Kinetics of Materials - R. Balluff_ S. Allen_ W. Carter (Wiley_ 2005) Episode 13 pptx

CHAPTER 21 CONCURRENT NUCLEATION AND A discontinuous transformation generally occurs by the concurrent nucleation and growth of the new phase i.e., by the nucleation of new particles an

EXERCISES 527

20.4 Find an expression for the thickness of the growing B-rich P-phase surface layer shown in Fig 20.3 as a function of time Details of the growth of this layer have been discussed in Section 20.1.2, and a strategy for determining the growth rate has been outlined Assume constant diffusivities in both phases Solution Solutions t o the diffusion equation in the a: and p phases, which match the boundary conditions, are

-

cp - CEO 1 - erf (A~/&Z)

Finally, the layer thickness is given by x = X I - x z = (A1 - Az)fi’

20.5 Using the scaling method, find an expression for the diffusion-limited rate of growth of a cylindrical B-rich precipitate growing in an infinite a-phase ma- trix Assume the same boundary conditions as in the analysis in Section 20.2.1 (Eqs 20.37-20.39) for the growth of a spherical particle Note that Fig 20.6, which applied to the growth of a sphericallarticle in Section 20.2.1, will also apply Use the scaling parameter 7 = r/(4Dat)lI2 You will need the integral

(20.90) which has been tabulated [29]

Solution Starting with the diffusion equation in cylindrical coordinates (see Eq 5.8)

and using the scaling parameter t o change variables, the diffusion equation in 7-space becomes

(20.91) This result, along with the boundary conditions given by Eqs 20.37-20.39, shows that the particle will grow parabolically according t o

528 CHAPTER 20: GROWTH OF PHASES IN CONCENTRATION AND THERMAL FIELDS

Integrating Eq 20.91 once,

dcg e-72

- = a 1 -

where a1 = constant Integrating again yields

Determining a1 from the condition cg = c"," when 17 = V R ,

The Stefan condition at the interface is

Finally, substituting Eq 20.95 into Eq 20.97, we have

for the determination of V R

20.6 Find an expression for the rate of thickening of a B-rich /3-phase precipitate platelet in an infinite a-phase matrix in a A I B binary system as in Fig 20.6 Assume diffusion-limited conditions and a constant diffusivity, Ea, in the a-phase matrix Also, assume that the atomic volume of each species is constant throughout so that there is no overall volume change and that the plate is extensive enough so that edge effects can be neglected Use the scaling method

Solution Let x be the distance coordinate perpendicular t o the platelet The boundary conditions will be the same as Eqs 20.37-20.39 if T - IZ: and 77 -+ x / m The method is basically the same as that used t o obtain the solution for the sphere in Section 20.2.1 In the present case, ~ ( t ) will be the half-thickness of the plate 77 will

be constant at the interface at the value vx = x(t)/&%, so that X ( t ) will increase parabolically with time according t o

We now determine qx by solving for c g and invoking the Stefan condition at the in- terface The difFusion equation was scaled and integrated in Cartesian coordinates in Section 4.2.2 with the solution given by Eq 4.28 When this solution is matched t o the present boundary conditions,

(20.100)

EXERCISES 529

The Stefan condition at the interface is

(20.101) Use o f Eqs 20.99 and 20.100 in Eq 20.101 then yields the desired expression for vx,

20.7 Consider again the problem posed in Exercise 20.6, whose solution had the form of a transcendental equation A simple and useful approximate solution

can be found by using the linear approximation to the diffusion profile shown

in Fig 20.14 Find the solution based on this approximation

diffusion-limited thickening of a slab of thickness Approximate composition vs distance profile for determination 2x of the

Solution The Stefan condition at the interface is

(20.103)

Using the linear approximation in Fig 20.14, &/ax = (cgm - c; ') / ( Z - x) Also,

the conservation of B atoms requires that the two shaded areas in the figure be equal Therefore, ( c g - cSm) x = (cgm - CEO) (2 - x ) / 2 Putting these relationships into

