Mateials cience Foum Online: 2010-06-02 IN: 1662-9752, ol. 653, pp 55-75 doi:10.4028/www.scientific.net/mf.653.55 2010 Tans Tech Publications, witzeland Pediction of Phase Diagams in Nano-sized inay lloys Toshihio Tanaka Division of Mateials and Manufactuing, Gaduate chool of Engineeing, Osaka Univesity, 2-1 Yamadaoka, uita, Osaka 565-0871, Japan. E-mail: tanaka@mat.eng.osaka-u.ac.jp Keywods: Nano paticles, Phase Diagams, inay lloys, Themodynamics, uface Tension bstact. The authos have evaluated the suface popeties as well as phase diagams of alloys on the basis of themodynamic databases. Extending these techniques, we developed a new system to estimate phase equilibia of metals and alloys in small paticle systems. The pesent pape descibes ou tial to evaluate the phase diagams of binay alloys in nano-sized paticle systems though themodynamic databases. Intoduction aious themodynamic databases have been compiled to be mainly applied to the calculation of phase diagams of alloys, salts and oxides [1]. The accumulation and assessment of themodynamic data and phase equilibia infomation to establish those databases is sometimes called CPHD (Compute Calculation of Phase Diagams) appoach [2] in Fig.1. Themodynamic Databases CPHD ppoach Phase Diagams of lloys, alts & Oxides etc. Phase Equilibia of Nano-sized Paticles uface Popeties (uface Tension, Intefacial Tension etc.) Fig.1. Concept fo the pediction of nano-paticles alloy phase diagams. The CPHD appoach has been ecognized to be useful in vaious aspects of mateials science and engineeing [1,2]. If it would be possible to use the themodynamic databases to evaluate vaious physico-chemical popeties as well as phase diagams, we could not only widen the applicability of those themodynamic databases but also futhe the undestanding of the physico-chemical popeties. The authos have applied those themodynamic databases to the evaluation of the suface tension of liquid alloys, molten ionic mixtues, molten slag and the intefacial tension between liquid steel and molten slag [3-17]. These woks ae aimed to undestand the themodynamic popeties of a mateial system including its suface o inteface as well as the bulk. ince the effect of its suface on the total themodynamic popeties can not be negligible in small paticles of metals and alloys, the phase elations in these metals and alloys ae dependent upon the size of the paticle and its suface popety. The themodynamics of solid-liquid phase equilibia in a small paticle system was fistly discussed by Pawlow in 1909 [18]. ince Takagi [19] obseved that the melting points of some pue metals decease with deceasing the size of these metallic paticles, a lot of studies have been This is an open access aticle unde the CC-Y 4.0 license (https://ceativecommons.og/licenses/by/4.0/)
56 Themal and Themodynamic tability of Nanomateials caied out on the effect of the paticle size on the melting point of pue metals [20-24]. s descibed above, the authos have evaluated the suface popeties as well as phase diagams of alloys on the basis of themodynamic databases. Extending these techniques, we have evaluated some phase equilibia of metals and alloys in nano paticle systems [25-30]. Haja and chaya have epoted the calculation of phase diagam in Cu-g nano-paticle alloy [31]. In the pesent pape, we descibe ou tial to evaluate the phase diagams of binay alloys in nano-sized paticle systems though themodynamic databases [25-28]. Themodynamics in mall Paticle ystems. ince Pawlow[18] poposed the themodynamics of small paticle system, a lot of wok have been done on the effect of the paticle size on the melting point of small paticles. lthough the above analysis by Pawlow[18] was based on the macoscopic themodynamic point of view, the equation deived by him can be applied to the micoscopic melting phenomena of nano-sized paticles. ccoding to Pawlow[18] the chemical potentials of pue element in liquid and solid small iquid, Paticle olid, Paticle paticles with adius, and, ae given in Eqs.(1) and (2): iquid, Paticle iquid, ulk 2 iquid iquid (1) olid, Paticle olid, ulk 2 olid olid (2) whee iquid, ulk and olid, ulk ae chemical potentials of the component in liquid and solid phases iquid of the bulk, and ae the suface tensions of pue element in solid and liquid phases. olid olid and iquid ae the mola volumes of pue element in solid and liquid phases. Fom the equilibium condition of liquid and solid phases (3) iquid, Paticle olid, Paticle the following equation is obtained: iquid, ulk olid, ulk 2 olid olid iquid iquid ( ) (4) iquid, ulk olid, ulk ince is the Gibbs enegy change of fusion of the element, the following appoximation can be given: iquid, ulk olid, ulk T H m, (1 ) (5) ulk T m, ulk whee H m, and T m, ae the heat of fusion and the melting point of the element, espectively. Thus, fom Eqs.(4) and (5), T T ulk m, 1 2 1 H m, olid olid iquid iquid ( ) (6) The tempeatue T satisfied with the above equation (6) coesponds to the melting point of the element in the small paticle system with the adius. lot of measuements on the melting of
