Heterogeneous nucleation and adsorption

10.1098/rsta.2002.1137 Heterogeneous nucleation and adsorption By B. Cantor University of York, York YO10 5DD, UK Published online 27 January 2003 This paper discusses the heterogeneous nucleation of solidification, treating the fundamental process as taking place by adsorption on the surface of the heterogeneous nucleant. A theoretical framework for adsorption and nucleation catalysis is described, and some related experimental results are discussed. Keywords: nucleation; catalysis; solidification; adsorption; surface energy 1. Introduction Solidification is almost always nucleated heterogeneously. The solid forms initially by catalysis on extraneous impurities which are present within the parent liquid. We are usually incapable of removing heterogeneous nucleating catalysts and are often ignorant of their nature. These facts have drastic consequences for the progress of most solidification reactions. The solid forms with a grain size, shape and distribution determined by the uncontrolled initial distribution of heterogeneities. It is, therefore, near impossible to predict or control the progress of the solidification process ab initio, and it can only be understood semi-empirically via forensic investigation after solidification is complete. This paper discusses the heterogeneous nucleation of solidification, concentrating on its description as an adsorption process. 2. Classical nucleation theory (a) Homogeneous nucleation In classical homogeneous nucleation theory (Christian 1975; Turnbull 1950), solid particles form initially throughout the bulk of an undercooled liquid. For a small particle, the reduction in energy caused by its formation is insufficient to overcome the energy barrier associated with its surface. Small particles increase the free energy of the material and can only form via random thermal fluctuations which are more likely to decay than grow. Beyond a critical size of particle, however, the reduction in energy caused by its formation is sufficient to overcome the energy barrier, and the reaction proceeds via continued particle growth. For a spherical particle, the critical particle radius r and energy barrier G are given by r = 2γ G, (2.1) One contribution of 15 to a Discussion Meeting Nucleation control. G = 16πγ3 3 G 2, (2.2) 361, 409 417 409 c 2003 The Royal Society

410 B. Cantor Table 1. Nucleation undercoolings, contact angles and nucleation site densities obtained from kinetic analysis of the solidification of liquid droplets embedded in a catalytic matrix using classical spherical-cap heterogeneous nucleation theory catalyst undercooling contact angle site density droplets matrix (K) (deg) (per droplet) Al ZrAl 3 0 0 Al NiAl 3 0 0 Pb Cu 0.5 4 10 12 Pb Al 22 21 10 6 Pb Zn 30 23 10 8 In Al 13 27 10 7 Cd Al 56 42 30 Sn Al 104 59 100 where γ is the solid liquid interfacial energy, G = L T/T m is the driving force for solidification, L is the latent heat, T m is the melting point, T = T m T is the undercooling, and T is the temperature. The rate of nucleation J is proportional to the probability of forming solid nuclei with excess energy G ( ) G J = J 0 exp, (2.3) kt where k is Boltzmann s constant and the pre-exponential factor J 0 depends upon the density of available nucleation sites (Cantor & Doherty 1979; Kim et al. 1991). (b) Heterogeneous nucleation In classical heterogeneous nucleation theory (Christian 1975; Turnbull 1950), the nucleation process is aided by a catalyst surface which reduces the magnitude of the energy barrier. Solid nuclei form as spherical cap-shaped particles at the catalyst liquid surface. The energy barrier G is then given by G = 16πγ3 f(θ) 3 G 2, (2.4) where f(θ) = 1 4 (2 3 cos θ + cos3 θ) is the catalytic efficiency, θ is the contact angle at the catalyst solid liquid triple point, cos θ =(γ cl γ cs )/γ, and γ cl and γ cs are the catalyst liquid and catalyst solid surface energies, respectively. (c) Efficient catalysis There are considerable problems with classical heterogeneous nucleation theory, particularly when the catalytic efficiency is high, i.e. when θ, f(θ) and G approach zero (Kim & Cantor 1994). The nucleus is only a few atoms thick when θ is less than 20 and falls below a monolayer when θ is less than 10. A spherical capshaped nucleus becomes impossible, and f(θ) is no longer an adequate measure of catalytic efficiency. Calorimetric measurements of latent heat given out during the solidification of liquid droplets embedded in a surrounding catalytic solid matrix do not agree well with the kinetics in equation (2.3), and, for efficient catalysis, give

