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Experimental and computer simulation determination of the structuralchanges occurring through the liquid-glass transition in Cu-Zr alloysM. I. Mendeleva; M. J. Kramera; R. T. Otta; D. J. Sordeleta; M. F. Bessera; A. Kreyssiga; A. I. Goldmana; V.Wesselsb; K. K. Sahub; K. F. Keltonb; R. W. Hyersc; S. Caneparic; J. R. Rogersd

a Ames Laboratory USDOE and Iowa State University, Ames, Iowa 50011, USA b Department ofPhysics, Washington University, St. Louis, Missouri 63130, USA c University of Massachusetts,Amherst, Massachusetts 01003, USA d NASA Marshall Space Flight Center, Huntsville, Alabama 35812,USA

First published on: 06 July 2010

To cite this Article Mendelev, M. I. , Kramer, M. J. , Ott, R. T. , Sordelet, D. J. , Besser, M. F. , Kreyssig, A. , Goldman, A. I. ,Wessels, V. , Sahu, K. K. , Kelton, K. F. , Hyers, R. W. , Canepari, S. and Rogers, J. R.(2010) 'Experimental and computersimulation determination of the structural changes occurring through the liquid-glass transition in Cu-Zr alloys',Philosophical Magazine, 90: 29, 3795 — 3815, First published on: 06 July 2010 (iFirst)To link to this Article: DOI: 10.1080/14786435.2010.494585URL: http://dx.doi.org/10.1080/14786435.2010.494585

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Philosophical MagazineVol. 90, No. 29, 14 October 2010, 3795–3815

Experimental and computer simulation determination of the

structural changes occurring through the liquid–glass

transition in Cu–Zr alloys

M.I. Mendeleva*, M.J. Kramera, R.T. Otta, D.J. Sordeleta, M.F. Bessera,A. Kreyssiga, A.I. Goldmana, V. Wesselsb, K.K. Sahub, K.F. Keltonb,

R.W. Hyersc, S. Caneparic and J.R. Rogersd

aAmes Laboratory USDOE and Iowa State University, Ames, Iowa 50011, USA;bDepartment of Physics, Washington University, St. Louis, Missouri 63130, USA;

cUniversity of Massachusetts, Amherst, Massachusetts 01003, USA; dNASAMarshall Space Flight Center, Huntsville, Alabama 35812, USA

(Received 25 September 2009; final version received 15 May 2010)

Molecular dynamics (MD) simulations were performed of the structuralchanges occurring through the liquid–glass transition in Cu–Zr alloys. Thetotal scattering functions (TSF), and their associated primary diffusescattering peak positions (Kp), heights (Kh) and full-widths at half maximum(KFWHM) were used as metrics to compare the simulations to high-energyX-ray scattering data. The residuals of difference between the model andexperimental TSFs are �0.03 for the liquids and about 0.07 for the glasses.Over the compositional range studied, Zr1�xCux (0.1� x� 0.9), Kp, Kh andKFWHM show a strong dependence on composition and temperature. Thesimulation and experimental data correlate well between each other. MDsimulation revealed that the Cu–Zr bonds undergo the largest changesduring cooling of the liquid, whereas the Cu–Cu bonds change the least.Changes in the partial-pair correlations are more readily seen in the secondand third shells. The Voronoi polyhedra (VP) in glasses are dominated byonly a few select types that are compositionally dependent. The relativeconcentrations of the dominant VPs rapidly change in their relativeproportion in the deeply undercooled liquid. The experimentally determinedregion of best glass formability, xCu � 65%, shows the largest temperaturedependent changes for the deeply undercooled liquid in the MD simulation.This region also exhibits very strong temperature dependence for thediffusivity and the total energy of the system. These data point to a strongtopological change in the best glass-forming alloys and a concurrent changein the VP chemistry in the deeply undercooled liquid.

Keywords: molecular dynamics simulation; X-ray diffraction; liquid metal;amorphous alloy; liquid–glass transition

1. Introduction

Glass relaxation has remained one of the most enigmatic aspects of materials science[1,2]. The concept of fragility has helped to explain the dynamic as well as the

*Corresponding author. Email: [email protected]

ISSN 1478–6435 print/ISSN 1478–6443 online

� 2010 Taylor & Francis

DOI: 10.1080/14786435.2010.494585

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Downloaded By: [Goldman, Alan] At: 15:55 19 August 2010

thermodynamic aspects of the glass transition. The thermophysical propertiesof the liquid as it cools through the glass transition (Tg) or, conversely, the glass asit is heated above the Tg, have been investigated experimentally using a wide varietyof techniques such as thermal analysis, rheometry, dilatometry, Mossbauerspectroscopy, nuclear magnetic resonance as well as a number of scatteringtechniques. However, all of the techniques have only provided partial insight intothe collective atomic rearrangements that give rise to the changing chemicaland topological short range order that manifests at the glass transition.Scattering studies typically only provide information on the changes in totalstructure function. A combination of neutron and X-rays and anomalousscattering experiments or extended X-ray fine structure (EXAFS) studies canbe used to extract changes in the partial-pair correlations and theircoordination numbers. But even a combination of these methods provides onlyone-dimensional information about the system structure. Thus, computer simulationmethods, such as reverse Monte Carlo (RMC) or similar techniques, are required toextract higher order correlations, i.e. bond angle distribution or polyhedral typeseither through Anderson-Honeycutt [3] or Voronoi analysis. Unfortunately, thephysical basis of atomic models created by RMC without further constraints isquestionable.

