ADVANCES IN HEAT TRANSFER Volume 28
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Advances in
HEAT TRANSFER Guest Editor Dimos Poulikakos* Department of Mechanical Engineering University of Illinois at Chicago Chicago, Illinoh
*PresentAddress: Department of Mechanical and Process Engineering, Institute of Energy Technology, Swiss Federal Institute of Technology, ETH Center, Zurich, Switzerland.
Serial Editors James P. Hartnett
Thomas F. Irvine
Energy Resources Center Unwersity of Illinois at Chicago Chicago, Illinois
Department of Mechanical Engineering State Universiry of New York at Stony Brook Stony Brook, New York
Serial Associate Editors Young I. Cho
George A. Greene
Department of Mechanical Engineering Drewel University Philadelphia, Pennsylvania
Department of Advanced Technology Brookhaven National Laboratoly Upton, New York
Volume 28
ACADEMIC PRESS San Diego Boston New York London Sydney Tokyo Toronto
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Copyright 0 1996 by ACADEMIC PRESS, INC.
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International Standard Serial Number: 0065-27 17 International Standard Book Number: 0-12-020028-7 PRINTED IN THE UNITED STATES OF AMERICA
CONTENTS Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix xi
Heat Transfer and Fluid Dynamics in the Process of Spray Deposition DIMOS POULIKAKOS AND JOHNM. WALDVOGEL
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. TheSprayRegion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 3 A. Convective Cooling of a Single Liquid Metal Droplet . . . . . . . . . . . . . . . 3 B. In-Flight Solidification of a Liquid Metal Droplet . . . . . . . . . . . . . . . . 11 C. Studies of Sprays in Spray Deposition . . . . . . . . . . . . . . . . . . . . . . . 18 I11. The Impact Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A. Splat Cooling of a Single Liquid Metal Droplet . . . . . . . . . . . . . . . . . 23 B. Impact and Solidification of Multiple Liquid Metal Droplets and Sprays . . . 54 IV . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Heat and Mass Transfer in Pulsed-Laser-Induced Phase Transformations P . GRIGOROPOULOS. TEDD . BENWEIT.JENG-RONG Ho. COSTAS XIANFAN XU.AND XIANG ZHANG
I . Pulsed Laser Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..................................... B. ThermalModeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Experimental Verification of the Melting Process . . . . . . . . . . . . . . . . D . Ultrashallow p+-Junction Formation in Silicon by Excimer Laser Doping . .
A . Background
E . Topography Formation
..............................
I1. Pulsed Laser Sputtering of Metals
..................... .................................... B. Time-of-Flight Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . C. Considering Thermal and Electronic Effects . . . . . . . . . . . . . . . . . . I11. Computational Modeling of Pulsed Laser Vaporization . . . . . . A. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Modeling Description-Transparent Vapor Assumption . . . . . . . . . . . A. Background
V
75 75 76 80 96 101 109 109 112 116 123 123 125
vi
CONTENTS
IV. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135 136 138
Heat and Mass Transfer in the Extrusion of Non-Newtonian Materials YOGESH JALURIA
I . Introduction
.................................... .................................... ................................. I1. Material Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. Single-Screw Extruder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Tapered Screw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Residence-Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Mixing Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Twin-Screw Extruder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Background B . Literature Review
V . FlowinDies
....................................
.......................... ............................
A. Coupling of Extruder with Die B. Transport in Complex Dies
VI . Combined Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . .
................................
A . Moisture Transport B. Chemical Reaction and Conversion
.......................
VII . Additional Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
145 146 149 152 155 155 176 176 180 184 187 190 197 201 201 205 213 213 218 220 225 226 227
Convection Heat and Mass Transfer in Alloy Solidification PATRICK J . PRESCOTT AND FRANK P . INCROPERA
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Physical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
................. .............................. ............................... .....................................
A . Historical Perspective of Solidification Models B. Single Domain Models C. Micro/Macro Models D . Submodels
231 238 249 250 253 261 264
vii
CONTENTS
.......... ......................... .................................... V. Strategies for Intelligent Process Control . . . . . . . . . . . . . . . . VI. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IV. Theoretical Results and Experimental Validation A . Semitransparent Analog Alloys B. MetalAlloys
269 270 288 308 326 328 329
Transport Phenomena in Chemical Vapor-Deposition Systems Roop L. W A N
1. Introduction
..................................... ........................................ ..................... I1. Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Rate-Limiting Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Some Basic Transport Considerations . . . . . . . . . . . . . . . . . . . . . . I11. Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.Scope B. Common CVD Reactor Configurations
A . Equations for a Multicomponent Mixture . . . . . . . . . . . . . . . . . . . . B. Simplified Governing Equations
C. Transport Properties
. . . . . . . . . .. .. .. .. .. .. .. .. ...............................
IV. Solutions for Selected Reactor Configurations
............
................................ ................................... ..................................
A . Horizontal Reactors B. Barrel Reactor C. Pancake Reactor D . &symmetric Rotating.Disk. Impinging.Jet, and Planar Stagnation-Flow Reactors E. Hot-Wall LPCVD Reactors
339 339 344 346 346 351 353 353 356 362 365 365 375 386
...................................... ...........................
389 399 408 413 414 415
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
427
V . Artificial Neural Network Models for CVD Processes . . . . . . . VI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CONTRIBUTORS
Numbers in parentheses indicate the pages on which the authors' contributionsbegin.
TEDD. BENNETT (75), Department of Mechanical Engineering, University of California, Berkeley, California 94720. COSTASP. GRIGOROPOULOS (75), Department of Mechanical Engineering, University of California, Berkeley, California 94720. JENG-RONG Ho (751, Department of Mechanical Engineering, University of California, Berkeley, California 94720. FRANK P. INCROPERA (230, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907. YOGESHJALURIA (1451, Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, New Jersey, 08903. Roop L. W A (3391, N Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309. DIMOSPOULIKAKOS' (l), Mechanical Engineering Department, University of Illinois at Chicago, Chicago, Illinois 60607. PATRICK J. PRESCOTT (231), Department of Mechanical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802. JOHNM. WALDVOGEL (11, Motorola Inc., Schaumburg, Illinois 60196. XIANFANXu2 (751, Department of Mechanical Engineering, University of California, Berkeley, California 94720. XYWG ZHANG(75), Department of Mechanical Engineering, University of California, Berkeley, California 94720.
Present Address: Department of Mechanical and Process Engineering, Institute of Energy Technology, Swiss Federal Institute of Technology, ETH Center, Zurich, Switzerland. Present Address: School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907.
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PREFACE
The area of materials processing has progressively become one of the focal points of university and industrial research in the 1990s. Transport phenomena play a central role in a plethora of applications in materials processing and manufacturing. However, the flow of information and collaboration among the scientific communities in the areas of transport phenomena and materials processing are not yet optimal. The purpose of this volume is to present a representative sample of existing research efforts in the area of transport phenomena, directly related to materials processing. This task is accomplished through five review papers (which compose the present volume) selected to cover a wide spectrum of applications. Naturally, due to space limitations the volume is not all-inclusive. However, I feel that it will provide the reader with a good flavor of the many exciting research areas in materials processing and manufacturing in which the transport phenomena scientific commumnity can contribute significantly. I thank the series editors of Advances in Heat Transfer for sharing my viewpoint that there is a pressing need for this special volume and for giving me the opportunity to put it together. Finally, thanks are due to all the contributors who made this volume possible in a timely fashion. D. Poulikakos, Guest Editor
xi
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ADVANCES IN HEAT TRANSFER,VOLUME 28
Heat Transfer and Fluid Dynamics in the Process of Spray Deposition
DIMOS POULIKAKOS* Institute of Energy Technology, Swiss Federal Institute of Technology (ETH), ETH Centec Zurich, Switzerland
JOHN M. WALDVOGEL Motorola, Inc. Schaumbutg, Illinois
I. Introduction
Spray deposition (or spray casting) is a novel rapid solidification technol-
ogy for the creation of advanced metals and metal composites. This technology is particularIy attractive to manufacturing because it shows promise to provide materials and products that combine superior properties and near net shape. With reference to the former, the extremely high cooling rates present in the process of spray deposition (especially at the early stages) capture nonequilibrium states that cannot be captured by more conventional casting methods (foundry solidification, for example) because the atomic mobility in the liquid phase of a metal is far greater than that in the solid phase. To this end, the cooling rates at the early stages of the spray deposition process are of the order of (106-108YC/s. With reference to the latter, the spray deposition process has been shown to produce near net shape products which eliminate the need for additional finishing steps in the manufacturing process. Moreover, the fine and homogeneous grain microstructure that appears to result from the spray deposition process may eliminate the need for additional mechanical working [l, 21. In this paper, a review is presented of the existing knowledge base of the process of spray deposition, focusing on issues in which transport phenomena are relevant. ‘Present address: Department of Mechanical and Process Engineering, Institute of Energy Technology, Swiss Federal Institute of Technology, ETH Center, Zurich, Switzerland. 1
Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
2
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
The defining features of the process of spray deposition are shown in Fig. 1. This process involves four distinct regions. In the first region, the metal or metal alloy under processing is melted inside a crucible (often utilizing induction heating) and subsequently heated to a desired superheat temperature that ensures good fluidity. The melting occurs in an inert environment (e.g., argon or nitrogen) to limit oxidation. A stream of molten metal exits through the bottom of the crucible and enters the second of the four regions mentioned above, the atomization region, in which the liquid metal stream is blasted with an inert atomizing gas and disintegrates into a spray. In the third region of the spray deposition process, the spray region, liquid metal elements disintegrate further into droplets. Droplet coalescence also takes place in the spray region. The liquid metal droplets constituting the spray travel in an inert environment
FIG.1. Schematic illustrating the spray deposition process.
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
3
(to limit oxidation). Cooling occurs during this travel, which often results in the partial solidification of the droplets prior to impact. The impact and solidification of the droplets on the substrate constitutes the fourth region of the spray deposition process. From the preceding description it is clear that transport phenomena play a pivotal role on the spray deposition process. To exemplify, heat transfer and fluid dynamics phenomena take place in all four regions outlined above: the melting in the crucible, the breakup of the liquid metal stream into the spray, the transportation of the droplet in the form of a spray, and the splashing and solidification of the droplet on the substrate. Despite this fact, out knowledge base on the effect of transport phenomena in the spray deposition process is very limited. Most of the existing studies have been performed by materials scientists and focus on metallurgical aspects of the process, which are also of great relevance and importance. The mission of this work is to review the existing studies in the open literature that focus on the effect of transport phenomena on the spray deposition process. In doing so, the existing base of knowledge and the state of the art of this process from the standpoint of transport phenomena will be defined. In addition, the research needs in these areas will be identified. The presentation will be centered around the most challenging fluid dynamics and heat transfer aspects of the spray deposition process that occur in the spray region and the impact region. It is worth noting that the melting process in the crucible is rather well understood. For each of these regions both basic studies involving single droplets and droplet arrays as well as more applied studies involving sprays will be presented.
11. The Spray Region
A.
CONVECTIVE COOLING OF A SINGLE LIQUID
METALDROPLET
Basic studies on the convective cooling of a liquid metal droplet placed in a gas stream are the first step toward the investigation of the heat and fluid flow phenomena in sprays. In addition, these studies are relevant to dilute regions in the spray where the effect of interaction between droplets is not important. A significant base of knowledge in the general area of convective cooling of droplets already exists because of the wide use of sprays in many engineering applications exemplified by spray combustion. Most of this knowledge is summarized in a recent review paper by Sirignano [3]. It was not until recently, however, that a complete dedicated
4
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
numerical study focusing on the presolidification fluid dynamics and convection phenomena of the problem of a superheated liquid metal droplet placed in a uniform gas stream was published by Megaridis [4].This study is pertinent to the laminar flow regime and assumes axisymmetric flow conditions. In what follows immediately, the mathematical model and the main findings in Megaridis [4]are highlighted. A schematic of a liquid metal droplet in flight under laminar axisymmetric flow conditions is shown in Fig. 2 The study in Megaridis [41 simulates the acceleration and simultaneous cooling of a liquid metal droplet suddenly placed in a uniform stream of an inert gas. To this end, the flow field in the gaseous stream and the shear-induced flow field in the liquid metal droplet are considered simultaneously. The model relies on experience of droplet transport phenomena gathered from earlier combustion-related studies. The conservation equations in the gas phase are [4]. Continuity:
Radial momentum:
Liquid-Metal Droplet
FIG. 2. Schematic of axisymmetric flow inside and around a liquid metal droplet from Megaridis [4].
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
5
Axial momentum:
Energy:
In the preceding equations, ur and u, are the radial and axial velocity components, respectively; p is the pressure; a, and QZ contain the relevant viscous terms in the momentum equations; T is the temperature; and pg, k,, and cp, are the gas density, thermal conductivity, and specific heat at constant pressure, respectively. The conservation equations in the liquid phase modeling the flow in the liquid metal droplet are cast in the stream function-vorticity formulation after introducing the stream function in the usual manner: 1 r dz
1 a*
u l , z = - - -. r dr Thus, the conservation equations in the liquid metal region (phase) are Ul,r =
- -,
Vorticity equation: dt = --
Stream-function equation:
Energy equation:
6
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
In these equations the subscript 1 denotes the liquid metal region; +, the stream function; w , the vorticity; T , the temperature; t, the time; and c p , , p , , k , , and p I , the specific heat at constant pressure, viscosity, thermal conductivity, and density of the liquid metal, respectively. To complete the model formulation, the initial and boundary conditions utilized in Megaridis [4]are postulated. 1. Initial conditions a. Gas phase: Att=O:
u,=O,
u , = U ~ , ~p ,= p m , T = T m . (10)
The subscript m denotes incoming free-stream conditions, and Urn. is the initial relative velocity between the droplet and the free stream. b. Liquid phase +=w=O, T=To. (11),(12) Att=O: where To is the injection temperature of the droplet. c. Droplet surface u,=u,=O, p = p m , T = T o . (13)-(15) Att=O: 2. Boundary conditions a. Gas-liquid interface: The conditions at the gas-liquid in spherical coordinates and with n denoting the direction perpendicular to the interface are as follows: Shear stress continuity:
pg
[---
du,,,
=PI
a UI.0
-1
+ -1 d o ,
a
a
0
, (16)
&I
where a is the droplet radius and subscripts 1 and g denote the liquid and gas phases, respectively. Tangential velocity continuity: u I , Ols = ug,O l s .
(17)
Temperature and heat flux continuity:
b. Inflow and outflow boundaries of the computational domain Inflow boundary U, =
0,
U, =
U,, p = p a , T
=
T,.
(20)-(23)
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
7
Note that the free-stream velocity U, is time-dependent as a result of the relative deceleration of the droplet with respect to the free stream. Outflow boundary Du, Dt
Du, - _ _ _ = -DT Dt
Dt
- -Dp Dt
=
0,
where D/Dt denotes the total derivative with respect to time. c. Axis of symmetry a v , _ap= _dT - - u, = 0 , Gas phase: -(25) dr dr dr Liquid phase:
dT $ = o = - = 0. dr
Equations (1)-(26) constitute the theoretical model solved numerically in Megaridis [4] to study the presolidification convection phenomena in a single liquid metal droplet placed in a quiescent stream of inert gas. The base case of the numerical solution simulated the cooling of a superheated liquid aluminum droplet initially at 1000 K, suddenly injected in a stream of nitrogen at 400 K. The ambient pressure was 1 atm and the initial value of the Reynolds number [Re = d & / k ) ] was 100. The symbol d in the definition of the Reynolds number is the droplet diameter. The remaining symbols were defined earlier. The variable properties of the gas phase were obtained from standard correlations [5]. The thermophysical properties of the liquid metal phase were assumed constant [61. The first result of interest in Megaridis [4] was a temporal comparison of the drag coefficient to the well-known correlation for laminar flow over a solid sphere [7]:
c
-
24
-(I Re,
+ 0.1935Re:.6305);
20 I Re, I 260.
(27)
This equation utilizes the film-adjusted Reynolds number, which is based on the relative velocity between the droplet and the free stream, the droplet diameter, the free-stream density, and the gas velocity evaluated at the film temperature (average between the free stream and the droplet surface temperatures). The result of this comparison is shown in Fig. 3. Since the Reynolds number is based on the relative velocity between the free stream and the droplet, high values of the Reynolds number in this graph correspond to early times in the cooling process. Clearly, except for
8
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
1.56 1.52 1.48 -
75.0 77.5 80.0 82.5 85.0 87.5 90.0 92.5 95.0 97.5 100.0
Instantaneous Reynolds Number FIG.3. C, vs. Re from Megaridis [4].
the early stages of the process when the droplet is introduced into the liquid stream, Eq. (27) predicts well the drag coefficient and its use is recommended. A characteristic map of liquid isotherms obtained in Megaridis [4] is shown in Fig. 4a. The nondimensionalization of time was based on the , tU,/a2). Clearly, the viscous diffusion time scale in the gas phase ( T ~ = coldest location in the droplet is in the vicinity of the forward stagnation point. It is in this vicinity where the solidification process will be initiated. Hence, despite the fact that the maximum temperature difference in the droplet is small (1 K in Fig. 4a), the solidification will not be radially symmetric and should not be modeled as such. Note that solidification has been observed to take place under conditions of severe undercooling and in the presense of recalescence [8-101, as will be discussed later herein, which also renders the radically symmetric modeling of the process inappropriate.
a
LIQUID-PHASE ISOTHERMS
Contour Interval: 8.65E-02 K, Min: 990 K , Max: 991 K
Reynolds Number = 97.32 Ambient Temperature= 400 K Initial Droplet Temperahre= 1000 Initial Reynolds Number = 100
K
A. Liquid-Aluminum Dro let B. Ranr-Marshall, Film d3eynolds Number C. Ram-Marshall, Free-Stream Reynolds Number
10.0 9.0 8.0
7.0
6.0 5.0 4.0
3.0 2.0 1
.o
0.0
0.0 3.0
6.0
9.0
12.0 15.0
18.0 21.0 24.0 27.0
30.0
Gas Hydrodynamic Diffusion Time Scale FIG.4. (a) Map of isotherms in liquid metal droplet from Megaridis [4]. The gas flow is from left to right. The arrow indicates the direction of increasing temperature. (b) Nu vs. dimensionless time T from Megaridis [4].
10
DIMOS POULIKAKOS A N D JOHN M. WALDVOGEL
The final main result in [4] was to test whether the popular Ranz-Marshall correlation for laminar convection from a solid sphere is appropriate for the problem of interest. This correlation reads [ l l ] Nu
=
2
+ 0.6Re1/’ Pr1I3.
(28) The average Nusselt number (Nu) is defined on the basis of the droplet diameter, the surface-averaged heat transfer coefficient between the gas and the droplet, and the free-stream thermal conductivity. The Prandtl number (Pr) is that of the gas at free-stream conditions. The Reynolds number is based on the relative velocity between the droplet and the free stream. Figure 4b shows that comparison between Eq. (28) and the numerical predictions of the numerical model outlined earlier [4]. Curve A in Fig. 4b shows the results predicted by the model, curve B indicates the results of Eq. (28) utilizing the film-adjusted Reynolds number (Re,) defined earlier in connection with Eq. (27), and curve C shows the results of Eq. (28) utilizing a Reynolds number based on the droplet diameter, the relative velocity between the gas and the free stream, and the gas properties at free-stream conditions. As shown in Fig. 4b, curve C agrees better with the numerical results than does curve B, which implies that the free-stream properties should be used in Eq. (28) to estimate Nu in liquid metal droplets. Furthermore, the agreement between the numerical results and curve C can be described as fair (within 15%)with the Ranz-Marshall correlation underpredicting Nu. This may result in significant errors, especially if solid nucleation and partial solidification with recalescence occur during the droplet flight. A need for improved correlations exists in this area. Although the laminar flow results of Megaridis [4] improve our knowledge of the basic mechanisms in the cooling of a liquid metal droplet, they cannot be applied directly to the real spray deposition process because the relevant heat and fluid flow phenomena often are in the turbulent regime. Our literature review indicated that no study analogous to Megaridis [4] for liquid metal droplets in the turbulent regime exists in the open literature. Instead, lumped models combined with empirical correlations are utilized. A description of a typical model of this kind is given in Gutierrez-Miravete et al. [12]. The droplet velocity is obtained from a simple force balance on the droplet (Newton’s second law)
where pd, ud, 6 , A , , pg, u g , and g are the droplet density, velocity, volume, surface area, gas density, velocity, and gravitational acceleration, respectively. The drag coefficient is denoted by C,. The value of the drag
11
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
coefficient recommended in Lavernia et al. 112, 131 is the one obtained in the 1960s [14]: 6 21 C , = 0.28 + -+ -. 0.1 IRe I4000. (30) Re ' In this equation the Reynolds number is based on the gas properties, the droplet diameter, and the relative velocity between the droplet and the gas. When compressibility effects are important (the Mach number of the gas flow in the atomizer may be quite high [15]), they should be accounted for in the drag coefficient expression. To this end, drag coefficient relations for high-speed flow past a small sphere can be utilized. A correlation of this kind developed from research related to rocket nozzle design is [16, 171
CD
[:
r
(1
=
+ 0.15 Re0.687)[1 ( 1 + explexp ~ ~ 0 . 8 8 p- *
(31)
Re In this equation M is the Mach number and Re the Reynolds number, both of which are based on the relative velocity between the particle and the gaseous stream and the gaseous stream properties. Regarding cooling of the droplet in the transitional and turbulent regimes, simple, lumped models are commonly used [12-141:
where A is the droplet surface area, V the volume, h the heat transfer coefficient, T the temperature, and To the initial temperature. The radiative cooling of the droplet (which may or may not be important) is accounted for by the term Qrad.The thermophysical properties are those of the liquid metal. In order to use this equation to obtain the temperature history of the droplet, information on the heat transfer coefficient is needed. Despite its limited validity, the Ranz-Marshall correlation mentioned earlier [Eq. (28)] is commonly used to provide this information. A better alternative would perhaps be the correlation proposed by Whitaker [18, 191 for heat transfer from an isothermal spherical surface: Nu
hD k
= -=
2
+ (0.4Re1/*+ 0.06Re2/3)Pr0.4 3.5 < Re < 7.6
X
lo4. (33)
12
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
This equation has been tested for 0.71 < Pr < 380 and 1 < pJpS < 3.2. It should be used with caution outside these ranges. All the thermophysical properties in Eq. (33) are of the gaseous stream evaluated at free-stream conditions, except for the viscosity p,, which is evaluated at the sphere surface temperature. At this point it is worth stressing that Eqs. (28) and (33) are both applicable for solid spheres. Obviously, similar correlations must be developed for liquid metal droplets in the high-Reynolds-number regime.
B. IN-FLIGHTSOLIDIFICATIONOF
A
LIQUIDMETALDROPLET
The liquid metal droplet size in the spray deposition process varies between typically 10 and 300 pm. In addition, the flow of the inert gas causing the atomization process depends on location in the domain of the spray. Therefore, several scenarios are possible for each droplet during its flight. Some droplets (usually the smaller droplets) completely solidify during their flight and impact the substrate in solid form. Some droplets solidify only partially, with the degree of solidification depending on the droplet size. Finally, some droplets do not solidify at all during their flight (usually the largest droplets) and impact the substrate in the liquid state. As mentioned earlier, when solidification ensues during the flight of liquid metal droplets in the process of spray deposition, it does so in the presence of severe undercooling. From these observations, it is obvious that the study of solidification of a single liquid metal droplet flying in an inert-gas environment is very relevant to the process of spray deposition. Our literature search showed that such study has not been performed at a level of sophistication that would involve the solution of the Navier-Stokes and energy equations in the gas and liquid metal regions. This is true even for the case of axisymmetric laminar flow in the gas region. Instead, approximate (usually lumped) models are used to estimate the in-flight solidification process. Representatives of such models will be discussed next. The basic elements of a simple, spatially isothermal solidification model for a liquid metal droplet of initial temperature T , flying in a gas environment are contained in Dubroff [8] and Lavernia et al. [131. As discussed earlier, solidification takes place under severe undercooling. Four distinct regimes descriptive of the process can be defined as summarized in Fig. 5. In the first regime convective and radiative cooling takes place until a solid nucleation temperature (TN) is reached. Note that the solid nucleation temperature is lower than the equilibrium solid nucleation temperature (TL), which in the case of an alloy is the liquidus temperature. The difference TN - TL is the undercooling present at the initial
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
13
t
*'
TL
h
6
8
F
Time
FIG. 5. Illustration of the four distinct regimes of in-flight cooling and solidification: convective and radiative cooling, recalescence, slow solidification up to Ts, and cooling of solid sphere.
stage of freezing. The second regime is recalescence. Here the solidification progresses extremely fast and the latent heat released raises the droplet temperature to the recalescence temperature T R, which is often very close to the equilibrium liquidus temperature, TL. After recalescence, the third regime starts (Fig. 51, where solidification proceeds at a much slower rate and the droplet temperature continues to decrease (the heat removal from the droplet surface is larger than the latent heat of fusion in this regime) until the solidification process is completed and the droplet is at the solidus temperature, T,. In the last (fourth) regime the droplet is a solid sphere cooled convectively and radiatively by its gaseous environment. As discussed earlier, depending on droplet size and flow and temperature conditions, a liquid metal droplet can impact the substrate while in any of these four regimes.
1. Modeling of Convective Cooling and Radiatwe Cooling Regimes Since the regimes do not involve solidification they can be modeled in an identical manner [8, 131. An energy balance in a spatially isothermal
14
DlMOS POULIKAKOS AND JOHN M. WALDVOGEL
control volume defined by the outer surface of the droplet yields dT ~ VC dt
+ h,A( T - T,) + UEA(T 4 - T:)
=
0.
(34)
This equation is similar to Eq. (32) and accounts for graybody radiative cooling. The droplet volume is denoted by V , the surface area by A , the specific heat by c, the temperature by T , the time by t , the density by p, the gas free-stream temperature by T,, the convective heat transfer coefficient by h,, the emissivity by E , and the Stefan-Boltzmann constant by u.If the dependence of the liquid metal thermophysical properties on temperature is known (or if these properties are assumed for simplicity independent of temperature) and if T, is a known function of t or constant, Eq. (34) can be integrated forward in time, starting from an initial condition, to yield the temperature history in the droplet in regimes 1 and 4. Note that an equation like Eq. (28) or (33) needs to be used to calculate the convective heat transfer coefficient in the model. Utilizing Eq. (28), for example, yields
h
k -(2
,-D + 0.6Re'/2Pr'/3), -
(35)
where the droplet thermal conductivity is denoted by k and its diameter, by D. Since the Reynolds number is based on the relative velocity between the droplet and the free stream, the definition of this velocity in the present lumped model needs to be discussed. To this end, the gas velocity can either be assumed to be a function of the droplet velocity, or it can be approximately estimated as a function of the distance from the nozzle exit. With reference to the former, an example of a simple (albeit arbitrary) assumption is that the gas velocity is a constant percentage of the droplet velocity
where 4 is a number between zero and unity (e.g., 0.5). Using Eq. (361, we can obtain the absolute droplet velocity as a function of time. Subsequently, we can determine the relative droplet velocity (and the relevant Reynolds number) as a function of time. Aided by knowledge gained from the preceding observations, we can use Eqs. (34) and (35) to determine the temperature history of the droplet during its flight. With reference to the latter, Lavernia et al. [13], without offering a rigorous proof, assumed the gas velocity to decay from its initial nozzle exit
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
15
value according to
where z is the distance from the nozzle exit, zref is a reference distance, and uge represents the gas velocity at the nozzle exit. As discussed earlier, after the gas velocity is obtained from Eq. (37), the absolute droplet velocity results from Eq. (29) and the temperature history of the droplet, from Eq. (34). Another issue relevant to the preceding discussion is the value of gas density used in the velocity and temperature calculations. Veistinen et al. [20, 211 showed that in the case of argon as the atomization gas, a density value at the nozzle exit equal 2.7 times the gas density at room conditions yielded excellent agreement between calculated and measured [221 gas velocities. For the case of argon as the atomization gas again, regression analysis of experimental data in Beattle and Julien [231 for the variation of argon density with pressure yielded 1131 p g = 1.6317 x lO-’p
+ 1.0585,
(38)
where the gas density is expressed in kilograms per cubic meters (kg/m3) and the pressure is expressed in newtons per square meter (N/m2).
2. Modeling of Recalescence Regime The initiation of this regime is marked by the solid nucleation (Fig. 5). The exact value of the nucleation temperature (T,) for the various metals and alloys depends on a host of parameters exemplified by the metal purity, the cooling rates, and the flow conditions. Therefore, this value is seldom known and is assigned arbitrarily in models of the process. There is significant need to create a reliable database of nucleation temperatures for flying liquid metal droplets for a variety of materials. A simple lumped model of the recalescence regime can be constructed if it is assumed that solid nucleates at the outer surface and advances concentrically inward (radial symmetry). This assumption facilitates the analysis but as discussed earlier in connection with Megaridis [41 it is unlikely that radial symmetry exists in the freezing process. Solid nucleation is likely to occur first at the forward stagnation point of the droplet. This fact was recognized by Levi and Mehrabian [24], who presented a solidification model in which a solid nucleated at a single point of the outer surface and not the entire surface. The simple model presented below does assume radial symmetry of the
16
DlMOS POULIKAKOS AND JOHN M. WALDVOGEL
freezing front and should be viewed in the context of this limitation. If the solid shell after nucleation and the liquid region are concentric, the solid fraction on the droplet is given by r
3
X=l-(&
(39)
where R is the droplet radius and r is the radial coordinate. The speed of propagation of the solid front [U = (dr/dt)]is obtained by taking the time derivative of Eq. (39): dX 3(1 - x ) ~ / ~ dt R During recalescence heat transfer at the phase-change interface occurs under nonequilibrium conditions. To this end, a crystallization kinetics relationship for the freezing velocity is required at the freezing front. Assuming that curvature effects do not dominate the freezing process the following relation can be used at the freezing front [25]: - =
where dm is the molecular diameter taken as the molecular jump distance, DL is the diffusivity in the liquid phase, A H , is the latent heat of fusion per molecule, T is the temperature, AT is the undercooling, Tf is the fusion temperature, and K is the Boltzmann constant [expressed in joules per kelvin per molecule (JK-*molecule-')]. for low and up to moderate undercooling, a Taylor series expansion of Eq. (41) accurate to the first order yields the Wilson-Frenkel relation [26, 271:
U =KAT,
( 42)
where the kinetic coefficient is
Note the introduction of a correction factor P [25, 281. This correction factor was suggested [28] to account for the fact that the molecular jump distance across the solid/liquid interface may be smaller than that for diffusion in the bulk liquid. In addition it corrects for reorientation effects necessary with asymmetric molecules. Determining the value of the correction factor P is obscure and is commonly taken to be equal to unity in the literature. The value of the kinetic coefficient can be determined from Eq. (43) utilizing the relevant thermophysical properties. Errors will, of course,
17
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
arise from approximations in property variations (e.g., the dependence of
D, on AT [251). These errors usually do not drastically affect the nucleation estimates, which are much more sensitive to the effect of nucleants. In the model presented here, Eq. (42) wiIl be assumed to describe adequately the crystallization kinetics [8, 121. Combining Eqs. (40) and (42) yields
The volumetric heat generation rate in the droplet due to the solidification process is
and the energy equation [Eq. (34)] modified to account for this heat generation becomes dT dX ~ VC + pVAH, - h c A ( T - T,) u & A ( T 4- T,) = 0. (46) dt dt All the symbols in Eq. (46) have been defined earlier. Radiation cooling is taken into account for the sake of generality, unlike in Grant et al. [8, 12, 131, where radiation effects were justifiably neglected. Equations (44) and (46) can be solved simultaneously with simple numerical means to yield the temperature history in the droplet in the recalescence regime. The initial conditions for the numerical solution are that at t = 0 the solid fraction is equal to zero ( x = 0) and the temperature is equal to the nucleation temperature ( T = T N ) .The solution should be advanced in time until the recalescence temperature is reached, marking the end of the recalescence regime.
+
+
3. Modeling of Solidification Regime This regime involves conventional solidification starting at the recalescence temperature, T R .The solid fraction at the beginning of this regime is that obtained at the end of the recalescence ( x R ) .It is again assumed that the heat of fusion is released uniformly within the droplet as the temperature decreases from T R to T,. Therefore, the energy equation for this regime becomes pVc
+ pVAH, TR--XT,R
IdT
-
dt
+ h c A ( T - T,) + v & A ( T 4- T,)
=
0.
18
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
Equation (47) can be integrated in time numerically until the solidification is complete (the droplet temperature becomes the solidus temperature). Regime 4, discussed earlier, follows. At this point the description of the solidification process of a single liquid metal droplet in flight based on a simple, lumped model is complete. It should be reiterated that the assumption of radial symmetry in the solidification process inherent in the studies that formed the basis for construction of the model presented in this section [8, 12, 131 was relaxed in the work of Levi and Mehrabian [24]. These authors developed a mathematical model for the freezing process in the undercooled droplet from a single (point) nucleation site at its surface neglecting radiative cooling from the surface. They also discuss the implications of single versus multiple nucleation sites. Their results on the freezing process indicate the presence of two distinct solidification regimes. In the first regime the solidification interface velocities are high, the droplet absorbs most of the released latent heat of fusion, and the surface cooling plays a minor role. In the second regime the solid growth is much slower and depends greatly on the cooling of the droplet surface. The extent of rapid solidification was determined to be a function of the nucleation temperature, the particle size, the kinematic coefficient, and the heat transfer coefficient. In a recent theoretical study of solidification of metal drops, Bayazitoglu and Cerny [29] analyzed the process of conduction freezing using a lumped model as well as a radially symmetric nonisothermal model. In addition to the radial symmetry imposed in the model (the shortcomings of this convenient assumption were discussed earlier), the presence of recalescence resulting from the severe undercooling and the associated nonequilibrium phenomena was also neglected. It was found that for relatively slow cooling rates up to lo4 K/s, which the authors claimed to be relevant to powder production, the lumped model was sufficiently accurate and the assumption of constant temperature inside the droplet justified. Numerical results for nonisothermal freezing showed that the proper choice of convective heat transfer coefficient and the accurate determination of the thermal emissivity are important in the determination of the temperature field and the freezing velocity of the interface.
c. STUDIES
OF SPRAYS IN SPRAY
DEPOSITION
Not many studies have been performed on real liquid metal sprays focusing strictly on the process of spray deposition. Most of the studies of this kind are pertinent to powder production. In addition, the main goal in the majority of published investigations on powder production is the
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
19
determination of particle size and distribution and not the relevant complex heat and fluid flow phenomena in the liquid metal spray. Representatives of these studies will be reviewed next. The discussion will center around studies utilizing the popular inert-gas metal atomization. Biancaniello et al. [301 performed an experimental study of metal atomization utilizing a supersonic inert gas. Using Schlieren, shadowgraph, and flash photography methods, they studied qualitively the flow fields in the gas and liquid regions. Emphasis was placed on aspiration conditions and nozzle designs that promoted the atomization process. The aspiration condition is the variation of the nozzle exit pressure with the die plenum pressure. It was found that a characteristic wave structure dominated the gas flow field at the maximum aspiration operating conditions. Using water as the operating fluid, it was also shown that the shape on the inner nozzle bore was instrumental in producing a liquid sheet. In a subsequent paper, Biancaniello et al. [31] analyzed metal powders produced by supersonic gas metal atomization. They discussed the droplet fragmentation mechanisms in the spray leading to the powder size distribution and offered three possible mechanisms of secondary droplet formation. In the first scenario, the primary droplet is pulled into a ligament that pinches down to the classic dumbbell shape and eventually breaks into two droplets. In the second mechanism the gas flow distorts the primary droplet into an umbrella shape, and this droplet further divides into many small droplets. The third mechanism is a variation of the first mechanism, in which satellite droplets form from the ligament in addition to the two secondary droplets mentioned above. In a companion paper, Biancaniello et al. [32] performed a real-time particle size analysis during inert-gas atomization based on the principle of Fraunhofer diffraction. They found their method to be a suitable candidate for process feedback control. An important parameter in liquid metal sprays is the mean diameter of the droplets in the spray. To this end, for lack of a better alternative, results from the literature on metal powders are commonly used to characterize gas-atomized liquid metal sprays in casting processes [ 121. A popular correlation of this kind is the Lubanska correlation for the mass mean droplet diameter [33, 341:
where dm0,5is the mass droplet diameter. According to its definition [351, 50% (fraction 0.5) of the total mass of the spray contains droplets of diameters smaller than that given by Eq. (48). In other words, Eq. (48) defines the diameter of the holes of a screen that would allow only 50% of
20
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
the total mass of the spray to pass through it. The parameters K , v, , vg, M,, M g ,d , , and We denote an empirical constant, the liquid metal kinematic viscosity, the gas kinematic viscosity, the melt mass flow rate, the gas mass flow rate, the melt stream diameter prior to atomization, and the Weber number of the liquid metal, respectively. The liquid metal and gas flow rates mentioned above are given by
where A , A , , pm , pg , g, F , 1, y , and Po are the cross-sectional area of the metal jet, the effective area of the gas nozzle, the metal density, the gas density, the gravitational acceleration, the discharge coefficient from the crucible, the height of the melt in the crucible, the ratio of the specific heat for constant pressure to that for constant volume of the atomizing gas, and the plenum pressure of the atomizing gas, respectively. The ratio M g / M , is a major factor on gas atomization of liquid metals, since it appears explicitly in Eq. (48). The gas flow rate can be kept practically constant for a fixed atomizer design [36, 371 by maintaining a constant gas atomization pressure. This is not true for the metal flow rate. Conventionally, the molten metal stream is allowed to fall freely by gravity to enter the region where the atomization process takes place. Hence, the metal mass flow rate depends on the metal static pressure head [denoted by 1 in Eq. (49)l. Controlling this pressure head controls M,. Ando et al. [36] performed a study to determine the pressure at the exit of the metal delivery tube during gas atomization. Knowledge of this pressure is necessary for the determination of the metal mass flow rate. On the basis of Bernoulli's theorem, they developed a method for the determination of the above-mentioned exit pressure, They found it to differ considerably from the ambient pressure in confined gas atomization where the melt stream is confined in the close vicinity of the atomizing gas jets. They used an ultrasonic gas atomizer for their work and water as the atomized fluid. The main theoretical result of their work (based on Bernoulli's theorem) is the following equation on the pressure difference between the ambient pressure ( P , ) and the pressure at the delivery tube exit ( P 2 ) A P = P 2 - P , = ~2 ( 2 g l -
(5:fj;i.
All the symbols in this equation are defined either in Fig. 6 or earlier in
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
i, i I
21
0 0 0 0 0 0 0
FIG.6. Illustration showing details of device to generate liquid metal spray.
connection with Eqs. (481450). The discharge coefficient is defined as
where E, is the mechanical energy per liquid mass of the liquid metal lost as frictional heating, V, is the velocity at the exit of the metal delivery tube (position 2 in Fig. 61, and p2 is a factor correlating the velocity to the kinetic energy at the same location [37]. The value of F is usually defined experimentally [361. Ando eta[. also propose the following equation for the metal mass flow rate as a function of the metal static head and pressure difference
By moving the delivery slit up and down relative to the gas jet impingement point, the authors [36] also found that if the gas jets are deflected on the delivery slit (Fig. 7a), the pressure is lowered, resulting in aspiration of the metal stream. If the gas jets just miss the delivery slit, the pressure increases resulting in undesirable backpressure effects in the metal stream (Fig. 7b). Finally, if the gas jets impinge well below the delivery slit (Fig. 7c), only a slight decrease or no effect was observed in the pressure and the metal flows freely by free fall. Earlier works on pressure determination
22
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
FIG.7. Delivery slit and gas jet effect on spray Schetch prepared following the discussion in Ando et al. [36].
at the exit tube of ultrasonic gas atomizers (USGAs) using a circular gas atomizer without metal flow [21, 38, 391 have also shown that significant variations of the delivery slit pressure compared to the atmospheric pressure may result depending on the relative position of the delivery slit and the gas jets. Before closing this section it is worth reiterating that a significant need for heat transfer and fluid mechanics research in real liquid metal sprays in the process of spray deposition exists. The existing base of knowledge relies on findings more pertinent to the related process of powder metallurgy and does not address numerous issues unique to the spray deposition process.
111. The Impact Region
The impact region in the process of spray deposition is perhaps the most challenging from the standpoint of transport phenomena. Splashing of liquid metal droplets initially on the substrate at high speeds (up to 100 m/s) and, later, on the completely or partially solidified layer of the already deposited material in the presence of rapid heat transfer and solidification under nonequilibrium conditions are features of this region. A good description of the various mechanisms responsible for the evolution of the microstructure in the deposited layer in the duration of the process given in Annavarapu et af. [2] is as follows:
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
23
Splat solidification. Here, discrete droplets that impact on the substrate spread extremely fast (the spreading time scales can be as low as nanoseconds [40]) and solidify primarily by conduction through the substrate. A droplet is completely solidified before another droplet impinges on it. Solidification rates are very high. This mechanism is characteristic of early stages of the spray deposition process and of thin deposit layers in particular. It yields fine-grain structures. Growth ofnuclei. This mechanism is present when low heat extraction from the bottom of the deposited layer results in a buildup of energy within it. The top region of the deposit (on top of which the incoming droplets fall) is only partially solidified. The cooling rates are low [12, 411. Studies have indicated [41,42] that the growth and coarsening of solid-phase nuclei in the partially solidified layer is directly related to the production of fine equiaxed grains in the final microstructure of the solid. Incremental solidification. Incoming droplets impact on a thin, completely liquid layer at the top of the deposit. The solidified material underneath the liquid layer acts as a chill, causing the advancement of the freezing front and the growth of the deposited layer. Incremental solidification also occurs when the thickness of partially solidified layer mentioned above becomes constant because the energy input equals the energy extraction in this layer. The solidification rates in the incremental solidification mechanism are low.
On the basis of this discussion it is clear that studies of the heat transfer and fluid mechanisms of both single as well as groups of liquid metal droplets impacting and solidifying on a substrate are directly related to the process of spray deposition. Single-droplet studies are particularly relevant to the splat solidification mechanism of the process. In the following sections single as well as multiple droplet studies will be reviewed sequentially. A.
SPLAT COOLING OF A SINGLE LIQUID
METALDROPLET
The heat and fluid flow phenomena occurring during the impact of a single liquid metal droplet on a cold substrate are neither conventional nor easy to study. There are several reasons for this fact. The fluid dynamics of high-speed droplet spreading is a free-surface problem with dramatic domain deformations in the presence of surface tension, and with possible droplet breakup phenomena and three dimensional effects. The heat transfer process involves rapid solidification, possibly under nonequilibrium conditions, in the presence of convection in a severely deforming
24
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
domain, coupled with conduction in the substrate and, whenever important, radiation to the environment. It is for these reasons that theoretical models of splat cooling were (and still are) constructed on the basis of educated assumptions. The first generation of such models does not involve sophisticated modeling of the complex impact fluid dynamics. Such studies will be reviewed first. 1. Studies without Sophisticated Fluid Dynamics Modeling Attempts have been made to facilitate the study of the various mechanisms of the fluid dynamics of the splashing process based on order-ofmagnitude (scaling) arguments. Bennett and Poulikakos [43] reviewed the state of the art of these attempts and proposed appropriate criteria that define the effect of surface tension and viscous forces on the maximum spreading of a droplet impacting a solid surface in connection with the process of splat-quench solidification. They defined two domains: the viscous dissipation domain and the surface tension domain, which are characterized by the Reynolds number and the Weber number and are discriminated by the principal mechanism responsible for arresting the splat. It was found that correctly determining the equilibrium contact angle was important to the prediction of the maximum spreading. Conditions under which the solidification process should not be expected to affect the maximum spreading were also determined. Utilizing a combination of arguments published earlier in Madejski [44], Collings et al. [45], and Chandra and. Avedisian [46], Bennett and Poulikakos [43] proposed the following equation for the spread factor 6 (the ratio of the splat diameter to the droplet diameter)
( 6/1.2941)5 + 3[(1 - cosWe8 ) e 2 - 41 = 1, (54) Re where 8 is the equilibrium contact angle, Re is the Reynolds number (based on the droplet diameter, the impact velocity, and the liquid metal viscosity and density, p u d / p ) , and We is the Weber number based on the liquid metal surface tension, density, droplet diameter, and impact velocity, pu2d/a. For the purpose of illustrating the relative contribution of the viscous energy dissipation and surface tension in terminating the splat spreading, a typical value of the equilibrium contact angle 8 = 7r/2 was assigned in Eq. (54) in Bennett and Poulikakos [43] to yield ( 5/1.294q5 Re
2- 41 + 3[ tWe
=
1.
(55)
25
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
In the extreme when We (55) becomes
+
~0
(surface tension effects are negligible), Eq.
5, = 1.2941Re'/'.
(56) On the other hand, when the viscous energy dissipation is negligible, Re + m: '/2
5,=(?+4)
.
(57)
The subscripts v and s in Eqs. (56) and (57) represent viscous and surface tension effects, respectively. If Eq. (56) were used to estimate the spread factor, it would yield a value larger than the actual value because it completely neglects surface tension effects. To improve the predictions of Eq. (561, a correction factor C , was introduced in Bennett and Poulikakos [43] such that
6 = C,& = C,1.2941Re'l5. Eliminating the spread factor 6 between Eqs. (55) and (58) yields Re=
[
(58)
[(We/3)(1 - C:) + 41'" 1.2941C,
(59)
Similarly, introducing a correction factor C , to improve the predictions of Eq. (57), which neglects viscous effects, we obtain
Combining Eqs. (55) and (60) to eliminate the spread factor, we obtain Re
=
(Cs[We/3 + 4]1/2/1.2941)5 * (1 - C;)[I + 12/We]
Equations (59) and (61) were plotted together in Fig. 8 [431, which defines graphically the viscous dissipation and surface tension domains in the splat-quenching process. The border between the two domains is marked by the boldface curve C , = C, = 0.816. Observing Fig. 8, we see that even well into the viscous dissipation domain the surface tension effects are significant. On the other hand, the viscous dissipation effects disappear more rapidly in the surface tension domain. By curve-fitting the boldface border curve of Fig. 8, the following condition was proposed in Bennett and Poulikakos [43], under which surface tension effects dominate the
26
DlMOS POULIKAKOS AND JOHN M. WALDVOGEL
Cv=0.95
500
.-Cs=0.60
_ _ - - -_ _ - - Surface tension domain
.... '
0
0
*
*
;
'
2000
'
'
;
'
c
*
;
'
'
*
6000
4000
:
'
n
'
8000
10000
Re FIG.8. We vs. Re from Bennett and Poulikakos [431 showing the viscous and surface tension domains.
termination of the splat spreading We < 2.8
(62)
In the scaling arguments of Bennett and Poulikakos ([431, and references therein), the effect of solidification in the droplet spreading was neglected. In the great majority of existing heat transfer studies on splat-quench solidification the fluid dynamics aspects of the process are neglected for simplicity. It is assumed that the droplet after impact spreads first, cools down, and subsequently solidifies. A comparison of the relevant time scales indicates that such an assumption is perhaps justified, especially at a first attempt to study the complex problem of splat-quenching of liquid metal droplets. Investigations in this category (neglecting the fluid dynamics) will be reviewed first. Research efforts in the area of splat-quench solidification first became noticeable in the 1960s, after it was observed that certain alloys could yield new metastable crystalline phases and amorphous solid phases. These results were attributed to the very high. cooling rates (in excess of lo5 K/s). Most of the research in the 1960s and early 1970s focused primarily on metallurgical aspects of the process and it has been reviewed by Jones [47] and by Anantharaman and Suryanarayana [48]. Studies of
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
27
heat transfer aspects of splat-quenching solidification are more recent. Madejski [44, 491 performed analytical and experimental studies of heat transfer during splat cooling of metal droplets. Based on a unidirectional solution of the Stefan type that assumes that both the splat and the substrate are semiinfinite bodies he determined the dependence of the spread factor 6 (defined earlier) on the Weber, Reynolds, and Peclet numbers and a parameter, k, defined as
where the subscripts s and 1 denote density in the solid and liquid regions, U is a freezing constant, and E = R,/D is the ratio of the liquid metal disk radius at the time of initial contact to the droplet diameter. Note that in the analysis of Madejski 1441 it was assumed that a droplet of diameter D deforms to a disk of radius R, when it contacts the substrate. Madejski [49] postulated that:
0
In the limit k
=
(1/Re)
In the limit k
=
(1/We)
6,
=
=
=
0, We > 100:
0:
1.2941(Re
+ 0.9517)”5.
(65) In these equations 6, is the maximum spread factor. In the general case concerning the flattening of a droplet without freezing ( k = 0) if We > 100 and Re > 100, the following equation was recommended for the maximum spread ratio [44]:
Results for the case where freezing was present in the spreading process
( k > 0) were obtained numerically and showed an insensitivity of the value of the maximum spread factor on k for small values of the Weber number. In the opposite extreme (Weber and Reynolds numbers approach infinity) the dependence of the maximum spread ratio on k was given by
6, = 1.5344 k-0.395. (67) The agreement between theoretical predictions and experiments in [49] were deemed to be “not bad” by Madejski. Numerical investigations of heat transfer and solidification aspects of a splat-cooled liquid metal droplet without accounting for the associated fluid dynamics phenomena and by modeling the heat transfer in the
28
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
substrate and the splat as unidirectional were performed by Wang and Matthys [50-521. They presented results of the interface velocity as a function of propagation distance with and without undercooling of the melt. With undercooling, the freezing interface velocity was shown to decrease rapidly as the freezing front advances. Without melt undercooling, the freezing interface velocity is heat-transfer-limited. This resulted in less drastic changes in the interface velocity with the propagation of the freezing front. The quality of the thermal contact between splat and substrate was influential is sustaining the interface velocity in an undercooling melt and critical in the absence of undercooling. Relevant to the splat-quenching process is the work of Shingu and Ozaki 1531, who investigated numerically rapid solidification occurring by conduction cooling. Rosner and Epstein [54] studied theoretically the simultaneous kinetic and heat transfer limitations in the crystallization of highly supercooled melts. Evans and Greer [55] developed a one-dimensional numerical solution to the rapid solidification of an alloy melt in order to study the solute trapping. They employed a two-equation model relating the interface velocity and solid composition to the temperature and liquid composition at the interface. In a recent paper, Bennett and Poulikakos [56] presented a combined theoretical and experimental study of the splat-quenching process. Although they did not consider the fluid mechanics of the process, they presented an extensive conduction-based model of the freezing process accounting for axisymmetric conduction in the substrate. In this respect, their model is more general than what has been presented in previous studies of similar nature [44, 49-52] and will be discussed in detail below. On the basis of order-of-magnitude arguments Bennett and Poulikakos [56] stated that as a first approximation it is sometimes reasonable to assume that in the process of splat-quenching the liquid metal droplet spreads first and solidifies subsequently. To this end, the splat was modeled as a thin liquid metal disk initially at uniform temperature, T,, which was suddenly brought into contact with a large (by comparison) substrate of initial temperature, To considerably lower than the freezing temperature of the splat material, Tf (Fig. 9). Heat was conducted away from the splat into the substrate. Solidification ensued and progressed until the entire splat was solidified. The heat conduction cooling of the splat continued after solidification was completed, until the splat temperature reached the substrate temperature. The heat conduction process was modeled as two-dimensional in both the splat and the substrate. To this end, the range of validity of previous one-dimensional models was explored in Bennett and Poulikakos [561. The conduction equation describing the transport of heat in the splat with
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
29
Liquid
i Splat
*
1 2
FIG.9. Schematic of the “disk” model in Bennett and Poulikakos [43].
respect to the cylindrical coordinate system (r, z ) of Fig. 9 is [561 j
dt
=
l,s,
(68)
where the subscript j takes on the values 1 or s when Eq. (68) is applied to the liquid or to the solid portion of the solidifying splat, respectively. The temperature is denoted by T,, the time by t , and the thermal conductivity, density, and specific heat of the splat by k j , p j , and c j , respectively. Equation (68) reflects the independence of the heat transport on the angular position from symmetry considerations. The heat conduction equation in the substrate is dT [d2T 1 dT p c -at= k : + d- r- + - ,r d r
]:lt (69)
in which the notation is analogous to that defined earlier, following Eq. (68). To complete the model formulation, the relevant initial, boundary, and matching conditions need to be discussed [56].The initial conditions of the problem are that both the substrate and the splat were isothermal prior to making contact with one another: At t = 0: T, = T,, T = To. (7017(71) The boundary conditions at the top and at the lateral surface of the splat are
’
dT,
Atz=0:
k .= h,(T, dz
Atr=R:
- k .I = ha(? dr
-
Ta), j
=
~,1,
(72)
-
Ta), j
=
s,1,
(73)
d q
30
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
where ha is the heat transfer coefficient between the splat surface and the ambient, T, is the ambient temperature, and the symbols s and 1 respectively, denote solid and liquid. These boundary conditions account for the convective removal of heat from the splat surface. A similar boundary condition to Eq. (73) accounts for convection of heat from the top surface of the substrate. Since freezing takes place in the splat, the solid and the liquid regions are separated by a freezing interface. The matching conditions for the temperature field at this interface are
where the subscript i denotes the position of the freezing interface and U, its velocity. Conditions (74)and (75) stand for the temperature continuity and for heat flux discontinuity because of the heat released on solidification. The negative sign in the left-hand side of Eq. (75) reflects the fact that the freezing interface velocity is pointing to the negative z direction. Note that in writing Eq. (75), the radial conduction was neglected for simplicity. This approximation is appropriate within the context of this model since the splat thickness is at least two orders of magnitude smaller than the splat diameter. The matching conditions at the splat-substrate interface are as follows:
This condition can alternatively be written for the substrate side of the interface: dT Atz=H: -- k - = h , AT, (77) dz These matching conditions account for the presence of a contact thermal resistance at the splat-substrate interface. To this end, matching condition (76) states the fact that the heat flux leaving the splat at the interface equals the product of a heat transfer (resistance) coefficient descriptive of the imperfect thermal contact at the interface, multiplied by the temperature jump across the interface (AT,) defined as the difference between the interface temperatures at the splat and the substrate sides. Condition (77) is analogous to Eq. (76) written for the substrate side of the interface. The temperature of the substrate far away from the interface is not affected by the presence of the splat
As z
+
a:
T + To.
(78)
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
31
The last issue to be discussed before completing the description of the heat conduction model is the undercooling present in the splat at the initiation and subsequent development of the solidification process. In the classic treatment of a freezing front, the front is defined by the freezing temperature of the material and its propagation velocity is limited by the rate at which heat can be conducted away from this front into the liquid and solid regions [Eq. (731. However, this treatment does not account for the presence of undercooling in the melt prior to the initiation of solidification. Such undercooling is a common occurrence in the splat-quenching process and other rapid solidification processes and results in the freezing front being at a temperature below the equilibrium freezing temperature. As discussed earlier in connection with Eqs. (41) and (42) to account for this fact, a freezing kinetics relationship between the amount of undercooling and the velocity of propagation of the freezing interface is needed. The equation used in Bennett and Poulikakos [56] is identical to Eq. (42):
U
= K(Tf-
Ti),
(79)
where K is the freezing kinetics coefficient, T, is the equilibrium freezing temperature of the solid-liquid interface, and Ti is the actual temperature of this interface. No details of the finite-difference method used for the numerical solution of the model in Bennett and Poulikakos [56] are given herein for brevity. Figures 10 and 11 present comparisons of theoretical and experimental results for the temperature and quenching rate histories, respectively, at the splat-substrate interface. A copper substrate was used in Fig. 10 and a Pyrex substrate was employed in Fig. 11. The initial temperature of both substrates was 25°C. The numerical model for the results in Fig. 10 duplicated the experimental condition of a 3-mm lead droplet released from 30 cm above the substrate with temperature at a release time of 468°C. The temperature at impact time was estimated to be 460°C. The experimentally measured spread factor was 4.1. The heat transfer coefficient at the splat-substrate interface that defines the contact resistance was assumed to be h , = 15 kW/m2 K. This value is within the ranges reported in the literature and was chosen so as to yield the best agreement between the theoretical model and the experiment. The conditions for the results of Fig. 11 were similar. This time the lead droplet diameter was measured to be 2.7 mm, the droplet impact temperature 486"C, and the spread factor 4.4. The contact heat transfer coefficient used was h , = 100 kW/m2 K. Examining Figs. 10 and 11, we conclude that the predictions of the model are satisfactory, especially if one takes into account the relative simplicity of the model. The temperature of the splat-substrate interface
a
500 400
300
200
100
o t 0
0.02
I
I
0.04
0.06
0.08
0.1
Time (s)
40,000 35,000
-9
30,000
v)
25,000 Numerical
Y
a
I
m c& 20,000
.-C
r 0
5
15,000
0 10,000
5,000
0 0
0.01
0.02
0.03
0.04
Time (s)
0.05
0.06
0.07
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
33
decreases monotonically in both Figs. 10a and l l a , except when solidification is initiated in the splat, resulting in temporary increase in temperature. The rate of cooling is very rapid initially, and slower after the completion of the freezing process. Figures 10b and l l b present a more critical comparison between experimental and numerical results in the form of quenching rate/time curves. It can be seen from these figures that after the quenching rates have become relatively small, the numerical and experimental results compare very favorably. At early times qualitative agreement is present but the sluggish response of thermocouples prohibits good quantitative agreement between theory and experiment. The numerical model was used to further explore and establish quantitatively the effect of the contact thermal resistance. Figure 12 shows the numerical results on the thermal history of splats quenched with varying degrees of thermal contact quality. The effect is substantial. The thermal histories of the bottom-center surface node of the splat for interface heat transfer coefficients ( h , ) ranging from 10 to 100 kW/m2 K are shown. It can be seen that as the interface heat transfer coefficient becomes small (poor thermal contact), its effect dominates the cooling rate of the splat (Newtonian cooling). Conversely, as the heat transfer coefficient becomes large, its effect becomes less influential to the cooling rate of the splat (ideal cooling). It is also clear that the interface heat transfer coefficient has significant influence over the length of time required to initiate freezing as well as over the duration of the freezing process. A characteristic feature of the numerical results of Figs. 10-12 is the undercooling disappearing just prior to the onset of freezing. This is an interesting detail that experimental results were unable to detect clearly. Referring back to Fig. 10a, it is apparent that the heat extraction is insufficient to sustain the undercooling achieved prior to solidification; that is, the kinetics of crystalline formation is so rapid that the rate of latent heat released is sufficient to substantially reheat the splat. This is an observation worth further consideration, because it indicates that despite the original undercooling, the interface freezing temperature may quickly
FIG.10. Comparison of experimental (circles) and numerical results of splat quenching of lead on a copper substrate from Bennett and Poulikakos [43]: (a) splat-substrate interface temperature vs. time; (b) splat-quenching rate vs. time. Experimental results: initial droplet temperature 468"C, initial substrate temperature 2 5 T , droplet diameter 3.0 mm, free-fall distance 30 cm, spread factor 4.1. Numerical results: initial splat temperature 460°C, initial substrate temperature 25"C, droplet diameter 3.0 mm, free-fall distance 30 cm, spread factor 4.1, heat transfer coefficient 15 kW m-' K-',undercooling, 40°C.
a
5001
Numerical
0
b
0.02
0.04
0.06
0.08
Time (s)
- Numerical
Time (s)
0.1
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
35
rise to the melting temperature. Note that microstructural features of the resulting solid are determined largely by the rate of solidification, which is dictated by thermal conditions at the freezing interface. Details of the transient thermal conditions within the splat during its solidification obtained numerically are presented in Fig. 13. For the sake of clarity, there are relatively few time steps presented. It can be seen that the freezing interface temperature rises to within 1°C of the melting temperature after this interface has propagated only 10 p m into the splat ( t = 5.13 ms). This result informs us that recalescence is confined to a relative small region adjacent to the contact surface. Hence, the enhanced solidification speed afforded by undercooling the melt is largely ineffectual at influencing the gross properties of the splat when thermal contact with the substrate is poor ( h , = 15 kW/rn2 K in Fig. 13) and the splat thickness is relatively large (156 p m in Fig. 13). One distinctive attribute of the numerical solution to the splatquenching problem in Bennett and Poulikakos [56] is that it accounts for two-dimensional conduction into the substrate. Previous investigations into this system have assumed that the conduction of heat into the substrate can be treated as one-dimensional. Figure 14 shows the dissipation of heat into a copper substrate assuming close to ideal thermal contact between the splat and substrate ( h , = 100 kW/m2 K). The evolution-decay of thermal gradients in the substrate is very rapid. The substrate surface temperature reaches a peak of approximately 95"C, during the period in which latent heat is released while the splat freezes. The elevation of temperature much beyond the outside radius of the splat (5.37 mm) is marginal, especially in the initial period of quenching. For most of the very early period, in which heat is being removed from the splat, isotherms developing beneath the splat are very flat, indicating one-dimensional conduction in the substrate material. The long-term transfer of heat away from the vicinity of splat, however, becomes significantly two-dimensional as demonstrated by the curvature of the isotherms.
FIG. 11. Comparison of experimental (circles) and numerical results of splat quenching of lead on a Pyrex substrate [43] (a) splat-substrate interface temperature vs. time; (b) splat-quenching rate vs. time. Experimental results: initial droplet temperature 494°C initial substrate temperature 2 5 T , droplet diameter 2.7 mm, free-fall distance 30 cm, spread factor 4.4. Numerical results: initial splat temperature 486"C, initial substrate temperature 2 5 T , droplet diameter 2.7 mm, free-fall distance 30 cm, spread factor 4.4, heat transfer coefficient 100 kW m-* K - ' , undercooling, 40°C.
36
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
450 400 350 300 250
200
150
100
50
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Time (s) FIG.12. Effect of thermal contact resistance between lead splat and copper substrate on the thermal history of the bottom-center location of the splat. Initial splat temperature 460°C, initial substrate temperature 25”C, droplet diameter 3.0 mm, spread factor 4.1, heat transfer coefficient 15 kW m-’ K - ’ [431.
2. Studies with Sophisticated Fluid L$narnics Modeling The work reported by Madejski and others [44, 49-561 represents a “fist generation” of studies of transport phenomena in the process of impact and solidification of a liquid metal droplet on a substrate. In all these studies the fluid mechanics of the process, for all practical purposes, was not taken into account in order to circumvent associated difficulties. Very recently, studies have been performed to explore the fluid dynamics of a liquid metal droplet impact. Examples of such studies are the works of Fukai et al. 157, 581, Trapaga and Szekely 1591, Lui et al. [60, 611, Marchi et al. [62], and Tsurutani et al. [63]. These and other relevant works will be reviewed in this section,
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
-
37
= 4.74ms
-t
= 5.13111s
-t
- - = 5.31111s - - - - - = 5.65ms - - - I = 6.60ms - - - .- = 9.08ms 1-1
t
t
-..--t = 13.08ms .....t = 14.54111s
t
\y, ’
0
20
295
60
40
Distance
from
80
Splat/Substrate
100
120
Interface
140
160
( pm)
FIG.13. Temperature variation with distance from the splat-substrate interface for a host of times I431.
Fukai et al. [57] published a theoretical study of the deformation of a spherical liquid metal droplet impinging on a flat surface. The study accounts for the presence of surface tension during the spreading process. The theoretical model is solved numerically utilizing deforming finite elements and grid generation to simulate accurately the large deformations, as well as the domain nonuniformities characteristic of the spreading process. The results document the effects of impact velocity, droplet diameter, surface tension, and material properties on the fluid dynamics of the deforming droplet. Two liquids with markedly different thermophysical properties, water and liquid tin, are utilized in the numerical simulations. The occurrence of droplet recoiling and mass accumulation around the splat periphery are standout features of the numerical simulations and
38
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
Time = 5 . 0 ~ 1 0 -(s).~
IO'C
.
0
pcr Isolhcrm. Ouler moat Isotherm = 30.C.
b ? l\
-..- T
3uu
/
I
1,000 lull 1,500 2.000 2,500
6000
8000
4000
2000
"
urn
;
'
l 6000
"
' 4000
6000
4000
8000
I O T per Isotherm. Outer most Isotherm = 30%.
Time = 2 . 0 ~ 1 0 .(s~).
8000
2000
0 wm
l
'
'
(
l 0
2000
~
~ 2000
'
l
~
4000
~
~
6000
~
'
8000
wm 10.C pcr Isotherm. Outer most Isotherm = 30.C.
Time = 5 . 0 ~ 1 0 -(~ s).
2.000 2,500
8000
6000
4000
0
2000
2000
4000
6000
8000
P Time = I.OXIO-~(s).
IO'C
per Isotherm. Outer most Isotherm = 30%.
0 500 1,000
wm 1.soo 2,000 2.500 8000
6000
4000
2000
0
2000
4000
6000
8000
P
FIG.14. Transient isotherms in the copper substrate. Initial splat temperature 46o"C, initial substrate temperature 2 5 T , droplet diameter 3.0 mm, spread factor 4.1, heat transfer coefficient 100 kW m - 2 K - ' undercooling, 40°C [43].
~
'
~
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
a
t"
39
bZ
0
X
FIG.15. Deforming droplet coordinate system in Fukai et al. [57].
yielded a nonmonotonic dependence of the maximum splat radius on time. The work in Fukai et al. [57] is a departure from earlier attempts at similar problems [59, 631 that were based on finite differences (marker and cell and control volume methods) with fixed grids. The mathematical model for the droplet spreading in Fukai et al. [57] is presented next. The model was formulated to simulate the impact of a liquid droplet on a solid substrate, starting at the instant that the droplet comes into contact with the substrate and proceeding until the droplet comes to rest after the splashing process is completed. A Lagrangian approach was adopted [64] because it facilitated the accurate simulation of the motion of the deforming free surface. In the Lagrangian axisymmetric conservation equations within an initially spherical droplet impacting on a solid surface (Fig. 151, r, z , and 6 are respectively the radial, axial, and azimuthal coordinates (Fig 19, p is the density, u is the radial velocity component, u is the axial velocity component, t is the time, p is the pressure, p is the viscosity, Y is the kinematic viscosity, g is the gravitational acceleration, c is the speed of sound, and y is the surface tension. The stresses are denoted by ui,, the uniform droplet velocity at the time of impact by u,-, and the radius of the droplet at impact by r,,. The conservation equations in dimensionless form are
dV
1
d
1
40
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
The dimensionless initial and boundary conditions of the problem are [57]
U=O, V = - l ,
AtT=O:
At 2
=
2 P = - We .
(83)
U = V = 0.
0:
(85)
At the free surface
-
Grrnr
+
H CrZnz= - 2 -n,, We -
C,,n,
+ q z n z= - 2
H
-n,. We
The nondimensionalization was carried out according to the following definitions: r R=z = -Z u = -U T / = - U r0 ro UO uo
The mean curvature of the free surface was defined as
+ [(rrl2+ (zt)2]nr 2 r ~ [ ( r ‘+) (~z r ) 2 ] 3 ’ 2
r2(r’.zr! - z’rrr)
H=
(89)
In these equations primes denote differentiation with respect to the arc length along the free surface s (Fig. 15). The nondimensionalization process created the following dimensionless groups (Reynolds, Weber, Froude, and Mach numbers, respectively): R e = - ,uoro U
We=--pr,u,2 Y
F r = - ,4 ro g
M = -UO. C
(90)
Note that the time derivative has been maintained in the continuity equation to facilitate the numerical solution of the model, as explained in
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
41
detail in Fukai et al. [57]. This solution was based on deforming finite elements and grid generation in connection with the artificial compressibility method and the Galerkin method [571. Background on the several issues involved in the artificial compressibility method is contained elsewhere [65-761. Herein, key results from Fukai et al. [57] will be discussed next. A numerical simulation was conducted involving a liquid tin droplet of radius r = 12 p m impinging on a stationary flat plate with a velocity ug = 25 m/s. These conditions were chosen as typical to spray coating applications. It is important to note, however, that solidification was not modeled in Fukai et al. [57]. For liquid tin, the following values were used: surface tension coefficient y = 0.554 N/m, density p = 7000 kg/m3, and kinematic viscosity v = 2.6 x lo-’ m2/s. These above conditions resulted in the following values of the relevant dimensionless numbers: Re = 1200, We = 100, and Fr = 5.6 X lo4. Figure 16 depicts a sequence of frames corresponding to different instances of the metal droplet impact process. Immediately after contact, a thin film is formed at the periphery of the splat, which propagates radially at velocities substantially higher than the impact velocity. These velocities varied as a function of time, but in general, their temporal peaks occurred at early stages, and were 2-3 times larger than the droplet impact velocities. The formation of a ring-shaped tip of the laterally propagating sheet is present in the tin simulation. However, it appears that the impeding effects of surface tension to the overall spreading process take longer to dominate compared to numerical simulations for water droplets [57]. This trend is expected, due to the higher value of We for the tin droplet compared to the water droplet (100 vs. 10 or 1.4). As seen in Fig. 16, the droplet stretches to a significant degree before its spreading is halted by the dominance of the surface tension mechanisms ( T > 4). It is important to note the significant mass accumulation around the periphery of the liquid tin splat, which is most pronounced after T = 2 (see Fig. 16). Its existence was verified with simple laboratory experiments by Fukai et al. [57]. The lateral flow direction of the edge of the splat is eventually reversed, as clearly shown in the late stages of the simulation. The tin simulation was carried to rather long times as a result of properly resolved splat thicknesses in the vicinity of the axis. The longer simulation times resulted in the capturing of the splashing event occurring when the reversed flow reached the axis of symmetry. The computation was terminated at T = 12.4 (5 ps after impact) due to grid generation limitations to adequately follow the upward motion of the apex formed in the center of the splat and the subsequent possible breakup of the flow into ligaments and/or droplets.
42
DlMOS POULIKAKOS AND JOHN M. WALDVOGEL
time=2.000
t
time=4.001
t
rs
time=l.200
time=6.801
tirne=l1.601
FIG.16. Splashing sequence of a liquid tin droplet with Re
=
1200 from Fukai et al. [57].
Substantial insight may be gained on the splashing dynamics by monitoring representative splat parameters as a function of time. For example, the splat radius as well as the splat thickness on the axis of symmetry are two parameters whose temporal variation can elucidate the relevant mass and momentum transport processes. Several additional simulations were performed in Fukai et af.[57], and selected results are discussed below.
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
43
Figure 17 displays the effect of Reynolds number on the nondimensional splat thickness Z,,,,, at the axis of symmetry of the flow domain. Four different runs are compared, characterized by the same Weber number (We = 801, and Reynolds numbers of 120, 1200, 6000, and 12,000, respectively. It is immediately apparent that the initial stage of impact (up to T = 2) is almost identical for all four cases, when the splat thickness at the axis of symmetry is employed as a measure of the deforming droplet dynamics. This result, which is in agreement with the predictions of Fukai et al. [58], suggests that the initial rate of change of the splat thickness at r = 0 is directly proportional to the impact velocity (u,,). The slope of the linear portion of the graph displayed in Fig. 17 is approximately equal to 0.88, a value slightly different from the values 0.7 to 0.84 reported in Trapaga and Szekely [591. Subsequent stages of the simulations are markedly different, displaying thicker splats corresponding to lower values of Re. Figure 17 also demonstrates that the splat thickness approaches an asymptotic value that increases with decreasing Re. In all four cases
Tin
(
We = 80 )
[a] - Re = 12,000 [b] - Re = 6,000 [c] - Re = 1,200 [d] - Re = 120
0
2
4
Dimensionless time
6
8 T
10
(tv I ro)
FIG. 17. Effect of the Reynolds number on the dimensionless splat thickness from Fukai et al. [57].
44
DIMOS POULlKAKOS AND JOHN M. WALDVOGEL
depicted in this figure, the final splat thickness at r = 0 is smaller than 10% of the preimpact diameter of the droplet. In fact, almost all tin droplet simulations completed showed that the final thickness at the center of the splat represents only a small fraction of initial droplet diameter (typical value around 5%). Figure 18 shows the effect of Reynolds number on the time variation of the dimensionless splat radius R,,, (the distance of the outermost liquid element from the axis of symmetry). Three simulations considered previously (We = 80 and Re = 120, 1200, and 6000) are compared in this figure, which demonstrates some important features of the spreading process. Initially, the splat radius increases with nondimensional time up to a maximum value. The splat radius subsequently decreases, and for the lowest Reynolds number approaches an asymptotic value. As expected, the maximum splat radius depends on the Reynolds number, since larger values of Re would naturally result in larger spreading. In addition, the simulations showed that the time at which the maximum occurs shifts
3.5 Tin
(
We = 8 0 )
3 .O
2.5
1
/, ‘4
\
K\ 1’ \ .A.
\ \
\
2.0
[a] - Re = 6,000 [b] - Re =1,200 [c] - Re = 120
1.5
1 .o 0
2
4
Dimensionless time
8
6 7
10
(tv / r ) 0
0
FIG. 18. Effect of the Reynolds number on the dimensionless splat radius from Fukai et al. [57].
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
45
toward later stages as the Reynolds number increases. The reason for this finding is as follows: The maximum of the curves shown in Fig. 18 corresponds to the time at which the outward spreading of the fluid has terminated. After this time, flow reversal and recoiling of the droplet occur. As the Reynolds number (flow inertia) increases, it takes more time for the combined action of viscosity and surface tension to slow down and eventually arrest the flow, thus explaining the shift in the maximum of the curves mentioned above. Figure 19 documents the effect of Weber number on the dimensionless splat radius R,,,. Three different runs are compared, characterized by the same Reynolds number (Re = 1200) and Weber numbers of 500,1000, and 5000, respectively. In all cases, the splat radius initially increases with nondimensional time, then achieves a maximum, and eventually decreases before it approaches an asymptotic value. Since larger Weber numbers correspond to lower surface tension forces, the maximum splat radius at high We values is expected to be higher (a trend clearly displayed in Fig.
4.0
I
I
3.5
3.0
x
A
I2
2.5
2.0
1.5
1 .o 0
2
4
6
Dimensionless time r
8 (tvo/
10
ro)
FIG. 19. Effect of the Weber number on the dimensionless splat radius from Fukai et al. [571.
46
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
19). In addition, the instant where the maximum splat radius is achieved is shifted toward later times with increasing values of We. The current study clearly shows that the final splat spread is affected by the corresponding Weber number. This trend does not agree with the results reported in Trapaga and Szekely [59], where a relative insensitivity of the rate of spreading on Weber number was found for the range of flow conditions investigated therein; We = 200-2000 and Re = 100-105. The effect of droplet size on the splat spreading rate was investigated by performing two additional simulations, each of a liquid tin droplet impinging on a stationary flat plate with velocity uo = 4 m/s. The two droplets, however, were characterized by different radii: ro = 375 and 37.5 pm, respectively. These conditions resulted in the following values of dimensionless numbers: Re = 6000, We = 80, Fr = 4600 for the ro = 375 p m droplet, and Re = 600, We = 8, Fr = 46,000 for the ro = 37.5 pm droplet. Figure 20 shows the temporal variation of splat radius for the two splatting events. The lower values of Re and We for the smaller droplets suggest that surface tension effects are more important than in the larger
3.5
j /
2.5
[a]
- ro =
I
I
&
vo = 4 m/s
Tin
.375 mm
Re=6,000 We=80 Fr=4.6E+3
i I
2.0
[b] - ro = .0375 mm
/
Re=600 Fr=.6E+4
1.5
1 .o
0
2 Dimensionless time
4 T
6 (tvJ ro)
FIG.20. Effect of droplet size on the splat spreading rate from Fukai el al. [57].
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
47
droplet case; therefore, the corresponding splat radius curve remains consistently lower than that of the larger droplet throughout the simulation. The modeling of the splat cooling of a liquid metal droplet accounting for the associated fluid mechanics phenomena as described in Fukai et al. [57] but in the absence of solidification was completed recently by Zhao et al. [77]. These authors used the Lagrangian formulation and extended the fluid dynamics model of Fukai et al. [57] to account for the heat transfer process in the droplet and the substrate. Following the notation in Fig. 15, the dimensionless heat transfer model accompanying the fluid dynamics model of Fukai et al. [57] in Zhao et al. [77] is
$1 2)
Energy equation in the splat: dT =
dt
-(
- [dR R e P r R dR 1
1
+
Energy equation in the substrate: dT, --
dt
-(-
A
1
d
- [ R g ]
R e P r R dR
+
z). d=T,
(92)
Initial and boundary conditions: T=l,
Art=0: At the droplet surface:
dT
T,=O. dT
-n ,
dR
(93)Y (94)
+n, = 0 dZ
(95)
dT
At the substrate surface prior to the splat arrival: In the substrate far from the interface ( 2 + - m) : At the splat-substrate interface ( Z
=
- = 0 (96) dZ
T, = 0 (97)
0) :
The nondimensionalization was carried out according to the following definitions:
The nondimensionalization of all additional quantities was defined earlier [in Eq. (SS)].
