2011-internetallics-Modeling hardness of polycrystalline materials and bulk metallic glasses

Modeling hardness of polycrystalline materials and bulk metallic glassesXing-Qiu Chen *,Haiyang Niu,Dianzhong Li *,Yiyi LiShenyang National Laboratory for Materials Science,Institute of Metal Research,Chinese Academy of Sciences,Wenhua Road No.72,Shenyang 110016,Liaoning,PR Chinaa r t i c l e i n f oArticle history:Received 24February 2011Accepted 23March 2011Available online 2June 2011Keywords:E.Mechanical properties,theory B.Elastic propertiesa b s t r a c tThough extensively studied,hardness,de fined as the resistance of a material to deformation,still remains a challenging issue for a formal theoretical description due to its inherent mechanical complexity.The widely applied Teter ’s empirical correlation between hardness and shear modulus has been considered to be not always valid for a large variety of materials.The main reason is that shear modulus only responses to elastic deformation whereas the hardness links both elastic and permanent plastic properties.We found that the intrinsic correlation between hardness and elasticity of materials correctly predicts Vickers hardness for a wide variety of crystalline materials as well as bulk metallic glasses (BMGs).Our results suggest that,if a material is intrinsically brittle (such as BMGs that fail in the elastic regime),its Vickers hardness linearly correlates with the shear modulus (H v ¼0.151G ).This correlation also provides a robust theoretical evidence on the famous empirical correlation observed by Teter in 1998.On the other hand,our results demonstrate that the hardness of polycrystalline materials can be correlated with the product of the squared Pugh ’s modulus ratio and the shear modulus (H v ¼2ðk 2G Þ0:585À3where k ¼G /B is Pugh ’s modulus ratio).Our work combines those aspects that were previously argued strongly,and,most importantly,is capable to correctly predict the hardness of all hard compounds known included in several pervious models.Ó2011Elsevier Ltd.All rights reserved.1.IntroductionDespite the great efforts,to understand the theory of hardness and to design new ultrahard materials are still very challenging for materials scientists [1e 4].During the past few years,several semi-empirical theoretical models [5e 9]have been developed to estimate hardness of materials based on:(i )the bond length,charge density,and ionicity [5],(ii )the strength of the chemical bonds [6],(iii )the thermodynamical concept of energy density per chemical bonding [7],and (iv )the connection between the bond electron-holding energy and hardness through electronegativity [8],and (v )the temperature-dependent constraint theory for hardness of multicomponent bulk metallic glasses (BMGs)[9].Experimentally,hardness is a highly complex property since the applied stress may be dependent on the crystallographic orientations,the loading forces and the size of the indenters.In addition,hardness is also characterized by the ability to resist to both elastic and irreversible plastic deformations and can be affected signi ficantly by defects (i.e.,dislocations)and grain sizes [10].Therefore,hardness is not a quantity that can be easily determined in a well-de fined absolutescale [1].It has been often argued [13]that hardness measurements unavoidably suffer from an error of about 10%.All these aspects add huge complexity to a formal theoretical de finition of hardness [5e 9].Within this context,to find a simple way to estimate hardness of real materials is highly desirable.Unlike hardness,the elastic properties