Additional insight into processes occurring during glass relaxation can be gainedfrom the molecular dynamics (MD) simulation [4]. However, there are at least twoproblems associated with the application of this method. First, the interatomicinteraction has to be accurately described. Whereas the most sophisticated method isab initio calculations, this method requires very long computational time and iscurrently limited to simulation cells containing no more than several hundreds ofatoms with the real time of several picoseconds. Models that incorporate a largernumber of atoms and longer time scales require semi-empirical potentials. The cleardisadvantage of using semi-empirical interatomic potentials is that the results candepend on the quality of those potentials. However, this approach allows forsimulations of systems containing up to several millions of atoms [5]. But asimulation time which is several nanoseconds is still relatively short, �109K/s [6], sothat there is no guarantee that the model system reaches a local energy minimum thatclosely approximates experiments since experimental cooling/heating rate aretypically three to seven orders of magnitude slower.

Even with these limitations, MD studies have been very fruitful in furthering ourunderstanding of kinetic processes in the liquid preceding the glass state [3,7].Correlating experiments to MD simulations is complicated by the intrinsic disorderof the liquid whose instantaneous state is a probability function dictated by anenergy landscape that is temperature dependent. The structure of the glass state isdependent on both the temperature from which the liquid is cooled and the rate atwhich it is cooled. Therefore, directly comparing the results of experiments to MDsimulations is problematic at best. However, similarity in trends of an experimentallyrelaxed glass and a simulation using appropriate interatomic potentials wouldindicate that the phenomena observed in the simulations are realistic even thoughthey are on a time scale not directly achievable experimentally. The simulations, inturn, provide insight into the structural and thermodynamic changes that accompanythe glass relaxation.

3796 M.I. Mendelev et al.


The Zr1�xCux system is an ideal system for a combined experimental andmodeling study. A very wide range of compositions can be made amorphous,(0.25 x5 0.7) [8] yet this range spans a number of eutectics and congruently-melting compounds [9]. Moreover, the liquidus temperatures of these compositionsare reasonably low, which allows us to perform wide-angle high-energy X-raydiffraction (HEXRD) experiments in the liquid state, the results of whichcomplement data obtained from heating the glass from the amorphous throughthe super-cooled liquid state.

The motivation for this work was a desire to gauge the accuracy of the three-dimensional (3D) atomic models derived from MD simulations, using the resultsof experiments. Aside from the obvious thermodynamic and mechanical propertycomparisons which have already been reported, in this paper we correlate thetemperature dependent changes in the total scattering function (TSF) on a series ofCu–Zr amorphous and liquid compositions. As we pointed out above, a directcomparison between an experimental study of glass relaxation and a MD simulationis not practical even if the semi-empirical interatomic potentials used were absolutelycorrect since the configurational changes are time/temperature dependent. Forexample, pulsed laser heating would be required to achieve the cooling rates of theMD simulations. However, there should be uniformity in the overall trends inthe TSF between the experiments and simulations. One measure of the change in thestructure of a disordered system is the position of the primary diffuse scattering peak(Kp). The rate of change in the volumetric thermal expansion should undergo aninflection at Tg [10]. There should also be a corresponding change in both the height(Kh) and full-width at half-maximum (KFWHM). Whereas the accuracy of the MDsimulations for modeling various thermophysical parameters can be readily checked,what is uncertain is whether the subtle changes in the atomic configurations areaccurate. As will be discussed later, Kp, Kh and their KFWHM appear to be verysensitive to composition and structure with respect to temperature and the effectivecooling profile used in the simulation. It should be emphasized that these metrics arebeing used as qualitative comparisons since extracting quantitative measure of thestructural changes from such limited data, for even a simple liquid, is problematic[11]. The correlation of any one metric at any given composition and temperaturein and of itself is not significant, but if the trends of all the metrics for a givencomposition over a wide range of temperatures correlate well between experimentsand simulations, then a strong argument for the reliability of the simulations canbe made.

2. Experimental and computational methods

2.1. Experiments

Two sets of high-energy X-ray scattering experiments were performed. In order toobtain the TSF of the alloys through the glass transition, rapidly solidified alloyswere heated through their crystallization temperature [12]. Obtaining the liquidstructure is a more complex experiment, especially obtaining the liquid structure inthe deeply undercooled state without crystallization [13]. Levitation techniques arepreferred since eliminating a crucible minimizes interference with the scattering and

Philosophical Magazine 3797


provides deeper undercoolings than can be achieved in typical castings [14,15].As will be discussed below, the cooling rate of the levitation was insufficient to coolthe alloys through the Tg but we were able to heat some of the amorphous samplesthrough Tg before the onset of crystallization.

The starting alloys of CuxZr1�x were prepared by arc melting mixtures ofhigh-purity Zr (99.95 mass %) and Cu (99.99 mass %) metals in an ultra-high purityAr atmosphere. The alloy was melted several times in order to increase thecompositional homogeneity. Amorphous ribbons were obtained by melt spinningwith a tangential wheel speed of 25m/s at a constant over pressure of 1.6� 104 Pa withultra-high purity He. The melt spun ribbons were approximately 2mm in width and�80 mm in thickness. In order to make �2mm diameter spheres for the levitationexperiments, the arc-melted ingots were inductively melted in fused silica crucibles andinjection cast into 1.5mm diameter Cu molds. These ingots were cut into appropriatelengths and remelted in the same arc melter to form �30mg spheres.

