Journal of Non-Crystalline Solids

Transcription

1 Journl of Non-Crystlline Solids 357 (11) Contents lists ville t ScienceDirect Journl of Non-Crystlline Solids journl homepge: locte/ jnoncrysol Therml expnsion nd recrystlliztion of morphous Al nd Ti: A moleculr dynmics study John J. Chu, Crig A. Steeves University of Toronto Institute for Aerospce Studies, 95 Dufferin Street, Toronto, Ontrio, Cnd M3H 5T rticle info strct Article history: Received 7 April 11 Received in revised form 9 June 11 Aville online 18 August 11 Keywords: Amorphous metls; Therml expnsion; Recrystlliztion; Moleculr dynmics simultion In this study, the therml expnsion nd recrystlliztion ehvior of morphous Al nd Ti re investigted using moleculr dynmics simultions. Amorphous phses re otined vi rpid quenching from liquid stte nd re susequently heted t rte of 1 K/ps. Using the chnge in simultion size over the course of heting, the therml expnsion coefficients of morphous Al nd Ti re clculted nd compred to their crystlline counterprts. From similr set of simultions, the recrystlliztion tempertures of Al nd Ti re determined y nlyzing their potentil energy profiles. In ddition, the chnge in volume s result of the phse trnsition is quntified y compring the tomic volumes of Al nd Ti in oth their morphous nd crystlline sttes. 11 Elsevier B.V. All rights reserved. 1. Introduction Bi-mteril lttices with tilorle therml expnsion [1] re pplied t the micro-scle to fricte n opticl surfce with zero therml expnsion. Such optics re pplicle to telescopes in spce where lrge temperture grdients nd vritions crete therml strins nd ffect geometric stility. To this end, thin films of Al nd Ti re fricted using e-em deposition nd photolithogrphy to mnufcture the proposed micro-scle lttice. As result of the deposition process, it hs een determined y electron cksctter diffrction nd X-ry diffrction tht the films of oth Al nd Ti hve n morphous tomic structure. Consequently, knowledge of the therml properties nd recrystlliztion ehvior of morphous Al nd Ti re required for the successful design of the micro-scle lttice. As elucidted y Steeves et l. [1], the therml expnsion of imteril lttice is governed y the geometry of the unit cell nd the therml properties of the selected constituents. It hs een determined tht the coefficient of therml expnsion (CTE) of Al nd Ti produce n idel rtio to crete lttice with nerly zero therml expnsion. The CTEs of polycrystlline Al nd Ti hve een thoroughly reserched nd documented in the literture [ ], however there hve een no studies on the therml properties of morphous Al nd Ti to the uthors' knowledge. In order to ccurtely design nd tilor the CTE of the proposed micro-scle lttice, the therml expnsion of the morphous films must e known. Corresponding uthor. E-mil ddresses: john.chu@utoronto.c (J.J. Chu), csteeves@utis.utoronto.c (C.A. Steeves). Amorphous mterils re metstle nd recrystlliztion cn e initited y numer of methods such s therml nneling [5]. In the ppliction of n opticl surfce in spce, the micro-scle lttice my e sujected to sufficiently high tempertures to cuse devitrifiction of the morphous thin films. It is therefore crucil to know the temperture t which recrystlliztion occurs in order to predict the ehvior of the i-mteril lttice. Two studies hve een found tht investigte the recrystlliztion temperture (T x ) of morphous Al using moleculr dynmics (MD) simultions, ut their results re conflicting. Utilizing non-locl pseudopotentil theory, Lu nd Szpunr [] gives T x of Al to e in the rnge of 55 3 K. In contrst, Shimono nd Onoder [7] finds T x to e pproximtely 3 K using the intertomic potentil y Oh nd Johnson [8]. Given the discrepncy etween these reported vlues, further investigtion on the recrystlliztion of morphous Al is wrrnted. Shimono nd Onoder [7] lso exmined the devitrifiction of morphous Ti nd gives T x to e roughly 8 K. The recrystlliztion tempertures reported y Shimono nd Onoder [7] for oth Al nd Ti re elow 93 K, contrdicting the oservtion tht the thin films re stle t room temperture. Another chrcteristic of morphous mterils is tht they re generlly less dense thn their crystlline counterprts [9]. As result of devitrifiction, n morphous mteril will thus contrct in volume nd exhiit negtive chnge in length. It is therefore importnt to quntify the mount of shrinkge due to recrystlliztion in order to model the lttice ehvior. The difference in volume etween the morphous nd crystlline sttes of Al nd Ti, however, re scrcely studied in the literture. For Al, one report y Becqurt et l. [1] gives volume chnge of 1% for crystlline to morphous trnsition t 3 K (or roughly 9% for devitrifiction), lthough the system studied ws under n pplied stress. No studies hve een found tht -393/$ see front mtter 11 Elsevier B.V. All rights reserved. doi:1.11/j.jnoncrysol

