Title Page
 Front Matter
 Table of Contents
 List of Tables
 List of Figures
 Theoretical and experimental...
 Experimental apparatus and...
 Pure Ga growth kinetics
 Summary and conclusions
 Appendix I: Gallium
 Appendix II: Ga-In system
 Appendix III: Heat transfer at...
 Appendix IV: Interfacial stability...
 Appendix V: Printouts of computer...
 Appendix VI: Supersaturation and...
 Biographical sketch

Title: Growth kinetics of faceted solid-liquid interfaces
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00090206/00001
 Material Information
Title: Growth kinetics of faceted solid-liquid interfaces
Physical Description: xxi, 340 leaves : ill. ; 28 cm.
Language: English
Creator: Peteves, Stathis D., 1957-
Publication Date: 1986
Copyright Date: 1986
Subject: Crystal growth   ( lcsh )
Materials Science and Engineering thesis Ph.D
Dissertations, Academic -- Materials Science and Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph.D.)--University of Florida, 1986.
Bibliography: Bibliography: leaves 318-339.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: Stathis D. Peteves.
 Record Information
Bibliographic ID: UF00090206
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000931593
oclc - 16243779
notis - AEP2535
oclc - 016243779


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Table of Contents
    Title Page
        Page i
        Page ii
    Front Matter
        Page iii
        Page iv
        Page v
        Page vi
        Page vii
    Table of Contents
        Page viii
        Page ix
        Page x
        Page xi
    List of Tables
        Page xii
    List of Figures
        Page xiii
        Page xiv
        Page xv
        Page xvi
        Page xvii
        Page xviii
        Page xix
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        Page xxi
        Page 1
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    Theoretical and experimental background
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    Experimental apparatus and procedures
        Page 117
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    Pure Ga growth kinetics
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    Summary and conclusions
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    Appendix I: Gallium
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    Appendix II: Ga-In system
        Page 278
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    Appendix III: Heat transfer at the solid/liquid interface
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    Appendix IV: Interfacial stability analysis
        Page 299
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    Appendix V: Printouts of computer programs
        Page 305
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    Appendix VI: Supersaturation and supercooling
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    Biographical sketch
        Page 340
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Full Text







To the antecedents of phase changes: Leucippus, Democritus,
Epicurus and, the other Greek Atomists, who first realized that a
material persists through a succession of transformations (e.g.


The assumption of the last stage of my graduate education at the

University of Florida has been due to people, aside from books and good

working habits. It is important that I acknowledge all those individ-

uals who have made my stay here both enjoyable and very rewarding in

many ways.

Professor Reza Abbaschian sets an example of hard work and devotion

to research, which is followed by the entire metals processing group.

Although occasionally, in his dealings with other people, the academic

fairness is overcome by his strong and genuine concern for the research

goals, I certainly believe that I could not have asked more of a thesis

advisor. I learned many things through his stimulation of my thinking

and developed my own ideas through his strong encouragement to do so.

His constant support and guidance and his unlimited accessibility have

been much appreciated. I am grateful to him for making this research

possible and for passing his enthusiasm for substantive and interesting

results to me. At the same time, he encouraged me to pursue any side

interests in the field of crystal growth, which turned out to be a very

exciting and "lovable" field. Finally, I thank him for his understand-

ing and his tolerance of my character and habits during "irregular"

moments of my life.

Professors Robert Reed-Hill and Robert DeHoff have contributed to

my education at UF in the courses I have taken from them and discussions

of my class work and research. Their reviews of this manuscript and


their insight to several parts of it was greatly appreciated. Professor

Ranganathan Narayanan has been very helpful with his expertise in fluid

flow; his suggestions and review of this work is very much acknowledged.

I thank Professor Tim Anderson for many helpful comments and for critic-

ally reviewing this manuscript. My thanks are also extended to Profes-

sor Robert Gould for his acceptance when asked to review this work, for

his advice, and for his continuous support.

Julio Alvarez deserves special thanks. We came to the University

at the same time, started this project, and helped each other in closing

many of the "holes" in the crystal growth of gallium story. He intro-

duced me to the world of minicomputers and turned my dislike for them

into a fruitful working tool. He did the work on the thermoelectric

effects across the solid-liquid interface. His collaboration with me in

the laboratory is often missed.

The financial support of this work, provided by the National

Science Foundation (Grant DMR-82-02724), is gratefully acknowledged.

I am also grateful to several colleagues and friends for their

moral support. I thank Robert Schmees and Steve Abeln for making me

feel like an old friend during my first two semesters here. Both hard-

core metallurgists helped me extend my interest in phase transforma-

tions; I shared many happy moments with them and nights of Mexican

dinners and "mini skirt contests" at the Purple Porpoise. With Robert,

I also shared an apartment; I thank him for putting up with me during my

qualifying exams period, teaching me the equilibrium of life and making

the sigma phase an unforgettable topic. Joselito Sarreal, from whom I

inherited the ability to shoot pictures and make slides, taught me to

stop worrying and enjoy the mid-day recess; his help, particularly in my

last year, is very much acknowledged. Tong Cheg Wang helped with the

heat transfer numerical calculations and did most of the program writ-

ing. From Dr. Richard Olesinski I learned surface thermodynamics and to

argue about international politics. Lynda Johnson saved me time during

the last semester by executing several programs for the heat transfer

calculations and corrected parts of the manuscript. I would also like

to thank Joe Patchett, with whom I shared many afternoons of soccer, and

Sally Elder, who has been a constant source of kindness, and all the

other members of the metals processing group for their help.

I have had the pleasure of sharing apartments with George Blumberg,

Robert Schmees, Susan Rosenfeld, Diana Buntin, and Bob Spalina, and I am

grateful to-them for putting up with my late night working habits, my

frequent bad temper, and my persistence on watching "Wild World of

Animals" and "David Letterman." I am very thankful to my friends, Dr.

Yannis Vassatis, Dr. Horace Whiteworth, and others for their continuous

support and encouragement throughout my graduate work.

I would also like to thank several people for their scientific

advice when asked to discuss questions with me; Professors F. Rhines (I

was very fortunate to meet him and to have taken a course from him), A.

Ubbelohde, G. Lesoult, A. Bonnissent, and Drs. N. Eustathopoulos (for

his valuable discussions on interfacial energy), G. Gilmer, M. Aziz, and

B. Boettinger. Sheri Taylor typed most of my papers, letters, did me

many favors, and kept things running smoothly within the group. I also

thank the typist of this manuscript, Mary Raimondi.

My very special thanks to Stephanie Gould for being the most im-

portant reason that the last two years in my life have been so happy. I

am so grateful to her for her continuous support and understanding and

particularly for forcing me to remain "human" these final months.

I also especially thank my parents and my sister for 29 and 25

years, respectively, of love, support, encouragement, and confidence in




ACKNOWLEDGEMENTS ................................................... iv

LIST OF TABLES .................. ................................... xii

LIST OF FIGURES ............................. ........................xiii

ABSTRACT ................................ ... ....................... xxi


INTRODUCTION ................................ ....................... 1



The Solid/Liquid (S/L) Interface ................................... 6

Nature of the Interface ............................. ............. 6
Interfacial Features .................... ......................... 8
Thermodynamics of S/L Interfaces .............................. .. 10
Models of the S/L Interface .............. ....................... 14
Diffuse interface model ....................... ..... .......... 14
The "a" factor model: roughness of the interface .............. 22
Other models ................................................... 25
Experimental evidence regarding the nature of the S/L interface 30

Interfacial Roughening ................................. ............ 36

Equilibrium (Thermal) Roughening .................. ................. 36
Equilibrium Crystal Shape (ESC) .................................. 46
Kinetic Roughening .............................................. 48

Interfacial Growth Kinetics ........................................ 53

Lateral Growth Kinetics (LG) ................... ................. .. 53
Interfacial steps and step lateral spreading rate (u ) ......... 54
Interfacial atom migration ...................... .............. 57
Two-dimensional nucleation assisted growth (2DNG) .............. 58
Two-dimensional nucleation ........................ ........... 59
Mononuclear growth (MNG) ..................................... 62


Polynuclear growth (PNG) ..................................... 64
Screw dislocation-assisted growth (SDG) ........................ 68
Lateral growth kinetics at high supercoolings ................... 72
Continuous Growth (CG) ........................................... 73
Growth Kinetics of Kinetically Roughened Interfaces .............. 78
Growth Kinetics of Doped Materials ............................... 83

Transport Phenomena During Crystal Growth .......................... 87

Heat Transfer at the S/L Interface ............................... 88
Morphological Stability of the Interface ......................... 93
Absolute stability theory during rapid solification ............ 98
Effects of interfacial kinetics ................................ 99
Stability of undercooled pure melt ............................. 100
Experiments on stability ....................................... 101
Segregation .................. .................................. 102
Partition coefficients ......................................... 102
Solute redistribution during growth ............................ 10.
Convection ..................................................... 106

Experimental S/L Growth Kinetics ................................... 112

Shortcomings of Experimental Studies ............................. 112
Interfacial Supercooling Measurements ............................ 113


EXPERIMENTAL APPARATUS AND PROCEDURES .............................. 117

Experimental Set-Up ..................... ........................... 117

Sample Preparation .................... ............................. 120

Interfacial Supercooling Measurements .............................. 125

Thermoelectric (Seebeck) Technique ............................... 125
Determination of the Interface Supercooling ...................... 129

Growth Rates Measurements .......................................... 134

Experimental Procedure for the Doped Ga ............................. 10


RESULTS ............................................................ 146

(111) Interface .......................... .......................... 146

Dislocation-Free Growth Kinetics ................................. 150
MNG region .............................. ....................... 155
PNG region ....................................... .............. 156

Dislocation-Assisted Growth Kinetics ............................. 159
Growth at High Supercoolings, TRG Region ......................... 161

(001) Interface .................................................... 164

Dislocation-Free Growth Kinetics ................................. 166
MNG region ..................................................... 166
PNG region ....................................... .......... 172
Dislocation-Assisted Growth Kinetics ............................. 173
Growth at High Supercoolings, TRG Region ......................... 174

In-Doped (111) Ga Interface ........................................ 175

Ga-.01 wt% In .................................................... 175
Ga-.12 wt% In .................................................... 187


DISCUSSION ......................................................... 194

Pure Ga Growth Kinetics ............................................ 194

Interfacial Kinetics Versus Bulk Kinetics ........................ 194
Evaluation of the Experimental Method ............................ 197
Comparison with the Theoretical Growth Models at Low Supercoolings 203
2DNG kinetics .................................................. 204
SDG kinetics ................................................... 209

Generalized Lateral Growth Model .................................. 213

Interfacial Diffusivity .......................................... 218
Step Edge Free Energy ............................................ 220
Kinetic Roughening ....................... ......................... 230

Disagreement Between Existing Models for High Supercoolings
Growth Kinetics and the Present Results ............................ 235

Results of Previous Investigations ................................. 242

In-Doped Ga Growth Kinetics ........................................ 246

Solute Effects on 2DNG Kinetics ................................. 246
Segregation/Convection Effects .................................. 249


CONCLUSIONS AND SUMMARY ............................................ 258


I GALLIUM ........................................................ 263

II Ga-In SYSTEM ................................................... 278

III HEAT TRANSFER AT THE S/L INTERFACE ............................ 280

IV INTERFACIAL STABILITY ANALYSIS ................................. 299

V PRINTOUTS OF COMPUTER PROGRAMS ................................ 305

VI SUPERSATURATION AND SUPERCOOLING ............................... 316

REFERENCES ......................................................... 318

BIOGRAPHICAL SKETCH ................................................ 340



TABLE 1 Mass Spectrographic Analysis of Ga (99.9999%) ........... 122

TABLE 2 Mass Spectrographic Analysis of Ga (99.99999%) .......... 123

TABLE 3 Seebeck Coefficient and Offset Thermal EMF of the (111)
and (001) S/L Ga Interface .............................. 131

TABLE 4 Typical Growth Rate Measurements for the (111) Interface. 137

TABLE 5 Analysis of In-Doped Ga Samples ......................... 141

TABLE 6 Seebeck Coefficients of S/L In-Doped (111) Ga Interfaces 142

TABLE 7 Experimental Growth Rate Equations ...................... 176

TABLE 8 Experimental and Theoretical Values of 2DNG Parameters .. 205

TABLE 9 Experimental and Theoretical Values of SDG Parameters ... 210

TABLE 10 Growth Rate Parameters of General 2DNG Rate Equation .... 213

TABLE 11 Calculated Values of g .................................. 238

TABLE 12 Solutal and Thermal Density Gradients ................... 252

TABLE A-i Physical Properties of Gallium .......................... 265

TABLE A-2 Metastable and High Pressure Forms of Ga ................ 267

TABLE A-3 Crystallographic Data of Gallium (a-Ga) ................. 271

TABLE A-4 Thermal Property Values Used in Heat Transfer
Calculations ............................................ 289



Figure 1 Interfacial Features. a) Crystal surface of a sharp
interface; b) Schematic cross-sectional view of a
diffuse interface. After Ref. (17) ................... 9

Figure 2 Variation of the free energy G at Tm across the
solid/liquid interface, showing the origin of asz.
After Ref. (22) ........................................ 13

Figure 3 Diffuse interface model. After Ref. (6). a) The sur-
face free energy of an interface as a function of its
position. A and B correspond to maxima and minima con-
figuration; b) The order parameter u as a function of
the relative coordinate x of the center of the inter-
facial profile, i.e. the Oth lattice plane is at -x .... 16

Figure 4 Graph showing the regions of continuous (B) and lateral
(A) growth mechanisms as a function of the parameters
P and y, according to Temkin's model.7 ................ 21

Figure 5 Computer drawings of crystal surfaces (S/V interface,
Kossel crystal, SOS model) by the MC method at the
indicated values of KT/0. After Ref. (112) ............ 42

Figure 6 Kinetic Roughening. After Ref. (117). a) MC inter-
face drawings after deposition of .4 of a monolayer on
a (001) face with KT/4 = .25 in both cases, but differ-
ent driving forces (Ap). b) Normalized growth rates of
three different FCC faces as a function of Au, showing
the transition in the kinetics at large supersaturations 50

Figure 7 Schematic drawings showing the interfacial processes for
the lateral growth mechanisms a) Mononuclear. b) Poly-
nuclear. c) Spiral growth. (Note the negative curva-
ture of the clusters and/or islands is just a drawing
artifact.) .............................. ............... 63

Figure 8 Free energy of an atom near the S/L interface. QL and
Qs are the activation energies for movement in the
liquid and the solid, respectively. Qi is the energy
required to transfer an atom from the liquid to the
solid across the S/L interface ........................ 74


Figure 9 Interfacial growth kinetics and theoretical growth rate
equations .............................................. 79

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Figure 22

Transition from lateral to continuous growth according
to the diffuse interface theory;25 no is the melt
viscosity at Tm ........................................ 81

Heat and mass transport effects at the S/L interface.
a) Temperature profile with distance from the S/L
interface during growth from the melt and from solution.
b) Concentration profile with distance from the interface
during solution growth .................................. 90

Bulk growth kinetics of Ni in undercooled melt. After
Ref. (201) ............................................. 92

Solute redistribution as a function of distance solid-
ified during unidirectional solidification with no con-
vection ................................................ 105

Crystal growth configurations. a) Upward growth with
negative GL. b) Downward growth with positive GL. In
both cases the density of the solute is higher than the
density of the solvent ................................. 109

Experimental set-up .................................... 118

Gallium monocrystal, X 20 .............................. 124

Thermoelectric circuits. a) Seebeck open circuit, b)
Seebeck open circuit with two S/L interfaces ........... 126

The Seebeck emf as a function of temperature for the
(111) S/L interface .................................... 132

Seebeck emf of an (001) S/L Ga interface compared with
the bulk temperature ................................... 133

Seebeck emf as recorded during unconstrained growth of a
Ga S/L (111) interface compared with the bulk supercool-
ing; the abrupt peaks (D) show the emergence of disloca-
tions at the interface, as well as the interactive
effects of interfacial kinetics and heat transfer ...... 135

Experimental vs. calculated values of the resistance
change per unit solidified length along the [111]
orientation vs. temperature ............................ 139

Seebeck emf vs. bulk temperature as affected by dis-
location(s) and interfacial breakdown, recording during
growth of In-doped Ga .................................. 144


Figure 23

Figure 24

Figure 25

Figure 26

Figure 27

Figure 28

Figure 29

Figure 30

Figure 31

Figure 32

Figure 33

Figure 34

Figure 35

Dislocation-free and Dislocation-assisted growth rates
of the (111) interface as a function of the interface
supercooling; dashed curves represent the 2DNG and SDG
rate equations as given in Table 7 ..................... 149

Growth rates of the (111) interface as a function of
the interfacial and the bulk supercooling .............. 151

The logarithm of the (111) growth rates plotted as a
function of the logarithm of the interfacial and bulk
supercoolings; the line represents the SDG rate equation
given in Table 7 ........................................ 152

The logarithm of the (111) growth rates versus the
reciprocal of the interfacial supercooling; A is the S/L
interfacial area ........................................


