mass transport at the solid/liquid interface of...

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Mass transport at the solid/liquid interface of growing composite in situ

Waldemar WołczyńskiIMMS PAS

2METRO – MEtallurgical TRaining On-line Copyright © 2005 Waldemar Wołczyński - IMMS PAS

Microstructure of the growing composite in situ

FIG. 1

Oriented growth of the (Pb) – (Cd) composite in situa/ frozen solid/liquid interface (s / l interface)b/ typical structure of growing α / β composite in situ


Conditions applied to the diffusion equation by Jackson-Hunt theory

02 =∂∂

+∇zC

DvC

0=∂∂

xC

βα SSx +=at and0=x

DvC

zC α

0−=∂∂

αSx <≤0for

DvC

zC β

0+=∂∂ βαα SSxS +≤<for

C – solute concentration in the liquid at the solid/liquid interfaceD – coefficient of diffusion in the liquid

K.A. Jackson, J.D. Hunt, Trans. AIME, 236, 1129-1142, (1966)


Solution to the diffusion equation by Jackson - Hunt theory

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛

+−⎟

⎟⎠

⎞⎜⎜⎝

⎛

+++=

∞

=∞

1expcos

nnE SS

nzSS

nxBCCCβαβα

ππ

βα

ββ

αα

SSSCSC

B+

−= 00

0

with

FIG. 2

Composite in situ growth:a/ J-H planar s/l interfaceb/ adequate J-H phase diagram



A solid / liquid interface behaviour according to Jackson-Hunt theory


Imperfections of the J-H theorya/ no mass balance:

C(x,0) - CE for α phase lamellais not equal to CE - C(x,0) for β phase lamella

b/ mass balance is satisfied for Sα = Sβand C0

α = C0β, only, in J-H theory

c/ undercooling greater than ∆T = T* – TE assumed in concept ofideally coupled growth

d/ discontinuity of temperature at the α / β inter-phase boundary

e/ non-realistic curvature of the solid / liquid interface shapeFIG.3



according to the J-H scheme (FIG. 3a) some parts of α - phase should grow from the liquid of the solute concentration adequate to the formation of β - phase, rather it could lead to the changes in inter-lamellar spacing and instability of a solid/liquid interface

FIG. 4

instability at the solid / liquid interfacea/ observed during composite in situ growthb/ concluded from the J-H theory (as above)




according to J-H solution of diffusion equationconcentration profile is common for bothlamellae of the composite in situ

^assumption of non separation of concentrationmicro-field (in the J-H theory) together with concept of ideally coupled growth∆Tα

* = ∆Tβ* = ∆T results in discontinuity

of undercooling at α / β inter-phase boundary^moreover, some parts of the α as well as β phaselamellae should grow outside of the regime

undercooling∆TD results from changes of the solute concentration∆TR results from the s / l interface curvature

FIG.5


Application of Jackson-Hunt theory to the phase diagram

total undercooling: ∆T = T* - TE

C solute concentrationCE solute concentration at eutectic pointSα half the width of the α phase lamellaSβ half the width of the β phase lamellaT equilibrium temperatureTα(x,z) equilibrium temperature corresponding

to changes in solute concentrationat the α phase solid / liquid interface (z = 0)

Tβ(x,z) equilibrium temperature correspondingto changes in solute concentrationat the β phase solid / liquid interface (z = 0)

T* real temperature of the solid/liquid interfaceΤE temperature of eutectic transformation

FIG.6

undercooling – phase diagram K.A. Jackson, J.D. Hunt, Trans. AIME, 236, 1129-1142, (1966)


An initial correction of Jackson-Hunt theory

FIG. 7

now

C0α Sα = C0

β Sβ

thus

a/corrected planar solid / liquid interfaceb/corresponding arbitrary phase diagram 000

0 =+

−=

βα

ββ

αα

SSSCSC

B

∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛

+−⎟

⎟⎠

⎞⎜⎜⎝

⎛

+++=

∞

=∞

1expcos

nnE SS

nzSS

nxBCCCβαβα

ππ


Initially corrected Jackson - Hunt theorySolute concentration and undercooling

FIG. 8 FIG. 9solute concentration and undercooling undercooling – phase diagram

no significant improvement of the J-H theory ! - as it is seen in FIG. 9(to be compared with the scheme in FIG. 6)


