Citation Zeitschrift für Metallkunde. 92(11)

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1 Title Themodynamic valuation of Phase Diagams Nano-Pa utho(s) Tanaka, Toshihio; Haa, shigeta Citation Zeitschift fü Metallkunde. 92(11) Issue 21 Date Text Vesion publishe URL DOI Rights Cal Hanse Velag, München Osaka Univesity

2 T. Tanaka, S. Haa: Themodynamic valuation of Nano-Paticle inay lloy Phase Diagams Toshihio Tanaka, Shigeta Haa Depatment of Mateials Science and Pocessing, Gaduate School of ngineeing, Osaka Univesity, Osaka, Japan Themodynamic valuation of Nano-Paticle inay lloy Phase Diagams Nano-paticle binay alloy phase diagams have been evaluated fom the infomation on the Gibbs enegy and the suface tension of the bulk size on the basis of the egula solution model. s the size of the paticle deceases, the liquidphase egion is enlaged in those binay phase diagams, in othe wods, the liquidus tempeatue deceases. ffect of the size of the paticle on the phase equilibia is emakable when the excess Gibbs enegy is positive and its absolute value is lage in solid and liquid phases. Keywods: Suface, tension; Regula solution model; Solidliquid equilibia; Nano-paticle; Themodynamics 1 Intoduction It has been known that the melting point of pue metals deceases as the size of the metal paticles becomes smalle [67Wo, 71Sam, 72Coo, 76uf, 8ll, 86ll, 88Sak, 91Sas] since Takagi found this phenomenon fo pue Pb and i [54Tak]. Themodynamic evaluation has been also caied out to investigate the effect of the paticle size on the melting point of pue metals [9Paw, 48Rei, 6Han, 77Cou]. In ou pevious wok [1 Tan2], we evaluated the binay phase diagams of small-size paticle systems when the phase diagams consist of liquid phase and pue solid phases, fo example, Cu-Pb, Cu-i and u-si alloys. Fom the infomation on the composition and tempeatue dependence of the suface tension of the liquid phase, the nano-paticle binay phase diagams can be calculated with the themodynamic databases, which contain a lot of infomation on the Gibbs enegy to evaluate the phase diagams of the bulk size. On the suface tension, the authos had aleady applied those themodynamic databases to the evaluation of the suface tension of liquid alloys and molten ionic mixtues, and the intefacial tension between liquid steel and molten slag [94Tan, 96Tan, 98Tanl, 98Tan2, 98Tan3, 99Tanl, 99Tan2, 99Tan3, 99Ued, OlTanl, 1Tan2]. Fom the above evaluations on the small-paticle binay phase diagams [1 Tan2], we found 1. The liquid phase egion of phase diagams is enlaged as the size of the paticle becomes smalle. The change in the phase diagams with the paticle size is emakable when the size of the paticle is below 1 nm. 2. In alloys, of which the composition dependence of the suface tension of the liquid phase shows lage downwad cuvatue, such as Cu-Pb and Cu-i alloys, the size of the paticle influences lagely the phase elations. On the othe hand, in u-si alloys, of which the suface tension of the liquid phase changes smoothly with the composition, the effect of the size on the phase elations is not so lage. The composition dependence of the suface tension is mainly detemined fom the excess Gibbs enegy of the liquid phase. In the pevious wok [1Tan2], we focused on some binay alloys, of which phase diagams consist of liquid phase and pue solid phases, and the solid solutions have not been consideed yet. We, howeve, need to make clea the effect of the excess Gibbs enegy on the nano-paticle binay phase diagams including solid solutions moe deeply. Pelton