Grain refinement of alloys by inoculation of melts

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Grain refinement of alloys by inoculation of melts Phil. Trans. R. Soc. Lond. A 2003 361, 479-495 doi: 10.1098/rsta.2002.1147

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This journal is © 2003 The Royal Society


10.1098/ rsta.2002.1147

Grain re¯nement of alloys by inoculation of melts By A. L. G r e e r Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK Published online 28 January 2003

Recent progress in understanding the inoculation of aluminium melts is reviewed. Transmission electron microscopy of inoculant particles in a metallic glass reveals details of the mechanism of nucleation of aluminium grains. While such studies de ne some of the conditions under which inoculation is e¬ective or not, they do not permit a prediction of grain size. Unusually for a nucleation-related phenomenon, quantitative prediction is possible. For potent inoculation such as is practised in aluminium alloys, grain initiation is limited by inoculant particle size, occurring rst on the largest particles. Simple thermal models can then describe the dependence of grain re nement on alloy content and processing conditions, and enable consideration of inoculant design. Keywords: aluminium alloys; grain re¯nement; ino culation; metallic glass; nucleation; solidi¯ cation

1. Introduction Inoculation is the addition of solid particles to a metallic melt to act as nucleant catalysts for the formation of ne equiaxed, rather than columnar, grains. Grain re nement brings many bene ts in the solidi cation process itself and in the properties of the as-cast material. Inoculation is almost universally practised in casting of aluminium and its alloys (McCartney 1989); this is its widest industrial use. Recent signi cant progress in understanding the inoculation of aluminium may also have relevance for inoculation of melts or solutions in a wide range of systems. Heterogeneous nucleation on a solid substrate is normally treated using the spherical cap geometry in the classical theory (this geometry with contact angle ³ is shown in gure 3, inset (i)). The nucleation kinetics are very dependent on ³ , and cannot be predicted reliably, as ³ and the underlying interfacial energies are not accurately measured or calculated. Furthermore, as discussed by Cantor (2003) and earlier by O’Reilly & Cantor (1995), the classical theory breaks down for potent nucleation catalysis (small ³ ), which is exactly the case of interest for melt inoculation. Lastly, heterogeneous nucleation often involves a variety of particles with ill-de ned characteristics. The nucleation stage in solidi cation modelling has therefore been treated with adjustable parameters; this paper reviews recent work making quantitative predictions from independently determined parameters. One contribution of 15 to a Discussion Meeting `Nucleation control’. Phil. Trans. R. Soc. Lond. A (2003) 361, 479{495

479

° c 2003 The Royal Society


A. L. Greer

480 2000

(iii)

grain size (µm)

1500

zirconium addition

1000 (ii)

(iv) 500

0

(i) 1

10

100

1000

holding time (min) Figure 1. Grain diameter of CP-Al inoculated with 1 part per thousand (ppt) Al{5Ti{1B as a function of holding time before casting. (i) A standard re¯ner shows little fade with holding time at 800 ¯C. (ii) A poor re¯ner (held at 760 ¯C) converges on the standard behaviour. (iii) Poisoning of re¯ner action at 800 ¯C after addition of 0.05 wt% zirconium (after 10 min). (iv) A tantalum-modi¯ ed re¯ner (held at 800 ¯C) is resistant to the poisoning action of added zirconium. (Data from Bunn (1998).)

Di¬erent types of re ner can be more or less e¬ective, and even for one type there can be better and worse batches. E¬ectiveness may be considered in terms of suppression of columnar growth or of the degree of re nement of equiaxed grains. The re ning action is degraded (poisoned) in the presence of certain solutes in the melt. The grain size (diameter) in the as-cast product varies with melt chemistry and processing conditions. For each of these aspects of grain re nement by melt inoculation it would be good to have some predictability.

2. Grain re¯ners for aluminium The most widely used re ners for aluminium are based on the system Al{Ti{B; a typical composition is Al{5 wt% Ti{1 wt% B (designated Al{5Ti{1B). Measurements are more easily made in small melt volumes rather than in full-scale castings. The TP-1 test with a melt volume of ca. 100 cm3 (Boone et al. 1991) is widely used to assess re ners. When the test is modi ed to include temperature measurement, the maximum undercooling ¢Tm ax of an inoculated melt is found to be ca. 0.2 K (Greer et al . 2000), and even smaller values have been suggested (Johnsson et al . 1993); such small ¢Tm ax is a sign of potent catalysis. The Al{5Ti{1B re ner consists of particles of Al3 Ti and TiB2 in an ¬ -Al matrix. When added to the melt, the matrix melts and the dilution of the titanium content leads to rapid dissolution of the Al3 Ti, while the TiB2 remains stable. It is known that Al3 Ti, when present, is a much better nucleant than TiB2 , so the potent action of Al{Ti{B re ners has been taken to imply that some Phil. Trans. R. Soc. Lond. A (2003)


Grain re¯nement of alloys by inoculation of melts

481

Al-based glassy matrix

Al

TiB2

50 nm Figure 2. Bright-¯ eld transmission electron micrograph showing ¬ -Al crystallites formed on a TiB2 particle embedded in a glassy Al85 Y8 Ni5 Co2 (at.%) matrix and annealed. The labelled crystallite sits on a (0001) ledge on the (10¹10) prismatic face of the hexagonal platelet particle. (Reproduced with permission from Schumacher et al . (1998).)

