Materials and Design 108 (2016) 60–67

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Materials and Design

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First principles studies on structural, elastic and electronic properties ofnew Ti\\— Mo\\— Nb\\— Zr alloys for biomedical applications

Paul S. NnamchiDepartment of Metallurgical and Materials Engineering, University of Nigeria, 410001, Nsukka, Enugu state, Nigeria

E-mail addresses: paul.nnamchi@unn.edu.ng, nnamch

http://dx.doi.org/10.1016/j.matdes.2016.06.0660264-1275/© 2016 Elsevier Ltd. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 6 March 2016Received in revised form 7 June 2016Accepted 16 June 2016Available online 23 June 2016

Stress shielding phenomenon has become a major drawback to the use of metallic biomaterials for orthopaedicimplants applications. In this study, a Ti\\Mo\\(Nb, Zr and Nb+ Zr) alloy systemwas investigated to design anddevelop novel low elastic Young's modulus Ti based alloy for implant application. The development and applica-tion of predictivemodelling and simulation are transforming thematerials engineering discovery process. To thisend, ab initio calculationwas used to evaluate the effects of composition on structural, elastic and electronic prop-erties of the materials. The data obtained from both theory and experiment were analysed and compared witheach other. Notable findings include low elastic Young's modulus values of 70.2 GPa, 80.6 GPa, 76.5 GPa,59.1 GPa and 32.3 GPa for the Ti\\6Mo\\6Zr, Ti\\6Mo\\6Nb, Ti\\6Mo\\6Nb\\2Zr, Ti\\6Mo\\5Nb\\3Zr andTi\\6Mo\\4Nb\\4Zr alloys, respectively ascribed to the unique elastic softening of their C′ and C44 shear moduli.The consistency in both results is discussed in terms of the sensitivity of the physical and electronic properties tothe alloying additions. Thus, the result indicates the approach can enhance the reduction of elastic Young's mod-ulus of metallic biomaterials for replacing some commonly used high modulus materials and prevent stressshielding in orthopaedic implants.

© 2016 Elsevier Ltd. All rights reserved.

Keywords:First principleLow Young's modulus Ti-alloysCASTEPImplant materialsBiomaterialsElastic properties

1. Introduction

Titanium and Ti-based alloys are now the most widely used bioma-terials for implant applications owing to their unique combination ofgood properties such as, high corrosion resistance, excellent biocompat-ibility and goodmechanical property [1,2]. Although, the elastic Young'smodulus of most Ti based alloys (100–110 GPa) ismuch less than stain-less steel (210 GPa), it is notably more than that of natural bone whichhas elastic Young'smodulus in the range of 10–40GPa [3,4,5]. This leadsto loosening and eventual premature failure of the implants, thus caus-ingmore pain to the sufferers [3,6]. This problem can be solved, if alloyswith bone matching elastic modulus are used [7].

In consequence, most recent research in the field have been focusedon the design of biocompatible, low elastic Young'smodulus Ti-alloys tosubdue this long-term health problem caused by the release of the toxicions from the alloys as well as the stress shielding effect [2,8,9]. Owingto this, soluble nontoxic, non-allergy metallic additives that tend to im-prove both strength and lowYoung'smodulus of Ti alloys are particular-ly desired [7,10]. At least one of such soluble additive are known toexist: Molybdenum. Molybdenum has been judged to be non-toxic,non-allergic and as a safe alloying element for developing low elasticmodulus Ti alloys with good strength for bone implant applications

i.paul@gmail.com.

[11–16]. It is the most effective β-phase stabiliser preferable for the Tibiomaterials. Three problems arise however: (1) lack of low tempera-ture ductility, indicating brittle behaviour may be inherent in Mo;(2) potent for improving high strength, but the element Mo belongsto refractory elements with very high melting points of ≈2896 K [17];and (3) they have heavy densities of 10.28 g/cm3 [17]. This implies anincrease in their alloying contents tends to increase both the meltingpoints and densities of obtained Ti alloys, which will make it unstablefor biomedical applications [5]. These considerations make Mo unsuit-able for this application. Nevertheless the existence of at least one ortwo ductilizing solute elements such as Nb and/or Zr, which havebeen judged to be non-toxic, non-allergic and safe to improve the inher-ent ductility and density of Ti\\Mo alloy system, are particularly de-sired, as the effectiveness of low Young's modulus strategy might notexplicitly be dependent on microstructure stability. Zr is added to in-crease the strength, while Nb which is a β stabilising element is expect-ed to improve corrosion resistance, mechanical performance andincrease hotworkability also [18,19]. This therefore raises the possibilityof developing biocompatible, low elastic modulus Ti alloys with highstrength and low elastic modulus materials for bone implant applica-tions [5–20].

