the structure factor inclusive of Debye-Waller factor and site occupancy,   is the multiplicity of (ℎ) reflection,   is the usual Lorentz-polarization factor, and   is the cell volume.
The profile shape function is the result of the convolution of instrumental and sample broadenings: where   is the sample broadening and   is the instrumental broadening.The latter can be written as follows: where  and  are, respectively, the symmetric and asymmetric components of the instrumental profile, determined for particular geometry of the diffractometer.The asymmetric component is given by where   is the asymmetry parameter and  0 is the Bragg angle of  1 peak.The Pseudo-Voigt function is defined as the contribution of a pure Lorentzian and a pure Gaussian (which will be corrected for the asymmetry of the peaks): where   is the scale parameter of the  function,  is the Gaussianity of X-ray peaks, and (2  ) and (2  ) are Gaussian and Lorentzian components, respectively: where 2 0 is peak position,  0 is the peak intensity, and FWHM is the full-width at half-maximum of reflections.
The FWHM is given by the Caglioti formula [32]: where , , and  are coefficients of quadratic polynomial or the coefficients of Caglioti.
The sample broadening   is the convolution of broadening due to the finite size of crystallites ⟨⟩, the r.m.s microstrain ⟨ 2 ⟩ 1/2 , and planar defects expressed in terms of stacking faults probability (SFP): where   ,   are the intrinsic and extrinsic deformation faults, respectively, and  is the twin faults.The Delft model for sample broadening gives the integral breadths for the Gaussian and Lorentzian components: eff is an effective size computed from the crystallite size ⟨⟩ and  is the radiation wavelength.
The MAUD software is based on the Rietveld method combined with a Fourier analysis.The detailed analysis was performed considering Fourier coefficients   ; after Stoke's corrections the size coefficients    and distortion coefficients    , which are related to   in the following equation: were separated from the log plot of   with the square root of the quadratic sum of the Miller indices ℎ 2 0 where [33] The reflections affected by stacking faults correspond to peaks satisfying the conditions: The microstructural parameters like effective size,  eff , and the r.m.s.microstrain ⟨ 2 ⟩ 1/2 are related, respectively, to the initial slope of    with  and slope of log   with ℎ 2 0 : where  denotes a normal length to the reflecting planes.
The SFP like intrinsic   , extrinsic   , and deformation twin fault probability  were evaluated through the following relation: where  is the lattice parameter,  is unbroadened by faulting,  is the number of reflections broadened, and  0 is the third coordinate replacing (ℎ) indices ( 0 = ℎ +  + ) [33].Peak asymmetry is obtained by measuring the intensity value on either side of the peak at two positions which are equidistant from the peak center ( 1 and  2 ) and derived from the extrinsic deformation and twin faulting probabilities.The mathematical expression for asymmetry is evaluated as follows: where the coefficient  2 depends on the peak 2 0 position: where  is the area of the peak,  1 is the peak intensity in the diffraction angle ( 0 +  2 ), and  0 is the completely symmetrical profile center [31].
The peak shift can be written, in terms of intrinsic and extrinsic deformation faulting probability, as follows [33]: From the values of lattice parameters, crystallite size, and microstrain, the dislocations density () can be calculated using the following formula: where  V is Bürgers vector, which is equal to  √ 6/3 for the hexagonal close-packed (HCP) structure and  √ 2/2 for the face-centered cubic (FCC) structure.The calculated profile yields scaling factors for each phase to fit the intensity of the observed pattern.These scale factors are related to the respective relative weight fractions by the following equation [34]: where  is the number of formula units in the unit cell,  is the molecular mass of the formula unit,  is the unit cell volume,  is the Rietveld scale factor, and   is the weight fraction of phase : the index "" in the summation covering all phases that are included in the model [34].The standard procedure is to determine the amorphous fraction by the internal standard method.The amount of the amorphous phase is obtained as the complement to one of the total amounts of crystalline phases [31].The weight percentage of the amorphous phase in a sample can be given by the following relationship [35]: where   is the amorphous fraction,   is the weighted internal standard fraction, and   is the refined fraction of the internal standard.
After adjusting the parameters of the instrument, the peak positions are corrected by successive refinements to eliminate systematic errors by taking into account the errors of the angular offset (Δ2) and the displacement of the sample.The background is adjusted by a polynomial of degree (2 to 5), for the first milling time.On the other hand, for longer milling times, the increase in the intensity of the background requires the use of higher degree polynomials.Then, the parameters of the crystal structure are refined, namely, the crystal parameters, the atomic positions, the Debye-Waller factor, and the percentage of phases.In the last step, we proceed to the refinement of crystallite size ⟨⟩, microstrain ⟨ 2 ⟩ 1/2 , and the probability of all three types of planes defects   ,   , and .
To evaluate the refinement, several factors have been introduced in order to know the agreement between calculated and observed models [34].
The minimization was carried out by using the reliability index parameter,   (weighted residual error), and  exp (the expected error) defined as follows [27]: where  and  are the number of experimental points and the number of fitting parameters, respectively.It is possible to calculate a statistical parameter, which must tend towards the unity for a successful refinement; the goodness of fit " 2 " is defined as the ratio between   and exp : The profile refinements continue until convergence is reached; the value of the quality factor  2 (GOF) is approaching one.

