Growth of the solid

Solidification structure

5.2 Development of matrix structure

5.2.3 Growth of the solid

It seems, however, that the action of an effective

I t I I L

Front

I L

Temperature regime

Plane growth

Cell growth

Dendrite growth

Figure 5.17 shows how detailed consideration of the phase diagram can explain the relatively complicated effects of segregation during freezing.

It is worth examining the logic carefully.

The original melt of composition C, starts to freeze at the liquidus temperature TL. The first solid to appear has composition kCo where k is known as the partition coefficient. This usually has a value less than 1 (although the reader needs to be aware

Figure 5.16 Transition of growth morphology from planac to

~ cellular, to dendritic, as compositionally induced

undercooling increases (equivalent

~ to Gn/ being reduced).

of the existence of the less common but important cases where k is greater than 1). For instance, for k = 0.1 the first solid has only 10 per cent of the concentration of alloy compared to the original melt;

the first metal to appear is therefore usually rather pure.

In general, k defines how the solute alloy partitions between the solid and liquid phases.

Thus:

I I I I I I

-

_I Distance

kCo

co

^{C d k}

Composition Composition

t

Effectively

I

undercooled I

Solute-rich zone built up ahead of advancing front as a result of solute rejection ( k 4 ) on

solidification Figure 5.17 Link between the

constitutional phase diagram for a b i n a y allo?; and constitutional undercooling on

I ,freezing.

Solidification structure I33 description to quuntitutive prediction. Computers have encouraged an acceleration of this new thinking.

Figure 5.16 illustrates how the progressive increase in constitutional undercooling causes progressive instability in the advancing front, so that the initial planar form changes first to form cells, and with further instability ahead of the front will be finally provoked to advance as dendrites.

Notice that the growth of dendrites is in response to an instubiliry condition in the environment ahead of the growing solid, not the result of some influence of the underlying crystal lattice (although the crystal structure will subsequently influence the details of the shape of the dendrite). In the same way stalactites will grow as dendrites from the roof of a cave as a result of the destabilizing effect of gravity on the distribution of moisture on the roof. Icicles are a similar example; their forms being, of course.

independent of the crystallographic structure of ice.

Droplets running down windowpanes are a similar unstable-advance phenomenon that can owe nothing to crystallography. There are numerous other natural examples of dendritic advance of fronts that are not associated with any long-range crystalline internal structure. It is interesting to look out for such examples. Remember also the converse situation that the planar growth condition also effectively suppresses any influence that the crystal lattice might have. It is clear therefore that the constitutional undercooling, assessed by the ratio G/R, is the factor that measures the degree of stability of the growth conditions, and so controls the type of growth front, not, primarily, the crystal structure.

Figure 5.18 shows a transition from planar, through cellular, to dendritic solidification in a low- alloy steel that had been directionally solidified in a vertical direction. The speeding up of the solidification front has caused increasing instability.

Figures 5.19 and 5.20 show different types of dendritic growth. Both types are widely seen in metallic alloy systems. In fact, dendritic solidification is the usual form of solidification in castings.

A columnar dendrite nucleated on the mould wall of a casting will grow both forwards and sideways, its secondary arms generating more primaries, until an extensive ‘raft’ has formed (Figure 5.21). All these arms will be parallel in terms of the internal alignment of their atomic planes. Thus (5.16)

For solidus and liquidus lines that are straight, k is accurately constant for all compositions. However, even where they are curved, the relative matching of the curvatures often means that k is still reasonably constant over wide ranges of composition. When k is close to I , the close spacing of the liquidus and solidus lines indicates little tendency towards segregation. When k is small, then the wide horizontal separation of the liquidus and solidus lines warns of a strongly partitioning alloying element.

On forming the solid that contains only kCo amount of alloy, the solute remaining in the liquid has to be rejected ahead of the advancing front.

Thus although the liquid was initially of uniform composition Co, after an advance of about a millimetre or so the composition of the liquid ahead of the front builds up to a peak value of C&. The effect is like that of a snow plough. This is the steady-state condition shown in Figure 5.17.

In common with all other diffusion-controlled spreading problems, we can estimate the spread of the solute layer ahead of the front by the order-of- magnitude relation for the thickness d of the layer.

If the front moves forward by d in time t , this is equivalent to a rate V,. We then have:

d = (Dt)”’ where V , = d / t

SO:

(1 = D N , (5.17)

where D is the coefficient of diffusion of the solute in the liquid. It follows that constitutional undercooling will occur when the temperature gradient, G , in the liquid at the front is:

(5.18) or

GIVS 5 -(TL - Ts)/D (5.19)

from Figure 5.17, assuming linear gradients. Again, from elementary geometry which the reader can quickly confirm, assuming straight lines on the equilibrium diagram, we may eliminate TL and Ts and substitute C,, k and m, where m is the slope of the liquidus line, to obtain the equivalent statement:

rnCo ( 1 - k ) k D

on solidificati& the arms will ‘knit’ tbgether with almost atomic perfection, forming a single-crystal lattice known as a grain. A grain may consist of thousands of dendrites in a raft. Alternatively a grain may consist merely of a single primary arm.

or, in the extreme, merely an isolated secondary arm.

