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1 Hghly monodsperse core-shell partcles created by sold-state reactons Supplementary Materals Velmr Radmlovc 1,2, Coln Ophus 1,, Emmanuelle A Marqus 3, Marta D Rossell 4, Alfredo Tolley 1,5, Abhay Gautam 1, Mark Asta 6 and Ulrch Dahmen 1 1 Natonal Center for Electron Mcroscopy, Lawrence Berkeley Natonal Laboratory, Berkeley, CA 2 Nanotechnology and Functonal Materals Center, Unversty of Belgrade, Belgrade, Serba 3 Department of Materals ence and Engneerng, Unversty of Mchgan, Ann Arbor, MI 4 Department of Materals, ETH Zürch, Zürch, Swtzerland 5 Comsón Naconal de Energía Atómca, San Carlos de Barloche, Río Negro, Argentna 6 Department of Materals ence and Engneerng, Unversty of Calforna, Berkeley, CA 1 AlL Thermodynamcs To apply nucleaton and growth models, we must frst construct a thermodynamc model for the Al-rch porton of the AlL ternary alloy. The two phases of nterest are the α-al fcc sold soluton and the L1 2 Al 3 (L,) ntermetallc, whch we wll refer to as the β phase. Very lttle expermental thermodynamc nformaton s avalable for the ternary Al-L- system [1]. Instead, we wll use the equaton gven by Mugganu et al. to nterpolate the Gbbs energy G of the α-al phase from CALPHAD studes of the consttuent bnary systems: [2] G = x A G 0 A + x B G 0 B + x C G 0 C + RT (x A log x A + x B log x B + x C log x C ) + x A x B L ν A,B(x A x B ) ν + x A x C L ν A,C(x A x C ) ν + x B x C L ν B,C(x B x C ) ν + x A x B x C G ABC (x A,x B,x C ) (1) where G 0 s the Gbbs energy of the th pure element wth a concentraton of x, L ν,j are the Redlch-Kster coeffcents of the excess Gbbs energy for each bnary system j, G ABC s an excess energy functon dependng on all concentraton varables and R s the gas constant. In our system, A, B and C are taken to be Al, L and respectvely. Snce we are modelng the Al-rch corner of the ternary phase dagram, we can assume that x L x Al and x x Al and therefore neglect the fnal two terms n Equaton (1). We set the reference states to the Gbbs energes of the pure materals at temperature T. Because we are descrbng an fcc sold soluton, these reference cophus@gmal.com NATURE MATERIALS 1

2 2 states are 0 for Al, the transformaton energy G bcc fcc L from bcc to fcc for L and the transformaton energy G hcp fcc from hcp to fcc for. Therefore, Equaton (1) becomes G α = x α LG bcc fcc L + x α G hcp fcc + RT [x α Al log xα Al + xα L log x α L + x α log x α ] + x α Al xα L L ν Al,L (xα Al xα L) ν + x α Al xα L ν Al, (xα Al xα ) ν (2) The requred thermodynamc parameters are taken from taken from phase dagram calculatons by Hallstedt and Km for Al-L [3] and Murray for Al- [4]. To calculate the drvng forces for nucleaton and growth, we requre the chemcal potentals of all consttuent elements. The Hllert equaton states that a partal molar property w of element correspondng to a molar property W n an m component system can be determned by [5] ( ) W w = W + x P,T,x j = m j=1 ( ) W x j (3) x j P,T,x =j where the x j and x j subscrpts ndcate that for the dervatves all concentratons other than the dfferentaton varable are held constant. Applyng Equaton (3) to Equaton (2) yelds the chemcal potentals of each element n the α phase: µ α Al = RT log x α Al L ν Al,L (xα Al xα L) ν 1 [(1 + ν)(2x α L + x α )x α Al xα L] +x α L + x α L ν Al, (xα Al xα ) ν 1 [(1 + ν)(x α L +2x α )x α Al xα ] µ α L = G bcc fcc L + RT log x α L x α Al L ν Al,L (xα Al xα L) ν 1 [(1 + ν)(2x α Al + xα )x α L x α Al ] x α Al xα L ν Al, (xα Al xα ) ν (1 + ν) µ α = G hcp fcc L + RT log x α x α Al xα L L ν Al,L (xα Al xα L) ν (1 + ν) x α Al L ν Al, (xα Al xα ) ν 1 [(1 + ν)(2x α Al + xα L)x α x α Al ] (4a) (4b) (4c) The β phase precptates n AlL have an L1 2 structure. Snce we are nterested n modelng the formaton of precptates wth hgher than L concentratons, and the growth of nearly stochometrc Al 3 L shells, we wll model the β phase as a pseudobnary Al 3 ( 1 x βl x β) compound wth a fxed Al concentraton of 75 atomc percent. In ths model the compostonal varable x β represents the amount of L on the L1 2 superlattce n the β phase. Note that Al 3 s a stable 2 NATURE MATERIALS