of materials can be measured and calculated in a highly accurate manner.Therefore,it has been historically natural to seek a correlation between hardness and elasticity.The early linear correlation between the hardness and bulk modulus (B)for several covalent crystals (diamond,Si,Ge,GaSb,InSb)was successfully established by Gilman and Cohen since 1950s [10,11].Nevertheless,successive studies demonstrated that an uniformed linear corre-lation between hardness and bulk modulus does not really hold for a wide variety of materials [1,12,13],as illustrated in Fig.1(a).Subsequently,Teter [12]established a better linear correlation between hardness and shear modulus (G ),as illustrated in Fig.1(b).This correlation suggests that the shear modulus,the resistance to reversible deformation under shear strain,can correctly provide an assessment of hardness for some materials.However,this correla-tion is not always successful,as discussed in Refs.[5,13,14].For instance,tungsten carbide (WC)has a very large bulk modulus (439GPa)and shear modulus (282GPa)but its hardness is only 30GPa [15],clearly violating the Teter ’s linear correlation [see*Corresponding authors.E-mail addresses:xingqiu.chen@ (X.-Q.Chen),dzli@ (D.Li).Contents lists available at ScienceDirectIntermetallicsjou rn al homepage:/locate/intermet0966-9795/$e see front matter Ó2011Elsevier Ltd.All rights reserved.doi:10.1016/j.intermet.2011.03.026Intermetallics 19(2011)1275e 1281Fig.1(b)][5].Although the link between hardness and elastic shear modulus can be arguable,it is certain to say that the Teter ’s correlation grasped the key.In this manuscript,following the spirit of Teter ’s empirical correlation,we successfully established a theoretical model on the hardness of materials through the introduction of the classic Pugh modulus ratio of G /B proposed in 1954[16].We found that the intrinsic correlation between hardness and elasticity of materials correctly predicts Vickers hardness for a wide variety of crystalline materials as well as BMGs.Our results suggest that,if a material is intrinsically brittle (such as BMGs that fail in the elastic regime),its Vickers hardness linearly correlates with the shear modulus (H v ¼0.151G ).This correlation also provides a robust theoretical evidence for the famous empirical correlation observed by Teter in 1998.On the other hand,our results demonstrate that the hardness of crystalline materials can be correlated with the product of the squared Pugh ’s modulus ratio and the shear modulus (H v ¼2ðk 2G Þ0:585À3where k is Pugh ’s modulus ratio).This formula provides the firm evidence that the hardness not only correlates with shear modulus as observed by Teter,but also with bulk modulus as observed by Gilman et al.Our work combines those aspects that were previously argued strongly,and,most importantly,is capable to correctly predict the hardness of all compounds included in Teter ’s [12],Gilman ’s [4,10],Gao et al.’s [5]and Sim u nek and Vacká r ’s [6]sets.Also,our model clearly demonstrates that the hardness of bulk metallic glasses is intrin-sically based on the same fundamental theory as the crystalline materials.We believe that our relation represents a step forward for the understanding and predictability of hardness.2.Model and resultsAccording to Vicker [10],the hardness of H v is the ratio between the load force applied to the indenter,F ,and the indentation surface area:H v ¼2F sin ðq =2Þd 2;(1)where d and q are the mean indentation diagonal and angle between opposite faces of the diamond squared pyramid indenter,respectively (Fig.2).In