Al3 Ti must be stabilized by the TiB2 particles (Schumacher et al. 1998). Indeed, the re ning action of Al{Ti{B re ners is potent only when the overall titanium content of the system is in excess of that needed to combine fully with the boron in the TiB2 particles. With melt stirring to avoid settling of particles, there is no degradation of re ner action with prolonged holding in the melt ( gure 1, curve (i)). Indeed, a re ner with poor performance can improve with holding ( gure 1, curve (ii)). The major problems with Al{Ti{B re ners are susceptibility to particle agglomeration and to poisoning; the e¬ect of zirconium is shown in gure 1, curve (iii). Newer Al{Ti{C re ners (with a typical composition Al{3Ti{0.15C) overcome some of these problems and are becoming more widely used (Schneider et al . 1998). However, the TiC nucleant particles in these dissolve slowly in the melt, precluding long holding times (Vandyousse et al . 2000).

3. Microscopical studies of nucleation As-cast microstructures do not readily reveal details of the nucleation stage of solidi cation, as these can be obscured by subsequent growth. In the case of Al3 Ti acting as a nucleant for ¬ -Al, the nucleant Al3 Ti may even be removed by peritectic reaction with the melt. These problems would be solved were growth to be stopped at an early stage; this is possible by using an Al-based metallic glass as an analogue of the liquid (Schumacher et al . 1998). An Al85 Y8 Ni5 Co2 (at.%) alloy with added Al{5Ti{1B re ner was melt-spun to a glassy ribbon. Figure 2 is a transmission electron micrograph of one of the TiB2 particles dispersed in the glassy matrix. For this sample the quench rate (105 to 106 K s¡1 ) was followed by a 30 min anneal at 483 K before thinning for transmission electron microscopy (TEM). Aluminium crystallites Phil. Trans. R. Soc. Lond. A (2003)


A. L. Greer

482

have started to form, but only on the f0001g basal faces of the TiB2 , which is in the form of hexagonal platelets. The ¬ -Al nucleates even on small f0001g-orientation ledges on the prismatic faces of the TiB2 . Re-entrant corners are not favoured nucleation sites, as contact with the prismatic faces is avoided. The f0001g-faces of the TiB2 appear to be coated with a thin layer of Al3 Ti (Cantor 2003; Schumacher et al . 1998), which is the actual nucleation substrate for ¬ -Al; when in melts of di¬erent chemistry it is absent, there is no nucleation of ¬ -Al, even in the metallic glass, where the driving force for crystallization is large. Electron di¬raction (Schumacher et al . 1998) shows that there are well-de ned orientation relationships: in all three phases, the close-packed planes and directions are parallel, and the in-plane repeat distances match within 6%. The layer of Al3 Ti may be stabilized on the TiB2 by adsorption e¬ects. The improvement seen in gure 1, curve (ii) can be attributed to the development of a suitable layer on initially poor nucleant particles. In the presence of zirconium in the melt, similar TEM studies of the metallic glass show that the TiB2 particles are converted to ZrB2 ; this has a signi cantly larger repeat distance in the f0001g-plane, is not coated with an aluminide, and does not act as a nucleation substrate for ¬ -Al (Bunn et al . 1999). TEM studies have also shown that in the presence of tantalum in the melt a very stable Al3 Ta coating is formed on TiB2 . A re ner exposed to tantalum in the melt is remarkably resistant to zirconium poisoning ( gure 1, curve (iv)). These studies show the importance of both crystallography and chemistry for e¬ective nucleation. For commercial re ners, at best only 1% of the TiB2 particles succeed in nucleating grains. The TEM studies o¬er no explanation for this ine¯ ciency, as all the particles observed in the metallic glass are successful nucleants.

4. Modelling of grain size (a) The free-growth criterion The ine¯ ciency suggests that the growth of grains on early nucleation events must in some way sti®e later nucleation. Maxwell & Hellawell (1975) pointed out that this happens through latent heat release limiting the undercooling of the melt. This e¬ect is signi cant because typical thermal di¬usion lengths in solidifying inoculated aluminium are 102 {103 times the grain diameter. For quantitative predictions of re ner e¯ ciency it is necessary to understand the conditions for grain initiation on nucleant particles. Maxwell & Hellawell assumed classical heterogeneous nucleation on nucleants with uniform characteristics (most importantly, the contact angle ³ ). The temperature dependence of the nucleation rate is so strong that there is e¬ectively instantaneous nucleation at a well-de ned undercooling ¢Tn (Hunt 1984). Similarly, a sharp onset undercooling is predicted by the adsorption model for heterogeneous nucleation (Cantor 2003). Whether classical nucleation or adsorption is assumed, the known potency of inoculants implies that ¬ -Al nuclei forming on the nucleants are thin and ®at. In such a case, nucleation of ¬ -Al may be insu¯ cient to ensure initiation of a grain. For nucleation on the f0001g-faces of TiB2 particles, the sequence leading to grain initiation is shown in gure 3, inset (ii). Any initial nucleus can readily grow across the face of the nucleant particle to form a thin coating, but can then grow further only by reducing the radius of curvature of its interface with the melt. This radius cannot go below the critical value r ¤ for nucleation at the instantaneous temperature. If the diameter d of the particle is such that d < 2r ¤ , then free Phil. Trans. R. Soc. Lond. A (2003)



483

1.0

undercooling for grain initiation (ºC)

(i)

q

0.8

0.6

0.4

(ii) minimum-radius hemispherical cap

TiB2

D Tfg

TiB2 d

0.2

0

D Tn

2

6 4 particle diameter (µm)

8

10

Figure 3. The bold line indicates the undercooling necessary for grain initiation. The free-growth undercooling is calculated; the nucleation undercooling is schematic only. Inset (i) shows the classical spherical-cap model for heterogeneous nucleation. Inset (ii) shows a cap of ¬ -Al growing on an inoculant particle through the critical hemispherical condition.