3 3 compound whereas Al 3 L s only metastable. For smplcty we have modeled the system as only contanng the α and β phases and thus our model wll not apply to calculatons of the equlbrum phase stablty on the L-rch sde of the ternary dagram. The Gbbs energy of the β phase s modeled as G β = x β G Al3 L + (1 x β )G Al3 + G mx Al 3 L x 1 x (x β ) (5) where G Al3 L and G Al3 are the Gbbs energes of pure Al 3 L and Al 3 and G mx Al 3 L x 1 x s the excess Gbbs energy between of a dsordered mxture of the two pure phases. The Gbbs energes of the pure phases are agan taken from phase dagram calculatons of Al-L [3] and Al- [4]. The Gbbs excess energy of mxng has been calculated n ths study usng a pseudobnary cluster expanson of Al 3 L-Al 3 employng the approach descrbed n the Methods secton of the manuscrpt. The results of Monte-Carlo smulatons for mxng free energes were ft to the followng form: G mx Al 3 L x 1 x = RT 4 [ ] x β log x β + (1 x β ) log(1 x β ) + x β (1 x β ) ν L ν Al 3 (L,) (1 2xβ ) ν (6) where L ν Al 3 (L,) are Redlch-Kster coeffcents for the Al 3L-Al 3 excess Gbbs energy and the factor of 1 4 occurs because only one of the four L1 2 superlattces are a random mx of L and. We also desre to know the chemcal potentals n the β phase. Because our Gbbs energy expresson for ths phase s only defned along the Al 3 L x β 1 x β compostonal lne, the chemcal potental µ β Al can only be fxed by equlbrum wth another phase, or multple phases. However we can stll treat theβ phase as a bnary compound wth respect to the composton x β and apply Equaton (3). Ths wll gve the correct rato between the chemcal potentals of L and, equal to µ β L µ β = G Al3 L + RT 4 log xβ + (1 x β ) 2 v Lv Al 3 (L,) (1 2xβ ) v 1 (1 2x β 2vx β ) G Al3 + RT 4 log(1 xβ )+(x β ) 2 v Lv Al 3 (L,) (1 2xβ ) v 1 [1 2x β +2v(1 x β )] (7) The necessary thermodynamc parameters for our calculatons are G bcc fcc L = T G hcp fcc = T G Al3 L = T G Al3 = T L 0 Al,L = T L 1 Al,L = T L 2 Al,L = T L 0 Al, = T L 0 Al 3 (L,) = 2033 L 1 Al 3 (L,) = 855 L 2 Al 3 (L,) = 1587 NATURE MATERIALS 3