2 37 J.J. Chu, C.A. Steeves / Journl of Non-Crystlline Solids 357 (11) nlyze nd report the volume differences etween crystlline nd morphous Ti. The purpose of the reserch presented in this pper is twofold. First, the therml expnsion of morphous Al nd Ti re investigted nd their temperture dependent CTEs re quntified. The second spect of this work studies the recrystlliztion of morphous Al nd Ti. In doing so, the recrystlliztion temperture nd contrction in volume due to devitrifiction re determined. To this end, MD simultions re used to explore the therml properties nd recrystlliztion ehvior of morphous Al nd Ti. Informtion otined from the MD simultions will ultimtely led to greter understnding in the properties nd ehvior of the proposed imteril lttice.. Methodology All moleculr dynmics simultions in this study re performed using LAMMPS (Lrge-scle Atomic/Moleculr Mssively Prllel Simultor) [11], n open source code developed nd mintined y Sndi Ntionl Lortories under the United Sttes Deprtment of Energy. LAMMPS is n efficient prllel MD code which implements sptil decomposition lgorithm, thus giving optiml scling for lrge nd lnced systems [11]. MD simultions re performed on the University of Toronto's SciNet Consortium [1] to tke dvntge of the sclility of LAMMPS nd the ville computtionl resources. For ll MD simultions descried in this pper, n orthogonl simultion ox is used with velocity-verlet integrtion scheme nd timestep of 1 fs..1. Intertomic potentil selection Approprite potentil functions must first e chosen to ensure good results from the MD simultions. Mny-ody intertomic potentils sed on the emedded-tom method (EAM) [13] nd Finnis-Sinclir pproch [1] re considered ecuse they re well suited for modeling metllic systems [15]. For oth Al nd Ti, numer of good qulity potentils exist in the literture to choose from. A study is therefore conducted to select the most suitle intertomic potentil for ech mteril. To model Al, three potentil functions re considered those y Liu et l. [1] (improved version of originl potentil y Ercolessi nd Adms [17]), Mendelev et l. [18], nd Zope nd Mishin [19]. For Ti, the intertomic potentils y Acklnd [] nd Zope nd Mishin [19] re evluted. The ojective of this pper is to investigte the therml expnsion nd recrystlliztion of morphous Al nd Ti, thus the est potentil is one tht most ccurtely predicts these physicl properties nd phenomen. Since experimentl therml expnsion dt is redily ville for crystlline Al nd Ti, the intertomic potentils tht est reproduce their ehvior re selected s the most suitle cndidtes. Using ech intertomic potentil, the CTE of Al nd Ti re determined vi MD simultions nd compred ginst their experimentl vlues. Crystlline Al hs fce-centered cuic (fcc) structure, while Ti exists in two crystl forms: hexgonl close-pcked (hcp) nd odycentered cuic (cc). The crystl structure of Ti t room temperture is hcp, while the cc phse of Ti is only stle t tempertures greter thn 1155 K [1]. Since the expected operting rnge of the i-mteril lttice is well elow 1155 K, this study concentrtes on the hcp form of Ti. To simulte the therml expnsion of crystlline Al, lock of fcc cells corresponding to 3, toms is creted in LAMMPS. Similrly for Ti, rry of hcp unit cells (lso 3, toms) is constructed. Lttice constnts of.5 Å nd.95 Å re used for Al nd Ti respectively [19]. Periodic oundries re enforced in ll directions to eliminte surfce effects, thus emulting ulk mteril. At the eginning of ech simultion, the system is equilirted t 5 K for 1 picoseconds (ps) under the isoric-isotherml (NPT) ensemle. The x, y, nd z dimensions of the simultion ox re llowed to vry independently from one nother under zero externl pressure. After equilirtion, the temperture of the system, T, is incresed from 5 K to 1 K t rte of 1 K/ps. During this process, the size of the simultion ox in the x, y, nd z directions ( lx, ly, nd lz) re recorded nd susequently used to clculte the therml expnsion of the system. The CTE is clculted s the verge therml expnsion in ech direction nd is given y the following formul: αðtþ = 1 h i 3 α x ðtþ + α y ðtþ + α z ðtþ " # = 1 1 d l xðtþ 3 lx ðtþ + 1 d l yðtþ dt ly ðtþ + 1 ð1þ d l zðtþ dt ð Þ dt Note tht the derivtives of the ox lengths with respect to temperture re required to compute α in Eq. (1). To otin this informtion, cuic polynomils re used to pproximte the simultion dt. Given polynomil interpolnt, it is thus strightforwrd to evlute iðtþnd d l iðtþ= dt t ny given temperture, where i is x, y, orz... Amorphous sttes Amorphous phses of Al nd Ti re creted y rpidly quenching ech mteril from its liquid stte s other MD studies hve done [,]. As efore, 3, toms re creted in unit cell rry with periodic oundry conditions in ll directions. To otin liquid Al nd Ti, the system is equilirted under NPT dynmics t 15 K nd 5 K respectively (well ove their melting points) for 1 ps. After equilirtion, the metllic liquids re cooled t rte of 1 K/ps [] to 5 K. Rpid cooling prevents the nucletion nd formtion of crystlline tomic structures, thus morphous sttes re otined. Immeditely fter cooling, the system is equilirted for 5 ps. Following equilirtion of the quenched stte, the potentil energy of the system is minimized to remove ny internl stress in the mteril cused y the rpid cooling process [3]. Minimiztion is performed using uilt-in function in LAMMPS tht itertively djusts the tomic coordintes to minimize the potentil energy using conjugte grdient optimiztion lgorithm. Since the tomic rrngements re