1. Basics of Segregation
|Phase Diagram and Macroscopic Segregation|
|What is segregation? Wikipedia
(English) offers nine different kinds of segregation. We have, for example,
racial, religious, or sex segregation - but none of the nine is what I want to
go into here. You need Wikipedia (German) to find it:
"Segregation in Materials Science: Separations of defects or impurities in a solid by diffusion" (my translation).
That's it. Not a particularly good definition, and certainly not very helpful. Let's go for a better one:
|Just to be on the safe side, let's
first make sure what is not segregation.
Consider a phase transformation from some liquid L to a mixed phase L
+ a. A state point inside the phase field of the
mixed phase decomposes along a tie line
into two state points at the phase field boundaries. You now have some a phase (typically at the container walls) and some
still liquid L phase in equilibrium. Both phases do not have the composition of the system but whatever the phase
diagram demands at the state point given. When salt water solidifies, the first
pieces of ice do not contain any salt, the remaining liquid is saltier.
Now would be a good time to brush up on your phase diagram background if the text above wasn't crystal clear.
|But let's go on. Even so the minority component of the system is now distributed inhomogeneously between the solid and the liquid phase, it is in equilibrium and we do not call this segregation. Segregation, as we like to define it, only takes place within one phase, not between phases! If the a phase would be austenite, for example, and the distribution of carbon inside the austenite at room temperature would be permanently inhomogeneous or non-uniform, we have carbon segregation. We also do not have equlibrium anymore.|
|Now let's look at segregation from a practical (and somewhat limited) viewpoint by listing the features and properties that are of interest to us.|
| Let's keep it easy by considering for starters
only segregation resulting from the
liquid-solid phase transformation. Here is a list of some salient if sometimes
trivial points linked to segregation:
|In other words: it is almost hopeless! Well, not
quite. Some basics can be readily understood, and the basic behavior of some
real systems is also not too difficult to understand. In what follows, I'll go
But be warned: What kind of segregation takes place in wootz steel, leading eventually to the pattern on wootz blades, is not all that clear to me at present.
|For starters, let's
look at some major points from the list above in some detail. First, let's find
out (again) how a phase diagram shows and quantifies the difference of the atom
concentration in different phases.
Of course, we take our old friend, the iron - carbon phase diagram for illustrating this point. We start, for example, with a composition of 2.7 % carbon, and a temperature well within the liquidus part as shown below.
|Now we lower the temperature of the system with some rate, e.g. 0.001 oC per minute, or 100 oC per minute, or whatever we like. As long as we do brain work, there are not restrictions for the cooling rate employed. If we would do a real experiment, very slow cooling rates tend to be extremely boring (even to graduate students), and very fast cooling rates are simply impossible.|
|Whatever we do, sooner or later we hit the line
separating the fully liquid phase (L) from the two-phase region L +
g. A little further down we reach the state
indicated by "1". The state point then decomposes into the two
state points shown.
What we will find in equilibrium is: a little bit of the g austenite phase, and a lot of the liquid phase. With the lever rule we could deduce how much, exactly, " a little bit" and "a lot" would be, but that is not all that important here.
|Important is that the phase diagram
tells us that inside the L + g field the
composition of the liquid and the solid must be quite different. The solid at state point 1,
for example, has about 1.3 % carbon, the liquid around 3 %. Those are the
nirvana or equilibrium values for these phases at the chosen temperature.
The fact that the carbon concentration must and will be different in the two phase is not what we call segregation. It is, however, the reason or cause for segregation to happen as soon as we deviate from equilibrium. The larger the difference in the two concentrations, the bigger the possible segregation effects.
It is therefore useful and customary to use the ratio of these two numbers to define an (equilibrium) segregation coefficient k, sometimes also called partition coefficient or distribution coefficient:
|Since the liquid typically contains the higher
concentration of the atoms in question, we have typically
|Now let's see how segregation works.
We start from a perfect equilibrium situation at point 1 and lower the
temperature until we arrive at state point No. 2. We have now about equal
amounts of solid and liquid and still different equilibrium concentrations of
carbon in the two phases. However, the carbon concentration in the two phases
are now also different from what they were
in state No. 1!
|The solid phase and the liquid phase now need
larger carbon concentrations compared to
what they had in state 1 to be in equilibrium. The segregation coefficient also
changes somewhat but not all that
If we want to keep equilibrium, the carbon concentration in the parts already crystallized while in state 1 now must go up since in equilibrium the concentration has to be the same everywhere.