Dislocation-free (111) low growth rates versus the inter-
facial supercooling for 4 samples, two of each with the
same capillary tube cross-section diameter .............. 157

The logarithm of the MNG (111) growth rates normalized
for the S/L interfacial area plotted versus the recip-
rocal of the interface supercooling .....................

Polynuclear (111) growth rates versus the reciprocal of
the interface supercooling; solid line represents the
PNG rate equation, as given in Table 7 ..................

Dislocation-assisted (111) growth rates versus the inter-
face supercooling; line represents the SDG rate equation,
as given in Table 7 .....................................

Dislocation-free and Dislocation-assisted growth rates
of the (001) interface as a function of the interface
supercooling; dashed curves represent the 2DNG and SDG
rate equations, as given in Table 7 .....................

The logarithm of the (001) growth rates versus the log-
arithm of the interface supercooling; dashed line rep-
resents the SDG rate equation, as given in Table 7 ......

Growth rates of the (001) and (111) interfaces as a
function of the interfacial supercooling ................

The logarithm of the (001) growth rates versus the
reciprocal of the interface supercooling ................

The logarithm of dislocation-free (001) growth rates
versus the reciprocal of the interface supercooling for
10 samples; lines A and B represent the PNG rate equa-
tions, as given in Table 7 ..............................









Figure 36

Figure 37

Figure 38

Figure 39

Figure 40

Figure 41

Figure 42

Figure 43

Figure 44

Figure 45

Figure 46

The logarithm of the (001) low growth rates (MNG) nor-
malized for the S/L interfacial area plotted versus the
reciprocal of the interface supercooling ................

Growth rates as a function of distance solidified of
Ga-.01 wt% In at different bulk supercoolings; (t )
indicates interfacial breakdown .........................

Photographs of the growth front of Ga doped with .01
wt% In showing the entrapped In rich bands (lighter
region) X 40 ............................................

Initial (111) growth rates of Ga-.01 wt% In as a func-
tion of the interface supercooling; ('. ---) effect of
distance solidified on the growth rate, and (--) growth
rate of pure Ga .........................................

Effect of distance solidified on the growth rate of
Ga-.01 wt% In grown in the direction parallel to the
gravity vector (a,b), and comparison with that grown in
the antiparallel direction (a) ..........................

Initial (111) growth rates of Ga-.01 wt% In grown in the
direction parallel to the gravity vector; ('--C-) effect
of distance solidified on the growth rate, and (--)
growth rate of pure Ga ..................................

Comparison between the growth rates of Ga-.01 wt% In in
the direction parallel ( 0) and antiparallel ( 0 ) to
the gravity vector as a function of the interface super-
cooling; line represents the growth rate of pure Ga .....


behavior of Ga-.12 wt% In (111) interface; a)
rates as a function of distance solidified, b)
front of Ga-.12 wt% In, X 40; solid shows as
regions .......................................... 188

Initial (111) growth rates of Ga-.12 wt% In as a function
of the interface supercooling; (*-0-") effect of distance
solidified on the growth rate, and (- ) growth rate of
pure Ga ................................................. 189

Initial (111) growth rates of Ga-.01 wt% In ( 0 ) and
Ga-.12 wt% In ( < ) as a function of the interface
supercooling; line represents the growth rate of pure
Ga ...................................................... 191

Initial (111) growth rates of Ga-.12 wt% In growth in the
direction parallel to the gravity vector as a function of
the interface supercooling; (*D0-") effect of distance
solidified, and (--) growth rate of pure Ga ........... 192

Figure 47

Figure 48

Figure 49

Figure 50

Figure 51

Figure 52

Figure 53

Figure 54

Figure 55

Figure 56

Figure 57

Figure 58

Initial (111) growth rates of Ga-.01 wt% In ( [ O )
and Ga-.12 wt% In ( X 0 ) grown in the direction
parallel ( X 0 ) and antiparallel ( 0 0 ) to
the gravity vector; continuous line represents the
growth rate of pure (111) Ga interface ..................

The logarithm of the (111) rates versus the reciprocal
of the interfacial (open symbols) and bulk supercooling
(closed symbols) for two samples sizes ..................

Absolute thermoelectric power of solid along the three
principle Ga crystal axes and, liquid Ga as a function
of temperature ..........................................




Comparison between optical and "resistance" growth rates;
the latter were determined simultaneously by two inde-
pendent ways (see programs #2, 3 in Appendix IV) ........ 202

Comparison between the (111) experimental growth rates
and calculated, via the General 2DNG rate equation, as
a function of the supercooling .........................


Comparison of the (001) experimental growth rates and
those calculated, using the General 2DNG rate equation,
growth rates as a function of the supercooling; note that
the PNG calcu lated rates were not formulated so as to
include the two observed experimental PNG kinetics ...... 215

The step edge free energy as a function of the inter-
facial supercooling. a) oe (AT) for steps on the (001)
interface. b) oe (AT) for steps on the (111) interface


The (111) and (001) growth rates as a function of the
interfacial supercooling. The dashed lines are calcu-
lated in accord with the general 2DNG rate equation "cor-
rected" for Di and supercooling dependent oe ............ 226

Comparison between the (111) dislocation-assisted growth
rates and the SDG Model calculations shown as dashed
lines ................................................... 227

Experimental (001) dislocation-assisted growth rates as
compared to the SDG Model calculated rates (dashed lines)
as a function of the interface supercooling ............. 229

The (111) growth rates versus the interface supercooling
compared to those determined from CS on the solid/vapor
interface (Ref. (117)) ..................................

The (111) growth rates versus the interface supercooling
compared to the combined mode of 2DNG and SDG growth
rates (dashed line) at high supercoolings ..............




Figure 59

Figure 60

Figure 61

Figure A-i

Figure A-2

Figure A-3

Figure A-4

Figure A-5

Figure A-6

Figure A-7

Figure A-8

Figure A-9

Figure A-10

Figure A-11

Comparison between the (001) growth curves and those
predicted by the diffuse interface model.6 .............. 236

Normalized (111) growth rates as a function of the nor-
malized supercooling for interface supercoolings larger
than 3.5C; continuous line represents the universal
dendritic law growth rate equation.336 .................. 243

Density gradients as a function of growth rate .......... 253

The gallium structure (four unit cells) projected on the
(010) plane; triple lines indicate the covalent (Ga2)
bond .................................................... 272

The gallium structure projected on the (100) plane;
double lines indicate the short covalentt) bond distance
dl. Dashed lines outline the unit cell ................. 273

The gallium structure projected on the (001) plane;
double lines indicate the covalent bond and dashed lines
outline the unit cell ................................... 274

Ga-In phase diagram ..................................... 279

Geometry of the interfacial region of the heat transfer
analysis; Lf is the heat of fusion ...................... 282

Temperature correction 6T for the (111) interface as a
function of Vri for different heat-transfer conditions,
Uiri; --- Analytical calculations (KL = Ks = K), --
Numerical calculations .................................. 290

Temperature correction 6T for the (001) interface as a
function of Vri for different values of Uiri; --- Anal-
ytical, -- Numerical calculations ...................... 291

Temperature distribution across the S/L (111) and (001)
interfaces as a function of the interfacial radius; ---
Analytical model calculations, -- Numerical calcula-
tions ................................................... 292

Ratio of the Temperature correction at any point of the
interface to that at the edge as a function of r' for
different values of Uiri/Ks ............................. 294

Comparison between the (111) Experimental results ( O )
and the Model (--- Analytical, -- Numerical) calcula-
tions, at low growth rates (V < .2 cm/s) ................ 295

Comparison between the (111) Experimental results (0,0)
and the Model (--- Analytical, -- Numerical) calcula-
tions as a function of Vri for given growth conditions .. 296


Figure A-12

Figure A-13

Comparison between the (001) Experimental results ( O )
and the Model (--- Analytical, -- Numerical) calcula-
tions as a function of Vri for given growth conditions .. 298

The critical wavelength Xcr at the onset of the insta-
bility as a function of growth rate; hatched area indi-
cates the possible combination of wavelengths and growth
rates that might lead to unstable growth front for the
given sample size (i.d. = .028 cm) ...................... 303

Figure A-14 The stability term R(w) as a function of the perturba-
tion wavelength and growth rate ......................... 304


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




December 1986

Chairman: Dr. Gholamreza Abbaschian
Major Department: Materials Science and Engineering

A novel method based on thermoelectric principles was developed to

monitor in-situ the interfacial conditions during unconstrained crystal

growth of Ga crystals from the melt and to measure the solid-liquid

(S/L) interface temperature directly and accurately. The technique was

also shown to be capable of detecting the emergence of dislocation(s) at

the crystallization front, as well as the interfacial instability and


The dislocation-free and dislocation-assisted growth kinetics of

(111) and (001) interfaces of high purity Ga, and In-doped Ga, as a

function of the interface supercooling (AT) were studied. The growth

rates cover the range of 10-3 to 2 x 104 m/s at interface supercoolings

from 0.2 to 4.60C, corresponding to bulk supercoolings of about 0.2 to

53C. The dislocation-free growth rates were found to be a function


of exp(-1/AT) and proportional to the interfacial area at small super-

coolings. The dislocation-assisted growth rates are proportional to

AT2tanh(l/AT), or approximately to ATn at low supercoolings, with n

around 1.7 and 1.9 for the two interfaces, respectively. The classical

two-dimensional nucleation and spiral growth theories inadequately des-

cribe the results quantitatively. This is because of assumptions treat-

ing the interfacial atomic migration by bulk diffusion and the step edge

energy as independent of supercooling. A lateral growth model removing

these assumptions is given which describes the growth kinetics over the

whole experimental range. Furthermore, the results show that the fac-

eted interfaces become kineticallyy rough" as the supercooling exceeds a

critical limit, beyond which the step edge free energy becomes negli-

gible. The faceted-nonfaceted transition temperature depends on the

orientation and perfection of the interface. Above the roughening

supercooling, dislocations do not affect the growth rate, and the rate

becomes linearly dependent on the supercooling.

The In-doped Ga experiments show the effects of impurities and

microsegregation on the growth kinetics, whose magnitude is also depend-

ent on whether the growth direction is parallel or antiparallel to the

gravity vector. The latter is attributed to the effects of different

connective modes, thermal versus solutal, on the solute rich layer ahead

of the interface.


Melt growth is the field of crystal growth science and technology

of "controlling" the complex process which is concerned with the forma-

tion of crystals via solidification. Melt growth has been the subject

of absorbing interest for many years, but much of the recent scientific

and technical development in the field has been stimulated by the in-

creasing commercial importance of the process in the semiconductors in-

dustry. The interest has been mainly in the area of the growth of crys-

tals with a high degree of physical and chemical perfection. Although

the technological need for crystal growth offered a host of challenging

problems with great practical importance, it sidetracked an area of re-

search related to the fundamentals of crystal growth. The end result is

likely obvious from the common statement that "crystal growth processes

remain largely more of an art rather than a science." The lack of in-

depth understanding of crystal growth processes is also due, in part, to

the lack of sensors to monitor the actual processes that take place at

the S/L interface. Indeed, it is the "conditions" which prevail on and

near the crystal/liquid interface during growth that govern the forma-

tion of dislocations and chemical inhomogeneities of the product crys-

tal. Therefore, a fundamental understanding of the melt growth process

requires a broad knowledge of the solid-liquid (S/L) interface and its

energetic and dynamics; such an understanding would, in turn, result in

many practical benefits.


Crystal growth involves two sets of processes; one on the atomic

scale and the other on the macroscopic scale. The first one deals with

the attachment of atoms to the interface and the second with the trans-

port of heat and mass to or from the growth front. Information regard-

ing the interfacial atomistic process, both from a theoretical and tech-

nical point of view, can be obtained from the interfacial growth kinet-

ics. Growth kinetics, in turn, express the mathematical relationship

between the growth rate (V) and the thermodynamic driving force, as re-

lated to the supercooling (AT) or supersaturation (AC), the analytical

form of which portrays a particular growth mechanism related to the

nature of the interface.

The main emphasis of this dissertation is to study the atomistic

processes occurring in the S/L interfacial region where the atoms or

molecules from the liquid assume the ordered structure of the crystal,

and to evaluate the effects of different factors, such as the structure

and nature of the interface, the driving force, and the crystal orienta-

tion, physical defects, and impurities on the growth behavior and kin-

etics. Another aim of the work is to obtain accurate and reliable

growth kinetics that would a) allow further insight to the growth mech-

anisms and their dependence on the above mentioned factors and b) pro-

vide accurate data against which the existing growth models can be test-

ed. In this respect, the growth behavior at increased departures from

equilibrium and any possible transitions in the kinetics is of prime


A reliable kinetics determination, however, cannot be made without

the precise determination of the interface temperature and rate. This


investigation plans to overcome the inherent difficulty of measuring the

actual S/L interface by using a recently developed technique during a

conjunct study about thermoelectric effects across the S/L interfaces.'

As shown later, this technique will also provide the means of a sensi-

tive and continuous way of in-situ monitoring of the local interfacial

conditions. The growth rates will also be measured directly and corre-

lated with the interfacial supercoolings for a wide range of supercool-

ings and growth conditions, well suited to describe the earlier men-

tioned effects on the growth processes.

High purity gallium, and gallium doped with known amounts of In

were used in this study because, a) it is facet forming material and has

a low melting temperature, b) it is theoretically important because it

belongs to a special class of substances which are believed to offer the

most fruitful area of S/L interfacial kinetics research, and c) of prac-

tical importance in the crystal growth community. Furthermore, detailed

and reliable growth rate measurements at low rates are already available

for Ga;2 the latter study is among the very few conclusive kinetics

studies for melt growth which provides a basis of comparison and a chal-

lenge to the present study for continuation of the much needed remaining

work at high growth rates.

