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Bifurcation and morphological instability

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Bifurcation and morphological instability
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Nadarajah, Arunan, 1959-
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vii, 120 leaves : ill. ; 28 cm.

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Ampoules ( jstor )
Boundary conditions ( jstor )
Convection ( jstor )
Crystal growth ( jstor )
Crystal morphology ( jstor )
Crystals ( jstor )
Liquids ( jstor )
Solidification ( jstor )
Solids ( jstor )
Velocity ( jstor )
Bifurcation theory ( lcsh )
Crystal growth ( lcsh )
Crystallization ( lcsh )
Solidification ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Arunan Nadarajah.

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BIFURCATION AND MORPHOLOGICAL INSTABILITY


By

ARUNAN NADARAJAH
























A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA

1988


f OMERS ITOF FLORIDA LIRAIES




BIFURCATION AND MORPHOLOGICAL INSTABILITY
By
ARUNAN NADARAJAH
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
r. OF FLORIDA LIBRARIES


ACKNOWLEDGEMENTS
In the course of the three and a half years of effort that went
into this dissertation, I received help from numerous people in various
ways that contributed to its completion. Enumerating them all would
make this note be at odds with the spirit of conciseness of the rest of
the document and so, reluctantly, I resolved to include only those who
had made a direct contribution. Needless to say, I still remember with
gratitude my debt to the rest.
First mention should be made to my erstwhile mentor, Professor M.S.
Ananth at the Indian Institute of Technology, Madras, who first kindled
an enduring interest in theoretical transport phenomena and encouraged
my proclivities toward graduate study. The greatest help came from my
advisor Dr. R. Narayanan, who apart from getting me involved in morpho
logical instability and giving many suggestions also imparted to me a
solid background in mathematics and hydrodynamic stability theory, not
to mention helping me in numerous other ways. Professor L.E. Johns,
Jr., first introduced me to linear operator theory and gave suggestions
too regarding my research, but more importantly, he was my "chemical
engineering conscience," broadening my vision when I tended to
specialize too much and keeping the objectives in perspective when I got
wrapped up in abstract theoretical points. It is not an exaggeration to
say that I probably would not have progressed this far academically
without these three individuals.


I would also like to thank Drs. S.R. Coriell and G.B. McFadden of
the National Bureau of Standards for many discussions and suggestions
regarding the subcritical nature of morphological instability; the
members of my supervisory committee: Professor U.H. Kurzweg, Dr. G.K.
Lyberatos, Dr. W.E. Lear, Jr., and Dr. S.A. Svoronos for their time and
effort on my behalf; the Department of Chemical Engineering for provid
ing a research assistantship during the first two years of my Ph.D. work
and the Department of Mathematics for a lectureship during the last one
and a half. My thanks also go to my uncle Dr. R.S. Perinbanayagam for
being a role model and helping me evolve a "meaningful philosophy of
life" and cope with the stress of graduate school.
Finally, I wish to express my gratitude to my colleague S. Pushpa-
vanam for many "enlightening" discussions and to Debbie Hitt for doing a
superb job of typing the manuscript and "correcting" my Queen's English
spelling!


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
ABSTRACT vi
CHAPTERS
1 DESCRIPTION OF THE PROBLEM 1
2 PREVIOUS WORK ON MORPHOLOGICAL INSTABILITY 8
2.1 Early Work 8
2.2 Later Research 10
2.3 Inclusion of Other Effects 12
2.4 Experiments in Morphological Instability .... 14
2.5 Limitations of Existing Models and
Unaddressed Issues 15
3 A UNIFORM FORMULATION 17
3.1 The Formulation 17
3.2 The Linear Stability Problem 23
3.3 The Adjoint Problem and Exchange of
Stabilities 30
3.4 Finite Containers and the Most Dangerous
Wavenumber 35
4 SUBCRITICAL BIFURCATION 38
4.1 Theory 38
4.2 The Second Order Problem 41
4.3 The Third Order Problem 46
4.4 Calculations and Comparisons 49
5 COMPARISONS WITH RAYLEIGH-MARANGONI CONVECTION .... 61
5.1 Rayleigh-Marangoni Convection in Brief 61
5.^- The Augmented Morphological Problem 65
5.3Comparison of Morphological Instability
with Rayleigh-Marangoni Convection 70
6 BIFURCATION BREAKING IMPERFECTIONS 74
6.1 Nature of Imperfections 74
6.2 Imperfection Due to Heat Loss 76


6.3 The Outer Expansions 78
6.4 The Inner Expansions 83
6.5 Imperfection Due to Advection in the Melt ... 87
6.6 Nonexistence of the Planar State 93
6.7 Asymptotic Solution 94
6.8 Controlling Imperfections 99
7 NEW DIRECTIONS 102
7.1 Transition to Dendritic Growth 102
7.2 Extension to Semiconductor Materials 103
7.3 Inclusion of Microscopic Effects 104
7.4 Numerical Methods 104
7.5 Experiments 105
APPENDICES
A NOMENCLATURE 107
B PHYSICAL PROPERTIES 113
REFERENCES 115
BIOGRAPHICAL SKETCH 120


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
BIFURCATION AND MORPHOLOGICAL INSTABILITY
By
Arunan Nadarajah
August, 1988
Chairman: Dr. Ranganathan Narayanan
Major Department: Chemical Engineering
Morphological instability refers to the tendency towards spatial
pattern formation on the liquid-solid interface when a dilute binary
mixture is solidified or fused. The importance of this phenomenon is in
the growth of metal alloy and semiconductor crystals from their melts,
where it influences the solute or dopant concentration resulting in
nonuniform physical and electrical properties.
Previous formulations of morphological instability have involved
several simplifying assumptions which restricted it to the study of a
region immediately surrounding the interface. The models have limited
validity and they require separate treatments for different situations
like freezing and melting. In this study a new uniform approach is
presented which considers the entire melt and crystal domain and is
applicable to all situations. Earlier formulations are shown to be
approximations of this and exchange of stabilities is proven
asymptotically.
vi


The model is then used with a weakly nonlinear technique to predict
the shape of the bifurcation diagram for various cell patterns. The
subcritical nature of morphological instability is shown and regions of
its prevalence are determined over the entire domain of experimental
parameters. This was used to compare with experimental results and to
determine optimal crystal growth regions.
A comprehensive comparison of morphological instability with con
vective instability was undertaken and this phenomenon was shown to
resemble Marangoni convection in its mathematical and physical fea
tures. This was done in order to introduce some of the multitudinous
mathematical techniques employed in convective instabilities into mor
phological instability and specifically was used here to complete the
eigenspace of the linearized problem.
Two imperfections which reside in the domain, heat loss at the
container wall and advection in the melt, were considered and shown to
be bifurcation breaking imperfections. Solutions to the problem were
obtained in both cases by matched asymptotic expansions and based on
these results a practical way of minimizing the effect of these imper
fections was suggested.
vii


CHAPTER 1
DESCRIPTION OF THE PROBLEM
Morphological instability refers to the process of spatial pattern
formation at the liquid-solid interface when a binary mixture is
solidified or fused. This is a problem of hydrodynamic instability and
like all other problems of this nature, for this phenomenon too there is
an onset point where the initially planar interface first begins to
deform and forms cellular patterns. These grow into deeper finger-like
shapes and eventually forming side branches and tree-like dendritic
structures.
The importance of this phenomenon is in the growth of metal alloy
and semiconductor crystals from their melts. Morphological instability
affects not just crystal shape but also the solute or dopant
concentration, resulting in nonuniform physical and electrical
properties in the crystal. This is especially perfidious in
applications where superfine crystals with very consistent properties
are required. Recently there have been some indications that crystal
quality can be significantly improved by growing them in low gravity as
this reduces other problems associated with crystal growth like natural
convection, but unfortunately not morphological instability. Growing
the crystal at very high temperature gradients or at very low growth
rates avoids morphological instability but crystals grown at high
temperature gradients are of poor quality due to thermal stresses and
the very low growth rates makes the process very expensive. Hence this
1


becomes a problem not only of avoiding crystal surface deformations but
one of optimization of the process as well.
There are several methods for growing crystals from the melt,
distinguished by the hydrodynamics of growth. The three basic ones are
Bridgman, Czochralski and float zone and most techniques are variations
of these, like horizontal and vertical Bridgman. A typical Bridgman
experiment is shown in Fig. 1-1. The material is usually in a quartz
ampoule and is melted and then recrystallized in the Bridgman furnace.
The upper part of the ampoule is maintained at a higher temperature than
the lower and solidification proceeds upwards and the ampoule is pulled
downwards at the same velocity v. The top of the melt is protected by a
liquid encapsulant like 620^. At the end of the process the ampoule is
broken to retrieve the crystal.
In the float zone technique shown in Fig. 1-2, the ampoule or the
material itself is pulled through a circular furnace and the melting and
recrystallization proceeds simultaneously. As this technique can be
done even without a container it avoids the problem of impurities from
the ampoule entering the crystal, but its more difficult to maintain a
uniform temperature gradient. Figure 1-3 shows the Czochralski method
where the crystal is rotated and pulled from a melt pool. As the
emphasis here is on the liquid-solid interface, the modelling of the
crystal growth process will be kept as general as possible but will
resemble the Bridgman technique the most.
The temperature and concentration profiles during typical crystal
growth conditions are shown in Fig. 1-4. The temperature profiles in
the liquid and solid are virtually straight lines and solute
concentration in the solid is virtually constant. But the solute


3
Figure 1-2. Float zone
Figure 1-3. Czochralski


4
concentration in the liquid changes sharply near the interface
because of solute rejection on solidification. This in turn has an
effect on the freezing point depression in the liquid as shown in the
figure. Figure 1-5 shows the same profiles but in a situation where the
freezing point in the liquid TM now exceeds the actual liquid
temperature (this change can be brought about by either reducing the
liquid temperature gradient or by making the change in near the
interface even sharper by increasing the growth velocity). This is
referred to as constitutional supercooling and the system responds to
this unstable situation by interface deformation. Countering this is
the interfacial tension which always acts to minimize the surface area
which in this case is the planar surface. When this balance is upset,
or in other words when the onset conditions are exceeded, the interface
loses planarity and forms cellular patterns. Figure 1-6 shows the
profiles for the fusion case, where we could have TM in the solid being
less than the actual solid temperature and once again interfacial
deformation is the system's response, balanced by interfacial tension.
The only difference here is that since solid diffusivities tend to be
several orders of magnitude lower than liquid diffusivities, the solute
concentration profile in the solid near the interface will vary even
more sharply resulting in lower onset conditions for morphological
instability.
In the paper of Trivedi and Somboonsuk (1984) there is a series of
photographs from an experiment (see Fig. 1 in their paper) where
succinonitrile/acetone crystals were grown. The first photograph shows
the liquid-solid interface just after onset and has a discernible
cellular pattern. Later ones show the cells becoming deeper, forming


5
Figure 1-4. Concentration and temperature profiles during
solidification
Figure 1-5. Concentration and temperature profiles during
solidification
Figure 1-6. Concentration and temperature profiles during fusion


6
fingers and ending up as dendrites. The arrows in the first photograph
mark the initial perturbations that eventually become dendrites. In
this thesis we will concentrate only on the region near the onset point,
shown by the first two photographs, though dendrites will be mentioned
in discussions.
In the papers by Morris and Winegard (1969) and Tiller and Rutter
(1956) we see another aspect of morphological instability, the variety
of cellular patterns, finger-like shapes, hexagonal cells and variations
of these. Other possible shapes are cylindrical rolls and rectangular
cells but by far the commonly encountered pattern is the hexagonal
one. The choice of the cell pattern is extremely important and factors
determining this choice will be discussed later. Their figures also
show that, unlike other forms of hydrodynamic instability, the number of
cells on a single crystal is in the hundreds.
Though it is customary to model morphological instability in terms
of the temperature and concentration profiles, in reality this
phenomenon seldom exists in isolation; it is usually coupled with fluid
flow in the melt. There are two kinds of flows that occur. The first
is buoyancy driven solutal convection which is caused by the sharp
solute concentration gradients in the melt. The other is the result of
density change during solidification. When solidification occurs there
is a constant rate of volume decrease which causes the melt to move in
to fill the vacated space. This motion is referred to as advection and
the rigid side walls of the ampoule will cause closed streamlined flow
in the melt as a result (see Fig. 6-2). In addition there will be flows
in the melt in Czochralski growth due to rotation and other kinds of
flows in special growth techniques.


7
Apart from these, several other parameters affect this phenomenon,
the most important of which are due to the fact that most crystals are
faceted; that is, they have a crystal lattice structure. Hence whether
the lattice axis is aligned or not with the growth direction is
extremely important as can be seen from the experiments of Heslot and
Libchaber (1985). Other important considerations are grain boundaries,
wetting of the ampoule wall and the presence of impurities. Also in
rapid solidification, the system will not be at thermodynamic
equilibrium and kinetic undercooling of the melt becomes significant.


CHAPTER 2
PREVIOUS WORK ON MORPHOLOGICAL INSTABILITY
2.1. Early Work
The first successful attempt at explaining morphological
instability qualitatively was by Rutter and Chalmers (1953). They
coined the word "constitutional supercooling" to describe the existence
of unstable melt regions near the interface where the freezing
temperature can be higher than the liquid temperature itself and
correctly identified this as the cause of interface deformation.
Tiller, Rutter, Jackson and Chalmers (1953) quantified
constitutional supercooling and for instability came up with the
condition
mGHo>-Gt
where m is the absolute value of the liquidus slope, G^ the liquid
temperature gradient at the interface and G^c the solute concentration
gradient in the liquid at the interface. The negative sign is caused
by G^ and G^ being in opposite directions (see Fig. 1-5). As can be
seen these simple thermodynamic explanations do not take into account
the stabilizing effect of interfacial tension. To do so would require
casting the problem as one of hydrodynamic instability and obtaining the
onset conditions from a linear stability analysis. This is exactly what
8


9
Mullins and Sekerka (1963, 1964) did when they considered the problem
with temperature and concentration equations in the liquid and solid and
boundary conditions at the interface. Their criterion for instability
to an infinitesimal disturbance was
mG > Gt + a2TA/L, (2.1-2)
c T M h
where GT is the weighted temperature gradient
Gt = (k^ + kgGgVik^ + ks) (2.1-3)
Here ks and k^ are the solid and liquid thermal conductivities. Gc is a
modified liquid concentration gradient given by
Gc = GJlc(ajl v/2D)/(ajl (1/2 kJv/D^) (2.1-4)
? 2 21/2
a£ = (a + vV4D p (2.1-5)
where a is the wavenumber of the disturbance, k the solute distribution
coefficient, TM the melting temperature of the pure solvent, Lh the
latent heat of fusion and A the interfacial tension. This analysis laid
the foundation for all further work in morphological instability.
Following them Woodruff (1968) did the linear stability analysis along
the same lines for the melting problem and came up with the same
criterion as (2.1-2) but with
G = G (na v/2D)/(nka + a + (1 k)v/2D) (2.1-6)
C SC S £ S X, X,


10
, 2 A 2 ,2.1 /2
a = (a + v /4D )
s s
(2.1-7)
where Gsc is the solid concentration gradient at the interface and n is
Ds/Dr the ratio of solute diffusivities.
2.2. Later Research
The next major contribution to the problem was made by Wollkind and
Segel (1970) who proved "exchange of stabilities" for this problem for
most parameter ranges. Proving exchange of stabilities is equivalent to
showing the existence of the onset of steady state nonplanar
solutions. They also considered the weakly nonlinear regime after the
onset of instability and using the method of Stuart (I960) and Watson
(I960) analyzed the problem for the case of two-dimensional rolls,
showing the existence of subcritical bifurcation at most growth
velocities.
The method of Stuart and Watson is essentially the theory of Landau
(see Drazin and Reid (1981)) and follows the dominant mode of
instability into the weakly nonlinear regime. In this form it is
applicable only to disturbances of one cellular pattern at a time, but
Segel and Stuart (1962) extended this theory to the prediction of the
preferred pattern by considering the interaction of two specified modes
of disturbance. Depending on the way these two modes were combined it
was possible to obtain two dimensional rolls or hexagonal cells and they
showed that the experimental parameters would determine the stability of
these patterns. Sriranganathan, Wollkind and Oulton (1983) adopted this


11
method for morphological instability and gave parameter ranges where
each type of cell was stable. The limitation of the method is that it
considers hexagonal and two dimensional roll patterns but not
rectangular cells or cylindrical rolls and ignores the effect of
container shape and size which have been shown to be important in wave
pattern selection (see Koschmeider (1967)).
Ungar and Brown (1984a) considered the highly nonlinear problem and
after making several simplifications obtained solutions using the finite
element method. Finite elements can handle highly nonlinear problems
and give very accurate numerical solutions but are extremely time
consuming. Solving the full morphological problem is a very expensive
proposition by this method and hence Ungar and Brown simplified the
problem by ignoring the latent heat and solid diffusivity and assuming
that thermal conductivities in liquid and solid were equal. This
allowed them to reduce the problem to a "one-sided model" consisting of
variables in the liquid region only, considerably simplifying the
algebra and saving computer time. Such a model will have only limited
validity in highly nonlinear regions and this was borne out when Ungar,
Bennett and Brown (1985) solved the complete problem. But their most
extensive calculations were done only for the one sided model and hence
this is of chief interest. These were done only for the case of two
dimensional roll disturbances and here they showed that contrary to that
reported by Wollkind and Segel, there were multiple regions of
supercritical and subcritical bifurcation. More importantly they showed
that at large deformations of the interface secondary bifurcations
occurred. Ungar and Brown (1985) also modelled the formation of deep
cells in an attempt to follow the transition to dendritic growth.


12
Nonlinear finite difference calculations were done in a more
limited way by McFadden and Coriell (198*0 for the two dimensional
case. Later McFadden, Boisvert and Sekerka (1987) extended the
calculations for the three dimensional patterns of hexagons and cross
rolls. In both cases the enormous expenses involved restricted
calculations to a few parameter values.
2.3. Inclusion of Other Effects
While these workers were investigating the basic problem others
were busy trying to incorporate various influences. The most important
concern was the effect of fluid flow. Delves (1968, 1971 and 1974) in
attempting to approximate the influence of advection and stirring in the
melt, studied the influence of plane Couette flow on the problem. He
showed that two dimensional roll disturbances in the flow direction were
stabilized but there was no effect on disturbances perpendicular to the
flow. Coriell, McFadden, Boisvert and Sekerka (1984) modelled Couette
flow more systematically and came to the same conclusion. Recently
McFadden, Coriell and Alexander (1988) examined the effect of plane
stagnation flow on two dimensional disturbances perpendicular to the
flow and here too the flow as found to be stabilizing.
In another very important development Coriell, Cordes, Boettinger
and Sekerka (1980) studied morphological instability with solutal
convection. They showed that the two instabilities were essentially
decoupled with the melt being unstable to convective disturbances of
long wavelengths and the interface unstable to nonplanar disturbances of
small wavelengths. Also at low growth rates the dominant instability


13
was convective and the interface was not easily disturbed. At high
growth rates the roles were reversed and at an intermediate velocity the
two instabilities became comparable. It was only at this rate the two
instabilities interacted and the result was the prevalence of
oscillatory instabilities. Their conclusion was that, except at this
particular growth rate, it is usually sufficient to study only the
dominant instability near its onset.
Following Coriell et al. several workers have looked at special
aspects of these two instabilities and their work has been reviewed by
Glicksman, Coriell and McFadden (1986). They all confirmed or refined
the work of Coriell et al. but all the main conclusions mentioned above
still hold.
Several other influences apart from fluid flow have been
incorporated into the model but only a few relevant ones will be
considered here. Coriell and Sekerka (1972, 1973) tried to include the
effect of grain boundaries on morphological instability by assuming that
its only effect was to shift the onset conditions. They failed to
observe that in the presence of grain boundaries there could be no
planar solutions to the problem and that the interface will be nonplanar
at all times. Ungar and Brown (1984b) obtained the solutions to this
problem by matched asymptotic expansions for small grain angles and
using finite elements for solutions of large grain angles.
In rapid solidification kinetic undercooling of the melt is
significant and Seidensticker (1967) included this and showed that it
caused a shift in the onset conditions. The significance of this was
shown by Hardy and Coriell (1968, 1969 and 1970) when they observed
morphological instability in the growth of ice crystals. Constitutional


14
supercooling was not a factor here and it was shown that kinetic
undercooling was the primary cause. This dual cause for morphological
instability is somewhat analogous to the situation in natural convection
where we find that the variations of density and surface tension with
temperature can both cause convective instability.
2.4. Experiments in Morphological Instability
The early work on modelling morphological instability was prompted
by experimental observations but beyond that very few quantitative
experiments have been done near the onset conditions. This is an
unfortunate state of affairs and experimental verifications of
theoretical predictions are badly needed if further concrete progress on
the theoretical front is to be made. The work of Morris and Winegard
(1969), Trivedi and Somboonsuk (1984) and of Heslot and Libchaber (1985)
have already been mentioned. Recently de Cheveigne, Guthman and Lebrun
(1985, 1986) have attempted to verify the weakly nonlinear and strongly
nonlinear theoretical predictions and one hopes that more work along
these lines will follow.
De Cheveigne et al. performed their experiments on
succinonitrile/acetone and CBr¡j/Br2 systems. (These organic mixtures
are much easier to work with than metal alloys as they are generally
nonfaceted, transparent and require small temperature gradients and
hence they have been very popular with experimentalists.) They found
that the cell pattern formed and its dimensions were strongly dependent
on geometry of the container. More importantly when they ran the
experiments for two dimensional roll patterns, they found only
subcritical instability.


15
2.5. Limitations of Existing Models and Unaddressed Issues
In Chapter 1 the cause of constitutional supercooling was explained
as being due to the sharp solute concentration gradient in the liquid
near the interface, while elsewhere in the liquid and the solid the
solute concentration was practically a constant. It would seem then
that the only region of interest is the interface and a liquid "boundary
layer" adjacent to it. This has prompted all previous workers in
morphological instability to consider D^/v as the characteristic length
of the problem and to ignore solid diffusion. A typical value
of D^/v is 100 microns and this means that the far ends of the melt and
crystal are infinitely far away and the domain of the problem is
effectively confined to the liquid boundary layer mentioned above. For
the melting problem a characteristic length of D/v is used and the
domain becomes an even smaller boundary layer in the solid.
These assumptions considerably simplify the algebra involved and
hence their popularity. But they constrain the validity of the model in
several ways. The most obvious one is that they necessitate the melting
and solidification problems to be studied separately, even though they
only differ in the direction of the growth velocity. Besides this
assumption fails for very small growth velocities, as it introduces a
singularity at v = 0. Later we will show that neglecting solid
diffusion also introduces a singularity and makes the model fail in the
nonlinear regime.
Finally, any effect which resides in the entire domain, not merely
the boundary layer, cannot easily be incorporated into the model, which
is why all influences on morphological instability studied so far are


16
either boundary layer effects (e.g., solutal convection) or interfacial
effects (e.g., grain boundaries and kinetic undercooling). Phenomena
that span the entire domain, like advection in the melt or imperfect
insultion of the ampoule walls, have been either inadequately treated or
ignored completely. Hence there is a need for a model that includes the
entire liquid and solid domains which would be applicable for all growth
velocities. This model should also dispense with the separate
treatments accorded so far to the solidification and fusion problems
with one uniform formulation.
In Section 2.2 it was mentioned in connection with the work of
Wollkind and Segel (1970) and of Ungar and Brown (1984a), that this
problem oscillates between subcritical and supercritical instabilities
for the case of two dimensional roll disturbances. They did not,
however, compute the ranges of each type of instability for the
experimental parameters involved. This is necessary in light of the
experiments of de Cheveigne et al. (1986) who observed no supercritical
instability. Also the extension of these predictions to three
dimensional disturbances like hexagonal and rectangular cells is yet to
be done.
It would not be an exaggeration to state that the inspiration for
all the theoretical work done so far in morphological instability has
come from Rayleigh-Benard convective instability. A comparison between
the two problems would be invaluable as a source of continued
inspiration and as a way to draw conclusions and conjectures about
morphological instability from the vast published literature on
Rayleigh-Benard convection. Hurle (1985) has attempted this but his
work can only be regarded as perfunctory and there exists a need for a
more rigorous treatment of the issue.


CHAPTER 3
A UNIFORM FORMULATION
3.1. The Formulation
Since we are not making the assumption that the liquid and solid
are very deep, the problem has to be formulated very carefully,
especially with regard to the outer boundaries, if we are to avoid an
intractable moving boundary problem.
A typical crystal growth set up is shown in Fig. 3-1. The ampoule
is heated by the heating coils surrounding it and they keep the melt
region at a temperature T1 and the crystal at T2. The temperatures T,
and T2 are maintained constant by means of thermocouples located at z =
s and z = -l. The ampoule is pulled towards the cooler end at the same
velocity V at which the crystal grows, thus keeping the interface
stationary. The region near the interface is protected by an insulating
shield and it is this region that becomes the domain in our model.
So in this model the outer liquid and solid boundaries become fixed
at z =-l and z=s respectively and the solid will be moving with a bulk
velocity v and the liquid with a bulk velocity v/Y, where Y is the ratio
of densities p /p.. In this section we will assume that Y=1 and
S A#
consider the effects of Y not being unity in Chapter 6 as this would
cause advection and fundamentally alter the basic problem. Also we will
assume that the melt concentration at the outer liquid boundary is a
constant C1 .
17


13
Thermocouple Thermocouple
Figure 3-1. Experimental set up


19
The domain equations in the liquid melt are
5, 9T*
at + v az
ac, ac^
at + v az
ag*\
D
In the solid region the equations are
(3.1-1)
(3.1-2)
3T 3T
s ^ s
at V 3z
a V2T
s s
(3.1-3)
+ v
ac
s
az
D V2c
s s
(3-1-^)
where T and C are the temperature and solute concentration, D and a the
diffusivity and thermal diffusivity, with the subscripts l and s
referring to the liquid and solid.
The boundary conditions are
T* V c* C1 at z =
(3.1-5)
T =T. at z=s
s 2
(3.1-6)
At the liquid-solid interface we will use £ to denote the departure
from planarity and write the boundary conditions at z=£
T0 = T = T mC0 Tm H
l S M i ML.
h
(3.1-7)
Vh ksVTs Lh(v *
(3.1-8)


20
(3.1-9)
(3.1-10)
where TM is the melting point of the pure solvent, \ the interfacial
tension, the latent heat, m the absolute value of the liquidus
slope, k. and k the liquid and solid thermal conductivities, k the
x* S
distribution coefficient, n the normal at the interface directed into
the solid and H is the curvature of the liquid-solid interface and in
Cartesian co-ordinates is given by
(3.1-11)
In cylindrical co-ordinates, if we assume circular symmetry, it
becomes
+ (3jL)2,-3/2
' 3r J
(3.1-12)
It is assumed that the side walls are sufficiently far apart and
well insulated to enable us to impose periodic boundary conditions in
that direction.
To convert these equations into the dimensionless form we will use
the liquid depth l as the characteristic length and the diffusive time
2
D^/v as the characteristic time.