Heterogeneous nucleation and adsorption 411 unphysical values for the density of heterogeneous nucleation sites (Kim & Cantor 1994). Table 1 shows typical results (Cantor & O Reilly 1997), with nucleation-site densities many orders of magnitude less than one per droplet when θ is low, which is clearly impossible. 3. Adsorption (a) Surface energy When catalysis is efficient, it seems more reasonable to regard heterogeneous nucleation as an adsorption process. Nucleation takes place by dynamic atom-by-atom adsorption at the catalyst surface, rather than by the formation of a bulk spherical cap. Coudurier et al. (1978) and Kim & Cantor (1994) have shown that the surface energy γ of a catalyst A adsorbing atoms from a solidifying liquid B is given by γ = x ij(µ 0 ij µ ij )+RT x ij ln x ij + δ ijkl mx ijx klw ijkl + δ ijkl nx ij x klw ijkl δ ijkj mx ij x kj w ijkj, (3.1) where the summation is over all i, k =A, B and j, l =S,L;µ 0 and µ are the standard state and equilibrium chemical potentials, δ ijkl = 0 for i = k and j = l, δ ijkl =1 otherwise; w ijkl = 1 2 zn 0(2e ijkl e ijij e klkl ) are interaction parameters; e ijkl are bond energies between ij and kl atoms; R is the gas constant; N 0 is Avogadro s number; z is the coordination number and m and n are fractional coordination numbers parallel and perpendicular to the surface, respectively. Equation (3.1) assumes pairwise atomic interaction energies, regular surface entropy, and cellular catalyst, liquid and solid phases with the same coordination number. The first term on the right-hand side is the chemical energy of removing surface atoms from the catalyst and liquid phases; the second term is their entropy of mixing; the third term is their internal bond energy; the fourth term is the bond energy between the surface atoms and the catalyst and liquid phases; and the fifth term is the broken bond energy needed to create a surface within the catalyst and liquid phases. (b) Immiscible systems When the catalyst A and the solidifying liquid B are immiscible, equation (3.1) can be simplified to (Cantor 1994) γ φr T + RT {φ ln φ +(1 φ) ln(1 φ)} + 3 2 φ(1 φ)rt m + 1 4 {φ(w S +3RT m )+(1 φ)(w L +3RT ma )}, (3.2) where φ = x is is the fraction of a monolayer adsorbed on the surface, w S and w L are A B interaction parameters in the solid and liquid, respectively, and T ma is the catalyst melting point. (c) Faceting and adsorption Figure 1 shows schematically the variation of surface energy γ with adsorption φ, from equation (3.2) with (a) T ma T m and w S >w L, and (b) T ma >T m and

412 B. Cantor (a) surface energy high T (b) surface energy high T low T low T adsorption adsorption Figure 1. Variation of surface energy with adsorption: (a) T ma T m and w S >w L; (b) T ma >T m and w S w L. w S w L. The second and third terms on the right-hand side of equation (3.2) are symmetrical in φ, and control the extent of surface faceting. A minimum at φ = 1 2 corresponds to a rough surface, and a maximum at φ = 1 2 and two minima at low and high φ correspond to a faceted surface. Faceting takes place when the temperature falls below T facet 3 4 T m. (3.3) The first and fourth terms on the right-hand side of equation (3.2) are linear in φ, corresponding to a straight line connecting the points γ(φ = 0) and γ(φ = 1), and control the extent of surface adsorption. Liquid B atoms are favoured at the surface for γ(φ =0)<γ(φ = 1) since the minimum surface energy is then at low φ, and solid B atoms are favoured for γ(φ =0)>γ(φ = 1) since the minimum surface energy is then at high φ. Adsorption of solid B always takes place at sufficiently low temperatures because of the first term in T which is the driving force for solidification. Adsorption takes place when the undercooling increases above T ads 1 (w S w L ) 3 4 R 4 (T ma T m ). (3.4) Equation (3.4) shows that a small nucleation undercooling and efficient nucleation catalysis are promoted by a large difference between the melting points of A and B, and a small difference between the solid and liquid immiscibilities of A and B. In figure 1a, T ma T m and w S >w L. From equation (3.4), T ads is large and A is a poor catalyst for heterogeneous nucleation of B. From equation (3.3), adsorption takes place below T facet, with two minima in γ at high and low φ, so the transition is sharp. In figure 1b, T ma >T m and w S w L. From equation (3.4), T ads is small and A is a good catalyst for heterogeneous nucleation of B. From equation (3.3) adsorption takes place above T facet, with a single minimum in γ at φ = 1 2, so the transition is less sharp. (d) Ternary dopants When a ternary element C is added, equation (3.1) for the surface energy must be summed over all i, k =A, B, C and j, l =S, L. When the catalyst and solidifying liquid are immiscible, and C dissolves only in the catalyst, equation (3.2) remains valid, with the catalyst melting point replaced by its liquidus temperature.

