www.elsevier.com/locate/jnoncrysol

Journal of Non-Crystalline Solids 353 (2007) 3565–3569

A comparison of crystal–melt interfacial free energiesusing different Al potentials

James R. Morris a,b,*, M.I. Mendelev c,d, D.J. Srolovitz d

a MS&T Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6115, USAb Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996-2200, USA

c Materials and Engineering Physics, Ames Laboratory, Ames, IA 50011, USAd Department of Physics, Yeshiva University, New York, NY 10033, USA

Available online 13 July 2007

Abstract

We have calculated the crystal–melt interfacial stiffnesses using simulations with three different interatomic potentials for Al, and fromthese derived the anisotropic crystal–melt interfacial free energies. We find that there is a strong dependence of the results on the potential,and that this dependence cannot be explained by the usual Turnbull relation between the interfacial free energy and the latent heat. Thepotentials which produce liquid structures in closer agreement with experiments give free energies in good agreement with nucleation data.� 2007 Elsevier B.V. All rights reserved.

PACS: 61.20.Ja; 64.70.Dv; 68.08.�p; 61.25.Mv

Keywords: Liquid alloys and liquid metals; Crystal growth; Nucleation; Neutron diffraction/scattering; Electrical and electronic properties; Modeling andsimulation; Molecular dynamics; Fluctuations; Surfaces and interfaces

1. Introduction

The crystal–melt interface is of interest to many research-ers, due to its importance in both crystal nucleation, and insolidification [1–13]. For metallic systems, the interface isusually rough, due to the weak anisotropy in the interfacialtension. However, this anisotropy, though weak, is impor-tant for rapid alloy solidification, as it determines the den-drite growth direction, and is necessary for stable dendritegrowth. The weak anisotropy for Al is demonstrated bythe observation of different dendrite growth directionsand morphologies, depending upon the alloy additions [14].

Recently, there has been a renewed push in measuring orcalculating the crystal–melt interfacial free energies and itsanisotropy [15]. Two new approaches have been recentlyproposed [16,17] and their mutual agreement has been dem-

0022-3093/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.jnoncrysol.2007.05.116

* Corresponding author. Address: MS&T Division, Oak Ridge NationalLaboratory, P.O. Box 2008, Oak Ridge, TN 37831-6115, USA. Tel.: +1865 576 7094; fax: +1 865 576 6298.

E-mail address: morrisj@ornl.gov (J.R. Morris).

onstrated for the Lennard-Jones system [18,19] and for thehard-sphere system [20]. These are the first approaches suf-ficiently accurate to treat the problem of anisotropy.

The primary interest is in predicting materials specific

properties. Why should solidification in Ni be differentfrom Al? What is the role of alloying? The solid–liquidinterfacial free energies have now been calculated for anumber of FCC metals, including Ni and Al [17,21–27].To a large extent, the results of these systems (and theLennard-Jones systems) show very close trends. First, theaverage interfacial free energy is approximately predictedby the Turnbull relation [1,2]:

c0 ¼ CTLq2=3s ; ð1Þ

where L is the latent heat, qs is the number density in thesolid phase, and the Turnbull coefficient CT has a magni-tude of approximately 0.5 [1,2,25,28]. Secondly, all of thesystems show anisotropies such that the interfacial freeenergies satisfy

c111 < c110 < c100: ð2Þ

3566 J.R. Morris et al. / Journal of Non-Crystalline Solids 353 (2007) 3565–3569

All anisotropies are on the order of 1–2%. This similarity isnot surprising, but raises the important issue of whether wecan really answer material specific questions, or whetherour results are essentially that of a more generic behavior.This is particularly of concern, given that the interatomicpotentials can give very different melting behaviors thanthe materials that they model, including melting tempera-ture, latent heats, liquid structures, and changes of densityat melting.

For Al, there are some experimental data. Turnbullfound a value of 92 mJ/m2 from nucleation experiments[1,2]; later work by Kelton [6] from nucleation gave avalue of 107 mJ/m2. However, more direct grain-boundarygroove experiments [29] give a significantly larger value of163 mJ/m2. The latter experiments benefit from the factthat they are a more direct measurement, balancing thecurvature of the interface against a thermal gradient; how-ever, the experiments use small amounts of Cu, which bothdirectly affects the interface, and also lowers the tempera-ture at which the measurement is made. Therefore, it is dif-ficult to resolve the experimental differences.

In a previous work [27], we examined these propertiesusing a potential for Al. The general trends appeared tofollow the above. In the present work, we examine the roleof the potential, by considering three different Al poten-tials. The potentials were all chosen to have propertiestuned specifically for the liquid behavior; in particular, allhave good agreement for the melting temperatures. Wedemonstrate that there are significant differences betweenthe different potentials, in the average interfacial freeenergy, the anisotropy in the interfacial free energy, andin the Turnbull coefficient. We find that most of these prop-erties correlate with the liquid structure, and that the sys-tems with more realistic liquid structures have freeenergies closer to the nucleation results.