21
The temperature will be made dimensionless by T = (T-Tw)/G2, with
M T
G? = (kG + kgGg)/(ks+ V (3.1-13)
where G. and G are the temperature gradients in the liquid and solid
X/ s
in the quiescent planar state.
Similiarly the concentration will be made dimensionless by
C = C/G l with
c
G
c
D G )/(D, D )
s sc 2. s
(3.1-14)
where G. and G are the concentration gradients in the liquid and
X* c sc
solid in the quiescent planar state.
So in dimensionless form, if we neglect the Lewis numbers D^/a in
the temperature equations, we have
(3.1-15)
3C __ 3C
+ v
3t 3z
V2f = 0
(3.1-16)
(3.1-17)
3C
3t
3C
+ v
- = nv2c.
3z
where v =
o ani Ds/Dt
The boundary conditions are
(3.1-18)


22
at z= -1,
V'V Ti
c* W ci
(3.1-19)
at z=s/4=s,
T = (T. tJ/gtA To (3.1-20)
S 2 M T 2
at the interface z=£
T4 Tg = -SeC^ -A H (3.1-21)
VT n 2VT n = L(v + ^) (3.1-22)
* 3 Dt
kC = Cg (3.1-23)
VC n (v + ^)C. = nVC n (v + ^)C (3.1-24)
* Dt 16 S Dt S
where is the total derivative, 3 the ratio of thermal conductivities
k /k. and Se, A and L are the Sekerka, capillary and latent heat numbers
Af
respectively.
Se
mG
c
A
V
W
LhDa
kiGT^
(3.1-25)
(3.1-26)
L
(3.1-27)


23
At this point a discussion of the choice of a critical parameter
becomes imperative. The experimentally variable parameters for this
problem are G^, and v or their equivalents in this formulation
G^, Gc and v. Most previous workers have adopted G^ or C1 as their
critical parameter but Hurle (1985) has proposed Se, by analogy with the
Rayleigh-Marangoni problem. Recently de Cheveigne et al. (1986) have
advocated the use of v from an experimentalist's perspective. Although
this is a valid choice the reason no one else has used it so far is
probably because v occurs in the domain equations and will give rise to
an infinite number of eigenvalues in the linearized problem. We second
Hurle's suggestion and choose Se as this seems to be the naturally
occurring coupling factor between temperature and concentration and the
fundamental cause for morphological instability. Besides it includes
both G^ and Gq But the suggestion of de Cheveigne et al. still
remains a valid one.
3.2. The Linear Stability Problem
From now onwards the " for dimensionless variables will be
dropped. The steady state planar solution to this problem occurs when
Cc(x,y) or Cc(r) = 0
(3.2-1)
The solution is then
(T1 + SeCjlc(o))z Secuto)
(3.2-2)


24
Tsc(z) = (T2 + SqC1o(z/s ~ SeC^to)
(3.2-3)
Cic(z)
(l-k)exp(vz)
l]/[-
(1-k )exp(-v)
k-exp(-v(1+s/n)) J Lk-exp(-v(1+s/n))
1]
(3.2-4)
C (z)
SC
St
(l-k)exp(vz/n)
kexp(v( 1+s/ti) )-1 J'Lk-exp(-v(1+s/n))
1 ]/[ttt
(l-k)exp(-v)
1]
(3.2-5)
To write the equations of the linear stability problem we will
impose an infinitesimal disturbance on the steady state solution.
T = T + T
4 ic a
(3.2-6)
with similar expansions for T C., C and £
S X s
Considering the linear stability problem, in order to separate
variables we will assume a horizontal cell pattern. This pattern for
two dimensional rolls is given by
, v 2imx ... 2irn
d> (x) = Cos i, with wave number a =
Tn L n L
(3.2-7)
for cross rolls (x,y) = Cos + Cos a = (3.2-8)
n Li Li n Li
for rectangular cells
<|>n(x,y) = Cos Cos a = 2im(1/L^ + \/\}) (3.2-9)
n Li Li n 1 .
for hexagonal cells


25
n(x,y) = 2Cos Cos 2y + Cos
4imy 4im
3L an = 3L
(3.2-10)
for cylindrical rolls (r) = J (a r/R)
non
(3.2-11)
where an are the zeros of and JQ and J-j are the zeroth order and
first order Bessel's functions of the first kind. We can now write
A A ,
T^(x,y,z,t) = T41(z) 1 (x,y)e0t
(3.2-12)
with corresponding forms for the other variables, where a is the
eigenvalue of the linear stability problem.
The linear stability problem becomes in the domain
(D2 a2)T41 0 (3.2-13)
oCai= (D2 vD a2 )C^1 (3.2-14)
(D2 a2)Tgl= 0 (3.2-15)
s1
Here we have
The boundary
used D to denote
dz'
conditions are
(3.2-16)
T,1 = * cu = 0 at z -1 (3.2-17)
T
s 1
= 0
at z = s (3.2-18)


26
At the interface, z=0
s1
(Gt V
T11 SeCt, il(Gt SeGIc- > A ) 0
DT1 8DTs1 oLAl
s1
(kG
la
G )
sc
0
DCtl nDG3l VC11 VC3l
v(cto"
c
sc
Solving the system we obtain a general equation for
instability
Se[G
aC^c(1-k)tanhasstanha^
c. k(na + v/2 tanha s)tanha + (a -v/2 tanha.)tanha s'
s s l l i s
2. oLtanhatanhas
+ a a(k.tanhas + k tanha)
where
2 o v -1V2
r 0 v t
a [a *
a = [a2 + a + v2/4] /2
kGtanhas + k G tanha
l ¡L s_s
k.tanhas + k tanha
, s
(3.2-19)
(3.2-20)
(3.2-21)
(3.2-22)
(3.2-23)
morphological
(3.2-24)
(3.2-25)
(3.2-26)
(3.2-27)


27
G. (a v/2 tanha.)tanha s + G (qa v/2 tanha s)tanha.
i X#C3C a/ S SC S S lo
c (a v/2tanha)tanha s + k(na + v/2tanha s)tanha.
x, l s s s il
(3.2-28)
If we can show exchange of stabilities for this problem, then for
neutral stability.
Se
o
Gj + a A
(3.2-29)
Here we have already used the fact that the Se obtained from the neutral
stability curve will be the same as SeQ defined later in eqn. (4.2-2).
In eqn. (3.2-29) if we let v become very large the critical wave number
ajjj^ (for which SeQ is a minimum) also becomes very large and we can
approximate all the tanh terms to unity. Further if we also neglect
solid diffusion, n and G are zero and the equation reduces to the well
s c
known results obtained by Mullins and Sekerka (1964).
r !to(at ~ l V) Ctc'k)
,, 1 +
H + v(k 2}
v(k -
H = 1 + a A -
aL
a(V ks}
(3.2-30)
For the case of neutral stability this becomes
Se
o
(1 + a2A Ma^/v + k ^)

(3.2-31)
It must be pointed out that setting all the tanh terms to unity is
equivalent to using the diffusive length as the characteristic length.
This is a boundary layer approximation not unlike that used in the study
of pipe flow at high Reynold's numbers.


28
In our derivations we did not restrict v to be positive and hence
eqn. (3.2-24) is also valid for negative velocities, that is the fusion
problem. If we replace v with -v and here too assume that v is very
large and set the tanh terms to unity we obtain the result of Woodruff
(1968).
G (na V? v)
Se [ 3 2
ocioa-)
0 nka + a. + ^ vCI-k) nka ao i v(1-k)
] 1 a2A
oL
a(k,+ k )
x. s
(3.2-32)
and for o = 0
Se =
o
(1 + a2A)(nkag + a + j v(1-k))
G (na
sc s
iv>
(3.2-33)
To compare this model with the approximations of Mullins and
Sekerka and that of Woodruff, SeQmin was calculated for various growth
velocities for the Pb-Sn system and the results are shown in Fig. 3~2.
As can be seen the approximations hold up very well for most growth
velocities but begin to fail for small velocities. (The thermophysical
data for the Pb-Sn system were those of Coriell et al. (1980).) The
ratio n for the Pb-Sn system is of the order 10 but for systems with
much smaller values of n like the C-Austenite system (see Clyne and Kurz
(1981) and Wolf, Clyne and Kurz (1982)) the approximations begin to fail
at higher velocities.


1200
800
400
VELOCITY (urns*1)


Figure 3-2. Comparison of previous approximations with general equation


30
3.3. The Adjoint Problem and Exchange of Stabilities
Exchange of stabilities refers to the nonexistence of time periodic
infinitesimal perturbations. Time-dependent infinitesimal perturbations
about the planar state will generally have periodic and nonperiodic
components, with a = + io^. For some problems it is possible to show
that ck is zero and this is called exchange of stabilities (see Iooss
and Joseph (1980) for details).
We still have to prove exchange of stabilities for this problem but
before we can do that it is necessary to obtain the adjoint problem. To
accomplish this we will define a column vector and a matrix operator
L
(3.3-D
2 2
(D a;
0
0
(D2- vD-a2- a)
0
0
0
0
2 2.
(D -a )
0
0
0
Se, 2 2 ,
(nD -pa -o-vD)
(3.3-2)
s
# r # r *
and an inner product <#,*>= I (T T. + C.C.)dz + J (T T + C C )dz
_1 1 ^ o 3 S 3 S
(3.3-3)


31
where T denotes the complex conjugate of T, T* the adjoint function of
T and
x
G
sc
)/ (3.3-4)
Then the domain equations can be written as = 0 (3-35)
In this inner product the adjoint problem becomes in the domain
2 2
(D -a )Tjl1 = 0
(3.3-6)
2 2
(D -a )Tal = 0
(3.3-7)
(D + vD -
2
a )cu
Si
(3.3-8)
(D^ +
n
2 ~*
a )CS1
2 C*
n si
(3.3-9)
Subject to the boundary conditions
Si Si 0
at z = -1
(3.3-10)
T = 0
s1
at z=s
(3.3-1D
At the interface z=0
(3.3-12)


32
nC£1 = Cs1
kDGs, DG^ '* 0
Ds', DL ' (GH s> 0
DI¡1 SeDG3*, % (Gt SeG£o- a'A > wfrj
2, s
aC (1-k)
(kG0 G ) 6eUJl1
2,c sc
(3.3-13)
(3.3-14)
(3.3-15)
(3.3-16)
So far we have been unable to show exchange of stabilities for this
problem directly. But Wollkind and Segel (1970) have proved it when the
boundary layer approximation is valid and we will prove exchange of
stabilities by performing an asymptotic analysis around the boundary
layer solution, which corresponds to n=0 and Pe=0, where Pe is the
Peclet number given by D^/lv.
'2,1
= T
00
2,1
r)T
10
2,1
+ PeT
01
2,1
(3.3-17)
00 10 01
a = o + no + Pea + ....
(3.3-18)
The other variables are expanded similarly.
If we use L00 to designate the operator L in (3.3-2) when n and Pe are
zero, then the linear perturbed systems become
00 210
J *1
.10
oo 201
J *1
= f
01
(3.3-19) & (3.3-20)


33
where

0
0
0
10"00
0 cm
* f01 =
0
01:00
o C ,
s1
0
0
p
(3.3-21) & (3.3-22)
With boundary conditions
at z =
(3.3-23)
at z=
(3.3-24)
at z=0 the equations are
boo(;;0,
> =
10
n00 "10
B (*1 ,
"01 ,
C, ) =
01
(3.3-25) & (3.3-26)
where B00 is the boundary operator defined by eqns. (3.2-19) (3.2.23)
when n and Pe are zero and
10
;>i- o
<>i- O
10r "00
o Lc1
"00,. .10 -10,
51 10.00,, *00 00.10,, .00
0 ct0O-k)t) 0 Cto(1-k)0,
(3.3-27)


34
h01 will have a similiar form with 01 superscripts replacing the 10. It
we let L00* and represent the adjoint operator and adjoint
function of and respectively, we can use the solvability
condition on the above system.
<#1
loo;;> =
<$
00*
f10>
(3.3-28)
00*200* 21 Ox
-
(3.3-29)
Subtracting we get
J(*
00* "00*
1
h10)
= <;* foi>
z=0 r
(3.3-30)
where the lefthand side of eqn (3.330) is the bilinear concomitant
evaluated at z=0.
Similiarly J(*
00* "00*
1
h1)
z=0
= <4>
00*
f01>
(3.3-3D
"00* "00
We note that # and ^ are real as they correspond to a state where
exchange of stabilities has been proved, while h1<\ and f^1 are
also made up of real quantities except for and Hence we
conclude from eqn (3 330) that is real and from (3-331)
that a01 is real; i.e. exchange of stabilities holds for the generalized
morphological instability problem upto 0(n) and 0(Pe). So we are
justified in using the neutral stability curve (3.2-29) to calculate SeQ
at least up to such order even though we often extend it further. Even
when the boundary layer approximation is valid, exchange of stabilities


35
does not hold for all growth conditions (see Coriell and Sekerka (1983))
and care must be exercised when such extensions are made.
3-4. Finite Containers and the Most Dangerous Wavenumber
Appealing to the proof of Section 3.3, henceforth we shall only
consider steady state solutions. A typical SeQ versus a diagram is
shown in Fig. 3~3 and SeQmin is the minimum value of SeQ and the
wavenumber at which this occurs is amin. If we can maintain the growth
conditions such that Se < Seom^n we can at least say that the planar
interface is stable to infinitesimal perturbations.
As SeQmin is the least value of Se at which the planar solution
loses the stability, am^n is the wavenumber of the disturbance which is
most likely to occur. Hence this wavenumber is commonly regarded as the
most dangerous and is the wavenumber at which morphological instability
is usually studied. In the derivations and discussions that follow this
is the wavenumber employed and it becomes our "operating wavenumber."
The wavenumber is a 2tt multiple of the reciprocal of the
wavelength. If the domain being considered is regarded as being finite
with periodic lateral boundary conditions, the wavelength R (or L
depending on the wave pattern used) in its dimensionless form is now the
aspect ratio, the ratio of the ampoule radius to the melt depth. For
this situation SeQ takes on different values depending on the number of
cells formed on the crystal surface as shown in Fig. 3-i1 (see also
Rosenblat, Homsy and Davis (1982)). For each value of R there is a
fixed number of cells which is the pattern that is most easily
disturbed, except at certain values of R where two different patterns


36
WAVENUMBER a
Figure 3-3. Se^ vs. wavenumber diagram


37
are equally dangerous. These "horizontal multiple points" are very
important and will be discussed later. At other values of R the most
dangerous number of cells is denoted by N and the corresponding
wavenumber of each cell and the value of SeQ are denoted as a¡^ and SeQN
respectively. For the analysis in Chapter 6 where the effect of a
finite container on morphological instability is considered, R is chosen
such that SeoN is as close as possible to SeQmin so that the worst
possible case can be examined.


CHAPTER 4
SUBCRITICAL BIFURCATIONS
4.1. Theory
The linear stability analysis will only give the onset conditions
for morphological instability. Nonlinear calculations are necessary to
determine the behavior beyond this point. Ideally numerical
calculations should give the most amount of information by being
applicable for small and large deformations, but as can be seen from the
work of Ungar and Brown these calculations are very expensive to carry
out and they were forced to simplify the nonlinear problem and perform
calculations for very few experimental conditions. McFadden et al.
(1987), even though they did not attempt calculations for larger
deformations, were faced with the same restrictions.
This then is the case for weakly-nonlinear methods. They are
generally valid only in a small region very close to the onset
conditions but they can be used to predict the shape of the nonlinear
curve for larger deformations and for several applications this
information is sufficient. More importantly due to the analytical
nature of the techniques they can be used to predict the weakly
nonlinear behavior for all experimental conditions. A case in point is
the work of McFadden et al., most of whose predictions could have been
obtained more cheaply for all parameter values from weakly-nonlinear
theories.
38


39
Probably the most useful information generated by these theories is
the subcritical behavior of the nonlinear curve. Some typical
bifurcation diagrams are shown in Figs. 4-1 to 4-4. The e = 0 axis
corresponds to the planar solution and initially for small values of Se
the planar solution is stable and usually the only possible solution.
For Se £ SeQN the Planar solution becomes unstable and a nonplanar
solution "bifurcates" from the planar one. Figure 4-1 shows a symmetric
bifurcation diagram where nonplanar solutions do not exist for Se <
SeoN. For Se > SeQjy, even an infinitesimal perturbation will make the
solution jump from the unstable planar solution to the stable planar one
while for Se < SeQ^ the planar solution is stable to all
perturbations. This behavior is referred to as supercritical
bifurcation and for these curves Se^ = 0, Se2 > 0 (where Se^ = dSe/de
2 2
and Se2 = d Se/de at e = 0). Figure 4-2 is nonsymmetric and as can be
seen stable and unstable nonplanar solutions exist for Se < SeoN, making
the planar solution stable to infinitesimal perturbations in this
region, while a large perturbation can make it jump to the stable
nonplanar branch. This is called a subcritical bifurcation diagram and
is characterized by Se-| 0. In this situation it is obvious that
growing the crystal at Se < SeQN is no guarantee of avoiding
morphological instability.
Figures 4-1 and 4-2 display the behavior usually seen in most
problems of hydrodynamic instability. Morphological instability is
unusual in having nonlinear curves shown by Figs. 4-3 and 4-4 as well.
These curves have been labelled "backward bending" to distinguish them
from the usual "forward bending" curves. (Actually they are Janus-like
in appearance bending backwards and forwards.) From the point of view


40
Figure 4-1. Forward bending, locally
symmetric, supercritical
bifurcation diagram
Figure 4-2. Forward bending,
unsymmetric sub-
critical bifur
cation diagram
Figure 4-3. Backward bending, locally
symmetric, subcritical
bifurcation diagram
Figure 4-4. Backward bending,
unsymmetric, sub-
critical bifurca
tion diagram


41
of the crystal grower this is unfortunate as their subcritical nature
increases the occurrence of subcritical bifurcation. Figure 4-3 is the
symmetric case with Se-j = 0 and Se2 < 0 and Fig. 4-4 the nonsymmetric
one with Se^ 0.
The symmetry or nonsymmetry of the bifurcation diagram in
hydrodynamic stability is dependent on the cell pattern (see Joseph
(1976)). For morphological instability it will be shown that two
dimensional rolls, rectangular cells and cross rolls produce locally-
symmetric bifurcation diagrams while cylindrical rolls and hexagonal
cells produce nonsymmetric diagrams. It will also be shown that both
backward bending and forward bending will occur for all cellular
patterns depending on the experimental parameters used. Hence
bifurcation can be subcritical or supercritical for two dimensional
rolls, rectangular cells and cross rolls but for hexagonal cells and
cylindrical cells bifurcation is always subcritical.
4.2. The Second Order Problem
Here we begin our weakly nonlinear analysis and in this section we
will calculate Se^, the first derivative of Se with respect to e.
Considering the neutrally stable nonplanar solution near the bifurcation
point SeQ, we will expand the variables around the planar steady state
solution.
2 3
T =T +eT +eT + £ T +
1 1 Alc lo 11 12
Se = Se + eSe. + e Se_ +.
o 1 2
(4.2-1)
(4.2-2)


42
V,
where e = <$
(4.2-3)
We have already obtained the linear perturbed solution
in Section
3.2. Substituting these expansions into the steady state versions of
eqns. (3.1-15) (3.1-24) and collecting the terms of order
2
e we get
the second order perturbed problem.
If the first order perturbed variables are written, following eqn.
(3.2-12), as
Vx,y,z,Seo) = TSl01(zSeoH1(xy)
(4.2-4)
Then the solution to the second order problem becomes
CO
Til1 = E. T£1n(z,Ser T01)<()n(x,y)
n=1
(4.2-5)
Substituting (4.2-4) and (4.2-5) into the second order problem and
taking Fourier transforms horizontally, we get
Ln '0
(4.2-6)
an TtU CI11 0 at z -1
(4.2-7)
T .. = 0 at z =s
s 11
(4.2-8)
B($11 ?11) = at z=0
(4.2-9)
at z=0
(4.2-9)


43
where L and B are the same as that used in (3.3~2) and (3.3-25) but with
o=0 and Se = Se and
o
11
-r (DT DT )I
^or o sor 11
f ;01(DT£01 + SeoDC£01) 2 vG£cSeoi01^11 Sei(Ci
~a ?01(T£01" BTs01 5111 + WVf eTs01):i12
^01(k DC£01 WW -?V4(kG£c W
"a C01(C£01
nGs01)i:i1 + ?01(C£01 nCs01)i:i2
(4.2-10)
L2 L1
I I ^(x,y)dxdy
o o
11
L2 L1
I ¡ <>1 (x.y)dxdy
o o
(4.2-11)
G2 G1 94. 94. -
I I ^(ir) + (*r}^dxdy
9y
"12
L2 L1
I I 4>^(x,y)dxdy
o o
(4.2-12)
It can easily be seen that for two dimensional rolls, cross rolls
and rectangular cells that 1^ = I12 = 0. Hence from the solvability
condition for these patterns Se^ will be zero; that is, the bifurcation
will be locally symmetric. But for hexagonal cells and cylindrical
rolls I-|i and will be non-zero and hence Se^ will also be nonzero
and we have nonsymmetric bifurcation and the existence of subcritical
instabilities.


44
We will analyze the case of Se^* 0 by considering hexagonal cells
as an example, but the results obtained are applicable to cylindrical
rolls as well. It should be noted here that all the horizontal cell
patterns we have been using are possible solutions to Helmholz's
equation
= 0
(4.2-13)
and the solutions always come in pairs. For hexagonal cells the
complementary solution to given in (3.2-10) is i|<
<|i. = 2 Cos
2tt
73:
X Sin
2ir
3LJ
4tt
Sln 3Ly
<4.2-14)
So the general form of eqn. (5.4) will be
T£01 (lf1
+ P^)
(4.2-15)
To determine p the procedure outlined by Joseph (1976) will be
used. We will proceed the same way as above but using (4.2-15) instead
of (4.2-4) and multiplying by ^ and integrating horizontally. This
will give the same set of eqns. (4.2-6) (4.2-10) but with
I
11
2
+ pi^1 ) (J^dxdy
+ pip1 )4>1dxdy
(1-p)2
(4.2-16)


45
3L /3L 3 dip dip dip. .
I I !(ir*p-3i) <-37 p-57>
O 0
'12 = 3L Al
ff
O 0
a /. 2
= (1-p )
(4'1 + H^xdy
(4.2-17)
which can be used to calculate Se^ We then repeat the process
with ^ and equate the two Se1's obtained. This will result in a cubic
equation for p.
P
3
3p = 0
(4.2-18)
and
Se1(1-p2)
(4.2-19)
Se G (kG G ) (a. + v/2tanha.)B (qa v/2tanha s)
1 c = lc SC r r l l_ + 3 S i
eo \2 * tanha qtanha^s J
01
(k + B)
11
(1+qB)tanha s
(na + v/2tanha s) X12J 1
s s
-Iiol + *1
(4.2-20)
6(Gi V [_^n + a(Gl + SeoGlc a A)(1/B-1 ) I
1 (B+A) Ltanha
(G -G )tanha
SL s
11
(1+A)tanhas
(B+A)
h2 > h2
(4.2-21)
Se I.. a(B-1)(kG. G )
_ 2_LL[ ^ (bg. + g /n)]
2 (k+B) L (B+A)tanha K-c sc
1"U =
(4.2-22)


46
where B = (a. v/2tanha)tanha s/(na + v/2tanha s)tanna. (4.2-23)
x. x. s s 3 x.
and A = tanhas/tanha (4.2-24)
There are three solutions 0, /3 and -/3 for p, but it can be seen
that Se1p for /3 and -/3 coincide. It is also obvious that while any
two of the solutions are independent the third is a linear combination
of the other two. Here is an instance where the problem exhibits a
multiplicity of solutions for the same eigenvalue, and could cause
secondary bifurcations further along the bifurcation curve.
The next obvious question is to ask if there is any point at which
Se1 in (4.2-20) goes to zero and hence causing Selp to go to zero.
Calculations done for the C-Austenite (i.e. steel) system did give such
a curve (see Fig. 4-5) though for most growth conditions this curve lies
far away from the critical (or most dangerous) wavenumber a^^
intersecting it only at high velocities.
4.3. The Third Order Problem
In Section 4.2, it was shown that for 2-D rolls, cross rolls and
rectangular cells Se^ was zero, which implied a symmetric bifurcation
diagram. But to learn more about the nature of this diagram it is
necessary to go to the next order. If we repeat the procedure described
for the second order problem for the terms of order e^ we will obtain
the third order problem.
L*
21
0
(4.3-1)


WAVENUMBER
47

Figure 4-5. Growth velocity vs. wavenumber chart for
C-Austenite system for hexagonal cells


48
TZ21 = GZ21 0
at z = -1
(4.3-2)
Ts21 0
at z=s
(4.3-3)
B(*2i C21) = h21
at z=0
(4.3-4)
21
(T T )I
2 ^01 WZ01 s01 '21
2
'I21^ 1~C01(TZ01 + SeoC.01) + 2501Seo DCZ01 + 6V C01SeoG,J
(GZ0 ?01G,c)Se2 Ac01(3/2 a I22~ I23)
2
_I21 [~2~ C01(kCZ01 Cs01) + 2C01(k DCZ01 DCs01/n)
+ ^01(kG*c Gso/ri2)]
a2 2 2
2 C01V(DCZ01 r|DCs01 )J21 + C01(DCZ01
nDCs01)J22
L2 L1
/ / 4>1 dxdy
o o
21
L2 L1
/ I 4>2dxdy
o o
(4.3-5)
(4.3-6)
LZ S 3*, 3*,
/ / +l[(k> ()2]dxdy
o o
9y
22
L2 L1
/ / 2dxdy
o o
(4.3-7)


49
L.i L2 a*, a*. ,
I I [-4 <-^>2
oo 3x
2 v 3y
34> 1 3^ 2 <(>^
2 3x 3y 3x3y
324>
dy
dp
1 ()2l*1dXdy
23
L2 L1
/ J o o
Once again using the solvability condition we obtain
(4.3-8)
Se_ 2 v(kG. -G )(1+B)tanha s
yVh,V (i22 r i2,)Seo <4-3-9)
01
(k+B) (na +1/2vtanha s)
s s
where
h1 ~^2a I22+I23^A~ I21Se0V
vG
Ic
(kG. -G )(a + v/2tanha.)/tanha
x.c sc x. x, i -
6(k+B)
*21 ''(Gtc-Gsc>
(k+B)
k(a^ + v/2 tanha^)
tanha.
(4.3-10)
(na -v/2 tanha s)B
3 2 1 *2,
n tanha s
s
Se v(kG. -G /n)
o to sc
6(k+B)
(4.3-11)
and B = (a -1/2 vtanha)tanha s/(na + 1/2 vtanha s)tanha. (4.3-12)
x, x, s s s l


50
4.4. Calculations and Comparisons
Using the results of the previous section, Se2 was calculated for
various values of the experimental parameters and the results are shown
in Figs. 4-6 to 4-9. Figure 4-6 was drawn in order to check the
derivation with that of Ungar and Brown (1984a) and shows calculations
done for 2-D rolls with Dg = 0 and 3 = 1 but is a more complete
calculation for the experimental parameter ranges involved than
theirs. The calculations agree with those of Ungar and Brown but it is
an unexpected result nevertheless, showing multiple regions of
subcritical and supercritical instability. It also seemed to contradict
the earlier calculations of Wollkind and Segel (1970) who did not see
any supercritical instability but this was resolved in the paper of
Wollkind and Wang (1988) and hence Fig. 4-6 agrees with their
calculations as well. As argued in Section 3.4 if we treat the amin
curve as the experimental "operating line," the bifurcation is mostly
supercritical which also seems to contradict the experimental results of
de Cheveigne et al. (1986) who observed only subcritical bifurcation.
In Fig. 4-7 calculations were done for more realistic values of Ds
and 3 and the results are significantly different with only one region
of supercritically and another of subcriticality. Ignoring solid
diffusion introduces a singularity to the problem and this accounts for
the distortions observed in Fig. 4-6. In Fig. 4-7 if we move along the
operating line for a fixed liquid temperature gradient, initially for
small growth velocities the bifurcation is supercritical, until at a
critical velocity a transition point is reached and the bifurcation
becomes subcritical.
Hence for every imposed liquid temperature


WAVENUMBER
51
Figure 4-6. Velocity vs. wavenumber chart for the one-sided
approximations of Ungar and Brown for 2-D rolls