Mateials cience Foum ol. 653 57 small paticle metals have been epoted so fa afte Takagi [19] obseved the decease of the melting points of some metallic paticles by an electon micoscope. Fig.2. Change in melting point of pue u with adius of a paticle. Fo example, Fig.2 shows the effect of the paticle adius on the melting point of u epoted by Coombes [32] and ambles [33]. The solid cuve in this figue indicates the calculated esults fom the above equation (6). s shown in Fig.2, the calculated esults give elatively good ageement with expeimental values although the calculated melting point shows lage values than the expeimental data fo small adius, especially unde 5nm. On this diffeence between the calculated melting point and the expeimental data, vaious discussions have been epoted, and in addition many impovements have been poposed so fa. oel [34] summaized the themodynamic investigations [18,21-23,32, 33, 35-40] on the melting of small paticles epoted hitheto as shown in Table 1. Table 1. Citeia and Phenomenological models fo the melting of nano-sized paticles. Citeia of fusion Phenomenological models P1(1 phase) P2(2Phases) P3(3 Phases) C1: chemical equilibiium Reiss & Wilson[35] Palow[18] Wonsky[21] Hanzen[36] Coombes[32] uffat & oel[22] ambles[33] Rie[37] C2 : liquid phase gemination Couchman & Jesse[23] at the suface C3 : fee enegies equality of Ross & ndes[38] solid and liquid of the same mass C4 : indeman hypothesis Couchman & Ryan[39] He classified the fusion citeia to some categoies as follows: (C1) (C2) (C3) Equality of the chemical potentials. Gemination of the liquid phase at the suface of the solid. Equality of the fee enegy of the solid paticle and that of the liquid paticle with the same mass.
58 Themal and Themodynamic tability of Nanomateials (C4) indemann hypothesis : the fusion happens when the mean quadatic atomic displacement eaches a cetain faction of the neaest neighbo distance. In the above list, we adopted only macoscopic themodynamic appoaches although the micoscopic eseaches, fo example, molecula dynamics appoach etc., wee also listed in oel s oiginal pape [34]. The above each citeion is elated to how many phases ae consideed as shown below: (P1) (P2) (P3) One phase model : only the stability of the solid phase is consideed. Two phase model : a solid phase and a liquid phase ae supposed to be in equilibium. Thee phase model : the thee phases, solid, liquid, vapo ae supposed to be in equilibium(tiple point type theoy) lthough a lot of wok have been done so fa as descibed above, we have not yet eached the completed undestanding of the melting of pue metals in small paticle systems because we do not have detail infomation on the atomic stuctue and the atomic inteactions in a nano-sized paticle. When the micostuctue was analyzed in the nano-sized paticles in the above studies, some physical paametes wee intoduced. Fo example, in the above (P2) Two phase model, the liquid phase laye appeas aound the solid phase coe to make the inteface between the solid and the liquid phases. This physical desciption is ealistic coesponding to some expeimental obsevations, but the thickness of the liquid laye is an unknown paamete, which we can not detemine deductively at the pesent stage. In the futue, the full compehension may be obtained by combining of the micoscopic analysis with the macoscopic themodynamic appoach on the melting of small paticles. lthough the above studies have been done fo mainly pue metals, some investigations on phase tansfomation in alloys have been epoted, of which phase equilibia have not been elucidated enough yet. The aim of the pesent pape is to investigate some fundamental issues on the application of themodynamic databases to the evaluation of phase diagams of nano-sized paticles. Hee we select the classical Palow s citeia fo the pesent tials because his appoach can be combined diectly with the macoscopic pocedue used in the themodynamic databases without any exta micoscopic paametes. In the pesent pape, we focus on binay alloy phase diagams in nano-sized paticle systems to know how liquidus cuves change with the size of the paticle, and how the themodynamic inteactions among the components, e.g. activity, excess Gibbs enegy etc. affect the phase diagams in nano paticle systems of binay alloys. inay lloy Phase Diagams of mall Paticle ystems consisting of Pue olid Phases and iquid Phase. t the beginning, we pay attention to some binay alloys, of which phase diagams consist of liquid phase and pue solid phases such as Cu-Pb, Cu-i and u-i alloys. In these alloys, thei phase diagams can be evaluated by only the infomation on the Gibbs enegy in the bulk and the suface tension of liquid phase, which can be obtained as functions of tempeatue and compositions fom the themodynamic databases. Gibbs Enegy of mall Paticle ystems. If a phase diagam consists of liquid phase and pue solid phases, and in addition, pue solid phase is selected as the efeence state of Gibbs enegy, the total Gibbs enegy of an alloy system in a small paticle with its adius is descibed in the following equations (7)~(10) [40]: Total ulk uface G G G (7) The Gibbs enegy of the bulk of - binay alloyδg ulk in Eq.(1), which coesponds to ΔG Total with =, is expessed in Eqs.(8) and (9).