The TSF for both sets of experiments were obtained at the Advanced PhotonSource (APS) at the 6ID-D beamline at APS in collaboration with the MidwestUniversities Collaborative Access Team (MUCAT), Argonne National Laboratory.The diffraction patterns during heating of the amorphous alloys through devitrifi-cation were obtained using an energy of �98.1 keV, which corresponds to awavelengths (�) of 0.126(4) A. For the levitation experiments, 129 keV, (0.096(6) A),X-rays were used. Silicon double-crystal monochromators were employed to selectthe wavelength in both experiments. The experiments performed on the melt-spunribbons and the levitated spheres will be referred to as the in situ heating and coolingexperiments, respectively. The forward scattered X-rays form Debye rings that wererecorded with a GE area detector using a data acquisition rate of 1 to 10 frames persecond or a MAR CCD having a frame-rate of 1 frame �20 s. The distance from thesample to the detector was calibrated using NIST Si 640C standard providing auseable Q range of 20 A�1 and 15 A�1 for the heating and cooling experiments,respectively. The Debye rings were integrated into a single 1D plot using FIT2D [14],which also provided a geometric correction for the flat plate geometry. For the in situheating, �10 of the ‘ribbon like’ samples were placed into a very thin walled �2mmdiameter silica tube and sealed in Ar. The samples were heated at a constant rate of5 and 20K/min through the crystallization temperature. Only the obviouslyamorphous data was used in the analysis presented here. The onset of nucleationof the crystalline phases was noted as a sharp increase in the asymmetry of theprimary diffuse scattering peak even before any other crystalline peaks were visuallyobvious. The sharp increase in the volumetric thermal expansion is consistent withprevious reports on both metallic and silicate glasses [10,15]. Tg occurs at a highertemperature for the faster heating rate and corresponds very well to the Tg obtainedby thermal analysis. Due to the limited memory buffer of the GE detector, onlyscattering data near Tg could be obtained for the slower heating rate. Therefore onlythe transient from the glassy state to the liquid state was captured. The CCD, whileslower, did not have this memory limitation so a more continuous record of the TSFas a function of temperature was obtained.

The in situ cooling experiments were preformed as previously described [16].One complication with the Cu alloys is its rather high vapor pressure, which leads toCu loss which had to be accounted for. The initial (samples not run but from the



same ingot) and final compositions were checked using electron microprobe analysisof a cross section of the entire sphere as well as by measuring their mass change.In addition, the volumes of the samples were measured in situ using a high-speedcamera in order to determine their density. When the maximum temperature was lessthan �1500K, the mass loss was minimal. For the sample with a startingcomposition of Cu64.5Zr35.5, approximately 7 at. % Cu was lost after multipleheating and cooling cycles, whereas the Cu33.3Zr66.7 alloy lost �6 at.% after multiplecycles. Scattering data presented for the nominal Cu64.5Zr35.5 are for the first of theseheating/cooling cycles, whereas the nominal Cu57Zr43 is from the last cycle.

Figure 1 shows a compendium of the position of the first diffuse scattering peak(Kp) of the as-collected raw intensity data using only the fast GE detector data forthe glasses on heating and for the liquid obtained on cooling. The position of Kp

exhibits a strong compositional dependence which is in agreement with the earlierdata [17]. A sharp inflection in the temperature dependence of Kp is observed at theTg. The extrapolation of the Kp in the supercooled liquid region does align well withthe data obtained from the deeply undercooled liquid. For reference, the liquidustemperature of the alloy is indicated with a large closed symbol while the nearlyvertical solid line connecting the Tg points of the respective compositions is shown asa reference. The scattering data corresponding to temperatures and compositions ofthe MD simulations discussed below, were then further processed to obtain the TSFusing standard methods [18,19].

2.2. Simulations

In the present study, we used two semi-empirical potentials to simulate the structureof Cu–Zr disordered alloys. Both potentials are of the Finnis–Sinclair type [20] and

2.5

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2.7

2.8

2.9

T (K)

Kp (

Å–1

)

70.064.561.458.856.050.045.540.030.0

15001200900600300

Figure 1. A compendium of the high-energy scattering experiments showing the position ofthe first diffuse scattering peak, Kp, as a function of temperature for the range of compositions(Cu content indicated in legend) in this study. The curved vertical line indicates the Tg,whereas the closed symbols indicate the liquidus temperature.



share the same potentials for pure Cu and Zr taken from [21] and [22]. For thepresent study, it is important that both semi-empirical potentials for pure elementsprovide good agreement with liquid diffraction data. The cross-functions were fittedto first principles data on lattice parameter and formation energy at T¼ 0K.In Mendelev et al. [23], the cross-functions were also fitted to reproduce the totalforces acting on atoms in a probe liquid configuration (force matching method firstused in Ercolessi and Adams [24]). We will refer to this potential below as MSK. Theauthors of [25] used diffraction data instead of the force matching method along withthe liquid density and formation enthalpy data. We will refer to this potential belowas MKOSYP.

The MD simulation scheme for creation of liquid and glass models that we usedin the present study is described in detail in Mendelev et al. [25]. Here we note thatthe models consisted of 5000 atoms and were obtained by cooling of liquid alloysequilibrated originally at T¼ 2000K with a rate of 4.9� 1010K/s. Snapshots weresaved every 50K and equilibrated at constant temperature for a period of 4.1 ns.

The liquid–glass transition temperature as a function of concentration is shownin Figure 2 (see [25,26] for details of determination of Tg from MD simulation).The MSK potential considerably overestimates Tg but provides the correct trend forthe composition range where it has been experimentally determined [17]. TheMKOSYP potentials provide very good agreement with experimental data for Tg inthe composition range near the Cu64.5Zr35.5 alloy, but at low Cu concentrationspredict a wrong trend (which is not very surprising since the data only for theCu64.5Zr35.5 alloy were used in the potential development procedure).

