Tài liệu Solidification processing (materials science & engineering) merton c. flemings

SOL~ICATlOr~ PROCESSING TN 630 ,f-53 McGRAW-HILL SERIES IN MATERIALS SCIENCE AND ENGINEERING Editorial Board MICHAEL B. BEVER M. E. SHANK CHARLES A. WERT ROBERT F. MEHL, Honorary Senior Advisory Editor A VITZUR: Metal Forming: Processes and Analysis AZAROFF: Introduction to Solids BARRETTAND MASSALSKI:Structure of Metals BLATT: Physics of Electronic Conduction in Solids BRICK, GORDON, AND PHILLIPS: Structure and Properties of Alloys BUERGER: Contemporary Crystallography BUERGER: Introduction to Crystal Geometry DE HOFF AND RHINES: Quantitative Microscopy DRAUGLIS, GRETZ, AND JAFFEE: Molecular Processes on Solid Surfaces ELLIOTT: Constitution of Binary Alloys, First Supplement FLEMINGS: Solidification Processing GILMAN: Micromechanics of Flow in Solids GORDON: Principles of Phase Diagrams in Materials Systems GUY: Introduction to Materials Science HIRTH AND LOTHE: Theory of Dislocations KANNINEN, ADLER, ROSENFIELD,AND JAFFEE: Inelastic Behavior of Solids MILLS, ASCHER, AND JAFFEE: Critical Phenomena in Alloys, Magnets, and Super-conductors MURR: Electron Optical Applications in Materials Science PAUL AND WARSCHAUER:Solids under Pressure ROSENFIELD,HAHN, BEMENT,AND JAFFEE: Dislocation Dynamics ROSENQVIST:Principles of Extractive Metallurgy RUDMAN, STRINGER, AND JAFFEE: Phase Stability in Metals and Alloys SHEWMON: Diffusion in Solids SHEWMON: Transformations in Metals SHUNK: Constitution of Binary Alloys, Second Supplement WERT AND THOMSON: Physics of Solids McGRAW-HILL BOOK COMPANY New York St. Louis San Francisco Dusseldorf Johannesburg Kuala Lumpur London Mexico Montreal New Delhi Panama Rio de Janeiro Singapore Sydney Toronto MERTON C. FLEMINGS Abex Professor of Metallurgy Massachusetts Institute of Technology Solidification Processing This book was set in Times Roman. The editors were B. J. Clark and Michael Gardner; the production supervisor was Joan M. Oppenheimer. The drawings were done by John Cordes, J & R Technical Services, Inc. The printer and binder was The Maple Press Company. Library of Congress Cataloging in Publication Data Flemings, Merton C 1929Solidification processing. (McGraw-Hill series in materials science and engineering) Includes bibliographical references. I. Title. 1. Solidification. 2. Alloys. 73-4261 TN690.F59 669'.9 ISBN 0-07-021283-x SOLIDIFICA PROCESSING nON Copyright © 1974 by McGraw-HilI, Inc. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,. without the prior written permission of the publisher. 567 89-MAMM-76 54321 CONTENTS .. ~~ T ~:J.>fJ~~:\"",1 - ,. '''' ~-, ~OTr.. clr.:.H.:J~~1;;;.f " V· ..···· ~ Preface 1 (JH) IX Heat Flow in Solidification r;( 2 !:J.J~~ . l.r 1 5 6 Growth of Single Crystals Solidification of Castings and Ingots' Casting Processes Employing Insulating Molds Casting Processes in which Interface Resistance is Dominant 12 Analytic Solutions for Ingot Casting Solidification of Alloys Problems in Multidimensional Heat Flow 17 21 24 Plane Front Solidification of Single-phase Alloys 31 Introduction 31 Equilibrium Solidification No Solid Diffusion 33 Limited Liquid Diffusion, No Convection Effect of Convection Czochralski Growth (Crystal Pulling) 34 36 41 44 vi Cellular Solidification Plane Front Solidification of Polyphase Alloys Solidification of Castings and Ingots CONTENTS Zone Melting 115 107 167 172 134 154 146 104 127 120 114 135 112 105 93 66 87 141 157 117 160 94 77 58 75 73 85 83 64 46 49 53 51 Nucleation Kinetics Fluid Flow andof Interface Thermodynamics Solidification Polyphase of Solidification Alloys: Castings and Ingots CONTENTS 290 246 244 234 214 252 279 275 286 284 239 229 264 263 274 295 200 215 267 208 224 219 273 272 177 188 183 207 203 193 191 187 180 vii viii Tabulation ofand Error Functions Tables of Approximate Thermal Data Processing Properties CONTENTS Growth 349 335 328 344 347 309 308 341 338 331 305 301 319 318 312 356 359 357 PREFACE This book has grown largely out of a lecture course given to senior-level and graduate students at Massachusetts Institute of Technology. It is intended for use in courses of this type, and also for the practicing engineer and research worker. The essential