530 CHAPTER 20: GROWTH OF PHASES IN CONCENTRATION AND THERMAL FIELDS

Figure 20.15: Binary phase diagram

The system is at the temperature T* and the p phase is essentially pure

B , while the CY phase contains a moderately low concentration of B so that Henry's law is obeyed The average radius of the shell is ( R ) and the thickness

of the thin shell is 6R, where 6R << (R) Assume that the diffusion rate of

B in the ,8 phase is extremely slow and can be neglected in comparison to its diffusion rate in the CY phase Find an expression for the shrinkage rate of the shell and show that the shell will shrink at a rate inversely proportional

to (R) Assume that the concentration of B in the CY phase is maintained

in local equilibrium with the ,B phase at both the inner and outer interphase boundaries between the shell and the p phase Also, neglect any small volume changes that might occur

Solution The outer interface is concave and the inner interface is convex with respect

t o the p phase The concentrations of B maintained in equilibrium in the a phase at the outer interface, cE,$, and a t the inner interface, , cy! are given by Eq 15.4 The concentration difFerence across the shell is therefore

Since Aca << c a ( m ) , the equations of continuity (for Stefan conditions, see Sec- tion 20.1.2) at the inner and outer interfaces can be expressed as

winc:fi = wincp + J (20.108)

EXERCISES 531

where 2)in and vout are the velocities of the inner and outer interfaces, respectively Using these relationships and neglecting small differences between ckp, cz$, and ca(oo), the average shell velocity is then

CHAPTER 21

CONCURRENT NUCLEATION AND

A discontinuous transformation generally occurs by the concurrent nucleation and growth of the new phase (i.e., by the nucleation of new particles and the growth of previously nucleated ones) In this chapter we present an analysis of the resulting overall rate of transformation Time-temperature-transformation diagrams, which display the degree of overall transformation as a function of time and temperature, are introduced and interpreted in terms of a nucleation and growth model

Consider homogeneous nucleation in a three-dimensional system In the simplest model, these nuclei form at random locations The nucleation rate, J , specifies the number of nuclei forming per unit volume per unit time These nuclei form

at locations that have not already been transformed by growth of any previously formed nuclei Once nucleated, a particle grows at a rate R = dR/dt and the untransformed volume decreases However, no particle can grow indefinitely A

particle nucleated near a surface can grow to impingement with the surface, and the transformation at that location will cease Similarly, two particles that nucleate near each other grow until they impinge and transformation ceases Alternatively, growth can be driven by supersaturation and individual nuclei could have their growth limited by the decreasing supersaturation in the untransformed volume

In previous chapters we have developed models for discontinuous transforma- tions that treat nucleation and growth processes independently However, when

Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter 533

Copyright @ 2005 John Wiley & Sons, Inc

534 CHAPTER 21: CONCURRENT NUCLEATION AND GROWTH

these processes occur concurrently, the overall transformation rate (i.e., the volume transformed per unit time) and microstructural characteristics such as particle or grain size depend on the interplay of nucleation and growth processes

The theory of the kinetics of concurrent nucleation and growth reactions has a rich history that includes work by Kolmogorov [l] , Johnson and Mehl[2], Avrami [3-

51, Jackson [6], and Cahn [7] Cahn's time-cone method for treating a class of these problems is the most general of these, with the most transparent assumptions, and

is presented here The method of Johnson, Mehl, and Avrami is covered in Section

4 of Christian's text [8]

21.1.1

The key to obtaining exact solutions to the transformation kinetics is to make explicit assumptions about the statistical homogeneity of nucleation and growth processes in the system Following Cahn, we denote homogeneous nucleation on randomly dispersed sites in a volume by volume nucleation; heterogeneous nucle- ation on randomly dispersed sites on a surface or interface in a volume by surface nucleation, and heterogeneous nucleation on randomly dispersed sites along a linear feature in a volume by line nucleation Exact solutions for transformation kinetics can be obtained when nucleation and growth rates are spatially homogeneous at any instant within the system of interest The method is applicable to finite samples with a wide range of geometries and can yield position-dependent transformation kinetics

