PHYSICAL REVIEW B 84, 064101 (2011)

Finite-temperature elasticity of fcc Al: Atomistic simulations andultrasonic measurements

Hieu H. Pham,1 Michael E. Williams,2 Patrick Mahaffey,3 Miladin Radovic,2,4

Raymundo Arroyave,2,4 and Tahir Cagin1,2,4,*

1Department of Chemical Engineering, Texas A&M University, College Station, TX 77843, USA2Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA

3Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843, USA4Materials Science and Engineering Program, Texas A&M University, College Station, TX 77843, USA(Received 13 November 2010; revised manuscript received 30 March 2011; published 8 August 2011)

Though not very often, there are some cases in the literature where discrepancies exist in the temperaturedependence of elastic constants of materials. A particular example of this case is the behavior of C12 coefficientof a simple metal, aluminum. In this paper we attempt to provide insight into various contributions to temperaturedependence in elastic properties by investigating the thermoelastic properties of fcc aluminum as a function oftemperature through the use of two computational techniques and experiments. First, ab initio calculations basedon density functional theory (DFT) are used in combination with quasiharmonic theory to calculate the elasticconstants at finite temperatures through a strain-free energy approach. Molecular dynamics (MD) calculationsusing tight-binding potentials are then used to extract the elastic constants through a fluctuation-based formalism.Through this dynamic approach, the different contributions (Born, kinetic, and stress fluctuations) to the elasticconstants are isolated and the underlying physical basis for the observed thermally induced softening is elucidated.The two approaches are then used to shed light on the relatively large discrepancies in the reported temperaturedependence of the elastic constants of fcc aluminum. Finally, the polycrystalline elastic constants (and theirtemperature dependence) of fcc aluminum are determined using resonant ultrasound spectroscopy (RUS) andcompared to previously published data as well as the atomistic calculations performed in this work.

DOI: 10.1103/PhysRevB.84.064101 PACS number(s): 62.20.de, 07.05.Tp

I. INTRODUCTION

Development and selection of new materials requiresaccurate knowledge of their properties as well as reliableexperimental and theoretical tools for their characterization.In the particular case of high-temperature structural materials,the component designer needs (at the very least) reliable infor-mation on their thermoelastic behavior over a wide temperaturerange.1 Although one may think that this information is readilyavailable, the truth of the matter is that there are significantdiscrepancies among different experimental studies, even forthe most widely used materials. The problem is even moredaunting in the case of novel materials currently underdevelopment for next-generation high-temperature structuralmaterials, for example.

As an example, we summarize in Fig. 1 published ex-perimental data on the adiabatic C12 elastic constant of fccaluminum as a function of temperature from four differentstudies.2–5 In these studies, the elastic constants were de-termined by identifying the resonant frequencies of single-crystalline specimens within the kHz-to-MHz range. In thisfigure, quantitative and even qualitative differences can be seenamong the three different experimental data sets. While themeasurements by Sutton,4 Kamm and Alers,3 and Gerlich andFisher2 show a softening of this shear constant with increasingtemperature, Tallon5 (the most recent—and one would expectthe most accurate—experimental work on elastic propertiesof aluminum) reports an actual increase of the C12 elasticconstant with temperature. Although some of the discrepanciescan be explained by the use of different frequency ranges(lower frequencies are used in the earlier study by Sutton4),

the qualitative differences, that is, softening vs hardening,observed indicate significant systematic problems in at leastone of the experimental investigations.

Given the fact that aluminum has been one of the mostwidely characterized and simulated metals,2,4–12 these resultsare rather surprising. However, it is important to note that theaccurate determination of elastic constants through resonancetechniques is far from trivial,13 and the actual results aresubject to nonnegligible degrees of interpretation. Based onthe published experimental results2–5 alone, it is impossible todetermine a priori which of the published data most accuratelyrepresent the actual thermoelastic behavior of fcc aluminum.Moreover, the puzzling hardening of C12 reported by Tallon,5

if true, may indicate unexpected significant anharmonicphenomena in a metal that is normally considered to beslightly anharmonic at most. In this work, we try to addressthese important issues by using two different computationaltechniques—electronic structure calculations based on densityfunctional theory14–16 as well as classical molecular dynamics(MD)—to determine the temperature dependence of theelastic tensor of aluminum and complement these simulationswith state-of-the-art resonant ultrasound spectroscopy (RUS)measurements.13,17 In the following sections, we will describethe different methods used as well as the results obtained.It is expected that the computational methodology presentedin this work can in turn be used to assess the qualityof published thermoelastic data for other important high-temperature structural materials and to reliably predict theseproperties in cases where no experimental information isavailable.

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HIEU H. PHAM et al. PHYSICAL REVIEW B 84, 064101 (2011)

FIG. 1. (Color online) Experimental2–5 adiabatic C12 elasticconstant as a function of temperature.

