An Equivalent Time Approach for Scaling the Mechanical Alloying Processes

In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 1 An equi alen ime app oach o scaling he mechanical alloying p ocesses J. J. Ipus1, J. S. Blázquez1, V. F anco1, M. Millán1, A. Conde1, D. Oleszak2, T. Kulik2 1 Dp o. Física de la Ma e ia Condensada, ICMSE-CSIC, Uni e sidad de Se illa, P.O. Box 1065, 41080, Se illa, Spain. 2 Facul y o Ma e ials Science and Enginee ing, Wa saw Uni e si y o Technology, ul. Woloska 141, 02-507, Wa saw, Poland. ABSTRACT. Dynamics o a single ball in o a plane a y ball mill is analyzed leading o a cubic dependence o he powe ans e ed du ing milling wi h he o a ional speed, . This leads o he de ini ion o an equi alen ime o desc ibe he s a e o ball milled powde s independen ly o . Mechanical alloying o Fe75Ge20Nb5 composi ion is s udied by a combina ion o expe imen al echniques (di e en ial scanning calo ime y, scanning elec on mic oscopy, ene gy dispe si e x- ay spec ome y, x- ay di ac ion, Mössbaue spec ome y and ib a ing sample magne ome y) and esul s e idence a good ag eemen wi h he p edic ions o he equi alen ime app oach. Keywo ds: C. Mechanical alloying and milling; C. Nanoc ys als Co esponding au ho : P o . A. Conde Depa amen o de Física de la Ma e ia Condensada. Uni e sidad de Se illa. Apa ado 1065, 41080 Se illa (Spain). Phone: (34) 95 455 28 85 Fax: (34) 95 461 20 97 E-mail: [email p o ec ed] In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 2 1. In oduc ion Ball milling has been shown as a e y e sa ile echnique o p oduc ion o me as able sys ems: nanoc ys alline, amo phous, supe sa u a ed solid solu ions, quasic ys als, e c [ 1 ]. Unlike ul a as cooling echniques, which can eeze high empe a u e mic os uc u es a low empe a u es in a na ow composi ional ange a ound he eu ec ic, ball milling is applied o a wide composi ional ange. Mechanical e olu ion induced by ball milling is due o he ene gy ans e ed om he milling media o he powde pa icles, con inuously submi ed o ac u e and cold welding p ocesses which will de ine hei inal mo phology. Among he di e en ypes o ball mills a ailable, plane a y ball mills a e widely used o p oduce such me as able ma e ials. In o de o unde s and he dynamics o plane a y ball mills, se e al au ho s ha e used he app oach o single ball dynamics and ha e ex apola ed hei esul s o ac ual milling p ocesses using se e al balls [ 2 , 3 , 4 , 5 , 6 , 7 , 8 ]. These s udies analyze he ene gy ans e ed pe collision be ween he ball and he ial wall and he dependence o he ball ajec o y on di e en pa ame e s such as a io be ween he equencies o main disk and ials, ball o powde mass a io and o he s [1,2,3,4,5,6,7,8, 9 , 10 ]. Those esul s yield a pa abolic dependence o he in ensi y o a single ball-powde in e ac ion wi h he o a ional speed. Mo eo e , he ajec o y desc ibed by his single ball was ound independen o he o a ional speed [2]. These wo ea u es, as i will be shown in he ollowing sec ion o his pape , migh lead o a cubic law o he es ima ed powe ans e ed o he powde unde a single ball app oach. The eliabili y o his p edic ion is es ed in he hi d sec ion o he pape by s udying he e olu ion o a mechanically alloyed Fe75Nb5Ge20 powde a di e en milling in ensi ies du ing di e en imes. In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 3 2. App oach o he dynamics o he plane a y ball mill 2.1 Mo emen o a single ball in o he ial Single ball app oxima ion has been used in he li e a u e o p edic he ene gy ans e o powde du ing milling. In his sec ion, a b ie