In addition, in the case of β stability, past studies have shown thatconcentration of alloying elements, such as Mo, Nb, and Ta to beretained in β after quenching from β phase field are 5, 15, and 20 at.%(hereafter, atomic will be omitted), for binary Ti\\Mo [20], Ti\\Nb

Fig. 1.An illustration of the arbitrary distribution of the chemical species over the sites of acubic structuremodel of Ti\\6Mo\\xNb\\xZr used in the calculation.Here Ti andMo atomsare shown in grey and deep green, while Zr and Nb are represented in red and lightergreen colours, respectively. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

61P.S. Nnamchi / Materials and Design 108 (2016) 60–67

[21], and Ti\\Ta alloys [22], respectively. According to the above view-point, it can be expected that Mo is the most effective β stabiliser andthe β type Ti\\Mo alloys are more suitable than the other β Ti alloysfor biomedical applications [23]. Nonetheless, most previous studieshave focused on Ti\\Nb alloys. Thus far, various ternary and quaternaryβ-phase Ti alloys composed of biocompatible elements have been de-veloped; e.g. Ti–Nb–Ta–Zr [24], Ti–Nb–Zr–Sn [17], Ti–Nb–Sn [25], etc.The Young's modulus of these alloys in the range of 60–80 GPa, whichis still higher than that of natural human bone — 20–40 GPa. Up tonow, Ti\\15Mo alloy [26], Ti–7.5Mo alloy [27], Ti\\10Mo and Ti\\20Mo[28] have been developed for biomedical applications in binary Ti\\Moalloys. Such effort notwithstanding, a review of the literature indicatesthat multicomponent β Ti\\Mo alloys have received considerably lessattention [21,26,27,28], even as the Ti\\Mo system has proved to be agood substitute for developing considerably safe Ni-free biomedical Tialloys due to their being non-toxic elements that do not cause any ad-verse effects on the human body [11,12,13,14,15,16]. Nevertheless, theYoung's moduli of the Ti\\Mo alloy systems reported so far are notlow enough [27,28,29], and there has been little research effort intothe Young's moduli of multicomponent Ti\\Mo alloys. Moreover, a cen-tral theoretical framework of lowmodulus determining factors in alloysremains elusive and our understanding is not nearly as sharp in the area.A much better understanding could be obtained by taking into accountthe effect of the elements on elasticmodulus using ab initio calculations.With the above background in mind, present theoretical investigationwas undertaken to predict and design low Young's modulus multicom-ponent Ti alloys with none toxic alloying elements for orthopaedicapplication.

1.1. Calculation method

The first-principles calculations in the present work are performedby employing the pseudo-potential plane wave (PP-PW) approachbased on density functional theory (DFT) [30] and implemented in theCambridge Serial Total Energy Package (CASTEP) [31,51]. In this pack-age, the density functional theory [32] and the Kohn-Sham approachare used to calculate the fundamental eigenvalue [33]. Interactions ofvalence electrons with ion cores are represented by the Vanderbilt-type ultra-soft pseudo-potential [34] for Ti, Mo, Nb and Zr atoms. Theexchange-correlation potential is treated within the generalized gradi-ent approximation (GGA) of Perdew, Burke and Ernzerhof (PBE) [35].One of the most important quantity in first principles calculations isthe ground state energy [36]; this quantity is very affected by the energycut off and the Brillouin zone sampling [29]. After careful tests, we haveset the energy cut off at 450 eV, achieved with the k point separation inBrillouin zone of the reciprocal space equivalent to 0.04 nm−1. Thesevalues ensure a good convergence with respect to reliable results.Throughout this study, the maximum tolerance on the total energywas chosen as 10−6 eV/atom and the Hellmann-Feynman force on allatoms was b10−2 eV/Å. The atomic coordinates and lattice parametersused for the Im3m (β), P63/mmm (α), Cmcm (α″) and P6/mmm (ω)space groups/phase investigated for the Ti\\6Mo\\xNb\\xZr xZrmulti-component alloys were taken from CASTEP library. In every case, thegeometrical optimization is alwaysmade, based uponwhich the config-urational/electronic structures and elastic properties are thencalculated.

Previously, it has been shown that some important elastic responses(such as Young's modulus), non-linear superelasticity and very lowwork hardening rate are sensitive functions of composition, requiringMo contents lower than a critical level of about 6 at.%, thus requiringcomposition of around the least stable β phase in the Ti\\Mo alloy sys-tem [37,38]. Here, we studied the effect of Zr and Nb micro-additions(biocompatible elements) on the thermodynamic phase stability, elasticand electronic properties of the Ti\\6Mo alloy (i.e. initial least stable βcomposition) by ab initio calculation. In addition to the lattice parame-ter optimization, we performed an arbitrary distribution of the chemical

species over the sites of the structures by replacing Ti atomswithMo, Zror Nb atoms (as illustrated Fig. 1), but had no effect on the system. Nband Zr atoms are added to binary Ti\\6Mo supercell consisting of asixty four atom to attain an average of 0, 6, 6, 5 and 4 at.% of Nb and 6,0, 2, 3 and 4 at.% of Zr, respectively. Theoretically predicted structuralparameters and thermodynamic properties of the multicomponentTi\\6Mo\\x-Nb\\XZr alloys are presented in Table 1.

1.2. Experimental validation methods

In order to compare the predictions with experimental data sixTi\\6Mo\\xNb\\xZr alloys (namely, Ti\\6Mo, Ti\\6Mo\\6Zr,Ti\\6Mo\\6Nb, Ti\\6Mo\\6Nb\\2Zr, Ti\\6Mo\\5Nb\\3Zr andTi\\6Mo\\4Nb\\4Zr) were prepared from commercially pure Ti, Mo,Nb and Zr metals of high quality (~99.97% purity) by vacuum arc-melting method. The actual compositions (See Table 2) and e/a valuesof the resulting ingots were analysed by inductively coupled plasmaatomic emission spectrometry (ICP-AES) at the lab of Metallurgicaland Materials Engineering, University of Nigeria, Nsukka, Enugu State.(Atomic percent are used here and throughout the text unless other-wise specified). To ensure good standard, the ingots (30 g) wereinverted and remelted at least six times to ensure chemical homogene-ity, before casting. The cast rods were heat treated at 960 °C under highvacuum for 24 h inside a tubular furnace, followed by quenching inwater at room temperature. As a result, there are no variation betweenthe bulk and surface compositions. The phase fractions were quantifiedusing a wide angle X-ray diffraction method.