Mateials cience Foum ol. 653 59 ulk Excess, G N G N G G RT N ln N N ln N (8) G Excess, 0 1 2 N N N N N N N N (9) 2 whee ΔG and ΔG ae Gibbs enegies of pue liquid phases elative to those of pue solid phases, of which tempeatue dependence ae listed in Table 2 [41, 42]. G Excess, is the excess Gibbs enegy of liquid phase in the - alloy. The inteaction paametes i (i=0~3) of G Excess, in Eq.(9) ae also listed in Table 3 [41, 43, 44]. N and N ae mole factions of components and. 3 3 Table 2. Data of Gibbs enegies of pue components Element Gibbs enegies of pue components / J mol -1 Ref. Cu G Cu =-48.7-(-142.53101) T+31.380005 T (1.0-ln T) [41] G Cu =-8044.1-(-110.40401) T+24.852997 T (1.0-ln T) -0.0037865 T 2 /2-(-138909)/2/T ΔG Cu =G Cu - G Cu Pb G Pb =-7347.8-(-133.83501) T+36.112106 T (1.0-ln T) -(-0.0097362) T 2 /2-(-279073)/2/T-3.2384 10-6 T 3 /6 [41] G Pb =-7608.7-(-75.81465) T+24.221176 T (1.0-ln T) -0.0087111 T 2 /2 ΔG Pb =G Pb - G Pb i G i =7900.3-(-52.22951) T+23.359299 T (1.0-ln T) -0.0031380 T 2 /2-1660630/2/T-(-7.1965 10-7 ) T 3 /6 [41] G i =-7817.7-(-100.00800) T+28.409393 T (1.0-ln T) -(-0.0246772)T 2 /2-5.0 10-5 T 3 /6 ΔG i =G i - G i u ΔG u =12552.0-9.385866 T [42] i ΔG i =50696.36-30.099439 T+2.0931 10-21 T 7 (298.15<T<1687) [42]
60 Themal and Themodynamic tability of Nanomateials Table 3 Data of excess Gibbs enegy of liquid phase G Excess, =N N { 0 + 1 (N -N )+ 2 (N -N ) 2 + 3 (N -N ) 3 } / J mol -1 Cu-Pb [43] Cu-i [41] u-i [44] 0 =27190.2-4.21329 T 1 =2229.2-0.53584 T 2 =-7029.2+6.48832 T 3 =-7397.6+5.07992 T 0 =24105.9-9.93287 T 1 =-2584.5+0.00906 T 0 =-23863.9-16.23438 T 1 =-20529.55-6.03958 T 2 =-8170.5-4.2732 T 3 =-33138.25+26.56665 T ΔG uface in Eq.(7), the effect of the suface onδg Total, is assumed as follows [40]: G uface 2 2 N N (10) whee σ is suface tension of liquid alloy, : mola volume of liquid alloy, : adius of a paticle, σ and σ : suface tensions of pue solid and, and : mola volumes of pue solid and. s shown in Eq.(10), we need the value of the suface tension σ of pue solid, but the pecise infomation on the value of σ and its tempeatue dependence ae insufficient [20, 45, 46]. Fom the data epoted in some efeences[20, 45, 46], the value of σ of pue solid at the melting point is found to be 25% lage than the suface tension of pue liquid on the aveage. Equation (11) is, theefoe, assumed to expess the suface tensionσ of pue solid in the pesent wok: T T o 1.25, mp, mp (11) T whee σ, mp is the suface tension of pue liquid at its melting point T,mp. The tempeatue dependence of σ in Eq.(11) is assumed to be the same as that of σ, which ae summaized in Table 4 [47]. In addition, the effects of cystal faces on σ is ignoed in the pesent wok. The mola volume of liquid alloy in Eq.(10) is assumed to be obtained fom the following equation: N N (12) whee and ae mola volumes of pue liquid and. Thei tempeatue dependence ae given in Table 4 [47]. The mola volume of pue solid is evaluated in Eq.(13), which is obtained by consideing the volume change due to the fusion at the melting point of each component.
