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Hardness of Al-based quasicrystalsevaluated via cluster-plus-glue-atommodelHua Chena, Lingjie Luoa, Jianbing Qianga, Yingmin Wanga &Chuang Donga

a Key Laboratory of Materials Modification (Ministry of Education),Dalian University of Technology, Dalian 116024, ChinaPublished online: 09 Apr 2014.

To cite this article: Hua Chen, Lingjie Luo, Jianbing Qiang, Yingmin Wang & Chuang Dong (2014)Hardness of Al-based quasicrystals evaluated via cluster-plus-glue-atom model, PhilosophicalMagazine, 94:13, 1463-1477, DOI: 10.1080/14786435.2014.887863

To link to this article: http://dx.doi.org/10.1080/14786435.2014.887863

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Hardness of Al-based quasicrystals evaluated viacluster-plus-glue-atom model

Hua Chen, Lingjie Luo, Jianbing Qiang, Yingmin Wang and Chuang Dong*

Key Laboratory of Materials Modification (Ministry of Education), Dalian University ofTechnology, Dalian 116024, China

(Received 9 October 2013; accepted 19 December 2013)

In this paper, the hardness of ternary Al-based quasicrystals was assessedthrough an application of the cluster-plus-glue-atom model. In this model, anystructure is decomposed into a first-neighbour strongly bonded cluster partand second-neighbour weakly bonded glue-atom part so that the overallstructural information is condensed into a local structural unit [cluster](glue atom)x. For quasicrystals, the averaged local units are formulated as[icosahedron] TM0,1(Transition Metal) and could be visualized as single icosa-hedron packing. Then, the hardness of quasicrystals was related to the ruptureof weak inter-cluster bonds. Typically, theoretical hardness values of 8–9 GPawere obtained using 19 broken inter-cluster bonds, which accounts for abouthalf of all the surface bonds of an icosahedron in the Mackay-type environ-ment. The unit cluster formulas would act as rigid units during deformationand cracking.

Keywords: quasicrystals; cluster-plus-glue-atom model; hardness

1. Introduction

Although hardness (H) has been widely accepted as an essential parameter of materials,it remains far from being completely understood because the measurements involveelastic, plastic and even crack processes [1–4].

Extensive studies have been made to correlate hardness with elastic moduli, such asbulk modulus (B) [5–7], shear modulus (G) [8–11] or c44 [12] and Young’s modulus(E) [11,13]. In fact, the hardest material diamond (~96 GPa) possesses the highest bulkmodulus (443 GPa) and shear modulus (535 GPa) in existence [14]. It should be notedthat the dependence of hardness on elastic moduli was mainly derived from single-setcovalent-bonding materials, and the situations involving complex bonding are not clearyet.

Hardness also depends strongly on plastic deformation, which is mediated by themotion, multiplication and interactions of dislocations [3]. As established by Tabor[15], the Vickers hardness (Hv) is a measure of the plastic yield properties of metals byHv ≈ 3Y, where Y is the yield strength. A simple linear relation also holds for metallicglasses (see Figure 1 in Ref. [16]), although the proportionality constants are somewhatlower than 3 [17,18].

*Corresponding author. Email: dong@dlut.edu.cn

© 2014 Taylor & Francis

Philosophical Magazine, 2014Vol. 94, No. 13, 1463–1477, http://dx.doi.org/10.1080/14786435.2014.887863

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Since cracks are closely involved in the hardness measurements, the theoretical frac-ture toughness, which is proportional to (γsG)

1/2, has been suggested to characterize thehardness of brittle materials, where γs is the surface energy and G the shear modulus[19]. It has been verified that the Vickers hardness of polycrystalline materials showeda good correlation with Pugh’s modulus ratio k =G/B [20]: Hv = 2(k2G)0.585 − 3 [21].The lowest brittle cleavage stress [22] and atomization energy [23] could also serve asdirect indicators of hardness by referring to density functional theory calculations andthermodynamic properties.

Gilman explored the origin of hardness for covalent substances from their electronicstructures, linking hardness directly with the LUMO (lowest unoccupied molecular orbi-tal)-HOMO (highest occupied molecular orbital) gap [24]. The microscopic process of ahardness test for covalent solids involves bond breaking and then re-linking, whichmeans that two electrons are excited from the valence band to the conduction band, andthus, the resistance to the applied force can be characterized by energy gap Eg [25].Based on Gilman’s theory, the hardness of typical covalent crystals was calculated byGao et al. using a semi-empirical expression H (GPa) = ANaEg, where Na is the cova-lent bond number per unit area and A is a proportional coefficient [26]. By introducingthe concept of bond strength, Šimůnek and Vackář calculated the hardness of covalentand ionic crystals by the first principles calculations [27]. Li et al. explored the intrinsicorigin of hardness from bond electronegativity, which characterizes the electron-holdingenergy of a bond [28]. The reference energy in Šimůnek and Vackář’s scheme and the

Figure 1. Schematic presentation of the cluster-resonance model, where the shell atoms are onthe average arranged in a spherical manner as governed by Friedel oscillation.

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common scale of electronegativity in Li et al.’s are essentially the same despite of somedifferences in the determination of atomic radius. It should be noted that in all the mod-els above, hardness of multi-component compounds is described as a weighted geomet-ric average of the hardness of all binary subsystems, each of which contains only onetype of bonding. From this point of view, none of these models give a complete pictureof the hardness at the atomic level, and the dependence of hardness on atomic structureremains an open issue.

In this paper, we attempt to assess the hardness of Al-based quasicrystals (QCs)with the aid of our cluster-plus-glue-atom model [29]. In the following, the cluster-based model will be first introduced to interpret compositions of Al-based QCs, andthen, a novel atomic-level hardness model for QCs is established by considering therupture of the weakest bonds within the framework of the cluster-plus-glue-atom model.