growth of the crystal from the particle is not possible. It becomes possible when the undercooling is increased, thus reducing r ¤ . The critical condition for free growth of the crystal through the minimum-radius hemispherical shape is when d = 2r ¤ . The undercooling for free growth ¢Tfg and the nucleant particle diameter d are simply related by 4¼ ; (4.1) ¢Tfg = ¢Sv d where ¼ is the solid{liquid interfacial energy and ¢Sv is the entropy of fusion per unit volume. (Equation (4.1) is readily derived from the classical expression for the critical nucleus radius at small undercooling.) The undercooling ¢Tfg constitutes a barrier for the e¬ective initiation of a new grain. When a nucleus is formed, it has an interface with the liquid, but this interface must break free from the inoculant particle in order to have free growth of the new grain. Either ¢Tn or ¢Tfg may be the critical undercooling for grain initiation, depending on which is greater. The ¢Tfg curve in gure 3 shows how this undercooling varies with particle diameter. As will be seen later, particle diameters in typical re ners vary over a wide range, but many are of the order of 1 m m. This diameter would correspond to a free-growth undercooling of the order of 0.5 K; since this is similar to or greater than maximum measured undercoolings, nucleant particles are small enough for the barrier to free growth to be signi cant. Indeed, recent work has taken the free-growth barrier to be controlling (Bunn 1998; Greer et al . 2000), i.e. it is assumed that the condition ¢Tfg > ¢Tn applies within the known size range of inoculant particles, though this could not remain so for inde nitely large particles. In e¬ect, it is assumed that nucleation is so potent (i.e. ¢Tn is so small) that this Phil. Trans. R. Soc. Lond. A (2003)


A. L. Greer

484

liquidus D Tmax

(b)

temperature

temperature

(a)

D Tz

recalescence

time

distance

Figure 4. Schematics of the thermal conditions during solidi¯cation. In case (a), a small, spatially isothermal melt shows recalescence. In case (b), a directionally solidi¯ed melt shows a quasi-isothermal zone. In each case the undercooling limiting the grain initiation is indicated.

is not the limiting step in initiating grains (schematically illustrated in gure 3). To test this hypothesis it is necessary to have a quantitative model for the sti®ing of further grain initiation by latent heat release. A complete model for melt solidi cation would include solute partitioning and transport, heat transport and ®uid ®ow in addition to crystal nucleation and growth. Fortunately, simpli cations can be made, and two models have been proposed for the analysis of grain re nement: one based on an isothermal melt and one on directional solidi cation. The isothermal-melt model was proposed by Maxwell & Hellawell (1975). The approximation that the melt is spatially isothermal throughout solidi cation is based on the large thermal di¬usion length and is good for small melt volumes as in the TP-1 test. As the melt is cooled below the liquidus temperature, grains are initiated and grow, releasing latent heat. As more grains grow, the latent heat release rate increases and eventually exceeds the rate of external heat extraction, giving recalescence ( gure 4a). In the solidi cation of larger melts there are clear temperature gradients, but microstructural modelling (Vandyousse & Greer 2002) shows that there is a quasi-isothermal zone in which grains grow without signi cant impingement ( gure 4b). In this zone, the latent heat release balances the heat extraction. In either case, there is a well-de ned maximum undercooling of the melt, ¢Tm ax or ¢Tz , which limits grain initiation and controls re ner e¯ ciency. (b) Isothermal-melt model Given the small measured values of ¢Tm ax , the volume fraction crystallized at recalescence must be small (less than 0.1%). As far as solute is concerned, the grains can be considered to grow in isolation from each other. It is also reasonable to assume, and has been veri ed numerically (Greer et al . 2000), that the growth of the crystals at recalescence has not yet become dendritic. Accordingly, solute-di¬usion-limited growth of spherical grains is assumed. As shown by Maxwell & Hellawell (1975), to Phil. Trans. R. Soc. Lond. A (2003)



485

number of grains (m- 3)

1013

predicted 1012

Maxwell & Hellawell (B)

1011

(A)

measured

1010 1010

1011

1012 1013 number of particles (m- 3)

1014

1015

Figure 5. The number of grains per unit volume as a function of the number of re¯ner particles per unit volume, showing a general trend to lower e± ciency at higher addition level. Data from grain diameters measured in TP-1 tests (closed circles) are compared with predictions of the free-growth model (open circles) (Greer et al . 2000). The predictions are qualitatively di® erent from those of Maxwell & Hellawell (1975) and are a much better ¯t to the data. (Reproduced with permission from Greer et al . (2000).)

a good approximation, the growth rate (dr=dt, where r is radius and t is time) at a given undercooling is proportional to the di¬usivity of the solute in the liquid and inversely proportional to the growth-restriction parameter, Q, given by Q = m(k ¡

1)C0 ;

(4.2)

where m is the liquidus slope in the phase diagram of the solute with aluminium, k is the equilibrium partition coe¯ cient and C0 is the solute content in the alloy melt. For low-solute contents the Q values for the di¬erent solutes in an alloy are additive. Cooling curves ( gure 4a) are calculated by dividing time into a series of isothermal steps; the number of grains initiated in each step is calculated, as is the total latent heat release from the incremental growth of all the grains initiated in earlier steps. This latent heat release, together with the imposed external heat extraction during the time increment, is used to calculate the cooling rate and thereby the temperature during the next increment. Once recalescence starts, there are no more initiation events and the number of grains per unit volume Nv is determined. In the original application of this numerical model, Maxwell & Hellawell (1975) assumed classical heterogeneous nucleation on particles of uniform size, with only one nucleation event permitted per particle. Their prediction for the variation of Nv with the number of inoculant particles per unit volume for commercial-purity aluminium (CP-Al) under TP-1 test conditions is shown in gure 5. In regime A, particles are so sparse that there is time for nucleation to occur on every particle, i.e. 100% e¯ ciency. In regime B, Nv saturates as more particles are added and the e¯ ciency falls dramatically. Input parameters to the model are adjusted roughly to match the experimental data, but the predicted plateau in grain size is not observed. Phil. Trans. R. Soc. Lond. A (2003)