48
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
In addition to the usual Prandtl number (Pr = v / a ) , the following dimensionless groups appeared in the heat transfer model:
In the preceding equations T is the temperature, k is the thermal conductivity, and CY is the thermal diffusivity. The subscripts s, *, and 0 denote substrate, dimensional quantity, and initial state, respectively. The remaining quantities were defined earlier in connection with the fluid dynamics model of Fukai et al. [571. In addition to the theoretical model outlined above, an experimental study was presented in Zhao et al. [78]. The experimental findings verified the numerical predictions adequately. The intricate details of both the numerical and the experimental procedures are contained in Zhao et al. [77, 781 and will not be repeated here for brevity. Representative results, shown in Figs. 21 and 22, will be discussed next. The first set of results examines the effect of the substrate material on the cooling of a molten metal droplet in low-speed spray coating applications. The temperature distribution is represented by the contour lines denoting the isotherms (Fig. 21). Instantaneous streamlines and the velocity vectors are also plotted in the left half of the droplet region to better illustrate the fluid flow effects on the thermal development history of the droplet. A tin droplet of radius ro = 9 p m was considered to impinge on three different substrates at a velocity uo = 29.4 m/s. For molten tin, the following property values were used: surface tension coefficient y = 0.544 N/m, density p = 7000 kg/m3, kinematic viscosity v = 2.64 X m2/s, thermal diffusivity CY = 1.714 X m2/s, and thermal conductivity k = 30 W/m * K. The values of the relevant dimensionless groups are Re = 1000, We = 100, and Fr = lo7. The substrate materials examined are copper, steel, and glass, respectively, to cover a large portion of the spectrum of the thermal diffusivity and the thermal conductivity. The thermal diffusivity and thermal conductivities for the substrate materim2/s and als used in the simulations are correspondingly 1.17 x 401 W/m K for copper, 3.95 X m2/s and 14.9 W/m . K for steel [American Iron and Steel Institute (AISI) 304 stainless steel], and 7.47 X lo-’ m2/s and 1.4 W/m * K for glass. It is immediately apparent from Fig. 21a-c that the cooling of the impinging droplets occurs practically simultaneously with the spreading. In the entire droplet spreading process the droplet temperature field demonstrates convective and two-dimensional features. In all cases, the fluid temperature is higher in the center region and lower around the spreading front. This is because high-temperature fluid is continuously supplied to
-
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
49
FIG.2Na). Tin droplet spreading on different substrates from Zhao et al. [77]: (a) T = 1.0.
50
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
FIG.21(b). Tin droplet spreading on different substrates from Zhao el al. [77]: (b) T
=
2.0.
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
FIG.2").
51
Tin droplet spreading on different substrates from Zhao el al. [77]: (c) T = 4.0.
52
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
10
:
9 -
Ink Droplet
ink experimental - - - - - - - ink numerical - - - - - . solder experimental solder numerical
r, = 1.48 mm v, = 1.945 m/s Re = 2860 We = 76 Fr = 262
8 7 -
5 h
6 -
Solder Droplet (TinnRad 50/50) ro = 1.6215 mm v, = 1.7 m/s Re = 10894 We = 83 Fr = 183
v
J
5: 4 -
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Time (nu)
FIG.22. Comparison of theoretical and experimental results on the maximum splat radius from Zhao et af. [78].
the center region, and the splat periphery is continuously cooled down by contacting the low-temperature surface of the substrate as it spreads outward. In the cases of steel and glass substrates, at late stages of the droplet spreading (7 = 4.0, Fig. 2 1 4 the temperature distribution within the splat is largely one-dimensional. The temperature gradients within the splat occur in the radial rather than the axial direction despite the fact that the splat thickness is much smaller than the splat diameter. It appears that at this time the center of the splat is cooled by the splat periphery rather than the substrate. This is a result of the strongly convective nature of the cooling process at the earlier stages and implies that approximate modeling attempts using an axial heat conduction model for a thin disk to simulate the heat transfer within the fully spread droplet are not accurate when the thermal conductivity of the substrate is low. The premise that “the droplet spreads first and cools down later” associated with the early analyses of splat cooling and thermal spray deposition may not always be valid. As shown in Fig. 21a-c, the copper substrate temperature changes very little during the entire process of spreading, whereas the glass substrate temperature increases rapidly in the neighborhood of the
HEAT TRANSFER AND FIUID DYNAMICS IN SPRAY DEPOSITION
53
splat-substrate interface. The droplet impacting on the copper substrate cools down the fastest compared to the droplets impacting on the steel and glass substrates under identical conditions. The flow structure is dominated by the inertial force at the initial stages of spreading. The flow structure experiences a drastic change at the late stages of spreading when the inertial force decreases as the fluid spreading slows down by the action of surface tension forces. Noteworthy is the mass accumulation around the periphery of the splat. A secondary flow vortex emerges near the contact line (magnified in the detail of Fig. 21c). The vortex grows larger and moves up to the top of the free surface in the splat periphery region. The theoretical modeling was quantitatively validated by performing numerical simulations with the same conditions present in experimental investigations performed using a photoelectric technique [78]. The temporal variation of spread ratios measured with the photoelectric method were compared with the numerical predictions [781. Representative results of such comparisons are shown in Fig. 22. The agreement between theory and experiment is good in the case of a liquid solder droplet. This is due to the fact that wetting effects are not as important in this case because liquid solder does not significantly wet the glass surface. The agreement deteriorates after the initiation of recoiling primarily because the effect of contact angle hysterisis is not modeled in Zhao et al. [77]. In a recent paper Liu et al. [60] presented a numerical investigation of micropore formation during the impact of molten droplets on a substrate in a plasma deposition process. Their work extends the earlier paper of Marchi et al. [62]. They utilized a fixed-grid finite-difference model for the fluid mechanics component of the problem, and they adopted the model of Madejski [44], which follows the unidirectional conduction (Stefan) approach for the heat transfer and freezing components of the problem. Although they dealt with velocities as high as 400 m/s, their model was axisymmetric. Note that earlier experiments on plasma-sprayed niobium particles on a substrate [79] showed severe droplet fragmentation (particle crushing) and a splat configuration that is far from axisymmetric. Subject to these approximations and to experimental verification, Liu et al. [601 postulate that following flattening and solidification the splat edge may separate from the solid-liquid interface, causing the formation of micropores in the fringe region of the splat. In the limits of high and low impact velocity and substrate temperature, they report low microporosity. If the impact velocity or substrate temperature is between these two extremes, the deformation and solidification velocities become comparable and most of the voids formed due to separation are fixed in the solidification layer, yielding large microporosity. Liu et al. [60] summarize that the combination of a liquid or mushy droplet condition at high impact velocity with a
54
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
solid or mushy surface condition improves contact and adhesion and reduces microporosity. Liu et al. [61] reported a numerical study of molten droplet impingement on a nonflat surface. They conclude that during the impingement onto an axisymmetric wavy surface, a single droplet spreads and eventually forms a thin, nonflat splat. If the surface wavelength is larger than the droplet diameter, the spreading process features an acceleration-deceleration cycle, which results in a violent breakup of the liquid. If the surface wavelength is smaller than the droplet diameter, the surface hinders the spreading process. The normal stress introduced by the curved surface affects the spreading process. As expected, decreasing the roughness size of the deposition surface and increasing the roughness spacing improved the splat flattening and reduced the liquid breakup. Waldvogel and Poulikakos [85] recently performed numerical simulations of the impact and solidification of picoliter size solder droplets on a substrate. They extended the numerical and theoretical tools described in Fukai et al. [57, 581 by improving the grid generation methodology and by modeling the solidification processing. At the time this present chapter was written, the results produced in Waldvogel and Poulikakos [SO] were typical of low impact velocities not common in the process of spray deposition. Despite this fact, it was clear that the final shape of the solidified splat depended greatly on the process parameters. Figure 23 shows three timeframes of the process of impact and solidification of a 50-pm-diameter solder (63/37 Sn/Pb) droplet impacting with 2-m/s velocity on an FR4 substrate (FR4 is a composite material commonly used in the electronics manufacturing industry). The droplet temperature at impact is 200°C and the initial substrate temperature 25°C. The droplet is completely solidified in the last frame (after 50 p s ) . In this case, the solidified splat has a doughnut shape. The velocity vectors in Fig. 23 populate the liquid region, which has not solidified yet, and visualize the fluid motion. The solidified portion of the droplet lies below the liquid region. A host of splat shapes were produced in Waldvogel and Poulikakos [80] depending on the operating conditions.
B. IMPACTAND SOLIDIFICATION OF MULTIPLELIQUIDMETAL DROPLETSAND SPRAYS 1. Studies without Sophisticated Fluid Dynamics Modeling The problem of heat transfer in the splat cooling of two liquid metal droplets impacting sequentially on a substrate was studied recently by Kang et al. [Sl]. The theoretical part of the study was focused on the heat
0.0
I .II
FIG.23. Representative stages of solidification of the 63/37-Sn/Pb droplet impacting on an FR4 substrate (from Waldvogel and Poulikakos [80]).
56
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
transfer aspects of the solidification process and the difference in the behavior of the solidification of the first and second droplet. The experimental part of the study aimed at the characterization of the structure of solidified splats composed of one or two droplets. It was found that the solidification of the second droplet exhibited drastically slower cooling rates compared to the first droplet. As a result, the grain structure of the top of a two-droplet splat was considerably coarser than the structure of the top of a single-droplet splat. The findings of Kang et al. [811 implied that in splat cooling severe limitations need to be imposed on the thickness of the resulting solid layer to ensure rapid solidification and fine-grain structure. In addition, it was shown that the temperature field in the substrate is two-dimensional and radial conduction in the substrate should not be neglected in the modeling of the process. The theoretical model of Kang et al. [81] is presented below. Following the simple approach used for single splats (discussed earlier), the difficulties associated with the fluid dynamics of the process were circumvented in Kang et al. [81]. In the resulting simple and easy-to-use model of the solidification process, each droplet in the splat was viewed as a thin metal disk initially in the liquid phase and finally, after the solidification was completed, in the solid phase. It was assumed that the second disk (splat) contacted the first disk (splat) a short time after the solidification process of the first splat was completed. Figure 24 shows a schematic of the model of a single-droplet splat, and Fig. 24 shows a schematic of the model of a two-droplet splat according to the aforementioned simplifications. Owing to the fact that the thickness to diameter ratio of the splat is very small (typically on the order of 0.05-0.15) the heat conduction process in the splat is modeled as unidirectional [44, 49-561. However, the present model will account for both radial and axial conduction in the substrate following the recommendations of Bennett and Poulikakos [56] for single splats. The first droplet in the splat is assumed to be a thin metal disk at uniform temperature that is suddenly brought into contact with a large (by comparison) substrate of initial temperature considerably lower than the freezing temperature of the disk metal, Tf.Heat is conducted away from the splat to the substrate. Solidification ensues and progresses until the entire splat is solidified. After a short time, the second droplet of the splat is deposited on top of the first droplet. The modeling and the initial conditions of the second droplet are identical to those of the first droplet. Heat is conducted from the second splat to the first splat and eventually to the substrate. Solidification of the second splat ensures and progresses until the entire region solidifies. Following the preceding discussion, the
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
57
b
1 s t Splat
FIG. 24. Schematic of single and double droplet impact (from Kang ef al. [Sl], with permission from ASME).
heat conduction equation in both the first and the second droplet in the splat reads
where the subscript j takes on the values 1 or 2 corresponding to the first (bottom) or the second (top) disk and the superscript m takes on the values s or 1 denoting solid or liquid region. The temperature is denoted by T ; the time, by t ; the axial coordinate, by z (Fig. 24); and the density, specific heat, and the thermal conductivity of the solidifying material, by r, c, and k , respectively. The heat conduction equation in the substrate is dT dt
where the notation is analogous to what was defined above with the added clarification that r stands for the radial coordinate (Fig. 24).
58
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
The initial conditions for the splat and the substrate are
To, Ti( z , t' = 0) = T o , R ( r , z , t = 0) = T, . T : ( z , t = 0)
(104)
=
(105)
( 106) It is worth clarifying that t is the time from the contact of the first splat with the substrate and t ' is the time from the contact of the second splat with the first splat. The boundary and matching condition accompanying the preceding equations are T ( r -+ m, z T ( r , z -+
-k;"
-kT -k
=
0,t)
-m,t)
-
dz
=
( 107) ( 108)
T,, T,,
=h,[T;"(z=H,t) -T,];
-
")
=
=
h , [ T , " ( z = 2 H , t ) - T,];
m=sorI,
(109)
m = s or I, (110)
= h , [ T ( r > R , z = O , t ) -T,] r > R , r=0,1
Boundary conditions (107) and (108) simply state the fact that the temperature inside the substrate far away from the surface as well as the temperature of the substrate surface remote from the splat region remain unaffected by the solidification process and retain their initial value. Boundary conditions (109)-(111) account for convective losses through the top of the first splat, the top of the second splat, and the top of the substrate, respectively. Clearly, boundary condition (109) is in effect prior to the deposition of the second splat. After the second splat is deposited on top of the first splat boundary condition (109) is replaced by
m = s or I, (112) where h, is the contact heat transfer coefficient (the reciprocal of the contact resistance) between the two splats. A matching condition analogous to Eq. (112) is utilized at the interface to couple the substrate with the first splat. -k
fl) dz
=
h,[T(r < R,z=O,t) - T;"(z=O,t)];
r
m
=s
or 1.
(113)
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
59
It has been found in studies of single droplet splats [52, 561 that it is important to account for the presence of contact resistance in the heat transfer modeling of the splat cooling process. The next issue to be discussed is the freezing interface separating the solid from the liquid region. At this interface the following energy balance is satisfied:
where subscript i denotes the freezing interface, U is the velocity of this interface, and L is the latent heat of fusion. The issue of nonequilibrium phenomena at the interface was treated in a manner identical to that of Bennett and Poulikakos [56] and to the discussin leading to Eq. (42). Hence, the freezing interface velocity was given by Eq. (79). The details of the numerical solution methodology of the model described above can be found in Kang et af. [81]. Only the key results will be presented here. Lead splats were utilized in all numerical solutions. The temperature distribution across the splat thickness for both the bottom and the top droplet in the splat in case of h, = 1.5 X lo5 W/m2 K is shown in Fig. 25. The temperature field of the bottom droplet differs markedly from the temperature field in the top droplet. Note that the times t = 0 or t ' = 0 correspond to the moment at which the first or the second disk simulating the flattened droplets come into thermal contact with the substrate (in the case of the bottom disk) or the bottom disk (in the case of the top disk). For example, the solid curve in Fig. 25a corresponds to the temperature distribution in the bottom disk 0.12 ms after the contact of this disk with the substrate. On the other hand, the solid curve in Fig. 25b represents the temperature distribution in the top disk 0.9 ms after this disk comes into contact with the bottom disk. Throughout [81] it was assumed that the second disk contacts the first disk in the splat very shortly (11 ms for h, = 1.5 x lo4 W/m2 K, 0.1 ms for h , = 1.5 X lo5 W/m2 K) after the solidification of the first disk is terminated. Returning to Fig. 25a, we observe that the cooling of the first droplet is initially localized in the vicinity of the splat-substrate interface in the form of a thermal boundary layer ( t = 0.12 ms). Subsequently, the cooling effect propagates through the splat until solidification ensues. A solid-liquid interface exists in the splat at t = 15.86 ms and is marked by the minimum in the temperature curve. This minimum is a direct result of the undercooling existing in the system. This undercooling was 40°C at the initiation of the freezing and it is about 30°C at t = 15.86 ms as shown in Fig. 25a
60
DIMOS POULIKAKOS A N D JOHN M. WALDVOGEL
a 1.0
.
- --------
1
0.12 p s 2.15 pe 15.86 pe
I
1.73 me
;0.6
#
-
I I
I
'
0.0
. ;0.6
-
'
'
"
'
'
. ' - _- .
0.90 p s ' 4 2 . 8 8 ~ ~j - - - - - - - - 93.15 p s ; -1.69 ms 3. 35 ms
/' .
. .
I
'1
.
,
I
I
I
,'
:I i
;I i /I i I
0.0
I . . ' -
. .
(recall that the equilibrium freezing temperature of lead is T, = 327.5'0 As time progresses, the freezing front propagates upward and the undercooling diminishes. At t = 0.77 ms, the freezing interface (marked by the cusp in the respective temperature curve) is practically at the equilibrium freezing temperature. The rapid solidification process of the first droplet has been completed by t = 1.73 ms, and the temperature variation across the thickness of the splat is linear. The temperature distributions in Fig. 25b, corresponding to the top droplet in the splat, exhibit temperature gradients that are not as sharp as those of Fig. 25a. As a result, the cooling process of the second droplet is considerably slower than that of the first droplet. It takes 93.15 ms to observe a solid-liquid interface in the top
61
HEAT TRANSFER AND FLUID DYNAMICS I N SPRAY DEPOSITION
droplet (Fig. 25b) similar to the one observed in the bottom droplet at 15.86 ms (approximately one-sixth of the earlier time). Severe undercooling exists early in the freezing process. The undercooling diminishes later, and the interface temperature becomes identical to the equilibrium freezing temperature. As a result of the heat released on freezing, especially at the earlier stages, the splat warms up temporarily (compare the curves for t = 93.15 ms and 1.69 ms in Fig. 25b) before it cools down eventually and exhibits a linear temperature distribution at the termination of the freezing process ( t = 3.35 ms). The temperature history of the bottom of the first droplet as well as the second droplet (disk) in the splat for characteristic values of the contact coefficient is shown in Fig. 26. The onset of the solidification is marked by the cusps in the temperature/time curves. Prior to initiation of the freezing, the temperature of the splat-substrate interface decreased very rapidly with time. The same was true, to a lesser extent, for the temperature of the interface between the top and bottom droplets for the case of the large contact coefficient ( h , = 1.5 X lo5 W/m2 K). The results for the low contact coefficient exhibit a less drastic dependence of the temperature on time prior to the freezing initiation, markedly so for the interface between the top and the bottom droplet in the splat. After the freezing front has departed from the splat-substrate interface, or the interface
480
h, (W/m2K) 1 . 5 ~05, 1 Bottom of 1st splat 1 . 5 ~05, 1 Bottom of 2nd splat 1.5x105, Top of 2nd splat
______________ ---
1.5x104, Bottom of 1st splat 1.5x104, Bottom of 2nd splat 1.5x104, Top of 2nd splat
I
,
.-. -
0.00
0.15
0.30
0.45 Time (ms)
0.60
0.75
0.90
FIG. 26. The effects of the splat/substrate interface thermal resistance in the cooling process (from Kang et al. [81], with permission from ASME).
62
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
between the two droplets, the temperature dependence on time is very weak. In the case of the large contact coefficient ( h , = 1.5 X lo5 W/m2 K) the cooling of the two aforementioned locations continues (albeit in a very slow fashion). In the case of the small contact coefficient ( h , = 1.5 x lo4 W/m2 K) the temperature is practically independent of time. An additional important finding in Fig. 26 is that it takes much longer time for the freezing of the top droplet in the splat to ensue compared to the bottom droplet in the splat (a direct result of the different cooling rates prior to solidification). For example, in the case of h, = 1.5 X lo5 W/m2 K the freezing of the bottom droplet is initiated at t = 0.015 ms, whereas it takes about 0.09 ms for the freezing of the top droplet to occur from the time it is placed into contact with the bottom droplet. The experimental portion of the work in Kang et al. [811 aimed at examining the effect of the difference in cooling rates between the first and second droplets in the splat on the microstructure of the solidified metal. The details of the experimental procedure as well as the design of the liquid metal droplet generator used to create the splats is contained in Kang et al. [81]. Lead was used as the working metal in all the experiments. The substrate temperature was maintained at 295.9 K. The droplet temperature at impact was 629.1 K. The impinging speed of the droplet on the substrate was 2.15 m/s. The resulting spread factor for the two-droplet splat was 6 = 3.972. Figure 27 shows scalining electron micrographs of the top of a single droplet splat (Fig. 27a) and a double-droplet splat (Fig. 27b). The impact of the retardation on the cooling rate is immediately visible in the grain size of the top of the two-droplet splat (Fig. 2%), which is visibly larger than the grain size of the top of the single droplet splat. An order-of-magnitude estimation of the grain size of the splats was calculated as the diameter of a circle of area equal to that of a grain measured from the electron micrographs. The top surface grain size was typically 45 p m for a single droplet splat (Fig. 27a) and 85 pm for a double-droplet splat (Fig 27b). The grain shapes at the top surface of the double-droplet splat are primarily equiaxial (Fig. 2%). On the other hand, elongated grains are observed at the top surface of the single-droplet splat (Fig. 27a). The elongated grains were found to exist mainly near the rim of the top surface of a single-droplet splat. These elongated grains are indicative of the fact that some outward spreading is still in progress when solidification is initiated. They are not observed on the two-droplet splat (Fig. 27b, where, as a result of lower cooling rates, there was perhaps enough time for the outward spreading to be practically terminated before solidification of the rim region.
FIG. 27. Electron micrographs of the top of the splat (Kang et al. [81], with permission from ASME): (a) single-droplet splat; (b) double-droplet splat.
64
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
Representative studies on multiple liquid metal droplet impact and solidification on a substrate are the works of Annavarapu et al. [21 and Gutierrez-Miravete et al. [82]. To this end [82], the solidification heat transfer phenomena occurring in spray deposition was analyzed using the enthalpy method and a numerical formulation based on finite differences. No fluid dynamics phenomena were taken into account. It was found that the average enthalpy of the spray at the time of impact, the metal mass flow rate (deposition rate), and the heat transfer coefficient at the substrate-deposit interface are all important factors in the spray deposition process. In Annavarapu et al. ([2], and references cited therein), two different approaches to model the droplet impact and consolidation on a substrate are offered: A continuum approach and a discrete-event approach. The continuum approach consists of a mass balance to obtain the deposit geometry and an energy balance to determine the thermal history of the deposit. The heat flux of the spray is used as an input. In the discrete-event approach the spray is not viewed as a continuous heat-mass flux domain. Since consolidation in the substrate occurs by the successive splatting of individual droplets, the deposit formation is modeled as a superposition of discrete events. The deposit is modeled as a group of individual splats stacked vertically. An average droplet energy prior to impact is utilized, and an energy balance is conducted to calculate the temperature history of the deposit. In what follows, both the continuum and the discrete approach will be outlined according to the relevant presentation in Annavarapu et al. [2]. The average spray enthalpy prior to impact is needed in both cases. Hence, this item will be discussed first. An artistic depiction of a group of liquid metal droplets in the spray prior to impact is shown in Fig. 28. To obtain the average spray enthalpy according to the approach in [2, 41, 42, 831, it is assumed that the liquid metal droplets in the spray are spherical and move in a linear trajectory and that the heat exchange between the gas and the droplets is interface-controlled. The model focuses on a single droplet at a time undergoing undercooling, recalescence, and solidification. It utilizes two experimentally obtained variables: the degree of initial undercooling and the size distribution of the droplets in the spray. As shown in Fig. 29, the spray is assumed to be approximately conical in a quasi-steady state. The spray enthalpy Hspray and the deposition rate d are assumed to be independent of time but dependent on the axial distance from the spray axis. The enthalpy of a droplet of diameter d , at some flight instance is Hd, = / c P dT
+- AHf dfs,
where A H f and dfs are respectively the latent heat of fusion and the solid fraction increment in the droplet. The weighted average of the enthalpy of
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
65
Impinging Gas Flow
Spray Cone Periphery
FIG.28. Artistic depiction of group metal droplets prior to impact.
individual droplets at some flight distance defines the enthalpy of the spray at that distance
,
where fd, is the fraction of droplets in the range d , to d , , . This fraction is obtained experimentally. The average spray enthalpy can be converted to an average temperature that can be used to obtain the liquid fraction in the spray from the Scheil equation [2, 841. a. Heat Transfer Modeling-Continuum Approach A one-dimensional heat balance in the substrate accounting for the fact that the deposit thickness grows with time yields [2] (117)
This equation is accompanied by the following boundary conditions: At the bottom of the deposit (77
=
0):
k dT
- - =h,(T, L d77
-
q), (118)
66
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
Atomization
+-Spray Radius
4
I
/ I P(x, y)
R(x, y)
FIG.29. Schematic of a conical spray based on the discussion in Annavarapu et al. [2].
At the top of the deposit ( 7 7
=
1) :
k dT
dL
p(Flspray- H ) -
dt
=
- - + hg(Tg- T t ) . (119) L dq
In equations (117)-(119) H , t , r], L , T b , T,, T , , T g , h , , and h , stand for enthalpy, time, dimensionless deposit thickness (77 = z / L ) , deposit thickness, deposit bottom temperature, deposit top temperature, substrate top temperature, gas temperature, heat transfer (resistance) coefficient between the deposit bottom and substrate top, and heat transfer coefficient between the gas and the deposit top, respectively. The following expression corresponding to the surface temperature of a semiinfinite solid subjected to a constant surface heat flux Q was used in Annavarapu et al. [2] to
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
67
estimate the surface temperature of the substrate:
2Q at/ k
T,=-(
T)
”*.
The rather straightforward details of the numerical solution of the model in Eqs. (117)-(120) for the temperature distribution of the deposit are given with clarity in Annavarapu et al. [2]. b. Heat Transfer Modeling-Discrete Approach Annavarapu et al. [2] utilized a simple one-dimensional energy balance in the direction of the deposit growth ( z ) to obtain the enthalpy change in every splat:
At the bottom of every splat the following boundary condition holds: dT
k-
dz
=h,(Tb -
At the top of each splat it is assumed that the spray enthalpy (p,,,,,) is applied at discrete time intervals ( S t ) in a singular manner. Heat removal occurs continuously at the top surface through conduction downward in the splat and convection in the atomizing gas. From this discussion, the boundary condition at the top surface is [2]
where A is a binary constant defined as follows: 1
A = { 0
fort = i S t , i otherwise,
=
1 , 2 , 3,...,
where the values of the integer i denote the arrival of liquid metal droplets at the top surface of the deposit and S t denotes the time lapse between successive arrivals. Note that in contrast to the continuum model, the deposit is only cooled between successive droplet arrivals. The details of the numerical solution of the discrete model are contained in Annavarapu et al. 121. The value of the gas heat transfer coefficient used in both the discrete and continuum models was approximated to be that of flat plates being cooled convectively by gas jets. The following expression for the Nusselt
68
DIMOS POULIKAKOS AND JOHN M. WALDVOGEL
number was utilized [2, 851: Nu pr 0.42 [l
--
(W/R)[l - l.lW/R] + 0.1( E / W - 6)W/R]F(Re) '
(124)
where
F ( Re)
=
2
(
Re0.55
1
)
+200
0'5
.
In these equations W is the width of the gas jet at the nozzle, R is the radial distance from the spray axis, Pr is the Prandtl number of the gas, and Re is the Reynolds number based on the droplet diameter. The approximate value used in Annavarapu et al. [21 based on Eqs. (124) and (125) was h, = 500 W/m2 K. The main findings in the study of Annavarapu et al. [2] are summarized as follows: The dependence of metal deposition rate on radial distance from the spray axis is Gaussian. The variation of the deposit thickness can be estimated by combining the variation of the deposition rate across the spray cone with a description of the substrate motion. A great variation of the heat flux across the deposit interface was found over a very short time period: The heat flux dropped from 12 X lo7 W/m2 at the initial stage of the deposition to a steady value of lo6 W/m2 over a period of 10-15 s. The respective change in the value of the heat transfer coefficient varied from 7 X lo4 W/m2 K to 5 X lo3 W/m2 K. The cooling rate of the bulk of the deposit during solidification was estimated to be less than 100°C/s. At low substrate velocities the deposit solidifies from a partially liquid layer that forms on its surface during deposition. The solid fraction in this partially liquid layer is usually higher than 0.95. The predictions of the discrete model indicate the formation of a partially solidified layer after a fraction of a second. The predicted cooling rates in the chill zone (in the range 105-108"C/s) agree with the observed microstructure. The continuum model underestimated the cooling rates in the chill zone by approximately two orders of magnitude. After the liquid layer forms on top of the deposit, both models predict the same solidification rate. Separation of the deposit from the substrate reduces the solid deposit cooling down to 5-10"C/s. The nonuniformity of the deposit due to
HEAT TRANSFER AND FLUID DYNAMICS IN SPRAY DEPOSITION
69
the mass flux distribution in the spray can be eliminated by exposing every location to an identical spray cycle. The temperature of both the substrate and the spray affect the structure of the deposited layer. In two very recent papers [86, 871, the growth dynamics of spray-formed aluminum billets in both the steady-state and transient regimes is examined through a mathematical model that does not explicitly treat the relevant transport phenomena. The generic motions of the billet surface that occur during production are discussed over the relevant parametric domain. The effect of these motions on the production time and efficiency was also stressed.
2. Studies with Sophisticated Fluid Dynamics Modeling Studies of multiple droplet impact involving sophisticated modeling of the relevant fluid dynamics are scarce, indeed. First attempts to deal with this complex problem are contained in Trapaga and Szekely [59] and in Liu et al. [60]. More specifically, Trapaga and Szekely [59] presented limited (one figure) results on the fluid dynamics of two identical cylindrical (to maintain the two-dimensional features of the problem) “droplets” impacting simultaneously on a flat surface and separated by a horizontal distance equivalent to six droplet diameters. The resulting splat dimensions were clearly affected by droplet interaction effects. Lui et al. [60] investigated the formation of micropores during interaction of multiple droplets. They presented results for two axisymmetric cases: two droplets falling in tandem at the axis of symmetry, and an axisymmetric toroidal ring falling above a droplet at the axis of symmetry. With reference to the former, when the second droplet impinged on the splat underneath it, which was already jetting radially outward, it produced an upward and outward liquid ejection. The outward ejection was enlarged as time progressed. Solidification limits this ejection. The droplet interaction was found to be likely to increase the microporosity. With reference to the latter, Liu et al. [60] found that as the toroidal ring falls on top of the spreading first splat, the inside fringe of the ring and the first splat (moving in opposite directions) interact so as to cause an inward and upward liquid motion. This interaction leads to the formation of voids at the locations of vortices in the liquid. Some of these voids may be refilled with liquid during subsequent deformation (which may also create new voids). Overall, there is a long way to go before thorough fluid dynamics modeling involving two or more impacting droplets will lead to a good understanding of droplet interaction in the spray deposition process.
70
DIMOS POULIKAKOS A N D JOHN M. WALDVOGEL
IV. Summary This chapter presented a review of the existing knowledge base of the process of spray deposition, focusing on issues in which transport phenornena are relevant. It is apparent that in addition to the materials science community, the heat transfer and fluid dynamics communities can have a significant impact in improving our understanding of the fundamental mechanisms occurring in this important process. Focusing on the impact regime alone, there is much to be learned from the standpoint of transport phenomena. To exemplify, even the single liquid metal droplet impact process is not well understood, although progress has been made to this end for axisymmetric splats. This process involves complex fluid dynamics phenomena occurring within a severely deforming domain in the presence of conjugate convection-conduction heat transfer and solidification. Three-dimensional phenomena and splat breakup, which often occur in cases involving high impact velocities, have yet to be modeled. The same is true regarding the rigorous modeling (from the standpoint of transport phenomena) of more realistic situations involving droplet groups and sprays. In all, heat transfer and fluid dynamics researchers are strongly encouraged to focus their attention on the various challenges offered by the process of spray deposition.
References 1. Szekely, J. (1987). Can advanced technology save the U.S. steel industry? Sci. Am. 257, 34-41. 2. Annavarapu, S., Apelian, D., and Lawley, A. (1990). Spray casting of steel strip: Process analysis. MetalI. Trans. A 21A, 3237-3256. 3. Sirignano, W. A. (1992). 1992 Freeman Scholar Lecture: Fluid dynamics of sprays. J . Fluids Eng. 115, 345-378. 4. Megaridis, C. (1993). Presolidification liquid metal cooling under convective conditions. Atomization Sprays 3, 171-191. 5 . Reid, R. C., Prausnitz, J. M., and Shenvood, T. K. (1977). The Properties of Gases and Liquids. McGraw-Hill, New York. 6. Boyer, H. E., and Gall, T. L., eds. (1985). Metals Handbook. American Society for Metals, Metal Park, OH. 7. Clift, R., Grace, J. R., and Weber, M. E. (1978). Bubbles, Drops and Particles. Academic Press, San Diego, CA. 8. Dubroff, W., Program Coordinator (1990). Semiannual Program Report to DOE: Deuelopment of a Spray-Forming Process for Steel. Idaho National Engineering Laboratory. 9. Froes, F. H., Suryanarayana, C., Lavernia, E., and Bobeck, G. E. (1991). Innovations in light metals synthesis for the 1990s. SAMPE Q.22, 11. 10. Dubroff, W., Program Coordinator (1989). Semiannual Program Report to DOE: Deuelopment of a Spray-Forming Process for Steel. Idaho National Engineering Laboratory.
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11. Ranz, W. E., and Marshall, W. R. (1952). Evaporation from drops I and 11. Chem. Eng. frog. 48, 141-146, 173-180. 12. Gutierrez-Miravete, E., Lavernia, E. J., Trapaga, G. M., Szekely J., and Grant, N. J. (1989). Mathematical model of the spray deposition process. Metall. Trans., A 20A, 71-85. 13. Lavernia, E. J., Gutierrez, E. M., Szekely, J., and Grant, N. J. (1988). A mathematical model for the liquid dynamic compaction process. Part 1: Heat flow in gas atomization. Int. J . Rapid Solid$. 4, 89-124. 14. Kurten, H., Raasch, J., and Rumpf, H. (1966). Chem-lng. Tech. 38, 941-948. 1.5. Figliola, R. S., Anderson, I. E., Molnar, D., and Morton, H. (1989). Flow mechanisms affecting rapid solidification in high pressure gas atomization. HTD [Publ.] ( A m . Soch. Mech. Eng.) 111, 7-12. 16. Crowe, C. T., Babcock, W. R., and Willoughby, P. G. (1972). Drag coefficients for particles in rarefied low Mach number flows. Prog. Heat and Mass Transfer 6,419-431. 17. Carlson, D. J., and Hoglund, R. F. (1964). Particle drag and heat transfer in rocket nozzles. AZAA J . 2. 1980-1984. 18. Whitaker, S. (1972). Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres and flow in packed beds and tube bundles. AIChE J . 18, 361-171. 19. Bejan, A. (1993). Heat Transfer, p. 268. Wiley, New York. 20. Veistinen, M., Lavernia, E. J., Baram, J., and Grant, N. J. (1988). Jet behavior during ultrasonic gas atomization. Int. J . Powder Metall. 25, 89-92. 21. Veistinen, M., Lavernia, E. J., Abinante, M., and Grant, N. J. (1987). Jet behavior during ultrasonic gas atomization. Mater. Lett. 5 , 373-379. 22. Backmark U., Backstorm, N., and Amberg, L. (1985). Swedish Institute for Metals Research, Stockholm. 23. Beattle, J. A,, and Julien, H. P. (1954). Ind. Eng. Chem. 46, 1668-1669. 24. Levi, C. G., and Mehrabian, R. (1982). Heat flow during rapid solidification of undercooled metal droplets. Metall. Trans., A 13A, 221-234. 25. Clyne, T. W. (1984). Numerical treatment of rapid solidification. Metall. Trans., B 15B, 369-381. 26. Wilson, H. A. (1990). fhilos. Mag. [ S ] 50, 238-246. 27. Frenkel, J. (1932). fhys. 2. Sowjetunion 1, 498-503. 28. Cahn, J. W., Hillig, W. B., and Sears. G. W. (1964). Acta Metall. 12,914-922. 29. Bayazitoglu, Y., and Cerny, R. (1993). Solidification of spherical metal drops in metal powder production. J . Mater. Process. Manuf. Sci., 2,51-61. 30. Biancaniello, F. S., Espina, P. I., Mattingly, G. E., and Ridder, S. D. (1989). A flow visualization study of supersonic inert gas-metal atomization. Mater. Sci. Eng., A 119, 161-168. 31. Biancaniello, F. S., Conway, J. J., Espina, P. I., Mattingly, G. E., and Ridder, S. D. (1990). Particle size measurement of inert-gas-atomized powder. Mater. Sci. Eng., A 124,9-14. 32. Biancaniello, F. S., Presser, C., and Ridder, S. D. (1990). Real-time particle size analysis during inert gas atomization. Mater. Sci. Eng., A 124,21 -29. 33. Lubanska, H. (1970). J . Met. 22, 45-49. 34. Rai, G., Lavernia, E. J., and Grant, N. J. (1985). J . Met. 37, 22-26. 35. ASTM Subcommittee E29.04 on Liquid Particle Measurement (1992). ASTM Standards on Liquid Particles and Sprays, PCN 03-529092-41. ASTM, Philadelphia. 36. Ando, T., Tsao, C.-Y., Wahlroos, J., and Grant, N. J. (1990). Analysis and control of gas atomization rate. Int. J . Powder Metall. (Pnnceton, N . J . ) 26,31 1-318.
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37. Geiger, G. P., and Poirer, D. R. (1980). Transport Phenomena in Metalluw. AddisonWesley, Reading, MA. 38. Ayers, J. D., and Anderson, I. E. (1987). Very fine metal powders. J . Met. 37, 16. 39. Couper, N. J., and Singer, R. J. (1984). In Rapid& Quenched Metals (S. Seeb and H. Warlimont, eds.), p. 1737. North-Holland Publ., Amsterdam. 40. Jones, H. (1983). Rapid Solidifcation of Metals and Alloys, p. 43, Institution of Metallurgists, London. 41. Mathur, P. C., Apelian, D., and Lawley, A. (1989). Acta Metall. 37, 429-443. 42. Mathur, P. C. (1988). Ph.D. Thesis, Drexel University, Philadelphia. 43. Bennett, T., and Poulikakos, D. (1993). Splat-quench solidification: Estimating the maximum spreading of a droplet impacting a solid surface. J . Mater. Sci. 28, 963-970. 44. Madejski, J. (1976). Int. J . Heat Mass Transfer 18, 1009-1013. 45. Collings, E. W., Markworth, A. J., McCoy, J. K., and Saunders J. H. (1990). J . Mater. Sci. 25,3677-3682. 46. Chandra, S., and Avedisian, C. T. (1991). On the collision of droplet with a solid surface. Proc. R . Sac. London, Ser. A 432, 13 (1991). 47. Jones, H. (1973). Splat cooling and metastable phases. Rep. Prog. Phys. 48, 1425-1497. 48. Anantharaman, T. R., and Suryanarayana, C. (1971). Review: A decade of quenching from the melt. J . Muter. Sci. 6, 1111-1135. 49. Madejski, J. (1983). Int. J . Heat Mass Transfer 26, 1095-1098. 50. Wang, G.-X., and Matthys, E. F. (1991). Modelling of heat transfer and solidification during splat cooling: Effect of splat thickness and splat substrate thermal contact. Int. J. Rapid Solidif. 6, 141-174. 51. Wang, G.-X., and Matthys, E. F. (1992). Numerical modelling of phase change and heat transfer during rapid solidification processes: Use of control volume integrals with element subdivisions. Int. J . Heat Mass Transfer 35, 141-153. 52. Wang, G.-X., and Matthys, E. F. (1991). Int. J . Rapid Solidif. 6, 297-324. 53. Shingu, P. H., and Ozaki, R. (1975). Solidification rate in rapid conduction cooling. Metall. Trans., A 6A, 33-37. 54. Rosner, D. E., and Epstein, M. (1975). Simultaneous kinetic and heat transfer limitations in the crystallization of highly undercooled melts. Chem. Eng. Sci. 30, 511-520. 55. Evans, P. V., and Greer, A. L. (1988). Modeling of crystal growth and solute redistribution during rapid solidification. Mater. Sci. Eng. 98, 357-361. 56. Bennett, T., and Poulikakos, D. (1994). Heat transfer aspects of splat-quench solidification: Modeling and experiment. J . Muter. Sci. 29, 2025-2039. 57. Fukai, J., Zhao, Z., Poulikakos, D., Megaridis, C., and Miyatake, 0. (1993). Modelling of the deformation of a liquid droplet impinging upon a flat surface. Phys. Fluids A 5 , 2588-2599. 58. Fukai, J., Shiiba, Y., Yamamoto, T., Miyatake, O., Poulikakos, D.,.Megaridis, C. M., and Zhao, Z. (1995). Wetting effects on the spreading of a liquid droplet colliding with a flat surface: Experiment and modeling. Phys. Fluids A 7, 236-247. 59. Trapaga, G., and Szekely, J. (1991). Mathematical modeling of the isothermal impingement of liquid droplets in spray processes. Metall. Trans. E 22, 901-914. 60. Liu, H., Lavernia, E. J., and Rangel, R., (1994). Numerical investigation of micropore formation during substrate impact of molten droplets in plasma spray processes. Atomization Sprays 4, 369-384. 61. Lui, H., Lavernia, E. J., and Rangel, R. H. (1994). Modeling of molten droplet impingement on a non-flat surface. Int. Mech. Eng. Congr. Exhib. Winter Ann. Meet. Chicago, 1994.
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62. Marchi, S. C., Lui, H., Sickinger, A., Muhleberg, E., Lavernia, E. J., and Rangel, R. H. (1993). Numerical analysis of the deformation and solidification of a single droplet impinging onto a flat substrate. J . Mater. Sci. 28, 3313-3321. 63. Tsurutani, K., Yao, M., Senda, J., and Fujimoto, H. (1990). Numerical analysis of the deforming process of a droplet impinging on a wall. JSME Int. J. [2] 33, 555-561. 64. Bach, P., and Hassager, 0. (1985). An algorithm for the use of the Lagrangian specification in Newtonian fluid mechanics and aDDlications to free surface flow. J. Fluid Mech. .. 152, 173. 65, Hirt, C . W., and Nichols, B. D. (1980). Adding limited compressibility to incompressible hydrocodes. J . Comp. Physiol. 34, 390. 66. Zienkiewicz, 0. C., and Taylor, R. L. (1991). The Finire Element Method, 4th ed. Vol. 2, pp. 513-514. McGraw-Hill, London. 67. Kawahara, M., and Hirano, H. (1983). A finite element method for high Reynolds number viscous fluid flow using two step explicit scheme. Int. J. Numer. Methods Fluids 3, 137. 68. Zienkiewicz, 0. C., and Taylor, R. L. (1989). The Finite Element Method, 4th ed. Vol. 1, pp. 206-215. McGraw-Hill, London. 69. Haley, P. J., and Miksis, M. J. (1991). The effect of the contact line on droplet spreading. J . Fluid Mech. 223, 57. 70. Huh, E., and Scriven, L. E. (1971). Hydrodynamic model of steady movement of a solid liquid fluid contact line. J. Colloid Interface Sci. 35, 85. 71. Dussan, E. B., V, and Davis, S. H (1974). On the motion of a fluid-fluid interface along a solid surface. J . Fluid Mech. 65, 71. 72. Dussan, E. B., V (1979). On the spreading of liquids on solid surfaces: Static and dynamic contact lines. Annu. Reu. Fluid Mech. 11, 371. 73. Silliman, W. J., and Scriven, L. E. (1980). Separating flow near a static contact line: Slip at a wall and shape of a free surface. J. Comp. Physics 34, 287. 74. Hocking, L. M., and Rivers, A. D. (1982). The spreading of a drop by capillary action. J. Fluid Mech. 121, 425. 75. Sheng, P., and Zhou, M. (1992). Immiscible-fluid displacement: Contact-line dynamics and the velocity-dependent capillary pressure. Phys. Reu. 45, 5694. 76. Leger, L., and Joanny, J. F. (1992). Liquid spreading. Rep. Prog. Phys. 25, 431. 77. Zhao, Z., Poulikakos, D., and Fukai, J. (1996). Heat transfer and fluid dynamics during the collision of a liquid droplet on a substrate: Part I-modeling. Int. J . Heat Mass Transfer (in press). 78. Zhao, Z., Poulikakos, D., and Fukai, J. (1996). Heat transfer and fluid dynamics during the collision of a liquid droplet on a substrate: Part 11-Experiments. Inr. J . of Heat Mass Transfer (in press). 79. Moreau, C., Cielo, P., Lamontagne, M., Dallaire, S., Krapez, J.-C., and Vardelle, M. (1990). Temperature evolution of plasma sprayed niobium particles impacting on a substrate. Proc. Nat. Therm. Spray Confer. 3rd, Long Beach, CA, pp. 19-26. 80. Waldvogel, J., and Poulikakos, D. (1996). Solidification of picoliter size solder droplets impacting on a substrate. In preparation. 81. Kang, B., Zhao, Z., and Poulikakos, D. (1994). Solidification of liquid metal droplets impacting sequentially on a substrate. J. Heat Transfer 116, 436-445. 82. Gutierrez-Miravete, E., Lavernia, E. J., Trapaga, G. M., and Szekely, J. (1988). A mathematical model for the liquid dynamic compaction process. Part 2: Formation of the deposit. fnf. J . Rapid Solidsf. 4, 125-150. 83. Gillen, A. G., Mathur, P. G., Apelian, D., and Lawley, A. (1986). Prog. Powder Metall. 42, 753-773.
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84. Flemings, M. C . (1974). Solidifcation Processing. McGraw-Hill, New York. 85. Martin, H. (19771. In Adtmnces in Heat Transfer (J. P. Hartnett and T. F. Irvine, eds.), Vol. 13. Academic Press, New York. 86. Frigaard, I. A. (1995). Growth dynamics of spray formed aluminium billets. Part 1: Steady state crown shapes. J . Mater. Process. Manuf. Sci. 3, 173-193. 87. Frigaard, I. A. (1995). Growth dynamics of spray formed aluminium billets. Part 1: Transient billet growth. J. Mater. Process. Manuf. Sci. 3, 257-275.
ADVANCES IN HEAT TRANSFER, VOLUME 28
Heat and Mass Transfer in Pulsed-Laser-Induced Phase Transformations
COSTAS P. GRIGOROPOULOS, TED D. BENNETT, JENG-RONG HO, XIANFAN XU,* AND XIANG ZHANG Depaitment of Mechanical Engineering, University of California, Berkeley, California
I. Pulsed Laser Melting A. BACKGROUND In terms of heat transfer, the fundamental question raised in this chapter is whether essential features of pulsed laser melting and resolidification at the nanosecond time scale can be described using a thermal model. Of specific interest is the description of the transient melting front propagation as a function of the induced temperature field. Several techniques have been developed to probe the transient temperature field during pulsed laser processing. However, as will become apparent, every technique has inherent limitations suitable only for particular materials, temperature ranges, spatial constraints, and so forth. A standard time-offlight measurement was used to measure the lattice temperature of bulk crystalline silicon during pulsed ruby laser heating [96]. This method involves measurement of the kinetic energy of particles released from the surface of the material. A characteristic temperature is extracted by fitting equilibrium Maxwellian distributions to the translational kinetic-energy data. However, the derivation of the temperature is based on the assumption that the ejected particles are in a thermal equilibrium state. Such an assumption may not be valid, particularly when the sputtering mechanism * Present address: School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907. 75
Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form resewed.
76
COSTAS
P. GRIGOROPOULOS et al.
is nonthermal. Transient temperature during nanosecond Nd: glass laser irradiation was also measured using an iron-constantan thin-film thermocouple [6]. A similar thermistor technique 1551 was applied to study the interface temperature of SiAs alloys during planar-interface solidification induced by a pulsed XeCl excimer laser. The complicated sample preparation involved in these two methods limits their usefulness for direct noncontact temperature monitoring of materials. The optical reflectivity technique [46]is the most widely used method to study the pulsed-laserinduced phase transformation and probe the transient temperature field. To obtain the temperature information by the optical reflectivity method, the temperature dependence of the optical properties (i.e., the complex refractive index) must be known. The sensitivity of this optical technique is reduced as the variation of reflectivity with temperature is decreased. Numerical heat transfer computations are used to obtain the transient temperature field “5, 108-1101. In this section, the pulsed laser melting process induced by nanosecondlength ultraviolet (UV) excimer laser pulses is examined. The main advantage in using UV radiation is the short optical penetration depth, which for metals and semiconductors is of the order of tens of nanometers. The excimer beam spatial distribution exhibits inhomogeneities (i.e., hot spots) but can be made uniform using homogenizers. Beam uniformity is essential for quantitative experimentation and for applications such as mask projection machining. The size of a focused excimer laser spot in most applications is of the order of millimeters, i.e. much larger than the absorption penetration depth and the thermal penetration depth in the which is on the order of micrometers. Thus, the target solid, 6, transient melting process is usually understood as a one-dimensional phenomenon. This simplification may not be adequate for modeling micromachining processes, where the thermal penetration depth becomes comparable with the micromachined feature size. It should, however, be mentioned that the smallest feature achievable with excimer laser technology is, in principle, determined by the optical diffraction limit, which scales with the short UV wavelengths (0.19-0.30 pm).
- fi,
B. THERMAL MODELING
Two methods are commonly used for computational handling of heat transport with phase change: (1) the enthalpy method, which is suitable for problems not requiring precise information about the liquid-solid interface not requiring and (2) integace tracking method, which utilizes a change of coordinates that allows an exact boundary condition to be imposed on
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
77
the moving interface at all times. Both methods are founded on the Fourier law for heat transport. Instead of the classical Fourier heat diffusion model, the hyperbolic heat conduction [54, 75, 801 and the vibrational cooling model [22] have been proposed. However, the mean free time between electron collisions in a metallic conductor and the relaxation time of electronic events in semiconductors are in the subpicosecond regime. Thus, for the nanosecond-length laser pulses considered in this work, it may be assumed that the laser light energy is immediately redistributed and passed on to the lattice and converted into heat at the location of absorption. Therefore, it is reasonable to assume a local thermodynamic equilibrium state, and the conventional concept of temperature and thermal properties can be defined and used in this study. In the enthalpy formulation approach [5, 891, the position of the interface does not appear explicitly in the calculation. The enthalpy function is used to account for phase change. In the regions to either side of the melting temperature zone, where T < T, and T > T,, where T, is the melting temperature, the enthalpies are
For T where
=
H(T)
= /'p(T)C,( 0
H( T )
=
T < T,
T )dT;
l T p ( T)C,( T ) dT + L , ;
T > T,.
0
(1) (2)
T, the enthalpy function assumes values between H,, and HI,,
Hsm =
j 0T r n p ( ~ ) c pd~( ~ )
(3)
HI,
/" p( T ) C , ( T )dT + L , .
(4)
=
0
The enthalpy value H = H,, is assigned to solid material at the melting temperature, whereas H = HI, corresponds to pure liquid phase at the same temperature. Thus, there is a region of partial melting, defined by
H,, < H
HI, ; T = T,. (5) Each point within this region can be assigned a solid fraction fJx, t ) and a liquid fraction fl(x, t ) , for which f,(x,t) + f , ( x , t )
=
1.
Thus, the enthalpy function during melting at T
H
= H,,
+flL,.
(6) =
T, is given by (7)
78
COSTAS P. GRIGOROPOULOS e l al.
Using enthalpy as a dependent variable, along with the temperature, the heat conduction in the target material is written
The excimer laser absorption penetration depth in the thin film, dab = l / q , is of the order of 10 nm [91], where q is the absorption coefficient. Assuming that the target material thickness is at least a few times larger than dab,the energy absorption, Qab(x,t ) , follows an exponential decay: Q,i,(x,t)
=
(1 - R e x c ) l ( t ) r l e - " " .
(9)
The normal reflectivity of silicon surface is given by
The absorption coefficient q is calculated as
v=-.
4%xc Aexc
Incident laser intensity is typically high enough for convection and radiation losses from the top surface to be negligible. Also, for nanosecond time scales the temperature penetration is small, so that the bulk material is maintained at the ambient temperature, T, :
T(x =d,,,t)
=
T,.
( 12b)
Initially the target is isothermal, at the ambient temperature: T ( x , O ) = T,. (13) These equations can be easily solved by implementing an implicit or explicit finite-difference algorithm on a fked grid. In contrast to the enthalpy method, the heat conduction equation can also be cast using temperature as the sole dependent variable:
A system of coordinates attached to the moving interface can now be considered. The transient position of the moving interface, measured from the location of the exposed surface at t = 0, is given by the function
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
79
Xint(f).The heat conduction equation is written in this system of coordinates:
ax "1
( P C , ) ( T ) [ DT - dT
=
G ( k ( T ) g )+ Q , , ( x , t ) .
(15)
Interface tracking schemes apply the exact boundary conditions at the moving solid/liquid-phase boundary. The energy transfer is balanced across the moving interface, which is at temperature Tint: T,(Xint) =Tl(Xint)
=
Tint
7
(16a)
If no superheating or undercooling is assumed, the interface is at the equilibrium melting temperature, T, . Melt-solid-interface velocities in pulsed laser melting and recrystallization can exceed 15 m/s [45, 671. For such velocities, the liquid phase is expected to be highly overheated or undercooled and the assumption of a constant, thermodynamic equilibrium, phase-change temperature is no longer valid [431. According to the quasi-chemical formulation of crystal growth from the melt [43, 441 the rate at which atoms join the crystal is
where k , is Boltzmann's constant, Tintis the interface temperature, and Q is the activation energy for viscous or diffusive motion in the liquid. Similarly, the rate at which atoms leave the crystal is
where L , is the latent heat of fusion. Thus, the activation energy for melting is Q + L , . At equilibrium, the rates for solidification and melting are equal, R, = RM, and for the interface temperature, Tint= T,. Combining (17) and (18):
The velocity of recrystallization [V,,JTint) > 01 or melting [Knt(Tint)< 01 is given by Knt('int)
= RF - R,
*
(20)
80
COSTAS P. GRIGOROPOULOS et al.
Utilizing Eqs. (18)-(20), the velocity of resolidification is expressed
vnt(Tint)= C’ exp
[
-kB:in~]
{
L , AT -
- ‘BTintTm]}
3
(21)
where
Equation (21) can be approximated by a linear relation between the interface velocity and the superheating temperature at the interface:
The material constant C,,which quantifies the effect of the interface velocity on the superheating temperature ’AT, represents the degree of interface superheating. Kluge and Ray [56] assigned a numerical value to this constant for silicon, C , = 9.8 K/(m/s), by fitting Eq. (22) to direct molecular dynamics predictions of the interfacial velocity as a function of temperature for epitaxial silicon crystal growth from the liquid phase. This interface response function is adopted in the numerical modeling of rapid melt propagation. In the work of Xu et al. [1141, the position of the interface and the temperature field were solved by casting Eq. (13, together with the interfacial boundary conditions [Eq. (16)1, in an implicit finite-difference form. The solution was iterated to satisfy the response function given by Eq. (22). Direct experimental measurement was recently obtained by Xu et al. [1131 and will be discussed in the next section.
c. EXPERIMENTAL VERIFICATION
OF THE
MELTINGPROCESS
For comparison with prior published work, we will concentrate on the study of pulsed laser melting of semiconductors in bulk and thin-film form. As will be shown later, certain inherent advantages of specific physical properties of semiconductors allow direct experimental observations. In addition, there are considerable related practical applications, such as annealing of ion implantation surface damage, recrystallization of amorphous and polycrystalline films, and enhancement of dopant diffusion. Both experimental and computational investigations of pulsed laser interactions with semiconductor materials have been performed. Transient conductance measurements [31, 1001, nanosecond-resolution X-ray diffraction measurements [64], and time-resolved reflectivity measurements [46] have been applied to obtain quantities such as melt penetration, melt duration, and melt front velocity. The experimental results were
PULSED USER-INDUCED PHASE TRANSFORhlATIONS
81
interpreted by numerical simulations to show that the pulsed laser melting of semiconductors in the nanosecond time regime is a thermal phenomenon.
1. Conductance Experiment Recently Xu et al. [114] examined the transient heating and melting of thin polysilicon (p-Si) layers deposited onto fused-quartz wafers by lowpressure chemical vapor deposition (LPCVD), as well as bulk silicon under nonequilibrium conditions. A schematic drawing of the experimental setup is shown in Fig. 1, and the sample configuration is depicted in Fig. 2. The samples are irradiated by a pulsed ultraviolet (UV) KrF excimer laser with wavelength A = 248 nm and pulse duration T~ = 52 ns [full-width half-
KrF excimer laser __
Power meter Beam splitter
Fast
HeNe la
Oscilloscope
FIG. 1. Experimental setup for optical reflectivity, transmissivity, and electrical conductance measurements during excimer laser melting of polysilicon films ([114], reproduced with permission from ASME).
82
COSTAS P. GRIGOROPOULOS el a(.
Electric Conductance Measurement
i
A1
A1
Sample
top-view
FIG.2. Sketch of sample and conductance measurement circuit “1141, reproduced with permission from ASME).
maximum (FWHM) = 26 ns]. The interface tracking numerical method was coupled with experimental information to resolve the temperature field and investigate interface superheating. The constant C , is taken as 10 K/(m/s) in the calculation, unless noted otherwise. A continuous-wave (CW) unpolarized HeNe laser ( A = 633 nm) is used as a probing light source for the reflectivity and transmissivity measurement. The resistivity of liquid silicon is lower than that of solid silicon by a factor of five orders of magnitude. When the surface of silicon film starts to melt, the total resistance of the silicon is reduced by several orders of magnitude. The depth of the molten layer can be determined from the transient voltage signal across the p-Si sample when the sample is heated by the pulsed laser. In the transient electric resistance measurement, the photoconductance signal, which is the electric conductance caused by the laser-excited free carriers, could last for microseconds in pure silicon. To reduce the free-carrier lifetime, gold was evaporated on the polysilicon film surface and diffused into the polysilicon film by convective heating. The photoconductance signal in this gold-diffused sample is completely separated from the conductance signal caused by phase change. Typical transient reflectivity, transmissivity, and conductivity signals are shown in Fig. 3. In Fig. 3a, the laser fluence is just above the melting threshold. Because of the temperature dependence of optical refractive index of silicon at 633-nm wavelength, the reflectivity of solid silicon increases with temperature. The reflectivity reaches 0.73 at about 30 ns
83
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
J
L
12
sv
--C Reflectivity
+Voltage
0 0
200
100
500
400
300
700
Mx)
800
Time (ns)
b
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I
~
.
'
"
'
'
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1
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Photo-conductance signal
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.
.
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.
9 - 8
L
w 7 w m 7 3 6 c
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Laser pulse (in arbitrary units)
0
50
100
150
200
250
300
350
400
Time (ns)
FIG.3. (a) Reflectivity, transmissivity, and electric conductance signals at a laser fluence slightly above the melting threshold; (b) electric conductance signal at the fluence, F = 0.32 J/cm2 ([114], reproduced with permission from ASME).
when the p-Si is melted. The transition signal decreases when temperature increases and drops to zero when the sample is melted. The electric voltage signal has a photoconductance-induced drop between 0 and 10 ns. The second voltage drop at about 30 ns is due to the melting of the sample surface. In Fig. 3b, the laser fluence is much higher. The initial voltage drop caused by photoconductance lasts about 10 ns. The second voltage drop, which is much larger than the photoconductance signal, is converted to the melt front depth. It is clear that the photoconductance signal is completely separated from the electric signals caused by phase change. Once the p-Si film starts to melt, the total resistance of the film should be
84
COSTAS P. GRIGOROPOULOS et al.
calculated as a liquid film in parallel to a solid film. But since the resistivity of liquid silicon is much lower than that of solid silicon, the resistance of the solid silicon can be neglected. From Fig. 2, the resistance of the film SZ, and the melt depth, S,, can be calculated as
where 1 and w are respectively the length and width of the silicon strip, as shown in Fig. 2. The temperature dependence of the liquid silicon conducis given by Glazov ef al. [36]. tivity, q,, Figure 4 shows the maximum melting depths at different laser fluences. The melting threshold, determined by both the experiment and calculation, is 0.16 J/cm2. This value is significantly smaller than the melting threshold of undoped silicon. This is because the reflectance of the Au-doped sample at A = 248 nm (0.25) is much lower than that of undoped silicon (0.7 for crystal silicon). At higher fluences, the measured maximum melting depth agrees with the calculated melting depth. Figure 5a shows the melting duration determined by conductance measurement and numerical simulation. The long-lasting “tail” of melting in the transient voltage signal could indicate undercooling, as suggested by Palmer and Mariner0 [76]. At high fluence ( F > 0.4 J/cm2) the measured melting duration is longer than that of the calculated results. This can also be due
120
-E
100
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Laser Fluence (J/cm2) FIG.4. Comparison between measured and calculated maximum melting depth at different laser fluences ([1141,reproduced with permission from ASME).
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
a
.... ....,....
m t. . . .
50
0
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,...
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,...
85
. , , ,
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0.1
, ,
,
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0.4
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Laser Fluence (J/cm2)
b
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..,,,,...,,,,.
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c
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3
D
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Laser Fluence (J/cmZ)
FIG.5. Comparison between measured and calculated melting duration at different laser fluences. The measured results were determined from (a) electric conductance experiment and (b) transmissivity experiment ([114], reproduced with permission from ASME).
to the liquid undercooling effect. Melting duration can also be determined from transmissivity measurement (Fig. 5b). The melting duration is taken as the time period when the transmissivity stays at zero. This gives a lower value of the melting duration because the penetration depth, dab, of the liquid silicon at the HeNe laser light wavelength is about 10 nm [91]. When the liquid silicon solidifies to a thickness of less than dab,the transmissivity starts to rise. The error due to this fact is estimated to be 10%. Figure 6 compares the transient melting depths with the numerical results at three different laser fluences. The maximum melting depth and the melt duration agree well with the numerical results. The melting and resolidification velocity can be determined from the slope of the melting
86
COSTAS P. GRIGOROPOULOS er al.
Q
p
30
----- Measurement Calculation
o
1
lo 0 0
50
100
150
2w
m
300
350
400
b
-
I
E
c
C
Time (ns) FIG.6. Comparison between measured and calculated melting depth for different laser fluences: (a) F = 0.25 J/cm*, (b) F = 0.30 J/cm2, (c) F = 0.35 J/cm* (11141, reproduced with permission from ASME).
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
87
front. The maximum resolidification speed determined from Fig. 6 is higher than that of the numerical results. The resolidification speed gradually fades to zero, causing a longer solidification process. It is also apparent that the discrepancies between theoretical predictions and experimental results increase at higher fluences. Meanwhile, the calculated solidification speed is almost constant during the entire solidification process. The constants can vary due to different microstructure of the material. The effect of different overheating levels on the melting depth and melting duration was examined quantitatively. The transient melting front position at different overheating degrees is shown in Fig. 7a. The coefficient C, was varied form 0 (no overheating) to 20 K/(m/s). The predicted melting depths show no significant dependence on overheating levels. The melting durations differ by only about 20 ns from strong overheating to no overheating, whereas the experimental accuracy on determining the melting duration exceeds 20 ns (Fig. 5a, b). On the other hand, the maximum surface temperature rise is higher when there is strong overheating (Fig. 7b). 2. Pyrometry Experiment Another recent experiment by Xu et al. [113, 1151 has probed the nature of the rapid-phase-change phenomena in undoped p-Si thin films. Surface temperature information is obtained by multiwavelength near-infrared (IR) pyrometry (Fig. 8). The desired temperature range is between 1500 and 3500 K. In this temperature range, thermal emission is strongest at wavelengths between 1 and 2 pm. The major difficulties in fast spectral thermal emission measurements arises from the low signal-to-noise ratio (SNR). Large solid angles are therefore required to achieve sufficient energy collection efficiency. Bandpass filters in the wavelength range from 1.1 to 1.7 p m are used to resolve spectral thermal emission signals at four different wavelengths (1.2, 1.4, 1.5, and 1.6 p m ) to enhance the measurement accuracy. To measure the temperature of the solid-liquid interface during laser melting of the p-Si sample, thermal emission is collected from the back side of the sample, as shown in Fig. 9a. In the wavelength range between 1.1 and 1.7 pm, both the solid silicon and the quartz substrate are transparent, so that their emissivity is zero according to Kirchhoff's law. The thermal emission from the solid silicon film and the quartz substrate in this wavelength range is insignificant. However, when the surface of the p-Si film is melted by the pulsed laser beam, the thermal emission from liquid silicon is detectable, since the liquid silicon has an emissivity around
88
COSTAS P. GRIGOROPOULOS et a1
a
60 50
r, a
I------Cl-O
40
: 30 rn C
E
z"
20 10
n 0
80
40
120
160
209
80
100
Time (ns)
E
E 1710
I-
1690
1650
G
-
0
20
-
40
60
Time (ns)
FIG.7. (a) Calculated melting depth at different superheating levels, F = 0.35 J/cm*; (b) calculated surface temperature at different superheating levels, F = 0.35 J/cm2 ([114], reproduced with permission from ASME).
0.28 in the near-IR. Thus the measured thermal emission signal is ascribed to liquid silicon. At near-IR wavelengths, the radiation absorption depth in liquid silicon is less than 18 nm. Therefore, the thermal radiation is emitted from a thin liquid silicon layer just behind the solid-liquid interface. The temperature derived from the measured thermal emission is close to the solid-liquid interface temperature. To obtain temperature information from the measured thermal emission, the temperature dependence of the spectral emissivity is required. The emissivity is generally a function of temperature and wavelength. The surface conditions may alter the emissivity drastically. The emissivity of a real sample surface can be much different from the reported literature
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
89
KrF excimer laser
Triggering detector
I
0 ’
Power meter
Oscilloscope
0 ’ I 1
Ge diode
FIG.8. Experimental setup for transient thermal emission measurement [115].
data for “ideal” surfaces under strictly controlled experimental conditions. In this work, the emissivity of the sample is independently measured. Figure 9b illustrates the experimental setup for measurement of the transient reflectivity during pulsed excimer laser heating. The temporal evolution of the emissivity can be obtained from transient reflectivity measurement. In the transient reflectivity measurement, a quartz halogen (QH) lamp is used as the light source. The derivation of temperature from thermal emission measurement is based on Planck’s blackbody radiation intensity distribution law: 2TC, eAb =
As exp( C,/AT) - 1 ’
where eAbis blackbody emissive power, A is wavelength, T is temperature, and C, and C, are blackbody radiation constants. The strong UV radia-
90
COSTAS P. GRIGOROPOULOS et al.
Excimer Laser
a
b
Concave mirror
@ ::
Pulsed laser beam
Lenses
Monochromator Ge diode
IR mirror
Light source Lenses
I
I
I
!
Sample FIG.9. Experimental arrangements for (a) interfacial temperature measurement; (b) in situ spectral reflectivity measurement [115].
tion of the excimer laser could trigger the following radiative mechanisms: (1) fluorescence from the optical components and (2) stimulated emission from the sample itself. The effect of the fluorescence in the experiment has been examined and found to be negligible. On the other hand, the stimulated emission from silicon has spectral lines in the UV and visible ranges, well outside the near-IR measurement band.
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
91
The detector collects thermal radiation through a solid angle ( 8 , to 8,, to A,). In fact, the voltage signal recorded on the oscilloscope, V , represents an integration of the thermal emission in this solid angle and wavelength band, modified by the emissivity of the material, the transmission of the optical components, and the detector spectral response:
dl to 4,) and over a wavelength bandwidth (A,
where R is the impedance of the oscilloscope (50 R), ~ ( h is) the spectral transmittance of the lenses and filters in the optical path, D(A) is the responsivity of the germanium diode at different wavelengths [in units of amperes per watt (A/W)]; &;(A, 8, 4, T ) is the directional spectral emissivity, and dA is the area on the sample that is sensed by the germanium diode. The emissivity is deduced from transient reflectivity measurement. Invoking Kirchhoffs law, the spectral directional emissivity is equal to the spectral directional absorptivity:
4( A, 0 , 4 , T ) = 4( A, 0 , 4 ,T ).
(26)
For an opaque, nontransmitting material (such as liquid silicon thicker 40 nm), the directional spectral absorptivity can be expressed as than N
e , + , T ) = 1 - R;( A,
-o,+,
T ) - R ~ A,( (27) where R:(A, -8,+, T ) is the specular reflectivity and &(A, T ) is the diffuse reflectivity. The measured root-mean-square (RMS) su$ace roughness of the samples used in the experiment was within 50 A, implying optical smoothness. Also, the melting interface is assumed to be optically smooth. Consequently, it can be argued that the diffuse reflection from the sample surface or the melting interface is negligible compared with the specular reflection. In this case, the emissivity can be expressed as a;( A,
1-R;(A,
-e,4,T).
(28) The emissivity is derived using the measured transient reflectivity data in Eq. (28). Once the emissivity is determined and the thermal radiation emission of the target material is measured, the temperature can be obtained by solving Eq. (25). In addition, the effect of a temperature gradient at the solid-liquid interface is considered in deducing the interface temperature. The measured thermal radiation is emitted from a near-interface liquid layer whose thickness is only a few optical absorption depths. Since there exists a falling temperature gradient in the liquid layer toward the melting interface, the temperature assigned to the measured E ; ( A , ~ , ~ , T )=
92
COSTAS P. GRIGOROPOULOS
el
al.
thermal emission exceeds the interface temperature. Defining the temperature calculated from the thermal emission spectrum as the “effective” temperature, the relation between this effective temperature, T,,, , and the temperature distribution inside the liquid film, T ( x ) , is approximated by
X
exp(
-2)
dd,dOdAdx&l.
The left side of Eq. (29) has the form established in Eq. (25). The right side is the measured thermal emission that originates from a liquid layer of thickness equal to a number ( N = 4) of optical absorption depths ( d a b ) . The x coordinate has its origin at the solid-liquid interface, and is directed into the liquid layer. At the highest laser fluence of 0.95 J/cm2, the temperature gradient at the interface (about 3.3 K/nm) leads to an overestimation of the measured “effective” temperature by approximately 40 K with respect to the actual interface temperature. It must be pointed out that Eq. (29) does not represent a rigorous solution of the radiative transfer in the skin metal layer, but rather constitutes an approximation to the contribution of the temperature gradient to the aggregate thermal emission. By the same token, in the case of surface temperature measurement, the experimental signal overestimates the actual surface temperature and the same correction procedure is followed. The issue of the internal thermal gradient contributions to the thermal emission warrants further attention, particularly when steep temperature profiles are encountered. Figure 10 shows the transient thermal A = 1.5 p m emission signals measured from the back side of the p-Si/quartz sample at A = 1.5 p m for several laser fluences. The thermal emission (and thus the interface temperature) reaches the maximum value several nanoseconds after the initiation of melting. The thermal emission measurement also yields the melting duration, since only liquid silicon emits light in the wavelength range between 1.2 and 1.6 pm. The length of the arrows in Fig. 10 indicates the duration of the melting process at each laser fluence. One interesting feature of the thermal emission signals is the absence of a plateau that would be expected for nearly constant interfacial temperatures. No attempt has been made to extract the transient interfacial temperature; however, it appears that the recrystallization may be driven
PULSED USER-INDUCED PHASE TRANSFORMATIONS
7 1
'
1 ' S ' (22
" " " " " I '
"
J
5 '
-
F = 0.55 Jlcrn'
3 -
-
93
:
1 -
a
:
c n l -
-100
0
100
200
300
400
500
Time (ns)
FIG.10. Thermal emission signals from the back side of the p-Si film at the wavelength A = 1.5 pm. The p-.%/quartz sample is heated by excimer laser pulses of different fluences
([ 1131, reproduced with permission from AIP).
by considerable supercooling. Comparing the thermal emission signals at F = 0.55 and 0.65 J/cm2, it can be seen that the maximum interface temperature increases with the laser fluence. However, this trend is not observed for laser fluences higher than 0.65 J/cm2. The corresponding maximum interface temperatures at different laser fluences are calculated using thermal emission signals at four wavelengths. In addition, the effect of a temperature gradient at the solid-liquid interface is considered in deducing the effective interface temperature. The resulting maximum solid-liquid interface temperatures are shown in Fig. l l a . In Fig. l l b , the melting duration obtained from the thermal emission measurement is compared with the numerical simulation results. Surface optical reflectance probing is employed to independently confirm the melting duration measurement and also examine the possibility that
94
COSTAS P. GRIGOROPOULOS et al.
temperature, 1685 K
m 5 0.0 . ' 0.5
' ' '
1
' ' ' ' 1 ' ' ' '
'
' ' ' '
'
' ' '
0.6 0.7 0.8 0.9 Laser Fluence (J/cm*)
1
1
FIG. 11. (a) Maximum melting temperature of the p-Si/quartz sample irradiated by excimer laser pulses of different fluences; (b) comparison between measured and calculated melt durations of the p-Si/quartz sample irradiated by excimer laser pulses of different fluences; (c) calculated maximum melting velocity of the p-%/quartz sample irradiated by excimer laser pulses of different fluences; (d) calculated maximum melting depth of the p-Si/quartz sample irradiated by excimer laser pulses of different fluences. ([1131, reproduced with permission from AIP).
high-temperuturesolid silicon can contribute to the thermal emission. The surface reflectivity measurement yields the melting duration because of the large increase of silicon reflectance on melting. The surface reflectivity and the back-side thermal emission measurement results are in close agreement. Considering that any thermal emission from high-temperature solid silicon would have to last much longer than the melted phase, the possibility that the back-side thermal radiation is emitted from hightemperature solid silicon is safely dismissed. Figures l l c , d show that the calculated maximum melt front velocities and maximum melting depths increase with the excimer laser fluence. Numerical calculation also predicted that the maximum surface temperature does not reach the evapora-
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
95
tion temperature (2628 K) at the highest laser fluence used in the experiment. The relation between the interface velocity and the interface superheating temperature allows determination of the interface response function during the laser-induced melting process. The melting velocity achieved is on the order of tens of meters per second, suggesting a departure from equilibrium conditions. The measured interface superheating temperature combined with the calculated interface velocity provides an experimental study of the interface kinetics under high-power laser irradiation. Assuming that there is a linear relation between the interface superheating temperature and the interface velocity at fluences lower than 0.65 J/cm2 (more properly, when the interface velocity is below 20 m/s), the response function coefficient C , is determined to be about 6 K/(m/s). However, a constant interface temperature is measured at laser fluences higher than 0.65 J/cm2. The linear interface response function is therefore invalid in the high-laser-fluence regime. When the interface velocity is higher than 20 m/s, the interface superheating temperature is “saturated” at about 110 K. The measured transient surface temperature of a bulk-Si sample irradiated by an excimer laser pulse at the fluence, F = 0.9 J/cm2, is shown in Fig. 12. Also shown in this figure is the calculated surface temperature at the same laser fluence. The effect of superheating is illustrated in the calculation. The material constant C3 is taken as 6 K/(m/s), which is the measured value for the polycrystalline silicon samples. For comparison, the calculation results without considering superheating are also shown in Fig. 12. Compared with the calculated results, the measured maximum surface temperature is higher, and a longer time is needed to reach the 2500 , , . , I , , ~ , I , , , , I , . , , I , , ~ , , , , , , l , , , ,
End of melting :
Equilibriummelting : temperature, 1685 KConsidering superheating
0
10
20
30 40 lime (ns)
50
60
70
FIG. 12. Measured and calculated surface temperature histories during melting of a bulk, crystalline Si sample with an excimer laser pulse of fluence F = 0.9 J/cm2.
96
COSTAS P. GRIGOROPOULOS et al.
highest surface temperature. The discrepancy could result from a higher superheating surface temperature for single-crystal Si (c-Si). It is worth noting that Boneberg et al. [12] examined the variation of the optical reflectivity of a c-Si wafer during irradiation with two successive frequency-doubled neodymium: yttrium aluminum garnet (Nd: YAG) ( A = 532 nm) laser pulses of nanosecond duration. The first pulse melted the surface, increasing the reflection coefficient up to the value of the metallic liquid silicon. On further heating of the surface with a second, timedelayed pulse, a decrease of the reflection coefficient was observed by up to 9% at the probing HeNe wavelength ( A = 633 nm), resulting from the temperature-dependent dielectric function of molten silicon. Knowledge of the temperature field is essential for studying in detail the phase transformation mechanisms of pulsed-laser-induced amorphization and recrystallization of silicon films on SiO, substrates [14, 42, 851. The critical issue is the degree of supercooling driving the rapid solidification.
D. ULTRASHALLOWJUNCTION FORMATION IN SILICON BY EXCIMER LASERDOPING
Control of the temperature field and the melt depth can lead to formation of box-shaped ultrashallow doping profiles in semiconductors. Excimer-laser-driven doping from the gas phase has been demonstrated by Landi et al. [63] and Matsumoto et al. [68]. Promising results have been reported recently using a simple excimer-laser-induced method from a spin-coated dopant solid glass layer [117].oThec-Si wafer is coated with a boron-doped spin-on-glass layer of 3000-A thickness. Rapid boron diffusion in the liquid silicon is achieved during melting; however, on the contrary, the diffusion is limited by the phase boundary during the solidification process. In order to understand the heat and the mass transfer in the ultrashallow p+-junction formation under the excimer laser irradiation on the silicon wafer with a thin solid diffusion oxide film, a one-dimensional numerical analysis is developed to simulate the transient heat and mass diffusion of the boron atoms across the thin molten silicon layer. As a first-order approximation, the thermal properties are independent of the concentration and the mass diffusivity is independent of the temperature. Therefore, the mass diffusion is decoupled from the thermal equations. The heat transfer is treated using the enthalpy scheme discussed in Section I.B. The mass diffusion is solved numerically at each step after the temperature field and the melt-solid interface location are found. Transient boron mass diffusion in this work is modeled by the one-dimensional
PULSED USER-INDUCED PHASE TRANSFORMATIONS
97
Fick’s equation: dt
dx
The boundary conditions and the initial condition for the mass transport are given: C ( x = O , t ) =Co,
C(x=d,,,t) =0,
C(x,t=O) =O,
(31)
where the surface boron concentration C , is chosen as the experimental cm2/s in value. The boron diffusivity D is set as a constant 2.3 X liquid silicon [58] and zero is solid, due to the six orders of magnitude difference between the mass diffusivities in the two phases. The boron dopant concentration profiles are measured by secondary-ion mass spectrometry (SIMS) [106]. The profile obtained at a fixed excimer laser pulse fluence of 700 mJ/cm2 are shown in Fig. 13; p+ junctions with depth of 70 nm to 140 nm are successfully fabricated, and the boron concentration as high as 2 X 10’’ atoms/cm3 is obtained. This value is about the boron solubility in crystalline silicon. Such a high doping level is due mainly to the high boron mass diffusivity in the thin liquid silicon layer induced
7
10160
40
80
120
160
200
Depth (nm)
FIG. 13. Ultrashallow p+-junction dopant concentration depth profiles at different laser pulse numbers N with a fixed laser fluence of 0.7 J/cm2. The solid dots represent experimental data measured by the second ion mass spectroscopy (SIMS) and the lines are numerical results [117].