and Thompson [75Pel] have evaluated the binay phase diagams in the bulk size systems by using the egula solution model when vaious values of the inteaction paamete of the model in both liquid and solid phases wee selected to elucidate the effect of the excess Gibbs enegy on the binay phase diagams, in which the melting points of the components ae fixed. The pupose of the pesent wok is to examine the effect of the excess Gibbs enegy in both liquid and solid phases on the nano-paticle binay alloy phase diagams on the basis of the egula solution model. 2 Themodynamic quations -solid phase equilibia ae evaluated in the pesent wok. When a pue solid phase is selected as the efeence state of Gibbs enegy, the total Gibbs enegies in liquid and solid phases, ilgtotal,liq and ilgtotal,soi, of an alloy system in a small paticle with its adius ae descibed in the following qs. (1)-(6) [1Tan2]: LlGTotal,Liq = LlGulk,Liq + LlGSuface,Liq (1) LlGTotal,Sol = LlGulk,Sol + LlGSuface,Sol (2) The Gibbs enegies of the bulk of an - binay alloy in liquid and solid phases, ilgulk,liq and ilgulk,soi in qs. (1) and (2), which coespond to ilgtotal,p (P = Liq o Sol) with = oo, ae expessed in qs. (3) and (4). LlGulk,Liq N ilgs + N LlGs + axcess,liq LlGulk,Sol = axcess,sol + RT(N lnn + N lnn) (4) 1236 Cal Hanse Velag, Mtinchen Z. Metallkd. 92 (21) 11

3 whee ilgs and ilgsae Gibbs enegies of pue components and in liquid phases elative to those of pue solid phases, in othe wods, the Gibbs enegy of the melting. axcess,liq and axcess,sol ae the excess Gibbs enegies of liquid and solid phases in the - alloy, N and N ae the mole factions of components and, and R is the gas constant. ilgsuface,liq and ilgsuface,sol in qs. (1) and (2), the effect of the smface on ilgtotai,p, ae assumed as follows [1Tan2]: 2a.Liq yliq ilgsuface.liq = (5) CJiq = (T- T,mp)/Nm- 1 (T,mp = 12K) (13) ciq =.6-.1 (T- T,mp)/Nm- 1 (T,mp = 6 K) (14) Fo the suface tension c 1 of pue solid X, the following equation was used in the pevious wok [1Tan2]: "' Liq Sol Liq ucjx ( ) cx = 1.25 cx,mp +---at T- Tx,mp (X= o ) (15) Z. Metallkd. 92 (21) 11 (12) whee ci,i;,p and Vx,mp ae the suface tension and the mola volume of the element X at its melting point. When we select Vx,mp = 1 X 1-6 m\nol- 1, as descibed below, the following ough elation is obtained fom q. (11): (JLiq (Nm-1) = 4.8 X 1-8R Tx,m/K (11) X,mp {Vx,mp/m3 moi-l} 2/3 whee Tx,mp and Sx,mp ae the melting point and the entopy of fusion fo pue substance X (X= o ). In the pesent wok, Sx,mp is oughly assumed to be 1 J K- 1 mol- 1 accoding to Richad's ule [95Gas]. In addition, we select hee T,mp = 12 K and T,mp = 6 K. Fo the suface tension of pue liquid metals at thei melting points, the following appoximation has been epoted [88Iid] : ilgs = T,mp S,mp - T S,mp (1) (9) whee Q and Qi ae the inteaction enegies in liquid and solid phases. Since Pelton and Thompson [75Pel] assumed the simple equation of the Gibbs enegy of the melting in thei evaluation of phase diagams, we also use the following equations as they applied: G xcess,sol _ N N QSoJ - (8) 2a.Sol ysol ilgsuface.sol = 2(NCJolyiol + NCJolyol) whee is the adius of a paticle, cliq and csoi ae the suface tensions of liquid and solid alloys, yliq and ysoi ae the mola volumes of liquid and solid alloys, c 1 and ifs 1 ae the suface tensions of pue solid and, and Vi 1 and V 1 ae the mola volumes of pue solid and. lthough the excess Gibbs enegies stoed in themodynamic databases, which ae used fo the calculation of phase diagams of the bulk, wee used in the pevious wok [1 Tan2], the egula solution model is applied in the pesent wok because vaious inteaction enegies can be selected easily as follows: G xcess,liq _ N N QLiq - (6) (7) T. Tanaka, S. Haa: Themodynamic valuation of Nano-Paticle inay lloy Phase Diagams 1237!