Statistical homogeneity is necessary for application of the time-cone method Statistical homogeneity is not a valid assumption during precipitation from super- saturated solution because the untransformed regions develop concentration gradi- ents around growing particles and hence the nucleation rate becomes nonuniform Also, in a material with thermal gradients undergoing a discontinuous transfor- mation (e.g , during continuous cooling) , the nucleation rate will be nonuniform However, the theory is applicable to discontinuous transformations in which the parent and product phases have the same composition and in which the tempera- ture is essentially uniform at any instant Examples include recrystallization, first- order order-disorder transformations, massive transformations, and crystallization during vapor-deposition processes

Time-Cone Analysis of Concurrent Nucleation and Growth

Probability that a Point r'will not be Transformed a t Time t The probability that a point Fin the sample will be untransformed at the time t is obtained by computing the probability that no nuclei had formed at any location r' and any previous time r that could have grown and led to prior transformation at r' and t This is accomplished in three steps:

i The set of points that could possibly affect a given point grows with time; therefore, to specify this set of points, a time coordinate is required in addition

to the spatial coordinates of the sample For nucleation and growth in a

sample of dimensionality 3, the augmented space has Cartesian coordinates

x, y, z , and t; more generally, coordinates r' and t for any spatial dimension Emanating from the point (.',t) and extending to earlier times is a domain that is the set of all points in the augmented space that would have caused transformation at (r', t ) if nucleation had occurred at a nearby point and earlier

21.1: OVERALL RATE OF DISCONTINUOUS TRANSFORMATION 535

time ( F , T) This subset of points is called V, For the d = 2 case, the domain

V, is a cone of height t; Cahn refers to this domain as the time cone The cross

section of the time cone at any time depends on the growth rate function &t) Figure 21.1 is a representation of the transformation by random nucleation along a one-dimensional sample for the constant-growth-rate case

Distance, x

Figure 21.1: Nucleation and growth along a one-dimensional specimen ( a ) Growth cones The apex of each cone coincides with the time and location of a nucleation event Once nucleated, constant linear growth transforms the region surrounding a nucleus Two nuclei have formed at tl; at t z the sample is approximately half transformed; at t 3 it is fully transformed Transforming regions impinge where growth cones intersect ( b ) Time cone for the point (z, t) For the point (2, t ) to be untransformed at time t , no prior nucleation can have occurred within the time cone volume, V,

11

iii

The nucleation rate can be integrated over the time cone to obtain the number

of nuclei expected in V,, denoted by ( N ) ,

The untransformed fraction 1 -C is obtained from the stochastic independence

of the nucleation events-that is, any particular nucleation event in untrans- formed material is not influenced by any other nucleation event Under these conditions, the theory can be formulated using the Poisson probability equa- tion [9], which states that if p = the mean rate at which events occur, then the probability, p ( k ) , that exactly Ic events occur in time t is

(21.1) The probability that exactly zero events occurs is given by Eq 21.1 as

536 CHAPTER 21: CONCURRENT NUCLEATION AND GROWTH

Equation 21.4 forms the basis of the theory for the kinetics of concurrent nu- cleation and growth transformations Specific cases can be formulated by deriving appropriate expressions for the quantity (N),

Time Cone V, for Isotropic, Time-Dependent Growth Rate R(t) The time cone's geometry is given by simple relations For isotropic (i.e., radial) growth, at time t the radius of a transformed region nucleated at an earlier time T is given by

rates J (t) and isotropic growth rates R (t) (such as in nonisothermal transforma- tions under conditions in which thermal gradients can be neglected), the number

of nuclei in V, is given for the d = 3 case as

t

( N ) , (t) = $ / J (t') [R (t, t')I3 dt'

Expressions for Transformation Rate when Nucleation and Growth Rates are Constant

If the growth velocity R is isotropic and constant, Eq 21.5 can be integrated and the time cone is the set of points ?that obey

