lecture 5: solidification of single phase alloyscc.sjtu.edu.cn/upload/20160301103754768005.pdf ·...

Lecture 5: Solidification of Single Phase Alloys1

Lecture 5: Solidification of Single Phase

Alloys

What is a single phase alloy?

An alloy which consists of one

phase in both liquid state and solid

state.


Equilibrium Solute Partition Ratio


• When a piece of alloy solid is growing with a plane front from an alloy liquid with an overall alloy composition, C0,the solute concentration in the liquid at the solid/liquid interface is CL*, and the solute concentration in the solid at the solid/liquid interface is CS*.

• The temperature at the solid/liquid interface is T*.

Lecture 5: Solidification of Single

Phase Alloys4

Phase diagram Solute concentration profile

across the solid/liquid interface

• During crystal growth, the solid and liquid at the interface are in thermodynamic equilibrium, so CL* and CS* are defined by the phase diagram.

• In most cases, CL* and CS* are not equal. The ratio between CS* and CL*, k, is called the equilibrium solute partition ratio:



• When the liquidus line and solidus line of the binary phase

diagram of the alloy are straight lines, k is a constant.

• In many cases the liquidus and solidus lines go downward,

and k<1.

• In some cases, the liquidus and solidus lines go upward, and

k>1.

Analysis of Solute Concentration Distribution During

Solidification of Single Phase Alloys

Different solidification conditions can cause different distribution

of solute concentration in the liquid and in the solid.

There are four situations:

Situation 1: Fully equilibrium solidification

Situation 2: No solute diffusion in solid, complete solute diffusion and

mixing in liquid.

Situation 3: No diffusion in solid, normal diffusion in liquid,

no convection.

Situation 4: No diffusion in solid, some convection in the liquid


Situation 1: Fully Equilibrium Solidification


Phase Alloys8

• To facilitate a fully equilibrium solidification situation, the

growth rate of the solid phase has to be very small, so that

the following criterion is met:

L = the grown length of the solid crystal within time, t.

Ds = solute atom diffusivity in the solid.

• In this situation, the solute diffusion in the liquid and solid is

complete.


• That is:

CS = CS* (2a)

CL = CL* (2b)

• This is fully equilibrium solidification, and in this case, there is no segregation.


Solidification starting

During solidification, T = T*

Calculation of solute concentration in liquid and in

solid as functions of the fraction of solid

• From mass balance during solidification, we have:

CSfS + CLfL = C0 (3a)

fS + fL = 1 (3b)

(fS = fraction of solid, fL = fraction of liquid.)

From Equations (1)-(3), we can get:


From these Equations (4a) and (4b), we can observe:

• When fS=0, CS= kC0. This means that at the beginning of the solidification, the solute concentration of the solid can be very different from the composition of the alloy.

• When fS=1, CS=C0. This means that once the solidification is complete, the solute concentration of the solid is equal to the composition of the alloy.


Situation 2: No solute diffusion in solid, complete

solute diffusion and mixing in liquid.


• In this situation, we have:

CS CS* (5a)

CL = CL* (5b)

• The distribution of solute concentration of the solid is not

uniform. That is: there is solute segregation in the solid.


Solidification starting

During solidification, T = T*

Calculation of CL and CS* as functions of fS

• When k<1, CS*<CL*. This means that when a fraction of the

liquid is turned into solid, solute atoms are ejected into the liquid,

making it richer in solute atoms.

• From mass balance during solidification, we have:

(CL-CS*)dfS = (1-fS)dCL (6)


• From Equations (1) and (5b), we have:

CS* = kCL (7)

• By combining equations (6) and (7) and re-arranging, we

have:



• By integrating equation (8) and using the boundary condition, fS=0 when CL=Co, we get:

CL = Co(1-fS)(k-1) (9a)

• Based on Eq. (7) and fL + fS = 1, we have:

CS* = kCofL(k-1) (9b)

• Equation (9a) or (9b) is called “Scheil equation”.


The Scheil equation is often used to calculate solute concentration

distribution in a solid piece formed by solidification in Situation 2.