_ G,P,u1k(T N ) ' (22b) RT NSuf 1 CJP = ifn +-ln--+-g,p,suf(t NSuf) N ' 1 G,P,ulk(T N ) ' (22a) The suface tensions ofliquid and solid alloy, cp (P = Liq o Sol), ae evaluated fom utle's equation [32ut] as follows [96Tan, 98Tan1, 98Tan2, 98Tan3, 99Tan1, 99Tan2]: RT NSuf 1 if= CJP +-In--+_ G,P,Suf(T NSuf) N ' (21) (2) It is assumed that the mola volumes of liquid alloy yliq and solid alloy ysol in qs. (5) and (6) ae obtained fom the following simple additivities: (19) Fo the mola volume, we used the following values by assuming that the tempeatue dependence of the mola volume and the volume change due to the melting ae neglected because the effect of the excess Gibbs enegy and the suface tension on the phase equilibia is focused in the pesent wok. v;iq = Viq = 1 X 1-6 jm 3 moc 1 (18) c 1 = 1.25 x.6-.1 (T- T,mp)/Nm- 1 (17) In q. (15), we assumed the following: 1. The value of c 1 of pue solid metals at the melting point is found to be 25 % lage than that of pue liquid metals on the aveage. 2. The tempeatue pependence of c 1 is assumed to be the same as that of cx 1 q. 3. The effects of cystal faces on c 1 ae ignoed. Fom qs. (13)-(15), the following elations ae obtained: ci 1 = 1.25 x (T- T,mp)/Nm- 1 (16) In addition, the tempeatue coefficient of the suface tension of pue liquid has been epoted to be about.1 Nm - 1 K -I [881id]. Thus, we assumed the following equation fo ciq and ciq in the pesent wok:

4 T. Tanaka, S. Haa: Themodynamic valuation of Nano-Paticle inay lloy Phase Diagams utle deived the above equation, assuming that the outemost monolaye of a mateial is the hypothetical "suface". In qs. (22a) and (22b), Nuf is the mole faction of element in the suface consideed by utle [32ut]. x = LN6 13 (Vkf/ 3 (No =vogado numbe; X = o; L = 1.91) is the mola suface aea of Rue X, and this is obtained fom the mola volume vk. Gx'P,ulk(T, N) and G,P,Suf (T, Nuf) ae the patial excess Gibbs enegies of component X in the bulk and the suface, espectively, as functions of T and N o Nuf (P = Liq o Sol). GP,ulk(T, N) can be obtained fom the following elations: agxcess,p agxcess,p G,P,ulk(T N ) = Gxcess,P + (1 _ N ) ' an (23a) (23b) Fo the excess Gibbs enegy in the suface G P,Suf (T, Nuf), we deived the following equations [96Tan, 98Tanl, 98Tan2, 98Tan3, 99Tan1, 99Tan2, 99Tan3, 99Ued, 1Tan1, 1Tan2] based on the model poposed by Yeum et al. [89Yeu]. (24) quation (24) means that G P,Suf (T, Nuf), which has the same fomula as GP,ulk(T, N), is obtained by eplacing N by Nuf in G,P,ulk(T, N) (X= o ), and then multiplying pmix to G,P,ulk(T, Nuf). pmix is a paamete coesponding to the atio of the coodination numbe in the suface to that in the bulk consideing the suface elaxation [96Tan,98Tan1,98Tan2,98Tan3,99Tan1,99Tan2,99Tan3, 99Ued]. Fo the solid solutions, we assume the following value as pmxby assuming the close-packed stuctue in the pesent wok because the coodination numbe in the bulk is 12 and that in the suface is 9: Table 1. Paametes used in the calculation of phase diagams. Substance Melting point ntopy of fusion K JK- 1 mol QLiq kj 3 2 Fig. 3a 1 Fig. 3b 15 The suface tension ap of liquid o solid alloy can be calculated fom qs. (22), (24)-(26) as follows: 1. Setting tempeatue Tand composition N of a solution. 