II. THERMODYNAMIC DEFINITIONS OF ELASTICCONSTANTS

The adiabatic (CSijkl) and isothermal (CT

ijkl) second-orderelastic constants can be defined as second derivatives of inter-nal energy E and the Helmholtz free energy F, respectively,with respect to the homogeneous deformation of the unit cell:18

CSijkl = 1

V0

∂2E

∂εij ∂εkl

, and CTijkl = 1

V0

∂2F

∂εij ∂εkl

, (1)

where V0 and εij are the reference volume and strain tensorof the system, respectively.

In turn, the total free energy of a system is described bya Hamiltonian H, which is the sum of kinetic energy andpotential energy; for a two-body form of interactions betweenparticles it can be written as

H =N∑

a=1

p2a

ma

+∑a<b

U (rab), (2)

where pa and ma correspond to momentum and mass ofparticle a, while rab corresponds to the interatomic separationbetween atom pairs a-b and U is the potential energy function.

Using the definition just given, the statistical fluctuationformula for second-order elastic constants can be derived andpresented as follows19,20:

Cijkl = 2NkBT

V0(δikδlj + δilδkj )

+〈χijkl〉 − V0

kBT(〈σijσkl〉 − 〈σij 〉〈σkl〉) (3)

where N is the number of particles, δ is the Kronecker delta,and σ is the microscopic stress tensor.

The first term has a direct connection with temperature andcorresponds to the kinetic energy contribution to the elastictensor. The second one is called the Born term and is relatedto the strain derivative of the interaction potentials U:

〈χijkl〉 = 1

V0

∂2U

∂εij ∂εkl

, (4)

in which the 〈〉 sign denotes ensemble averaging at thereference volume of the system. The averaging over a micro-canonical (NVE) or canonical (NVT) ensemble will correspondto adiabatic or isothermal elastic constants, respectively.

The kinetic and Born terms contribute to the intrinsicstiffness of the crystal. The last term in Eq. (3) corresponds tocontributions from fluctuations in the microscopic stress tensorof the crystal. While the Born term can certainly be affectedby anharmonic contributions to the free energy, it is the thirdterm in which these contributions become more apparent. Aswill be shown in this work, stress fluctuation contributionscan contribute significantly to the temperature dependence ofthe elastic tensor even in relatively simple systems such asaluminum.

III. DETERMINATION OF THERMOELASTICPROPERTIES THROUGH DFT-BASED STATIC

CALCULATIONS

A. Approach

At the most fundamental level, the elastic constants arerelated to the variation of interatomic forces with respect toatomic displacements. Over the past decades, many differentapproaches have been developed to predict the elastic prop-erties of crystals through the use of atomistic calculations.Dynamic methods make full use of the fluctuation-based defi-nition for the stiffness of the crystal described in Eq. (3). Staticmethods, on the other hand, can only consider contributions tothe stiffness tensor due to variations in the internal stress stateof the crystal with respect to uniform crystal deformationswithout considering anharmonic atomic displacements awayfrom equilibrium or (explicitly) fluctuations in the stresstensor. In these static techniques, only the so-called Borncontribution to the elastic constants can be calculated directly.The individual components of the elastic constant tensor can bedetermined either by establishing relationships between strainand resulting internal stresses21 or between the energy of thecrystal and imposed strains.22 Both static approaches can betrivially implemented within the context of DFT calculations.

In our DFT calculations, the elastic energy per crystal unitvolume is expanded in terms of the strain state. Specifically, wecalculate the variation in the energy of a crystal as a functionof strain, under constant volume constraints:

E(ei) = E0 + 12V

∑Cij eiej + O

[e3i

]. (5)

By judiciously selecting the proper volume-conservingstrain tensor, it is possible to establish a relationship betweenthe difference in lattice energies with respect to unstrainedstate and the corresponding strain.23 For example, in thecase of crystals of cubic symmetry,23 the volume-conservingorthorhombic strain tensor,

ε1 = −ε2 = x,

ε3 = x2/(1 − x2), (6)

ε4 = ε5 = ε6 = 0,

can be used to establish a relationship between the change inlattice energy and the shear elastic constant C11 − C12:

�E(x) = V (C11 − C12)x2 + O[x4]. (7)

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Likewise, the C44 elastic constant can be obtained fromvolume-conserving monoclinic strains:

ε6 = x,ε3 = x2/(4 − x2),

ε1 = ε2 = ε4 = ε5 = 0. (8)

The corresponding lattice energy-strain relationship can inturn be used to determine C44:

�E(x) = �E(−x) = V C44x2/2 + O[x4]. (9)

This expression is only applicable at 0 K, where contri-butions due to thermal excitations of vibrational, electronic,configurational, and/or magnetic degrees of freedom areneglected. At finite temperatures, however, one has to takethese contributions into account because straining the latticewould have an effect over thermally excited degrees offreedom, particularly those related to lattice vibrations. Inrecent work, Ackland et al.24 have used a similar approachto investigate the elastic properties of high-temperature metalsand intermetallics.