Results and Discussion
3.1.Structural Analysis.The X-ray diffraction (XRD) patterns of Ni 50 Ti 50 powders (0, 1, 3, 6, 12, 24, 48, and 72 h), shown in Figure 4, clearly indicate that MA introduces significant changes.There is a gradual peaks' broadening with a decrease in peaks' intensity as a function of the milling time, demonstrating the great impact of milling on the crushed starting elemental Ni and Ti starting powders, which can be associated with several processes occurring progressively [36]: (i) particles (grains) refinement (reduction of crystallite size); (ii) introduction of crystalline defects (interstices, dislocations, and grain boundaries); (iii) increase in the rate of microdeformations (large amount of energy); (iv) fragmentation of crystallites and/or effect of atomic disorder.
The as-mentioned phenomena result from welding and disordering effects, occurring because of repeated collisions (ball-powder-ball and ball-powder-wall of bowl).
The overlapping of XRD reflections arising from different phases results in an enlargement of peaks' width accompanied with a slight shift towards lower angles, indicating a slight increase in interatomic distances leading to unit cell expansion.
After few hours of milling, one can see the appearance of halo around 35-50 ∘ , which overlaps with reflections belonging to HCP-Ti and FCC-Ni.This can be attributed to the formation of an amorphous phase, which is explained by the significant structural disorder induced by the severe plastic deformations occurring during milling process.
Figure 5 shows the refined XRD patterns for elemental Ni and Ti powders, based on initial structural models: (i) face-centered cubic (FCC) for Ni with lattice parameter  = 0.3524 nm; (ii) hexagonal close-packed (HCP) for Ti with lattice parameters  = 0.2952 nm and  = 0.4686 nm.All the results are reported in Table 1.
The best Rietveld refinements of XRD patterns (1, 3, and 6 h) were obtained with three components: FCC-Ni, HCP-Ti, and an amorphous phase (Figures 6(a), 6(b), and 6(c)).The nanometer scaled diffusion couples are produced by high energy mechanical milling which involves fracture and coldwelding of milled powders.Thus, the atomic diffusivity is improved through the creation of a large amount of structural defects.As a result, metastable phases may form at early stage of the milling by solid-state reaction (SSR) process.
After 12 h of milling, the diffraction peaks of elemental FCC-Ni, HCP-Ti, and the amorphous phase remain while new diffraction peaks related to NiTi-martensitic (Figure 6(d)) emerge.Further milling up to (24 h), leads to a complete disappearance of HCP-Ti peaks associated with the formation of NiTi-austenitic (Figure 6(e)), while Ni-SS remains present with small amount (15.38%).Furthermore, it is well-known that the crystallite size decreases with the increase of negative mixing enthalpy (ΔH) [36].In addition, NiTi phase could be the product of the reaction between Ni and Ti due to their negative mixing enthalpy (ΔH = −67 kJ/mol) [37,38]:  Beyond 48 h of milling, XRD patterns reveal the same phases, NiTi-austenite (B2) and NiTi-martensite (B19  ), but with the dominance of the amorphous phase (Figures 6(f) and 6(g)) having a relative proportion of about 89.24%.
In order to investigate the phase's stabilities during mechanical milling, we calculated the phase proportions of the identified phases as a function of milling time.The results obtained are shown in Figure 7. Accordingly, the milling process was divided into four main stages (I, II, III, and IV) as shown in Figure 7.
In the first stage I (0-1 h), a decrease in the proportions of Ti and Ni was observed, which may be due to the mutual diffusion and to the formation of amorphous phase and solid solutions FCC-Ni (Ti) and HCP-Ti (Ni).
During the second stage II (1-12 h), which represents a postformation of amorphous phase, we note that Ti-SS and Ni-SS ratios decrease inversely to the proportion of the amorphous phase.
Then, in the third stage III (12-24 h), the percentage of initial Ni-SS continues to decrease, meanwhile Ti-SS has disappeared completely, thus leading to the continuous increase in the amount of the amorphous phase and the formation of martensitic phase.The formation of martensitic phase may be due to the negative mixing enthalpy [37].The complete transformation of the severely deformed phases into an amorphous structure is achieved upon further milling, through the mechanically enhanced solid-state amorphisation, which requires the existence of chemical disordering, point defects, vacancies, interstitials, lattice defects, and dislocations.
The fourth and final stage IV (24-72 h) is characterized by the emergence of new NiTi-austenite phase, while the initial Ni-SS entirely disappears favoring also to the formation of NiTi-austenite.The continuous increase in the proportion of the amorphous phase can be explained evidently along with the reduction in the amount of NiTi-martensite.
After 48 h, the fraction of the amorphous phase decreases due to its crystallization into more stable crystalline phases (NiTi-austenite and NiTi-martensite), which may be attributed to strain energy and the temperature increase during MA due to severe plastic deformations as reported by Amini et al. [19].However, the amorphous phase is predominant with a relative proportion of about 89.23%.
Indeed, the heavily plastic deformations induce strong distortion of the unit cell structures hence becoming less crystalline.The powdered particles are subjected to continuous defects that lead to a gradual change in the free energy 8 Journal of Nanomaterials  of the crystalline phases compared to the amorphous phase resulting into disorder in atomic arrangement.