98
COSTAS P. GRIGOROPOULOS et al.
by the pulsed laser irradiation. The diffusion profiles are driven deeper into the thin molten silicon as the laser pulse number increases, although the highest concentration limit is the boron solubility. It is interesting that, as the laser pulse number increases, the dopant profile shape is more “box-like” rather then the gradual decrease observed in most diffusion cases. The abrupt or box-like dopant profiles render the ideal p+-junction properties [105]. The art in the information of this box-like shape lies on the fact that, as the pulse number increases, the boron diffusion is limited by the maximum melting depth determined by the pulsed laser energy and pulse width. After the boron diffusion reaches the maximum melting depth, more boron atoms accumulate in the molten layer instead of crossing the melt-solid interface, because of the very low boron diffusivity in the solid silicon. At 20 laser pulses, almost the entire thin molten layer of 140-nm thickness is saturated with boron dopant. It is also demonstrated that the p+-junction depth can be incrementally changed by varying the laser pulse number at a fixed laser fluence. Numerical simulation based on the one-dimensional transient thermal and mass diffusion model predicts similar p+-junction profiles. Both the experimental and numerical dopant profiles at larger pulse numbers coincide to a certain depth about 140 nm, the maximum melting depth. The computational results fit the experimental profile well for the first laser pulse, but deviate substantially as the pulse number N increases. Numerically predicted dopant concentrations are generally lower than the experimental ones. It is believed that the boron diffusivity may depend on the boron concentration if the region is heavily doped, compared with the silicon intrinsic carrier of the order of 10’’ cm-3 at the melting temperature [107]. For the first few pulses, the heavily doped region is so small that the effect from the concentration dependence of the boron diffusivity is not significant. Boron diffusivity can be treated as a constant in the computations. As the pulse number increases, a larger fraction of the molten region becomes heavily doped, leading to a change in the boron diffusivity. A larger boron diffusivity could result in higher computed concentration for N = 10 and N = 20 in Fig. 13. There are other factors possibly causing the differences between the experimental profiles and numerical profiles at higher pulse numbers. The measured laser fluence is averaged over the pulses for the case of multipulse laser irradiation. Pulse-to-pulse fluence fluctuations generated by the excimer laser can be as large as 10%. The fluctuations can result in not only changes in the thermal process but also in the mass diffusion. The final dopant profiles are affected more strongly by the higher fluence pulses than the lower ones during the multipulse doping. Second, the microstructure in the resolidified thin layer is also of importance during the multipulse laser irradiation. Assuming epitaxial recrystallization in the
99
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
thin liquid silicon layer, the numerical simulation in this work may not be accurate enough to predict the thermal transport because of possible changes in physical properties such as the thermal diffusivity in the recrystallized silicon. Finally, the thermal conductivity of solid silicon layer can be decreased by a factor of 2-5 in the temperature range from room temperature to 600 K because of the heavy doping with boron [95, 981. At higher temperature, the thermal conductivity becomes less affected by the high dopant concentration. The decrease in thermal conductivity can lead to deeper and longer melting in silicon. The 10% increase in the liquid silicon thermal conductivity from 1685 to 1980 K during the pulsed laser doping may also influence the transient thermal and mass transport [ l l l ] . Figure 14 shows boron dopant depth profiles for different laser fluences at a fixed pulse number N = 20. It is demonstrated that the p+ junctions are box-like for 0.6, 0.7, and 0.8 J/cm2, but not for 0.9 J/cm2. The difference can be explained by the maximum melting depth. For the fixed pulse number N = 20, the boron diffusion depths already exceed the maximum melting depth at the laser fluences 0.6, 0.7, and 0.8 J/cm2. The accumulation of the dopant atoms is blocked by the liquid-solid interface
0.9 J/crn2 0.8 J/crn2 0.7 J/crn2 0.6 J/crn2 -0.9 J/crn2 - 0.8J/cm2 - . -0.7 J/cm2 -. 0.6 Jcm2 A
.
L ’
1016
0
‘
100
200
b
.
300
400
500
600
Depth (nrn) FIG. 14. Ultrashallow p+-junction dopant concentration depth profiles at different laser fluences with a fixed pulse number N = 20. The solid dots represent experimental data measured by SIMS, and the lines are numerical results [117].
100
COSTAS P. GRIGOROPOULOS el a [ .
and results in the box-like dopant distributions. However, for higher laser fluences such as 0.9 J/cm2, the diffusion length at N = 20 is smaller than the maximum melting length, and therefore, no dopant accumulation occurs. The p+-junction depth increases with the laser fluence because of the increase in both the melting depth and the melting duration. By defining the p+-junction depth dj as the distance from the surface to the position where the concentration drops below 10l8 ~ m - the ~ , dependence of the junction depth versus the laser fluence is shown in Fig. 15 for two different pulse numbers, N = 1 and N = 20. The computed junction depths are plotted for comparison. Gradual saturation of the junction depths with the pulse number N at fixed laser fluences is observed in Fig. 16, implying melt-solid interface-limited diffusion. The ultrashallow p + junctions of 30-nm depth at laser fluence 0.6 J/cm2 are successfully fabricated with this spin-on-glass (SOG) pulsed laser doping technique. It is demonstrated that the incremental depth achieved by varying the pulse number N can be as small as 20-30 nm. However, the incremental depth achieved by varying the laser fluence at an experimentally controllable level (about 0.1 J/cm2) is larger than that obtained by varying the pulse number. This is because the melt depth is mainly dominated by the larger
-1001 ' ' ' '
-0.5
' 0.6
'
'
'
'
I
0.7
'
'
'
'
I
'
'
0.8
'
'
'
0.9
'
'
'
1
Laser Fluence (J/cm2)
FIG. 15. Ultrashallow p+-junction depth dependence on the laser fluence for N = 1 and N = 20. The solid dots indicate experimental data measured by SIMS, and the lines show numerical results [117].
101
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
0
~
-5
~
0
~
5
"
~
10
~
~
15
"
"
20
~
25
'
~
30
~
Laser Pulse Number N
FIG. 16. Ultrashallow p+-junction depth dependence on the laser pulse number for laser fluences 0.7 and 0.8 J/cm2. The solid dots represent experimental data measured by SIMS and the lines, numerical results, [117].
melting depth and melting duration. The p+-junction sharpness is an important factor in the device performance. By defining the relative junction sharpness as dj/Adj, where Adj is the junction transition depth, it is found that the p+-junction sharpness increases rapidly as the pulse number N increases for the lower laser fluences of 0.6, 0.7, and 0.8 J/cm2, as shown in Fig. 17. For the laser fluence of 0.9 J/crn2, the p+-junction sharpness grows much slower than the lower fluence cases because the melting depth is longer than the diffusion depth up to 20 pulses, as mentioned earlier. Again, the melt-solid interface-limited diffusion is confirmed in Fig. 17. The optimal fluence range in pulsed laser SOG doping is therefore positioned at about 0.6-0.8 J/cm2. The transient pulsed laser induced melting and diffusion in silicon is a complex phenomenon. Many factors, especially the physical properties of this silicon layer, can be altered during the multiple laser pulses. It should be pointed out that the boron diffusivity dependence on the concentration and the temperature in the thin liquid silicon layer is not known yet.
E. TOPOGRAPHY FORMATION In the studies of pulsed laser melting of silicon described in the previous two sections, the irradiated surface did not produce pronounced topo-
"
~
~
102
COSTAS P. GRIGOROPOULOS et al.
-v
-4
0
4
8
12
16
20
24
28
Laser Pulse Number N
FIG. 17. Relative p+-junction sharpness dependence on the laser pulse number at laser fluences as 0.6, 0.7, 0.8, 0.9 J/cmZ [117].
graphic features on the completion of the resolidification process. In those experiments, the semiconductor material was subjected to irradiation by a single pulse, or a limited number of pulses. It is known, however, that periodic wave structures may be induced by laser irradiation on the surface of semiconductors, metals, polymers, dielectric materials, and liquids. This phenomenon was observed for a variety of laser beam parameters, ranging from continuous-wave (CW) beams or Q-switched pulses at intensities even below the damage threshold for some materials to picosecond pulses [28] and femtosecond UV pulses [37]. A theory explaining periodic patterns scaling with the incident laser light wavelength based on the concept of radiation “remnants” scattered from irregular surface structures was proposed by Sipe et d.[94] and Young et al. [112]. Other studies attributed the formation of surface structures of periods longer than the incident laser light wavelength to melt flow instabilities. Tokarev and Konov 1991 presented a theoretical study of thermocapillary waves induced by surface shear stress variations in laser melting of metals and semiconductors. It has also been hypoythesized that thermoelectric body forces exerted on melted semiconductors can also be destabilizing [24]. On metals, studies of target topography have shown formation of columnar structures [61, 62, 711, which have been observed to roughly align coaxially with the incident laser beam. The interrelation between melting, topography formation, and
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
103
sputtering mechanisms in excimer laser melting of gold was examined by Bennett et al. [9]. Gold was selected as a model system, because it does not introduce oxidation, which would alter the surface characteristics. Figure 18a-d show the influence of the laser beam angle of incidence on steady-state surface topography on a gold target. The surface droplet formations are seen to increase as the laser beam angle of incidence decreases. There is very little ridge formation at angles close to O”, but as the laser beam angle of incidence increases, ridge formation evolves-first almost in a parallel pattern, then becoming more chaotic at larger angles of incidence. The laser fluence was maintained at 1 J/cm2 in this sequence, and topographies represent steady-state conditions. Figure 19 shows in more detail under higher magnification the surface feature formations for a 5” angle of laser beam incidence. This micrograph shows that the nearly spherical droplets have a radius of 1 p m and are arranged in a regular square pattern with a characteristic spacing of about 4 pm. In addition, a visual examination of the ablation surface showed that as the laser fluence increased, so did the presence of surface topographies. At least four important effects of surface topography on the ablation process can be recognized: (1) the roughening of the surface provides greater surface area for molecular flux leaving the surface, (2) roughened surface topography supports surface droplet formation, (3) surface roughening will enhance the net laser to surface energy coupling by introducing multiple reflections, and (4) the surface topography may enhance local radiative heating, for example, in “valley” formations, due to surface reflections. Two possible mechanisms affecting topographic development are instability waves in the transiently existing thin melt and locally enhanced radiation heating and/or shadowing coupled to thermocapillary forces. In a laser ablation process, the formation and hydrodynamic removal of large droplets increases the net material removal rate. However, the introduction of droplets into the ablation plume becomes undesirable when sputtering thin films, and is one of the principal disadvantages in using laser ablation for deposition of thin metal films. Past research has established connections between the droplet density and such factors as surface topography [70], target material density [87], laser fluence and laser intensity homogeneity [SS], laser angle of incidence [loll, and incident radiation wavelength [59]. Van de Riet et al. [loll demonstrated a 10 fold reduction, by percent volume, in droplet production, accompanied by a reduction in droplet size, when the ablation target was scanned continuously to provide a virgin surface for the ablation process. Additionally, they showed that the volume percent of droplet production decreased, surprisingly, as the fluence increased for a scanning ablation process.
104
COSTAS P. GRIGOROPOULOS el al.
FIG. 18. Scanning electron micrographs showing the influence of the laser beam angle of incidence on the steady-state surface topography at a constant excimer laser fluence F = 1 J/cm2: (a) 8, = 5" produces an abundance of droplets with reduced ridge formations; (b) 8, = 15" produces highly columnar ridge formations; (c) for Oi = 25", a reduction in the number of droplets and loss of ridge columns is observed; (d) 8, = 45" produces higher surface agitation with reduced number of droplets ([9], reproduced with permission from AIP).
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
105
FIG.19. High-magnification scanning electron micrograph image of surface droplets shown in Fig. 18a ( F = 1 J/cm*, Oi = 5”) demonstrating characteristic length scales for droplet growth ([9], reproduced with permission from AIP).
Some of the characteristics of the ablation plume, including droplet production, were investigated by sputtering a thin film. The measurement of the thin-film thickness distribution determined that the ablation plume peaks at 12” off-axis from the normal and in the direction of the incident laser beam. The surface topography is indeed responsible for “deflecting” the plume. However, the deflection simply reflects a net change in the mean normal direction of surface area contributing to the ablation flux. The angular yield distribution is found to be highly forward-peaking, approximately fitting a C O S ’ ~8’ curve where the angle 8’ is measured from the centerline of the plume instead of the normal to the target surface. It has been suggested that strong forward-peaking plumes could indicate that
106
COSTAS P. GRIGOROPOULOS et al.
the molecular flux from the ablation surface is not liberated as “freely” as the classic evaporation model assumes [131. The percentage of material in the ablation plume made up of large droplets was also estimated. Kelly and Rathenberg [51] suggested that the majority of metallic material removed by laser ablation detaches hydrodynamically as droplets. However, from a survey of SEM pictures of the sputtered gold film it was recently concluded [91 that less than 10% of the bulk film deposited consisted of micrometer-size droplets. This study of the gold film deposition revealed that droplets around 1 p m in diameter were most preferentially liberated from the surface. Also, no significant change in percent volume of droplet present on the deposition was found as the film was surveyed radially outward from the peak. In order to develop a comprehensive description of the laser sputtering, some idea of the transient thermal conditions of the target is required. A one-dimensional model of the transient heat transfer with phase change in a gold target has been formulated [91 using the previously discussed enthalpy model. As argued before, the laser beam spot size is two orders of magnitude greater than the thermal penetration depth, suggesting that the surface conduction of heat is one-dimensional. Nevertheless, the onedimensional modeling of the surface ablation process is recognized as a simplification of surface conditions that arise over the course of many irradiation pulses. Despite the flat initial condition of the ablation target, experimental observations demonstrate topographic development of the target over several hundred laser pulses, which diminishes the assumed one-dimensional geometry. Consequently, the development of onedimensional effects in the heat conduction and the local surface irradiation represent the most significant shortcoming of the model. However, the one-dimensional model provides sufficient insight into the thermal condition of the target to warrant its consideration. Results of the numerical model are presented in Fig. 20 for the laser fluence F = 1 J/cm2. Of specific concern are the thermal penetration depth in the target solid S,, the melting depth&,, the melting duration T ~ and acceleration of the liquid-solid interface in the target qnt.During most of the period in which the surface melt exists, the acceleration of the liquid/solid interface is found to be laintl= 4.6 X lo9 m/s2, and is in the direction pointing outward normal to the target surface. Inertial forces acting on the melt are responsible for development of surface topography. Two effects lead to acceleration of the surface melt. The first effect is a direct consequence of phase change, when the densities of the liquid and solid are not equal. The second effect arises from the thermal expansion of the target material. The time scale for mechanical
,
107
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
-E L
U
L
: -
8
0 10
0
20
1 30
1 40
Time Ins)
d, 3000 i5.
1
i k,
-E
F=1.0Jlcm2
!!
2000
5
1000
c v)
0
8
16 24 Time ( n s )
32
40
0
0
20
40 60 Time ( n s )
80
FIG.20. Numerically calculated thermal conditions of the gold target during sputtering at F = 1 J/cm2: (a) temperature spatial distribution in the target on completion of melt resolidification;(b) surface temperature temporal behavior; (c) melt depth temporal behavior; (d) liquid-solid interface velocity behavior “91, reproduced with permission from AIP).
expansion of the solid target, T,,,,,, is evaluated with respect to the thermal penetration depth ( S , ( - 5 pm) and the propagation velocity of dilatational waves u, (3200 m/s) in the target solid. It is determined that T~~~~ is an order of magnitude smaller than the melting duration T ~ : ST Tmech =
“e
-
2 ns < r,,, = 31 ns.
From this observation, it is arguable that, due to the relatively short time scale of expansion, the target expands nearly to the extent it would if thermal conditions occurred “quasi-statically.” Within this approximation, it follows that thermal expansion is prescribed by a knowledge of the temperature field (and the linear expansion coefficient) in the target
108
COSTAS P. GRIGOROPOULOS er al.
material. It is therefore asserted that
represents an order-of-magnitude estimate of the acceleration experienced by the melt due to thermal expansion of the target solid. In Eq. (331, ATp (520 K) represents an average rise in temperature over the thermal penetration depth 8, (5 p m ) occurring within the time scale of the melt duration T,,, (31 ns). In ascertaining the melt acceleration due to phase change, it will be assumed that the acceleration discontinuity at the interface is “communicated” rapidly throughout the liquid (2500 m/s) with nearly the same velocity as through the solid. Melt acceleration due to phase change is related to the liquid/solid density ratio ( p s / p , = 1.055) at the melting temperature, as well as the local acceleration of the liquid-solid interface, aint. By invoking mass conservation across the liquid-solid interface, we can estimate the melt acceleration due to phase change as
Since acceleration of the melt resulting from thermal expansion is an order of magnitude smaller than that resulting from phase change, the former can be neglected. Therefore it is estimated that the melt is accelerated toward the target with an acceleration of the order of magnitude la,,,(
- 2.5 x lo8 m/s2.
(35)
The spatial distribution between surface droplets demonstrated for normal incident irradiation is governed by instability at the melt surface. Because of the change in density occurring during phase change, the acceleration of the liquid-solid interface (during both melting and resolidification) accelerates the surface melt and imposes an inertial force from the body of the melt. Since acceleration of the melt is into the target, the apparent body force experienced by the melt is directed away from the target. The inertial force is destabilizing the melt surface. Surface tension, however, is stabilizing and exerts a force that attempts to maintain a flat surface. The instability of an interface between two phases is a typical problem that has been treated for more general conditions than will be required here [17]. The instability arising from a motionless liquid overlaying a motionless vapor, in the presence of a gravity field, is known as a Rayleigh-Taylor instability. The instability experienced by the melt is also Rayleigh-Taylor, since in the reference frame of the melt, the inertial force due to acceleration acts as a gravitational body force.
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
109
The initially flat interface is subjected to spatially periodic disturbances. The wavelength of the disturbance that grows most rapidly is termed “the most dangerous wavelength.” For the conditions of the surface melt, it is found that the most dangerous wavelength is
which agrees remarkably well with the periodicity demonstrated by droplet growth on the target surface, as seen in Fig. 19. If on the other hand, the relative magnitudes of liquid and solid densities are reversed (i.e., ps < p,), the melt would experience a restraining body force back into the target, and there would be no destabilizing event to sponsor topographic growth. Silicon has a liquid density greater than the solid and produces no topographies comparable to the gold target on irradiation with excimer laser pulses. To estimate the requirement leading to the formation of a droplet than can ultimately be liberated from the surface, the forming droplet is modeled as a hemispheric dome. If the inertial force acting on the hemisphere is greater than the restraining force, the droplet will grow. This condition implies that
This dimension is higher than the diameter of droplets seen on the surface of the target. This discrepancy may indicate that the melt experiences an acceleration even greater than originally calculated. The agreement obtained by invoking this simple stability analysis is remarkable, but has to be viewed considering the assumptions made: perfect initial surface flatness, and no contamination modifying the surface characteristics. It is also conceivable that other modes of instability may appear in the thin melt film. 11. Pulsed Laser Sputtering of Metals
A. BACKGROUND Many mechanisms contributing to the net surface material removal rate have been identified with the laser ablation process. References to “thermal” or “photothermal” surface ablation in the literature generally embrace a model in which, on absorption, laser light energy is converted to lattice vibrational energy before bond breaking liberates material from the
110
COSTAS P. GRIGOROPOULOS
et
al.
bulk surface. The thermal mechanism is distinct from a “photochemical” or “electronic” process in which laser-induced electronic excitations lead directly to bond breaking before an electronic-to-vibrational energy transition has occurred. Both thermal and electronic ablation mechanisms liberate molecular size material from the surface, and either one or the other (or possibly both) are likely to be present to some degree in all laser ablation systems. This is distinct from two other ablation mechanisms, identified in the literature as hydrodynamic ablation and exfoliation, which lead to the introduction of large clusters, fragments, or droplets into the ablation plume. The “hydrodynamic” mechanism refers to the formation and liberation of micrometer-size droplets from the melt at the surface. The term hydrodynamic arises from the belief that bulk motion of the melted material is involved in the formation and ejection of droplets. Exfoliation, on the other hand, refers to an erosion-like mechanism by which bulk material is removed from the surface as an intact solid “flake.” Separation of the flake from the surface is thought to occur along energyabsorbing defects in the material. Kelly and Rothenberg [51] estimated that thermal evaporation could account for less than 1% of the total material removed, and concluded that hydrodynamic sputtering must account for the remainder. Regardless of the presence of hydrodynamic sputtering, first-order principles demand that the accumulation of sufficient thermal energy in the lattice will result in thermal vaporization of atomic material from the surface. The classic modeling of thermal evaporation is developed from a kinetic description that generates a couple of familiar laws: The velocity distribution is Maxwell-Boltzmann, and the angular distribution of the flux is isotropic, following the Knudsen cosine law. However, interpretation of the relative success or failure of the thermal description with respect to experimental data has always been a highly subjective, even controversial topic. Gibert et al. [341 produced a very interesting study of near-threshold sputtering of a Fe target irradiated with a N, laser beam ( A = 337 nm, rP = 10 ns, F = 50-320 mJ/cm2). They found (1) Maxwellian velocity distributions for both Fe’ ions and Fe atoms, (2) a broad agreement between the translational and internal excitation energies of the monatomic plume constituents-indicating thermal equilibrium between these modes of energy storage, and (3) that temperatures corresponding to translational and internal excitation energies were consistent with melting and evaporation of Fe. The principal conclusion of Gibert et al. concerning the mechanism of laser sputtering of Fe was that pulsed laser sputtering of Fe is a thermal phenomenon. The tacit assumption made in the conventional description of evaporation (and sublimation) is that the nascent atomic vapor flux leaves the
PULSED USER-INDUCED PHASE TRANSFORMATIONS
111
surface with the same energy distribution with which a uniform motionless vapor, in thermal equilibrium with the surface phase, would condense to that surface. In short, the vapor phase leaves the surface with a halfMaxwellian velocity distribution. In the literature, the region immediately adjacent to the irradiated surface is called the Knudsen layer. Within this region, interparticle collisions impart the trajectories and energy to the vapor particles recondensing on the surface. Thus, the Knudsen layer is characterized by a nonequilibrium gas kinetics description. The collisions between vapor particles leaving the surface eventually establish thermodynamic equilibrium in the gas and a finite extent to the Knudsen layer. The thickness of the Knudsen layer is generally recognized to be in the order of a few mean-free paths from the surface. Under conditions of large material flux from the surface, the mean-free path between collisions is small, resulting in a very thin Knudsen layer. In the case of a highvacuum environment and low material fluxes, the vapor plume is largely collisionless. Several studies have focused on the role of the Knudsen layer formation in laser vaporization, sputtering, and decomposition. Kelly and Dreyfus [49] showed that under the condition that the particle emission is driven by a thermal mechanism, the ejection velocities at the target surface are described by a half-range Maxwellian distribution. For as few as three collisions per particle, a Knudsen layer forms, which is confined within a few mean-free paths from the solid surface. Within this Knudsen layer, the density distribution function evolves to a full-range Maxwellian in a center of mass coordinate system. Moreover, Kelly and Dreyfus [50]showed that the Knudsen layer formation leads to forward peaking of the kinetic energy distributions of the ejected particles. Angular profiles following c0s4 B-cos'o 8 functions were thus expected, in general agreement with experimental measurements. For evaporation yields exceeding, for instance, half of a monolayer per a 10-ns laser pulse, Kelly [47, 481 showed that the resulting gas-phase intractions cause the Knudsen layer to evolve into an unsteady, adiabatic expansion. An explicit solution for the speed of sound and the gas velocity was obtained, emphasizing the importance of the Mach number and the specific-heat ratio ( A = C , / C , ) in the interpretation of experimental time-of-flight (TOF) measurements. Comparison of the gas dynamics model with numerical solutions of the flow equations and with direct simulations of the Boltzmann equation by Sibold and Urbassek [92] and Knight [57] showed reasonable agreement, despite the inherent assumption of local thermodynamic equilibrium (LTE). Finke et al. [30] and Finke and Simon 1291 treated the steady-state formation of the Knudsen layer numerically by solving the Boltzmann equation using an integral approach. Their solution yielded temperature, mass density, and
112
COSTAS P. GRIGOROPOULOS et a l .
velocity distributions in the Knudsen layer, as well as the decrease of pressure along the ejected vapor stream.
B. TIME-OF-FLIGHT MEASUREMENTS Stritzker et al. [96] and Pospiesczyk et al. [79] used a quadrupole mass spectrometer (QMS) in a TOF arrangement to determine the kinetic energies of evaporated Si and GaAs targets by 20-ns long pulsed ruby laser irradiation. On the basis of these energy distributions, they extracted the temperature of the evaporated atoms by fitting Maxwellian distributions from gas kinetic theory. Furthermore, by assuming that this temperature represented the lattice temperature, they concluded that the process was thermal. It is tempting to conclude that the TOF measurements offer an appealing means of characterizing the thermal conditions of the ejected plume. At this point, it is worthy to recall the basic principles of such measurements. The expection of a Boltzmann energy distribution in TOF measurements can obtain from formal gas dynamics. The number density of vapor particles having velocities between u = (u,, u y,u , ) and u + d u = ( u , + du,, u, + duy,u, + du,) liberated from the surface is dn,(u)
=
d n , ( u ) / u , = n , f ( u ) dux du, d u , ,
(38)
where n, is the total vapor number density at the surface and the Maxwellian velocity distribution function is given by
m
m( u:
+ uy’ + u : ) 2k,T
When the surface flux temporal behavior is approximated by a delta function, the number density velocity distribution becomes spatially resolved as higher velocity particles move farther from the surface than lower velocity particles in a given period of time. If the surface flux is approximated by a point source, and the particle stream is collisionless, the following TOF approximations can be made U, = - x / t uy
=Y/t
u, = z / t
dux = X t C 2 dt, du, = dy/t - y t C 2dt
=
dy/t
du,
=
dy/t(ly/xl << l),
du,
=
dz/t - Z t C 2 dt
du,
=
d z / t ( Iz/xl << l ) ,
=
dz/t
+ (Y / X ) dux, + (Z / X )
dux,
PULSED USER-INDUCED PHASE TRANSFORMATIONS
113
where x , y , and z are spatial coordinates and t is the time of flight. Since the number density of particles sweeping by the small area of the particle detector is of interest, it may be assumed that ly/xl << 1, Iz/xl -=K 1 and that Iuyl, Iu,I -=K Iu,I at the detector. Consequently, it may be anticipated that the detector signal is
N d ( t ) dt
=
/ / [ n , f ( u ) x t - 4 d t ] dydz,
(40)
which, when integrated over the surface area of the detector, A , yields
N d ( t ) dt
= A,nsf(u)xtC4d t ,
recalling that at the detector u: >> u i , uf and u,
&( f )
= A,n,(
(41)
=x/t,
m/2TkBT)3’2Xt-4eXp{ - ( m / 2 k ~ T )X (/ f ) ’ )
or
The measured density signal, N,(t), can be converted into the physically more meaningful flux distribution, as a function of the translational energy P( E ) using the following relation:
P ( E ) dE Recognizing that ( d E / d t )
=
u,N,(t) dt.
(43)
- t - 3 and - t - ’ , we can express (43) as P(E) (44) u,
t2N(t).
The translational energy is related to the Aight time as
E
( m / 2 ) (x / t ) 2 . (45) Combining Eqs. (42), (441, and (45), the TOF signal inverted into energy space is expected to be Boltzmann: =
i
P ( E ) - E e x p -k:Ti In the limit of a small surface vapor flux, the vapor expansion into vacuum is collisionless and the mean translational energy should be indicative of the surface phase temperature, E = 2k,T. The characteristics of the translation energy distribution in the ablation plume for a gold target have been explored [9] using the QMS T O F system describe< in [901. In that study, the material removal rates were kept small ( I10 A per pulse), and it was verified that the laser intensity was not sufficient to ignite plasma effects in the vapor plume. Figures 21a-d demonstrate the effect of laser
114
COSTAS P. GRIGOROPOULOS et al.
0
10
5
15
E (eV)
FIG.21. Effect of laser fluence on the translational energy distribution of neutral Au atoms 45", 0, = 0"). The circles correspond to the inverted TOF signal; the solid line represents a Boltzmann distribution having the same mean energy as the inverted TOF signal ([9], reproduced with permission from AIP).
(Oi =
fluence on the kinetic energy distribution in the plume. The laser fluence has been varied from a near-threshold fluence ( = 0.68 J/cm2) to about 1.0 J/cm2 over the course of this experiment. For this series of measurements, the detector was located normal to the surface and centered over the ablation area, and the laser beam was incident at 45" from the surface normal. Depicted in Fig. 21 are inverted TOF measurements in comparison with the theoretically predicted Boltzmann distribution. The first observation to be made from panels a-d (Fig. 21) is the apparent success
PULSED LASER-INDUCED PHASE TRANSFORMATTONS
115
of the Boltzmann distribution in describing the energy distribution in the plume. The second finding is that the measured mean kinetic energies (up to several electronvolts) correspond to temperature far exceeding the thermal expectations (1 eV 11,620 K). The meaning of the “temperature” derived from the mean translational energy is further explored. It was shown before that when near surface collisions are rare, the density sensitive TOF signal should vary as
-
&(t)
1
=
t4 exp( - r 2 X 2 / t 2 } ,
(47)
where r2= m/(2kBT). However, with the development of a stream velocity ii, the flight distance relative to coordinates moving at ii becomes I = x - iit, so the Eq. (47) becomes
&(t)
1
=
- exp{ --F2(x - i i r 1 2 / t 2 } , t4
where f 2= m / ( 2 k B f ) . Using Eq. (441, the temporal behavior of the density signal can be converted into a flux-sensitive energy distribution:
P(E)
- Eexp( - E / k B f + 2Fii(E/kBf)’12
-
f2P2).
(49)
Kelly and Dreyfus [50]have argued that ii is at least the sonic velocity at the outer edge of the Knudsen layer. If this is assumed, then for a monatomic perfect gas, ii = (5/6)’12F, and Eq. (49) becomes
p(E)
- Eexp( - E / k B f + 2(5E/6kBf)1’2
-
5/6}.
(50)
The mean kinetic energy of the above distribution is E = 3.67kBf. It is noted that the temperature in Eqs. (49)-(50) implicitly refers to the vapor temperature rather than the surface temperature. Nevertheless, Fig. 22 shows that the Boltzmann distribution fits better the experimental results than the “stream-velocity-corrected” distribution. The implication is that an order of 10 or 20 collisions per particle that is obtained by a simple estimate for the material removal rates considered is not sufficient to impart a significant stream velocity. The energy coupling between the incident laser beam and the target surface evolves with surface topography as the ablation of the surface proceeds. To investigate this affect, the plume energy has been measured while an initially “virgin” surface is repeatedly pulsed with laser light. The number of counts per pulse registered by the detector was analyzed as the total number of accumulated pulses grew (Fig. 23a). Figure 23b shows the development of plume energy distributions (fitted by Boltzmann distribu-
116
COSTAS P. GRIGOROPOULOS et al.
0
10
5
15
E (eV)
FIG.22. A direct comparison between experiment, a Boltzmann energy distribution, and a Knudsen layer perturbed energy distribution, as suggested by Kelly and Dreyfus [50] ([91, reproduced with permission from AIP).
tions) as the number of pulses increases. The energy distribution demonstrate increasing mean energy, 3.9-5.7 eV, with increasing accumulation of laser pulses. It is evident that energy coupling eventually reaches a steady-state condition-apparently at 300 laser pulses for the case illustrated here. Kelly et al. [52] observed that the ablation rate decrease with increasing surface roughness. Krebs and Bremert [62] found that as surface topography developed, the ablation rate decreased by 30-40% before reaching its steady-state value.
c. CONSIDERING THERMAL AND
ELECTRONIC EFFECTS
Until recently, the thermal nature of pulsed laser sputtering from metals has been questioned by relatively few researchers. The wisdom of a thermal description is supported by the conventional picture of how metals absorb radiative energy. Photons are thought to interact almost exclusively with the conduction band electrons of metals, which often are modeled as a free electron gas irrespective of whether the phase is liquid or solid. In contrast to insulators and semiconductors, incident photons excite conduction band electrons in metal without perceptible lattice relaxation, and there is no energetic electron-hole recombination event in the absence of
117
PULSED USER-INDUCED PHASE TRANSFORMATIONS
0
200
400 600 000 1000 Accumulated number of laser pulses
1200
14 10
0
100
200 300 400 500 Accumulated number of laser pulses
600
700
7
FIG.23. (a) Sputtering yield and (b) mean translational energy of neutral gold atoms vs. accumulated number of laser pulses on an initially virgin surface ([9], reproduced with permission from AIP).
a bandgap. Consequently, there is no conventional mechanism to localized electronic energy in metals. Instead, energetic electrons relax through scattering events with phonons, thereby transferring energy to the lattice. On the time scale of a nanosecond-range laser pulse, the cascade of energy is continuous with thermal heating of the lattice. Electron thermalization in gold occurs in the time scale of -500 fs (femtoseconds) [27] and electron-phonon energy relaxation occurs on the time scale of 1 ps. Energy transport (conduction) in metals is accomplished principally by electron-electron scattering and lattice heating, by electron-phonon scattering. Because of momentum conservation, energetic electrons are prohibited from localized decay during a scattering event. As heat stored in the condensed phase rises, the high-energy tail of the phonon distribution provides sufficient energy for surface atoms to overcome their binding
-
118
COSTAS P. GRIGOROPOULOS
el
al.
energy and escape to the vapor phase. The energy and momentum requirements for desorption are met by the lattice. Recent experimental results, however, have challenged the conventional view that nanosecond pulsed laser vaporization of metal can always be absorbed to thermal mechanisms [40, 53, 65, 901. These studies have explored nonthermal desorption from metal at low laser fluences from the solid phase. The apparent consensus is that the creation and local decay of a surface plasmon (a collective excitation of electrons) is responsible for discharging energetic atoms from metals. The principal issues concerning the role of plasmons in desorption are (1) the conditions required to excite plasmons and (2) how the plasmon energy is converted into the translational energy a single atom. Most attention has been given to the first question. The probability of exciting a surface plasmon with an incident photon is reduced by the typical mismatch in phase velocities of a plasmon and a photon having the same energy. Consequently, specialized geometries have been used to investigate the role of surface plasmons in desorption. To enhance coupling with surface plasmons, prisms have been used to slow the phase velocity of photons, whereas geometries leading to total internal reflection orient the wavevector parallel to the surface. Kim and Helvajian [53] and Lee et al. [65] have both reported energetic desorption from thin metal films using this geometry. Surface plasmons can also be excited on small metallic spheres, as realized classically by Mie theory. Coupling is strongest when the sphere is small compared to the photon wavelength. Hoheisel et al. [40] have observed resonantly enhanced desorption rates from metallic clusters (Na, K) of radii roughly 10% of the photon wavelength. Observations of energetic desorption from fairly generic conditions have led some researchers to relax the usual anticipated conditions for surface plasmon excitation [53]. It has been experimentally demonstrated that the conversion of photon energy to surface plasmon energy can be facilitated by surface roughness. The purest experimental demonstration of this is found using diffraction gratings. The reciprocal space vector of the grating can augment the plasmon wavevector to match the incident photon wavevector (parallel to the surface). In this manner, both energy and momentum can be conserved. A conceptually straightforward extension of this can be made to surfaces of arbitrary roughness. The surface roughness provides a characteristic momentum spectrum that can selectively furnish the momentum requirements of the photon-to-surface plasmon energy conversion. A detailed account of coupling between surface plasmons and phonons is given in a review article by Ritchie “1. The second question, of how the plasmon energy is coupled to desorption, is more difficult to resolve. Some theoretical consideration has been given to the problem [82], however, the mechanism for localizing the
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
119
plasmon energy to a single atom remains unclear. Even the most energetic plasmons carry very little momentum-necessitating a three-body collision process between the desorbing atom, plasmon, and the lattice (which provides the required momentum) to satisfy the conservation laws. Recent appeals to plasmon induced desorption have emphasized various and sometimes contradictory features of the process. For example, the energy of desorption is insensitive to fluence [40, 65, 901; the desorption energy is equal to the plasmon energy [90]; the desorption yield has a linear dependent on fluence [40, 651; the desorption yield has a superlinear dependence on fluence [82, 901; and desorption is resonant at photon energies near the plasmon energy [40, 901. An experiment has been constructed [lo] that extends previous notions of the nature of pulsed laser sputtering from metals. In this experiment, laser fluences are sufficiently high to study vaporization from the liquid phase. In desorption processes from the solid state it is not always clear how to specify cohesive energy, bonding state, and electronic structure of surface atoms. The high atomic mobility of liquid metals, however, suppresses any distinction between surface states and, for metals, the first-order electronic structure remains the same as for the solid phase. Another benefit of surface melting is that surface structures with dimensions smaller than the laser wavelength have very short damping periods while in the liquid phase. Consequently, small structures most amenable to photon coupling with surface plasmons are removed rapidly by the liquid surface capillary action. An additional degree of freedom introduced in the study by Bennett et al. [lo] allows the total energy delivered to the target to be partitioned between steady-state electron-beam heating of the back surface and pulsed laser heating of the front surface. This configuration permits investigation of the effect of variable electronic energy density (provided by the laser) for the same thermal conditions, illuminating what effect this has on the mean translational energy of vaporization. The target steady-state temperature is controlled by electron-beam (e-beam) heating. A resistively heated tungsten filament, held at ground potential behind the target, thermally emits electrons that are accelerated into the target, which is held at +lo0 V. As little as 2 W of e-beam heating is sufficient to raise the target temperature 800 K above the ambient temperature. Pulsed laser melt annealing is performed at high temperature and relatively low laser fluences to minimize the hydrodynamic formation of surface topography during annealing. After melt annealing, the mean variation in local surface normals is about +3" (measured by atomic force microscopy (AFM)). Figure 24 shows yields and translational energies of Au" for initial target temperatures of 375 and 1100 K. The two ranges of investigated fluences
120
COSTAS P. GRIGOROPOULOS et al.
600
t
0.5
0.7
0.9
1.1
1.3
F (J/cmz) FIG.24. Au" yield and mean translational energies in the first 10 ( 0 )and the second 10 ( 0 ) laser pulses on the annealed target surface. Data for two initial target temperatures (To) are shown; (a) shows yields and (b) depicts translational energies [lo].
are offset to keep the peak thermal conditions comparable for the different initial temperatures. The upper bound on the fluence range investigated is maintained below the onset of gas-phase plasma formation. For both temperature cases the mean translational energy correlates positively with laser fluence. There is no evidence of a single preferred desorption energy as has been suggested in connection with the local decay of a surface plasmon. In addition, the yield demonstrates a superlinear dependence on laser fluence. This suggests that the desorption process is not a single-photon process. Researchers investigating plasmon-induced desorption have been divided on this point. The scenario in which a plasmon decay couples with an antibonding electronic transition favors a linear dependence between yield and fluence [401. However, it has also been suggested that desorption can be mediated by a thermal process, such that the local decay of a surface plasmon served only to enhance the desorption energy [82, 901. In this case, yields should be superlinear with respect to laser fluence. However, there is only a finite probability for thermally ejected atoms to be recipients of the plasmon energy-leading to the
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
121
prediction of a bimodal energy distribution. No condition investigated in the present work produced a bimodal energy distribution. The mean translational energy of gold atoms leaving the surface can be compared with anticipated thermal energies based on numerically calculated surface temperatures. The numerical calculation uses the enthalpy method described in Section I, with a change in the surface boundary condition. The surface boundary condition is now prescribed by a kinetic relation between the surface temperature and vapor flux [see Eq. (5111. There are several salient features to this comparison, as shown in Fig. 25. The first is that actual mean translational energies are at least a factor of 2-3 higher than the thermal expectation. In addition, the slope in energy with respect to temperature is approximately 15kB instead of 2kB as would be predicted for a thermal process. Certainly, the most perplexing question is how the desorption process produces such high kinetic energy. Is the electronic energy density in the metal more or less important than the thermal “lattice” energy with respect to the energetic desorption process? To address this question, the energy delivered to the target surface has been partitioned between steady-state electron-beam heating of the back surface and pulsed laser heating of the front surface. The laser heated surface achieves the same peak temperature (approximately 2760 K) for all experimental conditions shown in Fig. 26. However, the laser contribution to the peak surface energy increases from 60 to 87%, progressing from left to right in the figure. Figure 26 shows no appreciable change in yield of Au” or mean translation energy. The constant yield is strong evidence that this aspect of
0.0
2200
2600
3000
3400
Peak Surface Temperature (K)
FIG.25. Measured and thermally anticipated ( 2 k , T ) translational energies as a function of calculated peak surface temperature. The data pertain to the first 10 laser pulses on the annealed target surface [lo].
122
COSTAS P. GRIGOROPOULOS et al.
OF- + -
87% laser
59% larcr
I 0.7
0.8
0.9
1.0
1.1
1.2
F(J/CIII~)
FIG.26. Au" yield (a) and mean translational energies (b) in the first 10 ( 0 )and I.--: seconi 10 ( 0 ) laser pulses on the annealed target surface. The peak surface temperature ( - 2680 K) is approximately the same for all initial target temperature-laser fluence combinations [lo].
the desorption process is governed by thermal conditions at the target surface. The fact that the mean translational energy is also nearly constant is a very interesting result indicating that desorption energy is most sensitive to the peak surface temperature and does not appear to be a strong function of the electronic energy density alone. A remarkable characteristic of the measured translational energy distributions is the closeness to Boltzmann form. Figure 27 shows two energy distributions for the initial target temperature of 375 K. The shape of the energy distribution is surprising only because it is reminiscent of a thermal process-despite the fact that energies are much too high to reconcile with the classic thermal model. However, the shape of the energy distribution does suggest that the source of desorption energy is not connected with a quantized value, such as the plasmon energy. If a decaying surface plasmon somehow transfers its energy to desorption, one might expect the energy distribution to be either a delta function at the plasmon energy or perhaps a Boltzmann function offset by the plasmon energy. Neither of these possibilities is suggested by the observed energy distribution.
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
123
I
-Bollzmann h
5 a
0 0
2
4
6
8
E (eV)
FIG.27. Comparison of measured energy distributions with Boltzmann fits for the first 10 laser pulses on the annealed surface. The initial target temperature for both cases is 375 K [lo].
The fact that our irradiation geometry is not conducive to plasmon excitation (at least by design), the fact that yields are superlinear with laser fluence, and the fact that Au translational energy distributions are neither bimodal nor peaked about a quantized value all suggest that the plasmoninduced desorption picture is deficient, at least with respect to terms suggested by earlier researchers. However, it is also clearly premature to rule out involvement of surface plasmons, particularly since the supertherma1 translational energy of desorption makes transparent the inadequacy of the conventional description. The theoretical basis for surface plasmoninduced desorption is perhaps too underdeveloped to formulate solid expectations for this process.
111. Computational Modeling of Pulsed Laser Vaporization A. BACKGROUND Various theoretical models have been proposed to describe material removal from a solid heated by laser irradiation. The thermal models of Afanas’ev and Krokhin [3], Anisimov [4], and Olstad and Olander [74] represent early theoretical contributions to this problem. Chan and Mazumder [ 161 developed a one-dimensional steady-state model describing the damage caused by vaporization and liquid expulsion due to laser-material interaction. Much of the work in those studies was driven by laser applications such as cutting and drilling, and was thus focused primarily on the target morphology modification, with no particular inter-
124
COSTAS P. GRIGOROPOULOS et al.
est in the detailed description of the properties and dynamics of the evaporated and ablated species. Moreover, these models dealt with continuous-wave (CW) laser sources, or relatively long (millisecond-range) time scales. During the first stage of interaction between the laser pulse and the solid material, part of the laser energy is reflected at the surface and part of the energy is absorbed within a short penetration depth in the material. The energy absorbed is subsequently transferred by heat conduction deeper into the interior of the target. At a later stage, if the amount of laser energy is large enough (depending on the pulse length, intensity profile, wavelength, the thermal and radiative properties of the target material), melting occurs and vaporization follows. The vapor generated can be ionized, creating a high-density plasma that further absorbs the incident laser light. The physical picture of laser energy interaction with evaporating materials at high fluence ( F > 1 GW/cm2) then becomes quite complicated. Of interest are the descriptions of the vaporization and ionization processes, the associated fluid motions and gas dynamics phenomena (including the vapor-plasma expansion and possible shock wave formations against the ambient environment pressure), and the intense electromagnetic fields generated [77]. Phipps el al. [78] developed a simple model to predict the ablation pressure and the impulse exerted on laser irradiated targets for laser intensities exceeding the plasma formation threshold. The model was shown by Phipps and Dreyfus [77] to follow the experimental trends for the mass loss rate and the ablation depth within a factor of 2. In their computational studies, Aden et al. [ l , 21 dealt with the laser-induced expansion of metal vapor against a background pressure. Using a compressible gas dynamics numerical model, they were able to capture the development of shock discontinuities in the vapor phase, in agreement with experimental observations. Vertes et al. [102, 1031 developed a onecomponent, one-dimensional model to describe the expansion of lasergenerated plasmas. The model incorporated the conservation equations for mass, momentum, and energy. Singh and Narayan [931 proposed a theoretical model for simulation of laser-plasma-solid interactions, assuming that the plasma formed initially undergoes a three-dimensional isothermal expansion followed by an adiabatic expansion. This model yielded athermal, non-Maxwellian velocity distributions of the atomic and molecular species, as well as thickness and compositional variations of the deposited material as functions of the target-substrate distance and the irradiated spot size. Further research is needed to enable direct prediction of pulsed laser interactions with materials. In particular, it is necessary to address in detail
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
125
the heat transfer and fluid flow phenomena that occur in the associated phase-change transformations from the solid to the liquid, and from the liquid to the vapor phases. B. MODELING DESCRIPTION-TRANSPARENT VAPORASSUMPTION It is sought to construct a numerical model for the computation of the heat transfer in the substrate, including the melting transformation and the following vaporization. It is assumed that a metal target is subjected to pulsed laser radiation and that the ejected vapor plume expands against an inert gas of reduced pressure. Figure 28a depicts a laser pulse incident on a bulk substrate and shows the generated melt region and the ejected vapor plume. The laser beam spot shape is assumed to be circular, of radius rlas= 1 mm and of uniform intensity distribution profile. In the thermal description of the phase transition from liquid to the vapor, it is necessary to cross the vaporization dome. This is accomplished by introducing a discontinuity layer just above the liquid surface. The physical origin and significance of the discontinuity layer are discussed next. In the classic kinetic model of evaporation, the vapor particles escaping from a hot liquid surface possess a half-Maxwellian distribution, corresponding to the liquid surface temperature. The velocity vectors of these particles point away from the liquid surface. The anistropic velocity distribution is transformed into an isotropic one by collisions among the vapor particles within a few mean-free paths (typically of the order of few micrometers) from the surface in a discontinuity region known as the Knudsen layer (Fig. 28a). Some of the particles experience large-angle collisions and are scattered back to the surface. Beyond the discontinuity layer, the vapor reaches a new internal equilibrium at a temperature different from the surface temperature (usually lower than the liquid surface temperature). Appropriate boundary conditions are needed for the vapor phase on the vapor side of this discontinuity layer (Fig. 28b). In order to derive these boundary conditions, the conservation equations for mass, momentum, and energy are applied across the discontinuity layer. There are two models for describing laser-induced vaporization: the solid/vapor-phase transition and the liquid/vapor-phase transition. If the intensity of the incident radiation exceeds a certain threshold, then the temperature of the target material within the penetration depth (m) exceeds the normal boiling temperature of the metal and is certainly higher than the melting temperature. Thus, the liquid/vaporphase transition model is used in this study. Also, the difference between the two modes is fundamental. Only within the framework of the liquid-vapor transition is it possible to consider the intensity region
126
COSTAS P. GRIGOROPOULOS ef al.
FIG. 28. (a) Diagram of laser-induced vaporization from a metal surface showing the heat-affected zone (HAZ), the various phases, and the coordinates of the computational domain; (b) diagram showing the variables across the discontinuity layer between the evaporating liquid surface and the vapor phase (1391, reproduced with permission of AIP).
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
127
corresponding to a metal surface temperature approaching the critical temperature [7]. It is also assumed that the vaporization mechanism is “surface vaporization,” that is, that no bubbles are formed prior to or during the liquid evaporation process. These two mechanisms are substantially different. For media that are free from impurities, entrapped gases and structural microdefects such as microcracks and pores, and that absorb radiation to a depth of m, it is estimated that volumetric radiation can be an effective means of material removal in comparison with surface vaporization only at temperatures close to 0.3L,/kB [841. For common metals, particularly Al, Au, and Cu, which are considered in this work, this value exceeds lo4 K. Another reason for the assumption of “surface evaporation” is that the surface tension coefficients of liquid metals are large during the lifetime of the liquid-metal layer [71. The heat transfer in the substrate is calculated by applying the enthalpy scheme presented in Section I.B. As long as the vapor is considered transparent to the incident laser radiation, nonemitting, and non-heatconducting, the heat transfer in the substrate is not affected by the vapor plume. Hence, considering that the beam radius is again much larger than the thermal penetration depth, it is inferred that the heat transfer in the substrate is one-dimensional. Assuming that the liquid behaves like a dense gas, the rate of evaporation from the liquid surface, j,,, can be derived using kinetic theory [15, 971 as ’
1,
=
-)”*
n l ( kBT,
2~m,
exp( -
5) - B,n,( -)”*, kBT, 2~m, kBTv
(51)
where the first term in the right side represents the evaporation rate from the liquid surface at temperature T , . The second term represents a damping of this evaporation rate due to the return of vapor molecules to the liquid surface. According to the analyses by Anisimov 141, Batanov et al. [7] and von Allmen [104], the upper limit for the entrapment of vapor molecules returning to the liquid surface is below 20%. The parameter 0, , the “effective sticking coefficient,” represents the probability of a vapor atom returning to the liquid surface from equilibrium conditions at the edge of the discontinuity layer manages to penetrate this layer to finally be adsorbed on the liquid surface. This parameter is chosen so as to yield a recondensation to evaporation rate in the range from 15 to 20%. The conservation equations of mass, momentum, and energy applied to the discontinuity layer are as follows: Mass conservation: ~v(-uv
- uint) = ~
l ( ul Uint).
(52)
128
COSTAS P. GRIGOROPOULOS el al.
Momentum conservation: Pv - PI = ~
v -uv (
-
Uint>(UI
+ uv).
(5 3 )
Energy conservation: HI + L ,
= H,
+ ;(u:
-
u;).
(54)
An equation of state for the liquid is also needed for the calculations. A
possible choice is the van der Waals equation of state. This equation is representative of a class of equations of state that are reasonably accurate, embodying the main features of most condensable gases [SI. Unfortunately, however, the van der Waals constants are not available from experiments to represent the liquid metal behavior from melting to the critical state. In this study, it is assumed that (1) the temperature rise and phase transitions take place under conditions of local quasi-equilibrium, allowing the use of classical thermodynamics relations; and (2) the liquid expansion takes place along the liquidus line. The first assumption simplifies the physical description of the problem. The choice of the liquidus line as the thermodynamic path for expansion is somewhat arbitrary, although it appears to be reasonable [S]. Thus, the liquid pressure, according to experimental values [41], can be expressed as log PI = - + u 2 log TI + u 3 . a1
TI
(55)
It is also found experimentally that the temperature dependence of the density for liquid metals is linear and is accurately represented by the equation PI = P m - N T , - T,). (5 6 ) Since the ejected vapor is modeled as an ideal gas, the enthalpy of the vapor phase, H,, is a function of the vapor temperature only, H , = H,(T,). The liquid surface temperature, TI,and the energy content associated with the liquid, HI, are directly calculated from the heat conduction in the substrate. One more assumption needed to close this system of equations is that the regression velocity of the liquid phase is very small, u , 0, when compared with the regression velocity of the vapor-liquid interface and the forward expansion velocity of the vapor. Because the liquid layer is thin, and since a one-dimensional model is applied for the heat transfer in the substrate, this assumption is reasonable. The vaporized material ejected from the liquid surface is modeled as a compressible, inviscid ideal gas. For the millimeter size of the laser spot diameter, which is the characteristic dimension in this problem, the continuum approach is expected to be valid for an ambient pressure P, =
-
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
129
atm. At lower pressures, the continuum framework is no longer valid [ll] and a calculation approach suited for rarefied media, such as the direct particle collision simulation, is required. The dynamic state of the metal vapor phase is described by the compressible and nondissipative Euler conservation equations for mass, momentum and energy: Mass conservation: dp
-
dt
+ div( pV)
=
0.
(57)
-grad P .
(58)
Momentum conservation: p
dV
- + p(V * grad)V dt
=
Energy conservation: p
de
- + p(grad)V dt
=
- P divV.
(59)
Two more equations define the total energy per unit mass, E, and the equation of state for an ideal gas p = (Y - l ) p e , where y is the ratio of the specific heats, y = C , / C , . Equations (561461) are solved for p, u , L’, E,, P , and e in an axisymmetric (cylindrical) calculation domain using the MUSCL Eulerian compressible gas dynamics scheme [21]. It is assumed that vaporization takes place on a circular area, with no dependence on the azimuthal coordinate direction. Computations were done for aluminum, gold, and copper. Figure 29 shows the temperature-time history of the surface of a gold metal substrate for different values of the laser fluence. Initially, the surface temperature increases rapidly during the laser pulse duration (20 ns). It then decreases rapidly, back to room temperature. The profiles show a “flat” region at T = T,, (1338 K) during which the surface temperature is equal to the melting temperature for solidification and beyond which no more liquid exists on the substrate. As shown in the first section of this chapter, the heat transfer in the target material and the melting process are governed by heat diffusion. Thus, Figs. 30a, b show that the maximum liquid surface temperature as well as the melting depth vary linearly with the laser pulse fluence. Considering this trend, and keeping in mind that the evaporation rate increases exponentially with temperature, as Eq. (51)
130
COSTAS P. GRIGOROPOULOS et al.
8000
E' W
2 3
Ha, 4000
a.' , c a, 0 (d
't
3
a
Time (ns)
-
FIG.29. Time histories of the surface temperature (ar x 0 ) of a gold substrate subjected to excimer laser pulses of 26-11s duration and of different fluences. The ambient pressure is P, = 1 atm, and the laser spot radius is rlan= 0.5 mm ([39], reproduced with permission of AIP).
shows, it is expected that the ablation depth increases exponentially with fluence as depicted in Fig. 30c. The dense liquid-solid substrate is subject to recoil forces exerted by the evaporating material. The total recoil momentum experienced by the irradiated body is
The term P,,, defined as h e V u v / r r & , is due to the momentum change at the liquid-vapor interface. Figure 30c presents the recoil momentum, divided by the total energy E, carried by the laser radiation pulse, as a function of the laser fluence. Figures 31a-c shows normalized pressure contours in the vapor phase for a laser fluence F = 1.75 J/cm2 incident on a bulk gold surface. The background pressure is set at P, = atm, and the laser spot radius on the sample surface, rlas = 0.5 mm. The shock front, pushed by the high vapor pressure (hundreds of atmospheres)
a9000
"
'
~
I
'
'
'
'
~
~
'
'
'
3
c
a,
E
L
P)
45000
3008
b
10 20 Laser fluence (J/cm2)
104'
30
'
'
'5
'
'
'
'10' . ' '15' ' '20'' '25' Laser fluence (J/cm*) '
'
'
'
,Jo
, . .
'
c -
T
.
4
L
.
:
1°fOO
10'
Laser fluence [ J/cm2)
FIG.30. (a) Maximum liquid surface temperature; (b) melting depth; (c) ablation depth; (d) dependence of vapor recoil momentum on laser fludnce for Au, Cu and Al substrates subjected to excimer laser pulses of 26-11sduration and of different fluences. The arrows in panel b mark the vaporization thresholds (VTs). The ambient pressure is P, = 1 atm, and the laser spot radius is rlas = 0.5 mm ([39], reproduced with permission of AIP).
132
COSTAS P. GRIGOROPOULOS et al.
d 0.010,
-T N
16.9 15.2 13.5 11.8 1a.2 8.5 6.8 5.1 3.4
'
0.005 .
1.7
.
.
I
I
I :
E
1000 (WS)
N
0.005
'
, '
e
b 0.01 0
E
N
0.005
0.000 r Iml
FIG.31. (a-c) Normatized pressure contours and (d-f) velocity vectors in the computational domain and for an Au substrate subjected to excimer laser pulses of 26-11s duration and of fluence F = 1.75 J/cm2 at (a-d) t = 1 ps, (b-e) t = 5 ps, and (c-f) t = 10 ps. The atm, and the laser spot radius is rlaS= 0.5 m m ([39], ambient pressure is P, = reproduced with permission of AIP). The coordinate z is here normal to the substrate.
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
133
produced during the laser pulse duration, propagates with an average speed of about 700 m/s into the ambient medium. For reference, it is noted that the sound speed for the ambient pressure and temperature is about 330 m/s. Figures 31d-f shows the corresponding velocity plots; they depict a situation similar in nature to the sudden explosion of the metallic target. The vaporized material is pushed upward and laterally simultaneously. As the vapor leaves the surface at high speed, it induces a recirculating zone of entrained fluid around it. Figures 32a-d show the normalized density ( p * = p / p J the normalized temperature (T,* = TJT,), the vapor speed, and the local Mach number contours at a time t = 5 ps. It can be seen that most of the high-kinetic-energy material is concentrated in a jet-like core region moving at supersonic speeds normal to the surface. The outer region is an essentially motionless, isothermal, low-temperature zone. The evaporating particles and the ambient gas have been assumed to be transparent to the incoming laser light. The validity of this assumption depends on the local plasma density. The plasma density determines the laser light-absorption coefficient. The mechanisms for absorption include inverse Bremsstrahlung and photoionization [ 1161. Computations of these effects on the plasma ignition and the laser energy coupling to the target were performed by Rosen et al. [831 and Duzy et al. [23] for microsecondlength UV laser pulses on aluminum. The degree of ionization can be estimated by invoking the Saha equation [181
n: - = 2.4 X lO2lT3I2exp( - U , / k , T ) , nn
where n , and ni represent the number of neutral and ionized particles per unit volume, respectively. For gold, the neutral particle density n, is 0(10261, whereas the temperature of the evaporating particles is 0(103) during the vaporization period. The ionization potential, Q ,for gold is 9.22 eV. Hence, the corresponding ni is 0(1O2l) and the estimated absorption coefficient is much smaller than 0(10-*) (m-'). Thus, the assumption of an optically thin vapor phase is reasonable, at least with respect to the particular absorption mechanism in a gold vapor plume and for the range of fluences examined. For a thorough treatment, the densities of the neutral, excited, and ionic species must be calculated using rate equations for the radiative transitions and considering the electronic structure of the target materials. The absorption coefficient can then be derived and the laser energy transfer to the vapor plume as well as the radiation loss by thermal emission calculated by solving the radiative transfer equation. This is a brief outline of the computational procedure that is being implemented by Ho et al. [381.
. a
P* 8 7 6
h
E
v
.N_
5.04 4.47 3.89
Y
.
N
I
0.005
0.005
r (m) 0.01 0
I
"
'
.
I
.
"
.
I
'
0.0101
'
b
3 2 1
E
v
N
0.005
n
-
13.70 12.01 10.31
5
8.62
'
'
I
'
0.005
0.000 r (m)
0.005
.
'
'
I
1
'
'
'
.
Ma 8 7 6 5
h
E
v
N
'
1
.d
T" 8 7 6
591.47 517.53 443.60 369.67 295.73 221.80 147.07 73.93
4
0.000
0.000
0.000
8 7 6 5
E
t
0.005
4
h
7.53 6.59 5.65 4.71 3.77
.
0.005
FIG.32. Contour plots for an Au substrate subjected to excimer laser pulses of 26-ns duration and of fluence F (a) normalized density; (b) normalized temperature; (c) vapor speed; (d) Mach number. The ambient pressure is P, = radius is ria. = 0.5 mm [391, reproduced with permission of AIP. The coordinate z is here normal to the substrate.
=
1.75 J/cm2 at t = 5 ps: atm, and the laser spot
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
135
To compare with the computational results and characterize the laserablated plume, one would need to identify the various species (neutrals, ions, electrons, clusters, etc.) and measure their concentration and rates of generation and consumption, as well as their velocity and energy distributions. In addition, information about chemical reaction kinetics, as well as cross sections for a variety of processes in the laser-generated plasma, would be required. Comprehensive reviews of experimental studies of pulsed laser-induced ablative processes have been presented in Miller [69] and Chrisey and Hubler [20a]. Kinetic energies of the ejected particles can be measured using TOF [118, 1191. Laser-induced fluorescence (LIF) is another very useful diagnostic for laser ablation plasmas, providing number densities, kinetic energies, rotational temperatures and vibrational temperatures [72, 731. Fast photography and other imagining techniques provide useful information for understanding the complex flow phenomena in pulsed laser ablation. The results of high-speed framing photography [Ma], streak photography [26], and intensified charge-coupled device array (ICCD) photography of plume emission [32] as well as techniques using slower cameras and pulsed lasers for delayed imaging in Schlieren or shadowgraph arrangements, have been reviewed by Geohegan [33]. Twodimensional, species resolved holographic interferometry was used by Lindley et al. [66] and Gilgenbach et al. [35] to measure absolute-linedensity profiles of KrF laser-generated aluminum plumes in vacuum and argon atmospheres. Direct acquisition of emission spectra led to derivation of vibrational temperatures and estimates of radical concentrations [20]. Density gradients in a laser-generated plume can be monitored by the deflection of a probe laser in the far field, as shown by Enloe et al. [25] and Chen and Yeung [19]. With sophisticated numerical modeling and detailed experimental, it will be possible to advance important applications such as the pulsed laser deposition (PLD). This technique has been drawing significant attention in the scientific community, having enabled fabrication of novel thin film materials of high quality and superior properties as compared to conventional manufacturing techniques.
IV. Conclusions In this chapter, recent research on pulsed-laser-induced phase-change transformations at the nanosecond time scale was reviewed. The melting of semiconductor materials was probed by optical reflectance, transmittance, electrical conductance, and infrared pyrometry. The experimental results were in general agreement with the thermal model, except for the high-irradiance regime. Direct measurements of the solid-liquid interface
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COSTAS P. GRIGOROPOULOS et al.
in melting of polysilicon films on glass substrates showed superheat by over 100°C. The rapid mass transfer in the liquid silicon phase allowed successful fabrication of well-controlled and sharply defined box-shaped ultrashallow dopant junction profiles. Annealing of gold was examined next. Utilizing heat transfer and stability analysis, the generation of surface morphology was attributed to Rayleigh-Taylor instability triggered by the reduction of density on melting. Time-of-flight measurements of the kinetic energy distribution in near-threshold sputtering of gold produced values exceeding the thermal model expectations. A computational heat transfer analysis of the heat transfer and the vapor gas dynamics adopting the transparent vapor assumption was discussed. More indepth experimental and theoretical studies are needed to resolve the fundamental mechanisms of ablation from the liquid or the solid phases.
Acknowledgment Support to the work presented in this chapter by the National Science Foundation, under Grants CTS-9210333 and CTS-9402911, is gratefully acknowledged. The authors are especially thankful to Douglas J. Krajnovich of the IBM Almaden Research Center for his contributions to collaborative research on the pulsed laser sputtering problem under a UC Berkeley-IBM Joint Study Agreement. The contributions of Richard E. Russo of the Lawrence Berkeley Laboratory and of Professor Joseph A. C. Humphrey are also acknowledged.
Nomenclature acceleration area particle detector area constants in the vapor pressure-temperature relation [Eq. (5511 dopant concentration constant in phase-change kinetic relation specific heat for constant pressure sonic speed specific heat for constant volume dopant concentration at the surface blackbody radiation constant,
C , = 3.7420 X 10’ W/pm4/m2 blackbody radiation constant, C, = 1.4388 X lo4 p m K material constant for the kinetic rate of solid-liquid transformation mass diffusivity junction depth junction transition depth optical absorption depth substrate thickness area on the laser-heated spot whose thermal emission is detected by the detector droplet diameter specific internal energy
PULSED LASER-INDUCED PHASE TRANSFORMATIONS
translational energy of ejected particles total energy per unit mass energy of the laser pulse blackbody emissive power distribution of molecules per unit volume laser fluence Planck's constant enthalpy laser intensity profile imaginary unit evaporation rate thermal conductivity Boltzmann constant extinction coefficient length of silicon strip latent heat of fusion latent heat of vaporization evaporation mass flow rate mass per atom recoil momentum real part of refractive index laser pulse number number density signal at the detector number of ions per unit volume number of liquid atoms per unit volume number of neutrals per unit volume number density of ejected particles at the surface number of vapor atoms per unit volume complex refractive index pressure recoil pressure [Eq. (6311 activation energy power absorbed coordinate in the radial direction laser beam radius reflectivity specular reflectivity diffuse reflectivity rate of melting rate of melting at equilibrium rate of solidification
z
137
rate of solidification at equilibrium time temperature interface temperature equilibrium melting temperature elevated target temperature velocity vector of ejected particles propagation velocity of dilatational waves voltage recorded on the oscilloscope velocity vector interfacial velocity width of silicon strip coordinate in the normal to the sample surface direction coordinates in a Cartesian system transient location of the interface from the surface of the solid layer at t = 0
SUBSCRIPTS ev exc int
I S
ss V m
evaporation excimer laser solid-liquid interface liquid solid substrate vapor phase ambient conditions
SUPERSCRIP~S
*
directional normalized variables
HATS quantity accounting for stream velocity formation at the edge of the Knudsen layer mean quantity
138
COSTAS P. GRIGOROPOULOS el a / .
GREEK ff (Yth
Y
C,/C,J
r am
6,
AT ATP &
77 0 6,
absorptivity thermal expansion coefficient ratio of the specific heats ( y = parameter in Eq. (47) [r2= m /(2k B T )I melting depth thermal penetration depth in the target solid interfacial undercooling (AT = Tm - Tint) average rise in temperature over the thermal penetration depth emissivity absorption coefficient polar angle detector angle
incident angle sticking coefficient laser light wavelength Constant in the densitytemperature relation [Eq. (5611 most dangerous perturbation wavelength light frequency density surface tension electric conductivity transmissivity of optics laser pulse duration melting duration time scale for mechanical expansion of target solid azimuthal angle electric resistance
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ADVANCES IN HEAT TRANSFER VOLUME 28
Heat and Mass Transfer in the Extrusion of Non-Newtonian Materials
YOGESH JALURZA Department of Mechanical and Aerospace Engineering Rutgers, The State Universiw of New Jersey, New Brunswick New Jersey
I. Introduction Materials processing has been an area of considerable interest to the heat transfer community in the recent years. This is mainly due to the advent of new materials and the crucial need to optimize processes in response to international competition. Heat transfer is the dominant consideration in a large variety of materials processing applications such as casting, thermoforming, crystal growing, welding, and heat treatment. Among the most important manufacturing processes are those concerned with the processing of polymers, which include materials such as plastics, composite materials, and food and pharmaceutical products. Since heating of these materials is needed for flow, forming, phase change, chemical conversion, and other changes associated with thermal processing, most of the relevant manufacturing processes are thermally based; thus, heat and mass transfer, thermodynamics, and fluid flow are the dominant mechanisms governing the process. Because of the tremendous importance of these materials in everyday life and in a wide variety of commercial applications, much work has been done on the development of processing techniques and the design of the relevant systems. Detailed information is also available in the literature on the processing of different materials and development of new products. However, these processes have received very little attention in the heat transfer literature, despite the importance of these materials and the fact that heat and mass transfer play a critical role in determining the behavior of the material and the characteristics of the final product. This chapter focuses on the important manufacturing 145
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process of screw extrusion of polymeric materials and discusses the underlying transport mechanisms and relevant practical issues that concern thermal transport in these materials. Therefore, this discussion is applicable to several other techniques for the thermal processing of such materials. A. BACKGROUND Extrusion is one of the most important manufacturing methods in many industries. This process is particularly useful when thermal and/or mechanical means are employed to obtain a uniformly processed product in a continuous operation. The screw extrusion process for polymeric materials usually includes the following stages: feeding, conveying, plasticizing, homogenizing, and pressurizing [62]. These processes take place within a special reactor consisting of one or more screws rotating inside a barrel, usually referred to as single-, fwin-, or rnultiscrew extruders (see Fig. 1). The pressure and temperature of the material rise as the material flows down the extruder channel toward the die. The cross-sectional shape of the product is obtained by pushing the molten material through a die. In order to obtain desired product quality and characteristics, a given set of operating parameters has to be maintained at certain values, for a particular design of the extruder. The main operating parameters are the screw rotational speed [revolutions per minute (rpm)], mass flow rate, and the barrel temperature; the extruder design parameters are screw diameter, screw profile, barrel diameter, extruder length, and other dimensions and materials used. The desired temperature level at the barrel is maintained by electrical heating, by using a hot fluid and/or by cooling of the barrel. Most of the materials of interest in polymer extrusion are nonNewtonian, which means that their viscosity is dependent on the applied strain rate, unlike Newtonian materials like air and water, whose viscosity does not vary with the strain rate. This considerably complicates the problem since the viscosity now depends on the flow that is itself strongly dependent on the viscosity. In addition, the viscosity and other fluid properties are strongly influenced by the temperature and the composition, such as moisture content or amount of converted material in food. This property variation couples the momentum and continuity equations with the energy and mass transfer equations through the constitutive equation for the material, again substantially complicating the analysis. The viscosity of these materials is often very high-many orders of magnitude greater than water, resulting in typically very small Reynolds numbers in the flow. A consequence of this feature is that the flow is dominated by viscous and
EXTRUSION OF NON-NEWTONIAN MATERIALS
147
FIG.1. Schematic of (a) a single-screw extruder and (b) a tangential corotating twin-screw extruder.
pressure effects and inertia terms are usually negligible. The geometry of the extruder is generally quite complex, particularly for twin-screw extruders, making it essential to develop specialized numerical techniques to simulate the flow and the heat and mass transfer. Clearly, many fundamental and applied aspects arise in this manufacturing process. Many of these have received detailed investigation as discussed later in this chapter, but several others are not very well understood and need a careful study. From a practical viewpoint, interest lies in
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YOGESH JALURIA
obtaining detailed information on the following aspects:
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Shear and temperature history of the material Residence-time distribution Mixing Pressure rise generated Temperature level reached Torque needed Transport in extruded material Die swell Distribution of species, conversion, and phase change System characteristics, desigp, and optimization
Most of these items refer to the extruder system and to the behavior of the material as it moves down the extruder channel. Residence-time distributions concern the time spent in the extruder by the material at various locations. The pressure and temperature at the die are important elements that determine the characteristics of the final product. Similarly, expansion of the material as it emerges from the die as a result of normal stresses and flashing-off of superheated fluids such as water are important concerns. There are also several basic aspects that are of particular interest to researchers in heat and mass transfer. Some of the important ones are 1. Non-Newtonian fluid behavior 2. Laminar mixing 3. Viscous dissipation 4. Conjugate transport 5. Variable properties 6. Combined heat and mass transfer 7. Chemical reactions 8. Phase change and conversion of material 9. Creeping flows 10. Adverse pressure flows 11. Flows in complex passages 12. Numerical simulation of transient and three-dimensional (3D) transport 13. Experimentation in rotating environments 14. Heat transfer in a flow of powder and solid particles
Many of these issues are not restricted to polymer processing and apply to other heat transfer processes, Certainly, a better understanding of the
EXTRUSION OF NON-NEWTONIAN MATERIALS
149
nature of these flows is highly desirable, particularly the effects due to strong property variations, conjugate or combined conduction-convection transport, and viscous dissipation. The nature of mixing and the effect of opposing pressure on the flow are interesting aspects. Several challenges are also posed for the numerical simulation. These include modeling complex geometries, chemically reacting processes, and large property changes. All these issues have received some attention in the literature and are discussed in this chapter.
B. LITERATURE REVIEW The extrusion technique has been increasingly used in many industries, such as those related to pharmaceuticals, food, and polymers [24, 62, 631. Extensive experimental and numerical studies have been carried out for the purpose of understanding the transport phenomena in the extrusion process, resulting in much scientific knowledge, primarily for single-screw extruders. Nevertheless, the design and operation of the extruder are still more art than science. The calculations for a single-screw extruder have resulted in important and relevant information that has been employed over many years by industry for the design of the system. Finite-difference methods are very well suited to this problem, particularly for screws of rectangular, trapezoidal, and other simple cross sections. For complicated shapes, finite-element methods are more appropriate and have been used. Some experimental validation of these methods has also been carried out in recent years. This effort has been extruded to dies and other flows in complicated passages. The first published analysis of the flow in a single-screw extruder is attributed to Rowel1 and Finlayson [55]. In this analysis, the screw channel is modeled as stationary while the barrel is considered to move in a direction opposite to that of the actual screw rotation. In one of the pioneering studies on the flow in a single-screw extruder, Griffith [20] solved for the fully developed flow of an incompressible fluid in a screw extruder, taking the material to be a power-law non-Newtonian fluid, as defined later. The velocity and the temperature profiles were found to be essentially the same as those in a channel of infinite width and length. The effects of curvature and leakage, across the flights, were ignored. Zamodits and Pearson [68] obtained numerical solutions for fully developed, twodimensional (2D) non-Newtonian flow of polymer melts in infinitely wide rectangular screw channels, taking into account the effects of transverse flow and superimposed steady temperature profile. Rauwendaal[53] devel-
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YOGESH JALURIA
oped an analytic expression for the throughput: the volumetric flow rate of the material through the extruder, and pressure rise for power-law fluids in single-screw extruders. Karian [32] developed an analytical expression for the mechanical power consumption for 2D flow of non-Newtonian fluids in single-screw extruders. Mitsoulis et al. [49] presented a 2D finite-element analysis of nonisotherma1 flow through dies and extruder channels for purely viscous and viscoelastic materials. Elbirli and Lindt [12] proposed a solution for the problem of thermally developing flow in a single-screw extruder where appreciable backflow exists because of the pressure gradient. Lindt [451 has presented a critical review on the work done by researchers in modeling the melting of polymers in a single-screw extruder. Later, Lindt [46] discussed the fully developed flow of temperature-dependent power-law fluid between parallel plates. An exact solution was developed for this flow in the absence of pressure gradients. Fenner [17] and Tadmor and Gogos [621 have solved the flow of a polymer in the feeding, compression, and metering sections of an extruder. Fenner [16] also solved the case of the temperature profile developing along the length of the screw channel. Agur and Vlachopolous [21 have studied the flow of polymeric materials in extruders, including a model for the flow of solids in the feed hopper, a model for the solid conveying zone, and a model for the melt conveying zone. Lawal and Kalyon [43] included wall slip in the numerical simulation. Karwe and Jaluria [33] have developed 2D models for the simulation of transport processes in single-screw extruders for non-Newtonian fluids. The extrusion process is simulated using the moving-barrel formulation. This method of formulation has been adopted by many investigators referred to earlier. In this case, the screw is mathematically treated as unwound and being held stationary with a Cartesian coordinate system attached to it, while the barrel is moved in a direction opposite to the actual screw motion. Using this method, the flow is much easier to visualize and simulate, whereas the resulting flow behavior would be the same as the actual flow. Esseghir and Sernas [14, 151 have carried out an extensive experimental work on the single-screw extrusion process using a fully instrumented single-screw extruder apparatus. Detailed measurements of the temperature profile across the extruder channel were obtained using a cam-driven traversing thermocouple. Some results obtained from this work are discussed later. Even though the basic transport in the extruder channel is threedimensional, not much work has been done on simulating the 3D flow. However, increasing effort on 3D modeling and simulation has arisen in the recent years. Gupta et al. [21] have developed a 3D finite-element model for incompressible flow of non-Newtonian fluids. This model can be
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151
applied to the simulation of single-screw extruders under isothermal conditions. Sastrohartono et al. [591 carried out a numerical study of the 3D flow and heat transfer in the channel of a single-screw extruder. Moisture transport is also important in many cases, particularly in food processing. Similarly, transport of volatiles and other gases is of interest in various polymers. Chemical reactions may also occur in reactive polymers. These aspects have not received much attention in the literature. However, the importance of including reactions and mass transfer in many practical processes has led to some interesting work in the recent years. The effect of mass transfer and of structural changes in the material on the flow and heat transfer in extruders, dies and other similar systems has been investigated in some recent studies [18]. Developed flow has been assumed in many cases, and the temperature field has been solved. The thermal field, as well as the concentration variation of, say, moisture in food materials, affects the flow through the effect on the fluid viscosity. Therefore, it is necessary to solve for the temperature, concentration, and velocity fields simultaneously. Twin-screw extruders have recently become very useful in a wide variety of applications, due to enhanced mixing characteristics, larger flow rates, and better control. In a typical counter- or corotating twin-screw extruder, the simulation of the transport processes is obviously very involved because of the complicated geometry that arises [28, 671. The mixing in the region between the two screws may be studied through a detailed finiteelement simulation. This may then be coupled with the modeling of the remaining, translation, region, which is similar to the flow in a single-screw extruder. This has been done to obtain the full simulation of a twin-screw extruder [38, 581. Typical results obtained from the numerical simulation, particularly those on mixing and on transport in the region between the two screws, are presented and discussed for a variety of twin-screw extruders. The coupling between the flow in a die and the upstream flow is also considered. This chapter presents the basic approach and the results from numerical studies on the thermal transport associated with the extrusion of polymers and other non-Newtonian materials. Heat and mass transfer mechanisms are of crucial importance in this manufacturing process, since the quality of the product is a very strong function of the thermal and shear history of the material. It is important to simulate the transport processes that arise and to determine the resulting flow, pressure, and temperature distributions obtained during the process. Experimental results obtained on laboratory and full-scale systems are also discussed and compared with the numerical predictions. The current state of the art in this field is discussed and areas that need a detailed investigation in the future are outlined.
