Dendritic Solidification - ITESCAM

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Dec 1, 2013 (3 years and 4 months ago)


Dendritic Solidification

H. K. D. H. Bhadeshia


A liquid when cooled solidifies. Alternatively, it may solidify when the pressure is
decreased or increased, depending on the sign of the density change. Once
nucleation has occurred, solidifi
cation proceeds by the movement of an
interface. The process may generate heat if the enthalpy of the solid is less than
that of the liquid. Similarly, solute may partition into the liquid if its solubility in
the solid is less than that in the liquid.

Computer simulated image of dendritic growth using
a cellular automata technique. Notice
the branching
on the dendrites. Photograph courtesy of the Institute
of Materials, based on the work of U. Dilthey, V.
Pavlik and T. Reichel, Mathematical Modelling of
Weld Phenomena III, eds H. Cerjak and H.
Bhadeshia, Institute of Materials, 1997.

accumulation of solute and heat ahead of the interface can lead to
circumstances in which the liquid in front of the solidification front is
supercooled. The interface thus becomes unstable and in appropriate
circumstances solidification becomes dendritic.

mechanism of this

is discussed elsewhere.

A dendrite tends to branch because the interface instability applies at all points
along its growth front. The branching give
s it a tree
like character which is the
orgin of the term dendrite.

Computer simulated image of the
dendritic solidification of pure nickel.
The simulation is of "free growth",

the solid is growing without contact with
anything but the liquid. The d
egree of
undercooling of the liquid in front of the
interface is indicated by the adjacent
scale. Photograph courtesy of the
Institute of Materials, based on the work
of U. Dilthey, V. Pavlik and T. Reichel,
Mathematical Modelling of Weld
Phenomena III, In
stitute of Materials,

Growth tends to occur along fast growth
directions which are generally <100> for
cubic metals.

Technological Consequences

We first consider the solidification of a "hardfacing alloy" which is
deposited as liquid on substrates which require wear resistance. Typical
applications include earth moving equipment, he
avy farm equipment and
rock crushers used in the mining industry.

The alloy has a chemical composition Fe
4.5C wt.%. During
cooling, niobium carbide dendrites are the first to solidify. Their shape
can be revealed by attacking the sample with an a
cid which removes the
matrix iron
rich phase
. In this particular case, the solid/liquid
interfacial energy is varies with orientation so the minimum energy shape
is that with cryst
allographic facets. The fast growth direction is still <100>
as can be deduced from the symmetry of the dendrite.

Scanning electron micrograph
of a niobium carbide (cubic
dendrite in an iron
hardfacing alloy. Photograph
by courtesy of Berit Gretoft,
Central Research
Laboratories, ESAB AB,

The second example deals with a cobalt
e alloy known as "Stellite".
This is much more expensive than the iron
base alloy described above
but is considerably tougher because it has a much finer microstructure.
Thus, it is necessary to use transmission electron microscopy to study
the fine struct

electron micrograph
of a cobalt
"blobby" dendrite
which is in a hard
eutectic (dark). The
eutectic is a mixture
of carbides and
matrix. The solid
liquid interface
energy is not very
anisotropic so the
dendrite adopts the
smooth shape.
theless, the fast
growth directions are
along <100>.
Micrograph by
courtesy of S.
Atamert and H. K. D.
H. Bhadeshia. See
also "Comparison of
the Microstructures
and Wear Properties
of Stellite Hardfacing
Alloys Deposited by
Arc Welding and
Laser Cladding"
Transactions A, Vol.
20A, 1989, pp. 1037
S. Atamert and
H. K. D. H.

Dendritic solidification frequently occurs under conditions which are far
from equilibrium. Given these circumstances, regions of solute
rich liquid
can be t
rapped between the dendrite arms, and solidify eventually to
rich solid regions. This in turn leads to the development of a
"banded" microstructure when the material is subsequently processed by
rolling or other mechanical fabrication methods. The b

can be detrimental to the mechanical properties.

An optical micrograph
showing a typical banded
microstructure in steel. The
depleted regions have
transformed in the solid state
into bainite whereas the
enriched regions are
artensitic. Photograph
courtesy of S. A. Khan and
H. K. D. H. Bhadeshia. See
also, "The Bainite
Transformation in Chemically
Heterogeneous 300M High
Strength Steel" Metallurgical
Transactions A, Vol. 21A,
1990, pp. 859
S. A.
Khan and H. K. D. H.