order to derive our model,we first assume that (i)the diamond squared pyramid indenter can be divided into four triangular based pyramid indenters and that (ii)the Vickers hardness is measured within the elastic scale.Then,for each triangular based pyramid,one can de fine the shear modulus G as,G ¼F4A tan ða Þ(2)which speci fies the ratio between shear stress and the shear strain.In terms of our model the exact shear area A on which the shear force (F )acts is unknown.But,the deformation area A *[A *¼1/8d 2tan(a )]delimited by the klO 0triangle is well de fined by the indentation geometry.Therefore,we can express the exact shear area (A )as:A ¼cA *¼c d 2tan ða Þ;(3)where c is the proportional coef ficient.It is clear that under elastic shear deformation the deformation area (A *)will be extremely small.However,upon real hardness measurements the deforma-tion area (A *)should be large enough so that the coef ficient c can be safely neglected and A z A *.Under this assumption,equation (2)can be revised as following,G ¼2F d 2tan 2ða Þ(4)Combining equations (1)and (4),the Vickers hardness readsH v ¼G tan 2ða Þsin ðq =2Þ¼0:92G tan 2ða Þ;(5)where the term sin(q /2)is intrinsically determined by the indenter itself,which can be considered as a constant (originated from the Vickers hardness,see equation (1)).For the diamond squared pyramid indenter with q ¼136 ,sin(q /2)is equal to 0.92for Vickers hardness measurement.In an ideal form of indentation,tan(a )¼0.404because of a ¼(p Àq )/2.0(c.f.,Fig.2).Therefore,equation (5)can be simpli fiedas,abFig.1.Correlation of experimental Vickers hardness (H v )with (a)bulk modulus (B )and with (b)shear modulus (G )for 39compounds (Table 2).Inset of panel (b):H v vs.G for 37BMGs (see Table 1).The solid line denotes empirical Teter ’s fitting values,whereas dashed lines correspond to the value derived from Eq.(6).The black and hollow squares denote data taken from Refs.[1,12].Fig.2.Illustration of indentation in terms of the squared diamond pyramid indenter.The red framework highlights one of four triangular based pyramid indenters.(For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)X.-Q.Chen et al./Intermetallics 19(2011)1275e 12811276H v¼0:151G:(6) Equation(6)represents a robust theoretical evidence of the linear correlation behavior observed by Teter[12],as reflected by the data shown in Fig.1(b).Residual discrepancies should be mainly attributed to the neglection of plastic deformation effects. Remarkably,we found that Eq.(6)is also valid for ing the experimental shear modulus G¼38.6(36.6)GPa[17]for Pd40Ni40P20BMG,the estimated Vickers hardness is5.83(5.53) GPa,in consistency with the experimental value of5.38GPa[17]. Similarly,for Fe41Co7Cr15Mo14C15B6Y2BMG by using the experi-mental G(84.3GPa[18])we obtained H v¼12.7GPa,in nice accord with the measured Vickers hardness of12.57Æ0.22GPa[18].As illustrated in the inset of Fig.1(b),the agreement with the experi-mental values is highly satisfactory for all37BMGs collected here (see Table1).Considering that BMGs are brittle materials without plastic fractures the data in the inset of Fig.1(b)strongly convey that our proposed formula(Eq.