aim of the book is to treat the fundamentals of solidification processing and to relate these fundamentals to practice. Processes considered include crystal growing, shape casting, ingot casting, growth of composites, and splat-cooling. The book builds on the fundamentals of heat flow, mass transport, and interface kinetics. Starting from these fundamentals, the basic similarities of the widely different solidification processes become evident. Problems at the end of each chapter relate principles to practice, illustrating important differences, as well as similarities between processes. Two years of college-level mathematics provides ample background for solving the problems given and for adequate comprehension of the text. In addition, it is desirable, though not necessary, that the student have a previous course in structure of materials. Emphasis of the book is on metallic alloys, but other materials are also considered. An essential element of all solidification processes is heat flow. This subject is treated in the first chapter, primarily to lend cohesiveness to the material to follow. It provides an excellent basis for description and comparison of solidification processes, and it can be treated with rather simple assumptions regarding the solidification mechanism. Chapter 2 deals with mass transport ("solute redistribution") in single-crystal growth. A quantitative description of transport in this type of solidification is"greatly simplified by the fact that the liquid-solid interface is single phase and planar. Equations derived in this chapter are also useful in describing dendritic solidification, except that they must be applied to tiny regions on the order of the dendrite arm spacmg. Chapter 3 deals with the important question of how to maintain a plane front in crystal growth, and of how solute redistribution occurs when the plane front breaks down to form "cells." Plane-front solidification is considered again in Chap. 4, this time for polyphase alloys, such as eutectics and off-eutectic "composites" solidified with an essentially planar liquid-solid interface. This chapter is the first to utilize the X PREFACE concept that the equilibrium melting point of a solid depends on its radius of curvature. Solidification as it occurs in usual castings and ingots is considered in Chaps. 5 and 6. More specifically, these chapters consider the microscopic aspects of such solidification, including dendritic growth, micro segregation, inclusion formation, and gas-pore formation. They draw heavily on the heat- and mass-transport concepts presented in earlier chapters. Fluid flow plays a larger role in solidification processes than is generally recognized. Flow is caused by introducing the metal to a mold, by density differences due to thermal or solute effects, or by solidification contractions. Fluid flow, treated in Chap. 7, has important effects on structure and segregation in solidification processes; many of these have been only recently recognized. An important part of this chapter deals with interdendritic fluid flow and its relation to porosity and segregation in castings and ingots. The major portions of the first seven chapters, and all quantitative treatments in these chapters, assume equilibrium at the liquid-solid interface. That is, they assume that the kinetic driving force necessary to advance a solidifying interface, is negligibly small. This assumption is not valid when facets form, but it appears to be an excellent approximation for the many alloys that solidify without facets. Implications of this assumption are considered in Chap. 8, which deals with the thermodynamics of liquid-solid equilibria. A portion of this chapter also deals with what is possible (and impossible) at the liquid-solid interface when conditions are such that equilibrium is not maintained. Kinetic effects at the liquid-solid interface, including nucleation, are discussed in Chap. 9. An understanding of growth kinetics, however qualitative, provides a basis for understanding the faceted growth morphologies observed in many real systems, and for understanding such solidification processes as growth of "ribbon crystals" by a twin-plane, reentrant growth mechanism. The