CHAPTER 10

MICROSEGREGATION ANDHOMOGENIZATION

10.1 INTRODUCTION

The preceding chapters have described the formation of various types ofmicrostructures, including cells, dendrites, eutectics and peritectics. Themicrostructure can be thought of as a combination of morphology andlength scales. These factors are important for determining material prop-erties, both in service and during processing. The present chapter exploresthe latter, focusing on how the solidification path relates to the evolutionof solid fraction and composition in the microstructure. These results are,in turn, coupled to the evolution of latent heat, which is the essential linkbetween microstructure and macroscale heat transfer, which is the subjectof Chap. 11.

We focus on the segregation profile that develops within the micro-structure due to the interaction of solute rejection by solidification withsolute diffusion in the liquid and solid phases. One such example has al-ready been encountered in Chap. 5, where the steady state, and the initialand final transients in planar growth of a binary alloy were considered. Inthat case, however, the main concern was segregation at the length scaleof an entire sample, whereas the present chapter is focused on segregationat the scale of the microstructure.

Key Concept 10.1: Transport at the microstructural scale

We will demonstrate that, for almost all cases of interest, both heatand solute transport at the scale of the secondary dendrite arms aresufficiently rapid for the temperature and liquid composition to be con-sidered uniform. Solute transport in the solid state, however, can varysignificantly depending on the local solidification conditions.

446 Microsegregation and homogenization

The first part of this chapter describes relatively simple one-dimensionalmodels for solute transport for binary alloys in the solid state. We thenmove on to cases of higher complexity, including multi-component alloysand more realistic morphologies.

10.2 1-D MICROSEGREGATION MODELSFOR BINARY ALLOYS

Consider the segment of a dendritic array shown in Fig. 10.1. The mi-crograph on the left-hand side comes from a directional solidification ex-periment in a succinonitrile – acetone alloy [20]. An idealization of thestructure suitable for modeling is presented on the right. The region ofthe secondary dendrite arms is modeled as a 2-D periodic array of unitcells, with lateral dimension of the secondary dendrite arm spacing �

2

andlength L. We neglect any variation in the third dimension for this partof the development. It is also assumed (for now) that there are only twocomponents and that the densities of the solid and liquid are constant andequal.

x

zL

�2

Fig. 10.1 (Left) A micrograph from a directionally solidified succinonitrile-acetonealloy. (Right) An idealized segment, displaying a periodic arrangement of sec-ondary dendrite arms.

For the sake of simplicity, the thermal properties of the solid and liq-uid are taken as equal. Scaling analysis will show that the temperaturecan be safely considered to be uniform at the scale of this segment of themicrostructure. The governing equations for heat and solute transport inthe unit cell under these assumptions are

@T

@t= ↵

✓

@2T

@x2

+@2T

@z2

◆

(10.1)

@C`

@t= D`

✓

@2C`

@x2

+@2C`

@z2

◆

(10.2)

@Cs

@t= Ds

✓

@2Cs

@x2

+@2Cs

@z2

◆

(10.3)

We now introduce the following scaled variables

⇠ =x

�2

/2; ⇣ =

z

L; ⌧ =

t

tf(10.4)

1-D microsegregation models for binary alloys 447

where tf is the local solidification time. Substituting the scaled variablesinto Eqs. (10.1)-(10.3) yields the semi-scaled equations

@T

@⌧=

4↵tf�2

2

✓

@2T

@⇠2+

�2

2

L2

@2T

@⇣2

◆

⇡ FoT

@2T

@⇠2(10.5)

@C`

@⌧=

4D`tf�2

2

✓

@2C`

@⇠2+

�2

2

L2

@2C`

@⇣2

◆

⇡ Fo`@2C`

@⇠2(10.6)

@Cs

@⌧=

4Dstf�2

2

✓

@2Cs

@⇠2+

�2

2

L2

@2Cs

@⇣2

◆

⇡ Fos

@2Cs

@⇠2(10.7)

We have invoked the assumption that �2

⌧ L, as suggested by the micro-graph in Fig. 10.1, which makes the problem one-dimensional. The dimen-sionless groups that appear in each equation are Fourier numbers, eachrepresenting the ratio of the local solidification time tf to the characteris-tic time for the diffusion of heat or solute. The meaning of the subscriptsshould be clear.