(6))is intrinsically connected to the shear modulus of materials if they fail in an elastic regime.It is highly difficult to realistically take plastic deformation into account in our modeling scheme.However,the indentation after a real hardness measurement shows the permanent plastic effect,which is,of course,reflected by the ratio l OO0=l OO00[namely, equal to tan(a)](see Fig.2).Note that the depth of the indenta-tion,l OO0,is parallel to the direction of shear deformation.We reasonably assume that its size should be closely correlated with the shear modulus of G,whereas the expansion wideness of the indentation,l OO00,is perpendicular to the direction of the loading force,hence,with almost little connection to the shear defor-mation.Therefore,the expansion wideness seems to reflect the ability to resist to compression effects,a property that should be related to bulk modulus,B.Accordingly,we proposed the following relation,tanðaÞf G=B:(7) Finally,combining equations(5)and(7),the hardness can be written as,H v f GðG=BÞ2:(8) It is interesting to note that in Eq.(8)the ratio of G/B is the famous modulus ratio proposed by Pugh in as early as1954[16].In his pioneer work,Pugh derived that the strain at fracture can be measured as3f(B/G)2.Indeed,hardness can be defined as the resistance to the applied stress at the critical strain of3that the system can sustain before yielding it to fracture.This clearly provides fundamental support for our model(Eq.(8)).Importantly, Pugh also highlighted a relation between the elastic and plastic properties of pure polycrystalline metals and stated that G/B is closely correlated to the brittle and ductility of materials[16]:the higher the value of G/B is,the more brittle the materials would be [16].Otherwise,the materials are expected to deform in a ductile manner with a low G/B value.This relation has been extensively accepted and applied not only to metal but also to high-strength materials[25e27].In principles,the covalent materials(such as diamond and c-BN)have the highest hardness but they are obvi-ously brittle with a larger Pugh modulus ratio.The strong covalent bonds indeed create a significant resistance to initialize the plastic flow to pin the dislocation,resulting in a quite high hardness. Conversely,ductile materials with a low Pugh’s modulus ratio are characterized by metallic bonding and low hardness.It is thus highly reasonable to establish a correlation between hardness and the modulus ratio G/B in Eq.(8).Thus,we revise further Eq.(8)in the following form,H v¼Ck m G n;ðk¼G=BÞ;(9)where H v,G and B are the hardness(GPa),shear modulus(GPa)and bulk modulus(GPa),respectively.The parameter k is the Pugh’s modulus ratio,namely,k¼G/B.C is a proportional coefficient.In order to derive the parameters C,m and n,wefirst selected ten materials with diamond-like(diamond,c-BN,b-SiC,Si and Ge), zinc-blende(ZrC and AlN)and rock-salt structures(GaP,InSb and AlSb)because their hardness,bulk and shear moduli are well-known(see Table2).By analyzing these data we found that C¼1.887,m¼1.171and n¼0.591.Hence,Eq.(9)becomes,H v¼1:887k1:171G0:591z1:887k2G0:585:(10) In order to assess the validity of this relation,we plot in Fig.3H v against k2G for a series of compounds.These data show a clear and systematic trend with k2G andfirmly establish a direct relation between hardness and k2G.Byfitting the data of Fig.3and revising further Eq.(10)we arrive to thefinal formula:Table1The comparison between the predicted Vickers hardness by Eq.