Mateials cience Foum ol. 653 61 (13) 1 whee α =(,m -,m )/,m, which is the atio of the volume change of solid due to the fusion, is listed in Table 4 [48].,m and,m ae the mola volumes of pue liquid and solid at the melting point. Table 4 Data of physical popeties uface tensionσ / Nm -1 of pue liquid components [47] / Nm -1 T -1 T Cu σ Cu =1.303-0.00023 (T-1356.15) -0.00023 Pb σ Pb =0.458-0.00013 (T-600.55) -0.00013 i σ i =0.378-0.00007 (T-544.1) -0.00007 u σ u =1.169-0.00025 (T-1336.15) -0.00025 i σ i =0.865-0.00013 (T-1687.15) -0.00013 Mola volume / m 3 mol -1 of pue liquid components [47] Cu Pb i u i Cu =7.94 10-6 {1.0+0.0001 (T-1356.15)} Pb =19.42 10-6 {1.0+0.000124 (T-600.5 i =20.80 10-6 {1.0+0.000117 (T-544.1)} u =11.3 10-6 {1.0+0.000069 (T-1336.15)} i =11.1 10-6 {1.0+0.00014 (T-1687.15)} σ,m.p. / Nm -1 [47] T,m.p. / K [47] α =(,m -,m )/,m [48] Cu 1.303 1356.15 0.0396 Pb 0.458 600.55 0.0381 i 0.378 544.1-0.0387 u 1.169 1336.15 0.055 i 0.865 1687.15-0.095 When we evaluate the composition and the tempeatue dependence of the suface tension σ of liquid phase in Eq.(10), we can calculate ΔG Total in Eq.(7) of liquid phase fom the above equations. s mentioned above, the authos have evaluated the suface tension of liquid - binay alloys on the basis of utle's equation [49] as follows [4-13]: (14a) RT N ln N 1 G E, 1 E, T, N G T, N RT N 1 1 E, T, N G T, N E, ln G (14b) N
62 Themal and Themodynamic tability of Nanomateials In Eqs.(14a) and (14b), σ and σ ae suface tensions of pue molten components. N and N ae mole factions of element in the suface and the bulk, espectively. = N 0 1/3 ( ) 2/3 (N 0 vogado numbe : = o, =1.091) is mola suface aea of pue, and this is obtained fom the mola volume. G E, (T,N ) and G E, (T,N ) ae patial excess Gibbs enegies of component in the bulk and the suface, espectively, as functions of T and N o N. ince the patial mola excess Gibbs enegy of component (= o ) in the bulk G E, (T,N ) in Eqs.(14a) and (14b) can be obtained diectly fom G Excess, in Eq.(9) by using Eqs.(15a) and (15b). G G E, E, T G Excess, Excess,, N G N (15a) N Excess, T N G 1 N Excess, G, (15b) N Fo the excess Gibbs enegy in the suface, we deived the following equations [4-13] based on the model poposed by peise,yeum et al. [50, 51]. E, MI E, T, N G T N G, (16) MI 0.83 :liquid alloys (17) Equation (16) means that G E, (T,N ), which has the same fomula as G E, (T,N ), is obtained by eplacing N by N in G E, (T,N ) (= o ) and then multiplying β MI to G E, (T,N ). β MI is a paamete coesponding to the atio of the coodination numbe in the suface to that in the bulk consideing the suface elaxation [4-13]. The suface tension σ of liquid alloys can be calculated fom Eqs.(14)~(17) as follows: 1) etting tempeatue T and composition N of a solution. 2) Inseting the values fo suface tension σ and mola volume of pue liquid substances at the above tempeatue in Eqs. (14a) and (14b). 3) Detemining excess Gibbs enegies in the bulk phase at the above tempeatue and composition, and substituting them in Eqs. (14a) and (14b). 4) Then, Eqs.(14a) and (14b) become the simultaneous equations with unknown N and σ. These equations ae solved fo those unknown N and σ numeically. Fom Eqs.(7)~(17), the Gibbs enegy ΔG Total in Eq.(7) of liquid phase elative to pue solid phase, which is selected as the efeence state in the pesent wok, can be obtained as a function of N at a given tempeatue. When the liquid phase is equilibated with pue solid, the chemical potential of the component is zeo as shown in Fig.3 because pue solid phase is selected as the efeence state. In othe wods, the liquidus composition N at tempeatue T in Fig.3 is detemined fom the composition which satisfies μ =0 on the Gibbs enegy cuve. When two liquid phases ae sepaated, the liquidus can be detemined fom the two intesections on a common tangent line with the Gibbs enegy cuve as shown in Fig.3.
