4. Segregation at High and Ambient Temperatures
|The Rules of the Game|
|After I have introduced you to the
influence of segregation on the final
microstructure, I will now concentrate on macro and microsegregation of
constituents and impurities in the final microstructure.
For simplicity in writing, I consider a melt that consists primarily of one constituent (e.g. 98 % iron), one alloying element (e.g. 1.95 % carbon) and some trace impurities (e.g. the rest of the periodic table with together 0.05 %). We also assume that the influence of the trace impurities on the melting point is negligible in comparison to the influence of the alloying element.
The simple way to look at this is to start with a few basic definitions plus simple truths:
|1. In a solid-liquid equilibrium, most elements prefer to remain in the
melt and thus have a
k < 1 that can be deduced from the respective phase diagram.
The (equilibrium) segregation coefficient of carbon in iron is thus
kC in Fe = 0.6 as a look on the iron - carbon
phase diagram demonstrates.
Some elements might prefer the solid to the liquid and then have segregation coefficients k > 1. This, while not uncommon, is rather the exception. It must be expected that the equilibrium segregation coefficient of the alloying element in some binary composition changes if some more elements are added. But as long as the concentrations of additional elements are low, it should not change very much.
|2. Full equilibrium is only maintained for interface velocities v » 0 mm/s. For noticeable interface velocities, we define an effective segregation coefficient keff that moves towards unity for increasing interface velocities v. This means that global segregation effects get smaller for increasing v! This is clear: A very rapidly moving solid-liquid interface will freeze the melt "as is"; there will be no difference in composition between solid and liquid anymore.|
|3. High temperature segregation is
the term I will use for the difference between the nominal concentration of alloy elements and
impurities and the actual concentration
right after solidification.
High temperature segregation depends pretty much only on the speed v(x, y, z, t) of the solid-liquid interface at some point in space and time, and the various concentrations of alloying elements and impurities in the melt. While the alloying element may influence v(x, y, z, t) to some extent (via supercooling and so on), trace impurities don't. For high-temperature segregation, however, it doesn't matter why the interface has the velocity it has.
|4. Room temperature segregation is
the term I will use for what kind of non-uniformities are left in the
concentration of of alloying element and impurities and / or caused by high temperature segregation.
In other words: At room temperature the concentration profile of alloying elements and impurities will not be the same as at high temperature since diffusion tends to equilibrate any differences on a scale given by how far atoms can diffuse until it gets too cold to move.
Local high concentrations of some atoms at high temperatures may increase the probability for nucleating precipitates or other large defects, causing an inhomogeneous distribution of those defects that does not mirror their high temperature segregation. Carbon in iron,. for example, might be distributed homogeneously at high temperature but precipitate very unevenly because the cementite precipitates nucleate at inhomogeneously distributed or segregated impurities.
|Now let's look at the most simple situation where a melt cast into a mould crystallizes close to equilibrium, implying a planar solid-liquid interface that moves slowly from the outside of the mould to the inside.|
|The relevant portion of the phase diagram for an alloying element with a segregation coefficient k < 1 (the rule) looks, slightly idealized, like this:|
|This could be, for example, a part of the iron -carbon phase diagram. The blue area then would stand for the (bcc) d -phase (or austenite, if one moves to somewhat higher carbon concentrations).|
|As long as we can approximate the liquidus and solidus line by straight lines with slopes ms and mL, simple geometry tells us that:|
|For some global or nominal concentration c0, we then can define the concentrations cs = kc0 at the temperature TL where first freezing will begin, and the concentration cL = c0/k at the temperature TS where freezing is completed.|
|This implies that the very first parts to solidify will have the alloying element incorporated with a concentration cs = kc0, and that is lower then the nominal concentration c0; always assuming equilibrium, of course!|
|High Temperature Macrosegregation|
|If we would remain at or very close
to equilibrium at all times, there would be no segregation at all. The
unavoidable differences in concentration between parts that solidified early
and those that crystallized late would be equilibrated by diffusion in the
solid already at high temperatures. We must therefore deviate from equilibrium
to some extent if we want some high-temperature segregation.
There are many ways of modelling this; one simple model was introduced by E. Scheil in 1942 1).
|Scheil simply assumed that there is no diffusion
in the solid, and arbitrarily fast diffusion in the liquid. In other words: the
solid is forced to preserve whatever concentrations the alloy element (and the
impurities) have right after solidification, and the liquid has a uniform
concentration everywhere and at all times.