Heterogeneous nucleation and adsorption 413 Table 2. Undercoolings and lattice mismatch for Pb and Cd solidification catalysed by Al doped with Ge Ge content undercooling mismatch droplets (ppm) (K) (%) Cd 0 56 4.03 5000 65 3.96 Pb 0 22 18.36 3100 32 18.34 4. Nucleation kinetics and microstructure (a) Droplet undercoolings Nucleation kinetics are difficult to investigate because they are sensitive to trace impurities. This problem can be overcome by dividing the liquid into many small droplets, to segregate the impurities into relatively few droplets and render them harmless (Turnbull 1950). Heterogeneous nucleation is studied by distributing the droplets throughout a solid catalytic matrix (Wang & Smith 1950; Southin & Chadwick 1978), with the kinetics monitored by calorimetry, and the solidified microstructures monitored by electron microscopy (Moore et al. 1987). Ab initio thermodynamic calculations using equation (3.1) predict that solidification of Pb should be catalysed very efficiently by Ag, with virtually no undercooling below the melting point, but that solidification of Sn should be catalysed only weakly by Al, with an undercooling of 140 K (Kim & Cantor 1994). Droplet studies give reasonable agreement, with measured undercoolings of 4 K and 104 K, respectively (Southin & Chadwick 1978; Kim & Cantor 1991). (b) Catalyst melting point Equation (3.4) predicts that adsorption and heterogeneous nucleation should become less efficient as alloying elements are added to lower the catalyst melting point. Droplet measurements give reasonable agreement, with undercoolings increasing linearly with falling catalyst melting point, by up to 10 K for solidifying Pb and Cd catalysed by a variety of different Al alloys (Cantor 1994). (c) Epitaxial nucleation Table 2 shows measured droplet undercoolings for solidification of Pb and Cd catalysed by Al doped with up to 5000 ppm Ge (Zhang & Cantor 1990; Ho & Cantor 1992). At these doping levels, Ge is insoluble in Pb and Cd but fully soluble in Al. In each case, droplet solidification is catalysed epitaxially by Al, producing faceted single crystal particles and a catalyst particle orientation relationship with parallel close-packed planes and directions. Table 2 includes values of atomic mismatch across the epitaxial close-packed planes. Ge additions increase the undercooling required to initiate Pb and Cd solidification, i.e. Ge dissolved in Al reduces its catalytic effect. Turnbull & Vonnegut (1952) predicted that catalytic efficiency is reduced by increasing lattice mismatch, but Ge in Al reduces the Al Pb and Al Cd atomic

414 B. Cantor Table 3. Undercoolings for Si solidification catalysed by Al doped with P and Na P content Na content undercooling (ppm) (ppm) (K) <0.25 0 60 2 0 9 0.5 80 7 2 850 48 (a) (b) Figure 2. (a) AlP layer on Al nucleating Si solidification. (b) TiAl 3 layer on TiB 2 nucleating Al solidification. mismatch, i.e. catalysis is dominated by chemical rather than structural compatibility at the catalyst surface. (d) Si modification Table 3 shows measured droplet undercoolings for solidification of Si catalysed by Al doped with up to 2 ppm P and 850 ppm Na (Ho & Cantor 1995a, b). High-purity Al is a poor catalyst, and multiple nucleation at a high undercooling of 45 60 K leads to a fine-scale submicrometre Si microstructure. Catalysis is improved dramatically by doping with P at levels as low as 0.5 2 ppm. P is adsorbed onto the Al surface to form a catalytic AlP layer, and at high P levels AlP particles precipitate within the liquid, nucleating large twinned single crystals of Si at a few degrees of undercooling. Nucleation takes place with cube cube orientation relationships between Al, AlP and Si. Figure 2a shows a high-resolution electron micrograph of an AlP adsorbed monolayer. Doping with O produces similar catalytic effects. Na additions poison the catalytic effect, probably by a preferential reaction to form Na 3 P, and high liquid undercoolings are restored, together with a modified fine-scale Si microstructure. Recent studies indicate that Sr removes P in a similar way. (e) Boride nucleation Nucleation microstructures are also difficult to investigate because of the difficulty of preventing rapid post-nucleation growth. Schumacher & Greer (1994, 1997)