2. 24-electron cluster formulas of Al-based quasicrystals

2.1. 24-electron rule of cluster formulas within the framework of thecluster-resonance model

It has been revealed that binary Al–TM (Transition Metal) QCs all conform to unifiedcluster formulas [icosahedron](glue)0,1 accommodating nearly 24 valence electrons perunit cluster formula (e/u) [30], in analogy to the newly developed cluster-resonancestructural model [31,32], as schematically depicted in Figure 1, which is a combinationof the cluster-plus-glue-atom model [29] and Häussler’s global resonance model [33].Here, the subscript 0, 1 refers to the number of glue atoms in the cluster formulas oficosahedral QCs and decagonal QCs, respectively. The former model has been devel-oped to decipher compositions of complex metallic alloys including QCs and bulkmetallic glasses [29], where a structure of any kind is dissociated into a first-neighbourcoordination cluster part and a glue-atom part situated at second-neighbour distance, orexpressed in cluster formula [cluster](glue atom)x. The cluster part should be the moststrongly bonded part so that in the presence of multiple clusters in a given structureonly the cluster possessing the densest atomic packing and the least cluster overlapenters into the cluster formula of the structure [34]. This model actually depicts theprincipal short-range-order feature that can be presumably inherited in relevant struc-tures. As stated below, the clusters for QCs are icosahedra adopted from crystalline ap-proximants [30]. In the global resonance model, a momentum-based resonance betweenthe electronic system with Friedel wavelength λFr = π/kF and the static atomic structurecharacterized by a wavelength 2π/Kp was established, kF being the Fermi vector and KP

the pseudo-Brillouin zone boundary, a consequence of which is spherical oscillation ofaveraged shells with respect to any centre atom following rn = (n + 1/4)λFr, where rn isthe radius of the nth shell, n is the shell number and λFr is the Friedel wavelength. Notethat the first nearest neighbour shell atoms of each cluster (n = 1) are located at an aver-age radius r1 = 1.25λFr in the cluster-resonance model. By introducing Fermi vectorkF ¼ ð3p2qa � e

aÞ1=3 into the resonance condition kF = π/λFr = 1.25π/r1, e/a is obtainedas

e

a¼ 1:253p

3� 1

qa � r31; (1)

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with ρa being the atomic density. After assuming ideal dense packing of the r1 clustersto assess ρa, the total number of valence electrons per unit cluster formula e/u wasobtained to be constant approaching 24 [32], irrespective of alloy compositions.

2.2. Determination of the principal icosahedral clusters in approximants

Compositions of binary Al–TM QCs have been well deciphered via the 24-electroncluster formulas [30] constructed by using the principal icosahedral clusters identifiedfrom the corresponding approximants. The principal cluster [34] is supposed to be themost strongly bonded part that represents the principal short-range order feature. Such astable local structure can be inherited after phase transition. Dense atomic packing andcluster isolation criteria have been raised to judge the right principal cluster from multi-ple clusters in a given structure. The former dense packing criterion tells that the princi-pal cluster should be the densest local part with reference to the average structure. Thedeviation from critical radius ratios R* [35], expressed by Δ = (R0/R1 − R*)/R*, wasemployed to simplify the description of the packing efficiency of a given cluster, withR0 and R1 being the atomic radii of the central atom and the averaged first-shell atoms[36]. For the ideally close-packed icosahedron, the R* ratio is 0.902. The latter isolationcriterion indicates that the principal clusters should be well separated from each other inthe crystal structures and represents the major structural feature. The isolation degree ofa certain cluster can be quantified by comparing the number of atoms of a cluster tothat after sharing. A large isolation degree should be preferred in distinguishing theprincipal cluster from multiple cluster candidates.

To illustrate the selection process, take the generally accepted approximant Al13Fe4phase as an example. There are 15 Al and 5 Fe non-equivalent sites in the unit cell(Pearson symbol mC102, space group C2/m). Note that one Al site has a partial occu-pancy of 0.7 [37] or 0.92 [38], and here for simplicity, we treat this occupancy as 1.Two types of icosahedra Al9Fe4 (the first element refers to the central atom) and Fe2Al11are present, as shown in Figure 2. The Goldschmidt atomic radii for Al (0.143 nm) andFe (0.127 nm) are used; thus, the packing efficiencies of Al9Fe4 and Fe2Al11 are (0.143/((0.143 × 8 + 0.127 × 4)/12) − 0.902)/0.902 = 15.2% and (0.127/((0.143 × 11 + 0.127 × 1)/12) − 0.902)/0.902 = −0.6%, respectively. Since these two icosahedra are reduced to thesame effective size of 9 atoms (Al9Fe4 reduced to Al7Fe2 and Fe2Al11 to Al8Fe) due tosharing between neighbouring ones, we conclude that Fe2Al11 should be the principalicosahedron cluster.

2.3. Cluster formulas and structural visualization of quasicrystals

The key points for describing Al–TM binary QCs, as stated in reference [30], aresummarized below:

(1) Al–TM binary QCs of the icosahedral type are averaged into an isolatedicosahedron itself [isolated icosahedron] and those of the decagonal type into[isolated icosahedron] TM1.

(2) The number of valence electrons per unit cluster formula e/u is close to 24, justlike in ideal metallic glasses [32]. The valence electron contribution of Al, e/aAl,is generally accepted as +3. By assuming the Al−3p1 electron being in complete

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Figure 2. (colour online) Two types of icosahedra Al9Fe4 (a) and Fe2Al11 (b) in the Al13Fe4phase structure, projected along the c-axis.