In order to proceed, experimental correlations between �2

and tf needto be introduced. Recall from our discussion of coarsening in Chap. 8, andEq. (8.115), that �

2

⇠ t1/3f . In particular, experimental measurements in

Al alloys, have shown that over a very wide range of solidification times,0.1 s < tf < 107 s, the following correlation provides a fairly accurate esti-mate for �

2

:�2

⇡⇣

10�5 m s�1/3⌘

t1/3f (10.8)

To compute the various Fourier numbers, the following property values forAl-Cu alloys are used as an example:

↵ ⇡ 4 ⇥ 10�5 m2 s�1; D` ⇡ 3 ⇥ 10�9 m2 s�1; Ds ⇡ 3 ⇥ 10�13 m2 s�1

(10.9)One should note the large disparity in the magnitude of the various diffu-sivities. Substituting these data into the expressions for the Fourier num-bers gives

FoT

⇡⇣

1.6 ⇥ 106 s�1/3⌘

t1/3f ; Fo` ⇡

⇣

120 s�1/3⌘

t1/3f ; Fo

s

⇡⇣

0.012 s�1/3⌘

t1/3f ;

(10.10)Thus, for all of the solidification times in the given range 0.1�107 s, we haveboth Fo

T

� 1 and Fo` � 1. Rewriting Eqs. (10.5) and (10.6), neglectingterms that are small compared to unity, we obtain

@2T

@⇠2⇡ 0 (10.11)

@2C`

@⇠2⇡ 0 (10.12)

The solution of these two equations with symmetry boundary conditionsat ⇠ = 0, 1 (x = 0, �

2

/2) is trivial, giving simply T = T (t) and C` = C`(t).In other words, both T and C` are independent of spatial position, but mayvary with time.

448 Microsegregation and homogenization

Key Concept 10.2: Scaling analysis results

Equation (10.10) shows that, for all of the solidification times in therange 0.1 � 107 s, both Fo

T

� 1 and Fo` � 1. This implies that atthe scale of the secondary dendrite arms, the temperature in bothphases, and the solute distribution in the liquid are spatially uniform,although they do vary in time.

There are two regimes for the composition in the solid. For tf > 1000 s,both sides of Eq. (10.7) are of comparable order, and the transient diffusionin the solid must be simulated. This case is called back diffusion, and sev-eral models for this phenomenon are described beginning in Sect. 10.2.1.For more typical casting conditions, where tf < 1000 s, we have Fo

s

⌧ 1and Eq. (10.7) becomes

@Cs

@⌧⇡ 0 (10.13)

This implies that the composition does not change with time in the solid, orthat the diffusion is negligible there. The result that was just obtained viascaling analysis of the transport equations was used as a starting pointfor a model first presented by Gulliver [6] and later by Scheil [12]. Thecompletion of the model requires an appropriate expression for the solutebalance at the moving interface.

Consider a solute balance over our model unit cell from Fig. 10.1. Sincethe densities of the solid and liquid phases have been assumed to be equal,and the unit cell is a closed system, we have

x⇤(t)

Z

0

Cs(x, t) dx +

�2/2Z

x⇤

C`(x, t) dx = C0

(10.14)

where x⇤ is the location of the solid-liquid interface. For all of the mi-crosegregation models that we will consider here, we assume the liquidcomposition to be uniform in space and equal to the interface compositionC⇤

` , so that the second integral can be evaluated directly, giving

x⇤(t)

Z

0

Cs(x, t) dx +

✓

�2

2� x⇤

◆

C⇤` = C

0

(10.15)

Differentiating Eq. (10.15) with respect to time gives

C⇤s

dx⇤

dt+

x⇤(t)

Z

0

@Cs(x, t)

@tdx +

✓

�2

2� x⇤

◆

dC⇤`

dt� C⇤

`

dx⇤

dt= 0 (10.16)

where we have used Cs(x⇤, t) = C⇤s . The remaining integral is resolved

by first applying the one-dimensional form of Eq. (10.3) to Eq. (10.16). We

1-D microsegregation models for binary alloys 449

C

x

Solid Liquid

x*

dx*

dCA*

�2 / 2

CA*

Cs*

Fig. 10.2 A construct demonstrating the partitioning of liquid from the interfaceto the liquid in the Gulliver-Scheil model of microsegregation.

also impose thermodynamic equilibrium at the interface, i.e, C⇤s = k

0

C⇤` ,

and combine the terms involving the interface velocity to obtain

� (1 � k0

)C⇤`

dx⇤

dt+

x⇤(t)

Z

0

Ds@2Cs(x, t)

@x2

dx + (�2

/2 � x⇤)dC⇤

`

dt= 0 (10.17)

The integration can be performed easily, with the result

� (1 � k0

)C⇤`

dx⇤

dt+ Ds

@Cs(x⇤, t)

@x+ (�

2

/2 � x⇤)dC⇤

`

dt= 0 (10.18)

The symmetry boundary condition at x = 0 was also imposed. Finally, wedivide by �

2

/2, and recognize that gs = x⇤/(�2

/2) to obtain

� (1 � k0

)C⇤`

dgsdt

+2Ds

�2

@Cs(x⇤, t)

@x+ (1 � gs)

dC⇤`

dt= 0 (10.19)

For values of Fos

where diffusion in the solid is negligible, as in theGulliver-Scheil model, Eq. (10.19) attains a simpler form by neglecting themiddle term on the left-hand side, leaving

(1 � k0

)C⇤`

dgsdt

= (1 � gs)dC⇤

`

dt(10.20)

This form of the boundary condition is illustrated in Fig. 10.2, and one cansee that the solute rejected by the moving interface in time dt is exactlybalanced by the increase in solute content of the liquid. Equation (10.20)can be rearranged for integration

g`

Z

1

dg`g`

= � 1

1 � k0

C⇤`

Z

C0

dC⇤`

C⇤`

(10.21)

where the initial conditions C⇤` = C

0

when f` = 1 has been invoked. Inte-gration and rearrangement of terms yields the Gulliver-Scheil equation,

C⇤` = C

0

gk0�1

` (10.22)

450 Microsegregation and homogenization

The microsegregation profile in the solid is obtained by setting C⇤s = k

0

C⇤` .