sufficiently disordered, minimiztion does not generte crystl lttices nd thus the energy of the system is driven to locl minimum. The resulting morphous tomic structure is verified y visulizing the tomic coordintes nd exmining the rdil distriution function (RDF) of the system. The RDF, lso known s g(r), is commonly used to chrcterize the tomic structure of mterils []. It mesures from ny given tom, the numer of other toms locted t distnce r, normlized y the numer of toms tht would e found in uniformly distriuted system. The coordintion numer (CN), which represents the numer of nerest neighors of given tom, is lso clculted from the RDF using the following eqution [5]: CN =πρ r 1 r gr ðþdr where ρ is the density of the system nd r 1 is the loction of the minimum of g(r) fter the first pek..3. Therml expnsion The morphous phses of Al nd Ti re heted t rte of 1 K/ps under the NPT ensemle to investigte their therml properties. As the temperture of the system is incresed, lx, ly, nd lz re recorded t regulr intervls s function of T. During the entire process, the size of the simultion ox is llowed to expnd nd contrct nisotropiclly under zero externl pressure. Using the sme method descried in Section.1, the CTEs of morphous Al nd Ti re clculted ccording to Eq. (1). To ccount for the metstle nture l z T l ðþ

3 J.J. Chu, C.A. Steeves / Journl of Non-Crystlline Solids 357 (11) of morphous structures nd to smple the sttisticl ensemle sufficiently, 1 simultions re conducted for ech mteril. Different initil configurtions re otined y further equilirting the minimized morphous sttes of Al nd Ti for vrying lengths of time. Thus, 1 sttisticlly equivlent smples re otined while eliminting the vriility in the quenching process. Dt collected from the 1 simultions re susequently used to determine men vlues of CTE nd the stndrd devition in the results... Recrystlliztion In studying the recrystlliztion ehvior of morphous Al nd Ti, morphous phses of Al nd Ti re otined vi the sme procedure s descried efore. Liquid Al nd Ti is rpidly quenched t rte of 1 K/ps to 5 K nd susequently equilirted. Following equilirtion, the potentil energy of the system is minimized. The recrystlliztion temperture of morphous Al nd Ti re determined y heting the mterils t rte of 1 K/ps under n isoric-isotherml ensemle. As the temperture of the system increses, the potentil energy nd volume of the system re recorded s function of temperture. A sudden drop in potentil energy will e noted when devitrifiction occurs s the metstle morphous phse trnsforms into more energeticlly fvorle crystlline stte [,]. The temperture t which n rupt decrese in potentil energy is oserved is thus denoted s T x. Confirmtion of recrystlliztion is provided y visulizing the tomic coordintes nd exmining the tomic structure vi the RDF nd coordintion numer of the system. The volumes of morphous Al nd Ti will ehve similrly to the potentil energy of the system. During initil heting, the volume of the mteril will expnd until T x is reched. When devitrifiction occurs, decrese in volume will e oserved s the morphous sttes trnsform into more compct crystlline stte. The mount of shrinkge due to recrystlliztion is therefore quntified y compring the volume of the morphous mteril t T x to the volume of its crystlline counterprt t higher temperture T x. The volume of the crystlline phse t T x is selected in order to emulte complete recrystlliztion over long period of time. A volumetric expnsion coefficient due to devitrifiction, β r, is thus defined to descrie the contrction in volume over the course of the phse trnsition: β r = 1 V ;x V c;x V ;x T x T x where V,x is the volume of the morphous mteril t T x, nd V c,x is the volume of the crystlline mteril t T x. Eq. (3) therefore gives liner pproximtion of the chnge in volume etween T x nd T x. Assuming isotropic mteril properties, liner expnsion coefficient due to recrystlliztion, α r, is relted to β r y the following eqution: = ð3α r β r Þ +3α r ðt x T x Þ + α 3 r ðt x T x Þ ðþ In order to sufficiently smple the sttisticl ensemle nd investigte the vrition in results, 1 sets of simultions re performed for Al nd Ti strting from the quenching process. Men vlues of T x nd V,x re thus otined for determining β r nd α r. Inputs into the 1 simultions re identicl except for the rndom numer seed which is used to generte the initil velocities of the toms under Boltzmnn distriution. For ech simultion, rndom numer seed is chosen etween 1 nd 1, using rndom numer genertor. 