2. Inseting the values fo suface tension a and mola volume Vk (P = Liq o Sol) of pue substances at the above tempeatue in qs. (22a) and (22b). 3. Detemining the excess Gibbs enegies in the bulk phase at the above tempeatue and composition, and substituting them in qs. (22a) and (22b). 4. Then, qs. (22a) and (22b) become the simultaneous equations with unknown Nuf and ap. These equations ae solved fo those unknown Nuf and ap numeically. 3 Calculation of Phase Diagams in Nano-Paticle Systems -solid phase equilibia ae obtained fom the following themodynamic conditions : Total,Liq Total,Sol li = li (27) Total,Liq Total,Sol li =li (28) ailgtotal,p Total,P - ilgtotal,p - N (P = Liq o Sol) li - an (29) ailgtotal,p Total,P = ilgtotal,p + ( 1 _ N ) (P = Liq o Sol) li an (3) Table 1 summaizes the paametes used fo the pesent calculation of the phase diagams. s shown in this table, the melting points of pue components and ae fixed to be 12 and 6 K, espectively. The entopy of melting is also fixed to be 1 J K- 1 mol- 1. The suface tension and the mola volume of pue components ae selected in qs. (13)-(19) as mentioned above. The inteaction paametes Q and Q ae changed fom -2 to 3 kj as shown in Figs. 2 and 3 which ae descibed below in detail. Suface Tension Mola volume Nm- 1 m 3 mol (T -12): fo Liq 1 X 1-6 : fo Liq 1.2 x l(T -12): fo Sol 1 X 1-6 : fo Sol.6-.l(T -6): fo Liq lox 1-6 : fo Liq.6 x l(T -6): fo Sol 1 X 1-6 : fo Sol Q11kJ Figs. 2c, 3c Fig. 2b Fig. 2a Fig. 3d Fig. 2d 1238 Z. Metallkd. 92 (21) 11 G,P,ulk(T N ) = Gxcess,P _ N ' an pmix =.83 : fo liquid alloys (25) pmix = 9/12 =.75: fo solid alloys (26)

5 T. Tanaka, S. Haa: Themodynamic valuation of Nano-Paticle inay lloy Phase Diagams ::J " ulk (a) Melting point of substance The change in the melting point of the pue substance with adius of the paticle can be evaluated fom the following q. (31), which is obtained fom qs. (1), (3) and (5) fo the pue substance X: GTotal,Liq = Gulk,Liq + GSuface,Liq (31) 6 58 ::J ttl Q; a Radius of paticle I nm ulk (b) Melting point of substance Radius of paticle I nm Fig. 1. Change in the melting point of pue substances (a) and (b) with the paticle adius. The tempeatue Twhich gives GTotal,Liq = in q. (31) is the melting point of pue X at a given adius of a paticle. The size dependence of the melting points of the substances and, of which physical popeties ae given in qs. (9)-(19), is shown in Figs. la and b. The change in the melting point with the paticle size fo pue u calculated fom the above q. (31) agees well with the expeimental esult [71Sam], as descibed in ou pevious wok [1 Tan2]. Usually, pue metals have the following ode of Gs, a and V (P = Liq o Sol): Gs: about J mol,.,.p. "X about 1 Nm- 1 Vx P. about 1 X X 1-6 m 3 mol-l Thus, when the adius of the paticle is aound 1 x X 1-9 m, in othe wods, is of nano-size ode, csuface,p has the same ode of Gulk,P, which means that the effect of the size of the paticle on the phase equilibia is noticeable when deceases below 1 nm. 12 (c) 1... ::I «i ,., 1 \.... \ ulk Cl,) I " :::l : nm L_ :.:... :::::: ! ulk (b) ulk (d)... ::I 8 6 f ::-.. ;: :: :::... : 1... ::I «i _... _.._.._.._::::: Fig. 2. Calculated esults ofthe phase diagams with systematic change of Q and Q on the basis of the egula solution model (a-d). Z. Metallkd. 92 (21)