It is also convenient to have the liquid fraction in terms of temperature.Making the usual assumption that the liquidus curve is given by the straightline T = Tf + m`C`, it is straightforward to show that

gs = 1 � g` = 1 �✓

T � Tf

Tliq � Tf

◆

1/(k0�1)

(10.23)

This equation can be compared to its counterpart derived for the lever rule,Eq. (3.11), which is repeated here for convenience

gs =1

1 � k0

T � Tliq

T � Tf(10.24)

Recall that the lever rule was derived as a solute balance assuming a uni-form spatial composition in each phase, and that, for a constant density,gs = fs.

Key Concept 10.3: Gulliver-Scheil model

An important special case that applies for many physically realisticcasting conditions corresponds to Fo

T

� 1, Fo` � 1, Fos

⌧ 1. The tem-perature and liquid composition in the control volume can be treatedas being spatially uniform, and the solid composition does not changein time. Applying local equilibrium and solute balance at the interfaceleads to the Gulliver-Scheil model, whose principal results relate thesolid composition to the solid fraction and temperature. For a binaryalloy, we have

C⇤s = k

0

C0

(1 � gs)k0�1 (10.25)

gs = 1 �✓

T � Tf

Tliq � Tf

◆

1/(k0�1)

(10.26)

Figure 10.3 compares the solid fraction calculated using the Gulliver-Scheil and lever rule equations for an Al-4.5 wt% Cu alloy. This systemforms a eutectic at a composition of 33 wt% Cu at 548�C. The Gulliver-Scheil model predicts that the solidification will be incomplete at the eu-tectic temperature. It is usually assumed that any liquid that remainsat the eutectic point will solidify as a eutectic. This would yield approx-imately 10% non-equilibrium eutectic for this alloy in the Gulliver-Scheilmodel. The lever rule, on the other hand, predicts that no eutectic willform. Figure 10.3 also includes data obtained by DTA experiments at var-ious cooling rates [8]. At a slow cooling rate of �5 K min�1, which corre-sponds to a solidification time tf of approximately 20 min, the measuredsolid fraction vs. temperature is close to that obtained using the lever rule.As the cooling rate increases to �10 K min�1, the solid fraction curve moves

1-D microsegregation models for binary alloys 451

(a) Solid fraction vs. temperature

Lever ruleGulliver-Scheil

Lever ruleGulliver-Scheil

Non-equilibrium eutectic

dT/dt = –5 K min–1

dT/dt = –10 K min–1

Solid fraction gs

Tem

pera

ture

[°C

]

gs(b) Microsegregation profiles

640

620

600

580

560

540

520

10

8

6

4

2

0

33.2

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Cs [w

t% C

u]

Fig. 10.3 (a) Computed fraction solid vs. temperature curves for an Al-4.5 wt% Cualloy, using the lever rule and the Gulliver-Scheil equation. The experimental datawere obtained by DTA experiments. [8] (b) The corresponding microsegregationpatterns for the two solidification models.

in the direction of the Gulliver-Scheil model. Note that the experimentaldata in both cases shows about 4% nonequilibrium eutectic.

The experimental observations are consistent with the scaling analy-sis presented earlier. The cooling rate of �5 K min�1 represents a solidifi-cation time tf ⇡ 1200 s, and a corresponding value of Fo

s

⇡ 14 � 1. Thus,one would expect to find a solid composition that is almost independent ofspatial position, i.e., the lever rule would apply. On the other hand, if thecooling rate were sufficiently high that Fo

s

⌧ 1, then the Gulliver-Scheilmodel would be valid. This represents the opposite extreme from the leverrule. For the intermediate cooling rate of �10 K min�1, Fo

s

⇡ 7 & 1. Forthis case, we need to consider diffusion in the solid state, which will beaddressed in the next section.

Before taking up this latter case, we note that solid fraction calcula-tions such as these can be used as input into macroscopic models. If thesolidification time tf is known to fall in the range where the Gulliver-Scheil model is valid (Fo

s

⌧ 1, Fo` � 1, FoT

� 1), then one can writethe enthalpy-temperature relationship used in Eq. (6.71) directly, using

h⇢h(T )i =

TZ

298

⇢cpdT + ⇢Lf (1 � gs(T )) (10.27)

as discussed in Chap. 6. If the values of the Fourier numbers are not appro-priate for the Gulliver-Scheil model, e.g. Fo

s

⇠ O(1), then a back-diffusionmodel is needed, as described in the next section.