In this work, we proceeded to impose volume-conservingstrains necessary to extract the C11 − C12 and C44 elastic con-stants. For each of the strained structures, the phonon densityof states (DOS) was determined through the direct force-constant approach.25 The resulting vibrational free energywas in turn calculated by using simple statistical mechanicalformulas.6 Although electronic contributions are not expectedto be significant in aluminum due to relatively low electronicdensity at the Fermi level, the electronic free energy wascalculated using the temperature-independent self-consistentone-electron approximation.26 In order to take into accountthe effects of lattice thermal expansion, the quasiharmonicapproximation was used. In this approximation, it is assumedthat the phonons (and electronic DOS) are only volumedependent.27

For each of the volumes considered, the total free energyas a function of strain was used to calculate the correspondingelastic constants [replacing �E by �FTotal [in expressions (7)and (9)] at different temperatures. Within the quasiharmonicapproximation, a Cij (V,T )surface—for C11–C12 and C44,respectively—was then constructed. The resulting elastictensor surface in V-T space was then represented mathe-matically using a two-dimensional (2D) spline (more detailsfollow). Using the volume expansion calculated from the abinitio total free energy calculations,6 the elastic constantsat the equilibrium volume at each temperature were thencalculated by evaluating the two-dimensional spline alongthe V-T path corresponding to the predicted volume thermalexpansion derived from the quasiharmonic approximation. Inorder to isolate C11 from C12, the relationship between theseelastic constants and the bulk modulus for cubic crystals wasused [B = (C11 + 2C12)/3]. The bulk modulus was in turncalculated by fitting an equation of state (EOS) of the form28

E(V ) = a + bV −1/3 + cV −2/3 + dV −1, (10)

where a, b, c, and d are fitting parameters. These parameterscan then be related to cohesive energy E0, equilibrium volumeV0, bulk modulus B0, and pressure derivative of the bulkmodulus, B′. For the detailed mathematical relations betweenthe fitting coefficients of the equation of state used and the

corresponding state variables the reader is referred to the workby Shang and Bottger.28 The particular choice of the equationof state did not greatly affect the results obtained in this workand comparisons with more conventional expressions (such asBirch and Birch-Murnaghan) resulted in no major differences.The particular reason for the selected EOS form was basedon mathematical convenience when fitting quasiharmonic freeenergies over the temperature range of interest.

The elastic constants calculated from this approach corre-spond to isothermal conditions. In contrast, resonance-basedexperiments are performed under adiabatic conditions and thusthe following thermodynamic conversion must be used:29

CSij = CT

ij + V λiλjT

Cv

. (11)

In cubic crystals, for C11 and C12, λi = λj =α(CT

11 + 2CT12).30 In the case of C44, the coefficients λ4 are

zero due to the symmetry of the thermal expansion coefficienttensor and thus there is no difference between the isothermaland adiabatic constants. This fact is highly useful when tryingto compare the results from calculations with experimentalmeasurements in a direct manner.

B. Results

The calculations were performed using projector-augmented plane-wave pseudopotentials (PAW) as imple-mented in the electronic structure code VASP.31,32 All thecalculations were performed within the generalized gradientapproximation (GGA) with the parametrization suggestedby Perdew and Wang (PW91).33 For these calculations, theelectrons in the configuration 3s23p1 were considered valencestates. In all calculations, the k-point mesh consisted of atleast 10,000 k points per reciprocal atom. The particularshape of the k-point mesh was adjusted depending on thesymmetry of the reciprocal lattice of the structures con-sidered. The energy cutoff used was 350 eV. The k-pointdensity and energy cutoff employed ensured a very goodconvergence (within <0.1 meV) of the total energies. Asmentioned earlier, the quasiharmonic approximation was usedto calculate the thermodynamic properties, including volumethermal expansion and bulk modulus, of fcc aluminum. Sevenvolumes (ranging from –1% to 5% lattice expansion) wereconsidered in our calculations. For each volume, we usedthe so-called “direct method” to obtain the phonon densityof states. The lattice dynamics calculations used to determinethe phonon density of states were performed using the ATAT

code,34,35 using supercells containing 32 atoms. From thesecalculations, the free energy (including contributions due toelectronic degrees of freedom) can be easily calculated usingthe standard statistical mechanics formulas.6 The reader isreferred to the article by one of the authors6 for a moredetailed comparison between calculated and experimentalthermodynamic properties of fcc Al.

As mentioned earlier, we used volume-conserving strainsto obtain the C11–C12 and C44 elastic constants. To calculateC11–C12 at each volume, we used seven volume-conservingtetragonal strains [see Eqs. (6) and (7)] with the strain parame-ter, x, ranging from –0.06 to 0.06. For C44, at each volume, weused five volume-conserving monoclinic strains [see Eqs. (8)

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TABLE I. Comparison between experimental and calculatedproperties of fcc aluminum.