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hybridization with the empty TM-d states and that e/a contribution from pureTMs is 1, the e/aTM assignment of a TM element in the binary Al–TM alloys isobtained, e/aTM = 1 − (100 − x)/x, x being the atomic percentage of TM [39].

Typical binary Al–TM QCs (see Ref. [40] and the references therein) are thenexplained using the cluster formulas containing nearly 24 electrons:

� Icosahedral QCs Al86(Fe,Mn,Cr)14, Al84V16 and Al83.5Ru16.5 are formulated as[TM2Al11] = Al84.6TM15.4, with e/u = 24.0 (e/aTM = 1 − (100 − 15.4)/15.4 = −4.5).

� Decagonal QCs A177.6Mn22.4 as [Mn2Al11]Mn1 = Al78.6Mn21.4 (e/u = 25.0, e/aMn

= −2.67); Decagonal QCs Al72–73Co27–28 and Al76–70Ni24–30 as [(Co,Ni)3Al10](Co,Ni) = Al71.4(Co,Ni)28.6 (e/u = 24.0, e/aCo,Ni = −1.5).

The above examples tell that only two kinds of icosahedra are involved in formulat-ing binary Al–TM QCs, namely TM2Al11 and TM3Al10.

In the present work, ternary QCs are addressed. We will see in the following thatternary QCs can be perfectly explained by the same base binary formulas after appro-priate third element substitutions. Most of these ternary QC compositions have beendocumented in Ref. [29].

(1) Ternary QCs developed from icosahedron TM2Al11:� [TM2(Al8Cu3)] = Al61.5Cu23.1TM15.4, explaining icosahedral QCs

Al62.5Cu24.5Fe13, Al65Cu20Mn15, Al65Cu24Cr11, Al65Cu20Os15 andAl61.6Cu24.8Ru13.6;

� [TM2Al9(Cu,Pd)2] = Al69.2(Cu,Pd)15.4TM15.4, explaining icosahedral QCsAl68.1Cu16.1Ru15.8 and Al70Pd15Mn15 [41];

� [(TM1Pd1)(Al9Pd2)] = Al69.2Pd23.1Mn7.7, explaining icosahedral QCAl68.7Pd21.7Mn9.6; and

� [TM1(TM0.5Pd0.5)Al10Pd1]Pd1 = Al71.4Pd10.7Mn17.9, explaining decagonal QCAl69.8Pd12.1Mn18.1.

(2) Ternary decagonal QCs developed from icosahedron TM3Al10:� [(Co1Ni2)Al10]Co1 = [(Ni1Co2)Al10]Ni1 = Al71.4Co14.3Ni14.3, explaining decag-

onal QC Al72Co15.5Ni12.5;� [(CoCu2)(Al9Cu)]Co1 = Al64.3Cu21.4Co14.3, explaining decagonal QC

Al64.3Cu21.9Co13.8 [42]; and� [Co(Cu0.5Co0.5)CuAl9Cu]Co1 = Al64.4Cu17.8Co17.8, explaining decagonal QC

Al66Cu17Co17 [43].

In these cases, the e/a contributions from TMs cannot be directly assignedbecause of the involvement of third elements. Alternatively, Equation (1)ea ¼ 1:253p

3 � 1qa�r31

can be used, which involves only atomic density ρa and clusterradius r1. ρa is calculated via an empirical expression ρa = NAρ/MA, whereNA = 6.02 × 1023 mol−1 is the Avogadro constant, MA =

PCiMi the average mole mass

of the alloy, Ci and Mi the atomic fraction and atomic mass of the ith element, ρthe mass density estimated by ρ =

P(CiMi)/

P(CiMi/ρi) [44], where ρi is the mass

density of a pure element i. In order to check the validity of this estimation, wehave calculated the theoretical atomic density of icosahedral QC-Al62.3Cu24.9Fe12.8

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using the expressions above (67.6 nm−3), which coincides well with the experimentalvalue of 66.7 nm−3 [45]. With the help of the estimated atomic density ρa and theradius of the cluster r1, e/a is then assessable. Knowing also the number of atomsin unit cluster formula Z, the total number of valence electrons per unit cluster for-mula is then obtained.

Take the ternary cluster formula [MnPdAl9Pd2] = Al69.2Pd23.1Mn7.7 (Z = 13), for exam-ple, which is constructed with one Mn (0.126 nm) and two Al (0.143 nm) in the shell of ico-sahedron Mn2Al11 replaced by three Pd (0.127 nm) in the base binary formulas [Mn2Al11]for icosahedral QC-AlMn. Icosahedron Mn2Al11 is extracted by a careful analysis of hexag-onal λ-Al4Mn (space group P63/m) [46] and μ-Al4Mn (space group P63/mmc) [47] phases,which have been shown to be closely related to icosahedral QC-AlMn [48]. Both crystalstructures possess giant unit cells with inherent disorder, where 108 Mn sites and 30 Al sitesin μ-Al4Mn and 102 Mn sites and 36 Al sites in λ-Al4Mn are surrounded by an icosahedralcoordination [30]. Note that all icosahedra centred by Mn atoms are nearly close-packed,with an average composition of Mn2.3Al10.6. Here, the most frequent icosahedron Mn2Al11was used to construct the icosahedral QC. By substituting atomic mass of Al (MAl = 26.982gmol−1), Pd (MPd = 106.42 g mol−1) and Mn (MMn = 54.938 g mol−1) along with the corre-sponding mass density (ρAl = 2.7 g cm−3, ρPd = 12 g cm−3 and ρMn = 7.47 g cm−3) into theempirical mass density equation, ρ=