A. L. Greer

486

Table 1. Parameters used in the free-growth calculations (The material parameters are mostly for pure aluminium. The data sources are cited by Greer et al. (2000).)

quantity

symbol

¼ ¢ Sv ¢ Hv Cpv D dT =dt

mJ m¡ 2 J K¡ 1 m¡ J m¡ 3 J K¡ 1 m¡ m 2 s¡ 1 K s¡ 1

value 3

3

158 1:112 £ 106 9:5 £ 108 2:58 £ 106 2:52 £ 10¡ 9 3.5

1.4

0.2

1.2

0 - 0.2

1.0

± 10% error

- 0.4

0.8

- 0.6 fitted log-normal distribution

0.6

- 0.8

error fraction

relative number of particles

solid{liquid interfacial energy entropy of fusion per unit volume enthalpy of fusion per unit volume heat capacity of melt per unit volume di® usivity in melt (Ti in Al) cooling rate in TP-1 test

units

- 1.0

0.4

- 1.2

0.2

- 1.4 0

1

2 3 4 particle diameter (µm)

5

6

Figure 6. The diameter distribution of TiB2 particles in a commercial Al{5Ti{1B re¯ner, ¯tted to a lognormal distribution with relative error as shown. The black bars show the fraction of the particles calculated by the free-growth model to be active for an addition level of 1 ppt in CP-Al cooled at 3.5 K s¡ 1 .

In contrast, Greer et al . (2000) assumed grain initiation was limited by the freegrowth criterion (4.1). Nucleation occurs on every particle, but grain initiation occurs only when ¢Tfg is exceeded, which on cooling occurs rst on the largest particles. The number of grains initiated in each undercooling increment depends on the size distribution of the nucleant particles. Fortunately, and in marked contrast to parameters such as the contact angle needed for a nucleation-based model, the size distribution is measurable, often by scanning electron microscopy of polished sections of re ner (Bunn 1998; Greer et al. 2000); this for a commercial Al{5Ti{1B re ner is shown in gure 6. When measured distributions are used as input, the free-growth model gives the predictions shown in gure 5; the spread of particle size eliminates the division into two regimes predicted by Maxwell & Hellawell. The free-growth predictions are in remarkably good agreement with experiment, considering that the input parameters (table 1) are all determined independently and are not adjustable. Phil. Trans. R. Soc. Lond. A (2003)



487

Recalescence sets a lower limit to the particle size at which grain initiation occurs. The black bars in the distribution in gure 6 show the active particles for typical conditions. The agreement with experiment gives strong support to the free-growth model, and further support comes from the close match between predicted and measured values of ¢Tm ax (Greer et al. 2000). However, the overall shapes of measured cooling curves are not well matched and suggest a need to consider temperature gradients. Figure 5 shows that free-growth predictions are better than those based on nucleation on a uniform set of particles. Predictions from a nucleation-based model could be tted to experiment by, for example, assuming a distribution of contact angle values; however, the adjustable input parameters would not be susceptible to independent determination. (c) Directional-solidi¯cation model For most castings, an isothermal-melt model cannot be applicable, and there is no overall undercooling of the melt before grain initiation. Solidi cation proceeds through the casting; locally the solidi cation front can be taken to be planar, and its velocity is antiparallel to the one-dimensional heat ®ow. In the steady state there can be no recalescence, but rather a temperature pro le with distance, of the kind in gure 4b, which migrates into the liquid. As the melt is cooled, grains are initiated and then grow dendritically. In the quasi-isothermal zone the latent heat release balances the heat extraction, but after the dendritic envelopes have impinged, the remaining interdendritic liquid, its composition altered by solute partitioning, solidi es more slowly and cooling resumes. A simple model can be based on the analysis of the quasi-isothermal zone (Greer et al . 2002). For an assumed ¢Tz , a free-growth analysis as used above gives the number of grains growing in the zone. This undercooling also sets the dendritic velocity; as for the di¬usion-controlled growth of spheres considered above, the velocity at a given undercooling is proportional to the di¬usivity of the solute in the liquid and inversely proportional to Q (equation (4.2)). Assuming that solute partitioning to and from the growing dendrites involves only the liquid within the dendritic envelopes, a Scheil analysis can be applied to calculate the fraction solidi ed within the envelopes from ¢Tz . Finally, a Johnson{Mehl{Avrami analysis is applied to estimate an average rate of transformation within the quasiisothermal zone. The rate of latent heat release derived from this solidi cation rate is matched to the external heat extraction. Using these relationships, the undercooling in the zone ¢Tz , and thereby the number of grains per unit volume formed in the solidi cation, can be derived from the alloy composition and solidi cation conditions. (d) Predictions of free-growth modelling The measured data and the predictions of the isothermal-melt free-growth model shown in gure 5 are shown again in gure 7, but in terms of the usual measure of grain size, the mean linear intercept ·l, rather than in terms of Nv (= 0:5=·l 3 ) (Greer et al . 2000). Also included in gure 7 are the predictions of the directionalsolidi cation model for the same conditions. The two free-growth models give similar predictions, not only as shown in gure 7, but over a wide range of conditions; no distinction is made between the models in the following comparison of prediction Phil. Trans. R. Soc. Lond. A (2003)


A. L. Greer

488 600

grain size (µm)

500 400 300 200 100 0

2

4 6 addition level of refiner (ppt)

