On the concentration dependence of the surface tension of liquid metallic alloys Theoretical basis Hungary, Miskolc, Egyetemvaros.

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1 On the cncentratn dependence the surace tensn lqud metallc allys heretcal bass 1 G.Kaptay, kmkap@gld.un-msklc.hu 2 Z.Papp zltanp@da.ed.ac.uk 1 Pressr at the Department Physcal Chemstry the Unversty Msklc 515 ungary, Msklc, gyetemvars 2 tudent at Unversty dnburgh, Dvsn nrmatcs ctland, UK bstract In the present paper a revsed versn Butler equatn s derved t calculate the surace tensn n mult-cmpnent lqud metallc slutns. he eects surace actve cmplees ntermetallc cmpunds n lqud metallc slutns s als dscussed. 1. Intrductn urace tensn s ne the basc thermdynamc prpertes lqud metallc allys, determnng ther behavr n derent technlgcal prcesses. urace tensn s determned by usual parameters state temperature and cmpstn. lthugh there are very ew epermental results n the cncentratn dependence the surace tensn multcmpnent allys, r cmple mdeg derent materals technlges ths nrmatn s bvusly needed. herere there s a need r a mdel, by whch the surace tensn mult-cmpnent metallc allys can be related t bulk thermdynamc prpertes. uch a mdel can serve as a bass r an nteracal mdule n cmple thermdynamc stwares. In the present paper such a mdel wll be derved rm basc thermdynamc prncples, and cmpared t the estng mdels n the lterature. Frst, let us cnsder the mdels gven n the lterature s ar, and then, let us derve ur wn equatn. 2. Mdels presented n the lterature he surace tensn mult-cmpnent slutns s tradtnally nterpreted n terms the Gbbs adsrptn stherms. Fr practcal use, hwever, the methd develped by Butler based n Gbbs s much mre user-rendly [1]. ccrdng t Butler, the partal surace tensn cmpnent n a mult-cmpnent slutn can be wrtten as: a 1 ω a where s the surace tensn pure cmpnent at the same temperature and pressure, gas cnstant, abslute temperature, ω partal mlar surace area cmpnent, a and a - the actvty cmpnent n the bulk and n the surace phase the slutn, respectvely.

2 he surace phase wll be n equlbrum, all the cmpnents wll have the same partal surace tensn values,.e. the llwng equalty s ullled between cmpnents and j: 2 j Fr an n-cmpnent slutn n-1 equaltes such as q.2 shuld be ullled. I the cmpstn the bulk slutn s gven, rm thse n-1 equatns the surace cmpstn can be btaned, as t s characterzed by n-1 ndependent varables. Fr perrmng such a prcedure, the llwng thermdynamc values are needed:. the surace tensn pure cmpnents as unctn temperature,. the actvtes n the bulk phase, epressed thrugh ts cmpstn and,. the actvtes n the surace phase, epressed thrugh ts cmpstn and, v. the partal mlar surace area values epressed thrugh the surace cmpstn and. Frm the abve lst ntal parameters ne can see that the new nrmatn needed r the evaluatn the cncentratn dependence the surace tensn s ω and a. Instead actvty, usually the actvty cecent cmpnent n the surace s epressed, dened as γ a /. hese values are usually epressed n the lterature usng the llwng apprmatns [2-10]: ω N 2 / 1 / v γ β γ 4 where s the gemetrcal actr, usually used as taken equal t the value derved r the 111 plane sld cc crystals, the partal mlar vlume cmpnent n the slutn r smplcty usually taken equal t the mlar vlume pure cmpnent, N v the vgadr number, β - a sem-emprcal parameter, usually dened as the rat the brken bnds at the surace. In derent mdels parameter β s taken equal t derent values r descrbng the surace tensn lqud metallc slutns. ar and Melrd [2] used a value between β 0.5 and t descrbe n-pb and Pb-In systems. Mnma and ud [] appled a cecent between β 0.80 and 0.84 r Cu-N and N-M systems. peser at al. [4-5] appled β 0.75 t descrbe Fe-Cu, Cu-Pb, n-pb, g-pb, Pb-In, B-g, Cu-l, Fe- and N- systems. he same cecents β 0.75 was used by Lee et al [6] r the Fe-N system. anaka et al. studed the nluence parameter β n the nterval between 0.5 and 1 n the nal results r g-pb, Cu-Pb, n-pb, Cu-Fe, Cu-l, N- and varus Fe-allys [7]. In the later paper anaka et al [8] the sem-emprcal cecent β 0.8 was suggested rm the analyss the dependency the surace tensn pure lqud metals n ther enthalpy evapratn, and ths cecent was appled t the Cu-Pn and Fe- systems. he cecent β 0.8 suggested by anaka et al [8] was successully appled by Mser et al. t descrbe the g-n [9] and the Pb-n [10] systems.

