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Lecture 3: Solidification
Driving Force
Solidification is undoubtedly the most important processing route for metals and alloys. Consider a pure metal (Fig. 1). At the fusion tem- perature T (^) f , ∆G = 0 so that
∆G = ∆H (^) f − T (^) f ∆S (^) f = 0 or ∆H (^) f = T (^) f ∆S (^) f
where ∆H (^) f is the latent heat of fusion and ∆S (^) f is positive for melting. For any temperature other than T (^) f ,
∆G = ∆H − T ∆S " ∆H (^) f − T ∆S (^) f = ∆S (^) f (T (^) f − T ) = ∆S (^) f (^) ︸︷︷︸∆T undercooling
The driving force is therefore proportional to the undercooling provided that the latent heat and the entropy of fusion do not vary much with temperature.
Fig. 1: Driving force for solidification
Nucleation and Growth Homogeneous nucleation occurs only at very large ∆T , for example, in containerless experiments where a pure liquid is isolated from its environment. In general, solidification is by heterogeneous nucleation, either on impurity particles or wherever the liquid comes into contact with the container surface. The velocity v of the transformation front is related to the differ- ence in the rate of liquid→solid atom jumps and the solid→liquid atom jumps:
v ∝ exp
− (^) kTQ
}[
1 − exp
− ∆kTG
}]
i.e. for small ∆G v ∝ ∆G v ∝ ∆T The last two proportionalities assume that the undercooling is small, in which case exp{x} " 1 + x.
Fig. 1: Barrier to interface motion
Geometry of Solidification Fig. 2 shows the grain structures possible. The chill zone contains fine crystals nucleated at the mould surface. There is then selective
CSL
Temperature CLS
liquid
solid C 0
solid liquid
CSL
CLS C 0
Concentration
concentration Distance
Alloys: Solute Partitioning Dendrite formation is extremely common in alloys, where solute partitions between the solid and liquid phases (Fig. 4). By convention, we shall label the composition of the solid phase which is in equilibrium with the liquid as C SL^ and a similar interpretation applies to C LS^. C 0 represents the average composition of the alloy. The partition coefficient k is written k = C^
SL C LS^ frequently^ <^1 Under equilibrium conditions the compositions of the solid and liquid at all stages of solidification are given by a tie–line of the phase diagram, and the proportions of the phases at any temperature by the lever rule.
Fig. 4: Solidification under equilibrium conditions
In practice, equilibrium will only be maintained at the interface (Fig. 5), where the compositions agree with the phase diagram. The composition of the solid follows the solidus. Steady state solidification
occurs when the temperature is T ∗^ , when C SL^ = C 0 and there is no further solute partitioned into the remaining liquid. From AH8, C = C SL^ + (C LS^ − C SL^ ) exp
− (^) D/vx
≡ C 0 + C^0 (1 k^ − k)exp
− (^) D/vx
Note that D/v has dimensions of length, approximately the diffusion distance into the liquid. It is typically about 0.5 mm but can be just a few micrometres in rapid solidification processing.
Fig. 5: Solidification under nonequilibrium conditions
1602, 1603 Levitation T607 dendrite, cast aluminium Video of succinonitrile
concentration (^0) f S 1
C 0 kC 0 concentration (^0) f S 1
C 0 kC (^0)
diffusion only convection
S L^ S moving furnace, zone refining
The redistribution of solute is larger when there is mixing in the liquid (Fig. 7). This can be used in zone refining, e.g. of silicon, with a repeated sweeping in one direction, leading to purification (elements for which k > 1 would be swept in the other direction). By moving the furnace to and fro, one can obtain zone levelling.
Fig. 7: Distribution of solute
Constitutional Supercooling (AH10) Solute is partitioned into the liquid ahead of the solidification front. This causes a corresponding variation in the liquidus temperature (the temperature below which freezing begins). There is, however, a positive temperature gradient in the liquid, giving rise to a supercooled zone of liquid ahead of the interface (Fig. 8). This is called constitutional supercooling because it is caused by composition changes. A small perturbation on the interface will therefore expand into a supercooled liquid. This gives rise to dendrites. It follows that a supercooled zone only occurs when the liquidus– temperature (T (^) L ) gradient at the interface is larger than the temperature
Fig. 8: Diagram illustrating constitutional supercool- ing.
gradient: ∂T (^) L ∂x
x=
∂ ∂Tx i.e., m ∂ ∂CxL
x=
∂ ∂Tx
where m is the magnitude of the slope of the liquidus phase boundary on the phase diagram. From AH8 we note that ∂CL ∂x
x=
= − C^
LS − C SL
D/v so that the minimum thermal gradient required for a stable solidification front is ∂T ∂x <
mC 0 (1 − k)v kD It is very difficult to avoid constitutional supercooling in practice be- cause the velocity required is very small indeed. Directional solidifica- tion with a planar front is possible only at low growth rates, for example
806 Cell formation 807 Plan view of cells 786 Dendrites 1281 Solute segregation 1287 Dendrite arm spacing 828,829 Al-Si, Al-Si-Na
Lecture 5: Solidification
Solidification Processing A key phenomenon in solidification processing is the transfer of heat by radiation, direct contact with the mould, conduction through air and convection in the air gap between the mould and ingot. Casting situ- ations may be divided according to whether or not significant thermal gradients are set up in the solidifying metal. The transfer of heat across a gap (an interface) is given by
q = h∆T
where q is the heat flux, h is the thermal conductance of the interface and ∆T the temperature difference between the source and the sink. Values of the heat transfer coefficient vary widely: 10 → 10 2 for large air gap; 10^3 → 10 4 for normal castings with gravity contact; 10^5 → 10 6 for excellent contact as in pressure casting. The temperature profile obtained depends on the a comparison of the heat conductance of the interface with that of the whole casting. The thermal conductance of a casting is written K/L where where K is the thermal conductivity of a casting of length L in the direction of heat flow. The ratio of the thermal conductance of the interface (h) to that of the casting, termed the Biot number:
Bi = (^) K/Lh = hLK
.
Fig. 11: Chill mould. The liquid is at a temperature higher than T (^) m , the melting temperature.
This cooling of the liquid will be followed by the second stage which is isothermal solidification (Fig. 12). For this case, the speed v of the solidification front can be obtained by balancing heat evolution against heat extraction:
q = h∆T = v∆H (^) F i.e., v = (^) ∆h∆HT F
where q is the heat flux and ∆H (^) F is latent heat released on solidification.
Fig. 12: Chill mould. The casting is now at the melt- ing temperature T (^) m
Permanent Mould PressureGravity^ diedie CentrifugalContinuous
Temporary Mould PlasterSand Lost wax
CASTING PROCESSES
Casting Processes
Fig. 12: Procedure for sand casting. This process is usually automated.
Sand Casting
In sand casting a mould is made by packing sand around a pattern (Fig. 13), removing the pattern and hardening the sand with polymer or silicate. The metal is then poured into the resulting mould in a controlled manner so that solidification proceeds in an orderly manner without leaving any holes or porosity. To avoid the blockage of chan- nels, chill blocks may be used to permit certain parst to solidify first. Risers are used to feed metal into the mould as it freezes and contracts.
process in the manufacture of steel and aluminium.
Others
Lost wax process, rapid solidification.
1444 Continuous casting Continuous casting video (Scunthorpe Steel) 865 Melt spinning