WAVENUMBER
52
>
Figure 4-7. Velocity vs. wavenumber chart for 2-D rolls
using the Pb-Sn system


53
gradient there will be a critical growth velocity below which the
bifurcation for 2-D roll disturbances is always supercritical. The
really surprising result here is that when these transition points for
various values of the gradient are joined, we get a straight line
through the origin. We now have a clearly demarcated supercritical
upper-triangular and subcritical lower-triangular one. It is well known
that the onset condition SeQ does not change much in the neighborhood of
a^H (see for example Coriell, McFadden and Sekerka (1985)) and so amin
is more of an interval than a unique point. It can also be seen from
Fig. 4-7 that the am^n curve practically hugs the line of transition
from sub to supercriticality and if we impose an interval for am^n it
would straddle the transition line. Thus for 2-D rolls it is unlikely
that we will ever see a sharp transition from subcritical to
supercritical bifurcation in experiments. More likely we will observe
subcritical behavior throughout as reported by de Cheveigne et al.
Proceeding to the three dimensional patterns, we obtained almost
identical results for square cells and cross rolls. Figure 4-8 is for
square cells and we see quite a change with the supercritical region
acquiring a characteristic balloon shape and having a sharp transition
to subcritical bifurcation along the amin curve. But here too if these
points of transition are joined, a straight line is the result,
demarcating a supercritical upper-triangular operating region and a
subcritical lower triangular one. Unlike the case of 2-D rolls these
should be visible to the experimentalist. So in order to avoid cross
rolls and square cellular instabilities not only should the crystal be
grown when Se < Seomin, but one should do so in the upper triangular
region. Figure 4-9 demonstrates the universality of our result in being


WAVENUMBER
54
VELOCITY (urns*1)
Figure 4-8. Velocity vs. wavenumber chart for square cells
using the Pb-Sn system


55
8000
6000
4000
2000
0
*
Figure 4-9. Velocity vs. wavenumber chart for the fusion
problem for square cells using C-Austenite system
WAVENUMBER


56
applicable for the melting problem as well and repeating the derivation
for this case separately as was done by Wollkind and Raissi (1974) is
unnecessary.
Finally, even though for hexagons Se-j was usually non-zero (and
hence Se2 cannot easily be calculated), in Section 4.2 we saw that there
were points at which Se-| did go to zero. If we attempt to evaluate Se2
for hexagons at these points I2i, I22 and ^23 become
(4.4-1)
5 2 2
!22 -f a(1+p)
(4.4-2)
(4.4-3)
and as can be expected Se2 will be in the form
(4.4-4)
where Se2 is that corresponding to p = 0. The other possible value is
when p is /3 or -/3. Here Se2p can be calculated from (4.3-9) (4.3-
12) and (4.4-1) (4.4-3). As mentioned in Section 4.2 these points are
usually far away from am^n and in Fig. 4-5 we found that along this
curve Se2 is positive at low values of "a." As "a" increases, at a
point above a=amin, Se2 becomes negative. Hence along this line the
bifurcation diagram is forward bending at low "a's" (including a^n) and
becomes backward bending at high values of "a." Even though this is


57
true only along the Se-|=0 line, by analogy with other cell patterns we
conjecture that this is valid when Se^O as well. In other words we
expect the bifurcation diagram to be forward bending along the line
and below (as shown in Figs. 4-10 and 4-11) but a transition to backward
bending along the line could occur at high velocities. Far above
the amin line the bifurcation diagram should be backward bending (see
Figs. 4-12 and 4-13).
To confirm these conjectures we turned to the nonlinear
calculations for hexagons of McFadden et al. (1987) but unfortunately
they were unable to complete the bifurcation diagram as their attempts
to compute the curve for e<0 failed. Also their calculations were done
only for the case of p=0 and not for p=/3. But they did confirm the
existence of subcritical instability.
To sum up, it has been shown that the Mullins and Sekerka and the
Woodruff models of morphological instability are of limited validity.
The uniform formulation is the more exact representation of the problem
and
* it is applicable for all growth velocities and not just the
relatively rapid solidification and fusion regions and provides a
single formulation from which all the different models for
various growth conditions can be obtained as limiting cases,
eliminating the duplication of derivations for different cases;
* it incorporates the whole crystal and melt regions into the
problem and not just a boundary layer region adjoining the
interface facilitating the study of various domain effects like
convection on morphological instability;


53
Figure 4-10. Bifurcation diagram
for hexagonal cells
Figure 4-11. Bifurcation diagram
for hexagonal cells
Figure 4-12. Bifurcation diagram
for hexagonal cells
Figure 4-13. Bifurcation diagram
for hexagonal cells


59
* it avoids the incorrect predictions of subcritical bifurcation
regions because of the singularities inherent in previous models.
* The principle of exchange of stabilities has been shown to be
applicable to this model as well even though only in an
asymptotic sense.
When the weakly nonlinear technique of Malkus and Veronis (1959) is
applied to this problem in a systematic way, it resulted in important
information about the shape of the bifurcation diagram for various
growth conditions. Some of these results are similar to those obtained
for Marangoni instability (see Joseph (1976)) which leads us to assert
that these results are valid for all hydrodynamic instability problems
in which the nonlinearity lies only on the boundary.
* If rectangular cells, cross rolls or two dimensional rolls are
the horizontal cell patterns then the bifurcation diagram will
always be locally symmetric. For hexagonal cells or cylindrical
rolls they are generally non-symmetric and hence the bifurcation
is subcritical.
* The problem can display a multiplicity of solutions for the same
eigenvalue. Specifically for hexagons there are two possible
solutions.
Considering the morphological instability problem in particular the
following were shown:
* for two-dimensional rolls there are two operating regions, one
subcritical (backward bending) and the other supercritical
(forward bending), but since the demarcation is not sharp its
probable that only subscritical bifurcation will be observed
experimentally.


60
* For rectangular cells the forward bending region has a
characteristic balloon-like shape and here too there is a
straight line dividing the operating region into subcritical and
supercritical zones, but here the transition is sharp and hence
probably observable experimentally.
* For hexagonal cells and cylindrical rolls the bifurcation diagram
shows both backward and forward bending behavior but the exact
regions of each can only be conjectured.


CHAPTER 5
COMPARISONS WITH RAYLEIGH-MARANGONI CONVECTION
5.1. Rayleigh-Marangoni Convection in Brief
When a horizontal layer of quiescent fluid is heated from below, on
account of thermal expansion, the fluid at the bottom will be lighter
than the fluid at the top. This unstable arrangement is maintained by
the viscosity of the fluid which inhibits any flow and suppresses
disturbances such that there will be a stable conduction profile in the
fluid. But when the adverse temperature gradient exceeds a critical
value, the viscous force is overcome and there will be cellular
convection. This instability is known as Rayleigh-Benard convective
instability.
There are several variations of this problem. Instead of an ad
verse temperature gradient, there could be an adverse solute concentra
tion gradient in the fluid causing once again an unstable top-heavy
arrangement. The convective instability arising from this is known as
solutal convection or the solutal Rayleigh problem. Another way to
cause convection is to have a very thin fluid layer heated from below,
but the top surface of the fluid instead of being kept at a fixed lower
temperature, is allowed to remain free. Here the thinness of the fluid
layer makes buoyancy effects negligible but convection will be caused by
surface tension variation on the free surface. This is known as Maran-
goni convection. Finally, there could be combinations of the above
61


62
three types of convective instability. When convection is caused by
thermal and concentration gradients it is known as double-diffusive
convection. Combination of either thermal or solutal convection with
surface tension driven flow is the Rayleigh-Marangoni problem.
In addition to these there are several other combinations possible
like Rayleigh-Benard convection with rotation or with a magnetic field
but for purposes of comparison with morphological instability it will be
seen that the three causes for convection mentioned above are the most
relevant.
In the discussion to follow it is desirable to consider the most
general form of this problem. Despite there being several interesting
features in the problem of double-diffusive convection, the causes for
convection there, the temperature gradient and the solute concentration
gradient are both bery similar and it is sufficient to look at the
effect of one gradient. The manner in which the surface tension effects
the problem is very different from the buoyancy effects and a general
formulation should include both. Accordingly we will examine the Ray
leigh-Marangoni problem with thermal convection. In the following
sections several other reasons for looking at this problem will become
apparent.
The equations for the Rayleigh-Marangoni problem are given by Sarma
(1987) and we will reproduce them here and refer the interested reader
to his paper for details. The steady state dimensionless Boussinesq
equations in the domain are
V-V = 0
(5.1-1)


63
2 Ra
V V Vp + ^ Tg V-VV = 0 (5.1-2)
V2T PrV*VT = 0 (5.1-3)
where p is the modified pressure, g the acceleration due to gravity, Ra
the Rayleigh number and Pr the Prandtl number. The boundary conditions
at the bottom of the fluid layer are
at z = 0, T = Tq and w = 0, 3w/3z = 0 (5.1-4)
where w is the vertical component of velocity.
The boundary conditions at the top are more complicated. Not only
will there be surface tension variation across the surface but the
surface is also free to deflect like the liquid-solid interface in
crystal growth.
at z = 1 + i, VT*n = BiT (5.1-5)
V*n = 0 (5.1-6)
- Bopn + Crn*[VV + VVT] = MaV T + Hn (5.1-7)
n
The dimensionless quantities are Bi the Biot number, Bo the Bond number,
Cr the Crispation number, Ma the Marangoni number and H the surface
curvature (see Scriven and Sternling (1964) for details).
Initially there will be a quiescent, linear, stable, conducting
solution to the problem with V =0. At a critical value of the charac-


64
teristic parameter (Ma or Ra) this conducting solution becomes unstable
to infinitesimal perturbations and we have a convective solution.
Performing a linear stability analysis about the conduction state,
separating variables and doing considerable manipulations we get in the
domain
(D2 a2)30 = a2Ra6 (5.1-8)
(D2 a2)3w = a2Raw (5.1-9)
where "a" is the wavenumber, w the Fourier coefficient of the vertical
component of velocity and 0 the Fourier coefficient of the temperature.
At the boundary at z = 0,
w = Dw = 0 = 0 (5.1-10)
At z = 1, w = 0 (5.1-11)
BiD2w = a2MaD0 (5.1-12)
BiCr(D3w 3a2Dw) = a2(Bo + a2)(D0 + Bi0) (5.1-13)
When the density variation with temperature is negligible or in the
absence of gravity, then Ra = 0 and we have the pure Marangoni problem
with all the nonlinearities only in the boundary. Even with this effect
in the boundary the important result of bifurcation, namely the fluid


65
velocity, effects only the domain, the deflections in the boundary being
only a secondary effect of convection. On the other hand, when the
upper boundary is also kept at a fixed temperature we have, Ma = Bo = Cr
= 0 and so
at z = 1 w = Dw = 0 = 0 (5.1-14)
This then is the pure Rayleigh-Benard problem with all the nonlin
earities only in the domain and the resulting nonquiescent solution also
manifests itself in the domain as convection. The Rayleigh-Marangoni
problem described by eqns. (5.1-8) (5.1-13) is a mixed problem with
nonlinearities in the domain and the boundary but the convective solu
tion resulting from these nonlinearities shows up mainly in the domain.
5.2. The Augmented Morphological Problem
As can be seen from Section 5.1, in the Rayleigh-Marangoni problem
there is a Ra-Ma domain-boundary duality which does not seem to exist in
morphological instability. From the problem description in Chapter 3 it
is easy to see that all the nonlinearities for this problem lie only in
the liquid-solid interface. This is a limitation because by virtue of
being on the boundary the Sekerka number is unique and hence also has a
unique eigenfunction and is insufficient when solutions to inhomogeneous
versions of the linearized morphological instability problem are needed,
as in Chapter 6 where "imperfections" are considered. This difficulty
also crops up in the pure Marangoni problem, but the Rayleigh-Marangoni
problem comes to the rescue, as there are countably many corresponding


66
values of Ra for each value of Ma and hence also countably many
eigenfunctions forming a complete set (see Rosenblat, Homsy and Davis
(1981)). The naturally occurring duality of Ma and Ra enables solutions
to inhomogeneous problems to be obtained in a straightforward manner.
In morphological instability there is no such obvious, naturally
occurring boundary-domain duality and it is necessary to create one. To
avoid confusion we will refer to the pure morphological problem of
Chapter 3 as the Sekerka problem, and (by analogy with the Rayleigh-
Marangoni problem) set up an eigenvalue problem, with the eigenvalue in
the domain, which we will call the "augmented morphological problem."
liquid:
solid:
n?2qs
(5.2-1)
(5.2-2)
(5.2-3)
(5.2-4)
where M is the eigenvalue which we label as the morphological number.
The boundary conditions are
at z = -1 ps, = = 0
(5.2-5)
at z = s, p = 0
*s
(5.2-6)
at z = 0, p. p + t(G. G ) = 0
X S X/ s
(5.2-7)


67
l SeQt ttOj SeGl0) A 0
9r
3pj, 3p
sr-6 5T 0
kq q + t(kG G ) = 0
. s £c sc
dql 3qs
3T n 3T vq* + vqs =
(5.2-8)
(5.2-9)
(5.2-10)
(5.2-11)
with periodic conditions at the lateral boundary at r = R. We can now
separate variables expressing the horizontal dependence as zeroth order
Bessel's functions of the first kind JQ(air/R), where a^ are the zeros
of the first order Bessel's functions of the first kind .
00 CO
p (r,z) = Z Z P9,,(z) J (a.r/R)
* i=i j=i 41J 0 1
(5.2-12)
In addition if we take Fourier transforms in the horizontal direc
tion and solve for q^, qg and the equations reduce to a system
in p. and p ... If we define a column vector
.1J sij
Q,
ij
P3ij
(5.2-13)
and a matrix differential operator Li
L.
l
)
BY.(D2 a2)
(5.2-14)


68
then the domain equations reduce to
L.Q.. = M..Q..
l 1J ij lj
(5.2-15)
where Y. = (G + Se 6 a2A )/(G + Se G a2A )
IX, Cl S Cl
(5.2-16)
G = (G B + G )/(B + k)
c 2, c sc
(5.2-17)
(a.. v/2tanha..)tanha .s
2.1 2d. si
(na + v/2tanha .s)tanha.
si si 2,1
(5.2-18)
, 2 x 2.. .1/2
Hi (5.2-19)
, 2 2 2,1/2
a = (a. + v /4q )
si 1
(5.2-20)
The boundary conditions become
at z piij 0
(5.2-21)
at z = s, p .. = 0
sij
(5.2-22)
at z = 0,
BiQij 0
(5.2-23)
where
B.
1
1
D
-Y.
l
-BD
(5.2-24)


69
Defining an inner product
dz + / p
0
s
(5.2-25)
where the refers to the adjoint eigenfunction and the complex
conjugate.
It can easily be seen that the system described by eqns. (5.2-13) -
(5.2-24) is self-adjoint in this inner product and so the eigenfunctions
Qij are complete. Solving the system we get
(5.2-26)
A ..Sin/M..- a^ (s z)
sij ij i
where Mj_j are solutions of the equation
(5.2-27)
lj i 3 3 ij
(5.2-28)
lj S3 ij i
Here A... and A .. can be determined from the normalizing condition
ij sij
(5.2-29)
where 6.. is the Kronecker delta,
ij


70
5.3- Comparison of Morphological Instability with
Rayleigh-Marangoni Convection
The principal aim of this section is to relate the mathematical
characteristics of the two problems so that we may introduce some of the
extensive mathematical techniques used to study Rayleigh-Marangoni
convection to morphological instability, but we will make some physical
comparisons as well. The augmented morphological problem described in
Section k.2 is similar to Rayleigh-Marangoni convection. The augmented
problem is self adjoint but the Rayleigh-Marangoni problem is nonself
adjoint. Both have an infinity of eigenvalues and corresponding eigen
functions, but while completeness of the eigenspace is assured for the
former, special theorems are required to show this for the latter (cf.
Nadarajah and Narayanan (1987)). It should also be noted that while the
Rayleigh-Marangoni problem attempts to describe a realistic situation,
the augmented morphological problem was artificially created in order to
solve inhomogeneous versions of the Sekerka problem described in Sec
tions 3*1 and 3.2.
This brings us to the question whether there is a practical situa
tion which is described by this mathematical concoction. The difficulty
in coming up with one stems from another important difference between
the two problems. In the pure Rayleigh-Benard problem (where Ma, Bo and
Cr are all zero) the nonlinearity is in the domain and the instability
too manifests in the domain as convection. Even in the pure Marangoni
problem (where Ra is zero) where the nonlinearity is in the boundary,
the instability is still mainly in the domain. In contrast, in the
Sekerka problem the nonlinearities and the resulting instability show up
in the boundary. Though there are other boundary effects (like kinetic


71
undercooling) which can cause morphological instability the only domain
effect which could give M physical significance is a heat source term in
the form MT or Me-E/RT (see Joseph (1965)). We do not know of any
experiment where morphological instability was observed as a result of a
heat source in the melt or the crystal but if one does exist it will
provide the true analogy to Rayleigh-Benard convection.
This is relevant as Hurle (1985) has attempted a comparison between
the Rayleigh-Benard problem and the Sekerka problem. It can now be seen
that the Sekerka problem can only be compared to the pure Marangoni
problem, with Se corresponding to Ma and A corresponding to the reci
procal of Bo. Besides, for periodic lateral boundary conditions, the
eigenvalues of both problems, Se and Ma, are unique.
Based on this comparison we can make an important conjecture.
Vrentas, Narayanan and Agrawal (1981) have shown that for the Marangoni
and the Rayleigh-Marangoni problems when the nonperiodic no-slip condi
tion for velocity is imposed at the sidewalls, the eigenvalue Ma is no
longer unique and has countably many values. In other words when the
walls are a finite distance apart Ma has many values, but as they are
gradually moved apart we have "spectral crowding" and in the limit when
they are sufficiently far apart to impose the periodic boundary condi
tion of total slip, all the values of Ma coalesce into a unique num
ber. Recently, following Coriell et al. (1980), several workers have
looked at the coupled problem of morphological instability with solutal
convection and all have assumed periodic boundary conditions. We sus
pect that here too if the no-slip condition for velocity at the side
wall is imposed, the Sekerka number will no longer be unique. All this
raises the question of completeness of the Marangoni and the Sekerka


72
eigenspaces and its probable that generalized eigensolutions (see Nai-
mark (1967)) are needed when Ma and Se are chosen as eigenvalues.
When these two problems are considered in a finite container we can
see yet another difference. Both problems have simple eigenvalues
except at certain aspect ratios of the container where two horizontal
modes can coexist (cf. Rosenblat, Homsy and Davis (1982)). In a typical
experiment of Rayleigh-Benard convection we would expect to see a dozen
or so convection cells (see for example Koschmieder (1967)) and increas
ing or decreasing the number of cells by one can significantly effect
the problem. Hence the multiple points in this problem are extremely
important and have been the subject of study. But in morphological
instability a single alloy crystal can contain hundreds of individual
cells and the addition or loss of one has hardly a noticeable effect on
the problem and consequently multiplicity of the lateral eigenfunctions
loses its significance. In addition unlike the Rayleigh number it is
well known that near the critical wave number a^, the critical value of
the Sekerka number SeQ hardly changes (see for example Coriell, McFadden
and Sekerka (1985)) and the choice of a^ has very little effect on
Seotj. Conversely, the choice of the operating Se will have a tremendous
impact on the resulting wavenumber (cf. Ramprasad and Brown (1987)).
Other differences have been mentioned in Chapter 4. Both the
Marangoni problem and the Sekerka problem have symmetric bifurcation
diagrams near the bifurcation point for two-dimensional rolls and rec
tangular cells and nonsymmetric curves for hexagonal cells and cylindri
cal rolls. But in morphological instability the curves can be "forward
bending" or "backward bending" depending on the operating conditions,
whereas in Marangoni convection the curves are forward bending every-


73
where. Hence the occurrence of subcritical instability is more wide
spread in morphological instability.


CHAPTER 6
BIFURCATION BREAKING IMPERFECTIONS
6.1 Nature of Imperfections
When the morphological instability problem was formulated in Chap
ter 3, several effects were ignored and the resulting problem is an
idealized or "perfect" one. Inclusion of these can alter the problem in
several ways, for example kinetic undercooling of the melt becomes an
important effect in rapid solidification, but all it does is alter the
onset condition for morphological instability. In the parlance of
bifurcation theory, an "imperfection" is an effect on the "perfect"
problem which alters it in a specific way. Such an imperfection will
cause the morphological instability problem not to have a planar solu
tion at all even below the onset condition, and these are known as
bifurcation breaking imperfections.
The effect of a typical imperfection on the bifurcation diagram is
shown in Fig. 6-1 The broken line is the solution in the presence of
imperfection and it can be seen that the interface will be nonplanar for
all nonzero values of Se. Obtaining solutions to the problem with
imperfections is extremely difficult and we will only seek asymptotic
solutions. Hence the problems to be considered should have very small
imperfections. Under such conditions the method of matched asymptotic
expansions of Matkowsky and Reiss (1977) can be employed and here it
will be used in a way similar to the work of Tavantzis, Reiss and Mat
kowsky (1978) for the Rayleigh-Benard problem.


75
Figure 6-1. Imperfect bifurcation diagram showing inner
and outer expansions


76
The method-is fairly straightforward. The variables are expanded
asymptotically with the imperfection parameter about the planar and the
nonplanar solutions and two outer expansions 0Q and 0-, are obtained as
shown in Fig. 6-1. At the bifurcation point SeoN these expansions break
down and it is necessary to have inner expansions I1 and I2 near SeoN
and to join the corresponding 0q and 0^, matching conditions have to be
specified.
The imperfections that can be analyzed in this fashion must, of
course, be small effects else a full-blown nonlinear solution will be
needed. Two of the most important effects which are habitually ignored,
namely imperfect insulation of the ampoule wall and advection in the
melt, are such imperfections and readily lend themselves to this type of
analysis. In Chapter 3 when the morphological instability problem was
modelled by a uniform formulation, it was mentioned that one reason for
this was to consider a finite crystal/melt region. This finiteness was
only in the vertical direction and in the horizontal direction the
imposition of periodic boundary conditions effectively meant that the
ampoule side walls were infinitely far apart. The two imperfections
that are to be considered are caused by nonperiodic lateral boundary
conditions and another way of looking at the effect of these
imperfections is to say that the container is now being considered to be
finite in the lateral as well as the vertical direction.
6.2. Imperfection Due to Heat Loss
As mentioned in the last section, it has been customary in this
problem to assume periodic boundary conditions laterally, which is


77
equivalent to assuming that the walls of the ampoule are perfectly
insulated or that they are so far apart that their effect can be ig
nored. In practice neither of these is likely to be achieved and here
we will examine the effect of a small amount of heat loss or heat gain
from the wall on the problem. If we take the ampoule to be cylindrical
with radius R,
3T
r = R, liquid: -k = 5f^(z) (6.2-1)
3T
solid: -k t = 5f (z) (6.2-2)
3 3r s
where f. and f are such that f.(5) = f (5) and f.(-1) = f (s) = 0.
X* ^ X/ S X/ s
If f^ and fg are positive it will mean heat loss and if they are nega
tive, heat gain. If we make the transformations
Vz)
T^(z,r) = 5 r + 0^(z,r) (6.2-3)
f (z)
and T (z,r) = 5 r + 0 (z,r) (6.2-4)
S K S
and substitute these in the steady-state versions of eqns. (3.1-15) -
(3.1-24), the temperature equations in the domain become
= 6rD2f^/k^ (6.2-5)
V20 = 5rD2f /k
s s s
(6.2-6)


78
The outer boundary conditions will remain unchanged but at the interface
eqn. (3.1-21) becomes
l % 5r(Vkt W <6-2-7)
8, SeCj A H - Srft/kt (6.2-8)
and eqn. (3.1-22) converts to
ve -n 8V0 -n = Lv 5r(Df Df ) (6.2-9)
X. S X, s
In order to solve this system we will be treating the heat loss as
an imperfection on the perfect problem (i.e. when 6=0). The perfect
problem is of course the Sekerka problem of Section 3.1.
6.3. The Outer Expansions
As this problem has been defined in a finite geometry, the number
of cells are fixed by the container size and the growth conditions. We
will choose these so that aN is very close to a_n the wavelength corre
sponding to Seom^n the least value of SeQ^. Another important decision
is the selection of the wave pattern and our analysis is done for cylin
drical rolls. An objection to this could be raised on the grounds that
in most experiments it is the hexagonal pattern which is observed. We
justify our assumption by once again making a comparison with Rayleigh-
Benard convection. It appears that in Rayleigh-Benard convection the
wave pattern selection is strongly influenced by the container size and


79
shape, with the hexagonal pattern prevailing for all shapes in wide
containers while in narrower ones the container shape determining the
pattern (e.g., cylindrical rolls for circular containers. See Kosch-
mieder (1967)). Since the principal aim of this paper is to study a
finite geometry effect on morphological instability, the cylindrical
roll pattern would be the logical choice. This is especially valid for
experiments such as those of Peteves (1986) where ampoules of radius
0.025 inches were used.
In Chapter 4 it was shown that the bifurcation diagram is unsymme-
tric for cylindrical rolls and the form of the outer expansions 0Q and
0^ are shown schematically in Fig. 6-1. If we use superscript o to
identify the problem with perfect insulation we can seek solutions by
means of asymptotic expansions about the perfect problem.
00
e0 = z ek 6k (6.3-1)
k=0
with similar expansions for 0 C., C and £. Substituting these expan-
S X/ s
sions into the problem and collecting the terms of order 6 we get an
inhomogeneous linear system. If we separate variables horizontally
CO
e! = Z 0¡ J (a.r/R) (6.3-2)
% X.. O 1
1 = 1 1
and eliminate cl C1 and we get for the expansion about the planar
X/ s
solution
L *1 = f1
i i i
(6.3-3)


80
where
1 =
x
.
i
f! = I.D2
l l
Vk*
f /k
s s
(6.3-4)
(6.3-5)
I.
i
r J (a.r/R)dr/
o x
rJ (a.r/R)dr
0 1
(6.3-6)
The boundary conditions are
at z = -1, 0^ = 0
at z = s, 01 = 0
si
at z = 0, B. 1 = h
li i
where
(6.3-7)
(6.3-8)
(6.3-9)
h. - I.
i i
Vk*
Df,
-Y.f /k
is s
Df
(6.3-10)
The eigenvalue problem of Section 5.2 has been shown to have a
complete set of eigenfunctions and hence can be used to solve the above
system.