3. Results 3.1. Intertomic potentils The typicl therml expnsion of crystlline Al is exemplified in Fig. 1 which plots the size of the simultion ox s function of ð3þ temperture using the intertomic potentil y Mendelev et l. [18]. Cuic polynomils used to fit the rw dt re lso illustrted s dshed lines for comprison. Therml expnsion in the x, y, nd z directions re prcticlly identicl due to its symmetricl fcc crystlline structure. lx, ly, nd lz re oserved to vry nd expnd smoothly s the temperture of the system is incresed. The dt otined from simultion re well represented y the polynomil interpolnts, which give n excellent fit. Although not shown here, similr oservtions re noted when using the potentils y Liu et l. [1] nd Zope nd Mishin [19] to simulte the therml expnsion of crystlline Al. When modeling crystlline Ti, it is found tht the therml expnsion in the x, y, nd z directions differ due to its nisotropic hexgonl crystl structure. However, the vrition in lx, ly, nd lz re still smooth functions of temperture nd descried ccurtely y cuic polynomils. The CTEs of crystlline Al nd Ti predicted y ech intertomic potentil vi MD simultions re illustrted in Fig.. Experimentl Length in x direction (Angstroms) Length in y direction (Angstroms) c Length in z direction (Angstroms) MD Cuic Fit MD Cuic Fit 5 1 MD Cuic Fit 5 1 Fig. 1. lx, ly, nd lz plotted s function of temperture in figures(),(),nd (c) respectively using the potentilymendelev et l. [18] to simultethetherml expnsion of crystlline Al. Rw nd fitted dt re shown s solid nd dshed lines respectively.

4 378 J.J. Chu, C.A. Steeves / Journl of Non-Crystlline Solids 357 (11) Coefficient of Therml Expnsion (ppm/k) Coefficient of Therml Expnsion (ppm/k) Liu et l. (MD) 1 Mendelev et l. (MD) Zope nd Mishin (MD) Wilson (Exp.) 5 Americn Institute of Physics Hndook (Exp.) Acklnd (MD) Zope nd Mishin (MD) Sirot nd Zhko (Exp.) Americn Institute of Physics Hndook (Exp.) dt from literture for crystlline Al [,] nd Ti [3,] re lso plotted in comprison to determine which potentil est reproduces the ctul oserved therml expnsion. Fig. () compres the selected potentils for Al. Simultions using the intertomic potentils y Liu et l. [1] nd Zope nd Mishin [19] re found to sustntilly underestimte the CTE of Al, the former more so thn the ltter. The resulting CTEs using the potentil due to Mendelev et l. [18] show etter correltion to the experimentl dt, predicting slightly higher therml expnsion. It is therefore determined tht the potentil y Mendelev et l. [18] est replictes the therml expnsion of Al, nd is selected s the potentil of choice in the susequent simultions. In Fig. (), the CTEs of Ti otined vi MD simultions re compred to those from experiments. It is evident from this plot tht the potentil due to Zope nd Mishin [19] produces results tht closely mtch the experimentl results y Sirot nd Zhko [3]. CTEs predicted y the intertomic potentil y Acklnd [] hve the correct overll trend, ut lrge devitions re noted when compred to the empiricl vlues. Thus the MD simultions revel tht the intertomic potentil y Zope nd Mishin [19] gives the est correltion to experimentl dt, nd is therefore used to model Ti in the following simultions. 3.. Amorphous sttes Fig.. Simulted (solid lines) nd experimentl (dshed lines) CTEs plotted s function of temperture for crystlline Al nd Ti in figures () nd () respectively. The tomic structures of liquid Al nd Ti cooled t rte of 1 K/ ps re visulized using Visul Moleculr Dynmics (VMD) [] to confirm the genertion of morphous metls. For oth mterils, the toms re found to hve disordered rrngement with no crystlline structures visile. In ddition to exmining the tomic coordintes, the RDFs of the morphous mterils re nlyzed. Typicl RDFs of morphous Al nd Ti t 3 K re shown in Fig. 3 () nd () respectively nd compred to the RDFs of their crystlline stte (shown in dshed lines). For crystlline mteril, numerous shrp nd well defined peks re oserved ecuse the toms sit nd oscillte out their lttice positions. The loctions of these peks in the RDF represent the distnces of the neighoring toms. Utilizing Eq. (), the coordintion numer of fcc Al nd hcp Ti re computed to e 1 s expected. With morphous mterils, the peks re roder nd locted t different vlues of r. The first pek of the RDF, which illustrtes the distnce of the nerest neighor, is shifted to the left when compred to the crystlline phse. This phenomenon hs een oserved y Celik et l. [7] in their study of locl structures in morphous Al. Another chrcteristic of g(r) for morphous mterils is the split second pek s seen in Fig. 3. The second pek is much roder thn the first, with two pexes oserved. This doule pek is common chrcteristic nd feture of morphous mterils nd hs een noted y other reserchers [8]. By integrting the first pek of g(r), the coordintion numer of morphous Al nd Ti re computed to e 1.7 nd 1. respectively. The morphous phses re therefore found to hve high coordintion, similr to tht of their crystlline sttes Therml expnsion The vrition in i s function of T in typicl simultion of morphous Al is illustrted in Fig.. Rw dt is illustrted y the thinner curves, while the cuic polynomil interpolnts re shown y the thicker lines. Informtion up until the temperture of recrystlliztion is used for fitting since only the CTE of the morphous sttes re of interest. Using the entire set of dt would lso result in poor fit due to the rupt chnges in size cused y the trnsition in phse. Prior to recrystlliztion, it is oserved from Fig. tht i does not vry smoothly s function of T. This is in contrst to the results in g(r) g(r) 8 l r (Angstroms) 1 8 Amorphous Crystlline Amorphous Crystlline r (Angstroms) Fig. 3. RDFs of morphous nd crystlline sttes of Al nd Ti t 3 K re shown in figures () nd () respectively. l