Figure 3 shows TSF for the Cu64.5Zr35.5 alloy as a function of temperature.This plot demonstrates a rather sharp change in the first peak as the model cooledthrough Tg. Similar plots were obtained for other compositions with both potentials.

600

700

800

900

1000

%Cu

Tg (

K)

MatternMSKMKOSYP

100%80%60%40%20%

Figure 2. Tg as function of concentration for both potentials and previously publishedexperimental data [17].



Figure 4 shows changes in Kp, Kh and KFWHM for all temperatures and compositionsfor both potentials. All show generally the same trend even though the Tg variesgreatly for the two potentials.

3. Results

3.1. HTXRD vs. MD

A qualitative comparison of the TSF determined by experiments and MDsimulations for four compositions at two temperatures is shown in Figure 5. Theresiduals are calculated using the standard approach of summing the square ofthe difference of the experimentally determined total scattering factors (TSFe) andthe calculated total scattering factor (TSFs) over the same Q range for n data points

Rp ¼ 1=nXni¼1

ðTSFe � TSFsÞ2

" #12

: ð1Þ

The Rp values were consistently less than 0.05 for the MKOSYP potential andtypically around 0.06–0.07 for the MSK potential. The largest residuals for theMKOSYP potential were for the 300K data sets, typically between 0.055Rp5 0.07and better than reported for comparisons between glasses and other simulations [27].Fitting of the liquid data was typically much better, less than 0.03 in most cases butalways better for the MKOSYP potential. A low Rp between experiments andsimulations is a necessary, but not sufficient condition for a reliable model.In addition, other metrics of the PDF and the TSF may be more sensitive to how wellthe simulation models the atomic structure of the alloy. In the experiments and in thesimulations, Kp is strongly composition and temperature dependent, in particular forthe higher Cu containing alloys (Figure 1). The experimental accuracy of the Kp iswithin 0.05%, based on multiple measurements of the same sample, and reproduc-ibility is better than 0.2% for measurements of different samples of the samecomposition when the detector to sample distance is calibrated as described above.

Figure 3. TSF of Cu64.5Zr35.5 models created with the MKOSYP potential as a functionof temperature. Note that there is a striking change in Kh as the model is cooled through Tg.The rapid change in the primary diffuse scattering peak just above Tg extends to the Kp

and KFWHM as well.



The general trends in the peak positions as functions of composition correspondquite well for both the glass (Figure 5a) and the liquid (Figure 5b). Moreover, thesubtleties of the sharpening of the higher Q peaks are captured in the amorphoussolid by the MD simulation, in particular the splitting of the second diffuse scatteringpeak during the cooling through Tg. Since the MKOSYP potential matches theexperimental data the best, the results of this potential will be the major focus of therest of this paper.

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2000150010005000

2000150010005000

2000150010005000

Figure 4. Changes in (a) Kp, (b) Kh and (c) KFWHM for all T values and compositions for bothmodels (MSK, lines; MKOSYP, open symbols). All show generally the same trend eventhough the Tg varies greatly for the two models.



A broader comparison of Kp for the entire composition range is shown inFigure 6 for the MKOSYP potential, which matches the experimental data muchbetter. Note that the Kp is slightly underestimated for the glass but quite accurate forthe liquid. More importantly, compositional dependencies of Kp are very consistentbetween experiments and simulations. The same trends in Kp, Kh and KFWHM

with composition and temperature as observed in the simulations are also reflectedin the experimental data (Figure 7). The systematic offset in Kp for Cu¼ 40%breaks the trend. However, this is most likely due to an uncertainty in the Cu contentdue to evaporative losses in the high-vacuum at high-superheats for this series

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pea

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–1)

MSKMKOSYP300 K1400 KXRD 300KXRD 1400K

100%80%60%40%20%

Figure 6. Kp vs. composition at 300 (blue online) and 1400K (red online), comparing the MDsimulation using MKOSYP potential (dashed lines) to experimental data (open symbols) overa wide range on glass compositions.

0

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Cu64Zr36Cu57Zr43Cu46Zr54Cu40Zr60Cu30Zr70

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TS

F

Cu64Zr36

Cu57Zr43

Cu46Zr54

Cu40Zr60

Cu30Zr70

(a) (b)

7654321 7654321

Figure 5. Comparison of the experimentally measured TSF for the CuxZr1�x, x¼ 0.64, 0.57,0.46, 0.40 and 0.30 data (open symbols) and the corresponding MD simulation usingMKOSYP potential (lines) at (a) 300K and (b) 1400K.



of alloys described above. The systematic differences in the Kp between the MDsimulations and experiments is not due to differences in density. The densitypredicted by the MKOSYP potential is in quite good agreement with previouslypublished densities of Cu–Zr glasses at room temperature [17] and nearly as well withthe liquid densities in this study. Whereas the Kp is loosely related to the numberdensity, its position in a multi-component alloy is dependent on the relativescattering factors of the partial functions and is not an intrinsic parameter. Thegeneral trends in the experimentally determined Kh and KFWHM also follow theMD simulations quite well. The systematic variation in these metrics is a reflection ofthe relative weights of the three well separated partial-pairs in this system; gCu-Cu,gCu-Zr and gZr-Zr and will be discussed more fully below.

The sharp inflection in the Kh at Tg was also observed for all alloys investigated,similar to previously reported for a quaternary Zr-based metallic glass [28]. Thedecrease in the peak height and its broadening is a reflection of the increase in atomicmobility as the glass is heated through Tg. The principle question is whether thesefeatures observed in the TSF as the glass is heated through Tg are representativeof the increasing thermal fluctuations of atomic configurations, which are alreadypresent in the liquid, or are they a reflection of changes in these configurations?Whereas the scattering studies cannot answer this question directly, they can help tovalidate the MD simulations, which can directly address this point.