(6)[H v¼0.151G]andthe experimental data for37bulk metallic glasses is shown,together with experi-mental Young modulus(E)and Shear modulus(G).The[XX];[YY]in thefirst columndenotes the reference numbers:XX is the reference for elastic constants and YY isthe reference for the Vickers hardness.Compounds E G H calc H expFe41Co7Cr15Mo14C15B6Y2[18]22684.312.7312.57Ni50Nb50[19]13248.27.268.93Ni40Cu5Ti17Zr28Al10[20]133.949.77.508.45Ni39.8Cu5.97Ti15.92Zr27.86Al9.95Si0.5[20]11743 6.498.13Ni40Cu5Ti16.5Zr28.5Al10[20]12245.2 6.837.84Ni45Ti20Zr25Al10[20]11442 6.347.76Ni40Cu6Ti16Zr28Al10[20]11140.9 6.187.65{Zr41Ti14Cu12.5Ni10Be22.5}98Y2[21]107.640.3 6.09 6.76Zr54Al15Ni10Cu19Y2[21]92.133.8 5.10 6.49Zr53Al14Ni10Cu19Y4[21]8631.5 4.76 6.44Zr41Ti14Cu12.5Ni8Be22.5C1[21]10639.5 5.96 6.13Zr46.75Ti8.25Cu7.5Ni10Be27.5[19]10037.2 5.62 6.1Zr48Nb8Cu14Ni12Be18[21]93.734.2 5.16 6.09Zr34Ti15Cu10Ni11Be28Y2[21]109.841 6.19 6.07Zr57Nb5Cu15.4Ni12.6Al10[19]87.331.9 4.82 5.9Zr48Nb8Cu12Fe8Be24[21]95.735.2 5.32 5.85Zr40Ti15Cu11Ni11Be21.5Y1Mg0.5[21]94.234.7 5.24 5.74Zr41Ti14Cu12.5Ni10Be22.5[19];[21]10137.4 5.65 5.97Zr41Ti14Cu12.5Ni10Be22.5[19];[22]10137.4 5.65 5.4Zr41Ti14Cu12.5Ni10Be22.5[19];[23]10137.4 5.65 5.88Zr41Ti14Cu12.5Ni10Be22.5[19]10137.4 5.65 5.23Zr65Al10Ni10Cu15[19]8330.3 4.58 5.6Zr65Al10Ni10Cu15[19]8331 4.58 5.6Zr57Ti5Cu20Ni8Al10[19]8230.1 4.55 5.4Cu60Hf10Zr20Ti10[19]10136.9 5.577Cu50Zr50[19]88.732.4 4.83 5.8Cu50Zr50[19]8532 4.83 5.8Cu50Zr45Al5[19]10233.3 5.03 5.4Pd40Ni40P20[19];[17]10838.6 5.83 5.38Pd40Ni40P20[17]e36.6 5.53 5.38Pd40Ni40P20[19]10838.6 5.83 5.3Pd40Ni10Cu30P20[19]9835.1 5.305Pd77.5Si16.5Cu6[19]92.932.9 5.25 4.5Pd77.5Si16.5Cu6[19]9634.8 5.25 4.5Pt60Ni15P25[19]9633.8 5.10 4.1Mg65Cu25Tb10[19]51.319.6 2.96 2.83Nb60Al10Fe20Co10[19]51.219.4 2.93 2.2Ce70Al10Ni10Cu10[19]3011.5 1.74 1.5Er55Al25Co20[24]70.7227.08 4.09 5.45Dy55Al25Co20[24]61.3623.52 3.55 4.7Tb55Al25Co20[24]59.5322.85 3.45 4.42Ho55Al25Co20[24]66.6425.42 3.84 4.14La55Al25Co20[24]40.915.42 2.33 3.48La55Al25Cu10Ni5Co5[24]41.915.6 2.363Pr55Al25Co20[24]45.917.35 2.62 2.58X.-Q.Chen et al./Intermetallics19(2011)1275e12811277H v ¼2 k 2G 0:585À3(11)from which we see that the hardness correlates not only with the shear modulus but also with the bulk modulus.Physically,the bulk modulus only measures isotropic resistance to volume change under hydrostatic strain,whereas shear modulus responses to resistance to anisotropic shear strain.Although the bulk modulus was thought to be less directly connected with hardness [5],the Pugh ’s modulus ratio k clearly contributes to the Vickers hardness.Equation (11)demonstrates that,if bulk modulus increases,hard-ness would decrease as long as the shear modulus remains unchanged,and vice versa.This behavior can be understood by the fact that if the Pugh ’s modulus ratio G /B gets smaller with increasing bulk modulus,the material would become more ductile.Its hardness can be thus expected to have a lower value.Taking the example,the compounds TiN and b -SiC have almost the same experimental shear moduli [28,29]of 187.2GPa and 191.4GPa,Table 2Comparison between theoretical values within our current model and experimental values as compared with available theoretical findings obtained through the modelsof Gao et al.[5]and of Sim u nek and Vack a r [6].Furthermore,the bulk and shearmoduli are compiled in this table.In the last column,“e ”and “c ”denote elastic data (G and B )from direct experimental and theoretical calculations,respectively.The Pugh ’s modulus ratio k is compiled in this table.For details,see text.The [XX];[YY]in the first column denotes the reference numbers:XX is the reference for elastic constants and YY is the reference for the Vickers hardness.Note that,as the experimentally measured hardness is a highly complex property,which depends on the loading force and the quality of samples.For instance,the hardness of diamond was reported in the range from 60GPa to 120GPa.For the sake of convenientcomparisons with the known Gao et al.