Dendrites of Zinc

The following photographs show dendrites of zinc obtained by withdrawing the
solid from a melt of impure zinc. The photographs are of samples collected by
Professor Paul Howell, Pennsylvania State University.

Dendrites of zinc rescued from partially
solidified melt.

Dendrites of zinc rescued from partially
solidified melt.

Dendrites of Ice

When the weather outside is
cold, moisture in a warm room can condense on
the inner surface to form a thin film of moisture. If the temperature outside is
sufficiently low, ice nucleates and grows. The region around the ice crystal
becomes depleted in moisture. Moisture then has to a
rrive to the ice crystal by
diffusion through the depleted zone, from the remaining moisture far from the
interface. Suppose a small part of the ice crystal accidentally advances further
then the rest of the interface. The diffusion distance for that pertu
decreases, and hence the perturbation grows faster. This leads to the formation
of a branch, and a
branching instability

is said to have formed. This leads to the
formation of ice dendrites as illustrated below. These pictures were taken at the
Harbin Institute of Technology


the temperature outside can be below
The mechanism described here is essentially how snow
flakes are supposed to
form, by the diffusion of water molecul
es through air on to the ice crystals.
flakes have the dendritic morphology in three dimensions.

Ice dendrites on inner surface of cold

Ice dendrites.


Ice dendrites.

Negative Dendrites

When a s
heet of ice undergoes internal melting, dendrites of water form inside
the ice. It is now the liquid which advances into the solid with an unstable
interface. Furthermore, since ice has a lower density than water, a bubble forms
inside each dendrite of the


Dendrites in Metallic Glass

The following transmission electron micrographs have kindly been provided by
Andrew Fairbank with copyright clearance from the University of Wollongong.
They show the early stages of dendrites of
(Fe,Si) growing in the solid
from the amorphous Fe

metallic glass during annealing at 433 °C for 60

Dendrites forming in Fe


Dendrites forming in Fe


Dendrites forming in Fe


Dendrites forming in Fe



forming in Fe


Interface Stability and Diffusion Bonding

Some materials cannot be welded by conventional techniques because the high
temperatures involved would destroy their properties. For such materials,
diffusion bonding is an

attractive solution because it is a solid state joining
technique, which is normally carried out at a temperature much lower than the
melting point of the material.

Diffusion bonding is a candidate process for joining many aluminium based
materials inclu
ding a variety of artificial composites. Unfortunately, the method
has been beset by difficulties, particularly that the bond line remains a plane of
weakness. This is because the bond plane is a site for impurity segregation,
where oxide particles may als
o be trapped. In addition, there can be problems
in ensuring the continuity of the metallic bond.

Equipment used for temperature gradient
diffusion bonding

zadi and Wallach (Materials Science and Metallurgy, University of
Cambridge) invented a disarmingly simple method of breaking up the planar
bond into an unstable interface which develops into a three
'sinusoidal' or cellular surface. A small
temperature gradient was applied at the
bond, causing the interface instability. This concept is taught in many
undergraduate courses but it took imagination and foresight on the part of
Shirzadi and Wallach to apply it to transient liquid phase bonding. T
he method
is incredibly successful, leading to a vast increase in bond strength, and has
been granted a UK patent, No. 9709167.2, the Granjon Prize of the
International Institute of Welding and the Cook
Ablett Award of the Institute of


Movies about dendritic and cellular solidification.

generated movies of dendrit
ic solidification.