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11. Material Characterization The most important concern in the screw extrusion process, as is true for most materials processing applications, is the material itself. The temperature and shear it undergoes as it moves down the extruder, changes in its chemical and physical characteristics that occur during the process, and the properties of the final extruded product are all of interest. Material properties vary due to temperature and pressure changes, chemical reactions, phase change and conversion, or structural changes in the material. Since the transport processes are strongly influenced by the material properties, it is important to employ accurate property data and appropriate property characterization in the analysis and in the experiments. A brief discussion of the material characteristics and commonly used constitutive equations is given here. For further details, the books on polymer processing mentioned earlier, as well as specialized books such as those by Han [231, Bird et al. [31, Pearson and Richardson [521, and Crochet et al. [lo], may be consulted. To differentiate between the flow behavior of a Newtonian fluid and that of a non-Newtonian fluid, let us consider the flow between two parallel surfaces with one surface stationary and the other moving at constant velocity. If the fluid is Newtonian, the velocity distribution is linear and the shear rate (defined below) du/dy, where u is the velocity parallel to the surfaces and y the distance from the stationary plate, is constant across the fluid layer. The shear stress T~~ is the force applied to the moving plate divided by its area. The rate of the shear stress variation with the shear rate is the coefficient of viscosity p of the fluid. For a Newtonian fluid, p is independent of the shear rate and depends only on the temperature and pressure. For non-Newtonian fluids, the plot rYx versus du/dy is not a straight line through the origin. The apparent viscosity, which is the ratio of the shear stress to the shear rate, may increase or decrease with du/dy, as shown in Fig. 2. If there is no recovery of deformation on removal of applied stress, i.e., there is no elastic response, the fluid is termed viscoinelastic or purely viscous. If it does have an elastic response, it is known as uzscoelastic. The former type of fluids are shown in Fig. 2. In time-independent fluids, the duration of the shear has no effect on the viscosity. Various types of polymeric materials are shown here. If the viscosity decreases with increasing shear rate, it is known as shear thinning or pseudoplastic. Most common plastics can be characterized as shear thinning materials. A common representation of polymeric materials is known as a generalized Newtonian fluid (GNF). The viscosity is then generally expressed as a function of the shear rate y, which is a scalar related to the second
153
EXTRUSION OF NON-NEWTONIAN MATERIALS
a 10
Fluids with a yield stress and a nonlinear flow curve Bingham plastic
5
;
10
Thixotropic
Pseudoplastic (shear thinning)
UI
Rheopectic
? 10
Newtonian
Y)
b
c Dilatant (shear thickening)
K
In
0 Shear rate,
du dy
0
du dy
FIG. 2. Shear stress variation with shear rate for purely viscous, viscoinelastic, nonNewtonian fluids: (a) time-independent; (b) time-dependent fluids (adapted from Skelland [601).
invariant of the rate of strain tensor, being simply du/dy for the simple shear flow considered for Fig. 2 [62].Several empirical models have been proposed to express the relation between the shear stress and the shear rate. Each of these models contains empirical parameters that are determined numerically to fit experimental data. Among the most commonly used models are the power-law model, which assumes a power-law variation of viscosity with shear rate; the modified power-law model, which is similar to the power-law model but allows the apparent viscosity to become a constant at low shear rates; the Ellis model, which approaches Newtonian behavior for small shear rates; and the Carreau model, which tends to represent the low and intermediate shear rate ranges well (see also Cho and Hartnett [9] and Irvine and Karni [25]). Considering the dependence on temperature T, the following power-law constitutive equation is often used to represent the dependence of p on T and on the shear rate i, for several fluids considered here
where b is the temperature coefficient of viscosity, n is the power-law index, and the subscript 0 denotes the reference conditions. Many fluids of practical interest can be approximated by the power-law relationship. Also, the exponential dependence on temperature may be replaced by the Arrhenius dependence, exp(b/T), for many fluids. If mass transfer is included, the viscosity will also depend on the species concentration c. Then, the viscosity expression includes a term such as exp[-b,(c - c,)] or exp(b,/c), where 6, is a constant, that multiplies the temperature-
154
YOGESH JALURIA
dependence term. Therefore, for starch, the constitutive equation is given as [411
A modified version of the Carreau model is used for a variety of materials, particularly Viscasil-300M, which is used for some experiments and numerical simulations [14]. The model is given as
a,
where C, b, and n are parameters that are determined by curve-fitting the experimental results. Figures 3 and 4 show experimental data on the viscosities of some Newtonian and non-Newtonian fluids. The approach to Newtonian behavior at small shear rates and a power-law dependence at larger values is clearly seen for the non-Newtonian fluids considered. For the CMC solutions, n is found to be 0.445 and 0.81 for pure and degraded fluids, respectively, whereas corn syrup is Newtonian ( n = 1). The values
-
102
e=
0
Legend
Globe Corn Syrup 1132 $= CMC 1.5% W. solution B= CMC 1.5% W. (degraded)
I cd
L
.g 101 0
In 0
5 .-0
s5.
n
1 00 CURVE #3
2'10-1 10-1
100
101
102
Shear Rate (l/sec)
FIG. 3. Viscosity measurements for light corn syrup and two carboxymethyl cellulose (CMC) solutions at room temperature (adapted from Sastrohartono et al. [571).
EXTRUSION OF NON-NEWTONIAN MATERIALS
155
.-
I
r-
0.LA exp (b/T)lS, 1+C[ i. exp(bm
+ in (sec
CB
-I),
p in (poise), T i n (K)
A = 0.72490, b= 2560.804 C= 7.4208605, n= 0.2671 h
.-% 0
-
.-
103-
v)
8 v)
5
101
102
103
Shear Rate (1Isec)
FIG.4. Viscosity measurements for dimethyl silicone fluid (Viscasil-300M) (adapted from Esseghir [13]).
of the constants for Viscasil-300M are given in Fig. 4 legend. Values for other fluids may be obtained from the references given.
111. Single-Screw Extruder
A. MODELING The geometry of a single-screw extruder is very complex, and the presence of screw flights with a rotating screw makes the problem timedependent. If the boundary conditions-such as barrel temperature, flow rate, and screw speed-are kept constant with time, a periodic behavior is expected to be obtained at a given point in the flow. Therefore, a time-dependent, 3D, problem arises with the additional complexity of non-Newtonian fluid with temperature-dependent properties. A solution of
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YOGESH JALURIA
the complete problem is, therefore, very involved, and simplified models have been developed to understand the basic phenomena and provide inputs that may be used for the design of practical extruders. We shall start with some of these simple models that apply under certain idealized conditions and gradually add different complexities to make them more realistic and to simulate different operating conditions, materials, and extruders. Let us first consider two-dimensional, steady-state, models that provide information on the basic characteristics of the extrusion process. 1. Two-Dimensional Transport
The simplified geometry of a single-screw extruder and the cross section of a screw channel are shown in Fig. 5. For ease of visualization and analysis, the coordinate system is fixed to the screw root and, thus, the barrel moves in a direction opposite to the screw rotation. This transformation is commonly employed in the literature [17]. Karwe et al. [36] have shown analytically that the results are independent of the coordinate system for a Newtonian fluid. The analysis applies to the metering section of the screw extruder. Curvature effects are important in sections with deep channels, such as the solid conveying zone. However, for the shallow metering section considered here, curvature effects are neglected. This allows the screw channel to be ‘opened out’, resulting in the rectangular channel shown in Fig. 5 with the barrel moving at velocity V,, at the screw helix angle 4. For steady, developing, 2D flow of a homogeneous fluid in a single-screw extruder with shallow and long channels, i.e., for H -=xB in Fig. 5 , after neglecting the inertia terms (creeping flow approximation), the equations for the conservation of momentum become
where p is the local pressure and 7 is the shear stress. The creeping flow approximation is valid because the Reynolds number Re based on channel height H is much smaller than unity, typically of order lop3,making the inertia terms much smaller than the viscous terms [50]. The clearance between the screw flights and the barrel is assumed to be small enough to neglect the leakage across the flights from one screw channel to the neighboring one. However, for a more realistic simulation of the process, the effects of leakage must be included, particularly the viscous dissipation generated in the clearance region. In the presence of strong viscous dissipation effects and/or heat addition from the barrel, the thermal convection along the z direction (along
157
EXTRUSION OF NON-NEWTONIAN MATERIALS
a I
Homer
Heated Barrel
r
screw
b Section A-A
I
.I C
Fully developed Isothermal Flow at the Inlet
\,
Hopper(Inlet)
Z,W
Isothermal Barrel at Tb
Die(out1et)
Adiabatic Screw
FIG.5. Simplified geometry of a single-screw extruder, along with the relevant boundary conditions.
the down-channel direction) is significant. Therefore, the temperature field develops along the z direction. The velocities will also change with the downstream position as a result of this change in temperature, if the fluid viscosity is dependent on its temperature. It is assumed that diffusion in the z direction is negligible in comparison with convection. This assumption can be shown to be valid for the extruder length L >> B , using a simple scaling analysis, and is borne out by experimental data. The temperature gradient in the y direction is expected to be much greater than that in the down-channel direction. The diffusion term in the y
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YOGESH JALURIA
direction is, therefore, retained. Thus, the energy equation becomes
where T is the local temperature, p the density, C p the specific heat at constant pressure, and k the thermal conductivity of the fluid. The first term on the right side is due to thermal diffusion in the y direction, and the last two terms are due to viscous dissipation. The shear stress T , , ~ and are given for this 2D flow by
where p is the molecular viscosity of the fluid. The shear rate for this 2D circumstance is given by the expression
with the viscosity p given by one of the various models, such as Eq. (l), whichever is appropriate for a given fluid. If the power-law model is used, the singularity at zero shear rate must be avoided, often by simply putting n = 1 as shear rate approaches zero, i.e., Newtonian behavior at small shear rates. The preceding energy equation, Eq. (51, is parabolic in the z direction and marching may be used to obtain the solution. A restriction to the flow is imposed by the presence of a die at the end of the extruder. The flow in the extruder is strongly coupled with that inside the die. For very narrow dies and large extruder speeds, a backflow may arise in the extruder channel in terms of the coordinate system described. This makes the problem elliptic, requiring a different approach for the solution. These considerations are discussed later in the chapter. The boundary conditions are also shown in Fig. 5. The temperature distribution at the barrel is specified as Tb(z),which, in many cases is a constant or has different values in different sections. The screw is taken as adiabatic. We may write these conditions as
159
EXTRUSION OF NON-NEWTONIAN MATERIALS
Since the energy equation is parabolic in z , boundary conditions are necessary only at z = 0 to allow marching in the z direction and, thus, obtain the solution in the entire domain. The boundary conditions at z = 0 are provided in terms of the developed velocity profiles, denoted by subscript dev, at T = T i . These are obtained by solving the momentum equations, keeping the temperature constant at T i ,by means of an implicit finite-difference scheme [331. These equations may be nondimensionalized with channel height H , barrel velocity component in the z direction Vb, , barrel and inlet temperatures Tb and T i ,and p o as characteristic quantities. The resulting dimensionless variables are
W
w * = -, 'b
fj=
T - Ti
-
p * = =P ,
Tb - Ti '
I
jj=j+
'b z
P
Pe
'bz
= -, CY
G=
k ( Tb - Ti) '
where Pe is the Peclet number, G is the Griffith number, and parameter p represents the dependence of viscosity on temperature. The dimensionless equations thus obtained are
The continuity equation for the conservation of mass is dU*
-d +X * -
dW* dZ*
=
0.
(13)
However, the constraints on the flow are generally written in integral form,
160
YOGESH JALURIA
given in dimensionless terms as
[u* dy*
=
0,
[w*dy*
=
Q/B qv = -, Hvb,
where the first condition ensures that the net flow across the channel is zero, if the leakage flow across the screw flights is negligible, and the second condition gives the down-channel flow rate. Therefore, the parameter qv is the dimensionless volumetric flow rate, generally called the throughput, emerging from the extruder. If the screw channel is not rectangular, qv is defined as Q/AVbz,where A is the channel cross section. The integral form of the continuity equation is generally used since the throughput can be specified as an operating condition. A similar nondimensionalization is used for an Arrehenius temperature dependence of viscosity, i.e., p varying as exp(b/T). For this case, two parameters, p1 and &, are defined as
p1 = Tb/Ti
and
Pz = b/Ti,
as employed by Gopalakrishna et al. [18]. Thus, the governing equations are nondimensionalized for different circumstances. Similarly, the boundary conditions are also obtained in dimensionless form. However, results are often presented in physical terms because of complicated property variation characteristics. The governing dimensionless equations are conveniently solved by means of finite-difference techniques for simple geometries. The computational domain is the rectangular channel shown. The iterative Newton-Raphson method [27] may be used to satisfy the conditions on the flow rates. The iteration is terminated when the pressure gradients satisfy a chosen convergence criterion. Using the boundary conditions in terms of u , w,and T at any upstream z location, the energy equation is solved to obtain the temperature distribution at the next downstream z location. The same approach is extended to solve the mass transfer problem. The numerical scheme is validated by comparisons with experimental data on actual, full-size, extruders, using both Newtonian and non-Newtonian fluids, as discussed later. For further details on the numerical scheme, see Karwe and Jaluria [33] and Gopalakrishna et al. [HI. 2. Fulb Developed Flow The simplest solution is that of the fully developed circumstance for which the temperature and the velocity fields are assumed not to vary downstream. The barrel is taken as isothermal and at the inlet tempera-
EXTRUSION OF NON-NEWTONIAN MATERIALS
161
ture of the fluid. Convective transport of heat is neglected, but the viscous dissipation effects are considered, so that the energy generated is lost to the barrel. Although analytic solutions can be obtained for certain channel flows driven by pressure or shear, the present circumstance, with combined pressure and shear effects, screw helix and non-Newtonian fluids, requires a numerical solution of the governing equations. Figure 6a shows the calculated w* velocity profiles for different values of q, . The characteristic curves, in terms of the throughput and the dimensionless pressure gradient, are shown for n = 0.5 and different Griffith numbers in Fig. 6b. When the pressure gradient is zero, the flow is due only to the viscous effect of the moving barrel and is termed as drugflow. For Newtonian flow, the velocity profile is linear for this circumstance and q v = 0.5. This situation is similar to the Couette flow between two parallel plates in the absence of a pressure gradient. For a favorable pressure gradient, the throughput exceeds 0.5, and for an adverse pressure gradient, it is less. The velocity profile bulges outward, with the velocity exceeding the linear variation, for the favorable case. The opposite occurs for the adverse pressure case. Similarly, for non-Newtonian flow, the profiles are seen to be strongly dependent on the throughput, although drag flow does not arise at q, = 0.5 but at a value that depends on the fluid, temperature, and other conditions. The screw channel is assumed to be completely filled with the non-Newtonian fluid. Therefore, a decrease in the throughput at a given screw speed implies a smaller-diameter die, with greater obstruction to the flow. This results in a greater pressure rise downstream and an increased adverse pressure gradient, which is reflected in decreased velocity levels. Reverse flow may also arise at very small throughputs in this coordinate system. Larger throughputs are obtained with a favorable pressure gradient, specifically, pressure decreasing downstream. In an extruder, the obstruction provided by the die and by a tapered screw in many cases increases the pressure, resulting in an opposing pressure circumstance, as seen for small values of q,. A higher Griffith number implies greater viscous heating. This results in higher temperatures and lower viscosity. This also gives rise to a smaller pressure gradient at a given throughput, in the positive-pressure-gradient range, which is of interest in extrusion. 3. Developing Flow
The results presented here are based on the coordinate axes fixed to the rotating screw. The ratio of axial screw length L to channel height H is taken as about 70, corresponding to practical extruders. This results in a dimensionless down-channel distance of around 200. Once the numerical
YOGESH JALURIA
/ / 5
-0.3
-0.1
I
I
I
I
0.1
0.3
0.5
0.7
I
I
1.1
0.9
I
I
1.3
1.5
W*
Screw helix angle = 16.0
LEGEND *=:G=0.0 a =:G=2.5 a = : G = 5.0
x( -5.0
, -4.0
1
-3.0
I
-20
I
-1.0
r
1
0.0
1.0
20
dimensionless pressure gradient
I
3.0
g aZ
I
4.0
I
5.0
FIG.6. Computed results for the fully developed case at n = 0.5 and p = 2.0: (a) profiles of the w * velocity component at G = 2.5 and different values of the throughput q v ; (b) characteristic curves showing variation of pressure gradient with throughput at various values of the Griffith number G (adapted from Kwon et al. [38]).
163
EXTRUSION OF NON-NEWTONIAN MATERIALS
results for the velocity and temperature fields are obtained, various other quantities of interest, such as the heat input to and from the barrel, local Nusselt number Nu,, bulk temperature, shear stress, and pressure at various downstream locations, including that at the die, are calculated. Only a few typical results are presented here for conciseness. Figure 7 shows the results in terms of isotherms and constant velocity lines along the unraveled screw channel. The temperature and velocity profiles at four downstream locations are also known. The temperature of the fluid far downstream is seen to increase above the barrel temperature. Therefore, beyond a certain downstream location, heat transfer occurs from the flow to the barrel if the barrel is maintained at a fixed temperature. This
0
SCREW ( adiabatic )
0
40.0
20.0
60.0
80.0
100.0
120.0
1eo.o
140.0
1~0.0
z.= z/H
.; 2
0
-0
a 0 0 l O A O 8 0 m LO 1s
-OD 00 10 4 0 so a I o ia
W.
40.0
200
60.0+
80.0
-0s 0 0 1 0 4 0 S O 8 1 0 1 I
-0800I0~08001011
W*
W*
=.
= 1000 z/H
1200
140.0
W*
160.0
180.0
0
00
00
10
9
I6
I0
0
o e 01 i e 9
10
10
00 06
I0
e
I0
LO
FIG.7. Calculated velocity and temperature fields in the extruder at n Pe = 3427, p = 1.61, and 4, = 0.3 (adapted from Kwon et al. [38]).
00
00
I0
I0
9
=
0.5, G
=
10.0,
LO
164
YOGESH JALURIA
implies initial heat transfer to the fluid by the barrel followed by heat removal from the fluid further downstream. This effect is due to the viscous heating of the fluid and varies strongly with the Griffith number. At larger Griffith numbers, the fluid temperature may be much higher than the barrel temperature. It is seen from these results that the constant velocity lines are almost parallel to the barrel, indicating very little convective mixing in the fluid, which is largely stretched as it goes through the extruder. To enhance mixing, reverse screw elements and breaks in the screws are often employed in practical single-screw extruders. Mixing is also substantially increased in twin-screw extruders, as discussed later in the chapter. It is also seen that large temperature differences exist in the fluid, from the barrel to the screw. This is due largely to the low thermal conductivity of typical polymeric materials. At relatively high values of the dimensionless throughput q v , the backflow is small and much of the fluid near the screw root remains unheated. Viscous dissipation is important and affects the thermal field substantially. Figure 8a shows the variation of the dimensionless pressure p* and the pressure gradient dp*/dz* along the screw channel. Figure 8b shows the corresponding variation of the bulk temperature 6 b u l k and the local Nusselt number Nu, at the inside surface of the barrel. These trends agree with the physical behavior in actual systems. The pressure rises toward the outlet, as does the bulk temperature. The actual values attained vary strongly with q v , as discussed earlier. The bulk temperature and the Nusselt number Nu, are defined as
where qin is the heat flux Input into the fluid at the barrel. Far downstream, the value approaches zero, indicating the small amount of energy transfer needed to maintain the barrel temperature at a given value. It may also be negative if the fluid loses energy to the barrel. 4. Three-Dimensional Transport
The basic transport processes in the extruder channel are threedimensional, although two-dimensional models, as outlined above, have been used extensively to model the flow and generate data needed for design. The main problem with such 2D models is that the effect of the flights is brought in by mass conservation considerations only, approximating the flights as being very far apart. Thus the recirculating flow gener-
. !
-1
/
3 34
3' > am
-0 d-
4 xY
SfE a 2-
i2-
4.8 a-
$1-
2 2I
0-
x7a.0
0.0
40.0
M.O
110.0
100.0
im.0
Dimensionless distance along m
110.0
1w.o
IM.O
:
w helix ZC
1' I-
Mrnenaionlen distance along arer helix
+
FIG.8. Variation of the pressure, pressure gradient, bulk temperature, and local Nusselt number in the down-channel direction for the conditions of Fig. 7 (adapted from Kwon et al. [381).
166
YOGESH JALURIA
ated in a screw channel, between two flights separated by a finite distance, is not simulated. Although this is applicable for shallow channels, 3D models are needed for deep channels and for a more realistic modeling of practical extruders. However, 3D modeling is fairly involved and not much work has been done on it [21]. A relatively simple model to simulate 3D flows has been developed by Sastrohartono et al. [59] and is outlined here. In the unwrapped extruder channel, it is reasonable to assume that the velocity vector V does not change significantly along the screw channel direction, i.e., z-coordinate direction, d V / d z << d V / d y or d V / d x . Based on these assumptions and using the coordinate system sketched in Fig. 5, the velocity vector V can be represented as
V = u( x , y ) i + u( x , y ) j + w( x , y ) k , ( 16) where i, j , k are unit vectors in the three directions. The momentum and continuity equations for the isothermal, steady-state, creeping flow are given as
du
du
dx
dy
-+-=o,
where the pressure gradient in Eq. (19) is prescribed to obtain the given flow rate and is thus distinguished from the pressure in Eqs. (17) and (18). The local shear rate is given by
(
2 ::)2
(:;
+-+-
(:;) 2
+ 2 -
Thus, a three-dimensional flow analysis for the screw channel can be carried out in terms of inplane and down-channel flows, using a 3D finite-element model, with marching in the z direction. This approach is simpler than the full 3D elliptic solution and requires much less computer
EXTRUSION OF NON-NEWTONIAN MATERIALS
167
storage and computational time. It can be used for typical extruder conditions, except when the throughput qv is of order 0.1 or less. As far as the inplane flow is concerned, the principle of virtual velocity is used to formulate the finite-element method for solving the x- and y-momentum equations, together with the continuity equation. Six-node triangular elements have been used here, with a quadratic and a linear interpolation for the velocity and pressure fields, respectively. The x - and y-momentum equations and the z-momentum equation are coupled to each other through the viscosity function in case of a non-Newtonian fluid. However, in case of a Newtonian fluid, they are independent of each other. Because of the coupled momentum equations, an iteration scheme is required to solve these equations for u, u, and w velocity components and pressure p. This is similar to the approach outlined by Kwon et al. [39]. The screw channel is assumed to be fully filled with the material. The computational domain is taken as the screw channel cross section, and any profile can easily be modeled using the finite-element method (FEM). Mesh discretizations of two examples of screw profiles are shown in Fig. 9 for a rectangular screw profile and a self-wiping screw profile, respectively. The latter is used in twin-screw extruders to allow one screw to wipe the surface of the other as it rotates, as discussed later. The two screw profiles have the same maximum width and depth, but the rectangular screw profile is assumed to have no gap between the screw and the barrel,
a
b
FIG. 9. Mesh discretization of single-screw extruder channels in the FEM model: (a) rectangular; (b) self-wiping.
168
YOGESH JALURIA
whereas the other one does. The results for the rectangular profile are close to those obtained from the finite-difference approach outlined earlier. Therefore, only the results for the self-wiping screw profile are presented here. The flow in the screw channel is simulated in terms of the inplane and the down-channel flows. Figure 10 shows typical inplane ( u , u ) and downchannel (w) velocity fields, and the temperature and pressure contours for pressure increasing in the down-channel direction. Velocity and temperature profiles are also shown. As can be seen from this figure, a recirculat-
At z'
= 90.05
Mesh discretization
u-v
vrlnsity field
\ T
w velocity contours
Temperature contours
W
SZ
Pressure contours
S 1 = barrel = 6O.W"C
SZ = licrew = adiabatic Malerid e Viscpsil qr = 0.3200 Screw speed = 60.0 rpm Maximum temperature = 6O.M)"C Minimum temperature = 32.74OC Bulk tempcrnture = 38.48"C Lalet temperature I 24.00"C
''on! ::m%:: u
velocity profile
w
Temperature profile
velocity profile
f. 0 5
0 -0.5
-0.2s
0 U*
0.25
4.5
a
0.5
W*
1.0
0
0.5
1.0
' T
FIG.10. Cross-sectional flow and temperature field from 3D modeling for Viscasil-300M at z* = 90.05, with 7; = 24"C, Tb = 40/60/80"C in the three sections, qv = 0.32, and N (rpm) = 60 (adapted from Sastrohartono et al. [591).
EXTRUSION OF NON-NEWTONIAN MATERIALS
169
ing flow arises in the channel and conveys fluid from near the barrel to the bottom of the channel. Therefore, this recirculation provides a mechanism for convective mixing of the fluid. Hot fluid near the barrel is conveyed downward in the channel, resulting in larger temperature near the bottom than that predicted by the 2D model, which is unable to capture this recirculating flow. The effect of this recirculation on the temperature field is seen very clearly in Fig. 11, where the fluid near the screw root is heated almost up to the barrel temperature by the recirculation. Therefore, it enhances uniformity of fluid temperature across the cross section. As expected, the effect increases if the throughput is reduced, resulting in grater pressure rise and recirculation. Although not shown here for purposes of brevity, it was found that this flow is significantly affected by leakage at the gaps between the screw and the barrel. Additional results from this 3D model are shown in Figs. 12 and 13. The temperature and w velocity contours are shown at the midplane of the channel, and the corresponding profiles are shown at four locations along the channel. Again, the effect of the recirculating flow on the temperature profiles is clearly seen. The barrel is divided into three sections at different temperatures in order to model the experimental study, described later. The fluid temperature increases downstream as a result of heat input at the barrel and viscous dissipation. The effect of the step changes in barrel temperature are seen in terms of sharp variations in the bulk temperature
5 -
4
E E 1M. 3 Y
.”
B -
2
B 6 1 0 65.0 67.5
70.0 72.5 75.0 77.5
80.0 82.5 85.0
Temperature [“C] FIG.11. Calculated temperature profiles from 3D modeling at the midplane of the channel for different flow rates of Viscasil-300M at 60 rpm. Cross-sectional temperature profile plotted at the end of a single screw extruder with a self-wiping screw profile: n = 0.2671 and N = 60 rpm (adapted from Sastrohartono et al. [591).
170
YOGESH JALURIA
Fluid: Viscasil.
screw speed * 60.0 Q m
Temperature contours at the center of a selfwiping screw channel
Baml: T, = 40.Do°C/60.000C/80.000C
qr = 0.3200
Ti = 24.W'C
-
Screw mot: adiabatic
Z* = 0.0
Z*
z* = 157.1
=90.1
z*
> 0.5
05
0
0 5 1 . 0
0
h
0.5
0.5
0
= 247.1
Lon
Lom
*A
z*=dH
h 0.5
0.5
1.0
0.5
0
T*
T'
T.
W*
velocity contours at the center of a selfwiping acrew channel
1.0
T'
Barrel: T, = 4O.OO'U6O.OO"C/8O.OO"C
l
. ., ,
... .
....
....
..
..... ... .. ....... ....... ..........._....... ....... .......................... .
-.
Scrcw mot: adiabatic 2.
f.
= 0.0
2'
0.5
0
h
05
10
W*
3
90.1
0.5
0
0.5
W*
1.0
2. I
I r * = d
157.1
I*
Lon
Lon
0.3
0.5
0
0.5 W'
1.0
0
= 247.1
0.5
1.0
W'
FIG.12. Velocity and temperature fields in the extruder channel from 3D modeling for the conditions of Fig. 10 (adapted from Sastrohartono et al. [W]).
and the local Nusselt number. Again, depending on the operating conditions, the barrel may have to be heated or cooled to maintain a given temperature. The pressure variation is similar to that from a 2D model. In actual practice, of course, such step changes in barrel temperature do not exist and more gradual changes are usually employed. 5. Axial Formulation
The marching procedure outlined above for the 2D and 3D modeling is quite satisfactory for a wide range of operating conditions. However, at very small values of q,, typically of order 0.1 or less, appreciable backflow
EXTRUSION OF NON-NEWTONIAN MATERIALS
.17.5
x 10‘ 5.0
2.5
-
171
x 10’
-P a
- 7.5 tx
-
e
2
e
L
P
P
5.0
2,s
Channel distance [m]
Channel distance [ml Average h = 5 I .475 [Wm-’Kl Average Nu = 3.8997 x 10’
Channel distance [ml
FIG. 13. Variation of pressure, pressure gradient, bulk temperature, local heat transfer coefficient, and Nusselt number along the screw channel from 3D modeling for the conditions of Fig. 10 (adapted from Sastrohartono et al. [59]).
due to opposing pressure arises in the extruder channel, in the movingbarrel coordinate frame employed here for analysis, and the numerical schemes used earlier do not work. This is due largely to flow and convection in the negative z direction resulting from the reverse flow. As discussed by Chiruvella et al. [71, two different approaches may be employed to obtain a solution under these circumstances. The first one solves the elliptic problem instead of the parabolic problem, in which marching is used, by including diffusion of energy in the down-channel direction. However, this solution requires boundary conditions at the end of the screw channel, and the numerical computation is quite involved. The flow through a die may be solved and then coupled with the extruder to obtain the needed downstream boundary conditions, or developed conditions may be imposed at the end of an extended computational domain. The second
172
YOGESH JALURIA
approach uses a different coordinate frame, involving the axial direction parallel to the screw axis. At each point in the extruder channel, the axial and tangential directions are considered. This is known as the a i d formulation, which can be visualized as the flow along the screw axis as one looks at the extruder from the front. Since there is no backflow occurring in the direction parallel to the screw axis;one can choose to march along this direction from the hopper to the die for essentially any throughput. Both these approaches were found to yield results that are fairly close for typical extruders. However, the axial formulation is preferable because of the much smaller computational effort needed for the marching scheme as compared to the elliptic formulation. Figure 14 shows the geometry and coordinate system for the axial formulation. The axes are in the direction of the screw axis and along the tangential direction; the third direction is along the channel height, as before. It can be shown analytically that, for an isothermal Newtonian fluid, the down-channel and axial formulations yield the same pressure and velocity distributions [36]. The same concept is used for nonisothermal, non-Newtonian flow. The governing equations are derived as before to yield
where x and 5 are the tangential and axial coordinates. The velocity components in these two directions are
<
=
w cos #J - u sin 4,
i$
=
w sin
#J
+ u cos I$
(24)
The integral equations for mass conservation are obtained as
L H B cdy
=
Q cos 4,
L H B Q dy
=
Qsin 4.
The shear rate is given by
[(%)i. (32] 1/2
y =
and the energy equation in these coordinates is
(25)
EXTRUSION OF NON-NEWTONIAN MATERIALS
173
* !
Screw Rotation
B Barrel
Screw
FIG. 14. Geometry and coordinate system for the axial formulation (adapted from Chiruvella et at. [7]).
The governing equations can be solved by the marching procedure outlined earlier. The velocity components are zero at y = 0 and at H, except for which is simply V, there, and the thermal conditions are the same as before. Figures 15a,b show the results from the elliptic formulation, indicating the recirculating flow in the down-channel coordinate system. The corresponding streamlines in the axial formulation are shown in Fig. 15c. It is seen that the velocity component in the direction parallel to the screw axis is always positive. The variation of pressure in the axial direction is shown in Fig. 16a, indicating expected trends. A comparison
c,
174
YOGESH JALURIA
Fluid:Viscasil-300M,T = 80'C,T, b
a
= 25'C, N = 80, M = 8.21 Isotherms
z/H
Streamlines
1.0;
I
02501
0.2 7
0.0825
I O-O b.0
42.5
85.0
127.5
170.0
255.0
z/H FIG. 15. Isotherms and streamlines along the screw channel obtained for Viscasil-300M at Ti = 2YC, Tb = 80"C, N = 80 rpm and rn = 8.21 kg/h, with (a), (b) from the elliptic formulation and (c) from the axial formulation (adapted from Chiruvella et al. [71).
between the results for pressure at the die from the two formulations, along with experimental data from Esseghir [13] for Viscasil-300M7 is shown in Fig. 16b. Close agreement is found between all three, lending strong support to the two models. The axial formulation is particularly attractive because of the applicability of the marching procedure, which has much small computer storage and computational time requirements than the elliptic code.
..
25.0
.
I
I . , ,
I
x
I
I . .
1
,..
I
I , .
I . .
s
..I..
1
.
,
7
8 20.0
-
!2
T x15.0
p!
010.0
d
-
A
-
8
-
-
Q 5.0
-
0.0
”
-
Experiment Elliptic Axial ~
”
~
~
”
~
~
~
~
~
~
FIG.16. (a) Variation of pressure with axial distance; (b) prediction of pressure at the die by the elliptic and axial formulations, along with experimental results (adapted from Chiruvella et al. [7]).
~
~
176
YOGESH JALURIA
B. TAPERED SCREW Single-screw extruders are often tapered to increase the pressure generated in the extruder channel. The' barrel is typically cylindrical with grooves on the inside surface to promote mixing and the screw root is continuously tapered from the feed section to the die. This taper angle corresponds to a change in channel depth from H I at the inlet to H2 at the outlet just before the die. The height H is then a function of z . The outlet height H, may be taken as the characteristic dimension in the nondimensionalization given by Eq. (9) so that the throughput qv becomes q v = (Q/Z3)/(H2Vbz). The local height H(z)may be used to nondimensionalize the local shear rate and the y coordinate (see Gopalakrishna et al. [18]). An additional parameter H , / H , , or the taper angle JI = arctan[(H, - H 2 ) / L ] ,where L is the helical length of the screw channel, arises in the analysis. The results of the simulation for a typical taper angle of 0.1432" are shown in Fig. 17. In the figure, the taper section is shown enlarged for clarity. There is a maximum in the pressure profile at a distance of about two-thirds of the total channel length. This arises because with a narrowing channel, the pressure must decrease to maintain the given flow rate. This results in a maximum in the pressure upstream of the die. As the non-Newtonian material flows in a tapered channel, it is squeezed into the decreasing gap. The spacing of the streamlines is seen to become closer as the fluid moves toward the die. The velocity profiles at the four downstream locations show a markedly different behavior for tapered channels as compared to the case with no taper. It may be noted that the initially curved velocity profile becomes linear (corresponding to drag flow) as the section becomes shallower. The temperature profile becomes almost completely uniform across the channel depth in the regions close to the die for a tapered channel. Therefore, the pressure rise is quite different in the case of a tapered screw and greater uniformity arises in the temperature field, as compared to a nontapered screw.
C. RESIDENCE-TIME DISTRIBUTION A very important consideration in the extrusion process is the residence-time distribution (RTD). This is an indication of the amount of time spent by a fluid particle in the extruder from the hopper to the die. If the material spends an excessive amount of time, it may be overprocessed, overcooked, if it is food, or degraded. Similarly, too small a time may lead to underprocessing. The final product is, therefore, strongly dependent on the RTD since structural changes due to thermal processing and chemical
EXTRUSION OF NON-NEWTONIAN MATERIALS
177
reactions are usually time-dependent. The residence time is experimentally obtained by releasing a fixed amount of color dye or tracer in the material at the hopper and measuring the flow rate of the dye material as it emerges from the extruder at the other end. The time it takes for the dye to first appear is the minimum residence time and related to the fastestmoving fluid. Similarly, an average residence time may be defined. The experimental determination of residence time may be numerically simulated by considering the flow of a slab of a dye as it moves from the hopper to the die, as sketched in Fig. 18a. If the velocity field is known from the solution of the governing equations, a fluid particle may be traced by numerically integrating the velocity over time. The axial component of the velocity is used to trace the particles. As expected, the particles near the barrel and the screw take a very long time to come out, as compared to the particles near the middle portion of the screw channel. This yields the amount of color dye emerging from the extruder at a function of time and may be used to obtain the minimum, average and the distribution of the residence time. Figure 18b shows these results in terms of the dye flow rate, normalized by the total flow rate. Clearly, most of the dye emerges over a short time interval, with the extended regions representing fluid near the barrel and the screw root. The calculated cumulative function F( t ) , which indicates the cumulative fraction of the total amount of dye emerging up to time t , is defined as
where f(t> dt is the amount of material that has a residence time between + df. The average residence time 1 is then calculated from
t and t
1 = /oatf(t) dt
V =
7,
Q
where I/ is the total internal volume of the extruder and Q is the flow rate. F ( t ) is plotted as a function of time in Fig. 18c, along with the distributions for a few other flows. It is seen that, although the basic trends are similar, the RTD is affected by the fluid and the flow configuration. It was found that the residence time distribution was only slightly affected by the barrel temperature. It was mainly affected by the throughput q v ,which substantially influences the flow field. Results at different operating conditions have been obtained in the literature and used for selecting the appropriate conditions for a given material or thermal process. The RTD in extrusion, including the three-dimensional circulatory flow, has been investigated by Joo and Kwon [29].
a
~ ~ O ~ H E R M Sn= 0.31, qv= 0.70, Pe= 1192.0, G= 0.000037
BARREL ( T = Tb)
100.0
200.0
400.0
"f"
700.0
600.0
z* = zm2
R
o
n
i
n
a
o
u
4 4
FIG. 17. Calculated results for a tapered extruder channel at n angle
= 0.1432":(a)
isotherms and temperature profiles.
=
0.31, qv (based on exit channel height)
=
0.7, Pe
=
1192, G
=
3.7
X
lo-' and taper
b STREAMLINES
BARREL ( T = Tb)
........ ............................. ................................. .................................. .....................................
-
1
100.0
0.0
t
itiz
300.0
200.0
t
500.0
z* = zm2
600.0
700.0
800.0
t
t
a
3
z - 0 3 0.0
0 3 0 4 0.4 0 1 19 13
W*
FIG.17. Continued. Calculated results for a tapered extruder channel at n taper angle = 0.1432": (b) streamlines and velocity profiles.
=
0.31, qv (based on exit channel height)
=
0.7, Pe
=
1192, C
=
3.7 x
and
180
YOGESH JALURIA
C
1,
FIG. 17. Continued. Calculated results for a tapered extruder channel at n = 0.31, qy (based on exit channel height) = 0.7, Pe = 1192, G = 3.7 x lo-' and taper angle = 0.1432": (c) down-channel variation of the pressure and pressure gradient.
D. MIXINGCHARACTERISTICS
It is important to study the flow undergone by the material particles as the fluid moves downstream. This leads to a better understanding of the mixing inside the screw channel. By studying inplane and axial velocity fields obtained from the FEM model, one can introduce particles inside the screw channel and follow the movement of these particles along the channel [56]. A very small time interval is taken for the numerical integration of the velocity field to obtain particle paths. The results of such particle tracings are presented in Fig. 19a in different views, for the case of an adverse pressure gradient. Four representative particles were traced for a given channel length of 1.5 screw turns. As can be seen in this figure, the particles undergo spiral movements, except particles near the barrel surface (e.g., particle A), which go straight across the flight gap into the
-
b
.- %Y
U Y
f
C
p9
0
M
d h 3
t: 9
LEGEND
0
0 0 0 W
p!
0
= Single-Screw Extruder = Pipe Flow (Newtonian) = Channel Flow (Newtonian)
= Plug Flow
E 0.0
0.5
1.0
1.5
2.0
2.5
3.0
tl T FIG. 18. Residence-time distribution (RTD) calculations: (a) schematic diagram showing the dye slab and the computational domain for RTD calculations; (b) variation of the dye flow rate, normalized by the total flow rate, with time for the conditions in Fig. 7 with Pe = 5000, (c) variation of the cumulative distribution function F ( t ) for different flow configurations, with ? as the average residence time (adapted from Kame and Jaluria [33]).
-* 182
YOGESH JALURIA
a
A
D
L
C
X
A
i" z
z
Y
FIG.19. Mixing characteristics in a single-screw extruder channel: (a) particle tracings of coupled cross- and down-channel flows in the screw channel with an adverse pressure gradient for a non-Newtonian material.
EXTRUSION OF NON-NEWTONIAN MATERIALS
b
183
Distributive Mixing Time Interval =0.15 sec.
a
b
C
d
e
f
9
h
i
i
k
I
FIG. 19. Continued. Mixing characteristics in a single-screw extruder channel: (b) time sequence of distributive mixing inside the screw channel (adapted from Sastrohartono and Kwon [56]).
184
YOGESH JALURIA
adjacent channel. The backflow is represented by particles C and D, which flow backward at the bottom of the screw channel. The spiral movement of the particles inside the screw channel clearly promotes mixing within the single-screw extruder. This recirculating flow is not captured by the simpler 2D model. The distributive mixing inside the channel may also be represented in terms of mixing between two different types of materials, shown as white and black portions in Fig. 19b with each initially occupying one half of the channel. These are followed with time as the fluid moves in the channel. Clearly, the process is a slow one, although the materials are eventually mixed with each other as time elapses. Several other measures of mixing have been considered in the literature. Kwon et al. [40] studied kinematics and deformation to characterize mixing. Substantial work has also been done on mixing in twin-screw extruders [30].
E. EXPERIMENTAL RESULTS Extensive experimental data on the extrusion process are available in the literature. However, most of these concern the practical issues in extrusion such as temperature and pressure at the die, residence-time distribution, total heat input, characteristics of the extrudate, distributive mixing in the screw channel, total torque exerted on the screw, and flow rate. Much of this information is reviewed in books such as those by Tadmor and Gogos [62], Harper [24], and Rauwendaal [54]. However, very few studies have focused on the temperature and velocity distribution in the channel, temperature and pressure variation with axial distance, viscous dissipation effects, and local heat transfer rates. Over the past decade, Sernas and coworkers [13-15] have carried out well-designed, accurate, controlled, and innovative experiments on single- and twin-screw extruders. These results have been used for the validation of the analytical and numerical models presented here, as well as for providing a better understanding of the basic heat transfer and fluid flow processes associated with extrusion. The experimental systems used and the results obtained are briefly discussed in this section and in Section IV. Figure 20 shows a schematic of the single-screw extruder used for these experiments. A 30.645-mm-diameter screw with the self-wiping profile of a Werner & Pfleiderer ZSK-30 twin-screw extruder is employed, with a helix angle of 16.2'. The various parts of the experimental system, including the die, which is interchangeable, are shown. A detailed sketch of the heating jackets and measuring stations is also given. A Plexiglas window can be fitted at any one of the measuring ports to provide optical access to the flow to observe the extent of fill in the screw channel. The three sections of
EXTRUSION OF NON-NEWTONIAN MATERIALS
185
7. Torque arm & force cell 8. Die system
4. Cam & cam shaft 10. Bar-coded wheel .. 5. Steel channel support 11. Timing b i t 6. Measuring sites 12. Forced-feed tube (1 of 2)
b
Heating Fluid Channels
\ DIE
3rd
2nd
1st’
Measuring Stations
FIG. 20. (a) Schematic three-dimensional view of the single screw experimental facility; (b) detailed sketch of the heating jackets and measuring stations (adapted from Esseghir and Semas [15]and from Chiruvella et al. [7]).
the barrel can be maintained at different uniform temperatures by the use of circulating water jackets. The pressure and temperature are measured at the measuring stations. For further details on the experimental facility and the instrumentation, see Esseghir and Sernas [14, 151. Measurement of the temperature profile in the screw channel is complicated because of the rotating screw. A cam-driven thermocouple system, as shown in Fig. 21a, is installed on the extruder to allow the thermocouple
186
YOGESH JALURIA
a
Inside of Barrel Wall
Barrel Wall
-. FIG.21. (a) Schematic view of the cam-driven thermocouple for temperature measurements in the screw channel; (b) schematic representation of the locus of points where temperature data are collected (adapted from Esseghir and Sernas [15]).
EXTRUSION OF NON-NEWTONIAN MATERIALS
187
probe to travel in and out of the channel in a synchronized motion linked to the screw rotation. The probe moves into the channel to a preset distance while the flights traverse as the result of screw rotation. Figure 21b gives a schematic representation of the locus of points where data are collected. As expected, considerable amount of care is involved in extracting the appropriate data for the temperature profile D51. Some characteristic experimental results for Viscasil-300M are shown in Figs. 22 and 23, along with numerical results from the 3D finite-element calculations. Figure 22 shows the variation of the pressure measured at the die with the mass flow rate and also the axial variation of the pressure. A close agreement between the experimental and numerical results is observed, providing strong support to the model. The die pressure rises with the flow rate, which is increased by raising the screw speed in this figure. The die is kept unchanged, and a higher rotational speed leads to a higher pressure, as expected. A closer agreement is obtained when the actual, self-wiping, profile of the screw is simulated, rather than a rectangular screw. Also, the pressure increases as the fluid moves from the inlet toward the die. An essentially linear variation is obtained. Good agreement is also obtained between the measured and predicted die temperatures, as shown in Fig. 22b. A few typical measured temperature profiles are shown in Fig. 23, along with numerical predictions. The effect of recirculation in the screw channel is seen in the temperature near the screw root being closer to the barrel temperature, than that predicted by the 2D model. Clearly, the 3D model captures this recirculation and provides much better agreement with the experimental data than the 2D model. Similar trends were observed for measurements and calculations for several other operating conditions.
lV. Twin-Screw Extruder Twin-screw extruders are used extensively in the processing of polymeric materials and in operations that include pumping, polymer blending, and distribution of pigments and reinforcing materials in molten polymers. Many applications are found in the processing of food, pharmaceuticals, paints, and many other highly viscous materials. In twin-screw extruders two screws lie adjacent to each other in a barrel casing whose cross section is in a figure-eight pattern. Twin-screw extruders are of many types, such as intermeshing, nonintermeshing, corotating, counterrotating, to name only a few. Screws that rotate in the same direction are called cororaring; screws rotating in opposite directions are known as countemorating twinscrew extruders. Depending on the separation between the axes of the two
188
YOGESH JALURIA
30
,-
Single-screwextruder characteristic experimental and FEM results: T m l = 80°C. fluid: Viscasil-300M
o Experimental results o FEM,rectangular screw E M . selfwiping screw
0
I
I
3
I
I
1
6 9 12 Mass flow rate [kg h"]
I
15
Axial pressure distribution: non-Newtonian fluid (Viscasil). barrel temperature= 25°C. inlet temperature = 18.9"C. qv = 0.34. N = 10 rpm, die diameter = 5.5 mm 54-
3-
m,Tdie = 25.1 "c Experimental, Tdi. = 24.2"C 0
200
400
600
800
IWO
I200
Channel length [mm]
FIG.22. Comparisons between 3D (FEM) predictions and experimental results for a single-screw extruder, using Viscasil-300M as the fluid: (a) variation of die pressure with mass flow rate for Ti = T,, = 80"C, and cylindrical die of 2.4-mm diameter; (b) axial variation of pressure for T, = 18.9"C, Tb = 25.0"C, N = 10 rpm, 5.5-mm-diameter die, and qv = 0.34 (adapted from Sastrohartono et al. [59]).
screws, twin-screw extruders are classified as intermeshing or nonintermeshing extruders. If the distance between the screw axes is less than the diameter at the tip of the screw flight, then one screw intermeshes with the other and thus yields an intermeshing twin-screw extruder. Otherwise, it is known as a nonintermeshing twin-screw extruder. When the distance be-
189
EXTRUSION OF NON-NEWTONIAN MATERIALS
a 5.-
0; *'b,
* q
4-
o
z-
h
0
4
30
8
;
.Q
4d
o f )
2-
0 0
il b
1-
0.
Temperature ["C]
b
8
4l 3
I
0
- Numerical
5
hpn-
o~~~pm~iccion
Experimental data
I
:I
I
I
5-
4-
***\
s
3-
L
x
I
x
2-
0
*
I-
2-
*
I-
0
0
I
I
I
I
J
0
I
l
l
I
]
FIG.23. Comparisons between numerical and experimental results on temperature profiles for Viscasil-300M, with (a, c) from the 3D (FEM) model and (b, d) from the 2D (FDM) model. For (a, b): T, = 20.3"C. Tb = 12.2"C, N = 20); for (c, d): T, = 18.8"C, T , = 22.3"C, N = 35 (adapted from Sastrohartono et al. [59]).
tween the screw axes is equal to the radius at the screw root plus that at the screw tip, so that the flights of one screw wipe the root of the other screw, the extruder is known as a filly intermeshing, self-wiping twin-screw extruder. Booy [41 proposed a mathematical model for the isothermal flow of a Newtonian liquid through corotating twin-screw extruders. Both completely filled and partially filled cases were investigated in that study.
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YOGESH JALURIA
Denson and Hwang [ l l ] developed expressions for the volumetric flow rate, or throughput, as a function of the axial pressure gradient using the actual channel geometry. They included the effects of both the channel width and depth on the flow. Maheshri and Wyman 1471 have also investigated the combined cross- and down-channel flow in an idealized leakproof intermeshing twin-screw extruder. Szydlowski and White [611 have formulated a composite model for the intermeshing corotating twin-screw extruder. The formulation is also applicable to single-screw and nonintermeshing, counterrotating twin-screw extruders. Todd [64] correlated the heat transfer coefficients in a corotating twin-screw extruder. Larsen and Jones [421 also studied the heat transfer in twin-screw extruders. They assumed the polymer to be mixed adiabatically in the intermeshing region and transferred to the other screw. Wang and While [66] have modeled the flow of non-Newtonian fluids in self-wiping corotating twin-screw extruders using the flow analysis network technique. Sastrohartono et at. [571 conducted a numerical and experimental study of the flow in the nip region of a partially intermeshing corotating twin-screw extruder. The mixing patterns in a corotating twin-screw extruder were studied by Gotsis and Kalyon [191 to gain fundamental insight into the dynamics of the flow during mixing. The numerical simulation of the transport process in a partially intermeshing tangential twin-screw extruder was carried out by Kwon et al. [38]. Sastrohartono et al. [58] also studied the nonisothermal flow in tangential and self-wiping twin-screw extruders using the finiteelement method. A. MODELING 1. Translation and Intermeshing Regions
The flow domain of a twin-screw extruder is a complicated one, and the simulation of the entire region is very involved and challenging. In order to simplify the numerical simulation of the problem, the flow is divided into two regions: the translation (TI region, and the intermeshing or miring (M) region, as sketched in Fig. 24. This figure schematically depicts a section taken normal to the screw axis of a tangential corotating twin-screw extruder, as sketched in Fig. lb, and the two regions. The counterrotating case is also shown. Because of the nature of the flow and geometric similarity, the flow and heat transfer analysis in the translation region is carried out in a manner identical to that for a single-screw extruder. Therefore, this region is approximated by a channel flow. The intermeshing, or mixing, region is represented by the geometrically complex central portion of the extruder, between the two screws. The two regions are
EXTRUSION OF NON-NEWTONIAN MATERIALS
191
Mixing Region Translation Region
Screw Root Barrel Wall Counter-rotating
Mixing Region Translation Region
-
1 . Screw Barrel Wall Root
Corotating
Translational Region
Mixing Region
Translational Region
FIG. 24. Schematic diagram of the cross section of a tangential twin-screw extruder, showing the translation (T) and intermeshing (M) regions.
192
YOGESH JALURIA
separated by a hypothetical boundary used for numerical calculations only. The location of the interface is chosen such that numerical results are essentially independent of its location. This approach can also be used for other types of extruders, particularly the self-wiping twin-screw extruder, in which one screw wipes the other as it rotates, causing the entire flow from one channel to move to the other after the intermeshing region. A sequence of T-M-T-M- . .. regions represents the flow in a twin-screw extruder, as depicted in Fig. 25 for corotating tangential and self-wiping twin-screw extruders. The two regions are solved for alternately and these are coupled at the interfaces between the two regions, using proper boundary conditions. For the tangential screw, a flow division ratio xf may be defined where this ratio represents the fraction of the flow from one screw channel that goes to the other channel after the M region. Thus, xf is unity for the self-wiping case, if leakages are neglected, and close to zero if the two screws are far apart and do not intermesh. For further details on this model for the twin-screw
a Translation Region lntermeshing Region Translation Region
b Translation Region Intermeshing Region Translation Region
FIG. 25. Flow sequence in the multiple computational domains of corotating tangential (a) and self-wiping (b) twin-screw extruders.
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193
extruder and on the numerical scheme, see Kalyon et al. [311, Kwon et al. [38], and Sastrohartono et al. [58].
2. Finite-Element Approach A finite-element model was developed by Sastrohartono et al. [58] to simulate the flow inside the intermeshing zone of a corotating tangential or self-wiping twin-screw extruder. It is assumed that there is no gap between the screw flight and the barrel. The boundary conditions for the M region are as follows: 1. At the inlets, a chosen velocity profile is applied. 2. At the outlets, a zero-traction component in the down-channel direction is applied, along with zero velocity in the direction normal to it. 3. At the barrel surface, the no-slip boundary conditions are applied. 4. At the screw root surface, the surface velocity for a given screw rotational speed is applied. Several combinations of boundary conditions have been used here. As for the boundary conditions at the inlet, two velocity profiles are of particular interest. The first one is a linear velocity profile; the other is the velocity profile obtained in an annulus with a rotating inner surface and a stationary outer surface, at a given level of annular pressure gradient. In the plane of the intermeshing region considered, the x coordinate is taken along the line joining the axes of the two screws and the y coordinate perpendicular to it. Some of the typical results for a tangential corotating twin-screw extruder intermeshing region are shown in Fig. 26, along with the mesh discretization employed. The typical inplane and down-channel velocity fields, and temperature, shear rate, and pressure contours are shown for a linear velocity profile applied at inlet. It is clearly seen that in the region away from the central part, the inplane velocity field is very uniform and the pressure contours show an essentially constant pressure gradient. The location where the velocity and pressure fields begin to deviate from the uniform distribution represents the boundary between the T and M regions. This boundary is found to be located very close to the central part. It is also to be seen that the material from one screw channel is divided into two different flow paths in the M region. Material near the screw flows in the same channel, whereas material from the other part, near the barrel, flows into the other channel. This flow behavior represents the
194
YOGESH JALURIA
Mesh Discretization
u-v Velocity Field
Temperature Contours
Shear Rate Contours
w Velocity Contours
Pressure Contours
FIG.26. Mesh discretization for the mixing region in a corotating tangential twin-screw extruder, along with typical computed results for low-density polyethylene (LDPE) at n = 0.48, Tb = 320°C, Ti = 22OoC,N = 60 rpm, qv = 0.3.
Mesh Discretization
Temperature Contours
u-v Velocity Field
Shear Rate Contours
w Velocity Contours
Pressure Contours
FIG.27. Mesh discretization for the mixing region in a corotating self-wiping twin-screw extruder, along with typical computed results for the conditions of Fig. 26.
EXTRUSION OF NON-NEWTONIAN MATERIALS
195
mixing that occurs inside the M region of a corotating tangential twin-screw extruder. Clearly, very little mixing occurs in the T region, as seen from the pressure contours, and large mixing occurs in the intermeshing zone. For the self-wiping extruder, shown in Fig. 27, the entire flow from one screw channel goes to the other. However, the large changes in the shear rate and pressure contours indicate the increased amount of mixing in the intermeshing region. To obtain a complete solution for the twin-screw extruder, the model for the translation region, which is the same as the model for a single-screw extruder channel, and the model for the intermeshing zone may be coupled together at appropriate locations. The translation region may be modeled by the simpler 2D or the more realistic 3D model, using finitedifference or finite-element methods. Employing the former, Fig. 28a shows the variation of the resulting pressure variation with axial distance in the extruder including both the translation and mixing regions. Both types of extruders are considered. As expected, it is found that the throughput in the screw channel is an important parameter. Figure 28b shows the corresponding variation of the bulk temperature in the fluid flow. Only a small portion of the screw channel is shown here. However, the approach may be employed over the entire extruder to obtain the overall pressure and bulk temperature rise. The pressure and temperature variations are quite similar to those for a single-screw extruder. However, the mixing is improved and the fluid is subjected to greater shear in the intermeshing region. In addition, a twin-screw is easier to maintain and control, because of greater flexibility provided by the presence of two screws. The self-wiping extruder is particularly useful for chemically reacting materials such as food since the wiping action ensures that excessive time is not spent by any material in the extruder. 3. Finite-DifferenceApproach
The finite-element method is appropriate for the modeling of the complex geometry of the intermeshing region. However, this approach requires substantial computer storage facilities and computational time, making it difficult to carry out the simulation on small computers such as workstations or personal computers, which are of considerable interest for industrial use. The computational effort can be considerably simplified if finite-difference or finite-volume methods can be used. This is possible if the intermeshing region can be approximated to allow easy discretization with uniform or nonuniform grids. Chiruvella et al. [5] approximated the intermeshing region of a self-wiping, corotating twin-screw extruder to develop a control-volume-based numerical scheme similar to the
196
YOGESH JALURIA
a mo 20 0
Legend
r
0.0
0.2
0.4 0.6 Axial Distance, [m]
0.8
1 .o
FIG. 28. Results of the coupling of the two regions: (a) variation of pressure and (b) variation of the bulk temperature, in tangential and self-wiping corotating twin-screw extruders for different flow rates under the conditions of Fig 26.
SIMPLER algorithm [51]. Figure 29 shows a cutaway view of the extruder considered and the cross section of the screws in a plane that winds along the helical path of the screw channel. The fluid flow in the screw channel is also shown along with the translation and intermeshing regions. The moving-barrel formulation is again used. Figure 30 shows the intermeshing
197
EXTRUSION OF NON-NEWTONIAN MATERIALS
region in greater detail and the approximation of this domain to simplify the calculations. The flow in the intermeshing region is three-dimensional with the flow shifting by the flight width as it goes from one channel to the other (Fig. 29) and gives rise to a velocity increase in the x direction. The governing equations in terms of the coordinate system of Fig. 30 are
2 dz
=
z( +( ; F$)
FZ), dW
dy du
-+ dx
du -
dy
dw +=o, dz
(33)
d
-(uT) dy
+ = [(;)+’
+ -(wT) dz (3+2(;)’+*(2)*+
;(
+
$)’I
I/’ ‘
(35) As discussed earlier, solutions are obtained for the translation and intermeshing regions and linked at the interface, or overlapping region, as shown in Fig. 30. The screw channels are assumed to be completely filled and leakage across the flights is neglected. For further details on the numerical scheme, see Chiruvella et al. [5]. Figure 31 shows calculated isotherms and pressure contours in the intermeshing region at various locations in the axial direction. The flow turns from one channel to the other with mixing resulting from the change in geometry. Figure 32 shows the pressure and temperature rise when the translation and intermeshing regions are coupled. A good agreement with earlier finite-element results is observed. Additional results at different operating conditions and materials were obtained by Chiruvella et al. [5].Viscous dissipation was found to increase the temperature in the fluid above the barrel temperature for typical operating conditions. This model is much simpler than the finiteelement model, requiring much less storage space and central processing
198
YOGESH JALURIA
t s
TS
FIG.29. A corotating twin-screw extruder with a self-wiping screw profile: (a) cutaway view; (h) cross-sectional view of the screw channels along the helical path (adapted from Chiruvella el al. [5]).
unit (CPU) time, thus making it attractive in modeling practical problems and in design for industrial applications.
B. EXPERIMENTAL RESULTS As discussed earlier for single-screw extruders, extensive experimental data are available for twin-screw extruders as well. But much of this information is on flow rates, residence time, torque exerted, extrudate temperature, and die pressure. Detailed results on temperature and velocity fields, temperature, and pressure rise along the screw axis, mixing characteristics in the intermeshing region and heat transfer rates have been of interest in very few studies. Some of these results are discussed here. Sastrohartono et al. 1573 carried out a numerical and experimental study of the flow in the nip region of a twin-screw extruder. Two rotating Plexiglas cylinders were driven by a variable-speed motor and the flow in
EXTRUSION OF NON-NEWTONIAN MATERIALS
199
b
,.... ...... ~
A
Section AA
FIG.29. Continued.
the region between the two cylinders was observed. Corn syrup and carboxymethyl cellulose (CMC) solutions were used as the fluids; the former was Newtonian and the latter, non-Newtonian. Their measured viscosities were given in Fig. 3. Air bubbles and dyes were used for visualization. Figure 33 shows the experimentally obtained streamlines in the region between the two cylinders, along with the numerical predictions. Clearly, good agreement is seen between the two. It is also seen that some of the fluid flowing adjacent the left cylinder continues to flow adjacent to it while the remaining goes to the other cylinder. This process is similar to the movement of fluid from one screw channel to the other in a tangential twin-screw extruder. A flow division ratio xf may be defined in terms of Fig. 34a as the fraction of the mass flow that crosses over from one channel to the other. A dividing streamline that separates the two fluid streams was determined from the path lines and used to determine the flow division ratio. Figure 34 shows a comparison between experimental and numerical results, indicating good agreement at small Reynolds numbers. As the Reynolds number increases, the assumption of negligible inertia terms in the momentum equation is expected to become invalid and a deviation between numerical and experimental results was observed for Reynolds numbers greater than around 1.0. These experiments indicate
200
YOGESH JALURIA
lntenneshingor Nip Region
FIG.30. Simplified domain of the intermeshing region (adapted from Chiruvella et al. [S]).
the basic features of the mixing process in the intermeshing region, even though only cylinders are considered and the effect of the flights is not included. The flow division ratio may be taken as a measure of mixing. Velocity measurements are quite involved because of the complex geometry and rotating screws. Kame and Sernas [34, 351 have carried out measurements of the fluid velocity field for heavy corn syrup that is transparent. A Plexiglas window was used for visual access to the twin screws in the extruder. A two-component laser Doppler anemometer (LDA) in the backscatter mode was used to measure the local velocities in
EXTRUSION OF NON-NEWTONIAN MATERIALS
20 1
the extruder. As expected, the flow was found to be very complicated and three-dimensional. The flow field in the translation region could be compared with the numerical prediction and is shown in Fig. 35. A fairly good agreement is observed, lending support to the model and the experimental procedure. Tangential and axial velocity components were also measured in the intermeshing region. Leakage across the flights was found to be significant, and the 3D nature of the flow field was evident in the results. V. Flow in Dies The heat transfer and flow in an extrusion die are very important in determining the characteristics of the extrudate and the operating conditions of the extruder since the flow rate through the die is fixed by the pressure difference across the die. The pressure generated in the extruder must match the pressure needed to force the fluid through the die at the given flow rate. Therefore, the flow rate, or throughput, which is a very important parameter in extruder analysis, is fixed by the die. At a given screw speed, a smaller-diameter die causes a greater obstruction to the flow. It, thus, reduces the flow rate and increases the pressure rise so that the resulting pressure can force the fluid through the narrower die. Different geometries of die are used in industry to obtain different extrudate characteristics, such as shape, density, homogeniety, and expansion after extrusion. In addition, complicated dies are used for coextrusion to combine materials, for making composites, and for extruding different materials. A considerable amount of literature exists on different types of dies, their heating and cooling, design, and analysis I481. A few important aspects are outlined here. A.
COUPLING OF
EXTRUDER WITH DIE
The analysis of the flow and heat transfer in the extruder channel yields the pressure at the entrance to the die, whereas an analysis of the die yields the pressure needed for the given flow rate. A correction procedure may be used to match the two, as done by Chiruvella et al. [61 using a Newton-Raphson correction scheme. Different dimensions were used for two geometries of dies, as shown in Fig. 36. The relationship between the pressure drop A p across a cylindrical region and the flow rate for a non-Newtonian fluid was taken from Kwon et al. [39]. The expression given is
R
4n
prR3
a
Iml-
in Ihe Frd Inln-mhinp A
m
Imlhemu in Ihe 11'" Ima-fneshng Regan
FIG.31. Temperature and pressure contours in the intermeshing region at different axial locations, with polystyrene as the fluid, the barrel at 250°C and the inlet 220°C (adapted from Chiruvella et al. [51).
FIG.31. Continued.
204
YOGESH JALURIA
.
. 2.2E-2 . .
-
f
P'=P'pv2avg
. 1.1E-2-
non-Newtonian fluid = 2000 eXp(-O.Ol(T-(20+273.15)) (y) -Os2 p = 750 kg/m3
Cp = 2500 J/kgK Flow Rate = 9.27 kg/h Screw Speed = 60 rpm Barrel at 145'C, Inlet at 45'C qv = Q/(AVb,) = 0.3
3
A = 4.102e-5m2
/Y/
,*'
O.OE-O?.~.~.-.-C
, / / '
.
.
I
I
/
. .
~
-
///
0
.? .
. .
. .
I
I
. .
'
'
. .
.
ZJH Bulk Temperature Profile in ZSK-30 I
.
"
'
I
~
.
~
~
,
'
0.0
'
I
~
'
c -
4
/ -
,/--0
-0.2
@' / ,
-
/' non-Newtonian fluid p = 2000 exp(-o.O1(T-(20+273.15)) (7)
//4
p = 750 kg/m3 Cp = 2500 J/kgK
Flow Rate = 9.27kg/h Screw Speed = 60 rpm Barrel at 145'C. Inlet at 45'C qv z Q/(AVb2) = 0.3 A = 4.102~-5 m2 .
0
I
100
.
.
.
.
I
.
.
_
.
300
200
(
,
.
.
.
400
,
,
.
.
.
500
ZJH
FIG. 32. Comparison between the results obtained from finite-volume and finite-element approaches, with the latter shown as points, for a corotating, self-wiping, twin-screw extruder.
where R is the radius and L is the length of the cylindrical pipe, n is the power-law index of the fluid, and &TI is a temperature-depe9dent coefficience in the viscosity expression, which is given as p = C ( T ) ( + Y n . Similarly, expressions for a conical die and for an orifice are given by Kwon et al. [39]. Using these expressions, Chiruvella et al. [6] investigated
'
~
EXTRUSION OF NON-NEWTONIAN MATERIALS
205
FIG. 33. Streamlines in the nip region for CMC solution at 16 rpm: (a) experimental results; (b) numerical predictions (adapted from Sastrohartono et al. [57]).
the characteristics of an extruder channel with a die. Figure 37 shows some typical results obtained in this study for two fluids and different dies. Die 1 is the most restrictive one; die 6, the least restrictive. The die characteristics and the screw characteristics are shown in terms of the variation of the pressure at the entrance to the die and the mass flow rate. As expected, the pressure at the die increases with flow rate for a given die, with the largest pressure obtained for the most restrictive die. However, at a given screw speed N , a decrease in flow rate results in an increase in pressure since this implies going to a smaller-diameter die. Therefore, for a given die and given screw speed, the operating point of the extruder may be determined from the intersection of the two characteristics in these graphs. This information is useful in the design of a system for the extrusion process. If the die is kept fixed and speed is increased, the flow rate increases as well as the pressure. All these trends are physically expected. However, it must be remembered that it is assumed that the screw channel is completely filled with fluid in all cases. Heat transfer and bulk temperature results were also obtained by Chiruvella et al. [6].
B. TRANSPORT IN COMPLEX DIES For more complicated dies, a numerical analysis may be carried out to obtain the pressure drop across the die for a given throughput. This may then be coupled with the analysis of the extruder channel, as discussed above. The flow and temperature fields may also be determined for
b
a
1.0
Qb
Qt
L2
.-0
0.8
_ _ _ __ ___ -
0.6 ..-u) .->
--
-
.
Experimental Numerical
- - --
.-
- - ---
-a
-
-
- b
_.
.__
n
g
0.4 ..
ii 0.2
.
-- ”
-
-
.
-.
--
-
- - ---
-
__-
_.
0.0+
FIG.34. (a) Coordinate system and volume flow rates in the nip region; (b) comparison of flow division ratio obtained from experimental and numerical results for CMC solution (adapted from Sastrohartono ef al. [571).
207
EXTRUSION OF NON-NEWTONIAN MATERIALS
a
Barrel
-
f f 1
1
-
-Numerical Prediction .
0.0
0.6
UXN b z
,
1.2
.
.
.
.
I
.
.
.
.
1.8
b Barrel
0.0
0.45
u i vbz
0.9
1.35
FIG. 35. Comparison between calculated and measured tangential velocity profiles for isothermal heavy corn syrup at 26.5"C,with (a) mass flow rate of 6 kg/h and screw speed of 30 rpm; (b) mass flow rate of 7.5 kg/h and screw speed of 41 rpm (adapted from Chiruvella et al. [5]).
208
YOGESH JALURIA
IYi.
ta R,1
Die#l - 4
Die # 5 & 6 Die Dimensions
FIG.36. Different die geometries and dimensions considered for matching the flow rate and pressure with the results for the extruder channel (adapted from Chiruvella et af. [61).
improving the design to obtain better control on the process and uniformity of the extrudate. A 3D finite-element model was developed by Gupta er al. [22] to simulate isothermal flow in a die. Figure 38 shows the geometry of a particular die, with the rotating conical screw on the left side and the cylindrical outflow on the right. This is a fairly complicated flow geometry, due to the rotating screw and the changing channel geometry as the fluid flows toward the outlet. The finite element method is particularly useful in the simulation of such a flow. The discretization used and the corresponding calculated pressure drop in the axial direction are shown in Fig. 38, for different operating speeds. The number of revolutions per minute (rpm) affects the flow rate, which, in turn affects the pressure drop, being highest at the maximum speed. The variation of pressure at the die inlet with the flow rate is also shown at different temperatures. The viscosity is lower at higher temperature, resulting in smaller pressure drop across the die. A comparison with experimental results obtained from the experimental
209
EXTRUSION OF NON-NEWTONIAN MATERIALS
0
10
20
30
40
Flow Rate [Kg/hr] 4
Flow Rate [Kghr] 4
FIG.37. Variation of pressure at the die with mass flow rate at different screw speeds using different dies during extrusion of (a) LDPE and (b) Viscasil-300M at the operating conditions shown in the figure (adapted from Chiruvella et af. [h]).
210
YOGESH JALURIA
a
LEGEND LEGEND Temperature = 26'C Temperature = 28'C 2 Temperature = 3O'C L Temperature = 32'C L TemDerature = 34'C
o
-
C
FIG.38. (a) Finite-element discretization of the circular die with the conical end of the screw rotating in the entrance region of the die; (b) pressure variation in the axial direction; (c) variation of pressure at the die with the flow rate.
EXTRUSION OF NON-NEWTONIAN MATERIALS
d
0
211
Numerical results Experimental results
FIG.38. Continued. fd) comparison between numerical and experimental results. The fluid for these results is heavy corn syrup (adapted from Gupta et nl. [22]).
facility of Fig. 20 is also shown. A good agreement is obtained over the range of flow rates considered. The simulation of another extrusion die system is shown in Fig. 39. In this die, a long cylindrical region is used as flow path of the fluid to help in eliminating unsteadiness in the flow that arises due to the rotating screw and for improving the uniformity of the material. A conjugate heat transfer problem results here because of conduction in the thick wall of the die and convection in the flow [44].Numerical modeling is used for obtaining the velocity and temperature fields, as well as the pressure variation, under a range of operating conditions. Most of the temperature variation arises near the boundaries due to the low thermal conductivity of the material. The flow is very well behaved with no separation or recirculation arising from the step changes in the cross section. This is expected from the large fluid viscosity and resulting low Reynolds number in the flow. Most of the pressure drop occurs in the final region of the die where the diameter is the smallest. Very little pressure variation occurs in the relatively large-diameter cylindrical flow path region. Thus, the flow and heat transfer in the die may be simulated to determine the pressure drop, which is then matched with the pressure rise in the extruder channel. The flow and temperature fields are used to indicate whether any stagnation or recirculation arises in the flow and for improving the design to obtain better quality of the extrudate.
Die section 1 2 3
Flow path
___-
Inlet ..........
Outlet
A
1.5
1.5
I
Streamlirles Near the Die
1
Isotherms Near the Die
I
e
1.o
,a35
0.5 I
0.0
1 16
Z'=dR
18
20
EXTRUSION OF NON-NEWTONIAN MATERIALS
213
M. Combined Heat and Mass Transfer
Combined thermal and mass transport mechanisms are important in many extrusion processes, particularly those dealing with food, reactive polymers, and several other materials with multiple species. Extrusion is an important manufacturing technique in thermal processing of food materials. It is used in the processing of many snack foods, cereals, pasta, confectionary, pet foods, and bread substitutes. Various starches, wheat, rice flour, and other materials, along with a chosen amount of water, are fed into the hopper and cooked through the input of shear and heat to obtain different extruded products (see Haper [241 and Kokini et al. [371). Chemical reactions also occur in food materials and other chemically reactive materials to substantially alter the structure and characteristics of the material. Some of these considerations are outlined here. A. MOISTURE TRANSPORT
Moisture transport is very important in the extrusion of food materials both during the extrusion process and after the material emerges from the extrusion die. The viscosity is a strong function of moisture concentration, and, therefore, the transport of moisture in the extruder affects the flow and the heat transfer process. As the material comes out of the extruder, unbound or free water evaporates and flashes off as a result of the much lower ambient pressure, as compared to that in the die. This can cause a large amount of extrudate expansion and, thus, considerable reduction in the density of the extrudate. This, in turn, affects the nature and quality of the final product. Therefore, it is important to study the transport of moisture in the extruder and also the effect of chemical reactions. For mass transfer in the extruder, for example, when moisture transport in the extrusion process for food materials is considered, the governing equation may be written, for small concentration levels, as
where c is the concentration of the species under consideration, 0, is the mass diffusivity, S is the reaction rate, and m is the order of the reaction: zero if the reaction rate is constant and one if the reaction rate varies linearly with concentration. Here Sc" represents a source term. However, for moisture it behaves as a sink, with S taking on negative values, to FIG.39. Numerical modeling of a particular extrusion die system to obtain the flow and temperature fields, as well as the pressure variation in the axial direction, using cornmeal at 21% moisture as the fluid.
t
t ,
a
C'
3 3
J 3
w
Lu
u an C*
w
1n
C*
UI
a s
LII
in
C'
FIG.40. Lines of constant moisture concentration c* and concentration profiles at four downstream locations in a single screw extruder for a non-Newtonian fluid with n = 0.3, 4 = 16.54", qv = 0.25, Pe = 7050, G = 0.005, p, = 1.134, p2 = 10.0, b, = 1.0, rn = 0, Le = 0.001, Oge, = 0.5, for viscosity varying with both temperature and moisture. (a) S = - 500.
, ro
c’o
, F ,
20
, 90
H / K = ,K
t DO
$
0
i4
0
x
3
--$
216
YOGESH JALURIA
" Y
I
0.0
65.0
I
130.0
I
195.0
I
260.0
Dimensionless Down-channel Distance z* FIG.41. Effect of the strength of the moisture sink S on the variation of the dimensionless pressure p* along the screw channel length z*, for the conditions of Fig. 40 (adapted from Gopalakrishna er al. [18]).
indicate moisture removal from the flow because of the chemical reaction. The concentration may be nondimensionalized with the inlet concentration of the species c i . The ambient concentration is taken as zero here, and the sink term comes into effect only when the temperature T is greater than a threshold temperature Tge, needed for gelatinization to occur. Thus, this equation may be solved by the same numerical procedure adopted for heat transfer, with Om,m, and S as additional variables. The viscosity change with moisture is included as given by Eq. (2). Figure 40 shows the moisture concentration contours and profiles for m = 0 for two different values of S. As seen here, the effect of the sink term, S,due to the reaction, is manifested in the form of a decrease in the moisture concentration due to bonding of water for gel formation as the material reaches the dimensionless gelatinization temperature Oge,. This loss of moisture occurs at a rate specified by the sink term S and is obviously more rapid for larger S. For the chosen conditions, the screw becomes hotter than the barrel and moisture removal occurs there first. Figure 41 shows the effect of varying the strength S of the sink on the pressure rise along the channel. As compared to the case of S = 0, the dimensionless pressure at the die is seen to be lower for higher sink strengths. The temperature rises at higher sink strengths and causes the
217
EXTRUSION OF NON-NEWTONIAN MATERIALS
viscosity to decrease despite the moisture reduction. This results in the material flowing more easily and causing smaller pressure rise. However, the results are very sensitive to the material properties and operating conditions, making it difficult to predict the trends. The convergence of the numerical scheme is also not assured for widely varying materials and conditions. Figure 42 shows the effect of changing the throughput qv on
a
..._ -10.0
!