Crystallite Size and Microstrain.
A diffraction peak broadening, observed with increasing milling time, is usually associated with grain size reduction and increase of internal strain.The variation in the average crystallite size ⟨⟩ and microstrain ⟨ 2 ⟩ 1/2 obtained from Rietveld refinements is plotted as a function of milling time in Figure 8.
In the early stage of milling, one can observe an important decrease of the crystallite size reaching 7 nm (24 h) and 29 nm (12 h) for Ni-SS and Ti-SS, respectively.It is also evident that the reduction in Ni-SS crystallite size is faster than Ti-SS, which is probably related to its smaller initial particle size of Ni (<45 m) compared to Ti (<150 m) [38].
But in the second stage, after 24 h, the crystallite size becomes less dependent on the milling time.This can be explained by the fact that mechanical energy delivered during milling is not sufficient to plastically deform the two phases NiTi-austenite and NiTi-martensite.The average crystallite size of NiTi-martensite after 72 h is of about 19 nm while that of NiTi-austenite reaches 42 nm (Figure 8(a)).The reduction in crystallite size is mainly due to severe deformations of powders during the milling, as well as the increase in probability of nucleation sites during crystallization, provided by higher defects density [36].
The as-obtained values of the crystallite size are in good agreement with the values previously reported on binary alloys prepared by MA [20,28,29].
From Figure 8(b), it is clear that the internal microstrain ⟨ 2 ⟩ 1/2 increases with increasing milling time.Thus during the first stage of milling, ⟨ 2 ⟩ 1/2 of Ni-SS and Ti-SS reaches about 0.7% and 1.16% after 12 h, respectively.The increase of the microstrain can be due to the increase of dislocations density induced by severe plastic deformations [28].For NiTi-martensite, ⟨ 2 ⟩ 1/2 increases to about 0.9%, after 48 h, and then decreases; this behavior may be explained by the fact that, beyond this time, the crystallite size reaches a stationary value.
It is important to mention that NiTi-austenite is characterized by higher microstrain comparatively to NiTimartensite, which may be due to a high concentration of stacking faults and a high dislocations density.In fact, actually, the austenite phase is metastable at room temperature and becomes unstable when an external mechanical or thermal energy is introduced.Generally, the values of microstrain achieved during milling for similar systems are around 1.5% [18]. 2 shows the variation of lattice parameters of Ni, Ti, NiTi-austenite, and NiTi-martensite for each milling time as obtained from Rietveld analysis.The decrease of the lattice parameters of Ni and Ti after 1 h of milling is probably due to the compressive forces caused by collisions [27], and therefore reducing the distance between neighboring atoms.It can be noticed that Ni and Ti lattice parameters increase with milling time reaching 0.3574 nm, after 24 h, and  = 0.2979 nm and  = 0.4708 nm, after 12 h of milling, respectively.This can be attributed to the severe plastic deformations and accumulation of large amount of structural defects such as stacking faults, grain boundaries, and enhanced dislocations density during the milling process [23].For the austenite phase, the lattice parameter increases up to 48 h and then remains nearly stationary until 72 h where it reaches a value of 0.3134 nm.The relative deviation of the lattice parameter (Δ/ 0 ) of the austenite phase after 48 hours of milling is 3.96% which is high compared to the relative deviation of the lattice parameter of martensitic phase Δ/ 0 = 1.397%,Δ/ 0 = 0.663%, and Δ/ 0 = 1.962%.This relatively high value can be associated with the severe plastic deformations, which introduces different types of defects such as dislocations, grain boundaries, gaps, and stacking faults and it can be also explained by the formation of a nonstoichiometric composition of the austenite phase.