P(0.692 × 26.982 + 0.231 × 106.42 + 0.077 ×

54.938)/P

(0.692 × 26.982/2.7 + 0.231 × 106.42/12 + 0.077 × 54.938/7.47) = 5.0 g cm−3 forternary icosahedral QC [MnPdAl9Pd2] = Al69.2Pd23.1Mn7.7 is obtained, which agrees wellwith the measured density of 5.1 g cm−3 for icosahedral QC-Al68.7Pd21.7Mn9.6 [49]. Theatomic density ρa = 6.02 × 1023 × 5.0/(0.692 × 26.982 + 0.231 × 106.42 + 0.077 × 54.938) =63.9 nm−3. It should be pointed out that there are differences in the average r1’s of icosahe-dron Mn2Al11 centred by different Mn sites, among which the smallest one (0.2634 nm)derived from approximant μ-Al4Mn should be used to approach a dense packing of atoms.The total number of valence electrons per unit cluster formula e/u is then assessable:e=u ¼ ðe=aÞ � Z ¼ 1:253p

3 � 163:9�0:26343 �13 ¼ 23:0.

Further analysis shows that the total electrons per unit cluster formula e/u of theseternary Al-based QCs are all close to 24. The e/u values of Al–Cu–Fe and Al–Cu–RuQCs are relatively low, about 20, which may be attributed to the relatively wide rangeof centre-shell atomic distances in the clusters which lead to large errors in determiningthe cluster radius r1. For instance, the 12 shell atoms of the Fe2Al11 icosahedron aredistributed on twelve distances 0.2304, 0.2462, 0.2476, 0.2515, 0.2575, 0.2575, 0.2580,0.2616, 0.2619, 0.3185, 0.3445, 0.3476 nm, and the real first resonance distance r1 maynot fall exactly at the averaged radius 0.2736 nm. An r1 value of 0.2538 nm givesexactly 24 electrons. In contrary, a small distribution of centre-shell distances signifies aperfect cluster [30], and the averaged radius can be reliably taken to be the first reso-nance distance r1. Table 2 summarizes the measured Vickers hardness values and theproposed 24-electron cluster formulas of ternary Al-based QCs. The radius of the clus-ter r1 in ternary Al-based QCs is also influenced by substitutions of third elements ofdifferent radii.

In accordance with the above QC composition formulism and local structuredescription, Al-based QCs are described with cluster formulas using isolated clustersplus a fixed number of glue atoms, so that they can be visualized as isolated icosahe-dron packing. This is of course not the real structure but only represents the averagelocal structure. However, this visualization tells that only three kinds of bonds are

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presumably involved in a cluster formula unit, namely centre-shell, shell–shell andshell-glue (refer to Figure 1). Also, the unit cluster formulas share the same number ofvalence electrons close to 24, irrespective of the alloy compositions, thus mimicking themolecular formulas for common chemical substances. As seen below, such a structuraldescription provides the key bonding information necessary for the hardness calculation:a hardness indentation can be simplified into the breaking of the weakest bondsassociated with unit cluster formula.

3. Theoretical hardness model in association with the rupture of the weakestinter-cluster bonds

Quasicrystals are notorious for their brittleness and may experience microscopic fractureduring indentation hardness measurements, which is surely associated with the ruptureof atomic bonds. In this sense, hardness of QCs should be directly governed by thebond strength. Semiconductor-like electronic transport properties and covalent-bondingnature have been extensively demonstrated in Al-based QCs [50–53]. Followingthe suggestion that hardness is related to the dissipated energy per unit volume of theindentation impression [54], we propose here that hardness of QCs corresponds tothe breakage of the weakest bonds per unit volume:

H ¼ New=V ; (2)

where N is the number of ruptured weakest bonds in the corresponding volume V andεw the bond enthalpy. It is obvious that only partial weakest bonds distributed in the Vvolume should be involved since the facture occurs on a surface rather than over theentire volume. Note that this expression implies a crucial role of the weakest bonds indetermining the hardness of QCs among various types of bonds, which can be recog-nized as the “Cask Effect”. This has been verified by the fact that the calculated hard-ness of the weakest bonds is much closer to the experimental value [55]. The cluster-plus-glue-atom model actually provides all necessary information about the types andthe numbers of the bonds.

By introducing the volume of each cluster formula expressed as Z/ρa into Equation(2), Z being the number of atoms in unit cluster formula and ρa the atomic density(number of atoms per unit volume), we obtain:

H ¼ new=ðZ=qaÞ; (3)

n being the number of the ruptured weakest bonds per unit cluster formula. Hardness is thenaccessible via parameters of the cluster formula (n, ρa, and Z) and the bonding strength εw.

After establishing the cluster formulas and henceforth the local first- and second-neighbour structural models for QCs, it is possible to assess the bonding enthalpies andthe number of the weakest bonds, presumably residing in the linkage between adjacentclusters or shell–shell bonding. Indeed, clusters have been proved to be intrinsic obsta-cles to cracks in QCs by molecular dynamics simulations [56]. We further notice thatthe radius of the elementary cluster (about 0.4–0.5 nm) observed by in situ cleavage ofsingle icosahedral QC-Al70.5Pd21Mn8.5 [57] is in agreement with the cluster outer radiusin our model, being equal to r1 plus the average atomic radius of the shell atoms R1, or0.2634 + 0.2634/1.902 = 0.4019 nm for the icosahedral QC-AlPdMn formulated as[MnPdAl9Pd2] = Al69.2Pd23.1Mn7.7.