The remainder of this introduction will briefly describe the fol-

lowing chapters of this thesis. Chapter II is a critical overview of

the theoretical and experimental aspects of crystal growth from the

melt. This subject demands an unusually broad background since it is a

truly interdisciplinary one in the sense that contributions come from

many scientific fields. The various sections in the chapter were

arranged so that they follow a hierarchal scheme based on a conceptual

view of approaching this subject. The chapter starts with a broad dis-

cussion of the S/L interfacial nature and its morphology and the models

associated with it, together with their assumptions, predictions, and

limitations. The concept of equilibrium and dynamic roughening of

interfaces are presented next, which is followed by theories of growth

mechanisms for both pure and doped materials. Finally, transport phe-

nomena during crystal growth and the experimental approaches for deter-

mination of S/L interfacial growth kinetics are presented.

In Chapter III the experimental set-up and procedure are presented.

The experimental technique for measuring the growth rate and interface

supercooling is also discussed in detail.

In Chapter IV the experimental results are presented in three sec-

tions; the first two sections are for two interfaces of the pure mater-

ial, while the third one covers the growth kinetics and behavior of the

doped material. Also, in this chapter the growth data are analyzed and,

whenever deemed necessary, a brief association with the theoretical

models is made.

In Chapter V the experimental results are compared with existing

theoretical growth models, emphasizing the quantitative approach rather

than the qualitative observations. The discrepancies between the two

are pointed out and reasons for this are suggested based on the concepts

discussed earlier. The classical growth kinetics model for faceted

interfaces is also modified, relying mainly upon a realistic description

of the S/L interface. Finally, the effects of segregation and fluid

flow on the growth kinetics of the doped material are interpreted.


Final comments and conclusions are found in Chapter VI. The Appen-

dices contain detailed calculations and background information on the Ga

crystal structure, Ga-In system, morphological stability, heat transfer,

computer programming, and supercooling/supersaturation relations.


The Solid/Liquid (S/L) Interface

Nature of the Interface

The nature and/or structure of interfaces between the crystalline

and fluid phases have been the subject of many studies. When the fluid

phase is a vapor, the solid-vapor (S/V) interface can easily be des-

cribed by associating it with the crystal surface in vacuum,3'4 which

can be studied directly on the microscopic scale by several experimental

techniques.s However, this is not the case for the S/L interface, which

separates two adjacent condensed phases, making any direct experimental

study of its properties very difficult, if not impossible. In contrast

with the S/V interface, here the two phases present (S and L) have many

properties which are rather similar and the separation between them may

not be abrupt. Furthermore, liquid molecules are always present next to

the solid and their interactions cannot be neglected, as can be done for

vapors. The S/L interface represents a far more peculiar and complex

case than the S/V and L/V interfaces; therefore, ideas developed for the

latter interfaces do not properly portray the actual structure of the

solid/liquid interface. In the following section, the conceptual des-

cription of the various types of S/L interfaces will be given, and each

type of interface will be briefly related to a particular growth mechan-


Two criteria have been used to classify S/L interfaces. The first

one, which is mainly an energetic rather than a structural criterion,



considers the interface as a region with "intermediate" properties of

the adjacent phases, rather than as a surface contour which separates

the solid and the liquid side on the atomic level. According to this

criterion, the interface is either diffuse or sharp.6-10 A diffuse

interface, to quote,6 "is one in which the change from one phase to the

other is gradual, occurring over several atom planes" (p. 555). In

other words, moving from solid to liquid across the interface, one

should expect a region of gradual transition from solid-like to liquid-

like properties. On the other hand, a sharp interfaces-10 is the one

for which the transition is abrupt and takes place within one inter-

planar distance. A specific feature related to the interfacial diffuse-

ness, concerning the growth mode of the interface, is that in order for

the interface to advance uniformly normal to itself (continuously), a

critical driving force has to be applied.6 This force is large for a

sharp interface, whereas it is practically zero for an "ideally diffuse"


The second criterion8-12 assumes a distinct separation between

solid and liquid so that the location of the interface on an atomic

scale can be clearly defined. In a manner analogous to that for the S/V

interface, the properties of the interface are related to the nature of

the crystalline substrate and/or macroscopic thermodynamicc) properties

via "broken-bonds" models. Based on this criterion, the interface is

either smooth (singular, 13 faceted) or rough (non-singular, non-

faceted). A smooth interface is one that is flat on a molecular scale,

represented by a cusp (pointed minimum) in the surface free energy as a

* Sometimes these interfaces are called F- and K-faces, respectively.13


function of orientation plot (Wulff's plot"4 or y-plot15). In contrast,

a rough interface has several adatoms (or vacancies) on the surface

layers and corresponds to a more gradual minimum in the Wulff's plot.

Any deviation from the equilibrium shape of the interface will result in

a large increase in surface energy only for the smooth type. Thus, on

smooth interfaces, many atoms (e.g. a nucleus) have to be added simul-

taneously so that the total free energy is decreased, while on rough

interfaces single atoms can be added.

Another criterion with rather lesser significance than the previous

ones is whether or not the interface is perfect or imperfect with re-

spect to dislocations or twins.11 In principle this criterion is con-

cerned with the presence or absence of permanent steps on the interface.

Stepped interfaces, as will become evident later, grow differently than

perfect ones.

Interfacial Features

There are several interfacial features (structural, geometric, or

strictly conceptual) to which reference will be made frequently through-

out this text. Essentially, these features result primarily from either

thermal excitations on the crystal surface or from particular interfa-

cial growth processes, as will be discussed later. These features which

have been experimentally observed, mainly during vapor deposition and on

S/L interfaces after decanting the liquid,16 are shown schematically in

Fig. la for an atomically flat interface. (Note that the liquid is

omitted in this figure for a better qualitative understanding of the

structure.) These are a) atomically flat regions parallel to the top-

most complete crystalline layer called terraces or steps; b) the edges

Terraces, Steps

Edge (ledge)


Figure 1 Interfacial Features. a) Crystal surface of a sharp
interface; b) Schematic cross-sectional view of a
diffuse interface. After Ref.(17)


(or ledges) of these terraces that are characterized by a step height h;

c) the kinks, or jogs, which can be either positive or negative; and d)

the surface adatoms or vacancies. From energetic considerations, as

understood in terms of the number of nearest neighbors, adatoms "prefer"

to attach themselves first at kink sites, second at edges, and lastly on

the terraces, where it is bonded to only one side. With this line of

reasoning, then, atoms coming from the bulk liquid are incorporated only

at kinks, and as most crystal growth theories imply,18 growth is

strongly controlled by the kink-sites.

Although the above mentioned features are understood in the case of

an interface between a solid and a vapor where one explicitly can draw a

surface contour after deciding which phase a given atom is in, for S/L

interfaces there is considerable ambiguity about the location of the

interface on an atomic scale. However, the interfacial features (a-c)

can still be observed in a diffuse interface, as shown schematically in

Fig. lb. Thus, regardless of the nature of the interface, one can

refer, for example, to kinks and edges when discussing the atomistics of

the growth processes.

Thermodynamics of S/L Interfaces

Solidification is a first order change, and, as such, there is dis-

continuity in the internal energy, enthalpy, and entropy associated with

the change of state.19 Furthermore, the transformation is spatially

discontinuous, as it begins with nucleation and proceeds with a growth

process that takes place in a small portion of the volume occupied by

the system, namely, at the interface between the existing nucleus (crys-

tal seed or substrate) and the liquid. The equilibrium thermodynamic


formulation to interfaces, first introduced by Gibbs20 forms the basis

of our understanding of interfaces. The intention here is not to review

this long subject, but rather to introduce the concepts previously high-

lighted in a simple manner. If the temperature of the interface is

exactly equal to the equilibrium temperature, Tm, the interface is at

local equilibrium and neither solidification nor melting should take

place. Deviations from the local equilibrium will cause the interface

to migrate, provided that any increase in the free energy due to the

creation of new interfacial area is overcome so that the total free

energy of the system is decreased. On the other hand, the existence of

the enthalpy change, AH = HL HS, means that removal of a finite amount

of heat away from the interface is required for growth to take place.

At equilibrium (T = Tm) the Gibbs free energies of the solid and

liquid phases are equal, i.e. GL = GS. However, at temperatures less

than Tm, only the solid phase is thermodynamically stable since GS < GL.

The driving force for crystal growth is therefore the.free energy dif-

ference, AGv, between the solid and the supercooled (or supersaturated)

liquid. For small supercoolings, AGv can be written as

G, LT (1)

where L is the heat of fusion per mole and Vm is the solid molar volume.

The S/L interfacial energy is likely the most important parameter des-

cribing the energetic of the interface, as it controls, among others,

the nucleation, growth, and wetting of the solid by the liquid. Accord-

ing to the original work of Gibbs, who considered the interface as a

physical dividing surface the S/L interfacial free energy is related to


the "work done to create unit area of interface." Analytically Oas can

be given by

Osz = UsT TSs 1 + PVi = Us TSsZ (2)

where UsZ is the surface energy per unit area, SsZ is the surface en-

tropy per unit area, and the surface volume work, PVi, is assumed to be

negligible. A further understanding of the surface energy, as an excess

quantity for the total energy of the two phase system (without the

interface), can be achieved by considering Fig. 2. Here the balance in

free energy across the interface is accomodated by the extra energy of

the interface, Ost.

The step edge (ledge) free energy is concerned with the effect of a

step on the crystal surface of an otherwise flat face. As discussed

later, this quantity is a very important parameter related to the exist-

ence of a lateral growth mechanism versus a continuous one and the

roughening transition. In order to understand the concept of edge free

energy, consider the step (see Fig. 1) as a two-dimensional layer that

perfectly wets the substrate. In this particular case, the extra inter-

facial area created (relative to that without the step) is the periph-

ery; the energetic barrier for its formation accounts for the step edge

energy. Based on this concept, the step edge free energy is comparable

to the interfacial energy and, in some sense, the values of these two

parameters are complementary. For example, it has been stated21 that

for a given substance and crystal structure, the lower the surface free

energy of an interface, the higher the edge free energy of steps on it

and vice-versa. However, such a suggestion is contradictory to the

traditionally accepted analytical relation given as6






--TT S
m L

S / L

Figure 2 Variation of the free energy G at T across the solid-
liquid interface, showing the origin of a s. After
Ref. (22).


oe = Os9 h (3)

where oe is the edge energy per unit length of the step and h is the

step height. However, this relation, as discussed later, has not been

supported by experimental results.

Models of the S/L Interface

As may already be surmised, the most important "property" of the

interface in relation to growth kinetics is whether the interface is

rough or smooth, sharp or diffuse, etc. This, in turn, will largely

determine the behavior of the interface in the presence of the driving

force. Before discussing the S/L interface models, one should disting-

uish between two interfacial growth mechanisms, i.e. the lateral (step-

wise) and the continuous (normal) growth mechanisms. According to the

former mechanism, the interface advances layer by layer by the spreading

of steps of one (or an integral number of) interplanar distance; thus,

an interfacial site advances normal to itself by the step height only

when it has been covered by the step. On the other hand, for the con-

tinuous growth mechanism, the interface is envisioned to advance normal

to itself continuously at all atomic sites.

Whether there is a clear cut criterion which relates the nature of

the interface with either of the growth mechanisms and how the driving

force affects the growth behavior are discussed in the following sec-


Diffuse interface model

According to the diffuse interface growth theory,6 lateral growth

will take over "when any area in the interface can reach a metastable

equilibrium configuration in the presence of the driving force, it will


remain there until the passage of the steps" (p. 555). Afterwards, ob-

viously, the interface has the same free energy as before, since it has

advanced by an integral number of interplanar spacings. On the other

hand, if the interface cannot reach the metastable state in the presence

of the driving force, it will move spontaneously. This model, which

involves an analogy to the wall boundary between neighboring domains in

ferromagnets,23 assumes that the free energy of the interface is a peri-

odic function of its mean position relative to the crystal planes, as

shown in Fig. 3a. The maxima correspond to positions between lattice

planes. The free energy, F (per unit area), of the interface is given


F = a E {f(un) + Ka-2(un .n+1)2} (4)

where a is the interplanar distance and the subscripts n, n + 1, repre-

sent lattice planes and K is a constant; u is related to some degree of

order, and f(un) is the excess free energy of an intermediate phase

characterized by u, formed from the two bulk phases (S and L). The

second term represents the so-called gradient energy,24 which favors a

gradual change (i.e. the diffuseness) of the parameter un. Leaving

aside the analytical details of the model, the solution obtained for the

values of u's which minimize F are given as

u(z) = tanh (z) (5)

where z is a distance normal to the interface and the quantity



a) ai




Position of interface

-3 -2 -1



___ __ ___ I____hjluuI



Figure 3 Diffuse interface model. After Ref. (6). a) The
surface free energy of an interface as a function
of its position. A and B correspond to maxima and
minima configuration; b) The order parameter u as
a function of the relative coordinate x of the
center of the interfacial profile, i.e. the Oth
lattice place is at -x.

l v

n = (2/a) (K/f)1/2 (6)

signifies the thickness of the interface in terms of lattice planes. As

expected, the larger diffuseness of the interface, the larger is the co-

efficient K characterizing the gradient energy and the smaller the quan-

tity fo which relates to the function f(un). The interesting feature of

this model is that the surface energy is not constant, but varies peri-

odically as a function of the relative coordinate x of the center of the

interface where the lattice planes are at z = na -x (see Fig. 3b).

Assuming the interface profile to be constant regardless of the value of

x we have

o(x) = o, + g(x)oo (7)

where oo is the minimum value for a, and cog(x) represents the "lattice

resistance to motion" and g(x) is the well known diffuseness parameter

that for large values of n is given as

-4 4 3 2nirx t n
g(x) = 2 4 n (1 cos --) exp (- ) (8)
a 2

Note that g(x) decreases with the increasing diffuseness n. Its limits

are 0 and 1, which represent the cases of an ideally diffuse and sharp

interface, respectively.

In the presence of a driving force, AGv, if the interface moves by

6x, the change in free energy is given as

6F = (AG + o d(x)) 6x (9)
v o dx

For the movement to occur, 6F must be negative. The critical driving

force is given by

-AG = dg(x) Trogmax(10)
v dx max a
2 3 2
n an
max 8 exp (- ) (11)
max 8 2

Thus, if the driving force is greater than the right hand side of eq.

(10), which represents the difference between the maxima and minima in

Fig. 3a, the interface can advance continuously. The magnitude of the

critical driving force depends on g(x), which is of the order of unity

and zero for the extreme cases of sharp and ideally diffuse interfaces,

respectively. In between these extremes, i.e. an interface with an

intermediate degree of diffuseness, lateral growth should take place at

small supercoolings (low driving force) and be continuous at large AT's.

Detailed critiques from opponents and proponents of this theory

have been reported elsewhere.25-27 A summary is given next by pointing

out some of the strong points and the limitations of this theory: 1)

The concept of the diffuse interface and the gradient energy term were

first introduced for the L/V interface,24 which exhibits a second order

transition at the critical temperature, Tc, where the thickness of the

interface becomes infinite.28 Since a critical point along the S/L line

in a P-T diagram has not been discovered yet, the quantities f(un) and

the gradient energy are hard to qualify for the solid-liquid interface.

The diffuseness of the interface is determined by a balance between the

energy associated with a gradient, e.g. in density, and the energy re-

quired to form material of intermediate properties. The concept of the

diffuseness was extended to S/L interfaces6 after observing29 that the

grain boundary energy (in the cases of Cu, Au, and Ag) is larger than

two times the OsZ value. 2) The theory does not provide any analytical


form or rule for prediction of the diffuseness of the interface for a

given material and crystal direction. However, the model predicts6 that

the resistance to motion is greatest for close-packed planes and, thus,

their diffuseness will comparatively be quite small. 3) The theory,

which has been reformulated for a fluid near its critical point30 (and

received experimental support24,31), provides a good description of

spinodal decomposition32'33 and glass formation.3

The present author believes that this theory's concept is very rea-

sonable about the nature of the S/L interface. Indeed, recent studies,

to be discussed next, indirectly support this theory. However, there

are several difficulties in "following" the analysis with regard to the

motion of the interface, which stem primarily from the fact that it a)

does not explicitly consider the effect of the driving force on the dif-

fuseness of the interface, and b) conceives the motion of the interface

as an advancing averaged profile rather than as a cooperative process on

an atomic scale, which is important for smooth interfaces.