Widmanstätten ferrite

Cast iron


Allotriomorphic ferrite



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Stainless Steels



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Morphological instabilities during growth:

linear stability


By Guru

Morphological instabilities

Typically, when a liquid alloy solidifies, as heat is extracted and the solid
nucleates and grows, even if the initial solid
liquid interface is planar, pretty
soon it breaks up

resulting i
n cellular and

structures; see
this page
for some samples of such structures and videos

(both experimental and
ulated). Similar break
up of planar interfaces can also happen when a solid
grows in another that is supersaturated, purely by diffusion, at isothermal

A rigorous mathematical study of these kinds of instabilities of interfaces during
growth we
re pioneered by Mullins and Sekerka in a couple of classic papers
[1,2]; as the quote below shows, this work is considered as one of the key steps
in the general area of study known as pattern formation:

A determination of the stability of simple solutions

to moving
equations with respect to shape perturbations is an important step in the
investigation of a wide range of pattern
formation processes. The pioneering
work of Mullins and Sekerka on the stability of the growth of solidification fronts
nd of Saffman and Taylor on moving fluid
fluid interfaces were major
advances. The basic approach is to analyze the initial, short time, growth and/or
decay of an infinitesimally small perturbation as a function of the characteristic
length scale or wavele
ngth of the perturbation. Although the linear stability
approach, exemplified by this work, is not always sufficient, it is a basic tool in
theoretical morphogenesis.

Paul Meakin in Appendix A of his
Fractals, scaling and growth far from

In this blog post, I will talk about the papers of Mullins and Sekerka; the work of
Saffman and Taylor as well as the insufficiency of

linear stability analyses
mentioned in the quote above deserve their own posts; and, may be someday I
will write them.

Point effect of diffusion

Here is a schematic showing growth during a phase transformation regulated by
(a) flow of heat and (b) diffusi
on. In case (a), the heat is getting extracted from
the left hand side, resulting in the growth of the solid into the liquid. In case (b),
it is the diffusion of solute from the supersaturated solid on the right hand side
that results in the growth of the
solid on the left hand side. In both cases, the
interface is shown by the dotted lines in the schematic; though the interface is
shown to be planar, as we see below, in most of the cases, it would not remain

In the second case, wherein one solid (say, 1) is growing into a supersaturated
solid (say, 2), the schematic composition pr
ofile will look as shown here:

With the above composition profile, it is e
asier to see as to why one can expect
the interface, shown to be planar in Fig. 1 (b) above can be expected not to
remain planar: suppose there is a small protrubation on the planar interface; the
sharper the disturbance, the larger the area (or volume) of

material ahead from
which, by diffusion, the material can be ferried to the interface, resulting in faster
growth. This is known as the point effect of diffusion.

The figure below explains the point effect of diffusion:

as opposed to a case where a planar interface would result in a half
circle of
, where,
is the diffusion distance, if there is a pro
trubation with a
sharp end, that sharp end can ferry material from a(n almost) circular region of
. Thus, it is favourable for the interface
to break into a large number of
such jagged edges purely from a point of view of growth; however, such jagged
interfaces lead to higher interfacial areas and hence higher interfacial energies.
Thus, the actual shape of the interface is determined by these
two opposing

namely, interfacial energy considerations and the point effect of

Even though the above explanation was in terms of diffusion, a similar effect
can be shown to operate in the case of heat extraction also. In fact the gener
way of looking at both is to consider the gradients in these fields

be it
composition or temperature. In Chapter 9 of the book
Introduction to nonlinear

(edited by Lui Lam), L M Sander explains the mechanism behind
Sekerka instability using a schematic of equipotential lines ahead of a

since they are bunched up ahead of a protuberance, it grows (
p. 200

Fig. 9.4
). Of course,
this explanation is a visual version of what Mullins and
Sekerka have to say in their paper [1]:

The isoconcerntrates are then bunched together above the protuberances and
are rarified above the depressions of the perturbation. The corresponding
of diffusion flux away from the depressions onto the protuberances
increases the amplitude of the perturbation; we may view the process as an
incipience of the so
called point effect of diffusion.

The analysis of Mullins and Sekerka

The mathematical analys
is of Mullins and Sekerka [1] is aimed at understanding
the morphology of the interface; as they themselves explain:

The purpose of this paper is to study the stability of the shape of a phase
boundary enclosing a particle whose growth during a phase trans
formation is
regulated by the diffusion of the material or the flow of heat. … The question of
stability is studied by introducing a perturbation in the original shape and
determining whether this perturbation will grow or decay.