Mateials cience Foum ol. 653 63 Fig.3. Detemination of phase diagam fom Gibbs enegy cuve. Calculation of Phase Diagams in mall Paticle ystems In the pesent wok, we have evaluated the phase diagams of Cu-Pb, Cu-i and u-i binay systems, of which phase diagams consist of liquid phase and pue solid phases. The data which ae used fo the calculations ae summaized in Tables 2-4 [41-44,47,48]. Figs.4(a),(b) and (c) show the calculated esults of the suface tension of liquid Cu-Pb, Cu-i and u-i alloys fo a given tempeatue with activity cuves in the bulk of liquid phase. Ideal Ideal Ideal Ideal Expe. Ideal Calc. Ideal Fig.4. ctivities of components in bulk and suface tension of liquid alloys.
64 Themal and Themodynamic tability of Nanomateials s descibed in the pevious wok [4-13], the calculated suface tension of liquid alloys obtained fom Eqs.(14)~(17) agee well with the expeimental esults although only Cu-Pb alloy in Fig.4(a) shows the compaison of the calculated values with the expeimental esults [52, 53] in the pesent pape. It is well known that when the alloy, such as Cu-Pb o Cu-i systems, has the lage diffeence between σ and σ of pue liquid components, the composition dependence of σ shows lage downwad deviation fom the lineaity of σ and σ. In addition, we found fom these calculations that in alloys with positive deviation of activity fom ideal solution in the bulk (Cu-Pb & Cu-i alloys), the suface tension deviates negatively fom ideal solution. On the othe hand, in alloys with negative deviation of activity fom ideal solution in the bulk (u-i alloy), the suface tension has the tendency to show the positive deviation fom ideal solution. Thus, as shown in Figs.4(a),(b) and (c), Cu-Pb and Cu-i liquid alloys indicate the lage downwad cuvatue of the composition dependence of the suface tension. On the othe hand, in u-i alloy, the suface tension of liquid phase changes smoothly with the composition. Fom Eqs.(7)~(10), the Gibbs enegy of pue component including the suface is obtained as follows: Total ulk uface G G G G G 2 ( 2 [ {1.25 ), mp T ( T T, mp )} 1 The tempeatue T which gives ΔG Total =0 in Eq.(18) is the melting point of pue at a given adius of a paticle. Figs.1 & 5(a)-(d) show the change in the melting point of pue u, Pb, Cu, i and i with the adius of the paticle calculated fom Eq.(18). ] (18) (a) (b) (c) (d) Fig.5. Change in melting points of Pb, Cu, i and i with adius of a paticle.
Mateials cience Foum ol. 653 65 Figs.6(a),(b) and (c) show the phase diagams of Cu-Pb, Cu-i and u-i alloys fo =20nm, 10nm and 5nm as well as the bulk. ince it has been epoted that the value of the suface tension σ, σ, etc. is influenced by the cuvatue of the suface in a small paticle below =5nm[54-56], the pesent appoach can not be extended to the evaluation of the phase diagams in the small paticle systems with < 5nm. Fom these esults, we found that 1) iquid phase egion in the phase diagams is enlaged as the size of the paticle becomes smalle. 2) Two liquid phases sepaation egion speads as the size of the paticle becomes smalle. 3) In alloys, of which composition dependence of the suface tension of liquid phase shows lage downwad cuvatue, such as Cu-Pb and Cu-i alloys, the size of the paticle influences lagely on the phase elations. On the othe hand, in u-i alloy, of which suface tension of liquid phase changes smoothly with the composition, the effect of the size on the phase elations is not so lage. Fig.6. Phase diagams of Cu-Pb, Cu-i and u-i systems in nano-sized binay alloys. inay lloy Phase Diagams with olid olutions in mall Paticle ystems. In the peceding sections, 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. The pupose of this section 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.