In a later development7 about the nature of the S/L interface, many

aspects of the original diffuse interface theory were reintroduced via

the concept of the many-level model." Here the thickness of the inter-

face, i.e. its diffuseness, is considered a free parameter that can ad-

just itself in order to minimize the free energy of the interface (F);

the latter is evaluated by introducing the Bragg-Williams35 approxima-

* As contrasted to other models where the transition from solid to
liquid is assumed to take place within a fixed and usually small num-
ber of layers, e.g. two-level or two-dimensional models.


tion,* and depends on two parameters of the model, namely B and y, given

v 4W
S= and y 4W

here W = Es (Ess + EZg)/2 is the mixing energy, EsZ is the bond

energy between unlike molecules and Ess, Ezz are the bond energies

between solid-like and liquid-like molecules, respectively; K is the

Boltzman's constant.

Numerical calculations show that the interface under equilibrium is

almost sharp for y > 3 and increases its diffuseness with decreasing y.

It can also be shown that the roughness of the interface defined as10i36
S = U (12)

where Uo is the surface energy of a flat surface and U that of the act-

ual interface. The latter increases with decreasing y, with a sharp

rise at y -2.5. This is expected since U is related to the average num-

ber of the broken bonds (excess interfacial energy).37

When the interface is undercooled, AGv < 0, the theory shows a pro-

nounced feature. The region of positive values of the parameters B and

y can be divided into two subregions, as shown in Fig. 4. In region A

there are two solutions, each corresponding to a minimum and a maximum

of F, respectively, while in region B there are no such solutions. In

* The Bragg-Williams or Molecular or Mean Field approximation35 of stat-
istical mechanics assumes that some average value E can be taken as
the internal energy for all possible interfacial configurations and
that this value is the most probable value. Then, the free energy of
the interface becomes a solvable quantity. Qualitatively speaking,
this approximation assumes a random distribution of atoms in each
layer; therefore, clustering of atoms is not treated.


10-2 A



10-5 l
0 1 2 3 Y

Figure 4 Graph showing the regions of continuous (B) and lateral
(A) growth mechanisms as a function of the parameters
B and y, according to Temkin's model.7


this region, F varies monotonically so that the interface can move con-

tinuously. On the other hand, in region A the interface must advance by

the lateral growth mechanism. Moreover, depending on the y value, a

material might undergo a transition in the growth kinetics at a measur-

able supercooling. For example, if y = 2, the transition from region A

to region B should take place at an undercooling of about .05 Tm (assum-

ing that L/KT, 1, which is the case for the majority of metals). How-

ever, to make any predictions, W has to be evaluated; this is a diffi-

cult problem since an estimate of the EsZ values requires a knowledge of

the interfaciall region" a-priori. It is customarily assumed that Egs =

EZ which leads to a relation between W and the heat of fusion, L. But

this approximation, the incorrectness of which is discussed elsewhere,

leads, for example, to negative values of asZ for pure metals.38 Never-

theless, if this assumption is accepted for the moment, it will be shown

that Temkin's model stands somehow between those of Cahn's and Jackson's

(discussed next).

The "a" factor model: roughness of the interface

Before discussing the "a" factor theory,8'9 the statistical mechan-

ics point of view of the structure of the interface is briefly des-

cribed. The interfacial structure is calculated by the use of a parti-

tion function for the co-operative phenomena in a two-dimensional lat-

tice. Indeed, the change of energy accompanying attachment or detach-

ment of a molecule to or from a lattice site on the crystal surface can-

not be independent of whether the neighboring sites are occupied or not.

A large number of models39 have been developed under the assumptions i)


the statistical element is capable of two states only and ii) only

interactions between nearest neighbors are important.

The "a" factor theory, introduced by Jackson,8 is a simplified

approach based on the above mentioned principles for the S/L interface.

This model considers an atomically smooth interface on which a certain

number of atoms are randomly added, and the associated change in free

energy (AG) with this process is estimated. The problem is then to

minimize AG. The major simplifications of the model are a) a two-level

model interface: as such it classifies the molecules into "solid-like"

and "liquid-like" ones, b) it considers only the nearest neighbors, and

c) it is based on Bragg-Williams statistics.

The main concluding point of the model is that the roughness of the

solid-liquid interface can be discriminated according to the value of

the familiar "a" factor, defined as

a= L (13)

where E represents the ratio of the number of bonds parallel to the

interface to that in the bulk; its value is always less than one and it

is largest for the most close-packed planes, e.g. for the f.c.c. struc-

ture (111) = .5, (100) = 1/3, and (110) = 1/6. It should be noted

that the a factor is actually the same with y in Temkin's theory. For

values of a < 2, the interface should be rough, while the case of a > 2

may be taken to represent a smooth interface. Alternately, for mater-

ials with L/KTm < 2, even the most closely packed interface planes

should be rough, while for L/KTm > 4 they should be smooth. According

to this, most metallic interfaces should be rough in contrast with those

of most organic materials which have large L/KTm factors. In between


these two extremes (2 and 4) there are several materials of considerable

importance in the crystal growth community, such as Ga, Bi, Ge, Si, Sb,

and others such as H20. For borderline materials (a = 2), the effect of

the supercooling comes into consideration. For these cases, this model

qualitatively suggests26,40 that an interface which is smooth at equil-

ibrium temperature may roughen at some undercooling.

Jackson's theory, because of its simplicity and its somewhat broad

success, has been widely reviewed in many publications.25,26,27,'3 The

concluding remarks about it are the following:

a) In principle, this model is based on the interfacial "roughness"

point of view.10'36 As such, it attempts to ascribe the interfacial

atoms to the solid or the liquid phase, which, as mentioned elsewhere,

is likely to be an unrealistic picture of the S/L interface. Thus, the

model excludes a probable "interface phase" that forms between the bulk

phases so that its quantitative predictions are solely based on bulk

properties (e.g. L).

b) The model is essentially an equilibrium one since the effect of

the undercooling on the nature of the interface was hardly treated.

Hence, it is concluded that a smooth interface will grow laterally, re-

gardless of the degree of the supercooling. A possible transition in

the nature of the interface with increasing AT is speculated only for

materials with a 2. Indeed, it is for these materials that the model

actually fails, as will be discussed later.

c) The anisotropic behavior of the interfacial properties is lumped

in the geometrical factor E, which could be expected to make sense only


for flat planes or simple structures, but not for some complex struc-


d) In spite of the limitations of this model, the success of its

predictions is generally good, particularly for the extreme cases of

very smooth and very rough interfaces.26'27'34

Other models

The goal of most other theoretical models of the S/L interface is

the determination of the structural characteristics of the interface

that can then be used for the calculation of thermodynamic properties

which are of experimental interest; the majority of these models follow

the same approaches that have been applied for modeling bulk liquids.

Therefore, these are concerned with spherical (monoatomic) molecules

that interact with the (most frequently used) Lennard-Jones, 12-6,

potential.42 The L-J potential, which excludes higher than pair contri-

bution to the internal energy, is a good representation of rare gasses

and its simple form makes it ideal for computer calculations. The model

approach can be classified into three groups:

a) hard-sphere,

b) computer simulations (CS); molecular dynamics (MD), or Monte

Carlo (MC), and

c) perturbation theories.

In the Bernal model (hard-sphere),43 the liquid as a dense random

packing of hard spheres is set in contact with a crystal face, usually

with hexagonal symmetry (i.e. FCC (111), HCP (0001)). Computer algor-

ithms of the Bernal model have been developed4 based on tetrahedral

packing where each new sphere is placed in the "pocket" of previously


deposited spheres on the crystalline substrate. Under this concept, the

model44,'4 shows how the disorder gradually progresses with distance

from the interface into the liquid. The beginning of disorder, on the

first deposited layer, is accounted by the existence of "channels""4 (p.

6) between atom clusters, whose width does not allow for an atom to be

placed in direct contact with the substrate. As the next layer is de-

posited, new sites are eventually created that do not continue to follow

the crystal lattice periodicity, which, when occupied, lead to disorder.

However, the very existence of the formed "channels" is explained by the

peculiarity of the hcp or fcc close-packed crystal face that has two

interpenetrating sublattices of equal occupation probabilities.4 The

density profiles calculated at the interface also show a minimum associ-

ated with the existence of poor wetting; on the other hand, perfect wet-

ting conditions were found when the atoms were placed in such a way that

no octahedral holes were formed.46 Thermodynamic calculations from

these models allow for an estimate of the interfacial surface energy

(oUs), which are in qualitative agreement with experimental findings.

In conclusion, these models give a picture of the structure of the

interface which seems reasonable and can calculate asg. However, they

neglect the thermal motion of atoms and assume an undisturbed crystal

lattice up to the S/L interface, eliminating, therefore, any kind of

interfacial roughness.

Computer simulation of MC and MD techniques are linked to micro-

scopic properties and describe the motion of the molecules. In contrast

with the MD technique, which is a deterministic process, the MC tech-

nique is probabilistic. Another difference is that time scale is only


involved in the MD method, which therefore appears to be better suited

to study kinetic parameters (e.g. diffusion coefficients). From the

simulations the state parameters such as T, P, kinetic energy, as well

as structural interfaciall) parameters, can be obtained. Furthermore,

free energy (entropy) differences can be calculated provided that a ref-

erence state for the system is predetermined. The limitations of the CS

techniques are4 a) a limited size sample (-1000 molecules), as compared

to any real system, because of computer time considerations; the small

size (and shape) of the system might eliminate phenomena which might

have occurred otherwise. b) The high precision and long time required

for the equilibriation of the system (for example, the S/L interface is

at equilibrium only at T,, so that precise conditions have to be set-

up). c) The interfacial free energy cannot be calculated by these tech-


MD simulations of a L-J substance have concluded47 for the fcc

(100) interface that it is rather diffuse since the density profile nor-

mal to the interface oscillates in the liquid side (i.e. structured

liquid) over five atomic diameters. Similar conclusions were drawn from

another MD48 study where it was shown that, in addition to the density

profile, the potential energy profile oscillates and that physical prop-

erties such as diffusivity gradually change across the interface from

those of the solid to those of the liquid. Note that none of these

studies found a density deficit (observed in the hard sphere models) at

the interface. However, in an MC simulation49 of the (111) fcc inter-

face with a starting configuration as in the Bernal model, a small defi-

cit density was observed in addition to the "channeled-like" structure


of the first 2-3 interfacial layers. A more precise comparison of the

(100) and (111) interfaces concluded50 that the two interfaces behave

similarly. Interestingly enough, this study also indicates that the

L S transition, from a structural point of view, as examined from mol-

ecular trajectory maps parallel to the interface, is rather sharp and

occurs within two atomic planes, despite the fact that density oscilla-

tions were observed over 4-5 planes. However, these trajectory maps, in

terms of characterizing the atoms as liquid- or solid-like, are very

subjective and critically depend on the time scale of the experiment;51

an atom that appears solid on a short time could diffuse as liquid on a

longer time scale.

The perturbation method of the S/L interface52 has not yet been

widely used to determine the interfacial free energy or the structure of

the liquid next to the solid, but only to determine the density profiles

at the interface. The latter results are shown to be in good agreement

with those found from the MD simulations, but do not provide any add-

itional information. In a study of the (100) and (111) bcc inter-

faces,51'53 calculations suggest that the interfacial liquid is "struc-

tured," i.e. with a density close to that of the bulk liquid and a

solid-like ordering. The interfacial thickness was estimated quite

large (10-15 layers) and the observed density profile oscillations were

less sharp than those observed47-50 for the fcc interfaces. This was

rationalized by the lower order and plane density (area/atom) for the

bcc interfaces. Despite the differences in the density profiles among

the (100) and (111) interfaces, the interfacial potential energies and

S/L surface energies were found to be nearly equal (within 5%).51

The interfacial phenomena were also studied by a surface MD

method,4',55 meant to investigate the epitaxial growth from a melt. It

was observed that the liquid adjacent to the interface up to 4-5 layers

had a "stratified structure" in the direction normal to the interface

which "lacked intralayer crystalline order"; intralayer ordering started

after the establishment of the three-dimensionally layered interface

regions. In contrast with the previously mentioned MD studies, non-

equilibrium conditions were also examined by starting with a supercooled

melt. For the latter case, the above mentioned phenomena were more pro-

nounced and occurred much faster than the equilibrium situation. These

results are supported by calculations56 of the equilibrium S/L interface

(fcc (001) and (100)) in a lattice-gas model using the cluster variation

method. In addition, it was shown that for the nonclose-packed face

(110), the S L transition was smoother and the "intermediate" layer

observed for the (001) face was not found for the (110) face. However,

despite these structural differences, the calculated interfacial ener-

gies for these two orientations differed only by a few percent.57

Most of the methods presented here give some information on the

structure and properties of the S/L interface, particularly of the

liquid adjacent to the crystal. In spite of the fact that these models

provide a rather phenomenological description of the interface, their

information seems to be useful, considering all the other available

techniques for studying S/L interfaces. In this respect, they rather

suggest that the interfacial region is likely to be diffuse, particu-

larly if one does not think of the solid next to the liquid as a rigid

wall. Such a picture of the interface is also suggested from recent


experimental works that will be reviewed next. These simulations re-

sults then raise questions about the validity of current theories on

crystal growth58'59 and nucleation60 which, based on theories discussed

earlier, such as the "a" factor theory, assume a clear cut separation

between solid and liquid; this hypothesis, however, is significantly

different from the cases given earlier.