2

The radius of the time cone, I?- Jl, is linear in time, and hence the time cone will

be a right circular cone of height t - T The volume of the time cone, which Cahn calls the nucleation volume V,, for transformation in a system of dimensionality d

21.1: OVERALL RATE OF DISCONTINUOUS TRANSFORMATION 537

can be generalized to systems of arbitrary dimensionality, d, by use of Eq 21.9,

giving

J B t

Substituting the appropriate factors from Eq 21.10 into Eqs 21.11 and 21.4 gives

expressions for the fraction transformed in one, two, and three dimensions for the case of constant nucleation and growth rates J and R The resulting expressions for the untransformed volume are

21.1.2 Transformations near the Edge of a Thin Semi-Infinite Plate

Consider a semi-infinite thin plate that is effectively two-dimensional, lying in the zy-plane, with a single edge along x = 0 It is assumed that there is no heteroge- neous line nucleation at the edge of the sheet For constant-volume nucleation and growth rates and isotropic growth, points lying within the near-edge region x < Rt require special consideration In the bulk of the plate away from this region, the time cone will be a right circular cone of height t, and the fraction transformed will be given by the d = 2 form of Eq 21.12 Close to the edge, the time cone will

be truncated by the plane x = 0, its volume will be less than in the bulk, and the number of nuclei ( N ) c contained in the truncated cone will decrease as x + 0.l The transformation rate in the near-edge region will thus be slower than in the

'Nuclei cannot form outside the sample and hence cannot influence the transformation anywhere inside it

538 CHAPTER 21: CONCURRENT NUCLEATION AND GROWTH

bulk, and the grain size after completion of the transformation will be larger near the edge

To apply Eq 21.4, an expression for the volume of a right circular time cone of

height t having its axis located at (s, y) is required When s > Rt, the cone volume

V, is given by Eq 21.10 for the d = 2 case, and the edge s = 0 has no influence

on the transformation kinetics When s < Rt, the time cone is truncated as in

Fig 21.3a Its volume may be expressed in terms of quantities shown in Fig 21.3b and c: its base radius, Rt, height, t , and the distance of the truncated face from the cone axis, s

(21.13)

Figure 21.3a compares the volume fraction transformed, <, vs time for a location

s 5 Rt where Eq 21.12 applies with that calculated using Eq 21.13 evaluated at

the specimen edge, s = 0 This plot reveals the extent that the transformation rate

is reduced near the specimen edge compared to the specimen bulk

Figure 21.3: (a) Truncated time cone (b) Vertical cross section of time cone (c)

Horizontal cross section through truncated portion of time cone

Useful insights into the kinetics of a phase transformation that proceeds by nucle- ation and growth can be obtained by observing the fraction transformed, 5, under isothermal conditions at a series of different temperatures This is usually done

by undercooling rapidly to a fixed temperature and then observing the resulting isothermal transformation The kinetics generally follows the typical C-shaped behavior described in Exercise 18.4 If a series of such curves is obtained at differ-

ent temperatures, the time required to achieve, for example, < = 0.01, 0.50, and

to the factor exp[-(AGc + Gy)/(kT)] At T = Teq, there is no undercooling,

AG, + 00, and the rate of nucleation is zero The time thus required for any transformation is infinite at Teq As the transformation temperature is decreased, the undercooling and the transformation driving force increase, which causes a rapid decrease in AG, while G F remains constant This rapid decrease in AG, causes [AG, + G y ] to decrease more rapidly than kT, which, in turn, causes the factor exp[-(AG, + Gp)/(lcT)] to increase and the nucleation rate to increase However,

on further cooling, the rate of decrease of AGc slows and eventually kT decreases more rapidly than [AG, + G p ] , causing the factor exp[-(AG, + Gy))/(kT)] to decrease and the rate of nucleation to decrease Eventually, the entire system becomes “frozen in” and the time required for any significant transformation again becomes essentially infinite