Property Experiment Calculations Units

a 4.046a 4.050 AB0K 79b 72 GPaB298K 76c 67 GPaC11,0K 107b 105d,103e GPaC12,0K 61b 54d,56e GPaC44,0K 28b 26d,30e GPaα298K 23f 25 ×10−6/KCp ,298K 24.3g 24.4 J/(mol K)S298K 28.4g 29.5 J/(mol K)Debye temperature, 394h 381 K

aReference 56.bReference 57.cReference 2.dElastic constants were calculated using stress-strain relations(Reference 21).eElastic constants were calculated using energy-strain fits(Reference 23).fReference 58.gReference 59.hReference 60.

and (9)] with the strain parameter, x, ranging from 0 to 0.06.In both sets of calculations, the strain parameter was equallyspaced. For each strain state, we calculated its vibrational andelectronic free energies as previously described. By combiningthe elastic constants calculated at different temperatures andvolumes, we created a V-T surface for C11–C12 and C44. TheV-T surface was represented mathematically through the useof a 2D smooth B spline (using the interpolate module inSCIPY36). The temperature dependence of the elastic constantswas then obtained by evaluating the 2D spline along the V-Tpath resulting from the predicted thermal expansion for fccaluminum. Finally, we applied the conversion [see Eq. (11)]from isothermal to adiabatic elastic constants in order tocompare our calculations with experiments. We would liketo note that the use of the smooth 2D spline also contributedto the reduction in the noise associated with the calculation ofthe elastic tensor and this is the main reason for the low levelof noise in the temperature dependence of the elastic tensorcalculated through DFT + quasiharmonic approximations.

Table I shows comparisons between calculated and ex-perimental structural and thermodynamic properties of fccAl. In general, the calculated structural and thermodynamicproperties agree very well with experiments. In the table, wealso present the calculated elastic constants (at 0 K) usingtwo different approaches: one based on energy-strain fits23

used in the rest of the present paper—and the other one basedon stress-strain relations.21 The table show that both sets ofcalculations (using the same k-point mesh density, energycutoff, and convergence criteria already described) yieldedresults relatively close to experiments (within ∼5 GPa) aswell as to one another (within ∼2 GPa). Since these twocalculation schemes are fundamentally independent of oneanother, the agreement suggests the calculations are alreadywell converged with respect to k-point mesh and energy cutoff.Other calculations performed at different strain levels (for

FIG. 2. (Color online) Experimental2–5 and DFT calculated C11

adiabatic elastic constant. For the curve labeled “cold curve” (dashed,red line), the elastic constants where calculated from the strain-energy relations at 0 K at different volumes. Ab initio-calculatedvolume expansion was then used to convert volume to temperaturedependence. The curve labeled “quasiharmonic” (solid, blue line)was obtained from strain-free energy relations.

the stress-strain calculations) and maximum strains (for theenergy-strain fits) suggest convergence within ±1 GPa.

After applying the corrections outlined in this paper, weproceeded to compare the resulting temperature dependenceof the elastic constants with the published data. Figure 2compares C11 as obtained from experiments2–5 with the DFTcalculations. In the figure, we include the calculated C11

obtained simply by finding the value of this elastic constantat different volumes through the 0 K energy-strain relations(cold curve), as well as the C11 in which we used the freeenergy-strain relations (quasiharmonic). From the figure it isseen that only by considering the vibrational contributions tothe total free energy of the strained structures it is possibleto closely reproduce the temperature dependence of C11 asreported by Gerlich and Fisher2 and Kamm and Alers,3 aswell as (in a limited manner) the results by Tallon.5

In Fig. 3, the (cold curve) C12 shows a softening closeto that reported by Gerlich and Fisher2 and Kamm andAlers.3 Surprisingly, our calculations suggest that vibrationalcontributions lead to a stiffening of C12. In this case, thequasiharmonic C12 remains almost constant as temperatureincreases, in contrast with what Gerlich and Fisher2 and Kammand Alers3 suggest. Our results seem to suggest that Sutton4

overestimated the temperature dependence of this elasticconstant. The hardening of C12 with temperature reported byTallon5 is not supported by our calculations, even consideringthe relatively small softening resulting from our quasiharmoniccalculations. It is likely that the underestimation of softeningin our quasiharmonic calculations results from the small super-cells used to calculate the strain-free energies. As will be seenin the next section, fluctuations in the microscopic stress tensorhave very limited effects on the softening of this componentof the stiffness tensor. Thus, this softening originates fromthe weakening of the interatomic force constants as volumeincreases, and is thus mostly due to Born-like contributions.

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FIG. 3. (Color online) Experimental2–5 and DFT calculated C12

adiabatic elastic constant. For the curve labeled “cold curve” (dashed,red line), the elastic constants where calculated from the strain-energy relations at 0 K at different volumes. Ab initio-calculatedvolume expansion was then used to convert volume to temperaturedependence. The curve labeled as quasiharmonic (solid, blue line)was obtained from strain-free energy relations.

In Fig. 4, we present the comparison between calculatedand experimental adiabatic elastic constants including onlythe results published by Gerlich and Fisher2 and Kamm andAlers3. The figure shows that the temperature dependence of(quasiharmonic) C11 agrees rather well with the experiments(in this figure we exclude the results reported by Sutton4 andTallon5). The figure shows that the quasiharmonic calculationsdiffer from experiments by an almost constant amount. Thisdiscrepancy is likely due to the underestimation of thebulk modulus calculated within the GGA approximation, asdiscussed earlier.

