Heat Flow and Interface Stability Elemental metals - solidification rate controlled by rate at which latent of fusion can be conducted away from the solid/liquid interface Heat conduction can be either into the solid OR liquid depending on temperature gradient near the interface Case I: Solid growing into superheated liquid Solid/liquid interface growing with a velocity v Temperature gradients in both solid and liquid with interface at melting temp. T m Planar solid/liquid interface moving with velocity v, heat flow balance K S T S ' = K L T L ' + vl v K s are the thermal conductivities T s are the thermal gradients L V is the latent heat of fusion The above equation holds for growth of any planar interface even when heat is conducted into the liquid (T L < 0) Consider a small solid protrusion at the solid/liquid interface growing into the liquid Locally at the protrusion - temp. gradient into the liquid is increased, temp. gradient in solid decreases More heat flow from the liquid into the protrusion, slower rate of heat Phase extraction Transformations into the solid 1
Therefore, slower rate of growth of the solid at the protrusion as compared with the planar regions - solid protrusion is removed with time Interface protrusion is unstable Case II: Solid growing into undercooled or supercooled liquid Prostrusion at solid/liquid interface - negative temperature gradient at the interface becomes more negative - steeper gradient in liquid Faster rate of heat extraction into the liquid near the protrusion Solid protrusion grows faster Protrusions at interface are stable Planar solid/liquid interface growing into a supercooled liquid is inherently unstable Case I type situation (heat conduction through the solid) typically takes place when the solidification takes place from the mould walls - cooler than the molten liquid 2
Case II type situation (heat conduction into the liquid during solidification) - takes place during the initial stages of solidification for nucleation taking place on heterogeneous nucleation sites within the melt (e.g. impurity particles) - substantial undercooling (or supercooling) of the liquid might be required Solid nucleus forming on an impurity particle will be growing into a undercooled liquid - latent heat conducted into the liquid Solid particle will develop many stable protrusions in different directions during growth - arms will grow from the solid particle Initially spherical nucleus Interface becomes unstable Starts developing primary arms in many directions - usually crystallographic (e.g. for cubic metals along <001> directions along the cube axes) As primary arms grow and elongate, the interfaces become unstable and break up forming secondary arms, then tertiary arms and so on Shape of this type of solid - Dendrite (comes from the Greek word for tree) Dendritic growth morphology In pure metals these dentrites are referred to as thermal dendrites 3
Solidification of Single Phase Alloys Consider a binary alloy with the phase diagram as shown below: Liquidus and solidus are considered straight lines Define partition function as: k = X S X L X S and X L are the equilibrium solute contents at the ends of the tie line at a certain temperature k - independent of temp. Solidification of such an alloy is a rather complex process - depends on a combination of temp. gradients, cooling rates, growth velocities Simplified case: Planar growth of liquid/solid interface Unidirectional heat flow or heat extractions along a single direction Special furnace in which heat flow occurs only along a single direction 4
Three limiting cases of solidification: 1. Infinitely slow (equilibrium) solidification. 2. Solidification with no diffusion in the solid but perfect mixing in the liquid. 3. Solidification with no diffusion in the solid and only diffusional mixing in the liquid Equilibrium solidification: easiest case Alloy of composition X 0 starts solidifying at temperature T 1. First solid to form has composition, kx 0. Cooling further below T 1, more solid is formed. Cooling rate is sufficiently slow for diffusion to take place in the solid phase allowing for equilibrium homogeneous solid composition to be achieved Relative amounts of solid and liquid at any temp. - lever rule Since solidification is 1D, solute conservation implies that the shaded areas in the adjacent figure are equal Solidification is completed at T 3 and the last solidifying liquid has composition X 0 /k 5
No Diffusion in Solid, Perfect Mixing in Liquid: Often cooling rate is too rapid for substantial diffusion in the growing solid phase - assume no diffusion in the solid phase Liquid composition is kept homogeneous during cooling by continuous stirring 1D heat extraction as previous case: Solidification starts at temp. T 1 First solid to form has composition kx 0 < X 0, purer than the liquid from which it forms Solute rejected into the liquid - raises its composition above X 0 Temp. of interface has to be reduced before further solidification occurs - next layer of solid to form has a composition slightly richer in solute than kx 0. As process continues, liquid gets progressively richer in solute and solidification takes place at progressively lower temperatures At any temperature local equilibrium is maintained at the interface - solid and liquid compositions given by end points of tie line But since no diffusion in the solid is permitted the layers of solid forming retain their original compositions Therefore, mean composition of the solid is always lower than the interface solid composition (given by the phase diagram) - shown by the dashed line below: 6