An example of such T T T kinetics is solidification According to Eq 12.2,

(21.14)

if it is assumed that the entropy and enthalpy of solidification are each independent

of temperature below T, Here, AT is the undercooling and A H is the enthalpy

of solidification According to Eq 19.4, AG, then has the form AG, = A/(AT)’, where A is a constant independent of temperature The nucleation rate below T, then has the temperature dependence

Time Figure 21.4: TTT diagram for a nucleation and growth phase transformation At

T = Teq the parent phase is at equilibrium, and no undercooling is present The three

curves indicate the attainment of C = 0.01,0.50, and 0.99 transformation, respectively

540 CHAPTER 21: CONCURRENT NUCLEATION AND GROWTH

The nucleation rate will therefore be a maximum at the temperature where the

exponent in Eq 21.15 is a minimum, which occurs at the temperature

M Avrami Kinetics of phase change-I J Chem Phys., 7(12):1103-1112, 1939

M Avrami Kinetics of phase change-11 Transformation-time relations for random distribution of nuclei J Chem Phys., 8(2):212-224, 1940

M Avrami Kinetics of phase change-111 Granulation, phase change, and microstruc- ture J Chem Phys., 9(2):177-184, 1941

J.L Jackson Dynamics of expanding inhibitory fields Science, 183(4123):446-447,

1974

J.W Cahn The time cone method for nucleation and growth kinetics on a finite domain In Thermodynamics and Kinetics of Phase Transformations, volume 398 of Materials Research Society Symposia Proceedings, pages 425-438, Pittsburgh, PA, 1996 Materials Research Society

J.W Christian The Theory of Transformations zn Metals and Alloys Pergamon Press, Oxford, 1975

E Parzen Modern Probability Theory and Its Applications John Wiley & Sons, New York 1960

EXERCISES

21.1 Consider transformation kinetics in one dimension, such as recrystallization (see Section 13.1) in a narrow wire For a finite wire of length L , the proba- bility that a region will have transformed will depend on its proximity to the end of the wire Investigate the end effects on transformation kinetics on a finite length of wire 0 < x < L Assume that the nucleation occurs uniformly

in the unrecrystallized regions at a constant rate, J , and that the growth rate

R is constant Calculate the probability that a point x will have transformed

at time t

0 The solution should be symmetric around x = L/2 There are three separate cases to consider; one of them is x < Rt and L - x > Rt (see Fig 21.5)

EXERCISES 541

L

-Figure 21.5: Length-time area of influence for a point 2

Solution Poisson statistics apply when events are random and mutually independent, which is assumed to be the case both in time and along the wire The probability

p(n, A ) that n events occur in an “area” (length x time) A with event rate J is given

bv the Poisson distribution

(21.17)

Therefore, the probability that no event occurs is p ( 0 , A ) = e - J A , and the probability that some (one or more) events occur is 1 - p ( 0 , A ) , which is equal to the probability that a region will have transformed Therefore, the problem depends only on the area

of the time cone illustrated in Fig 21.5

Case 1: Very short times or effectively infinite L There is no interference from the boundaries The condition for this case is x > Rt The area of the time cone is

A1 = Rt x t = Rt2 The probability that a point x will have transformed is independent

of z when Rt < z < L/2:

Case 2: Near the end of a finite wire There is interference only from the boundary a t

x = 0 The condition for this case is L > Rt and z < Rt (or, in a slightly different form, x < Rt and z < L - k t ) The area of the time cone, A2, is A1 minus the area where z < 0:

Case 3: Very short wire or long times There is interference from both boundaries The

condition for this case is L < Rt (or x < Rt and x < L - k t ) The area of the time cone, As, is A1 minus the area where X > L:

Rt - (L - z) Rt - (L - x) - - 2(LRt + Lz) - (L2 + 222) (21.21)

542 CHAPTER 21: CONCURRENT NUCLEATION AND GROWTH

21.2 Consider recrystallization and grain growth in an infinite thin sheet Assume

that the nucleation rate of recrystallized grains is a linear function of tem- perature above a critical temperature, T,, and the nucleation rate is zero for