FIG. 4. (Color online) DFT calculated and experimental2,3 adia-batic elastic constants of fcc Al. For the curves labeled “cold curve”(dashed lines), the elastic constants where calculated from the strain-energy relations at 0 K at different volumes. Ab initio-calculatedvolume expansion was then used to convert volume to temperaturedependence. The curves labeled “quasiharmonic” were obtained fromstrain-free energy relations.

Figure 4 shows that the agreement is also quite good forC44. This elastic constant is not affected by the underestimationof the bulk modulus and in this case, the cold curve and quasi-harmonic calculations show virtually no difference. Despitethese promising results, our calculations still underestimate thesoftening in C12 reported by Gerlich and Fisher2 and Kammand Alers.3

Given the good agreement between the DFT calculationsand at least two experimental data sets, one would be temptedto conclude that full consideration of dynamic, anharmoniccontributions to the stiffness tensor are not important. Infact, even though fcc aluminum appears to be a weaklyanharmonic crystal, Forsblom et al.9 have in fact shownthat aluminum is highly anharmonic at even moderate tem-peratures. What contributes to the apparent nonanharmonic-ity of this crystal is the almost complete cancellation ofanharmonic contributions of opposite signs. If anharmoniccontributions in fact are important in fcc aluminum, the goodagreement between the DFT calculations and the experimentsby Gerlich and Fisher2 could be due to fortuitous errorcancellations. To further investigate this possibility, in the nextsection we employ dynamic methods to estimate the elasticstiffness tensor through the fluctuation formulas presentedin Eq. (3).

IV. DETERMINATION OF THERMOELASTICPROPERTIES THROUGH MOLECULAR DYNAMICS

CALCULATIONS

A. Approach

We performed molecular dynamics (MD) calculations onthe elastic constants of Al using tight-binding potentialssuggested by Rosato et al., which is often referred to as RGL(Rosato, Guillope, and Legrand).37,38 Although simpler pairpotentials for fcc Al have been successfully developed, theystill fail to fully reproduce the physical properties of crystallinematerials, as the role of electron density and the atomicbonding in pair potentials are underestimated. For instance,pure pair potentials imply the Cauchy relation between elasticconstants C12=C44, which is not necessarily true in real metalsand alloys. Also, stacking fault energies, surface structure, andrelaxation properties cannot be accurately estimated while us-ing pair potentials.37 Many-body models overcome these lim-itations by properly treating the essential band character of themetallic bonding. Over the past decades, a collection of many-body potentials has been developed, including those based oneffective medium theory,39 embedded atom method,40 as wellas those based on the tight-biding approach, such as the Finnisand Sinclair,41 Sutton and Chen,42 and the RGL model.37,38

The tight-binding methods describe interatomic interac-tions as a combined effect of a short-range pairwise repulsionand a many-body density-dependent cohesion. The functionalform of RGL interaction potential for an atom a can bedescribed as follows37,38:

Ea = AV (rab) − ξρa, (12)

in which

V (rab) =∑b �=a

exp(

−p

(rab

r0− 1

)),

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HIEU H. PHAM et al. PHYSICAL REVIEW B 84, 064101 (2011)

ρa =[∑

b �=a

ϕ(rab)

]1/2

,

ϕ(rab) = exp

[−2q

(rab

r0− 1

)],

where r0 is the first-neighbor atomic distance, and A, ξ , p, andq are empirical parameters whose values are obtained by fittingto 0 K properties such as cohesive energy, elastic coefficients,and structure stability; they are all published in the work ofCleri and Rosato.37

Compared to the Sutton-Chen scheme, which has beensuccessfully utilized to study various bulk properties of metaland metal alloys,43–48 the RGL model is fairly similar interms of its functional expression and number of fittingparameters. The first term in Eq. (12) indicates the atomicrepulsions that take into account the increase in kineticenergy of bonding electrons when two ions get close toeach other. The Sutton-Chen model introduces a power forminstead of an exponential expression to describe these ion-ionrepulsions. The second term, having a general formulation of

the type ξ [∑

b �=a ϕ(rab)]1/2, specifies a many-body cohesionthat accounts for the nature of the effective band energy and itbalances with repulsion forces in order to stabilize the crystals.The function ϕ(rab) corresponds to the local electronic chargedensity induced at site a from atoms at site b, and is alsodescribed by a power function in the case of the Sutton-Chenmodel.

Our MD simulation model consists of 500 Al atoms,in which the cutoff distance covers up to the fifth-nearestneighbor49 (corresponding to a

√5/2 in fcc crystals), and

the simulation time step is chosen to be 1 fs. At first,a thermalization process is conducted by slowly heatingthe system from 0.001 K (with a temperature incrementof 1 K per step) until it reaches the desired temperature. Thezero-strain state is determined from constant temperature andstress ensembles, by changing the volume of the supercell.Once the reference volume has been obtained, a strict velocityscaling of 50,000 steps is performed, followed by a preliminarymolecular dynamics run of 20,000 steps to equilibrate thesystem at the temperature of interest. The elastic constants arethen derived from the second derivative of the total energy

TABLE II. Molecular dynamics and quasiharmonic DFT calculations of aluminum second-order elastic constants.