5 J.J. Chu, C.A. Steeves / Journl of Non-Crystlline Solids 357 (11) Fig. 1 which show smooth chnges for crystlline Al. Pressure oscilltions re inherent to MD simultions due to the use of sttisticl ensemles to control mcroscopic quntities nd cuse the simultion size to vry. This effect is mplified y the fct tht morphous mterils re metstle. When morphous Al is heted from 5 K, the system in generl expnds in the x, y, nd z directions. Aove certin temperture however, sudden overll decrese in the simultion size is oserved. This phenomenon is consequence of morphous Al trnsitioning into denser crystlline stte. Once the morphous phse hs crystllized, the therml expnsion resumes tht of crystlline mteril. It cn e seen in Fig. 1 tht pst K, the lengths of the simultion ox grow stedily with incresing temperture. Although not shown here, dt collected from the simultion of morphous Ti hs similr chrcteristics nd ptterns s descried here for Al. In this study, the CTEs of morphous Al nd Ti re clculted etween 1 K nd T x t 5 K intervls. For Al nd Ti, T x hs een estlished to e roughly 5 K nd 55 K respectively detils in determining their recrystlliztion tempertures re given lter in this pper. Using Eq. (1), the CTE t given temperture is clculted nd then verged over the 1 simultions to produce the dt plotted in Figs. 5 nd for Al nd Ti respectively. The vrition in CTEs is shown y error rs out the men vlues, illustrting plus nd minus one stndrd devition. Differences in results rise due to vrying initil configurtions which yield diverse trjectories. The CTEs of crystlline Al nd Ti from literture nd MD simultions in this work re lso plotted in Figs. 5 nd for reference nd comprison. Fig. 5 illustrtes the CTEs of morphous nd crystlline Al. Experimentl dt for the therml expnsion of crystlline Al re tken from the study y Wilson [] nd the Americn Institute of Physics Hndook []. The two experimentl sources give results tht re in very close greement with ech other. No studies hve een found tht give the CTE of morphous Al. From this work, the computer simulted therml expnsion of crystlline Al produces results tht re in ccordnce with experimentl dt. The trend of incresing therml expnsion with temperture is correctly reproduced y the intertomic potentil, nd the predicted CTEs re slightly higher thn those reported from experiment. A difference of pproximtely 3 prts per million per Kelvin (ppm/k) t room temperture is oserved, which decreses with higher tempertures. This devition is miniml in comprison to previous MD study y Alper nd Politzer [9] which overestimtes the therml expnsion of crystlline Al y fctors of The disprity etween MD nd experimentl results is explined y the limited description provided y empiricl intertomic potentils. The CTEs of Ti in oth crystlline nd morphous phses re illustrted nd contrsted in Fig.. Two sources in the literture hve 9 Coefficient of Therml Expnsion (ppm/k) Fig. 5. The clculted CTE of morphous Al is plotted s function of temperture. Crystlline CTEs from simultion nd literture re lso shown for comprison. een found tht give the experimentl CTE of crystlline Ti s function of temperture those from study y Sirot nd Zhko [3] nd the Americn Institute of Physics Hndook []. Once gin, there hve een no studies found which give the therml expnsion of morphous Ti. The two experimentl sets of dt for the CTE of crystlline Ti re in good greement with one nother. Both give CTEs of similr mgnitude nd n incresing trend with higher temperture. Sirot nd Zhko [3] reports vlues tht re slightly lower thn those from the Americn Institute of Physics Hndook, ut the differences re no greter thn 1. ppm/k. Using the potentil y Zope nd Mishin [19], the therml expnsion of crystlline Ti shows excellent conformnce with oth experimentl sources s seen in Fig.. Results re very close to the experimentl CTEs given y Sirot nd Zhko [3], ut re slightly lower thn those from the Americn Institute of Physics Hndook []. The disgreement etween the simulted nd experimentl CTEs is once gin ttriuted to the limited ccurcy of the intertomic potentil. 3.. Recrystlliztion The devitrifiction of morphous Al nd Ti is first confirmed y exmining the evolution of their RDFs nd vi visuliztion. Fig. 7 () nd () illustrtes g(r) of Al nd Ti respectively t 3 K nd higher temperture lter on in the simultion. The RDFs t 3 K clerly revel n morphous structure s noted y the rod nd split second pek. At 8 K nd 9 K, the RDFs of Al nd Ti hve chnged significntly with the ppernce of new peks, indicting crystlline phse. To nlyze the tomic structure further, the coordintion numer of Al nd Ti t 8 K nd 9 K respectively re clculted ccording to Length (Angstroms) (MD) (Fit) (MD) (Fit) (MD) (Fit) Fig.. Vrition in simultion cell size s morphous Al is heted from n initil temperture of 5 K t rte of 1 K/ps. The simultion ox is free to expnd nd contrct nisotropiclly with zero externl pressure. Coefficient of Therml Expnsion (ppm/k) Experimentl Crystlline (Sirot nd Zhko) Experimentl Crystlline (AIPH) MD Simultion Crystlline MD Simultion Amorphous Fig.. The clculted CTE of morphous Ti is plotted s function of temperture. Crystlline CTEs from simultion nd literture re lso shown for comprison.