4 4 where all energes are gven n J/mol and temperatures are n K. The last requred component s an expresson for the nterfacal energy γ between the precptate and matrx. Unfortunately a complete expresson would depend on solute chemcal potentals, temperature, excess entropy, and adsorpton values, as gven from an ntegraton of the Gbbs adsorpton theorem. Due to the lack of nformaton concernng the excess thermodynamc quanttes n ths system, we assume a lnear dependence on the precptate composton x β : γ(x β )=x β γ L + (1 x β )γ (8) where γ L and γ are the nterfacal energes of Al Al 3 L and Al-Al 3. The value of the Al Al 3 nterfacal energy s taken to be 100 mj/m 2 [6]. We wll assume γ = 10 mj/m 2, a rough average of varous lterature estmates [7]. Ths value for the Al-Al 3 L nterface and the assumed lnear dependence are very approxmate, but ths wll have a mnor effect on the conclusons gven below, concernng precptate nucleaton. Ths s because nucle n AlL are nearly stochometrc Al 3, placng the value we use for nterfacal energy very close to the more accurate value for the Al-Al 3 nterface. 2 AlL Precptate Core Nucleaton The nucleaton and growth of Al 3 (L,) precptate cores can be modeled usng the formalsm developed by Kuehmann and Vorhees to descrbe Ostwald rpenng n ternary alloys [8]. Ther formalsm consders the quasstatonary approxmaton to growth of precptates n a matrx wth known compostons x α at an nfnte dstance from each precptate. Durng growth a concentraton gradent for all elements present exsts on the matrx sde of the precptate-matrx boundary. The composton at the precptate-matrx nterface s set by a local equlbrum condton accountng for the Gbbs-Thompson effect on the chemcal potentals at the curved nterface. The dependence of ths concentraton gradent on radal dstance from the precptate center r s obtaned by the solvng the Laplace equaton 2 x α = 0 n sphercal coordnates whch gves x α (r) =x α +(x α β x α ) r 0 r (9) where x α β s the composton of the th element on the matrx sde of the nterface and r 0 s the precptate radus. Kuehmann and Vorhees gve the multcomponent Gbbs-Thompson relatonshps µ α β = µ β + 2γ(xβ )V m r 0 for = Al, L, (10) where µ α β s the chemcal potental on the matrx sde of the boundary for element and V m s the molar volume (formally, V m n Equaton (10) should be the partal molar volume for speces, but n the applcaton below we wll assume that all partal molar volumes are equal to the molar 4 NATURE MATERIALS

5 5 volume). In the α phase the chemcal potentals are gven by Equatons (4a) to (4c) usng the approprate compostons. Durng a nucleaton event, the matrx s assumed to have constant mean composton as n Fgure 1. Because there s no compostonal gradent at the boundary, µ α β = µ α. The composton and crtcal radus r of the ntal nucle as a functon of matrx composton (x α L,xα ) can be found by smultaneously solvng the L and forms of Equaton (10) along wth Equaton (7). Fgure 1: hematc of precptate nucleaton n AlL. To model nucleaton knetcs, we frst consder the change n the grand potental of the system arsng from the formaton of a β nucleus n the α matrx. Ths change n grand potental wll be denoted Ω β and s gven as follows: Ω β = [F 4πr3 β 34 xβ µαal 3V m 4 µα L 1 ] xβ µ α +4πr 2 γ(x β ) (11) 4 where F β s the Helmholtz energy of the β phase. At atmospherc pressure, we approxmate F β as Gbbs energy G β. The crtcal radus r s determned from the expresson Ω/ r = 0, gvng the classcal expresson r 2γ(x β )V m = G β 3 4 µα Al xβ 4 µα L = 2γ (12) 1 xβ 4 µ α G v where G v s the change n Gbbs energy between the β and α phase per unt volume. Insertng Equaton (12) nto (11) gves the energy barrer to nucleaton Ω= 256πVmγ(x 2 β ) 3 3 [ 4G β 3µ α Al xβ µ α L (1 ] xβ )µ α 2 (13) NATURE MATERIALS 5

6 6 The precptate nucleaton rate J s equal to J = A exp ( Ω ) k B T (14) where A s a prefactor contanng the Zeldovch factor and knetc terms related to the attachment rate of monomer speces, and k B s the Boltzmann constant n an Arrhenus temperature dependent term. The A term certanly depends on the concentraton varables x α L and xα, but the exponental dependances wll domnate the nucleaton rate. Thus, we express the change n nucleaton rate as a rato of the nucleaton rate of Al 3 L x 1 x to the nucleaton rate J Al3 of pure Al 3 n an Al- alloy, wth x α L = 0,.e. ( ) J ΩAl3 Ω = exp (15) k B T J Al3 where Ω Al3 s the barrer to nucleaton of pure Al 3. The effect of ncreasng lthum n the AlL matrx (and also therefore n the precptate) on the rato of nucleaton rates and energy barrer of L1 2 precptates n AlL for a fxed alloy composton of x β = s summarzed n Fgure 2. Ths fgure shows that only a small amount of L s ncorporated nto ntal Al 3 nucle and that the energy barrer decreases monotoncally wth ncreasng matrx/precptate L composton. Ths decrease n nucleaton barrer leads to an ncreased nucleaton rate by approxmately sx orders of magntude for x L =8.5% relatve to nucleaton of Al 3 n an 0.11% Al- alloy. Ths dervaton of relatve nucleaton rate has employed several approxmatons and thus there s a wde range of uncertanty; however we conclude that ncreasng matrx L content certanly leads to a dramatc ncrease n precptate nucleaton rate. Fgure 2: The energy barrer and nucleaton rate of L1 2 precptates n AlL as a functon of matrx L content x α L. Precptate composton xβ as a functon of matrx composton nset. 6 NATURE MATERIALS

