Phase Transformation in Materials
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1 2015 Fall Phase Transformation in Materials Eun Soo Park Office: Telephone: Office hours: by an appointment 1
2 Contents for previous class Solidification: iquid Solid < Nucleation > & < Growth > Equilibrium Shape and Interface Structure on an Atomic Scale Thermal Roughening Growth of a pure solid 1) Continuous growth : Atomically rough or diffuse interface 2) ateral growth : Atomically flat of sharply defined interface a) Surface (2-D) nucleation b) Spiral growth Kinetic roughening When the growth rate of the singular Interface is high enough, it follows the ideal growth rate like a rough interface. 2
3 Removal of latent heat Heat Flow and Interface Stability 1) Superheated liquid : Extraction of latent heat by conduction in the crystal 2) Supercooled liquid : conduction of latent heat into the liquid T S & T v T S 0 & T v 3
4 Q: Alloy solidification? 1. Solidification of single-phase alloys Three limiting cases 1) Equilibrium Solidification: perfect mixing in solid and liquid 2) No Diffusion in Solid, Perfect Mixing in iquid 3) No Diffusion on Solid, Diffusional Mixing in the iquid - Planar S/ interface unidirectional solidification - Superheated liquid - Cellular and Dendritic Solidification 4 - Constitutional supercooling
5 1. Solidification of single-phase alloys k = S < 1 Fig A hypothetical phase diagram. k = S / is constant. k : partition coefficient : mole fraction of solute In this phase diagram of straight solidus and liquidus, k is const. (independent of T). Planar S/ interface unidirectional solidification 5
6 1. Solidification of single-phase alloys Three limiting cases 1) Equilibrium Solidification (perfect mixing in solid & liquid) 2) No Diffusion in Solid, Perfect Mixing in iquid 3) No Diffusion on Solid, Diffusional Mixing in the iquid 1) Equilibrium Solidification (perfect mixing in solid & liquid) low cooling rate : infinitely slow solidification Fig A hypothetical phase diagram. k = S / is constant. k = S partition coefficient - Sufficient time for diffusion in solid & liquid - Relative amount of solid and liquid : lever rule - Solidification starts at T 1 ( s =k 0 ) and ends at T 3 ( = 0 /k). 6
7 Composition vs x at T 2 A A S A S = A Fig Unidirectional solidification of alloy 0 in Fig (a) A planar S/ interface and axial heat flow. (b) Corresponding composition profile at T 2 assuming complete equilibrium. Conservation of solute requires the two shaded areas to be equal. A S = A 7
8 1) Equilibrium Solidification : perfect mixing in solid and liquid s = k T 1 -ΔT 0 +α k 0 +α T 2 Conservation of solute requires the two shaded areas to be equal. * Equilibrium solute concentration k 0 s 0 A S A A S = A 0 0 /k < E T 3 +ΔT 0 /k-α 0 -α iquid α
9 2) Non-equilibrium Solidification: No Diffusion in Solid, Perfect Mixing in iquid : high cooling rate, efficient stirring - Separate layers of solid retain their original compositions mean comp. of the solid ( S ) < s - iquid become richer than 0 /K E at the last part of solidification. - Variation of s : solute rejected to the liquid solute increase in the liquid ( S < s ) solid x < x s liquid > 0 k E s T 1 -ΔT local equil. at S/ interface T 2 Fig Planar front solidification of alloy 0 in fig assuming no diffusion in the solid, but complete mixing in the liquid. (a) As Fig. 4.19, but including the mean composition of the solid. (b) Composition profile just under T 1. (c) Composition profile at T 2 (compare with the profile and fraction solidified in Fig.4.20b) (d) Composition profile at the eutectic temperature and below. T E 9
10 2) No Diffusion in Solid, Perfect Mixing in iquid : high cooling rate, efficient stirring - Separate layers of solid retain their original compositions - mean comp. of the solid ( ) < s S T 1 -ΔT T 2 iquid Primary α + Eutectic solid x < x liquid > 0 k E s s T 3 10 T E
11 Mass balance: non-equilibrium lever rule (coring structure) When cooled by dt from any arbitrary T, determine the followings. : solute ejected into the liquid =? solute increase in the liquid Ignore the difference in molar volume between the solid and liquid. f s : volume fraction solidified solute ejected into the liquid=? proportional to what? df s ( S ) solute increase in the liquid=? proportional to what? (1-f s ) d ( ) df = (1 f ) d S S S Solve this equation. when f S = 0 S,? Initial conditions S = k 0 and = 0 11
12 df d d d = = = f k k fs S (1 ) S S 0 f S (1 k)( 1) d ln(1 f ) = S O d ln ln O = ( k 1) ln(1 f S ) = f ( k 1) O s = k = k f ( ) ( k 1 1 ) S O S : non-equilibrium lever rule (Scheil equation) quite generally applicable even for nonplanar solid/liquid interfaces provided here, the liquid composition is uniform and that the Gibbs-Thomson effect is negligible. If k<1: predicts that if no diff. in solid, some eutectic always exist to solidify. ( s < ) 12
13 3) No Diffusion on Solid, Diffusional Mixing in the iquid : high cooling rate, no stirring diffusion - Solute rejected from solid diffuse into liquid with limitation - rapid build up solute in front of the solid rapid increase in the comp. of solid forming (initial transient) - if it solidifies at a const. rate, v, then a steady state is finally obtained at T 3 - liquid : 0 /k, solid: 0 local equil. at S/ interface Composition profile at T 2 < T S/ < T 3? Steady-state profile at T 3? at T E or below? 13
14 Interface temperature * Steady-state at T 3. The composition solidifying equals the composition of liquid far ahead of the solid ( 0 ).