final chapter deals with relations between the structure and properties of cast materials and with properties of wrought material produced from cast structures. An essential aim of many solidification processes is to obtain optimum properties in the resultant material. This chapter gives examples showing how the principles presented in earlier chapters can be utilized to produce structures with improved mechanical or physical properties. The book draws heavily on research conducted over the last decade at Massachusetts Institute of Technology by students and associates of the author. A special note of thanks is due them. Critical comments and suggestions of John Cahn have been received and acted on with pleasure. The bulk of the book was written while the author was on sabbatical leave as Overseas Fellow at Churchill College, Cambridge University, England. He is grateful for the unique combination of stimulation and relaxation provided by that environment, and by his colleagues there. MERTON C. FLEMINGS 1 HEAT FLOW IN SOLIDIFICATION GROWTH OF SINGLE CRYSTALS A variety of different techniques are employed to produce single crystals from melts. These can be grouped in three categories as those in which the entire charge is melted and then solidified from one end, a large charge is melted and a small crystal withdrawn slowly from it, and only a small zone of the crystal is melted at anyone time. Figure 1-1 shows the methods schematically. The first category of crystal-growing techniques is termed normal freezing. A commonly used normal freezing method for low-melting-point metals is growth in a horizontal boat. Here, a charge of metal is contained within a long crucible of small cross section open at the top. A seed crystal may be placed at one end of the boat to obtain a crystal of predetermined orientation. The charge and part of the seed are first melted in a suitable furnace. Next, the furnace is withdrawn slowly from the boat so that growth proceeds from the seed; alternatively, the boat is withdrawn slowly from the furnace and the solid-liquid interface moves until the whole charge is solid. In a similar crystal-growing method, the crucible is vertical and open at the top; this is often termed the Bridgeman method. In a minor modification of these techniques, neither the furnace nor the crucible moves. The charge is melted and 2 HEAT FLOW IN SOLIDIFICATION 000000000 TO INERT GAS SOURCE OR VACUUM COILS (a) TO t INERT VACUUM SOURCE TO INERT GAS SOURCE OR VACUUM o o o o o o o o o o o o o o GAS OR "FLOATING" ZONE LIQUID 0 0 0 ~HEATING o COILS o LIQUID o o 0-- HEATING COILS o o FIGURE 1-1 Examples of crystal-growing (c) floating zone. methods. (a) Boat method; (b) crystal pulling; equilibrated in a furnace constructed so that one end of the furnace is substantially colder than the other end (temperature-gradient furnace). The temperature gradient is maintained constant in the furnace, and crystal growth is obtained by slowly lowering overall furnace temperature. In growing single crystals, it is not necessary that the entire charge be molten. For some purposes, it is desirable to melt initially only a portion of the charge and move this molten zone slowly through the charge (zone melting and zone freezing). Many types of heat sources are used for zone melting, including induction, resistance, electron beam, and laser beam. The zone is moved either by mechanically moving the power source with respect to the crystal or vice versa. Zone melting is done either with or without crucible. The latter type, crucibleless zone melting, or floating zone melting, is widely used for reactive and high-melting-point materials. The molten zone is h,eld in place by surface tension forces sometimes aided by a magnetic field. GROWTH OF SINGLE CRYSTALS 3 Another single-crystal-growing technique, used widely for growing single crystals of silicon, germanium, and nonmetals, is the crystal-pulling, or Czochralski, technique. In this case, the charge material is placed in a crucible and melted. A seed crystal is attached to a vertical pull rod, lowered until it touches the melt, allowed to come to thermal equilibrium, and then raised slowly so that crystallization proceeds from