66 Themal and Themodynamic tability of Nanomateials Gibbs Enegy in mall Paticle ystems. When a pue solid phase is selected as the efeence state of Gibbs enegy, the total Gibbs enegies in liquid and solid phases, ΔG Total,iq and ΔG Total,ol, of an alloy system in a small paticle with its adius ae descibed in the following equations (19)~(24) [25-27]: Total, iq ulk, iq uface, iq G G G (19) Total, ol ulk, ol uface, ol G G G (20) The Gibbs enegy of the bulk of - binay alloy in liquid and solid phases, ΔG ulk,iq and ΔG ulk,ol in Eqs.(19) and (20), which coesponds to ΔG Total,P (P=iq o ol) with =, ae expessed in Eqs.(21) and (22). ulk, iq Excess, iq G N G N G G RT N ln N N ln N (21) ulk, ol Excess, ol G G RT N ln N N ln N (22) whee ΔG and ΔG ae Gibbs enegies of pue liquid phases elative to those of pue solid phases, in othe wods, the Gibbs enegy of the melting. G Excess, and G Excess,ol ae the excess Gibbs enegies of liquid and solid phases in the - alloy. N and N ae mole factions of components and. ΔG uface,iq and ΔG uface,ol in Eqs.(19) and (20), the effect of the suface on ΔG Total,P, ae assumed as follows [25-27]: G G uface, iq uface, ol iq 2 ol 2 iq ol 2 N 2 N ol ol ol ol N N ol ol whee the symbols ae explained just below in Eq.(10) but hee we conside the suface tension of solid alloy σ ol In ode to evaluate the excess Gibbs enegy, the egula solution model is applied in the pesent wok because vaious inteaction enegies can be selected easily as follows : ol ol (23) (24) Excess, iq G N N W iquid (25) Excess, ol G N N W olid (26) whee W iquid and W olid ae the inteaction enegies in liquid and solid phases. ince Pelton and Thompson [57] assumed the simple equation on the Gibbs enegy of the melting in thei evaluation of phase diagams in Ref. [57], we also use the following equations as they applied. (27) G T, mp, mp T, mp (28) G T, mp, mp T, mp
Mateials cience Foum ol. 653 67 whee T,mp and,mp ae the melting point and the entopy of fusion fo pue substance (= o ). In the pesent wok,,mp is oughly assumed to be 10J K -1 mol -1 accoding to Richad's ule [46]. In addition, we select hee T,mp =1200 K and T,mp =600 K. On the suface tension of pue liquid metals at thei melting points, the following appoximation has been epoted [47] : T 2 3 1 m mol 3, mp K iq 1 8, mp, mp Nm 4.810 R (29) wheeσ,mp iq and,mp ae the suface tension and the mola volume of the element at its melting point. R is the gas constant, 8.314 JK -1 mol -1. When we select,mp =10 10-6 m 3 mol -1, as descibed below, the following ough elation is obtained fom Eq.(29): T iq, mp 1, mp Nm (30) 1000 In addition, the tempeatue coefficient of the suface tension of pue liquid has been epoted to be about 0.0001Nm -1 K -1 [47]. Thus, we assumed the following equation fo σ iq and σ iq in the pesent wok. 1 T T / Nm T K (31) iq 1.2 0.0001, mp, mp 1200 1 T T / Nm T K (32) iq 0.6 0.0001, mp, mp 600 When the peceding equation (11) can be applied fo the suface tension σ ol of pue solid, the following elations ae obtained fom Eqs.(11), (31) and (32): 1 T T Nm 1.251.2 0.0001 (33) ol, mp / 1 T T Nm 1.25 0.6 0.0001 (34) ol, mp / On 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. 1010 / m mol iq iq 6 3 1 (35) 1010 / m mol ol ol 6 3 1 (36) It is assumed that the mola volumes of liquid alloy iq and solid alloy ol in Eqs.(23) and (24) ae obtained fom the following simple additivities: iq N N (37) iq iq ol N N (38) ol ol
68 Themal and Themodynamic tability of Nanomateials lthough the suface tension of liquid and solid alloy σ P (P=iq o ol) ae evaluated fom utle's equation [49] in Eq.(14) as mentioned above, the excess Gibbs enegy in the suface fo liquid and solid solutions ae obtained fom the following equations (39)-(41). E, P, uf uf MI E, P, ulk uf T, N G T N G, (39) MI 0.83:liquid alloys (40) MI 9 0.75 :solid solutions (41) 12 In paticula, fo the solid solutions, we assume the value in Eq.(41) as β MI by assuming the closed-packed stuctue in the pesent wok because the coodination numbe in the bulk is 12 and that in the suface is 9: Calculated Results of the nano-sized phase diagams with solid solutions. iquid -solid phase equilibia ae obtained fom the following themodynamic conditions : Total, iq Total, ol (42) Total, iq Total, ol (43) Total, P Total, P Total, P G G N P iq o ol (44) N Total, P Total, P Total, P G G 1 N P iq o ol (45) N Table 5: Inteaction enegies of W iquid and W olid used in the calculation of phase diagams. olid W / 30 15 0-15 iquid W / 20 Fig.8(a) 10 Fig.8(b) 0 Fig.7(c) &Fig.8(c) Fig.7(b) Fig.7(a) -20 Fig.8(d) Fig.7(d) Figs.7(a)-(d) show the calculated esults of the phase diagams by using the inteaction enegies in Table 5. In these figues, the solid cuves indicates the phase equilibia in the bulk( = ). On the othe hand, the chain and the dotted cuves ae the calculated esults fo =10nm and 5nm, espectively.