6 T. Tanaka, S. Haa: Themodynamic valuation of Nano-Paticle inay lloy Phase Diagams Figues 2 and 3 show the calculated esults of the phase diagams by using the paametes in Table 1. In these figues, the solid cuves indicate the phase equilibia in the bulk ( = oo ). On the othe hand, the chain and the dotted cuves ae the calculated esults fo = 1 nm and 5 nm, espectively. Figue 2a indicates the phase diagams fo the ideal solutions in both solid and liquid phases, in othe wo d s, Q L!q =. QSol =. Wh en QSol mceases f om to 3 kj with Q'i =, the phase diagams change as shown in Figs. 2b and Figue 2d shows the phase diagams fo Qt = -2 kj and Ql = -15 kj as an example fo Q'i < and Ql <.. When Ql is fixed to be 3 kj and Q'i changes fom +2 to -2 kj, the phase diagams ae obtained as shown in Figs. 3a-d. s can be seen in Fibo. 3, when the inteaction L" S paametes ae Q 'i and Q, the effect of the paticle size on the phase diagams is emakable. s descibed in the pevious woks [94Tan, 96Tan, 98Tan1, 98Tan2, 98Tan3, 99Tanl, 99Tan2, 99Tan3, 99Ued, 1Tan1, 1 Tan2], the liquid o solid alloys indicate the lage downwad cuvatue of the composition dependence of the suface tension in the conditions of Q'i and Ql. This composition dependence of the suface tension affects the contibution of,1gsuface,p to,1gtotal,p. specially, when Ql, in othe Y.ods, the excess Gibbs enegy of the solid solution is positive and its absolute value is lage, the solid solutions do not appea in some of the phase diagams of the bulk as shown in Figs. 2c, 3c and 3d. Howeve, when the paticle size deceases, the contibution of,1gsuface,sol to,1gtotal,sol in the solid phase cannot be ignoed. Consequently, as shown in Figs. 2 and 3, the solid solution appeas in the small-paticle systems even when the bulk phase diagams do not show the solid solutions. In addition, the solid solution egion is enlaged as the size of the paticle becomes smalle. Theefoe, the smallpaticle binay phase diagams in Cu-Pb, Cu-i and u-si alloys obtained in ou pevious wok [1Tan2] might be coected by consideing the solid solutions although the solid solutions ae not consideed in the pevious evaluations of those phase diagams. Howeve, we need the exact infomation on the excess Gibbs enegy of the solid solutions of those alloys, which can be ignoed fo the bulk phase diagams. In addition, we have to pay caeful attention to the pocedue to evaluate the suface tension a 8 of the solid solution although q. (26) is used to evaluate a 8 in the pesent wok. We have aleady confimed the validity of the pocedue to calculate the suface tension al of liquid alloys by compaing the calculated esults of al with expeimental values [94Tan, 96Tan, 99Tan1, OlTanl], but q. (26) fo a 8 is only a tial as an appoximation. s descibed above, the phase diagams of binay alloys in the nano-paticle systems can be evaluated fom the infomation on the Gibbs enegy and the suface tension of the bulk phase although the following ough assumptions have been still used in the pesent wok: 1. Quantum effect on the nano-paticle is not consideed. 2. The effect of the cuvatue of the paticle on the suface tension is neglected although it has been epoted that.._ ::::1 «i.._ 1-4 (a) 12 (c)... 1 \... ulk <I)... S... I "., ::::1, «i a; 8 : snm :::.:-....:::::: i (b).''-. 1 \I -,_ ulk ' ' :: -... :... 1nm 8 -sn;.:-----._ : i ::" 6. --: if=::::::-:.:::::-:::::::::::-:::::::::::.::-:::-::::::::.::-:::-:.::-:= ulk 1 ooo -... _,..._ \ 5nm.. 8 l -... (d) I 2oo o Fig. 3. Calculated esults of the phase diagams with systematic change of.q. and.q = 3 kj on the basis of the egula solution model (a-d). 124 Z. Metallkd. 92 (21) 11

