CHAPER 7 NUCLEAION 7.1 INRODUCION I this text, we focus our attetio o crystallie solids that form from the melt. he process begis with the creatio of a cluster of atoms of crystallie structure, which may occur due to radom fluctuatios. his chapter describes the processes by which clusters form, ad the coditios that affect the rate of their productio ad survival. Clusters that are too small to survive will be called embryos, whereas those that are sufficietly large to be stable are termed uclei. hermodyamics plays a importat role, with the clusters becomig more pletiful ad viable as the temperature decreases below the equilibrium freezig poit. he cocept of ucleatio also applies i other cotexts, such as the first appearace of voids associated with gas porosity ad other defects. his is discussed i Part III. We begi with the classical theory of homogeeous ucleatio, where the clusters are assumed to appear spotaeously i a melt free of ay impurities. his theory is the easiest to uderstad, but it leads to predictios for ucleatio rates that are far differet from those observed i practice. his leads to the cocept of heterogeeous ucleatio, itroduced i Sect. 7., i which the clusters form preferetially o foreig particles i the melt, or at iterfaces such as those betwee the melt ad its cotaier. Particles may be preset simply as a result of ievitable impurities, or they may be itetioally added i a process called ioculatio, which promotes the formatio of uclei just below the equilibrium meltig poit, i.e., at small udercoolig. Relatively recet experimetal results show that the heterogeeous ucleatio rate depeds, i this case, o the size distributio of foreig particles i the melt, as discussed i Sect. 7.4. 7.2 HOMOGENEOUS NUCLEAION 7.2.1 Embryos ad uclei Cosider a homogeeous, pure liquid at uiform temperature above the equilibrium meltig poit f. he structure of the liquid, which ca be
274 Nucleatio measured for example by eutro diffractio, has short rage order over distaces of a few atomic radii, but is disordered over loger distaces. he atomic mobility is also high. his permits the formatio of clusters of a few atoms with a crystallie structure, but it is ot eergetically favorable for the crystal structure to persist i such small clusters. his was discussed i Chap. 2, ad we refer the reader to Fig. 2.2 for a graphical descriptio. Ideed, for > f, the Gibbs free eergy of the solid G m s is higher tha that of the liquid G m`. Whe < f, the free eergy diagram shows that G m s <Gm`, however this simple balace does ot accout for the surface eergy cotributio associated with the formatio of the solid-liquid iterface (Eq. (2.61)). Nor does it reveal the rate at which solid uclei appear. his is the subject of the preset chapter. Despite the fact that we discuss clusters cosistig of a relatively small umber of atoms, the liquid ad solid phases are cosidered to be cotiuous media, ad the iterface betwee the two phases is cosidered to be a sharp surface of separatio. For the sake of simplicity i the presetatio, the solid cluster is assumed to be a sphere of radius R cotaiig N atoms. We treat the solid ad liquid as cotiua, which implies that R is much larger tha the actual dimesios of the iterface, which is typically a few atomic diameters thick. We assume that both phases are homogeeous, with molar free eergies G m s ad G m` i the solid ad liquid phases, respectively, ad that the iterfacial eergy is give by. he free eergy of a liquid cotaiig a solid particle was described i detail i Sect. 2.4.2, ad the results from those derivatios are reused here. he differece i free eergy betwee a system cotaiig a solid particle of volume V s ad surface area A i cotact with the liquid, ad a etirely liquid system is give by G m` G m s G = V s V m + A (7.1) It was show i Eq. (2.21) that for a small udercoolig, G m s G m` Sf m, where Sf m is the molar etropy of fusio. Furthermore, from Eq. (2.6) we have s f = Sf m/v m. Substitutig these results ito Eq. (7.1), ad replacig V s ad A by appropriate values for a sphere of radius R, yields G = 4 R s f +4 R 2 (7.2) Figure 7.1 shows the three terms i Eq. (7.2), for the particular case of pure Al, whe > 0, i.e., < f. he first term o the right had side is proportioal to R ad egative, idicatig that eergy is released by solidificatio whe >0. he secod term is quadratic i R ad positive, correspodig to the eergy pealty associated with the creatio of a surface. For small R, the surface eergy pealty exceeds the volumetric liberatio of eergy, whereas the volumetric cotributio domiates at larger radii. his creates a maximum i G called the homogeeous ucleatio barrier, as idicated i Fig. 7.1. he radius R c at which this occurs is determied by differetiatig Eq. (7.2) with respect to R ad