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In all the cases, the third element Cu, Pd, Co or Ni substitutes certain shell sites ofthe base icosahedron, whereas the second atom Fe, Ru, Mn, Co or Ni (by second herewe mean those TM’s that form binary QCs with Al) remains unchanged at the centre ofthe icosahedron and at the glue sites connecting adjacent icosahedra. As a consequence,the weakest inter-shell bonds in a QC are a mixture of Al–Al, Al–TM (TM = Mn, Fe,Co, Ni, Cu, Ru and Pd) and TM–TM bonds, a weighted average of which is used togive the average bond enthalpies of the weakest bonds.

On the assumption that the enthalpy of a condensed substance is adequately repre-sented by first-neighbour interactions only, bond enthalpy data in condensed states εwwere obtained from measured enthalpies of formation and crystallographic data [58].For a pure condensed element A, eelemA�A can be expressed as eelemA�A ¼ DHsub

NAðCN=2Þ, whereΔHsub is the heat of sublimation and CN is the coordination number of A. eelemA�A in pureAl, Mn, Fe, Co, Ni, Cu, Ru and Pd are 0.57, 0.45, 0.62, 0.74, 0.74, 0.58, 1.12 and0.65 eV, respectively. With the average coordination number in QCs being 13.385 [59],

the like-atom bond enthalpy in QCs is simply modified by eQCA�A ¼ eelemA�ACN=2

13:385=2

� �to take

into account the dependence of bond enthalpy on the coordination number. eQCA�A forAl–Al, Mn–Mn, Fe–Fe, Co–Co, Ni–Ni, Cu–Cu, Ru–Ru and Pd–Pd is thus obtained as0.51, 0.44, 0.64, 0.66, 0.67, 0.52, 1.00 and 0.58 eV, respectively.

The bond enthalpy values between unlike atomic pairs in QCs could be assessed byreferring to QC-related phases, especially the CsCl-type structures that contain mainlyunlike bonds, as listed in Table 1. This kind of phase has been revealed to be anapproximant of QCs [60].

Take the Al–Cu–Fe ternary icosahedral [Fe2Al8Cu3], for example, where an Fe-cen-tred icosahedron is composed of eight Al, three Cu and one Fe atoms in the shell sites.The probability (P) of bonding between any two atoms is correlated to their local chemis-tries on the icosahedron shell: PAl−Al = (8/12)2 = 0.44, PAl−Cu = (8/12) × (3/12) × 2 = 0.33,PAl−Fe = (8/12) × (1/12) × 2 = 0.11, PCu−Cu = (3/12)2 = 0.06, PCu−Fe = (3/12) × (1/12) × 2= 0.04, PFe−Fe = (1/12)2 = 0.01, the sum of which is 1. The bonding enthalpy informationis missing for Cu–Fe with a large positive enthalpy of mixing [68], but the probability isvery low and can be neglected. The average bond enthalpy is determined by a weightedaverage of all possible bonds, normalized by the bonding probability: εav = ΣPi × εi/ΣPi.By introducing the bonding probability and the corresponding bond enthalpy, the averagebond enthalpy of the weakest bonds in icosahedral QC-AlCuFe is obtained: εav = 0.55 eV.This bond enthalpy is larger than that of pure Al–Al bond 0.51 eV due to the introduction

Table 1. Bond enthalpy data between unlike atomic pairs εA−B in QC-related phases, with ΔfHbeing the enthalpy of formation.

QC-related phases Structure type ΔfH (kJ mol−1) εA−B (eV)

AlMn AuCu −47.4 [61] 0.55AlFe CsCl −53.4 [62] 0.62AlCo CsCl −106.8 [63] 0.70AlNi CsCl −116.6 [64] 0.72Al2Cu Al2Cu −46.5 [65] 0.58AlRu CsCl −124.1 [66] 0.89AlPd CsCl −175.3 [67] 0.75

Note: εA−B has an average uncertainty of about ±4% [58].

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of stronger Al–TM bonds. It should be noted that the standard deviation of εA−B in differ-ent A−B phases is less than 8% and typically much lower. Also, the Al–TM bonds occupya smaller proportion as compared to those of Al–Al, leading to an even smaller deviationin εav, typically about 3%. Considering that the condensed bond enthalpy εw has an aver-age uncertainty of about ± 4% [58], it is then assumed that the calculated hardness followsthe same uncertainty.

The last step towards calculating hardness is to count n, the number of the rupturedweakest bonds between the clusters. By employing Equation (3) and using the hardnessdata above for ternary Al-based QCs to fit measured values as documented [69–78], themean value of n is quite close to 19 as shown in Table 2.

Regarding the total number of the weak bonds between icosahedral clusters, herethe total outer bonds of the icosahedron in the Mackay-type environment are examinedbecause all ternary Al-based QCs in this work belong to this type. Figure 3 depicts thetwo-shell Mackay icosahedron, where the second shell can be divided into two sub-shells: in addition to the 12 atoms located above the vertices of the central first-neigh-bour shell icosahedron, 30 atoms sited above the edge centres, making a total 42second-shell atoms surrounding the first-shell icosahedron. There are different numbersof bonds formed between the first-shell icosahedron and the two outer subshells, 1 and2, respectively. Then, the total number of the bonds belonging to the central icosahe-dron is deduced as (12 × 1 + 30 × 2)/2 = 36. Since the fitted broken bond number is closeto 19, it is concluded that about half of the total inter-cluster bonds per unit cluster for-mula rupture during hardness indentation testing.