Mateials cience Foum ol. 653 69 W iquid 0 W olid 30 W iquid 0 W olid 0 W iquid 0 W olid 15 W iquid 20 W olid 15 Fig.7. Phase diagams nano-sized binay alloys fo vaiousw iquid and olid W values. W iquid 0 W olid 30 W iquid 20 W olid 30 W iquid 10 W olid 30 W iquid 20 W olid 30 Fig.8. Phase diagams nano-sized binay alloys fo a fixed olid W. Figue 7(a) indicates the phase diagams fo the ideal solutions in both solid and liquid phases, in othe wods, W iq =W ol =0. When W ol inceases fom 0 to 30 with W iq =0, the phase
70 Themal and Themodynamic tability of Nanomateials iq diagams change as shown in Figs. 7(b) and (c). Figue 7 (d) shows the phase diagams fo W = -20 and W ol iq =-15 as an example fo W <0 and W ol <0. When W ol is fixed to be 30 and W iq changes fom -20 to +20, the phase diagams ae obtained as shown in Figs.8(a)-(d). s can be seen in Figs.7 and 8, when the inteaction paametes ae W iq >0 and W ol >0, the effect of the paticle size on the phase diagams is emakable. s descibed above, the liquid o solid alloys indicate the lage downwad cuvatue of the composition dependence of the suface tension in the conditions of W iq >0 and W ol >0. These composition dependence of the suface tension affect the contibution of ΔG uface, P to ΔG Total, P. Especially, when W ol >0, the solid solutions do not appea in some of the phase diagams of the bulk as shown in Figs.7(c), 8(c) and 8(d). Howeve, when the paticle size deceases, the contibution of ΔG uface,ol to ΔG Total,ol in the solid phase can not be ignoed. Consequently, as shown in Figs. 7 and 8, 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. s descibed above, the phase diagams of binay alloys in the small paticle systems can be evaluated fom the themodynamic infomation usually stoed in some databases, although the following ough appoximations have been used in the pesent wok: 1) The mass balance of atoms between the bulk and the suface ae not consideed though the suface segegation occus in alloys. 2) The effects of cystal faces on the suface popeties of solid phases ae ignoed. 4) The tempeatue dependence of the suface tension of pue solid phase is assumed to be the same as that of pue liquid phase. Compaison of Calculated Results on Nano-sized Phase Diagam with Expeimental Obsevation of Melting ehavio in inay lloy s one of the examples fo the compaison of calculated esults on nano-sized phase diagams with expeimental obsevations, Figs.9 and 10 shows the melting behabio of i-n binay alloy small doplet and the phase diagam of i-n system fo the bulk and nano-sized paticle, which was conducted by ee et al. [28]. Fig.9. Obsevation fo melting behabio of i-n binay alloy in a nano-sized doplet at 80 0 C (353K). [28]
Mateials cience Foum ol. 653 71 Fig.10. Calculated esults on the phase diagam in i-n binay alloy fo bulk and nano-sized paticle. [28] In the expeiment, fistly pue i was vapoized in TEM chambe at 80 0 C( 353 K), and nano-sized pue solid i paticle was deposited on gaphite substate as shown in Fig.9(a). Then, Pue n was vapoized and the n vapo penetated into i nano-sized paticle to make liquid alloy phase coexisted with pue i as indicated in Fig.9(b) and (c). In these figues, the inteface between soid i and liquid phases was obseved, and finally liquid alloy was fomed completely as shown in Fig.9(d). When we compae the above melting behavio of i-n alloy in Fig.9 with the phase diagam in Fig.10, fistly pue solid i existed, and then solid i was coexisted with liquid alloy phase accoding to the incease of n content. Finally, the composition of the alloy eached eutectic point to make liquid nano-sized doplet. i-n alloy does not make liquid phase at 80 0 C( 353 K) in the bulk size as shown in the phase disgam in Fig.10, but in the nano-sized paticle, liquid phase may exist as suggested in the calculated esult in Fig.10 and it was demonstaed as indicated in Fig.9. The detail of the above expeimental pocedue has been descibed in Refs. [28, 58,59]. J-G. ee and H. Moi have caied