3.2. Partial-pair correlations

The MD simulations for both potentials in this study are in accord with previousexperimental studies showing that all of the amorphous alloys in the Cu–Zr systemexhibit short Cu–Cu, intermediate Cu–Zr and longer Zr–Zr bonds in the firstcoordinating shell [29,30]. The compositional dependence of the Cu–Cu bond lengthis very small, less than 0.02 A over the entire compositional range simulated for theamorphous structure and no discernable difference in the liquid state in the range ofthe glass-forming compounds (0.25 x5 0.7) (Figure 8a). There is a sharp change

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Figure 7. Comparison between characteristics of the first peak of TSF obtained fromexperiment (open symbols) and MD simulation using MKOSYP potential (lines).



in the bond lengths at Tg, with a parabolic temperature dependence of the bondlength. The Zr–Zr bond length shows a very large compositional dependence butprimarily for compositions that cannot be made amorphous through liquidquenching (Figure 8b). There is also a change in the Zr–Zr bond length at Tg.The Cu–Zr bond length shows the largest variability over the composition andtemperature range simulated (Figure 8c). These bonds show very little temperaturedependence in the amorphous state, a strong compositional dependence and adistinct inflection at Tg. The temperature dependence in the liquid state for theCu–Zr bond length is largest for the best glass formers (0.55� x� 0.65), i.e. thosewhich could be readily cast as 1 to 2mm diameter rods, and weakest for the high

Cu-Cu

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T (K)

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Figure 8. The peak position of the first shell pair correlations for (a) gCu-Cu, (b) gZr-Zr and(c) gCu-Zr for theMD simulation using the MKOSYP potential. Each of these graphs has thesame span for the y-axis to emphasize the range of change for the peak positions with bothcomposition and temperature.



Zr containing alloys, which could only be made amorphous by rapid quenching.This would suggest that there is considerable bond reordering in the liquid state forthe best glass formers prior to vitrification. Whereas it appears to be a contradictionthat average bond length is shortening as the alloy is heated through Tg and thedensity is decreasing, this can be attributed to the formation of less topologicallydense packing where the shortest atomic separation is less than that in the glass.

The longer range pair correlations (second and third atomic shells) show thatthere are substantial differences in the partial-pairs between the liquid and glassstates. Whereas the partial-pairs in the liquid at 1400K show a surprising degree ofsimilarity across most compositions, the 300K data shows considerable variationin the longer range pair correlations (Figure 9). For instance, the second shell forCu–Cu and Cu–Zr partials show a large degree of peak splitting while the Zr–Zrshow an enhanced asymmetry when comparing the glassy to the liquid state.Both the separation of the peaks and their relative intensities are stronglycompositional dependent. The height of the fluctuations in the partial pairsbeyond 8 A are also higher for the glass. This is clear evidence that the structuralre-ordering of the glass compared to the liquid is mostly manifested in changes in thesecond and third atomic shells.

3.3. Voronoi tessellation and physical properties

Voronoi tessellation is one way of describing the local distribution of atoms ina disordered structure [31–34]. In this method, a coordination polyhedronsurrounding a center atom can be differentiated by specific Voronoi indices hnii.Here ni denotes the number of i-edged faces of the Voronoi polyhedron (VP) andP

i ni is the coordination number (CN) of the center atom in the polyhedron. Thenumber of edges of each polyhedral face denotes the number of common neighborsshared by the center atom and one of its neighboring atoms. The CN are based oni¼ 3� 10 but only four numbers hn3, n4, n5, n6i need to be reported. We analyze howthe concentration of individual VP types changed with temperature and composi-tion. For the MD simulations, the VP distributions were determined by averagingover 10,000 snapshots to sufficiently sample the numerous configurations.

0.00.51.01.52.02.53.03.54.04.5

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121086420

Figure 9. The gCu-Cu, gZr-Zr and gCu-Zr for the glass (300K) (top plots) compared to the liquid(1400K) (bottom plots) using the MKOSYP potential.



These data were then categorized based on frequency of the various VP types for

both the Cu and Zr centered VPs. All compositions and temperatures were

dominated by �5 types of Cu-centered polyhedra, h0,2,8,1i, h0,2,8,2i, h0,0,12,0i,

h0,1,10,2i and h0,3,6,4i, and �5 types of larger Zr polyhedra; h0,2,8,4i, h0,2,8,5i,

h0,1,10,4i and h0,1,10,5i and h0,2,8,6i (Figure 10). As expected, the most common VP

in liquid alloys make up a smaller fraction of the total, 515%, while for the glasses

they comprise more than 50% in some cases. The three most common of the Cu

centers and the two most common of the Zr centers change from low Cu containing

alloys to the more Cu rich alloys. The transition occurs near the mid-point of the

phase diagram but is not sharply delineated. The icosahedral h0,0,12,0i is only a

small fraction of the liquid but dominates the intermediate to high Cu containing

glasses. These results are qualitatively similar to those obtained by Cheng et al. [35]

using a different interatomic potential; however, their population of h0,0,12,0i was

considerably higher and the next most common polyhedron differ.Figure 11 shows the fraction of most popular VPs in the Cu64.5Zr35.5 as a

function of temperature. Below Tg, about 50% of all Cu-centered VPs and 30% of

the Zr-centered belong to only four VP types. Figure 11 demonstrates a change in the

300 K

0%

5%

10%

15%

20%

%Cu

% (

Cu

cen

ters

)