Turning off diffusion in the solid prevents equalization of the concentration in the solid, and thus assures that segregation effects at high temperatures are preserved at low temperatures.
This is not realistic at all for room temperature segregation, and not too good for high temperature segregation either! However, with diffusion coefficients "fixed" in this simple way, a mathematical analysis is possible, resulting in curves like the following one (I spare you the derivation and the equations):
|The figure illustrates what one would expect for
some concentration c0 = 0.3, and a segregation
The first part to solidify has an alloying element concentration of kc0 = 0.2. With increasing fraction of solidified material the concentration in the melt and solid increases; first slowly and then rapidly. The curves, however, diverge (running up to infinity) for 100 % or complete crystallization. That is obviously nonsense and simply reflects the shortcomings of the model. So we must ignore the prediction of the model for a high percentage of crystallization.
The model, however, describes the basic trends for the first two-thirds or so of "normal" high temperature crystallization rather well. As long as we look at average quantities, it even doesn't depend very much on the nature of the crystallization front. Planar, cellular, dendritic - it makes no difference for averages taken over areas and times large enough.
|The Scheil model, in other words, describes macrosegregation. There are several other models with increasing sophistication but they all describe only macrosegregation at high temperatures.|
|All models give some ideas about the influence of the segregation coefficients and atom movement (by diffusion and /or convection) in the melt. The larger the difference of the segregation coefficient to unity, the larger the segregation effects. That is not only true for the alloying element but also for the trace elements that follow the same basic equations - as long as they are incorporated as single atoms.|
|We can abuse the Scheil model or the other models to some extent also for larger deviations from equilibrium and thus larger interface velocities v. All we need to do is to replace the equilibrium segregation coefficient k with the "effective" segregation coefficient keff(v) that depends on the interface velocity and approaches unity for large v|
|That leads to a little paradox: macro segregation
effects are only present for non-equilibrium but seem to get smaller in all macrosegregation models if the
deviations from equilibrium get larger,
since the effective segregation coefficients are then closer to 1!
Well, yes, the effects of macrosegregation do tend to get smaller for larger interface velocities as outlined above. More important, however, is that the effects of microsegregation get larger for larger interface velocities!
So let's look at high temperature microsegregation now. It is far more exciting than macrosegregation but also far more difficult to understand and model.
|High Temperature Microsegregation|
|The easiest way to consider
microsegregation is to ponder the following points.
|In a very schematic way, this can be illustrated by enlarging on the figure in the backbone::|
|On the left
the variation of the lead and tin concentration due to
macro segregation, coming right from the
phase diagram, is shown schematically. That is the figure in the backbone.
On the right we look at just the lead concentration in a small region (millimeter or smaller) The concentration fluctuates more or less periodically around the average values (dotted line) given by macro segregation, due to micro segregation effects.
Increasing the growth speed decreases macrosegregation effects, but increases microsegregation
|It is clear by now that the average interface velocity vav and its fluctuations Dv are the decisive parameters. It is also clear that during casting, vav and Dv cannot be controlled directly. In other words: there aren't any buttons somewhere on the apparatus that you can turn to a desired value of v and Dv, ensuring that these values are now firmly established until further notice.|
|As far as the fluctuations Dv are concerned, it's even worse. They will tend to increase with increasing vav, but not in a simple straight-forward way. For example, when you crank up the interface velocity, dendritic growth will start at some point, and fluctuations get much larger. The local growth speeds at the tips of the dendrites can be rather large for a while in comparison to vav, for example.|
|Obviously, what I would need to do
now is to establish all the reason for velocity fluctuations, followed by some
general fluctuation theory that is then applied to microsegregation.
I'm not going to do that here, for the simple reason that I don't know of such a general theory. I will give you my personal "theory" for that topic in this link; and I will have much to say about the topic in the next module of this series.
Here we simply assume that dendritic growth will cause some fluctuations and only look in a more empirical way at what that can do to microsegregation.
|If we have dendritic growth, the dendrites, by definition, grew considerably faster for a while than the solid they are attached to. It is thus clear that the liquid in between the already solidified dendrite arms will tend to be enriched with alloy /impurity atoms that have an (effective) segregation coefficient <1. The dendrite itself will be enriched with atoms that have segregation coefficients > 1. This happens, indeed, and can be observed (with some luck) at room temperature. The picture below shows particularly nice examples.|
|The pictures were taken with an
SEM in the
Colors encode concentrations of the elements named in the picture, and there is always a precise scale relating colors to numbers. I omitted that scale; it is sufficient to know that the concentration increases going from dark colors (black, blue,...) to bright colors (..., yellow, red).