Heterogeneous nucleation and adsorption 415 have shown how nucleation at a catalyst surface can be frozen by replacing a highdiffusivity solidifying liquid with a low-diffusivity crystallizing glass. Solidification of Al alloys is well known to be nucleated heterogeneously by TiB 2 particles. In Albased glasses, crystallization of primary Al is nucleated by excess Ti adsorbed onto the TiB 2 particle surfaces to form a catalytic TiAl 3 layer, as shown in figure 2b. Zr additions poison boride nucleation of Al by substituting for Ti in the boride to form (Ti,Zr)B 2, and by co-adsorbing with the excess Ti to replace the catalytic TiAl 3 layer with an ineffective (Ti,Zr)Al 3 layer (Bunn et al. 1999). Si additions poison boride nucleation, by co-adsorbing with the excess Ti to replace the catalytic TiAl 3 layer with an ineffective Ti 2 Si layer (McKay 2002). (f ) Nucleation of secondary intermetallics In 1xxx series Al Fe Si alloys, addition of 500 ppm V is effective in nucleating metastable AlFe m particles instead of the stable AlFe 3 phase (Allen et al. 1998, 1999, 2001). This seems to be associated with a catalytic V-rich adsorbed layer at the primary Al interface, but direct evidence has not yet been obtained. When TiB 2 particles are present, the critical V content is reduced to below 100 ppm, but the mechanism for this is unclear. TiB 2 particles are also effective catalysts for secondary intermetallic phases in other Al alloys (Hsu 1999; Sha 2001). In 6xxx series Al Cu Mg alloys, TiB 2 particles are effective at nucleating metastable β-alfesi particles instead of the stable α-alfesi phase (Fe and Si are usually present as impurities in 6xxx alloys). 5. Conclusions There have been important recent advances in understanding the mechanisms and kinetics of heterogeneous nucleation of solidification. Weak catalysis can be described by classical macroscopic spherical-cap heterogeneous nucleation theory, but the theory fails for efficient catalysis with contact angles below 10 20 and undercoolings below 10 20 K. Efficient catalysis is better described as a microscopic atom-by-atom adsorption process at the catalyst surface. A pairwise interaction energy, regular entropy, adsorption model of the catalyst surface gives good ab initio predictions of nucleation undercooling and also predicts correctly the poisoning effect of alloying by reducing the catalyst melting point. Droplet and glassy alloy experimental techniques confirm that efficient catalysis usually proceeds by adsorbing a catalytic nucleating near-monolayer on a substrate. Notable examples include Si nucleation by a monolayer of AlP on Al, and Al nucleation by a monolayer of TiAl 3 on TiB 2. References Allen, C. M., O Reilly, K. A. Q., Cantor, B. & Evans, P. V. 1998 Intermetallic phase selection in 1xxx Al alloys. Prog. Mater. Sci. 43, 89 170. Allen, C. M., O Reilly, K. A. Q., Evans, P. V. & Cantor, B. 1999 The effect of vanadium and grain refiner additions on the nucleation of secondary phases in 1xxx Al alloys. Acta Mater. 47, 4387 4404. Allen, C. M., O Reilly, K. A. Q. & Cantor, B. 2001 Effect of semisolid microstructure on solidified content in 1xxx Al alloys. Acta Mater. 49, 1549 1564.