Characterization of the elastic Young's modulus was carried out byusing an ultrasonic resonance frequency method (GrindoSonic). Thecombined error of the method are ±0.2. To ensure good standard, theobtained Young's moduli values were an average of eight (8) measure-ments. For comparison, the Young's modulus of the commonly usedbiomaterial- Ti\\6Al\\4 V alloy as the reference was also determinedunder the same condition. The details of the equation for the dynamicYoung's modulus (E), has been explained elsewhere [39,40].

2. Results and discussion

2.1. Structural and lattice properties

Material properties very often depend directly or indirectly on thestructure of a substance [41]. The phase transformations which can

Table 1Theoretically predicted structural parameters and thermodynamic properties of the mul-ticomponent Ti\\6Mo\\x\\Nb\\XZr alloys.

Formation energy, EFm (MeV) of the phases a,(Å)(Cal/Exp.)

Compounds Im3m(β) P63/mmm(α)

Cmcm(α″)

P6/mmm(ω)

Ti\\6Mo −48 −28 −41 −35 3.24/3.29Ti\\6Mo\\6Zr −55.3 −35.3 −45.3 −45.1 3.2/3.31Ti\\6Mo\\6Nb −84.1 −44.1 −67.1 −47.1 3.11/3.20Ti\\6Mo\\6Nb\\2Zr −72.7 −22.7 −56.7 −42.1 3.19/3.21Ti\\6Mo\\5Nb\\3Zr −69.1 −27.1 −57.1 −42.7 3.23/3.26Ti\\6Mo\\4Nb\\4Zr −48.9 −28.8 −48.9 −38.9 3.24/3.29

62 P.S. Nnamchi / Materials and Design 108 (2016) 60–67

occur in titanium and other 3d metals with elements such as Nb, Zr, Ta,V, Fe, Mo and Cr form an area of exploration for physical–chemical fac-tors of structural stability. This is largely because, while they possess abcc solid solution (called the β phase) extending over large compositionranges at high temperatures, metastable phases are formed during thedecomposition of β phase, which has a marked effect on the physicalproperties of these alloys. For a property driving design of novel materi-al, it is therefore extremely important to know the structure and to un-derstandwhy a particular structure is adopted. For solids, the formationenergy can be very instructive in this context.We studied the effect of Zrand Nb micro-additions (biocompatible elements) on the relative sta-bility and elastic modulus of the Ti\\6Mo alloy (i.e. initial least stableβ composition) by ab initio calculation. Formation energy, represent ameasure of the energy needed to bind together or dissociate theatoms in a particular crystal structure, and can be used, for instance, toidentify the equilibrium structure of a crystal at low temperature. Thestructure with the highest negative formation energy will be adoptedto reveal thermodynamic stability and understand metallurgical trendsin the alloy. In the present work, the equilibrium formation energies ofthe (omega)ω, (hexagonal)α, (orthorhombic)α″ and (bcc) β structur-al phases for the Ti\\6Mo\\xNb\\xZr xZr alloys (where x = 0–0,2,3,4and 6) were calculated in a similar way to Eq. (1) based on the workby [42]:

E TixMo6þUð Þð ÞFM ¼ Ebulktot TixMo6 þ Uð Þð ÞX

iN

−κμ TixMo6þUð Þð Þ�E TixMo6þUþGð ÞN i¼1ð Þ

− 1−κð Þ:μ TixMo6ð Þð ÞðE TixMo6ð ÞN i¼1ð Þ

�ð1Þ

Here, EFMis the total formation energy of the unit cell of the alloy at aparticular crystal structure used in the present calculation, U representsone or combined atoms of Zr and Nb; N is the total number of atoms persupercell; Etotbulk(TixMo6+U) is thefirst principle calculated total energiesof the respective alloys; μ is the chemical potential of the alloys in itscorresponding bulk phase, and κ is the alloy composition. From thepoint of view of thermodynamics, lower formation energymeans betterstability of a particular crystal structure over another. The formation en-ergies EFM are normalised by the number of atoms on the supercell andthe neighbouring atoms were relaxed as to minimize the total energy.

From the data presented in Table 1, it can be seen that the β (Bcc)phase has the lowest formation energy at the equilibrium volume. Al-though, the orthorhombic α″ phase and bcc β phase exhibited similar

Table 2Nominal composition of the cast Ti\\6Mo\\xNb\\xZr ingots and analysed using inductive-ly coupled plasma atomic emission spectrometry (ICP-AES) (all in atomic %).

S/N Mo Nb Zr Ti balance e/a

1 5.78 – – Ti-balance 4.122 5.92 5.76 – Ti-balance 4.123 5.80 – 6.03 Ti-balance 4.134 6.02 6.04 2.04 Ti-balance 4.105 6.02 5.05 3.02 Ti-balance 4.146 6.01 4.01 4.01 Ti-balance 4.16

stability trends with close values of EFm, the values for the β (bcc)-phase are lower than those of theα″ phase by≈±17.52meV/atoms, in-dicating their nearly equal probability to occur in the samples. In linewith the report by [43]. They suggested that such indicated the averagechemical bondbetween atoms in theβ phasewas stronger than in otherphases. This trend was confirmed experimentally, as explained below.