Relative amounts of solid and liquid given by level rule applied to X S and X L. Liquid composition can become much richer than X 0 /k during solidification and even reach the eutectic composition X E. Soldification can finish by the final liquid undergoing eutectic solidification and forming a two phase (α + β) structure Average composition after complete solidification is still X 0 but the profile looks very different as compared to equilibrium solidification 7
Variation of X S along solidified bar obtained by equating solute rejected by small (infinitesimal) layer of solid formed to solute increase in the liquid ( X L X S )df S = ( 1 f S )dx L X S = kx 0 1 f ( ( S ) k 1 ) X L = X 0 f ( k 1) L f S is the volume fraction of solid Integrating using the boundary condition X S = kx 0 when f S = 0 Above equations are known as the non-equilibrium lever rule or Scheil equations 8
Cellular and Dendritic Solidification Diffusion of solute away from growing solid into liquid during alloy solidification - analogous to conduction of latent heat of fusion into liquid during pure metal solidification Complication - solute gradient ahead of solid/liquid interface - variation in equilibrium solidification temperature (liquidus line) Actual liquid temperature gradient can follow any line, e.g., T L At steady-state, at the interface, T L = T e = T 3 Temp. gradient T L lies below the critical T e value - so liquid is supercooled below its equilibrium solidification temp. - constitutional supercooling (arising from compositional or constitutional effects) 9
Stable protrusions can form on a planar interface during alloy soldification provided a region of constitutional supercooling exists ahead of the growing solid For the T L shown, the temp. of the tip of the protrusion will be higher than that of the surrounding interface - unlike pure metal solidification all the interface need not be isothermal in alloy solidification As long as tip temp. is < local liquidus temp., protrusion can still grow If temp. in liquid (T L ) > critical temp. gradient (T e ), then the tip temp. will exceed the local liquidus temp. and the protrusion will melt back Under steady-state growth, the critical temp. gradient can be calculated as follows: T L ' ( ) ( ) > T 1 T 3 D / v Where T 1 and T 3 are the liquidus and solidus temps. for comp. X 0 Regrouping, condition for no constitutional supercooling is, T L ' ( ) v > T 1 T 3 D T 1 -T 3 is the equilibrium freezing range of the alloy Planar front solidification is most difficult in case of alloys with a large equilibrium freezing range and high rates of solidification Usually, alloys never solidify with a planar solidification front - only under very specially controlled conditions Temp. gradients and growth rates are not individually controllable but depend in a complex manner on the rate at which heat is conducted away from solidifying alloy 10
Process of breakdown of an initially planar solidification front can be shown as below: Formation of first protrusion - lateral rejection of solute - solute pileup at the base of protrusion - lowers equilibrium solidification temp. locally - causes formation of recesses - triggers the formation of other protrusions Protrusions develop into long arms, or cells, parallel to the direction of heat flow - cellular solidification Lateral solute rejection concentrates at the cell walls - last liquid to solidify at lowest temperature Tips of the cells grow into the hottest liquid - therefore have minimum solute concentration Liquid between the cells - enriched in solute may even reach the eutectic composition and then the cell walls may have a second phase Cells have the same orientation as their neighbors and can join together to form single grain 11
Cellular solidification is stable only for certain range of temp. gradients - if gradient is too small, the cells or primary arms start developing secondary arms, tertiary arms.etc. - dendritic solidification Reason from cells to dendrites is not fully understood Probably associated with the creation of constitutional supercooling within the liquid entrapped between the walls of growing cells - causes interface instabilities in the lateral direction (perpendicular to the growth direction of primary cellular arms) For unidirectional solidification to continue, there should be minimal temp. gradient that develops in the lateral direction - perpendicular to growth direction Therefore, typically the cell arm spacing is such that this lateral gradient is reduced as much as possible - experimentally it is observed that with faster cooling rates the spacing decreases - consistent since faster cooling rates would allow less time for solute diffusion into the liquid between the cellular arms - reducing the spacing reduces the probability of developing constitutional supercooling 12