Dislocations Density.
The variation of the dislocations density can be associated with the ball-to-powder mass ratio that characterizes the collision number, the energy impact, and the number of defects introduced within the lattice.
Figure 9 illustrates the evolution of dislocations density of Ni and Ti as a function of milling time and can be divided into two stages:  I (0-12 h) and  II (12-24 h).It is obvious that  increases slightly in the first stage  I from about 0.0014 × 10 16 to 0.4791 × 10 16 m −2 for Ni-SS and 0.0027 × 10 16 m −2 to 0.5691 × 10 16 m −2 for Ti-SS.The as-obtained  values are comparable to the values of dislocations density limit in metals achieved by plastic deformations (10 13 m −2 for screw dislocations and 10 16 m −2 for edge dislocations) [39].During the final stage  II , for Ni-SS, the dislocations density increases considerably up to 2.6851 × 10 16 m −2 with the disappearance of Ti.This increase as compared to the first stage  I , could be due to the formation of NiTi-martensite phase.Indeed, the Ti radius is slightly larger than that of Ni ( Ti = 0.176 nm >  Ni = 0.149 nm), so the diffusion of Ti into Ni leads to the distortion of Ni crystal structure and hence further increase in the dislocations density.The as-obtained values of the  It can be noticed that, for both HCP-Ti and Ni-FCC, SFP increases with increasing milling time; reaching after 24 h a higher value of 0.0364 (HCP-Ti) compared to 0.0189 (FCC-Ni).The increase of SFP for both Ni and Ti with extensive milling time reflects an important increase in the dislocations density (Figure 9).It has been noticed that twining and deformation stress depend on the initial crystallite size and milling intensity [18].
The parameter 1/(sfp) indicates the average number of compacted layers between two stacking faults, either deformation or twin-type, according to Warren's formulae [40].In fact, there are 27 and 43 ordered planes between the two consecutive stacking faults in HCP-Ti and FCC-Ni after 24 and 48 h of milling, respectively.As it is clear, the values of the average number of compacted layers between two stacking faults in HCP-Ti(Ni) are found to be higher compared to FCC-Ni(Ti) and thus can explain that Ti facilitates the formation of the amorphous phase more than Ni.Meanwhile, Ni contributes to the formation of martensitic and austenitic phases.

Conclusion
A mixture of Ni and Ti elemental powders with a nominal composition Ni 50 Ti 50 was mechanically alloyed in a planetary ball mill under argon atmosphere.The final products were nanocrystalline NiTi-martensite (B19) and NiTi-austenite (B2) phases with an average crystallite size in the range of a few nanometers (19 nm∼42 nm) with a large internal strain (1.02%∼1.48%)and a dominant amorphous phase (89 wt.%).
The induced heavy plastic deformations lead to crystallite size refinement and accumulation of structural and point defects, resulting in a significant contribution of stacking faults (SFs) resulting in the destabilization of crystalline structures.As a consequence, we demonstrated that mechanical alloying is a nonequilibrium process, where phase composition and transformation of binary NiTi can be modified from that predicted using thermodynamic equilibrium phase diagram.

Figure 1 :Figure 2 :
Figure 1: The illustration of Gaussian and Lorentzian peak-shape functions.

Figure 3 :
Figure 3: Contribution of crystalline and amorphous phases to an XRD pattern.

Figure 4 :
Figure 4: XRD patterns of the Ni 50 Ti 50 powders at different milling time.

Figure 7 :
Figure 7: Evolution of the proportions of the identified phases as a function of milling time.

Figure 9 :Figure 10 :
Figure 9: Evolution of dislocations density of Ni and Ti as a function of milling time.

Figure 10
Figure10shows the variation of stacking faults probability (SFP) of HCP-Ti and FCC-Ni as deduced from the Rietveld refinements of XRD patterns of Ni and Ti.In both Ni and Ti crystal structures, two different types of stacking faults can be found, namely, deformation and twin stacking faults.It can be noticed that, for both HCP-Ti and Ni-FCC, SFP increases with increasing milling time; reaching after 24 h a higher value of 0.0364 (HCP-Ti) compared to 0.0189 (FCC-Ni).The increase of SFP for both Ni and Ti with extensive milling time reflects an important increase in the dislocations density (Figure9).It has been noticed that twining and deformation stress depend on the initial crystallite size and milling intensity[18].The parameter 1/(sfp) indicates the average number of compacted layers between two stacking faults, either deformation or twin-type, according to Warren's formulae[40].In fact, there are 27 and 43 ordered planes between the two consecutive stacking faults in HCP-Ti and FCC-Ni after 24 and 48 h of milling, respectively.As it is clear, the values of the average number of compacted layers between two stacking faults in HCP-Ti(Ni) are found to be higher compared to FCC-Ni(Ti) and thus can explain that Ti facilitates the formation of the amorphous phase more than Ni.Meanwhile, Ni contributes to the formation of martensitic and austenitic phases.

Table 1 :
Structural and microstructural properties of unmilled Ni and Ti using the Rietveld refinements.

Table 2 :
Lattice parameters of Ni, Ti, NiTi-austenite, and NiTi-martensite as function of milling time.