0.0
I
65.0
130.0
195.0
260.0
Dimensionless Down-channel Distance z*
b 1.0-
3 0.8-
a
5E
0.6-
a,
Q
$ I-
0.4-
* 0.2Y
3
0.0 Y 0.0
I
65.0
130.0
195.0
260.0
Dimensionless Down-channel Distance z* FIG.42. Effect of the throughput qv on the variation of the dimensionless pressure p* and bulk temperature Obulk in the down-channel direction for n = 0.5, G = 0.0, and other conditions as in Fig. 40 (adapted from Gopalakrishna et al. [181).
218
YOGESH JALURIA
the pressure and the bulk temperature. The trends are similar to those discussed earlier for single-screw extruders. The results are strongly influenced by the throughput. For a q, value of 0.3, there is essentially no pressure variation in the extruder channel, indicating a situation with no obstruction because of a die. At lower values of q,, the pressure rises because of obstruction, simulating the circumstance in typical extrusion processes. At q, values larger than 0.3, the pressure drops in the axial direction, indicating that the material must be pushed by a higher pressure at the inlet to maintain such a high flow rate. The bulk temperature rises faster for lower qv since the material stays in the extruder channel for longer time, i.e., residence time is longer, allowing the material to heat up. Thus, this approach may be used to compute transport of moisture or other species in the extruder. B.
CHEMICAL RFACnON AND CONVERSION
It is also important to model the chemical conversion process in reactive materials, such as food, to determine the nature and characteristics of the extruded material. A simple approach to analyze these processes is discussed here. The governing equation for chemical conversion may be given as [65] d -(1 - X ) = -K(1 -X)", dt where t is time and X is the degree of conversion, defined as
X=
Ma - Mt Ma - Mf '
(39)
where M,, is the initial amount of unconverted material, taken as starch here, Mf is the final amount of unconverted starch, and Mt is the amount of unconverted starch at time t . The order of the reaction is m and K is the reaction rate. The order of the reaction m in Eq. (38) has been shown to be zero and the rate of the reaction K given as [65] K
=K,
-k K,,
( 40)
where K,
= K T o exp( - E , / R T ) ,
(41)
EXTRUSION OF NON-NEWTONIAN MATERIALS
219
is a parameter obtained experimentally for the given material. A and one-dimensional approximation may be applied to model the degree of conversion defined in Eq. (381, as given by [l, 81
dx w -dZ= K . (43) As defined earlier, w is the velocity in the down-channel direction, X is the degree of conversion defined in Eq. (391, and K is the reaction rate defined in Eq. (40). Thus, numerical results on conversion are obtained by integrating this equation. Typical results on the conversion of amioca, which is a pure form of starch, at constant screw speed and throughput, are presented in Fig. 43. The degree of conversion depends on the axial velocity and the reaction rate constant K , which varies with the local temperature of the extrudate. For smaller axial velocities, the degree of conversion up to a given axial location is higher. This is clear in the region near the screw root and the barrel in all the conversion contours. As the temperature increases down-
Axial Distance [m]
b
T,r120°C
Axial Distance [m]
FIG.43. Isotherms and constant conversion contours at an inlet temperature of 90°C and a barrel temperature of 120°C for pure starch (amioca) at 30% moisture, with m = 30 kg/h, N = 100 rpm.
220
YOGESH JALURIA
stream, the degree of conversion increases, although the velocity distribution does not change very much. At a barrel temperature of 150°C, the local temperature is much higher than that at 120°C. Since the reaction rate constant is higher when the local temperature is higher, the material is converted in a shorter distance than when the temperature is lower. Several other results were obtained under different temperature levels and throughputs. This figure shows the typical trends observed. Some additional results are shown for a tapered single-screw extruder in Figs. 44 and 45. The isotherms and the conversion contours are shown, indicating trends similar to those seen earlier. With increasing flow rate, the degree of conversion decreases since the residence time is smaller, allowing less time for the chemical reactions to occur. A comparison with experimental results also shows good agreement. However, in these cases the screw is not filled with a rheologic fluid throughout. Only a fraction of the channel length is completely filled, with the remaining portion either partially full or containing powder which cannot be treated as a fluid. It is evident that, depending on the property and chemical kinetics data, the models may easily be used to determine chemical conversion and other changes in the extrusion process. VII. Additional Considerations
There are several additional aspects that have not been brought out in the discussion given in the preceding sections. Some work has been done on all these aspects. However, further work is needed to understand and model the basic transport mechanisms in order to obtain detailed results for the extrusion process. Some of the important aspects are 1. Powder flow and heat transfer 2. Polymer melting 3. Filled length and partially full screws 4. Conjugate transport 5. Complicated screw elements, e.g., reverse elements and kneading blocks 6. Other practical aspects, e.g., grooved barrel and postextrusion transport
The material is generally fed through the hopper as solid pieces or as powder. This material is conveyed by the rotating screw, compacted, and then melted or chemically converted as a result of the thermal and mechanical energy input. Figure 46a shows a schematic of the overall process for food extrusion. The solid-conveying region is generally mod-
221
EXTRUSION OF NON-NEWTONIAN MATERIALS
Isotherms Tb=115' C , T,=QO"C,m=10 Kgh.. N=l 00 rpm
0.0030 0.0025
-
0.0020
-
0.0015
-
0.0010
-
0.0005
-
Y (m)
0.0
0.2
0.4
0.6
= (m)
0.8
1.o
1.2
0.8
1.o
1.2
Conversion 0.0035
-
0.0030 0.0025
-
0.0020
-
0.0015
-
0.0010 0.0010
-
0.0005
-
O.oo00 o . o o 0 0 ~ ' " " " " " " " ' " ' ' ' ' " " ' ' 0.0 0.2 0.4 0.6
z(m)
FIG.44. Isotherms and conversion contours while extruding amioca in a tapered single screw extruder, with Tb = 115"C,Ti = WC, N = 100 rpm, rir = 10 kg/h, moisture = 30% (adapted from Chiruvella et al. [8]).
30 Scnw spnd 0 7 5 rpm 0100 rpm 'A 125 rpm
25
-E
0 150 rpm X 1 7 5 rpm
al
0 al
s
20
c 0
E
0
5 15 t 0
0
r
5
1
2
1.5
2.5
3
3.5
4.5
4
5
Flow Rate [kg/h]
b Materlal:Amioca Moisture: 30% (wb) Screw Speed: 200 rpm lnkt Tempersturn: 90C
0
0
0
0
110
116
120
126
130 135 140 Baml Temperatun [q
0
Expl [Lsi B Kokini, lBW]
o
Numerical
145
110
166
160
FIG.45. (a) Predicted variation of mean degree of conversion at the die with the Aow rate at different screw speeds for amioca with 30% moisture, Tb = Ti = 90°C; (b) comparison between numerical predictions and experimental results for amioca at 30% moisture at various barrel temperatures, with m = 3.0 kg/h, N = 200 rpm, Ti = 90°C (adapted from Chiruvella et al. [81).
EXTRUSION OF NON-NEWTONIAN MATERIALS
223
FIG.46. (a) Schematic of various regions in food extrusion; (b) modeling of powder flow in a single-screw extruder.
eled as a plug flow with friction and slip at the boundaries, as sketched in Fig. 46b. Friction factors have been measured for different materials and are available in the literature. The force balance yields the pressure variation in this region [62]. In food processing, the material is generally fed as a powder with a high level of porosity. This material is compacted, due to the increase in pressure downstream. For modeling the compaction process, information is needed on the variation of density or porosity of the material with pressure. Also, accurate friction factors are needed for the given powder and barrel and screw materials. Very little work has been done on powder conveying and compaction even though this process is expected to have a substantial effect on the heat transfer and conversion processes downstream. Clearly, further detailed work is needed on this problem. We have assumed that the extruder channel is completely filled with the material being extruded. This is rarely the case for twin-screw extruders, and even for single-screw extruders the channel may be partially full, or starved. The flow and heat transfer mechanisms are expected to be quite
224
YOGESH JALURIA
different for this circumstance as compared to that for a completely filled channel. Pressure is not expected to rise as long as the channel is starved. The heat transfer rate is also expected to be much less. Therefore, it is important to model the starved-fed region and determine the length of the filled zone. Experimental inputs have been used in many cases to determine this length in order to calculate the pressure and temperature rise. Further work is needed to understand and model the transport in this region. As the polymeric material moves downstream, it melts or converts. Work has been done on the modeling of the melting process in polymers. A simple model developed by Tadmor and Klein [63] is shown in Fig. 47. The flow in the pool of molten polymer and the solid region are clearly indicated. The transition region in which the material goes from a solid to a melt has been modeled in several studies. However, further work is needed on conversion mechanisms and on detailed modeling of the transition region in order to obtain a global model for the entire extrusion process. Most of these processes have not been studied in detail for twin-screw extrusion. The screw has been taken as isothermal or adiabatic in most studies, with the barrel at a specified temperature distribution. In actual practice, there is conductive transport in both the screw and the barrel. The conjugate problem that couples the conduction process with the flow needs to be solved to determine the actual temperature distribution that arises. This is a complicated problem that has not been considered in detail. However, it is an important consideration and must be included in the
FLIGHT
CIRCULATORY FLOW OF PREVIOUSLY MELTED POLYMER
FIG.47. Sketch of a model for studying melting of a polymer in the screw channel (adapted from Tadmor and Klein [63]).
EXTRUSION OF NON-NEWTONIAN MATERIALS
225
model for a realistic simulation of the extruder. Similarly, many practical aspects such as grooves in the barrel, different types of dies, die swell, and different types of elements, particularly elements with a reversed screw and kneading elements used for mixing, have received some attention in the literature because of their importance in real extrusion processes. However, further detailed work is needed on most of these in order to provide a better understanding of the underlying physical phenomena and to allow these to be coupled with other aspects of extrusion, particularly in twin screw extrusion.
VIII. Conclusions The work done on the fluid flow and the heat and mass transfer in the screw extrusion of non-Newtonian fluids is presented, considering different types of single- and twin-screw extruders. The basic mechanisms that govern the thermal transport in this important manufacturing process are considered in detail, starting with the characterization of common materials employed for extrusion. Different mathematical models and formulations that may be applied under different conditions and for different extruders are discussed. The relevant numerical procedures for obtaining the velocity and temperature fields in the conveying, heated, section of the screw channel are outlined, along with important results obtained for typical extruders. Practical issues such as the residence-time distribution, mixing characteristics, heat input, and pressure and temperature rise are also considered. Tapered single-screw extruders are often employed in industry and are modeled to yield the resulting pressure distribution. Experimental results are also presented on the various aspects to validate the numerical models as well as to indicate the basic features of the extrusion process. It is shown that the flow is strongly influenced by the dimensionless flow rate qv in the extruder. The temperature field is found to be strongly affected by the dimensionless parameters G, the Griffith number, and Pe, the Peclet number. However, the corresponding changes in the velocity field are small. It is also found that at small values of q,, a significant amount of reverse flow arises in the moving-barrel formulation that is commonly used for modeling. This leads to problems with numerical convergence. Methods to avoid these are discussed. In the presence of strong viscous dissipation, heat transfer is observed to occur from the fluid to the barrel downstream, in the portion of the extruder near the die. It is found that the residence-time distribution is not significantly affected by the values of power law index n or the barrel temperature distribution for
226
YOGESH JALURIA
the range of parameters considered here. It is dependent mainly on the dimensionless volumetric flow rate q v. An approach for the numerical modeling of twin-screw extruders has been presented. The flow inside the extruder is divided into translation and mixing regions. The simulation of the translation region is very similar to that for a single-screw extruder having the same barrel and screw channel design. The complete simulation of a twin-screw extruder is obtained by coupling the simulations of the two regions at a numerically determined interface between these. Different types of twin-screw extruders, particularly corotating tangential and self-wiping extruders, are considered, and basic trends, in terms of the pressure and temperature rise and flow field in the intermeshing region, are presented. The flow through dies is another important consideration since this determines the operating conditions, particularly the flow rate, in the extruder. A three-dimensional finite-element model and a simpler twodimensional control volume model for simulating complicated die geometries have been discussed. The extruder is coupled with the die to obtain the resulting flow rate and pressure rise. Combined heat and mass transfer is important in several applications, particularly in reactive polymers and food materials. The numerical modeling of moisture transport and the effect of chemical reactions on the transport in the extruder and on the final product are discussed. It is shown that these effects are very important and must be included for an accurate modeling of the thermal processing of food and other such chemically reactive materials. Thus, interesting and important results obtained from numerical simulation and experimentation on this materials processing technique are presented and discussed. Some important areas that need further work are also indicated. Acknowledgments The author would like to thank Professors T. H. Kwon, V. Sernas, and M. V. Kame for several discussions during the course of this work and several students and postdoctoral fellows who contributed to this effort. This work was supported by a grant from the N.J. Center for Advanced Food Technology.
Nomenclature A
B b
channel cross-sectional area width of the screw channel temperature coefficient of viscosity, Eq. (1)
bm CP C
concentration coefficient of viscosity, Eq. (2) specific heat at constant pressure of the fluid concentration
EXTRUSION OF NON-NEWTONIAN MATERIALS
mass diffusivity barrel inner diameter Griffith number, G = G Pv&/MTb - Ti) screw channel height H channel heights at inlet and HI7 H2 outlet of a tapered screw channel convective heat transfer h coefficient K chemical reaction rate, Eq. (40) k thermal conductivity of the fluid L axial or down-channel length of extruder Le Lewis number, Le = a/Dm M f , M " , M , final, initial, and local amount of unconverted starch, Eq. (39) m order of chemical reaction Nusselt number, Nu, = h H / k Nu, n power-law index of fluid Pe Peclet number, Pe = VbzH / a local pressure P total volumetric flow rate Q heat transfer rate to the fluid qin from the barrel dimensionlessJolumetric flow 4v rate, qv = Q/AVhz Reynolds number, Re = Re
z direction velocity components in x , y, and z directions, respectively degree of conversion, Eq. (39) coordinate distances flow division ratio in twinscrew extruder
Drn
Db
VbzH/V
S T Tge I
t b'
I
strength of sink for removal of free moisture from the flow, Eq. (37) local temperature temperature at which gelatinization or conversion occurs time velocity of the barrel in the
227
u , L', w
X x, Y, z
.
GREEKSYMBOLS a
P PI7
P2
AP
4 i. P
*
P 7
O
x. t
thermal diffisivity of fluid dimensionless parameter in Eq. (9) dimensionless parameters, PI = Tb/q; p2 = b / T pressure drop across an extrusion die screw helix angle shear rate coefficient of viscosity fluid density taper angle shear stress dimensionless temperature tangential and axial coordinates, Eqs. (22) and (23)
SUBSCRIPTS b I
0 m
barrel inlet reference value moisture
SUPERSCRIPTS
*
dimensionless quantity
References 1. Abib, A H., Jaluria, Y., and Chiruvella, R. V. (1993). Thermal urocessine. of food materials in a single screw extruder. HTD [Pubf.](Am. SOC. Mech. kng.) 254,>7-67. 2. Agur, E. E., and Vlachopoulos, J. (1982). Numerical simulation of single screw plasticating extruder. Polym. Eng. Sci. 22, 1084-1094. 3. Bird, R. B., Armstrong, R. C., and Hossager, 0. (1977). Dynamics of Polymeric Liquids, Vol. I. New York.
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YOGESH JALURIA
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MATERIALS
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27. Jaluria, Y., and Torrance, K. E. (1986). Computational Heat T m n ~ e rTaylor . & Francis, Washington, D.C. 28. Janssen, L. P. B. M. (1978). %in Screw Exmtswn. Elsevier, Amsterdam. 29. Joo, J. W., and Kwon, T. H. (1993). Analysis of residence time distribution in the extrusion process including the effect of 3D circulatory flow. Polym. Eng. Sci. 33, 959-970. 30. Kalyon, D. M., and Sangani, H. N. (1989). An experimental study of distributive mixing in fully intermeshing, co-rotating twin screw extruders. Polym. Eng. Sci. 29, 1018-1026. 31. Kalyon, D. M., Gotsis, A. D., Yilmazer, U., Gogos, C., Sangani, H., Aral, B., and Tsenoglou, C. (1988). Development of experimental techniques and simulation methods to analyze mixing in co-rotating twin screw extrusion. Adu. Polym. Technol. 8, 337-353. 32. Karian, C. (1986). Power consumption for extrusion of nowNewtonian melt. SPE ANTEC Tech. Pap. 32,53-58. 33. Karwe, M. V., and Jaluria, Y. (1990). Numerical simulation of fluid flow and heat transfer in a single screw extruder for non-Newtonian fluids. Numer. Heat Transfer 17, 167-190. 34. Kame, M. V., and Sernas, V. (1995). Velocity measurements in the nip region of a co-rotating twin screw extruder using Laser Doppler Anemometry. SPE ANTEC Tech. Pap. 41, 150-156. 35. Kame, M. V., and Sernas, V. (1996). Application of laser Doppler anemometry to measure velocity distribution inside the screw channel of a twin screw extruder. J. Food. Process. Eng. (to be published). 36. Kame, M. V., Chiruvella, R. V., and Jaluria, Y. (1995). Coordinate system independence of shear rate during isothermal single screw extrusion of a Newtonian fluid. J. Food. Process Eng. 18, 55-69. 37. Kokini, J. L., Ho, C.-T., and Kame, M. V., eds. (1992). Food Extrusion Science and Technology. Dekker, New York. 38. Kwon, T. H., Jaluria, Y., Karwe, M. V., and Sastrohartono, T. (1991). Numerical simulation of the transport processes in a twin screw polymer extruder. In Progress in Modeling Polymer Processing (A. I. Isayev, ed.), Chapter 4, pp. 77-1 15. Hanser Publishing, New York. 39. Kwon, T. H., Shen, S. F., and Wang, K. K. (1986). Pressure drop of polymeric melts in conical converging flow: Experiments and predictions. Polym. Eng. Sci. 26, 214-224. 40. Kwon, T. H., Joo, J. W., and Kim, S. J. (1994). Kinematics and deformation characteristics as a mixing measure in the screw extrusion process. Polym. Eng. Sci. 34, 174-189. 41. Lai, L. S., and Kokini, J. L. (1990). The effect of extrusion operating conditions on the online barrel viscosity of 98% Amylopectin (Amioca) and 70% Amylose (Hylon-7) corn starches during extrusion. J. Rheol. 34, 1245-1266. 42. Larsen, H., and Jones, A. (1988). Heat transfer in twin screw extruders. SPE ANTEC Tech. Pap. 34, 67-70. 43. Lawal, A,, and Kalyon, D. M. (1993). Incorporation of wall slip in non-isothermal modeling of single screw extrusion processing. In Transport Phenomena in Processing (S. I. Guceri, ed.), pp. 985-996. Technomic Pub. Co., Lancaster, PA. 44. Lin, P., and Jaluria, Y. (1995). Heat transfer in polymer melts flowing in constricted channels. Proc. ASME /JSME Therm. Eng. J. Con$, 4th, Hawaii, 1995, vol. 4, pp. 23-31. 45. Lindt, J. T. (1985). Mathematical modeling of melting of polymers in a single screw extruder. A critical review. Polym. Eng. Sci. 25, 585-588. 46. Lindt, J. T. (1989). Flow of a temperature dependent power-law fluid between parallel plates: An approximation for flow in a screw extruder. Polym. Eng. Sci. 29, 471-478. 47. Maheshri, J. C., and Wyman, C. E. (1980). Mixing in an intermeshing twin screw extruder chamber; Combined cross and down channel flow. Polym. Eng. Sci. 20 (91, 601-607.
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48. Michaeli, W. (1992). Extrusion Dies for Plastics and Rubber. Hanser Publishing, Munich. 49. Mitsoulis, E., Vfachopoulos, J., and Mirza, F. A. (1984). Finite element analysis of Row through dies and extruder channels. SPE ANTEC Tech. Pap. 30, 53-58. 50. Panton, R. L. (1984). Incompressible Flow. Wiley, New York. 51. Patankar, S. V. (1980). Numerical Heat Transfer and Fluid Flow. Taylor & Francis, Washington, DC. 52. Pearson, J. R. A., and Richardson, S. M., eds. (1983). Computational Aiialysis of Polymer Processing. Appl. Sci. Publ., London. 53. Rauwendaal, C. (1985). Throughput-pressure relationships for power-law fluids in single screw extruders. SPE ANTEC Tech. Pap. 31, 30-33. 54. Rauwendaal, C. (1986). Polymer Extrusion. Hanser Publishing, New York. 55. Rowell, H. S., and Finlayson, R. D. (19221, Engineering 114, 606. 56. Sastrohartono, T., and Kwon, T. H. (1990). Finite element analysis of mixing phenomena in tangential twin screw extruders for non-Newtonian fluids. Int. J. Numer. Merhods. Eng. 30, 1369-1383. 57. Sastrohartono, T., Esseghir, M., Kwon, T. H., and Sernas, V. (1990). Numerical and experimental studies of the flow in the nip region of a partially intermeshing co-rotating twin screw extruder. Polym. Eng. Sci. 30, 1382-1398. 58. Sastrohartono, T., Jaluria, Y., and Kame, M. V. (1994). Numerical coupling of multiple region simulations to study transport in a twin screw extruder. Numer. Heat Transfer 25, 541-557. 59. Sastrohartono, T., Jaluria, Y., Esseghir, M., and Sernas, V. (1995). A numerical and experimental study of three-dimensional transport in the channel of an extruder for polymeric materials. Znr. J. Heat Mass Transfer 38, 1957-1973. 60. Skelland, A. H. P. (1967). Non-Newtonian Flow and Heat Transfer. Wiley, New York. 61. Szydlowski,W., and White, J. L. (1988). Simulation of flow of polymer melts in intermeshing co-rotating twin screw extruders. SPE ANTEC Tech. Pap. 34, 89-92. 62. Tadmor, Z., and Gogos, C. (1979). Principles of Polymer Processing. Wiley, New York. 63. Tadmor, Z., and Klein, I. (1978). Engineering Principles of Plasticating Extrusion. Kleiger Publishing Co., Huntington, NY. 64. Todd, D. B. (1988). Heat transfer in twin screw extruders. SPE ANTEC Tech. Pap. 34, 54-58. 65. Wang. S. S., Chiang, C. C., Yeh, A. I., Zhao, B., and Kim, 1. H. (1989). Kinetics of phase transition of waxy corn starch at extrusion temperatures and moisture contents. J. Food. Sci. 54, 1298-1301. 66. Wang. Y., and White, J. L. (1989). Non-Newtonian flow modeling in the screw regions of an intermeshing co-rotating twin screw extruder. J . Non-Newtonian Fluid Mech. 32, 19-38. 67. White, J. L. (1990). Twin Screw Elrtncsion. Hanser Publishing, New York. 68. Zamodits, H. J., and Pearson, J. R. A. (1969). Flow of polymer melts in extruders. Part I. Effect of transverse temperature and of a superimposed steady temperature profile. Trans. SOC. Rheol. 13, 357-385.
ADVANCES IN HEAT TRANSFER. VOLUME 28
Convection Heat and Mass Transfer in Alloy Solidification
PATRICK J. PRESCOTT Department of Mechanical Engineering, The Pennsylvania State Uniiiersiy, University Park, Pennsyhjania
FRANK P. INCROPERA School of Mechanical Engineering, Purdue Uniuersity, West Lafa.yette, Indiana
I. Introduction
Heat, mass, and momentum transport within a solidifying alloy are important in several industrial processes, such as casting, welding, and growth of single crystals, In studying grain structures, Cole and Bolling [l]concluded that natural convection played an important role in the columnarto-equiaxed transition in alloy castings. Another set of theoretical and experimental studies by Flemings et al. [2-61 connected macrosegregation, a maldistribution of alloy constituents, to shrinkage- and buoyancy-driven convection during solidification. Technological advances in solidification processing have been thwarted by an insufficient understanding of these phenomena. The intent of this chapter is to review the current understanding and models of convective phenomena that occur during solidification and to identify areas in need of additional research. Following this introductory section, the nature and relevance of physical phenomena associated with alloy solidification will be discussed. Section 111 covers mathematical models of convection and related phenomena, and Section IV reviews results of simulations based on such models and comparisons of predictions with experimental observations. Strategies for intelligent control of solidification processes are discussed in Section V, and a summary is given in Section VI. 23 1
Copyright 0 1996 hy Academic Press, lnc. All rights of reproduction in any form resewed.
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One feature that distinguishes the solidification of an alloy from that of a pure substance is the morphology of the solid-liquid interface. Since pure substances change phase isothermally, the interface between their phases is smooth and coincident with the isotherm corresponding to the phase-change temperature Tf(Fig la). In contrast, solid and liquid phases of an alloy may exist in equilibrium with each other over a given temperature range (Fig. lb). Although an alloy can solidify with a planar solid-liquid interface if the temperature gradient at the interface is sufficiently large and the growth rate is small, most alloy solidification processes involve a two-phase region known as a mushy zone [7]. The mushy zone is composed of solid dendrites and interdendritic liquid, and it separates fully solidified and melted regions during solidification. Den-
a
PURE SUBSTANCE
BINARY ALLOY SOLIDUS INTERFACE
INTERFACE
LIQUID
FIG.1. Solidification systems for (a) a pure substance and (b) an alloy.
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IN ALLOY SOLIDIFICATION
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drites grow naturally with a very large specific surface area and dendrite arm spacings on the order of 10-100 p m for most industrial applications. Figures 2a, b show dendrite structures in NH,CI-H,O and Pb-Sn systems, respectively. In the macroscopic sense, such growth renders the solid-liquid interface highly irregular and virtually irresolvable. Hence, the mushy zone is commonly treated as a porous (dendritic) solid structure that is saturated with interdendritic liquid. The permeability of the mushy zone is nonuniform and likely anisotropic, depending on the specific microstructural features of the dendritic array. The range of temperatures over which an alloy changes phase is provided by its equilibrium phase diagram. Figure 3 shows such a diagram for a generic binary eutectic system. An equilibrium phase diagram defines regions of temperature and composition in which two or more phases may exist in thermodynamic equilibrium with each other. For temperatures above the liquidus lines, a single liquid phase exists as a solution of constituents A and B. In regions between the liquidus and solidus lines, a solid-liquid mixture exists, and at the eutectic point, a three-phase mixture of liquid and two solid phases exists. The two solid phases are designated as a and p, each of which may exist alone in certain ranges of temperature and composition, or they may coexist as a composite, as they do at the eutectic point. Pure substances are represented along the left and right edges of the equilibrium phase diagram (i.e., 0% B and 100% B), where solidus and liquidus temperatures coincide. The solidus and liquidus temperatures also coincide at the eutectic composition C,, but for all other alloy compositions, freezing will occur over a temperature range defined by the corresponding liquidus and solidus temperatures. A representative simple eutectic system is the lead-tin system, and its equilibrium phase diagram is shown in Fig. 4a. However, there are other types of alloy systems. For example, the copper-nickel phase diagram, Fig. 4b, represents an isomorphous system, with only one solid phase of unlimited solubility. In contrast, the aluminum-copper phase diagram of Fig. 4c represents a complex system, with several solid phases of limited solubility, some of which are intermetallic compounds. Moreover, the AI-Cu system features peritectic reactions, where a solid-liquid mixture freezes isothermally to form a single solid phase (different than the original solid phase), in addition to eutectic reactions, wherein two solid phases precipitate simultaneously from a single liquid phase. Many alloys contain more than two components and, thus, have more complicated equilibrium phase diagrams. However, a finite freezing range and the existence of a mushy zone during solidification are common in all of these systems. Hence, simple binary systems have been selected by most researchers who have studied convection during alloy solidification.
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FIG.2. Dendritic structures for (a) NH4CI-H20 and (b) Pb-Sn.
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A
6
L
Composition (“Awt 6)
FIG.3. Equilibrium phase diagram for a generic binary eutectic system.
Advancement of the solidification front and local solidification rates are affected by natural convection occurring within the melt during solidification, even in highly conductive metals. For a side-chilled mold, such as that illustrated in Fig. 5 , the convection heat transfer coefficient between the solid-liquid interface and the bulk liquid is larger at the top of the cavity, where the thermal boundary-layer thickness is relatively small and energy is advected from the melt to the interface. Thus, solidification progresses more slowly at the top than at the bottom. Convective transport phenomena can influence the solidification process in other ways, which ultimately affect properties such as the strength, hardness, and corrosion resistance of the final casting. For example, dislocations and the distribution of impurities are important features that affect the quality of semiconducting crystals and depend on convection in the melt from which the crystals are grown. Convection is also responsible for nonuniformly redistributing alloy constituents during solidification of cast ingots, billets, and blooms, as well as affecting the size, orientation, and distribution of grains in the casting. Nonuniform distributions of chemical composition and physical structure in an alloy casting can significantly affect the design and reliability of mechanical components, especially those used for aerospace or other vehicular applications where weight is of critical importance. The inhomogeneity of a casting is manifested as statistical variations in material properties such as yield strength. Hence, to design reliable components,
!$
1g
t
Solid
Boo600-
400 -
200-
o ~ " " " " " " " ' " "
Al
Weight Percenl Copper
CU
FIG.4. Equilibrium phase diagram for (a) Pb-Sn, (b) Cu-Ni, and (c) AI-Cu (adapted from Binary Alloy Phase Diagrams (T.B. Massalski, ed.), 2nd ed., Vols. 1, 2, and 3. ASM International, 1990).
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237
FIG.5. Partially solidified binary alloy, chilled from the side, with thermally driven natural convection in the melt.
engineers must use properties that differ from nominal values by as many as three standard deviations, thereby incurring a significant weight penalty. Aircraft and other vehicles could be made lighter, and inspection intervals for critical parts could be lengthened by improving the homogeneity of the alloys presently in use. Moreover, further design improvements could be achieved by using materials that presently cannot be mass-produced because of inadequate process control. Viewed differently, with better control of solidification processes, materials with highly consistent properties can be produced at a much lower cost. However, in order to control such processes, they must be well understood. Other defects in solidified alloys that are affected by convection include porosity and hot tears. Convection in the bulk melt affects the solidification rate, which, in turn, governs the strength of shrinkage-driven flow within the mushy zone of an alloy. Hence, the evolution of dissolved gases and the development of pores, which depend strongly on shrinkage-driven flow and the associated pressure drop within the mushy zone, also depend on convection conditions. Furthermore, residual stresses in a casting are established by thermal conditions during solidification, and are responsible for the development of internal cracks, known as hot tears, during processing. Left uncontrolled, natural convection will contribute to nonuniform distributions of alloy constituents and grain structures, as well as to porosity and hot tears. Without a thorough understanding and control of convective transport phenomena during alloy solidification, materials with
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acceptable defect levels can be produced only by trial-and-error process development and/or by costly inspection and selection processes, in which defective portions of the material are rejected and reprocessed. Since solidification processes are energy intensive, increasing their yield has obvious economic and strategic benefits. Improving the yield depends on thoroughly understanding macroscopic transport phenomena and their interactions with microscopic (or even atomic scale) events, so that mathematical models can be developed to accurately simulate and control the processes [7]. Adaptability in materials processing requires that processes be designed and controlled intelligently, rather than by trial-and-error iteration [8]. Numerical simulation based on valid mathematical models offers opportunities to gain insights to process phenomena that are difficult, if not impossible, to extract from measurements. Hence, mathematical models and numerical simulation procedures must be advanced and substantiated with experimental data, in order to establish the understanding and tools required to design, modify, and control materials processing operations for efficient and competitive production. Because of the multiple, and often disparate, length and/or time scales that influence solidification processes, the challenge of numerically simulating solidification phenomena is formidable.
11. Physical Phenomena
The physical richness of convection heat and mass transfer in alloy solidification may be attributed to many phenomena playing important, interacting roles with different length and/or time scales. The various phenomena and their effects on convection are discussed in this section. Since solidification is induced by cooling through one or more boundaries, temperature gradients are established within a superheated molten alloy. Since fluid density is sensitive to temperature, the corresponding density gradients yield buoyancy forces, which may induce fluid motion. Thermal convection is certain to occur in a side-chilled casting, because the thermally induced density gradient is perpendicular to the gravitational body force and generates vorticity within the fluid. On the other hand, a density gradient aligned with the gravity vector (increasing density in the downward direction) is unconditionally stable and induces no fluid motion. However, if the density gradient is positive in the upward direction, a fluid may become unstable. In this case, local horizontal temperature gradients associated with perturbations in the temperature field would generate
CONVECTION IN ALLOY SOLIDIFICATION
239
small vortices, which would further disturb the temperature field and give rise to large-scale convection patterns [9]. Another important consequence of the temperature gradient is the establishment of a two-phase region (the mushy zone) during solidification of an alloy. The size and extent of the mushy zone is determined by the magnitude of the temperature gradient, the alloy composition and the equilibrium phase diagram, which defines liquidus and solidus temperatures as functions of alloy composition. Furthermore, the liquidus line on the equilibrium phase diagram indicates that, within the mushy zone, the temperature gradient is accompanied by a gradient in liquid composition. That is, for a hypoeutectic alloy, C < C, (Fig. 3), the liquid composition increases as the temperature decreases, and since the liquid density depends on solute concentration, as well as temperature, solutaf buoyancy forces are also established within the mushy zone. Solutal and thermal buoyancy forces may either augment or oppose each other, depending on the relative densities of the alloy constituents and the particular constituent with which the interdendritic liquid becomes enriched as solidification proceeds. For example, as an Al-4.5 wt% Cu alloy (Fig. 4c) solidifies, the primary solid phase is mostly aluminum, whereas the interdendritic liquid becomes enriched with copper, the denser of the two constituents. Therefore, solutal and thermal buoyancy forces augment each other in this situation. In contrast, because the interdendritic liquid is enriched with tin (the lighter constituent) as it cools, solutal and thermal buoyancy forces oppose each other when a Pb-19 wt% Sn alloy (Fig. 4a) solidifies. The relative strengths of the solutal and thermal buoyancy forces within the mushy zone are represented by a buoyancy parameter N,
where Ps and PT are, respectively, the solutal and thermal expansion coefficients of the fluid and m is the slope of the liquidus line on the equilibrium phase diagram ( m = AT/AC,). For N > 0 (Fig 6a), solutal buoyancy augments thermal buoyancy. Conversely, thermal and solutal buoyancy forces oppose each other when N < 0. For -1 < N < 0 (Fig. 6b), solutal buoyancy partially offsets thermal buoyancy, which still exerts the dominant influence on flow in the mushy zone. However, if N < - 1 (Fig. 6c), solutal buoyancy forces oppose and dominate over thermal buoyancy forces within the mushy zone. Hence, conditions in the mushy zone generate vorticity which opposes that generated by thermal buoyancy outside of the mushy zone.
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a
PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
N>O
-1
N<-1
FIG. 6. Thermosolutal convection patterns in the melt and mushy zone of a solidiljmg alloy: (a) solutal buoyancy forces augmenting thermal buoyancy forces; (b) solutal buoyancy forces opposing thermal buoyancy forces, with thermal buoyancy dominating in mushy zone; and (c) solutal buoyancy forces opposing and dominating thermal buoyancy forces in the mushy zone.
Although solutal buoyancy forces arise within the mushy zone, their effect is not confined to this region, and the resulting flows interact with those that are thermally driven in the bulk melt. The resulting transport is termed double-difiswe convection [101. A variety of double-diffusive flows have been studied during the solidification of transparent solutions, which freeze dendritically and permit direct flow visualization. These organic or aqueous solutions are known as analog alloys. Figure 7 [111 shows double-diffusive convection patterns at increasing times during the solidification of a H,O-2 wt% Na,CO, (hypoeutectic) solution, wherein solutal buoyancy augments thermal effects. Solid and mushy layers grow from the right sidewall, while the left wall is heated. Early on (Fig. 7a), thermal convection occurs in the bulk melt and retards solid growth near the top of the cavity. With time, solid salt-enriched liquid is ejected from the mushy zone, and, because this liquid is both solutally and thermally dense, it forms a layer along the bottom of the cavity (Fig. 7b). However, as a result of heating from the left wall, the fluid is recirculated and mixed within a layer of limited vertical extent. Subsequently, additional layers form beneath the original layer (Fig. 7c, d). Although the solutal stratification of the layers is stable, recirculation within each layer is driven by a horizontal temperature gradient. Shear occurs across the interfaces between layers, and the recirculation within
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L
C
d
FIG. 7. Schematic rendition of double-diffusive convection during solidification of an H20-2 wt% Na2C02 solution at various times following the initiation of solidification: (a) 10 min, (b) 30 min, (c) 75 min.. and (d) 150 min (adapted from Thompson and Szekely [ll]).
each layer is weak. Hence, heat transfer between the left wall and the solidification front is relatively small in the lower portion of the cavity, yielding mushy and solid zone thicknesses that are larger. Solidification is retarded slightly at the very bottom due to salt enrichment, which depresses the local liquidus temperature. The more vigorous convection occurring above the double-diffusive layers impedes advancement of the liquidus interface near the top of the cavity. When heating from the left sidewall was reduced, the layer that formed along the bottom was virtually stagnant [HI. Furthermore, instead of forming multiple layers of decreasing (with height) salt concentration, a stagnant layer grew vertically with a stable salt stratification. As shown in Fig. 8, conditions differ if the cold interdendritic liquid is depleted of salt (a hypereutectic solution) and ascends to the top of the mushy zone, where it is ejected into the melt [12-141. Initially, some of the dendrites which form at the chilled right wall are broken as a result of shear forces imposed by the clockwise, thermally driven recirculation cell in the melt and descend to the bottom of the test cell. After approximately 3 min, however, solidification proceeds from the chilled wall (Fig. 8a) and
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PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
FIG. 8. Shadowgraphs showing mushy region growth and double-diffusive convection effects with opposing thermal and solutal buoyancy forces during solidification of an H20-30 wt% NH,CI solution after (a) 3 min, (b) 15 min, and (c) 100 min (adapted from Beckermann and Viskanta [12]).
compositionally lighter (water-rich), but thermally heavier (colder), fluid begins to accumulate in a horizontal layer at the top of the test cell. The layer, in which there is thermally driven convection, is separated from underlying compositionally heavier, but thermally lighter, fluid by a diffusive interface. As the top layer grows, this interface descends and a second weaker interface begins to form above the first (Fig. 8b). However, the weaker interface eventually erodes, while the original interface continues to descend (Fig. 8c), until it, too, begins to erode at t > 100 min. Accumulation of water-rich fluid above the diffusive interface depresses the local liquidus temperature, rendering solidification more difficult and causing dendrites formed earlier in the process to be remelted. This effect is
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CONVECTION IN ALLOY SOLIDIFICATION
manifested by the stunted development of the liquidus interface in the upper regions of the test cell. Thermosolutal convection is also responsible for macrosegregation and freckle formation in castings. Macrosegregation refers to a maldistribution of alloy constituents, and several common forms ate shown schematically in Fig. 9. Within the mushy zone of a solidifying alloy of composition C (Fig. 31, local compositions of the solid and liquid phases, C,and C,, differ, and the requirement of thermodynamic equilibrium acts to segregate the alloy constituents on the microscopic scale of dendrite arm spacing. Large-scale (macr0)segregation results when relative motion between the liquid and dendrites occurs along (or against) the liquid concentration gradient. Hence, although macrosegregation begins with local (i.e., microscopic) segregation of constituents between phases, it is ultimately a consequence of species advection in the liquid relative to that in the solid. Freckles represent a particularly severe form of macrosegregation, generally extending along the length of a directionally solidified ingot (Fig. 10). They are the last regions to solidify and tend to have increased porosity, as well as dendritic structures aligned obliquely with the direction of growth [15]. The origin of freckles was first related to solutally buoyant jets of interdendritic fluid ascending from the mushy zone of aqueous ammonium chloride unidirectionally solidified from below [ 161. The jets are discharged from vertically oriented channels, which are believed to nucleate at the liquidus interface and to propagate into the mushy region due to a remelting phenomenon 117, 181. Figure 11 reveals a channel that formed
V-segregates
A-segregates
Mold
Casting
Bottom negative cone segregate
FIG.9. Illustration of common macrosegregation patterns in cast ingots.
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PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
FIG. 10. Freckles distributed around the periphery of a cylindrical ingot of Mar-M200 (reprinted with permission from Giamei and Kear [15], 1970, ASM International).
during solidification of a H20-27 wt% NH,Cl solution and extended between the liquidus and solidus interfaces [18]. The figure also reveals dendritic structures of different orientation. Several phenomena associated with double-diffusive convection during unidirectional solidification are apparent in Fig. 12, which was obtained during solidification of a H20-27 wt% NH,Cl solution [18]. Fluorescein dye injected into two of the channels tracks the corresponding plumes of cold, water-rich fluid that emerge from the mushy region and collect at the top of the test cell. An unstable situation is created, in which colder (thermally heavier), water-rich (compositionally lighter) fluid overlays warmer (thermally lighter), saltier (compositionally heavier) fluid, The double-diffusive instability yields the vertical array of convection layers which are seen to extend from the sidewalls of the test cell. The figure also reveals extended mushy region growth around the plumes, giving the appearance of volcanos. This morphology was initiated early in the process, when vigorous salt fingers [lo, 171 advected dendrite fragments from the mushy zone, some of which descended to form small rings of debris about the more prominent fingers [17], which became precursors to the development of channels and the associated plumes. Mechanisms other than thermal and solutal buoyancy may also influence fluid flow and convective transport during alloy solidification. Forced convection may exist as a result of filling a mold for ingot casting. In continuous casting, a stream of molten alloy descends into a mold, and
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FIG. 11. Channel formation associated with unidirectional solidification of a H,O-27 wt% NH,C1 solution
“1.
inertia and shear forces may establish a recirculation pattern that interacts with buoyancy-driven flows within and near the mushy zone growing from the sidewalls. Within a mushy zone, flow is also induced by shrinkage associated with phase change, which may be significant for large solidification rates or very small permeabilities [19, 201. Although shrinkage-induced flow is not likely to significantly affect solute redistribution [20], it plays an important role in the evolution of gas pores during solidification [21, 221. Pores tend to form when the flow resistance, due to low permeability, is large, thereby induc-
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PATRICK J. PRESCOTI' AND FRANK P. INCROPERA
FIG.12. Flow visualization using fluorescein dye injection during unidirectional solidification of H,O-27 wt% NH,CI: results reveal solutally driven helical plumes, double-diffusive layers, and volcano-like liquidus interface morphology [ 181.
ing large pressure gradients and low pressures deep within the mushy zone, which enhance the evolution of dissolved gases from the melt. Other mechanisms for convection are capillary, inertia-acceleration, and electromagnetic induction. Capillary effects exist in systems with free surfaces. Since surface tension depends on temperature and liquid composition, gradients in these field variables along the free surface of the melt will drive convection. The existence of surfactants, dirt, and/or foreign particles may also affect surface tension-driven flows. Inertial forces induced by rotation, oscillation, or vibration provide means of generating or altering convection conditions, as do electromagnetic body (Lorentz) forces induced in electrically conducting media by static or alternating magnetic fields. The effects of externally applied magnetic fields on grain structure were studied by Cole and Bolling [l,231, who found that large columnar grains form when a static direct-current (dc) magnetic field is applied. In this case the field dampens convective motion. In contrast, small equiaxed grains form when a stirring effect is created by passing an electric current perpendicular to the magnetic field. Vives and Perry [24-261 observed the same trends when they applied static (dc) and alternating-current (timeharmonic) (ac) magnetic fields. Specifically, their temperature measurements revealed the damping and stirring influences, respectively, of the two types of magnetic fields.
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Depending on the particular process and conditions, some of the aforementioned mechanisms for convection may be inherently significant, whereas others may require active measures to effect. For example, in an arc welding process, Lorentz forces will act on the weld pool, along with surface tension forces. However, in a Czochralski crystal growth application, Lorentz forces will not exist without externally applying a magnetic field on the melt. It is important that process designers recognize and understand the naturally occurring phenomena in a given process and synthesize means of augmenting the beneficial features of convection, while reducing or eliminating its detrimental effects. Both the rate and the direction of dendrite growth are affected by the temperature field in the melt and by solute transport within the mushy zone, both of which are, in turn, influenced by convection. Since convection significantly affects the development of microstructural features in a solidifying alloy, the evolution of microstructure can be neither well understood nor controlled without a good understanding of convective transport phenomena. However, macroscopic and microscopic phenomena are coupled, as convection is strongly influenced by the exchange of momentum, heat, and mass between solid and liquid phases at the microscale. For example, because the permeability is anisotropic in a mushy zone with large, well-ordered columnar grains, a columnar structure will have preferred flow directions. In contrast, a mushy region consisting of small equiaxed grains is better characterized by an isotropic permeability. In either case, the permeability depends on the morphology, which determines the interfacial surface area per unit volume, as well as the spacing between dendrite branches. In turn, these conditions affect the viscous shear stress acting on the interfacial area. The structure of a mushy zone is nonhomogeneous, with large liquid fraction and permeability near the liquidus interface and relatively small liquid fraction and permeability near the solidus interface. Depending on conditions, the permeability may be virtually zero in the mushy zone near the solidus interface. Matters are complicated further by the fact that dendrites may become detached as a result of local remelting and/or fracture under flow-induced stresses. The resulting fragments can be transported through high-permeability regions of the mushy zone and into the bulk melt. This condition is shown in Fig. 13, which includes a shadowgraph and a particle tracking photograph taken during solidification of a H,O-27 wt% NH,CI solution from one sidewall of a rectangular test cell [18]. Ten minutes into the experiment, detached crystals began to appear in an upper corner of the melt, just below a newly formed convection cell (Fig. 13a). The slurry contained a large number of dendrite fragments, which were believed to have been fractured from the dendritic
FIG.13. Shadowgraph (a) and particle-tracking photograph (b) of slurry formed during solidification of a H,O-27 solution from one sidewall of a rectangular test cell [18].
wt% NH,CI
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structure of the stationary mushy zone by strong solutal upflow occurring within this zone. While being dragged a short distance into the melt, the crystals descended slowly and were eventually entrained in the thermally driven downflow of fluid adjacent to the liquidus front. Some of the crystals became lodged in the liquidus front of the mushy zone, whereas others fell to the cavity floor. Figure 13b also reveals A-segregate-type channels, which were the principal sources of fragments discharged from the mushy region. For such conditions the effects of neither convection on grain multiplication nor free-floating grains on convection are well understood. Local nonequilibrium effects may also occur during solidification and are governed by heat and mass transfer at the microscale. For example, the growth of a dendrite tip is limited by the transport of solute from the phase interface, which, in turn, affects the rate of latent heat release and the temperature field. The rate of solute rejection at the solid-liquid interface is affected by mass diffusion in the solid dendrite, as well as by the local solidification rate. If the slope of the solidus line separating the two-phase (solid/liquid) region from the solid phase region on the equilibrium phase diagram is not infinite (Fig. 31, the solute concentration in precipitated solid gradually changes as solidification proceeds. Most solidification processes of industrial importance occur rapidly enough to outpace the rate of species diffusion in a solid dendrite, thereby causing dendrite branches to become cored, with lower solute concentrations in their center regions. Since this nonequilibrium condition affects the microscopic partitioning of solute between solid and liquid phases, the liquid volume fraction, solutal buoyancy force, flow conditions, and macrosegregation in the mushy zone are also affected. Convection both affects and is affected by microscopic phenomena that occur during the solidification of alloys. In addition to influencing microstructural features of the resulting grains, convection plays important roles in redistributing solute, the evolution of dissolved gas and pore formation, and the initiation and propagation of cracks in the solid. Several mathematical models of convective transport are reviewed in the following section. 111. Mathematical Models
The development of reliable mathematical models for transport phenomena occurring during the solidification of alloys is important for two reasons. Since metal alloys are opaque, convection patterns are difficult to discern experimentally, especially in buoyancy-driven flows [27-291. In
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PATRICK J. PFESCOTT AND FRANK P. INCROPERA
many cases, the only option involves numerical simulation. Furthermore, experimentation is expensive, especially at the industrial scale, rendering numerical simulation attractive for exploring novel processing schemes. Discussion of the current generation of solidification models is preceded by a historic account of model developments over the past three decades. Numerical results representative of the capabilities and limitations of recently developed solidification models are discussed in Section IV.
A. HISTORICAL PERSPECTIVE
OF SOLIDIFICATION
MODELS
Early solidification models that included convection effects were highly simplified and neglected the coupling that exists between flows in the solid, mushy, and melted zones. Szekely and Chhabra [30] circumvented consideration of the mushy zone by modeling the effects of natural convection in the melt during the solidification of pure lead from a verti.ca1 sidewall. They demonstrated that, even in highly conducting metals, natural convection significantly affects local heat transfer rates between the melt and solid-liquid interface and, thus, local solidification rates. Their model involved using a convection heat transfer correlation for free convection from a vertical plate as a boundary condition to a conduction problem with phase change, and predictions of the solidification rate and the shape of the solid-liquid interface agreed well with experimental results. Although local heat transfer considerations are important in alloy solidification, fluid flow velocity distributions are deemed critical, as they affect the columnar-to-equiaxed (CET) transition [ll. Szekely and Stanek [311 developed a model for predicting transient vertical velocity distributions across the width of a melted region during the solidification of an alloy in a side-chilled mold. The analysis was highly simplified and limited to lowRayleigh-number flows (Ra < 5001, but is represented the first attempt to quantify flow conditions in the melt during solidification of an alloy. In a related study, Szekely and Todd [32] studied transient natural convection in a rectangular cavity. A transient, two-dimensional numerical model was developed for predicting temperatures and flow patterns, but moving boundaries or interfaces were not considered. However, they did consider two “phases” (both fluid) with a stationary interface, across which shear forces and heat were transmitted. Although greatly simplified, the early works of Szekely et at. [30-321 were significant because they marked a beginning of the development of sophisticated models for predicting transport phenomena during alloy solidification processes. Eventually, other numerical models were developed for predicting natural convection flow and heat transfer in the melted
CONVECTION IN ALLOY SOLIDIFICATION
25 1
region during solid-liquid phase change with moving interfaces, although the existence of a mushy zone was not considered [33-3-51. Flow and transport in the mushy zone is important for predicting macrosegregation in solidified alloys. Among the earliest models that link macrosegregation to interdendritic fluid flow are those of Flemings and coworkers [2, 31, who ignored convection in the bulk liquid and developed a differential solute redistribution equation for application in the mushy zone. Interdendritic fluid flow was assumed to be induced by solidification shrinkage, and solute was transported by fluid advection. However, the analysis required prescription of the temperature and fluid velocity fields. Although the model had limited utility, it did demonstrate how interdendritic fluid flow is responsible for macrosegregation and was verified experimentally [41. Later, Mehrabian et af. [61 extended the solute redistribution model [2] by incorporating an equation for buoyancy-driven flow in the mushy zone. Using Darcy's law, the mushy zone was modeled as a porous medium and its permeability was a prescribed function of the liquid volume fraction. Calculations for a solidifying Al-4.5 wt% Cu alloy were in good agreement with experimental observations [4]. The solute redistribution model was used by Kou et af. [36] to predict conditions for rotating, axisymmetric ingots, and Fujii el al. [37], who considered the effects of various alloy compositions on macrosegregation during the solidification of steel, refined the model by coupling the momentum and energy equations. However, a shortcoming of the solute redistribution model is that it does not account for coupling which exists between the mushy and fully melted zones. The first model to incorporate coupling between mushy and bulk liquid zones was reported by Szekely and Jassal [38], who predicted thermal convection in the melt and mushy regions. Transport equations were developed for a solid-liquid mixture in the mushy zone, and traditional single-phase equations were used in the melt. However, the effects of solutal buoyancy were ignored, and macrosegregation was not predicted. The first model to couple the bulk melt and mushy regions and to include macrosegregation was reported by Ridder et af. [39], who, in the manner of Szekely and Jassel, used a two-domain approach in which separate equations were solved in the mushy and all-liquid zones. The iterative procedure involved matching pressures and velocities at the boundary separating the two regions, and use of a remeshing procedure was dictated by the fact that the liquidus interface location could not be determined a priori. Although good agreement was obtained between predicted and measured macrosegregation results, phenomena such as remelting, channel formation, and double-diffusive interfaces were not predicted. These models [38, 391 have been dubbed rnultidornain models, because the mathematical
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PATRICK J. PRESCOTT AND FRANK P. INCROPERA
solution domain is distinctly divided according to the physical domains (solid, mushy, and melt zones). Because of practical considerations related to their numerical solution algorithms, multidomain models are not well suited for predicting irregular interface shapes. Single-domain models, which overcame many of the limitations of the multidomain models, emerged in 1983 [40,41] and later showed promise of becoming useful tools for simulating solidification processes [42]. Bennon and Incropera [431 presented a set of equations for momentum, energy, and species transport in binary, solid-liquid, phase change systems, which concurrently applied in all regions (solid, mushy, and liquid) and required only a single, fixed numerical grid and a single set of boundary conditions to effect a solution [44]. Hence, the solid, mushy, and liquid regions were implicitly coupled. In their formulation, which was later clarified by Prescott et al. [45], the mushy zone was viewed as solid-liquid mixture with macroscopic properties. Individual phase conservation equations were summed to form a set of mixture conservation equations, which were also valid in the single-phase regions. Limiting assumptions were invoked (a nondeforming solid phase, T, = TI, and no macroscopic species diffusion through the solid phase) to reduce the number of dependent variables, and the solidus and liquidus interfaces, as well as individual phase variables, were implicitly determined by solving the mixture equations. Other models similar to the continuum model [43] were developed independently. Voller and Prakash [46] presented a model for heat and momentum transport, but did not consider species transport and hence macrosegregation. Later, Voller et a1 [47] extended a continuum formulation (with species transport) to include microsegregation (coring) effects and conditions in which the solid and liquid phases moved with equal velocities. Their objective was to explore the envelope in which single-phase models could be applied for binary solidification systems. Another singledomain, fixed-grid model was presented by Beckermann and Viskanta [12], who used the technique of volume-averaging as the basis for deriving macroscopic conservation equations for individual phases. Simplifications were made by invoking the assumptions of thermal and chemical equilibrium between phases and by assuming that the solid phase was stationary. More recently, Prakash [48] and Ni and Beckermann [49] have presented models in which individual phase conservation equations are solved separately with the aid of interphase transport models. Such models permit the relaxation of assumptions pertaining to a nondeforming solid phase, thermal equilibrium (T, = TI), and negligible species diffusion through the solid phase, at the expense of solving twice as many partial-differential equations. At present, efforts are being made to incorporate microscopic phenomena [50] in macroscopic convection models [49, 51-53].
253
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B. SINGLEDOMAIN MODELS
1. Mixture Models Continuum and volume-averaged models [12, 43, 47, 54-591 have been and continue to be used for simulating solidification processes and are undergoing further development and refinement based on experimental measurements. The salient features of these models are discussed in this section, as well as differences that exist between the models. Single domain models are based on governing equations that apply in all regions of a solidification system. Thus, the equations can be integrated across the entire domain without the need to explicitly subdivide the domain into solid, liquid, and mushy regions. Instead, these regions are implicitly defined within the system by the distributions of energy (enthalpy or temperature) and composition determined from the solutions of the model equations. The model equations developed by Bennon and Incropera [431 and clarified by Prescott et al. [45] are as follows in Cartesian coordinates: Continuity: dP dt
-
+ V * (pV) = 0.
x- Momentum:
4 PU) + v dt
*
( pVu)
Energy:
4 Ph) dt
dt
1
+V*(pVh)=V*
+ v -( p V C ) = v . ( pDVC) + v *[ pDV(C,
-
C)]
It is assumed that the mushy zone is coherent (the dendrites form one continuous rigid structure), solid and liquid phases are in local thermal
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PATRICK J. PRJ?SCOTT AND FRANK P. INCROPERA
equilibrium (TI = q),and there is no macroscopic transport of species through the solid phase ( D = flD,). The field variables (V,u, h, C) in these equations are mixture average values, where the mixture contains solid and liquid phases, each of which consists of two constituents [43]. For example, the mixture velocity is defined as the mass fraction-weighted average of the liquid and solid velocities.
v=
fSVS
+ fIV, *
(6)
In the development of Eq. (3), several terms were deemed negligible in all regions (solid, mushy, and liquid) and discarded [45]. The terms remaining in Eq. (3) are those that are significant in at feast one region ofthe entire domain during solidification. For example, the third term on the right side of Eq. (3) is a Darcy damping term, which accounts for momentum exchange between solid and liquid phases at the microscopic (dendrite arm spacing) scale. This is a dominant term that renders the advection and viscous terms negligible within the mushy zone. However, in the fully liquid region (g, = 01, where a permeability model prescribes K = 03, the Darcy term vanishes. Thus, the model for permeability (see Section III.D.l) provides an automatic conversion of the momentum equation for the different zones in the solidification system, so that the equation can be applied in all regions and across their boundaries. The dependent variable in the energy Eq. (4) is the mixture enthalpy ( h = f , h , + f , h , ) , and by combining the first two terms on the right side, with h: = c:T, Fourier’s law of heat conduction is recovered. The third term on the right side of Eq. (4) is a phase enthalpy dispersion term, which, along with the second (advection) term on the left side of the equation, was found by adding solid- and liquid-phase advection terms. The dispersion term is significant only within the mushy zone, where f,(V - V,) # 0. A similar term was derived for the momentum equation but was discarded because it is negligible in comparison to the Darcy term. In the energy equation, however, this term cannot be neglected, because ( h , - h,) is usually quite large. In a fashion similar to that used for the energy equation, the species mass conservation equation is written in terms of a mixture species concentration (C =f& +f,Cl), and the third term on the right side of Eq. (3, along with the second term on the left side, arises from adding solid and liquid advection terms. The second term on the right side of Eq. (5) is a diffusion-like source term, which arises because mass diffusion is assumed to be significant only in the liquid phase. Thus, by adding the first two terms on the right side of Eq. (51, Fick’s law of diffusion is recovered for the liquid phase. Diffusion in the solid phase has been discarded, since it is negligible.
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255
The continuum model equations [Eqs. (2)-(5)J are mutually coupled and must be used with a closure model to determine the local temperature, liquid composition, and mixture mass and volume fractions. Assuming local thermodynamic equilibrium between phases, these quantities can be found from the mixture enthalpy h , mixture composition C , and an equilibrium phase diagram [42]. The procedure involves finding a temperature that satisfies both the enthalpy and composition requirements of the mixture. If it is assumed that (1) mixtures are ideal, (2) thermodynamic properties are constant, and (3) the liquidus and solidus lines on the equilibrium phase diagram are straight, the problem reduces to finding the roots of a quadratic equation, only one of which is physically plausible (T, < T < T , ) [42]. These assumptions can be relaxed, but a numerical root-finding technique would have to be employed [60]. The assumption of local thermodynamic equilibrium can also be relaxed to allow local supersaturation in the liquid, although a suitable model for supersaturation would have to be introduced [61]. Another approach to developing a single-domain model utilizes volumeaveraging. Compared to the continuum formulation, the volume-averaging approach is preferred by some modelers because it forces the development and incorporation of physical insights related to the different scales of the problem [62, 631. Beckermann and Viskanta [ 121 used volume-averaging theorems to develop a set of governing equations for convective transport phenomena in binary solidification systems. A detailed development of continuity and liquid momentum equations based on volume-averaging theorems is presented by Ganesan and Poirier [55], and Prescott et al. [45] delineate the similarities and differences between the continuum and volume-averaged formulations. The working versions of these formulations are equivalent in that they both account for momentum transport in the liquid phase and invoke virtually the same set of assumptions. A significant feature of the Bennon and Incropera formulation [43] is that it readily admits to solution using a control-volume-based, finitedifference method [441 based on the SIMPLER algorithm [64]. The numerical model of Bennon and Incropera [44] involved a fully implicit time-marching scheme, which is unconditionally stable and does not restrict the size of the time step. An explicit time-marching scheme was used by Xu and Li I591 to solve continuum-model equations. Other single-domain, mixture models differ primarily according to constitutive models (e.g., Darcy terms), numerical implementation, or the inclusion of special terms. Amberg [57] simplified the model of Hills et al. [41] and numerically predicted macrosegregation in an FeC casting. Xu and Li [19, 591 developed an explicit finite-difference method for solving a continuum model, which includes nonequilibrium freezing (coring) and solidification shrinkage effects. Zeng and Faghri [651 formulated a contin-
256
PATRICK J. PRESCOTT AND FRANK P. INCROPERA
uum solidification model with the energy equation based on temperature. Poirier and coworkers formulated model equations using a combination of the volume-averaging [55] and continuum theory [56] approaches and solved them using finite elements [66]. 2. Two-Phase Models The single-domain models discussed in the previous section are mixture models, in which the dependent variables represent a mass fractionweighted average of the corresponding solid and liquid quantities (e.g., h = fshs + f,h,). The essential assumptions in formulating such models are (1) the solid-phase motion is prescribed, (2) the solid and liquid are in local thermal equilibrium ( T , = TI), and (3) the macroscopic diffusion of species in the solid phase is negligible. These assumptions are relaxed in complete two-phase models, which include mass, momentum, energy, and species conservation equations for both solid and liquid phases, all of which are valid throughout the physical domain. Prakash [48] presented a two-phase model for binary solid-liquid phase change based on the following governing equations: Mass:
Momentum:
Energy:
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257
In Eqs. (7) and (81, the term hs represents the mass rate of solid formation, which is equal but opposite to the rate of liquid formation. The last terms on the right side of Eqs. (9)-(14) are phase volumetric source terms representing solid-liquid interactions due to both phase change and diffusion at the interface, and they provide the coupling between solid and liquid equations for the respective conserved quantity (momentum, energy, or species). Using volume-averaged (macroscopic) conservation equations for individual phases, Ni and Beckermann [491 developed a two-phase model for transport phenomena occurring during alloy solidification. They further used volume-averaging theorems [62, 63, 671 to mathematically describe solid-liquid interactions and suggested ways to evaluate such terms. Volume-averaging is a method of incorporating smaller-scale phenomena in a macroscopic model through a formal integration of continuum equations, which are valid at the microscopic scale. Integration is performed over a representative elemental volume (REV), which is much smaller than the macroscopic system, but larger than the microscopic scale at which details are considered important. For example, a REV centered about a point within the mushy zone will include both solid and liquid phases. The following model equations (where the subscript k refers to either phase) were presented by Ni and Beckermann [49]:* Mass: d
-( dt
Ek P k )
+ v .(Ek Pk(Vk)k)
= rk.
(15)
Momentum: d -( E k Pk(Vk ) k )
dt
+v
= -EkV(Pk)k
+ &k(Bk)k
'
+
( E k Pk(Vk
'
((Tk))
f M k .
>k(vk )k) -
'
(< Pk+k+k>) ( 16)
*Volume-averaged quantities are enclosed in brackets, ( ), and the su erscript k is used to denote volume-averaged quantities for phase k. Thus, (V,) = c k ( V kP), where &k is the volume fraction of phase k within the REV h e . , ck = g,).
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PATRICK J. PRESCO’IT AND FRANK P. INCROPERA
Energy:
Species:
=
-v
*
(&))
-
v
*
(
+ J,.
( 18)
The term on the right side of Eq. (151, r,, represents the rate of mass formation of phase k per unit L-olume, and the last terms on the right side of Eqs. (16)-(18) represent the effects of solid-liquid interactions due to both phase change and diffusion at the interface. The third term on the right side of Eq. (16) and the second terms on the right side of Eqs. (17) and (18) are dispersion terms. In the momentum Eq. (161, for example, the dispersive “strFsses” are analogous to Reynolds stresses in turbulent flow, and the term V, represents the spatial fluctuation of the velocity of phase k within a REV. Dispersion terms do not appear in Prakash-model equations [Eqs. (7)-(14)], and an advantage of the volume-averaging technique is that it reveals terms, such as dispersion, which would otherwise be overlooked. However, the significance of dispersion is not well understood, and the effect is usually neglected [49, 551. Additional research is needed to determine whether neglecting dispersion is justified in solidification problems and under what conditions, if any, it must be considered. The third term on the right side of Eqs. (9) and (10) and the fourth term on the right side of Eq. (16) account for body forces, such as buoyancy or an electromagnetic field. The model of Prakash included source terms on the right sides of the energy and species conservation equations, whereas such terms were not included in either Eq. (17) or (18). Another difference between the models of Prakash [48] and Ni and Beckermann [49] related to the diffusion terms. In Eqs. (16)-(18), they are expressed in terms of fluxes and require substitution of appropriate constitutive relations [49]. In contrast, the Prakash-model equations [Eqs. (9)-(14)], explicitly include the laws of Newton, Fourier, and Fick for the stress, heat flux, and species mass flux, respectively. Another advantage of the volume-averaging technique is that it provides clear guidance for developing proper constitutive relations [49, 55, 62, 631. For example, the
CONVECTION IN ALLOY SOLIDIFICATION
259
volume-averaging technique suggests the following models for heat and species fluxes [49] ( q k ) =
-k$. -DZ
(19)
(&kV(Tk)’),
(Pk&kv(Ck)k)?
(20) where k: and DZ are, in general, anisotropic tensors that depend on the volume fraction of phase k and microscopic morphologic features of the solid-liquid interface (tortuosity), as well as the molecular thermal conductivity and mass diffusivity, respectively, of the solid and liquid phases. However, since complete models for k; and Dt have yet to be developed and will require significant empirical inputs, the simplified models of Prakash currently represent acceptable simplifications. Nevertheless, volume-averaging clarifies the approximations associated with simplified constitutive relations, and research directed to the development of improved constitutive relations is clearly needed. Based on volume-averaging theorems, Ni and Beckermann [491 suggest the following model for the stress tensor: (jk) =
( T k ) = p *k ( v ( & k ( v k ) k )
*
+ [ v ( & k ( V k ) k ) ] f - (V,)”&k
-
v&*(vS)’), (21)
where the viscosity pz depends on whether the phase is liquid or solid, the local solid fraction E , , and the tortuosity of the solid-liquid interface. Additional discussion of this topic is provided in Section III.D.2. Constitutive relations must also be substituted for the phase interaction terms (the last terms on the right side of each transport equation). Although volume-averaging provides useful insights to the origin and nature of these terms, both Ni and Beckermann [49] and Prakash [48] suggest models that are essentially the same. In all cases, the solid/liquidphase interaction terms are equal and opposite to each other and are composed of two components, one due to phase change and another due to microscopic nonequilibrium effects. Following the notation of Ni and Beckermann 1491 M k = ML -I-M i , Qk=Qkr+Qf,
(22)
(23)
Jk =J$ +JL, (24) where the superscript r designates transport due to phase change and d, q, and j designate transport due to local nonequilibrium. The rate of momentum transfer to phase k through the solid-liquid interface due to phase change within a REV ( M : ) can often be neglected because the density difference between solid and liquid phases is usually
260
PATRICK J. PRESCO’IT AND FRANK P. INCROPERA
small. However, the effect should be considered in rapid solidification processes [49]. The viscous interaction term (M;) depends strongly on the physical conditions being modeled and requires the introduction of empirical data for closure [49]. Additional discussion of solid-liquid momentum exchange is provided in Section III.D.l. The transport of energy and species across the phase interface is based on the assumption that the solid and liquid phases exist in equilibrium with each other at the interface. Hence, thermodynamic relationships can be applied at the interface to couple the solid and liquid macroscopic transport equations [48, 491. The exchange of energy between solid and liquid phases due to phase change may be expressed as
Qf
= %kirk
7
(25)
where hkirepresents the average enthalpy of phase k about the solid-liquid interface. The heat exchange due to thermal undercooling is expressed as
where S, is the interfacial surface area per unit volume, and as illustrated in Fig. 14a, lf is the diffusion length scale for phase k . Application of Eq. (26) requires knowledge of S, and I f , which depend on local conditions, such as the solidification rate, fraction solid, and interdendritic flow. The rate equations for interfacial species transport are Jk
r--
- Ckirk
for the contribution due to phase change and
for the contribution due to constitutional undercooling [49]. Figure 14b illustrates the diffusion length scale l i , which accounts for microscale phenomena and must be modeled through an appropriate empirical relationship [49]. Accurate modeling of microscale phenomena is essential to coupling the macroscopic transport equations through the phase interaction terms, and models that provide coupling between macro- and microscales are discussed in the next section. Equally important, however, is the numerical implementation of such models. Prakash [48] suggests a type of “upwind” numerical implementation of interfacial transport terms, which prevents physically unrealistic results (e.g., c k > 1 or c k < 0) and converges quickly.
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261
0 liquid
- interface
b
interface
0 solid
FIG. 14. Illustration of interfacial diffusion length scales for (a) heat transfer and (b) species transfer (reprinted with permission from Ni and Beckermann [49], 1991, ASM International).
C. MICRO/MACRO MODELS As discussed in Sections I1 and III.B.2, microscopic and macroscopic phenomena are coupled, and a research area of considerable importance relates to the development of solidification models that simultaneously predict the evolution of macroscopic and microscopic phenomena. However, the current understanding of micro/macro interactions is primarily qualitative, and macroscopic convection models that also predict features such as microsegregation, grain size and orientation, columnar-to-equiaxed
262
PATRICK J. PRESCO'R AND FRANK P. INCROPERA
transition (CET), primary and secondary arm spacings, and porosity are not yet well developed. A shortcoming of the original single-domain mixture models [12, 42, 431 is the manner in which solute is partitioned between solid and liquid phases. At a given temperature, the solid phase is assumed to be fully saturated with solute according to the equilibrium phase diagram. That is, the h e r law is applied under the assumption that freezing occurs through a series of equilibrium states. In order for equilibrium freezing to be a reasonable assumption, solidification must occur very slowly or the primary dendrites must be pure (there is complete rejection of solute by the dendrites). In most casting processes, the distribution of solute in single dendrite arm is nonuniform, due to the changing concentration of precipitated solid (Fig. 3) and the very low rate of diffusion in the solid phase. The condition is termed microsegregation. The manner in which solute is partitioned microscopically affects the calculation of solid fraction and liquid concentration, which, in turn, affect permeability and buoyancy forces, respectively. It also affects the amount of eutectic material formed in the final casting and the properties of the casting. Microsegregation has been considered by invoking the Scheil assumption [68], which presumes interdendritic liquid to be locally well mixed and negligible diffusion within solid dendrites. However, use of the assumption is not a matter of applying the Scheil equation [68] for partitioning the solute and determining the local solid fraction, since the Scheil equation does not account for the advection of solute into and out of a volume element. Rappaz and Voller 1691 discuss implementation of the Scheil assumption in modeling microsegregation in a macroscopic solidification model. The average concentration of solute in the solid phase is represented as
where the integral, which accounts for the solidification history, is incrementally evaluated over each time step and the integrand over a time step is simply the solid composition taken from the equilibrium phase diagram. A difficulty with this method is that it requires maintaining a complete record of the solidification history at each spatial grid node in order to property account for microsegregation during remelting [69]. For problems with no remelting, the equation works well without special consideration 147, 701. Remelting can be considered with Eq. (29) by approximating the composition of remelted solid to be [71], which is an underestimation for hypoeutectic alloys. Felicelli et al. [72] realized, however, that the Scheil
c,
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263
assumption and Eq. (29) can be applied and used to accurately account for remelting without requiring excessive memory to record the Cs -fs history by discretizing solid fraction. That is, Eq. (29) is expressed as
Acs,l
where the increments are calculated, recorded, and adjusted (if remelting occurs). The detailed solute accounting procedure of Eqs. (29) and (30) becomes significantly more complicated when solid grains are no longer stationary, and the Scheil assumption has not been applied in simulations that account for advection of equiaxed grains. The lever law and Scheil assumption represent two limiting dendrite conditions, namely, those of infinite and zero diffusion, respectively. In some cases, diffusion occurs in dendrites during solidification at a rate that renders neither the lever law nor the Scheil assumption valid, and the magnitude of uncertainty associated with invoking either one of the two limiting assumptions can be assessed by comparing results of simulations based on each limiting condition [711. A model for microscopic diffusion would have to be incorporated for greater precision, and Sundarraj and Voller 1731 developed a dual-scale model, which combined a onedimensional microsegregation model [74, 751 with a macrosegregation model, to treat solute transport during solidification of an AI-Cu alloy. Mo [76] proposed a model utilizing internal variables to account for microsegregation in a macrosegregation model. This model agrees with predictions made using the Scheil and lever law (equilibrium) assumptions for short and long solidification times, respectively. It is also able to simulate conditions between the two limiting cases and accommodates remelting in a straightforward manner [761. Another microscopic feature of importance is the actual microstructure, which affects terms like S, and 29 in Eq. (26) and ultimately determines the mechanical properties of casting. Phenomena such as nucleation, growth, impingement, and coarsening of grains [681 are influenced by thermal, solutal, and flow conditions, and models relating microstructural features to microscopic and macroscopic transport phenomena were reviewed by Rappaz [50]. However, many of the recently developed micro/macro models that incorporate nucleation, growth, and impingement of grains neglect convection in the melt [77-801. Nucleation and grain growth models [50, 681 were included in a volumeaveraged two-phase solidification model [491 for predicting final-grain-size distributions in solidified alloys, as well as convective transport phenomena during solidification [81]. Similar extensions have been proposed for continuum models [52, 531, although calculated results are not yet available.
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PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
Although two-phase models [48, 491 are less restrictive than mixture models [12, 43, 471, they are still limited in their ability to resolve microscopic phenomena, for which three different length scales can be identified [82]: (1) the overall radius of a grain, (2) the dendrite tip radius, and (3) the secondary arm spacing. In view of these considerations and with the goal of predicting microstructural details in a macroscopic transport model, including convection, Wang and Beckermann have proposed a volume-averaged multiphase model [51, 83, 841, which distinguishes interdendritic liquid from extradendritic liquid by modeling them as different phases (Fig. 15). The multiphase model is an extension of the two-phase model [49], and it considers two interfaces, one between solid dendrites and interdendritic liquid and another between the interdendritic and extradendritic liquids (i.e, the grain envelope). Each phase interface has a characteristic length scale, requiring two sets of interfacial species diffusion length scales (Fig. 16). The model has been used to predict columnar to equiaxed transition [85, 861, but without convection. Micro/macro modeling of alloy solidification is a relatively new approach to simulating casting processes, and it is only beginning to be included in models that account for convection. Micro/macro considerations of convective transport during alloy solidification are complex and computationally intensive, and much additional model development is needed. However, as computational power continues to increase, verified micro/macro solidification models are likely to become useful engineering tools for process design and analysis. D. SUBMODELS The single-domain models described in Section 1II.B require the introduction of several submodels. For example, a model based on experimental data for permeability must be used in the mixture momentum Eq. (31, rendering it semiempirical. Similarly, drag coefficients and solid- and liquid-phase viscosities are required for the two-phase momentum Eq. (161, and models must be used for transport coefficients such as thermal conductivity. The purpose of this section is to review various models that have been proposed for use in single-domain formulations of convection during alloy solidification. 1. Permeability and Dray Coeficients
Darcy's law is used in momentum equations to account for momentum exchange between interdendritic liquid and solid dendrites. The model assumes that the rate of momentum exchange between phases is propor-
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265
dendrite envelope
phase s
phase d
phase 1
a dendrite envelope phase I
phase: d
phase S
b FIG.15. Schematic illustration of physical conditions considered by the multiphase model: (a) columnar dendritic growth; (b) equiaxed dendritic growth (reprinted with permission from Wang and Beckermann 1831, 1993, Elsevier Science Inc.).
tional to the difference in their respective velocities and inversely proportional to permeability, which represents the square of an appropriate microscopic, viscous length scale. With dendrite arm spacings being on the order of l o w 5rn, permeabilities on the order of lo-" to lop9 m2 would seem appropriate.
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PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
s-d interfnce
% d-I
interface
Liquid
FIG.16. Illustration of diffusion length scales associated with interfacial species transfer in the multiphase model (reprinted with permission from Wang and Beckermann [841, 1993, ASM International).