desc ip ion o he o a ional speed dependence o bo h he ene gy in ol ed in di e en ball-powde in e ac ions and he equency o hese e en s is gi en. Figu e 1 shows a scheme o he plane a y ball mill, whe e  R is he posi ion o he ial cen e and i s modulus he adius o he main disk;  , he posi ion o he ball espec o he cen e o he ial, which modulus is he adius o he ial minus ha o he ball;   Rp is he posi ion o he cen e o he ball; = +  0 is he angle o a ed by he main disk a ime and  i s angula speed; = +  0 is he angle o a ed by he ial a ime and  i s angula speed. In his scheme, he o a ions o he ial and ha o he main disk a e opposi e. Th ee o hogonal e e ence sys ems ha e been de ined o posi ion he ball a poin P; a Ca esian ine ial sys em   ,,i j k and wo pola and non-ine ial sys ems: ,, R ee        , which ollows he o a ion o he main disk, and ,, uu       , which ollows he o a ion o he ial. In he simpli ied s udied sys em, he pa icle mo emen mus ul ill some equi emen s, also conside ed by o he au ho s [3,4]:  Only he mo emen o one ball is conside ed.  The mo emen occu s in a plane. In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 4  Only wo o ces a e conside ed; he no mal eac ion o he ial wall o e he ball,  N , and he ic ion o ce,   F .  The pa icle keeps ixed a a poin o he ial wall since N>0.  When N=0, he ball de aches om he ial wall and mo es eely un il i impac s again wi h he ial wall.  A e he collision, he ball mo es again s uck o he wall. A e analyzing he dynamics o he sys em and conside ing only he o ce componen pa allel o  u an exp ession o N pe mass uni o ball is ound, )cos( 22   R m N (1) This exp ession is meaning ul only o N>0, i N=0 he ball will de ach om he ial wall, hus a de achmen ime, d, can be ob ained as:                    00 2 2 a ccos 1    R d (2) Analogously o exp ession (1), conside ing he componen pa allel o   u , an exp ession o F  pe mass uni o ball is ob ained, )sin( 2    R m F (3) This solu ion will apply i he ball is assumed o be ixed a a poin o he ial wall. Once he ball de aches om he ial wall i mo es eely wi h cons an eloci y un il i collides again wi h he ial wall. In o de o calcula e he ime o ligh , , he sepa a ion dis ance be ween he ball and he ial wall has been analyzed. In ac , he ball will In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 5 collide when pR equals he adius o he ial, . Figu e 2 shows he cu e o he di e ence be ween he squa e o pR and he squa e adius o he ial as a unc ion o ime om he de achmen o he plane a y mill used in sec ion 3, wi h R=0.125 m, =0.017 m (i has been conside ed ha he ball has a adius o 0.005 m), a =150, 250 and 350 pm, being he alues o d and o he same o de o magni ude. By using di e en alues o , keeping cons an he a io /, i can be obse ed ha bo h d and a e p opo ional o 1/ (see exp ession (2) and inse o igu e 2, espec i ely) and, he e o e, he numbe o de achmen and collision e en s emains cons an along a pe iod o o a ion o he main disk. Fo > d, N=0 (as well as F  ) and he ball is no in con ac wi h he ial wall. A e he ball collides wi h he wall, N will eco e a non ze o alue. The ajec o y o he ball, o bo h ine ial and non-ine ial e e ence sys ems a e shown in igu e 3. I can be obse ed ha he poin o collision, PC, is di e en o he posi ion o he de achmen poin a he ime o collision, PD’. The e o e, a phase angle,  , mus be included in he a gumen o he cosine unc ion o equa ion (1). A e he collision, he p ocess is epea ed con inuously and igu e 4 shows he alue o N, solid line, du ing a o a ional pe iod o he main disk along wi