[5]and Sim u nek and Vack a r ’s [6]models,here we also quoted the same values as previous models quoted [5,6,7,8,9].If compounds were not discussed in the previous theoretical models [5,6,7,8],here the experimental Vicker ’s hardness was selected,mainly based on the saturated Vickers hardness value (namely load-invariant indentation hardness)at a relatively large loading force.For instance,for ReB 2,we selected the Vicker ’s hardness of 30.1GPa at the large loading forces of 4.9N as discussed in Ref.[46].Compounds G B k H calc H exp H Gao H SimunekDiamond [28];[5]535.5442.3 1.21195.79693.695.4e Diamond [28];[5]548.3465.5 1.17893.99693.695.4c Diamond [35];[5]520.3431.9 1.20593.59693.695.4c Diamond [12];[5]535.0443.0 1.20895.49693.695.4e BC 2N [36];[37]446.0403.0 1.10776.9767871.9c BC 2N [12];[37]445.0408.0 1.09175.4767871.9e BC 5[38];[32]394.0376.0 1.04866.771c c-BN [39];[5]405.4400.0 1.01465.26664.563.2e c-BN [28];[5]403.4403.70.99963.86664.563.2c c-BN [28];[5]382.2375.7 1.01763.16664.563.2c c-BN [40];[5]404.4384.0 1.05368.26664.563.2c c-BN [12];[5]409.0400.0 1.02366.26664.563.2e g -B 28[41];[33]236.0224.0 1.05449.050c B 60[12];[42]204.0228.00.89536.438e b -SiC [28];[5]191.4224.70.85232.83430.331.1e b -SiC [28];[5]196.6224.90.87434.53430.331.1c b -SiC [43];[5]190.2209.20.90935.53430.331.1c b -SiC [44];[5]186.5220.30.84632.03430.331.1e b -SiC [12];[5]196.0226.00.86734.13430.331.1e SiO 2[12];[12]220.0305.00.72129.03330.4e ReB 2[45];[46]273.0382.00.71532.930.1e WC [47];[5]301.8438.90.68833.43026.421.5e WC [5]282.0439.00.64229.33026.421.5e B 4C [48];[12]192.0226.00.85032.830a e VC [This work];[6]209.1305.50.68526.22927.2c ZrC [49];[30]169.7223.10.76126.325.8e ZrC [50];[30]182.5228.30.79929.425.8c ZrC [50];[30]185.9228.00.81530.525.8c ZrC [51];[30]169.6223.30.75926.225.8e ZrC [12];[30]166.0223.00.74425.225.8e TiC [49];[6]182.2242.00.75327.124.718.8e TiC [52];[6]176.9250.30.70724.524.718.8c TiC [15];[6]198.3286.00.69325.824.718.8c TiC [53];[6]187.8241.70.77728.824.718.8e TiC [12];[6]188.0241.00.78029.024.718.8e TiN [54];[30]183.2282.00.65022.52318.7c TiN [29];[30]187.2318.30.58820.02318.7e TiN [55];[30]205.8294.60.69926.72318.7c TiN [56];[30]207.9326.30.63723.82318.7c RuO 2[57];[5]142.2251.30.56615.72020.6c RuO 2[58];[5]173.0248.00.69823.72020.6c AlO 2[40];[5]161.0240.00.67121.52020.6c AlO 2[40];[5]160.0259.00.61819.22020.6c AlO 2[59];[5]164.0254.00.64620.72020.6e AlO 2[12];[5]162.0246.00.65921.12020.6e NbC [60];[6]171.0333.00.51315.51818.3c NbC [51];[6]171.7340.00.50515.21818.3e AlN [40];[5]134.7206.00.65418.41821.717.6c AlN [61];[5]130.2212.10.61416.51821.717.6c AlN [62];[5]123.3207.50.59415.21821.717.6c AlN [63];[5]132.0211.10.625171821.717.6e AlN [12];[5]128.0203.00.63116.91821.717.6e NbN [64];[6]155.9292.00.53415.41719.5e NbN [12];[6]156.0315.00.49513.91719.5e HfN [28];[30]186.3315.50.59120.017c HfN [28];[30]164.8278.70.59118.417c GaN [65];[5]105.2175.90.59813.715.118.118.5e GaN [12];[5]120.0210.00.57114.115.118.118.5eTable 2(continued )Compounds G B k H calc H exp H Gao H SimunekZrO 2[40];[5]88.0187.00.4718.41310.8c ZrO 2[66];[5]93.0187.00.4979.51310.8e Si [49];[5]66.697.90.68011.81213.611.3e Si [67];[5]64.097.90.65410.91213.611.3c Si [67];[5]63.290.70.69711.81213.611.3c Si [68];[5]61.796.30.64010.21213.611.3c Si [68];[5]61.789.00.69311.51213.611.3c GaP [51];[5]55.788.20.6319.39.58.98.7e GaP [49];[5]55.888.80.6289.29.58.98.7e GaP [47];[5]56.188.60.6339.49.58.98.7e GaP [70];[5]61.989.70.69011.59.58.98.7c AlP [71];[5]49.086.00.5707.19.49.67.9e AlP [70];[5]51.890.00.5757.59.49.67.9c AlP [72];[5]48.886.00.5677.09.49.67.9c InN [65];[6]55.0123.90.444 5.1910.48.2c InN [71];[6]77.0139.60.5529.7910.48.2c Ge [67];[6]53.172.20.73611.38.811.79.7c Ge [67];[6]43.860.30.7269.58.811.79.7c