The permeability of a dendritic array depends on several factors, including the local volume fraction solid and its structure. Regions within the mushy zone near the liquidus interface have relatively small local volume fractions of solid and a relatively large permeability. Conversely, the permeability of the dendritic array is relatively small near the solidus interface. Furthermore, the permeability of a dendritic mushy zone may depend on flow direction (anisotropic), as in the case with a columnar structure, or it may be isotropic when grains are equiaxed and small. The original macrosegregation models [5, 36, 371, which considered transport only within the mushy zone, used the following isotropic model to account for the variation of permeability with volume fraction of solid:
K = KO(1 - g , ) 2 = Keg:, (31) where KO is a model constant chosen to fit permeability data [21]. Although this model is appropriate within regions of the mushy zone for which g, > i, it is not valid near the liquidus interface. Furthermore, Eq. (31) is not suitable for use in the mixture momentum Eq. (31, because Eq. (31) yields K = KO when g, = 0 (gl = 1). That is, Eq. (31) fails to eliminate the third term on the right side of Eq. (3) when g, = 1 and, hence, would impose a non-physical damping on flow in the fully melted region. A model more suitable for the continuum momentum equation is the Kozeny-Carman (or Blake-Koseny) equation for permeability [12, 42,461:
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CONVECTION IN ALLOY SOLIDIFICATION
where the permeability constant KO is an empirical constant that depends on dendrite arm spacings [87]. This model assures that the permeability is infinite in the fully melted region (g, = 0, g , = l), and K varies continuously with g, to a value of zero in the fully solid region where g, = 1 ( g , = 0). However, the Blake-Kozeny equation is not intended for use with g, less than about 0.50 [88]. West [89] proposed a piecewise continuous permeability model that differentiates regions in the mushy zone according to their proximity to the liquidus interface. The model has been adopted by Amberg and coworkers in their simulations of binary metal alloy solidification [57, 701. West's model assumes capillary behavior in regions far from the liquidus, and it provides a transition to dispersed particle behavior in regions with large liquid volume fractions. The model is of the form
(8,
> 4).
(35b)
m2 and K 2 = 8.8 X The suggested model constants of K , = 6.4 X lo-'' m2 were chosen to fit experimental data [21]. The reliability of permeability models is significantly limited by a lack of permeability data. Of the limited data that have been published [21, 90-921, none report values of permeability for solid fractions less than approximately 30% [93]. However, the corresponding region of the mushy strongly influences macrosegregation, since the relative motion between solid and liquid is largest for small solid volume fractions. Such data are not available because of difficulties associated with their measurement, and numerical simulations of flow through tortuous paths have recently been performed in efforts to provide permeability data in the low-g, regions of mushy zones [94, 951. The two-phase solidification models described in Section III.B.2 require submodels for interfacial momentum exchange, and for flow through a coherent mushy zone, such as columnar dendritic regions or regions with packed equiaxed crystals, Darcy's law may be applied [49]. In regions with
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PATRICK I. PRESCOTT AND FRANK P. INCROPERA
free-floating crystals, the rate of momentum exchange between solid and liquid phases can be expressed as [491 (V,)'),
(36)
where Ad/Vo represents the total projected area of the solid phase per unit volume and C, is a drag coefficient that depends on a two-phase Reynolds number. Experimental data are required to correlate C , with two-phase flow conditions and microstructural features, and correlations based on a limited amount of data have been used [81, 96-981. In a proposed extension of the continuum model [52] to account for the motion of free-floating dendrites in the melt or slurry region, it is argued that the only significant relative motion between free-floating dendrites and liquid is due to buoyancy. The difference between solid and liquid velocities is based on a balance between local (microscopic) viscous and buoyancy forces
where d, is a characteristic grain diameter and pmis the effective viscosity of the slurry [53]. Instead of calculating the solid velocity from a separate solid momentum equation [e.g., Eq. (lo)], V, is related to the liquid velocity and used in the energy and species transport equations.
2. Transport Properties Transport properties, such as thermal conductivity, viscosity, and mass diffusivity, must be prescribed or modeled in order to solve the governing equations for either a mixture or a two-phase model. The most common modeling technique is to use independent sets of constant transport properties for the liquid and solid phases, with the assumption that neither the coexistence of phases nor the morphology of the interface affects the macroscopic properties. Hence, conventional, single-phase property data are used, and the transport properties of the mixture (continuum) are allowed to vary according to the local volume fractions of phases. The thermal conductivity of the mixture is expressed as
CONVECTION IN ALLOY SOLIDIFICATION
269
The mixture mass diffusivity is modeled in a similar fashion, and with Ds = 0 and g, p , = f,p , the mass diffusion coefficient appearing in Eq. ( 5 ) is
D = filll. (39) As discussed by Beckermann and Viskanfa [12] and Ni and Beckermann [491, the volume-averaging technique suggests that conventional singlephase values for properties such as k , and k , can be used only as approximations in lieu of available data that suggest a relationship between microstructural features and individual phase transport coefficients. In general, the macroscopic diffusion transport coefficients for a phase within a two- or multiphase system can be expected to be different from their microscopic versions and even anisotropic, despite isotropic diffusion at the microscale [49]. However, sufficient data are not yet available to justify detailed models that account for tortuosity. Liquid, solid, and mixture (slurry) viscosities are also important parameters in solidification models with convection. The solid phase within a coherent mushy zone may be regarded as having an infinite viscosity, meaning that it can withstand any stress without deforming. Because of the absence of experimental data from dendritic flows or a general theory [82], the liquid-phase viscosity in the mushy (or slurry) zone is typically prescribed to be the actual (microscopic) liquid viscosity, despite the presence of the dendritic structure. In a slurry region, the solid viscosity can be modeled in the following manner [49, 821:
where ps0is the solid viscosity in the limit when E, = 0, E , , ~is a critical value of solid fraction at which the solid becomes rigid, and a is an empirical constant (typically a = 2.5). Alternative models for solid viscosity can be adopted [49, 821.
IV. Theoretical Results and Experimental Validation Continuum and volume-averaged solidification models discussed in Section I11 have been used to predict convective phenomena for several different conditions. Many studies have focused on solidification of aqueous salt solutions, since these systems can readily be used in the laboratory and the convection patterns can be visualized. Metal alloys are receiving increased attention, but with fewer comparisons of predictions with experi-
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PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
mental results. Representative model predictions are reviewed in the following subsections, along with assessments based on comparing predictions with experimental results. A. SEMITRANSPARENT ~INALOG ALLOYS
Semitransparent analog alloys are used because they permit flow visualization. The most common analog alloy used in laboratory experiments is aqueous ammonium chloride (NH,Cl-H,O), with NH,Cl concentrations between 25 and 32 wt%. This system is popular because its freezing range corresponds to readily controllable conditions (7'' = - 15°C) and NH,Cl crystallizes with a dendritic morphology (i.e., with primary, secondary, and higher-order branches), which is similar to that of most metals [99, 1001. A disadvantage of using aqueous ammonium chloride is that it is highly corrosive and, thus, requires that appropriate measures be taken in the laboratory. Another analog alloy used by experimenters is aqueous sodium carbonate (NaC0,-H,O), which is less caustic than aqueous ammonium chloride. Its freezing range is also convenient for laboratory studies, but the NaCO, crystals which form in the mushy zone are devoid of secondary dendrite branches. Hence, the mushy zone is microstructurally different than those associated with most metal alloys. 1. Solidijication from the Sidewall of a Rectangular Cavity
The continuum model of binary solid-liquid phase change 1431 was first used to simulate convection heat, mass, and momentum transfer during solidification of a NH,Cl-H,O solution from one sidewall of a rectangular cavity of aspect ratio (height/width) A = 4 [42]. Two cases were considered, both of which involved opposing thermal and solutal buoyancy forces. In one case, a sidewall was cooled and maintained at a temperature below the solidus temperature, while the opposite wall was maintained at the initial temperature of the melt (above the liquidus temperature). Plots of the velocity, stream function, temperature, and liquid composition fields were presented at various times during solidification. Conditions were conduction dominated at the beginning of the process, and the simulation showed the transient development of convection patterns. Convection in the melt was thermally driven by heating from the hot sidewall and cooling through the solid and mushy zones, and solutally driven flow was predicted in the mushy zone. The coupled flows induced significant deviation from conduction dominated conditions, and the thicknesses of the fully solidified and mushy regions varied with vertical position. Furthermore, as a result of variations in thermal and solutal conditions near the liquidus
271
CONVECTION IN ALLOY SOLIDIFICATION
interface separating the mushy and melted zones, the shape of the liquidus interface was highly irregular. Figure 17 shows conditions that were predicted after 360 s of cooling. A thin layer of fully solid material covers the left sidewall, and mushy (s + I), and liquid (1) zones are indicated on the velocity plot (Fig. 17a). The streamlines in Fig. 17b reveal more clearly the thermally driven (counterclockwise) cell in the melt and the solutal (clockwise) cell in the mushy zone, and Figs. 17c,d show the temperature and liquid compositions fields, which have been distorted significantly by advection effects. Eventually, as the solidification rate diminished to near-zero, thermal-convection-dominated conditions in the melt, whereas solutal convection was confined to a relatively small region near the bottom of the mushy zone.
a
b
C
d
+ 8.1 mm/s
FIG.17. Predicted conditions at f = 360 s for solidification of aqueous ammonium chloride in a differentially heated rectangular cavity: (a) velocity vectors, (b) streamlines, (c) isotherms, and (d) liquid isocomps. (Reprinted from Bennon and Incropera [42]. Copyright 1987, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, U.K.).
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PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
A second simulation case involved an adiabatic sidewall opposite the chilled surface. Following the initial conduction-dominated period, thermal and solutal convection cells again developed in the melted and mushy zones, respectively. However, in this case, thermal convection in the melt was not sustained by a heated sidewall. As solidification progressed, temperature gradients in the melt became small and flow conditions in both the melt and mushy zones were increasingly influenced by solutal buoyancy. The evolution of macrosegregation patterns was also studied [loll. Figures 18a,b show, respectively, velocity vectors and liquid compositions predicted at t = 390 s, with solid (s), mushy (s + I), and liquid (1) zones identified in Fig. 18a. In addition, four regions in the mushy zone are outlined with dashed lines. Within these regions, interdendritic fluid is channeled, and the subsequent effects are revealed by the mixture composition plot of Fig. 18c, which corresponds to t = 870 s. Channels are created when water-rich interdendritic fluid flows upward into a warmer region of the mushy zone, displacing saltier fluid and partially remelting or dissolving the existing dendrites. Hence, a high-permeability region is formed, providing a preferred path for buoyant fluid. As interdendritic fluid is drawn into a channel, it precipitates NH,Cl in a region to the right and below the channel, leading to the A-segregate pattern (Fig. 18c) of alternating positive and negative segregation layers that slant upward and to the right from the chilled (left) wall. The aforementioned simulations were the first to account for coupled heat, mass, and momentum transport in solid, mushy, and melted regions and to predict features such as an irregular liquidus interface, the channeling of interdendritic fluid, and the formation of distinct A segregates in the final casting. The effects of microsegregation (coring) [69] and solid movement on macrosegregation were studied by Voller et al. [47]. It was found that when the solid was assumed to move with the liquid (no relative motion), macrosegregation was small. This result is expected because, without relative motion between solid and liquid, macrosegregation can occur only by species diffusion in the liquid, which occurs at a very low rate relative to advection. Although microsegregation does not physically occur in NH,CI dendrites, which are pure, it was simulated [47] and found to increase the severity of macrosegregation slightly, without changing the overall pattern of segregation. Continuum model predictions were compared with experimental observations in another study [102], in which a H,O-31 wt% NH,CI solution was solidified in a rectangular cavity. The model successfully predicted several qualitative features of the process, including liquidus interface irregularities, channel formation, and the formation and subsequent ero-
273
CONVECTION IN ALLOY SOLIDIFICATION
a
b
C ( t = 870s)
( t = 390s)
Ve Ioc it y
- 0.55
Macrosegregation
Liquid Composition
1 mm/s
f, HzO = 0.70
f
HzO
00.47- 0.56 0.56- 0.64 B 0.64- 0.72 C_ 0.72- 0.80
FIG. 18. Solidification of aqueous ammonium chloride in a side-chilled rectangular mold: (a) velocity vectors at t = 390 s, (b) liquid isocomps at t = 390 s, and (c) macrosegregation at t = 870 s (reprinted with permission from Bennon and Incropera [loll, 1987, ASM International).
sion of a double-diffusive interface. Figure 19 shows predicted convection conditions after 480 s of cooling through the left sidewall. A layer of cold but solutally buoyant (water-rich) fluid has formed at the top of the melted region (Figs. 19a,b) and is separated from the bulk melt by a doublediffusive interface (Figs. 19c,d). Thermal buoyancy forces act to recirculate
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PATRICK J. PRESCOTT AND FRANK P. INCROPERA
C
-
5.69mm/s
FIG. 19. Predicted conditions after 480 s of solidification of 31 wt% NH,CI-H,O in a differentially heated rectangular cavity: (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps. (Reprinted from Christenson et ul. [102]. Copyright 1989, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB,
U.K.)
the fluid in this layer, as well as the bulk liquid beneath it (Fig. 19b). Interdendritic fluid ascends within the mushy zone as a result of solutal buoyancy forces (Figs. 19b, d), and completely remelted channels were predicted in the upper part of the mushy zone (Fig. 19a). With time, the rate of solidification decreases, as does the flow of interdendritic fluid from the mushy zone to the top layer. Hence, the double-diffusive interface could not be sustained, and at t = 720 s, the interface was eroded. Although the predicted transport phenomena were in qualitative agreement with experiments, predictions did not compare well quantitatively [1021. An example of the discrepancy between predicted and measured results is shown in Fig. 20, where the solidus and liquidus interface
CONVECIION IN ALLOY SOLIDIFICATION
a
275
--- Measured
-
Predicted
b
7 \
I
I I FIG.20. Measured and predicted interface locations at (a) t = 180 s, (b) t = 660 s, and (c) t = 1200 s. (Reprinted from Christenson et at. [102]. Copyright 1989, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, U.K.)
locations are plotted at three different times during solidification. The model significantly underpredicted the rate of advancement of the solidification fronts, with much of the discrepancy attributable to uncertainties in prescribed model parameters. It was subsequently determined [1031 that the value prescribed for the solid thermal conductivity in the early singledomain model simulations [12, 42, 101, 1021 was incorrect (due to a typographical error in the source from which it was obtained [38]) and that the actual value was approximately six times larger. Other sources of disagreement include uncertainties in the permeability of the mushy zone and the effects of temperature and concentration on all properties. A model consisting of equations for momentum and species transport in the liquid phase (assuming a stationary solid phase with no diffusion), along with an equation accounting for heat transfer in a solid-liquid
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PATRICK J. PRESCOTT AND FRANK P. INCROPERA
mixture, was used to simulate experimental conditions for which an aqueous ammonium chloride solution was solidified from one side of a square cavity [12,104]. Although the model was developed from volume-averaging theorems and its form is slightly different than that of the continuum model [43, 451, the two models contain the same essential features. Predicted results were in qualitative agreement with measurements, but quantitative agreement was only fair [12, 1041. The prescribed value of the solid thermal conductivity was taken from Szekely and Jassal [38], which, as discussed previously, was much too small. Because the simulation did not consider a layer of fully solidified material, the effect of this erroneous input was, perhaps, less noticeable than in the study of Christenson et al. [1021. Discrepancies between predicted and measured results were also attributed to uncertainties in other model inputs and assumptions, including that of an isotropic permeability for the mushy zone. The solidification of aqueous ammonium chloride in square and tall rectangular ( A = 4) cavities was simulated recently by Zeng and Faghri [lo51 using a continuum model with a temperature-based energy equation [651. Thermal and solutal buoyancy effects were examined for solidification in the square cavity by considering (1) only thermal buoyancy, (2) only solutal buoyancy, and (3) thermosolutal convection. The conditions considered by Christenson et al. [13, 1021 were simulated in a tall rectangular cavity, with similar results. It appears as though Zeng and Faghri [lo51 also used an unrealistically low value for the solid thermal conductivity. The effect of anisotropic permeability was considered by Yo0 and Viskanta [106]. A continuum model was used with an anisotropic permeability model to simulate some of the experimental conditions reported by Beckermann [104]. Permeability in the mushy zone depended on whether the flow was parallel, perpendicular, or oblique to the growth direction of primary dendrites, which was assumed to be aligned with the local temperature gradient. The ratio of the principal permeabilities (parallel or perpendicular to temperature gradient) was varied from 0.5 to 2, and it was found that predicted convection conditions were very sensitive to this ratio. Figures 21a-c show predicted streamlines for R = 0.5, 1.0, and 2.0, respectively, after 20 min of solidification. Since the temperature gradients are primarily horizontal, R represents the ratio of permeability for flow in the horizontal direction to that in the vertical direction (KJK,,). The geometric mean of K x and K , was kept constant. When the permeability to flow in the vertical direction is relatively small (R = 2.0) (Fig. 21c), the flow of water-rich interdendritic fluid to the top of the cavity is impeded, thereby retarding development of a double-diffusive layer along the top of the liquid region. In contrast, when the permeability to flow in the vertical direction is relatively large ( R = 0.5) (Fig. 21a), the growth of a
(dw)
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a
277
b
FIG. 21. Streamlines predicted at t = 20 min for solidification of aqueous ammonium chloride in a square cavity with anisotropic permeability in the mushy zone: (a) R = 0.5; (b) R = 1.0; (c) R = 2.0. (Reprinted from Yo0 and Viskanta [1061. Copyright 1992, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB,
U.K.)
water-rich layer along the top of the liquid pool is accelerated. The significant influence of anisotropy on solidification suggests that the effect should be considered to achieve accurate simulations.
2. Solidificationfrom the Bottom Wall-Unidirectional Solidification Double-diffusive convection leading to freckle formation in unidirectionally solidified ingots has been observed experimentally in aqueous ammonium chloride solutions [16, 107-1111 but had not been predicted
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PATRICK J. PRESCOTT AND FRANK P. INCROPERA
until a continuum model was applied [112, 1131. In one of these investigations [113], the continuum model was used to study the effects of permeability and cooling rate on the convective transport phenomena and macrosegregation during unidirectional solidification in a vertically aligned cylindrical mold. The melt was assumed to be initially quiescent, homogeneous (fONHbC' = 0.32), and isothermal with an 8°C superheat (To = 50°C). The axisymmetric simulation began with a sudden reduction in temperature of the bottom circular surface of the mold (T,,), and three cases were considered. Case 1 used a permeability constant of K O = 5.556 X m2 in Darcy's law and a chill plate temperature of T, = -30°C. Conditions for cases 2 and 3 corresponded to KO = 5.556 X lo-.'' m2, T, = - 30°C and K O = 5.556 x lo-'' m2, T, = - 100°C, respectively. For case 1, conditions during the first 6 min of solidification were conduction-dominated, with the advancing solidification front, isotherms, and liquid isocomps remaining planar and horizontal. However, the density gradient within the mushy zone is unstable because the NH,Cl concentration increases with temperature, according to the equilibrium phase diagram. Figure 22a shows velocity vectors, streamlines, and isotherms at t = 6 min, when fluid motion is initiated. The attendant disturbances amplify, and by t = 7 min (Fig. 22b), the effect of fluid motion induces
FIG. 22. Predicted flow field and isotherms for unidirectional solidification from below (case 1): (a) t = 6 min; (b) 1 = 7 min; (c) t = 8 min. (Reprinted from Neilson and Incropera [113].Copyright 1992, with permission from Springer-Verlag.)
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small perturbations in the isotherms near the liquidus interface. At t
=
8 min (Fig. 2 2 4 fluid velocities have become significantly larger and the
array of recirculation cells along the liquidus interface is exchanging fluid between the melted and mushy zones. Cold fluid ascending from the mushy zone is solutally buoyant and extracts heat from the surrounding, warmer fluid of nominal composition, which is descending to the mushy zone. As solidification continues, convection becomes stronger and continues to distort otherwise planar isotherms and liquid isocomps. Fluid motion extends throughout the melt, and the complex flow pattern changes continuously. The perturbed temperature and liquid composition fields along the liquidus interface affect the growth of the interface, and because the rate of species diffusion is much smaller than the rate of heat diffusion, channels were eventually predicted to originate at the liquidus interface [113]. Cold, water-enriched (solutally buoyant) liquid that is generated in the mushy zone by the precipitation of solid ammonium chloride, ascend to warmer, water-deficient regions of the mushy zone, acquiring energy far more effectively than ammonium chloride. The net effect is an increasing potential to remelt or dissolve dendrites and to thereby induce the downward growth of fully melted channels [112]. The upward flow of fluid through a channel and into the bulk melt is augmented by the fact that it is heated (gains thermal buoyancy) without significant loss in solutal buoyancy. For the conditions of case 1 [113], two prominent channels develop and provide for the discharge of interdendritic fluid into the bulk melt, while bulk fluid seeps across the liquidus interface and into the mushy zone around the channel openings. When the permeability of the mushy zone is increased by one order of magnitude (case 21, the aforementioned transition from conduction- to convection-dominated conditions occurs much sooner, indicating that Darcy damping reduces the rate at which perturbations can grow. Also, because of the lower resistance to interdendritic fluid flow, the increased permeability enhances exchange of fluid between the melt and mushy zones, with a corresponding increase in the number of fully melted channels that are predicted to form following the onset of convection [113]. Figure 23 corresponds to f = 25 min for case 2 and reveals five channels, one of which is at the outer radius, or left edge, of the plot. When the solidification rate was increased by lowering the bottom surface temperature to -100°C (case 3), the onset of convection was delayed and the amount of fluid exchanged between the melted and mushy zones decreased significantly. The effects of permeability and cooling rate on macrosegregation are shown in Fig. 24. Liquidus and solidus lines are superimposed on the
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PATRICK J. PRESCOIT AND FRANK P. INCROPERA
FIG. 23. Predicted flow field for unidirectional solidification from below (case 2) at t = 25 min: (a) velocity vectors; (b) streamlines. (Reprinted from Neilson and Incropera [113]. Copyright 1992, with permission from Springer-Verlag.)
macrosegregation plots. Macrosegregation in fully solidified regions is permanent, whereas macrosegregation in the mushy zones changes slightly, as solidification continues past 50 min. Case 2 provides the most severe macrosegregation (Fig. 24b), since it corresponds to the earliest onset of convection and the largest fluid velocities. Since solidification was conduction-dominated for the longest period of time in case 3, macrosegregation is least severe. The foregoing simulations of unidirectional solidification [112, 1131 were performed under the assumption of axial symmetry. Therefore, the channels and freckles were predicted to be rings, rather than the discrete, pencil-shaped regions observed experimentally [ 16, 107-1 111. However, despite this obvious difference, the axisymmetric calculations are consistent with experimental observations in several important respects. Specifically, channels have been observed to initiate at the liquidus interface and to grow downward [107,108].Furthermore, the number of active channels decreases with time, and terminated channels have been revealed in etched sections of unidirectional solidified metal castings [15, 1141. Transient, three-dimensional calculations in cylindrical coordinates were performed [1151, and the predictions revealed discrete channels, while confirming the conclusions drawn from axisymmetric simulations [112, 1131.
N 00
Y
FIG.24. Macrosegregation patterns predicted at r = 50 min for unidirectional solidification: (a) case 1; (b) case 2; (c) case 3. (Reprinted from Neilson and Incropera [113]. Copyright 1992, with permission from Springer-Verlag.)
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PATRICK J. PRESCOTT AND FRANK P. INCROPERA
The problem of unidirectional solidification of a binary alloy from below continues to be the subject of many experimental [17, 116-1191 and theoretical [ 120, 1221 studies. Numerical simulations of unidirectional solidified metal alloys are discussed in Section 1V.B.
3. Multidirectional Solidification Despite being a common casting configuration, multidirectional solidification has received much less attention than unidirectional solidification from a chilled bottom wall or sidewall. Solidification of aqueous ammonium chloride in a square cavity chilled from the bottom wall and vertical sidewalls was recently studied both numerically and experimentally [18, 1231. The numerical calculations were based on a continuum model for momentum, heat and species transport during binary solid-liquid phase change [43] and simulated experimental conditions in a square test cell (101.6 x 101.6 mm). The nominal composition of the alloy was 27 wt% NH,Cl, and the initial temperature of the melt was uniform at 25°C which is 6.4"C above the nominal liquidus temperature of 18.6"C. At t = 0, the left and right vertical and bottom walls of the mold were chilled to - 30"C, which is 14.6"C below the eutectic temperature of -154°C. Details regarding the properties of the aqueous NH,CI system, experimental methods, and numerical procedures can be found elsewhere [18, 1231. Predictions during the early stages of solidification indicated a strong influence of thermal convection in the melt, causing the mushy zone thickness along the vertical sidewalls to increase with increasing distance from the top. The mushy zone thickness along the bottom wall was uniform and relatively large. At t = 10 min (Fig. 2 9 , conditions within the melt are asymmetric (despite the symmetry of the imposed boundary conditions). This result is attributed to the somewhat random locations at which two channels formed in the mushy zone over the bottom wall (Fig. 25b). Thermally driven downflows along the vertical liquidus interfaces turn inward at the liquidus interface along the bottom wall, causing solutally buoyant fluid emerging from the channels to coalesce near, but slightly to the right of, the cavity midplane (Figs. 25a, b). Interdendritic liquid in the bottom mushy zone is drawn into the two channels, in a manner similar to that for unidirectional solidification from below [ 17, 1121, while interdendritic fluid in the sidewall mushy regions ascends and is discharged into the melt along the top of the cavity. Figures 25c, d indicate that cold-water-enriched fluid is accumulating at the top of the mold, and, due to the depression of the liquidus temperature with water enrichment, dendrites are remelted (Fig. 25b).
CONVECTION IN ALLOY SOLIDIFICATION
a
2-D (102x66) t=lO.O min
-
b
Wmar = 1.51 x 1 0 2
t= 600.0 sec
3.40 mm/s
283
ymin= -1.31 x 10.2
Y
f
I 1 1 W
d
+ x
X
t= 600.0 sec
W
X
FIG.25. Predicted conditions at t = 10 min for multidirectional solidification of aqueous ammonium chloride in a rectangular cavity; (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps [18].
Experiments revealed physical conditions much more complex than those predicted by the model during early stages of solidification [123]. Shadowgraphs (Fig. 26) revealed two counterrotating thermal convection cells that developed immediately after the mold walls were chilled. Although dendritic crystals began forming on the sidewalls within 15 s of
284
PATRICK J. PRESCOm AND FRANK P. INCROPERA
FIG.26. Shadowgraphs taken during multidirectional solidification of aqueous ammonium chloride in a rectangular cavity: (a) r = 1 min; (b) I = 2.5 min; (c) t = 5 min; (d) t = 7.5 min [181.
cooling, the vigor of the thermal convection cells caused most of the crystals to be detached and to be advected with the liquid. Since solid NH,Cl is relatively dense, many of the equiaxed crystals collected in two piles on the bottom of the cavity near the sidewalls (Fig. 26b). Solidification along the bottom wall generated cold but solutally buoyant interdendritic liquid, and a fine-scale salt finger convection pattern was observed (Figs. 26a, b). As solidification continued, there was a transition from many salt fingers to fewer, well-defined plumes (Figs. 26a-d) and regions along the bottom, where equiaxed crystals collected, were especially conducive to pluming and channel formation. Eventually, the plumes emanating from these regions were enveloped by the mushy zone advancing from the sidewalls, which essentially transformed the plumes into A-segregate channels in the sidewall mushy regions [18, 1231. Within 2 min, coherent mushy regions formed along the sidewalls 118, 1231, and interdendritic liquid ejected from the mushy zone was entrained in thermally driven downflows along the vertical liquidus interfaces. How-
CONVECTION IN ALLOY SOLIDIFICATION
285
ever, because of solutal buoyancy forces associated with water-enriched liquid emerging from the sidewall mushy region, flows along the vertical liquidus interfaces turned abruptly inward and then upward, exhibiting the fishhook structures shown in Fig. 26b. In Fig. 26d, water-enriched liquid is seen to be accumulating from the top left and right regions of the cavity. Eventually these pockets joined along the top and formed the first of several double-diffusive convection cells, which significantly affected solidification conditions in the adjacent mushy regions [18, 1231. These double-diffusive convection cells also interacted with plumes ascending from the middle of the bottom mushy layer. Within 10 min, small detached dendritic crystals were observed in the upper region of the melted zone, near the liquidus interface. This slurry region and the A-segregate channels, from which the dendrite fragments were believed to have originated [18, 1231, are shown in Fig. 13. Equiaxed crystals were also discharged from the bottom mushy region and advected by ascending plumes, The advection of equiaxed dendritic crystals has been observed by others [124-1271, and it is not yet clear whether fragments are fractured by flow-induced drag, remelted due to coarsening, or created by a combination of fracture and remelting effects. The numerical model [18] did not account for the advection of crystallites and hence could not predict the formation of mounds of equiaxed crystals on the floor of the mold. At a vertical position of approximately 75 mm, the model also predicted extended mushy zone growth from each sidewall and an associated merging of liquidus interfaces to form a “bridge” of dendrites across the melted zone. The predicted macrosegregation pattern at t = 30 rnin (Fig. 27) shows severely segregated freckles near the bottom, where vertical channels formed during early stages of solidification, and an A-segregate pattern along the sidewalls near the top. Although the model is able to predict several key trends, it is limited by its inability to account for slurry behavior and nonequilibrium conditions that existed in the experiments [18, 1231. Furthermore, the numerical simulations were unable to resolve many fine-scale features of the doublediffusive convection that occurred during solidification. Models that account for two-phase, nonequilibrium behavior [49,52,53] would have to be implemented with a much finer numerical discretization to capture such details. The computational effort required for such improvements is presently prohibitive.
4. Other Configurations The continuum model [43] has been used to simulate solidification of semitransparent analog alloys for conditions involving mixed convection
286
PATRICK .I. PRESCOTT AND FRANK P. INCROPERA
A segregate
NHqCI-Rich
Water-Rich
19.9-22.4 22.4-24.9 24.9-27.3 FIG.27. Macrosegregation predicted at chloride [18].
f =
=
27.3-29.7 29.7-32.1 32.1-34.6
30 min for solidification of aqueous ammonium
[ 1281, combined buoyancy- and surface-tension-driven convection [ 129, 1301, and buoyancy-driven convection in a horizontal annular region [ 1311. Solidification of an aqueous NH,C1 solution flowing downward in a vertical rectangular channel was studied numerically [ 1281, and predictions have been compared with experimental results [1321. Mixed convection conditions resulted from the interaction of thermosolutal buoyancyinduced flows with the forced through flow. For a flow Reynolds number of Re = 200, solutally buoyant interdendritic liquid was able to penetrate the liquidus interface, thereby discharging water-enriched liquid into the main flow near the top of the channel. These conditions favored the development of distinct channels in the mushy zone, facilitating the exchange of liquid between mushy and melted zones. With Re = 800, interdendritic flow was confined to the mushy region, and channel develop-
CONVECTION IN ALLOY SOLIDIFICATION
287
ment was reduced significantly. The effects of chill wall temperature and nominal alloy composition were also studied, and the system was found to be most sensitive to composition. Although experimental trends were predicted by the simulations, good quantitative agreement was precluded by (1) the inability to experimentally effect a sudden change in chilled wall temperature, (2) uncertainty in the nominal alloy composition, and (3) uncertainties associated with model assumptions, such as equilibrium freezing, constant thermophysical properties, and mushy zone permeability [132]. The effects of thermo/diffusocapillary convection were delineated in numerical simulations of solidification for aqueous NH,CI in a 20 X 20-mm cavity with a top-free surface [1291. Without buoyancy effects (a zerogravity environment), conditions associated with thermocapillary convection were compared with those for thermo/diffusocapillary convection and found to be virtually identical, indicating that diffusocapillary effects were negligible. In the absence of gravity, surface tension significantly affected solidification by retarding solid growth near the top of the cavity, where surface tension driven flow is strongest. In a 1-g environment, surface tension driven flow along the top free surface inhibits the discharge of interdendritic liquid from the mushy zone into the melt during early stages of solidification. Soon thereafter, however, solutal buoyancy forces discharge interdendritic liquid from the mushy zone into a region just below the top-free surface, thereby subdividing the melt into a top cell driven by capillary forces and a bottom cell driven by thermal buoyancy. Eventually, conditions approached those associated exclusively with buoyancy-driven convection [1291. Differences between predicted and experimental results were associated with the times at which significant events occurred and with the thicknesses of the solid and mushy layers [130]. The continuum model [43] was also used to simulate the solidification of an aqueous sodium carbonate solution (Na,CO,-H,O) from the inner surface of a horizontal annulus [131]. To assess the accuracy of predicted results, the conditions of the simulation were chosen to match those of a related experimental study [133]. Although solutal upflow from the top portion of the mushy zone was predicted and observed, there was a significant difference in the flow structures. In the simulation, a single thin plume of cold but solutally buoyant fluid ascended from the uppermost region of the mushy zone. In contrast, shadowgraphic and dye injection flow visualization revealed a wide band (across the top quadrant of the mushy zone) of buoyant plumes with a very complex, perhaps turbulent, flow structure. Also, whereas the numerical model predicted the development, downward movement and eventual erosion of a single doublediffusive interface, multiple interfaces were observed in the experiments.
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PATRICK J. PRESCOTT AND FRANK P. INCROPERA
Moreover, the actual evolution of double-diffusive convection patterns occurred much more slowly than did conditions predicted by the model. Finally, while good agreement was obtained between measured and predicted results for the thickness of the fully solid layer, the thickness of the mushy zone was overpredicted. Discrepancies were attributed to numerically induced false diffusion, the lack of a turbulence model to account for the complex mixing that occurred at the top of the cavity, and uncertainties associated with the mushy zone permeability, which was thought to be highly anisotropic in the experiments [131]. The disagreement with mushy zone thickness may also be attributed to uncertainties in the actual alloy composition and, perhaps, to constitutional undercooling at dendrite tips 1681. B. METAL ALLOYS
Although the solidification models discussed in Section I11 are able to simulate solid-liquid phase change in many physical systems (e.g., freezing and melting in Arctic waters, food processing, and magma chambers), they were developed specifically for simulating the casting of metal alloys. Metals are unique by virtue of their low Prandtl numbers. The Prandtl number, which is the ratio of diffusivities for momentum and heat, is an important parameter in buoyancy-induced convection, since it affects the relative thicknesses of velocity and thermal boundary layers, which, in turn, determine the regions over which thermal buoyancy and viscous forces are significant. The Prandtl number of most molten metals is on the order of lo-' to lo-*, which is two to three orders of magnitude smaller than values typical of aqueous solutions. Amberg [S7] simulated the solidification of an Fe-1 wt% C alloy (Pr = 0.174) in a rectangular cavity (100 mm tall by 200 mm wide), cooled through its vertical sidewalls with a constant and uniform heat flux (60 kW/m2). Approximately one hour was required to completely solidify the casting. The simulation was based on model equations from Hills et al. [41], which were integrated numerically using a finite-volume approach with an explicit time-marching scheme for most terms in the governing equations and an implicit scheme for heat diffusion terms. Thermal convection within the melt affected the shape of the liquidus interface during early stages of solidification, whereas solidification within the mushy zone wa6 primarily conduction-dominated. During intermediate stages of solidification, the entire mold cavity was occupied by a mushy zone, and a weak solutally driven convection cell provided solute transport that enriched the top and depleted the bottom region of carbon. However, the
CONVECTION IN ALLOY SOLIDIFICATION
289
final carbon concentration was between 0.97 and 1.03 wt% throughout most of the casting. Amberg’s modeling procedures [57] were also employed to simulate experiments in which Sn-10 wt% Pb and Pb--15 wt% Sn alloys were solidified in a square (100 X 100-mm) mold [70]. The experiments were performed with a constant temperature difference maintained between the vertical walls, while the temperatures of both walls were gradually reduced from values exceeding the liquidus temperature to values below the solidus temperature. The measured wall temperatures were used as boundary conditions in the simulations, Thermal and solutal buoyancy forces augmented each other during the solidification of the Sn-10 wt% Pb alloy (Pr = 0.0151, and predicted fluid velocities, along with lines of constant solid fraction, are shown in Figs. 28a-c at times of 3,6, and 9 min, respectively [70]. At t = 3 min (Fig. 28a), a strong, counterclockwise thermally driven convection cell in the melt is responsible for retarding the growth of the mushy zone near the top of the mold. Since solutal buoyancy forces augment thermal buoyancy forces, interdendritic fluid descends through the mushy zone and is discharged into the melt along the bottom of the cavity, and at later times, dendrites are remelted along the bottom as a result of the accumulation of Pb-rich fluid (Figs. 28b, c). Figures 29a, b show predicted macrosegregation patterns at 9 and 45 min, respectively, and reveal significant Pb enrichment along the bottom of the casting. Good agreement was obtained between predicted and measured macrosegregation results (Figs. 30a-c). Because of opposing thermal and solutal buoyancy forces during solidification of the Pb-15 wt% Sn alloy [70], convection and macrosegregation patterns are more complex than those of the Sn-10 wt% Pb alloy. Fluid velocities are shown in Figs. 31a-c for 12, 24, and 36 min, respectively. At t = 12 min, the mushy zone thickness is highly nonuniform as a result of strong, counterclockwise thermal convection in the melt. However, with increasing time, the thermally driven cell vanishes, as solutally driven Convection becomes dominant (Figs. 31b, c). The clockwise solutal convection cell in the mushy zone is also responsible for the highly irregular distributions of solid fraction in the mushy zone (Fig. 31) and the irregular macrosegregation pattern (Fig. 32). Although fully melted channels in the mushy zone were not predicted, Fig. 32 displays an A-segregate pattern of slanted adjacent regions of positive (Sn-rich) and negative (Pb-rich) segregation. Also, a wedge (analogous to a cone) of Sn-rich material was predicted to form in the top right region of the casting (Fig. 32b) and is attributed to solutal convection during the intermediate and later stages of solidification. Although macrosegregation measurements were made at
290
PATRICK J. PRESCO’IT AND FRANK P. INCROPERA
x=o
x=lOcm
x= 0
x= 10 cm
x=O
x= i o c m
FIG.28. Velocity vectors and liquid volume fraction distributions predicted for solidification of a Sn-10 wt% Pb alloy: (a) t = 3 min; (b) t = 6 min; (c) 9 min. Velocity arrow lengths equal to the grid spacing correspond to 5 cm/s and 0.1 mm/s in the fully melted and mushy zones, respectively. Isopleths are drawn for liquid volume fractions of 1.0, 0.8, and 0.6. (Reprinted with permission from Shahani et al. [70], 1992, ASM International.)
only a few locations, reasonably good agreement was obtained between predicted and measured results (Figs. 33a-d). Detailed predictions of the evolution of convection and macrosegregation during solidification of a Pb-19 wt% Sn alloy in an experimental test cell are provided by Prescott and Incropera [1341. Solidification occurred in axisymmetric, annular mold of stainless steel, cooled along its outer vertical wall. The heat flux was expressed as
E z
a
9%
II
%
13% 0 II
1 5%
x
x =o
x = lorn
b
x =o
x=lOn
FIG.29. Macrosegregation patterns predicted for a Sn-10 wt% Pb alloy at (a) t = 9 min and (b) t = 45 min (reprinted with permission from Shahani et al. [70], 1992, ASM International).
29 1
292
PATRICK J. PRESCOTT AND FRANK P. INCROPERA
a Wt. % Pb
20 -
vertical section VII, at xm9.47 cm
18 -
0
experiment
-elmulation 16-
~
0
87 0
'
I
I
I
1
2
4
6
8
10
Y (cm) FIG.30. Measured and predicted macrosegregation patterns for a Sn-10 wt% Pb alloy at (a) x = 9.47 cm, (b) x = 8 cm, and (c) x = 3.5 cm. The cooled surface corresponds to x = 0. (Reprinted with permission from Shahani el al. [70], 1992, ASM International.)
where T'Jr) is the local wall temperature, T, is the coolant temperature, and U is an overall heat transfer coefficient. Values of T, = 13°C and U = 35 W/m2 K were prescribed to match experimental conditions [135, 1361. According to predictions, a thermal convection cell is established in the melt once cooling is initiated [134]. The cell is driven by a radial temperature gradient confined within 10 mm of the mold wall, and the central portion of the melt becomes thermally stratified during the initial 90 s of cooling. Moreover, convective mixing reduces the temperature gradient throughout the melt, thereby delaying the onset of solidification relative to conduction-dominated conditions. At t = 120 s, solid dendrites begin precipitating at the bottom of the cooled mold wall, thereby forming a two-phase (mushy) zone. As cooling continues, the mushy zone grows, with the liquidus interface moving vertically upward and radially inward. h il result of phase equilibrium requirements, the precipitation of solid is accompanied by Sn enrichment of interdendritic liquid, which ifiduces solutal buoyancy forces acting upward on the interdendritic liquid. f i e
-
293
CONVECTION IN ALLOY SOLIDIFICATION
b Wt. Yo Pb 20
-
18
-
, 16
-
14
-
12
-
vertical section VI, at x=8 cm
0
0 experiment -simulation
o
10 -
0
8
I
I
0
0
0
0
-
1
1
20 vertical section 111, at xz3.5 cm 0 experiment -simulation
-o r
8
0
1
I
1
2
4
6
1
8
10 Y (cm)
FIG.30. Continued.
294
PATRICK J. PRESCOlT AND FRANK P. INCROPERA
z
1 -
x =o
x=lOm
h
1 -
,
x =o
x=lOm
C
x
,
x =o
I
*
x=lOm
FIG.31. Velocity vectors and liquid volume fraction distributions predicted for solidification of a Pb-15 wt% Sn alloy: (a) t = 12 min; (b) t = 24 min; (c) t = 36 min. Velocity arrow lengths equal to the grid spacing correspond to 5 cm/s and 0.1 mm/s in the fully melted and mushy zones, respectively. Isopleths are drawn for liquid volume fractions of 1.0, 0.8,0.6, 0.4, and 0.2. (Reprinted with permission from Shahani et al. [70], 1992, ASM International.)
solutal buoyancy forces oppose thermal buoyancy forces caused by the radial temperature gradient, and because the density of Sn is significantly less than that of Pb, solutal forces dominate within the mushy zone ( N = -14). Hence, solidification within the mushy zone can be regarded as providing an upward momentum source for the interdendritic liquid. As the mushy zone continues to grow, the influence of solutal buoyancy increases, and Figs. 34 and 35 show predicted convection conditions at t = 155 s and t = 195 s, respectively. Field plots of velocity vectors, streamlines, isotherms, and liquid isocomposition lines are drawn on r-z planes, with the outer and inner radii represented by the left and right
a
18% 16%
12%
I
14%
x=o
1
x=l Ocm
b 24%
22% 20% 1 8%
16%
1 2%
1 4%
I
I
x=o
x=l Ocm
FIG.32. Macrosegregation patterns predicted for a Pb-15 wt% Sn alloy at (a) t = 30 min and (b) f = 120 min. (Reprinted with permission from Shahani el al. [70], 1992, ASM International.)
295
35
:.
30-. 25
vertical section VII, at x-9.47cm 0 experiment -simulation
'
0
-
20 -
15-+
10
= I
1
35 -.
:.
30-.
25
vertical section VI, at x=8 cm
(
0 experiment -simulation
-
20 -
15
-4
10 1
I
I
I
I
C Wt. X Sn 35
'
vertical section 111, atx=3.5cm
.
0 experiment -simulation
30-. 25
-
0
d Wt. % Sn 35
.
30-. 25
vertical section II. atx-2cm 0 experiment -simulation
-
FIG.33. Continued.
297
298
PATRICK .I.PRESCOTT AND FRANK P. INCROPERA
edges of each plot, respectively. Also, the bottom and vertical mold walls are included in these plots. At t = 155 s (Fig. 34), the mushy zone covers approximately 75% of the inside surface of the outer mold wall. Fluid is exchanged between the mushy and melt zones in a relatively confined region near the top of the mushy zone (Fig. 34b), where a strong, solutally driven upflow, emerging from the mushy zone, interacts with thermally driven downflow in the bulk melt. The interaction turns both flows radially inward, thereby constricting the thermal cell and returning discharged interdendritic fluid to the mushy zone. Since the Pb-Sn system is characterized by a large Lewis number (Le = Sc/Pb = 8600), fluid within the solutal convection cell readily exchanges energy with the bulk liquid but largely retains its composition. Hence, warm, Sn-enriched fluid from the melt is advected toward the mushy zone, establishing conditions conducive to the development of channels within the mushy zone. A channel, although not fully melted, is aligned vertically and is located along the mold wall for z* 2 0.5. It is delineated by a thick dashed line in Fig. 34a. As solidification continues, the channel along the outer mold wall grows and continues to ingest interdendritic fluid, which is advected to the top of
a -+
b 18.7 mmls
d
C Wrniw4.04254 Wrnax= 0.03687
Trni,-j=279.2T Tmax=289.0°C
f%li"=O.1900
FIG.34. Convection conditions in a solidifying Pb-19 wt% Sn alloy after 155 s of cooling: (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps (reprinted with permission from Prescott and Incropera [134], 1994, ASME).
CONVECTION IN ALLOY SOLIDIFICATION
299
the mold cavity. In addition, thermosolutal interactions along the liquidus interface create small recirculation cells in which interdendritic and bulk liquid are exchanged. This flow pattern was identified as the genesis of an A-segregate pattern in the final casting [134]. The momentum associated with the thermal convection cell gradually decreases, as radial temperature gradients in the melt diminish and opposing solutal buoyancy forces increase. At t = 195 s, the thermal cell is confined to the bottom half of the fully melted zone (Fig. 35b), while the solutal cell encompasses the mushy zone and the top half of the bulk melt, which has become solutally stratified (Fig. 35d). The thermal cell is completely extinct by 240 s, beyond which a large solutally driven cell occupies the entire mold cavity and provides for the recirculation of interdendritic liquid. Figure 36 shows the predicted evolution of macrosegregation between 240 and 600 s. At t = 240 s, macrosegregation is characterized by a pattern of A segregates [7, 1371 (Fig. 36a), which formed in the outer periphery of the ingot as a result of thermosolutal convection during the early stages of solidification. The overall mass fraction solid is 10.4% at 240 s, and
b
a -
A
-
C
d ffLn=O. 1900
9.43 mmls
FIG.35. Convection conditions in a solidifying Pb-19 wt% Sn alloy after 195 s of cooling: (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps (reprinted with permission from Prescott and Incropera [134], 1994, ASME).
a
b
C
d
L
FIG. 36. Predicted macrosegregation patterns in a Pb-19 wt% Sn alloy after (a) 240 s, (b) 300 (reprinted with permission from Prescott and Incropera [134],ASME).
I
(c) 360 s, (d) 480 s, and (e) 600 s
CONVECTION IN ALLOY SOLIDIFICATION
301
macrosegregation intensifies with time, as solidification and solutal convection in the mushy zone enrich the lower and outer regions of the ingot with Pb, while transporting Sn to the top and downward along the inner radius. During the intermediate stages of solidification, a large cone of Sn-rich material forms on the top of the mold and extends deep into the ingot. The development of this cone segregate is illustrated in Figs. 36b-e. By 600 s, the overall mass fraction solid is 45.4%, and although the solid fraction in the top part of the cone segregate is less than lo%, solid fractions exceed 35% throughout the rest of the ingot, with Pb-rich zones near the outer radius being 60% solidified. At this time the permeability is sufficiently small throughout the cavity to inhibit fluid recirculation and hence further development of macrosegregation. Since discrete channels form in actual ingots and interdendritic flow patterns are, therefore, three-dimensional [136], local composition variations may exceed those shown in Fig. 36, which are predicted for axisymmetric conditions. In a related study, the effect of cooling rate on convection conditions and macrosegregation was studied by comparing simulations with five different values of the overall heat transfer coefficient between 35 and 3000 W/m2 K [601. Because the extent of the mushy zone and time over which the permeability is large are reduced with increased cooling rate, macrosegregation is less severe. The macrosegregation predicted for U = 3000 W/m2 * K is shown in Fig. 37, and except for small regions near the top and bottom of the casting, the Sn concentration is within 0.5% of the nominal (19%) concentration. The predicting thermosolutal convection patterns and the macrosegregation trends were verified through comparisons with measurements and metallographic examinations [ 1361. However, predicted and measured results were characterized by several significant differences. Figures 38a-c show predicted and measured cooling curves at three different vertical locations within the mold cavity during the first 600 s of cooling [136]. Overall, the predicted cooling rate exceeds the measured cooling rate, and differences may be attributed to uncertainties in prescribed thermodynamic properties, which, for example, affect the thermal capacitance of the system, and to uncertainties in the overall heat transfer coefficient, which governs the rate of heat transfer from the system. However, the most conspicuous disagreement relates to the absence of the undercooling and recalescence effects in the predicted cooling curves between 150 and 300 s. The measured cooling curves in Figs. 38a-c reveal a period of time, immediately after the commencement of solidification, during which the temperature increases for r* 5 0.8. This nonequilibrium phenomenon, termed recalescence [68], occurs following nucleation in an undercooled melt. Because the model assumed local thermodynamic equilibrium, such
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PATRICK J. PRESCOTT AND FRANK P. INCROPERA
20-21.5%
19.5-20°/o
U19-19.5%
m
18.5-19%
FIG. 37. Final macrosegregation pattern predicted for a Pb-19 wt% Sn alloy with = 3000 W/m2 . K (reprinted with permission from Prescott and Incropera [60],1991, ASM International).
U
phenomena could not be predicted. Furthermore, inferences drawn from the measured recalescence patterns indicated the existence of a convecting slurry [136], whereas the model assumed the mushy zone to be stationary. Although attempts have been made to account for nonequilibrium phenomena and solid-particle transport in the melt using the continuum model [61], there is a clear need to incorporate more sophisticated models for such phenomena [52, 531. Finally, temperature fluctuations were observed in measured cooling curves prior to and just after the onset of solidification. Such fluctuations are indicative of vigorous convection occurring within the molten metal and disappear following recalescence (Fig. 38), since the mushy zone dampens fluid motion. Predicted cooling curves are smooth, even before solidification, indicating failure to resolve pertinent features of the flow in the simulation. Measured macrosegregation patterns from six experimental ingots [1361 are compared with model predictions [134] in Figs. 39a-c. Zero A%Sn
CONVECTION IN ALLOY SOLIDIFICATION
303
corresponds to no macrosegregation, positive values represent Sn enrichment, and negative values represent Sn depletion. The agreement between predicted and measured macrosegregation patterns is reasonable, and although the data are scattered, they confirm general trends predicted by the model. That is, the concentration of Sn increases with increasing height and, at z * = 0.5 and 0.83, with decreasing radius. However, since the uncertainty in the measured results is only +0.40%Sn [135, 1361, it does not account for the scatter, which is as large as 4%. The circumferential variation of macrosegregation in an experimental ingot [1361 is plotted in Fig. 40, which shows %Sn as a function of 8 at ( r * , z*) = (0.3,0.83). The plot indicates clearly the three-dimensional nature of the macrosegregation field. If conditions were axisymmetric, the variation of %Sn with 8 would be within the measurement uncertainty interval. Since the data in Fig. 39 were taken at different circumferential positions, the scatter is largely attributable to existence of a threedimensional macrosegregation pattern. Locations in Fig. 39 that exhibit pronounced scatter are believed to be within segregated regions associated with channeling during solidification. Metallographic examination of ingot sections revealed pockets of eutectic material, providing further evidence that macrosegregation is three-dimensional, due to the development of discrete channels in the mushy zone during solidification 11361. The volume-averaged two-phase model of solidification [49] was used to simulate equiaxed solidification of an Al-4 wt% Cu alloy in a 50 X 50-mm cavity cooled from one vertical sidewall [81]. Since the interdendritic liquid is enriched with copper, thermal and solutal buoyancy forces augment each other. The model accounted for nucleation, growth, and advective transport of grains. However, the floating or settling of dendrites was not modeled, because the solid density was prescribed to be equal to the liquid density. Results demonstrated the model’s ability to predict recalescence, macrosegregation, and grain size distribution. Solidification of a metal alloy, chilled from below, has been simulated by Heinrich et af. [138, 1391 using a finite-element formulation [721. The Pb-10 wt% Sn alloy was characterized by unstable solutal gradients when cooled from below, and the resultant solutal upwelling produced channels in the mushy zone and freckles in the final casting [15, 108, 1141. Calculations were performed for small physical domains (e.g., 5 X 10 mm and 2.5 X 4.5 mm) [138, 1391 so that solidification and transport behavior could be resolved within the channel region using a grid of 20 X 30 elements. The effect of lateral cooling or heating through sidewalls was investigated, and it was concluded that channels tended to form along the sidewalls, especially with lateral cooling. Sidewall heating had the effect of moving the channels inward slightly [139].
304
PATRICK J. PRESCO’IT AND FRANK P. INCROPERA
a310 305 300 295 T
290
(“C:) 285 280 275 270 0
60
120
180
240
300
360
420
480
540
600
Time ( 5 )
FIG.38. Measured and predicted cooling curves during the solidification of a Pb-19 wt% Sn alloy in a cylindrical annular mold: (a) z* = 0.083 (near bottom); (b) z* = 0.50 (midheight); (c) z* = 0.83 (near top) (reprinted with permission from Prescott et al. [136], 1994, ASME).
Solidification of Al-Cu alloys from below was simulated by Diao and Tsai [140, 1411, and their predictions agreed very well with experimental data [4, 1421. However, the density gradient was both thermally and solutally stable, and thus, no buoyancy-induced convection occurred. Fluid flow was driven solely by shrinkage, rendering the problem one-dimensional. In a subsequent study, Diao and Tsai considered under-riser macrosegregation [143]. Overall, stable buoyancy conditions were imposed, and fluid flow was driven mainly by shrinkage associated with phase change. However, because of sudden expansion experienced by downward flow from the riser into the main mold cavity, recirculation cells were induced in the top comer regions of the cavity and were responsible for macrosegregation in the under-riser region. The effects of shrinkage-driven convection on solidification were considered by Chiang and Tsai [%I, and an attempt was made to compare the effects of shrinkage to those of buoyancy [144] in a 1% Cr steel, chilled from the bottom and sidewalls of a rectangular cavity. However, their calculations were based on an unrealistically large contraction ratio, [( ps - p , ) / p , ] X loo%, of 20% and a thermal expansion coefficient more than four orders of magnitude smaller than an appropriate value. Further-
305
CONVECTION IN ALLOY SOLIDIFICATION
120
180
240
300
360
420
480
540
600
360
420
480
540
600
Time (s)
C
0
60
120
180
240
300 Time (s)
FIG.38. Continued.
more, solutal buoyancy was completely ignored. Combined shrinkage- and buoyancy-driven convection during the solidification of an AI-Cu alloy was also simulated by Xu and Li [19] using a continuum model [59]. However, the effects of shrinkage and buoyancy were not delineated in a systematic manner.
306
PATRICK J . PRESCOTT AND FRANK P. INCROPERA
I
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The effects of shrinkage and buoyancy on macrosegregation were compared recently for the solidification of a Pb--19.2 wt% Sn alloy from the side of a rectangular cavity [20]. Nine numerical simulations were performed, covering a matrix of conditions involving three cooling rates and (1) thermosolutal convection without shrinkage, (2) thermosolutal convection with shrinkage, and (3) shrinkage-driven flow without buoyancy effects. Results demonstrated that buoyancy exerts the dominant influence on macrosegregation and that shrinkage effects are important only near the solidus interface under extreme cooling conditions. A scaling analysis reinforced conclusions drawn from the numerical simulations [20] and suggested similar behavior for an Al-Cu alloy. Scaling also suggested that, for a Cu-Sn system, shrinkage effects may be important relative to buoyancy, since thermal and solutal buoyancy forces nearly offset each other in the mushy zone. Although continuous or direct-chill (DC) casting is of great practical importance and the subject of many investigations, simulations of convection heat and mass transfer in the melt and mushy zones are sparse. Existing models are somewhat restrictive, treating, for example, only conduction [145], only transport within the mushy zone [146- 1481, or transport only within the melted region [149]. Convection and solidification
307
CONVECTION IN ALLOY SOLIDIFICATION
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FIG.39. Continued.
in a twin-roll continuous casting process were simulated by Ha et al. [150] using a continuum model [431, although solute transport was not considered and buoyancy was neglected. These effects were also ignored by Farouk et al. [151,152], who simulated solidification in a twin-belt continuous caster, with turbulence treated using a low-Reynolds-number model
308
PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
24
23
22
s
8 21
20
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e (deg.1 FIG.40. Circumferential variation of measured Sn concentration at r* (reprinted with permission from Prescott et al. [136], 1994, ASME).
5
0.30 and Z*
90
-
0.83
[1531. Shyy et al. [154] modeled steady-state continuous ingot casting, with electron-beam melting (EBM), and included the effects of strand motion, thermal buoyancy, surface tension, and turbulence on convection in the melt and mushy zones. The simulation incorporated an adaptive grid over which continuum equations were integrated. However, the effects of solute transport on macrosegregation and buoyancy were not considered. Finally, a single-domain model, which accounted for thermal and solutal buoyancy in both liquid and mushy zones, was used to simulate DC casting of an A1-4.5 wt% Cu alloy [155]. The effects of thermal and solutal buoyancy forces and casting speed were studied, and although the results of the investigation were considered to be preliminary, they indicated that solutal buoyancy contributed significantly to increased copper concentration along the centerline of the ingot.
V. Strategies for Intelligent Process Control
Buoyancy-induced fluid flow, which is primarily responsible for most forms of macrosegregation and affects grain structure in castings, is not directly controllable. One means of eliminating buoyancy effects is to
CONVECTION IN ALLOY SOLIDIFICATION
309
perform solidification processes in a zero-gravity environment, and although studies have indicated that such measures would prevent the formation of freckles in directionally solidified castings [109,156,157], they are not practical for large-scale industrial operations. Macrosegregation can also be reduced by increasing the cooling rate, but the cooling rate is often limited by uncontrollable factors such as contact resistance between the mold and ingot. Moreover, because of the thermal resistance imposed by the outer solid shell as it forms, the effective cooling rates in interior regions of an ingot are necessarily smaller than those near the mold wall. Hence, macrosegregation is more difficult to control in large ingots. In addition to gaining a better understanding of the mechanisms by which macrosegregation occurs, some researchers have explored process control options for which macrosegregation is inhibited. Possible options include the use of inertial, centrifugal, or electromagnetically induced forces to activity control macrosegregation. Kou et al. [36] performed experiments in which a cylindrical mold was rotated about its axis during the solidification of a Sn-Pb alloy. A n axially moving cooling jacket surrounded the mold, and immersion heaters maintained a melt above the mushy zone. The entire system was mounted on a turntable, and thermocouples connected through a slip ring were used to monitor temperatures during the solidification process. Without rotation, Pb-rich segregates formed in the center of the ingot (positive centerline segregation) as a result of the upward concavity of the liquidus and solidus interfaces. The shape of the interfaces allowed Pb-rich interdendritic liquid to flow downward and radially inward, along a path that was nearly parallel to the liquidus interface. Centrifugal forces due to rotation therefore inhibited denser Pb-rich fluid from migrating toward the center of the ingot, and a more uniform radial composition profile was achieved. However, it was reported that if rotational speeds became too large, segregates formed along a ring between the axis and outer edge of the ingot. Petrakis et al. [158] also investigated rotation as a means of controlling macrosegregation. The apparatus was a modification of that used by Kou et al. [36], and again, the phase-change material was tin alloyed with lead. In these experiments the ingot remained stationary, but swirl was introduced into the melt (above the mushy zone) by rotating immersion heaters. They found that with sufficient swirling of the melt, the shape of the mushy zone could be changed from concave to convex upward. Depending on rotational speed and solidification rate, rotation was found to have a beneficial effect on macrosegregation. The aforementioned studies involved Sn-Pb alloys for which thermal and solutal buoyancy forces augmented each other. The objectives of introducing swirl were to (1) oppose the effect of the buoyancy forces with a
310
PATRICK J. PRESCOTI AND FRANK P. INCROPERA
centrifugal force and/or (2) stabilize the liquid density gradient by enhancing heat transfer in the melt, rendering the isotherms, liquid isocomps, and liquidus and solidus interfaces nearly planar (horizontal). However, when solutal and thermal buoyancy forces oppose each other during solidification, the liquid density gradient cannot be completely stabilized, and, when the system is cooled from below, double-diffusive finger and plume convection may occur. Hence, in contrast to the aforementioned studies, Sample and Hellawell [lo71 found that steady mold rotation had little or no effect on freckle formation in unidirectionally solidified ingots of NH,Cl-H,O and Pb-Sn that produce solutally unstable density gradients when solidified from below. However, it was discovered that, by slowly rotating an ingot about an axis making a small angle with the direction of gravity (i.e., a precessional movement), freckle formation could be suppressed in directionally solidified ingots [107, 1081. Although the original motivation for this work was to continuously change the orientation of the gravitational field relative to the ingot, it was surmised that suppression of channel formation was due primarily to the shearing action associated with movement of the bulk melt along the liquidus interface [107]. The shearing action was induced by the precessional motion and minimized perturbations in the liquid density gradient [108]. The findings of Sample and Hellawell [107, 1081 are supported by the numerical simulations and experiments of Neilson and Incropera [112, 159, 1601. Both numerical simulations [ 1591 and experiments [ 1601 indicate that relatively slow, steady rotation of the mold during unidirectional solidification of a binary alloy has a negligible effect on the formation of channels and freckles, although steady rotation does have the effect of “organizing” the flow in the melt overlying the mushy zone [159]. Furthermore, the hypothesis of Sample and Hellawell [ 1081 regarding the physical basis of channel formation is consistent with additional calculations [1591, for which intermittent rotation was applied during unidirectional solidification. When a stationary mold holding a quiescent liquid is suddenly rotated, an initial “spinup” period follows, during which the liquid gradually approaches steady, solid-body rotation. Angular momentum is transferred from the mold walls to the liquid by shear forces in boundary layers along the mold wall known as Ekman layers. Shear forces in Ekman layers are responsible for minimizing perturbations in field variables along the liquidus interface, thereby eliminating a prominent mechanism for channel nucleation [112]. Although the Ekmann layers would eventually vanish as the liquid achieves solid-body rotation, mold rotation can be terminated, creating a “spindown” period during which the liquid gradually loses its angular momentum.
CONVECTION IN ALLOY SOLIDIFICATION
311
Simulations were performed [159] using the continuum equations for binary solid-liquid phase change 1431 in axisymmetric coordinates. In addition to equations for conservation of total mass, axial and radial momentum, energy, and species, an equation was introduced for conservation of swirl ( a = o r 2 ) in a system containing liquid, mushy, and solid zones [159]. The equation is of the form d 2CL CL -(pn)fV~(pVR)=V~(~vn)-----(n-n,) dt r ar K
where the mixture swirl is a mass fraction-weighted average of solid- and liquid-phase swirls, = f,n, + fin,.The third term on the right side of Eq. (42) is a Darcy damping term. The fourth term on the right side of Eq. (42) is an advection-like source term, which arises in a manner similar to advection-like source terms in Eqs. (4) and (5). The last term on the right side of Eq. (42) accounts for solid phase acceleration [45], which must be prescribed. Figures 41-43 show predicted convection conditions at various times during solidification of an aqueous NH,C1 solution from below with intermittent rotation [159]. The end of the first spinup period occurs at t = 90 s, for which conditions are shown in Fig. 41. Swirl (Fig. 41c) is responsible for centrifugal forces that drive counterrotating convection cells in the melt (Fig. 41b). The Ekman layer along the liquidus interface (Figs. 41a, b) diminishes perturbations in temperature (and salt concentration) along the interface (Fig. 41d). However, a solutally driven plume emerges from the most elevated region of the mushy zone, which corresponds to the outer mold wall (Figs. 41a, b). The melt achieved a near state of solid-body rotation at t = 90 s, at which point mold rotation was halted to reduce the potential for eventual channel formation. Figure 42 shows predicted conditions during the subsequent spindown period ( t = 130 s). The centrifugally induced convection cell above the liquidus interface has reversed direction, restoring nearly horizontal isotherms. The simulation continued with alternating spinup and spindown periods, and conditions at t = 19 min (Fig. 43) reveal the existence of channels at the centerline and outer radius, but not at intermediate radii. The aforementioned numerical results suggest that intermittent rotation, with a half-period approximately equal to the theoretical spinup time,
312
PATRICK J. PRESCOTI' AND FRANK P. INCROPERA
-
2 . 3 8 mn/s
,x=l .000
b
An* = 0.100
C
r
-
r-++ FIG.41. Predicted convection conditions at t = 90 s (end of spinup) during unidirectional solidification of an aqueous ammonium chloride solution with intermittent rotation (10 rpm): (a) velocity vectors; (b) streamlines; (c) isoswirls; (d) isotherms. (Reprinted from Neilson and Incropera [159]. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, U.K.)
suppresses the formation of channels, and therefore freckles, throughout most of a unidirectionally solidified casting. This conclusion was validated experimentally [160]. Figures 44a-c are photographs taken from above a cylindrical mold, in which aqueous ammonium chloride was solidified from below. Figure 44a shows that channels formed only along a ring near the outer radius of the mold during intermittent rotation with a half-period of 90 s. Without rotation (i.e., a static casting), channels formed at random
CONVECTION IN ALLOY SOLIDIFICATION
313
b
&,=0.413
An*= 0.041
r+-I
FIG. 42. Predicted convection conditions at t = 130 s (spindown) during unidirectional solidification of an aqueous ammonium chloride solution with intermittent rotation (10 rpm): (a) velocity vectors; (b) streamlines; (c) isoswirls; (d) isotherms. (Reprinted from Neilson and Incropera [159]. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, U.K.)
locations on the horizontal plane shown in Fig. 44b. Figure 44c shows that when the mold is rotated intermittently at a frequency twice that of Fig. 44a, channels form along two concentric rings. One ring, containing nine channels, formed along the outer mold wall, and the other ring, containing five channels, formed within a radial interval from 60-65% of the outer mold radius.
314
PATRICK J. PRESCOTT AND FRANK P. INCROPERA
b
r
<+
FIG.43. Predicted convection conditions at t = 19 min (during spinup for the seventh cycle of the intermittent rotation process) during unidirectional solidification of an aqueous ammonium chloride solution with intermittent rotation (10 rpm): (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps. (Reprinted from Neilson and Incropera [159]. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, U.K.)
Inertially induced convection can also be effected by rocking a mold. Such effects were investigated experimentally for unidirectional solidification in a rectangular mold using a 27% aqueous NH,Cl solution [161]. Oscillation frequencies of 1-2 cycles per minute were considered, and the associated inertial forces induced fluid motion that enhanced mixing in the melt. Inertially driven, oscillatory convection in the melt also swept the
CONVECTION IN ALLOY SOLIDIFICATION
315
FIG. 44. Top view of liquidus interface showing channel distribution after 90 min of solidification of an aqueous ammonium chloride solution: (a) intermittent rotation with a half-period of 90 s; (b) static solidification;(c) intermittent rotation with a half-period of 45 s. (From Neilson and Incropera [la]. Reproduced with permission. All rights reserved.)
liquidus interface of perturbations in temperature and solute concentration that induce channel formation in statically cast ingots. Although still present, channels that formed under rocking conditions were fewer in number and smaller than those associated with static solidification. Moreover, the thick overgrowth of dendrites that extended from the liquidus
316
PATRICK J. PRESCO'IT AND FRANK P. INCROPERA
interface around a channel under static solidification conditions was absent under rocking conditions. That is, the volcano-like morphology shown in Fig. 12 did not evolve during solidification with mold oscillation. Garimella et al. [162] recently conducted experiments in which vibration was imposed on an NH,CI-H,O system, solidified from below. It was found that with sufficiently strong vibration (0.6 mm in amplitude and a frequency of about 130 Hz), channels could be completely suppressed for alloys with nominal compositions less than about 26% NH,Cl, and partially suppressed for alloys of higher salt content. Although the mechanism for channel suppression is not completely understood, it was suggested that vibration altered the microstructure of the mushy zone by causing secondary and/or tertiary dendrite arms to break from main branches and to settle within the mushy zone [162]. Although it was observed that finger convection persisted during vibration, rather than vanishing after plumes and channels developed, it was not reported whether the discharge of dendrite fragments accompanied finger convection with vibration, as it does during early stages of solidification without vibration [ 1621. Although not discussed, vibration can induce fine-scale mixing, thereby altering the effective diffusivities of species and heat, and hence the effective Lewis number, which is an important parameter in double-diffusive convection [lo]. Another means of influencing convection in liquid metals is through the use of magnetic fields, and reviews of efforts to develop electromagnetic materials processing technologies are available [ 163, 1641. However, neither experimental nor theoretical studies of the effects of magnetic fields on convection and macrosegregation during solidification are abundant. Furthermore, although the effects of both steady (dc) and timeharmonic (ac) magnetic fields on solidified grain structure have long been recognized [ l , 231, the complex flow field and its interaction with a dendritic mushy zone are not understood well enough to be modeled accurately. Because body (Lorentz) forces are induced that are proportional, but in opposition, to fluid flow transverse to a magnetic field, convection in a melt may be dampened with application of a dc magnetic field. That is, the locally induced Lorentz force is
F,
=
a,(V X B) X B,
(43) where a, is the electrical conductivity of the fluid, V is the fluid velocity, and B is the magnetic induction field. The use of dc magnetic fields in Czochralski crystal growth has been investigated [165-1681, and results indicate that melt convection is reduced and axial dopant distribution is more homogeneous with application of a magnetic field. Theoretical mod-
CONVECTION IN ALLOY SOLIDIFICATION
317
els for transport phenomena in the presence of magnetic fields have been developed for crystal growth applications [169-1761, but these models were developed specifically for melted regions, and not for two-phase zones. Vives and Perry [261 experimentally considered magnetically damped convection for the solidification of pure tin and two different aluminum alloys (1050 and 2024). It was found that the applied magnetic field reduced or eliminated temperature oscillations, while suppressing convection and possibly turbulence. A metallographic study also showed that the magnetic field yielded coarse-grain structures, which is consistent with previous studies [l]. However, interdendritic fluid flow and macrosegregation were not addressed for the aluminum alloys. Boettinger et al. [177] directionally solidified Pb-Sn in a very smalldiameter (3-mm) crucible. Their study showed significant macrosegregation caused by solutally driven interdendritic flow, both with and without the application of vertical and horizontal magnetic fields. The ineffectiveness of the magnetic field at reducing macrosegregation was attributed to the relatively small field strengths applied in their experiments, and a linear stability analysis [1781 showed that extremely large magnetic fields would be required to prevent instabilities and solutal convection. The effects of magnetic damping on thermosolutal convection and macrosegregation during the solidification of a Pb-Sn alloy in an axisymmetric annular mold (cooled through its outer vertical mold wall) has been studied numerically by Prescott and Incropera [ 1791. The applied magnetic field was assumed to be purely axial, rendering the Lorentz force purely radial. Hence, the effect of the magnetic field was treated by including the term - me B,'u on the right side of the radial momentum equation, where u is the radial velocity component. It was found that an applied magnetic field significantly affects thermally driven flows and the interaction between thermally and solutally driven flows during early stages of solidification. Because magnetic damping is negligible compared to small-scale viscous (Darcy) damping during intermediate stages of solidification, interdendritic flows and macrosegregation are not significantly altered by magnetic fields of moderate strength. Scaling analysis revealed that extremely strong fields would have to be applied in order to effectively dampen convection patterns that contribute to macrosegregation. In contrast to steady magnetic fields, time-harmonic or moving magnetic fields induce stirring forces in a molten alloy. Therefore, electromagnetically induced forces (Lorentz forces) drive convection within a molten metal, the pattern of which will depend on several factors, including whether single- or multiphase inductors are used. Several experimental investigations of induction stirring have been performed during solidification or casting processes, although interdendritic
318
PATRICK J. PRESCOTT AND FRANK P. INCROPERA
flow and macrosegregation have not been addressed. Vives and Ricou [ 1801 studied electromagnetic induction and associated fluid motion during the electromagnetic casting of aluminum alloy 7049. Search coils and potential sensor probes were used to measure the magnitude, direction, and phase of the magnetic induction and electric current density fields, and a magnetic velocity probe [181] was used to measure fluid velocities. The effect of an electromagnetic screen was also studied in a laboratory model of an electromagnetic caster using mercury [182], and it was found that fluid motion was very sensitive to the vertical position of the screen (positioned between the inductor and the strand). Meyer et af. [1831 studied the flow of molten aluminum in a rectangular test cell driven electromagnetically by a vertical linear motor. In contrast to thermal convection, the flow direction and velocity were easily controlled through the inductor. Flow in a shallow, rectangular bath of mercury, induced by a linear motor beneath the bath, was studied by Dubke et af. [184] in an effort to validate a mathematical model for electromagnetically induced turbulent flow in the central melt region of a continuous cast strand. A Hall-effect probe was used to measure the induction field, whereas velocities were measured by a drag probe and by particle tracing on the top-free surface of the test section. Laboratory experiments simulating electromagnetic casters were also performed by Li et af. [185], who used a Wood’s metal to simulate the melt and two different inductors, powered by a 30-kW, 3-kHz power supply, to induce fluid motion in the melt. Search coils and potential sensor probes were used to measure induction and current density, respectively, and a magnetic velocity probe [181] was used to measure fluid velocities. The effects of inductor size and current, as well as the position of an electromagnetic shield, on meniscus shape and flow conditions were studied, and measured quantities were compared to model predictions [1851. More recently, Vives [186,1871 has reported results of an experimental study of electromagnetic rheocasting of metal alloys and metal matrix composites. The process involves mechanically rotating a set of permanent magnets about a model and is capable of refining the grain structure with less electricity consumption than needed with induction stirring [186,187]. Theoretical models for transport phenomena, including ac magnetic field effects, have been developed for induction furnaces [ 188-1921, in which turbulent fluid motion is induced. In addition, fluid flow models with electromagnetic stirring have been reported by researchers studying continuous casting [184, 193, 1941 and electromagnetic casting [185, 195, 1961. However, since all of these models were developed for single-phase systems, flows in a mushy zone or interactions between interdendritic and bulk liquid flows were not considered. Ilegbusi and Szekely considered
CONVECTlON IN ALLOY SOLIDIFICATION
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electromagnetic stirring of a slurry [197, 1981 and solidification of a solid-liquid slurry with electromagnetic stirring (rheocasting) [ 1991. The emphasis of this work was on modeling the non-Newtonian behavior of a solid-liquid slurry (with solid fractions between 15% and 60%) subjected to strong electromagnetic agitation. Models that consider the effects of ac magnetic fields on coupled interdendritic and melt convection, as well as macrosegregation, during the solidification of binary alloys were not developed until very recently. Prescott and Incropera [200, 2011 considered electromagnetic induction as a means of altering convection during solidification of a Pb-19 wt% Sn alloy in an axisymmetric annular mold. Numerical simulations were performed using a continuum model for binary solidification [43, 451, with extensions to account for Lorentz forces and turbulence. The electromagnetic field induced by the induction coils was characterized by its angular frequency w and wavenumber k,, as well as by the RMS field strength, Br,R M S , at the outer mold wall. A dimensionless grouping
represents the relative influences of Lorentz and thermal buoyancy forces; a second grouping
accounts for both thermal and solutal effects. The study [201] considered conditions for which G , = 3.2 and G, = -0.25, indicating that thermal convection, which occurs exclusively before the onset of solidification, is augmented significantly by Lorentz forces, whereas the combined influence of thermal and solutal buoyancy forces within the mushy zone is attenuated slightly (25%) by the Lorentz forces. A low Reynolds number k--E model [153, 2021 was used to account for turbulence. The turbulence kinetic energy equation was modified by adding a Darcy damping term [ - ( p , / K ) k e ] as a source that renders the turbulence kinetic energy k, negligible in the mushy zone. The Darcy term vanishes in the fully liquid region where the permeability K is infinite [201]. The rationale for this model extension is found elsewhere [201], along with other model details. Simulations were performed with and without (laminar flow) the turbulence model. Both simulations accounted for undercooling and recalescence using a supersaturation model [61, 2001. Results of the laminar flow simulation with electromagnetic stirring (EMS) are summarized as follows.
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PATRICK J. PRESCOTT A N D FRANK P. INCROPERA
On initiation of cooling and application of electromagnetic stirring forces, a vigorous convection cell develops in the mold cavity, and predicted velocities exceed 60 mm/s. Hence, convective heat transfer between the melt and the cooled mold wall is increased, delaying the onset of Solidification by approximately 10 s relative to conditions without EMS [611. During initial stages of solidification ( t < 170 s), the complex temperature field associated with thermally and electromagnetically driven convection yields a mushy zone of highly nonuniform thickness along the cooled mold wall [200]. Because the melt becomes undercooled prior to the onset of solidification, the mushy zone grows quickly and by t = 180 s, the mold cavity is occupied almost entirely by a mushy zone. Since the 1%), velocities through much of volume fraction is very small (less than the mushy zone are large ( 27 mm/s), and although solid dendrites are assumed to remain stationary, conditions are extremely conductive to formation of a slurry. Near the outer mold wall, a radial gradient in liquid Sn concentration drives a convection cell that opposes and interacts with the convection cell driven primarily by Lorentz forces. The interaction occurs near the top of the cavity and yields significant distortions in the temperature and liquid concentration fields, creating conditions that strongly favor local remelting and the development of a channel in the mushy zone. A partially melted channel forms by t = 195 s and is outlined with a thick dashed line in Fig. 45a. The radial component of the liquid Sn concentration gradient (Fig. 45d) is increasing and is responsible for growth of the solutal convection cell (Fig. 45b). However, growth of this cell is slowed by opposing Lorentz forces, which remain constant during the procqss. As solidification continues past 195 s, the solutal convection cell grows, and the interaction between counterrotating convection cells continues to promote channel development by advecting warm fluid from the interior of the cavity toward the channel site. As the mushy zone and liquid Sn ooncentration gradients grow, flow conditions become dominated by solutal buoyancy, and the electromagnetically driven cell decays and eventually disappears at t = 240 s. The final macrosegregation pattern is established within 600 s and is shown in Fig. 46a. The channel that developed in the mushy zone is manifested as a severely segregated region. With electromagnetic stirring, the predicted macrosegregation is significantly larger ( M R M S = 1.61%) then that predicted without stirring (MRMs= 1.34%) [134]. This increase is attributed to the prominent channel that formed in the electromagnetic stirring simulation, and the result contradicts experimental findings [1351. The discrepancy is believed to be due largely to the assumption of laminar flow. N
N
321
CONVECTION IN ALLOY SOLIDIFICATION
a -,
b Wmi,p-O. 02606 13.4mmls
C
Tmir+276.8'C ...... Tm~284.8'C
ff&=0.1906
FIG.45. Predicted convection conditions at t = 195 s during solidification of a Pb-Sn alloy with electromagnetic stirring. Laminar flow conditions: (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps. (Reprinted with permission from Prescott and Incropera 12011, 1995, ASME).