Experimental evidence regarding the nature of the S/L interface

Apparently, the large number of models, theories, and simulations

involved in predicting the nature of the S/L interface rather illus-

trates the lack of an easy means of verifying their conclusions. In-

deed, if there was a direct way of observing the interfacial region and

studying its properties and structures, then the number of models would

most likely reduce drastically. However, in contrast to free surfaces,

such as the L/V interface, for which techniques (e.g. low-energy dif-

fraction, Auger spectroscopy, and probes like x-rays61) allow direct

analysis to be made, no such techniques are available at this time for

metallic S/L interfaces. Furthermore, structural information about the

interface is even more difficult to obtain, despite the progress in

techniques used for other interfaces.62 Therefore, it is not surprising

that most existing models claim success by interpreting experimental re-

sults such that they coincide with their predictions. Some selected

examples, however, will be given for such purposes that one could relate

experimental observations with the models; emphasis is given on rather

recent published works that provide new information about the interfa-

cial region. A detailed discussion about the S/L interfacial energies

will also be given. Indirect evidence about the nature of the


interface, as obtained from growth kinetics studies, will not be covered

here; such detailed information can be found, for example, in several

review papers25,26,63 and books.64,65

Interfacial energy measurements for the S/L interface are much more

difficult than for the L/V and S/V interfaces.62 For this reason, the

experiments often rely upon indirect measurement of this property; in-

deed, direct measurements of asz are available only for a very few cases

such as Bi,66 water,67 succinonitrile,68 Cd,"69 NaCI and KCl1,70 and

several metallic alloys.62 However, even in these systems, excepting

Cd, NaCI, and KC1, information regarding the anisotropy of asz is lack-

ing.71"76 Nevertheless, most evaluations of the S/L interfacial ener-

gies come from indirect methods. In this case, the determinations of

as deal basically with the conditions of nucleation or the melting of a

solid particle within the liquid. For the former, that is the most

widely used technique, Osz is obtained from measured supercooling

limits, together with a crystal-melt homogeneous nucleation theory in

which asZ appears as a parameter60'77 in the expression
M o
J = K exp (- ) (14)

Here J is the nucleation frequency, Ky is a factor rather insensi-

tive to small temperature changes, and M is a material constant. On the

* Strictly speaking, only these measurements are direct; the rest, still
considered direct in the sense that the S/L interface was at least ob-
served, deal with measurements of grain boundary grooves or intersec-
tion angles (or dihedral angles) between the liquid, crystal, and
grain boundary.7174 The level of confidence of these measurements75
and whether or not the shape of the boundaries were of equilibrium or
growth form76 remain questionable.


other hand, the latter method, i.e. depression of melting point of small

particles (spherical with radius r) by AT, is based on the well known

Gibbs-Thomson equation78
2o T
AT = s m (15)

Homogeneous nucleation experiments were performed by subdividing

liquid droplets and keeping them apart by thin oxide films, or by sus-

pending the particles in a suitable fluid in a dilatometer and measuring

the nucleation rates (J) and associated supercoolings (AT).77,79 The

determined values were correlated with the latent heat of fusion with

the well known known relation77,80*

ao .45 L (units of g-at).
sz g-atom

However, more recent experiments have shown that much larger supercool-

ings than those observed earlier are possible,81 and the ratio AT/Tm

considerably exceeds the value of .2 T, 77,79 which is often taken as

the limiting undercooling at which homogeneous nucleation occurs in pure

metals. As a consequence, many of the experimentally determined values

are in error by as much as a factor of 2. The main criticism of the OsZ

values determined from nucleation experiments includes the following:

a) the influence of experimental conditions (e.g. droplet size, droplet

coating, cooling rates, and initial melt superheat) on the amount of

maximum recorded undercooling,8lb b) whether a crystal nucleus (of

atomic dimensions, a few hundred atoms)/melt interface can be adequately

described with asz of an infinite interface, which is a macroscopic

* A slope of .45 has also been proposed80 for the empirical relation of
the ratio o s/agb (ogb is the grain boundary surface tension).
s2. gb gb


quantity,76 c) whether the observed nucleation is truly homogenous or

rather if it is taking place on the surface of the droplets,82 d) the

assumption that the nucleus has a spherical shape or that asZ is

isotropic," and e) the fact that the values obtained represent some

average interfacial energy over all orientations. In spite of these

limitations, the asz values deduced from nucleation experiments still

constitute the major source of S/L interfacial energies; if used with

skepticism, they provide a reference for comparison with other inter-

facial parameters. Moreover, it should be mentioned that these values

have been confirmed in some cases using other techniques or theoretical

approaches which have not been reviewed here. However, the theoretical

approaches84-87 have also been criticized because they assume complete

wetting, atomically smooth interfaces, and that the liquid next the

interface retains its bulk character.

Experimental attempts to find a critical point between the solid

and the liquid by going to extreme temperatures and pressures (high or

low) have always resulted in non-zero entropy or volume changes at the

limit of the experiment, suggesting that a critical point does not

exist. Similar conclusions are drawn from MD studies,88 despite the

wide range of T and P accessible to computer simulations. Theoretical

studies,89 which disregard lattice defects, also predict that no crit-

ical point exists for the S/L transition because the crystalline sym-

metry cannot change continuously. In contrast to these results, a

critical point was found in the vicinity of the liquidus line of a K-Cs

* Note that the temperature coefficient of asZ has also been neglected
in most studies.


alloy;90 also, a CS of a model for crystal growth from the vapor found

that the phase transition proceeds from the fluid phase to a disordered

solid and afterwards to the ordered solid.91

Strong molecular ordering of a thin liquid layer next to a growing

S/L interface has been suggested92 as an explanation of some phenomena

observed during dynamic light scattering experiments at growing S/L

interfaces of salol and a nematic liquid crystal.93 In an attempt to

rationalize this behavior, it was proposed that only interfaces with

high "a" factors can exert an orienting force on the molecules in the

interfacial liquid; however, such an idea is not supportive of the ob-

servation regarding the water/ice (0001) interface (a = 1.9).94-96 The

ice experiments94'95 have shown that a "structure" builds up in the

liquid adjacent to the interface (1.4-6 pm thick), when a critical

growth velocity (-1.5 pm/s) is exceeded, that has different properties

from that of the water (for example, its density was estimated to be

only .985 g/cc, as compared to 1 g/cc of the water) and ice, but closer

to that of water. Interpreting these results from such models as that

of the sharp and rough interface, of nucleation (critical size nuclei)

ahead of the interface and of critical-point behavior, as in second-

order transition* were ruled out. Similar experiments performed on

salol revealed97 that the S/L interface resembles that of the ice/water

system, only upon growth along the [010] direction and not along the

[100] direction. The "structured" (or density fluctuating) liquid layer

* It should be noted they95 determined the critical exponent of the
relation between line width and intensity of the scattered light in
close agreement with that predicted29'30 for the diffuse liquid-vapor
interface at the critical point.


was estimated to be in the order of 1 pm. An explanation of why such a

layer was not formed for the (100) interface was not given. Still,

these results agree in most points with the ones mentioned earlier92 and

are indirectly supported by the MD simulations54'56 discussed earlier.

However, despite the excellence of these light scattering experiments

for the information they provide, there is still some concern regarding

the validity of the conclusions which strongly depend on the optics

framework. 9

Aside from the computer simulations and the dynamic light-scatter-

ing experiments, experimental evidence of a diffuse interface is usually

claimed by observing a "break" in the growth kinetics V(AT) curve; this

is associated with the transition from lateral to continuous growth kin-

etics. As such, these will be discussed in the section regarding kin-

etic roughening and growth kinetics at high supercoolings.

Confirmation of the "a" factor model has been provided via observa-

tions of the growth front (faceted vs. non-faceted morphology) for sev-

eral materials.26 Although experimental observations are in accord with

the model for large and small "a" materials, there are several materials

which facet irrespective of their "a" values. These are Ga,2,63,99

Ge'100o', Bi,63 Si,102 and H20,103 which have L/KTm values between 2 and

4 and P4'04 and Cd69 whose L/KTm values are about 1. Other common fea-

tures of these materials are a) complex crystal structures, oriented

molecular structure; b) semi-metallic properties; c) some of their

interfaces have been found to be non-wetted by their melts; and d) their

S/L interfacial energies do not follow the empirical rule of ost .45

L. Hence, these materials belong to a special group and it would be

difficult to imagine that simple statistical models could be adequate to

describe their interfaces. However, these materials are of great theor-

etical importance in the field of crystal growth, as well as of techni-

cal importance referring to the electronic materials industry.

Next, the effect of temperature and supercooling on the nature of

the interface is discussed.

Interfacial Roughening

For many years, one of the most perplexing problems in the theory

of crystal growth has been the question of whether the interface under-

goes some kind of smooth to rough transition connected with thermody-

namic singularities at a temperature below the melting point of the

crystal. This transition is usually called the "roughening transition"

and its existence should significantly influence both the kinetics dur-

ing growth and the properties of the interface. The transition could

also take place under non-equilibrium or growing conditions, called the

"kinetic roughening transition," which differs from the above mentioned

equilibrium roughening transition. These subjects, together with the

topic of the equilibrium shape of crystals, are discussed next.

Equilibrium (Thermal) Roughening

The concept of the roughening transition, in terms of an order-

disorder transition of a smooth surface as the temperature increases was

first considered back in 1949-1951.10,36 The problem then was to calcu-

late how rough a (S/V) interface of an initially flat crystal face

(close-packed, low-index plane) might become as T increases. This was

possible after realizing that the Ising model for a ferromagnet could be


adapted to the treatment of phase transformations (order-disorder,

second-order phase transformation) by recognizing that the equilibrium

structure of the interface is mathematically equivalent to the structure

of a domain boundary in the Ising model for magnetism.

Statistical mechanics,39 as mentioned previously, have long been

associated with co-operative phenomena such as phase transition; more-

over, in recent years, the important problem of singularities related

with them has been a central topic of statistical mechanics. Its appli-

cation to a system can be reduced to the problem of calculating the par-

tition function of the system. One of the most popular tractable models

for applications to phase changes is the Ising or two-dimensional lat-

tice gas model.* The Ising model is a square two-dimensional array of

magnetic atomic dipoles. The dipoles can only point up or down (i.e. an

occupied and a vacant site, respectively); the nearest neighbor inter-

action energy is zero when parallel and p/2 when antiparallel. Thus,

this model restricts atoms to lattice sites and assumes only nearest

neighbor interactions with the potential energy being the sum of all

such pair interactions. This simple model has been rigorously solved'06

to obtain the partition function and the transition temperature Tc

(Curie temperature) for the ferromagnetic phase transition paramagneticc

- ferromagnetic). Hoping that this discussion provides a link between

the roughening transition and statistical mechanics, the earlier discus-

sion about roughening continues.

* Strictly speaking, the two models are different, but because of their
exact correspondence,105 they are considered similar.

Burton et al.10 considered a simple cubic crystal (100) surface

with (/2 nearest neighbor interaction energy per atom. Proving that

this two level problem corresponds exactly to the Ising model, a phase

transition is expected at Tc. This transition then is related to the

roughening of the interface ("surface melting") and the temperature at

which it takes place is related to the interaction energy as
exp (- ) = 1, or-- .57
2KT (

where TR is the roughening temperature. For a triangular lattice, e.g.

(111) f.c.c. face KTR/p is approximately .91. The authors also consid-

ered the transition for higher (than two) level models of the interface

using Bethe's approximation. It was shown that, with increasing the

number of levels, the calculated TR decreases substantially, but remains

practically the same for a larger number of levels. Although this study

did not rigorously prove the existence of the roughening transition,i07

it gave a qualitative understanding of the phenomenon and introduced its

influence on the growth kinetics and interfacial structure. The latter,

because of its importance, motivated in turn a large number of theoret-

ical works'08 during the last two decades. This upsurge in interest

about interfacial roughening brought new insight in the nature of the

transition and proved59'109'110 its existence from a theoretical point

of view. In principle, these studies use mathematical transformations

to relate approximate models of the interface to other systems, such as

* Exact treatments of phase transitions can be discussed only for
special systems and two dimensions, as discussed previously. For more
than two dimensions, approximate theories have to be considered.
Among them are the mean field, Bethe, and low-high temperature expan-
sions methods.


two-dimensional Coulomb gas, ferroelectrics, and the superfluid state,

which are known to have a confirmed transition. As mentioned prev-

iously, it is out of the scope of this review to elucidate these

studies, detailed discussion about which can be found in several


At the present time, the debate about the roughening transition

seems to be its universality class or whether or not the critical behav-

ior at the transition depends on the chosen microscopic model. Based on

experiments, the physical quantities associated with the phase transi-

tion vary in manner IT-TcIP when the critical temperature Tc is ap-

proached. The quantities such as p in the above relation that charac-

terize the phase transition are called critical exponents. They are

inherent to the physical quantities considered and are supposed to take

universal values (universality class) irrespective of the materials

under consideration. For example, in ferromagnetism, one finds as

T Tc (Curie temperature):

susceptibility, x a (T Tc)-Y
(T > Tc)
specific heat, C(T) = (T Tc)-a

Another important quantity in the critical region is the correla-

tion length, which is the average size of the ordered region at temper-

atures close to Tc. In magnetism, the ordered region (i.e. parallel

spin region) becomes large at Tc, while in particle systems the size of

the clusters of the particles become large at Tc. The correlation

length also obeys the relation'05

IT TcI- (T > Tc)
T (16)(T < T
|Tc TI-V (T < Tc)


or, according to a different model, E diverges in the vicinity of TR

R 1/2
= exp (C/( TR) (T < TR)
C = m (T > TR)

where C is a constant (about 1.5i13 or 2.1114). The above mentioned

illustrates that the universality class can be different depending on

the model in use. To be more specific, the difference in behavior can

be realized by comparing the relations (16) vs. (17); the former, which

belongs to the two-dimensional Ising model, indicates that E diverges by

a power law, while the latter of the Kosterlitz-Thouless113 theory shows

that diverges exponentially.

One, however, may wonder what the importance of the correlation

length is and how it relates, so to speak, to "simpler" concepts of the

interface. In this view, E relates to the interfacial width;59 hence,

for temperatures less than the roughening transition, the interfacial

width is finite in contrast with the other extreme, i.e. for T's > TR; E

also corresponds to the thickness of a step so that the step free energy

can then be calculated from E. Indeed, it has been shown that oe is re-

lated to the inverse of &.110,115 Thus, these results predict that the

step edge free energy approaching TR diverges as

T -T
o e exp (-C/( ) 1/2) (18)
e TR

and is zero at temperatures higher than TR.116 Hence, the energetic

barrier to form a step on the interface does not exist for T's higher

than TR.


In summary, the key points of the roughening transition of an

interface between a crystal and its fluid phase (liquid or vapor) are

the following: a) At T = TR a transition from a smooth to a rough

interface takes place for low Miller index orientations. At T < TR the

interface is smooth and, therefore, is microscopically flat. The edge

free energy of a step on this interface is of a finite value. Growth of

such an interface is energetically possible only by the stepwise mode.

On the other hand, for T > TR, the interface is rough, so it extends

arbitrarily from any reference plane. The step edge energy is zero, so

that a large number of steps (i.e. arbitrarily large clusters) is al-

ready present on a rough interface. It can thus grow by the continuous

mechanism. Pictorial evidence about the roughening transition effects

can be considered from the results of an MC simulation117 of the SOS

model* (S/V interface), shown in Fig. 5. Also, a transition with in-

creasing T from lateral kinetics to continuous kinetics above TR was

found for the interfaces both on a SC11 and on an fcc crystal'17 for

the SOS model, b) It is claimed that most theoretical points of the

transition have been clarified. Based on recent studies, the tempera-

ture of the roughening transition is predicted to be higher than that of

the BCF model. Furthermore, its universality class is shown to be that

of the Kosterlitz-Thouless transition. Accordingly, the step edge free

* If, for the ordinary lattice gas model in a SC crystal, it is required
that every occupied site be directly above another occupied site, one
ends up with the solid-on-solid (SOS) model. This model can also be
described as an array of interacting solid columns of varying heights,
hr = 0, 1, ..., -; the integer hr represents the number of atoms in
each column perpendicular to the interface, which is the height of the
column. Neighboring sites interact via a potential V = Klhr-hr'j. If
the interaction between nearest neighbor columns is quadratic, one ob-
tains the "discrete Gaussian" model.

Figure 5 Computer drawings of crystal surfaces (S/V interface,
Kossel crystal, SOS model) by the MC method at the
indicated values of KT/d. After Ref. (112).


energy goes to zero as T TR, vanishing in an exponential manner.

These points have been supported and/or confirmed by several MC simula-

tions results,19 in particular, for the SOS model.

As may already be surmised, the roughening transition is also ex-

pected to take place for a S/L interface. Indeed, its concept has been

applied, for example, in the "a" factor model;8'9 the "a" factor is in-

versely related to the roughening transition temperature TR, assuming

that the nearest neighbor interactions (p) are related to the heat of

fusion. Such an assumption is true for the S/V interface where only

solid-solid interactions are considered (Ess = p, Esv = Evv 0). Then,

for the Kossel crystal,120" Lv = 3( where Lv is the heat of evaporation.