<0,0,12,0><0,3,6,4><0,2,8,1><0,1,10,2><0,2,8,2>

1400 K

0%

5%

10%

15%

20%

% (

Cu

cen

ters

)

<0,0,12,0><0,3,6,4><0,2,8,1><0,1,10,2><0,2,8,2>

300 K

0%

5%

10%

15%

% (

Zr

cen

ters

)

<0,2,8,4><0,2,8,5><0,1,10,4><0,1,10,5><0,2,8,6>

1400 K

0%

5%

10%

15%

%Cu

% (

Zr

cen

ters

)

<0,2,8,4><0,2,8,5><0,1,10,4><0,1,10,5><0,2,8,6>

(c)

(a) (b)

(d)

9080706050403020

%Cu9080706050403020

%Cu80706050403020 80706050403020

Figure 10. Five most common Cu centered VPs at (a) 300 and (b) 1400K and the five mostcommon Zr centered VPs at (c) 300 and (d) 1400K based on the MKOSYP potential.Percentages based on average of 105 MD time steps for each of the cluster center types.



liquid structure as the alloy approaches Tg. The temperature dependence of thechemistry of the VPs types can be further analyzed to directly address the questionof whether structural changes observed in the liquid–glass transition are strictlytopological or involve chemical reordering in this system.

Figure 12 shows both the average Zr CN in the Cu-centered VPs (Figure 12a) andZr-centered VPs at each temperature as well as the Zr CN in each of the mostcommon VP types for the Cu- and Zr-centered VPs, respectively, for the Cu64.5Zr35.5alloy. Whereas the CN for Zr of the VPs are relatively unchanged in the glass, thevarious VP types undergo a rapid change in CN, i.e. their chemistry, as the liquidapproached Tg. Simply put, Figure 12 shows that as the liquid is cooled to the glassystate, the Cu and Zr tend to prefer to be nearest neighbors. Whereas this trend issmall for the average Cu-centered VPs, it is quite dramatic for the most populousVPs. The Zr-centered VPs in general show a rather large change for nearly all typesof VPs. This is also consistent with the high heat of mixing for Cu–Zr. The unlikeatoms tend to try to be nearest neighbors to the degree that the topological orderingwill allow.

One of the drawbacks of the Voronoi tessellation is that it can over-emphasizesmall topological changes. For highly disordered systems, such as a liquid, there canbe hundreds of VP types. One way of more broadly analyzing the VP types is togroup them by CN. This captures all of the data for each of the VP centers yet is lessambiguous than integrating the area under the partial-pair correlation functions.Figure 13 shows a dramatic difference in the CN for the Cu rich and Zr rich alloysas a function of temperature. The Cu rich alloy is dominated by 13 and 14 CNCu-centered VPs in the liquid but undergoes a rapid reordering to 12 and 13 CN VPjust above Tg. The Zr rich alloy shows a more subtle rearrangement of the

0

2

4

6

8

10

12

14

16

18

T (K)

VP

(%

)<0,0,12,0,0><0,3,6,4,0><0,1,10,2,0><0,2,8,2,0><0,3,6,3,0>Tg

0

2

4

6

8

10

12

14

16

18

T (K)

VP

(%

)

<0,1,10,5,0>

<02,8,6,0>

<0,1,10,4,0>

<0,2,8,5,0>

Tg

(b)

2000150010005000 2000150010005000

(a)

Figure 11. Fraction of most common VPs as function of temperature in Cu64.5Zr35.5alloy simulated with the MKOSYP potential. Percentages based on average over 10,000snapshots. The Cu-centered VPs (a) and Zr-centered VPs (b). The vertical line at Tg is addedas a guide to the eye.



Cu-centered VPs and a smaller change in the Zr-centered VPs. This would suggestthat the Cu centered VPs in particular are becoming more densely packed, especiallyfor the Cu rich samples.

The transition from the glass to liquid is clearly demarked by structural andcompositional changes at the local atomic level which should then be manifest in thekinetic and thermodynamic properties. Figure 14 shows the difference between thetotal energy, E, and the energy of harmonic vibrations (3kBT) as function oftemperature. If all atoms exhibited only harmonic vibrations with respect to their

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6.0

6.1

6.2

T (K)

CN

(C

uZ

r)

<0,0,12,0,0>

<0,3,6,4,0>

<0,1,10,2,0>

<0,2,8,2,0>

<0,3,6,3,0>

Average

Tg4.5

4.6

4.7

4.8

4.9

5.0

5.1

5.2

5.3

5.4

5.5

T (K)C

N (

ZrZ

r)

<0,1,10,5,0>

<02,8,6,0>

<0,1,10,4,0>

<0,2,8,5,0>

Average

Tg

(b)

2000150010005000 2000150010005000

(a)

Figure 12. The average Zr coordination number and the most common VPs and the averagefor (a) Cu-centered and (b) Zr-centered VPs as a function of temperature. Percentages basedon average over 10,000 snapshots. The vertical line at Tg is added as a guide to the eye.

0%

10%

20%

30%

40%

50%

T (K)

% o

f C

u o

r Z

r ce

nte

rs

12 13

14 15

16 Tg

0%

10%

20%

30%

40%

50%

T (K)

% o

f C

u o

r Z

r ce

nte

rs

12 1314 1516 Tg

(b)

2000150010005000 2000150010005000

(a)

Figure 13. Coordination numbers as a function of temperature for the (a) Cu64.5Zr35.5 and(b) Cu30Zr70 using the MKOSYP potential. The closed symbols are only the coordinationnumbers for the Cu-centered VPs and the open are for the Zr centered VPs. The vertical lineat Tg is added as a guide to the eye.



equilibrium positions, E� 3kBT would not depend on T and Figure 14 illustratesthat this is nearly the case for the glass at lower temperatures. The small increase ofthe value of E� 3kBT up to the stark inflection at Tg may be explained byanharmonicity of atomic vibrations. In the liquid state, E� 3kBT strongly dependson temperature and for very high temperatures, the slope decreases. The highestslope just above Tg can be explained by necessity of rearranging the more denselypacked VP, which are described by the most popular VP types (Figures 11 and 12).