Flow velocities as high as 60 mm/s were predicted in the laminar simulation, and with the hydraulic diameter of the mold cavity used as the length scale, the corresponding Reynolds number exceeds 2.5 X lo4, rendering the assumption of laminar flow suspect. Turbulence has the effect of increasing the effective viscosity, peff= p , + p t , and with a dimensionless viscosity defined as P* =
( PI + P t ) / P I
=
P,ff/PI
9
(46)
effective Prandtl and Schmidt numbers may be expressed as P,ff/P and
Pr p*
.-\
,
322
PATRICK J. PRESCOTT AND FRANK P. INCROPERA
a
Mlm = 1.61%
b
FIG. 46. Macrosegregation patterns at I = 600 s predicted by (a) the laminar EMS simulation and (b) the turbulent EMS simulation (reprinted with permission from Prescott and Incropera [201], 1995, ASME).
For the Pb-Sn system, Pr = 0.02 and Sc = 172 (Le = Sc/Pr = 8600). Hence, for p* = 100, Prt,e = 1.2, and Prt,s= 1.0, Preff= 0.75, and Sceff= 1. Therefore, the effective Lewis number is 1.33, and although the respective molecular diffusivities of momentum, energy, and species are highly disparate, turbulence has the effect of approximately equalizing the diffusion rates. Furthermore, by increasing the effective diffusion coefficients for all field variables, turbulence reduces the relative influence of advection (transport by the mean flow), rendering transport rates less sensitive to velocity vectors and more sensitive to gradients in the field variables. With cooling and application of electromagnetic stirring forces, a convection cell develops in the mold cavity and is initially similar to that predicted for laminar flow. However, within approximately 30 s, p* exceeds 200 and flow velocities begin to decease. Within 90 s, the electro-
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magnetically driven flow is nearly steady, with a maximum velocity of 15 mm/s and p;,, = 220. Solidification begins at t = 150 s, which is consistent with the laminar simulation. A mushy zone forms and its shape is determined largely by the shape of the isotherms, which are influenced significantly by the convection cell. Turbulence is dampened in the mushy zone, although some turbulent mixing still occurs near the liquidus interface and is responsible for reducing gradients in liquid Sn concentration. Turbulence can survive only in regions of the mush with a very small solid fraction ( < l%),where slurry conditions are likely to occur in the actual flow. Because of the undercooled melt conditions, the mushy zone grows quickly and fluid velocities decrease rapidly. At t = 180 s (Fig. 471, a mushy zone occupies almost the entire mold cavity. Although the turbulence intensity is significantly reduced (Fig. 47e), it still enhances mixing in the interior of the cavity, thereby maintaining nearly uniform temperatures and concentrations in much of the mushy zone (Figs. 47c, d), with nearly vertical isotherms and liquid isocomps existing closer to the cooled mold wall. In this region, the temperature and liquid composition fields are nearly diffusion-dominated, and with the absence of perturbations in these fields, channel formation is inhibited. Since 1G21< 1, the solutal convection cell is growing and eventually dominates flow conditions in the cavity. However, because Lorentz forces oppose solutal buoyancy forces, the clockwise convection cell in Fig. 47b grows less rapidly than it would without electromagnetic stirring. By 195 s, most of the turbulence has been dissipated by Darcy damping. However, convection is still influenced by prior turbulent mixing. Uniform temperatures and liquid compositions persist in much of the mushy region, and except at the top and bottom of the cavity, isotherms and liquid isocomps are nearly vertical in a region adjoining the outer mold wall. Hence, the propensity for a channel to develop is not enhanced by interaction of the two convection cells, and in contrast to conditions predicted for laminar flow, a channel has not developed and flow velocities are relatively small. The solutal convection cell continues to grow with time, while the electromagnetically driven cell gradually decays. The convection pattern during intermediate stages of solidification (210 < t < 600 s) supplies cool, Sn-rich liquid to the top interior region of the cavity, thereby establishing vertical gradients of temperature and liquid Sn concentration [2001. However, in the outer periphery of the mold cavity, gradients in temperature and liquid concentration remain primarily radial and nearly uniform in the vertical direction. The final macrosegregation pattern predicted by the turbulent simulation is shown in Fig. 46b. Although the solutal convection pattern during
- 1
FIG. 47. Predicted convection conditions at t = 180 s during solidification of a Pb-Sn alloy with
electromagnetic stirring. Turbulent flow conditions: (a) velocity vectors; (b) streamlines; (c) isotherms; (d) liquid isocomps; (e) effective viscosity (turbulence). (Reprinted with permission from Prescott and Incropera [201], 1995, ASME.)
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intermediate stages of solidification caused the formation of a cone of positive segregation at the top of the ingot, macrosegregation in the outer region of the ingot is small. The average macrosegregation hiRMs is 1.22%, which is less than that predicted without electromagnetic stirring (1.34%) [1341 and much less than that predicted by the laminar simulation (1.61%) (Fig. 46a). Turbulence was found to have a significant effect on convection conditions that lead to the formation of channels in the mushy zone and, therefore, on macrosegregation. Namely, turbulence reduced perturbations in the temperature and liquid concentration fields, thereby inhibiting the formation of channels in the mushy zone. With increased turbulent mixing, effective diffusion coefficients for momentum, energy, and species transfer were essentially equalized, decreasing the relative effect of advection, and causing gradients in temperature and liquid Sn concentration to remain primarily radial and nearly uniform in the vertical direction. With turbulence, counterrotating convection cells driven by solutal buoyancy and Lorentz forces occupied the outer and inner portions of the cavity, respectively, during the early stages of solidification. Moreover, mutual interactions between these cells were minimal, in contrast to the stronger and more complex interactions predicted for laminar flow with electromagnetic stirring. Consequently, severely segregated regions associated with channels were not predicted to occur in the turbulent simulation. The studies discussed in this section demonstrate that convection conditions during alloy solidification can be effectively controlled through external means. Centrifugal and other inertial forces were shown to be highly effective at minimizing channel formation or, at least, controlling the location at which they form in a casting. The flow of molten metals can also be influenced, without direct mechanical contact, by applying a time-harmonic magnetic field. Electromagnetic induction could, for example, be used to introduce swirl into a solidifying molten metal alloy in order to reduce perturbations in temperature and liquid concentration fields and hence the potential for channel segregation. Magnetic fields can also be used to enhance or dampen turbulence in the melt, which, depending on the application, may have beneficial effects on macrosegregation and/or grain structure. Turbulence is enhanced with electromagnetic stirring and has the effect of reducing perturbations in time-averaged field variables and also promoting equiaxial grain growth. In contrast, a dc (steady) magnetic field dampens fluid flow and turbulence and promotes columnar growth. A process-control strategy must depend on the specific objective of the solidification process, the characteristics of the alby system (e.g., whether thermal and solutal buoyancy forces oppose each other), other convection mechanisms inherent to the process (e.g., surface
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tension or Lorentz forces), and, of course, economic considerations. In any case, such strategies cannot be applied to their fullest potential without a thorough understanding of the macroscopic convective phenomena and their interrelationships with microscopic phenomena.
VI. Summary It is clear that internal convection strongly influences alloy solidification processes. Local cooling and solidification rates are affected by convection conditions, which, in turn, have a strong influence on the microstructural features of a casting. Moreover, advective transport of interdendritic liquid is responsible for macrosegregation. In order to improve solid/liquidphase-change processes involving multicomponent materials, it is vital that engineers understand the convection conditions associated with the processes. However, obtaining a reliable knowledge base for guiding process improvements is made difficult by several complicating features associated with alloy solidification. Single-domain models for binary solid/liquid-phase-change systems [12, 41, 43, 461 have contributed to enhancing the knowledge base, and the results of several computational studies were reviewed to demonstrate the potential of the models to predict convective transport phenomena for a wide variety of process conditions. However, it must be emphasized that, although these models represent significant improvements over previous treatments of binary solid-liquid phase change, they are not without weaknesses, nor have they been fully validated with experimental data. Like any model for transport phenomena, predictions depend on the prescribed thermophysical properties. Therefore, one weakness of the models relates to the paucity of property data required to simulate solidification processes. These properties include liquid viscosity, thermal conductivity, specific heat, and the latent heat of fusion. Moreover, since there can be large temperature and concentration differences present in the systems of interest, the temperature and concentration dependence of these properties may well be important. As discussed earlier, another important model parameter that possesses large uncertainty is the permeability associated with the mushy zone. A small amount of permeability data is currently available, and most of it is limited to regions of the mushy zone with relatively large solid fractions. In contrast, with respect to advection, the most important region of the mushy zone is that with relatively small solid fractions and correspondingly large permeabilities. Dendrite branches near the liquidus interface are also disposed to fragmentation, which further complicates transport phenomena. It is believed that fragmentation occurs because dendrite branches remelt at their
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bases and/or fracture under stress imposed by liquid flowing over them, but these mechanisms are not understood well enough to be modeled accurately. Furthermore, nonequilibrium phenomena such as undercooling and the nucleation and growth of grains are known to occur during solidification, and although they have been included in previous micro/macro models without convection [50, 771, micro/macro models that include convection are not well developed. An extensive database for interphase (microscopic) transport must be developed in order to realize the benefits from micro/macro solidification models that include convection. However, progress in this area is being made [49, 51-53, 841, and improved prediction capabilities appear to be forthcoming. Other issues related to simulating alloy solidification processes are in need of attention. The effects of shrinkage on flow and the formation of gas pores should be considered and included in single-domain models of convection during alloy solidification. Turbulence is also important in many processes, but has received little attention in binary phase-change systems. Turbulence can be induced directly by buoyancy or through vigorous fluid motion induced by buoyancy or other means, such as electromagnetic stirring. Turbulence has the effect of increasing effective diffusion coefficients for momentum, heat, and species, but the extent of its influence on overall process behavior is not well known. Also, efficient higher-order numerical algorithms are needed to resolve, in greater detail, the physical features, which occur at various times and locations during solidification. This need is driven by the fact that multiple length and time scales are associated with solidification phenomena, which also compounds the difficulty of the computational task, as it is difficult to efficiently resolve all physical details with a single fixed grid. Perhaps the most significant class of materials for which solidification models can be applied consists of metal alloys. However, there are very little experimental data with which numerical predictions can be compared. This condition is due partly to experimental difficulties associated with metal systems. Indeed, their opacity renders convection patterns difficult to discern, especially for buoyancy-driven flows. However, one advantage that metal systems have over their aqueous salt analogs is that they can be examined metallographically following an experiment. In addition, macrosegregation patterns can be quantified by chemically analyzing small samples from different locations with an ingot. It is important that experiments be performed with actual metal alloys, since many of their properties, such as the Prandtl number, are vastly different from those of transparent salt analogs. Possible means of intelligently controlling alloy solidification processes were discussed in Section V, and the role of numerical simulation in developing control strategies was reviewed. Both numerical simulations
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and experiments have demonstrated the utility of imposing centrifugal or other inertial forces on a solidifying alloy. Also, although the use of magnetic fields to control convection in metal alloys has long been recognized, electromagnetically driven convection in two-phase systems is not yet well understood. Accurate models, which account for magnetic field effects, must be developed so that magnetic fields can be used most efficiently in future solidification technologies.
Nomenclature A
B C C
c:
D
4 FL
f GI
G2
g
h h: I J j K k M m
Pr P
Q 4” R Re
area (m2) body force (N/m3); magnetic induction field ( T ) species mass concentration specific heat (J/kg. K) specific heat of solid at an arbitrary reference state (J/kg. K) binary mass diffusion coefficient (m2/s) grain diameter (m) Lorentz force (N/m3) mass fraction dimensionless grouping representing the ratio of thermal buoyancy to Lorentz forces dimensionless grouping representing the ratio of combined thermal and solutal buoyancy to Lorentz forces volume fraction; gravitational acceleration (m/s2) enthalpy (J/kg) reference enthalpy, c: T (J/kg) interfacial source term interfacial species source kg/m’ . s) species mass flux (kg/m* . s) permeability (m2) thermal conductivity (W/m. K) interfacial momentum source (N/m’) slope of liquidus line of the equilibrium phase diagram ( K ) Prandtl number pressure (N/m2) interfacial energy source (w/m3) heat flux (W/m‘) ratio of principal permeabilities Reynolds number
r
sc se
S” T I
U u, L1 V V x, y ,
radius (m) Schmidt number volumetric energy source (W/m’) volumetric interfacial surface area (m-’) temperature CC or K) time (s) overall heat transfer coefficient (W/m2. K) velocity components (m/s) volume (m3) velocity (m/s) Cartesian coordinates
GREEKSYMBOLS a
P PS PT
primary solid phase secondary solid phase solutal expansion coefficient thermal expansion coefficient
(K-’) &
rk
LL I** V
P-
Pk pc
* 7
n 0
volume fraction mass rate of formation of phase k (kg/m3 . s) viscosity (N . s/m2) dimensionless effective viscosity kinematic viscosity (m2/s) density (kg/m3) partial density of phase k , gk pk (kg/m’) electrical conductivity (A /V . m) stress tensor (N/mZ) streamfunction (kg/s. m) or (kg/s. rad) swirl ( m r Z ) angular velocity (rad/s)
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SUBSCRIPTS C
e eff I
k I max min r S
t V
cold; critical energy effective interface phase k liquid phase; liquidus maximum minimum radial component solid phase; solutal turbulent volumetric
W
wall
X
x component
Z
axial component
329
SUPERSCRIPTS d
i k 1 S
r
drag component species diffusion component within phase k within liquid within solid phase-change component
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ADVANCES IN HEAT TRANSFER, VOLUME 28
Transport Phenomena in Chemical Vapor-Deposition Systems ROOP L. MAHAJAN Department of Mechanical Engineering, University of Colorado, Boulder, Colorado
I. Introduction A.
SCOPE
Chemical vapor deposition (CVD) refers to the formation of a crystalline material on a substrate by the reaction of the chemicals from the vapor phase using an activation energy. A schematic presentation of the sequence of various steps involved in a CVD process is shown in Fig. 1 [l,21. After the initial reactants are transported to the reaction zone above the susceptor (step l),they undergo a series of chemical reactions and generate new reactive species and intermediates (step 2). These, along with the initial reactants, are transported to the surface (step 31, where they are adsorbed (step 4). Surface diffusion (step 9, followed by surface reaction, nucleation, and lattice incorporation (step 61, leads to the formation of a solid film. The gaseous reaction products desorb (step 7) and diffuse away from the surface to the main gas stream (step 81, where they are convected away outside the deposition chamber (step 9). In many CVD processes, the deposition reaction is considered to be fully heterogeneous since it occurs as a catalytic reaction in an adsorbed layer on the substrate surface and growing film [l].In such cases, step 2 does not take place. CVD differs from physical vapor deposition (PVD) processes in which thin films are deposited on a cold substrate either through the direct evaporation of a source material or by sputtering under vacuum condi339
Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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-
main gas flow
*@
mass transport to reaction zone
.($
gas phase reactions
‘(iiJDiffusion
rans sport away from the
* deposition chamber
to main gas stream
0-0
tions. In the former, resistance, induction, or electron-beam or laser-beam heating is used to cause evaporation of the source material that deposits on the cold surface of the substrate [31. In molecular-beam expitaxy (MBE), sometimes referred to as the refined uacuurn evaporation technique, collision-free beams from various sources interact chemically on the substrate to form an epitaxial film [4]. An advantage offered by MBE is that high quality layers can be grown at low temperatures. This permits sharp profiles and hetro-epitaxies. As a result, MBE is finding increased use in the growth of thin films of a range of materials, including groups III-IV and II-VI compounds, Si/SiGe, magnetic materials, and high-temperature superconductors [5]. In PVD by physical sputtering, positive ions, usually Ar’, accelerated by an applied electric field, bombard a source target, and eject (sputter) atoms from it. These atoms then deposit on the substrate. Much of the metallization and dielectric deposition in microelectronics processing industry is carried out by PVD processes. For a review of PVD, the reader is referred to Fraser [3], and Lee [6]. MBE is covered in a number of references [4-81. In addition to CVD and PVD, there are several other film-forming techniques. These include electrodeposition, liquid-base epitaxy, and vapor-phase epitaxy. Electrodeposition utilizes electroplating, electroless plating, or electrolytic anodization. In electroplating, the cathode (metal to
TRANSPORT PHENOMENA IN CVD SYSTEMS
34 1
be plated) is placed in an electrolytic solution, that contains the ions of the metal to be deposited. Under the action of the electric current, thin layers of metal are deposited on the cathode. The use of electroplating is limited to deposition of metals on conductive substrates. In solid-state technology applications, thick copper films are deposited using electroplating. In electroless plating, no external current source is used and a metallic layer is formed as a result of chemical reaction taking place in a plating solution. This technique, however, can be applied to deposit only a few metals, alloys, and semiconductor compounds on nonconducting substrates. A main application has been the deposition of Ni films on Si substrates [l]. In electrolytic anodization, an oxide coating is produced by means of the electrochemical oxidation of an anode that forms the substrate material. This process has been used to grow anodic oxide layers on the surface of 111-V compound semiconductors. Electrolytic deposition techniques are covered in Lawless et al. [9-141. Liquid-phase epitaxy consists of deposition from a liquid solution of a thin film with the same crystalline structure as its supporting single-crystal substrate. The process is relatively simple and has been used to grow many 111-V and 11-VI semiconductor compounds and magnetic materials. In solid-phase epitaxy (SPE), a metastable amorphous layer deposited by another technique, such as CVD or vacuum evaporation, is converted into an ordered crystalline solid having the same crystalline structure as the substrate. The conversion takes place in the solid phase and may be accomplished by thermal annealing or by ion bombardment. Some of the applications include formation of silicon-on-insulator (SOI) structures for multilevel and three-dimensional microelectronics applications, growth of silicide layers for electrical contacts, and low-temperature epitaxy for high-performance electronic and optoelectronic devices. An excellent review of both LPE and SPE is given in Small et al. [15] and Olson and Roth [ 161, respectively. Compared to other film formation techniques, CVD offers a number of unique advantages such as versatility, quality, reproducibility, and costeffectiveness. Other desirable features of CVD are its ability to provide conformal deposition and its relatively higher throughput. As a result, CVD has emerged as the dominant technology for producing films, especially in microelectronic applications, relegating the older PVD to only a secondary role [ll. A wide variety of materials is now produced using CVD. In microelectronics manufacturing, for example, CVD is used to produce highly uniform thin layers (0.01-10 pm) of semiconductors such as epitaxial silicon and gallium arsenide, dielectrics such as silicon dioxide and silicon nitride,
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ROOP L. MAHAJAN
and metallic conductors W, Al, and heavily doped polysilicon. Other applications include the production of metal films of W, Mo, Au, and Pt to provide protective coatings; ceramic materials such as Al,O,, Tic, Sic, B,C, and TiB, for hard coatings; anticorrosive coatings of BN, MoSi, , Sic, and B,C for turbine blades; powders (Si,N4, Sic) to fabricate complicated parts via sintering and hot pressing; fibers (B, B4C, Sic) to fabricate composite materials, and high-purity monolithic materials such as ZnSe, ZnS, CdS, and CdTe as infrared optical materials [17, 181. CVD processes can be classified according to the type of their activation energy, In thermally activated CVD,hereafter referred to as thermal CVD, thermal energy is used to produce gas-phase and surface reactions that result in the formation of a thin film on a substrate. Metal-organic CVD (MOCVD), also denoted as olganornetallic vapor-phase epitaxy (OMVPE), is a special form of CVD, in which at least one of the reactants is a metal-organic compound. Initially developed for the growth of AlGaAs films for solar cells, MOCVD is now widely used for growing epitaxial films of compound semiconductors [191. In plasma-enhanced CVD (PECVD), a glow discharge plasma produced in gaseous reactants supplies much of the energy for the reaction. The generated reactive species, assisted by the creation of a more active substrate surface from the impact of colliding ions, lead to the deposition of films on the substrate. The highly reactive species allow deposition at the comparatively low temperatures of 200-400°C. A low substrate temperature offers the advantage of reducing damage to the substrate and the device structure. However, PECVD reactions tend to be less efficient and unreproducible. PECVD has been successfully used for the deposition of amorphous and single-crystal Si, compound semiconductors, Sic and dielectrics. Some selected references are Lee et al. [6, 20-271. Photo-CVD (PCVD) utilizes photons for the excitation of reactant gases and gas species adsorbed on the substrate. Mercury lamps and UV/IR lasers in the wavelength range < 700 nm are generally used as photon sources. Through direct (photodisassociation) or indirect (photosensitization) dissociation of the precursor, highly reactive molecular intermediates such as free radicals and atoms are produced that allow the formation of films at very low temperatures (50-200°C). As a result, radiation device degradation, wafer warpage, defect generation, and other hightemperature-associated problems are minimized. Low deposition rate, nonuniformity among wafers, increased complexity, and low throughput are some of the disadvantages of this technique. A comprehensive review of photoassisted epitaxy is given in Irvine [28]. References [29-371 cover some applications and mechanisms of the photochemical reactions involved.
TRANSPORT PHENOMENA IN CVD SYSTEMS
343
Photothermal-assisted or laser-induced thermal CVD [28, 29, 38-43] depends on local substrate heating by means of infrared or visible laser light that is not absorbed by gas-phase molecules [l]. Laser CVD allows the localized deposition of various metals at high rates. Its use, however, has been limited by the relatively short lifetime of continuous lasers. Electron-beam-assisted CVD [44-481 utilizes a spatially confined plasma generated by the electron beam to promote reaction in a small volume of suitable reactive gases, causing deposition on a heated substrate (150-500"C) located directly beneath the region. Some applications of this technique include deposition of Si, S O , , and Si,N, . Finally, ion-beamassisted CVD [49, 501 uses a focus beam of ions to induce deposition. Recently, thin ceramic membrane layers supported by a highly porous ceramic substrate have been produced using a modified CVD process. The modified CVD process is generally operated in an opposing reactant geometry in which a porous substrate separates the reactor into two chambers. One of these is filled with an oxygen source, and the other has a metal source reactant. The two gas reactants from the two chambers interdiffuse into the porous substrate and react to form a solid product that is deposited in the internal pore surface of the substrate. For example, in preparation of ceramic membranes, one may use a coarse-pore ceramic substrate, water vapor as a source of oxygen and ZrCI, as the metal source reactant. The two gases would react to form ZrO,, which is deposited in the pores of the ceramic substrates. These supported ceramic membranes, with an average thickness of 3-6 p m and subnanometer average pore diameter, serve as excellent candidates for applications in separation and catalytic conversion. The desired selectivity and permeability of the membrane can be achieved by adjusting the pore size and thickness of the membrane. Examples of different films using this technique and the associated deposition analyses are given in Lin et al. [51-581. A majority of the CVD processes are thermally activated, and these are the focus of this chapter. After a brief description of the commonly used reactor configurations in Section I.B, basic transport considerations are discussed in Section 11. The governing equations of transport from the generalized to simplified formulation are given in Section 111. Section IV reveals the solutions for specific reactor geometries. Artificial neural network models trained with physical model solutions [physical artificial neural network (physico-ANN) models] are introduced in Section V as fast-response models for optimization and control of CVD manufacturing processes. A few concluding remarks follow in Section VI. Readers interested in CVD processes using other than thermal activation energy are referred to some of the literature cited above. Perhaps the best source of published material on the subject up to 1989 is a reference
344
ROOP L. MAHklAN
book by Morosanu [l] listing 5370 references related to CVD thin films. Other focused reviews on the subject are given in Jensen et al. [2, 59-611. B. COMMON CVD REACTOR CONFIGURATIONS To fulfill the requirements arising from numerous applications, a variety of reactors have evolved over the years. They may operate at atmospheric reduced pressures (APCVD), or low pressures (LPCVD). The operating pressure in the former is commonly taken to vary from one-tenth of an atmosphere to an atmosphere (0.1-1 atm; 76-760 torr). In the latter, the pressure is typically lop3atm. Both APCVD and LPCVD reactors, in turn, may operate as hot-wall reactors, where the entire process chamber is uniformly heated, or in a cold-wall environment, where only the depositing surfaces are heated. The commonly used “cold-wall” and “hot-wall” reactor configurations for various thermal CVD processes are shown in Fig. 2. The three reactors in Figs. 2a-c have been used in production over the years, especially for silicon epitaxy. Of these, the horizontal reactor (Fig. 2a) is perhaps the oldest configuration. A silicon-carbide-coated graphite susceptor is housed in a quartz chamber, and is generally tilted by = 3” to the horizontal. This tilt, as discussed later in Section IV.A, is necessary to ensure the uniformity of deposition along the length of the susceptor. The wafers are heated through direct contact with the inductively heated graphite susceptor. The quartz chamber walls, in comparison, are relatively cold. These reactors offer high throughput, but uniformity in deposition is difficult to achieve. However, the simplicity of the reactor and a vast amount of published literature makes this reactor a research tool for investigating new chemistries. In the pancake reactor (Fig. 2b), the carrier gas and the semiconductor material are injected through a nozzle over a slowly rotating graphite disk susceptor. As in a horizontal reactor, the susceptor is inductively heated. The wafers are placed peripherally on the susceptor. The thorough mixing of the gases above the susceptor and the susceptor rotation normally result in excellent uniformity, repeatability, and a lower defect density 1621. The throughput, compared to that of the horizontal reactor, is relatively low. Mahajan and Ristorcelli [63] proposed a cascade pancake reactor to increase throughput. In such a reactor, the graphite susceptors are stacked vertically in two or more layers and the susceptors are resistance-heated. The construction, however, is complex. The IR (infra-red) heated vertical barrel reactor (Fig. 2c) virtually replaced the horizontal and pancake reactors in the 1980s. It has since retained its dominance for large-volume production of silicon epitaxial
TRANSPORT PHENOMENA IN CVD SYSTEMS
345
wafers. In a reactor of this design, silicon wafers rest in shallow pockets in a multifaceted, slightly tapered susceptor that is rotated inside a quartz bell jar. The most common mode of heating is through an array of infrared lamps backed by reflectors. The mixture of gases enters through two nozzles at the top of the reactor, and is exhausted through a central hole at the bottom. The ambient working pressure may be either atmospheric or about 0.1 atm. As opposed to the horizontal and pancake reactors, the flow direction is designed to be aligned with the direction of gravity, suggesting a less destabilizing effect of buoyancy. Hot-wall tubular reactors (Fig. 2d) are commonly used to deposit polysilicon and other dielectric films. The reactor is heated from outside, and both reactor and wafers are assumed to be isothermal. The tdtal pressure in the heated zone varies from 0.1 to 10 torr, and the temperature ranges from 300 to 900°C[6]. The fluid distributes equally in the annular region between the periphery of the wafers and the inner wall of the reactor and provides the source gas to the interwafer region. The hot-wall LPCVD reactors are also available in vertical configuration [61, 641. The ease of temperature control and reduced particle contamination are cited as advantages over the horizontal reactor [61]. Exploiting the classical fluid dynamic result that the infinite planar, the impinging-jet and the infinite rotating-disk flows yield uniform boundary layers, several investigators have studied variations of these basic configurations for different applications. Figure 2e shows an impinging-jet singlewafer reactor that operates in a low pressure environment. A horizontal reactor with several impinging jets (Fig. 2f) allows large surface area of uniform deposition. The vertical pedestal, cold-wall single-wafer reactor (Fig. 2g) is one of the most commonly used configurations for MOCVD applications. The carrier gas containing the reactants enters through the top of the reactor and flows over a slowly rotating susceptor placed perpendicular to the flow. Other configurations have been investigated over the years as potential deposition systems. Some of these, including rotational-disk reactors and stagnation-flow reactors, are described in Section IV. With the continuing trend toward larger-diameter wafers and a decrease in minimum feature size in microelectronics manufacturing [the Semiconductor Industry Assoc. (SIA) Roadmap [65] calls for a wafer diameter of 300 mm and a minimum feature size of 0.18 pm by the year 20011, there is an increasing demand on the uniformity of the deposited films. For example, epitaxial layers will be required to achieve f 2 % uniformity in thickness. This has generated a keen interest in single-wafer reactors and placed new emphasis on the understanding of underlying transport phenomena for improving reactor designs.
346
ROOP L. M A W A N
b
a Flow
300 K
JTC
* Susceptor at T,
Gas
Outlet
FIG. 2. Common reactor configurations: (a) horizontal reactor; (b) pancake reactor; (c) barrel reactor; (d) hot-wall tubular LPCVD reactor; (el single-wafer impinging-jet LPCVD reactor; (f) multiple impinging-jet horizontal reactor; (g) vertical pedestal reactor.
11. Transport Phenomena
A. RATE-LIMITING STEPS Referring to Fig, 1, it is clear that many of the steps involved are transport-related. However, the extent and role of the transport phenomena are determined by the CVD process parameters such as substrate
TRANSPORT PHENOMENA IN CVD SYSTEMS
347
d Gas Injectors
Inlet Gas
Exhaust Gas
in boat
-
Sheath
e
f
I
11
Flow
1111
Flow
Substrate
RF Coils Susceptor
d
3
348
ROOP L. W
A
N
temperature, flow rate, and the reactant partial pressures, as well as the chemistries involved. Depending on these factors, the deposition process may be thermodynamically-, diffusion- or .kinetics-controlled. In a thermodynamically-controlledprocess, the mass transfer of species to and away from the deposition zone is much slower than either mass transfer between the main flow and the substrate or the mass transfer from the surface processes. As a result,’steps 1 and 9 in Fig. 1 are rate-controlling steps. The process is assumed to proceed under thermodynamic equilibrium, and the deposition rate is generally determined by the equilibrium values of the partial pressures of the system species. In a diffusion- or mass transport-controlled process, the rate determining step is the diffusion or transport of the reactant gases to the substrate surface (steps 3 and 8). The gas flow, heat transfer, and species diffusion play a dominant role in determining the deposition characteristics. Finally, in a kinetics-controlled system, the surface processes (steps 4-7) are not as fast as steps 1 and 9 or 3 and 8 and the rate-determining step is the slowest of the surface steps 4-7. As discussed below, substrate temperature then plays an important role and flow field has only a marginal influence. For some combination of temperature and concentration, the homogeneous gas-phase reactions play an important role and determine the deposition rate. Figure 3 schematically shows the effect of the substrate temperature in determining the dominant rate-controlling process. At low substrate temperatures, the surface reactions are generally the slowest steps. That is, the surface reaction rate is kinetically controlled and follows the Arrhenius relationship
t fe
c5
Kinetics Controlled
Mass Transport Limited
Thermodynamics Limited ~
FIG. 3. Different rate controlling domains in a CVD process involving an exothermic reaction [18].
TRANSPORT PHENOMENA IN CVD SYSTEMS
349
where r is the reaction rate constant, A is a constant, E is the activation energy, R is the gas constant, T is the substrate temperature, and c R , , is the concentration of the reactant gases at the substrate. Clearly, the dependence of the deposition rate on the substrate temperature is relatively steep. For a pth-order surface reaction, cR = cf,, where ci,,is the surface concentration of the reactant species i and p is an empirical constant. For a kinetically controlled process, all the reactant species arriving at the substrate are immediately replaced by fresh molecules from the bulk stream so that ci,s= c ~ , Dropping ~ . the subscript i for convenience, 9’ for a first-order kinetically controlled process is simply
9‘= c,r. (2) At higher temperatures, the surface reactions are very fast, and all the reactants reaching the surface are consumed by the surface reactions. The diffusion or transport of the reactant species to the substrate then becomes the limiting step. This is the mass transport, or difision-limited domain. In this case, the diffusion flux, J D , is governed by the gradient of the concentration of the reactant species at the surface and is given by
where D, is the effective diffusion coefficient of the species i in the gas mixture, c its concentration, and n is the direction normal to the surface. Taking the diffusion length in the deposition zone in Fig. 1 to be L, J D in Eq. (3) can be approximated as
where c, and c, refer to the concentration of the reactant species i in the gas stream and at the surface, respectively. Since all the reactant species arriving at the surface are immediately consumed, c, = 0, and J D is simply
The overall dependence of JD on temperature through D, and L is relatively weak. At still higher values of T, steps 1 and 9 become the slowest steps and a thermodynamically-limited regime results. In this domain, the deposition rate depends strongly on the temperature as exp( - A H / R T ) , where A H is the heat of reaction at constant pressure. For an exothermic reaction, A H is negative and the net reaction rate (forward rate - reverse rate), and hence the deposition rate, decreases with increasing temperature as shown
350
ROOP L. MAHkTAN
in Fig. 2. At a sufficiently high temperature, homogeneous gas-phase nucleation can take place, leading to the formation of powder deposits instead of a crystalline material [18]. For an endothermic process with positive A H , the net reaction rate and the resulting deposition rate increase with an increase in temperature. A comparison of Eqs. (1) and (3) provides a convenient way of determining whether a process is kinetically or mass-transport-controlled; the The nondimenformer results for J , >> W s and the latter, for J , << 9'. sional number, DaS, the surface Damkohler number, expresses this ratio:
Thus, Das is a measure of the time for the surface reaction to the time for diffusion to the surface. Its value varies from lop3to lo3 in CVD systems [2], with Da' >> 1 representing the diffusion controlled, and Da' << 1 the kinetically controlled domains. For a first-order reaction, this simply reduces to rL Da" -. (7) De The total gas flow rate can also be used to delineate the rate-controlling steps [l]. At low flow rates, the gas stream has sufficient residence time to equilibrate with the substrate surface. The deposition rate is then thermodynamically limited and increases linearly with the flow rate. At increased flow rates, the surface concentrations cannot quickly adapt to the flow of products toward or from the deposition zone, and the diffusional transfer becomes the controlling rate. In this domain, the deposition rate varies nonlinearly with the total flow rate. At sufficiently high flow rates, the decomposition becomes kinetically controlled and is independent of the total flow rate. The preceding discussion is intended to provide only a general guideline for determining the different reaction domains in a CVD system. The specific values of the surface temperature and the total flow rates at which the rate-controlling transitions take place depend on the particular chemistries involved. It should be noted that a vast majority of practical CVD processes operates in kinetically or mass-transport-controlled domains, and proceed in nonequilibrium conditions. The equilibrium thermodynamic analysis, therefore, does not provide accurate results. These calculations, however, serve a useful purpose in determining the feasibility of the chemical reaction under specified conditions to provide quantitative information about the process. The thermodynamic calculations can pre-
TRANSPORT PHENOMENA IN CVD SYSTEMS
35 1
dict the amount of a deposit and partial pressures of vapor species under given conditions. Computations of equilibrium compositions for several CVD systems have been reported in a number of publications [66-SO]. CVD phase diagrams can be drawn that predict deposition or etching of the given solid phase. The theoretically obtainable amounts of deposits under specified operating conditions and phase diagrams are discussed in Stringfellow et al. [81-84]. For kinetically controlled processes, a knowledge of the rate law, activation energy, and reaction order is necessary to perform a kinetic analysis of the surface reactions. It is common to perform an analysis of a supposed reaction sequence by including the homogeneous and heterogeneous equilibrium established in the system under steady-state conditions [ 11. Chemical kinetics of the different deposition systems have been reviewed, for example, in Shaw et al. [SS-881. In this chapter, only a few of the most commonly used chemistries are considered. Finally, in mass-transport- or diffusion-controlled processes, the deposition rate is profoundly affected by the fluid flow and transport of heat and species involved. These are discussed in detail for some of the representative reactor configurations discussed in Section IV. B. SOMEBASICTRANSPORT CONSIDERATIONS The ability to deposit thin films reproducibly and control their properties is critical to the acceptance and advancement of CVD. To this end, a comprehensive knowledge of the flow, thermal, and concentration fields, as well as gas and surface reactions, is necessary. Depending on the pressure, temperature, and the characteristic reactor dimensions, the flow in the reactor may be treated as continuum, free molecular, or transitional. The Knudsen number, Kn = h / L , a measure of the ratio of the mean molecular free path of the molecules to the characteristic length of a reactor, serves to identify these domains. For Kn < 0.01, the continuum description is applicable. The flow in atmospheric and reduced-pressure CVD reactors with large characteristic dimensions generally meets this criterion. For large values of Kn 2 10, the continuum assumption does not hold good and the transport is by free motion of the molecules between the surface and the remote fluid. Between the extremes of continuum and free-molecule regime is a transitional domain. The conditions under which the continuum model of flow first fails is called the “slip flow” regime, because it may be analyzed by assigning temperature and velocity “slip” at fluid-solid interfaces. The slip flow analysis is generally considered to yield accurate results from continuum conditions to Kn = 0.1. Beyond this value to Kn = 10, the
352
ROOP L. MAHkTAN
slip flow changes to free-molecule flow through a transition regime. Both the free molecular and transitional flows are encountered in very low pressure CVD reactors (I 0.1 torr) and also in the processing of the submicrometer-range features in atmospheric and reduced pressure CVD reactors [89-921. In continuum flow reactors, the gas flow may vary from laminar forced to mixed convection where the buoyancy-induced flow may give rise to recirculatory flows. In some configurations, such as in a pedestal rotating reactor, the buoyancy-induced and forced flows oppose each other and multiple steady flows arise. Under certain reactor configuration and operating conditions (e.g., a heated susceptor facing up in a cold-wall reactor), the flow may even become turbulent. The thermal field in CVD reactors varies from nearly isothermal conditions in hot-wall reactors to complex temperature distributions in cold-wall reactors. The temperature-flux boundary conditions prescribed on the bounding surfaces of the reactor significantly affect the nature of the flow field. In these cases, detailed thermal energy-balance models deploying conjugate heat transfer analysis may be required. The mass transport in CVD reactors can be quite complex. In addition to the commonly used convective and concentration-gradient-driven diffusional transport, thermal diffusion (the Soret effect) needs to be considered. The latter arises as a result of temperature gradients and can contribute significantly to the overall deposition rate, especially in cold-wall reactors, where the temperature gradients tend to be large. The gas-phase reactions, where they occur, provide a source of species, which, in turn, are transported by diffusion and convection to the surface. In contrast to the cold-wall APCVD reactors, the mass transport in hot-wall multiwafer LPCVD reactors is dominated by multicomponent diffusion. Gas and surface reactions play an important role. In the tubular LPCVD reactor shown in Fig. 2d, convection and diffusion of the reacting species are the dominant modes of transport in the annular region and only diffusive transport dominates in the interwafer region. The inference from the brief discussion above is that the transport phenomena, gas-phase, and surface chemical reactions need to be understood to optimize the CVD systems. Details vary from reactor to reactor and from one deposition chemistry to another. However, a fundamental understanding of underlying mechanisms can be obtained from a study of the few specific systems discussed in Section IV for the commonly used chemistries. Emphasis is on silicon processes since they constitute a significant majority of microelectronics applications. Metal-organic CVD is reviewed in some depth because of its growing applications in quantum-well devices and wires.
TRANSPORT PHENOMENA IN CVD SYSTEMS
353
111. Governing Equations
This section starts with the most general form of the equations of conservation of total mass, momentum, energy, and individual species for a multicomponent system [93]. Simplifications that arise in many of the CVD systems are discussed. In particular, simplified equations are provided for a binary system in both vector form and the familiar Cartesian and cylindrical coordinates. A. EQUATIONS FOR
A
MULTICOMPONENT MIXTURE
Continuity: dP dt
- + v . ( pv)
=
0
Momentum:
+
where the pressure tensor rr is given by rr = T PI, in which p is the static pressure, I is the unity tensor, and T is the shear stress tensor:
where the superscript t denotes a transposed vector. Note that Eq. (9) differs from that for a pure fluid only in the last term, where pX has been replaced by C p i x i to take into account the possibility that each chemical species may be acted on by a different external force per unit mass X i . Energy: dt
=
- ( V * q ) -(T:VV)
The energy flux q is composed of contributions due to the conductive energy, qc, and q', the Dufour energy flux. The latter represents
354
ROOF' L. MAHklAN
contribution due to concentration gradients:
For a multicomponent mixture, q" is given by [94]
1
DT
- V(ln x i ) ,
RT i=l
Mi
where Mi is the molecular weight of species i in the n-component mixture and x i and DT are its mole fraction and multicomponent thermal diffusion coefficient, respectively. The second and third terms on the right side of Eq. (11) represent heating due to viscous dissipation and pressure effects, respectively. The fourth term represents the rate of work done on the species by body forces per unit volume, if each chemical species were acted on by a different body force. The last term represents the contribution from gas-phase reactions. The species transport equations are given below in a formulation using mass reactions and diffusive mass fluxes relative to the mass-averaged velocity of the gas mixture. As pointed out in Kleijn [94], this procedure is more convenient for numerical simulations. The mass-averaged velocity can be obtained from Navier-Stokes (NS) equations, and the resulting equation is similar in form to the energy equation and to each component OF the NS equations. Chemical species: Poi)
dt
+v
- jj + c v6Mj9,F, ng
*
V(
poi) =
-V
(15)
j= 1
where the total diffusive mass flux j i of species i is composed of jc, the flux due to ordinary diffusion due to concentration gradients; the pressure diffusion jf, the forced diffusion jig; and the Soret or thermal diffusion jr :
+ jf
+ jr.
( 16) Finally, in a n-component gas mixture, since the mass fractions must sum up to 1: j i = jf
c
+jig
i=n
i=l
wi=l
TRANSPORT PHENOMENA IN CVD SYSTEMS
355
Therefore, only n - 1 independent species equations are implied in eq. (15). The last term in Eq. (15) represents the formation and destruction of the ith species due to ng reversible gas-phase reactions, where 9,! is the net reaction rate (forward - reverse)
9I:=@+
J
( 18)
-9E, J
where 9;+ and 3;-are given by [95]
The two reaction rate constants, rj+ and rj- , are related to each other through the chemical reaction equilibria as
r j + ( P , T ) RT Kj(T)
Z'=tv,,
(F) '
r j - ( P , T )=
where K j ( T ) is the thermodynamic equilibrium constant for the jth gas-phase reaction and is given by
K ~ ( T= ) e-AGT(T)/RT
(22)
The values of r j + , r j - , AGO depend on the specific deposition chemistries considered, where AGO is the standard Gibbs energy. In Eq. (16), the component jf indicates that there may be a net movement of the ith species in a mixture to separate under a pressure gradient imposed on the system. For pressure gradients generally encountered in CVD systems, this term is very small. The forced diffusion term j! is identically zero if gravity is the only external force. The diffusion fluxes jC in a multicomponent mixture are given by the Stefan-Maxwell equations [93]. In terms of massffactions and species fluxes, they are
(23) where is the average molecular weight, see Eq. (30). Equations (23), coupled with the additional equation n
C jC = 0, i= 1
(24)
356
ROOP L. W
A
N
form a closed set of equations from which the n diffusive mass fluxes jc can be obtained as [96] jc
= - pDei V o i -
pwiDeiV(ln
a)+ MuiDei
:C
Ji jzi
MjDij
9
(25)
where Dei is an effective diffusion coefficient for species i and is given by
Equations (23) and (24) can be solved iteratively, in conjunction with Eqs. (25) and (261, to obtain diffusion fluxes in terms of mass fractions. The Soret or thermal diffusion term, jr, which is a measure of the diffusion of the species due to a temperature gradient, is given by j;
-0: V(ln T ) ,
(27) where D,?is the multicomponent thermodiffusion coefficient for species i. An approximate expression for DT [93] is DT=
=
C
j+i
c2
-MiMiDiik$, P
(28)
where k c is the thermodiffusion ratio. The exact formulation for DT for a multicomponent mixture is given by Hirschfelder et al. [97]. For binary mixtures and multicomponent mixtures of isotopes, Eq. (28) is exact. Note that thermodiffusion flwr for all species must sum up to zero, i.e., C jr = 0. Finally, an equation of state for the density is required:
-
PM p=-, RT where
M
is given by
a -2Mixi. =
i= 1
B. SIMPLIFIED GOVERNING EQUATIONS For many of the CVD systems, a few simplifications can make the preceding set of equations more tractable. The body force X is normally the gravitational force due to gravity vector g. The bulk viscosity K = 0. The carrier gas is generally in great excess compared to the reactants, thus allowing dilute approximations to be applied. As a result, in the energy
TRANSPORT PHENOMENA IN CVD SYSTEMS
357
equation, contributions due to the viscous dissipation, the pressure term, the Dufour effect, and heat formation due to gas-phase reactions are negligible. Further, as discussed above, jp and jf can be ignored in the species transport formulation. The simplified momentum and energy equations are thus
a
-(
at
pv)
+ v * ( pw) = - v p + v . p [ v v + (VV)' - 3 I V . v ] + p g , (31)
+v.VT =V*(kVT). (aT 1
(32)
at
The species transport is given by
a -(
pwi)
+v
n8
*
V ( pw,)
=
-V
- (jf + j:) + C v d ~ ~ g f ;
at
j= 1
i
=
1, ..., n - 1, (33)
where jf and jT are given by Eqs. (24)-(28). A commonly used alternative formulation of species transport in mole fractions, x i , is given below. Neglecting the forced and pressure diffusion contributions as before, the species transport equation for a multicomponent mixture is dX;
c
-
- + c [ v V x , ] = V [cD,Vx, + cDiaixiV In T ] at
In some CVD applications, the deposition is governed by species transport of a single species from a binary mixture; see Section IV. For such binary systems, jf is simply the first term of Eq. (25). Equation (26) yields Dei = D,,so that jf
=
-pD,,Vw,.
(35)
The thermal diffusion effect in a binary system can be easily deduced from Eq. (28) and is simply c2
DT2 = - M , M , D , , k ~ , . P
Another commonly used representation of thermal diffusion in binary
358
ROOP L. M A W A N
systems uses the thermal diffusion factor a [93], where
kr2 = a x 1 x 2 .
(37) Finally, time scales for diffusion and convection in CVD reactors are short relative to the film growth. As a result, the time derivatives may be set equal to zero [21. Boundary conditions required to supplement the preceding set of equations are specific to a given reactor configuration and are given in detail in the following sections where different CVD systems are discussed. However, some generalizations can be made. The boundary conditions on velocity and temperature are conventional and unremarkable. The surface boundary conditions on species deserve a few general comments. On nonreacting surfaces in a CVD reactor, there is no net flux of species. Equation (33) then reduces to
( pwiv + jf
i-j r )
n
=
0.
(38)
For impermeable surfaces, this reduces to
+
(j? jr) * n = 0. (39) An inlet, prescribed species concentration and zero species diffusion are generally appropriate so that
and (jf + jr) * n = 0 . (40) For the exit plane sufficiently away from the deposition surface, the zero-gradient boundary condition is generally accurate: wi = w,,in
n Vwi = 0; i = 1,..., n . (41) For reacting surfaces, the net total mass flux of species i normal to wafer surface must equal to the rate of production of species i through n, surface reactions at the wafer surface: j=n,
( poiv
+ j? + jr) - n = - C
U$M;SY;.
(42)
j= 1
The total net mass flux due to all the species leads to a velocity component normal to surface, u,: 1 n "s U , = - C C U~"~M;SY;. (43) P i = I j=1 Note that for mass-controlled deposition processes, the reactant species reaching the surface is totally consumed. The boundary condition then simply is Wl
= 0.
(44)
TRANSPORT PHENOMENA IN CVD SYSTEMS
359
The surface reaction rate 9 :is generally expressed as the product of the reaction probability (or reactive sticking coefficient) yi and the effusive flux for the j t h species:
where the effusive flux is estimated from the kinetic theory of gases. The value of y, can be calculated if the heterogeneous thermal decomposition paths of the species involved are known. However, some adjustments are often necessary to match the experimental data. Finally, the growth rate for the film species can be calculated from the combined contribution from each of the film materials containing gaseous species: r=n Mf n, M= M, = lJ;qa;'m, (46)
c
I=
pf
1
cc
I=1
1-1
where alfi"" is the number of film atoms in species i and M , and pf refer to the molecular weight and density of the film, respectively.
1. Governing Equations in Three-Dimensional (3D ) Cartesian and Cylindrical Coordinates The simplified Eqs. (31)-(33), along with the continuity equation, are presented below in the familiar rectangular and cylindrical coordinates. These will be referred to frequently in Section IV for discussing the solutions for selected configurations. a. 3D Cartesian-Continuity Equation dP -
dt
d
+ -d(xp u )
d
d
+ -dY( p u ) + -d(z p w ) = .
Momentum equations: d
-(
dt
d
pu)
+ -[d X
d
pu2]
+ -(dY
d
puu)
+ -(d z
puw)
(47)
;)
.( -*-ey - + Je
:)
( -k- ey - +
( -x- ey - = j x e
Je
Je
e
e ie e
e
ze ( z M d )-
e
+ (Mad )-xe + (Mnd )-xe + (Md )-i e e
e
e
09E
NVWHVVJ '1dOOX
361
TRANSPORT PHENOMENA IN CVD SYSTEMS
flg
+c
i = 1 7 2 ,..., n
v$Mi9f;
-
1.
j= 1
b. 3D Cylindrical Coordinates
Continuity equation 1 d
dp
1
d
+ -r -(d0
a
pu)
+ -(d z
+pou) r -( dr p u 2 r ) + r -( de
+ -(d 2
+ -r -(dr dt
pur)
pw)
=
0.
(53)
Momentum equations: I
d
-( dt
PU)
=
d
1
d
pwu) -
PV2
r
_dr
dr
+-'rp d
-( at
d
[
r dB
du
1
PU)
"1 r
(54)
+pg,,
1 d d puur) + -( p v 2 ) + -( dz p w u ) r ae
d
+; (;
=-I? + -" [ p { - - d+' r r d0
dr
1 d +--
2p -
r de
PU"
+r
(')]I
dr r
du
-
r do[ r ( d e + u ) - ? v . v ]
{:: Z ) ] %(;)I
+-dZa [ p - + - -
d
1 d u
+ 2[.
-p
+
u
+ Pgo,
(55)
362
ROOP L. W
d
-(
1
dt
pw)
d
1
A
d
+ ;z ( r p u w ) + -r -(dB aw
+-dz
N
puw)
d
+ -(d z
pw2)
2p
2 p - - - V * ~ dz 3
Energy equation: d(PCpT) dt
1 d
1
d
+ ;z ( r p C p u T ) + -r -(do
pC,vT)
d
+ -(d z
pCpwT)
Species equation: d
-(
dt
i
pw,)
1
d
1
d
+ -r -d(rr p u w i ) + -r -(do
d
puy)
+ -(d z
pww,)
d
c *] + - -( MD,, r .C
-1 d [rMwiDei r dz 1
d
d
In T
T)
1
d DTr d InT dr
d
In T
i+.i
C. TRANSPORT PROPERTIES A knowledge of the thermophysical properties appearing in the governing equations for the various gas species and mixtures is required for obtaining solutions. In addition, since large temperature variations exist in
363
TRANSPORT PHENOMENA IN CVD SYSTEMS
many of the CVD systems, the temperature dependence of these properties must also be known. Also, values of binary diffusion coefficient and binary thermodiffusion ratio are needed for the gas pairs involved. Kleijn and Hoogendoorn [98] have tabulated these properties for some of the commonly used gas species in silicon and GaAs deposition systems (see in Table I>. The temperature dependence is given in the form of second-order polynomials. It might be noted that p, Cp,and k were based on experi-
TABLE I TRANSPORT PROPERTIES OF COMMONLY USEDGAS SPECIESIN CVD [95,98]
Property CL
k
Gas (pair) TMGa 44% H2 NZ SiH, TMGa AH3 HZ NZ SiH TMGa hH3 H2 N, SiH, SiH4tN2 SiH,, H, Nz, H, TMGa, H 2 TMGa, N, A H , , Hz A H 3 7 Nz TMGa, ASH, H 2 ,SiH, N,, SiH, H2,NZ TMGa, H, TMGa, N, h H 3 , H, ASH,, N, TMGa, ASH
.,
Cp
Dl,
kT2
Cn
-1.15 X -4.32 x 2.63 X 4.93 x 1.47 X -3.52 x -7.16 x 5.77 x 8.15 x -2.12 x 5.40 X 2.45 X 1.44 x 1.03 x 4.74 x -9.64 X -2.90 X -3.20 X - 1.87 X -4.17 X -2.26 X -6.15 X -2.26 X -2.74 X -5.15 X -2.71 X 1.32 X 6.36 X 8.86 X 3.09 X 1.94 X
CI
10-~ 10-6 10-3 10-3 10-2 10-3 10-2 10' 10' 104 103 102 lo-' 10" 10' 10' lo-' 10" lo-' lo-' lo-' lo-' lo-' 10" lo-' lo-' lo-' lo-'
3.35 x 5.94 x 2.22 x 4.55 x 3.66 X 3.85 x 6.53 x 4.43 x 6.24 x 1.45 x 1.60 X 1.08 X -2.61 X 4.58 x 3.26 X 6.25 x 2.06 X 2.44 X 1.64 X 2.89 x 1.73 X 4.57 x 1.27 x - 1.70 X - 1.69 X -1.61 X -1.54 X -1.58 X -1.57 X -1.55 X - 1.79 X
10-8 10-8 10-8 lo-' 10-5 10-5 10-4 10-5 10-4 10" 10" lo-' 10-3 10' 10-3 lo-' lo-' lo-' 10-3 lo-' 10-~ 10-3 10' 10' 10' 10' 10' 10" 10' 10'
-6.68 -1.46 -5.19 -1.08 -6.81 -3.84 -3.47 -7.54 -4.48 -1.31
lo-'' lo-" lo-'' x lo-" X lo-'' X X X
X
x 10-9 x 10-8 x 10-9 X
0 -4.24 x 10-4 8.67 x 1.34 x - 1.08 x 8.50 X 2.81 x 3.37 x 3.13 x 4.93 x 2.80 x 7.49 x 3.18 X -6.35 x -4.94 x -9.15 x - 3.57 x -3.36 x -4.35 x -4.06 x -1.91 x
10-4 10-4 10-3 10-5 10-5 10-5 10-6 10-5
10-3 10-3 10-3 10- 3 10-3 10-3 10-3 10-3
364
ROOP L. W
A
N
mental data in Maitland and Smith [99, 1001, and statistical mechanical calculations using the Chapman-Ensog theory [97, 1011. Whereas the values of p, C,, and k are well correlated by the expressions in Table I, there is some uncertainty about the binary and thermodiffusion data. For example, it was pointed out in Kleijn and Hoogendoorn [98] that the experimentally measured values of the binary diffusion coefficient trimethylgallium (TMG) and H, by Suzuki and Sato [lo21 were about 70% higher than those given in Table I. It is a common practice to express the temperature dependence of p, k , and D in the power-law formulation. Some of these are listed below for ready reference. 0.648
For H, :
?=(;) PO k
,
(59)
0.69 1
where po and ko refer to the properties of hydrogen at 300 K and are 8.96 X kg m-' s-l and 1.83 X lo-' W m-l K-', respectively [103]. The power dependence of the binary diffusion coefficient for Di for SiH,-H,, SiH,-H,, s i ~ H 6 - H[104], ~ and TMG-H, [lo51 is as follows:
For SiH4-H, :
D
1.70
DO
For SiH,-H, :
D
1.71
DO
For Si2H6-H, :
For TMG-H, :
D
1.70
DO
-=(;) D
1.73 I
DO
where Do (cm*/s') = 0.611,0.640,0.494, and 0.430, respectively. Note that all these properties are evaluated at atmospheric pressure. From the properties of the individual gas species, the properties of the
TRANSPORT PHENOMENA IN CVD SYSTEMS
365
gas mixture can be calculated as follows [106]:
where
and Mi and Mj are the molecular weights of species i and j , respectively.
IV. Solutions for Selected Reactor Configurations A. HORIZONTAL REACTORS Of the CVD reactors, the horizontal reactor is perhaps the most widely investigated configuration. In the late 1960s and early 1970s, it was the reactor of choice for high volume production of silicon epitaxial wafers [62], and it remains as an important configuration for research and low-pressure, low-temperature metal-organic CVD applications [1051. Over the years, much experience has been accumulated about understanding the transport phenomena and the decomposition chemistries involved. The two deposition systems for which chemical kinetics are known are the deposition of silicon from silane and of GaAs from trimethylgallium and arsine. These two are reviewed next.
1. Silicon CI/D Some of the earlier approximate models are due to Shepherd [1071, Bradshaw [108], Andrews et al. [1091, Eversteyn et af. [110], Rundle [ l l l , 1121, and Bloem 11131. Eversteyn et af. assumed a relatively stagnant boundary layer of gas adjacent to the heated susceptor and a thoroughly mixed gas in the region above. The boundary-layer thickness was taken to vary as square root of free-stream velocity. Assuming linear variation of temperature in the stagnant layer and a diffusion controlled deposition
366
ROOP L. MAHAJAN
process, they developed an expression for the growth rate of silicon that matched their experimental data for the horizontal as well as the tilted susceptor. Assuming constant temperature and velocity profile in the reactor tube, Rundle [112] developed an expression for the deposition rate that showed an exponential dependence on downstream distance x . The mass transport was assumed to be diffusion-controlled. On the basis of laminar flow boundary-layer theory and using an approximate relation for the diffusion flux of the reactive chemical species, Berkman et al. [114] reviewed the past models and derived expressions for the silicon deposition rate for horizontal and tilted susceptors. Takahashi et al. [1151 computed two-dimensional velocity and temperature profiles and a three-dimensional (3D) concentration profile. Their model assumes constant physical properties and neglects Soret effects. Coltrin et al. [116] published a detailed two-dimensional (2D) numerical model of the transport phenomena in a horizontal reactor. Their model used boundary-layer approximations and included gas-phase and surface chemical reactions, variable physical properties, and buoyancy effects. A set of 120 elementary homogeneous reactions was analyzed. Extensive sensitivity analysis indicated 20 of these to be most important. They found that the time evolution of all the species concentrations in the gas phase is determined by the initial decomposition of silane into dislylene:
Subsequent gas-phase reactions simply distribute the intermediate species to their constrained equilibrium value. Radicals formed by these gas-phase reactions diffuse to the surface and contribute to the decomposition. Chemical reactions at solid surfaces were included in the model through boundary conditions. The intermediate unsaturated reactive species were assumed to react with the solid surface with unit probability (sticking coefficient of 1). For the surface reaction of SiH4 given by
the surface reaction probability ySiH,= 5.45 exp( - 8556/T), based on the data in Farrow [1171, was used. The predicted growth rates showed good agreement with the experimental data of Eversteyn et al. [1101. In a later paper, Coltrin et al. [1181 expanded the model to include Soret effect and multicomponent diffusion. The surface reaction probability ySiH, was modified to match the experimental data and was taken as YSiH,
=
5.37 x 10-2e-9400/T.
(71)
367
TRANSPORT PHENOMENA IN CVD SYSTEMS
For susceptor temperatures between 550 and 750"C, the Soret effect reduced the growth rate by as much as 50%. The authors did not use their modified model to compare their computed results with the experimental data. Ristorcelli and Mahajan [119], in their study of dopant incorporation in silicon epitaxy, also modeled the deposition process in horizontal reactors. Making the usual assumptions of neglection of buoyancy, Soret, Dufour and property variation, they numerically solved the 2D laminar boundarylayer equations of continuity, momentum, energy and species transport. The deposition of silicon from silane in hydrogen was assumed to be diffusion-controlled. Excellent agreement was shown with the experimental data of Eversteyn et al. [110]. The effect of the susceptor tilt angle 8 on the deposition rate was computed (see Fig. 4). The silicon deposition rates, normalized by their values at x = x,,, are plotted for tilt angles of 8 = 0", 1.5", and 2.5" for Re = 35; and Sc = 2.7 where x o corresponds to the leading edge of the first silicon wafer on the graphite susceptor. First consider 8 = 0. As silicon deposits, the bulk flow is depleted of the silicon source and the bulk concentration decreases in the downstream direction. This decrease in the driving concentration force, along with the increasing boundary-layer thickness, results in decreases in the deposition rate in the streamwise direction. For nonzero values of 8, the flow accelerates along the length of the reactor, the boundary layer thins, and thus increases mass transfer. The depletion and the compensating effect due to flow acceleration result in an optimum value of 8 as shown in Fig. 4. Later, Mahajan and Wei [1031 relaxed the boundary-layer assumption and systematically investigated the effect of buoyancy, Soret, Dufour, and
1.1
.9 .7 .5
---
. I
5
I
I
15
20
1
10
Distance Downstream (xlh) FIG.4. Effect of tilt angle on axial deposition uniformity in a horizontal silicon epitaxy reactor [119].
368
ROOP L. MAHAJAN
property variation. They used a 2D form of Eqs. (471-61) and (34) with the Dufour term q x included in the energy equation: q" = aRT
-j, . M,
In their computational simulations, the physical dimensions and parameters used were similar to those of Eversteyn et al. [110]. The reactor height was 2.05 cm, the substrate was held at 1323 K, the top-wall temperature was set at the inlet gas temperature of 300 K, and inlet concentration of silane was 1241 dyn/cm2 (0.76 torr). The inlet flow velocity and partial pressure of the carrier-gas hydrogen were 17.5 cm/s and -760 torr, respectively. The physical properties, p, p , and k of hydrogen, and the diffusion of silane into hydrogen were taken as those given in Section
1v.c.
The following three cases were studied: (1) no buoyancy effect, no Dufour and Soret effects, and the transport properties evaluated at the film temperature = (T, + Tc)/2; (2) the same as case 1 with Soret and Dufour effects included individually and jointly; (3) the same as case 2 with the addition of variable property effects. Their results for 8 = 2.9", along with the data [IlOIare shown in Fig. 5, which shows a plot of deposition rate vs x. For case (a), very good agreement is seen with the experimental data of Eversteyn et al. 11101. A comparison with the numerical calculations of Ristorcelli and Mahajan [119] under identical assumptions showed identical results. Calculations for case (b) with the Soret effect included, clearly showed its significant effect on the deposition rate. The computed results are much lower than those observed experimentally. With inclusion of the Dufour term, no discernible difference was revealed. The computational results when property variations are taken into account, curve c, indicate close agreement again with the experimental data. The suggestion is that the variable property and the Soret effect cancel each other. The implication was that the good agreement between the approximate 2-D models and the experimental data is coincidental, and that improved analysis including Soret effect and property variation is needed. The 2-D models described have been successful in predicting the deposition rate in the axial direction of the reactor, for certain reactor parameters (e.g., low reactor height, carrier gases helium and hydrogen). However, it is now well established that the fluid flow in horizontal reactors can be quite complex, involving 3-D and/or unsteady flows. In the presence of buoyancy, secondary flow instabilities arise that vary from transverse, traveling waves to longitudinal rolls. These, in turn, can affect both the
369
TRANSPORT PHENOMENA IN CVD SYSTEMS
0.6
0.5 h
.E 2 .-g C
0.4
E
Y
3 0.3
B
3
8
0.2 0.1
0.0
0
5
10
15
20
25
30
x(cm) FIG. 5. Growth rate plots for horizontal reactor showing property variation. Soret and Dufour effects [103]: (curve a) -, average property; (curve b) -.-.-, average property + Soret, also average property + Soret + Dufour; (curve c) * * * *, variable property + Soret + Dufour; (curve d) X X X Eversteyn et al. [103].
deposition rate and uniformity [2, 104, 105, 120, 1211. A related question is: Under what flow circumstances does the buoyancy induced flow affect the forced convection flow in a horizontal reactor? Some of the earlier studies [114, 115, 1221 used the parameter Gr/Re2 and the criteria proposed by Sparrow and Gregg [123] for mixed convection adjacent to a vertical surface to characterize the transition of flow from forced to mixed convection. However, as demonstrated by the experiments in Giling et al. [120, 1241 and the numerical study in Moffat and Jensen [104], no correlation was found between the flow characteristics and the parameter Gr/Re2. This lack of correlation is not surprising since the reactor flow configuration is more akin to the classic Benard problem of thermal instability of a layer enclosed between two differentially heated parallel infinite plates. The appropriate parameter, then, is Ra = Gr. Pr, which, for a fluid layer between two rigid horizontal surfaces, has a value of Racr = 1708 for the onset of instability marked by longitudinal rolls.
370
ROOP L. MAHAJAN
The sidewalls and their thermal boundary conditions, and the fluid Pr affect the value of Racr [104, 105, 125, 1261. For typical gases used in the CVD reactors, Pr = 0.7, the effect of sidewalls is to generally stabilize the flow. On the basis of these and other related studies of fluid flow and heat transfer in a horizontal channel with the bottom wall heated [127-1361, a common conclusion is that stationary longitudinal rolls form for Ra > Racr. On the other hand, the threshold value of the critical Rayleigh number at which the time-dependent transverse rolls form, Racr,,, depends on Re. It approaches 1708 at low Re but is higher for higher Re. At sufficiently high flow rates, the transverse rolls are suppressed. This behavior is seen in Fig. 6, which shows the 3D stability curves for transverse (00) to longitudinal rolls (90"). At a given Re, flow for Ra above the stability curve is unstable. Figure 6 [136] suggests that flow is least stable to longitudinal waves. However, in the presence of sidewalls, it is possible in some parameter domain for the flow to be least stable to transverse rolls. This is especially true at low Re. Evans and Greif [129] presented numerical simulations for transient 3D mixed convection flow in a horizontal channel heated from below for Ra = 3333, L/H = 8, W/H = 2 and Re = GH/v of 5, and where L,W ,and H refer to the average inlet velocity, channel length, width, and height, respectively. The sidewalls were assumed to be adia-
u,
10 z --
1 10
I
I
lllllll
lo?
I
I 1 1 1 I 1 1 1
I
1
I
10'
314 Re FIG. 6. Three-dimensional neutral stability curves; curve labels represent orientation of instability, 0" for transverse waves and 90" for longitudinal rolls [129, 1361.
TRANSPORT PHENOMENA IN CVD SYSTEMS
371
batic. Both periodic transverse and longitudinal rolls were observed, and their interactions produced complex flow and temperature field patterns. The computed temperature and velocity field for Re = 5 in the vertical central plane are presented in Fig. 7. For positions corresponding to the crests and troughs of the traveling wave in Fig. 7, the fluid ascends or descends in the yz plane (not shown). At higher Re, the recirculation in the spanwise direction was more vigorous and asymmetric. The averaged value of the associated heat transfer rate was found to 15-40% higher than that without the instability. Another interesting observation was the effect of the sidewalls on the onset of instability. It was shown that at Re = 10, the presence of the sidewalls stabilized the flow; at Re = 5, the effect was the opposite. Three-dimensional numerical solutions of fluid flow and heat transfer in a horizontal reactor were coupled with the species transfer by Moffat and Jensen [lo41 to predict the epitaxial deposition rate of silicon. The governing Eqs. (47)-(51) and Eq. (34) were simplified using the fully parabolic flow approximation; that is, the diffusion terms in the axial direction were neglected. The sidewalls were assumed adiabatic. Two gas phase reactions, one given by Eq. (69) and another by Si,H, * SiH,
+ SiH,,
(73)
were considered. The forward and backward reaction rate constants for
0.0
2.0
0.0
2.0
4.0
6.0
8.0
4.0
6.0
8.0
b 1.0 ).
0.0 X
FIG. 7. Computed temperature and velocity fields in the vertical central plane in a horizontal channel heated from below at a time of maximum temperature at the center point: Re = 5, Gr = 5000: (a) velocity field; (b) isotherms [1291.
372
ROOP L. W
A
N
these reactions at atmospheric pressure were expressed in the form
(74)
~ ~ + , ~ - = A T ~- Ee ,x/ RpT[ ] .
The coefficients A , p, and E,, used for both forward and reversed reaction rate constants, are listed in Table 11. It was postulated that these reactions are in thermodynamic equilibrium and present a closed subsystem of the full kinetic mechanism. For surface reactions, it was shown that for the silane-hydrogen system considered, the deposition was governed by the diffusion of silane through a thin boundary layer to the surface, where it reacts to deposit silicon with a surface reaction rate modified by the presence of the silylene and the disilane flux at the surface. For the surface reaction of silane, ySiH,given in Eq. (71) was used. For both SiH, and Si,H,, it was taken as 1. The calculations indicated that the contribution of the surface reaction of silane to the total deposition rate was about 86% at the susceptor temperature of 1300 K and about 95% at 1400 K. The results serve to provide an explanation for the relatively good agreement obtained using the 2D models, which assumed that the deposition was mass-transfercontrolled and could be determined using only one surface reaction of silane, Eq. (70). The results, however, clearly demonstrated that although the past 2D models enjoyed a remarkably good success in predicting the axial deposition rate, they did not capture the transverse variation. For the base-case parameters of T, = 1373 K, T, = 300 K, h = 2 cm, reactor = 17.5 cm/s, and PSiH,= 0.76 torr in 1 atm H,, their width = 8 cm, qnlet calculations of the growth rate are shown in Fig. 8a. The corresponding flow trajectory and isotherms and isoconcentration (partial pressure) plots are shown in Figs. 8b, c, d, respectively. For the base case, the Rayleigh number is below the critical value for the onset of instability. Even in the absence of the buoyancy-induced rolls, however, the transverse variation is evident. When the height of the reactor was doubled from the base-case value of 2 cm, Ra increased to 1816 and the buoyancy-induced axial rolls TABLE I1 HOMOGENEOUS REACTIONRATE COEFFICIENTS FOR SILANE AND DISILANE DISSOCIATIONS [lo41 Forward A, (l/s)
Reverse
Pf Eaf unit(s) (kcal/mol)
Silane dissociation 0.610 x lo2’ -5.00 Disilane dissociation 0.213 x -6.47
58.83 56.40
A1
Ear
(cm2/mol. s)
PI
(kcal/mol)
0.528 X 0.179 X 10’’
-4.44 -4.50
3.41 3.07
TRANSPORT PHENOMENA IN CVD SYSTEMS
373
symmetric about the reactor midplane set in. The rolls rotate inward, causing an increased deposition rate around the midplane (see Fig. 9a). Changing the thermal boundary conditions on the sidewalls of the reactor produces pronounced effects. For example, if the sidewalls are cooled, the rolls rotate outward and correspondingly reduce the deposition fate around the midplane (see Fig, 9b). 2.
Mocm
Most of the discussion so far has been directed toward the epitaxial deposition of silicon. The horizontal reactor configuration has also been extensively used for growing epitaxial layers of 111-V compounds. In these reactors, the substrate temperature is generally lower ( - 973 K) than that used in silicon epitaxy ( - 1300 K). The most widely used reactants are trimethylgallium (TMG) and arsine. The carrier gas is typically hydrogen. The mathematical modeling of fluid flow and heat transfer [e.g., 105, 120, 121, 1371 has proceeded on lines similar to those discussed above. The gas-phase and surface reaction kinetics involved in MOCVD have been recently reviewed in Stringfellow [138]. However, in most of the modeling studies, it is assumed that the MOCVD of GaAs at atmospheric conditions is controlled by mass transfer of the Ga containing species. As a result, a single-species (TMG) diffusion model is used for mass transfer. Sat0 and Suzuki [137] used a boundary-layer concept, whereas van de Ven et al. [120] used a simplified 2D formulation to model the deposition rate of GaAs. The effects of buoyancy, susceptor taper, and thermal conditions of the reactor walls were investigated in detail. Stock and Richter 11211 experimentally investigated fluid flow and heat transfer for four different reactor configurations (Fig. 10) and showed that the chimney reactor (Fig. 10d) resulted in better uniformity. In the numerical computations performed at AT & T-Bell Laboratories, these four reactor configurations were analyzed for their deposition characteristics. Using the 2D formulation given in Mahajan and Wei [1031, it was shown that the chimney configuration provided the best uniformity (see Fig. 11). Calculations for different susceptor tilt angles determined 8 = 3.5 to be the best optimum angle for uniformity. Using the parabolized form of the 3D formulation discussed above for silicon epitaxy, Moffat and Jensen [1051 revealed the significant role played by buoyancy effects, aspect ratio, and reactor wall conditions. The conclusions are similar to those discussed for the silicon CVD in Moffat and Jensen [104]. At reduced operating pressures, the simplified one-species modeling approach discussed above is not accurate and both the gas-phase and
374
ROOP L. W
A
N
a
-El
h
5
0.0
5.0
10.0
15.0
20.0
15.0
20.0
Axial Distance (Cm)
b
c
o N
-E,
c
Ecn .-
2
z
0.0
5.0
10.0
Axial Distance (cm) FIG.8. The computed (a) growth rate ( pm/min), (b) flow trajectories, (c) isotherms, and (d) isoconcentration plots for a horizontal reactor 2 cm high X 8 cm wide. In panel (d), top-Psilane in torr; middle-PSily,,,, in torr X 10'; bottom-Pdislane in torr X lo4. Average inlet velocity = 17.5 cm/s, Psi", at inlet = 0.76 ton in 1 atm H,, T, = 1373 K, T, = 300 K, sidewalls adiabatic [104].
TRANSPORT PHENOMENA IN CVD SYSTEMS
375
d
0.4
_ _...__,__ ~ , _,. ......... .. . .o,*..- ... ........ ..... . ............. ......... ...... . . \
.
\
0 Axial Distance
(Cm)
FIG.8. Continued.
surface reactions need to be included [2, 1391. Using the reaction kinetics for the growth of GaAs from trimethylgallium and arsine (detailed in 1140, 14111, Jensen et al. 11391 showed that although the single-species model correctly predicts the growth rate trends, it overpredicts the absolute magnitude by 10-15%. The difference is higher at lower flow rates. These results again point out that for accurate predictions of deposition characteristics, the reaction kinetics and their interaction with transport phenomena need to be understood.
B. BARRELREACTOR As mentioned in Section I.B, the vertical barrel reactor (Fig. 2c) has been the reactor of choice for large-volume epitaxial deposition of silicon. As a result, there have been a number of experimental and numerical
376
ROOP L. M'4HAJAN
b
.O
FIG. 9. Effect of sidewall thermal conditions on longitudinal rolls and deposition in a horizontal reactor: (a) insulated sidewalls, (b) cooled sidewall [2].
studies of this configuration [142-1541. Many of these studies [142-1461 assume a simple downward flow of the incoming gas mixture uniformly around the susceptor. A 2D formulation is used, and the deposition is assumed to be mass-transport-controlled. However, the flow in a production barrel reactor is complex, and convective temperature asymmetries and fluctuations exist [148, 1491, and recirculating flow patterns exist in the reactor [150]. Lord [150] reported the flow measurements and mass transfer data in a specially constructed experimental apparatus that had the same configuration and controlling dimensionless parameters as in a production barrel reactor. The flow pattern, visualized by smoke tracer flow paths, showed a strong recirculating flow pattern driven by the momentum supplied by the flow through the inlet nozzle (see Fig. 12). The pattern persisted over the range of flow rates and inlet nozzle orientations commonly used in produc-
TRANSPORT PHENOMENA IN CVD SYSTEMS
a
377
-b
Flow
3
Flow j
Flow FIG.10. Reactor configurations (a-d) representing different flow orientations to gravity.
Configuration
-a
----b -.-.-.-.
.......... Cd
0
5
10
15
20
x (cm)
FIG.11. Growth rate for four reactor configurations (a-d) of Fig. 10 for a 3.5" tapering angle.