According to various references [44], the formationof themetastablephases namely.α´,α“andω in Ti\\Mo alloys can be predictedwhen thee/a ratio reaches ≈4.13, but e/a ratio larger than 4.2, β phase becomesthe dominant phase. In this study, the e/a ratio for theTi\\6Mo\\xNb\\xZr alloys is between 4.0 and and4.16 (See Table 2).Therefore, alloys are within the vicinity of martensitic transformationrange; absence of α´ and ω phases in the alloys can be attributed tothe highβ stability index of theNb alloy additions. Based on the findingsin Tables 1 and 2, it is obvious that the micro-addition of Nb and Zr ele-ments prevented the formation of α-phase as well as theω-phase, thatis known to be detrimental to the mechanical properties of Ti based al-loys, but stabilised the α″ and β-Ti phases. In effect, the elastic proper-ties of the alloys were modelled by homogenizing the polycrystallineaggregate of α″/β composite based on these experimental and theoret-ical outcomes. The result is consistent with experimental phenomena[27], as well as other previously observed first principles [45].

The equilibrium lattice constants, a (Å) of Ti\\6Mo\\xNb\\xZr mul-ticomponent alloys are comparedwith experimental data in Table 1 andvisualised in Fig. 2. However, because the trend indicate that the influ-ence of Zr element is not very obvious, to elucidate clearly the effect ofNb additions on the alloys, the lattice constant of Ti\\6Mo\\6Zr alloy(i.e. alloy with no Nb content), 3.2 Å (3.31 Å) for theory (experiment)respectivelywas not represented in Fig. 2. As evident, the calculated lat-tice parameter deceased linearly with increasing Nb concentration. Thex dependence of the theoretical lattice constant can be fitted with therelationship a = 3.3392–0.0396 × (Å), which is in agreement with theexperimental relationship a = 3.5241–0.0419 × (Å). The discrepancybetween the theoretical slope and experimental value is b2, 5%. It canbe seen that the calculated lattice constants match fairly closely withcorresponding experimental data, which indicates that these parame-ters are reasonable for the calculation. Furthermore, the decrease inthe lattice constant with increasing x (Nb) may be attributed to thefact that the atomic radius of Nb (1.46 Å) is slightly smaller than thatof Ti (1.47 Å), while the concentration of Zr (1.6 Å), is less comparedto the Nb content in the alloys. Therefore, the replacement of Ti by Nbmay have induced the shrinkage of the crystal lattice [41]. For the per-spective, it is possible that higher content of Zr in the materials could

Fig. 2. Theoretical equilibrium lattice constants (solid line) of the Ti\\6Mo\\x-Nb\\Xzr(x = 0, 6, 6, 5 and 4 at.% for Nb and x = 6, 0, 4, 3 and 2 at.% for Zr) alloys as a functionof the concentration x (Nb). For comparison, the experiment data are also shown(dashed line) [Ref. 34, 40]. The lines are for eye guidance.

63P.S. Nnamchi / Materials and Design 108 (2016) 60–67

have induced expansion of the crystal lattice and eventual increase oflattice parameter. However, the general trend in the present result hasnot supported fully the assumption. Perhaps, may be due to the neutralnature of Zr element in Ti (i.e., soluble in both α and β Ti phases butstabilisers none). As shown previously [43], the phase stabilising effectof Zr is associated with the phase stability of the alloy arising fromother alloying element other than Zr. Although, the reason for this re-mains unknown, one could infer that the observed trend reflects theco-addition of both elements in the alloys.

The result of the theoretical calculated composition of β phase andthe experimentally determined volumetric fractions are presented inTable 3. From both results, it can be seen that the volume fraction ofβ-phase increases with Nb content, with Ti\\6Mo\\6Nb having themaximum and Ti\\6Mo\\4Nb\\Zr having the least. The increment inthe other Ti\\6Mo\\6Nb N Ti\\6Mo\\6Nb\\2Zr N Ti\\6Mo\\5Nb\\3Zr N Ti\\6Mo\\4Nb\\4Zr shows that the volume fraction of β phaseincreased with increasing Nb content. This is in line with previous ex-perimental data [41], suggesting that Nb rates of approximately 6% al-most completely suppressed ω fraction in Ti\\7.5Mo alloy [27]. On thecontrary, it can be seen that β phase decreased with increasing Zr con-tent in the alloys. This is unsurprising, such an effect of Zr-on thephase stability and martensite start temperature Ms were reported forlow Zr-containing β type alloys, Ti\\Cr\\Zr [46], Ti\\Ta\\Zr [47],Ti\\Nb\\Ta\\Zr [48] and Ti\\Nb\\Sn\\Zr [49]. The fact is evident inalloy without Nb and Zr as shown in Table 3, (i.e. Ti\\6Mo), β phasewas 51% (61%) for theory (experiment), respectively, probably due tothe presence of Mo which is one of the strongest β-stabilising elementin it. In contrast when Zr was added (i.e., Ti\\6Mo\\6Zr), β-phase de-creased by≈5.9%. There are several reports suggesting thatβ stabilisingeffect of Zr is associated with stability arising from other alloying ele-ments other than Zr [43]. However, the reason for this remains unclear.