Figure 15 shows the diffusivity for both potentials. Examination of this figuredemonstrates that the diffusivity at low temperatures, just above Tg, has much higheractivation energy than at higher temperatures. This can be explained by the fact thatthe atoms belonging to stable complexes gives small contribution to diffusion but thefraction of such atoms decreases with increasing temperatures. The deviation froman Arrhenius behavior occurs well above Tg.

4. Discussion

Comparing the total scattering functions (TSF) and the corresponding metrics of Kp,Kh and KFWHM of the molecular dynamic simulations to high-energy X-rayscattering data shows that the overall trends and their absolute values correspondwell particularly for the MKOSYP potential. The Rp between the experimentallydetermined TSFs and the MKOSYP potential simulations is �0.03 for the liquids

–4.5

–4.5

–4.4

–4.4

–4.3

–4.3

–4.2

T (K)

E–3

KBT

(eV

/ato

m)

MKOSYP

MKS

2000150010005000

Figure 14. Energy of as function of temperature in Cu64.5Zr35.5 alloy. The inflection points arethe best indication of the Tg, which is 986 and 770K for the MKS and MKOSYP potentials,respectively.



and about 0.05 to 0.07 for the glasses and about 30 to 50% higher for the MSKpotential. This agreement is quite good for the MKOSYP potential given both theuncertainties of the experimental data, approximations in developing the interatomicpotentials and the large differences in cooling rates. For the in situ coolingexperiments, the small spheres radiantly cooled at a rate of not more than 102K/scompared to 4.9� 1010K/s used in the simulations. The amorphous samples weremade via melt-spinning with a cooling rate �106K/s [36]. So, although the experi-mental data on heating through Tg and the undercooled liquids are at very lowcooling rates compared to the simulations, the strong temperature dependence of theTSF metrics match well (Figure 7). Lower cooling rates would have a much morepronounced effect on the temperature dependence since the lower cooling rateprovides more time for the liquid to sample more configurations. Increased samplingof configuration allows for more of the atoms to find lower energy configurations.To test this, we ran simulated cooling rates from 1014 down to 1010K/s for the bestglass-forming composition, Cu¼ 64.5%, which also shows the largest temperaturedependence on all metrics measured. Whereas there is a clear trend in thesimulation’s properties with cooling rate, the effect on bulk modulus, energy/atomand density are quite small, 2.7%, 0.36% and 0.16%, respectively, over the fourorders of magnitude in the cooling rate [37]. Similar results were obtained in [38,39]using a different interatomic potential. The partial-pair correlation function arevirtually indistinguishable for cooling rates51012K/s, whereas the only VP type thatshows any significant change is the h0,0,12,0i [39] and [37] will warrant further study.Experimentally, we see little difference in either TSF between cast rods �2mm

–10

–9

–8

–7

–6

–5

–4

500100015002000

T (K)

Lo

g (

D)

Cu (MSKpotential)Cu (MKOSPpotential)Zr (MSKpotential)Zr (MKOSPpotential)

Tg (MKOSP)

Tg (MSK)

Figure 15. Diffusivity (D, cm2/s) as function of temperature in the simulated Cu64.5Zr35.5alloy. Regardless of the potential, the deviation of the diffusivity from an Arrhenius behavioroccurs well above Tg. The vertical lines at each of the Tgs are added as a guide to the eye.



in diameter and melt-spun ribbons for the Cu¼ 64.5% alloys. Thermal analyses,a much more sensitive measure of the energies of the systems, are also quite similarfor cast rods and ribbons. This is not surprising given that the calculated diffusivitiesat Tg indicate that the system would have enough time to sample a large numberof configurations for most liquid-quench experiments.

Over the compositional range studied, Zr1�xCux (0.1� x� 0.9), the Kp showsa strong dependence on composition and temperature in the simulations which arereflected in the experimental data. Changes in the temperature dependence of Kh andKFWHM in the region of the undercooled liquid to the glass transition (Tg) alsocorrelate well between simulations and where experimental data exist. We wouldargue that these three parameters are a more reliable metric to track the correlationsbetween the experiments and simulations, rather than the residuals alone, for thefollowing reasons: the residuals can be quite small even if the peak-positions do notcorrelate all that well for broad peaks like those seen in liquids and glasses. A MDsimulation with a shorter, broader peak compared to the experimental data can havethe same residual as one that matches height and width but not the peak positions.In fact, this appears to be the case with the MSK potential, the Rp for which is onlyslightly higher than the MKOSYP results when compared to the same experimentaldata. Figures 1, 4 and 6 show that Kp changes systematically with composition forboth experiments and MD simulations. Whereas the changes in Kh and KFWHM aremore complex functions of composition, these and Kp all show significanttemperature dependence in the liquid state and little or no change in the glassstate over a wide temperature range. The abrupt change in the temperaturedependence of these three metrics is consistent at their respective compositions andoccurs at Tg. Of the two potentials tested, the MKOSYP best matches theexperimental data over the entire composition and temperature range explored.Whereas the effect of cooling rate still needs further investigation, the excellentcomparison of the experimental data over such a broad range of composition andtemperatures provides a high degree of confidence in the 3D atomic modelsconstructed using the MKOSYP potential.