Homogeeous ucleatio 275 4 10 15 2 10 15 Volume term Area term G(R) Free eergy [J] 0 2 10 15 G homo R c 4 10 15 0 1 10 8 2 10 8 10 8 4 10 8 5 10 8 6 10 8 R [m] Fig. 7.1 Surface, bulk ad total free eergies of a spherical solid as fuctios of its radius for a fixed udercoolig =5K. Property data for Al are tabulated i able 7.1. settig the result equal to zero: R c = 2 s f = 2 (7.) where = /( s s f ) is the Gibbs-homso coefficiet. Substitutig this result ito Eq. (7.2) gives = 4 R 2 c = 16 ( s f ) 2 2 (7.4) A embryo of radius R c is called a critical ucleus, sice it is eergetically favorable for uclei with R<R c to melt, ad for R>R c to grow. Note that for < 0, correspodig to temperatures above f, both terms o the right had side are positive for all values of R, ad ay embryo of the solid that forms should always remelt. Let us ote that the exact geometry of the embryo is ot importat it chages the prefactor i Eq. (7.4), but ot i such a way as to affect the coclusios. able 7.1 Material properties for Al. Property Value Uits f 9 K 0.09 J m 2 s f 1.02 10 6 Jm K 1 9.12 10 8 Km V m 1.18 10 5 m mol 1
276 Nucleatio Key Cocept 7.1: Critical ucleus for homogeeous ucleatio he radius of a critical ucleus at udercoolig is give by: R c = he ucleatio barrier is give by: 2 s f = 2 = 4 R 2 c = 16 ( s f ) 2 2 We have actually already ecoutered this result twice i earlier chapters. First, i Chap. 2, we foud the relatioship betwee the udercoolig ad curvature at equilibrium, ad Eq. (7.) simply correspods to Eq. (2.62) with the mea curvature apple = Rc 1. We also foud this result i Sect. 5.4.1, where a aalysis of the growth rate of a spherical solid i a udercooled melt revealed that the sphere had a critical radius of R c =2 /. (See Eq. (5.187) ad the subsequet discussio.) he importat poit is that all of these aalyses are tied together: the cotributio of curvature to thermodyamic equilibrium leads to a coditio of equilibrium, where a particle of a give radius R grows or shriks, depedig o its size relative to R c. his equilibrium poit is ustable because d 2 ( G)/dR 2 < 0, idicatig that the Gibbs free eergy decreases for both R < R c ad R > R c. Clusters of atoms smaller tha the critical ucleus are called embryos, while those larger tha R c are called uclei. Example 7.1 calculates the critical ucleus size ad the magitude of the homogeeous ucleatio barrier as a fuctio of the udercoolig for pure Al. Example 7.1: Critical radius for ucleatio i Al Cosider a melt of pure Al, with the properties tabulated i able 7.1. Determie the critical radius R c, the umber of atoms N c, cotaied i a critical ucleus, ad the free eergy chage upo solidificatio as a fuctio of the udercoolig. he solutio is a straightforward applicatio of the equatios just derived. We have: R c = 2 = 2(9.12 10 8 Km) = 1.82 10 7 Km
Homogeeous ucleatio 277 N c = 4 N 0 V m R c = (4 )(6.02 102 atoms mol 1 )(1.82 10 7 Km) (11.8 10 6 m mol 1 ) N c = 1. 109 atoms K For coveiece i plottig, is ormalized by the characteristic eergy associated with the thermal fluctuatios of oe atom, k B beig Boltzma s costat: = 16 ( s f ) 2 2 = (16 )(0.09 J m 2 ) ()(1.8 10 2 JK 1 )(1.02 10 6 Jm K 1 ) 2 1 2 = 9.9 108 K 2 he results are plotted below. 0 0 20 20 R c D [K] 40 60 N c D / D [K] 40 60 80 80 100 10 2 10 4 10 6 10 8 N c or DG homo /kb 100 10 9 10 8 10 7 10 6 R c [m] A closer examiatio of the results leads us to doubt the validity of the homogeeous ucleatio theory. hese doubts become eve more apparet i the ext sectio. For udercoolig values up to about 20 K, the critical radius is larger tha 10 m, which is sufficietly large for the assumptios of cotiuum behavior ad a sharp iterface to be reasoable. However, a cluster of this size cotais more tha 100, 000 atoms, ad thus the formatio of a critical ucleus of eve this small size requires a ulikely amout of orgaizatio i the liquid. O the other had, if we cosider a cluster cotaiig a more probable umber of atoms, i.e. approximately 1000,