378
ROOP L. MAHAJAN
Acetate Film
7
direction indicators
FIG.12. Flow pattern observed in a simulated barrel CVD reactor. The arrows indicate the smoke tracer flow paths [150].
tion. Making use of heat-mass transfer analogy, 5°C and 2°C range, cholestic liquid films were used to study the effect of different nozzle orientations and flow rates on the deposition rate. With a change in temperature, the film passes through a specific color pattern, generally the color sequence of the rainbow “ROY G BIV.” Thus if an entire surface is covered by liquid crystal film (LCF), and is in the temperature range of the film, the color pattern indicates the temperature distribution over the surface. In this heat transfer setup, the ambient temperature and the heat flux supplied to the susceptor were uniform. It follows from the basic equation, q = h(t, - tJ, that variation in t, (the LCF temperature) corresponds to a variation in the heat transfer coefficient, h. Then, from the heat-mass transfer analogy, variation in LCF color is an indicator of the variation in the mass transfer coefficient. That is, the lower temperature regions on the film correspond to higher heat transfer coefficient, or equivalently to higher mass transfer coefficients and higher deposition rate. The color patterns observed for a 10-sided susceptor for 75 mm wafers revealed heavier deposition near the facet edges than near the
TRANSPORT PHENOMENA IN CVD SYSTEMS
379
center. Direct measurements of deposited layer thickness in a production reactor matched with the findings of the heat transfer study. Mahajan et al. [151] later expanded this study to include different orientations for the left and right nozzles: The effect of flow imbalance, from the left and the right nozzles, influenced the “left to right” variation. Photographs in Fig. 13 show the color patterns observed for (a) when the flow was mostly (80-100%) from the left nozzle and (b) when the flow from the left nozzle was 60% of the total flow. The corresponding line contours are also shown. For the flow condition in Fig. 13a, the temperature variation occurs from left to right and top to bottom where the left and top are cooler than the right and bottom portions of the film, respectively. At reduced flow imbalance (Fig. 13b), the region of nonuniformity is confined to the smaller region on the left top corner of the facet. For near balanced flow, the temperature variation is almost entirely eliminated. It was noted that minimum temperature variation was achieved not for the 50-50 flow, but for the left nozzle flow, slightly (- 2%) higher than from the right. The need for this imbalance was attributed to the rotation of the susceptor. As the susceptor rotates counterclockwise, an eddy is shed at the edge of each facet, which causes higher heat transfer closer to the right side of the facet. Therefore, a slightly higher flow is required from the left nozzle to compensate for this eddy effect. The results with the nozzle aiming positions indicated that the standard industry practice of keeping both the nozzles at the same orientation may not always be desirable. The skewed deposition patterns can be minimized using different orientations for the two nozzles. A comparison with the production data showed good agreement. In an attempt to closely simulate the flow and transport in a production barrel reactor, Yang et al. [152] developed a 3D model of an idealized barrel reactor geometry shown in Fig. 14. The various dimensions of the reactor are close to those in a production reactor, and are L = 80 cm, L , = 3.8 cm, L , = 22 cm, L, = 38 cm, L , = 6 cm, ri = 1 cm, r, = 15 cm, ro = 19 cm, and re = 6 cm. For computational ease, a circular cylindrical susceptor was considered instead of a polygonal susceptor. The susceptor taper was not included. Despite these deficiencies, the model is quite comprehensive and is presented below in somewhat detail. The flow, consisting of carrier-gas hydrogen and small amounts of silane, enters the reactor through the two nozzles located at the top of the reactor at 8 = 0 and 8 = 180”. The total inlet pressure is 760 torr and the inlet partial pressure of silane is 0.76 torr. The flow is assumed laminar and the governing equations are those given in Section III.C, Eqs. (53)-(58). As opposed to the two gas-phase reactions considered by Moffat and Jensen [lo41 for the silane chemistry, only one gas-phase reaction, partial pyrolysis
380
ROOP L. W
A
N
FIG.13. Liquid crystal films showing effect of flow imbalance on a “mockup” CVD barrel reactor: (a) 80-100% flow from the left nozzle; (b) 60% flow from the left nozzle [151].
TRANSPORT PHENOMENA IN CVD SYSTEMS
+
-
- _
4
..-
0
0 0 0 0 0 0 0 0
H2,siH4 Quartztube
( air cooled)
0 0
0
01IR lamps 0 0 0
Lt fr--
Heated susceptor
'1 1
exhaust gas
I 90"
180"
I 270"
FIG.14. Schematic of the idealized barrel reactor geometry [152].
381
382
ROOP L. MAHAJAN
of SiH, to SiH, [Eq. (6911, was considered. The rationale for selecting only one of the possible 20 important gas-phase reactions was the finding in Coltrin et al. [116] that the time evolution of all species in the gas phase is constrained by the initial decomposition of silane in silylene given above. The reaction rate given in Coltrin et al. [116] was used to calculate the rate of formation of SiH, . Two surface reactions, Eq. (70) and the decomposition of silylene through the following surface reaction, were considered:
The reaction in Eq. (70) was assumed to be diffusion-controlled, whereas that in Eq. (75) was taken to be kinetically controlled. For the latter, the reaction rate given by Jensen and Graves [155] was adopted:
where kl(molSi/m2 Si atm) = 1.25 x 109e-'8500/T,k,(atm-') = 1.75 X lo3, and k3(atm-') = 4 x lo4, and PH, and PSiH,are respectively the partial pressures of hydrogen and silane in atmospheres. The boundary conditions for velocity, temperature, and species concentration are conventional but modified for the specific reactor configuration. Specifically, a linear temperature profile was assumed along the top horizontal surface of the susceptor, and a zero-gradient temperature condition was imposed at the bottom. No deposition was assumed at these two surfaces. The susceptor stem was taken to be at the temperature of the inlet gases and considered free of deposition. Numerical solutions were obtained using the finite-difference SIMPLE algorithm [156] for the flow, thermal, and silicon deposition rates. The temperature and flow fields for the base case of T, = 1300 K, Tcold= 300 K, with an inlet velocity of uin = 10 cm/s, injection angle 4 = 0", PSiH, = 0.76 torr, and susceptor rotation R = 0 rpm are presented in Fig. 15 at two circumferencial locations of 8 = 0" and 90". The velocity vectors at every other grid point are shown on the right halves; the isotherms are presented in the left halves. Clearly, the strong buoyant flow induced by the hot susceptor (Gr = 2.6 X lo5) interacts with the incoming injected flow to generate complex flow structures. The characteristic length in Gr was taken as the inner radius. Along the 8 = 0 plane, the incoming jet reduces the strength of the buoyant vortex and streams past the hot susceptor in the annular space. A recirculating flow also exists near the exit as a result of sudden expansion from the narrow annular space to the wider space under the susceptor. For the 8 = 90" plane, which is in between the nozzles, the buoyancy-induced flow is dominant; and the axial flow along
TRANSPORT PHENOMENA IN CVD SYSTEMS
383
a
FIG. 15. Computed temperature and flow fields for an idealized barrel reactor for (a) = 1300K, T, = 300 K, injection angle 4 = O", suscep-
8 = 0" plane and (b) 8 = 90" plane; T, tor rotation fl = 0 [152].
the susceptor is upward. The axial velocity distributions for different values of 0 at z = 34 cm are shown in Fig. 16. The results confirm the progressively stronger influence of buoyancy-induced upward flow opposing the downward jet as one moves away from the nozzle location 8 = 0" (180") toward the 90" (270") plane. The distributions of deposition rate over the susceptor surface shown in Fig. 17 indicate a strong axial as well as circumferential variation. The former is due to the depletion effect discussed in Section 1V.A for a horizontal reactor; and the latter is related to flow asymmetries and the associated thermal fields. The effects of the injection angle and the susceptor rotation R on the silicon deposition characteristics were presented. It was shown that tilting
384
ROOF’ L. M A H N A N
.15
16
17
16
19
r (cm) FIG.16. Axial velocity distributions along the radius at z other simulation parameters are those in Fig. 15 [152].
=
34 cm for different 0 angles;
FIG.17. Silicon growth rate profiles along the susceptor for simulation parameters of Fig. 15 [152].
385
TRANSPORT PHENOMENA IN CVD SYSTEMS
inlet nozzles to 4 = 45" resulted in better mixing of the injected gases and more uniform deposition in the axial direction. The nonuniformities observed in the circumferential direction for the stationary susceptor case clearly confirmed the industry practice of rotating the susceptor to achieve uniform deposition rates. The rotation of the susceptor results in a total deposition thickness that circumferentially integrates the varying deposition rates in the 0 direction. The calculations performed for R = 20 and 40 rpm indicated a minimal effect of R on the axial deposition profiles. As indicated earlier, the most effective way of dealing with that nonuniformity is to taper the susceptor so as to provide higher flow rates to compensate for the depletion effect of SiH, . A direct comparison of these results with the measurements of Lord [150] is not possible because of some of the idealizations made in the model. However, the 3D model does capture the complexity of the flow and can be easily extended to include the susceptor taper and other deposition chemistries. Transport in a MOCVD vertical modified barrel reactor has been analyzed by Dilawari and Szekely [153, 1541. A conical rotating susceptor instead of the polygon-shaped susceptor in Fig. 14 was used. The axisymmetric formulation of Eqs. (53)-(58) was used. For the species transport, one homogeneous gas-phase reaction for the decomposition of TMG in a hydrogen carrier to gallium and methane was used: Ga(CH,),
+ $H2
-+
Ga
+ 3CH,.
(77)
Assuming the rate constant reported in Sat0 and Suzuki [137], the production of TMG, STMG, was calculated as STMG =
~ ~ ~ ~X ~10" 2exp. 3 9
).
(78)
The growth of GaAs was assumed to be controlled by mass transfer of Ga and undecomposed TMG; that is, the wTMG and wGa were taken to be zero each at the susceptor surface. A parametric study included the effect of the position of the substrate, the reactor pressure, gas flow rate, reactor wall temperature, reactant concentration, rotation of the susceptor, substrate temperature, and the reactor inlet shape. An upshot of their study was that different parameters need to be carefully adjusted to achieve uniform deposition. For example, at atmospheric condition, two circulation loops, similar to those shown in Fig. 15, were observed. An upper loop above the conical dome formed due to entrainment by the inlet stream and the lower one ,due to buoyancy, extending into the annular space were observed. The loops increase in strength with an increase in distance between the inlet and the substrate.
386
ROOP L. MAHkTAN
The corresponding deposition rates showed higher rates with increased circulation. The depletion effect caused axial nonuniformities. Lowering the reactor pressure eliminated the buoyancy-induced loop resulting in more uniform deposition over most of the reactor. The reactor wall temperature, through its effect on the driving force for buoyancy-induced flow, suggested a strong influence. As mentioned earlier, in actual production reactors, axial asymmetries generally arise as a result of the nonaxisymmetric introduction of gases. Fully time-dependent, 3D models are necessary. C. PANCAKEREACTOR Although barrel reactors have recently dominated the industry for high-volume production of silicon epitaxy, pancake reactors (Fig. 2b) continue to be used for niche applications. The published literature on mathematical modeling of this reactor has been very limited. This is partially due to the conventional view held among practitioners and researchers that the deposition in a pancake reactor is similar to that of a horizontal reactor. It is assumed that inlet flow issuing from the central nozzle streams radially outward, forming a boundary layer similar to that in a horizontal reactor. This picture is only partially valid, as shown in Suzuki et al. [157] and Oh et al. [158]. In an experimental study, Suzuki et al. [157] observed the gas flow patterns at selected cross-sections by projecting collimated parallel light through a slit into the transparent quartz reactor. Figure 18 shows the observed patterns at B = 0" and B = - 30". In Fig. Ma, the streamlines run horizontally over the susceptor. This flow is supplemented by the ascending flow caused by buoyancy. However, at B = -3O", the flow pattern is quite different. There is backflow toward the center nozzle. Streamlines observed on a horizontal cross section also suggested complicated gas flow patterns. Silicon tetrachloride (SiCl,) carried by hydrogen was used for silicon deposition. The measurements of concentration distribution using a quadrupole mass spectrometer showed the presence of SiHC1, , SiH,Cl, , and HCl and SiCI, , indicating homogeneous decomposition of SiCl, in the gas phase. Nonuniformities in the epitaxial deposited layer were observed. It was shown that by using supplemental gas flows at selected locations on the susceptor, uniform deposition could be achieved. Ignoring the azimuthal momentum equation (for very slow rotations typically of pancake reactors, this is a reasonable assumption), Oh et al. [ 1581 used the 2D cylindrical coordinate formulation [Eqs. (53)-(58)] modified for 2D analysis. A dichlorosilane-hydrogen gas mixture was used for deposition of silicon. The following reactions proposed by Morosanu
387
TRANSPORT PHENOMENA IN CVD SYSTEMS
a
b
--
FIG. 18. Experimental flow visualizations in a pancake reactor at two different vertical cross sections of (a) 0 = 0" plane and (b) 0 = -30" plane; T, = 8WC, flow rate = 27 liters/min [157].
et al. [159] were used for the intermediate substrate temperatures ( < 950°C < T < 1150°C)and at low hydrogen pressures (30 torr < p < 500 torr):
SiCl,
+s
c-,
+ H,, SiCI, - s,
SiH,Cl,
+s
c-,
SiH2C1, s,
SiH,Cl, * s
c-,
SiCI, * s
SiH,CI,
H,
SiCI, * s
+ 2s
c-,
SiCl,
2H *
+ H,,
S,
+ H s -, Sic1 s + Hcl + s, *
(79)
(80) (81)
(82)
($3) (84)
where s is an active site on the substrate surface. The reaction given in Eq. (84) was taken to be the limiting step. Depending on the operating conditions, the deposition rate G was calculated by the species boundary
388
ROOP L. W
A
N
condition on the surface:
-(Mi
pDj d q
G
=
G
=A
- + kT dz
dz
or exp( - E , / R T ) P i ,
where Pi is the partial pressure of SiH,CI,. Equation (86) was based on the kinetic model proposed by Morosanu et al. [159]. In that model, the analysis was carried out by including both the homogeneous and heterogeneous equilibria established in the system. The growth rate expressions were provided for different operating ranges of temperature and pressure. The calculated stream function plots from Oh et al. [158] are shown in Fig. 19 to demonstrate interesting flow behavior observed experimentally in pancake reactors. At low flow rates, (Fig. 19a), the buoyancy-induced
a
h
-A_-
FIG.19. Computed streamline lots in a pancake reactor, T, = 950°C, P, = 150 torr: (a) H, flow = 20 slm, SiH,Cl, flow = 0.73 slm; (b) H, flow = 60 slm, SiH2C12 flow = 0.22 slm
[158].
TRANSPORT PHENOMENA IN CVD SYSTEMS
389
flow is relatively strong and aids in forming a circulating cell that covers most of the reactor domain. The incoming flow reaches the top of the roof of the reactor, goes around the recirculating cell, and mostly leaves the reactor through the exit opening. However, a part of this flow is entrained in the recirculatory cell. Note that the flow adjacent to the susceptor is inward. The deposition in this case is expected to be strongly influenced by the gas flow field. With increase in flow rate (Fig. 19b), the buoyancy-aided cell is pushed to the top. The incoming flow detours the cell and moves radially parallel to the susceptor surface before exiting. Another recirculatory cell (shown by the dotted line), moving in a direction opposite to that of the other cell, is generated. In a sense, the radial outward flow isolates the susceptor from the recirculating cells, and has the appearance of flow in a horizontal reactor with its leading edge being closer to the center of the reactor. The isotherms and concentration profiles were consistent with the flow fields just presented. A comparison of the growth rate calculations with the experimental results in Morosanu et al. [159] showed good agreement between the two. For the low concentrations of SiH,CI, considered, the effect of the thermal diffusion on the growth rate was found to be less than 1%. D. AXISYMMETRIC ROTATING-DISK, IMPINGING-JET, AND PLANAR STAGNATION-FLOW REACTORS All these geometries ideally yield uniform boundary layers. The governing equations of mass, momentum and energy reduce to coupled ordinary differential equations and one-dimensional(1D) similarity solutions can be obtained [160-1651. Utilizing 1D solutions, Hitchman and Curtis [166] and Pollard and Newman [167] analyzed epitaxial deposition of silicon from SiCl, and GaAs from trimethylgallium and arsine, respectively. However, in actual CVD reactor configurations, the finite size of the disk, presence of the reactor walls, and buoyancy effects produce flows that depart from the ideal 1D solutions. Complex flows and nonuniformities arise as revealed by flow visualization experiments [168-1711, requiring 2D and 3D analyses. A brief summary of these and other analyses is given below for the slow-speed (l-50-rpm) and fast-speed (500- 1500-rpm) rotating-disk reactors, impinging-jet reactors, and stagnation-flow reactors. 1. Rotating-Vertical-Pedestaland Rotating-Disk Reactors The slow rotating-vertical-pedestal reactor, shown in Fig. 2g, is one of the common geometries used for the MOCVD of compound semiconductors, and has been analyzed by Moffat and Jensen and other authors [105,
390
ROOP L. MAHAJAN
171-1771. The carrier gas containing the reactants enters through the top of the reactor, in which a slowly rotating graphite susceptor is placed perpendicular to the flow of the reactants. Depending on the operating conditions and the chemistries used, the reactants and/or the reactive intermediates, formed as a result of homogeneous gas-phase reactions, convect and diffuse to the substrate, where they deposit the desired layer. Fotiadis et al. [171] used an axisymmetric formulation to analyze the flow structure and deposition of GaAs and AlAs from a dilute mixture of trimethylgallium (TMG), trimethylaluminurn, and arsine in hydrogen. Gas-phase reactions were ignored, and the deposition was assumed to be controlled by mass transfer of TMG. This allowed a single-species (TMG) diffusion model with a fast surface reaction. Computations were carried out for the base case conditions (5slm flow rate, 900-K susceptor temperature, 300-K inlet temperature, 100-torr pressure) and for a range of flow rates and susceptor temperatures. They also performed flow visualizations in an experimental reactor by seeding micrometer-sized TiO, particles into the carrier-gas helium. The reactor was run at atmospheric conditions, and the flow rate varied from 2 to 15 slm. The susceptor was maintained at a relatively small temperature difference ( < 60 K) above room temperature to minimize distortion of the flow patterns due to thermophoretic transport of the seed particles away from the hot susceptor. A comparison of the flow pattern observed with that predicted by the model showed an excellent agreement. This is shown in Figs. 20a, b for flow rates of 2.5 and 7.5 slm, respectively, with the susceptor held at 350 K. At the lower flow rate, the flow is buoyancy-dominated.The induced cell isolates the susceptor from the incoming flow. At 7.5 slm, the forced flow impinges on the susceptor, but it is pushed up the reactor by buoyancy before its exit. The numerical solutions for typical MOCVD reactor conditions showed the existence of multiple stable flows for the same parameter values. For example, in one sequence of calculations, the susceptor temperature was held constant, and the inlet flow rate was increased in increments from the no-flow condition. The buoyancy-induced cell dominated the reactor flow until at 5.4 slm, when it was flushed out of the reactor by the incoming flow. However, when the inlet flow was reduced from this rate, the recirculation cell does not appear until a flow rate of 4.8 slm. Thus, at flow inlet rates of 4.8-5.4 slm, two very different stable solutions were possible for the same inlet flow rate and susceptor temperature. Above and below these two values, the flow was either forced-flow-dominated (a desirable option for uniform deposition and abrupt junctions) or buoyancydominated, respectively. Patnaik et al. [ 1721 also observed this hysteresis phenomenon in their numerical simulations of a vertical pedestal MOCVD reactor. They observed multiple steady-state flows over a range of suscep-
TRANSPORT PHENOMENA IN CVD SYSTEMS
391
FIG. 20. Computed and experimental flow streamlines for a vertical pedestal reactor;
T, = 350 K, T, = 300 K (a) 2.5 slrn; (b) 7.5 slm 11711.
tor temperatures, as well as rotation rates. As many as two hysteresis loops and five possible steady-state flows were computed for rotation rates between 320 and 350 rpm. Figure 21 shows their Nu plot as a function of the susceptor rotation rate. The implication of these results is that the growth conditions in a MOCVD reactor may depend on the path taken to reach specific values of the rotation rate, susceptor temperature, flow rate, or pressure. Since the hysteresis effect is associated with the appearance and disappearance of the buoyancy-induced cell [ 1781, the hysteresis effect can be eliminated by operating the reactor at low pressure [172, 1761. For fast-rotating-disk flows, Evans and Greif [163] presented 2D axisymmetric flows that included the effects of finite geometry, buoyancy, and thermal conditions on the reactor walls. Only fluid flow and heat transfer results were presented. Using steady laminar, axisymmetric form of governing Eqs. (53)-(57) with d/dO terms omitted, they provided numerical solutions for a range of aspect ratios, ro/rd and L / r d , mixed convection parameter G T R ~ - ~Re, / ~and , disk surface temperature where ro, rd, and L refer to the radius of the reactor, radius of the disk, and distance from
392
ROOF’ L. MAHAJAN
I
100
-
I ! !
3
z
i
a,
z5
-a,
5-
c
ln
v)
.=
z’ 0
I
I
1
reactor inlet to rotating disk, respectively (see Fig. 22). It was shown that for values of GrRe-3/2 2 3, there is no recirculation of fluid in the region above the disk, and the temperature field is uniform over the disk except for the region close to the edge. For GrRe-3/2 < 3, a recirculation zone appears and heat transfer uniformity deteriorates. The computations indicated that in the absence of recirculation, the dependence of Nu on L/rd and ro/rd is insignificant. However, in its presence, this dependence is large. Decreasing ro/rd and increasing L / r d deteriorates the heat transfer uniformity. In a companion paper, Evans and Grief [179] studied the importance of reactor wall thermal boundary conditions and showed that an isothermal cooled reactor wall results in reduced circulation compared to that for an adiabatic wall. Increasing the reactor inlet velocity had also a beneficial effect. In Evans and Grief [180], the effect of introducing inlet flow through an unheated, porous, and stationary disk normal to its surface and toward the rotating disk was investigated. This flow combination allowed an investigation of the effect of the forced flow on that induced by an infinite rotating disk in an infinite medium. A similarity solution of the governing equations was shown to exist. Solutions were obtained for a range of spin Reynolds number, Re, = L2w/vL and forced Reynolds number Re, = LU/v,, and for temperature ratio, T, = (T, - TR)/TR,of 0.001 and 2.33, where w is the disk spin rate, L is the distance from inlet to rotating disk, U is the
TRANSPORT PHENOMENA IN CVD SYSTEMS
393
FIG.22. Schematic of a rotating-disk reactor [179].
inlet velocity, vL is the fluid viscosity evaluated at the inlet conditions, and T, and T R are the surface temperatures of the rotating and stationary disks, respectively. The inlet flow was found to have a strong effect on the extent and uniformity of heat flux from the rotating disk. The implication is that in rotating-disk CVD reactors, the inlet flow rate may be manipulated to affect deposition uniformity. As expected, the effects of variable properties were found to be important. Coltrin et al. [181] coupled the fluid flow and heat transfer model for a rotating-disk reactor with gas-phase chemical kinetics and surface reactions. Using the results from Coltrin et al. [116] for deposition chemistry of silane, they predicted the silicon deposition rate as a function of inlet-flow velocity temperature, susceptor rotation, and composition of the carrier gas. Consistent with the past CVD studies, the results showed that the deposition rate changes from being kinetically controlled at lower substrate temperature (800 K) to mass-transport-limited at higher temperature (1300 K). The deposition rate increases with an increase in spin rate in the mass transport domain but remains almost unaltered in the kinetically controlled domain. At an intermediate temperature of 950 K, the deposition rate decreases with increasing spin rate. A similar trend was observed with the inlet flow velocity.
394
ROOP L. W
A
N
2. Impinging-Jet Reactors The impinging-jet geometry has been explored in Kleijn et al. [95, 96, 182-1851. In Vandenbulcke and Vuillard [182] and Chin and Tsang [1831, the parameter domains where the deposition is controlled by either diffusion, convection, or reaction are identified. The 1D analysis of Michaelidis and Pollard [184] is very similar to that in [1671 for a rotating disk. A comprehensive model of a single-wafer impinging-jet reactor has been carried out by Kleijn et al. [95,96, 1851. A schematic of their modeled reactor is shown in Fig. 2e. The gases are introduced radially from the top into an injection tube, which is positioned perpendicularly above the resistively heated susceptor. The walls of the reactor are water-cooled to the ambient temperature. In their analysis, Kleijn et al. [96] assumed the flow to be laminar and axisymmetric. For the low pressures (0.1-10 torr) considered, the homogeneous gas-phase reactions were ignored. The deposition path of silicon from silane was assumed to be governed by the adsorption of silane at the surface, where it diffuses to a reaction site and decomposes according to Eq. (70). The surface reaction rate was assumed to be rate-limitipg and of the form given in Eq. (76). The continuum 2D axisymmetric equations of continuity, momentum’ energy, and species used are the same as developed in Section 111. Numerical solutions were obtained using a control-volume-based finite-element method. The simulations showed that, as in other cold-wall reactors, the Soret effect reduced the deposition rate and increased the radial nonuniformity. These effects were more pronounced for hydrogen than for nitrogen as a carrier gas. Increasing the total pressure in the system results in an increase in the deposition rate. However, the buoyancy effects can also become significant, and can induce recirculating cells, leading to nonuniformity. The model was also exercised to study the effect of other parameters such as total flow rate and inlet reactant concentration. In a later study [95], the model was extended to include homogeneous gas-phase reactions and additional heterogeneous surface reactions for the reactive intermediates. Of the 20 different homogeneous gas-phase reactions proposed in Coltrin et al. [116] for silane chemistry, five potentially dominant mechanisms were selected for analysis. Baseline calculations were performed for the susceptor temperature of 900 K, a total flow rate of 1000 sccm (standard cubic centimeters per minute), a total pressure of 133 Pa, and a silane inlet mole fraction of 0.1 in nitrogen carrier gas. The walls and the inlet flow were assumed to be at 300 K. The analysis showed that the contribution of the reactive intermediates to the total growth rate is less than 0.5% under normal LPCVD operating conditions in a cold-wall reactor. This is seen in Fig. 23, which shows the growth rates as a function
TRANSPORT PHENOMENA IN CVD SYSTEMS
,
1 o3
I
total
lo2 1 0'
1
1 oo
10.2 10 10
total gasphase -_--------------\ .1 8H2 7
/
10"
395
s i 2 H6
S'2H4 S13H8 I
1 I
FIG. 23. Growth rate profiles due to reactive intermediates in the gas phase in an impinging-jet LPCVD reactor, Fig. 2e; T, = 900K, T, = 293 K, p = 133 Pa, total flow rate = 100 slm, inlet mole fraction of silane in N, = 0.1 r9.51.
of radial position due to each of the silicon containing reactive intermediates in the gas phase. Note that the deposition due to the reactive intermediate, although very small, is nonuniform. Most of the growth rate comes from the heterogeneous surface reaction of silane [Eq. (7011 and is uniform along the radius. The simulations also suggested that increasing the substrate temperature from 900 to 1000 K increases the total growth rate by a factor of 8. The contribution due to gas-phase reactions remains small, and the resulting growth rate uniformity is still > 98%. Using the same reactor configuration as discussed above, Kleijn et al. [185] provided a detailed mathematical model for the deposition of tungsten from WF, and H, . The predictions matched well with the experimentally observed growth rates and uniformities. Multiple impinging-jet arrangements in an atmospheric horizontal MOCVD reactor (Fig. 2f) have been investigated at A T & T-Bell Laboratories for multiwafer deposition. Single or multiple 2D jets enter the reactor from the top wall, impinge on the heated substrate at the bottom wall, and exit from the two ends. In the AT & T-Bell Labs study, 1, 11, and 21 gas jets were tried. The formulation is the same as that discussed in Section 1V.A for horizontal silicon epitaxy [103]. The reactant species were arsine and TMG and the carrier gas was hydrogen. The reactor height and slot width were 2 cm and lmm, respectively.
396
ROOP L. MAHAJAN
The computed stream functions, isotherms, and isoconcentration lines are shown in Fig. 24 for 1- and 11-jet configurations. The left halves of the plots are for 1-jet configuration, and the right halves are for the 11 impinging jets. For the single-jet impingment, the incoming flow streams past the buoyancy-induced vertices and the isotherms show a dip at the center due to the incoming cool jet. TMG disperses as the gas flows toward the substrate, resulting in a Gaussian distribution. Clearly, uniform deposition is not attained. When the incoming jets are increased to 11, flow dips further down toward the bottom wall and pushes the buoyancy induced vortex. The net dispersion of TMG due to different incoming jets results in flatter but wavier concentration distributions. For the 21-jet inlet (not shown), the buoyancy induced vortex is totally eliminated and the concentration distributions are flat. The net result is a uniform growth rate over a large area of the heated substrate as shown in Fig. 25. The residence time of the reactants is relatively short, suggesting that sharp interfaces can be obtained.
a
2 cm 1 cm
0
1 cm-
0
f
10 cm
20 cm
30 cm
40 cm
FIG.24, Normalized stream function, temperature, and concentration profiles for the 1and 11-jet impinging-jet reactor; T, = 973.16 K, T, = 300 K, PTMCin H, at 1 atm = 50 Pa, average gas speed = 15 cm/s: (a) stream functions; (b) isotherms; (c) TMG isoconcentration lines.
TRANSPORT PHENOMENA IN CVD SYSTEMS
397
__--_ 11 Jets 21 Jets
x (cm) FIG.25. Growth rate for 1-, 11-, and 21-jet impinging-jet arguments; other parameters same as in Fig. 24.
3. Planar Stagnation-Flow CFD Reactors
Compared to the vertical pedestal reactors which have found extensive application for MOCVD, planar stagnation-flow-typereactors have evoked relatively low interest in industry and among researchers. Using 2D axisymmetric formulation, Wahl [1691 analyzed the flow in an axisymmetric stagnation flow. Both the downflowing and upflowing inlet-gas configurations were considered. The calculated streamlines matched well with flow visualizations. Using single species model and ignoring Soret effect, the deposition rates for SiO, and Si,N, were calculated and shown to be in reasonable agreement with the experimental data. Later, Wahl el al. [1861 deposited thin layers of Y 2 0 3 , ZrO, , and body-centered cubic yttria stabilized zirconia through direct chlorination of Y and/or Zr. Houtman et al. [187] also presented a 2D axisymmetric formulation for transport analysis of a vertical stagnation flow. Simplified energy and species transport equations given in Section I11 were described. However, the deposition chemistry (of GaAs) was not considered, and only limited flow and heat transfer results were provided. The heat-mass transfer analogy was invoked to calculate the mass transfer rates. The focus of the paper was on the limits of the conventional 1D similarity solution for a planar surface of infinite extent. The results showed that in certain ranges of reactor parameters (Reynolds number, aspect ratios, susceptor tempera-
398
ROOP L. MAHAJAN
ture), the 1D solution could be applicable over the central portion of the susceptor. However, there was clear indication that in actual production reactors, the edge effects due to the finite size of the susceptor, the presence of the reactor walls and their thermal conditions are likely to cause buoyancy-induced flow circulations and 2D or 3D flows. Lee et al. [188] conducted measurements on the growth of GaAs in an inverted (upflowing inlet gases) stagnation reactor and compared their data with the mass transfer calculations in Houtman et al. [187]. These calculations were made under the assumption that the deposition rate is limited by the mass transfer of the trimethyl gallium. The agreement between the predictions and the data was rather poor, suggesting the need for improved modeling incorporating thermal boundary conditions on the reactor and detailed kinetics of GaAs deposition. Recently, Calmidi and Mahajan [178] analyzed in detail the flow field and thermal characteristics in mixed convection in a related configuration (Fig. 26). The heated surface and the inlet are aligned on the same vertical axis. The flow enters the enclosure vertically and exits through the two openings on the sides. Similar to the results reported in Fotiadis et al. 11711 for rotating-pedestal reactors, the authors observed multiple steady-state solutions for certain combinations of Gr and Re. For a fixed Grashof number (constant susceptor temperature), as the forced flow is gradually increased, the buoyancy cell appears at a relatively high value of Reynolds number. On the other hand, when the flow is gradually decreased from that forced-convection domain, the cell does not appear at the same value of Reynolds number at which it disappeared. Instead, it reappears at a much lower Reynolds number. Figure 27 shows the stream-function contours for decreasing and increasing Reynolds number for Gr/Re2 = 2.5. They clearly indicate two different steady-state solutions. The net result is a hysteresis loop in the Nu-Gr/Re2 plot. Similar hysteresis characteristics were also observed for the case of constant flow over the heated surface with variable heating (constant Re and varying Gr). An interesting result of their study is the unsteady flow behavior observed at higher exit openings for 45 < Gr/Re2 < 60. An examination of the transients indicated a periodic forming and sweeping away of a buoyancy-induced plume near the edge of the heated surface. The corresponding change in the Nusselt number was as much as 50%. The implication of these results for CVD applications is twofold. For some combinations of parameters, for example, large exit openings, unsteady flows can arise that can cause significant nonuniformities in heat transfer or mass transfer rate. Also, fully time-dependent 3D models are needed to completely understand the transport phenomena.
TRANSPORT PHENOMENA IN CVD SYSTEMS
399
I
I
I I J‘ i I
AiI R I
I I
I I
I I I 1
L
HEATED PLATE
P
4
FIG. 26. Flow configuration and geometry for stagnation flow on a heated surface in a partial enclosure [1781.
E. HOT-WALLLPCVD REACTORS Multiwafer hot-wall LPCVD reactors (Fig. 2d) are extensively used for the deposition of oxide, nitride, and polysilicon films. In such reactors, 50-200 wafers are stacked side-by-side with a constant separation between them. The wafers are generally placed back-to-back, concentric to the tube cross-sectional area so that the center of the wafer coincides with the axis of the tube. As in the other CVD deposition systems described earlier, an important consideration in growing films in the LPCVD reactors is the inter- and intrawafer uniformity in the thickness of the layer deposited. Figure 28 shows the measured temperature and growth rate in a horizontal LPCVD reactor [189]. The drop in temperature at the ends is attributed to the additional radiative loss from the end wafers “seeing” the cold reactor
400 la
ROOP L. MAHklAN
Ib
I
FIG.27. Stream-function contours showing two entirely different flow solutions in mixed convection for the configuration in Fig. 26: (a) for decreasing Re; (b) for increasing Re [178].
walls. This temperature drop is accompanied by a drop in the film growth rate toward the front and back of the tube. A common industrial practice in response to this nonuniformity has been to load a series of “dummy” wafers on the two ends. Typically, these wafers are reiatively inexpensive substrates and are used repeatedly for a few runs before they are discarded. This ensures uniform deposition on the central wafers. There is, however, an attendant loss of reactor capacity by 5 2 5 % 11891. The film thickness uniformities between wafers (interwafer) and across a single wafer (intrawafer) in a batch are determined by the temperature field and transport of the species to the wafer surface. Another important consideration in LPCVD processing is the quality of the films grown; a particular concern is contamination by particle dropoff from hot walls. The kinetics of decomposition of reactants in the gas phase and wall deposits determine the particle counts on the wafers. The surface kinetics play an important role in determining the film quality.
401
TRANSPORT PHENOMENA IN CVD SYSTEMS
a 0
2 a,
U
Y
-10
g
'.. .-m> a,
W
-20
d
g -.-m
2 -30 Dimensionless Boat Position
b
-s
0.4
20
Y
v
0
0.3
.E
z
3 C
$
3 C
0.2
ii
e 9!
3
8
c
! I
C Y 0
z E
'E2. -0C
12
?L
0,
-s
0.1
4
E E I-"
0.0 '0.0
0.2
0.4
0.6
0.8
1.o
Dimensionless Boat Position
FIG.28. Measured (a) axial and (b) radial temperature and film thickness profiles for a hot-wall tubular LPCVD reactor [lSSJ. Reproduced with permission of the American Institute of Chemical Engineers. Copyright 0 1992 AIChE. All rights reserved.
402
ROOP L. MAHAJAN
Both the chemical reactions and transport phenomena in these reactors have been studied by a number of investigators [155, 189-2181. An excellent review of these is provided in Badgwell et al. [61]. 1. Heat Transfer Model In most of the analyses [155, 189-2071, the reactor and the wafers are assumed to be isothermal. However, as shown in Fig. 28, both axial and radial nonuniformities in temperature exist in the reactor. Motivated by these measurements, Badgwell et al. [2111 developed a diffuse radiation heat transfer model to predict temperature profile within a vertical hot-wall multiwafer reactor. Heat conduction and convection in the vapor phase were ignored. The temperature profile of the reactor tube was assumed to be set by the exterior heater elements. The wafer surfaces were taken to be of constant emissivity. Representing the interior reactor surfaces to be graybodies and the reactor doors as isothermal disks of stainless steel, they used the following energy-balance equations for the wafer and radiation heat transfer in the enclosure: dT = - k , V * VT,, PSC,,,
q; = Ei Equation (88) determines the net heat flux radiated away from the wafer surface i in radiation balance with a set of nf surfaces of area A j . Here dFdi-,, is the conventional differential view factor that gives the fraction of radiation emitted from the differential surface element &Ii, which is intercepted by the differential surface element d4,.The net radiative heat flux, in turn, is balanced by the net flux transported by conduction from within the wafer surface [Eq. (8711. The thermal boundary conditions for the wafer surface follow from the local energy balance at the surface and a symmetric condition at the center:
-ks VT, * n and
$l r = o
= qi
=
(89)
0.
As in Hu [213], they averaged the wafer energy balance axially (temperature drop across a wafer is expected to be small), and obtained a set of
403
TRANSPORT PHENOMENA IN CVD SYSTEMS
ordinary differential equations for radial heat transfer in the wafer by conduction. For the radiative heat flux, they obtained a set of equations for the discretized elements of the inlet and outlet sections, and the wafer-carrying section between each wafer. The equations were made dimensionless by using wafer radius as the reference dimension. The reference temperature was assumed to be equal to that of the outer-tube middle point. Wafer state solutions were presented for the average temperature Ti and radial temperature nonuniformity f i , where
The predicted temperature profiles are shown in Fig. 29, where the dimensional boat positions of 0.0 and 1.0 represent, respectively, the first and the last wafers in the boat. In these calculations, p s , C p , s ,k , , and E were taken to be 2.330 g/cm3, 932 J/(kg K), 35.7 W/(m K), and 0.65, respectively. An excellent agreement is seen between the model predictions and the experimental data for average wafer temperature. For radial temperature nonuniformity, however, the disagreement is quite large. This discrepancy was attributed to the assumption of reactor axisymmetry assumed in the model. In actual reactors, the boat structure generally rests on the bottom of the tube in such a way that the wafers are not centered directly in the tube. A parametric study was then conducted to evaluate the effect of different reactor design parameters and the outer-tube temperature profiles. The calculations revealed that wafer spacing and wafer radius have significant impact on both and i i .An increase in wafer radius worsens the axial uniformity. The radial temperature uniformity becomes worse at the ends but better in the center of the load. Likewise, the net effect of increasing the wafer spacing is to degrade the overall temperature uniformity. Increasing the thickness of the wafer improves the radial uniformity without affecting the axial temperature profile. For more details, the reader is referred to Badgwell et al. [211].
-
-
2. Deposition Model An examination of Fig. 28 reveals that the deposition rate closely parallels the temperature distribution. Coupled with the fact that temperature in the reactor is relatively low (= 6OO0C), the data suggest that the
404
ROOP L. W
6
3 3
A
N
Measured
585.
580 -
% 575 2
a,
570
b
'E 8C
1 C
e
Predicted 7 Measured
jot
0
0.2
0.4
0.6
0.8
1
Dimensionless boat position
FIG. 29. Measured and predicted temperature profiles in a hot-wall tubular LPCVD reactor: (a) average temperature and (b) radial temperature nonuniformity [211].
deposition is kinetically controlled. As a result, the flow effects are not expected to play a dominant role. Indeed, Middleman and Yeckel [204] and Hitchman et al. [191] showed that flow in the reactor is isobaric and exhibits a flat velocity profile. This allows a dropping of the momentum equation. If one further ignores the axial and radiation temperature variation and thus assumes the reactor to be isothermal, one sees the energy equation dropping off. The transport problem then simply reduces to one in which convection and diffusion dominate in the annular space whereas only diffusive transport dominates the interwave region. Many of the earlier deposition models of LPCVD reactors [155, 197, 1981 made use of these assumptions to gain an insight into the physics of
TRANSPORT PHENOMENA IN CVD SYSTEMS
405
the phenomena taking place in the reactor. However, as pointed out in Badgwell et al. “91 and Roenigk and Jensen [200], some of these assumptions are not strictly valid for most LPCVD processes of interest. For example, Roenigk and Jensen [200] extended an earlier model of LPCVD of polysilicon from silane by Jensen and Graves [155] and included multicomponent diffusion effects. Using Stefan-Maxwell relations, it was shown that the simplified model is valid only for high flow rates. For typical operating conditions, it overpredicted growth rates by about 10%. Badgwell et al. “91 presented a comprehensive model of hot-wall multiwafer LPCVD reactors. As in earlier studies, a flat velocity profile was assumed for the gas phase so that the momentum equations could be dropped. However, axial and radial variations in temperature were allowed so that the corresponding thickness variations could be calculated. The species conservation equations for a multicomponent system in the mass flux, mass fraction formulation [Eqs. (23)-(28)] were used for the whole reactor, which was divided into four regions. These are tubular inlet and outlet regions in front of and behind the wafers, the annular space surrounding the wafers, and the interwafer regions. For the inlet, outlet, and annular regions, equations were averaged radially, producing ordinary differential equations (ODES) in r. This procedure is justified since the diffusion in the gas phase is assumed to be rapid and the time scale associated with diffusion is much smaller than that for deposition [2001. Perfect radial mixing may therefore be assumed. Similarly, in the interwafer region, since the wafer spacing is small compared to the wafer radius, and the diffusion is rapid, the axial variations in this region are small. Assuming this, the species conservation equations in the interwafer were averaged axially to produce ODES in z. These equations for the different regions were coupled together through boundary conditions. The overall continuity equation was used to solve for the flow velocity. The model equations were numerically solved by orthogonal collocation on finite elements [219], and the results were presented for deposition of polysilicon from a gas stream of silane and hydrogen. On the basis of their calculations of a subset of gas-phase reactions and the surface reactions proposed by Kleijn [95], the gas-phase decompositions were shown to be unimportant and hence were neglected. The deposition was assumed to proceed through a rate-limiting surface decomposition of adsorbed silane. The surface reaction for decomposition of silane and the kinetic rate were those given in Eqs. (70) and (761, respectively. The experimentally measured temperature profile was used in the calculations. The overall agreement between the experimental and predicted growth rates in radial and axial direction was good.
406
ROOP L. MAHAJAN
3. Noncontinuum Transport The continuum analysis presented above for the LPCVD reactors breaks down in ultra-high-vacuum conditions where the operating pressure is in the millitorr (mtorr) range. The mean-free path is then likely to be larger than the reactor characteristic dimension of the reactor, resulting in a transition or free molecular flow. Also, as pointed out in Section 11, for deposition into micrometersize windows, the same situation may arise. In Lee [6], some representative calculations are given for LPCVD reactors for operating pressure down to lo-, torr. It is shown that for epitaxial silicon growth at lo-, ton- and 800°C in a 2.5-cm-radius reactor, the contribution of molecular flow to the total flow is about 96%. For Si,N, deposition, based on SM,Cl,/NH, in a reactor operating at a relatively higher pressure of 0.05 torr and 910"C, the contribution is still 13%. At 1 torr and 910°C for deposition of SO, from SiH,Cl,/N,O reactants, however, the deviation from continuum flow is small. (The contribution of molecular flow is less that l%.) At atmospheric pressure for silicon epitaxy, the flow is continuum. For micrometersize deposition features, the pressure does not have to be very low for molecular flow to dominate. In the example above, when the calculations were repeated for deposition in a 2-pm cavity, a complete molecular flow was indicated for all the cases except for the atmospheric epitaxial growth. Even in the latter case, the contribution due to molecular flow was calculated to be 59%. For purpose of analysis, the noncontinuum flow may be divided into three Knudsen-number domains, discussed in Section 11. Analysis of noncontinuum flows is based on a statistical description of a gas on the molecular level. Using Boltzmann's equation for distribution function for each species in a gas mixture, the transport quantities of interest can be derived [220]. In the limiting case of very high JSnudsen number, collisions between molecules are rare in the region of interest and may be ignored. In this collisionless or free-molecule limit, the analysis then becomes relatively simple. It requires only the specification of the molecular distribution functions at the boundaries, and the nature of the molecular interactions with any solid surfaces. For simple geometries, analytical solutions can generally be derived easily [220]. At the other end of the spectrum of the noncontinuum flow, slip flow, the Chapman-Enskog [2211 method can be used to provide a solution of the Boltzmann equation by perturbing the distribution function from its equilibrium value. However, these solutions are possible for only a restricted class of problems. A common approach is to use the Navier-Stokes continuum governing equations with some modifications, subject to velocity slip and temperature-jump boundary conditions at the fluid-solid
TRANSPORT PHENOMENA IN CVD SYSTEMS
407
interface. The velocity slip is characterized by a reflection or momentum coefficient a, ; and the temperature jump, by an accommodation coefficient, uT. Both these coefficients characterize the surface-gas interactions taking place in noncontinuum flow and are defined in terms of incident, reflected, and equilibrium molecular energy or moments. For example, the thermal accommodation coefficient uT at a point on a surface is defined as [220, 2221:
where ey, e;, and e: respectively represent the molecular energy flux incident on the surface, the energy flux of the molecules as they are actually reflected from the surface, and the energy flux that would have resulted if the molecules had been reflected in equilibrium at the wall temperature T, . For continuum flow, where multiple collisions occur at the surface, uT = 1. At the other end, if the molecules make no accommodation to the wall, uT = 0. For transitional flows, the direct simulation Monte Carlo technique, developed by Bird [220] and originally applied to many aerospace problems, may then be applied. In this technique, a gas flow is simulated by concurrently tracking thousands of representative molecules as they move through the flow field. As the molecules go through collisions and boundary interactions in simulated physical space, their position coordinates and velocity components are stored in the computer and modified with time. After a steady flow is achieved, the macroscopic flow properties are obtained by time-averaging the simulation conditions. Coronell and Jensen [223] successfully used this technique to analyze transitional flows in LPCVD reactors. For the reactor operating pressures in the range of mtorr, flow simulations were carried out for hydrogen, argon, and nitrogen. The pressure-drop calculations revealed that most of the drop occurs across the wafer stack and depends strongly on the diameter of the wafers. It was shown that for heavier gas molecules, the gas-phase collisions are reduced. As a result, pressure gradients are not dissipated and the pressure drop is larger. Particle trajectories and velocity vectors in a multiwafer (six 8-cm wafers placed 0.5 cm apart) LPCVD reactor operating at inlet ambient pressure of 10 mtorr are shown in Fig. 30, indicating that most of the molecules bypass the interwafer region, suggesting inefficiency of the reactor configuration for deposition. The interwafer region simulations were also performed for a reactor operating pressure of 0.1 torr and a mixture of silane and hydrogen. The radial growth uniformity was examined for different interwafer spacings and entering reactant composition. At higher spacings,
408
ROOP L. MAHkTAN
FIG.30. Particle trajectory and velocity field in transitional flow in an LPCVD reactor for upstream ambient pressure = 10 mT, flow rate = 48 sccrn [223].
reactant molecules diffuse deeper into the interwafer region before being depleted. This allows better radial uniformity. The reactant composition did not affect the uniformity, indicating the dominance of gas-surface interactions. A comparison with the continuum flow calculations pointed out the inaccuracies that arise without taking noncontinuum effects into consideration.
V. Artificial Neural Network Models for CVD Processes
The physical models discussed above are ideal for revealing the underlying physics of the CVD processes. However, the utility of these models for fast-response analysis and control is limited because of the generally large computational time required to simulate a run. To solve this difficulty, Mahajan and his coworkers [176, 2241 have recently proposed physiconeural modeling as an alternative approach to obtain accurate and yet fast-response models. Briefly, in this approach, the numerical simulations are used to generate input(s)-output(s) data set for a selected range of input variables. The numerical data so obtained are used to train and test an artificial neural network (A")model, which they term as a physiconeurul model. Such a model can then be used to predict the behavior of the reactor accurately and quickly for changing input conditions. It can also be used to identify the optimum process conditions for the desired output, such as, uniformity in the thickness of the deposited layer. Additionally, ANN models can provide the necessary algorithm for run-to-run and real-time control of the process. Before some of their results are presented, a brief introduction to ANN is provided below. Briefly, artificial neural networks are massively parallel, highly interconnected systems of computational nodes or neurons. A typical ANN struc-
409
TRANSPORT PHENOMENA IN CVD SYSTEMS
Input Layer
Hidden Layer j-th layer
Output Layer k
y2
yk
FIG.31. Schematic diagram of an artificial neural network (A").
ture is shown in Fig. 31. It consists of an input layer into which the input vector (independent variables) are fed, an output layer, and one or several hidden layers in between. The hidden layers link the input and output layers together and allow for complex, nonlinear interactions between the inputs to produce the desired outputs. Computations are done in the hidden layers and the output layer, but not the input layer. Consider a multiple input-output relation mapped by a neural network. It can be mathematically expressed as a recursive formula relating the output of a layer to its input:
where y: is the output of the jth neuron in the kth layer, is the weight on the connection from the ith neuron in the ( k - 11th layer to the jth neuron in the kth layer, BF is the bias connected to the jth neuron in the kth layer and Nk- are the number of neurons in the ( k - 0 t h layer. Note that yo0 = xi are the inputs and No are the number of inputs, F is the activation function, which may be thought of as providing a non-linear gain for the artificial neuron. It is typically a sigmoid function Eq. (95), and bounds the output from any neuron in the network.
F(u)
1 =
(1
+ e-')
*
(95)
410
ROOP L. W
A
N
To train an ANN for developing a process model, a set of input-output data is required that should be representative of the process. The first step of training is the forward pass, which consists of calculating the output vector by running the input vector through the network. This is followed by a backward pass, where the error derivatives are calculated for each weight. The error derivatives for a weight are summed until all the training data points have been run through the network once. This constitutes an epoch. The weights are updated after each epoch such that the network error decreases. Readers are referred to Wasserman 12251 and Smith [2261 for further details. The training process, briefly outlined above, can, however, be difficult and quite time-consuming. Training mainly depends on the network structure, transformation of data, and the search algorithm used. The choice of these factors plays a crucial role in the ANN model development. Recently a neural network modeling approach was proposed [2271 to alleviate some of these problems. For selecting the appropriate neural network architecture, the total data points are split into a training set with about three-fourths of the data points and a testing set with approximately one-fourth of the data points. The inputs are normalized between 0 and 1. The outputs, however, are normalized between 0.2 and 0.8 in order to avoid the saturation region of the sigmoid transfer function. In training and testing, different measures are used to evaluate the performance of the network and select its optimum structure. They are defined below: N
R~ = 1 -
RMSmonitor = wRMStest + ( 1 RMS
=
N
C (yi -yip)2/ iC (yi - j j ) 2 , i= 1 = 1
i"C i= 1
- w)RMStrain
r
( Y , -YP)~/N ,
,
(96)
(97) (98)
where yi and yp are the actual and predicted values, respectively, and j j is the averaged value. The correlation coefficient R2 measures the goodness of fit of the training data; R 2 of 100% means a perfect fit. RMSteStand RMStrainare the testing and the training errors, respectively, and w(0 5 w I 1) is the weighting factor representing the relative importance of testing performance. Since the prediction error is typically more important, w = 0.75 is chosen to ensure larger weightage to the test error. Building the architecture starts from the simplest network with one hidden layer having one hidden neuron. The network is trained with the training data set and R2 is monitored to check the adequacy of the learning of the network. If a prescribed value of R 2 ( = 0.8) is not
TRANSPORT PHENOMENA IN CVD SYSTEMS
41 1
achieved, another hidden neuron is added. After the prescribed value of R 2 of 0.8 is reached, testing data are simultaneously fed to the network. The network then calculates the outputs for both the training and testing sets, which are then compared with the actual outputs. RMS,,,,,,, is calculated, and training stops when the minimum RMS,,,,,,, is obtained. To ensure that the optimum architecture has been achieved, another hidden neuron is added and simultaneous training and testing are performed. This process is continued until the specified requirement of RMS,,,,,,, is satisfied or no further improvement in RMS,,,,,,, results. This procedure ensures that the network chosen is the simplest, requires the minimum training time, and avoids overfitting. Deploying the ANN modeling technique described above, Kelkar et al. [176] developed a physiconeural model for MOCVD in a vertical-pedestal reactor of the type shown in Fig. 2g. Using the fluid dynamics analysis package (FDAP) modified for inclusion of Soret effect, they obtained numerical solutions to the governing set of 2D axisymmetric model formulations for a range of processing parameters. These were AT = T, - T, (500-1000 K), R e (0-1001, rotation rate of the susceptor R (0-1001, and inlet mole fraction of TMG (0.001-0.05). Using a central composite statistically designed experiment, 25 simulations were used to train the neural network model. Six additional points acted as a test set. A neural network of (4,4,1) configuration was determined to be the best model, where the numbers in parentheses represent the number of inputs, the number of neurons in the hidden layer, and the number of outputs, respectively. It was shown that the neural network model predicted the deposition rate accurately (see Fig. 32a). A similar model using the radial distance as an additional variable was developed for the radial deposition rate. A total of 63 simulations were used to train an ANN model whose optimum configuration was found to be (5,4,1). Again, the match between the model predictions and the physical model simulations was good (see Fig. 32b). They demonstrated that the computational time required for one simulation run on a DEC 5000 workstation was 0.09 s for the trained ANN model. The corresponding time using the physical model was about 80 min. They also identified the process conditions for minimum variance of the deposition rate. Earlier Wang and Mahajan [228] used the same concept to build a physico-ANN model for silicon epitaxy in a horizontal reactor. For training, they used the input-output data simulated by the 2D numerical model of Mahajan and Wei [103], where the input data corresponded to the inlet velocity U, (17-51 cm/s), inlet concentration of silane C, (318.5-957.5 Pa), susceptor temperature (1250-1450 K), and downstream distance x from the leading edge of the susceptor. The output was the deposition rate
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a
0
.
7
4
7
b
0.321
0.181
0
0.5
1 1.5 2 2.5 Radius of Susceptor (an)
3
:
FIG.32. Comparison between ANN model and physical model predictions of MOCVD in a vertical pedestal reactor: (a) average deposition rate; (b) local deposition rate [0 = training and testing points; X = validation points; * *, ANN model; - = physical model].
of silicon. A total of 98 simulation runs were made for different values of the processing parameters. Using a number of combination of training and testing data points (49,49), (42,56), .. .,(21,771, they determined the optimum configuration to be (4,8,8, l), which contains two hidden layers of eight computing neurons each. Again, the agreement between the prediction of the physico-A” model and the computational model was within 2%. The conclusion was that the physiconeural models approach the accuracy of the computational models and that they could be used in
TRANSPORT PHENOMENA IN CVD SYSTEMS
413
applications where quick response to a “what if’ parametric scenario is required. It might be noted that ANN modeling approach is equally attractive when a physical modeling of the process may not be possible. For example, in a barrel production reactor (Fig. 2c), the input parameters are taken to be the valve readings of the left and right nozzles, nozzle aiming positions of the two nozzles, and the valve readings for the main and additional hydrogen flow. The additional flow, called the rotational flow, is provided independent of the main flow, vertically down through the hollow susceptor. This flow serves to prevent deposition on the inner surface of the susceptor. It is also sometimes taken as the control value. Clearly, these parameters are not easily relatable to the independent parameters in the governing set of equations. In such a case, an ANN model using the equipment control parameters themselves provides the input-output relationship that can be used to optimize and control the process. This approach was used in Wang and Mahajan [2281, and Bose and Lord [2291 to model the reactor. A review of neural network modeling approach and its applications is provided in Mahajan et al. [230, 2311.
VI. Concluding Remarks The past two decades of research and the availability of increased computer power have significantly increased our understanding of the transport phenomena of chemical vapor deposition for commonly used reactor configurations. For some deposition systems, such as SiH,-H, and TMG-arsine-H ,, detailed reaction kinetics and reaction pathways are available and have been coupled with the transport analyses, as described in Sections I11 and IV. However, such a level of kinetics detail is not available for many other deposition systems. With the continuing rise in the use of CVD for growing films of new materials, there is a need for sustained research in understanding both the homogeneous gas phase and the heterogeneous surface reactions that might take place for different reactant combinations and reactor operating conditions. The requirements of deposition uniformity and precise dopant control in many microelectronic applications are becoming increasingly stringent. To meet these demands, the reactors are operated at pressures of the order of a few millitorr. In addition, the minimum feature size is shrinking to submicrometer levels. At these operating pressures or feature sizes, there is a departure from the continuum analysis, and noncontinuum treatment,
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discussed briefly in Section IV,E, needs to be used. The direct simulation Monte Carlo technique will find a wider usage for treating such flows. Finally, to receive a wider acceptance in manufacturing, physical models must be tailored to the production equipment configurations and boundary conditions. Many of the idealized assumptions have to be relaxed. This, coupled with the demand for accurate and real-time process control, would see an acceleration in the development of physiconeural network models discussed in Section V. Acknowledgments The author gratefully acknowledges the professional assistance of Ms. Ann Geesaman of CAMPmode (The Center for Advanced Manufacturing and Packaging of Microwave, Optical and Digital Electronics) in preparing this manuscript. Mrs. Kavita Mahajan deserves a special mention for her support and critical review of the chapter. Additional help provided by Dr. Y. Yokoyama and graduate students V. Calmidi and A. Kelkar is also appreciated.
Nomenclature molar concentration of the mixture specific heat binary diffusion coefficient multicomponent thermal diffusion coefficient effective multicomponent diffusion coefficient for species i surface Damkohler number = rs L/Cre, * Dref Grashof number = gP(T, - T,)L3/vZ gravity vector, g, = -9.81 m/s2 reactor height en thalpy unity tensor mole flux vector mass flux vector thermodynamic equilibrium constant for jth gas-phase reaction thermodiffusion ratio Knudsen number = h/L characteristic length molecular weight of species i average molecular weight number of species
ng "s
Pi PO Pr r
ri+
riR Ra Re
9f
9; sc f
T U
V U
r;
W
Xi
Xi x, y,
number of gas-phase reactions number of surface reactions partial pressure of species i standard pressure Prandtl number, wC,,/k reaction rate constant, radial coordinate forward reaction rate constant reverse reaction rate constant universal gas constant Rayleigh number = Gr Pr Reynolds number = U L / u jth gas-phase reaction rate jth surface reaction rate Schmidt number = u / D time temperature velocity component in x direction velocity vector velocity component in y direction specific volume velocity component in z direction mole fraction of species i total body force per unit mass of component i Cartesian coordinates
TRANSPORT PHENOMENA IN CVD SYSTEMS
GREEKSYMBOLS
7r
P 0
P Y E
0 K
A I.L V
ve ,:v
thermal diffusion factor, Eq. (37) volumetric coefficient of thermal expansion sticking coefficient emissivity tilt angle, radial coordinate bulk viscosity mean molecular free path dynamic viscosity kinematic viscosity stoichiometric coefficient for the ith species in the jth gas-phase reaction stoichiometric coefficient for the ith species surface reaction j
415
pressure tensor, Eq. (9) density shear stress tensor.. Ea.. (10) species mass fraction
SUBSCRIPTS C
ref S
cold surface reference conditions susceptor or hot surface
SUPERSCRIP~S average S surface
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INDEX A Ablation, 109-110 Ablation plume, 105-107 Alloys, 26 macrosegregation, 243, 251,266, 327 double diffusive convection, 240-241 electromagnetic stirring, 320-325, 327 inhibition, 308-310, 316 semitransparent analog alloys, 272 shrinkage and buoyancy, 306,308 tin-lead alloys, 291-292, 295-304, 317 solidification, 231-238,326, 328 from bottom wall, 277-282 continuum model, 270-288,311 freckle formation, 243-244 mathematical models, 249-269 metal alloys, 288-308, 316, 327 multidirectional solidification, 282-285 physical phenomena, 238-249 process control strategies, 235, 237, 238, 308-326,327-328 semitransparent analog alloys, 270-288 from sidewall of rectangular cavity, 270-277 unidirectional solidification, 277-282 zero gravity, 309 Aluminum, laser vaporization, 129-131 Aluminum-copper alloys, 233, 236, 239, 304, 317 Amioca, conversion during extrusion, 219-220, 221, 222 Ammonium chloride analog alloy, 270, 286, 311-316 Analog alloys, 240 Anisotropic permeability, alloy solidification, 276 Artificial neural networks (A”), chemical vapor deposition modeling, 408-413 Atmospheric reduced pressure chemical vapor deposition (APCVD), 344 Atomization, spray deposition, 10, 11, 18-21
Axial momentum equation, liquid metal droplet, 4 Axis of symmetry, liquid metal droplet, 6
B Barrel chemical vapor deposition reactors, 346,375-386 Blackbody radiation, 89 Blake-Kozeny equation, 266 Boundary conditions chemical vapor deposition, 358,382 laser evaporation, 125 liquid metal droplet, 5 splat quenching, 29, 47, 57 Buoyancy parameter, 239
c Carbon-iron alloy, solidification, 288 Carreau model, 153, 154 Casting, alloy solidification, 244-245, 306 Channels, alloy solidification, 244-245, 310, 312-313,316 Chemical vapor deposition (CVD), 339-344, 413-414 equations multicomponent mixture, 353-356 simplified, 356-362 models, artificial neural network models, 408-4 13 reactors, 344-345 barrel reactor, 346, 375-386 horizontal reactor, 346, 365-375 hot-wall LPCVD reactor, 399-408 impinging-jet reactor, 345, 346, 394-395 pancake reactor, 346, 386-389 planar-stagnation-flow reactor, 397-399 rotating disk reactor, 389-394 transport phenomena, 346-353,362-365 Conductance, semiconductor melting, 81-87 Conduction freezing, metal drops, 17-18 Conservation equations, liquid metal droplet in flight, 3-5
427
428
INDEX
Continuity equation chemical vapor deposition, 353, 359, 361 liquid metal droplet, 3 Continuum model, alloy solidification, 270-288,311 Convection alloy solidification, 235, 237, 246-249 ammonium chloride solution, 311-316 double-diffusive convection, 240-241, 244,277 models, 264-269 thermosolutal convection, 239-243 Convective cooling modeling, 13-14 single liquid metal droplet, 3-11 Cooling spray deposition convective cooling, 3-11 in flight, 11-18 splat cooling, 23-54 Copper, laser vaporization, 129-131 Copper-aluminum alloys, 233, 236, 239, 304 Copper-nickel alloys, 233, 236 Corotating twin-screw extruder, 187, 189-190, 191 Counterrotating twin-screw extruder, 187, 190-191 Czochralski crystal growth, 247, 316
D Darcy's law, 264,317 Defects, solidified alloys, 237, 245 Dendrites, alloy solidification, 232-233, 234, 247-248,266,326-327 Density gradient, alloy solidification, 238-239 Developing flow, single-screw extruder, 161-164 Dies, flow in, 201-212, 226 Direct-chill (DC) casting, alloy solidification, 306, 308 Discharge coefficient, 20 Doping, ultrashallow p+-junction formation, 96-101 Double-diffusive convection, 240-241, 244, 277 Drag coefficient, liquid metal droplet, 7-8, 10
Droplets spray deposition convective cooling, 3-11 multiple liquid metal droplets, 54-69 single liquid metal droplets, 3-18,23-54 splat cooling, 23-54 sprays, 18-22 Dufour energy flux, 353
E Effective sticking coefficient, 127 Ekman layers, 310 Electrodeposition, 340 Electromagnetic rheocasting, 318 Electron-beam assisted chemical vapor deposition, 343 Electronic ablation, 110 Electroplating, 340-341 Ellis model, 153 Emissivity, 91 Energy equation chemical vapor deposition, 353, 360, 362 convective cooling, 13 laser vaporization, 128, 129 spray deposition liquid metal droplet, 4, 5 recalescence, 16 solidification, 17 splat cooling, 46 Enthalpy, 64, 76-79 Equilibrium freezing, 262 Equilibrium phase diagram, 233,235, 239 Eutectic system, 233 Extrusion, 146-151 chemical reaction and conversion, 218-220 flow in dies, 201-212 moisture transport, 213-218 screw extrusion of polymers, 145-151, 220-226 combined heat and mass transfer, 213-220 material characterization, 152-155 single-screw extruder, 147, 149, 155-187 twin-screw extruder, 151, 187-201, 225-226
F Film formation, 340-341 Finite-difference modeling, twin-screw extruder, 195-197
429
INDEX
Finite-element modeling, twin-screw extruder, 193-195 Flow alloy solidification, 270-288, 318 barrel chemical vapor deposition reactors, 378 extruder dies, 201-212, 226 non-Newtonian material, 146, 152 tapered channel, 176 Flow rate, gas atomization of liquid metals, 19-20 Fluid dynamics extrusion die, 201-212, 226 spray deposition, 1-3 impact region, 23-69 spray region, 3-22 Food extrusion, 220,223 Freckles, alloys, 243-244 Freezing kinetics coefficient, 31 Fully developed flow, single-screw extruder, 160-161
G Gallium arsenide, chemical vapor deposition, 373-375, 385,390 Gas pores, alloy solidification, 244-245, 327 Generalized Newtonian fluid (GNF), 152-153 Gold, laser vaporization, 129-131
H Heat conduction equation, phase change, 78-79 Heat transfer alloy solidification, 238 chemical vapor deposition, 402-403 extrusion of nowNewtonian materials, 145-151, 220-226 chemical reaction and conversion, 218-220 combined heat and mass transfer, 213-220 material characterization, 152-155 moisture transport, 213-218 single-screw extruder, 147, 149, 155-187 twin-screw extruder, 151, 187-201, 225-226
modeling continuum approach, 65-66 discrete approach, 66-68 phase change enthalpy method, 76-79 interface tracking method, 76, 79 pulsed-laser-induced phase transformations melting, 75-109, 135 sputtering, 109-123 vaporization, 123-135, 136 spray deposition, 1-3 impact region, 23-69 spray region, 3-22 Horizontal chemical vapor deposition reactors, 346, 365-375 gallium arsenide, 373-375 MOCVD, 373-375 silicon, 365-373 Hot tears, 237 Hot-wall chemical vapor deposition reactor, 344,345,399-408 Hydrodynamic ablation, 109-110 Hypereutectic solution, 241 Hypoeutectic alloys, 239, 262
I Impinging-jet reactors, 345, 346, 394-396 Incremental solidification, 22 Induction stirring, alloy solidification, 317-318 Inflow boundary, liquid metal droplet, 6 Interdendritic liquid, 232, 233, 239, 251, 326 Intermeshing region, twin-screw extruder, 190-192, 195-197 Iron-carbon alloy, solidification, 288 Isomorphous system, 233
K Kinetic energy, during laser ablation, 112-1 16 Knudsen layer, 111, 125 Knudsen number, 351 Kozeny-Carmen equation, 266
L Laminar convection, Rantz-Marshall correlation, 8, 11
430
INDEX
Laminar flow, mathematical model, 3-10 Laser ablation, 109-110 kinetic energies, time-of-flight, 112-1 16 surface topography, 101-109 Laser-induced thermal chemical vapor deposition, 342-343 Lasers pulsed laser melting, 75-109, 135 pulsed laser sputtering, 109-123, 136 pulsed laser vaporization, 125-135, 136 Lead-tin alloy, 236, 239 macrosegregation, 291-292, 295-304, 317 solidification, 288-304, 309, 322 Lever law, alloy solidification, 262, 263 Liquid metal sprays droplet formation, 18-21 splat quenching, 23-54 Liquid-phase epitaxy, 341 Lorentz forces, alloy solidification, 247, 316, 317 Low pressure chemical vapor deposition (LPCVD), 344,399-408 Lumped models, 10, 11
M Macrosegregation alloys, 243, 251, 266, 327 double diffusive convection, 240-241 electromagnetic stirring, 320-325, 327 inhibition, 308-310, 316 semitransparent analog alloys, 272 shrinkage and buoyancy, 306, 308 tin-lead alloys, 291-292, 295-304, 317 Magnetic fields, alloy solidification, 246, 3 16-3 19 Mass conservation equation, laser vaporization, 127, 129 Mass diffusion, ultrashallow p+-junction formation, 96-101 Mass transfer extrusion of non-Newtonian materials, 145-151, 220-226 chemical reaction and conversion, 218-220 combined heat and mass transfer, 213-220 material characterization, 152-155 moisture transport, 213-218 single-screw extruder, 147, 149, 155-187
twin-screw extruder, 151, 187-201, 225-226 pulsed-laser-induced phase transformations melting, 75-109, 135 sputtering, 109-123 vaporization, 123-135, 136 Mass transport, chemical vapor deposition, 349, 352 MBE, see Molecular-beam epitaxy Melting, pulse-laser-induced, 75- 109, 135 Metal alloys, solidification, 288-308,316,327 Metal drops, conduction freezing, 17-18 Metal-organic chemical vapor deposition (MOCVD), 342,373-375 barrel reactor, 385 impinging-jet reactor, 395 rotating-vertical-pedestal reactor, 346, 389 Metal powders, atomization, 18-19 Metals metal alloys, solidification, 288-308, 316, 327 pulsed-layer sputtering, 109-1 12 thermal and electronic effects, 116-123 time-of-flight measurements, 112-1 16, 136 Metal sprays, see Liquid metal sprays Microsegregation, alloy solidification, 262 MOCVD, see Metal-organic chemical vapor deposition Modeling alloy solidification, 249-250 dual scale model, 263 history, 250-252 micro- and macromodels, 261-264 multidomain model, 251-252 multiphase model, 264 single-domain models, 252, 253-261,326 submodels, 264-269 chemical vapor deposition artificial neural network models, 408-413 deposition, 403-406 heat transfer model, 402-403 pulsed laser melting, 76-80 pulsed laser vaporization, 123-135, 136 screw extrusion, 149-151 single-screw extruder, 155-176 twin-screw extrusion, 190-197
431
INDEX
spray deposition convective cooling, 13-14 heat transfer, 65-68 liquid metal droplet impact, 68-69 radiative cooling, 13-14 recalescence, 15- 17 solidification, 17- 18 splat quenching, 35-54 Modified power-law model, 153 Moisture transport, screw extrusion, 151, 213-218 Molecular-beam epitaxy (MBE), 340 Momentum conservation equation chemical vapor deposition, 353, 359-360, 361-362 laser vaporization, 128, 129 Multidirectional solidification, alloys, 282-285 Multidomain models, alloy solidification, 251-252 Mushy zone alloy solidification, 232-233,239-242, 245-246, 247,251 modeling, 251, 253 semitransparent analog alloys, 270
N Neural networks, chemical vapor deposition modeling, 408-413 Newtonian fluid, 152 Nickel-copper alloys, 233, 236 Nonintermeshing twin-screw extruders, 188 Nan-Newtonian materials, 146 material characterization, 152-155 screw extrusion, 146-151 combined heat and mass transfer, 213-220 flow in dies, 201-212 single-screw extruder, 15-187, 147, 149 twin-screw extruder, 151, 187-201, 225-226 Nozzle design, atomization, 10, 18 Nucleation temperature, 12-13 Nusselt number, 8
0 OMVPE, see Organometallic vapor-phase epitaxy
Organometallic vapor-phase epitaxy (OMVPE), 342 Outflow boundary, liquid metal droplet, 6
P Pancake chemical vapor deposition reactors, 346, 386-389 PCVD, see Photo-chemical vapor deposition PECVD, see Plasma enhanced chemical vapor deposition Peritectic reactions, 233 Permeability, alloy solidification, 264, 276 Phase-change transformations pulsed laser melting, 75-109, 135 pulsed laser sputtering, 109-123 pulsed laser vaporization, 123-135, 136 Photo-chemical vapor deposition (PCVD), 342 Physical vapor deposition, 339-340 Physiconeural model, chemical vapor deposition, 413 Planar-stagnation-flow chemical vapor deposition reactors, 397-399 Plasma deposition, 53 Plasma enhanced chemical vapor deposition (PECVD), 342 Plasmon decay, 120-123 Polymers flow in dies, 201-212 melting, 224 screw extrusion, 146- 151 combined heat and mass transfer, 213-220 material characterization, 152-155 single-screw extruder, 147, 149, 155-187 twin-screw extruder, 151,187-201, 225-226 Polysilicon films, excimer laser melting, 81-96 Power-law model, 153 Prandtl number, 8, 288 Process control, alloy solidification, 235, 237, 238,308-326,327-328 Pseudoplastic material, 152 p+-junction formation, heat transfer and mass transfer, 96-101 Pulsed laser melting, 75-76, 135 experimental verification of melting, 80-81 conductance, 81-87 pyrometry, 87-96
432
INDEX
Pulsed laser melting (Continued) heat transport, thermal modeling, 76-80 topography formation, 101- 109 ultrashallow junction formation, 96-101 Pulsed laser sputtering metals, 103, 106, 109-112 thermal and electronic effects, 116-123 time-of-flight measurements, 112-116, 136 Pulsed laser vaporization, computational modeling, 123-135, 136 Purely viscous material, 152 Pyrometry, semiconductor melting, 87-96
R Radial momentum equation, liquid metal droplet, 3 Radiative cooling, modeling, 13-14 Rantz-Marshall correlation, 8, 11 Recalescence, 13, 15-17 Recalescence temperature, 17 Refined vacuum evaporation, 340 Remelting, alloy solidification, 262, 263 Residence-time distribution, screw extrusion, 176- 180 Reynolds number, 8 Rheocasting, electromagnetic, 318 Rotating-vertical-pedestal MOCVD reactor, 346.389
S Salt fingers, 244 Scheil assumption, 262 Screw extrusion, modeling, 149-151 Self-wiping twin-screw extruders, 189, 192 Semiconductors chemical vapor deposition, 341-342 pulsed laser melting, 80-96 Semitransparent analog alloys, 270-288 Shear stress, 152 Shear stress continuity, liquid metal droplet, 6 Shear thinning material, 152 Shrinkage, alloy solidification, 305-306 Side-chilled mold, solidification, 235, 237 Silicon chemical vapor deposition, 365-373, 382-385,386
polysilicon film melting, 81-96 surface topography after laser ablation, 101-109 ultrashallow doping profiles, 96-101 SIMPLER algorithm, 195, 255, 382 Single-domain models alloy solidification, 250-252, 326 dual-scale model, 263 micro- and macromodels, 261-264 mixture models, 253-256 multiphase model, 264 submodels, 264-269 two-phase models, 256-261 Single-screw extruder experimental results, 184-187 heat transfer, 147, 149 mixing characteristics, 180-184 modeling, 155-176 axial formulation, 170-175 developed flow, 161-164 fully developed flow, 160-161 three-dimensional transport, 164-170 two-dimensional transport, 156- 160 residence-time distribution, 176-180 tapered screw, 176, 225 Slip flow regime, 351 Slurry region, 269 Sodium carbonate analog alloy, 270, 287 Solidification alloys, 231-238,326-328; see aho Macrosegregation from bottom wall, 277-282 continuum model, 270-288, 311 mathematical models, 249-269 metal alloys, 288-308, 316, 327 multidirectional solidification, 282-285 physical phenomena, 238-249 process control strategies, 235, 237, 238, 308-326,327-328 semitransparent analog alloys, 270-288 from sidewall of rectangular cavity, 270-277 unidirectional solidification, 277-282 zero gravity, 309 spray deposition incremental solidification, 22 liquid metal droplet, 11-18 modeling, 17-18 splat solidification, 22 Solid-liquid interface, alloys, 232
433
INDEX
Solid nucleation, 12, 15 Solid-phase epitaxy, 341 Solutal buoyancy, 239, 251 Solute redistribution model, 251 Soret term, 356 Special concentration equation, chemical vapor deposition, 354, 360, 362 Splat solidification, 22 Spray deposition, 1-3 impact region, 22-23 solidification of multiple liquid metal droplets and sprays, 54-69 splat cooling of single liquid metal droplet, 23-54 spray region convective cooling of single liquid metal droplet, 3-11 in-flight solidification of single liquid metal droplet, 11-18 sprays, 18-22 Sprays, spray deposition, 18-21 Spread factor, 24 Sputtering, pulse-laser-induced, 103, 106, 109- 123 Stream-function equation, liquid metal droplet, 5 Surface topography, laser ablation, 101-109
T Tangential corotating twin-screw extruder, 147, 193, 194 Tapered screw extruders, 176, 225 Thermal ablation, 110 Thermal chemical vapor deposition, 342 Thermodiffusion ratio, 356 Thermo/diffusocapillary convection, alloy solidification, 287 Thermosolutal convection, 239-243 freckle formation, 243-244 macrosegregation, 243 magnetic damping, 317 Three-dimensional transport, single-screw extruder, 164-170 Time-of-flight measurements, kinetic energies during laser ablation, 112-116 Tin-lead alloy, 236, 239 macrosegregation, 291-292, 295-304, 317 solidification, 288-304, 309, 322
Transient conductance, semiconductor melting, 81-87 Translation region, twin-screw extruder, 190-192 Transport phenomena alloy solidification, 238-248, 268-269, 326 magnetic field, 318-319 semitransparent analog alloys, 270-288 chemical vapor deposition, 346-353, 362-365 screw extrusion of polymers, 149-151, 213-226 chemical reaction and conversion, 218-220 in dies, 205-212 moisture transport, 213-218 single-screw extrusion, 147, 149, 155-187 tapered screw extruders, 176, 225 twin-screw extrusion, 151, 187-201, 225-226 spray deposition, 1-3, 22 impact region, 23-69 spray region, 3-22 Turbulence, alloy solidification, 318-325, 327 Twin-screw extruder, 187-190 experimental results, 197-201 heat transfer, 151, 225-226 modeling, 190-197 finite-difference approach, 195- 197 finite-element approach, 193-195 translating and intermeshing regions, 190-192
U Ultrashallow doping profiles, 96-101 Unidirectional solidification, alloys, 277-282
V Vaporization, pulse-laser-induced, 123-135, 136 Viscasil-300M, 154-155, 187, 188 Viscoelastic material, 152 Viscoinelastic material, 152 Vorticity equation, liquid metal droplet, 5
Z Zero gravity, alloy solidification, 309
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