the seed crystal. The crystal is rotated slowly as it is pulled, and crystal diameter is controlled by adjusting pull rate and/or heat input to the melt. Many variations of these crystal-growing techniques are described in the literature. Crystals are generally grown in vacuum but may be grown in air or inert atmosphere. Highly volatile materials are encapsulated and grown under pressure. Crystals with volatile species as alloy elements are also encapsulated, or grown under flux. In one rather old process, the liquid is carried by a plasma arc as small droplets from an electrode (Verneuil method). These and other techniques for growth of specific materials have been described. 1 The basic heat-flow objectives of all crystal-growing techniques are to (1) obtain a thermal gradient across a liquid-solid interface which can be held at equilibrium (e.g., stable with no interface movement) and (2) subsequently to alter or move this gradient in such a way that the liquid-solid interface moves at a controlled rate. A heat balance at a planar liquid-solid interface in crystal growth from the melt is written KsGs - KLGL where Ks = PsHR (1-1) = thermal conductivity of solid metal, cal/(cm)(DC)(s) KL = thermal conductivity of liquid metal, calj(cm)(DC)(s) Gs = temperature gradient in solid at the liquid-solid interface, gradient in liquid at the liquid-solid interface, DC/cm GL = temperature DC/cm R = growth velocity, cm/s Ps = density of solid metal, g/cm3 H = heat of fusion, cal/g Note from Eq. (1-1) that growth velocity R is dependent, not on absolute thermal gradient, but on the difference between KsGs and KLGL. Hence, thermal gradients can be controlled independently of growth velocity. This is an important attribute of single-crystal-growing furnaces since growing good crystals of alloys requires that the temperature gradients be high and growth rate be low. K" KL> H, and Ps are constants of the materials being solidified; GL is directly proportional to the heat flux in the liquid at the liquid-solid interface. 4 HEAT FLOW IN SOLIDIFICATION Growth velocity would be at a maximum when GL becomes negative (undercooled melt); however, good crystals cannot be grown in undercooled liquids, and so the practical maximum growth velocity occurs when GL -4 0, or from Eq. (1-1) = Ks~ R max (1-2) PsH G•• thermal gradient in the solid at the interface, is evaluated by experiment or heatflow calculations. As a simple illustrative example of calculation of solid gradient Gs, consider the case of floating-zone (crucibleless) crystal growth in which (1) crystal is of circular cross section, (2) heat transfer from the crystal to surroundings is by convection, (3) growth is at steady state, and (4) temperature gradients within the crystal transverse to the growth direction are low. Consider a cylindrical element in the solid crystal dx' in thickness, moving at the velocity R of the liquid-solid interface, Fig. 1-2. Then, for steady state, the temperature of the moving element remains constant and a heat balance is written (for unit time) Net heat change net heat change from net heat change from _ 0 from couduction + moving boundary + loss to surroundings as d 2T (pscsna2 dX') - R dT (pscsna2 dX') - h(T - To)(2na dX') dX'2 dx' = 0 (1-3) where x' = distance from liquid-solid interface (negative in solid), cm = specific heat of solid metal, cal/(g)(°C) a = radius of crystal, cm h = heat transfer coefficient for heat loss to surrounding, call Cs (cm2Wc)(s) T = temperature at x', °C To = ambient temperature, °C Ps = density of the solid crystal, g/cm3 as = thermal diffusivity of the solid crystal (KslPsc.), cm2 Is Now, integrating Eq. (1-3) with the boundary conditions that at x' = 0, T = Tm = - 00, T = To, the temperature in the solidifying metal is given by (Tm = melting point of metal), at x' T - To ~J (1-4) Gs = (dTldx')x =0 = exp {-[~ 2as - J(~)22ets + aKs Xl} The thermal gradient in the solid at the liquid-solid interface then is SOLIDIFICATION I SOLID 5 OF CASTINGS AND INGOTS fS LIQUID i i fu.I a: TM :J fCI: a: w a. a5 f- FIGURE 1-2 Temperature distribution growth (schematic). and when Rj2as in To o crystal DISTANCE FROM .INTERFACE, x' «1 Gs::::; ~ (TM- To) (1-6) aKs )1/2 (2h For