h he analy ical solu ion o equa ion (1), do ed line, o compa ison. 2.2 Es ima ion o he dependence o he ene gy ans e on he equency The simple model desc ibed abo e can be used o p opose a dependency o he ene gy ans e ed du ing milling om he balls and ial o he powde . Se e al In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 6 mechanisms can be conside ed as candida es o ene gy ans e om he balls and ial o he powde . Fi s , while he ball is in con ac wi h he ial wall, he powde which is in be ween is comp essed and shea ed and he wo k done o e he powde du ing his comp ession could be app oxima ed o: 2 1kdNdxNWN (4) whe e d is a dis ance o he o de o he powde pa icle size (mic ome e ) and k1 is a cons an . Fo =150 pm, he ene gy ans e ed by uni mass o ball can be es ima ed as ~10-4 J/kg o d~1 m. Analogously, i could be possible o es ima e a simila o de o magni ude o he shea wo k done by he ic ion o ce. On he o he hand, when he ball, a e being de ached, impac s agains he ial wall, a change in i s eloci y is necessa y o allow he ball o mo e wi h he ial again. This change implies a educ ion in he kine ic ene gy o he ball, which can be calcula ed as:   2 2 )( 2 1impdC mE  (5) whe e ( d) is he speed o he ball a he de achmen ime and imp is he modulus o he speed o he ial wall a PC:       2 2cossin)(         R R ddddd   (6)             2 2cossin         R R d d d dimp   (7) The alues o EC pe uni mass o ball, as i occu s o he es ima ed wo k o comp ession, and as expec ed by exp essions (6) and (7), ollows a 2 law. Fo =150 pm he o de o magni ude o he ans e ed ene gy is 1 J/kg pe collision e en . In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 7 Some au ho s ha e es ima ed he ene gy ans e ed o he powde , assuming a He zian collision app oxima ion and a e es ima ing he amoun o powde in be ween he ball and he ial wall [4,6,8]. Se e al pa ame e s a ec he inal esul s, as ball o powde mass a io and elas ic p ope ies o he milling media as well as o he powde , e c, which we e cons an along he expe imen s pe o med in his wo k. The e o e, he  dependence o he ene gy ans e ed du ing he milling p ocess is p ese ed. As i has been shown in igu e 4, du ing a pe iod o o a ion o he main disk, a cons an numbe o comp ession e en s and impac s occu . Assuming he wo k done du ing each comp ession is desc ibed by equa ion (4) and ha o each impac is desc ibed by equa ion (5), he a e age powe could be es ima ed, o long imes, as he whole wo k done du ing a pe iod o o a ion o he main disk di ided by his ime:   3 2  k T EWn PCN (8) whe e n is he cons an numbe o e en s pe pe iod and k2 is a cons an . The e o e, he ene gy ans e ed a e a ime would be p opo ional o 3 and, i se e al alues o  a e used (keeping cons an he a io /) an equi alen ime, eq, can be de ined as:   3 0 / eq (9) whe e 0 is a e e ence equency (in he ollowing s udy 0=150 pm, which is he lowes equency used). As milling ime inc eases, ene gy is ans e ed om milling media o powde yielding mic os uc u e e olu ion and di e en ans o ma ions [1]. Quan i ying expe imen ally his ene gy is a di icul ask as i s acquisi ion is no comple ely e e sible. In o de o o e come hese obs acles, some au ho s ha e p oposed ene gy maps o app oxima ely compa e mic os uc u es ob ained om di e en milling condi ions [3,8]. In his wo k, i has been assumed ha he s a e o he powde is In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 8 uni ocally de e mined by he amoun o ene gy ans e ed o he powde s. The e o e, powde o Fe75Nb5Ge20 composi ion has been cha ac e ized by using di