Then, n = 19 is used to calculate the theoretical hardness of quasicrystals in the fol-lowing, and the results are shown in Table 2. There is a good agreement between thecalculated and the measured hardness values, considering the relatively large experimen-tal errors in hardness tests (which may well arise from QC samples prepared throughdifferent routes such as sintering and casting). It is interesting to note that the calculatedhardness values are in the range of 8–9 GPa in the Al-based QCs, despite of the factthat they are constructed by different cluster formulas. Specifically, the reported Vickershardness values at ambient temperature for Al–Cu–Fe icosahedral QC are about 8 GPa[69,70] and 9.8 GPa [71], both being comparable with our calculated 8.7 GPa. TheVickers hardness of icosahedral QC Al68Cu15Ru17 is 10.5 GPa [72], which is somewhathigher than our calculated 8.8 GPa. The Vickers hardness of a single-grain icosahedralQC Al70Pd20Mn10 is 6.9–8.0 GPa at room temperature in symmetry orders of three-,five- and twofold [73]. These hardness values are slightly higher than those measuredfor a single-grain icosahedral QC-Al73Pd20Mn7 (5.9–7.1 GPa) [74] and are lower thanthat reported for polycrystalline icosahedral QC Al70Pd15Mn15 (9.3 GPa) [72]. Thecalculated values are 8.7 GPa for [Mn2Al9Pd2] = Al69.2Pd15.4Mn15.4 and 9.0 GPa for[MnPdAl9Pd2] = Al69.2Pd23.1Mn7.7.

There is only small anisotropy observed in the hardness values of decagonal QCAl70Co15Ni15 [72,75], where the hardness measured on the basal plane is about 8.8 GPawhile that on the prism plane is 9.4 GPa. Our calculated value is 8.3 GPa. Hardnesstests under different loads have been carried out on single-grain Al–Cu–Co–Si decago-nal QC grown from Al63Cu20Co15Si2, with hardness values of 7.8–9.0 GPa [76]. TheVickers hardness of decagonal QC Al63.5Cu18.3Co16.4Si1.8 was determined as 9.6 GPa[77]. Kang and Dubois [78] gave 8.1 GPa for decagonal QC Al63Cu17.5Co17.5Si2 with avolumetric porosity of about 12%. Our calculated values are 8.1 GPa for [Co

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Table2.

Theoretical

Vickers

hardness

ofAl-basedQCsfrom

thecluster-plus-glue-atom

mod

el.IQ

Cstands

foricosahedralqu

asicrystalsandDQCsfor

decago

nalon

es.

Experim

entalcompositio

nsCluster

form

ulas,

r 1(nm),e/u

ZEmpiricalρ a

(nm

−3)

εav(eV)

Hcal(G

Pa)

Hexp(G

Pa)

ndedu

ced

from

Hexp

IQC-A

l 64Cu 2

2Fe 1

4[Fe 2Al 8Cu 3]=Al 61.5Cu 2

3.1Fe 1

5.4,

0.27

36,19

.213

67.8

0.55

8.7

8.4[69]

18.2

IQC-A

l 63Cu 2

5Fe 1

27.8[70]

16.9

IQC-A

l 63.5Cu 2

4.0Fe 1

2.5

9.8[71]

21.3

PolyIQ

C-A

l 68Ru 1

7Cu 1

5[Ru 2Al 9Cu 2]=Al 69.2Cu 1

5.4Ru 1

5.4,

0.27

58,19

.713

64.3

0.58

8.8

10.5

[72]

22.7

IQC-A

l 70Pd 1

5Mn 1

5[M

n 2Al 9Pd 2]=Al 69.2Pd 1

5.4Mn 1

5.4,

0.26

34,22

.813

63.9

0.58

8.7

9.3[72]

20.3

IQC-A

l 70Pd 2

0Mn 1

0[M

nPdA

l 9Pd 2]=Al 69.2Pd 2

3.1Mn 7

.7,

0.26

34,23

.013

63.2

0.61

9.0

6.9–

8.0[73]

14.6–16.9

IQC-A

l 73Pd 2

0Mn 7

5.9–

7.1[74]

12.5–15.0

DQC-A

l 70Co 1

5Ni 15

[CoA

l 10Ni 2]Co 1

=Al 71.4Co 1

4.3Ni 14.3,

0.26

27,23

.714

66.7

0.57

8.3

8.8–

9.4[72]

20.1–21.5

DQC-A

l 71Co 1

3Ni 16

8.3–

9.3[75]

19.0–21.3

DQC-A

l 63Si 2Cu 2

0Co 1

5[CoC

u 2Al 9Cu]Co 1

=Al 64.3Cu 2

1.4Co 1

4.3,0.26

27,23

.314

67.7

0.59

8.6

7.8–

9.0[76]

17.2–19.8

DQC-A

l 63.5Cu 1

8.3Co 1

6.4Si 1.8

[Co(Cu 0

.5Co 0

.5)CuA

l 9Cu]Co 1

=Al 64.4Cu 1

7.8Co 1

7.8,0.26

27,23

.314

67.8

0.55

8.1

9.4–

9.8[77]

22.1–23.1

DQC-A

l 63Cu 1

7.5Co 1

7.5Si 2

8.1[78]

19.1

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(Cu0.5Co0.5)CuAl9Cu]Co1 = Al64.4Cu17.8Co17.8 and 8.6 GPa for [CoCu2Al9Cu]Co1 =Al64.3Cu21.4Co14.3.

The above results support the conjecture that rigid cluster formula units do exist inQCs, which serve as “unbreakable units” during hardness tests.

4. Conclusions

In this paper, we present an approach to assess the theoretical hardness of Al-basedQCs with the aid of the cluster-plus-glue-atom model. We found that ternary Al-basedQCs would always be interpreted by certain element substitutions on the first shell ofthe icosahedra in the corresponding binary cluster formulas [icosahedron](glue)0,1. Anovel atomic-level hardness model for QCs is established by considering partial rup-ture of the relatively weak inter-cluster bonds. Typically, theoretical hardness valuesof 8–9 GPa are obtained for all QCs using the number of inter-cluster bonds of 19,which accounts for about half of the total number of bonds as can be obtained in theMackay-type environment. It is then confirmed that the unit cluster formulas wouldact as rigid units during the Vickers hardness measurements, in addition to beingchemical and electronic units.