8

10

Figure 7. Grain diameter (mean linear intercept) for CP-Al inoculated with Al{5Ti{1B at various levels. The grain diameters measured in TP-1 tests (diamonds) are compared with free-growth model predictions based on an isothermal melt (circles) or on directional solidi¯ cation (squares). (Reproduced with permission from Greer et al . (2002).)

and experiment. Figure 7 illustrates that increasing the inoculation level re nes the grain size, which, however, tends to a limiting value. There are very similar trends of decreasing grain size with increasing cooling rate and with increasing growthrestriction factor Q. These are not illustrated here, but have been shown by Greer et al . (2000, 2002). The predicted e¬ect of the cooling rate is consistent with the sparse data available. In contrast, there are many data available on solute e¬ects, notably from Spittle & Sadli (1995), who studied various addition levels of Cr, Cu, Fe, Mg, Mn, Si, Zn and Zr. Their results show that Q is a good universal parameter for describing solute e¬ects. Furthermore, the free-growth predictions provide a good match to the measured decrease of grain size with increasing Q. Overall, the free-growth model agrees with the grain sizes measured in TP-1 tests, except when the measured sizes exceed 400 m m. Although not always evident from the standard TP-1 cross-section, such large sizes probably occur when there is directional columnar growth. The competition between equiaxed and columnar growth is not accounted for in the simple modelling performed so far. Although it has been noted that the temperature gradient appears not to be important in controlling the grain size in the equiaxed regime, high values certainly favour columnar growth (Hunt 1984). Free-growth modelling has also been applied with some success to the prediction of grain size in inoculated aluminium alloys undergoing directional solidi cation in a Bridgman furnace (Greer et al . 2002). It has, however, not yet been applied to larger-scale commercial castings, in which signi cant ®uid ®ow may contribute to grain initiation by transport of dendrite fragments.

5. Nucleation laws for solidi¯cation modelling As reviewed, for example, by Stefanescu et al. (1990), comprehensive computer simulation and modelling of solidi cation must include, among many factors, macromodPhil. Trans. R. Soc. Lond. A (2003)



489

elling of ®uid ®ow and heat transfer and micromodelling of phase nucleation and growth. As growth laws are often well established, it is likely that the derivation of the nucleation law is the greatest outstanding problem in developing a comprehensive predictive model. The early observations of Old eld (1966) showed that there can be a clear correlation between the population of grains and the maximum undercooling reached in the melt. In Old eld’s case, the population n was proportional to the square of the undercooling ¢T . This corresponds to a nucleation law for cooling in which the nucleation rate dn=d¢T is linearly proportional to ¢T . Such simple, empirical laws predict continuous nucleation rather than the e¬ectively instantaneous nucleation given by a simple classical nucleation analysis (Hunt 1984). They are easy to apply in solidi cation modelling and have been widely used. In the cellular-automaton{ niteelement modelling of Rappaz & Gandin (1993) now used in commercial software, for example, dn=d¢T has a Gaussian distribution about a maximum as a function of undercooling. Such laws envisage a spectrum of nucleation sites, each site becoming active instantaneously when its critical undercooling is reached. This approach is completely compatible with the free-growth model. With a measured particle size distribution as in gure 6, the nucleation rate dn=d¢T increases strongly, roughly exponentially, with increasing ¢T in the undercooling range of interest. The model that is now known as the free-growth model has been used for many years in di¬erent contexts. Turnbull (1953) found that small mercury droplets coated with a potent catalyst for nucleation of the solid exhibited athermal nucleation, in which the fraction of droplets solidifying on cooling depended only on the undercooling and not on time. Turnbull postulated the existence of surface patches, with a distribution of sizes, on the droplets, and noted that nuclei formed on the patches would become transformation nuclei on cooling, when the critical nucleation radius became less than the patch radius. This is equivalent to equation (4.1). Based on this analysis, Turnbull used the solidi cation kinetics to derive the size distribution of the patches, nding a roughly Gaussian variation of dn=d¢T with ¢T . Grain initiation controlled by a free-growth condition rather than a true nucleation barrier had earlier been invoked by Turnbull (1950) to analyse heterogeneous nucleation on mould walls and similar surfaces. The preservation of the solid phase in cavities in the mould wall even when the liquid is superheated, and the subsequent free growth from those cavities at su¯ cient undercooling, can be used to explain the dependence of the undercoolability of metallic melts on prior superheating.

6. Design of grain re¯ners In x 4 it was established that a free-growth model with the measured particle size distribution as input can be used to make quantitative predictions of grain re nement. It is then of interest to use the modelling to design re ners with optimized particle size distributions, for example to give minimum grain size for a given volume fraction of nucleant phase (i.e. TiB2 for Al{Ti{B re ners). Two types of ideal distribution in diameter have been used: normal (i.e. Gaussian) and lognormal. The former is mathematically more straightforward, but has the disadvantage of giving small numbers of particles with unphysical negative diameter; the latter is a better match to typical size distributions in metallurgical microstructures and ts well in the present case ( gure 6). Tronche & Greer (2000) have used normal distributions with Phil. Trans. R. Soc. Lond. A (2003)


A. L. Greer

490

100 80

grain size (µm)

200

60 150 40 100

20

. 50 0

1

2 3 4 5 6 mean particle diameter (m m)

7

8

number and volume efficiencies (%)

250

0

Figure 8. Calculations for CP-Al inoculated with a volume fraction of re¯ner particles corresponding to 2 ppt addition of Al{5Ti{1B and cooled at 3.5 K s¡ 1 , showing the grain diameter (bold line), volume e± ciency (closed squares) and number e± ciency (open squares) as a function of arithmetic mean particle diameter, for a ¯xed width of size distribution. The mean diameter in a commercial Al{5Ti{1B re¯ner (¯) is close to the minimum-grain-size optimum. (Reproduced with permission from Quested et al . (2002).)