7 7 3 AlL Precptate Core Growth Knetcs After the ntal nucleaton burst of AlL precptate cores, we assume that ntally the cores are small enough for the solute concentratons n the matrx phase to be unchanged. Durng the growth stage the growth-lmtng solute concentraton (whch s n ths case) wll asymptotcally decrease towards ts solublty lmt n the α matrx. For the general ternary alloy consdered by Kuehmann and Vorhees, there are 4 unknown varables; the two precptate and the two matrx-sde boundary solute compostonal varables (or alternatvely these 4 chemcal potentals). In our system, the precptate s assumed to be a pseudobnary phase defned along the composton lne connectng Al 3 L to Al 3 and therefore the number of unknowns reduces to three. The concentraton gradents gven by Equaton (9) around three dfferent szed β precptates are shown n Fgure 3. Fgure 3: Concentraton profles n an α-al matrx surroundng β phase precptates. The left partcle wll shrnk, the center partcle radus wll reman constant (when the precptate radus equals the crtcal radus) and the rght partcle wll grow. The three forms of Equatons (10) are not enough to solve the quasstatonary compostons of ths system. Ths s because n the ternary alloy the β pseudobnary phase has a constant Al composton. Therefore the value of µ β Al can be freely adjusted wth an nfntesmal change n β phase Al concentraton x β Al. Thus we must make use of the knetc relatonshp derved by Kuehmann and Vorhees [8], whch for our system s x β L xα β L x β xα β = ( DL,Al D,Al ) x α β L x α β x α L xα where D L,Al =0.96 exp( 130/RT ) and D,Al =5.31 exp( 173/RT ) are the dffusvtes n unts of cm 2 /s taken from Moreau et al. [9] and Fujkawa [10] respectvely. Ths equaton was derved by (16) NATURE MATERIALS 7

8 8 consderng the mass balance of the two solutes at the matrx-precptate nterface. Replacng the β phase compostonal varables wth ther x β dependence gves ( ) α β DL,Al x L x α L x β 4x α β L 1 x β 4x α β = D,Al x α β xα Another equaton relatng the three unknowns can be constructed by usng the fact that the β phase chemcal potentals are the partal molar quanttes of G β and nsertng all three forms of Equaton (10). Ths gves G β = 3 4 µα β Al + xβ 4 µα β L (17) + 1 xβ µ α β 4 2γ(xβ )V m (18) r 0 The total free energy G β can be evaluated as a functon of x β from Equaton (5). The fnal requred expresson can be found from the rato of L and chemcal potentals n Equaton (10), gven by µ β L µ β = r 0µ α β L 2γ(x β )V m r 0 µ α β (19) 2γ(xβ )V m The value of ths rato s defned by Equaton (7). Equatons (17) to(19) plus Equatons (4), (5) and (7) provde enough nformaton to solve for the boundary compostons x α β (or equvalently the boundary chemcal potentals µ α β ) and precptate composton x β as a functon of precptate radus r 0 and matrx composton. Equaton (17) can be used to reduce the problem to a set of two non-lnear equatons, whch can be solved solved numercally. Fgure 4 shows the dfference between boundary composton and matrx composton x α xα β for both L and as a functon of precptate radus r 0. The magntude of the dfference n L concentraton s much smaller owng to the larger dffuson rate of L than n Al (231 tmes larger at 450 C). Fgure 4: (left) Dfference between boundary and matrx composton of L and and (rght) precptate composton as a functon of precptate radus at T = 450 C. The equlbrum precptate composton as a functon of radus s also shown n Fgure 4. The β phase concentraton of s always hgher than the matrx composton and so f x α β x α s 8 NATURE MATERIALS