We look on top of dendrites oriented perpendicular to the viewing plane. It is clear that rhenium (Re) is enriched in the dendrites, tantalum (Ta) in the space in between.
Just for fun, here is the composition of that super alloy:
|Looking at the composition you get an idea that "super" alloys are not only super with respect to their properties, but also with respect to their complexity. So, next time when you are in an airplane look (with your brain) at the turbine blades made from superalloys in those jet engines. They are the ones that must take the highest temperatures in there, and thus are most prone to failure. You better give credit to the guys who developed super alloys to a point where they never fail! No airplane, to the best of my knowlegde, has ever crashed because turbine blades failed.|
|If you now start thinking that your next sword should be made from a super alloy - forget it. Except if you plan to go to hell, where you might be induced to wield your sword at temperatures above 1.000 °C ( 1.832 °F). Because that's what super alloys are good for: providing mechanical strength (and corrosion resistance and so on) at very high temperatures.|
|How about dendrite caused
microsegregation in steel? Pretty much the same thing - in general. As for
details, what kind of steel are we talking here? There are many hundreds if not
thousands of steels that have sufficiently different compositions to give
different segregation behavior in detail.
Here are two examples:
|These are EDX pictures as above, but with
concentrations coded in a grey scale instead of in a color scale. Concentration
increase from black to white. The view is in the direction of dendrite growth.
In the right-hand picture areas with more or less identical dendrite orientation (marked by crosses) are outlined; this might define (roughly) the grain structure. The left-hand picture essentially shows just one grain. In both examples the interdendritic areas are enriched in manganese. There is not much difference between the two steels with respect to room-temperature manganese microsegregation but with respect to carbon segregation (see below).
|Allright, now let's pause for a
minute and recapitulate what we have achieved so far:
|Before I tackle the really difficult third point in the next module of this series, let's finish this module by looking at a few essentials of room temperature segregation.|
|Room Temperature Microsegregation|
|So we have a freshly crystallized material at a temperature still close to TM, with some inhomogeneous distribution of alloying elements and impurities, and some microstructure related to this. What is going to happen if we now let the material cool down to room temperature?|
|A first and simple answer is: It all depends on:
|In short: A lot can and will happen - but in the
background there is always diffusion as the
decisive enabling and limiting factor. So
one way to look at room temperature segregation is to consider how far the atom
of interest can move during the cooling-down time in the given matrix.
I covered that already rather exhaustively in this module. From a practical point of view, and with carbon steel in the back of our minds, we are well advised to differentiate between interstitial atoms, typically fast diffusers, and substitutional atoms, typically slow diffusers. The first quantity to consider is the diffusion length L, the average distance between start and stop of a "random walker". The essential equations going with it were:
|The second version allows us to have an axis
showing the diffusion length after 1 second in the typical
Arrhenius plot of
Here is the relevant picture for diffusion in iron taken from this module:
|What we take from that picture (or
the more detailed ones) is the 1-second diffusion length
L1 at the temperature T0 of
interest. I don't need to tell you that the diffusion length goes with the
square root of the time, so after two
seconds it is not twice of what it is after 1 second, but only
2½ = 1.41 times larger Of interest here is the melting
temperature of iron / steel around 1500 0C.
What we see is that for interstitial carbon (C) in bcc iron we have an L1 » 70 µm at 1500 0C, in fcc iron it is around 30 µm. Carbon will cover distances of many micrometers within the first second after solidification. Substitutional manganese (Mn), on the other hand, will have moved less than 0.5 µm.
|During cooling the movements become sluggish and
essentially stops long before room temperature is reached. I have showed in
detail how one can compute the total diffusion length
Lto and come up with some estimates of what is going
on. What we need to know are the diffusion data of carbon and manganese and how
fast the specimen cooled down.
Let's apply this to the segregation shown in the above pictures for manganese steel. Before I do this, however, let's look at two more sets of data form this paper:
|The paper provides "line scans" of
concentrations. For a "line scan" you just move your probe along a
line across the sample and record the concentration of the elements you are
interested in. The spatial resolution of the line scan is a few µm. The
pictures above are just a lot of line scans with color-coded concentrations.