81
00 CO
1 = E 1 Q. J (a.r/R)
i-1 j=1 1J 1 1J 0 1
(6.3-11)
J(W
M. .
1J
z=0

U i
M. .
1J
(6.3-12)
where J(Q..,h. )| is the bilinear concomitant evaluated at z=0
ij i z=0
J(VV
z=0
- V01/ Dfs>vki Dfu(Vh hWh
(6.3-13)
If we assume that and are nonzero, then when Se = SeQN, Mj^
becomes zero and the outer expansion (6.3-11) (6.3-13) will fail.
When we expand about the nonplanar solution we end up with a much
more complicated system. But since we have expressed the solution to
the perfect nonplanar problem itself in terms of a perturbation series
in e, we can do the same for the imperfect problem.
i1 = 4>1 + £*] + ... (6.3-14)
o
If we take the zeroth order problem and once again separate vari
ables laterally,
*1 = Z *1. J (a.r/R) (6.3-15)
o .=1 o o 1
and eliminate
eqns. (6.3~3)
and we get the same system described by
Hoi soi oi
- (6.3-10) but with as the variable. Again using the
01
eigenfunctions of Section 5.2, the solution is


82
*1 = E E Q.. J (a.r/R)
i = i j=i 01 ij o l
(6.3-16)
with taking the same form as eqns. (6.3-12) and (6.3-13).
Since these equations are valid only for Se close to SeoN, we can write
an expansion for
dM
N1
M (Se) = M (Se ) + (Se Se ..)
N1 N1 oN oN dSe
. (6.3-17)
Se = Se
oN
which simplifies to
VSe)
0 +
gS1NMN1 + 0(G )
(6.3-18)
where
dM...
_ N1
MN1 dSe
Se = Se
oN
(6.3-19)
and Se1N has been derived in Section 4.2. Hence the outer expansion
about the nonplanar solution is
* = + _ J (a r/R) + 0(e ) +
c oN o N
( J) o o
6[ _-0 QM1 Jm(aMr/R) + 0(e )] + 0(6 )
£S6inm'ni
N1 o N
(6.3-20)


83
6.4. The Inner Expansions
The outer expansions described in the previous section fail at Se =
SeoN> requiring inner expansions in this vicinity. For Se very near
CO
Se (p) = Se
(6.4-1)
oN
k=2
<5(p) = pC
(6.4-2)
where b and c are integers to be determined such that the solution does
not fail at Se = SeQ^. We can now expand the solutions in orders of p.
CO
u(r,z,p) = e(r,z,Se,<5) = E uk(r,z)pk/k!
k=0
(6.4-3)
CO
v(r,z,p) = C(r,z,Se,) = E vk(r,z)pk/k!
k=0
(6.4-4)
CO
w(r,z,p) = c(r,Se,) = Z wk(r)pk/k!
k=0
(6.4-5)
Here the superscripts on u, v and w are not powers but indices.
Substituting these expansions into the problem set up in Section 6.2 and
collecting the terms of order p, we once again get an inhomogeneous
problem. Separating variables and eliminating v., v and w, the problem
36 S
reduces to
(6.4-6)


84
with boundary conditions
at z = -1 ,
uii 0
at z = s
u = 0
si
at z = 0,
B.*! = 5'h1
iTi i
where
and
d<5
5 '
Se' =
U=0
dSe
du
p=0
The variables and have been defined in Section 5
and h^ in Section 6.3. But Y^ now becomes
Y = (G, + Se G af A)/(Gq + Se G af A )
1 X, ON Cl s oNci
This means that the operator is singular when i=N,
for solutions to exist 6has to orthogonal to the null
Using this solvability condition with QN1 we get
z=0
- (6.4-7)
(6.4-8)
(6.4-9)
(6.4-10)
(6.4-11)
.2 and f1
i
(6.4-12)
j=1 and hence
space of Ln.
(6.4-13)


85
From this we can conclude that <5' = 0 (assuming J <0,,, .fl>) and
NI N
that c>1. This makes the system (6.4-6) (6.4-12) homogeneous and
so are constant multiples of .
o
* = A$
(6.4-14)
Proceeding to the next order we have
2 1
L.*r = s"f
ii i
(6.4-15)
and at z=0,
2 2
B.<|i7 = h
ii i
(6.4-16)
h2 = 6"h] 2ASe'
i i
c G /(G. + Se G -a.A )
oi cl oN c i
+ 2A I. Q
i1
(6.4-17)
where
? 3 ? 2
... = I rJ')(a. r/R)dr/ J rJ (a.r/R)dr
i 1 J o i ^oi
(6.4-18)
and Q is a complex function of the linear stability variables. The
solvability condition for i=N, j=1 now gives
J (Q
N1
(6.4-19)
Hence we can take 6" 0 and Se' 0 and so b=1 and c=2. Equation
(6.4-19) is a quadratic in A with two real roots A1 and A2 of opposite
signs, corresponding to two inner expansions I-] and I2.


86
Se = SeoN + £1/2 + 0(6)
(6.4-20)
i> = * + A^*o61/2 + 0(6)
(6.4-21)
In terms of y, the outer expansion 0Q about the planar state of
eqn. (6.3-11) can now be expressed by
in ( j)qni J0(V/R)/5H1 <6-',-22)
and the nonplanar expansion 01 of eqn. (6.3~20)
*il [0N1.fi> V/SeJQN, JC(V/R) <6-,,-23)
where we expressed as J (a^r/R). attempting to match these
with the inner expansions of eqn. (6.4-21) we get
lim A1 = lim A2 -( J)/5M' (6.4-24)
£- -00 £-00
lim A, = lim A- = DMC/Se... (6.4-25)
_ 1 2 N IN
Hence using
A1
in
eqn.
(6.4-21)
will
give the
inner expansion I1
for
matching 0Q
for
Se
< SeoN
with 0.|
for
Se > SeoN.
A2 will result in
I2,
matching 0Q
for
Se
> SeoN
with 0i
for
Se < SeofJ
as shown in Fig. 6-
1 .


87
6.5. Imperfection Due to Advection in the Melt
In modelling morphological instability during solidification it is
customary to neglect the difference in density between liquid and
solid. Though this difference is very small (e.g. for the Pb-Sn system,
the ratio of densities Pg/P^ is 1.035) the volume contraction upon
solidification results in closed streamlined flow in the melt which
fundamentally alters the planar state of this problem. Obtaining the
solution to the planar state with this flow is difficult enough even
without attempting the formidable task of solving the nonplanar prob
lem. In order to include the effect of the flow on morphological insta
bility previous workers have therefore relied on simplifying assumptions
or looked at limiting cases.
The scenario in a typical solidification experiment is schemati
cally shown in Fig. 6-2. The heating coils maintain constant tempera
tures at positions z = -l and z = s in the ampoule. As solidification
proceeds, the ampoule .is pulled in the positive z-direction at a velo
city v in order that the liquid-solid interface will remain stationary
at z = 0. To fill the space created by volume contraction upon solidi
fication, the melt will move towards the crystal with a bulk velocity
of v(p /p. 1) and this process is commonly referred to as "advec-
S 3C
tion." In the presence of the rigid ampoule walls, this flow also
resembles the bolus flow of slugs through a pipe, resulting in closed
streamlined flow in the melt.
In the literature we find three approaches to tackling this prob
lem, the least of which is the work of Caroli, Caroli, Misbah and Roulet
(1985b) who ignored the rigid side walls. Then the only effect of


88
Figure 6-2. Crystal growth with advection in the melt


89
advection is a negligible modification of the growth velocity in the
melt. The other two approaches are more substantial and analize
limiting cases of this phenomenon. Since the traditional formulation of
morphological instability concentrates only at a "boundary layer" region
of the liquid-solid interface, in such a model the closed streamlined
flow shown in Fig. 6-2 can be approximated by Couette flow in Region I
and stagnation flow in Region II. Delves (1968, 1971 and 197*0 and
Coriell, McFadden, Boisvert and Sekerka (1984) looked at the effect of a
forced plane Couette flow and showed that the onset of morphological
instability is somewhat suppressed for disturbances in the flow
direction, while disturbances perpendicular to the flow were
uneffected. Recently McFadden, Coriell and Alexander (1988) examined
the effect of a plane stagnation flow on disturbances perpendicular to
the flow and here too the flow was found to be stabilizing.
Based on these two limiting cases it might be tempting to conclude
that advection generally stabilizes the liquid-solid interface. In this
and the next few sections we will embark upon a more complete analysis
of advection than has been attempted so afar and show that the above
assertion is questionable at best. In our model we consider morphologi
cal instability with advection in the absence of natural convection. We
choose not to include natural convection as we wish to study the effect
of advection only and natural convection will only further complicate an
already nontrivial problem. Besides it has been shown very conclusively
by Coriell, Cordes, Boettinger and Sekerka (1980) and by Caroli, Caroli,
Misbah and Roulet (1985a) that the convective and morphological modes
are decoupled except at points where they become comparable. Hence our
model will be valid for the high growth velocity region where morpholo-


90
gical instability is dominant and also in low gravity environments where
natural convection can be neglected.
The experimental set up was briefly described earlier. The steady
state domain equations in dimensionless form are
V T* 0
(6.5-1)
2
V C* V 3T U'VC*
(6.5-2)
V T =0
s
(6.5-3)
2 3Cs
nVC v ~ 0
s 9 z
(6.5-4)
Here u is the velocity of the closed streamlined flow in the melt
caused by advection and 5 is (p /p. 1). The boundary conditions are
S X
at z -1, Tji T1 Cj, = C1
(6.5-5)
at z = s, Ts = T2
(6.5-6)
at z = T^ = Ts = SeC^ + A H
(6.5-7)
VT*n fJVT *n = Lv
x. s
(6.5-8)
kC. = C
l s
(6.5-9)
VC *n (1+6)vC. = nVC *n vC
X/ X/ s s
(6.5-10)


91
Before we can write the equations for u, further assumptions are
necessary. The domain of the velocity equations extend over the entire
melt region, not just the depth l that we have considered. In the float
zone technique this melt region will still have a constant depth as
solidification proceeds but in Bridgman growth the melt region will
continuously shrink. Since crystal growth velocities are usually so
small, even in the latter technique it takes awhile before there is an
appreciable change in the melt depth. Hence over a short time span a
constant depth assumption will be valid even for Bridgman growth. We
will also assume that the melt is bounded on all sides by rigid walls.
This will require a container for zone refining and a very viscous
encapsulant for the melt in Bridgman growth.
Under these assumptions the velocity problem has been solved by
Duda and Vrentas (1971) for low velocities and here we will only give
the solution and refer the interested reader to their paper for the
details.
u = (5vR/r)3ip/9z, u = 5v(1 (R/r)3^/9r) (6.5-11)
r* z
CO CO
ill = I A F (r)Sinnir( 1 + z/Y) + £ G (z)rJ,(b r/R) (6.5-12)
, n n n 1 n
n=1 n=1
F (r) = [rl. (mrr/Y)I0(nir/h) r^I. (mr/h)I0(nirr/Y)/R]/I^(mr/h) (6.5-13)
n 1 2 1 2-2
G (z) = 2B [sinhb (z+Y)/R (z/Y+1)exp(-b z/R)Sinhb h]
n nL n y n n J
+ 2C (z+Y)/R expb h Sinhb z/R
n *n n
(6.5-14)


92
2 2
8n J0(bnH1'V'exptinh (2expbnh exp(-bnh) exp3b h)/h] (6.5-15)
Bngn -Hbhexpbnh I AmQmn 2(1 exp2bnb) I AQ (-,)" (6.5-16)
m=1 m=1
hg C = 4(b hexpb h Sinhb h) Z A Q
n n n *n n m mn
m=1
(4b h+2 2exp2b h) Z A Q (-1 )
n n m mn
m=1
m
(6.5-17)
bn = \ nir^ I0^nir/h)I2^nir/l1^ I^(nir/h) ]/I2(mr/h)
(6.5-18)
mn
2bn(rmr/h)2I2(rmr/h) J^ib^)
I2(mir/h)[ (imr/h)2 + b 2]2
2 n
(6.5-19)
y = l /l, h = Y/R
m
(6.5-20)
where is the total melt depth, R the dimensionless radius of the
ampoule, ur and uz the radial and axial components of the velocity, Jp
the Bessel function of the first kind of order p and ID the modified
Bessel function of the first kind of order p. Duda and Vrentas have
determined the first twenty coefficients of An for various values of h
and we will use these in our calculations.


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BIFURCATION AND MORPHOLOGICAL INSTABILITY
By
ARUNAN NADARAJAH
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
r. OF FLORIDA LIBRARIES

ACKNOWLEDGEMENTS
In the course of the three and a half years of effort that went
into this dissertation, I received help from numerous people in various
ways that contributed to its completion. Enumerating them all would
make this note be at odds with the spirit of conciseness of the rest of
the document and so, reluctantly, I resolved to include only those who
had made a direct contribution. Needless to say, I still remember with
gratitude my debt to the rest.
First mention should be made to my erstwhile mentor, Professor M.S.
Ananth at the Indian Institute of Technology, Madras, who first kindled
an enduring interest in theoretical transport phenomena and encouraged
my proclivities toward graduate study. The greatest help came from my
advisor Dr. R. Narayanan, who apart from getting me involved in morpho¬
logical instability and giving many suggestions also imparted to me a
solid background in mathematics and hydrodynamic stability theory, not
to mention helping me in numerous other ways. Professor L.E. Johns,
Jr., first introduced me to linear operator theory and gave suggestions
too regarding my research, but more importantly, he was my "chemical
engineering conscience," broadening my vision when I tended to
specialize too much and keeping the objectives in perspective when I got
wrapped up in abstract theoretical points. It is not an exaggeration to
say that I probably would not have progressed this far academically
without these three individuals.

I would also like to thank Drs. S.R. Coriell and G.B. McFadden of
the National Bureau of Standards for many discussions and suggestions
regarding the subcritical nature of morphological instability; the
members of my supervisory committee: Professor U.H. Kurzweg, Dr. G.K.
Lyberatos, Dr. W.E. Lear, Jr., and Dr. S.A. Svoronos for their time and
effort on my behalf; the Department of Chemical Engineering for provid¬
ing a research assistantship during the first two years of my Ph.D. work
and the Department of Mathematics for a lectureship during the last one
and a half. My thanks also go to my uncle Dr. R.S. Perinbanayagam for
being a role model and helping me evolve a "meaningful philosophy of
life" and cope with the stress of graduate school.
Finally, I wish to express my gratitude to my colleague S. Pushpa-
vanam for many "enlightening" discussions and to Debbie Hitt for doing a
superb job of typing the manuscript and "correcting" my Queen's English
spelling!

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
ABSTRACT vi
CHAPTERS
1 DESCRIPTION OF THE PROBLEM 1
2 PREVIOUS WORK ON MORPHOLOGICAL INSTABILITY 8
2.1 Early Work 8
2.2 Later Research 10
2.3 Inclusion of Other Effects 12
2.4 Experiments in Morphological Instability .... 14
2.5 Limitations of Existing Models and
Unaddressed Issues 15
3 A UNIFORM FORMULATION 17
3.1 The Formulation 17
3.2 The Linear Stability Problem 23
3.3 The Adjoint Problem and Exchange of
Stabilities 30
3.4 Finite Containers and the Most Dangerous
Wavenumber 35
4 SUBCRITICAL BIFURCATION 38
4.1 Theory 38
4.2 The Second Order Problem 41
4.3 The Third Order Problem 46
4.4 Calculations and Comparisons 49
5 COMPARISONS WITH RAYLEIGH-MARANGONI CONVECTION .... 61
5.1 Rayleigh-Marangoni Convection in Brief 61
5.^- The Augmented Morphological Problem 65
5.3Comparison of Morphological Instability
with Rayleigh-Marangoni Convection 70
6 BIFURCATION BREAKING IMPERFECTIONS 74
6.1 Nature of Imperfections 74
6.2 Imperfection Due to Heat Loss 76

6.3 The Outer Expansions 78
6.4 The Inner Expansions 83
6.5 Imperfection Due to Advection in the Melt ... 87
6.6 Nonexistence of the Planar State 93
6.7 Asymptotic Solution 94
6.8 Controlling Imperfections 99
7 NEW DIRECTIONS 102
7.1 Transition to Dendritic Growth 102
7.2 Extension to Semiconductor Materials 103
7.3 Inclusion of Microscopic Effects 104
7.4 Numerical Methods 104
7.5 Experiments 105
APPENDICES
A NOMENCLATURE 107
B PHYSICAL PROPERTIES 113
REFERENCES 115
BIOGRAPHICAL SKETCH 120

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
BIFURCATION AND MORPHOLOGICAL INSTABILITY
By
Arunan Nadarajah
August, 1988
Chairman: Dr. Ranganathan Narayanan
Major Department: Chemical Engineering
Morphological instability refers to the tendency towards spatial
pattern formation on the liquid-solid interface when a dilute binary
mixture is solidified or fused. The importance of this phenomenon is in
the growth of metal alloy and semiconductor crystals from their melts,
where it influences the solute or dopant concentration resulting in
nonuniform physical and electrical properties.
Previous formulations of morphological instability have involved
several simplifying assumptions which restricted it to the study of a
region immediately surrounding the interface. The models have limited
validity and they require separate treatments for different situations
like freezing and melting. In this study a new uniform approach is
presented which considers the entire melt and crystal domain and is
applicable to all situations. Earlier formulations are shown to be
approximations of this and exchange of stabilities is proven
asymptotically.
vi

The model is then used with a weakly nonlinear technique to predict
the shape of the bifurcation diagram for various cell patterns. The
subcritical nature of morphological instability is shown and regions of
its prevalence are determined over the entire domain of experimental
parameters. This was used to compare with experimental results and to
determine optimal crystal growth regions.
A comprehensive comparison of morphological instability with con¬
vective instability was undertaken and this phenomenon was shown to
resemble Marangoni convection in its mathematical and physical fea¬
tures. This was done in order to introduce some of the multitudinous
mathematical techniques employed in convective instabilities into mor¬
phological instability and specifically was used here to complete the
eigenspace of the linearized problem.
Two imperfections which reside in the domain, heat loss at the
container wall and advection in the melt, were considered and shown to
be bifurcation breaking imperfections. Solutions to the problem were
obtained in both cases by matched asymptotic expansions and based on
these results a practical way of minimizing the effect of these imper¬
fections was suggested.
vii

CHAPTER 1
DESCRIPTION OF THE PROBLEM
Morphological instability refers to the process of spatial pattern
formation at the liquid-solid interface when a binary mixture is
solidified or fused. This is a problem of hydrodynamic instability and
like all other problems of this nature, for this phenomenon too there is
an onset point where the initially planar interface first begins to
deform and forms cellular patterns. These grow into deeper finger-like
shapes and eventually forming side branches and tree-like dendritic
structures.
The importance of this phenomenon is in the growth of metal alloy
and semiconductor crystals from their melts. Morphological instability
affects not just crystal shape but also the solute or dopant
concentration, resulting in nonuniform physical and electrical
properties in the crystal. This is especially perfidious in
applications where superfine crystals with very consistent properties
are required. Recently there have been some indications that crystal
quality can be significantly improved by growing them in low gravity as
this reduces other problems associated with crystal growth like natural
convection, but unfortunately not morphological instability. Growing
the crystal at very high temperature gradients or at very low growth
rates avoids morphological instability but crystals grown at high
temperature gradients are of poor quality due to thermal stresses and
the very low growth rates makes the process very expensive. Hence this
1

becomes a problem not only of avoiding crystal surface deformations but
one of optimization of the process as well.
There are several methods for growing crystals from the melt,
distinguished by the hydrodynamics of growth. The three basic ones are
Bridgman, Czochralski and float zone and most techniques are variations
of these, like horizontal and vertical Bridgman. A typical Bridgman
experiment is shown in Fig. 1-1. The material is usually in a quartz
ampoule and is melted and then recrystallized in the Bridgman furnace.
The upper part of the ampoule is maintained at a higher temperature than
the lower and solidification proceeds upwards and the ampoule is pulled
downwards at the same velocity v. The top of the melt is protected by a
liquid encapsulant like 620^. At the end of the process the ampoule is
broken to retrieve the crystal.
In the float zone technique shown in Fig. 1-2, the ampoule or the
material itself is pulled through a circular furnace and the melting and
recrystallization proceeds simultaneously. As this technique can be
done even without a container it avoids the problem of impurities from
the ampoule entering the crystal, but its more difficult to maintain a
uniform temperature gradient. Figure 1-3 shows the Czochralski method
where the crystal is rotated and pulled from a melt pool. As the
emphasis here is on the liquid-solid interface, the modelling of the
crystal growth process will be kept as general as possible but will
resemble the Bridgman technique the most.
The temperature and concentration profiles during typical crystal
growth conditions are shown in Fig. 1-4. The temperature profiles in
the liquid and solid are virtually straight lines and solute
concentration in the solid is virtually constant. But the solute

3
Figure 1-2. Float zone
Figure 1-3. Czochralski

4
concentration in the liquid changes sharply near the interface
because of solute rejection on solidification. This in turn has an
effect on the freezing point depression in the liquid as shown in the
figure. Figure 1-5 shows the same profiles but in a situation where the
freezing point in the liquid TM now exceeds the actual liquid
temperature (this change can be brought about by either reducing the
liquid temperature gradient or by making the change in near the
interface even sharper by increasing the growth velocity). This is
referred to as constitutional supercooling and the system responds to
this unstable situation by interface deformation. Countering this is
the interfacial tension which always acts to minimize the surface area
which in this case is the planar surface. When this balance is upset,
or in other words when the onset conditions are exceeded, the interface
loses planarity and forms cellular patterns. Figure 1-6 shows the
profiles for the fusion case, where we could have TM in the solid being
less than the actual solid temperature and once again interfacial
deformation is the system's response, balanced by interfacial tension.
The only difference here is that since solid diffusivities tend to be
several orders of magnitude lower than liquid diffusivities, the solute
concentration profile in the solid near the interface will vary even
more sharply resulting in lower onset conditions for morphological
instability.
In the paper of Trivedi and Somboonsuk (1984) there is a series of
photographs from an experiment (see Fig. 1 in their paper) where
succinonitrile/acetone crystals were grown. The first photograph shows
the liquid-solid interface just after onset and has a discernible
cellular pattern. Later ones show the cells becoming deeper, forming

5
Figure 1-4. Concentration and temperature profiles during
solidification
Figure 1-5. Concentration and temperature profiles during
solidification
Figure 1-6. Concentration and temperature profiles during fusion

6
fingers and ending up as dendrites. The arrows in the first photograph
mark the initial perturbations that eventually become dendrites. In
this thesis we will concentrate only on the region near the onset point,
shown by the first two photographs, though dendrites will be mentioned
in discussions.
In the papers by Morris and Winegard (1969) and Tiller and Rutter
(1956) we see another aspect of morphological instability, the variety
of cellular patterns, finger-like shapes, hexagonal cells and variations
of these. Other possible shapes are cylindrical rolls and rectangular
cells but by far the commonly encountered pattern is the hexagonal
one. The choice of the cell pattern is extremely important and factors
determining this choice will be discussed later. Their figures also
show that, unlike other forms of hydrodynamic instability, the number of
cells on a single crystal is in the hundreds.
Though it is customary to model morphological instability in terms
of the temperature and concentration profiles, in reality this
phenomenon seldom exists in isolation; it is usually coupled with fluid
flow in the melt. There are two kinds of flows that occur. The first
is buoyancy driven solutal convection which is caused by the sharp
solute concentration gradients in the melt. The other is the result of
density change during solidification. When solidification occurs there
is a constant rate of volume decrease which causes the melt to move in
to fill the vacated space. This motion is referred to as advection and
the rigid side walls of the ampoule will cause closed streamlined flow
in the melt as a result (see Fig. 6-2). In addition there will be flows
in the melt in Czochralski growth due to rotation and other kinds of
flows in special growth techniques.

7
Apart from these, several other parameters affect this phenomenon,
the most important of which are due to the fact that most crystals are
faceted; that is, they have a crystal lattice structure. Hence whether
the lattice axis is aligned or not with the growth direction is
extremely important as can be seen from the experiments of Heslot and
Libchaber (1985). Other important considerations are grain boundaries,
wetting of the ampoule wall and the presence of impurities. Also in
rapid solidification, the system will not be at thermodynamic
equilibrium and kinetic undercooling of the melt becomes significant.

CHAPTER 2
PREVIOUS WORK ON MORPHOLOGICAL INSTABILITY
2.1. Early Work
The first successful attempt at explaining morphological
instability qualitatively was by Rutter and Chalmers (1953). They
coined the word "constitutional supercooling" to describe the existence
of unstable melt regions near the interface where the freezing
temperature can be higher than the liquid temperature itself and
correctly identified this as the cause of interface deformation.
Tiller, Rutter, Jackson and Chalmers (1953) quantified
constitutional supercooling and for instability came up with the
condition
mGHo>-Gt
where m is the absolute value of the liquidus slope, G^ the liquid
temperature gradient at the interface and G^c the solute concentration
gradient in the liquid at the interface. The negative sign is caused
by G^ and G^ being in opposite directions (see Fig. 1-5). As can be
seen these simple thermodynamic explanations do not take into account
the stabilizing effect of interfacial tension. To do so would require
casting the problem as one of hydrodynamic instability and obtaining the
onset conditions from a linear stability analysis. This is exactly what
8

9
Mullins and Sekerka (1963, 1964) did when they considered the problem
with temperature and concentration equations in the liquid and solid and
boundary conditions at the interface. Their criterion for instability
to an infinitesimal disturbance was
mG > - G„ + a2TA/L, (2.1-2)
c T M h
where GT is the weighted temperature gradient
Gt = (k^ + kgGgVik^ + kg) (2.1-3)
Here ks and k^ are the solid and liquid thermal conductivities. Gc is a
modified liquid concentration gradient given by
Gc = GJlc(ajl - v^)/^ - (1/2 - kJv/D^) (2.1-4)
2 2 21/2
a£ = (a¿ + vV4D p (2.1-5)
where a is the wavenumber of the disturbance, k the solute distribution
coefficient, TM the melting temperature of the pure solvent, Lh the
latent heat of fusion and A the interfacial tension. This analysis laid
the foundation for all further work in morphological instability.
Following them Woodruff (1968) did the linear stability analysis along
the same lines for the melting problem and came up with the same
criterion as (2.1-2) but with
G = G (na - v/2D„)/(nka + a + (1 - k)v/2D„) (2.1-6)
C SC S £ S X, X,

10
, 2 A 2 ,2.1 /2
a = (a + v /4D )
s s
(2.1-7)
where Gsc is the solid concentration gradient at the interface and n is
Ds/Dr the ratio of solute diffusivities.
2.2. Later Research
The next major contribution to the problem was made by Wollkind and
Segel (1970) who proved "exchange of stabilities" for this problem for
most parameter ranges. Proving exchange of stabilities is equivalent to
showing the existence of the onset of steady state nonplanar
solutions. They also considered the weakly nonlinear regime after the
onset of instability and using the method of Stuart (I960) and Watson
(I960) analyzed the problem for the case of two-dimensional rolls,
showing the existence of subcritical bifurcation at most growth
velocities.
The method of Stuart and Watson is essentially the theory of Landau
(see Drazin and Reid (1981)) and follows the dominant mode of
instability into the weakly nonlinear regime. In this form it is
applicable only to disturbances of one cellular pattern at a time, but
Segel and Stuart (1962) extended this theory to the prediction of the
preferred pattern by considering the interaction of two specified modes
of disturbance. Depending on the way these two modes were combined it
was possible to obtain two dimensional rolls or hexagonal cells and they
showed that the experimental parameters would determine the stability of
these patterns. Sriranganathan, Wollkind and Oulton (1983) adopted this

11
method for morphological instability and gave parameter ranges where
each type of cell was stable. The limitation of the method is that it
considers hexagonal and two dimensional roll patterns but not
rectangular cells or cylindrical rolls and ignores the effect of
container shape and size which have been shown to be important in wave
pattern selection (see Koschmeider (1967)).
Ungar and Brown (1984a) considered the highly nonlinear problem and
after making several simplifications obtained solutions using the finite
element method. Finite elements can handle highly nonlinear problems
and give very accurate numerical solutions but are extremely time
consuming. Solving the full morphological problem is a very expensive
proposition by this method and hence Ungar and Brown simplified the
problem by ignoring the latent heat and solid diffusivity and assuming
that thermal conductivities in liquid and solid were equal. This
allowed them to reduce the problem to a "one-sided model" consisting of
variables in the liquid region only, considerably simplifying the
algebra and saving computer time. Such a model will have only limited
validity in highly nonlinear regions and this was borne out when Ungar,
Bennett and Brown (1985) solved the complete problem. But their most
extensive calculations were done only for the one sided model and hence
this is of chief interest. These were done only for the case of two
dimensional roll disturbances and here they showed that contrary to that
reported by Wollkind and Segel, there were multiple regions of
supercritical and subcritical bifurcation. More importantly they showed
that at large deformations of the interface secondary bifurcations
occurred. Ungar and Brown (1985) also modelled the formation of deep
cells in an attempt to follow the transition to dendritic growth.