416 B. Cantor Bunn, A. M., Schumacher, P., Kearns, M. A., Boothroyd, C. B. & Greer, A. L. 1999 Grain refinement by Al Ti B alloys in aluminium melts: a study of the mechanisms of poisoning by zirconium. Mater. Sci. Technol. 15, 1115 1123. Cantor, B. 1994 Embedded droplet measurements and an adsorption model of the heterogeneous nucleation of solidification. Mater. Sci. Engng A 178, 225 231. Cantor, B. & Doherty, R. D. 1979 Heterogeneous nucleation in solidifying alloys. Acta Metall. 27, 33 46. Cantor, B. & O Reilly, K. A. Q. 1997 The nucleation of solidification in liquid droplets embedded in a solid matrix. Curr. Opin. Solid. State. Mater. Sci. 2, 318 323. Christian, J. W. 1975 The theory of transformations in metals and alloys. Oxford: Pergamon. Coudurier, L., Eustathopoulos, N., Desre, P. & Passerone, A. 1978 Rugosite atomique et adsorption chimique aux interfaces solide liquide des systèmes metalliques binaires. Acta Metall. 26, 465 475. Ho, C. R. & Cantor, B. 1992 Effect of Ge on the heterogeneous nucleation of Cd solidification by Al. Phil. Mag. A 66, 141 149. Ho, C. R. & Cantor, B. 1995a Heterogeneous nucleation of Si solidification in Al Si and Al Si P alloys. Acta Metall. Mater. 43, 3231 3246. Ho, C. R. & Cantor, B. 1995b Modification of hypereutectic Al Si alloys. J. Mater. Sci. 30, 1912 1920. Hsu, C. 1999 Solidification of 6xxx series aluminium alloys. DPhil thesis, University of Oxford, Oxford, UK. Kim, W. T. & Cantor, B. 1991 Solidification of Sn droplets embedded in an Al matrix. J. Mater. Sci. 26, 2868 2878. Kim, W. T. & Cantor, B. 1994 An adsorption model of the heterogeneous nucleation of solidification. Acta Metall. Mater. 42, 3115 3127. Kim, W. T., Zhang, D. L. & Cantor, B. 1991 Nucleation of solidification in liquid droplets. Metall. Trans. A 22, 2487 2501. McKay, B. 2002 Heterogeneous nucleation in Al Si alloys. DPhil thesis, University of Oxford, Oxford, UK. Moore, K. I., Chatopadhyay, K. & Cantor, B. 1987 In situ transmission electron microscope measurements of solid Al-solid Pb and solid Al-liquid Pb surface energy anisotropy in rapidly solidified Al 5wt%Pb. Proc. R. Soc. Lond. A 414, 499 507. Schumacher, P. & Greer, A. L. 1994 Heterogeneously nucleated -Al in amorphous aluminium alloys. Mater. Sci. Engng A 178, 309 313. Schumacher, P. & Greer, A. L. 1997 On the reproducibility of heterogeneous nucleation in amorphous Al 85, Ni 10, Ce 5 alloys. Mater. Sci. Engng A 226 228, 794 798. Sha, G. 2001 Intermetallic phase selection in 6xxx series Al alloys. DPhil thesis, University of Oxford, Oxford, UK. Southin, R. T. & Chadwick, G. A. 1978 Heterogeneous nucleation in solidifying metals. Acta Metall. 26, 223 231. Turnbull, D. 1950 Formation of crystal nuclei in liquid metals. J. Appl. Phys. 21, 1022 1028. Turnbull, D. & Vonnegut, B. 1952 Nucleation catalysis. Ind. Engng Chem. Res. 44, 1292 1298. Wang, C. C. & Smith, C. S. 1950 Undercooling of minor liquid in binary alloys. Trans. Metall. Soc. AIME 188, 136 138. Zhang, D. L. & Cantor, B. 1990 Effect of Ge on the heterogeneous nucleation of Pb solidification by Al. J. Cryst. Growth 104, 583 592.

Heterogeneous nucleation and adsorption 417 Discussion F. R. N. Nabarro (School of Physics, University of the Witwatersrand, Johannesburg, South Africa). How do you distinguish a solid B atom and a liquid B atom in a surface. A B atom is just a B atom. Solid or liquid refer to its relation to its neighbours. B. Cantor. The solid and liquid are treated as cellular phases with the same coordination number, so the difference is expressed via different interaction energies with neighbouring atoms.