3 . he mdel develped n the present paper In ur mdel the Butler cncept q-s 1-2 wll be used as a bass, but t wll be altered by cnsderng separately the enthalpy and the entrpy parts. q. als remans vald, but the value cecent wll be altered. he thery wll be re-cnstructed rm the very begnnng, startng wth the mdel r the surace tensn ne-cmpnent lqud metals..1. he surace tensn ne-cmpnent lqud metals By dentn, the partal ecess surace Gbbs energy cmpnent n the pure phase can be btaned as the derence between the Gbbs energy the surace and bulk phases: ω G G 5 he mlar surace area pure cmpnent can be wrtten n an analgus way wth q.: 1 / 2 / N v ω 6 he enthalpy and entrpy derences between the surace and bulk phases n pure lqud metals can be wrtten as: 1 β 7 c 8 0 where c s the chesn enthalpy a pure lqud metal, beng apprmately equal t the cndensatn enthalpy r a mre advanced dentn see [11], β z/z, the rat the crdnatn numbers n the surace and bulk phases, and are the mlar vlumes pure cmpnent n the surace and bulk phases, respectvely. ccrdng t ur prevus wrks [12-1], pure lqud metals have a surace structure clse t the 111 plane the cc crystal,.e. z 9. Cmbnng ths value wth the average crdnatn number n pure lqud metals z 11, β 9/ Frm the same surace mdel the value parameter n q.6: 1.06 [12-1]. he entrpy derence n q.8 apprmately equals 4 J/mlK [12-1]. he cmbnatn these theretcally establshed parameters prvde a perect match wth the epermentally determned surace tensn values pure lqud metals [12-1]. Let us mentn, that these values are clse t thse appled by anaka et al [8] β 0.8, 1.091, entrpy cntrbutn neglected. ubsttutng q-s 6-8 nt q.5 the nal epressn r the surace tensn pure cmpnent can be wrtten as: 1 β c 9 2 / 1 / N v

4 .2. he partal surace tensn cmpnent n the mult-cmpnent lqud metallc slutn Fr a partal surace tensn cmpnent n the mult-cmpnent lqud slutn q.5 can be re-wrtten, but wth mttng the subscrpt, reerrng t the pure phase: G G ω 10 he partal mlar surace area cmpnent can be wrtten by q.. wever, we take 1.06, beng equal t the value derved r the pure lqud metal, and beng ndependent cncentratn. he partal mlar vlume s dened as: 11 where s the ecess partal mlar vlume cmpnent, havng the cncentratn dependence vald r the bulk phase, but beng a unctn the cmpstn the surace phase. he enthalpy derence s cmpsed that n the pure phase, takng nt accunt the partal mng enthalpy terms. he partal mng enthalpy the surace phase s a unctn the surace cmpstn, and s reduced by cecent β, t take nt accunt the reduced crdnatn number at the surace: 1 c β β 12 he entrpy term wll nclude the term gven abve r the pure phase, the cnguratnal entrpy term and the partal ecess mng entrpy term. Nn these quanttes shuld be multpled by a crrectn actr β r any ther cecent, at least, n the rst apprmatn: 0 1 ubsttutng q-s 11-1 nt q.10, and takng nt accunt q.9, the llwng nal result s btaned: 1 / 2 / 2 / 1 v N β 14 e-arrangng q.14 t be alke the Butler equatn: a a 2 / 1 ω 15