T < T, [i.e., at temperatures above T,, J = a(T - T,)] Also assume that the grain-growth rate, R, is constant and independent of temperature Suppose that at time t = 0 the sheet is heated at the constant rate T ( t ) = Tc/2 + Pt

Using Poisson statistics, the probability that exactly zero events occur in a time t is po = exp(-(N,))

Solution Nucleation begins when the sheet reaches the temperature T,; the time

t o reach T, is TC/(2,B) The time cone in this two-dimensional sample, for a constant growth rate, is the right circular cone V, given by

C = 1 - exp(-(N),)

( 2 1.24)

(21.25)

CHAPTER 22

S 0 L I D I F I CAT I 0 N

The mechanisms of the motion of liquid/crystal interfaces during solidification were discussed in Section 12.3, and aspects of the heat-conduction-controlled motion of liquid/solid interfaces and their morphological stability under various solidification conditions were treated in Chapter 20 This sets the stage for considering the entire process of the solidification of a body of liquid into a solid

Solidification results in a wide range of structures depending upon the type of material and the conditions under which the solidification occurs [l-31 Although

a huge literature describes and analyzes the many phenomena involved, we restrict ourselves to two cases: when the liquid/solid interface is stable and plane-front solidification is achieved, and when the interface is unstable and cellular or dendritic growth occurs The first mode of solidification has important uses as a method of removing impurities or producing uniform distributions of solute atoms in materials The second is prevalent in the casting of many alloys and has been the subject of

a vast amount of study

22.1 PLANE-FRONT SOLIDIFICATION IN ONE DIMENSION

22.1.1 Scheil Equation

In Fig 2 2 1 ~ one-dimensional solidification is depicted; a liquid binary alloy initially

of uniform composition co is placed in a bar-shaped crucible of length L The bar

is progressively cooled from one end, so it solidifies from one end to the other

Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter 543

Copyright @ 2005 John Wiley & Sons, Inc

544 CHAPTER 22: SOLIDIFICATION

Figure 22.1: (a) Bar-shaped specimen after plane-front solidification Grayscale

intensity indicates solute composition variation (b) Composition distribution along the bar after plane-front solidification with different effective partition ratios From Flemings [2]

with a stable and planar liquid/solid interface (see the discussion of stability in Section 20.3) The eutectic phase diagram in Fig 2 0 4 ~ shows the liquidus and solidus approximated as straight lines It is assumed that the interface is a good

source or sink (Section 13.4) and that the liquid and solid are essentially in local

equilibrium at the interface As described in Section 20.1.3, the first solid to form

is of composition coSL, considerably less than CO Excess solute is rejected by the solidifying material and deposited in the liquid because diffusion into the solid is usually negligible However, for the solute to enter the liquid and become dispersed,

it must first diffuse through the liquid boundary layer adjacent to the liquid/solid interface Because, in this layer, any flow in the liquid is lamellar flow parallel to the interface, solute atoms can pass through it only by diffusional transport Once the solute has diffused through the layer of thickness 6, it is quickly mixed into the remaining bulk liquid by convection, and it may be assumed that the concentration

in the liquid is uniform up to the boundary layer The boundary layer therefore acts as a “diffusion barrier” and limits the rate at which the rejected solute can be

dispersed throughout the liquid As a result, the solute concentration in the liquid builds up in front of the advancing interface However, at some time after the onset

of solidification, the distribution of solute in the boundary layer reaches a quasi- steady state, as shown in Fig 20.4b, where a “spike” of concentration has formed

It may be assumed that local equilibrium prevails at the liquid/solid interface, and

if the temperature there is TLIs, the phase diagram in Fig 2 0 4 ~ indicates that the solute concentration in the liquid directly at the interface, cLs, must be related

to the concentration of the solid being formed at the interface, csL, by csL = kcLs The quantity k, the ratio of the solidus concentration to the corresponding liquidus concentration, is known as the partition ratio When the slopes of the solidus and liquidus curves are constant, as in Fig 20.4a, the partition ratio is constant

As solidification continues and solute is continuously rejected into the remaining liquid, the concentration in the bulk liquid increases slowly and the quasi-steady- state solute distribution in the boundary layer evolves This, in turn, produces