Here is a "line scan" of the manganese (Mn) and carbon (C) concentration in a manganese iron sample with medium carbon level.
|What do we see? Let's list the important points:
|Here is the same kind of picture for a "TRIP" steel with low manganese concentration and very low carbon concentration (not specified).|
|What do we see here ? Let's list the important
|Note that all the segregation
pictures and data shown so far were taken at room
temperature and thus show room-temperature microsegregation, or what is left
from the primary high-temperature
segregation occurring during casting.
The pictures are remarkable and go a long way towards explaining the "water" pattern in wootz steel. Let's analyze them in the light of what must have happened during cooling down.
|First, we need to realize that the melting point
of the Fe - 25 % Mn alloy is about 1400 oC (2552 oF) and
that ist solidifies into fcc austenite that stays stable to almost room
temperature. Carbon thus might be
atomically dissolved throughout.
The interesting question is: How far can carbon (and Mn, Al, Si, ...) "go" during the cooling of the samples. In other words, What do we get for the total diffusion length Lto for cooling rates of 100 K/s at the surface and 1 K/s in the center of the cast?
Switching from cooling rates to the more convenient "cooling half-time" thalf, the time it takes to cool to about half the starting temperature as defined in the link, gives thalf = 1400 s or 14 s, respectively
|The 1 s diffusion length of carbon at 1400
oC (2552 oF) is
µm in the fcc austenite phase. Consult the tables
made for that
purpose we find that we need to multiply this number by a factor of roughly
20 for the large cooling half-time, and by about 2 for the small one.
We thus find that the total diffusion length for carbon is somewhere between the extremes of Lto(C) = 200 µm - 20 µm. Now this is very strange.
|Whichever way one looks at it, one conclusion is unavoidable:|
|This leads to another unavoidable conclusion: The carbon shown in the line scans above is not atomically dissolved! It must be precipitated, there is no other logical explanation|
high-manganese case where I suspected that "carbon thus might
be atomically dissolved throughout.", this is probably not the case.
Inspecting the proper ternary phase diagram
would give the answer but it's not so interesting, so I won't bother.
The interesting thing is the second figure with the peculiar carbon distribution. It is now important to note that element analysis with a scanned electron beam (i.e. EDX and its brethren) cannot distinguish between atomically dissolved atoms and atoms in a precipitate. Whenever the electron beam hits a Fe3C cementite particle, carbon would register at 25 %. However, since the volume probed by electron beam is much larger than that of typical precipitate (typically more then 10 µm3) it almost never hits only a cementite particle, but always a mixture of particle(s) and iron. It then registers the average of the carbon concentration in the volume probed - and we get numbers that fluctuate very much between close to zero (there is almost no carbon in the ferrite iron) and relatively large values.
|That's quite obviously what we see in the line scan of the low-carbon TRIP steel. The interesting part of what we see, however, is;:|
|That implies that carbon precipitation does not
happen everywhere and uniformly but mostly in "special" places. There
the precipitate density is high, while in between it is rather low. The reason
for this peculiar kind of room-temperature segregation is the high temperature
segregation of some slow diffusing atom tied to the formation of dendrites.
These atoms obviously provide the nuclei for carbon precipitation; it is prone
to start wherever the concentration is highest.
It is quite possible that the original manganese distribution caused the nucleation of the final carbon segregation by precipitation, since it belongs to the so-called "carbide formers". Since the manganese concentration varies on a scale given by the dendrite size, it can be relatively coarse.
|Now to the climax: That's exactly how
you would describe the cementite distribution for
wootz steel! What we learn here for the
making of wootz blades is:
|OK - now we know the secret of wootz blades, sort of? No, we don't! One major ingredient is still missing. This will be the topic of the next module in this series.|
|1) E. Scheil, Z. Metallkunde, 34, (1942), 70-72.|
|Back to Segregation Science|
|1. Basics of Segregation|
|2. Constitutional Supercooling and Interface Stability|
|3. Supercooling and Microstructure|
|4. Segregation at High and Ambient Temperatures This module|
|Segregation and Striations in CZ Silicon|
|Microsegregation and "Current Burst" theory|
Science of Welding Steel
Wagner's "The Ring of the Nibelung"
Science of Alloying
Diffusion in Iron
The Verhoeven - Wadsworth Jousting Tournament
11.5.2 Structure by Dendrites?
Segregation at Room Temperature
Phenomenological Modelling of Diffusion
Size and Density of Precipitates
Segregation in Silicon
1. Basics of Segregation
Scanning Electron Microscope
The Iron Carbon Phase Diagram
Microsegregation and "Current Burst" Theory
Constitutional Supercooling and Interface Stability
© H. Föll (Iron, Steel and Swords script)