How could that happen? At state point 1 you usually find a layer of the solid coating the walls of your vessel. When you go to state point 2, you add another layer but with a different composition. How can the two concentrations equilibrate? You guessed it: by diffusion! Carbon from the carbon-rich melt keeps diffusing into the crystallized parts until the right concentration is achieved everywhere. The carbon concentration in the liquid then will be automatically at the correct value, too.
|The key point is: This will take some time. It will actually take a lot of time - until hell freezes over or until the end of time, whatever comes first, if you go for absolute perfection. If you are practical and only request that the actual concentrations should be very close to, but not necessarily exactly at the equilibrium concentration, it still takes some time - but now you might be able to wait for it.|
|How much time it takes to stay sufficiently close
to equilibrium at every temperature depends mostly on the magnitude of the
coefficient of carbon in g-iron at the
chosen temperature. If we look at trace impurity segregation, the diffusion
coefficients of the various
also come into play. For our example, they tend to be far smaller than that of
At high temperatures the diffusion coefficients are large and it may not take very long to get close to equilibrium. However, since diffusion coefficients decrease exponentially with decreasing temperature, waiting times at lower temperatures get exponentially longer.
Before I come back to what that implies, let's cool down our sample a bit more.
|Eventually we hit the line separating the L +
g phase from the austentite (g) +
cementite (Fe3C) phase mixture. The system decomposes into
austenite and cementite, and the carbon concentration in the austenite changes
once again. This "once again" must happen once again by by diffusion.
As we go down the temperature inside the g + Fe3C phase field, the concentration of carbon in the austenite and the amount of Fe3C cementite changes.
Eventually we hit the line separating the (g) + (Fe3C) phase from the final ferrite (a) + cementite (Fe3C) phase mixture known as pearlite. The system decomposes into ferrite and cementite as shown above for one example state, and the carbon concentration in the ferrite changes by diffusion as we go down the temperature as before.
In the end, at room temperature (not included in the drawing above) we have ferrite and cementite. If we managed to maintain equilibrium at all times, the concentration of carbon and impurities in the two phase is what it should be and uniform. Let's emphasize this:
|True. If we manage to have equilibrium for the final structure at room temperature, there is no segregation because the relevant atoms (carbon and some minor impurities in this example) are uniformly distributed. However:|
|Achieving equilibrium for our example here
involves to move a lot of carbon atoms around all the time. To do so requires
extremely slow cooling rates even for
practitioners, and infinitely slow cooling rates for purists. In other words:
It can't be done under most circumstances. Some remainder of the effects taking
place at the three phase transformations involved are bound to be permanently
present at room temperature.
For slowly diffusing impurities (like vanadium) the final inhomogeneous distribution will tend to be more strongly related to the distribution right after freezing than for quick moving atoms (like carbon).
|So let's consider the other extreme. We go to some state, e.g. state No. 1 within some short time, not maintaining equilibrium all the time. Then we cool down extremely rapidly "freezing in" whatever we had as solid. Since we are only interested in the solid, that can be done most easily by just decanting the mix. Quickly pour off the liquid. This keeps not only the solid at whatever composition it had at that moment in time, it also allows a direct look at the morphology of the solid liquid interface. That's how the picture in the backbone was obtained.|
|The solid we obtain will have about the
composition indicated by the phase diagram for the temperature at which we
decanted the mix. This is neither the composition of the original liquid nor
what we would expect at room temperature. For our example it would not even
have the right structure (a + Fe3C
or pearlite) but would be austenite. That won't really happen because you just
can't cool down that fast.
In other words; it is not all that easy to assess what happened during freezing. What's left at room temperature is always different from what there was at freezing, and it is not always very clear what kind of changes occurred during (rapid) cooling.
|In real life we will be somewhere between the two extremes. We will cool down too fast for achieving near-equilibrium "everywhen" and everywhere, but since we solidify all of the original liquid, the average or global composition of the solid will be the same as that of the original liquid. That is not true for the local composition. The parts that solidified first (close to the container wall) will have a lower concentration, the last parts to solidify will have a higher concentration. We have macro segregation, in other words.|
|This just rehashed with some more details what I already stated in the backbone about "macroscopic segregation" in the extremely simple lead (Pb) - tin (Sn) system. The Fe-C system considered here is far more complex. It has two more phase transitions besides the liquid-solid "freezing". That doesn't make it easy to deal with but it doesn't add something qualitatively new either. Knowing about cooling rates, etc., we even could calculate the global concentration variations.|
|Before I go into this, let's
|That much is obvious for macro segregation, as we call concentrations differences or gradients on scale that is just a bit smaller than the sample size. You have macrosegregation if the concentration in a 10 cm sample changes on a scale of centimeters.|
|What about micro segregation, or concentration gradients on a micrometer scale and thus typically much smaller than the sample size? Or even worse, for scales in between "macro" and "micro" that become important, for example, for wootz steel.|
|Same thing in principle - but for far more complex reasons! In order to get a grasp on this topic, we first need to look at the morphology of the solid-liquid interface and on the way it moves. For doing this, I must first discuss a few basic about the currents flowing through a moving liquid-solid phase boundary during solidification.|
|Currents Through the Solid-Liquid Interface|
|One key to (micro) segregation
phenomena is the morphology or the
"shape" of the interface. I bet that if anybody would ask you what
the interface between the liquid and solid part of some freezing concoction
would look like, you would tend to say: "Probably planar, just like the
interface between some liquid and gas. If I pour some water into a pot, its
interface to the air is just a simple plane. Why should it be otherwise, when
it starts to freeze? I've actually seen that happen, and the interphase (note
the clever spelling!) between some ice on a lake and the water is usually quite
Not to mention that you, personally, emphasized many times that interfaces want to minimize their area. Between a liquid and a solid contained in some pot or vessel, nothing beats a planar interface, give or take a little global curvature, with respect to minimal area."