J = - D - - w c dX The diffusion equation in the liquid is

(22.1)

(22.2)

which has the general solution

where x is the distance from the interface The two constants a1 and a2 can be evaluated through use of the boundary conditions2 cL(0) = cLs and cL(S) = C O ,

with the result

is (ck - cSL)dx, where the slowly changing concentration in the bulk liquid is

'When this is not true, the kinetics of the formation of the transient must be taken into account [4]

2Note that the amount of solute in the spike is negligible compared to that in the bulk liquid, and therefore the concentration in the bulk liquid remains approximately co

546 CHAPTER 22: SOLIDIFICATION

represented by c& (rather than its initial value C O ) The change in concentration

in the liquid is then dck = [ ( c k - c s L ) / ( L - x ) ] dx Using csL = k'cL 3 0 1

which yields the Scheil equation,

csL(x) = k'co (1 - %) "-I

(22.8)

(22.9) Because any diffusion in the solid is neglected, csL(x) in Eq 22.9 represents the distribution of solute after the solidification

Equation 22.9 has two limiting forms When the parameter 6v/DL >> 1, k' = 1 according to Eq 22.7 This situation is encouraged by a lack of convection, a high solidification rate, and a slow rate of diffusion in the liquid The concentration spike in the liquid is then strong and cLs quickly reaches a level where, according

to Eq 22.9, csL = co and the composition of the solid being formed and the compo- sition of the bulk liquid are the same On the other hand, when 6v/DL << 1, k' = k

There is then rapid mixing in the liquid, the diffusion barrier is nonexistent, and there can be a large difference between the compositions of the solid being formed and the bulk liquid, depending on the factor k Some typical curves of cSL(x) vs x under these different conditions are shown in Fig 22.lb for the system in Fig 20.4 When k' < 1, the composition of the solid a phase increases continuously and even- tually reaches its maximum value, cSL(max), when the liquid reaches the eutectic composition After that, the solid forms as a eutectic and its average composition

is the eutectic composition, C E

As shown by Fig 22.lb, the concentrations of solute atoms are significantly reduced in the material that is solidified early in the solidification process when

k' < 1 One-dimensional plane-front solidification can therefore be used as a method

of purification However, purification is carried more effectively out by modifying the process and adopting a zone-melting technique

22.1.2

In zone melting, a bar-shaped specimen as shown in Fig 2 2 2 ~ is first melted at one end to form a melted zone of length 1 This zone is then moved along the entire specimen at a constant rate while keeping 1 constant As it moves, it picks up solute atoms and eventually deposits them near the other end of the bar, thereby purifying one end Each iteration of the process leads to increasing purity

The zone is generally much longer than the width of the liquid boundary layer (i.e., 1 >> 6) When the zone moves a distance dx, the amount of solute gained by the zone is (CO - csL) dx, and therefore

Zone Melting and Zone Leveling

22.2: CELLULAR AND DENDRlTlC SOLIDIFICATION 547

L Distance along bar, x -

b

0

Figure 22.2:

solute composition variation

zone-melting pass From Flemings [ 2 ]

(a) Bar-shaped specimen after zone melting Grayscale intensity indicates

(b) Composition distribution along the bar after a single

where the lower limit to the first integral occurs because the initial composition of the liquid in the zone is CO Integration then yields

cSL(x) = co[l - (1 - k’)e-”’’L] (22.12)

A characteristic plot of cSL(z) as a function of x is shown in Fig 22.2b

Considerable purification is achieved during zone melting The final transient at the end begins when the leading end of the zone reaches the end of the specimen

At that point, the solidification becomes very similar to plane-front solidification Additional passes produce further purification and very small solute concentrations

in the first part of the specimen An asymptotic limit exists, however, as taken up

in Exercise 22.2

When the zone length is relatively short, k’ is large, and when the number

of passes is small, the bulk of the specimen solidifies at very nearly a uniform composition corresponding to CO Zone solidification can be used in this manner to produce compositional uniformity, a technique known as zone leveling