12
Nonlinear finite difference calculations were done in a more
limited way by McFadden and Coriell (198*0 for the two dimensional
case. Later McFadden, Boisvert and Sekerka (1987) extended the
calculations for the three dimensional patterns of hexagons and cross
rolls. In both cases the enormous expenses involved restricted
calculations to a few parameter values.
2.3. Inclusion of Other Effects
While these workers were investigating the basic problem others
were busy trying to incorporate various influences. The most important
concern was the effect of fluid flow. Delves (1968, 1971 and 1974) in
attempting to approximate the influence of advection and stirring in the
melt, studied the influence of plane Couette flow on the problem. He
showed that two dimensional roll disturbances in the flow direction were
stabilized but there was no effect on disturbances perpendicular to the
flow. Coriell, McFadden, Boisvert and Sekerka (1984) modelled Couette
flow more systematically and came to the same conclusion. Recently
McFadden, Coriell and Alexander (1988) examined the effect of plane
stagnation flow on two dimensional disturbances perpendicular to the
flow and here too the flow as found to be stabilizing.
In another very important development Coriell, Cordes, Boettinger
and Sekerka (1980) studied morphological instability with solutal
convection. They showed that the two instabilities were essentially
decoupled with the melt being unstable to convective disturbances of
long wavelengths and the interface unstable to nonplanar disturbances of
small wavelengths. Also at low growth rates the dominant instability

13
was convective and the interface was not easily disturbed. At high
growth rates the roles were reversed and at an intermediate velocity the
two instabilities became comparable. It was only at this rate the two
instabilities interacted and the result was the prevalence of
oscillatory instabilities. Their conclusion was that, except at this
particular growth rate, it is usually sufficient to study only the
dominant instability near its onset.
Following Coriell et al. several workers have looked at special
aspects of these two instabilities and their work has been reviewed by
Glicksman, Coriell and McFadden (1986). They all confirmed or refined
the work of Coriell et al. but all the main conclusions mentioned above
still hold.
Several other influences apart from fluid flow have been
incorporated into the model but only a few relevant ones will be
considered here. Coriell and Sekerka (1972, 1973) tried to include the
effect of grain boundaries on morphological instability by assuming that
its only effect was to shift the onset conditions. They failed to
observe that in the presence of grain boundaries there could be no
planar solutions to the problem and that the interface will be nonplanar
at all times. Ungar and Brown (1984b) obtained the solutions to this
problem by matched asymptotic expansions for small grain angles and
using finite elements for solutions of large grain angles.
In rapid solidification kinetic undercooling of the melt is
significant and Seidensticker (1967) included this and showed that it
caused a shift in the onset conditions. The significance of this was
shown by Hardy and Coriell (1968, 1969 and 1970) when they observed
morphological instability in the growth of ice crystals. Constitutional

14
supercooling was not a factor here and it was shown that kinetic
undercooling was the primary cause. This dual cause for morphological
instability is somewhat analogous to the situation in natural convection
where we find that the variations of density and surface tension with
temperature can both cause convective instability.
2.4. Experiments in Morphological Instability
The early work on modelling morphological instability was prompted
by experimental observations but beyond that very few quantitative
experiments have been done near the onset conditions. This is an
unfortunate state of affairs and experimental verifications of
theoretical predictions are badly needed if further concrete progress on
the theoretical front is to be made. The work of Morris and Winegard
(1969), Trivedi and Somboonsuk (1984) and of Heslot and Libchaber (1985)
have already been mentioned. Recently de Cheveigne, Guthman and Lebrun
(1985, 1986) have attempted to verify the weakly nonlinear and strongly
nonlinear theoretical predictions and one hopes that more work along
these lines will follow.
De Cheveigne et al. performed their experiments on
succinonitrile/acetone and CBr¡j/Br2 systems. (These organic mixtures
are much easier to work with than metal alloys as they are generally
nonfaceted, transparent and require small temperature gradients and
hence they have been very popular with experimentalists.) They found
that the cell pattern formed and its dimensions were strongly dependent
on geometry of the container. More importantly when they ran the
experiments for two dimensional roll patterns, they found only
subcritical instability.

15
2.5. Limitations of Existing Models and Unaddressed Issues
In Chapter 1 the cause of constitutional supercooling was explained
as being due to the sharp solute concentration gradient in the liquid
near the interface, while elsewhere in the liquid and the solid the
solute concentration was practically a constant. It would seem then
that the only region of interest is the interface and a liquid "boundary
layer" adjacent to it. This has prompted all previous workers in
morphological instability to consider D^/v as the characteristic length
of the problem and to ignore solid diffusion. A typical value
of D^/v is 100 microns and this means that the far ends of the melt and
crystal are infinitely far away and the domain of the problem is
effectively confined to the liquid boundary layer mentioned above. For
the melting problem a characteristic length of D„/v is used and the
domain becomes an even smaller boundary layer in the solid.
These assumptions considerably simplify the algebra involved and
hence their popularity. But they constrain the validity of the model in
several ways. The most obvious one is that they necessitate the melting
and solidification problems to be studied separately, even though they
only differ in the direction of the growth velocity. Besides this
assumption fails for very small growth velocities, as it introduces a
singularity at v = 0. Later we will show that neglecting solid
diffusion also introduces a singularity and makes the model fail in the
nonlinear regime.
Finally, any effect which resides in the entire domain, not merely
the boundary layer, cannot easily be incorporated into the model, which
is why all influences on morphological instability studied so far are

16
either boundary layer effects (e.g., solutal convection) or interfacial
effects (e.g., grain boundaries and kinetic undercooling). Phenomena
that span the entire domain, like advection in the melt or imperfect
insultion of the ampoule walls, have been either inadequately treated or
ignored completely. Hence there is a need for a model that includes the
entire liquid and solid domains which would be applicable for all growth
velocities. This model should also dispense with the separate
treatments accorded so far to the solidification and fusion problems
with one uniform formulation.
In Section 2.2 it was mentioned in connection with the work of
Wollkind and Segel (1970) and of Ungar and Brown (1984a), that this
problem oscillates between subcritical and supercritical instabilities
for the case of two dimensional roll disturbances. They did not,
however, compute the ranges of each type of instability for the
experimental parameters involved. This is necessary in light of the
experiments of de Cheveigne et al. (1986) who observed no supercritical
instability. Also the extension of these predictions to three
dimensional disturbances like hexagonal and rectangular cells is yet to
be done.
It would not be an exaggeration to state that the inspiration for
all the theoretical work done so far in morphological instability has
come from Rayleigh-Benard convective instability. A comparison between
the two problems would be invaluable as a source of continued
inspiration and as a way to draw conclusions and conjectures about
morphological instability from the vast published literature on
Rayleigh-Benard convection. Hurle (1985) has attempted this but his
work can only be regarded as perfunctory and there exists a need for a
more rigorous treatment of the issue.

CHAPTER 3
A UNIFORM FORMULATION
3.1. The Formulation
Since we are not making the assumption that the liquid and solid
are very deep, the problem has to be formulated very carefully,
especially with regard to the outer boundaries, if we are to avoid an
intractable moving boundary problem.
A typical crystal growth set up is shown in Fig. 3-1. The ampoule
is heated by the heating coils surrounding it and they keep the melt
region at a temperature T1 and the crystal at T2. The temperatures T1
and T2 are maintained constant by means of thermocouples located at z =
s and z = -l. The ampoule is pulled towards the cooler end at the same
velocity V at which the crystal grows, thus keeping the interface
stationary. The region near the interface is protected by an insulating
shield and it is this region that becomes the domain in our model.
So in this model the outer liquid and solid boundaries become fixed
at z=-l and z=s respectively and the solid will be moving with a bulk
velocity v and the liquid with a bulk velocity v/Y, where Y is the ratio
of densities p /p0. In this section we will assume that Y=1 and
s
consider the effects of Y not being unity in Chapter 6 as this would
cause advection and fundamentally alter the basic problem. Also we will
assume that the melt concentration at the outer liquid boundary is a
constant C1 .
17

13
Thermocouple Thermocouple
Figure 3-1. Experimental set up

19
The domain equations in the liquid melt are
9T*
at + v az
ac, ac^
at + v az
D
In the solid region the equations are
(3.1-1)
(3.1-2)
3T 3T
s ^ s
at V 3z
a V2T
s s
(3.1-3)
+ v
ac
s
az
d v2c
s s
(3-1-^)
where T and C are the temperature and solute concentration, D and a the
diffusivity and thermal diffusivity, with the subscripts l and s
referring to the liquid and solid.
The boundary conditions are
T* - V c* - C1 at z =
(3.1-5)
T =T- at z=s
s 2
(3.1-6)
At the liquid-solid interface we will use £ to denote the departure
from planarity and write the boundary conditions at z=£
T0 = T = T - mC0 - Tm —H
l S M i ML.
h
(3.1-7)
Vh ■ " - ksVTs • " ■ Lh(v *
(3.1-8)

20
(3.1-9)
VC*
n - (v * Dú)ci ■
D VC
s s
n - <’ ♦ i)Cs
(3.1-10)
where TM is the melting point of the pure solvent, \ the interfacial
tension, the latent heat, m the absolute value of the liquidus
slope, k. and k the liquid and solid thermal conductivities, k the
x* S
distribution coefficient, n the normal at the interface directed into
the solid and H is the curvature of the liquid-solid interface and in
Cartesian co-ordinates is given by
«■ 1 A <’ *
3x
K K ¿L. + Ón
3x 3y 3x3y ^2 u
(|f)2)]«
0 * <|f)2 *
a^2i-3/2
(3.1-11)
In cylindrical co-ordinates, if we assume circular symmetry, it
becomes
H - [ <1
3r
(3C)2) _i£][1 + (i5.)2]"3/2
v3r; ‘ r3rJL ^3r; J
(3.1-12)
It is assumed that the side walls are sufficiently far apart and
well insulated to enable us to impose periodic boundary conditions in
that direction.
To convert these equations into the dimensionless form we will use
the liquid depth l as the characteristic length and the diffusive time
2
D^/v as the characteristic time.

21
The temperature will be made dimensionless by T = (T-Tw)/G„2, with
M T
GT = - (k¿G¿ + kgGg)/(ks+ V (3.1-13)
where G. and G are the temperature gradients in the liquid and solid
X/ s
in the quiescent planar state.
Similiarly the concentration will be made dimensionless by
C = C/G l with
c
G
c
(Dt°io
D G )/ (D. * D )
s sc 2. s
(3.1-14)
where G. and G are the concentration gradients in the liquid and
X* c sc
solid in the quiescent planar state.
So in dimensionless form, if we neglect the Lewis numbers D^/a in
the temperature equations, we have
(3.1-15)
3C __ 3C
—— + v —
3t 3z
V2f = 0
(3.1-16)
(3.1-17)
3C
3t
3C
+ v
- = nv2c.
3z
where v =
o ani " â–  VDt
The boundary conditions are
(3.1-18)

22
at z= -1,
V'V - Ti
c* - W - ci
(3.1-19)
at z=s/4=s,
T = (T. ' tJ/gtA - To (3.1-20)
S 2 M T 2
at the interface z=£
T4 - Tg = -SeC^ -A H (3.1-21)
VT • n - 2VT • n = L(v + ^) (3.1-22)
* 3 Dt
kC¿ = Cg (3.1-23)
VC • n - (v + ^)C. = nVC • n - (v + ^)C (3.1-24)
* Dt 16 S Dt S
where is the total derivative, 3 the ratio of thermal conductivities
k /k. and Se, A and L are the Sekerka, capillary and latent heat numbers
¿3 Af
respectively.
Se
mG
c
A
V
W
LhDa
k£GTl
(3.1-25)
(3.1-26)
L
(3.1-27)

23
At this point a discussion of the choice of a critical parameter
becomes imperative. The experimentally variable parameters for this
problem are G^, and v or their equivalents in this formulation
G^, Gc and v. Most previous workers have adopted G^ or C1 as their
critical parameter but Hurle (1985) has proposed Se, by analogy with the
Rayleigh-Marangoni problem. Recently de Cheveigne et al. (1986) have
advocated the use of v from an experimentalist's perspective. Although
this is a valid choice the reason no one else has used it so far is
probably because v occurs in the domain equations and will give rise to
an infinite number of eigenvalues in the linearized problem. We second
Hurle's suggestion and choose Se as this seems to be the naturally
occurring coupling factor between temperature and concentration and the
fundamental cause for morphological instability. Besides it includes
both G^ and Gq . But the suggestion of de Cheveigne et al. still
remains a valid one.
3.2. The Linear Stability Problem
From now onwards the " - " for dimensionless variables will be
dropped. The steady state planar solution to this problem occurs when
Cc(x,y) or Cc(r) = 0
(3.2-1)
The solution is then
(T1 + SeCjlc(o))z - Secuto)
(3.2-2)

24
Tsc(z) = (T2 + SqC1o(°»z/s ~ SeC¿c(o)
(3.2-3)
Cic(z)
(l-k)exp(vz)
l]/[-
(1-k )exp(-v)
k-exp(-v(1+s/n)) J Lk-exp(-v(1+s/n))
1]
(3.2-4)
C (z)
SC
(l-k)exp(vz/n)
kexp(v( 1 +s/ti) )-1 ' J' L k-exp(-v( 1 +s/n))
1]/tüT
(l-k)exp(-v)
1]
(3.2-5)
To write the equations of the linear stability problem we will
impose an infinitesimal disturbance on the steady state solution.
T = T + T
4 ic l
(3.2-6)
with similar expansions for T , C., C and
S X» s
Considering the linear stability problem, in order to separate
variables we will assume a horizontal cell pattern. This pattern for
two dimensional rolls is given by
, v _ 2imx ... , 2irn
d> (x) = Cos —:—, with wave number a = ——
Tn L n L
(3.2-7)
for cross rolls

n Li Li n Li
for rectangular cells
<|>n(x,y) = Cos Cos a = 2im(1/L^ + 1/L*) (3.2-9)
il Li - Li — n 1 C.
for hexagonal cells

25
n (x, y) = 2Cos Cos 2â„¢y + Cos
^imy _ 4irn
3L * an = 3L
(3.2-10)
for cylindrical rolls (r) = J (a r/R)
non
(3.2-11)
where an are the zeros of , and JQ and J-j are the zeroth order and
first order Bessel's functions of the first kind. We can now write
A A ,
T^(x,y,z,t) = T41(z) 1 (x,y)e0t
(3.2-12)
with corresponding forms for the other variables, where a is the
eigenvalue of the linear stability problem.
The linear stability problem becomes in the domain
(D2 - a2)Tai= 0 (3.2-13)
oCai= (D2 - vD - a2 )C^1 (3.2-14)
(D2 - a2)Tgl= 0 (3.2-15)
s1
Here we have
The boundary
used D to denote
dz'
conditions are
(3.2-16)
TÜ,1 = °* cu = 0 at z = -1 (3.2-17)
T
s 1
= 0
at z = s (3.2-18)

26
At the interface, z=0
s1
(Gt - V
T11 * SeCt, * il(0t * SeGIc- a A > â–  0
DTÍ1 - SDT3, ' °LS1
s1
(kG
ic
G )
sc
0
DCtl * nDG3l ' VC11 * VC3l
»v c
sc
Solving the system we obtain a general equation for
instability
Se[G
aC^c(1-k)tanhasstanha^
c. k(na + v/2 tanha s)tanha + (a„-v/2 tanha.)tanha s'
s s l l i s
2. _ oLtanhatanhas
+ a a(k.tanhas + k tanha)
where
2 . o . v
r ¿ a v t
a° -[a *
a¿ = [a2 + a + v2/4] '2
k„G„tanhas + k G tanha
l ¡L s_s
k.tanhas + k tanha
Í, s
(3.2-19)
(3.2-20)
(3.2-21)
(3.2-22)
(3.2-23)
morphological
(3.2-24)
(3.2-25)
(3.2-26)
(3.2-27)

27
G. (a - v/2 tanha.)tanha s + G (qa - v/2 tanha s)tanha.
i a/ S SC S S lo
c (a - v/2tanha„)tanha s + k(na + v/2tanha s)tanha.
x, l s s s X,
(3.2-28)
If we can show exchange of stabilities for this problem, then for
neutral stability.
Se
o
Gj + a A
(3.2-29)
Here we have already used the fact that the Se obtained from the neutral
stability curve will be the same as SeQ defined later in eqn. (4.2-2).
In eqn. (3.2-29) if we let v become very large the critical wave number
ajjj^ (for which SeQ is a minimum) also becomes very large and we can
approximate all the tanh terms to unity. Further if we also neglect
solid diffusion, n and G are zero and the equation reduces to the well
s c
known results obtained by Mullins and Sekerka (1964).
r °!to(at ~ l V) , °Ctc(1~k)
0L ,, 1 +
H + v(k ' 2}
♦ v(k -
H = 1 + a A -
aL
a(V k3}
(3.2-30)
For the case of neutral stability this becomes
Se
o
(1 + a2A Ma^/v + k - ^)

(3.2-31)
It must be pointed out that setting all the tanh terms to unity is
equivalent to using the diffusive length as the characteristic length.
This is a boundary layer approximation not unlike that used in the study
of pipe flow at high Reynold's numbers.

28
In our derivations we did not restrict v to be positive and hence
eqn. (3.2-24) is also valid for negative velocities, that is the fusion
problem. If we replace v with -v and here too assume that v is very
large and set the tanh terms to unity we obtain the result of Woodruff
(1968).
G (na - V? v)
Se [ — 3 2
ocioa-«)
0 nka + a. + ^ vCI-k) nka * ao * i v(1-k)
] - 1 ♦ a2A
oL
a(k,+ k )
x. s
(3.2-32)
and for o = 0
Se =
o
(1 + a2A)(nkag + a¿ + j v(1-k))
G (na
sc s
iv>
(3.2-33)
To compare this model with the approximations of Mullins and
Sekerka and that of Woodruff, SeQmin was calculated for various growth
velocities for the Pb-Sn system and the results are shown in Fig. 3~2.
As can be seen the approximations hold up very well for most growth
velocities but begin to fail for small velocities. (The thermophysical
data for the Pb-Sn system were those of Coriell et al. (1980).) The
ratio n for the Pb-Sn system is of the order 10 but for systems with
much smaller values of n like the C-Austenite system (see Clyne and Kurz
(1981) and Wolf, Clyne and Kurz (1982)) the approximations begin to fail
at higher velocities.

1200
800
400
VELOCITY (urns'1)
â—„
â–º
Figure 3-2. Comparison of previous approximations with general equation

30
3.3. The Adjoint Problem and Exchange of Stabilities
Exchange of stabilities refers to the nonexistence of time periodic
infinitesimal perturbations. Time-dependent infinitesimal perturbations
about the planar state will generally have periodic and nonperiodic
components, with a = + io^. For some problems it is possible to show
that ck is zero and this is called exchange of stabilities (see Iooss
and Joseph (1980) for details).
We still have to prove exchange of stabilities for this problem but
before we can do that it is necessary to obtain the adjoint problem. To
accomplish this we will define a column vector ♦ and a matrix operator
L
(3.3-D
L =
2 2
(D¿ - a;
0
0
— (D2- vD-a2- a)
0
0
0
0
2 2.
(D -a )
0
0
0
Se, 2 2 ,
—(pD -pa -o-vD)
(3-3-2)
and an inner product <#,♦>= J (T^ T^ + C?C?)dz + j (T^T^ + C^C^dz
IV
s s
s s
(3.3-3)

31
where T denotes the complex conjugate of T, T* the adjoint function of
T and
x
G
sc
)/ (3.3-4)
Then the domain equations can be written as = 0 (3-3—5)
In this inner product the adjoint problem becomes in the domain
2 2
(D -a )Tjl1 = 0
(3.3-6)
2 2
(D -a )Tsl = 0
(3.3-7)
(D + vD -
2
a >cn
° Si
(3.3-8)
(D^ + —
n
2 ~*
a )CS1
2 C*
n si
(3.3-9)
Subject to the boundary conditions
Si • Si •0
at z = -1
(3.3-10)
T = 0
s1
at z=s
(3.3-1D
At the interface z=0
(3.3-12)

32
nC£1 = Cs1
kDGs‘, - DG^ * '* ■ 0
DÍs', - DÍL * «' <°l - Gs> - 0
<1 * SeDG3*, * 'Í <°l * SeG£0- a'A > ■ (G^Vt ?il
2, s
aC (1-k)
(kG0 - G ) 6eUJl1
2,c sc
(3.3-13)
(3.3-14)
(3.3-15)
(3.3-16)
So far we have been unable to show exchange of stabilities for this
problem directly. But Wollkind and Segel (1970) have proved it when the
boundary layer approximation is valid and we will prove exchange of
stabilities by performing an asymptotic analysis around the boundary
layer solution, which corresponds to n=0 and Pe=0, where Pe is the
Peclet number given by D^/lv.
'2,1
= T
00
2,1
r)T
10
2,1
+ PeT
01
2,1
(3.3-17)
00 10 _ 01
a = o + no + Pea + ....
(3.3-18)
The other variables are expanded similarly.
If we use L00 to designate the operator L in (3.3-2) when n and Pe are
zero, then the linear perturbed systems become
00 210
J *1
.10
oo 201
J *1
= f
01
(3.3-19) & (3.3-20)

33
where
r
â– 
0
0
0
10"00
0 cm
* f01 =
0
01:00
o C ,
s1
0
0
. <
(3.3-21) & (3.3-22)
With boundary conditions
at z = - »
(3.3-23)
at z=“
(3.3-24)
at z=0 the equations are
boo(;;0,
> =
10
n00 "10
B (*1 ,
"01 ,
C, ) =
01
(3.3-25) & (3.3-26)
where B00 is the boundary operator defined by eqns. (3.2-19) - (3.2.23)
when n and Pe are zero and
10
;>i°- o
10r "00
o Lc1
"00,, .10 _10,
51 10.00,, , *00 00.10,, , .“00
0 ct0O-k)t) . 0 Cto(1-k)0,
(3.3-27)

34
h01 will have a similiar form with 01 superscripts replacing the 10. It
we let L00* and represent the adjoint operator and adjoint
function of and respectively, we can use the solvability
condition on the above system.
<#1 ’
loo;;°> =
<$
00*
f10>
(3.3-28)
^00*200* 210X
=
(3.3-29)
Subtracting we get
J(*
00* "00*
’1
h10)
= <;°°* foi>
z=0 ' r
(3.3-30)
where the lefthand side of eqn (3.3“30) is the bilinear concomitant
evaluated at z=0.
Similiarly J(*
00* "00*
’1
h°1)
z=0
= <$
00*
f01>
(3.3-3D
"00* "00
We note that # and ^ are real as they correspond to a state where
exchange of stabilities has been proved, while h1<\ and f^1 are
also made up of real quantities except for and . Hence we
conclude from eqn (3.3“30) that is real and from (3-3—31)
that a01 is real; i.e. exchange of stabilities holds for the generalized
morphological instability problem upto 0(n) and 0(Pe). So we are
justified in using the neutral stability curve (3.2-29) to calculate SeQ
at least up to such order even though we often extend it further. Even
when the boundary layer approximation is valid, exchange of stabilities

35
does not hold for all growth conditions (see Coriell and Sekerka (1983))
and care must be exercised when such extensions are made.
3-4. Finite Containers and the Most Dangerous Wavenumber
Appealing to the proof of Section 3.3, henceforth we shall only
consider steady state solutions. A typical SeQ versus a diagram is
shown in Fig. 3~3 and SeQmin is the minimum value of SeQ and the
wavenumber at which this occurs is amin. If we can maintain the growth
conditions such that Se < Seom^n we can at least say that the planar
interface is stable to infinitesimal perturbations.
As SeQmin is the least value of Se at which the planar solution
loses the stability, am^n is the wavenumber of the disturbance which is
most likely to occur. Hence this wavenumber is commonly regarded as the
most dangerous and is the wavenumber at which morphological instability
is usually studied. In the derivations and discussions that follow this
is the wavenumber employed and it becomes our "operating wavenumber."
The wavenumber is a 2tt multiple of the reciprocal of the
wavelength. If the domain being considered is regarded as being finite
with periodic lateral boundary conditions, the wavelength R (or L
depending on the wave pattern used) in its dimensionless form is now the
aspect ratio, the ratio of the ampoule radius to the melt depth. For
this situation SeQ takes on different values depending on the number of
cells formed on the crystal surface as shown in Fig. 3-i1 (see also
Rosenblat, Homsy and Davis (1982)). For each value of R there is a
fixed number of cells which is the pattern that is most easily
disturbed, except at certain values of R where two different patterns

36
WAVENUMBER a
Figure 3-3. Se^ vs. wavenumber diagram
Figure 3-4.
SeQ curves for various aspect ratios

37
are equally dangerous. These "horizontal multiple points" are very
important and will be discussed later. At other values of R the most
dangerous number of cells is denoted by N and the corresponding
wavenumber of each cell and the value of SeQ are denoted as a¡^ and SeQN
respectively. For the analysis in Chapter 6 where the effect of a
finite container on morphological instability is considered, R is chosen
such that SeoN is as close as possible to SeQmin so that the worst
possible case can be examined.