9 9 postve the precptate wll grow, and f t s negatve the precptate wll shrnk. As the precptate grows, ts composton x β wll asymptotcally approach the equlbrum value of approxmately 0.4, gven by the te-lne n the two-phase regon of the ternary phase dagram. Note that ths value gves an overall L concentraton n the cores of 10 at.%, very close to the expermentally measured value of 11 at.%. The precptate growth rate can be computed from the expresson [8] r 0 t = D x α x α β r 0 x β x α β (20) where = L or wll gve the same result due to the mass balance enforced by Equaton (17). We have evaluated ths functon for an alloy wth 0.11 at.% both wth and wthout 8.5 at.% L. These growth knetcs are plotted n Fgure 5 for several matrx concentratons. Qualtatvely, both Al and AlL knetcs have the same shape. Both show ncreasng growth rate wth ncreasng radus r 0 for small values, a maxmum and then a long decreasng tal at large r 0 where the 1/r 0 dependence n Equaton (20) begns to domnate. Durng precptate growth, the matrx solute wll necessarly decrease, and as a consequence the growth rate decreases for all precptate szes. The crtcal radus, where a partcle wll nether grow nor shrnk ( r 0 / t = 0), begns to ncrease wth decreasng matrx for both alloys. Ths asymptotc ncrease of the crtcal radus towards nfnty s essentally responsble for the phenomena of coarsenng [8]. Fgure 5: Precptate growth rates for an Al0.11 at.% alloy contanng (top left) 8.5 at.% L and (bottom left) 0 at.% L, as a functon of precptate radus. (rght) The crtcal radus as a functon of matrx concentraton for both cases. However, a few dfferences between AlL and Al are apparent. The growth rate for AlL s generally hgher, whch has two basc causes. The addton of L lowers the solublty of n the α-al matrx and ths ncreases the drvng force for precptate growth. The growng precptates also ncorporate more L as they grow larger; snce the system s knetcally lmted by the slower dffuson of n α-al, decreasng concentraton n the core leads to an ncrease n growth rate. NATURE MATERIALS 9

10 10 The second dfference between AlL and Al s the dependence of the crtcal radus on matrx composton. The addton of L to Al suppresses the crtcal radus, keepng t very low for most solute concentratons. Ths s shown graphcally on the rght n Fgure 5. If the crtcal radus s smaller than vrtually all precptates, coarsenng wll be decreased. However some coarsenng wll stll occur f any of the β partcles are smaller than the radus wth a maxmum growth rate, and at very late tmes (when matrx solute s close to the solublty lmt). 4 AlL Precptate Shell Growth Durng the thrd and fnal heat treatment of AlL, nearly stochometrc Al 3 L shells grow onto the prevously grown Al 3 (L,) cores. The ntal stages of ths growth do not actually nvolve any nucleaton; rather the Al 3 L unformly wets the surface of the cores, whch are more or less perfect heterogeneous nucleaton stes. Ths wettng s drven by a lowerng of nterfacal energy. Equaton (8) gves an estmate for the nterfacal energy of Al Al 3 L of 60 mj/m 2. From the pseudobnary cluster expanson we estmate the nterfacal energy of the Al 3 L Al 3 L nterface at zero temperature to be 10 mj/m 2. Our earler assumpton for the Al Al 3 L nterfacal energy was also 10 mj/m 2, gvng a combned total of 20 mj/m 2 for the Al Al 3 L Al 3 L nterfaces. Snce rasng the temperature can only lower the nterfacal energy, the Al 3 L wettng layer on the precptate cores wll lower the nterfacal energy by a mnmum of 40 mj/m 2. After the wettng stage, the shells begn to grow due to the supersaturaton of L n the matrx. Usng the same formalsm as the prevous secton, we have calculated the equlbrum concentraton gradents as a functon of total precptate radus and far-feld matrx composton. These composton gradents as a functon of precptate radus r 0 are shown n Fgure 6 for the estmated matrx composton at the start of shell growth, x α L =8.4 and xα =0.006 at.%. Fgure 6: (left) Dfference between boundary and matrx composton of L and and (rght) precptate composton as a functon of precptate radus at T = 190 C. 10 NATURE MATERIALS