2.2. Elastic properties

Elastic properties ofmaterials are of interest, because of the informa-tion they provide concerning fundamental solid state properties, such asinteratomic potentials, bonding characteristics, specific heat, thermalexpansion, equation of states. Additionally, the elastic constants deter-mine the response of the crystal to external forces, as characterized bybulk modulus, shear modulus, Young's modulus, and Poisson's ratio,and obviously play an important part in determining the strength ofthe materials. [50]. As mentioned in the introduction, the developmentand application of predictive modelling and simulation aretransforming the materials engineering discovery process. Thus, sys-tematic evaluation of these property via computational modelling cangive a good insight into the characteristic of the materials. In this case,Density functional theory (DFT) in materials studios such as CASTEP isone of the most useful technique to obtain the elastic constants and

Table 3Theoretically predicted polycrystalline constants.

Compounds VβTheory Vβ

Exp B Polycrystalline elastic properties(GPa)

G Eβα″th EExp υ G/B

αTi 0 0 111.3 39.4 131.6 – 0.33 0.35Mo 1 1 120.3 22.5 113.3 – 0.37 0.6Nb 1 1 136.3 58.6 150.8 – 0.31 0.43Zr 0 0 103.2 60 93.0 – 0.36 0.58Ti\\6Mo 0.51 0.61 101.2 41 102.1 109 0.24 0.41Ti\\6Mo\\6Zr 0.48 0.42 190.73 49.2 70.2 69.01 0.26 0.42Ti\\6Mo\\6Nb 0.88 0.83 100 42.18 80.6 86 0.18 0.26Ti\\6Mo\\6Nb\\2Zr 0.71 0.67 114.83 40.46 76.5 81.1 0.19 0.30Ti\\6Mo\\5Nb\\3Zr 0.68 0.63 118.2 46.12 59.1 63.8 0.20 0.32Ti\\6Mo\\4Nb\\4Zr 0.52 0.49 104.1 49.6 32.3 37.8 0.20 0.35Ti\\6Al\\4 V [12] – – – – – 110 – –

properties of new materials, and have been applied by several authors,including [51,52].

Because of the special significance of the polycrystalline bulk modu-lus, shear modulus, Young's modulus, and Poisson's ratio for technolog-ical and engineering applications, we calculated these quantities by T-Matrix hominization of the elastic constants [53]. From the precedinginvestigation, we observed that the new multicomponentTi\\6Mo\\xNb\\xZr alloys exhibit cubic and orthorhombic phases. Inorder to account for structural symmetry in a material consisting ofcubic and orthorhombic structures (See Section 3.1), the number of Cijin the elastic tensor can be reduced from 36 to just 9 for an orthorhom-bic lattice and to 3 for a cubic lattice, due to Cij=Cji, and there beingstrong symmetry in the two lattices [54]. A relationship among thesemeasures are given in several sources [55]. A cubic crystal is character-ized by three constants, C11, C12 and C44, while for orthorhombic crystalswith lower symmetry possessed nine independent elastic constants:C11, C22, C33, C44, C12, C55, C66, C23 and C13. The values of calculated elasticconstants are listed in Table 4. The studies by [55] and [56] have demon-strated that the appropriate stability criteria for a stressed lattice arethose based on enthalpy considerations. Under hydrostatic pressure,the three stability requirements for a cubic crystal are [57]: C11− |-C12, |N0; C11+2 C12N0; C44N0; C12N2 C11, while the requirement foran orthorhombic system leads to the following equations [58]:CijN0; C11+ C22N2 C12 ; C11+ C33N2 C13 ; C11+ C22 C33+2 C12+2C23+2 C13N0. Therefore, the calculated elastic constant of the ortho-rhombic α″-phase type and β-type phase satisfy their stability condi-tions (see Table 3), which are referred to as spinodal, shear and Borncriteria, respectively [54]. Since these compounds has been previouslysynthesized [58,59], the study confirms again the accuracy of the calcu-lated elastic properties. The result also reveal the variation features ofCij's with different Nb content. It can be seen that C12 of thisTi\\6Mo\\xNb\\XZr alloys were twice lower than, C11, but droppedrapidly with increasing addition of Zr, and C12 of Ti\\6Mo\\4Nb\\4Zrexhibited themaximum value. The result indicate that the lower C11 re-flect weak resistance to shear deformation. This is a very importantproperty of alloys that display shape memory capability. However, fur-ther studies is needed to evaluate this aspect of the work.

It is well known that, if the length scale of deformation is much larg-er than the grain size, the response of a material is that of an aggregate[54]. And the elastic properties of an aggregate can be uniquely calculat-ed following the methodology of the integral elastic response of multi-phase polycrystals originally applied by Zeller and Dederichs [55] to de-termine elastic properties of single phase polycrystals with cubic sym-metry. The concept was extended to determine the multiphasecomposites of hexagonal α phase and bcc β-phase in binary Ti\\Nb al-loys by [42]. Here, we apply this criterion to calculate (a) the elasticproperties of the homogenized polycrystalline aggregates of these mul-tiphase Ti\\6Mo\\xNb\\xZr alloys, consisting of orthorhombic mar-tensitic α″-phase and bcc β-phase. For materials with orthorhombicsymmetry, Eqs. (21) and (22) in [43] comes to Eqs. (2) and (3), respec-tively:

15τ44 ¼ a−bþ β 2d−2c−eð Þ þ 3γ d−cþ eð Þ þ ηβΔ0

1−αβ−9γ Kv−~B0

� �þ β β þ 2γð Þ c−dð Þ−2eβγ−

13

ηβ2Δ00

þ3C44−~μO

1−2k C44−~μO� �þ C55−~μO

1−2k C55−~μO� �þ C66−~μO

1−2β C66−~μO� �

0@

1A

ð2Þ

τ11 þ 2τ12 ¼9 Kv −~B

0� �þ 2β d−cþ eð Þ þ 3β2Δ0

3 1−αβ−9γ KV−~B0� �

þ β β þ 2γð Þ c−dð Þ−2eβγ−1ηβ2Δ0

3

" #

ð3Þ

Here, G and B replaces μ⁎ and B⁎ in the equations for β, η, and Δ′. Assoon as G and B have been determined, the homogenized

Table 4Theoretically calculated elastic constants of pure elements (Ti, Mo, Nb and Zr) and the multicomponentTi\\6Mo\\x-Nb\\XZralloys. C ′ is the modulus of basal plane shear {110}⟨110⟩,C44 is the modulus of non-basal plane shear {001}⟨100⟩, and anisotropy factor A = C44/C′ [65].