22.2 CELLULAR AND DENDRlTlC SOLIDIFICATION

22.2.1

When the liquid/solid interface is unstable according to the criteria discussed in Section 20.3.3, a cellular or dendritic structure is developed When the degree of instability is relatively low, an array of protuberances develops on the interface as shown in Fig 2 0 8 ~ These protuberances, called cells, advance perpendicular to the interface Their shapes vary depending upon the type of material, the orientation

of the interface, and other factors For (100) liquid/solid interfaces in cubic metals, equiaxed cells form like those in Fig 20.8b However, for a (110) interface, the cells take on a corrugated configuration of long hills and furrows When the degree of

Formation of Cells and Dendrites

Figure 22.3: Transition from cellular to dendritic growth as the growth velocity is

increased (a) Cellular growth at low velocities (b) Cellular growth deviated to the fast growing (100) direction ( c ) Appearance of ridges along the primary dendrites (d) Start

of secondary dendrite branch formation After Flemings [2]

22.2.2

During dendritic growth, extensive solute segregation occurs in the interdendritic spaces; this phenomenon is a serious problem in the casting of alloys The segrega- tion occurs because of the tendency of the solidifying solid to reject excess solute into the remaining liquid and can be understood using the model developed to an- alyze plane-front solidification However, the geometry of the dendritic liquid/solid interface and the adjacent diffusion field is complex

As a reasonable approximation, the dendritic structure may be represented by the diagram in Fig 22.4 The solidification in the interdendritic space can be

described by constructing the cell (shown dashed) and assuming that solidification proceeds in a manner similar to the plane-front solidification of a bar (as discussed in

Section 22.1) Under typical casting conditions, k’ = k Therefore, the segregation

Solute Segregation during Dendritic Solidification

t

c,,

Figure 22.4: Simplified cell model for analyzing interdendritic segregation

22.3: STRUCTURE OF CASTINGS AND INGOTS 549

profile along x follows Eq 22.9 approximately with k‘ = k and L equal to half the interdendritic spacing A typical profile is illustrated in Fig 22.1 for a system with the phase diagram shown in Fig 20.4a The formation of the eutectic in the last material to solidify can be expected When well-defined secondary dendrite branches are present, the solidification cell must be located between the branches rather than between the primary dendrites Much of the strong segregation present after solidification can be eliminated by solution treatments in which castings are annealed at elevated temperatures and the segregates are dispersed by solid-state diffusion [2]

22.3 STRUCTURE OF CASTINGS AND INGOTS

Castings are typically produced by pouring liquid into a relatively cold mold and allowing solidification to take place Heat is removed from the solidifying material

by conduction out through the mold The classic grain structure obtained after this type of solidification is illustrated in Fig 22.5

Three distinct zones are often (but not always) present The chill zone consists

of small equiaxed grains and results from the relatively rapid cooling rate due to the initially rapid outward flow of heat to the cold mold This produces considerable undercooling, and many small grains with random orientations are nucleated at,

or near, the mold surface These small grains grow until they impinge Further into the casting, the grains that grow most rapidly in the direction of the thermal gradient will advance most rapidly Grains that are oriented with slow mobilities in the growth direction will be assimilated by those with faster mobilities and result

in a columnar structure Finally, further into the casting in the liquid ahead of the advancing grains, a third zone consisting of equiaxed grains may form Here, nuclei for the growth of new grains are provided by small pieces of secondary den- drite branches that have become detached from the stalks of the oncoming primary dendrites This detachment occurs as a result of temperature fluctuations and con- vection For materials of high symmetry (such as cubic metals), these nuclei will grow into approximately equiaxed grains because they are isolated and growing in

a very slightly undercooled environment The latent heat of solidification that is released makes the growing crystals local “hot spots,” and the growth therefore

Chill zone Columnar zone Equixed zone

Figure 22.6:

zone, and equiaxed zone Classic grain structure of casting (or ingot) containing chill zone, columnar From Bower and Fleming [5]