Of course, in the case of
a dilute alloy, during solidification, the solid
interface is known to break
up and this break
up is more complicated since it
involves simultaneous heat flow and diffusion; and, in another paper published
shortly afterwards [2], Mullins and Sekerka

analyse the stability of such an

The purpose of this paper is to develop a rigorous theory of the stability of the
planar interface by calculating the time dependence of the amplitude of a
sinusoidal perturbation of infinitesimal initial amplit
ude introduced into the shape
of the plane; … the interface is unstable if any sinusoidal wave grows and is
stable if none grow.

Both these papers are models of clarity in exposition; Mullins and Sekerka are
very careful to discuss the assumptions they mak
e and the validity of the same;
they also show how these assumptions are physical in most cases of interest.

As noted above, the actual break
up of an interface is determined by two
competing forces

the capillary forces which oppose the break
up and the
point effect which promotes break
up; what Mullins and Sekerka achieve
through their analysis is to get the exact mathematical expressions (albeit under
the given assumptions and approximations) for these two competing terms.

The continuing relevance


are several limitations associated with the Mullins
Sekerka analysis; it is
a linear stability analysis; it assumes isotropic interfacial energies; it neglects
elastic stresses, if there be any.

Of course, there are many studies which try to rectify some
of these limitations;
for example, we ourselves have carried out
Sekerka type instability
analysis for stressed thin films
. Numerical studies and nonlinear analyses which
look at morphological sta
bility overcome the problems associated with the
assumption of linearity that forms the basis of Mullins
Sekerka analysis.

But what is more important is that in addition to being the basis for these other
studies, Mullins
Sekerka analysis, by itself, also
continues to be of relevance

both from a point of view of our fundamental understanding of some of these
natural processes and from a point of view of practical applications of industrial
importance. I can do no better than to quote from
this (albeit a bit old) news

Scientists in the 1940s and ’50s were well aware of instabilities and knew they
played a role in formation of dendrites. But until Mullins and Sekerka
their first paper in 1963, no one had ever been able to explain the mechanisms
that accounted for instabilities.

The Mullins
Sekerka theory provided a method that scientists and engineers
could use to quantify all sorts of instabilities, said Jor
ge Vinals, an associate
professor of computational science and information technology at Florida State
University and a former post
doctoral fellow who studied under Sekerka.

Understanding instabilities is the first step in controlling them, so this
ology is important for engineers who need to make industrial processes
as stable as possible, Vinals said. Physicists, on the other hand, find that
interesting things happen when systems become unstable and so have an
entirely different sort of interest in

the theory. Mathematicians, for their part,
have launched entire fields, such as non
linear dynamics and bifurcation theory,
that explore the underlying mathematical descriptions of instabilities.

One example of how the theory has been put to use is in th
e semiconductor
field, where computer chips are made out of large, single crystals of silicon that
are sliced into thin wafers. In the early years, these single crystals measured
just an inch in diameter; today, 12
inch diameter crystals are produced,
lting in wafers that each can yield hundreds of fingernail
size computer

“You don’t just walk into the lab and build a bigger [silicon crystal] machine
because in a bigger machine these instabilities can eat you alive,” Sekerka said.
But by understa
nding the instabilities that occur as liquid silicon crystallizes,
engineers have found ways to greatly reduce the formation of dendrites.

Sekerka, a Wilkinsburg native who earned his doctorate in physics from
Harvard University, said he and Mullins weren’
t thinking about such applications
40 years ago. Though working in a metallurgy department during Pittsburgh’s
steel and aluminum heyday, they weren’t especially inspired by the needs of the
metals industry, either.

“We were driven by intellectual curiosit
y more than the need to solve any
particular problem,” he said. “Some of the greatest discoveries come from
following intellectual curiosity.”

I will end this post with a link to the
obituary of W W Mullins

(by R F Sekerka H
Paxton) and that of his wife
June Mullins

to give an idea of the person behind
these works.


[1] W W Mullins and R F Sekerka,
Morphological stability of a particle growing
by diffusion or heat flow
, Journal of Applied Physics,
, 323

[2] W W Mullins and R F Sekerka,
Stability of a planar interface during
solidification of a dil
ute binary alloy
, Journal of Applied Physics,
, 444

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cellular solidification
dendritic solidification
linear stability
morphological stability analysis

This entry was posted on October 9, 2008 at 4:07 pm and is filed under
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