Table 1Some fitted properties for the potentials considered in this work

Target value(firstprinciples/experiment)

ErcolessiandAdams

Mei andDavenport;Sturgeon andLaird

Mendelevet al.

Unrelaxed vacancyenergy(eV/atom)

0.66 0.78 0.90 0.69

DE(BCC � FCC)(eV/atom)

0.100 0.057 0.034 0.074

DE(HCP � FCC)(eV/atom)

0.028 0.018 0.005 0.028

Tmelting (K) 933 933 931 939

2. Calculational techniques

The approach that we take is the same that we have usedin similar previous calculations [19,27] and we will notrepeat the details here. Essentially, a crystal–melt systemis simulated in coexisting equilibrium, with a well definedorientation of the crystal interface separating the crystalfrom the liquid. For a rough interface, the height of theinterface will fluctuate. In equilibrium, the Fourier trans-form of the fluctuations follows [17,27,30]

hjhðqÞj2i ¼ kBT~cq2

; ð3Þ

where ~cðhÞ is the interfacial ‘stiffness’, which depends uponthe orientation of the interface. Here, we will only considerrotations around a [001] axis, so that the orientation maybe specified by a single angle h. In this case, the stiffness isrelated to the interfacial free energy by

~cðhÞ ¼ cðhÞ þ c00ðhÞ: ð4Þ

For a four-fold rotational symmetry, the lowest orderanisotropic expression for the interfacial free energy is

cðhÞ ¼ c0ð1þ e4 cos 4hÞ; ð5Þ

and the corresponding stiffness is therefore

~cðhÞ ¼ c0ð1� 15e4 cos 4hÞ: ð6Þ

We expect that the anisotropy parameter e4 will be on theorder of 1%. Eq. (6) indicates that the stiffness (and thefluctuations, given in Eq. (3)) will be significantly moreanisotropic than this. This is also found in practice[17,19,24,27].

The simulations are performed using three different Alpotentials, each having similar form. The energy of atomi may be written as

Ei ¼ F ðqiÞ þ1

2

X

j 6¼i

uðrijÞ; qi ¼X

j 6¼i

f ðrijÞ: ð7Þ

The first potential is from Ercolessi and Adams [31] andwas explicitly fit using liquid-like configurations and cho-sen due to its accurate melting temperature. We will referto this as the EA potential. The second is originally dueto Mei and Davenport [32], and was modified by Sturgeonand Laird to have a good melting temperature [33]. We willrefer to this as MDSL. The final one was developed by us[34], and we shall refer to this as the MSAHM potential.Again, the potential was explicitly fit to have a good melt-ing temperature.

Several of the fitted properties for these potentials aregiven in Table 1. The MS potential clearly gives gener-ally better properties, in particular the energy differencesbetween the FCC, HCP and BCC phases. As indicatedabove, all give good melting temperatures (within 1% ofthe experimental value).

The simulations and analysis are similar to those in Ref.[27] and we will not repeat details here. Briefly, the inter-faces are equilibrated for each potential for 500000 molec-ular dynamics time steps, and then the next 1 million timesteps are used to calculate the height functions. Each MDtime step represents 0.53 fs. The values of hjh(q)j2i are thencalculated, and fit to the form of Eq. (3) to obtain the stiff-nesses. The simulation sizes are reported for the MSAHM

Table 2Simulation geometries for MSAHM potential

Interface Interfacenormal

System size (A) Number ofatoms

(100) [001] 132.51 · 274.28 · 16.56 32768(120) [001] 200.32 · 153.41 · 16.56 40960(110) [001] 140.55 · 291.07 · 16.56 36864(110) ½1�10� 132.51 · 291.07 · 23.43 49153(111) ½1�10� 202.87 · 148.54 · 23.43 38400(112) ½1�10� 143.448 · 210.07 · 23.43 38400

Simulations for EA and MDSL potentials were similar, except those withinterface normals along ½1�10� were half as large in this direction, and thenumber of atoms was similarly reduced by half.

J.R. Morris et al. / Journal of Non-Crystalline Solids 353 (2007) 3565–3569 3567

potential in Table 2. The EA results were reported in [27],and the MDSL potential was similar (scaled by the appro-priate densities at the melting temperature). The latter twodiffer from the MSAHM potential by a reduction by halfalong the short direction for simulations with a short direc-tion along the ½1�10� direction. This is due to the longerpotential cut-off for the MSAHM potential which madelarger simulations necessary.

3. Results

In Fig. 1, we show the stiffnesses for the (100), (21 0)and (110) planes, as a function of the orientation. Here,h is taken to be the angle between the normal and the[100] direction. We have plotted the stiffnesses vs. cos4h

Fig. 1. Interfacial stiffness vs. orientation, for three different potentials.Each data point represents the stiffness for one simulation, calculated byfitting the fluctuations to Eq. (3). Straight lines indicate fits to the formgiven in Eq. (6).

so that we may fit to Eq. (6). As we see, the stiffnessdepends sensitively to both orientation and on potential.The fits give the values of the anisotropy coefficient e4 aswell as the average interfacial free energy c0.