crystals of high melting point, where (TM - To) is large and the coefficient of heat transfer h is increased by radiation heat transfer, thermal gradients attainable are quite high, 100°Cjcm or more. For lower-melting-point crystals, other cooling is necessary to attain steep gradients. As an example, Mollard achieved gradients of the order of 500°Cjcm in i-in-diameter tin crystals by using a thin steel crucible, resistanceheating the crucible at just above the liquid-solid interface and water-cooling it just below the interface.2 Equation (1-6) is equally applicable to this arrangement, with h now representing the total resistance to radial heat flow from the metal crystal to the cooling water. For the arrangement employed, this resistance was primarily at the metal-crucible interface and was about 0.04 calj(cm2)CC)(s). In Prob. 1-1 at the end of this chapter we illustrate that calculated thermal gradients obtained using Eq. (1-6) are about those attained experimentally, 500°Cjcm. SOLIDIFICATION OF CASTINGS AND INGOTS In most casting and ingot-making processes, heat flow is not at steady state as in the above examples. Hot liquid is poured into a cold mold; specific heat and heat of fusion of the solidifying metal pass through a series of thermal resistances to the cold mold until solidification is complete. Figure 1-3 shows this process schematically for solidification of a pure metal. Thermal resistances which, in general, must be considered are those across the liquid, the solidifying metal, and the metal-mold interface 6 HEAT FLOW IN SOLIDIFICATION AIR I LIQUID SOLID UJ a: ::;) I« a: UJ 0.. AT. METAL-MOLD INTERFACE AT, MOLD-AIR INTERFACE :!; UJ IFIGURE 1-3 Temperature profile in solidification of a pure metal. TO DISTANCE and those in the mold itself. The problem is mathematically and physically complex and becomes even more so when anything other than simple geometries are considered, when thermal properties are allowed to vary with temperature, or when alloys are considered. Problems such as these are now usually handled by computer methods, and some examples will be considered later. There are, however, certain simplifying approximations that can be made for a number of cases of engineering interest. Some of these will be examined before considering the general problem in further detail. CASTING PROCESSES EMPLOYING INSULATING MOLDS Sand casting and investment casting are two processes for making shaped castings which employ relatively insulating molds. Both are very old processes, and both are important commercially today. 3 Figure 1-4 illustrates the sand-casting process used to make a segment of a household radiator. Three sand-mold segments are made separately and assembled to produce a mold cavity the shape of the final casting described. Patterns of wood or metal are employed to make the proper impression in the upper and lower mold halves; the sand is rammed in place over the pattern. The outer mold segments (cope and drag) retain the shape of the impression because the sand used contains a few percent of water and clay and sometimes other binding agents. The internal segment CASTING PROCESSES EMPLOYING INSULATING MOLDS 7 COMPLETED CASTINGBROKEN OPEN TO REVEAL INTERIOR MOLD SECTION A MOLD SECTION B FIGURE 1-4 Sketch of sand-casting process as used in manufacture of a household radiator (From Taylor, Flemings, and WuljJ.4) (core) is generally made of baked oil or resin-bonded sand to achieve greater strength and to reduce the amount of volatile components. In cope-and-drag investment casting, mold pieces are made, not by ramming a dry or nearly dry sand mixture, but by pouring a slurry of investment material. One such material widely used for nonferrous alloys is plaster. For ferrous materials, a suitable material is mullite bonded with ethyl silicate. In lost-wax casting, the pattern is originally made of wax and then invested with a suitable slurry which is subsequently baked at high temperature. During the hightemperature baking, the wax melts and drips out or volatilizes along with moisture in the mold. Two types of lost-wax casting are common. In the older process, the wax is placed in a box, or can, and a slurry poured to fill the box (Fig. 1-5). In the new shell-investment-casting process, the pattern is dipped successively in a slurry and then in a fluidized bed of fine particulate material until a shell of desired thickness is built up. The basic advantage of both types of lost-wax casting is that the process allows intricate parts to be made without regard to the problem of pattern removal from the mold. The major advantages of investment-casting processes as compared with sand castings are the greater complexity, thinner sections, and better dimensional accuracy and surface finish that can be obtained. The major disadvantage is their usually greater cost and size limitation. 8 HEAT FLOW IN SOLIDIFICATION Preparing a Mold for Investment Casting The "Lost Wax" or Precision Casting Process u Wax n.; is melted and injected into a metal die to form the In-gate disposable Sprue patterns. Pattern Metal die Pouring cup - palleZ:::' '''' Patterns are "welded" to wax gates and runners Ita form a "tree". The "tree" is precoated A metal flask by is nexl placed around the dipping in a refractory slurry and is then dusted "tree" and sealed to the pallet: then the investment, a coarser with refractory sand. refractory in a more viscous slurry is poured around the precoated "tree". , Wax IL.J--;'. Finally, before casting, the mold is placed in a furnace , I .\. dripping5'---....', j When the investment has and carefully fired to 1300-1900'F. to rf/1wve "set", all wax residue the mold is placed in an oven at 200'F. to investment and melt wax pattern. dry the out the The mold hot )Ready to is Pour. and free of any trace of wax. and reach the temperature at which it will receive the molten metal. FIGURE 1-5 Preparing a mold for investment casting. (From Taylor, Flemings, and WuljJ.4) From a heat-flow standpoint, the important characteristic of solidification of a metal in processes such as those discussed above is that the metal is a much better conductor of heat than the mold. Thus, solidification rate depends primarily on thermal properties of the mold. The thermal conductivity of the metal has practically no influence. Also, except in relatively heavy-section shell investment castings, the mold can be considered to be semi-infinite in extent; i.e., the outside of the mold does CASTING PROCESSES EMPLOYING INSULATING MOLDS 9 not heat up during solidification. The heat-flow problem sketched in Fig. 1-3 is now very much simplified, especially if we assume further that the metal is poured with no superheat, that is, exactly at its melting point TM' as shown in Fig. 1-6. Consider first the problem of unidirectional heat flow. Metal is poured exactly at its melting point against a thick, flat mold wall initially at room temperature To. Thus, the mold surface is heated suddenly to TM at time t = O. This is a transient, one-dimensional heat-flow problem, and the solution must conform with the partial differential equation -aT at = where am ax (1-7) = thermal diffusivity of mold, cm2(s Km = therl1}.al conductivity Pm = density of mold, g(cm3 t am a2T2 = time, of mold, cal(cmWC)(s) s x = distance from mold wall, cm (negative into the mold) The solution to this equation for the boundary conditions stated above gives the temperature T in the mold as a function of time t at distance from the mold surface x: T - TM To - TM = erf -x _ 2)amt (1-8) where erf denotes the error function. The error function of zero is zero, and the error function of infinity is unity. A list of tabulated error functions is given in Appendix A. The rate of heat flow into the mold at the mold-metal interface is given by A x=o = _Km(aT) ax (:L) x=o (1-9) where x increases positively from left to right in Fig. 1-6, q is rate of heat flow, and A is area of the mold-metal interface. By partial differentiation of Eq. (1-8) with respect to x, letting x = 0 and combining the results with Eq. (1-9), the rate of heat flow across the mold-metal interface is seen to be A x=o (CL) = (1-10) ret _JKmpmCm (TM - To) where em is specific heat of the mold material. Now, the heat entering the mold comes only from heat of fusion of the solidifying metal since the solid as well as the liquid metal is exactly TM (Fig. 1-6). Thus, A x=o (CL) at = _p S H as (1-11)