out many obsevations on nano-paticle phase tansfomation. The infomation on the matte can be obtained in the efeences quoted in Refs. [28,58,59]. Poblems in Themodynamics of mall Paticle ystems. s shown in Fig.1, the calculated esult of the dependence of the melting point of u on the paticle size is lage than that of the expeimental esults at aound <5nm. aious discussions on this diffeence have been epoted so fa as descibed above. In the pesent section, we discuss the above issue fom a little bit diffeent point of view. Effect of the uface Tension on the Calculated Result on the Melting Point of Nano-sized u paticle. In Fig.1, we calculated the melting point of u by using the data on the suface tension of the bulk liquid and solid u epoted in Ref. [60]. ince it is elatively easy to measue the suface tension of pue liquid u, which is not oxidized in the high tempeatue, a lot of expeimental esults of the suface tension of liquid u wee epoted so fa. The uncetainty of the expeimental esults is not so small even fo pue u. In addition, it is quite difficult to measue the pecise value on the suface tension of pue solid metals. When the value of the suface tensions of liquid and solid u ae changed in Eq.(4), the melting point of pue u with the adius smay shift. When we discuss, theefoe, the diffeence of the calculated melting point of the nano-sized paticle fom the
72 Themal and Themodynamic tability of Nanomateials expeimental data, we should conside the eliability of the data used in the above themodynamic evaluations in addition to the citeia. Effect of the uface Melting on the Calculated Result on the Melting Point of Nano-sized u Paticle. The suface melting has been known well in the bulk metals. In this phenomenon, the thin layes nea the suface of metals ae melted below the melting points of the bulk. Fo example, Kojima and usa [61] investigated the suface melting of pue Cu by using MD method. They epoted the suface melting occus at 900 K in the face [111] although the bulk Cu melts at 1356 K. The eason of the suface melting is supposed that the binding of atoms nea the suface is weake than that in the bulk. Theefoe, in the nano-sized paticles, of which suface aea pe unit volume is lage than that of the bulk systems, the suface phenomenon may occu moe emakably than the bulk system. When we conside the effect of the suface melting on the fusion of nano paticles, we should use smalle value of the heat of fusion than that of the bulk. Howeve, we do not have any idea on how much the heat of fusion should be educed in the evaluation of the popeties in the nano-paticles. Then, we evaluated the heat of fusion of u to satisfy the calculated value of the melting point in Eq.(4) with the expeimental data in Fig.1. Fig.11 shows the change in the heat of fusion of u with the adius [62]. s shown in this figue, the heat of fusion deceases with deceasing the adius, which means that the bonding between atoms ae weakened in small paticle size as deceasing the paticles size. When we investigate the themodynamic evaluation of the melting point of the nano-sized paticles moe pecisely, we should conside the effect of the suface melting on those small paticles. 14000 12000 ulk ΔH /Jmol -1 10000 8000 6000 4000 0 100 200 /nm Fig.11. Calculated esults on change in the heat of fusion of u with adius of a paticle. Concluding Remaks. In the pesent pape, the autho has descibed some fundamental pocedues to evaluate the solidliquid phase elations of nano-sized paticles in binay alloys fom the calculation of the suface popeties as well as the phase equilibia on the basis of themodynamic databases, which ae usually used fo the calculation of phase diagams of the bulk mateials. In ode to obtain quantitatively pecise values of the melting points and liquidus tempeatues in alloys, we should cay out futhe investigation on as follows: 1) Moe detail discussion on the citeia on the melting of nano-sized paticles.
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