6 377 J.J. Chu, C.A. Steeves / Journl of Non-Crystlline Solids 357 (11) g(r) g(r) r (Angstroms) K, Amorphous 8K, Crystlline 3K, Amorphous 9K, Crystlline 8 r (Angstroms) Fig. 7. RDF of Al t 3 K nd 8 K re depicted in figure (). Figure () illustrtes the RDF of Ti t oth 3 K nd 9 K. Eq. (). For the recrystllized sttes of Al nd Ti, CN=11.98 nd 1. respectively. The coordintion numers of oth systems re very close to the expected vlue of 1 for fcc Al nd hcp Ti (CN =8 for cc Ti). Thus, morphous Ti is found to recrystllize into its hcp form s expected since T x is well elow its phse trnsition temperture. Visulizing the tomic coordintes t these higher tempertures lso revels ordered crystl lttices, confirming the devitrifiction of Al nd Ti. Fig. 8 () nd () illustrtes the vrition in potentil energy of morphous Al nd Ti respectively s function of temperture. Ech solid line represents the dt otined from one of the ten simultions. As the morphous metls re heted from 5 K, their potentil energy grows t roughly liner rte until the point of recrystlliztion. In the cse of Al, n rupt drop in potentil energy is oserved t T x s the metstle morphous stte trnsforms into more energeticlly fvorle crystlline stte tht is of lower potentil energy. For Ti however, sutle chnges in potentil energy re seen s tempertures pproch T x. Unlike Al, the potentil energy of the Ti system does not exhiit extreme nd rupt decreses during devitrifiction. In generl, smll dip in potentil energy is found to occur t T x, followed y lrge nd sudden drop t elevted tempertures. For oth mterils, the potentil energy resumes liner trend with temperture fter recrystlliztion hs occurred. From the potentil energy profiles shown in Fig. 8, the recrystlliztion temperture is estimted for ech of the ten simultions nd n verge vlue of T x is computed. For morphous Al, T x is clculted to e 53. K with stndrd devition of.7 K. The men recrystlliztion temperture of morphous Ti is determined to e 55. K with stndrd devition of 8.1 K. T x for oth Al nd Ti re illustrted y the verticl dshed lines in Fig. 8. Considerle vrition Potentil Energy (ev/tom) Potentil Energy (ev/tom) in the recrystlliztion temperture is oserved mong the ten simultions for oth mterils. The discrepncy in results is explined y the metstle nture of morphous mterils nd the smpling of vrious configurtions within the sttisticl ensemle. The verge tomic volume profiles of morphous Al nd Ti re plotted in Fig. 9 with those of crystlline Al nd Ti illustrted for comprison. It is noted tht even fter recrystlliztion hs occurred, the volume of the morphous system does not rech tht of perfectly crystlline stte nd is slightly elevted. From the nlysis conducted erlier, T x is tken to e roughly 5 K nd 55 K for Al nd Ti respectively. Using this informtion, V,x is extrcted from the morphous tomic volume curves s depicted in Fig. 9. Fromthese profiles, it is estimted tht the process of recrystlliztion occurs over temperture rnge of 1 K, thus T x is selected to e equl to T x +1K for oth Al nd Ti. With the knowledge of T x, V c,x is determined from the crystlline tomic volume curves s shown in Fig. 9. Tle 1 summrizes the vlues identified from the procedure descried ove which re used to clculte β r for Al nd Ti. Sustituting the corresponding numers into Eq. (3) gives β r to e K 1 nd K 1 for Al nd Ti respectively. The vlues of β r re susequently used in Eq. () to solve for the liner expnsion coefficient due to recrystlliztion given T x T x =1K. Using the fsolve function in MATLAB [3], the non-liner expression is solved to give α r equl to ppm/k nd 3.3 ppm/k for Al nd Ti respectively.. Discussion Therml expnsion T = 53. K T = 55. K Fig. 8. Potentil energy of morphous () Al nd () Ti plotted s function of temperture. Results from ll ten simultions re shown nd the verge recrystlliztion temperture is illustrted y the dshed verticl line. A comprison etween the therml expnsion of morphous nd crystlline Al derived from MD simultions revels numer of different