5 wth the dentn γ : γ β 16 One can see that q-s are derent rm q-s 1, 4. I q-s are cmbned wth q.2, the cncentratn dependence surace tensn can be calculated, suppsng that the surace tensn and mlar vlume pure cmpnents are knwn as unctn temperature, and all the ecess unctns,, are knwn as unctn cmpstn the bulk lqud metal and temperature. Generally, n analytcal slutn ests r ths prblem, and thus usually numercal methds are appled see [14]. he analytcal slutn can be und nly r the bnary deal slutn -B, the tw cmpnents have the same mlar vlume. hen, the cmpstn the surace phase and the surace tensn can be calculated as: k 1 17.a [ k 1 ] ω 17.b ω k ep B 17.c B 1 0, ,5 1 B sgma J/m2 1,5 1, 1,1 0,9 0,7 0,5 0 0,5 1 B Fg.1. he cncentratn dependence the surace cmpstn let and surace tensn rght the bnary deal slutn 0.5 J/m 2, B 1.5 J/m 2, B 10-4 m /ml curves rm bttm t tp crrespnd t 500 K, 1,000 K and 1,500 K he cncentratn dependence the surace cmpstn and the surace tensn a bnary deal slutn are gven n Fg 1. One can see that even n an deal slutn there s a great tendency t segregatn a cmpnent wth a less surace tensn value cmpnent t the surace phase, leadng t a very nn-ear cncentratn dependence the surace tensn. s ne can see rm Fg.1, ths tendency decreases wth temperature, as at hgher temperature the rle bulk cnguratnal entrpy s ncreased.

6 4. On the rle ntermetallc cmpunds n the cncentratn dependence s s bvus rm phase dagrams, n many metallc phases ntermetallc cmpunds can rm. me thse cmpunds melt cngruently, and thus bvusly reman stable even n a lqud state, especally, there s a sgncant nn-metallc bnd ests between the cmpnents. uch systems are ten descrbed n the lterature usng the asscated slutn mdel [15-18]. I there s a sgncant part nc bndng n the ntermetallc cmpund, ts wn surace tensn s usually belw the surace tensns ts cnsttuent pure metals [12]. In these cases, mnmum pnts n the cncentratn dependence the surace tensn are epected. wever, n mnmum pnts can be descrbed by the rmalsm gven abve. In rder t descrbe mathematcally the systems wth asscates, an asscate slutn mdel shuld rst be appled. hen, n the bnary system, at least three quas-cmpnents the tw mnmers and at least ne asscate wll est. he cncentratn dependence such a system can be descrbed nly, the surace tensn the pure lqud ntermetallc s knwn, and the system s treated as a -cmpnent system. Fr ths rst the equlbrum bulk cncentratns the three cmpnents shuld be und rm bulk thermdynamc values see [15-18]. I the surace tensn the lqud ntermetallc s nt knwn, t can be estmated by a ttng prcedure, usng epermental data. Cnclusns he Butler equatn has been revsed n the present paper t descrbe mre crrectly the cncentratn dependence n mult-cmpnent lqud metals. Calculatns are perrmed n [14]. he hypthess s gven abut the pssble rle ntermetallc cmpunds n the mnmum pnts bserved n the cncentratn surace tensn curves. Numercal calculatns wll be perrmed n the uture t prve ths hypthess. Lterature 1. Butler J...: Prc.y.c., ar.p., Melrd D.., rans. Faraday c Mnma K., ud., J.Japan Inst. Metals, and peser., Prer D.., Yeum K., crpta Metall., Yeum K., peser., Prer D.., Metall. rans. B., 20 B Lee.K., Frhberg M.G., ajra J.P., teel research, anaka., Ida., teel research, anaka., ack K., Ida., ara., Z.Metallkunde, Mser Z., Gasr W., Pstrus J, J. Phase qulbra, Gasr W., Mser Z., Pstrus J., J. Phase qulbra, G.Kaptay, G.Cscsvszk, M..Yaghmaee, Materals Wrld e-jurnal, accessble at: July, G.Kaptay: hess r dr. habl, Unversty Msklc, 1998, 51 pp. 1. G.Kaptay,.Báder, L.Blyán, Materals cence Frum, vls Z.Papp, G.Kaptay see ths vlume 15. K.Wasa, K.Muka, J. Japan Inst. Metals, F.mmer, Z. Metallkunde, and G.Kaptay,.Berecz, Chemcal Papers, M..Yaghmaee, G.Kaptay, G.Jánsy, Materals cence Frum,