MD DFT

T, K EC, GPa Kinetic energy Born Fluctuation Total Cold curve Quasiharmonic

0 C11 0.00 95.02 0.00 95.02 103.89 103.35C12 0.00 74.40 0.00 74.40 55.88 56.16C44 0.00 37.16 0.00 37.16 29.51 29.90

50 C11 0.17 94.12 −0.71 93.58 103.59 102.71C12 0.00 73.70 −0.05 73.65 56.05 56.32C44 0.08 36.68 −0.42 36.34 29.20 29.60

100 C11 0.32 93.73 −1.40 92.65 103.08 101.80C12 0.00 73.30 −0.08 73.21 56.13 56.44C44 0.16 36.43 −0.90 35.69 28.82 29.10

200 C11 0.64 92.05 −2.64 90.06 101.54 99.21C12 0.00 71.96 −0.23 71.73 56.07 56.57C44 0.32 35.53 −1.72 34.13 27.86 27.64

300 C11 0.98 90.45 −4.27 87.15 99.41 95.79C12 0.00 70.56 −0.28 70.29 55.74 56.55C44 0.49 34.58 −2.78 32.29 26.68 25.72

400 C11 1.29 89.16 −5.90 84.55 96.82 91.76C12 0.00 69.39 −0.41 68.99 55.21 56.41C44 0.65 33.83 −3.54 30.93 25.33 23.56

500 C11 1.62 87.53 −7.42 81.73 93.93 87.35C12 0.00 67.94 −0.38 67.57 54.54 56.17C44 0.81 32.87 −4.60 29.07 23.85 21.35

600 C11 1.91 86.15 −9.05 79.02 90.85 82.79C12 0.00 66.66 −0.44 66.22 53.80 55.84C44 0.96 32.03 −5.39 27.59 22.28 19.30

700 C11 2.21 84.48 −10.97 75.71 87.74 78.28C12 0.00 65.13 −0.54 64.60 53.04 55.44C44 1.10 31.01 −6.29 25.82 20.66 17.61

800 C11 2.50 83.02 −12.99 72.53 84.72 74.07C12 0.00 63.62 −0.73 62.89 52.32 54.99C44 1.25 30.02 −7.27 23.99 19.04 16.50

900 C11 2.81 81.77 −15.16 69.42 81.95 70.37C12 0.00 62.08 −0.65 61.43 51.72 54.51C44 1.41 29.03 −8.54 21.89 17.47 16.16

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TABLE III. Average temperature dependence of elastic constants; comparison between MD and DFT calculations.

MD DFT

Kinetic Born Fluctuation Total Cold curve Quasiharmonic

�C11, GPa 2.8 –13.3 –15.2 –25.6 –21.9 –33.0dC11/dT 3.1×10−3 –1.5 × 10−2 –1.7 × 10−2 –2.8 × 10−2 –2.4 × 10−2 –3.7 × 10−2

�C12, GPa 0.0 –12.3 –0.7 –13.0 –4.2 –1.7dC12/dT 0.0 × 100 –1.4 × 10−2 –7.2 × 10−4 –1.4 × 10−2 –4.6 × 10−3 –1.8 × 10−3

�C44, GPa 1.4 –8.1 –8.5 –15.3 –12.0 –13.7dC44/dT 1.6 × 10−3 –9.0 × 10−3 –9.5 × 10−3 –1.7 × 10−2 –1.3 × 10−2 –1.5 × 10−2

with respect to the homogeneous deformation of the unit cell,as given in Eq. (3), by performing 250,000 steps of constantshape and constant energy simulations (EhN ensemble). Thenine elastic constants were calculated separately in order toensure convergence. It was then verified that the resultingstiffness tensor’s symmetry was cubic (i.e., nonvanishing C11,C12, and C44).

B. Results

The MD simulations show that the elastic constants ofAl decrease with temperature, as a softening in materials isexpected. This is in agreement with the DFT calculations aswell as with the experimental results by Gerlich and Fisher2

and Sutton4. The stability criteria for cubic crystals were heldfor Al, as C44 > 0, C11 > 0, and C11 > C12. As demonstrated inFigure 8, C44 declines and the divergence between C11 and C12

diminishes as temperature increases. Beyond the equilibriummelting point, vanishing C11–C12 and C44 would correspondto Born melting.

The kinetic energy, Born term, and fluctuation contributionsare listed (Table II) and plotted separately (Figs. 5–7) inorder to elucidate their individual contributions to the stiffnesstensor. As can be seen in Fig. 5, the kinetic contributions arerather small, compared to the other two terms, and even vanishfor C12. Although the kinetic term contributes to a stiffeningof C11 and C44, its relative contribution to the total elasticconstants (Table II and Fig. 8) is insignificant.