7 J.J. Chu, C.A. Steeves / Journl of Non-Crystlline Solids 357 (11) Atomic Volume (Angstroms 3 /tom) Atomic Volume (Angstroms 3 /tom) Amorphous 1.5 Crystlline Amorphous Crystlline Fig. 9. Atomic volume versus temperture profiles of morphous nd crystlline () Al nd () Ti. Dt points used to clculte β r re identified. fetures. From Fig. 5, it is pprent tht for oth phses of Al, the CTEs re similr t tempertures elow 5 K the CTE of morphous Al is however few ppm/k lower. At tempertures ove 5 K however, the predicted therml ehvior etween morphous nd crystlline Al is very different. For crystlline Al, the CTE shows n incresing trend with higher tempertures, while in the cse of morphous Al, the CTE drops off rpidly. At 3 K, the CTE of morphous Al is pproximtely 7 ppm/k lower thn tht of crystlline Al, nd decreses to 1 ppm/k t 5 K. This rupt decline in therml expnsion is ttriuted to thermlly ctivted recrystlliztion in which the morphous tomic structure trnsforms into more fvorle nd denser crystlline stte. The error in the clculted CTEs of morphous Al is found to increse s the temperture pproches T x. At 35 K nd elow, the stndrd devition is pproximtely 1.3. ppm/k. For K nd higher, the stndrd devition grows significntly s the CTEs otined from simultion re spred over lrger rnge of vlues. The incresed vriility t higher tempertures cn e explined y the unstle mnner in which morphous Al trnsitions to crystlline phse. Furthermore, pressure fluctutions inherent to MD simultions re found to crete prominent volume fluctutions in the system, prticulrly ner T x ecuse of the metstle morphous stte of Al. These fctors ffect the polynomil fitting of simultion dt, which cuses vritions in α since it is sensitive to the slopes of the cuic interpolnt. As result, lrge vritions in the predicted CTE ner T x re oserved. Tle 1 Tempertures nd tomic volumes for clculting β r for Al nd Ti. Mteril T x (K) V,x (Å 3 ) T x (K) V c,x (Å 3 ) Al Ti Compring the MD results illustrted in Fig. shows tht the CTE of morphous Ti is consistently higher thn tht of its crystlline stte. Differences etween 1.7 nd 3.3 ppm/k over the temperture rnge of 1 55 K re oserved. The CTE of morphous Ti remins more or less stedy s the temperture pproches T x, unlike morphous Al where drstic decline in therml expnsion is oserved. The difference in ehvior cn e explined y compring the volumes of Al nd Ti in oth their morphous nd crystlline sttes. At 3 K, the tomic volumes of morphous nd crystlline Al re 18. Å 3 nd 1.73 Å 3 respectively, corresponding to negtive 8% chnge in volume. This vlue is comprle to the 9% predicted y Becqurt et l. [1]. Amorphous nd crystlline Ti hve tomic volumes of 17.8 Å 3 nd 17.8 Å 3 respectively, difference tht is extremely smll. A significnt discrepncy in tomic volume, nd therefore density, is found etween the two phses of Al. This disprity is miniml when compring the two sttes of Ti. Thus s morphous Ti recrystllizes, the decrese in volume is much smller when contrsted with the phse trnsition of Al. This is evident from Fig. 1 which illustrtes the tomic volume versus temperture profiles of morphous Al nd Ti during recrystlliztion. In the cse of Al, negtive chnge of roughly. Å 3 is oserved, while very sutle differences re noted for Ti. Furthermore, the MD simultions show tht the process of devitrifiction for Ti tkes plce grdully over greter temperture rnge nd is less rupt thn tht of Al. As result of these two fctors, the therml expnsion of morphous Ti does not exhiit shrp drop ner T x. For morphous Ti, the stndrd devition in CTE rnges etween.1 nd.8 ppm/k, which is proportionlly lower thn tht of morphous Al. The reson for this my e due to the smll difference in tomic volume etween morphous nd crystlline Ti nd the gentler phse trnsition. The vrition in results is found to increse s Atomic Volume (Angstroms 3 /tom) Atomic Volume (Angstroms 3 /tom) Fig. 1. Atomic volume versus temperture profiles of morphous () Al nd () Ti during heting nd recrystlliztion. Ech line represents one of the ten simultions performed.