Figure 3. (colour online) Two-shell Mackay icosahedron, with the first-shell icosahedron(dark grey) surrounded by 12 atoms (small spheres) located above the vertices and 30 atoms(larger spheres) sited above the edge centres of the central first-neighbour icosahedron.

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FundingThis work was supported by the Natural Science Foundation of China under [grant number51131002].

References

[1] B.R. Lawn and V.R. Howes, J. Mater. Sci. 16 (1981) p.2745.[2] W.C. Oliver and G.M. Pharr, J. Mater. Res. 19 (2004) p.3.[3] J.J. Gilman, Chemistry and Physics of Mechanical Hardness, John Wiley, Hoboken, NJ,

2009.[4] B.R. Lawn and D.B. Marshall, J. Am. Ceram. Soc. 62 (1979) p.347.[5] A.Y. Liu and M.L. Cohen, Science 245 (1989) p.841.[6] C.-M. Sung and M. Sung, Mater. Chem. Phys. 43 (1996) p.1.[7] J.M. Léger, J. Haines, M. Schmidt, J.P. Petitet, A.S. Pereira and J.A.H. Da Jornada, Nature

383 (1996) p.401.[8] D.M. Teter, MRS Bull. 23 (1998) p.22.[9] H.-Y. Chung, M.B. Weinberger, J.-M. Yang, S.H. Tolbert and R.B. Kaner, Appl. Phys. Lett.

92 (2008) p.261904.[10] A.P. Gerk, J. Mater. Sci. 12 (1977) p.735.[11] X. Jiang, J.J. Zhao and X. Jiang, Comput. Mater. Sci. 50 (2011) p.2287.[12] S.-H. Jhi, J. Ihm, S.G. Louie and M.L. Cohen, Nature 399 (1999) p.132.[13] W.H. Wang, Prog. Mater Sci. 57 (2012) p.487.[14] R.A. Andrievski, Int. J. Refract. Met. Hard Mater. 19 (2001) p.447.[15] D. Tabor, Br. J. Appl. Phys. 7 (1956) p.159.[16] C.A. Schuh and T.G. Nieh, J. Mater. Res. 19 (2004) p.46.[17] F.J. Lockett, J. Mech. Phys. Solids 11 (1963) p.345.[18] M. Dao, N. Chollacoop, K.J. Van Vliet, T.A. Venkatesh and S. Suresh, Acta Mater. 49

(2001) p.3899.[19] Z. Ding, S. Zhou and Y. Zhao, Phys. Rev. B 70 (2004) p.184117.[20] S.F. Pugh, Philos. Mag. Ser. 7(45) (1954) p.823.[21] X.-Q. Chen, H.Y. Niu, D.Z. Li and Y.Y. Li, Intermetallics 19 (2011) p.1275.[22] P. Lazar, X.-Q. Chen and R. Podloucky, Phys. Rev. B 80 (2009) p.012103.[23] V.A. Mukhanov, O.O. Kurakevych and V.L. Solozhenko, J. Superhard Mater. 30 (2008)

p.368.[24] J.J. Gilman, Science 261 (1993) p.1436.[25] J.J. Gilman, Hardness – a strength microprobe, in The Science of Hardness Testing and its

Research Applications, J.H. Westbrook and H. Conrad, eds., ASM, Metals Park, OH, 1973,p.51.

[26] F.M. Gao, J.L. He, E.D. Wu, S.M. Liu, D.L. Yu, D.C. Li, S.Y. Zhang and Y.J. Tian, Phys.Rev. Lett. 91 (2003) p.015502.

[27] A. Šimůnek and J. Vackář, Phys. Rev. Lett. 96 (2006) p.085501.[28] K.Y. Li, X.T. Wang, F.F. Zhang and D.F. Xue, Phys. Rev. Lett. 100 (2008) p.235504.[29] C. Dong, Q. Wang, J.B. Qiang, Y.M. Wang, N. Jiang, G. Han, Y.H. Li, J. Wu and J.H. Xia,

J. Phys. D: Appl. Phys. 40 (2007) p.R273.[30] H. Chen, J.B. Qiang, Q. Wang, Y.M. Wang and C. Dong, Isr. J. Chem. 51 (2011) p.1226.[31] G. Han, J.B. Qiang, Q. Wang, Y.M. Wang, C.L. Zhu, S.G. Quan, C. Dong and P. Häussler,

Philos. Mag. 91 (2011) p.2404.[32] G. Han, J.B. Qiang, F.W. Li, L. Yuan, S.G. Quan, Q. Wang, Y.M. Wang, C. Dong and

P. Häussler, Acta Mater. 59 (2011) p.5917.[33] P. Häussler, J. Phys. Colloques 46 (1985) p.C8–361.

Philosophical Magazine 1475

Dow

nloa

ded

by [

Ston

y B

rook

Uni

vers

ity]

at 0

0:01

26

Oct

ober

201

4

[34] J.X. Chen, Q. Wang, Y.M. Wang, J.B. Qiang and C. Dong, Philos. Mag. Lett. 90 (2010)p.683.