the isothermal-melt model; Quested et al . (2002) have used lognormal distributions with the directional-solidi cation model. Qualitatively, the predictions are in agreement for the e¬ects of varying mean diameter and distribution width while holding constant the volume fraction of nucleant phase. The e¬ects of varying the arithmetic mean diameter are shown in gure 8, with the width of the lognormal distribution set to match that in a commercial Al{5Ti{1B re ner (geometric standard deviation, ¼ = 0:88). There is a clear minimum in grain size at intermediate values of mean particle diameter. Large particles (for xed volume fraction) are necessarily few and give large grain sizes. On the other hand, small particles, though numerous, become active only at greater undercooling, when the crystal growth rate is greater. The optimum re nement is at intermediate sizes. Interestingly, empirical development has brought the mean particle diameter in commercial re ners very close to this optimum. Figure 8 also con rms that, at the optimum re nement, the number e± ciency (i.e. number of grains per added particle) is low. Alternatively, the e¯ ciency can be evaluated in terms of the volume fraction of nucleant-phase (i.e. TiB2 )-initiating grains. As the active particles are the largest, this volume e± ciency is substantial (greater than 60%). While the behaviour in gure 8 has not been quantitatively tested, it nevertheless indicates that the low number e¯ ciencies usually quoted can be a misleading guide to re ner e¬ectiveness. In practice it may be more important to have a uniform grain size than a ne grain size. One reason why grain size might vary across a casting is non-uniformity in the cooling rate, the e¬ects of which are shown in gure 9. These calculations are for idealized re ners with a Gaussian particle-size distribution of xed width (standard deviation) 0.5 m m. As the average particle size is increased (again holding the total volume fraction of the nucleant phase constant), the e¯ ciency (measured as number or volume fraction) increases and the grain size becomes increasingly insensitive to Phil. Trans. R. Soc. Lond. A (2003)



491

600

commercial Al–5Ti–B refiner

grain size (µm)

500

l = 10.0 m m 400

l = 1.0 m m

300

l = 5.0 m m

200 100

l = 2.0 m m

0

2

4 6 8 cooling rate (K s- 1)

10

12

Figure 9. The calculated dependence of grain diameter on cooling rate for CP-Al inoculated with idealized grain re¯ners (with a Gaussian particle diameter distribution and di® erent average particle diameters ¶ ) or with a commercial re¯ner (closed circles) with size distribution as in ¯gure 6. (Reproduced with permission from Tronche & Greer (2000).)

cooling rate. As is also shown in gure 9, the calculations suggest that, with the particle size distribution in a commercial re ner, the grain size is rather sensitive to cooling rate in the range of greatest practical interest: 1{5 K s¡1 . The calculations yielding the results in gures 8 and 9 also predict that re ner e¯ ciency is greater for narrower particle size distribution. Surprisingly, this is a rather weak e¬ect. Within the free-growth model, in which larger particles initiate grains sooner than smaller particles, 100% e¯ ciency would be achieved for monosized particles but the deterministic model would break down in such a case. In an isothermal melt, for example, all the uniformly sized grains would grow until recalescence brought them simultaneously to the critical radius; at that point stochastic e¬ects would determine which grains would grow and which would re-melt. The calculations performed so far have all assumed a nucleant particle shape that is characteristic of a commercial Al{5Ti{1B re ner. As seen in TEM, these particles are hexagonal prisms. In the model these are approximated as discs with a thickness equal to 35% of their diameter. As the active surfaces are the hexagonal/circular (0001) faces, the particles would be equally e¬ective if they were thinner. The e¬ectiveness of re ners, measured in terms of the re ning achieved for a given volume fraction of nucleant phase, therefore could be increased if the particles were thinner. For Al{Ti{ B re ners, in which the nucleant phase is the hexagonal TiB2 , changing the aspect ratio of the particles is possible, at least in principle. In contrast, for Al{Ti{C re ners in which the nucleant phase is cubic TiC with octahedral morphology, changing the particle aspect ratio is not a design option. Perhaps the most important reason to inoculate melts is the suppression of columnar growth. As noted by Hunt (1984), equiaxed growth is favoured by a larger population of nucleant particles and by a smaller undercooling to activate them; the latter is the more important e¬ect. The free-growth model implies, therefore, that larger particles would be better for suppressing columnar growth. As yet, however, there Phil. Trans. R. Soc. Lond. A (2003)


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has been no quantitative application of the free-growth model that is able to predict the e¬ects of inoculation on the columnar-to-equiaxed transition in castings.