FIG. 5. (Color online) Kinetic energy contributions to C11, C12,C44 as a function of temperature in MD calculations.

By examining Table II, it can be seen that the Born term(Fig. 6) constitutes the most important contribution to thestiffness tensor. Table III also suggests that this term hasalmost the same temperature dependence for all three elasticconstants. Our calculations indicate that the Born term ofthe stiffness tensor softens as temperature increases, which isconsistent with weakening of interatomic bonds as interatomicdistance increases. The fluctuation term (Fig. 7) representsthe contributions to the stiffness tensor due to fluctuationsin the microscopic stress tensor. Since these fluctuationsare related to the amplitude of atomic displacements, astemperature increases fluctuation contributions to the stiff-ness tensor also increase. Table III shows that fluctuationcontributions to the temperature dependence are strongestfor C11, almost comparable to the temperature dependenceof the Born contribution. In the case of C12 and C44, thetemperature dependence of the fluctuation term is 100 timesand 10 times weaker than the corresponding Born term,respectively.

Since the temperature dependence of the Born term issimilar for all the elastic coefficients and the contributionsfrom kinetic energy terms are relatively small, the differencein the temperature dependence of the total elastic constantsis mostly determined by the behavior of the fluctuationcomponents. Those contributions increase continuously andget up to ∼18% and ∼29% of the Born terms for C11

and C44, respectively, at 900 K (which is slightly below

FIG. 6. (Color online) Born term contributions to C11, C12, C44

as a function of temperature in MD calculations.

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FIG. 7. (Color online) Fluctuation term contributions to C11, C12,C44 as a function of temperature in MD calculations.

the experimental melting point of aluminum). In contrast,fluctuation terms in the C12 constant are almost negligible.Figure 8 shows clearly a linear dependence of elasticity withrespect to temperature in the region after room temperatureup to 900 K. Also, the slopes of C11, C12, and C44 withrespect to temperature reported by these molecular dynamicscalculations are in relatively good agreement (see Table III)with those obtained from our earlier DFT calculations. In fact,the major disagreement between the classical MD and DFTcalculations corresponds to the low-temperature C12 elasticconstant, with C44 differing by <10% and C44 agreeing almostperfectly.

V. DETERMINATION OF ELASTIC MODULI BYRESONANT ULTRASOUND SPECTROSCOPY

A. Approach

RUS is a high-precision dynamic technique that allowsdetermination of up to nine elastic constants by measuringvibrational spectrum of the samples with known geometry—usually in the shape of parallelepipeds or cylinder—andmass.13,50

FIG. 8. (Color online) MD calculations on C11, C12, C44 atelevated temperatures.

A polycrystalline aluminum sample (Metalman, LongIsland City, NY) with a purity of 99.999% and a densityof 2.689 g/cm3 was used in the present study to determineelastic moduli and to compare to the results of DFT andMD calculations. The sample was precisely machined in theform of disks with an average diameter of 28.052 mm and anaverage thickness of 3.270 mm. The Young’s (E) and shear(μ) moduli of the sample were determined using resonantultrasound spectroscope (Magnaflux Quasar, Albuquerque,NM). The details of the experimental setup for RUS can befound elsewhere.13,51,52 Briefly, the disk-shaped sample wassupported by three piezoelectric transducers. One transducer(the transmitting transducer) generates an elastic wave ofconstant amplitude but of varying frequency (covering alarge number of vibrational eigenmodes of the sample). Theresonance response of the excited sample is detected by theother two transducers, i.e., the receiving transducers. In orderto study the variation of elastic moduli as a function oftemperature, the commercially available setup for RUS at roomtemperature was modified for high-temperature measurementsin which 4-in.-long SiC extension rods were used to transmitthe ultrasound waves to and from the sample at the desiredtemperature in the furnace, while keeping the piezoelectrictransducers out of the furnace and thus unaffected by hightemperature. The sample was heated at a ramping rate of10 K/min in an argon atmosphere and resonance spectra werecollected at intervals of 50 K, from room temperature up to773 K after an isothermal hold of 10 min.

The resonant spectra were collected in the 10–200-kHzfrequency range to cover the first 40 eigenfrequencies. Withknown dimensions, density, and a set of “guessed” elasticconstants C11 and C44, considering the material as an isotropicsolid—accurate elastic moduli of the solid can be determinedfrom collected RUS spectra using a multidimensional, iterativefitting approach (RUSpec, Magnaflux Quasar, Albuquerque,NM) that minimizes the root-mean-square error between themeasured and calculated resonant frequencies for a sampleof known mass and dimensions. The calculated resonant fre-quencies were determined by minimization of the Lagrangianequation corresponding to the free-body vibrations of an elasticsolid, using the Rayleigh-Ritz method.50,53