Materials Elastic constants/GPa

C11 C12 C13 C23 C33 C44 C55 C66 C11−C12 C′ A

Ti 172.6 96.6 70 75.3 179.8 47.9 37.0 38 38 – –Mo 259.7 161.1 131.2 71.6 160.2 57.3 26.2 49.3 49.3 – –Nb 247 143 72.9 130.5 131.9 24 27.5 52 52 – –Zr 141.1 70.2 82.6 85.6 94.3 25.3 34.3 35.1 35.45 – –Ti\\6Mo 183 156.7 42.1 62.5 53.2 103.2 31.5 37 13.15 6.58 15.7Ti\\6Mo\\6Zr 271 175 62.1 48.4 58.2 161 25.7 47.9 48 24 6.7Ti\\6Mo\\6Nb 191.3 145.7 42.1 40.6 53.2 123 27.1 22.7 22.8 11.4 10.8Ti\\6Mo\\6Nb\\2Zr 254 182.1 27.5 82.8 12.4 144 44.8 36 35.95 17.9 8Ti\\6Mo\\5Nb\\3Zr 258 187 23.5 64.4 12 114 38 40.1 35.5 17.8 6.4Ti\\6Mo\\4Nb\\4Zr 269 189.6 19 50.78 19 131 33.73 50.7 50.7 25.4 5.2

64 P.S. Nnamchi / Materials and Design 108 (2016) 60–67

polycrystalline Young's modulus (E), and Poison's ratio (ν) can be ob-tained using standard elastic relationships: The homogenized polycrys-talline Young's modulus and Poison's ratio are calculated using:

E ¼ 9BG3Bþ G

; ð4Þ

ν ¼ 3B−2G3 2Bþ Gð Þ ð5Þ

Our calculated results for B, G, E, ν and G/B are listed in Table 3. Theoutcome reveal a slight variation of featureswith elemental substitutionin the materials. The Young's modulus (E) deceases (increases) sharplywith increasing Nb (Zr) content. This implies thattheseTi\\6Mo\\xNb\\XZr alloys are responsive to the Nb and Zr addi-tions. The alloys with the highest Nb addition of 6%, namelyTi\\6Mo\\6Nb and Ti\\6Mo\\6Nb\\2Zr have the maximum elasticYoung's modulus ~80.6, 76.5 respectively, while Ti\\6Mo\\4Nb\\4Zralloy has a lower E of 32.3 GPa. The elastic Young's modulus of32.3 GPa is significantly low and comparable to natural bone andmuch lower than other alloys already been used for biomaterials im-plant applications.

There is a remarkable difference in the precursory elastic softeningbetween some Ti-(Mo, Nb) alloys and other β phase alloys. Most β-phase alloys are known to show softening only in elastic constant C′(i.e., C11 and C12) [60–64], with other elastic constants behaving normal.This yields an anisotropy factor A (=C44/C′) as large as 10–20 [65]. Thevalue of ~15.68 for Ti\\6Mo alloy is typical. In sharp contrast, due to Nband Zr micro-additions, the present Ti\\6Mo\\xNb\\xZr alloys showssoftening in both C′ and C44 (See Table 4); this leads to a very low ornear vanishing anisotropy, which further decreases with intended low-ering of elastic modulus. Such difference has been noticed by previousstudies [66–67]; but the underlying meaning has not been figured outuntil very recently [68]. By description, C′ correspond to the {110}⟨110⟩ shear modulus, which is naturally related to the Zener instabilityand to the formation of basal-plane-basedmartensite (α“). The bondingor phase transition in bcc is also inside this plane. To the contrary,C44 isthe {001} ⟨100⟩ shear modulus, which at first sight seems to have noth-ing to do with the formation of the orthorhombic α“martensite as ob-served in this Ti\\6Mo system. However, for cubic crystals, C44 is alsoequal to {001}⟨100⟩ shear modulus. This {001}⟨100⟩ is just what is re-quired for the orthorhombic distortion to form theα″ structure. The de-scription is consistent with the experimental trend and presence of theα″ phase in the alloy; which is said to presents an elastic young's mod-ulus about half of the β phase [47]. Therefore, we can assume that addi-tion of elements that softening both elastic constants, (C´ and C44) ismain reason for the unique elastic Young's modulus of theTi\\6Mo\\xNb\\xZr alloys studied.