Unfortunately, however, for the S/L interface all kinds of bonds (Ess,

Es9, EZ) are significant enough to be neglected so that one could not

assume a model that accounts only vertical or lateral (with respect to

the interface plane) bonds. Assumptions such as EZZ = EsZ cannot be

justified, either. Several ways have been proposed"21 to calculate Esz.

Their accuracy, however, is limited since both Es, and EZZ, to a lesser

extent, depend on the actual properties of the interfacial region which,

in reality, also varies locally. Nevertheless, such information is

likely to be available only from molecular dynamics simulations at the


Quantitative experimental studies of the roughening transition are

rare, and only a few crystals are known to exhibit roughening. Because

of the reversible character of the transition, it is necessary to study

* As Kossel crystal120 is considered a stacking of molecules in a primi-
tive cubic lattice, for which only nearest neighbor interactions are
taken into account.


a crystal face under growth and equilibrium conditions above and below

TR. That means the "a" factor, which is said to be inversely propor-

tional to TR, has to change continuously (with respect to the equilib-

rium temperature) or that L/KTm has to be varied. For a S/V interface,

depending on the vapor pressure, the equilibrium temperature can be

above or below TR, so that "a" can vary. The only exception in this

case is the He S/L superfluidd) interface, at T < 1.76 K. For this

system, by changing the pressure, the "a" factor can be varied over a

wide range, in a small experimental range (i.e. .2 K < T < 1.7 K), where

equilibrium shapes, as well as growth dynamics, can be quantitatively

analyzed.96 For a metallic solid in contact with its pure melt though,

this seems to be impossible because only very high pressure will influ-

ence the melting temperature. Thus, at Tm a given crystal face is

either above or below its TR;122 crystals facet at growth conditions

provided that Ti < TR, where Ti is the interface temperature. Thus, the

roughening transition of a S/L interface of a metallic system cannot be

expected, or experimentally verified.

In spite of the fact that most of the restrictions for the S/L

interface do not exist for the S/V one, most models predict TR's (for

metals) higher than Tm, thus defying experimentation on such interfaces.

The majority of the reported experiments are for non-metallic mate-

rials such as ice,123 naphthalene,124 C2C16 and NH4C1,125 diphenyl,126

adamantine,127 and silver sulphide;128 in these cases the transition was

only detected through a qualitative change in the morphology of the

crystal face (i.e. observing the "rounding" of a facet). The likely

conclusions from these experiments are that the transition is gradual


and that the most close-packed planes roughen the last (i.e. at higher

T). Also, it can be concluded that the phenomena are not of universal

character (e.g. for diphenyl and ice the most dense plane did not

roughen even for T = T,, while for adamantine the most close-packed

plane roughened below the bulk melting point) and that the theoretically

predicted TR's for S/V interfaces are too high (e.g. for C2C16 the

theoretical value of KTR/Lv is 1/16 compared with the theoretical value

of 1/8). It was also found that impurities reduce TR.127

The roughening transition for the hcp He crystals has been experi-

mentally found for at least three crystal orientations ((0001), (1100),

(1101)129,130). Moreover, a recent study130 of the (0001) and (1100)

interfaces, is believed to be the first quantitative evidence that

couples the transition with both the growth kinetics and the equilibrium

shape of the interface. Below TR the growth kinetics were of the lat-

eral type; that allowed for a determination of the relationship ce(T).

At TR it was shown that oe vanished as
exp (-C/( R )1/2)

in accord with the earlier mentioned theories. At T > TR the interfaces

advanced by the continuous mechanism.

As far as S/L interfaces of pure metallic substances are concerned,

the roughening transition is likely non-existent experimentally. A

faceted to non-faceted transition, however, has been observed for a

metallic solid-solution (other liquid metals or alloys) interface in the

Zn-In and Zn-Bi-In systems.'31',32 The transition, which was studied

isothermally, took place in the composition range where important


changes in Osz occurred. Evidence about roughening also exists for

several solvent-solute combinations during solution growth.133

Additional information about the roughening transition concept

comes from experimental studies on the equilibrium shape of microscopic

crystals. This topic is briefly reviewed in the next section.

Equilibrium Crystal Shape (ESC)

The dynamic behavior of the roughening transition can also be

understood from the picture given from the theory of the evolution of

the equilibrium crystal shape (ECS). In principle, the ECS is a geomet-

rical expression of interfacial thermodynamics. The dependence of the

interfacial free energy (per unit area) on the interfacial orientation n

determines r(T,n), where r is the distance from the center of the crys-

tal in the direction of n of a crystal in two-phase coexistence.14'15

At T = 0, the crystal is completely faceted.134" As T increases, facets

get smaller and each facet disappears at its roughening temperature

TR(n). Finally, at high T, the ECS becomes completely rounded, unless,

of course, the crystal first melts. As discussed earlier, facets on the

ECS are represented with cusps in the Wulff plot, which, in turn, are

related to nonzero free energy per unit length necessary to create a

step on the facet; 13 the step free energy also vanishes at TR(n), where

the corresponding facets disappear. Below TR, facets and curved areas

on the crystal meet at edges with or without slope discontinuity (i.e.

smooth or sharp); the former corresponds to first-order phase transition

and the latter to second-order transitions. The edges are the

* It is generally believed that macroscopic crystals at T = 0 are facet-
ed; however, this claim that comes only from quantum crystals still
remains controversial.134


singularities of the free energy r(T,n)136 that determines the ECS phase

diagram.137 The shape of the smooth edge varies

y = A(x xc)8 + higher-order terms

where xc is the edge position; x, y are the edge's curvature coordin-

ates. The critical exponent 8 is predicted to be as 8 = 2136 or 9 =

3/2.137,138 The 3/2 exponent is characteristic of a universality

class'39,140 and it is therefore independent of temperature and facet

orientation as long as T < TR. Indeed, the 3/2 value has been reported

from experimental studies on small equilibrium crystals (Xe on Cu sub-

strate141 and Pb on graphite134). For the equilibrium crystal of Pb

grown on a graphite substrate, direct measurements of the exponent 6 via

SEM yielded a value of 8 = 1.60, in the range of temperatures from 200-

3000C, in close agreement with the Pokrovsky-Talapov transition139 and

smaller than the prediction of the mean-field theory.137 Sharp edges

have also been seen in some experiments, as in the case of Au,142,143

but they have received less theoretical attention.

At the roughening transition, the crystal curvature is predicted to

jump from a finite universal value for T = TR+ to zero for T =

TR-,130,138144 as contrasted to the prediction of continuously vanish-

ing curvature.136 Similarly, the facet size should decrease with T and

vanish as T TR-, like exp (-C/V(TR T)),113 as opposed to the behav-

ior as (TR T)1/2.136 The jump in the crystal curvature has been ex-

actly related59 to the superfluid jump of the Kosterlitz-Thouless trans-

ition in the two-dimensional Coulomb gas.113,130'134'141 In addition,

the facet size of Ag2S crystals128 was found (qualitatively) to de-

crease, approaching TR, in an exponential manner.


Although the recent theoretical predictions seem to be consistent

with the experimental results, the difficulty of achieving an ECS on a

practical time scale imposes severe limitations on the materials and

temperatures that can be investigated. The only ideal system to study

these phenomena is the 4He (see an earlier discussion), for which sev-

eral transitions have already been discovered in the hcp phase. Whether

the superfluid 4He liquid resembles a common metallic liquid and how the

quantum processes affect the interface still remain unanswered.

Kinetic Roughening

In the last decade or so, MC simulations of SOS kinetic model" of

(001) S/V interface of a Kossel crystal have revealed117,145',46 a very

interesting new concept, the "kinetic roughening" of the interface; in

distinction with the equilibrium roughening caused by thermal fluctua-

tions, the kinetic roughening is due to the effect of the driving force

on the interface during growth. The simulations show that when a crys-

tal face is growing at a temperature below TR (T < TR) under a driving

force AG less than a critical value AGc, it is smooth on an atomic scale

and it advances according to a lateral growth mechanism. However, if

the crystal face is growing at T < TR, but at a driving force such that

AG > AGc, it will be rough on an atomic scale and a continuous growth

* This is an extension of the SOS model for (S/V) growth kinetics
studies. Atoms are assumed to arrive at the interface with an extern-
ally imposed rate K+. The evaporation rate K-, on the other hand, is
a function of the number of nearest neighbors, i.e. fn,m' which is the
fraction of surface atoms in the n/th layer with m lateral neighbors.
The net growth rate is then the difference between condensation and
evaporation rates in all layers. Unless some specific assumptions are
made concerning K-, and/or about fnm, the system cannot be solved.
Indeed, all the existing kinetic SOS models essentially differ only in
the above mentioned assumptions. (See, for example, references 117
and 119.)


mechanism will be operative. The transition in the interface morphology

and growth kinetics as a function of the driving force is known as kin-

etic roughening. Computer drawings of the above mentioned simulations,

shown in Figs. 6a and 6b, show the kinetic roughening phenomenon. It can

be seen that at a low driving force the growth kinetics are non-linear,

as contrasted with the high driving force region where the kinetics are

linear. These correspond respectively to lateral and continuous growth

kinetics, as discussed in detail later. It is believed that the high

driving force results in a relatively high condensation rate with re-

spect to the evaporation rate. In addition, the probability of an atom

arriving on an adjacent site of an adatom and thus stabilizing it, is

overwhelming that of the adatom evaporation. These result in smaller

and more numerous clusters, as contrasted to the low driving force case

where the clusters are large and few in number.

As far as the author knows, an experimental verification of kinetic

roughening for a S/L interface in a quantitative way is non-existent.

There are a few studies which identify the transition with morphological

changes occurring at the interface with increasing supercooling.133

Such conclusions are of limited qualitative character and under certain

circumstances could also be erroneous, because 1) there may be a clear-

cut distinction between equilibrium and growth forms of the interface,12

2) even when the growth is stopped, the relaxation time for equilibrium

may be quite long130 for macroscopic dimensions, and 3) a "round" part

of a macroscopically faceted interface does not necessarily have to be

rough on an atomic scale. Such microscopic detailed information can be

gained only from the standpoint of interfacial kinetics, which also


A--D (100)

0 1*1 1
0 3 4 5 6 7

A/kT 20

Figure 6 Kinetic Roughening. After Ref. (117). a) MC interface
drawings after deposition of .4 of a monolayer on a (001)
face with KT/4 = .25 in both cases, but different driving
forces (A. b) Normalized growth rates of three different

FCC faces as a function of Al, showing the transition in
the kinetics at large supersaturations.
B f o (100)

Figure 6 Kinetic Roughening. After Ref. (117). a) MC interface
drawings after deposition of .4 of a monolayer on a (001)
face with KT/p = .25 in both cases, but different driving
forces (AH). b) Normalized growth rates of three different
FCC faces as a function of Aj, showing the transition in
the kinetics at large supersaturations.


allow for a reliable determination of critical parameters linked to the

transition. There are a few growth kinetics studies which provide a

clue regarding the transition from lateral to continuous growth; these

will be reviewed next rather extensively due to the importance of the

kinetic roughening in this study.

A faceted (spiky) to non-faceted (smooth spherulitic) transition

was observed for three high melting entropy (L/KTm 6-7) organic sub-

stances, salol, thymol, and O-terphenyl. 47 The transition that took

place at bulk supercoolings ranging from 30-50C for these materials was

shown to be of reversible character; it also occurred at temperatures

below the temperature of maximum growth rate. An attempt to rational-

ize the behavior of all three materials in accord with the predictions

of the MC simulation results"17 was not successful. The difference in

the transition temperatures (20, 13, and -10C for the 0-terphenyl,

salol, and thymol, respectively) were attributed to the dissimilar crys-

tal structures and bonding.

Morphological changes corresponding to changes from faceted to non-

faceted growth form together with growth kinetics have been reported148

for the transformation I-III in cyclohexanol with increasing supercool-

ing. The morphological transition was associated with the change in

growth kinetics, as indicated by a non-linear to linear transition of

the logarithm of the growth rates, normalized by the reverse reaction

term [1 exp(- AG,/KT)], as a function of 1/T (i.e. log(V/1-

exp(- AGv/KT)) vs. 1/T plot); the linear kinetics (continuous growth)

* This feature will be further explained in the continuous growth sec-


took place at supercoolings larger than that for the morphological

change and also larger than the supercooling for the maximum growth

rate. The change in the kinetics was found to be in close agreement

with Cahn's theory.25 It should be noted that the low supercoolings

data, which presumably represented the lateral growth regime, were not

quantitatively analyzed; also, the "a" factor of cyclohexanol lies in

the range of 1.9-3.7, depending on the E value. It was also sug-

gested149 that normalization of the growth rates by the melt viscosity

at high AT's might mask the kinetics transition.

The morphological transition for melt growth has also been ob-

served133 for the (111) interface of biphenyl at a AT about .03C; the

"a" factor of this interface was calculated to be about 2.9. For growth

from the solution, the transition has been observed at minute supercool-

ing for facets of tetraoxane crystals with an "a" factor in the order of

2. 150

Based on kinetic measurements, it was initially suggested that P4

undergoes a transition from faceted to non-faceted growth at supercool-

ings between 1-9C.Isl However, this was not confirmed by a later study

by the same authors, who reported that P4 grew with faceted dendritic

form at high supercoolings."4

In conclusion, a complete picture of the kinetic roughening phe-

nomenon has not been experimentally obtained for any metallic S/L

interface. It seems that for growth from the melt because of the lim-

ited experimental range of supercoolings at which a change in the growth

morphology and kinetics can be accurately recorded, only materials with


an "a" factor close to the theoretical borderline of 2 are suitable for

testing. Even in such cases the transition cannot be substantiated and

quantified in the absence of detailed and reliable growth kinetics anal-


Interfacial Growth Kinetics

Lateral Growth Kinetics (LG)

It is generally accepted that lateral growth prevails when the

interface is smooth or relatively sharp; this in turn implies the fol-

lowing necessary conditions for lateral growth: 1) the interfacial

temperature Ti is less than TR and 2) the driving force for growth is

less than a critical value necessary for the dynamic roughening transi-

tion, and/or the diffuseness of the interface.

The problem of growth on an atomically flat interface was first

considered by Gibbs,20 who suggested that there could be difficulty in

the formation of a new layer (i.e. to advance by an interplanar or an

interatomic distance) on such an interface. When a smooth interface is

subjected to a finite driving force (i.e. a supercooling AT), the liquid

atoms, being in a metastable condition, would prefer to attach them-

selves on the crystal face and become part of the solid. However, by

doing so as single atoms, the free energy of the system is still not de-

creased because of the excess surface energy term associated with the

unsatisfied lateral bonds. Thus, an individual atom, being weakly bound

on the surface and having more liquid than solid neighbors, is likely to

"melt" back. However, if it meant to stay solid, it would create a more

favorable situation for the next arriving atom, which would rather take


the site adjacent to the first atom rather than an isolated site. From

this simplified atomistic picture, it is obvious that atoms not only

prefer to "group" upon arrival, but also choose such sites on the sur-

face as to lower the total free energy. These sites are the ones next

to the edges of the already existing clusters of atoms. The edges of

these interfacial steps (ledges) are indeed the only energetically

favorable growth sites, so that steps are necessary for growth to pro-

ceed (stepwise growth). The interface then advances normal to itself by

a step height by the lateral spreading of these steps until a complete

coverage of the surface area is achieved. Although another step might

simultaneously spread on top of an incomplete layer, it is understood

that the mean position of the interface advances one layer at a time

(layer by layer growth).