7. Summary and conclusions Inoculation of metallic melts can be used to give desirable equiaxed grain structures on casting. It is most widely practised for aluminium alloys. By embedding them in a metallic glass, the nucleant action of inoculant particles, relevant for their action in the melt, can be studied microscopically. The inhibition of growth in the glass permits unambiguous identi cation of nucleation sites. The particular case of Al{Ti{B inoculants used in aluminium alloys shows the importance of chemistry and crystallography for e¬ective nucleation. The nucleant particles are hexagonal platelets of TiB2 coated with an adsorbed layer of Al3 Ti. Nucleation of ¬ -Al occurs only on the f0001g-faces of the borides, with the close-packed planes and directions in TiB2 , Al3 Ti and ¬ -Al being parallel. The Al3 Ti layer appears to be essential for e¬ective nucleation. It is promoted by excess titanium in the melt, and its absence explains the poisoning of re ner action in the presence of, for example, zirconium. An understanding of layer stability has enabled the modi cation of re ners to make them more poison resistant. In successfully inoculated melts, at most 1% of the particles nucleate grains; this ine¯ ciency is caused by the release of latent heat, limiting the available melt undercooling, and cannot be directly related to nucleation. Commercial inoculants for aluminium are so potent that grain initiation is limited by the barrier to free-growth of nucleated ¬ -Al from the nucleant surface, rather than by nucleation itself. The undercooling necessary to initiate free growth is inversely proportional to the diameter of the particle surface, and the particle size distribution determines re ner performance. With measured particle size distributions as the input, quantitatively correct predictions can be made for grain size as a function of addition level of Al{Ti{B or Al{Ti{C inoculants, cooling rate and solute level in the melt. Such modelling allows the design of improved re ners. Commercial Al{Ti{B re ners have a characteristic particle size close to that giving a minimum grain size, and are e¯ cient, as measured by the volume (rather than number) fraction of added particles which initiate grains. The quantitative prediction of heterogeneous nucleation kinetics is rare. The success of quantitative predictions for inoculation of aluminium alloys relies on grain initiation being dominated by well-characterized nucleants so potent that grain initiation is limited only by the free-growth criterion. This condition is likely to be met in other cases, for example the catalysis of ice nucleation in some living systems (Franks 2003; Zachariassen & Kristiansen 2000). In such cases of potent nucleation or seeding, the size distribution of the catalyst substrates may play an important role in the overall kinetics. The contributions of past and present members of A.L.G.’ s research group are gratefully acknowledged, in particular those of A. M. Bunn, M. W. Meredith, T. E. Quested, P. Schumacher, A. Tronche and M. Vandyousse¯. Thanks are due to the EPSRC for ¯nancial support; Alcan International Ltd (Banbury Laboratory), the London & Scandinavian Metallurgical Co. Ltd (Rotherham), and Pechiney (Centre de Recherches de Voreppe) for ¯nancial support, access to research facilities and fruitful interactions; members of the EU Network `Microstructural Engineering by Solidi¯cation Processing’ for useful discussions; and Professor D. J. Fray for the provision of laboratory facilities. Phil. Trans. R. Soc. Lond. A (2003)



493

References Boone, G. W., Wilson, B. H., Carver, R. F. & Moody, R. R. 1991 Precision and reproducibility of the Aluminium Association standard test procedure for aluminium alloy grain re¯ners. In Light metals 1991 (ed. E. L. Rooy), pp. 1085{1093. Warrendale, PA: The Minerals, Metals & Materials Society. Bunn, A. M. 1998 Grain re¯nement in aluminium alloys. PhD thesis, University of Cambridge, UK. Bunn, A. M., Schumacher, P., Kearns, M. A., Boothroyd, C. B. & Greer, A. L. 1999 Grain re¯nement by Al{Ti{B re¯ners in aluminium melts: a study of the mechanisms of poisoning by zirconium. Mater. Sci. Technol. 15, 1115{1123. Cantor, B. 2003 Heterogeneous nucleation and adsorption. Phil. Trans. R. Soc. Lond. A 361, 409{417. Franks, F. 2003 Nucleation of ice and its management in ecosystems. Phil. Trans. R. Soc. Lond. A 361, 557{574. Greer, A. L., Bunn, A. M., Tronche, A., Evans, P. V. & Bristow, D. J. 2000 Modelling of inoculation of metallic melts: application to grain re¯nement of aluminium by Al{Ti{B. Acta Mater. 48, 2823{2835. Greer, A. L., Quested, T. E. & Spalding, J. E. 2002 Modelling of grain re¯nement in directional solidi¯cation. In Light metals 2002 (ed. W. Schneider), pp. 687{694. Warrendale, PA: The Minerals, Metals & Materials Society. Hunt, J. D. 1984 Steady state columnar and equiaxed growth of dendrites and eutectic. Mater. Sci. Engng 65, 75{83. Johnsson, M., Backerud, L. & Sigworth, G. K. 1993 Study of the mechanism of grain re¯nement of aluminium after additions of Ti-containing and B-containing master alloys. Metall. Trans. A 24, 481{491. McCartney, D. G. 1989 Grain re¯ning of aluminium and its alloys using inoculants. Int. Mater. Rev. 34, 247{260. Maxwell, I. & Hellawell, A. 1975 A simple model for grain re¯nement during solidi¯ cation. Acta Metall. 23, 229{237. Old¯eld, W. 1966 A quantitative approach to casting solidi¯cation: freezing of cast iron. Trans. ASM 59, 945{961. O’ Reilly, K. A. Q. & Cantor, B. 1995 Solidi¯cation behaviour of Al particles embedded in a Zr aluminide matrix. Acta Metall. 43, 405{417. Quested, T. E., Greer, A. L. & Cooper, P. S. 2002 The variable potency of TiB2 nucleant particles in the grain re¯nement of aluminium by Al{Ti{B additions. Mater. Sci. Forum 396{402, 53{58. Rappaz, M. & Gandin, Ch.-A. 1993 Probabilisitic modelling of microstructure formation in solidi¯cation processes. Acta Metall. Mater. 41, 345{360. Schneider, W., Kearns, M. A., McGarry, M. J. & Whitehead, A. J. 1998 A comparison of the behaviour of AlTiB and AlTiC grain re¯ners. In Light metals 1998 (ed. B. Welch), pp. 953{ 961. Warrendale, PA: The Minerals, Metals & Materials Society. Schumacher, P., Greer, A. L., Worth, J., Evans, P. V., Kearns, M. A., Fisher, P. & Green, A. H. 1998 New studies of nucleation mechanisms in Al-alloys: implications for grain-re¯nement practice. Mater. Sci. Technol. 14, 394{404. Spittle, J. A. & Sadli, S. B. 1995 E® ect of alloy variables on grain re¯nement of binary aluminium alloys with Al{Ti{B. Mater. Sci. Technol. 11, 533{537. Stefanescu, D. M., Upadhya, G. & Bandyopadhyay, D. 1990 Heat transfer: solidi¯cation kinetics modeling of solidi¯cation of castings. Metall. Trans. A 21, 997{1005. Tronche, A. & Greer, A. L. 2000 Design of grain re¯ners for aluminium alloys. In Light metals 2000 (ed. R. D. Peterson), pp. 827{832. Warrendale, PA: The Minerals, Metals & Materials Society. Phil. Trans. R. Soc. Lond. A (2003)