CHAPTER 4
SUBCRITICAL BIFURCATIONS
4.1. Theory
The linear stability analysis will only give the onset conditions
for morphological instability. Nonlinear calculations are necessary to
determine the behavior beyond this point. Ideally numerical
calculations should give the most amount of information by being
applicable for small and large deformations, but as can be seen from the
work of Ungar and Brown these calculations are very expensive to carry
out and they were forced to simplify the nonlinear problem and perform
calculations for very few experimental conditions. McFadden et al.
(1987), even though they did not attempt calculations for larger
deformations, were faced with the same restrictions.
This then is the case for weakly-nonlinear methods. They are
generally valid only in a small region very close to the onset
conditions but they can be used to predict the shape of the nonlinear
curve for larger deformations and for several applications this
information is sufficient. More importantly due to the analytical
nature of the techniques they can be used to predict the weakly
nonlinear behavior for all experimental conditions. A case in point is
the work of McFadden et al., most of whose predictions could have been
obtained more cheaply for all parameter values from weakly-nonlinear
theories.
38

39
Probably the most useful information generated by these theories is
the subcritical behavior of the nonlinear curve. Some typical
bifurcation diagrams are shown in Figs. 4-1 to 4-4. The e = 0 axis
corresponds to the planar solution and initially for small values of Se
the planar solution is stable and usually the only possible solution.
For Se £ SeQN the Planar solution becomes unstable and a nonplanar
solution "bifurcates" from the planar one. Figure 4-1 shows a symmetric
bifurcation diagram where nonplanar solutions do not exist for Se <
SeoN. For Se > SeQjy, even an infinitesimal perturbation will make the
solution jump from the unstable planar solution to the stable planar one
while for Se < SeQ^ the planar solution is stable to all
perturbations. This behavior is referred to as supercritical
bifurcation and for these curves Se^ = 0, Se2 > 0 (where Se1 = dSe/de
2 2
and Se2 = d Se/de at e = 0). Figure 4-2 is nonsymmetric and as can be
seen stable and unstable nonplanar solutions exist for Se < SeoN, making
the planar solution stable to infinitesimal perturbations in this
region, while a large perturbation can make it jump to the stable
nonplanar branch. This is called a subcritical bifurcation diagram and
is characterized by Se-| * 0. In this situation it is obvious that
growing the crystal at Se < SeQN is no guarantee of avoiding
morphological instability.
Figures 4-1 and 4-2 display the behavior usually seen in most
problems of hydrodynamic instability. Morphological instability is
unusual in having nonlinear curves shown by Figs. 4-3 and 4-4 as well.
These curves have been labelled "backward bending" to distinguish them
from the usual "forward bending" curves. (Actually they are Janus-like
in appearance bending backwards and forwards.) From the point of view

40
Figure 4-1. Forward bending, locally
symmetric, supercritical
bifurcation diagram
Figure 4-2. Forward bending,
unsymmetric sub-
critical bifur¬
cation diagram
Figure 4-3. Backward bending, locally
symmetric, subcritical
bifurcation diagram
Figure 4-4. Backward bending,
unsymmetric, sub-
critical bifurca¬
tion diagram

41
of the crystal grower this is unfortunate as their subcritical nature
increases the occurrence of subcritical bifurcation. Figure 4-3 is the
symmetric case with Se-j = 0 and Se2 < 0 and Fig. 4-4 the nonsymmetric
one with Se^ * 0.
The symmetry or nonsymmetry of the bifurcation diagram in
hydrodynamic stability is dependent on the cell pattern (see Joseph
(1976)). For morphological instability it will be shown that two
dimensional rolls, rectangular cells and cross rolls produce locally-
symmetric bifurcation diagrams while cylindrical rolls and hexagonal
cells produce nonsymmetric diagrams. It will also be shown that both
backward bending and forward bending will occur for all cellular
patterns depending on the experimental parameters used. Hence
bifurcation can be subcritical or supercritical for two dimensional
rolls, rectangular cells and cross rolls but for hexagonal cells and
cylindrical cells bifurcation is always subcritical.
4.2. The Second Order Problem
Here we begin our weakly nonlinear analysis and in this section we
will calculate Se^, the first derivative of Se with respect to e.
Considering the neutrally stable nonplanar solution near the bifurcation
point SeQ, we will expand the variables around the planar steady state
solution.
2 3
T =T +eT +eT + £ T +
11 Alc lo 11 12
Se = Se + eSe. + e Se_ +.
o 1 2
(4.2-1)
(4.2-2)

42
V,
where e = <$ ~ ¿
(4.2-3)
We have already obtained the linear perturbed solution
in Section
3.2. Substituting these expansions into the steady state versions of
eqns. (3.1-15) - (3.1-24) and collecting the terms of order
2
e we get
the second order perturbed problem.
If the first order perturbed variables are written, following eqn.
(3.2-12), as
Vx,y,z,Seo) = T2.01(z’SeoH1(x’y)
(4.2-4)
Then the solution to the second order problem becomes
CO
Til1 = E. WZ’SV Tü01)<()n(x,y)
n=1
(4.2-5)
Substituting (4.2-4) and (4.2-5) into the second order problem and
taking Fourier transforms horizontally, we get
L*i, â–  0
(4.2-6)
anú TtU ■ CI11 ■ 0 at z - -1
(4.2-7)
T .. = 0 at z =s
s 11
(4.2-8)
B($11 , ?11) = at z=0
(4.2-9)
at z=0
(4.2-9)

43
where L and B are the same as that used in (3.3~2) and (3.3-25) but with
o=0 and Se = Se and
o
11
-r (DT - DT )I
^or ioi sor 11
f 501(DTi,01 + SeoDC«,01) “ 2 vGi,cSeoi01^11 “ Sei(ci
~a C01(T«,0r BTs01 5111 + ?01(W eTs01):i12
c-wk dcíoi - DCsoi^ -¿v4(kG*c - ‘W
"a C01 (C«,01
nCs01)i:i1 + ?01(Ci,01 ” nCs01)i:i2
(4.2-10)
L2 L1
I I ^(x,y)dxdy
o o
’11
L2 L1
I j 4>1 (x.y)dxdy
o o
(4.2-11)
rl2 G1 94». 94». -
I I ^(ir) + ^ ldxdy
9y
"12
L2 L1
I I 4>^(x,y)dxdy
o o
(4.2-12)
It can easily be seen that for two dimensional rolls, cross rolls
and rectangular cells that 1^ = I12 = 0* Hence from the solvability
condition for these patterns Se^ will be zero; that is, the bifurcation
will be locally symmetric. But for hexagonal cells and cylindrical
rolls I-|i and will be non-zero and hence Se^ will also be nonzero
and we have nonsymmetric bifurcation and the existence of subcritical
instabilities.

44
We will analyze the case of Se^* 0 by considering hexagonal cells
as an example, but the results obtained are applicable to cylindrical
rolls as well. It should be noted here that all the horizontal cell
patterns we have been using are possible solutions to Helmholz's
equation
£± . i± . a2* . 0
9x¿ 9y
and the solutions always come in pairs. For hexagonal cells the
complementary solution to <$ given in (3.2-10) is t|>
Pit Pit Utt
<|»1 » 2 Cos JqX Sin -jj-y - Sin ^y (4.2-14)
So the general form of eqn. (5.4) will be
T£01 (lf1
+ P^)
(4.2-15)
To determine p the procedure outlined by Joseph (1976) will be
used. We will proceed the same way as above but using (4.2-15) instead
of (4.2-4) and multiplying by will give the same set of eqns. (4.2-6) - (4.2-10) but with
I
11
2
+ p*1 ) (J^dxdy
+ pij^ )1dxdy
(1-p)2
(4.2-16)

45
3L /3L 34) 3*. - 34»! H. ,
J / í ]*idxdy
O 0
'12 = 3ÍT73L
ff
O 0
a ,. 2*
= — (1-p )
(')>1 + P^ )4>1 dxdy
(4.2-17)
which can be used to calculate Se^ . We then repeat the process
with and equate the two Se^ 's obtained. This will result in a cubic
equation for p.
P
3
3p = 0
(4.2-18)
and
Se1(1-p2)
(4.2-19)
Se. G (kG„ - G ) (a. + v/2tanha.)B (pa - v/2tanha s)
1 C = „ lc sc ( r l l_ + 3 s -[
eo , „\2 ‘ *■ tanha„ qtanha^s J
’01
(k + e)'
11
(1+nB)tanha s
(na + v/2tanha s) X12J “1
s s
J + h,
(4.2-20)
6(GÜ " V [_fn + a(Gl + SeoGlc - a A)(1/g-1 ) I
1 (3+A) Ltanha
(G -G )tanha
SL s
11
(1+A)tanhas
(8+A)
*12 * * h2
(4.2-21)
Se I., a( 8-1 ) (kG. - G )
_ —2__U_[ — £c— - (bg„ + G /n) ]
2 (k+B) L (B+A)tanha fcc sc
tu =
(4.2-22)

46
where B = (a. - v/2tanha„)tanha s/(na + v/2tanha s)tanna. (4.2-23)
x. x. s s 3 x,
and A = tanhas/tanha (4.2-24)
There are three solutions 0, /3 and -/3 for p, but it can be seen
that Se1p for /3 and -/3 coincide. It is also obvious that while any
two of the solutions are independent the third is a linear combination
of the other two. Here is an instance where the problem exhibits a
multiplicity of solutions for the same eigenvalue, and could cause
secondary bifurcations further along the bifurcation curve.
The next obvious question is to ask if there is any point at which
Se1 in (4.2-20) goes to zero and hence causing Se^ to go to zero.
Calculations done for the C-Austenite (i.e. steel) system did give such
a curve (see Fig. 4-5) though for most growth conditions this curve lies
far away from the critical (or most dangerous) wavenumber amin,
intersecting it only at high velocities.
4.3. The Third Order Problem
In Section 4.2, it was shown that for 2-D rolls, cross rolls and
rectangular cells Se^ was zero, which implied a symmetric bifurcation
diagram. But to learn more about the nature of this diagram it is
necessary to go to the next order. If we repeat the procedure described
for the second order problem for the terms of order e^ we will obtain
the third order problem.
L*
21
0
(4.3-1)

WAVENUMBER
47
â–º
Figure 4-5. Growth velocity vs. wavenumber chart for
C-Austenite system for hexagonal cells

48
TZ21 = GZ21 " 0
at z = -1
(4.3-2)
Ts21 " 0
at z=s
(4.3-3)
B(*2i » C21) = h21
at z=0
(4.3-4)
21
• — (T - T )I
2 ^01 WZ01 s01 '21
2
'I21^ 1~C01(TZ01 + SeoCí.01) + 2501Seo DCZ01 + 6V C01SeoGi,J
(GZ0Í ?01G«,c)Se2 “ Ac01(3/2 a I22~ I23)
2
_I21 ^~2~ C01(kCZ01 " Cs01) + 2C01(k DCZ01 " DCs01/n)
+ ^01(kG*c ' Gso^2)1
a2 2 2
2 C01V(DCZ01 " r|DCs01 )J21 + C01(DCZ01
nDCs01)J22
L2 L1
/ / 4>1 dxdy
o o
’21
L2 L1
/ I 4>2dxdy
o o
(4.3-5)
(4.3-6)
LZ S 3*, 3*,
/ / +l[(—k> * (—)2]dxdy
o o
9y
’22
L2 L1
/ / 2dxdy
o o
(4.3-7)

49
L.i L2 a*, a*. ,
I I [-4 <-^>2
oo 3x
2 v 9y
34> 1 3^ 2 <(>^
2 9x 3y 9x3y
924>
dy‘
dp
1 (~d)2]*ldXdy
’23
L2 L1
/ J o o
Once again using the solvability condition we obtain
(4.3-8)
Se_ 2 v(kG. -G )(1+B)tanha s
TVhrV (I22 - r i2,)Seo rf-22 <4-3-9>
’01
(k+B) (na +1/2vtanha s)
s s
where
h1 ~^2a I22+I23^A~ I21Se0V
vG
Ic
(kG. -G )(a + v/2tanha.)/tanha„
x.c sc x. x, i -
6(k+B)
h„ = -
*21 ’"'W
(k+B)‘
k(a^+ v/2 tanha^)
tanha.
(4.3-10)
(na -v/2 tanha s)B
3 2 ' 1 G*o * *2,
n tanha s
s
Se v(kG. -G /n)
o to sc
6(k+B)
(4.3-11)
and B = (a -1/2 vtanha„)tanha s/(na + 1/2 vtanha s)tanha. (4.3-12)
l l s s s l

50
4.4. Calculations and Comparisons
Using the results of the previous section, Se2 was calculated for
various values of the experimental parameters and the results are shown
in Figs. 4-6 to 4-9. Figure 4-6 was drawn in order to check the
derivation with that of Ungar and Brown (1984a) and shows calculations
done for 2-D rolls with Dg = 0 and 3 = 1 but is a more complete
calculation for the experimental parameter ranges involved than
theirs. The calculations agree with those of Ungar and Brown but it is
an unexpected result nevertheless, showing multiple regions of
subcritical and supercritical instability. It also seemed to contradict
the earlier calculations of Wollkind and Segel (1970) who did not see
any supercritical instability but this was resolved in the paper of
Wollkind and Wang (1988) and hence Fig. 4-6 agrees with their
calculations as well. As argued in Section 3.4 if we treat the amin
curve as the experimental "operating line," the bifurcation is mostly
supercritical which also seems to contradict the experimental results of
de Cheveigne et al. (1986) who observed only subcritical bifurcation.
In Fig. 4-7 calculations were done for more realistic values of Ds
and 3 and the results are significantly different with only one region
of supercritically and another of subcriticality. Ignoring solid
diffusion introduces a singularity to the problem and this accounts for
the distortions observed in Fig. 4-6. In Fig. 4-7 if we move along the
operating line for a fixed liquid temperature gradient, initially for
small growth velocities the bifurcation is supercritical, until at a
critical velocity a transition point is reached and the bifurcation
becomes subcritical.
Hence for every imposed liquid temperature

WAVENUMBER
51
Figure 4-6. Velocity vs. wavenumber chart for the one-sided
approximations of Ungar and Brown for 2-D rolls

WAVENUMBER
52
Figure 4-7. Velocity vs. wavenumber chart for 2-D rolls
using the Pb-Sn system

53
gradient there will be a critical growth velocity below which the
bifurcation for 2-D roll disturbances is always supercritical. The
really surprising result here is that when these transition points for
various values of the gradient are joined, we get a straight line
through the origin. We now have a clearly demarcated supercritical
upper-triangular and subcritical lower-triangular one. It is well known
that the onset condition SeQ does not change much in the neighborhood of
a^H (see for example Coriell, McFadden and Sekerka (1985)) and so amin
is more of an interval than a unique point. It can also be seen from
Fig. 4-7 that the am^n curve practically hugs the line of transition
from sub to supercriticality and if we impose an interval for amin it
would straddle the transition line. Thus for 2-D rolls it is unlikely
that we will ever see a sharp transition from subcritical to
supercritical bifurcation in experiments. More likely we will observe
subcritical behavior throughout as reported by de Cheveigne et al.
Proceeding to the three dimensional patterns, we obtained almost
identical results for square cells and cross rolls. Figure 4-8 is for
square cells and we see quite a change with the supercritical region
acquiring a characteristic balloon shape and having a sharp transition
to subcritical bifurcation along the amin curve. But here too if these
points of transition are joined, a straight line is the result,
demarcating a supercritical upper-triangular operating region and a
subcritical lower triangular one. Unlike the case of 2-D rolls these
should be visible to the experimentalist. So in order to avoid cross
rolls and square cellular instabilities not only should the crystal be
grown when Se < SeQmin, but one should do so in the upper triangular
region. Figure 4-9 demonstrates the universality of our result in being

WAVENUMBER
54
VELOCITY (urns*1)
Figure 4-8. Velocity vs. wavenumber chart for square cells
using the Pb-Sn system

55
8000
6000
4000
2000
0
•*
Figure 4-9. Velocity vs. wavenumber chart for the fusion
problem for square cells using C-Austenite system
WAVENUMBER

56
applicable for the melting problem as well and repeating the derivation
for this case separately as was done by Wollkind and Raissi (1974) is
unnecessary.
Finally, even though for hexagons Se-j was usually non-zero (and
hence Se2 cannot easily be calculated), in Section 4.2 we saw that there
were points at which Se-| did go to zero. If we attempt to evaluate Se2
for hexagons at these points I2i, I22 and *23 become
(4.4-1)
5 2 2
!22 -§ a¿(1+p¿)
(4.4-2)
(4.4-3)
and as can be expected Se2 will be in the form
(4.4-4)
where Se2 is that corresponding to p = 0. The other possible value is
when p is /3 or -/3. Here Se2p can be calculated from (4.3-9) - (4.3-
12) and (4.4-1) - (4.4-3). As mentioned in Section 4.2 these points are
usually far away from am^n and in Fig. 4-5 we found that along this
curve Se2 is positive at low values of "a." As "a" increases, at a
point above a=amin, Se2 becomes negative. Hence along this line the
bifurcation diagram is forward bending at low "a's" (including a^n) and
becomes backward bending at high values of "a." Even though this is

57
true only along the Se-|=0 line, by analogy with other cell patterns we
conjecture that this is valid when Se^O as well. In other words we
expect the bifurcation diagram to be forward bending along the line
and below (as shown in Figs. 4-10 and 4-11) but a transition to backward
bending along the line could occur at high velocities. Far above
the amin line the bifurcation diagram should be backward bending (see
Figs. 4-12 and 4-13).
To confirm these conjectures we turned to the nonlinear
calculations for hexagons of McFadden et al. (1987) but unfortunately
they were unable to complete the bifurcation diagram as their attempts
to compute the curve for e<0 failed. Also their calculations were done
only for the case of p=0 and not for p=±/3. But they did confirm the
existence of subcritical instability.
To sum up, it has been shown that the Mullins and Sekerka and the
Woodruff models of morphological instability are of limited validity.
The uniform formulation is the more exact representation of the problem
and
* it is applicable for all growth velocities and not just the
relatively rapid solidification and fusion regions and provides a
single formulation from which all the different models for
various growth conditions can be obtained as limiting cases,
eliminating the duplication of derivations for different cases;
* it incorporates the whole crystal and melt regions into the
problem and not just a boundary layer region adjoining the
interface facilitating the study of various domain effects like
convection on morphological instability;

53
Figure 4-10. Bifurcation diagram
for hexagonal cells
Figure 4-11. Bifurcation diagram
for hexagonal cells
Figure 4-12. Bifurcation diagram
for hexagonal cells
Figure 4-13. Bifurcation diagram
for hexagonal cells

59
* it avoids the incorrect predictions of subcritical bifurcation
regions because of the singularities inherent in previous models.
* The principle of exchange of stabilities has been shown to be
applicable to this model as well even though only in an
asymptotic sense.
When the weakly nonlinear technique of Malkus and Veronis (1959) is
applied to this problem in a systematic way, it resulted in important
information about the shape of the bifurcation diagram for various
growth conditions. Some of these results are similar to those obtained
for Marangoni instability (see Joseph (1976)) which leads us to assert
that these results are valid for all hydrodynamic instability problems
in which the nonlinearity lies only on the boundary.
* If rectangular cells, cross rolls or two dimensional rolls are
the horizontal cell patterns then the bifurcation diagram will
always be locally symmetric. For hexagonal cells or cylindrical
rolls they are generally non-symmetric and hence the bifurcation
is subcritical.
* The problem can display a multiplicity of solutions for the same
eigenvalue. Specifically for hexagons there are two possible
solutions.
Considering the morphological instability problem in particular the
following were shown:
* for two-dimensional rolls there are two operating regions, one
subcritical (backward bending) and the other supercritical
(forward bending), but since the demarcation is not sharp its
probable that only subscritical bifurcation will be observed
experimentally.

60
* For rectangular cells the forward bending region has a
characteristic balloon-like shape and here too there is a
straight line dividing the operating region into subcritical and
supercritical zones, but here the transition is sharp and hence
probably observable experimentally.
* For hexagonal cells and cylindrical rolls the bifurcation diagram
shows both backward and forward bending behavior but the exact
regions of each can only be conjectured.

CHAPTER 5
COMPARISONS WITH RAYLEIGH-MARANGONI CONVECTION
5.1. Rayleigh-Marangoni Convection in Brief
When a horizontal layer of quiescent fluid is heated from below, on
account of thermal expansion, the fluid at the bottom will be lighter
than the fluid at the top. This unstable arrangement is maintained by
the viscosity of the fluid which inhibits any flow and suppresses
disturbances such that there will be a stable conduction profile in the
fluid. But when the adverse temperature gradient exceeds a critical
value, the viscous force is overcome and there will be cellular
convection. This instability is known as Rayleigh-Benard convective
instability.
There are several variations of this problem. Instead of an ad¬
verse temperature gradient, there could be an adverse solute concentra¬
tion gradient in the fluid causing once again an unstable top-heavy
arrangement. The convective instability arising from this is known as
solutal convection or the solutal Rayleigh problem. Another way to
cause convection is to have a very thin fluid layer heated from below,
but the top surface of the fluid instead of being kept at a fixed lower
temperature, is allowed to remain free. Here the thinness of the fluid
layer makes buoyancy effects negligible but convection will be caused by
surface tension variation on the free surface. This is known as Maran-
goni convection. Finally, there could be combinations of the above
61

62
three types of convective instability. When convection is caused by
thermal and concentration gradients it is known as double-diffusive
convection. Combination of either thermal or solutal convection with
surface tension driven flow is the Rayleigh-Marangoni problem.
In addition to these there are several other combinations possible
like Rayleigh-Benard convection with rotation or with a magnetic field
but for purposes of comparison with morphological instability it will be
seen that the three causes for convection mentioned above are the most
relevant.
In the discussion to follow it is desirable to consider the most
general form of this problem. Despite there being several interesting
features in the problem of double-diffusive convection, the causes for
convection there, the temperature gradient and the solute concentration
gradient are both bery similar and it is sufficient to look at the
effect of one gradient. The manner in which the surface tension effects
the problem is very different from the buoyancy effects and a general
formulation should include both. Accordingly we will examine the Ray¬
leigh-Marangoni problem with thermal convection. In the following
sections several other reasons for looking at this problem will become
apparent.
The equations for the Rayleigh-Marangoni problem are given by Sarma
(1987) and we will reproduce them here and refer the interested reader
to his paper for details. The steady state dimensionless Boussinesq
equations in the domain are
V-V = 0
(5.1-1)

63
2 Ra
V V - Vp + ^ Tg - V-VV = 0 (5.1-2)
V2T - PrV*VT = 0 (5.1-3)
where p is the modified pressure, g the acceleration due to gravity, Ra
the Rayleigh number and Pr the Prandtl number. The boundary conditions
at the bottom of the fluid layer are
at z = 0, T = Tq and w = 0, 3w/3z = 0 (5.1-4)
where w is the vertical component of velocity.
The boundary conditions at the top are more complicated. Not only
will there be surface tension variation across the surface but the
surface is also free to deflect like the liquid-solid interface in
crystal growth.
at z = 1 + i, VT*n = - BiT (5.1-5)
V*n = 0 (5.1-6)
- Bopn + Crn*[VV + VVT] = MaV T + Hn (5.1-7)
n
The dimensionless quantities are Bi the Biot number, Bo the Bond number,
Cr the Crispation number, Ma the Marangoni number and H the surface
curvature (see Scriven and Sternling (1964) for details).
Initially there will be a quiescent, linear, stable, conducting
solution to the problem with V =0. At a critical value of the charac-

64
teristic parameter (Ma or Ra) this conducting solution becomes unstable
to infinitesimal perturbations and we have a convective solution.
Performing a linear stability analysis about the conduction state,
separating variables and doing considerable manipulations we get in the
domain
(D2 - a2)30 = - a2Ra6 (5.1-8)
(D2 - a2)3w = - a2Raw (5.1-9)
where "a" is the wavenumber, w the Fourier coefficient of the vertical
component of velocity and 0 the Fourier coefficient of the temperature.
At the boundary at z = 0,
w = Dw = 0 = 0 (5.1-10)
At z = 1, w = 0 (5.1-11)
BiD2w = a2MaD0 (5.1-12)
BiCr(D3w - 3a2Dw) = a2(Bo + a2)(D0 + Bi0) (5.1-13)
When the density variation with temperature is negligible or in the
absence of gravity, then Ra = 0 and we have the pure Marangoni problem
with all the nonlinearities only in the boundary. Even with this effect
in the boundary the important result of bifurcation, namely the fluid

65
velocity, effects only the domain, the deflections in the boundary being
only a secondary effect of convection. On the other hand, when the
upper boundary is also kept at a fixed temperature we have, Ma = Bo = Cr
= 0 and so
at z = 1 , w = Dw = 0 = 0 (5.1-14)
This then is the pure Rayleigh-Benard problem with all the nonlin¬
earities only in the domain and the resulting nonquiescent solution also
manifests itself in the domain as convection. The Rayleigh-Marangoni
problem described by eqns. (5.1-8) - (5.1-13) is a mixed problem with
nonlinearities in the domain and the boundary but the convective solu¬
tion resulting from these nonlinearities shows up mainly in the domain.
5.2. The Augmented Morphological Problem
As can be seen from Section 5.1, in the Rayleigh-Marangoni problem
there is a Ra-Ma domain-boundary duality which does not seem to exist in
morphological instability. From the problem description in Chapter 3 it
is easy to see that all the nonlinearities for this problem lie only in
the liquid-solid interface. This is a limitation because by virtue of
being on the boundary the Sekerka number is unique and hence also has a
unique eigenfunction and is insufficient when solutions to inhomogeneous
versions of the linearized morphological instability problem are needed,
as in Chapter 6 where "imperfections" are considered. This difficulty
also crops up in the pure Marangoni problem, but the Rayleigh-Marangoni
problem comes to the rescue, as there are countably many corresponding

66
values of Ra for each value of Ma and hence also countably many
eigenfunctions forming a complete set (see Rosenblat, Homsy and Davis
(1981)). The naturally occurring duality of Ma and Ra enables solutions
to inhomogeneous problems to be obtained in a straightforward manner.
In morphological instability there is no such obvious, naturally
occurring boundary-domain duality and it is necessary to create one. To
avoid confusion we will refer to the pure morphological problem of
Chapter 3 as the Sekerka problem, and (by analogy with the Rayleigh-
Marangoni problem) set up an eigenvalue problem, with the eigenvalue in
the domain, which we will call the "augmented morphological problem."
liquid:
solid:
(5.2-1)
(5.2-2)
(5.2-3)
(5.2-4)
where M is the eigenvalue which we label as the morphological number.
The boundary conditions are
at z = -1 , p^ = q^ = 0
(5.2-5)
at z = s, p = 0
*s
(5.2-6)
Pü " ps + t(GX, ~ V = 0
at z = 0,
(5.2-7)

67
»l ♦ SeQt * ttOj * SeGl0) * A ¿Í , 0
9r
3pj, 3p
sr-6 ST â– 0
kq - q + t(kG„ - G ) = 0
Í. s ¿c sc
dql 3qs
ST " n 3T ‘ vq* + vqs = °
(5.2-8)
(5.2-9)
(5.2-10)
(5.2-11)
with periodic conditions at the lateral boundary at r = R. We can now
separate variables expressing the horizontal dependence as zeroth order
Bessel's functions of the first kind JQ(air/R), where a^ are the zeros
of the first order Bessel's functions of the first kind .
00 CO
p (r,z) = Z Z Pm(z) J (a.r/R)
* i=i j=i 41J 0 1
(5.2-12)
In addition if we take Fourier transforms in the horizontal direc¬
tion and solve for q^„, qg„ and the equations reduce to a system
in p„. . and p ... If we define a column vector
Í.1J sij
Q,
ij
«J
P3ij
(5.2-13)
and a matrix differential operator Li
L.
l
)
BY.(D2 - a2)
(5.2-14)

68
then the domain equations reduce to
L.Q.. = - M..Q..
l 1J ij lj
(5.2-15)
where Y. = (G. + Se 6 - a2A )/(G + Se G - a2A )
IX, Cl S Cl
(5.2-16)
G = (G„ B + G )/(B + k)
c X, c sc
(5.2-17)
(a.. - v/2tanha..)tanha .s
2.1 2d. si
(na . + v/2tanha ,s)tanha..
si si 2,1
(5.2-18)
, 2 x 2.. .1/2
Hi ' (5.2-19)
, 2 2 . 2,1/2
a . = (a. + v /4q )
si 1
(5.2-20)
The boundary conditions become
at z pUj â–  0
(5.2-21 )
at z = s, p .. = 0
sij
(5.2-22)
at z = 0,
BiQij * 0
(5.2-23)
where
B.
1
1
D
-Y.
l
-BD
(5.2-24)

69
Defining an inner product
dz + / p
0
s
(5.2-25)
where the refers to the adjoint eigenfunction and the complex
conjugate.
It can easily be seen that the system described by eqns. (5.2-13) -
(5.2-24) is self-adjoint in this inner product and so the eigenfunctions
Qjj are complete. Solving the system we get
(5.2-26)
A ..Sin/M..- a^ (s - z)
sij ij i
where Mjj are solutions of the equation
(5.2-27)
lj i 3 3 ij Í
(5.2-28)
lj Í S3 ij i
Here A... and A .. can be determined from the normalizing condition
¿ij sij
(5.2-29)
where 6.. is the Kronecker delta,
ij

70
5.3- Comparison of Morphological Instability with
Rayleigh-Marangoni Convection
The principal aim of this section is to relate the mathematical
characteristics of the two problems so that we may introduce some of the
extensive mathematical techniques used to study Rayleigh-Marangoni
convection to morphological instability, but we will make some physical
comparisons as well. The augmented morphological problem described in
Section k.2 is similar to Rayleigh-Marangoni convection. The augmented
problem is self adjoint but the Rayleigh-Marangoni problem is nonself
adjoint. Both have an infinity of eigenvalues and corresponding eigen¬
functions, but while completeness of the eigenspace is assured for the
former, special theorems are required to show this for the latter (cf.
Nadarajah and Narayanan (1987)). It should also be noted that while the
Rayleigh-Marangoni problem attempts to describe a realistic situation,
the augmented morphological problem was artificially created in order to
solve inhomogeneous versions of the Sekerka problem described in Sec¬
tions 3*1 and 3.2.
This brings us to the question whether there is a practical situa¬
tion which is described by this mathematical concoction. The difficulty
in coming up with one stems from another important difference between
the two problems. In the pure Rayleigh-Benard problem (where Ma, Bo and
Cr are all zero) the nonlinearity is in the domain and the instability
too manifests in the domain as convection. Even in the pure Marangoni
problem (where Ra is zero) where the nonlinearity is in the boundary,
the instability is still mainly in the domain. In contrast, in the
Sekerka problem the nonlinearities and the resulting instability show up
in the boundary. Though there are other boundary effects (like kinetic

71
undercooling) which can cause morphological instability the only domain
effect which could give M physical significance is a heat source term in
the form MT or Me E^RT (see Joseph (1965)). We do not know of any
experiment where morphological instability was observed as a result of a
heat source in the melt or the crystal but if one does exist it will
provide the true analogy to Rayleigh-Benard convection.
This is relevant as Hurle (1985) has attempted a comparison between
the Rayleigh-Benard problem and the Sekerka problem. It can now be seen
that the Sekerka problem can only be compared to the pure Marangoni
problem, with Se corresponding to Ma and A corresponding to the reci¬
procal of Bo. Besides, for periodic lateral boundary conditions, the
eigenvalues of both problems, Se and Ma, are unique.
Based on this comparison we can make an important conjecture.
Vrentas, Narayanan and Agrawal (1981) have shown that for the Marangoni
and the Rayleigh-Marangoni problems when the nonperiodic no-slip condi¬
tion for velocity is imposed at the sidewalls, the eigenvalue Ma is no
longer unique and has countably many values. In other words when the
walls are a finite distance apart Ma has many values, but as they are
gradually moved apart we have "spectral crowding" and in the limit when
they are sufficiently far apart to impose the periodic boundary condi¬
tion of total slip, all the values of Ma coalesce into a unique num¬
ber. Recently, following Coriell et al. (1980), several workers have
looked at the coupled problem of morphological instability with solutal
convection and all have assumed periodic boundary conditions. We sus¬
pect that here too if the no-slip condition for velocity at the side
wall is imposed, the Sekerka number will no longer be unique. All this
raises the question of completeness of the Marangoni and the Sekerka

72
eigenspaces and its probable that generalized eigensolutions (see Nai-
mark (1967)) are needed when Ma and Se are chosen as eigenvalues.
When these two problems are considered in a finite container we can
see yet another difference. Both problems have simple eigenvalues
except at certain aspect ratios of the container where two horizontal
modes can coexist (cf. Rosenblat, Homsy and Davis (1982)). In a typical
experiment of Rayleigh-Benard convection we would expect to see a dozen
or so convection cells (see for example Koschmieder (1967)) and increas¬
ing or decreasing the number of cells by one can significantly effect
the problem. Hence the multiple points in this problem are extremely
important and have been the subject of study. But in morphological
instability a single alloy crystal can contain hundreds of individual
cells and the addition or loss of one has hardly a noticeable effect on
the problem and consequently multiplicity of the lateral eigenfunctions
loses its significance. In addition unlike the Rayleigh number it is
well known that near the critical wave number a^, the critical value of
the Sekerka number SeQ hardly changes (see for example Coriell, McFadden
and Sekerka (1985)) and the choice of a^ has very little effect on
SeQfj. Conversely, the choice of the operating Se will have a tremendous
impact on the resulting wavenumber (cf. Ramprasad and Brown (1987)).
Other differences have been mentioned in Chapter 4. Both the
Marangoni problem and the Sekerka problem have symmetric bifurcation
diagrams near the bifurcation point for two-dimensional rolls and rec¬
tangular cells and nonsymmetric curves for hexagonal cells and cylindri¬
cal rolls. But in morphological instability the curves can be "forward
bending" or "backward bending" depending on the operating conditions,
whereas in Marangoni convection the curves are forward bending every-

73
where. Hence the occurrence of subcritical instability is more wide¬
spread in morphological instability.