Steps on an otherwise smooth interface can be created either by a

two-dimensional nucleation process or by dislocations whose Burgers vec-

tors intersect the interfaces; the growth mechanisms associated with

each are, respectively, the two-dimensional nucleation-assisted and

screw dislocation-assisted, which are discussed next. Prior to this,

however, we will review the atomistic processes occurring at the edge of

steps and their energetic, since these processes are rather independent

from the source of the steps.

Interfacial steps and step lateral spreading rate (us)

In both lateral growth mechanisms the actual growth occurs at

ledges of steps, which, like the crystal surface, can be rough or

smooth; a rough step, for example, can be conceived as a heavily kinked

step. For S/V interfaces it has been shown107'112 that the roughness of


the steps is higher than that of their bonding surfaces and it decreases

with increasing height; moreover, MC simulations find that steps roughen

before the surface roughening temperature TR. On the other hand, for a

diffuse interface, the step is assumed6 to lose its identity when the

radius of the two-dimensional critical nucleus, rc, becomes larger than

the width of the step defined as

w = h/(g)1/2 (19)

Note that the width of the step is thought to be the extent of its pro-

file parallel to the crystal plane; hence, the higher the value of w,

the rougher the step is and vice versa. Interestingly enough, even for

relatively sharp interfaces, i.e. when g ~ .2-.3, the step is predicted

to be quite rough. Based on this brief discussion, the edge of the

steps is always assumed to be rough.

Atoms or molecules arrive at the edge of the steps via a diffusive

jump across the cluster/liquid interface. Diffusion towards the kink

sites can occur either directly from the liquid or vapor (bulk diffu-

sion) or via a "surface diffusion" process from an adjacent cluster, or

simultaneously through both. For the case of S/L interfaces, however,

it is assumed that growth of the steps is via bulk diffusion only.152

Furthermore, anisotropic effects (i.e. the edge orientation) are ex-


The growth rate of a straight step is derived as152"
S= K D T- (20)
e hRTT E T

For detailed derivation, see further discussion in the continuous
growth section.


where D is the liquid self-diffusion coefficient and R is the gas con-

stant. Cahn et al.25 have corrected eq. (20) by introducing the phenom-

enological parameter B and the g factor as
-1/2 DLAT
e = 5(2 + g-1/2) DLAT (21)
e hRTT

Here B corrects for orientation and structural factors; it principally

relates the liquid self-diffusion coefficient to interfacial transport,

which will be considered next. B is expected to be larger than 1 for

symmetrical molecules (i.e. molecularly simple liquids for which "the

molecules are either single atoms or delineate a figure with a regular

polyhedral shape"''5) and less or equal to 1 for asymmetric molecules.

In spite of these corrections, the concluding remark from eqs. (20) and

(21) is that ue increases proportionally with the supercooling at the


When the step is treated as curved, then the edge velocity is de-

rived as17

= Ue (1 rc/r) (22)

where r is the radius of curvature. In accord with eq. (22), the edge

of a step with the curvature of the critical nucleus is likely to remain

immobile since u = 0.

If one accounts for surface diffusion, ue is given according to the

more refined treatment of BCF10 as

Ue = 2axsV exp (- W/KT) (23)

where a is the supersaturation, xs is the mean diffusion length, v is

the atomic frequency (v 1013 sec-l), and W is the evaporation energy.

For parallel steps separated by a distance yo, the edge velocity is

derived as


Ue = 2oxsv exp (- W/KT) tanh (yo/2xs) (24)

which reduces to (23) when yo becomes relatively large.

Interfacial atom migration

The previously given analytical expression (eq. (20)) for the edge

velocity can be written more accurately as

ue = c AGvexp(- AGi/KT) (25)

where c is a constant and AGi is the activation energy required to

transfer an atom across the cluster/L interface. This term is custom-

arily assumed 54 to be equal to the activation energy for liquid self-

diffusion, so that ue in turn is proportional to the melt diffusivity or

viscosity (see eq. (20)).

Before examining this assumption, let it be supposed that the

transfer of an atom from the liquid to the edge of the step takes place

in the following two processes: 1) the molecule "breaks away" from its

liquid-like neighbors and reorients itself to an energetically favorable

position and 2) the molecule attaches itself to the solid. Assuming

that the second process is controlled by the number of available growth

sites and the amount of the driving force at the interface, it is ex-

pected that AGi to be related to the first process. As such, the inter-

facial atomic migration depends on a) the nature of the interfacial

region, or, alternatively, whether the liquid surrounding the cluster or

steps retains its bulk properties; b) how "bonded" or "structured" the

liquid of the interfacial region is; c) the location within the

interfacial region where the atom migration is taking place; and d) the

molecular structure of the liquid itself. Thus, the combination and

the magnitude of these effects would determine the interfaciall


diffusivity," Di. Alternatively, suggesting that Di = D, one explicitly

assumes that the transition from the liquid to the solid is a sharp one

and that the interfacial liquid has similar properties to those of the

bulk. Although this assumption might be true in certain cases,25,153

its validity has been questioned25,153'155 for the case of diffuse

interface, clustered, and molecularly complex liquids. These views have

been supported by recent experimental works92'95',56 and previously dis-

cussed MD simulations of the S/L interface,5s0,s53,-s6 which indicate

that a liquid layer, with distinct properties compared to those of the

bulk liquid and solid, exists next to the interface. Within this layer

then the atomic migration is described by a diffusion coefficient Di

that has been found to be up to six orders of magnitude smaller92'95

than the thermal diffusivity of the bulk liquid; if this is the case,

the transport kinetics at the cluster/L interface should be much slower

than eq. (20) indicates. Moreover, if the interfacial atom migration is

3-6 orders of magnitude slower than in the bulk liquid, one should also

have to question whether atoms reach the edge of the step as well by

surface diffusion. As mentioned earlier, these factors are neglected in

the determination of ue. Finally, it should be noted that AGi also

enters the calculations of the two-dimensional nucleation rate via the

arrival rate of atoms (Ri) at the cluster, which is discussed next.

Two-dimensional nucleation-assisted growth (2DNG)

As indicated earlier, steps at the smooth interface can be created

by a two-dimensional nucleation (2DN) process, analogous to the three-

dimensional nucleation process. The main difference between the two is

that for 2DN there is always a substrate, i.e. the crystal surface,


where the nucleus forms. The growth mechanism by 2DN, conceived a long

time ago;157 can be described in terms of the random nucleation of two-

dimensional clusters of atoms that expand laterally or merge with one

another to form complete layers. In certain limiting cases, the growth

rate for the 2DNG mechanism is predominantly determined by the two-

dimensional nucleation rate, J, whereas in other cases the rate is

determined by the cluster lateral spreading velocity (step velocity), ue

as well as the nucleation rate. These two groups of 2DNG theories are

discussed next, succeeding a presentation of the two-dimensional nuclea-

tion theory.

Two-dimensional nucleation. The prevailing two-dimensional nucle-

ation theory is based on fundamental ideas formulated several decades

ago.158-161 These classical treatment, which dealt with nucleation from

the vapor phase, and the basic assumptions were later followed in the

development of a 2DN theory in condensed systems.

The classical theory assumes that clusters, including critical nuc-

lei, have an equilibrium distribution in the supercooled liquid or that

the growth of super-critical nuclei is slow compared with the rate of

formation of critical size clusters. It also assumes, as the three-

dimensional nucleation theory, single atom addition and removal from the

cluster, as well as the kinetic concept of the critical size nuc-

leus.162" The expression -for the nucleation rate is given as

1 1

* The validity of these assumptions has been the subject of great con-
troversy and continues to be so. For detailed discussion, see, for
example, ref. 162.


where w? is the rate at which individual atoms are added to the critical

cluster (equal to the product of arrival rate, Ri, and the surface area

of the cluster, S), ni is the equilibrium concentration of critical nuc-

lei with i" number of atoms, and Z is the Zeldovich non-equilibrium fac-

tor which corrects for the depletion of the critical nuclei when nuclea-

tion and growth proceed. Z has a typical value of about 10-2,163 and is

given as

i 1/2 1
Z = ( )
z= TaTKT

where AGI is the free energy of formation of the critical cluster. For

the growth of clusters in the liquid, it is assumed that the clusters

fluctuate in size by single atom increments so that the edge of the

cluster is rough. The arrival rate Ri is then defined as described pre-

viously for the growth of a step. Finally, the concentration of the

critical nuclei is given as
n. = n exp (- )K
1 KT

where n is the atom concentration. For a disk-like nucleus of height h,

the work needed to form it is given as
AG = e (26)
h AG

where oe is the step edge free energy per unit length of the step. For

small supercoolings at which the work of forming a critical two-dimen-

sional nucleus far exceeds the thermal energy (KT), the nucleation rate

per unit area can be approximately written, as derived by Hillig,164 in

the form of

N LAT 1/2 30D AG
J = ( -) exp (- ) (27)
m m o

where N is Avogadro's number and ao is the atomic radius. This expres-

sion, that confirmed an earlier derivation,165 is the most widely

accepted for growth from the melt. The main feature of eq. (27) is that

J remains practically equal to zero for up to a critical value of super-

cooling. However, for supercoolings larger than that, J increases very

fast with AT, as expected from its exponential form. Relation (27) can

be rewritten in an abbreviated form as

jT 1/2 AG AG
J KD( )12 exp (- --) Kn exp (- K-) (28)

where Ko is a material constant and Kn is assumed to be constant within

the usually involved small range of supercooling. Although theoretical

estimates of Kn are generally uncertain because of several assumptions,

its value is commonly indicated in the range of 10212.163 The very

large values of Kn, and the fact that it is essentially insensitive to

small changes of temperature, have made it quite difficult to check any

refinements of the theory. Indeed, such approaches to the nucleation

problem that account for irregular shape clusters166 and anisotropy

effects167 lead to same qualitative conclusions as expressed by eq.

(28). Also, a recent comparison of an atomistic nucleation theory from

the vapor145 with the classical theory leads to the same conclusion. In

contrast, the nucleation rate is very sensitive to the exponential term,

therefore to the step edge free energy and the supercooling at the clus-

ter/liquid (C/L) interface. The nature of the interface affects J in

two ways. First, in the exponential term, AG", through its dependence

upon oe and in the pre-exponential term through the energetic barrier


for atomic transport across the C/L interface. The assumptions of the

classical theory are simple in both cases, since oe is taken as con-

stant, regardless of the degree of the supercooling, and the transport

of atoms from the liquid to the cluster is described via the liquid

self-diffusion coefficient. These assumptions are not correct when the

interface is diffuse6 and at large supercoolings.32 These aspects will

be discussed in more detail in a later chapter.

Mononuclear growth (MNG). As was mentioned earlier, two-dimen-

sional nucleation and growth (2DNG) theories are divided into two

regions according to the relative time between nucleation and layer com-

pletion (cluster spreading). The first of these is when a single crit-

ical nucleus spreads over the entire interface before the next nuclea-

tion event takes place (see Fig. 7a). Alternatively, this is correct

when the nucleation rate compared with the cluster spreading rate is

such that

1/JA > I/ue or for a circular nucleus A < (ue/J)2/3 (29)

where A, 1 are the area and the largest diameter of the interface, re-

spectively. If inequality (29) is satisfied, each nucleus then results

in a growth normal to the interface by an amount equal to the step

(nucleus) height, h. Thus, the net crystal growth rate for this class-

ical mononuclear (and monolayer) mechanism (MNG) is given as164,168

V = hAJ (30)

In this region, the growth rate is predicted to be proportional to the

interfacial area (i.e. crystal facet size). The practical limitations

of this model, as well as the experimental evidence of its existence,

will be given later.


a) A


AT, k A h i




Figure 7 Schematic drawings showing the interfacial processes for


AT >0

Figure 7 Schematic drawings showing the interfacial processes for
the lateral growth mechanisms a) Mononuclear. b) Poly-
nuclear. c) Spiral growth. (Note the negative curvature
of the clusters and/or islands is just a drawing artifact.)

Polynuclear growth (PNG). At supercoolings larger than those of

the MNG region, condition (29) is not fulfilled and the growth kinetics

are described by the so called polynuclear (PNG) model.* According to

this model, a large number of two-dimensional clusters nucleate at ran-

dom positions at the interface before the layer is completed, or on the

top of already growing two-dimensional islands, resulting in a hill- and

valley-like interface, as shown in Fig. 7b. Assuming that the clusters

are circular and that ue is independent of the two-dimensional cluster

size, anisotropy effects, and proximity of neighboring clusters, the one

layer version of this model was analytically solved.169 This was poss-

ible by considering that for a circular nucleus the time, T, needed for

it to cover the interface is equal to the mean time between the genesis

of two nuclei (i.e. the second one on top of the first), or otherwise

given by

SJ(u t)2 dt = 1 (31)

Integration of this expression and use of the relation V = h/T yields

the steady state growth rate (for the polynuclear-monolayer model) given


V = h (rJu2/3)1/3 (32)

This solution has been shown by several approximate solutions164,168'170

and simulations168,171,172 to represent well the more complete picture

of multilevel growth by which several layers grow concurrently through

* It should be mentioned that the use of the term "polynuclear growth"
in this study should not be confused with the usually referred unreal-
istic model,18 which considers completion of a layer just by deposi-
tion of critical two-dimensional nuclei.

nucleation and spreading on top of lower incomplete layers. The more

general and accurate growth rate equation in this region is given by

V= ch (JUe2)1/3 (33)

where the constant c falls between 1-1.4. It is interesting that eq.

(32), being an approximation to the asymptotic multilevel growth rate,

has been shown to be very close to the exact value of steady state con-

ditions that are achieved after deposition of 3-4 layers.173 It was

also suggested from these studies that for irregularly shaped nuclei the

transient period is shorter than for the circular ones. Nevertheless,

the growth rate is well described by eq. (33).

The effect of the nucleus shape upon the growth rate has been con-

sidered in a few MC simulation experiments for the V/Kossel crystal

interface. 17 Square-like172 and irregular nuclei result in higher

growth rates. This increase in the growth rate can be understood in

terms of a larger cluster periphery, which, in turn, should (statistic-

ally) have a larger number of kink sites than the highly regular cluster

shapes assumed in the theory. This situation would cause a higher atom

deposition to evaporation flux ratio. Furthermore, surface diffusion

during vapor growth was found to cause a large increase in the growth


As indicated earlier, eqs. (32) and (33) were derived under the

assumption that the nucleus radius increases linearly with time. Al-

though this assumption does not really affect the physics of the model,

it plays an important role in the kinetics because it determines the 1/3

exponent in the rate equations. For example, assuming that the cluster

radius grows as r(t) tl/2 (i.e. the cluster area increases linearly


with time) as in a diffusion field, the growth rate equation is derived


V z c'h (JUe2)1/2 (34)

where c' is a constant close to unity. Indeed, growth data (S/V) of a

MC simulation study were represented by this model.176 Alternatively,

if the growth of the cluster is assumed to be such that its radius in-

creases with time as r(t) t + t1/2 (i.e. a combined case of the above

mentioned submodels), it can be shown that the growth rate takes the

form of

V = c"h (JUe2)2/5 (35)

where c" is a constant. Therefore, according to these expressions, the

power in the growth rate equation varies from 1/3 to 1/2.177

A faceted interface that is dislocation free grows by any of the

two previously discussed 2DN growth mechanisms. At low supercoolings

the kinetics are of the MNG mode, while at higher supercoolings the

interface advances in accord with PNG kinetics. The predicted growth

rate equations (eqs. (30) and (32)) can be rewritten with the aid of

eqs. (27), (26), and (20) as
AT 1/2 e (36)
(MNG) V = K A ( AT) exp (- ae (36)

AT 5/6 _e_
(PNG) V= K (i-) exp (- 3T) (37)
2 T 3TAT

Here, KI, K2, and M are material and physical constants whose analytical

expressions will be given in detail in the Discussion chapter. The

growth rates as indicated by eqs. (36) and (37) are strongly dependent

upon the exponential terms, and therefore upon the step edge free energy


and the interfacial supercooling. Although the pre-exponential terms of

the rate equations, strictly speaking, are functions of AT and T, prac-

tically they are constant within the usually limited range of supercool-

ings for 2DNG. The distinct features associated with 2DNG kinetics are

the following: a) A finite supercooling is necessary for a measurable

growth rate (-10-3 um/s); this is related to the threshold supercooling

for 2DN, mentioned earlier, and it is governed by oe in the exponential

term. The smaller ce is, the smaller the supercooling at which the

interfacial growth is detectable. b) Only the MNG kinetics are depend-

ent on the S/L interfacial area. c) Since the pre-exponential terms are

relatively temperature independent, both MNG and PNG kinetics should

fall into straight lines in a log(V) vs. 1/AT plot. d) From the slope

of the log(V) vs. 1/AT curve (i.e. Moe2/T), the step edge free energy

can be calculated,63177-181 provided that the experimental data have

been measured accurately. oe can then be used to estimate the diffuse-

ness parameter "g" via the proposed relation6

oe = osz h (g)1/2 (38)

e) Furthermore, in the semilogarithmic plot of the growth data, the

ratio of the slopes for the MNG and PNG regimes should be 3, according

to the classical theory; however, as discussed earlier, this ratio can

actually range from 2 to 3 depending on the details of the cluster

spreading process.