7 the value of the suface tension is influenced by the cuvatue of the suface in a small paticle especially below = 5 nm [49Tol, 79Mo, 89Gla]. 3. The effects of cystal faces on the suface popeties of solid phases ae ignoed. 4. The elation between the suface tension of solid at the melting point and that of liquid as well as the tempeatue dependence of the suface tension of solid is assumed to be common fo all kinds of substances. 4 Concluding Remaks ll llen, G.L.; Gi1e, W.W.; Jesse, W..: cta Metall. 28 (198) Kum Kumikov, V.K.; Khokonov, Kh..: J. ppl. Phys. 54 (1983) ll llen, G.L.; ayles, R..; Gile, W.W.; Jesse, W..: Thin Solid Films 144 (1986) Iid Iida, T.; Guthie, R.I.L.: The Physical Popeties of Metals, Claendon Pess, Oxfod (1988) 71 and Sak Saka, H.; Nishikawa, Y.; Imua, T.: Phil. Mag. 57 (1988) Z. Metallkd. 92 (21) 11 9Paw 32ut 48Rei 49Tol 54Tak 59 ll 6Han 62Swa 67Wo 71Sam 72Coo 72Wit 75Pel Pawlow, P.: Z. Phys. Chen. 65 (199) utle, J..V.: Po Roy. So 135 (1932) Reiss, H.; Wilson, L.: J. Colloid Sci. 3 (1948) Tolman, R.C.: J. Chim. Phys. 17 (1949) Takagi, M.: J. Phys. So Jpn. 9 (1954) llen,.c.; Kingey, W.D.: Tans. Metal!. So IM 215 (1959) Hanszen, K.J.: Z. Phys. 157 (196) Swalin, R..: Themodynamics of Solids, John Wiley, New Yok (1962). Wonski, C.R.M.: it. J. ppl. Phys. 18 (1967) Sambles, J.R.: Po Roy. So Lond. 324 (1971) Coombes, C.J.: J. Phys. F 2 (1972) Wittenbeg, L.J.; DeWitt, R.: J. Chen. Phys. 56 (1972) Pelton,.D.; Thompson, W.T.: Pog. Solid State Chen. 1 (1975) 119. Refeences The nano-paticle binay alloy phase diagams wee evaluated fom the infomation on the Gibbs enegy and the suface tension of the bulk phase on the basis of the egula solution model although some ough appoximations have been applied in the pesent wok. The suface tensions of liquid and solid alloys ae also calculated fom the Gibbs enegy of the bulk in the above evaluation. The effect of the size of the paticle on the phase equilibia is conspicuous when the excess Gibbs enegies in solid and liquid phases ae positive and thei absolute values ae lage. ven when the solid solutions ae not appaently found in the phase diagams of the bulk systems, the contibution of the Gibbs enegy of the solid solution to the phase equilibia cannot be ignoed in the small-paticle systems. T. Tanaka, S. Haa: Themodynamic valuation of Nano-Paticle inay lloy Phase Diagams 1241 Pof. D. T. Tanaka Depatment of Mateials Science and Pocessing 2-1 Yamadaoka, Suita, Osaka , Japan Fax: mail: tanaka@mat.eng.osaka-u.ajp Coespondence ddess (Received June 15, 21) 89Yeu 89Gla 91Sas 94Tan 95Gas 96Tan 98Tan2 98Tanl (1998) Tanaka, T.; Haa, S.; Ogawa, M.; Ueda, T.: Molten Salt Foum Tanaka, T.; Haa, S.; Ogawa, M.; Ueda, T.: Z. Metallkd Tan3 99Tan1 99Tan2 99Tan3 99Ued OlTanl 1Tan Yeum, K.S.; Speise, R.; Poiie, D.R.: Metal!. Tans. 2 (1989) Gladkikh, N.T.; Gigoeva, L.K.; Dukaov, S.V.: Sov. Phys. Solid State 31 (1989) Sasaki, K.; Saka, H.: Phil. Mag. 63 (1991) Tanaka, T.; Iida, T.: Steel Res. 65 (1994) Gaskell, D.R.: Intoduction to the Themodynamics of Mateials, Taylo & Fancis, Washington, DC (1995). Tanaka, T.; Hack, K.; Iida, T.; Haa, S.: Z. Metallkd. 87 (1996) (1998) Tanaka, T.; Haa, S.; Ueda, T. in: P.C. Tulove, H. C. De Long, G.R. Staffod, S. Deki (eds.), Po 11th Int. Symp. on Molten Salts, San Diego, C (1998) Tanaka, T.; Hack, K.; Haa,S.: MRS ulletin 24 (1999) Tanaka, T.; Haa, S.: lectochemisty 67 (1999) Tanaka, T.; Haa, S.: Z. Metallkd. 9 (1999) Ueda, T.; Tanaka, T.; Haa, S.: Z. Metallkd. 9 (1999) Tanaka, T.; Hack, K.; Haa,S.: Calphad 24 (21) Tanaka, T.; Haa, S.: Z. Metallkd. 92 (21) uf uffat, Ph.; oel, J-P.: Phys. Rev. 13 (1976) Cou Couchman, P.R.; Jesse, W..: Natue 269 (1977) Mo Mookhov, I.D.: Izv. kad. Nauk SSSR Metal!. No.6 (1979)