15 No Diffusion on Solid, Diffusional Mixing in the iquid During steady-state growth, (Interface liquid: Diffusion rate) Rate at which solute diffuses down the concentration gradient away from the interface = Rate at which solute is rejected from the solidifying liquid (iquid Solid: solute rejecting rate ) J Set up the equation. = DC x ' = v( C ( ) J = D = v S ( Solidification rate of alloy: excess solute control) K T = K T + v S S V ( Solidification rate of pure metal: latent heat control, 10 4 times faster than that of alloy) C S ) Solve this equation. S d = 0 0 c = ln( k 0 for = v D 0 0 ) all dx v ln( 0) = D x = 0; = / k x 0 x + c steady-state steady-state 15
16 ln 0 1 ( k 0 = 0 1) = 0 1 ( v D k k x ) e vx D = O [1 + 1 k k exp( x )] ( D / ν ) ( decreases exponentially from 0 /k at x=0, the interface, to 0 at large distances from the interface. The concentration profile has a characteristic width of D/v. ) - The concentration gradient in liquid in contact with the solid : ( ) J = D = v S = Dv S 16
17 Alloy solidification - Solidification of single-phase alloys * No Diffusion on Solid, Diffusional Mixing in the iquid When the solid/liquid interface is within ~D/v of the end of the bar the bow-wave of solute is compressed into a very small volume and the interface composition rises rapidly leading to a final transient and eutectic formation. 17
18 No Diffusion on Solid, Diffusional Mixing in the iquid Fig Planar front solidification of alloy 0 in Fig assuming no diffusion in solid and no stirring in the liquid. (a) Composition profile when S/ temperature is between T 2 and T 3 in Fig (b) Steady-state at T 3. The composition solidifying equals the composition of liquid far ahead of the solid ( 0 ). (c) Composition profile at T E and below, showing the final transient. D/v 18
19 Concentration profiles in practice : exhibit features between two cases Zone Refining 19
20 Q: Cellular and Dendritic Solidification by constitutional supercooling in alloy 20
21 2. Cellular and Dendritic Solidification Fast Solute diffusion similar to the conduction of latent heat in pure metal, possible to break up the planar front into dendrites. complicated, however, by the possibility of temp. gradients in the liquid. steady-state solidification at a planar interface What would be T e along the concentration profile ahead of the growth front during steady-state solidification? T temp. gradients in the liquid T e 21
22 * Constitutional Supercooling No Diffusion on Solid, Diffusional Mixing in the iquid Steady State * Actual temperature gradient in iquid T * equilibrium solidification temp. change T equil. At the interface, T = T equil. (not T E ) = T 3 T ' > (T 1 -T 3 )/(D/v) : the protrusion melts back - Planar interface: stable T ' /v < (T 1 -T 3 )/D : Constitutional supercooling cellular/ dendritic growth 22
23 Q: Planer Cell structure Dendrite? by constitutional supercooling in superheated liquid 23
24 Cellular Solidification: formation by constitutional supercooling in superheated liquid If temperature gradient ahead of an initially planar interface is gradually reduced below the critical value, (constitutional supercooling at solid/liquid interface) Solute pile up ower Te : ocal melting Formation of other protrusions First protrusion Break down of the interface: formation of cellular structure Heat flow Protrusions develop into long arms or cells growing parallel to the direction of heat flow Fig The breakdown of an initially planar solidification front into cells 24
25 Cellular Solidification: formation by constitutional supercooling in superheated liquid If temperature gradient ahead of an initially planar interface is gradually reduced below the critical value, (constitutional supercooling at solid/liquid interface) S Heat flow Fig Supercooling ahead of planar interface <The breakdown of an initially planar solidification front into cells>
26 Cellular Solidification: formation by constitutional supercooling in superheated liquid If temperature gradient ahead of an initially planar interface is gradually reduced below the critical value, (constitutional supercooling at solid/liquid interface) Convexity First protrusion Break down of the interface: formation of cellular structure Fig Solute diffusion ahead of a convex interface <The breakdown of an initially planar solidification front into cells>
27 Cellular Solidification: formation by constitutional supercooling in superheated liquid If temperature gradient ahead of an initially planar interface is gradually reduced below the critical value, (constitutional supercooling at solid/liquid interface) Solute pile up T e T v Convexity Solute pile up ower T equil : ocal melting First protrusion Break down of the interface: formation of cellular structure Heat Balance Equation K T = K T + v S S V K: thermal conductivity <The breakdown of an initially planar solidification front into cells>