Detailed 2DNG kinetics studies are very rare, in particular for the

MNG region, which has been found experimentally only for Ga2 and Ag.182

The major difficulties encountered with such studies are 1) the necess-

ity of a perfect interface; 2) the commonly involved minute growth


rates; 3) the required close control of the interfacial supercooling

and, therefore, its accurate determination; and 4) the problems associ-

ated with analyzing the growth data analysis when the experimental range

of AT's is small or it falls close to the intersection of the two MNG

and PNG kinetic regimes for a given sample size. Nevertheless, there

are a couple of experimental studies which rather accurately have veri-

fied the 2DN assisted growth for faceted metallic interfaces.2,63,99,182

Screw dislocation-assisted growth (SDG)

Most often crystal interfaces contain lattice defects such as screw

dislocations and these can have a tremendous effect on the growth kinet-

ics. The importance of dislocations in crystal growth was first pro-

posed by Frank,183 who indicated that they could enhance the growth rate

of singular faces by many orders of magnitude relative to the 2DNG

rates. For the past thirty years since then, researchers have observed

spirals caused by growth dislocations on a large variety of metallic and

non-metallic crystals grown from the vapor and solutions,16 and on a

smaller number grown from the melt.'84

When a dislocation intersects the interface, it gives rise to a

step initiating at the intersection, provided that the dislocation has a

Burgers vector (t) with a component normal to the interface.185 Since

the step is anchored, it will rotate around the dislocation and wind up

actually in a spiral (see Fig. 7c). The edges of this spiral now pro-

vide a continuous source of growth sites. After a transient period, the

spiral is assumed to reach a steady state, becoming isotropic, or, in

terms of continuous mechanics, an archimedian spiral. This further

means that the spiral becomes completely rounded since anisotropy of the


kinetics and of the step edge energy are not taken into account. How-

ever, it has been suggested119 that on S/V interfaces sharply polygoni-

zed spirals may occur at low temperatures or for high "a" factor mater-

ials. Nonrounded spirals have been observed during growth of several

materials,186''87 as well as on Ga monocrystals during the present


Most theoretical aspects of the spiral growth mechanism were first

investigated by BCF in their classical paper,10 which presented a revo-

lutionary breakthrough in the field of crystal growth. Interestingly

enough, although their theory assumes the existence of dislocations in

the crystal, it does not depend critically on their concentration. The

actual growth rate depends on the average distance (yo) between the arms

of the spiral steps far from the dislocation core. This was evaluated

to be equal to 4nrc; later, a more rigorous treatment estimated it as

19rc.188 The curvature of the step at the dislocation core, where it is

pinned, is assumed to be equal to the critical two-dimensional nucleus

radius rc. On the other hand, for polygonized spirals, the width of the

spiral steps is estimated186 to be in the range of 5rc to 9rc.

According to the continuum approximation, the spiral winds up with

a constant angular velocity w. Thus, for each turn, the step advances

Yo in a time yo/ue = 2nr/. Then the normal growth rate V is given aso1

V = bw/27 = byo/ue (39)

where b is the step height (Burgers vector normal component). According

to the BCF notation, from eq. (24) where yo = 4rrc 47Ye/KTo (here Ye

is the step edge energy per molecule), one gets the BCF law

V = f-v exp (- W/KT) (02/01) tanh (ol/o) (40)

S= x and f is a constant.
1 KTx

BCF also considered the case when more than one dislocation merges

at the interface. For instance, for a group of S dislocations, each at

a distance smaller than 2nrc from each other, arranged in a line of

length L, eq. (40) holds with a new yo = Yo/S when L < 4Arc and yo

2L/S when L > 47rc. Nevertheless, the growth rate V can never surpass

the rate for one dislocation, regardless of the number and kind of dis-

locations involved.

For growth from the melt, the rate equation for the screw disloca-

tion growth (SDG) mechanism has been derived as152,189

V = (41)
41rT RTo V
m sZ m

Canh et al.25 have modified eq. (41) for diffuse interfaces with a

multiplicity factor B/g. The physical reason for this parabolic law is

that both the density of spiral steps and their velocity increases pro-

portionally with AT. Models for the kinetics of nonrounded spirals also

predict a parabolic relationship between V and AT.190 However, another

model that accounts for the interaction between the thermal field of the

dislocation helices has shown that a power less than two can be found in

the kinetic law V(AT).191

The influence of the stress field in the vicinity of the disloca-

tion has shown to be significant on the shape of growth and dissolution

(melting) of spirals in several cases.192 It can be shown'88 that the

effect of the stress field extends to a distance rs from the core of the

dislocation given as

b c 1/2
s 2-

where p is the shear modulus. Nevertheless, corrections due to the

stress field are usually neglected since most of the time rs < Yo.

In conclusion, dislocations have a major effect on the kinetics of

growth by enhancing the growth rates of an otherwise faceted perfect

interface, as it has been shown experimentally for several materi-

als.2,25,26,34,63 Predictions from the classical SDG theory describe

the phenomena well enough, as long as spiral growth is the dominant pro-

cess.145 As far as growth from the melt is concerned, most experimental

results are not in agreement with the commonly referred parabolic growth

law, eq. (41); indeed, the majority of the S/L SDG kinetics found in the

literature are expressed as V ATm with m < 2.

In contrast with the perfect (and faceted) interface, a dislocated

interface is mobile at all supercoolings. Moreover, the SDG rates are

expected to be several orders of magnitudes higher than the respective

2DNG rates, regardless of the growth orientation. Like the 2DNG kin-

etics, the dislocation-assisted rates can fall on two kinetic regimes

according to the BCF theory. This can be understood by considering the

limits of SDG rate equation, eq. (40), with respect to the supersatura-

tion a. It is realized that when a < a0, i.e. low supersaturation, then

one has the parabolic law

V 02

and for o D o0 the linear law

V o

For the parabolic law case, yo is much greater than xs and the reverse

is true for the linear law. In between these two extreme cases, i.e. at

intermediate supersaturations, the growth rates are expected to fall in

a kinetics mode faster than linear but slower than parabolic; such a

mode could be, for example, a power law, V = ATn, with n such that 1 < n

< 2.

For growth from the melt, the BCF rate equation can be rewritten


V = N AT2 tanh (P/AT) (42)

where N and P are constants. Equation (42) reduces to a parabolic or to

a linear growth when the ratio P/AT is far less or greater, respective-

ly, than one.

Lateral growth kinetics at high supercoolings

According to the classical LG theory, the step edge free energy is

assumed to be constant with respect to supercooling, regardless of poss-

ible kinetics roughening effects on the interfacial structure at high

AT's. Based on a constant oe value, the only change in the 2DNG growth

kinetics with AT is expected when the exponent AG*/3KT (see eq. (37)) is

close to unity. In this range, the rate is nearly linear (-ATn, n =

5/6). An extrapolation to zero growth rates from this range intersects

the AT axis to the right of the threshold supercooling for 2DN growth.

For SDG kinetics, based on the parabolic law (eq. (40)), no changes in

the kinetics are expected at high AT's. However, the BCF law (eq.

* For detailed relations between supersaturation and supercooling see
Appendix VI.

(39)), as discussed later, for large supercoolings reduces to an equa-

tion in the form

V = A' AT B' (43)

where A' and B' are constants. Note: if eq. (43) is extrapolated to

V = 0, it does not go through the origin, but intersects the AT axis at

a positive value.

It should be mentioned that none of the above discussed transitions

has ever been found experimentally for growth from a metallic melt. The

parabolic to linear transition in the BCF law has been verified through

several studies of solution growth.181,193

Continuous Growth (CG)

The model of continuous growth, being among the earliest ideas of

growth kinetics, is largely due to Wilson194 and Frenkels95 (W-F). It

assumes that the interface is "ideally rough" so that all interfacial

sites are equivalent and probable growth sites. The net growth rate

then is supposed to be the difference between the solidifying and melt-

ing rates of the atoms at the interface. Assuming also that the atom

motion is a thermally activated process with activation energies as

shown in Fig. 8, and from the reaction rate theory, the growth rate is

given as154,196

V = V exp (- -) [1 exp (- )] (44)

where Vo is the equilibrium atom arrival rate and Qi is the activation

energy for the interfacial transport. As mentioned earlier, for practi-

cal reasons, Qi is equated to the activation energy for self-diffusion

in the liquid, QL, and Vo = avi where a is the jump distance interlayerr

spacing/interatomic distance) and vi is the atomic vibration frequency.





Scc L

_______ X -

Figure 8 Free energy of an atom near the S/L interface. QL and
Q are the activation energies for movement in the liquid
and the solid, respectively. Qi is the energy required to
transfer an atom from the liquid to the solid across the
S/L interface.


Hence, aviexp (- Qi/KT) = D/a where D is the self-diffusion coefficient

in the liquid. A similar expression can be derived based on the melt

viscosity, n, by the use of the Stokes-Einstein relationship aDn = KT.

Therefore, eq. (44) can be rewritten as

V = F(T) [1 exp (- )] (45)

where F(T) in its more refined form is given as197

F(T) Da f -
X2 n

in which f is a factor (5 1) that accounts for the fact that not all

available sites at the interface are growth sites and A is the mean dif-

fusional jump distance. Note that if A =a, then F(T) = Df/a. Further-

more, for small supercoolings, where LAT/KTmT < 1, eq. (45) can be re-

written as (in molar quantities)25

V = AT = KAT (46)
aRTT c

which is the common linear growth law for continuous kinetics. For most

metals the kinetic coefficient Kc is of the order of several cm/seco'C,

resulting in very high growth rates at small supercoolings. Because of

this, CG kinetics studies for metallic metals usually cover a small

range of interfacial supercoolings close to Tm; in view of this, most of

the time linear and continuous kinetics are used interchangeably in the

literature. However, this is true only for small supercoolings, since

for large supercoolings the temperature dependence of the melt diffusiv-

ity has to be taken into account. Accordingly, the growth rate as a

function of AT is expected to increase at small AT's and then decrease

at high AT's. On the other hand, a plot of the logarithm of

V/[l exp (- )T
kT T

as a function of 1/T should result in a straight line, from the slope of

which the activation energy for interfacial migration can be obtained.

Indeed, such behavior has been verified experimentally25,26,63,198 in a

variety of glass-forming materials and other high viscosity melts.

An alternative to eqs. (45) and (46) was proposed by suggesting

that the arrival rate at the interface for simple melts might not be

thermally activated;199,200 the kinetic coefficient Kc then was assumed

to depend on the speed of sound in the melt. This treatment was in good

agreement with the growth data for Ni,201 but not with the data of

glass-forming materials. Another approach suggested that the growth

rate is given as202
a 3KT 1/2 LAT
V = KT)1/2 f [1 exp (- )]T
where the atom arrival rate is replaced by (3KT/m) which is the

thermal velocity of an atom. This equation was in good agreement with

recent MD results on the crystallization of a Lennard-Jones


Other approaches for continuous growth are mostly based on the kin-

etic SOS model for a Kossel crystal in contact with the vapor.117,145

As mentioned elsewhere, the basic difference among these models is the

assumption concerning clustering (i.e. number of nearest neighbors),

which strongly effects the evaporation rate and, therefore, the net

growth rate.204 In addition, these MC simulations only provide informa-

tion about the relative rates in terms of the arrival rate of atoms.

For vapor growth, the latter is easily calculated from gas kinetics.


For melt growth, however, the arrival rate strongly depends on the

structure of the liquid at the interface, which is not known in detail.

Therefore, these models cannot treat the S/L continuous growth kinetics

properly. Some general features revealed from these models are dis-

cussed next to complete this review.

All MC calculations for rough interfaces indicate linear growth

kinetics. The calculated growth rates are smaller than those of the W-F

law, eq. (44). This is understood since the latter assumes f = 1.

Interestingly enough, the simulations show that some growth anisotropy

exists even for rough interfaces. For example, for growth of Si from

the melt, MC simulations predicted205 that there is a slight difference

in growth rates for the rough (100) and (110) interfaces. The observed

anisotropy is rather weak as compared to that for smooth interfaces, but

it is still predicted to be inversely proportional to the fraction of

nearest neighbors of an atom at the interface (5 factor). Nevertheless,

true experimental evidence regarding orientation dependent continuous

growth is lacking. If there is such a dependence, the corresponding

form of the linear law would then be

V = Kc(n) AT (47)

This is illustrated by examining the prefactor of AT in eq. (46). Note

that the only orientation dependent parameter is (a), so that the growth

rate has to be normalized by the interplanar spacing first to further

check for any anisotropy effect. If there is any anisotropy, it could

only relate to the diffusion coefficient D, otherwise Di to be correct,

and, therefore, to the liquid structure within the interfacial region.

At present, the author does not know of any studies that show such


anisotropy. In contrast, it is predicted'17 that there is no growth

rate difference between dislocation-free and dislocated rough inter-

faces. This is because a spiral step created by dislocation(s) will

hardly alter the already existing numerous kink sites on the rough


A summary of the interfacial growth kinetics together with the

theoretical growth rate equations is given in Fig. 9. Next, the growth

mode for kinetically rough interfaces is discussed.

Growth Kinetics of Kinetically Roughened Interfaces

As discussed earlier, an interface that advances by any of the lat-

eral growth mechanisms is expected to become rough at increased super-

coolings. Evidently, the growth kinetics should also change from the

faceted to non-faceted type at supercoolings larger than that marking

the interfacial transition.

In accord with the author's view regarding the kinetic roughening

transition, the following qualitative features for the associated kinet-

ics could be pointed out: a) Since the interface is rough at driving

forces larger than a critical one, its growth kinetics are expected to

resemble those of the intrinsically rough interfaces. Thus, the growth

rate is expected to be unimpeded, nearly isotropic, and proportional to

the driving force. Moreover, the presence of dislocations at the inter-

face should not affect the kinetics, b) It is clear that the faceted

interface gradually roughens with increasing AT over a relatively wide

range of supercoolings. The transition in the kinetics should also be a

gradual one. c) In the transitional region the growth rates should be

faster than those predicted from the lateral, but slower than the

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