GaAs [47];[5]46.575.00.6207.87.58.07.4e GaAs [69];[5]46.775.50.6197.87.58.07.4e GaAs [49];[5]46.775.40.6197.87.58.07.4e YO 2[28];[5]72.5166.00.437 6.37.57.7c YO 2[28];[5]62.7146.50.428 5.37.57.7c YO 2[73];[5]66.5149.30.445 6.07.57.7e InP [69];[5]34.371.10.483 3.8 5.4 6.0 5.1e InP [47];[5]34.472.50.475 3.6 5.4 6.0 5.1e AlAs [74];[5]44.877.90.575 6.758.5 6.8e AlAs [69];[5]44.678.30.569 6.558.5 6.8e GaSb [69];[6]34.256.30.607 5.8 4.5 6.0 5.6e GaSb [49];[6]34.156.40.606 5.8 4.5 6.0 5.6e GaSb [47];[6]34.356.30.608 5.8 4.5 6.0 5.6e AlSb [75];[5]31.556.10.561 4.64 4.9 4.9c AlSb [69];[5]31.958.20.549 4.54 4.9 4.9e AlSb [49];[5]32.559.30.548 4.64 4.9 4.9e AlSb [47];[5]31.958.20.549 4.54 4.9 4.9e InAs [69];[5]29.557.90.509 3.6 3.8 5.7 4.5e InAs [49];[5]29.559.10.499 3.4 3.8 5.7 4.5e InSb [47];[5]23.046.90.490 2.4 2.2 4.3 3.6e InSb [69];[5]22.946.50.492 2.4 2.2 4.3 3.6e InSb [49];[5]22.946.00.498 2.5 2.2 4.33.6e ZnS [49];[6]32.878.40.418 2.5 1.8 2.7e ZnS [47];[6]31.577.10.408 2.3 1.8 2.7e ZnSe [47];[6]28.863.10.456 2.7 1.4 2.6e ZnTe [47];[6]23.451.00.459 2.11 2.3e ZnTe [49];[6]23.451.00.4592.112.3eaB 4C was suggested to be very hard in Ref.[76].Mukhanov et al.recently pre-dicted that the Vickers hardness of B 4C was 45.0GPa [85]in agreement with the reported experimental data of 45GPa in Ref.[77](see Table 1in Ref.[85]).However,we also noted that Teter [12]ever summarized the hardness of B 4C with a value of 30Æ2GPa.In addition,the experimental value of 32e 35GPa was recently summarized in the handbook [78].Therefore,here we quoted the experimental Vickers hardness of 30GPa,as summarized by Teter in Ref.[12].X.-Q.Chen et al./Intermetallics 19(2011)1275e 12811278respectively.However,the experimental bulk modulus of TiN (318.3GPa)[28,29]is larger by about 42%than that of b -SiC (224.7GPa).In terms of our formula,b -SiC is found to be harder than TiN,in agreement with the experimental observations [5,30][Expt (Calc in this work):TiN with H v ¼23(20)GPa and b -SiC with H v ¼34(33)GPa].To further assess the performance of our model (Eq.(11))we show in Fig.4a comparison between the estimated and experi-mental values for a series of compounds (see Table 2),con firming a good agreement.Also WC,which is wrongly found to be a superhard (49GPa)material within Teter ’s linear correlation,is now predicted to have a Vickers hardness of 29.3GPa in very good accordance with experimental value (30GPa [15]).In particular,Figs.3and 4convey that our proposed formula reproduced very well the Vickers hardness for all well-known superhard materials (Diamond [5,6],BC 2N [5,6,12,31],c-BN [5,6],c-BC 5[32],and g -B 28[33,34]).The interesting case is the compound of ReB 2,which was thought to be superhard [79].Although its Vickers hardness was debated extensively [79e 81],there is now a wide-accepted consensus that its Vickers hardness of 30.1GPa at the large loading force of 4.9N [45,46].Using the experimentally measured bulk and shear moduli [45](B ¼273GPa and G ¼382GPa)and interms of our Eq.(11),the Vickers hardness is derived to be 32.9GPa,in nice agreement with the experimental data [45,46].In addition,we also noted that Qin et al.claimed that the Vickers ’hardness of ReB 2is only about 18GPa for a densi fied compact sample [82].However,this value was very recently demonstrated experimen-tally not to re flect well the intrinsic hardness of ReB 2[83]because that the amorphous boron,which is not detectable only by powder X-ray diffraction (Fig.2in Ref.[82]),exists in the synthetic samples in the Qin et al.’s experiment [82].As reported in Ref.