28 Cellular Solidification: formation by constitutional supercooling in superheated liquid If temperature gradient ahead of an initially planar interface is gradually reduced below the critical value, (constitutional supercooling at solid/liquid interface) Convexity First protrusion Solute pile up ower Te : ocal melting Formation of other protrusions an array of cells : most of cells having 6 neighbers Break down of the interface: formation of cellular structure Heat flow Protrusions develop into long arms or cells growing parallel to the direction of heat flow 28 <The breakdown of an initially planar solidification front into cells>
29 Tips of the cells grow into the hottest liquid and therefore contain the least solute. iquid Primary α + Eutectic Even if 0 << max Solute file up eutectic solidification formation of 2 nd phases at the cell wall Fig Temperature and solute distributions associated with cellular solidification. Note that solute enrichment in the liquid between the cells, and coring in the cells with eutectic in the cell walls. 29
30 * Cellular microstructures Note that each cell has virtually the same orientation as its neighbors and together they form a single grain. (a) A decanted interface of a cellularly solidified Pb-Sn alloy (x 120) (after J.W. Rutter in iquid Metals and Solidification, American Society for Metals, 1958, p. 243). (b) ongitudinal view of cells in carbon tetrabromide (x 100) (after K.A. Jackson and J.D. Hunt, Acta Metallurgica 13 (1965) 1212). 30
31 * Temp. and solute distributions associated with cellular solidification. 1) Note that solute enrichment in the liquid between the cells, and coring in the cells with eutectic in the cell walls. T 3 < T 1 2) 3) Tips of the cells grow into the hottest liquid and therefore contain the least solute. Even if 0 << max Solute file up eutectic solidification formation of 2 nd phases at the cell wall 31
32 The change in morphology from cells to dendrites * Cellular microstructures are only stable for a certain range of temp. gradients. Sufficiently low temp. gradients Creation of constitutional supercooling in the liquid between the cells causing interface instabilities in the transverse direction (although, No temp. gradient perpendicular to the growth direction) Develop arms, i.e. dendrites form & Change in the direction of the primary arms away from the direction of heat flow into the crystallographically preferred directions i.e. (100) for cubic metals. Fig Cellular dendrites in carbon tetrabromide. ( After.R. Morris and W.C. Winegard, Journal of Crystal Growth 6 (1969) 61.) Fig Columnar dendrites in a transparent organic alloy. (After K.A. Jackson in Solidification, American Society for Metals, 1971, p. 121.) 32
33 Cellular and Dendritic Solidification At the interface, T = T e (not T E ) = T 3 T, liquid = T 1 : T = T 1 -T 3 (superheating) Criterion for the stable planar interface: T ' > (T 1 -T 3 )/(D/v) : the protrusion melts back_steeper than the critical gradient ( T 1 -T 3 : Equilibrium freezing range of alloy) T ' /v > (T 1 -T 3 )/D arge solidification range of T 1 -T 3 or high v promotes protrusions. need to well-controlled experimental conditions (temp. gradient & growth rate) Constitutional supercooling: T ' /v < (T 1 -T 3 )/D Formation of Cell and Dendrites Structures Solute effect : addition of a very small fraction of a percent solute with very small k ( k = S ) (T 1 -T 3 ) promotes dendrites. Cooling rate effect : Higher cooling rate allow less time for lateral diffusion of the rejected solute and therefore require smaller cell or dendrite arm spacings to avoid constitutional supercooling. 33
34 Solidification of Pure Metal : Thermal gradient dominant Solidification of single phase alloy: Solute redistribution dominant a) Constitutional supercooling Planar Cellular growth cellular dendritic growth Free dendritic growth 응고계면에조성적과냉의 thin zone 형성에의함 Dome 형태선단 / 주변에 hexagonal array Nucleation of new crystal in liquid T 조성적과냉영역증가 Cell 선단의피라미드형상 / 가지들의 square array/ Dendrite 성장방향쪽으로성장방향변화 성장이일어나는 interface 보다높은온도 성장하는 crystal 로부터발생한잠열을과냉각액상쪽으로방출함에의해형성 Dendrite 성장방향 / Branched rod-type dendrite b) Segregation : normal segregation, grain boundary segregation, cellular segregation, dendritic segregation, inversegregation, coring and intercrystalline segregation, gravity segregation 34
K S T S ' = K L T L ' + vl v
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