11 11 The calculated equlbrum precptate composton s also plotted n Fgure 6. We see at 190 C the growng shells are vrtually pure Al 3 L. Ths phase can exst n equlbrum wth the -rch L1 2 phase already present n the cores, as shown n the pseudobnary Al 3 -Al 3 L phase dagram plotted n the manuscrpt. Because the shells contan almost no, ther growth s controlled only by the local concentraton of L, as s evdent n Fgure 6. The concentraton at the boundary dffers only nfntesmally from the far-feld matrx composton. Usng Equaton (20) we can compute the β precptate growth radus for a gven α matrx composton. The growth rates for varous L matrx concentratons are plotted n Fgure 7. Fgure 7: Al 3 L Shell growth rates for an AlL alloy contanng varous amounts of L n the α matrx. The crtcal radus as a functon of matrx concentraton for both cases. Shell growth nvolves only L dffuson, and thus the growth rate curves resemble those of Al- n Fgure 5. There s however one major dfference; the AlL cores act as seeds for the Al 3 L shells, wth the ntal sze dstrbuton centered at approxmately 9 nm. Ths means that all of the precptates are ntally n the large r growth regon, where the growth rate scales wth 1/r. Ths regon s referred to as sze focusng [11] because smaller precptates wll grow faster than larger ones. The net result s that the AlL partcle sze dstrbuton wll be more monodsperse after shell growth than the core dstrbuton. Ths s the prmary cause of the monodspersty n AlL precptates and why the overall partcle radal dstrbuton s more monodsperse than the dstrbuton of core rad. The larger and more monodsperse AlL partcles wll resst coarsenng more effectvely than a wder dstrbuton wth a smaller ntal mean typcal of a bnary alloy precptates. NATURE MATERIALS 11

12 12 5 3D atom probe tomography of AlL The chemcal composton of the precptates n AlL was measured by 3D atom probe tomography (APT). As shown n Fgure 8, L s concentrated n the shell, but sgnfcant amounts of L reman n the matrx and the core. By comparson, s entrely confned to the core. The mean relatve core compostons of and L are approxmately 60% and 40% respectvely, agreeng wth the predctons of the AlL CALPHAD thermodynamc model. Fgure 8: Concentraton profle measured wth APT across a representatve core-shell precptate of an AlL alloy wth nsets showng and L dstrbuton. 12 NATURE MATERIALS

13 13 References [1] V Raghavan. Al-L- (Alumnum-Lthum-andum). Journal of Phase Equlbra and Dffuson, 30(2): , [2] YM Mugganu, M Gambno, and JP Bros. Enthalpes of formaton of lqud B-Ga-Sn tn alloys at 723 K The analytcal representaton of the total and partal excess functons of mxng. Journal de Chme Physque, 72(1):83 88, [3] B Hallstedt and O Km. Thermodynamc assessment of the Al-L system. Internatonal journal of materals research, 98(10): , [4] JL Murray. The Al- (alumnum-scandum) system. Journal of Phase Equlbra, 19(4): , [5] J Ganguly. Thermodynamcs n earth and planetary scences. Sprnger Verlag, [6] J Røyset and N Ryum. Knetcs and mechansms of precptaton n an Al-0.2 wt.% alloy. Materals ence and Engneerng A, 396(1-2): , [7] B Noble and SE Bray. Use of the gbbs-thompson relaton to obtan the nterfacal energy of δ precptates n al-l alloys. Materals ence and Engneerng A, 266(1-2):80 85, [8] CJ Kuehmann and PW Voorhees. Ostwald rpenng n ternary alloys. Metallurgcal and Materals Transactons A, 27(4): , [9] C Moreau, A Allouche, and EJ Knystautas. Measurements of the dffuson rate of lthum n alumnum at low temperature by elastc recol detecton analyss. Journal of appled physcs, 58(12): , [10] SI Fujkawa. Impurty Dffuson of andum n Alumnum. In Defect and Dffuson Forum, volume 143, pages , [11] Y Yn and AP Alvsatos. Collodal nanocrystal synthess and the organc norganc nterface. Nature, 437(7059): , NATURE MATERIALS 13