As we can see in Table 3, the Shear modulus of theTi\\6Mo\\xNb\\xZr alloys studied is less than the Bulk modulus. This

indicates that the alloys exhibited weaker resistance to shape changethan to volume change, in line with the low resistance to Shear defor-mation, C44 (See Tables 3 and 4). By considering the force required topropagate a dislocation is proportional to Gbwhere b is the Burgers vec-tor, Pugh suggested that the ratio G/B indicates the intrinsic ability of acrystallinemetal to resist fracture and deform plastically [69]. Empiricalobservations implicate G/B as indicating well brittleness (when G/B) N 1.75) or tough behaviour (when G/B b 1.75). The results indicatethat the values of G/B for the alloys are lower than the critical value sep-arating brittleness from ductility. From this perspective, we can con-clude that these Ti\\6Mo\\x\\Nb\\XZr compounds have goodmechanical behaviours. Furthermore, the lower and the upper limitsPoison's ratio (ν) are given as 0.25 and 0.5 for central force solids, re-spectively [55]. The calculated Poison's ratio of these multicomponentTi\\6Mo\\xNb\\xZr alloys is mainly between 0.18 and 0.26. Chenet al. in the 1970s [70,71] established a link between a low ν and easeof “atomic regroupings” and plasticity is predicted as being higher inmaterials with low ν, where non-central bonding was believed toprevail.

2.3. Electronic and bonding properties

The total decomposed DOS of Ti\\6Mo\\x\\Nb\\xZr (where x = 0to 6) and pure Nb/Zr are shown in Figs. 3 and 4, respectively. From thesefigures, we can see that there is a deep valley close to the Fermi level(EF) and this valley is referred to as a pseudogap. This pseudogap indi-cates the presence of covalent bonding in the multicomponentTi\\6Mo\\x\\Nb\\xZr alloys. Quite often the pseudogap separatesthe bonding states from the antibonding/nonbonding states [72]; andthere exists experimental evidence that links the theoretical electronicstructure/position of (EF) with respect to the pseudogap [73] to the sta-bility of a broad class of Ti-based alloys using DFT ab initio calculations[37,46]. The vertical dotted line in the figure is the Fermi level (EF).From Figs. 3 and 4, we observe that the Fermi level falls below thepseudogap in Ti\\6Mo\\xNb\\xZr. Thus, not all the bonding statesare filled and some extra electrons are required to attain maximum sta-bility in these compound. The result is in line with the e/a ~ 4.0–4.16value for the alloy, which lies within the vicinity of martensitic transfor-mation range. This can be the reason for the presence of an orthorhom-bic α“metastable phase in the alloys. However, the decomposed DOSshows that the density of states at the Fermi level is dominated by Ti dstates [74]. Even though considerable amount of s and p states are pres-ent in the occupied part of the DOS, their contribution at the Fermi levelis very small. The DOS curves also show that the larger part of the broadTi d DOS is above, EF.

By comparing the DOS of the multicomponent Ti\\6Mo\\xNb\\xZrwith the pure elements (Nb and Zr), a clear difference in their d-statesare evident, indicating that change in bonding originates mainly fromthe contribution of 4d states in the alloys. Thus, affirming the correlationbetween the micro-alloying additions and electronic properties of the

Fig. 3. Calculated total densities of states of the Ti\\6Mo and the Ti\\6Mo\\xMo\\xZr multicomponent alloys (the dotted lines are the Fermi level). (a) Ti\\6Mo; (b) Ti\\6Mo\\6Zr(c) Ti\\6Mo\\6Nb; (d) Ti\\6Mo\\6Nb\\2Zr; (e) Ti\\6Mo\\5Nb\\3Zr; (f) Ti\\6Mo\\4Nb\\4Zr.

Fig. 4. Calculated total densities of states of the pure (a) Nb and (b) Zr elements.

65P.S. Nnamchi / Materials and Design 108 (2016) 60–67

66 P.S. Nnamchi / Materials and Design 108 (2016) 60–67

materials. This is much revealing in Fig. 3b and f belonging toTi\\6Mo\\6Zr and Ti\\6Mo\\4Nb\\4Zr alloys (high Zr content) re-spectively. When comparing the pseudogap, it is evident that only thepseudogap of the two compounds seem to fall below their Fermi level,thus indicating that not all the electrons are filled and extra electronsare needed for the two compounds to attain maximum stability. Thisnot surprising as Zr is a less β stabiliser element than Nb, although,the exact reason remains unclear.

A further look on the energies higher than −10 eV (close to theFermi) or antibonding features for the different multicomponent alloysshows that aside from Ti\\6Mo\\6Zr (i.e., highest Zr content) that ex-hibited the lowest energy of 14.2 eV, other multicomponent alloys arebetween 20 and 24 eV. It is also striking that the formation energies ofTi\\6Mo\\6Zr and Ti\\6Mo\\4Nb\\4Zr alloys (Section 3.1) are lessnegative, thus indicating less stability when comparedwith other alloysand reliability of calculated result. This outcome is also in linewith theire/a range ~4.0–4.16, which lies in the vicinity of metastable martensiticphase transformation. Thus, might be related to the higher content of Zrand rise in exothermic energy (i.e. negative formation energy)/Ti ratioin the alloy is caused by the addition of neutral elements, viz.: Zr. (I.e.completely miscible in bothα and β phase in the Ti\\Zr binary diagram[43]. Perhaps, it is Nb not Zr addition that boosted the d-hybridisationand indirectly the stability of the alloys studied. This may be the reasonfor the high volume of orthorhombic α″metastable phase in theTi\\6Mo\\6Zr and Ti\\6Mo\\4Nb\\4Zr alloys with high Zr content.From this viewpoint, the results indicate that the structural stabilitychange in the Ti\\6Mo system is connected with the lowering ofN(EF). This is due to the hybridisation of the d-electronic states of thealloying atoms, but with also the statistical weight of Ti d-states at EF.In order to understand the intrinsic mechanism by which alloying ele-ments influence thermodynamic phase stability, we compared by inte-grating the DOS multiplied by the energy E up to the energycorresponding to the highest occupied states for Ti\\6Mo\\6Zr withthe other multicomponent alloys. In the case of themulticomponent al-loys, only Ti\\6Mo\\4Nb\\4Zr alloy is shown here, because their inte-grating the DOS energy obtained from all the multicomponent alloysare similar. The result is presented in Fig. 5. We can see that the energyis lowered by the alloying additions. The observation is in a good agree-ment with the DOS of the bcc phase obtained by [75], characterized bysimilar existence of a deep valley or a pseudogap in the Fermi levels,separating out bonding from antibonding electron states [76,77].