278 Nucleatio the correspodig udercoolig is almost 100 K. Such large udercooligs are rarely achieved i practice, uless very rapid solidificatio techiques are used. Further, sice the critical ucleus size the approaches atomic dimesios, the assumptios made i this theory are ulikely to apply. hese icosistecies i the theory of homogeeous ucleatio lead us to propose alterative mechaisms for ucleatio i practical applicatios. 7.2.2 Nucleatio rate he relatioship betwee cluster size ad free eergy is importat as it allows a estimatio of the rate at which clusters of a give size will appear. We have see that i a udercooled melt ( >0), the formatio of a cluster or embryo of size R requires a additioal free eergy G(R), which exhibits a maximum of height at R = R c, as show i Fig. 7.1. O the other had, G(R) is a icreasig fuctio of R for a superheated melt ( <0). We ow proceed to compute the rate of formatio of critical uclei as a fuctio of temperature. Deote the desity of atoms i the liquid as `, ad the desity of clusters of radius R i equilibrium with the liquid as R. he cluster of radius R cotais N atoms, so we also refer to the desity of such clusters as N. Most ucleatio models posit that ucleatio is a thermally activated process, i.e., the eergy of atoms follows a Maxwell-Boltzma distributio, such that R ` G(R) or N ` G(N) (7.5) As G(N = 1) = 0 for a cluster cosistig of a sigle atom, the limitig case N=1 = ` is appropriate. Strictly speakig, Eq. (7.5) is ot exact sice the formatio of dimers, trimers, etc., aturally depletes the melt of sigle atoms. However we assume that the desity of such clusters is small. Further, we make a more drastic assumptio: we assume that this relatioship cotiues to be valid below the meltig poit, whe clusters of critical size R c could possibly form. Ideed, the formatio of such clusters which will the grow clearly should distort the equilibrium distributio give by Eq. (7.5). A more detailed discussio of this poit ca be foud i the text by Christia [1]. For the curret discussio, we oetheless assume that the desity of clusters of critical size R = R c is small with respect to `, ad give by Eq. (7.5). he distributio of embryo sizes is show schematically i Fig. 7.2. For values of R>R c, the growth of uclei ivalidates Eq. (7.5), idicated i the figure by a dashed lie that cotiues the distributio i this regio. hus, we are iterested i fidig c, the desity of embryos that reach the critical radius R c, which ca be foud by settig G(R) = i Eq. (7.5), ad usig Eq. (7.4) to obtai c ` 16 ( s f ) 2 (7.6)
Homogeeous ucleatio 279 R 1 R c R G R Fig. 7.2 he desity of embryos as a fuctio of their radius. A ucleus of critical size R c will grow if it maages to add oe more atom. he rate at which this occurs is proportioal to the atomic vibratio frequecy 0 ad the probability of capturig a atom at the surface, p c. Combiig these elemets, we ca deduce the ucleatio rate, I homo. Key Cocept 7.2: Homogeeous ucleatio rate he ucleatio rate i a homogeeous pure liquid is give by: 16 I homo = 0 p c c = 0 p c ` exp {z } ( s f ) 2 I0 homo (7.7) he prefactor I0 homo is ot strogly temperature-depedet, ad its value ca be estimated from its three compoet terms. As a example, we use the data for Al give i able 7.1 to compute ` = N 0 /V m 6 10 28 atoms m. he atomic vibratio frequecy beig about 10 1 s 1 at f, let us assume that there is o difficulty i attachig a atom to the surface of the ucleus, i.e., p c 1. Substitutig these values ito Eq. (7.7) yields I homo = 6 10 41 m s 1 exp 16 ( s f ) 2 (7.8) he ucleatio rate depeds strogly o temperature through the two competig terms i the deomiator of the expoetial i Eq. (7.8). he decrease i the ucleatio eergy barrier cotributes the term 2, which teds to make the ucleatio rate icrease very strogly with icreasig udercoolig. he Maxwell-Boltzma distributio cotributes the term, correspodig to the decreasig mobility of atoms at low temperatures.