B. Results

Figure 9 shows the measured Young’s and shear moduliand Poisson’s ratio of the polycrystalline aluminum sampleas determined by RUS over the 297–773 K temperaturerange. In order to compare these new measurements with theelastic constant tensor calculated in this work and measuredpreviously,2–5 the single-crystal elastic tensor was transformedto polycrystalline elastic constants through Voigt-Reuss-Hill54

averaging.Figures 9 and 10 clearly show a very good agreement

between the DFT calculations and the experimental mea-surements by Gerlich and the RUS results obtained in thiswork. The RUS measurements for polycrystalline aluminumin this work differ by <∼2 GPa at the highest tempera-tures measured. The (cold curve) DFT calculations slightlyoverestimate the Young’s and shear modulus (compared tothe experimental results) and underestimate Poisson’s ratio,

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FINITE-TEMPERATURE ELASTICITY OF fcc Al: . . . PHYSICAL REVIEW B 84, 064101 (2011)

(a)

(b)

FIG. 9. (Color online) Polycrystalline (a) Young’s and (b) shearmodulus. The averaged elastic constants from experimental results byGerlich and Fisher2 as well as those measured in this work throughRUS are compared to DFT and classical MD calculations.

although the calculated temperature dependence agrees verywell with the experimental data sets used for comparison. The(quasiharmonic) DFT calculations yield Young’s and shearmodulus within the ranges measured by Gerlich as well as withthe measurements reported in this work. However, an inflectionat ∼600 K takes these polycrystalline elastic constants closerto those predicted by our MD calculations.

Figure 9 shows that the molecular dynamics simulationsexhibit the largest discrepancy with the polycrystalline elasticmoduli and Poisson’s ratio derived from the measurementsby Gerlich and this work. As previously described, themajor reason for this discrepancy is an overestimation (of∼20%) of the low-temperature C12 coefficient. Despite thisdiscrepancy, Fig. 9 shows that the temperature dependence ofthe averaged polycrystalline elastic constants derived from theMD calculations agree very well with the experiments (as wellas with the DFT calculations).

VI. CONCLUSIONS

This paper was motivated by the significant quantitativeand qualitative discrepancies in the finite temperature elastic

FIG. 10. (Color online) Polycrystalline Poisson’s ratio. The av-eraged elastic constants from experimental results by Gerlich andFisher2 as well as those measured in this work through RUS arecompared to DFT and classical MD calculations.

constants of fcc Al measured by three different groups. Giventhe fact that aluminum is one of the most technologicallyimportant and well characterized metals, the discrepancieswere very surprising. While a priori it is not possible to judgethe reliability of a particular data set, we have demonstrated inthis work how we can use atomistic simulation techniques toassess the reliability of experimental data.

Our DFT calculations exhibit surprisingly good agreementwith the measurements by Gerlich and Fisher2 and by thisstudy, even though these calculations neglected to consideranharmonic contributions to the stiffness tensor. Althoughthis good agreement seems to suggest that fcc Al is at mostweakly anharmonic, results by others9 suggest that whatseems to happen is that anharmonic contributions at leastpartially cancel out.

Our MD calculations consider all the possible contribu-tions to the stiffness tensor. These calculations suggest thatanharmonic contributions (manifested in the fluctuation term)seem to contribute significantly to the temperature dependenceof the elastic constants, particularly C11. These anharmoniceffects are partly cancelled by the intrinsic kinetic stiffnessof the crystal. While the calculated C12 obtained through MDsimulations is significantly higher than the DFT-calculated andexperimentally measured elastic constant, C11 and C44 agreewell with the DFT calculations (errors of ∼10%). In fact, MDand DFT calculations predict a change between 300 and 900 Kof ∼17, ∼9, and ∼10 GPa for C11, C12, and C44, respectively.More importantly, this is also what we can extract from thedata reported by Gerlich.

Finally, the calculated and averaged elastic constants in thiswork agree very well (at least with regards to the temperaturedependence) with the experiments by Gerlich as well as themeasurements performed in this work through RUS. Discount-ing the discrepancies in the MD-derived polycrystalline elasticmoduli due to overestimation of the low-temperature C12, thiswork illustrates a sound methodology for the assessment ofthe quality of published thermoelastic data for other importanthigh-temperature structural materials.

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ACKNOWLEDGMENTS

H.H.P. and T.C. would like to acknowledge partial supportfor this research through ARO MURI (Anthenien) and ONR(Grant No. N000140811054). Molecular dynamics simula-tions have been carried out at the facilities of Labora-tory of Computational Engineering of Nanomaterials. R.A.acknowledges the support from NSF through Grants No.CMMI-0758298, No. 1027689, No. 0953984, No. 0900187,

No. DMR-0805293, and No. CBET-0932249. First-principlescalculations were carried out in the Chemical EngineeringCluster and the Super-Computer Facility of Texas A&MUniversity as well as in the Ranger Cluster at the TexasAdvanced Computing Center. Preparation of the input files andanalysis of the data have been performed within the frameworkAFLOW/ACONVASP55 developed by Stefano Curtarolo as wellas with the ATAT package34,35 developed by Axel van de Walle.

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