e en s uc u al and magne ic echniques a e milling a di e en equencies du ing di e en imes, keeping cons an he / a io. Resul s om di e en expe imen al echniques ha e been escaled using an equi alen ime de ined by equa ion (9) wi h 0=150 pm. 3. Applica ion o expe imen al esul s 3.1 Expe imen al echniques Fe75Nb5Ge20 composi ion was p oduced om pu e powde s (pu i y99 %) by ball milling in a F i sch Pul e ise e 5 plane a y ball mill using ha dened s eel balls (10 mm diame e ) and ials. The ini ial powde mass was 5 g and he ball o powde a io 10:1. The a io be ween he o a ional speed o he ial () and ha o he main disk () was ixed o /=-2. Th ee di e en alues o  we e used: 150, 250 and 350 pm. Some powde was aken ou a e selec ed imes ( om 1 h o 150 h), opening and closing he ials in a gon a mosphe e o a oid oxygen and humidi y con amina ion. Size and mo phology o he powde pa icles we e s udied by scanning elec on mic oscopy (SEM) in a Jeol JSM-6460 LV and ene gy dispe si e X- ay (EDX) analyses we e pe o med using an Incax-sigh o Ox o d Ins umen s. The c ys alline s uc u e was s udied by X- ay di ac ion (XRD) using Cu-K adia ion in a B uke D8I di ac ome e and he local en i onmen o Fe a oms was analyzed by Mössbaue spec ome y (MS). Mössbaue spec a we e eco ded a oom empe a u e in a ansmission geome y using a 57Co(Rh) sou ce. The alues o he hype ine pa ame e s we e ob ained by i ing wi h NORMOS p og am [ 11 ]. The mal s abili y o he samples was s udied by di e en ial scanning calo ime y (DSC) using a Pe kin-Elme DSC7 In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 9 unde a gon low. Speci ic sa u a ion magne iza ion,  S was measu ed in a Lakesho e 7407 ib a ing sample magne ome e (VSM), applying a maximum ield o 1.5 T. 3.2 Calo ime y Figu e 5 shows he DSC scans o he alloy a e milling 150 h a di e en o a ional speeds. Bo h samples milled a 250 and 350 pm exhibi simila cu es, showing a b oad exo he mic maximum gene ally asc ibed o elaxa ion and c ys al g ow h e ec s [1]. This ac implies ha he ene gy eleased a e hea ea men is no p opo ional o he amoun o ene gy supplied by he milling media, as milling a 350 pm mus be mo e ene ge ic han milling a 250 pm. On he o he hand, he sample milled a 150 pm exhibi s, along wi h he b oad exo he m, an endo he mic peak. This endo he m is asc ibed o he e ogenei ies obse ed in he powde ; as i will be shown la e by XRD and MS echniques, se e al c ys alline phases coexis in he powde o med a e 150 h milling a 150 pm. A de ailed discussion on he he mal e olu ion o milled samples as a unc ion o he milling ime will be epo ed elsewhe e [ 12 ] as i is ou o he scope o he p esen pape . 3.3 Powde pa icle size and composi ion SEM images we e used o measu e he a e age powde pa icle size, <d>, (s a is ic o e an a e age o 200 pa icles pe sample) as a unc ion o ime and o a ional speed. Figu e 6.a shows he alues o <d> as a unc ion o ime (uppe panel) and he equi alen ime de ined by equa ion (9) (lowe panel). Fo samples milled a 150 pm, powde pa icle size keeps almos cons an (~15 m) o low milling imes o dec ease o longe imes down o a cons an alue o ~5 m. Fo samples milled a 250 pm, <d> s a s dec easing a e y low milling imes and he same cons an alue, ~5 In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 16 Table 2 Milling ime anges o he h ee ypes o MS i ings pe o med: a) single si e a HF=33 T, b) si e a 33 T plus wo magne ic hype ine ield dis ibu ions and c) using a single magne ic hype ine ield dis ibu ion.  ( pm) [/0]3 a) single HF=33 T b) HF=33 T +HF dis ibu ion c) single HF dis ibu ion (h) eq (h) (h) eq (h) (h) eq (h) 150 1 up o 10 20 o 150 >150 250 4.6 up o 1 4.6 5-20 23-92 50 230 350 12.7 none 2-5 25.4-63.5 10 127 In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 17 Figu e cap ions Figu e 1. Scheme o he h ee e e ence sys ems used. Figu e 2. De achmen dis ance be ween ball and ial wall du ing ligh o 150, 250 and 350 pm, he c osses show he momen o collision. The inse shows he linea ela ionship be ween he ime o ligh , , and he o a ional pe iod o he main disk, T. Figu e 3. Le ; ine ial e e ence sys em: solid line, ball ajec o y, dashed line, ajec o y o he cen e o mass o he ial, do ed ci cles show he ial posi ion a he de achmen and collision e en s. Righ , non-ine ial sys em: Ci cle ep esen s he ial; hick line, ball ajec o y; s aigh solid lines indica e he de achmen poin o he ball om he wall a he de achmen ime, PD, and a he collision ime, PD’; PC indica es he collision poin o he ball;  is he angle be ween PD’ and PC. Figu e 4. No mal o ce exe ed o e he ball (con inuous line) and analy ical solu ion o equa ion 6 (do ed line) du ing one pe iod o he main disk o a ion calcula ed o =150 pm. Figu e 5. DSC scans o Fe75Ge20Nb5 samples a e 150 h milling a di e en equencies. Figu e 6. a) Powde pa icle size as a unc ion o ac ual ime and equi alen ime. b) A e age C con en as a unc ion o ac ual ime and equi alen ime. Figu e 7. XRD pa e ns o samples a e di e en milling imes a di e en milling in ensi ies. C osses, bcc-Fe; squa e, cc-Ge; ci cles, bcc-Nb; iangle, in e me allic. Figu e 8. a) Angula posi ion, 2, and b) ull wid h a hal maximum, FWHM, o he (110) di ac ion maximum o he -Fe(Ge,Nb) phase as a unc ion o ac ual ime (abo e) and equi alen ime (below). c) A ea ac ion o cc Ge and bcc Fe phases Figu e 9. Mössbaue spec a o samples a e di e en milling imes a di e en equencies. In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 18 Figu e 10. a)A e age hype ine magne ic ield, <HF>, and b) sa u a ion magne iza ion o he di e en s udied samples as a unc ion o ac ual ime (abo e) and equi alen ime (below). In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 19 Figu e 1     e eR u u R P j i p In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 20 Figu e 2 0.0 0.5 1.0 0.0 0.1 0.2 0.00 0.02 0.04 0.06 -4 -3 -2 -1 0 1 2 3 4 5 350 pm 250 pm |R-p|2- 2 (10-4 m2) ime (s) 150 pm (s) T (s) In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 21 Figu e 3. 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 Y (m) X (m) -0,02 0,00 0,02 -0,02 0,00 0,02 PD'PC PD Y (m) X (m)  In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 22 Figu e 4. 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 -10 0 10 20 30 40 50 60 N/m (N/kg) ime (s) ime/T In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 23 Figu e 5. 400 500 600 700 800 900 350 pm Tempe a u e (K) 250 pm Exo (2 W/g) 150 pm dH/d In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 24 Figu e 6. 110 100 1000 0 5 10 15 110 100 1000 0 5 10 15 150 pm 250 pm 350 pm <d> (m) milling ime (h) a) <d> (m) ·[]3 (h) 110 100 1000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 110 100 1000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 b) C con en (%) milling ime (h) C con en (%) ·[]3 (h) In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478 h p://dx.doi.o g/10.1016/j.in e me .2007.12.011 25 Figu e 7. 30 40 50 60 70 1 h 2 (deg ees) 5 h 10 h 20 h 50 h 100 h 150 pm 150 h 30 40 50 60 70 1 h 2 (deg ees) 5 h 10 h 20 h 50 h 100 h 250 pm 150 h 30 40 50 60 70 2 h 2 (deg ees) 5 h 10 h 20 h 50 h 100 h 350 pm 150 h