Fundamentals of the current analysisConcept of the coupled growth

coupled growth isa new concept toimprove J-H theory

∆T*α = T*

α – TE

∆T*β = T*

β – TE

desired relation: undercooling – phase diagram**βα TT ∆≠∆FIG. 10 but generally


Undercooling of the s/l interface due to the concept of coupled growth

curvature undercooling

( ) ( )0,0, * xTTxTRα

ααδ −=

( ) ( )0,0, * xTTxTRβ

ββδ −=

undercooling – phase diagram

( ) ( ) *0,0, ααα δδ TxTxT R ∆=+

total undercooling( ) ( ) *0,0, β

ββ δδ TxTxT D ∆=+FIG. 11


Solute concentration dueto the concept of coupled growth

( ) ( ) ECxCxC −= 0,0, ααδ

( ) ( ) ECxCxC −= 0,0, ββδ


( ) ( ) 00,0,0 <−= ES CSCSC αα

αα

consequentiallyFIG. 12 ( ) ( ) 00,0,0 >−= ES CSCSC α

βα

β


Fundamentals of the new solutiondue to the concept of coupled growth

FIG. 13( ) ( ) 00,0,0 <−= ES CSCSC α

αα

α

a/ planar s / l interface b/ corresponding

arbitrary phase diagram ( ) ( ) 00,0,0 >−= ES CSCSC αβ

αβ


Diffusion equation due to the concept of coupled growth

FIG. 14


diffusion equation is:

02

2

2

2=

∂∂

+∂

∂+

∂

∂zC

Dv

zC

xC δδδ

( ) ( ) 0,, =−= ECzSCzSC ααδwith


Assumptions for the new solution to diffusion equation



02

2

2

2=

∂∂

+∂

∂+

∂

∂zC

Dv

zC

xC δδδ

it is required to solve the diffusion equation in such a way to:a/ satisfy the solid / liquid interface undercooling behaviour

defined in FIG. 15b/ obtain the solution separately for the α – phase lamella

and β – phase lamella

FIG. 15


General solution to the diffusion equation


02

2

2

2=

∂∂

+∂

∂+

∂

∂zC

Dv

zC

xC δδδ

general solution to diffusion equation formulated in accordance with the concept of coupled growthis as follows:

)()(),( zZxXzxC =δ


Solution to the diffusion equationDetailed formulations

general solution to diffusion equation formulated in accordance with the concept of coupled growthis as follows:

)()(),( zZxXzxC =δwhere

)(sin)(cos)( xBxAxX ωω +=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−−= z

Dv

DvzZ 2

2

2

42exp)( ω

A, B, ω - parameters are to be defined


Definition of the unknown parameters

A, B, ω - parameters are to be defined definitions are to be given separately

[ ]αSx ,0∈ 0≥za/ for the α phase lamella

the values of B and ω parameter yield from conditions a/ and b/:

0),(

0=

∂∂

=xxzxCδ

a/ 0),( =zSC αδand b/

0=Bit yieldsfrom a/ 0)0cos()0sin( =⋅+⋅− ωωωω BA

0)cos( =αωSAfrom a/ and b/K,2,1,

2)12(

12 =−

== − nS

nn

α

πωω

and


Detailed solutionto the diffusion equation


the detailed solution to diffusion equation is:

∑∞

=− ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

1

2

2

2

12 2)12(

42exp

2)12(cos),(

nn z

Sn

Dv

Dv

SxnAzxC

αα

ππδ

12 −nA are constantswhere


Slow solidificationSolution to diffusion equation

Dv

Sn

22)12(

>>−

α

πit is evident that for slow solidification:


∑∞

=− ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

1

2

2

2

12 2)12(

42exp

2)12(cos),(

nn z

Sn

Dv

Dv

SxnAzxC

αα

ππδ

reduces to

∑∞

=− ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

112 2

)12(exp2

)12(cos),(n

n zS

nS

xnAzxCαα

ππδ


A2n-1 parameters

the values of A2n-1 parameters are calculated applying the condition:

],0[,0)();(),(

0ααα

δ Sxxfxfz

zxC

z∈<=

∂∂

=for

a/ micro-filed of solute concentration for rapid solidification

∑∞

=−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−=

∂∂

1

2

2

2

120 2

)12(cos2

)12(42

),(n

nz S

xnS

nDv

DvA

zzxC

αα

ππδ

b/ micro-filed of solute concentration for slow solidification

∑∞

=−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

∂∂

112

0 2)12(

cos2

)12(),(n

nz S

xnS

nA

zzxC

αα

ππδ


Application of the f(x) function

additionally, new function f(x) is to be introduced:

)()2(),()(,22),( xfSxfxfxfSxSxf −=+=−≤≤− ααα

the following property of the f(x) is to be applied

∑∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛+≈

1

0

2cos

2)(

nn S

xnaa

xfα

π

dxS

xnxf

Sa

S

n ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

α

αα

π2

0 2cos)(1

where


f(x)

since:

it yields:

)()2( xfSxf −=+ α kn 2=for K,2,1,0=k

02

2cos)(2

2cos)(1

)(22cos)(

22cos)(1

22cos)(

22cos)(1

22cos)(

22cos)(1

2)2(2cos)2(

22cos)(1

22cos)(

22cos)(1

22cos)(1

0 0

0 0

0 0

0

0

0

0

0

2

2

02

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−+⎟⎟

⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ +++⎟⎟

⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫

−

−

−

α α

α α

α α

α

α

α

α

α α

α

α

ααα

ααα

ααα

ααα

α

αα

αα

ααα

αα

ππ

ππ

ππ

ππ

ππ

ππ

π

S S

S S

S S

S

S

S

S

S S

S

S

k

dxS

xkxfdxS

xkxfS

xdS

xkxfdxS

xkxfS

dxS

xkxfdxS

xkxfS

dxS

xkxfdxS

xkxfS

dxS

SxkSxfdxS

xkxfS

dxS

xkxfdxS

xkxfS

dxS

xkxfS

a


f(x)

forsince:

it yields:

)()2( xfSxf −=+ α K,2,1,12 =−= kkn

∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛ −

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++−−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ +−++⎟⎟⎠

⎞⎜⎜⎝

⎛ −

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

−

−

−

−

−

α

α α

α α

α

α

α

α

α

α

α α

α

α

αα

ααα

ααα

ααα

ααα

α

αα

αα

ααα

αα

π

ππ

ππ

ππ

ππππ

ππ

ππ

π

S

S S

S S

S

S

S

S

S

S

S S

S

S

k

dxS

xkxfS

xdS

xkxfdxS

xkxfS

dxS

xkxfdxS

xkxfS

dxS

xkxfdxS

xkxfS

dxkS

xkxfdxS

xkxfS

dxS

SxkSxfdxS

xkxfS

dxS

xkxfdxS

xkxfS

dxS

xkxfS

a

0

0 0

0 0

0

0

0

0

0

0

0

2

2

012

2)12(cos)(2

)(2

)12(cos)(2

)12(cos)(1

2)12(cos)(

2)12(cos)(1

2)12(cos)(

2)12(cos)(1

22

)12(cos)(2

)12(cos)(1

2)2()12(cos)2(

2)12(cos)(1

2)12(cos)(

2)12(cos)(1

2)12(cos)(1


Result of the application of the f(x) function

)()2( xfSxf −=+ αassuming:

kn 2=forK,2,1,0=k02 =kait yields

and

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=−

α

αα

πS

k dxS

xkxf

Sa

012 2

)12(cos)(2

K,2,1,12 =−= kknfor

∑∞

=− ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −≈

112 2

)12(cos)(

kk S

xkaxf

α

πfinally, the Fourier series of the f(x) is:


Final definition of the A2n-1 parameter

for rapid solidification:

dxS

xnxf

SSn

Dv

DvA

S

n ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛ −

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−=

−

−

α

αα

αα

ππ

0

12

2

2

12 2)12(

cos)(22

)12(42

K,2,1=nfor slow solidification:

dxS

xnxf

nA

S

n ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

−=−

α

αα

ππ 0

12 2)12(

cos)()12(

4K,2,1=n


Properties of the solution to diffusion equation

0),(),(

20=

∂∂

=∂

∂

== α

δδ

Sxx xzxC

xzxC

[ ]αSx ,0∈

00

),2()2()()(),(

== ∂+−∂

−=+−−=−==∂

∂

zz zzSxCSxfxfxf

zzxC α

ααααδδ

according to assumption:

)()2(),()( xfSxfxfxf ααααα −=+=−


Solution to the diffusion equation

[ ]βαα SSSx +∈ , 0≥zb/ for the β phase lamella

∑∞

=−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−⎟

⎟⎠

⎞⎜⎜⎝

⎛ +−−

=

1

2

2

2

12 2)12(

42exp

2)()12(

cos

),(

nn z

Sn

Dv

Dv

SSSxn

B

zxC

ββ

βα ππ

δ

dxS

SSxnxf

S

Sn

Dv

DvB

S

SS

n

∫−

−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++−−=

α

βα β

βαβ

β

β

π

π

2)()12(

cos)(2

2)12(

42

12

2

2

12with

rapid solidification


Solution to the diffusion equation

[ ]βαα SSSx +∈ , 0≥zb/ for the β phase lamella

∑∞

=− ⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−⎟

⎟⎠

⎞⎜⎜⎝

⎛ +−−=

112 2

)12(exp

2)()12(

cos),(n

n zS

nS

SSxnBzxC

ββ

βα ππδ

dxS

SSxnxf

nB

S

SSn ∫

−− ⎟

⎟⎠

⎞⎜⎜⎝

⎛ +−−

−−=

α

βα β

βαβ

ππ 2

)()12(cos)(

)12(4

12

K,2,1=n],0[),(

),(

0ββ

δSxxf

zzxC

z∈=

∂∂

=andslow solidification


Presentation of solution to the diffusion equation

a/ for the α phase lamella

[ ]αSx ,0∈ 0=z

[ ]βαα SSSx +∈ ,

b/ for the β phase lamella

0=zFIG. 16

micro-field of solute concentrationundercooling


Total mass balance withinmicro-field of solute concentration

0)12()12(

)12()1(4),(),(

22222

212

22222

212

1

1

0 00

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−++−

−++

−−

=++

−−

∞

=

−∞ ∞ +

∑∫ ∫ ∫∫

ππ

πδδ

ββ

β

αα

α

βα

α

α

DnSvSv

SB

DnSvSv

SA

nDdzdxzxCdzdxzxC

nn

n

nSS

S

S

( )( ) K,2,1,

)12(

)12(222222

22222212

12 =−++

−++=

−− n

DnSvSvS

DnSvSvSAB n

nπ

π

ααβ

ββαwhere

for rapid solidificationK,2,1,

2

1212 =⎟⎟⎠

⎞⎜⎜⎝

⎛= −− n

SSAB nn

β

αfor slow solidification


Visualization of total mass balance within micro-field of solute concentration

FIG. 17

mass balance calculated forplanar solid / liquid interface


Local mass balance within micro-field of solute concentration

local mass balance is satisfied at z = 0 for α phase lamella and z = d for β phase lamella

0),()0,(0

=+ ∫∫+

dxdxCdxxCSS

S

S βα

α

αδδ

after some rearrangements

02

)12(exp

)12()1(2

)12()1(2

22222

1

1

121

1

12 =⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −++−∑

−

−−∑

−− ∞

=

−

−

∞

=

−

− dSD

DnSvvS

nS

Bn

SA

n

n

nn

n

nβ

βββαπ

ππ

d - phase protrusion


Visualization of local mass balance within micro-field of solute concentration

FIG. 18

protrusion, d, is the transient phase;it has property of the liquidbut structure of the solid


Definition of the phase protrusion

rapid solidification

( )( )

0

2

)12(exp

)12(

)12(1

)12()1(

22222

22222

22222

1

1

12

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −++−

−++

−++−

×−

−∑∞

=

−

−

dSD

DnSvvS

DnSvvSS

DnSvvSS

nA

n

n

n

β

ββ

ααβ

ββα π

π

π

02

)12(exp1)12(

)1(1

1

12 =⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−

−−

∑∞

=

−

− dS

nSS

nA

n

n

nββ

α πslow solidification


Confirmation for the existenceof the phase protrusion

FIG. 19

oriented growth ofthe (Pb) – (Cd) composite in situ

phase protrusion visible for (Cd) - leading phase, FIG. 19a


α / β inter-phase mass balance

the s / l interface mass balance requires

( ) ( ) ( )0,10, xCkDvS

zxCS α

αα

α

αδ

−=∂

∂ [ ]αSx ,0∈

[ ]βαα SSSx +∈ ,( ) ( ) ( )dxCkDvS

zdxCS ,1, β

ββ

β

βδ

−=∂

∂

mass balance at the α / β inter-phase

( ) ( )

( ) ( ) ( ) ( )( ) 0,0,,0,

,lim0,lim

0000 =+=+

=∂

∂+

∂∂

+→−→

dSCSSCSDvdSC

DvSSC

DvS

zdxCS

zxCS

SxSx

αβ

βαα

ααβ

βαα

α

ββ

α

αα

α

δδ


Triple point of the s / l interface

not only mechanical equilibriumbut thermodynamic equilibrium at the triple pointis satisfied as well

solute concentration micro-field exists within the transition phase, d, (over leading phase) as if it wasthe liquid phase

solute concentrationundercooling FIG. 20


Concluding remarks

at slow solidification, typical for composite in situ growth

K,2,1,2

1212 =⎟⎟⎠

⎞⎜⎜⎝

⎛= −− n

SSAB nn

β

α ( )2

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

β

αα

α

ββ S

SxfSS

xf

the current description of micro-filed of solute concentrationcan be mathematically reducedto the J-H equation, however under condition that phase diagram would be symmetrical onethat is C0

α = C0β and consequentially Sα = Sβ , additionally,

phase protrusion becomes d =0,CE comes back to the α / β inter-phase boundaryand mass balance is satisfied at each co-ordinate, z

METRO MEtallurgical TRaining On-line

Education and Culture

Mass transport at the solid/liquid interface of growing composite in situ

End of the lecture

mass transport at the solid/liquid interface of...

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