The fitted values are given in Table 3, along with some ofthe liquid and melting properties. We compare to the exper-imental values for the latent heats, density of the liquid andsolid phases at melting. We also compare the average inter-facial free energy with the nucleation results of Kelton [6]and with the grain-boundary groove experiments of Gun-duz and Hunt [29]. We see that the EA potential givesresults quite close to the latter experiments, as noted in[27]. However, the other potentials give closer agreementwith the nucleation data. The Turnbull coefficients comparesimilarly; thus, the discrepancies cannot be accounted forby different densities and latent heats. Note that the ‘typical’Turnbull value of 0.54 [25] is between these values; the lowvalue for the nucleation data was apparent even in theoriginal data of Turnbull [1,2]. We also note that the aniso-tropy coefficients are small, as expected; both the EA andMSAHM potential give good agreement with equilibriumshape measurements [35].

To explore the differences further, we show the correla-tion function G(r) from the different potentials in Fig. 2, incomparison with neutron scattering results. The correlationG(r) is related to the usual pair distribution function g(r) by

GðrÞ ¼ r½gðrÞ � 1�: ð8Þ

This quantity is directly related to the coherent scatteringfunction S(q). In Fig. 2, the bottom curve and the dashedcurves are the neutron scattering results [36]. Clearly, noneof the potentials are in very close agreement; however, theMSAHM potential is clearly closest to the experimentalvalues, while the EA potential is particularly poor. In par-ticular, the EA potential predicts a split second peak, indi-cating a bi-modal second-nearest neighbor distance. Thishas been interpreted as being an indication of ‘icosahedral’

Table 3Calculated values related to melting and crystal–melt interfacial freeenergies, calculated using the three potentials and compared to experi-mental values

ErcolessiandAdams

Mei,Davenport,Sturgeon,and Laird

Mendelevet al.

Experiment

qsol (A�3) (Tm) 0.058 0.056 0.056 0.056qliq (A�3) (Tm) 0.053 0.052 0.052 0.053Lm (eV/atom) 0.103 0.096 0.104 0.112c0 (meV/A2) 9.4 6.1 6.4 10.1 (Gunduz

and Hunt)6.7 (Kelton)

c0/(Lmq2/3)(Turnbullscaling)

0.60 0.43 0.41 0.62 (G&H)0.41 (Kelton)

e4 0.009 0.022 0.009 0.0089(Napolitanoet al.)

Fig. 2. Pair correlation function G(r) for the three different potentials,compared to neutron scattering results [36]. Experimental results areshown on the bottom graph, and as dashed lines with the simulationresults. Simulation results have been vertically offset for clarity.

3568 J.R. Morris et al. / Journal of Non-Crystalline Solids 353 (2007) 3565–3569

structure of the liquid. On the other hand, the other poten-tials are typical of hard-sphere-like liquids; this is demon-strated explicitly for the MDSL potential in Ref. [37].

4. Conclusions

We have performed simulations of three different crys-tal–melt interfaces, and three different potentials for Al,to calculate the average interfacial free energy and theanisotropy in the (100) plane. All potentials have goodmelting temperatures, and thus temperature effects havebeen eliminated. The MSAHM potential appears to bebest, in terms of ordering energies, liquid structure, andunrelaxed vacancy energy [34]. This potential gives an aver-age interfacial free energy and anisotropy in good agree-ment with nucleation results and equilibrium shapemeasurements. The value of the Turnbull coefficient alsocompares well to the nucleation data.

The large differences between the different potentialssuggests that material specific predictions for these quanti-ties will require accurate potentials. It is not immediatelyapparent why the different potentials give such differingresults. Most apparent is the large interfacial free energyfrom the EA potential. This potential gives a very differentliquid structure from the experiment and from the otherpotentials. This makes the very reasonable point that thestructure of the liquid has a strong effect on the crystal–melt interfacial free energies [4]. We have also calculatedthe diffusion for these potentials, and again the EA isanomalous, having a much slower diffusion rate in the

liquid. We will discuss this more thoroughly in a laterpublication.

We also note that there is a large experimental discrep-ancy for the interfacial free energy. One origin of this maybe due to the fact that the grain-boundary groove experi-ment was performed with Cu in the liquid [29], in orderto image the curvature of the interface. Preliminary calcu-lations show that Cu strongly affects the liquid struc-ture. Noting that many quasicrystal systems are based onAl–Cu alloys, we speculate that Cu makes the liquid moreicosahedral. The EA potential appears to be more consis-tent with icosahedral order, and gives results in good agree-ment with the values of Gunduz and Hunt [29].

Acknowledgements

This research has been sponsored by the Division ofMaterials Sciences and Engineering, Office of Basic EnergySciences, US Department of Energy under contractDE-AC05-00OR-22725 with UT-Battelle and contractW-7405-ENG-82 with Iowa State University of Scienceand Technology. We also acknowledge partial fundingfrom the Department of Energy’s Computational Materi-als Science Network project, ‘Microstructural EvolutionBased on Fundamental Interfacial Properties’, and com-puter time from the Department of Energy’s NationalEnergy Research Scientific Computing Center.

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