[83],the ReB 2sample synthesized by spark plasma sintering exhibits a Vickers hardness of 27.0Æ 4.7GPa.Through a tri-arc crystal-growing furnace the synthesized high-purity ReB 2crystal with grains oriented with respect to the c-axis has a high hardness value of 39.5Æ2.5GPa [83].Again,our estimated Vickers hardness value is still within these experimental values.Another attention has to be paid to the case of B ing the experimental bulk and shear moduli (B ¼230GPa and G ¼206GPa)[84],its Vickers hardness is calculated to be 36.7GPa within our current model.This value is well within the scale of the experimentally measured results from 32to 38GPa [42,84]for polycrystalline boron suboxide sintered samples,although a Vick-ers hardness of 45GPa was reported for the single crystals under a loading force of 0.98N [42].Indeed,the light loading force of 0.98N is not large enough to obtain a real hardness.It is thus ex-pected to have a smaller hardness if a loading force larger than 0.98N is applied.Our estimated value for B 6O is also in good agreement with the derived value of 37.3GPa through a very recent thermodynamic model of hardness [83].We further estimated two more phases of carbon (C 4and M -carbon),which were suggested to be superhard [86e 90].Utilizing elastic shear and bulk moduli obtained in Ref.[88],the Vickers hardness of C 4is calculated to be 69.0GPa (c.f.,Fig.3)that is comparable to the superhard c-BN.Moreover,through using the calculated bulk and shear moduli (B ¼415GPa and G ¼468GPa [91])for the M -carbon phase,we obtained its Vickers hardness of 81.0GPa (c.f.,Fig.3),placing M-carbon in between BC 2N and dia-mond,agreeing well with the value (83.1GPa)obtained by Simunek ’s model [87].In addition,from Fig.4all estimated data are in good agreement with those obtained from pervious models [5,6].Nevertheless,we would like to emphasize that,although our proposed model canreproduce well the results obtained by Gao et al.’s [5]and Sim u nekand Vack a r [6]models,the underlying mechanism is substantiallydifferent.Gao ’s and Sim u nek and Vack a r ’s models are based onbond properties such as bond length,charge density,ionicity and their strengths and coordinations in crystalline materials.Differ-ently,our model depends totally on the so-called polycrystalline moduli (bulk and shear moduli as well as Pugh ’s modulus ratio),which indeed response directly to the abilities of resistance under loading forces for polycrystalline materials.As demonstrated above,for polycrystalline materials the introduced Pugh ’s modulus ratio in our model plays a crucial role in elucidating plastic deformation,which is intrinsically different from all known semi-empirical hardness models [5e 9].3.Discussion and remarksThe hardness of a material is the intrinsic resistance to defor-mation when a force is applied [1].Currently,a formal theoretical de finition of hardness is still a challenge for materials scientists.The need for alternative superhard and ultrahard materials for modern technology has brought a surge of interest on modeling and predicting the hardness of real materials.In particular,in recent years several different semi-empirical models for hardness of polycrystalline covalent and ionic materials have beenproposed.Fig.4.Correlation between experimental and theoretical Vickers hardness (H v )for 39compounds,as compared with the estimated data from the models [5,6](see Table 2).Fig.3.Experimental Vickers hardness as a function of the product k 2G (k ¼G /B ).All data are collected from literature (see Table 2).X.-Q.Chen et al./Intermetallics 19(2011)1275e 12811279。