|Good arguments. You are absolutely
right - as long as we consider systems at, or not to too far from equilibrium.
But now we consider systems far from
equilibrium. Those systems are always characterized by the fact that there are
large current (densities) of something flowing through, towards, or away from
Why? Because the system always wants to get into equilibrium. Since it is far off equilibrium, something needs to change. But you only can make changes if something suitable moves from here to there - and that constitutes a current of something! Here are some examples of currents that we have used before:
|If you ponder that for a while, you
realize that there are two kinds of
currents that must be flowing through, towards, or from a liquid-solid
interface during freezing:
|Heat, a form
of energy, needs to flow through the
interface and eventually out of the system
for two reasons:
|The heat current density jH, i.e. how much heat flows out per square centimeter and second, is exclusively determined by the temperature gradient dT/dx or how much the temperature changes per length unit, and the thermal conductivity k of the materials. In three dimensions we have T(x, y, z) (use this link for the meaning of the "napla" operator ) for the temperature gradient, and the fundamental equation writes|
|From a mathematical point of view this is exactly
the same kind of equation as the one we used for describing
why not? We have the same underlying physical phenomena, after all. The only
difference is that something a bit more abstract then particles flows: energy!
In principle we should also write the thermal conductivity k under the napla sign for reasons explained below. We also need to consider everything as a function of temperature and thus time.
|Atoms, or more precisely,
impurities and alloying atoms, need to flow towards, "through", and
away from the interface because the equilibrium concentration in the solid and
the liquid is, as a rule, not the same. What the respective concentrations will
be at some temperature T is given by the phase diagram as
outlined above. Looking at phase
diagrams for a while makes clear that the only exceptions to the rule are
eutectic (or eutectoid)
Here are a few points of interest:
|We know already that the particle (= atom) current density jA, i.e. how much atoms flow out per square centimeter and second, is exclusively determined by the concentration gradient dc/dx, or how much the concentration changes per length unit, and the diffusion coefficient D of the atoms. In three dimensions we have c(x, y, z) (use this link for the meaning of the "napla" operator ) and the fundamental equation writes|
|That is our old diffusion equation or Fick's first law, except that we now have the diffusion coefficient under the Napla sign since it is no longer constant but depends on the temperature, which in turn depends on the coordinates and the time t. In the liquid, for example, diffusion is always much faster than in the solid.|
|To make things a bit more complicated
or realistic, it is necessary to realize:
|What a mess! Let's see if we remember
what we wanted to learn about? Ah - yes: segregation!
Well, just use the two equations above, add boundary conditions, and specify them for whatever you have in mind. Then solve them, deriving the concentration and distribution of some impurity at room temperature, and you have segregation covered.
|Before you now rush off, firing up your computer
to do this, let me give you a piece of advice: Forget
Simulating crystal growth is far beyond the ken of most if not all people. I certainly couldn't do it. With present-day computers experts can do a lot for some "simple" problems, but in the words of one such expert (Jeffrey J. Derby, Department of Chemical Engineering & Materials Science University of Minnesota):
|We need to take a different approach to get closer to understanding segregation. To do so we need to keep in mind: it is the interface and all the the currents tied to it that determine what will happen.|
|Back to Segregation Science|
|1. Basics of Segregation (This module)|
|2. Constitutional Supercooling and Interface Stability|
|3. Supercooling and Microstructure|
|4. Segregation at High and Ambient Temperatures|
|Segregation and Striations in CZ Silicon|
|Microsegregation and "Current Burst" theory|
Diffusion in Iron
11.5.2 Structure by Dendrites?
Segregation at Room Temperature
Phenomenological Modelling of Diffusion
Segregation in Silicon
Some Phase Diagrams
Segregation at Room Temperature
Microsegregation and "Current Burst" Theory
Constitutional Supercooling and Interface Stability
© H. Föll (Iron, Steel and Swords script)