The boundaries formed between rafts of different orientation, originating from different nucleation (5.20)

which is the solution derived from more rigorous diffusion theory by Chalmers in 1953, nicely summarized by Flemings (1974). This famous result marked the breakthrough in the history of the understanding of solidification by the application of physicc. It marked the revolution from qualitative

Figure 5.18 Structure of a low-alloy steel subjected to accelerating freezing from bottom to top, changing from planar, through cellular; to dendritic growth.

events, are known as grain boundaries. Sometimes these are called high-angle grain boundaries to distinguish them from the low-angle boundaries that result from small imperfections in the way the separate arms of the raft may grow, or suffer slight mechanical damage, so that their lattices join slightly imperfectly, at small but finite angles.

Given a fairly pure melt, and extremely quiescent conditions, it is not difficult to grow an extensive dendrite raft sufficient to fill a mould having dimensions of 100 mm or so, producing a single crystal. Nordland (1967) describes a fascinating experiment in which he solidifies bismuth at high

Figure 5.19 Transparent organic alloy showing dendritic solidification. Columnar growth ( a ) and equiaxed growth (b) with a modification to the alloy by the addition of a strongly partitioning solute, with k << 1, which can be seen to be segregated ahead of the growing front (courtesy J. D. Hunt; see Jackson et al. 1966).

Solidification structure I35 Primary

-

Figure 5.20 Rather irregular dendrites common in al~iminium alloys at ( a ) SO and ( b ) 90 per cent solidijkd.

The secondary arms spread laterally, joining to ,form continuous p1nte.s (after Singh et al. 1970).

undercooling and high rates, but preserves the fragile dendrite in one piece. He achieves this by adding weights to the furnace that contained his sample of solidifying metal, and suspends the whole assembly in mid-air, using long lengths of polypropylene tubing from the walls and ceiling of the room. In this way he was able to absorb and dampen any outside vibrations.

In a review of the effects of vibration on solidifying metals, Campbell ( 1 98 1) confirms that

yy#///,

Grain size w

H Primary dendrite arm spacing w Secondary dendrite arm spacing Figure 5.21 Schematic illustration of the formution of LI

raft qf dendrites to make grains. The dendrite steins within uny one raft or grain are all cn.stalloRraphicnl1~

related to a common nucleus.

Figure 5.22 Grain re$nement threshold ^(I.\

a ,function of amplitude and,frequenc.v (?/' rfbration (Campbell 1981).

Nordland's results fall into a regime of frequency and amplitude where the vibrational energy is too low for damage to occur to the dendrites (Figure 5.22).

5.2.3.1 Dendrite arm spacing (DAS)

In the metallurgy of wrought materials, it is the grain size of the alloy that is usually the important structural feature. Most metallurgical textbooks therefore emphasize the importance of grain size.

In castings, however, grain size is sometimes important (as will be discussed later), but more often it is the secondary dendrite arm spacing (sometimes shortened merely to dendrite arm spacing, DAS) that appears to be the most important structural length parameter.

The mechanical properties of most cast alloys are seen to be strongly dependent on secondary arm spacing. As DAS decreases, so ultimate strength, ductility and elongation increase. Also, since homogenization heat treatments are dependent on the time required to diffuse a solute over a given average distance d, if the coefficient of diffusion in the solid is D, then from the order-of-magnitude relation we have

d = (Dt)'/* (5.21)

Thus finer DAS results in shorter homogenization times, or better homogenization in similar times;

the cast material is more responsive to heat treatment, giving better properties or faster treatments.

It is now known that DAS is controlled by a coarsening process, in which the dendrite arms first grow at very small spacing near the tip of the dendrite. As time goes on, the dendrite attempts to reduce its surface energy by reducing its surface area. Thus small arms preferentially go into solution while larger arms grow at their expense, increasing the average spacing between arms. The rate of this process appears to be limited by the rate of diffusion of solute in the liquid as the solute transfers between

c n l

dissolving and growing arms. From an equation such as 5.21, and assuming the alloy solidifies in a time t f , we would expect that DAS would be proportional to ti1*, since tf is the time available for coarsening. In practice it has been observed that DAS is actually proportional to

t"

where n usually lies between 0.3 and 0.4. Figure 5.23 shows the magnificent research result, that the relation between DAS and tf continues to hold for A l 4 . 5 C u alloy over eight orders of magnitude. Interestingly, however, a plot of grain size on the same figure shows that grain size is completely scattered above the DAS line. Clearly, a grain cannot be smaller than a single dendrite arm, but can grow to unlimited size in some situations.

In summary, DAS is controlled by solidification time. Grain size, on the other hand, is controlled by a number of quite different processes, some of which are discussed further in the following section.