CHAPTER 6
BIFURCATION BREAKING IMPERFECTIONS
6.1 . Nature of Imperfections
When the morphological instability problem was formulated in Chap¬
ter 3, several effects were ignored and the resulting problem is an
idealized or "perfect" one. Inclusion of these can alter the problem in
several ways, for example kinetic undercooling of the melt becomes an
important effect in rapid solidification, but all it does is alter the
onset condition for morphological instability. In the parlance of
bifurcation theory, an "imperfection" is an effect on the "perfect"
problem which alters it in a specific way. Such an imperfection will
cause the morphological instability problem not to have a planar solu¬
tion at all even below the onset condition, and these are known as
bifurcation breaking imperfections.
The effect of a typical imperfection on the bifurcation diagram is
shown in Fig. 6-1 . The broken line is the solution in the presence of
imperfection and it can be seen that the interface will be nonplanar for
all nonzero values of Se. Obtaining solutions to the problem with
imperfections is extremely difficult and we will only seek asymptotic
solutions. Hence the problems to be considered should have very small
imperfections. Under such conditions the method of matched asymptotic
expansions of Matkowsky and Reiss (1977) can be employed and here it
will be used in a way similar to the work of Tavantzis, Reiss and Mat¬
kowsky (1978) for the Rayleigh-Benard problem.

75
Figure 6-1. Imperfect bifurcation diagram showing inner
and outer expansions

76
The method-is fairly straightforward. The variables are expanded
asymptotically with the imperfection parameter about the planar and the
nonplanar solutions and two outer expansions 0Q and 0-, are obtained as
shown in Fig. 6-1. At the bifurcation point SeoN these expansions break
down and it is necessary to have inner expansions I1 and I2 near SeoN
and to join the corresponding 0q and 0^, matching conditions have to be
specified.
The imperfections that can be analyzed in this fashion must, of
course, be small effects else a full-blown nonlinear solution will be
needed. Two of the most important effects which are habitually ignored,
namely imperfect insulation of the ampoule wall and advection in the
melt, are such imperfections and readily lend themselves to this type of
analysis. In Chapter 3 when the morphological instability problem was
modelled by a uniform formulation, it was mentioned that one reason for
this was to consider a finite crystal/melt region. This finiteness was
only in the vertical direction and in the horizontal direction the
imposition of periodic boundary conditions effectively meant that the
ampoule side walls were infinitely far apart. The two imperfections
that are to be considered are caused by nonperiodic lateral boundary
conditions and another way of looking at the effect of these
imperfections is to say that the container is now being considered to be
finite in the lateral as well as the vertical direction.
6.2. Imperfection Due to Heat Loss
As mentioned in the last section, it has been customary in this
problem to assume periodic boundary conditions laterally, which is

77
equivalent to assuming that the walls of the ampoule are perfectly
insulated or that they are so far apart that their effect can be ig¬
nored. In practice neither of these is likely to be achieved and here
we will examine the effect of a small amount of heat loss or heat gain
from the wall on the problem. If we take the ampoule to be cylindrical
with radius R,
3T
r = R, liquid: -k¿ = 5f^(z) (6.2-1)
3T
solid: -k t——■ = 6f (z) (6.2-2)
3 3r s
where f. and f are such that f.(5) = f (?) and f.(-1) = f (s) = 0.
Xi ^ X/ S X/ s
If f^ and fg are positive it will mean heat loss and if they are nega¬
tive, heat gain. If we make the transformations
Vz)
T^(z,r) = 5 — r + 0^(z,r) (6.2-3)
f (z)
and T (z,r) = 5 — r + 0 (z,r) (6.2-4)
S K S
and substitute these in the steady-state versions of eqns. (3.1-15) -
(3.1-24), the temperature equations in the domain become
= - 6rD2f^/k^ (6.2-5)
V20 = - 5rD2f /k
s s s
(6.2-6)

78
The outer boundary conditions will remain unchanged but at the interface
eqn. (3.1-21) becomes
°l - % - - 5r(Vkt ' W <6-2-7)
8, * SeCj ♦ A H - - Srft/kt (6.2-8)
and eqn. (3.1-22) converts to
ve -n - 8V0 -n = Lv - 5r(Df„ - Df ) (6.2-9)
X. S X, s
In order to solve this system we will be treating the heat loss as
an imperfection on the perfect problem (i.e. when 6=0). The perfect
problem is of course the Sekerka problem of Section 3.1.
6.3. The Outer Expansions
As this problem has been defined in a finite geometry, the number
of cells are fixed by the container size and the growth conditions. We
will choose these so that aN is very close to amin the wavelength corre¬
sponding to Seom^n the least value of Se^. Another important decision
is the selection of the wave pattern and our analysis is done for cylin¬
drical rolls. An objection to this could be raised on the grounds that
in most experiments it is the hexagonal pattern which is observed. We
justify our assumption by once again making a comparison with Rayleigh-
Benard convection. It appears that in Rayleigh-Benard convection the
wave pattern selection is strongly influenced by the container size and

79
shape, with the hexagonal pattern prevailing for all shapes in wide
containers while in narrower ones the container shape determining the
pattern (e.g., cylindrical rolls for circular containers. See Kosch-
mieder (1967)). Since the principal aim of this paper is to study a
finite geometry effect on morphological instability, the cylindrical
roll pattern would be the logical choice. This is especially valid for
experiments such as those of Peteves (1986) where ampoules of radius
0.025 inches were used.
In Chapter 4 it was shown that the bifurcation diagram is unsymme-
tric for cylindrical rolls and the form of the outer expansions 0Q and
0^ are shown schematically in Fig. 6-1. If we use superscript o to
identify the problem with perfect insulation we can seek solutions by
means of asymptotic expansions about the perfect problem.
00
9,, = Z 0, 6k (6.3-1)
k=0
with similar expansions for 0 , C0, C and £. Substituting these expan-
sions into the problem and collecting the terms of order 6 we get an
inhomogeneous linear system. If we separate variables horizontally
CO
0¡ = I 0¡ J (a.r/R) (6.3-2)
¡C . , X,. o 1
1 = 1 1
and eliminate cl , C1 and we get for the expansion about the planar
X/ s
solution
L *1 = f1
i i i
(6.3-3)

80
where
í1 =
x
í.
i
f! = - I.D2
l l
Vk*
f /k
s s
(6.3-4)
(6.3-5)
I.
i
r J (a.r/R)dr/
o x
rJ (a.r/R)dr
0 1
(6.3-6)
The boundary conditions are
at z = -1, 0^ = 0
at z = s, 91 . = 0
si
at z = 0, B. í1 = hi
li i
where
(6.3-7)
(6.3-8)
(6.3-9)
h. - - I.
i i
Vki
Df,
-Y.f /k
is s
â– Df
(6.3-10)
The eigenvalue problem of Section 5.2 has been shown to have a
complete set of eigenfunctions and hence can be used to solve the above
system.

81
00 CO
♦1 = E 1 Q. . J (a.r/R)
i-1 j=1 1J 1 1J 0 1
(6.3-11)
J(Qij»hi)
M. .
1J
z=0

U i
M. .
1J
(6.3-12)
where J(Q..,h. )| _ is the bilinear concomitant evaluated at z=0
ij i z=0
J(VV
z=0
- VDf)t - Dfs>Vki * DPii (6.3-13)
If we assume that and are nonzero, then when Se = SeQN, Mj^
becomes zero and the outer expansion (6.3-11) - (6.3-13) will fail.
When we expand about the nonplanar solution we end up with a much
more complicated system. But since we have expressed the solution to
the perfect nonplanar problem itself in terms of a perturbation series
in e, we can do the same for the imperfect problem.
i1 = 4>1 + £*] + ... (6.3-14)
o
If we take the zeroth order problem and once again separate vari¬
ables laterally,
*1 = Z *1. J (a.r/R) (6.3-15)
o .=1 oí o 1
and eliminate
eqns. (6.3~3)
and , we get the same system described by
Hoi soi oi
- (6.3-10) but with . as the variable. Again using the
01
eigenfunctions of Section 5.2, the solution is

82
*1 = E E Q.. J (a.r/R)
° i = i j=i 01 ij o l
(6.3-16)
with taking the same form as eqns. (6.3-12) and (6.3-13).
Since these equations are valid only for Se close to SeoN, we can write
an expansion for
dM
N1
M (Se) = M (Se ) + (Se - Se ..)
N1 N1 oN oN dSe
. (6.3-17)
Se = Se
oN
which simplifies to
VSe)
0 +
eSeiNMN1 + 0(G )
(6.3-18)
where
dM...
i. * _ N1
MN1 dSe
Se = Se
oN
(6.3-19)
and Se1N has been derived in Section 4.2. Hence the outer expansion
about the nonplanar solution is
* = * + _í J (a r/R) + 0(e ) +
c oN o N
«Q ,fJt> “ J) o o
i[ cSe1NM'N1'" QN1 J0(aNr/R) * 0
(6.3-20)

83
6.4. The Inner Expansions
The outer expansions described in the previous section fail at Se =
SeoN> requiring inner expansions in this vicinity. For Se very near
CO
Se (p) = Se
(6.4-1)
oN
k=2
<5(p) = pC
(6.4-2)
where b and c are integers to be determined such that the solution does
not fail at Se = Se^. We can now expand the solutions in orders of p.
CO
u(r,z,p) = e(r,z,Se,<5) = E uk(r,z)pk/k!
k=0
(6.4-3)
CO
v(r,z,p) = C(r,z,Se,ó) = E vk(r,z)pk/k!
k=0
(6.4-4)
CO
w(r,z,p) = c(r,Se,ó) = Z wk(r)pk/k!
k=0
(6.4-5)
Here the superscripts on u, v and w are not powers but indices.
Substituting these expansions into the problem set up in Section 6.2 and
collecting the terms of order p, we once again get an inhomogeneous
problem. Separating variables and eliminating v., v and w, the problem
36 S
reduces to
(6.4-6)

84
with boundary conditions
at z = -1 ,
Uu â– 0
at z = s
u . = 0
si
at z = 0,
B.*! = 5'h1
iTi i
where
and
d<5
5 '
Se' =
U=0
dSe
du
p=0
The variables and have been defined in Section 5
and h^ in Section 6.3. But Y^ now becomes
Y = (G, + Se G„ - af A)/(Ge + Se „ G - af A )
1 X, ON Cl s oNci
This means that the operator is singular when i=N,
for solutions to exist 6has to orthogonal to the null
Using this solvability condition with QN1 we get
z=0
- (6.4-7)
(6.4-8)
(6.4-9)
(6.4-10)
(6.4-11)
.2 and f1
i
(6.4-12)
j=1 and hence
space of Ln.
(6.4-13)

85
From this we can conclude that S' = 0 (assuming J * <0*,, .f1,>) and
NI N
that c>1. This makes the system (6.4-6) - (6.4-12) homogeneous and
so are constant multiples of * .
o
*1 = A$o (6.4-14)
Proceeding to the next order we have
L.*2 = S"fJ (6.4-15)
and at z=0, B.^2 = h2 (6.4-16)
ill
h2 = 6"h? - 2ASe'
i i
C . G /(G. + Se „ G -a.A )
oi c i oN c i
+ 2A I. Q
l1
(6.4-17)
where
? 3 ? 2
... = I rJ^ía.r/R)dr/ J rJ (a.r/R)dr
i1Joi Joi
(6.4-18)
and Q is a complex function of the linear stability variables. The
solvability condition for i=N, j=1 now gives
J (Q
N1
(6.4-19)
Hence we can take 6" * 0 and Se' * 0 and so b=1 and c=2. Equation
(6.4-19) is a quadratic in A with two real roots A1 and A2 of opposite
signs, corresponding to two inner expansions I-] and I2.

86
Se = SeoN + £ó1/2 + 0(6)
(6.4-20)
i> = *° + A^*o61/2 + 0(6)
(6.4-21)
In terms of y, the outer expansion 0Q about the planar state of
eqn. (6.3-11) can now be expressed by
♦in ■ - j)qni J0(V/R)/5HÑ1 <6-',-22)
and the nonplanar expansion 01 of eqn. (6.3~20)
♦ill ■ [«0N1.fi> - * V/Se1N]QN, J0(aNr/R> (6.1-23)
where we expressed as J (a^r/R). In attempting to match these
with the inner expansions of eqn. (6.4-21) we get
lim A1 = lim A2 -( - J)/5M' (6.4-24)
£-» -00 £-»00
lim A, = lim A- = D„£/Se,„ (6.4-25)
_ 1 2 N IN
Hence using
A1
in
eqn.
(6.4-21)
will
give the
inner expansion I1
for
matching 0Q
for
Se
< SeoN
with 0.|
for
Se > SeoN.
A2 will result in
I2,
matching 0Q
for
Se
> SeoN
with 0^
for
Se < SeofJ
as shown in Fig. 6-
1 .

87
6.5. Imperfection Due to Advection in the Melt
In modelling morphological instability during solidification it is
customary to neglect the difference in density between liquid and
solid. Though this difference is very small (e.g. for the Pb-Sn system,
the ratio of densities Pg/P^ is 1.035) the volume contraction upon
solidification results in closed streamlined flow in the melt which
fundamentally alters the planar state of this problem. Obtaining the
solution to the planar state with this flow is difficult enough even
without attempting the formidable task of solving the nonplanar prob¬
lem. In order to include the effect of the flow on morphological insta¬
bility previous workers have therefore relied on simplifying assumptions
or looked at limiting cases.
The scenario in a typical solidification experiment is schemati¬
cally shown in Fig. 6-2. The heating coils maintain constant tempera¬
tures at positions z = -l and z = s in the ampoule. As solidification
proceeds, the ampoule .is pulled in the positive z-direction at a velo¬
city v in order that the liquid-solid interface will remain stationary
at z = 0. To fill the space created by volume contraction upon solidi¬
fication, the melt will move towards the crystal with a bulk velocity
of v(p /p. - 1) and this process is commonly referred to as "advec-
S 3C
tion." In the presence of the rigid ampoule walls, this flow also
resembles the bolus flow of slugs through a pipe, resulting in closed
streamlined flow in the melt.
In the literature we find three approaches to tackling this prob¬
lem, the least of which is the work of Caroli, Caroli, Misbah and Roulet
(1985b) who ignored the rigid side walls. Then the only effect of

88
Figure 6-2. Crystal growth with advection in the melt

89
advection is a negligible modification of the growth velocity in the
melt. The other two approaches are more substantial and analize
limiting cases of this phenomenon. Since the traditional formulation of
morphological instability concentrates only at a "boundary layer" region
of the liquid-solid interface, in such a model the closed streamlined
flow shown in Fig. 6-2 can be approximated by Couette flow in Region I
and stagnation flow in Region II. Delves (1968, 1971 and 197*0 and
Coriell, McFadden, Boisvert and Sekerka (1984) looked at the effect of a
forced plane Couette flow and showed that the onset of morphological
instability is somewhat suppressed for disturbances in the flow
direction, while disturbances perpendicular to the flow were
uneffected. Recently McFadden, Coriell and Alexander (1988) examined
the effect of a plane stagnation flow on disturbances perpendicular to
the flow and here too the flow was found to be stabilizing.
Based on these two limiting cases it might be tempting to conclude
that advection generally stabilizes the liquid-solid interface. In this
and the next few sections we will embark upon a more complete analysis
of advection than has been attempted so afar and show that the above
assertion is questionable at best. In our model we consider morphologi¬
cal instability with advection in the absence of natural convection. We
choose not to include natural convection as we wish to study the effect
of advection only and natural convection will only further complicate an
already nontrivial problem. Besides it has been shown very conclusively
by Coriell, Cordes, Boettinger and Sekerka (1980) and by Caroli, Caroli,
Misbah and Roulet (1985a) that the convective and morphological modes
are decoupled except at points where they become comparable. Hence our
model will be valid for the high growth velocity region where morpholo-

90
gical instability is dominant and also in low gravity environments where
natural convection can be neglected.
The experimental set up was briefly described earlier. The steady
state domain equations in dimensionless form are
?v°
(6.5-1)
2 ^i
V C2 - V 3T - U'VCH " °
(6.5-2)
V T =0
s
(6.5-3)
2 3Cs
nV~C - v T-2- = 0
s 9z
(6.5-4)
Here u is the velocity of the closed streamlined flow in the melt
caused by advection and 5 is (p /p. - 1). The boundary conditions are
S X»
at z - -1, Tji " T1 * Cj, = C1
(6.5-5)
at z = s, Ts = T2
(6.5-6)
at z =5, T^ = Ts = - SeC^ + A H
(6.5-7)
VT„*n - fJVT *n = Lv
x. s
(6.5-8)
kC. = C
2, s
(6.5-9)
VC *n - (1+<5)vC. = nVC *n - vC
X/ X/ s s
(6.5-10)

91
Before we can write the equations for u, further assumptions are
necessary. The domain of the velocity equations extend over the entire
melt region, not just the depth l that we have considered. In the float
zone technique this melt region will still have a constant depth as
solidification proceeds but in Bridgman growth the melt region will
continuously shrink. Since crystal growth velocities are usually so
small, even in the latter technique it takes awhile before there is an
appreciable change in the melt depth. Hence over a short time span a
constant depth assumption will be valid even for Bridgman growth. We
will also assume that the melt is bounded on all sides by rigid walls.
This will require a container for zone refining and a very viscous
encapsulant for the melt in Bridgman growth.
Under these assumptions the velocity problem has been solved by
Duda and Vrentas (1971) for low velocities and here we will only give
the solution and refer the interested reader to their paper for the
details.
u = (5vR/r) 3ip/9z, u = 5v(1 - (R/r)9

r* z
CO CO
ill = Z A F (r)Sinnir( 1 + z/Y) + £ G (z)rJ,(b r/R) (6.5-12)
, n n . n 1 n
n=1 n=1
F (r) = [rl, (nirr/Y)I0(mr/h) - r^I. (mr/h)I0(nirr/Y)/R]/I^(mr/h) (6.5-13)
n 1 2 1 2-2
G (z) = 2B [sinhb (z+Y)/R - (z/Y+1)exp(-b z/R)Sinhb h]
n nL n y n n J
+ 2C (z+Y)/R expb h Sinhb z/R
n *n n
(6.5-14)

92
2 2
gn = Jo^bn^ 4bnhexpbnh + ^2exPbnh " exp(-bnh) - exp3t>nh)/h] (6.5-15)
Bngn " -%bexpbnh * AmQmn + 2(1 - exP2bnh) Z ^Qmn^f (6.5-16)
m=1 m=1
hg C = 4(b hexpb h - Sinhb h) Z A Q
nn n*n n . m mn
m=1
(4b h+2 - 2exp2b h) Z A Q (-1 )
n n , m mn
m=1
m
(6.5-17)
bn = \ nir^ I0(nir/h^I2^nir/b^ ” !^(nir/b) J/I^nir/h)
(6.5-18)
mn
2bn(rmr/h)2I2(rmr/h) J^ib^)
I2(mir/h)[ (mir/h)2 + b 2]2
2 n
(6.5-19)
y = l /%, h = Y/R
m
(6.5-20)
where is the total melt depth, R the dimensionless radius of the
ampoule, ur and uz the radial and axial components of the velocity, Jp
the Bessel function of the first kind of order p and ID the modified
Bessel function of the first kind of order p. Duda and Vrentas have
determined the first twenty coefficients of An for various values of h
and we will use these in our calculations.

93
6.6 Nonexistence of the Planar State
The key assumption made in the two previous approaches to solving
the advection problem was that a stable planar state was still possible
even in the presence of melt flows. But this is true only if the velo¬
city profile very close to the interface is assumed to be in a special
form, the vertical component of velocity being independent of any hori¬
zontal coordinates and the horizontal component of velocity at most a
linear function of that horizontal coordinate. As can be seen by eqns.
(6.5-11) - (6.5-19) these assumptions can only be viewed as limiting
cases and in general up and uz will be complex functions of both r and
z. In this section we will use this to prove the nonexistence of planar
solutions for this problem. This proof can also be used to show the
nonexistence of planar solutions for the case of imperfect insulation
considered in Sections 6.6 - 6.4.
In Section 3.1 where we assumed that 6 was zero, a planar solution
was possible with the temperatures and concentrations functions of z
only. Then the energy balance condition at the interface (given by eqn.
(6.5-8)) could be used to determine the constant growth velocity v. In
other words the choice of T^, T2 and fixed v. For the case of non¬
zero 5 we will show that planar solutions do not exist by contradic¬
tion. If there is a possible planar solution eqn. (6.5-2) specifies
that cannot have radially independent solutions. The solution will
be in the form
C (r.z) - I C n=1
(6.6-1)

94
where JQ is the zeroth order Bessel function of the first kind and an
are zeros of the first order Bessel function of the first kind. Then
the boundary conditions (6.5-7), (6.5-9) and (6.5-10) will require
that , Ts and Cs be expressed in similar expansions in terms of Bessel
functions. If we now return once more to eqn. (6.5-8) to calculate v,
we can see that v too should be expressed in terms of Bessel functions
radially, but this contradicts our assumption of a planar solution.
Hence in the presence of advection there can be no planar solutions to
this problem.
6.7. Asymptotic Solution
A complete nonplanar solution of the equations described in Section
6.6 will require tedious numerical calculations. But such calculations
may be unnecessary, for as mentioned before for most systems 6 is an
extremely small number and hence an asymptotic solution should be pos¬
sible about the 6=0 solution. This would be similar to the method
used for analyzing heat loss at the ampoule wall and here too we will
show that advection is a bifurcation breaking imperfection on the mor¬
phological instability problem.
But unlike the problem with imperfect insulation, here the addi¬
tional terms are in the concentration equations, not in the temperature
ones. Hence as done earlier we cannot eliminate the concentrations and
reduce the problem to only the temperature equations and so the eigen¬
value problem defined in Section 5.2 has to be modified to include the
concentrations in the column vector. Correspondingly the differential
operator Li, the boundary operator and the inner product will have to
be changed to those used in Chapter 3.