[35] D.B. Miracle, W.S. Sanders and O.N. Senkov, Philos. Mag. 83 (2003) p.2409.[36] Q. Wang, C. Dong, J.B. Qiang and Y.M. Wang, Mater. Sci. Eng., A 449–451 (2007) p.18.[37] P.J. Black, Acta Crystallogr. 8 (1955) p.43.[38] J. Grin, U. Burkhardt, M. Ellner and K. Peters, Z. Kristallogr. 209 (1994) p.479.[39] M. Stiehler, J. Rauchhaupt, U. Giegengack and P. Häussler, J. Non-Cryst. Solids 353 (2007)

p.1886.[40] H. Chen, Q. Wang, Y.M. Wang, J.B. Qiang and C. Dong, Philos. Mag. 90 (2010) p.3935.[41] A.-P. Tsai, A. Inoues, Y. Yokoyama and T. Masumoto, Philos. Mag. Lett. 61 (1990) p.9.[42] S. Taniguchi and E. Abe, Philos. Mag. 88 (2008) p.1949.[43] J. Guo, E. Abe, T.J. Sato and A.-P. Tsai, Jpn. J. Appl. Phys. 38 (1999) p.L1049.[44] O.N. Senkov, D.B. Miracle, V. Keppens and P.K. Liaw, Metall. Mater. Trans. A 39 (2008)

p.1888.[45] C. Dong, A. Perrot, J.-M. Dubois and E. Belin, Mater. Sci. Forum 150–151 (1994) p.403.[46] G. Kreiner and H.F. Franzen, J. Alloys Compd. 261 (1997) p.83.[47] C.B. Shoemaker, D.A. Keszler and D.P. Shoemaker, Acta Crystallogr. 45 (1989) p.13.[48] L.A. Bendersky, Mater. Sci. Forum 22–24 (1987) p.151.[49] M. Boudard, M. de Boissieu, C. Janot, G. Heger, C. Beeli, H.-U. Nissen, H. Vincent,

R. Ibberson, M. Audier and J.M. Dubois, J. Phys.: Condens. Matter 4 (1992) p.10149.[50] G. Trambly de Laissardière, D. Nguyen-Manh and D. Mayou, Prog. Mater Sci. 50 (2005)

p.679.[51] K. Kirihara, T. Nagata, K. Kimura, K. Kato, M. Takata, E. Nishibori and M. Sakata, Phy.

Rev. B 68 (2003) p.014205.[52] U. Mizutani, J. Phys.: Condens. Matter 10 (1998) p.4609.[53] E. Belin-Ferré, J. Phys.: Condens. Matter 14 (2002) p.R789.[54] M. Sakai, Acta metall. mater. 41 (1993) p.1751.[55] F.M. Gao, Phy. Rev. B 73 (2006) p.132104.[56] C. Rudhart, P. Gumbsch and H.-R. Trebin, Crack propagation, in quasicrystals, in

Quasicrystals: Structure and Physical Properties, H.-R. Trebin, ed., Wiley-VCH, Weinheim,2003, p.484.

[57] P. Ebert, M. Feuerbacher, N. Tamura, M. Wollgarten and K. Urban, Phys. Rev. Lett. 77(1996) p.3827.

[58] D.B. Miracle, G.B. Wilks, A.G. Dahlman and J.E. Dahlman, Acta Mater. 59 (2011) p.7840.[59] C.L. Henley and V. Elser, Philos. Mag. B 53 (1986) p.L59.[60] C. Dong, Scr. Metall. Mater. 33 (1995) p.239.[61] D. Nguyen-Manh and D.G. Pettifor, Intermetallics 7 (1999) p.1095.[62] K. Rzyman, Z. Moser, A.P. Miodownik, L. Kaufman, R.E. Watson and M. Weinert, Calphad

24 (2000) p.309.[63] S.V. Meschel and O.J. Kleppa, NATO ASI Ser. 256 (1994) p.103.[64] O.J. Kleppa, J. Phase Equilib. 15 (1994) p.240.[65] S.G. Fries and T. Jantzen, Thermochim. Acta 314 (1998) p.23.[66] W.-G. Jung and O.J. Kleppa, Metall. Trans. B 23 (1992) p.53.[67] R.E. Watson, M. Weinert and M. Alatalo, Phys. Rev. B 65 (2001) p.014103.[68] A. Takeuchi and A. Inoue, Mater. Trans. 46 (2005) p.2817.[69] L. Bresson and D. Gratias, J. Non-Cryst. Solids 153 & 154 (1993) p.468.[70] E. Giacometti, N. Baluc, J. Bonneville and J. Rabier, Scr. Mater. 41 (1999) p.989.[71] U. Köster, W. Liu, H. Liebertz and M. Michel, J. Non-Cryst. Solids 153 & 154 (1993)

p.446.[72] S. Takeuchi, H. Iwanaga and T. Shibuya, Jpn. J. Appl. Phys. 30 (1991) p.561.[73] Y. Yokoyama, A. Inoue and T. Masumoto, Mater. Trans., JIM 34 (1993) p.135.

1476 H. Chen et al.

Dow

nloa

ded

by [

Ston

y B

rook

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vers

ity]

at 0

0:01

26

Oct

ober

201

4

[74] M.H. Wollgarten Saka, Mechanical properties of quasicrystalline surfaces, in New Horizonsin Quasicrystals: Research and Applications, A.I. Goldman, D.J. Sordelet, P.A. Thiel andJ.M. Dubois, eds., World Scientific, Singapore, 1997, p.320.

[75] N.K. Mukhopadhyay, A. Belger, P. Paufler and P. Gille, Philos. Mag. 86 (2006) p.999.[76] N.K. Mukhopadhyay, G.C. Weatherly and J.D. Embury, Mater. Sci. Eng., A 315 (2001)

p.202.[77] R. Wittmann, K. Urban, M. Schandl and E. Hornbogen, J. Mater. Res. 6 (1991) p.1165.[78] S.S. Kang and J.M. Dubois, Philos. Mag. A 66 (1992) p.151.

Philosophical Magazine 1477

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