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Turnbull, D. 1950 Kinetics of heterogeneous nucleation. J. Chem. Phys. 18, 198{203. Turnbull, D. 1953 Theory of catalysis of nucleation by surface patches. Acta Metall. 1, 8{14. Vandyousse¯, M. & Greer, A. L. 2002 Application of cellular automaton: ¯nite element model to the grain re¯nement of directionally solidi¯ed Al-4.15 wt.% Mg alloys. Acta Mater. 50, 1693{1705. Vandyousse¯, M., Worth, J. & Greer, A. L. 2000 E® ect of instability of TiC particles on the grain re¯nement of Al and Al-Mg alloys by addition of Al{Ti{C inoculants. Mater. Sci. Technol. 16, 1121{1128. Zachariassen, K. E. & Kristiansen, E. 2000 Ice nucleation and antinucleation in nature. Cryobiol. 41, 257{279.

Discussion J. H. Perepezko (Department of Materials Science and Engineering, University of Wisconsin-Madison, WI, USA). Earlier, Professor Cantor stressed the importance of adsorption in heterogeneous nucleation. Could you clarify the possible role of adsorption in your model of grain re nement? A. L. Greer. The point I wanted to make was that, when nucleation on grain-re ner particles is very e¬ective (i.e. when nucleation occurs at very small undercooling), the nucleation itself ceases to be the controlling factor in initiating grains. In that case, the undercooling for grain initiation is related only to particle size, according to the free-growth model. The mechanism by which such potent heterogeneous nucleation occurs, however, could well involve adsorption rather than classical nucleation. Indeed, it is clear from Cantor’s work that the classical spherical-cap model for heterogeneous nucleation could not apply. F. Franks (BioUpdate Foundation, London, UK ). It was implied that the e¯ ciency of nucleators is low and depends only on the particle size. In the case of ice nucleation by AgI, the method of preparation of the catalyst is more important. The most e¯ cient catalysts are those prepared in situ. A. L. Greer. You are correct that the condition of the catalyst is important. Only when the catalyst is very good does the e¬ectiveness of nucleation relate only to particle size. It is essential that the catalyst is very potent: when it is not, the condition of the catalyst, rather than particle size, is critical. Your example of AgI has parallels with what is found for grain-re ning catalysts for aluminium. Speci cally, the grain re ners work better or worse depending on the chemistry of the re ner (i.e. catalyst) particles’ surfaces. As mentioned in the paper, poor re ners can improve with holding in the melt, an e¬ect explicable in terms of the formation of an aluminide layer on the boride particles, and not in terms of particle size. Furthermore, the e¬ectiveness of a re ner can be destroyed (`poisoned’) by the action of certain solutes in the melt. This again is related to the condition of the re ner particles. Your point, about the most e¬ective catalysts being prepared in situ, is also found for grain re ners for aluminium. The titanium diboride particles which act as nucleation catalysts are most e¬ective when prepared (by reaction of salts) in situ in an aluminium melt. B. Cantor (University of York, UK ). It seems very surprising that the mean nucleating particle size is much lower than the e¬ective nucleating particle size. I suspect this comes from assuming a lognormal distribution. A top-hat distribution would perhaps put the mean nucleating particle size equal to the e¬ective nucleating particle size. Phil. Trans. R. Soc. Lond. A (2003)



495

A. L. Greer. You are correct that the mean particle size is below the typical size of the particles that are e¬ective. This arises when the particle size distribution has a tail extending up to large sizes, a feature of the lognormal distribution found experimentally. We have made calculations of the e¬ects of di¬erent shapes of size distributions and found that the e¯ ciency of a re ner is indeed higher when the size distribution is narrower. Perhaps surprisingly, this does not seem to be a strong e¬ect. It should be noted that, as the particle size distribution becomes narrower, in the extreme, if it were a delta function, our simple model would break down. For a deltafunction distribution, all particles would become active at the same undercooling, but it would still be the case that only some would initiate grains. Depending on local thermal ®uctuations rather than particle size, some of the nuclei would grow and others would re-melt. R. W. Cahn (Department of Materials Science and Metallurgy, University of Cambridge, UK ). Does the population of inoculant particles frozen into a casting or ingot cause any problems (e.g. embrittlement)? If so, special e¬orts by the inoculant makers are needed to eliminate the ine¬ective `tail’ of particles by some process (such as a cyclone) to separate out a narrow size range. A. L. Greer. The added particles can indeed cause problems. For cast ingots rolled into a thin sheet, for example, large particles or aggregates of particles are clearly a problem if their diameter approaches or exceeds the sheet thickness. The problems are most likely to occur with the largest particles and, as emphasized, these are also the most e¬ective nucleants|so compromise may be needed. Embrittlement induced by the particles is not a problem for aluminium alloys. The addition of hard particles can even have bene cial e¬ects on the mechanical properties. The ine¬ective particles are the smaller ones and they end up at grain boundaries, swept there by the growing grains. There, they may play a role (not yet clearly identi ed) in in®uencing the selection of intermetallic phases appearing in the nal stages of solidi cation; these phases may in turn have important e¬ects on the nal product. The use of a cyclone or suchlike is an intriguing possibility, but it must be remembered that low-cost production is essential.

Phil. Trans. R. Soc. Lond. A (2003)