95
L.Q.. = - M..Q..
i ij ij ij
(6
.7-1)
r
M*
II
(D2-a2)
0 0
0
•
0
— (D2-vD-a2) 0
X 1
0
0
0 (D2-a2)
0
0
0 0 ^
X
(nD2-vD-na2)
O
(6
.7-2)
„ 0 „
<* ,*> = J (TI
-1 *
. + C*C.)dz + í (T*T + C*C )dz
X. i i q 3 3 SS
(6
.7-3)
B. =
i
(1-Yi)
Se Y.
i
i
0
D
0 -6D
0
x/Y.
X
(xSe/Y.-k) 0
1
(6
.7-4)
-vx/Y.
i
•
[D-v(xSe/Y.+1-k)] 0
-nD
where
$ has been
expanded to (T0, C., T , C ) and
36 Yj S S
*1 • x • (l<0lo - Gsc)/ The eigenvalues M^j are the roots of
Se = (- gt + a.A )/ GQ
(6.7-7)

96
Gp is the same as that defined in eqn. (5.2-8) but Gc becomes
G = (G„ B + G )/(B+k)
c He sc
(6.7-8)
B =
(a.. - v/2tana„.)tana .s
ill Zi si
(na . + v/2tana .tana„.
si si Hi
(6.7-9)
2 2 .. . 1 / 2
a.. = (M.. - a. - v /4)
Hi ij l
(6.7-10)
/ .y, 2 2..,. 1/2
a . = (M.. - a. - v /4)
31 ÍJ i
(6.7-11)
The modified eigenvalue problem defined above is nonself adjoint
and therefore its completeness is not assured. The adjoint problem will
have the same form as that obtained in Section 3.3.
Now if we write the outer expansions
00
* = I Ík 5k (6.7-12)
k=0
and collect the first order terms in <5, the solution can be obtained in
the same form as in Section 6.3. For Se < Se0jj we have
I1
i
f| = (0, -lJvDC°, 0, 0)
? ? 2
= J R(3 J oi i o i
(6.7-13)
(6.7-14)
0
(6.7-15)

97
The outer expansion about the planar state then becomes
" " 2
* = *+61 E ——— Q. .j (a.r/R) + 0(5 ) (6.7-16)
c .... M. . 1J O 1
i = 1 J=1 ij
*
where Q. . is the adjoint eigenfunction of Q. . . Similarly the outer
i J l J
expansion about the nonplanar solution is
+ e* J (a r/R) + 0(e )
ON o N
r
6 -cf!' Qv„J U^/R) +
eSe,,.M.t, Ni o N
in N1
0(e°)]
+ 0(62)
(6.7-17)
Once again it can be seen that both solutions will fail as
Se + Se0N and inner expansions are needed.
Se(y) = Se + £yb + Z £kykb/k! (6.7-18)
k=2
6(y) = y°
(6.7-19)
with corresponding expansions for the variables T., T , C., C and £.
Jv S X» s
The first order problem in y then becomes
(6.7-20)

98
where fj has been defined in eqn. (6.7-13). Here all the boundary
conditions will
be homogeneous and the solvability condition gives
6' < QN1 ’fly = ° (6.7-21)
Hence 6' = 0, C
> 1 and = A* . In the next order
T 0
L.*2 = 5"f] (6.7-22)
li i
and at z = 0,
B.*2 = h2 (6.7-23)
h2 = -
i
-2 ASe' [0, ioiGc, 0, 0] + 2A2I.1Q (6.7-24)
I., = 1 rJ^(a.r/R)dr/f rJ2(a.r/R)dr (6.7-25)
11 0 0 1 0 ° 1
The solvability
condition now gives
J(QN1’hi} “ 6" < QN1’fN > = 0 (6.7-26)
So here too we can take 6" * 0 and Se' * 0
Se = SeQN + 5 * = VO
where A refers
V
to A1 and A2, the two solutions of the quadratic equa-
tion in A, eqn.
(6.7-26). The positive solution A1 will give the inner

99
expansion matching the outer expansion eqn. (6.7-16) for Se < SeoN with
the outer expansion eqn. (6.7-17) for Se > SeoN. The negative solution
will result in the inner expansion matching eqn. (6.7-16) for Se >
SeoN with eqn. (6.7-17) for Se < SeoN.
6.8. Controlling Imperfections
In order to calculate the imperfection due to heat loss/gain the
form of the functions f^(z) and fg(z) must be known. A possible form
for these are
Vz) = (Tüc(z) ' V + (T1 " Ta)(z " c)/(1 + ?) (6.8-1)
f (z) = (T (z) - T ) - T - T )(z - c)/(s - z) (6.8-2)
s sc a 2 a
where T_ is the ambient temperature and T. and T are T„ and T in the
planar state defined by eqns. (3.2-2) and (3.2-3). This form was chosen
to satisfy the experimental conditions and to resemble Newton's law of
cooling. Hence 6/k. and <5/k will represent Biot numbers for the liquid
X/ s
and solid sections of the ampoule.
The resulting curves for the two imperfections are shown in Figs.
6-3 and 6-4. Figure 6-3 corresponds to the case of heat gain and
advection in the melt and Fig. 6-4 corresponds to heat loss. In Chapter
4 we concentrated on identifying suitable growth conditions in order to
avoid morphological instability. But as we have shown these
inperfections undermine our earlier effort by causing the liquid-solid
interface to be nonplanar at all times. All this makes the situation

100
Figure 6-3. Imperfect bifurcation diagram for heat loss
and advection
Figure 6-4. Imperfect bifurcation diagram for heat gain

101
seem hopeless for the crystal grower but it should be noted that even
though these imperfections cause nonplanarity right from the beginning,
large deviations from planarity still occur only for Se > SeQN. Still
for those growers concerned even with the small amount of nonplanarity
at Se < SeoN, we suggest a possible way to reduce the effect of
imperfections by mutual cancellation.
With regard to the imperfect insulation case, the important obser¬
vation to be made is that this is an imperfection that can be controlled
to some extent. This can be done either by adjusting the amount of
insulation or by varying the ambient temperature. If this were the only
imperfection in the problem, its effect can be minimized by good insula¬
tion and by keeping Ta between and T2. Coupled as it usually will be
with the imperfection due to advection in the melt, this control can be
used to reduce their effects by mutual cancellation. Total elimination
is of course not possible but by maintaining the ambient temperature
above that of the ampoule it should be possible to minimize the effect
of these imperfections.

CHAPTER 7
NEW DIRECTIONS
There are several important issues still waiting to be addressed in
morphological instability in crystal growth. In this Chapter some of
these which are important and deserve attention will be briefly dis¬
cussed.
7.1. Transition to Dendritic Growth
This is a very open question and it will probably be years before
even preliminary results can be obtained. Ungar and Brown (1985) have
made an attempt in this direction by modelling the formation of deep
cells but were unable to proceed to the point where the cells begin to
coalesce. Another way of looking at their work is to say that they
tried to portray the transition to dendritic growth as a smooth one and
they failed in this.
This brings up the question, is the transition really smooth? If
it is smooth then it can only mean that the failure of Ungar and Brown
was only due to shortcomings of their deep cell model and improving the
model should rectify the problem. But if the transition is similar to
transition to turbulence in fluid flow or transition to chaos in dynami¬
cal systems, then we have a much more complex problem which defies easy
solutions. The experiments of Heslot and Libchaber (1984) indicate that
the coalescing of cells is catastrophic rather than smooth.
102

103
A possible way to determine whether the transition is smooth or not
is to look at secondary bifurcations for the steady state case. Second¬
ary and cascading bifurcations would indicate a more complex transi¬
tion. When Ungar and Brown (1984a) initially did their nonlinear calcu¬
lations with their simplified one-sided model they observed subcritical
bifurcations but when they repeated their calculations for a more accu¬
rate model (Ungar, Bennet and Brown (1985)) they failed to see any.
Unfortunately their calculations were performed for only two sets of
experimental conditions and their results are by no means conclusive.
I have already talked about the multiplicities in the problem and
these could lead to secondary bifurcations and hence examining the
multiplicities in the manner of Reiss (1983) would be a starting
point. If secondary bifurcations are observed an attempt could be made
to show that dendritic growth is a result of chaos in the system.
7.2. Extension to Semiconductor Materials
The theory of morphological instability was developed for binary
alloys but its biggest potential application is in the growth of semi¬
conductor materials from the melt. But these materials are different in
two important respects. Firstly they are not simple binary mixtures
like alloys but complex compounds in the crystal form and secondly all
binary semiconductors like gallium arsenide have a one-to-one ratio of
the components and hence even the melt is not a dilute mixture. The
effect of these is that the equilibrium relations at the liquid-solid
interface have to be reformulated. For example, the liquidus slope can
no longer be assumed to be a straight line and at least a quadratic
relationship will be required.

104
7.3. Inclusion of Microscopic Effects
Until now the researchers in morphological instability have been
fairly successful in studying the problem in the presence of the various
macroscopic influences that beset it while microscopic effects have been
almost completely ignored, except for the influence of grain boundaries
(see Ungar and Brown (1984b)). Anisotropy, that is the influence of the
lattice structure of the crystal on morphological instability, cannot be
ignored and this view has been reinforced by the experiments of Heslot
and Libchaber (1984). But how to include a microscopic effect on a
macroscopic model?
This is a fundamental question not just in crystal growth but in
all fluid mechanics. Molecular dynamics and statistical mechanics show
great promise in bridging the gap between molecular theory and continuum
mechanics (see Bitsanis, Magda, Tirrell and Davis (1987)). Even though
this would require tremendous effort and might seem like an overkill, it
is unavoidable if we hope to come up with an accurate model for the
influence of anisotropy on morphological instability. More importantly
it will be invaluable in understanding the mechanics of solidification
on a molecular level.
7.4. Numerical Methods
Almost all results in this thesis have been obtained by means of
asymptotic expansions. The advantages of this have already been dis¬
cussed in earlier chapters but there were numerous instances where the
effects of large deviations from the basic problem are necessary,

105
requiring numerical solutions. Ideally both types of analysis should be
done, one complementing the other.
Of course if molecular dynamics are employed, a numerical approach
is the only avenue, but even less demanding problems require it. In
this work some calculations were done for a weakly nonlinear analysis
with hexagonal and cylindrical cells but definite conclusion about the
bifurcation diagram could not be reached due to the absence of numerical
nonlinear calculations. Similarly the influences of large ampoule wall
heat losses and natural convection with rigid walls require numerical
solutions not to mention looking for secondary bifurcations along the
nonlinear curve.
7.5. Experiments
As mentioned several time in this thesis, there is a strong need
for quantitative experiments near the onset points. Recently de
Cheveigne et al. (1985 and 1 986) and Heslot and Libchaber (1984) have
made some efforts in this direction but a lot more needs to be done for
experiments to even catch up with theoretical work. Some of the ques¬
tions that need to be answered by experiments are:
* Subcritical and supercritical bifurcations: As discussed in Section
4.4 theory predicts there are subcritical and supercritical bifurca¬
tion regions for this problem and experimental verification of these
would provide very useful information for the practical crystal
grower.
* Wavepattern and wavenumber selection: In this work (as well as in
the work of others) wherever a choice had to be made with regard to

106
wavepattern or wavenumber, the decision was based on theoretical
considerations or analogy with Rayleigh-Benard convection. In some
instances, notably in wavepattern selection, the issue was skirted by
considering all possibilities. Experiments need to be done to verify
the validity of criteria proposed or to establish new criteria in
selections.
* Reducing imperfections: In Section 6.8 a way of reducing the effect
imperfections by playing off one against the other was proposed.
Only experiments can tell if this is a viable solution to the problem
of imperfections.
* Transition to dendritic growth: Heslot and Libchaber have shown that
oscillatory solutions precede dendritic growth. Cell disappearance
and cell splitting are also reported precursors. More work is needed
to establish the thread that leads from the cellular pattern near
onset to dendritic growth.

APPENDIX A
NOMENCLATURE
Latin
A
Ratio defined in eqn. (4.2-24)
A
V
Constant multiple defined by (6.4-14)
a
Wavenumber or aspect ratio
an
Wavenumber for n^h horizontal eigenvalue
V as
Modified wavenumbers defined in (3.2-25) and (3.2-26)
aU' asi
Modified wavenumbers defined in (5.2-19) and (5.2-20)
amin
The wavenumber for which SeQ is a minimum
B
Boundary operator
B
Ratio defined in eqn. (4.2-23)
Bi
Biot number
Bo
Bond number
b
Exponent defined by eqn. (6.4-1)
C1
Boundary concentration at z = -l
crcs
Solute concentrations
c
Exponent defined by eqn. (6.4-2)
Cr
Crispation number
D
d_
dz
dn
Constant used in eqn. (6.4-23)
DrDs
Solute diffusivities
f
Inhomogeneous column vector in the domain
107

103
V fs
Functions defined in (6.2-1) and (6.2-2)
GrGs
Temperature gradients at the interface in the planar
state
Gi,c’Gsc
Concentration gradients at the interface in the
planar state
Gc
(D.G. + D G )/(D0 + D )
l Ic s sc l s
G»p
(k*GJl + ksGs)/(kX. + ks}
g
Gravitational vector
Interface curvature
h
Inhomogeneous column vector at the boundary
h
Y/R
Ji
Ratio of integrals defined by (6.3— 6)
XP
Modified Bessel's function of the first kind of order
P
XiJ
Ratios of integrals for various cell patterns used in
Chapter 4
J
Bilinear concomitant
JP
Bessel's functions of the first kind of order p
k
Distribution coefficient
krks
Thermal conductivities
L
Matrix differential operator for linear problem
L
Latent heat number, L^D^/k^G,^
L,Li,L2>R
Wavelengths for various horizontal cell patterns
Lh
Latent heat
l
Liquid melt depth
l
m
Total melt depth
M
Eigenvalue of augmented morphological problem

109
m
Ma
n
P
Pe
Pr
Q
q
v
R
Ra
s
Se
Se
omin
W1
Ta
T1’T2
t
a
V US
ur» u
v
Absolute value of liquidus slope
Marangoni number
Unit normal vector at the interface, pointing into
the solid
Constant defined in (4.2-15) or modified pressure
Temperature eigenfunctions in augmented morphological
problem
Peclet number, D^/vi.
Prandtl number
Eigenvector of augmented morphological problem
Concentration eigenfunctions in augmented
morphological problem
Ampoule radius
Rayleigh number
Solid depth
Sekerka number, mGc/GT
The minimum value of SeQ
Temperatures
Ambient temperature
Freezing temperature of pure solvent
Boundary temperatures at z = -l and z = s
Deviation from planarity in augmented morphological
problem
Advective velocity
Temperatures in inner expansions
Components of u
Convective velocity
Crystal growth velocity

110
v
V
v
s
w
X
Yi
Concentrations in inner expansions
Horizontal factor of the z component of V or
deviation from planarity in inner expansions
(kG. - G )/(G. - G )
1C SC X. S
Ratio defined in eqn. (5.2-16)
Greek
B
y
ó
e
C
n
9
\
u
5
prps
a
*
Thermal diffusivities
k /k
s
l
l /l
m
Imperfection parameter
Surface Laplacian
Pertubation variable about the planar state
Departure of interface from planarity
Fourier coefficient of temperature
Transformed temperatures defined by (6.2-3)
and (6.2-4)
Interfacial tension
61/C
Stretching parameter defined by eqn. (6.4-1)
Densities
Eigenvalue for linear stability problem
Column vector (T^.C.,! ,C ) or (T.,T )
X» 36 S S 36 S
nth horizontal eigenfunction

Ill
*P Column vector in inner expansion
Adjoint eigenfunction of

Q Inhomogeneous column vector in eqn. (6.4-17)
and (6.7-24)
Gothic
2
A Capillary number,
Gq Modified concentration gradient defined in (3.2-28)
Gj Modified temperature gradient defined in (3.2-27)
Subscripts
c Refers to the planar state
mn m refers to order in the perturbation series in e
n refers to the nth horizontal Fourier coefficient
l Refers to liquid phase
s Refers to solid phase
Superscripts
Dimensionless quantity
Adjoint vector or operator
Complex conjugate

112
Variable in linear stability problem
n Order of terms in perturbation series for n, Pe,
ó or p

APPENDIX B
PHYSICAL PROPERTIES
Properties of Alloy Systems
Pb-Sn
Pb-Sb
C-Austenite
D (cm2 s 1)
3.0x10-^
2.0x10-^
2.0x10-^
1
1
CO
1
e
o
o*
0.159
0.147
0.15
k (J cm 1 s 1 K 1 )
s
0.297
0.275
0.29
m (K/wt %)
2.33
5
65.3
T„ (K)
600
600
1485
\ (J cm 2)
4.27x10-6
2.30x10-6
2.04x10~5
Lh (J cm-^)
256
280
2013
k
0.3
0.4
0.36
-3
pa (g cm )
5
10.66
10.66
7.4
113

114
Properties of Organic Systems
Succinonitrile/Acetone
CBrjj/B^
(cm2 s 1)
1.27x10“5
1 .2x10“5
(J cm ' s 1 K S
2.23x10"3
k (J cm ^ s 1 K 1)
s
2.25x10~3
1 .1x10"2
ra (K/wt %)
3.02
2.9
tm (K)
331
93
\ (J cm"2)
8.94x10-7
7.00x10~7
Lh (J cm-3)
47.8
34.45
k
0.1
0.16
-3
p„ (g cm )
s
1 .02
3.42

REFERENCES
Bitsanis, I., J.J. Magda, M. Tirrell and H.T. Davis, Molecular dynamics
of flow in micropores, J. Chem. Physics, 87, p1733, 1987.
Caroli, B., C. Caroli, C. Misbah and B. Roulet, Solutal convection and
morphological instability in directional solidification of binary al¬
loys, J. de Physique, 46, p401 , 1985a.
Caroli, B., C. Caroli, C. Misbah and B. Roulet, Solutal convection and
morphological instability in directional solidification of binary al¬
loys. II. Effect of the density difference between the two phases,
J. de Physique, 46, pi 657, 1985b.
Cheveigne, S. de, C. Guthman and M.M. Lebrun, Nature of the transition
of the solidification front of a binary mixture from a planar to a
cellular morphology, J. Crystal Growth, 73, p242, 1985.
Cheveigne, S. de, C. Guthman and M.M. Lebrun, Cellular instabilities in
directional solidification, J. de Physique, 47, p2095, 1986.
Clyne, T.W. and W. Kurz, Solute redistribution during solidification
with rapid solid state diffusion, Metall. Trans. A, 12A, p965, 1981.
Coriell, S.R., M.R. Cordes, W.J. Boettinger and R.F. Sekerka, Convective
and interfacial instabilities during unidirectional solidification of
a binary alloy, J. Crystal Growth, 49, pi 3* 1980.
Coriell, S.R., G.B. McFadden, R.F. Boisvert and R.F. Sekerka, Effect of
a force Couette flow on coupled convective and morphological
instabilities during unidirectional solidification, J. Crystal
Growth, 69, p. 15, 1984.
Coriell, S.R., G.B. McFadden and R.F. Sekerka, Cellular growth during
directional solidification, Ann. Rev. Mater. Sci., 15, p. 119, 1985.
Coriell, S.R. and R.F. Sekerka, Morphological stability near a grain
boundary groove in a solid-liquid interface during solidification of
a pure substance, J. Crystal Growth, 19, p90, 1972.
Coriell, S.R. and R.F. Sekerka, Morphological stability near a grain
boundary groove in a solid-liquid interface during solidification of
a binary alloy, J. Crystal Growth, 19, p285, 1973.
Coriell, S.R. and R.F. Sekerka, Oscillatory morphological instabilities
due to non-equilibrium segregation, J. Crystal Growth, 61, p499,
1983.
115

116
Delves, R.T., Theory of the stability of a solid-liquid interface during
growth from stirred melts, J. Crystal Growth, 3, p562, 1968.
Delves, R.T., Theory of the stability of a solid-liquid interface during
growth from stirred melts II, J. Crystal Growth, 8, pi3, 1971.
Delves, R.T., Theory of interface stability, in Crystal Growth, ed. B.R.
Pamplin, Pergamon, Oxford, 1974.
Drazin, P.G. and W.H. Reid, Hydrodynamic Stability, Cambridge University
Press, Cambridge, 1981.
Duda, J.L. and J.S. Vrentas, Steady flow in the region of closed stream
lines in a cylindrical cavity, J. Fluid Mech., 45, p247, 1971.
Glicksman, M., S.R. Coriell and G.B. McFadden, Interaction of flows with
the crystal-melt interface, Ann. Rev. Fluid Mech., 18, p307, 1986.
Hardy, S.C. and S.R. Coriell, Morphological stability and the ice-water
interfacial free energy, J. Crystal Growth, 3, 4, p569, 1968.
Hardy, S.C. and S.R. Coriell, Morphological stability of cylindrical ice
crystals, J. Crystal Growth, 5, p329, 1969.
Hardy, S.C. and S.R. Coriell, Morphological stability of ice cylinders
in aqueous solution, J. Crystal Growth, 7, pl47, 1970.
Heslot, F. and A. Libchaber, Unidirectional crystal growth and crystal
anisotropy, Physica Scripta, T9, pi 26, 1985.
Hurle, D.T.J., On similarities between the theories of morphological
instability of a growing binary alloy crystal and Rayleigh-Benard
convective instability, J. Crystal Growth, 72, p738, 1985.
Iooss, G. and D.D. Joseph, Elementary Stability and Bifurcation Theory,
Springer-Verlag, New York, 1980.
Joseph, D.D., Non-linear heat generation and stability of the tempera¬
ture distribution in conducting solids, Int. J. Heat and Mass Trans.,
8, p28l, 1965.
Joseph, D.D., Stability of Fluid Motions II, Springer-Verlag, Berlin,
1976.
Koschmeider, E.L., On convection under an air surface, J. Fluid Mech.,
30, p9, 1967.
Malkus, W.V.R. and G. Veronis, Finite amplitude cellular convection, J.
Fluid Mech., 4, p225, 1958.
Matkowsky, B.J. and E.L. Reiss, Singular perturbations of bifurcations,
SIAM J. Appl. Math., 33, p230, 1977.

117
McFadden, G.B., R.F. Boisvert and R.F. Sekerka, Nonplanar interface
morphologies during unidirectional solidification of a binary alloy
II: three dimensional computations, J. Crystal Growth, 84, p371,
1987.
McFadden, G.B. and S.R. Coriell, Nonplanar interface morphologies during
unidirectional solidification of a binary alloy, Physica, 12D, p253,
1984.
McFadden, G.B., S.R. Coriell and J.I.D. Alexander, Hydrodynamic and free
boundary instabilities during crystal growth: The effect of a plane
stagnation flow, to be published in J. Crystal Growth, 1988.
Morris, L.R. and W.C. Winegard, The development of cells during the
solidification of a dilute Pb-Sn alloy, J. Crystal Growth, 5, p36l,
1969.
Mullins, W.W. and R.F. Sekerka, Morphological stability of a particle
growing by diffusion or heat flow, J. Appl. Physics, 34, p323, 1963.
Mullins, W.W. and R.F. Sekerka, Stability of a planar interface during
solidification of a dilute binary alloy, J. Appl. Physics, 35, p444,
1964.
Nadarajah, A. and R. Narayanan, On the completeness of the Rayleigh-
Marangoni and Graetz eigenspaces and the simplicity of their eigen¬
values, Quart. Appl. Math., 55, p8l, 1987.
Nadarajah, A. and R. Narayanan, Morphological instability in dilute
binary systems: a uniform approach, Submitted to Physico-Chemical
Hydrodynamics, 1988.
Naimark, M.A., Linear Differential Operators, Part I, Frederick Ungar,
New York, 1967.
Peteves, S.D., Growth kinetics of faceted solid-liquid interfaces,
Doctoral dissertation, University of Florida, Gainesville, 1986.
Ramprasad, N. and R.A. Brown, Spatial wavelength dependence of direc¬
tional solidification cells with finite depth, Paper 84j, AIChE
Annual Meeting in New York, November 1987.
Reiss, E.L., Cascading bifurcations, SIAM J. Appl. Math., 43, p57, 1983.
Rosenblat, G., G.M. Homsy and S.H. Davis, Eigenvalues of the Rayleigh-
Benard and Marangoni problems, Phys. Fluids, 24, p2115, 1981.
Rosenblat, G., G.M. Homsy and S.H. Davis, Nonlinear Marangoni convection
in bounded layers. Part 1. Circular cylindrical containers, J. Fluid
Mech., 120, p91, 1982.
Rutter, J.W. and B. Chalmers, A prismatic substructure formed during
solidification of metals, Can. J. Physics, 31, p15, 1953.

113
Sarma, G.S.R., Interaction of surface-tension and buoyancy mechanisms in
horizontal liquid layers, J. Thermophys. Heat Trans., 1, p1 29, 1987.
Scriven, L.E. and C.V. Sternling, On cellular convection driven by sur¬
face-tension gradients: effects of mean surface tension and surface
viscosity, J. Fluid Mech., 19, p321, 1964.
Segel, L.A. and J.T. Stuart, On the question of the preferred mode in
cellular thermal convection, J. Fluid Mech., 13, p289, 1962.
Seidensticker, R.G., Stability considerations in temperature gradient
zone melting, in Crystal Growth, ed. H.S. Peiser, Pergamon, Oxford,
1967.
Sriranganathan, R., D.J. Wollkind and D.B. Oulton, A theoretical
investigation of the development of interfacial cells during the
solidification of a binary alloy, J. Crystal Growth, 62, p265, 1983.
Stuart, J.T., On the nonlinear mechanics of wave disturbances in stable
and unstable parallel flows, J. Fluid Mech., 9, p353, I960.
Tavantzis, J., E.L. Reiss and B.J. Matkowsky, On the smooth transition
to convection, SIAM J. Appl. Math., 34, p322, 1978.
Tiller, W.A. and S.W. Rutter, The effect of growth conditions upon the
solidification of a binary alloy, Can. J. Physics, 34, p96, 1956.
Tiller, W.A., J.W. Rutter, K.A. Jackson and B. Chalmers, The redistribu¬
tion of solute atoms during the solidification of metals, Acta
Metall., 1, p428, 1953-
Trivedi, R. and K. Somboonsuk, Constrained dendritic growth and spacing,
Mat. Sci. and Eng., 65, p65, 1984.
Ungar, L.H., Directional solidification from a bifurcation viewpoint,
Doctoral dissertation, M.I.T., Cambridge, 1984.
Ungar, L.H., M.J. Bennett and R.A. Brown, Cellular interface morphol¬
ogies in directional solidification. III. The effects of heat
transfer and solid diffusivity, Physical Rev. B, 31. p5923, 1985.
Ungar, L.H. and R.A. Brown, Cellular morphologies in directional solidi¬
fication: the one-sided model, Physical Rev. B, 29, pi 367, 1984a.
Ungar, L.H. and R.A. Brown, Cellular interface morphologies in direc¬
tional solidification. II. The effect of grain boundaries, Phys.
Rev. B, 30, p3993, 1984b.
Ungar, L.H. and R.A. Brown, Cellular interface morphologies in direc¬
tional solidification. IV. The formation of deep cells, Physical
Rev. B, 31, P5931, 1985.

119
Vrentas, J.S., R. Narayanan and S.S. Agrawal, Free surface convection in
a bounded cylindrical geometry, Int. J. Heat and Mass Trans., 24,
pi 513» 1981.
Watson, J., On the nonlinear mechanics of wave disturbances in stable
and unstable parallel flows, J. Fluid Mech., 9, p371, I960.
Wolf, M., T.W. Clyne and W. Kurz, Microstructure and cooling conditions
of steel solidified in the continuous casting mold, Arch. Eisenhut-
tenwesen, 53, p3, 1982.
Wollkind, D.J. and S. Raissi, A nonlinear stability analysis of the
melting of a dilute binary alloy, J. Crystal Growth, 26, p277, 1974.
Wollkind, D.J. and L.A. Segel, A nonlinear stability analysis of the
freezing of a dilute binary alloy, Phil. Trans. Roy. Soc. London A,
268, p351, 1970.
Wollkind, D.J. and S. Wang, A nonlinear stability analysis of a model
equation for liquid phase electro-epitaxial growth of a dilute binary
substance, SIAM J. Appl. Math., 48, p52, 1988.
Woodruff, D.P., The stability of a planar interface during the melting
of a binary alloy, Phil. Mag., 17, p83, 1968.

BIOGRAPHICAL SKETCH
The author was born on the 3rd of March, 1959, in Jaffna, Sri
Lanka, but moved shortly thereafter to Colombo where he had his early
education. In July, 1978, he joined the chemical engineering program at
the Indian Institute of Technology, Madras, and received his bachelor's
degree in June, 1983. In August of the same year he began his graduate
studies in chemical engineering at the University of Florida and did
research on Rayleigh-Benard convection on his way to a master's degree
in August, 1984. Since then he has been a doctoral student in chemical
engineering at the same institution.
l on

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
/V,
Ranganathan Narayanan, Chairman
Associate Professor of Chemical
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
c4 2/'
Ulrich H. Kurzweg
Professor of Engineering Sciences
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Gerasimos K. Lyberatos
Assistant Professor of Chemical
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
William E. Lear, Jr.
Assistant Professor of Mechanical
Engineering

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
5
A. s'.
Spyros A. Svoronos
Associate Professor of Chemical
Engineering
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
August 1988
Cl. Hjuh*
Dean, College of Engineering
Dean, Graduate School

UNIVERSITY OF FLORIDA
1262 08556 7823




PAGE 1

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