8 377 J.J. Chu, C.A. Steeves / Journl of Non-Crystlline Solids 357 (11) tempertures pproch T x, similr to the ehvior noted for Al. As explined efore, this is consequence of the metstle nture of the morphous stte nd the pressure oscilltions which influence the polynomil fits... Recrystlliztion As morphous Al undergoes devitrifiction, its potentil energy is found to decrese in multiple steps nd in discontinuous fshion s seen in Fig. 8. The potentil energy of the system is lso found to vry fter recrystlliztion hs occurred. This ehvior is ttriuted to the sudden nucletion nd growth of the crystlline phses nd the formtion of multiple grins in the system. Grin oundries introduce dditionl energy to the system ecuse the onds in this region re stretched nd the toms re in suoptiml stte [31]. Further heting elimintes these oundries to crete lrger grins, thus the potentil energy continues to decrese with higher tempertures. Due to different initil configurtions, recrystlliztion nd grin growth vries for ech simultion. Thus, the potentil energies fter devitrifiction re not necessrily the sme. In the cse of Ti, the chnges in potentil energy re very sutle ner T x, while lrge nd sudden decreses re oserved t higher tempertures. This phenomenon cn e explined y the fct tht morphous Ti tends to crystllize into lrge numer of smll grins, evident through visuliztion of the tomic coordintes. Fig. 11 illustrtes slice of simultion cell t 8 K where numerous crystl lttices re visile, ech with different directionlities. The cluster of toms elonging to one specific orienttion represents grin. As explined previously, grin oundries re non-idel tomic configurtions which dd potentil energy to the system. As the temperture of the system is incresed, the grin oundries re eliminted to form lrger grins, thus explining the drop in potentil energy oserved t tempertures ove T x. Grin growth is found to vry nd initite t different tempertures due to the diversity etween MD simultions. The recrystlliztion of morphous Al nd Ti into multiple grins cn lso explin the discrepncies in volume oserved in Fig. 1, which plots the tomic volume of Al nd Ti in oth their morphous nd crystlline sttes. Although the tomic volumes of morphous Al nd Ti shrink due to recrystlliztion, they re still oserved to e lrger thn tht of single crystlline system. This is explined y the presence of multiple grins nd grin oundries, which occupy more spce thn perfectly crystlline stte. From the MD simultions performed in this work, the predicted vlue of T x for morphous Al is pproximtely 5 K. Studies y Lu nd Szpunr [] nd Shimono nd Onoder [7] report T x of Al to e 55 3 K nd 3 K respectively, which re widely conflicting vlues. The estimted recrystlliztion temperture from this work is lso found to differ from the previous investigtions, ut is closer to the rnge given y Lu nd Szpunr [] eing roughly 1 K lower. The recrystlliztion temperture of morphous Ti derived from this study is round 55 K, pproximtely 1 K higher thn the vlue predicted for Al. MD simultions y Shimono nd Onoder [7] predict T x to e pproximtely 8 K for Ti, which is much lower thn the vlue otined from this reserch. However, the investigtion y Shimono nd Onoder [7] nd the study conducted in this pper oth predict T x of Ti to e greter thn tht of Al. This trend is in ccordnce with intuition ecuse the cohesive energy of Ti is higher thn Al, thus more energy (higher temperture) is required to rek the tomic onds. The liner expnsion coefficient due to recrystlliztion determined in this study for Al nd Ti differ drsticlly from one nother. For Al, α r is clculted to e ppm/k, which is very lrge nd negtive vlue. In the cse of Ti, modest vlue of 3.3 ppm/k is computed from the simultion dt. Given the extreme difference in volume etween the morphous nd crystlline phses of Al, lrge negtive vlue of α r is required to descrie the chnge in volume during devitrifiction. The opposite is noted for Ti, where slightly negtive expnsion coefficient is sufficient to descrie the smll trnsformtion in volume due to recrystlliztion. 5. Conclusions MD simultions hve een used in this work to investigte the therml expnsion nd recrystlliztion ehvior of morphous Al nd Ti. Glssy metls were creted vi quenching nd susequently heted t rte of 1 K/ps under isoric-isotherml conditions. The therml expnsion of morphous Al hs een found to e similr to tht of crystlline Al t tempertures elow 5 K, ut flls off rpidly s the morphous microstructure egins to recrystllize. Amorphous Ti on the other hnd is found to show reltively stle ehvior nd hve consistently higher CTEs thn crystlline Ti. Using the potentil energy versus temperture profiles, the recrystlliztion tempertures of morphous Al nd Ti hve een determined to e pproximtely 5 K nd 55 K respectively. Dt from MD simultions lso showed tht the process of recrystlliztion occurs over temperture rnge of roughly 1 K. The devitrifiction of morphous Al hs een found to e ssocited with n extreme negtive chnge in volume, wheres sutle differences re oserved for morphous Ti. As result, the liner expnsion coefficient due to recrystlliztion hs een clculted to e ppm/k nd 3.3 ppm/k for Al nd Ti respectively. Acknowledgments Funding for this reserch hs een provided y the Keck Institute for Spce Studies. Computtions were performed on the GPC supercomputer t the SciNet HPC Consortium. SciNet is funded y: the Cnd Foundtion for Innovtion under the uspices of Compute Cnd; the Government of Ontrio; Ontrio Reserch Fund Reserch Excellence; nd the University of Toronto. References Fig. 11. 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