Furthermore, to gainmore insight into the influence of the additionson the bonding behaviour of these newly developed alloys, we havegiven considerable attention to their electron density characteristics.The electron localisation functions (ELF) in the (110) plane which is

Fig. 5. Integrated total density of state of (a) Ti\\6Mo alloys (a) and(b) Ti\\6Mo\\4Nb\\4Zr, revealing the effect of alloying element. The vertical dotted lineindicates the Fermi level.

the plane of stability in cubic crystals for Ti\\6Mo alloy (i.e. initialmate-rial) and Ti\\6Mo\\4Nb\\4Zr alloys are shown in Fig. 6(a) and (b) re-spectively. Since all the ternary and quaternary alloys charge densitiesplots have exhibited similar electron localization function,Ti\\6Mo\\4Nb\\4Zr alloy. Thus, Fig. 6 (b) is shown as the representa-tive. It is evident from the electron charge densities that significant elec-tronic sensitivity was apparent due to Nb and Zr micro-additions. It canbe seen that each Ti atom is surrounded by near–neighbour eight(consisting of Mo, Nb and Zr) atoms. Thus, forming network of metallicbonds around Ti atoms, owing to the interaction between Ti atoms andthe other near-neighbour transition atoms. The electron charge densityof Ti\\6Mo alloy is symmetric with less substantial changes in theTi\\Mo bonding having an electron density of 2.05 electrons/Å3,which is very similar to pure Ti. This imply that the influence of 6atom of Mo alloying elements on the electronic structures of Ti is notas strong. The calculated charge densities for the multicomponentTi\\6Mo\\xNb\\xZr from our GGA calculation is within 2.15–2.5 elec-trons/Å3. This value is found to be in good agreement with the valueof 2.4 electrons/Å3 obtained by [80] in binary Ti\\Nb. The high ELFvalue in Fig. 6(b) compared to 2.05 for the binary Ti\\6Mo indicatethat the Nb and Zr alloying additions led to an obvious anisotropy inthe bonds between ions in the electron localization function (ELF) of al-loys. In effect, the solute atoms were mainly distributed around the Tiatom, forming a ‘triangular’ shape structure, with Ti atoms located inthe middle. This indicates that the influence of alloying elements onthe charge density is quite localised, and only the Ti atoms locatedclose to the vicinity of the alloying atom were slightly more localisedat the centre. Previously result by [73] suggest that this type of metallicbond distribution can enhance resistance to shear deformation and im-prove β phase stability.

3. Conclusions

We have used the CASTEP method to perform a set of first principletotal energy calculations to determine the equilibrium structural pa-rameters, elastic and electronic properties aiming to develop novellow elastic Young's modulus, biocompatible Ti\\Mo\\Nb\\Zr alloy fororthopaedic implant applications. In order to evaluate relative thermo-dynamic stability,we calculated the formation energies of the geometricoptimized structures that occur in titanium based alloys, namely Im3m(β), P6/mmm (α), Cmcm (α″) and p3 ̅m1 (ω) structures for the fiveTi\\6Mo\\XNb\\Xzr alloys using GGA (PBE).

The results reveal a slight variation of features with elemental sub-stitution in the materials. For instance, the polycrystals composite waspredicted to consist of mainly cubic β-phase with a fairly high Nb con-tent (~6 at.%) and a matrix of orthorhombic α″ phase and cubic β-phase with lower Nb and a fairly high Zr content (~4 at.%). Despite thefact that our theoretical thermodynamic analysis overestimates the vol-ume fractions of β phase compared with those experimentally found,the predicted compositional trend was qualitatively correct. Thus, theoverall the ground state equilibrium phase and elastic constants obtain-ed from the GGA (PBE) calculations are found to be in good agreementwith other experimental data.

The Young's modulus of the alloys were obtained based on the inte-gral elastic response of the homogenized polycrystals consisting of α″and β phases. The Young's modulus (E) deceases (increases) sharplywith increasing Nb (Zr) content. This implies thattheseTi\\6Mo\\xNb\\XZr alloys are responsive to the Nb and Zr addi-tions. The alloys with the highest Nb addition of 6%, namelyTi\\6Mo\\6Nb and Ti\\6Mo\\6Nb\\2Zr have the maximum elasticYoung's modulus ~80.6, 76.5 respectively, while the lower E ofTi\\6Mo\\4Nb\\4Zr alloy (32.3 GPa) can be ascribed to the sensitivityof the elasticmoduli E to the elastic constants,C11, C12,C44 softening, con-sistentwith the presence of theα″ phase in the alloy; which presents anelastic young's modulus about half of the β phase.

Fig. 6. Theoretically calculated ELF on the (110) surface of electron density of (a) binary Ti\\6Mo alloy and (b) Ti\\6Mo\\6Nb\\2Zr alloy.

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