Assured that the MKOSYP potential is a reasonable model for the Cu–Zrsystem, we can now use this model to explore some of the key structural changes thatoccur in this deeply undercooled liquid alloy that are not currently accessible byexperiments. The key observations are: Cu–Zr bond length, as measured by the peakposition in the partial-pair correlation functions, undergoes the most profoundchanges with temperature, Cu–Cu the least. These changes in the first shell peakposition as a function of temperature are more significant the higher the Cu contentof the alloy, including the Zr–Zr pairs. This suggests that the topology of the moreCu-rich alloys is more temperature sensitive, indicating that these alloys undergoa higher degree reordering than the Zr-rich alloys. A similar analysis of thetemperature dependency of the Cu- and Zr-centered VPs for Cu50Zr50 show muchsmaller changes with the deeply undercooled liquid than were seen in the Cu64.5Zr35.5alloy (Figure 12). Changes in the partial-pair correlations are more readily apparentin the second and third shells than in the first shell (Figure 9). This supports that thechanges in the undercooled liquid topology also involve the medium-range order(5–10 A). Quantification of the change in the medium-range order will requiretechniques that go beyond analyzing relationships of the nearest neighbors.



Even though the changes in the first shell of the partial pairs are more subtle, theVoronoi tessellation does show profound changes in the short-range order (SRO) asthe liquid is deeply undercooled. Most significant is that the VPs are dominated byonly a few select types which are compositionally dependent. These dominant VPsrapidly increase in their frequency in the deeply undercooled liquid state. The regionof best formability, x � 65%, shows the largest temperature dependent changes inthe deeply undercooled liquid. This composition range also shows a very significantchange in the Zr CN in the most abundant VP in the deeply undercooled liquid. Thispoints to a combination of topological and chemical reordering in the deeplyundercooled liquid preceding glass formation. This region also has a very strongtemperature dependence on diffusivity and the total energy of the system.The temperature range over which the deviation in the diffusivity clearly undergoesthe most profound decrease also strongly correlates with the region where theCu-centered VPs also undergo a significant change in CN (see Figure 13a andFigure 15). This strongly supports some form of mode-coupling occurring in thedeeply undercooled liquid. The composition of the best glass forming alloy coincideswith a deeply undercooled liquid where the Cu-centered polyhedra form a densenetwork of icosahderal clusters consistent with previous reports of MD simulationsin this alloy system [35] and experimentally demonstrated in a similar Zr-basedsystem [40]. The Zr-rich alloys show a much less significant change in the VP typeswith temperature and the most populous are h0,2,8,1i, h0,2,8,2i and h0,3,6,4i. Thissuggests that the Zr-rich glass is much more ‘liquid-like’ than the Cu-rich glasses.It should be noted that other semi-empirical potentials also lead to the similarconclusions [41]. This would be consistent with the observation that the Zr-richalloys, approximately (0.255 x5 0.45) can only be made amorphous by rapidlyquenching from the liquid. Thermal analysis also shows that these alloys have amuch narrower supercooled liquid region (Tx�Tg) and a lower reduced glasstemperature [42].

5. Conclusion

We used the total scattering functions (TSF) and the associated primary diffusescattering peak’s position (Kp), height (Kh) and full-width at half-maximum(KFWHM) as metrics with which to compare the molecular dynamic simulations tohigh-energy X-ray scattering data. The residuals between the experimentallydetermined TSFs and the simulations are �0.03 for the liquids and about 0.07 forthe glasses for the MKOSYP potential. Over the compositional range studied,Zr1�xCux (0.1� x� 0.9), the Kp shows a strong dependence on composition andtemperature in the simulations which are reflected in the experimental data. Changesin the temperature dependence of Kh and KFWHM in the region of the undercooledliquid before the glass transition (Tg) correlate well between simulations and theresults of experiments where data exist. An updated potential using liquiddiffraction data along with the liquid density and formation enthalpy data appearsto fit the experimental data, both scattering as well as Tg, better than previoussemi-empirical potentials. The resulting 3D atomic models were analyzed usingVoronoi tessellation to elucidate the key configurational changes observed in



cooling the liquid through Tg. These are: Cu–Zr bonds undergo the most profoundchanges, Cu–Cu the least. Changes in the partial-pair correlations are more readilyseen in the second and third shells indicative of development of the medium-rangeorder. The Voronoi polyhedra (VPs) in glasses are dominated by only a few selecttypes that are compositionally dependent. These dominant VPs rapidly increase intheir proportion of the total VP types in the deeply undercooled liquid state. Theregion of best formability, x � 65%, shows the largest temperature dependentchanges in the deeply undercooled liquid. This region also has a very strongtemperature dependence of diffusivity and the total energy of the system. These datapoint to a strong topological change in the best glass-forming alloys in the deeplyundercooled liquid where the Cu-centered polyhedra form a dense network oficosahderal clusters.

Acknowledgements

Work at the Ames Laboratory was supported by the Department of Energy, Office of BasicEnergy Sciences, under Contract No. DE-AC02-07CH11358. The high-energy X-ray work atthe MUCAT sector of the APS was supported by the US Department of Energy, Office ofScience, Basic Energy Sciences under Contract No. DE-AC02-06CH11357. Ames Laboratoryis operated for the US Department of Energy by Iowa State University under Contract No.DE-AC02-07CH11358. The work at Washington University was partially supported by theNational Science Foundation under Grant Nos. DMR-0606065 and DMR-0856199, and byNASA under Contract No. NNX07AK27G.

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