The Scheil equation shows that CL increases with increasing fS, and CL

can become infinitely large when fS approaches 1.

However, CL does have its limit.


For a hypoeutectic alloy, the limit is the eutectic

composition CE.

When CL=CE, eutectic solidification occurs, and the liquid solidifies

like a pure metal. From this point, the Scheil equation is not valid.

For an isomorphous alloy, the limit is 100%.

When CL=100%, the remaining liquid becomes a pure metal, and

solidifies as a pure metal. At this point, the Scheil equation is not

valid.

Situation 3: No diffusion in the solid, normal

diffusion in the liquid, no convection.


First we look at what happens in the initial transient period.

From the mass balance, we have the following differential equation:

R = growth rate

x’= the distance of the point of interest from the solid/liquid interface

DL = diffusion coefficient of solute atom in the liquid.


Phase Alloys21

The boundary conditions for this differential equation are:

CL=C0 at t=0 for x’ > 0

CL = C0 at x’ = for t > 0

When k is small, the solution of this differential equation is:

x is the distance from the starting point of solidification to the S/L

interface.


Solute concentration distribution in the solid in the initial transient stage

In Situation 3, there exists a steady state in

which the solid growth does not cause any

change of the distribution of solute

concentration in the liquid.



From equation (11), we can see that CS*≤C0, and CS* increases

with x.

Since CL*=CS*/k and k<1 , the solidification of the liquid at the

S/L interface causes ejection of the excess solute atoms into the

liquid in front of the S/L interface.

The flux of the ejected solute atoms can be calculated using the

following equation:

q1 = R(CL* - CS*) = RCL*(1-k) (12)

R is growth rate.


Phase diagram

Initial point

Transient stage Steady state stage

The figure is from 徐洲, 姚寿山主编,《材料加工原理》科技出版社,

2003.

p62

Initial

transient

stage

Steady

state

stageFinal

transient

stage

Solu

te c

oncentr

atio

n

Tem

pera

ture

Solute concentration

Calculation of the distribution of CL in liquid

To maintain the overall mass balance, CS* must be equal to

C0 in the steady state.

In the steady state, the differential equation (10) can be

changed into:

The boundary conditions are:

CL=C0/k at x’=0,

CL=C0 at x’ = .


(14)


There is also a further constrain set by the mass balance at the

S/L interface in the steady state.

The solution of differential equation (12) is:

(15)

(16)


In the final stage, the steady state cannot be maintained,

because of the limited distance from the S/L interface to the

wall of the container.

In this stage, the solute concentration of the liquid at the S/L

interface increases continuously.

At some point, CL* reaches the eutectic composition CE or

100%, and at this point, the remaining liquid solidifies like an

eutectic liquid or pure metal liquid.

The final stage is also called the final transient stage.

Situation 4: No diffusion in solid, some

convection in the liquid


When convection is present in the solidifying liquid,

there is a layer in front of the S/L interface.

Outside this layer, the distribution of solute

concentration is uniform.

Inside this layer, the distribution of the solute

concentration is controlled by solute atom diffusion in

the liquid.


The figure is from 徐洲, 姚寿山主编,《材料加工原理》科技出版社, 2003. p64


The solution of differential equation (14) for this situation is:

If we just look at the two sides of the layer, it is very similar to

Situation 2.

We can define an effective solute partition ratio, k’, as:

(17)


From equation (17), we can derive an equation which can be

used to calculate k’.

When d is very small, the Scheil equation can be slightly

modified and used to calculate the solute concentration in

the solid at the S/L interface as a function of solid fraction:

(18)


Solute concentration distribution of the solid formed by solidification

with no solute diffusion in the solid and limited amount of convection


It is important to note the following points:

When the convection is very strong, δ≈0 k’ k we have

situation 2.

When the convection is very weak, δ≈ k’ ≈ 1 CS*≈CL’≈

C0 we have situation 3.

lecture 5: solidification of single phase alloyscc.sjtu.edu.cn/upload/20160301103754768005.pdf ·...

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